Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Lecture notes 

Numerical Methods in 

Quantum Mechanics 

Corso di Laurea Magistrale in Fisica 

Interateneo Trieste – Udine 

Anno accademico 2012/2013 

Paolo Giannozzi 

University of Udine 

Contains software and material written by 

Furio Ercolessi 1 and Stefano de Gironcoli 2 

1 Formerly at University of Udine 

2 SISSA - Trieste 

Last modified May 23, 2013

Contents 

Introduction 1 

0.1 About Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

0.1.1 Compilers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

0.1.2 Visualization Tools . . . . . . . . . . . . . . . . . . . . . . 2 

0.1.3 Mathematical Libraries . . . . . . . . . . . . . . . . . . . 2 

0.1.4 Pitfalls in C-Fortran interlanguage calls . . . . . . . . . . 3 

0.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 

1 One-dimensional Schrödinger equation 5 

1.1 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . 5 

1.1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

1.1.2 Exact solution . . . . . . . . . . . . . . . . . . . . . . . . 6 

1.1.3 Comparison with classical probability density . . . . . . . 8 

1.2 Quantum mechanics and numerical codes: some observations . . 9 

1.2.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 9 

1.2.2 A pitfall: pathological asymptotic behavior . . . . . . . . 9 

1.3 Numerov’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 10 

1.3.1 Code: harmonic0 . . . . . . . . . . . . . . . . . . . . . . . 12 

1.3.2 Code: harmonic1 . . . . . . . . . . . . . . . . . . . . . . . 13 

1.3.3 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 15 

2 Schrödinger equation for central potentials 16 

2.1 Variable separation . . . . . . . . . . . . . . . . . . . . . . . . . . 16 

2.1.1 Radial equation . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.2 Coulomb potential . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.2.1 Energy levels . . . . . . . . . . . . . . . . . . . . . . . . . 20 

2.2.2 Radial wave functions . . . . . . . . . . . . . . . . . . . . 20 

2.3 Code: hydrogen radial . . . . . . . . . . . . . . . . . . . . . . . . 21 

2.3.1 Logarithmic grid . . . . . . . . . . . . . . . . . . . . . . . 21 

2.3.2 Improving convergence with perturbation theory . . . . . 22 

2.3.3 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 24 

3 Scattering from a potential 25 

3.1 Short reminder of the theory of scattering . . . . . . . . . . . . . 25 

3.2 Scattering of H atoms from rare gases . . . . . . . . . . . . . . . 27 

3.2.1 Derivation of Van der Waals interaction . . . . . . . . . . 27 

i

3.3 Code: crossection . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

3.3.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 30 

4 The Variational Method 32 

4.1 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . 32 

4.1.1 Demonstration of the variational principle . . . . . . . . . 32 

4.1.2 Alternative demonstration of the variational principle . . 33 

4.1.3 Ground state energy . . . . . . . . . . . . . . . . . . . . . 34 

4.1.4 Variational method in practice . . . . . . . . . . . . . . . 35 

4.2 Secular problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 

4.2.1 Expansion into a basis set of orthonormal functions . . . 36 

4.3 Plane-wave basis set . . . . . . . . . . . . . . . . . . . . . . . . . 38 

4.4 Code: pwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 

4.4.1 Diagonalization routines . . . . . . . . . . . . . . . . . . . 40 

4.4.2 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 41 

5 Non-orthonormal basis sets 42 

5.1 Non-orthonormal basis set . . . . . . . . . . . . . . . . . . . . . . 42 

5.1.1 Gaussian functions . . . . . . . . . . . . . . . . . . . . . . 44 

5.1.2 Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . 44 

5.2 Code: hydrogen gauss . . . . . . . . . . . . . . . . . . . . . . . . 44 

5.2.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 46 

6 Self-consistent Field 47 

6.1 The Hartree Approximation . . . . . . . . . . . . . . . . . . . . . 47 

6.2 Hartree Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 48 

6.2.1 Eigenvalues and Hartree energy . . . . . . . . . . . . . . . 49 

6.3 Self-consistent potential . . . . . . . . . . . . . . . . . . . . . . . 50 

6.3.1 Self-consistent potential in atoms . . . . . . . . . . . . . . 50 

6.4 Code: helium hf radial . . . . . . . . . . . . . . . . . . . . . . . . 51 

6.4.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 52 

7 The Hartree-Fock approximation 53 

7.1 Hartree-Fock method . . . . . . . . . . . . . . . . . . . . . . . . . 53 

7.1.1 Coulomb and exchange potentials . . . . . . . . . . . . . . 55 

7.1.2 Correlation energy . . . . . . . . . . . . . . . . . . . . . . 56 

7.1.3 The Helium atom . . . . . . . . . . . . . . . . . . . . . . . 57 

7.2 Code: helium hf gauss . . . . . . . . . . . . . . . . . . . . . . . . 57 

7.2.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 59 

8 Molecules 60 

8.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . 60 

8.2 Potential Energy Surface . . . . . . . . . . . . . . . . . . . . . . . 61 

8.3 Diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 62 

8.4 Code: h2 hf gauss . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

8.4.1 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . 63 

8.4.2 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 64 

ii

9 Electrons in a periodic potential 66 

9.1 Crystalline solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 

9.1.1 Periodic Boundary Conditions . . . . . . . . . . . . . . . 67 

9.1.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . 68 

9.1.3 The empty potential . . . . . . . . . . . . . . . . . . . . . 68 

9.1.4 Solution for the crystal potential . . . . . . . . . . . . . . 69 

9.1.5 Plane-wave basis set . . . . . . . . . . . . . . . . . . . . . 70 

9.2 Code: periodicwell . . . . . . . . . . . . . . . . . . . . . . . . . . 72 

9.2.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 73 

10 Pseudopotentials 74 

10.1 Three-dimensional crystals . . . . . . . . . . . . . . . . . . . . . . 74 

10.2 Plane waves, core states, pseudopotentials . . . . . . . . . . . . . 75 

10.3 Code: cohenbergstresser . . . . . . . . . . . . . . . . . . . . . . . 76 

10.3.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 78 

11 Exact diagonalization of quantum spin models 79 

11.1 The Heisenberg model . . . . . . . . . . . . . . . . . . . . . . . . 79 

11.2 Hilbert space in spin systems . . . . . . . . . . . . . . . . . . . . 80 

11.3 Iterative diagonalization . . . . . . . . . . . . . . . . . . . . . . . 81 

11.4 Code: heisenberg exact . . . . . . . . . . . . . . . . . . . . . . 82 

11.4.1 Computer Laboratory . . . . . . . . . . . . . . . . . . . . 83 

A From two-body to one-body problem 85 

B Accidental degeneracy and dynamical symmetry 87 

C Composition of angular momenta: the coupled representation 88 

D Two-electron atoms 90 

D.1 Perturbative Treatment for Helium atom . . . . . . . . . . . . . . 91 

D.2 Variational treatment for Helium atom . . . . . . . . . . . . . . . 92 

D.3 ”Exact” treatment for Helium atom . . . . . . . . . . . . . . . . 93 

D.3.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 95 

E Useful algorithms 96 

E.1 Search of the zeros of a function . . . . . . . . . . . . . . . . . . 96 

E.1.1 Bisection method . . . . . . . . . . . . . . . . . . . . . . . 96 

E.1.2 Newton-Raphson method . . . . . . . . . . . . . . . . . . 97 

E.1.3 Secant method . . . . . . . . . . . . . . . . . . . . . . . . 97 

iii

Introduction 

The aim of these lecture notes is to provide an introduction to methods and 

techniques used in the numerical solution of simple (non-relativistic) quantummechanical 

problems, with special emphasis on atomic and condensed-matter 

physics. The practical sessions are meant to be a sort of “computational laboratory”, 

introducing the basic ingredients used in the calculation of materials 

properties at a much larger scale. The latter is a very important field of today’s 

computational physics, due to its technological interest and potential applications. 

The codes provided during the course are little more than templates. Students 

are expected to analyze them, to run them under various conditions, to 

examine their behavior as a function of input data, and most important, to 

interpret their output from a physical point of view. The students will be asked 

to extend or modify those codes, by adding or modifying some functionalities. 

For further insight on the theory of Quantum Mechanics, many excellent 

textbooks are available (e.g. Griffiths, Schiff, or the ever-green Dirac and Landau). 

For further insight on the properly computational aspects of this course, 

we refer to the specialized texts quotes in the Bibliography section, and in 

particular to the book of Thijssen. 

0.1 About Software 

This course assumes some basic knowledge of how to write and execute simple 

programs, and how to plot their results. All that is needed is a fortran or C 

compiler and some visualization software. The target machine is a PC running 

Linux, but other operating systems can be used as well (including Mac OS-X 

and Windows), as long as the mentioned software is installed and working, and 

if you know how to use it in oractise. 

0.1.1 Compilers 

In order to run a code written in any programming language, we must first 

translate it into machine language, i.e. a language that the computer can 

understand. The translation is done by an interpreter or by a compiler: the 

former translates and immediately executes each instruction, the latter takes 

the file, produces the so-called object code that together with other object codes 

and with libraries is finally assembled into an executable file. Python, Java (or at 

1

an higher level, Matlab, Mathematica) are examples of “interpreted” language. 

Fortran, C, C++ are “compiled” languages. 

Our codes are written in Fortran 90. This is a sophisticated and complex 

language offering dynamical memory management, arrays operations (e.g. 

matrix-vector products), modular and object-based structure. Fortran 90 however 

can be as efficient as Fortran 77 and maintains a wide compatibility with 

existing Fortran 77 codes. It is worth mentioning that the first applications 

of computers to physics go back to well before the birth of modern computer 

languages like C++, python, or even C: there is still a large number of codes 

and libraries written in Fortran 77 (or even Fortran 66!) widely used in physics. 

Fortran 90 (or even Fortran 77, in this respect) is not a well known language. 

There are however many available resources (see for instance the web page 

mentioned in the bibliography section) and the codes themselves are very simple 

and make little usage of advanced language features. In any case, there are no 

objections if a student prefers to use a more widespread language like C. A 

version of all codes in C is also available (no warranty about the quality of the 

C code in terms of elegance and good coding practice). 

In all cases, we need a C or Fortran 90 compiler. In PCs running Linux, the 

C compiler gcc is basically part of the operating system and is always present. 

Recent versions of gcc also include a Fortran compiler, called gfortran. If this 

is absent, or it is not easy to install it, one can download the free and rather 

reliable compiler, g95 1 . It is possible to install on Mac OS-X and on Windows 

either gcc with gfortran or g95. 

0.1.2 Visualization Tools 

Visualization of data produced by the codes (wave functions, charge densities, 

various other quantities) has a central role in the analysis and understanding 

of the results. Code gnuplot can be used to make two-dimensional or threedimensional 

plots of data or of analytical expressions. gnuplot is open-source 

software, available for all operating systems and usually found pre-installed on 

Linux PCs. An introduction to gnuplot, with many links to more resources, 

can be found here: http://www.gnuplot.info/help.html. 

Another software that can be used is xmgrace 2 . This is also open-source 

and highly portable, has a graphical user interface and thus it is easier to use 

than gnuplot, whose syntax is not always easy to remember. 

0.1.3 Mathematical Libraries 

The usage of efficient mathematical libraries is crucial in “serious” calculations. 

Some of the codes use routines from the BLAS 3 (Basic Linear Algebra Subprograms) 

library and from LAPACK 4 (Linear Algebra PACKage). The latter 

1 http://www.g95.org 

2 http://plasma-gate.weizmann.ac.il/Grace 

3 http://www.netlib.org/blas 

4 http://www.netlib.org/lapack 

2

is an important and well-known library for all kinds of linear algebra operations: 

solution of linear systems, eigenvalue problems, etc. LAPACK calls 

BLAS routines for all CPU-intensive calculations. The latter are available in 

highly optimized form for many different architectures. 

BLAS and LAPACK routines are written in Fortran 77. BLAS and LA- 

PACK are often available in many operating systems and can be linked directly 

by the compiler by adding -llapack -lblas. In the case of C compiler, it 

may be needed to add an underscore ( ) in the calling program, as in: dsyev , 

dgemm . This is due to different C-Fortran conventions for the naming of “symbols” 

(i.e. compiled routines). Note that the C compiler may also need -lm to 

link general mathematical libraries (i.e. operations like the square root). 

0.1.4 Pitfalls in C-Fortran interlanguage calls 

In addition to the above-mentioned potential mismatches between C and Fortran 

naming conventions, there are a few more pitfalls one has to be aware of 

when Fortran 77 routines are called by C (or vice versa). 

• Fortran passes pointers to subroutines and functions; C passes values. In 

order to call a Fortran routine from C, all C variables appearing in the 

call must be either pointers or arrays. 

• Indices of vectors and arrays start from 0 in C, from 1 in Fortran (unless 

differently specified in array declaration or allocation). 

• Matrices in C are stored in memory row-wise, that is: a[i][j+1] follows 

a[i][j] in memory. In Fortran, they are stored column-wise (the other 

way round!): a(i+1,j) follows a(i,j) in memory. 

An additional problem is that C does not provide run-time allocatable matrices 

like Fortran does, but only fixed-dimension matrices and arrays of pointers. 

The former are impractical, the latter are not usable as arguments to pass to 

Fortran. It would be possible, using either non-standard C syntax, or using 

C++ and the new command, to define dynamically allocated matrices similar 

to those used in Fortran. We have preferred for our simple C codes to “simulate” 

Fortran-style matrices (i.e. stored in memory column-wise) by mapping them 

onto one-dimensional C vectors. 

We remark that Fortran 90 has a more advanced way of passing arrays to 

subroutines using “array descriptors”. The codes used in this course however do 

not make use of this possibility but use the old-style Fortran 77 way of passing 

arrays via pointers. 

3

0.2 Bibliography 

J. M. Thijssen, Computational Physics, Cambridge University Press, Cambridge, 

1999. A second edition has appeared: 

http://www.cambridge.org/gb/knowledge/isbn/item1171410. 

F. J. Vesely, Computational Physics - An Introduction: Second Edition, Kluwer, 

2001. Also see the author’s web page: 

http://www.ap.univie.ac.at/users/Franz.Vesely/cp0102/serious.html, 

containing parts of the accompanying material. 

S. E. Koonin e D. C. Meredith, Computational physics - Fortran Version, 

Addison-Wesley, 1990. See also Dawn Meredith’s web page: 

http://pubpages.unh.edu/%7Edawnm/. 

Timothy H. Kaiser, A quick introduction to advanced Fortran-90: 

http://www.sdsc.edu/%7Etkaiser/f90.html. 

4

Chapter 1 

One-dimensional Schrödinger 

equation 

In this chapter we will start from the harmonic oscillator to introduce a general 

numerical methodology to solve the one-dimensional, time-independent Schrödinger 

equation. The analytical solution of the harmonic oscillator will be first 

derived and described. A specific integration algorithm (Numerov) will be used. 

The extension of the numerical methodology to other, more general types of 

potentials does not present any special difficulty. 

For a particle of mass m under a potential V (x), the one-dimensional, timeindependent 

Schrödinger equation is given by: 

− ¯h2 d 2 ψ 

+ V (x)ψ(x) = Eψ(x), (1.1) 

2m dx2 where ψ(x) is the wave function, in general complex, and ¯h is the Planck constant 

h divided by 2π. In the following we are focusing on the discrete spectrum: 

the set of isolated energy values for which Eq.(1.1) has normalizable solutions, 

localized in space. 

1.1 The harmonic oscillator 

The harmonic oscillator is a fundamental problem in classical dynamics as well 

as in quantum mechanics. It represents the simplest model system in which 

attractive forces are present and is an important paradigm for all kinds of vibrational 

phenomena. For instance, the vibrations around equilibrium positions 

of a system of interacting particles may be described, via an appropriate coordinate 

transformation, in terms of independent harmonic oscillators known as 

normal vibrational modes. The same holds in quantum mechanics. The study 

of the quantum oscillator allows a deeper understanding of quantization and of 

its effects and of wave functions of bound states. 

In this chapter we will first remind the main results of the theory of the 

harmonic oscillator, then we will show how to set up a computer code that 

allows to numerically solve the Schrödinger equation for the harmonic oscillator. 

The resulting code can be easily modified and adapted to a different (not simply 

5

quadratic) interaction potential. This will allow to study problems that, unlike 

the harmonic oscillator, do not have a simple analytical solution. 

1.1.1 Units 

The Schrödinger equation for a one-dimensional harmonic oscillator is, in usual 

notations: 

d 2 ( 

ψ 

dx 2 = −2m ¯h 2 E − 1 ) 

2 Kx2 ψ(x) (1.2) 

where K the force constant (the force on the mass being F = −Kx, proportional 

to the displacement x and directed towards the origin). Classically such an 

oscillator has a frequency (angular frequency) 

√ 

K 

ω = 

m . (1.3) 

It is convenient to work in adimensional units. These are the units that will 

be used by the codes presented at the end of this chapter. Let us introduce 

adimensional variables ξ, defined as 

ξ = 

( mK 

¯h 2 ) 1/4 

x = 

(using Eq.(1.3) for ω), and ɛ, defined as 

( ) mω 1/2 

x (1.4) 

¯h 

ε = Ē hω . (1.5) 

By inserting these variables into the Schrödinger equation, we find 

d 2 ( ) 

ψ 

dξ 2 = −2 ε − ξ2 

ψ(ξ) (1.6) 

2 

which is written in adimensional units. 

1.1.2 Exact solution 

One can easily verify that for large ξ (such that ε can be neglected) the solutions 

of Eq.(1.6) must have an asymptotic behavior like 

ψ(ξ) ∼ ξ n e ±ξ2 /2 

(1.7) 

where n is any finite value. The + sign in the exponent must however be 

discarded: it would give raise to diverging, non-physical solutions (in which the 

particle would tend to leave the ξ = 0 point, instead of being attracted towards 

it by the elastic force). It is thus convenient to extract the asymptotic behavior 

and assume 

ψ(ξ) = H(ξ)e −ξ2 /2 

(1.8) 

where H(ξ) is a well-behaved function for large ξ (i.e. the asymptotic behavior 

is determined by the second factor e −ξ2 /2 ). In particular, H(ξ) must not grow 

like e ξ2 , or else we fall back into a undesirable non-physical solution. 

6

Under the assumption of Eq.(1.8), Eq.(1.6) becomes an equation for H(ξ): 

H ′′ (ξ) − 2ξH ′ (ξ) + (2ε − 1)H(ξ) = 0. (1.9) 

It is immediate to notice that ε 0 = 1/2, H 0 (ξ) = 1 is the simplest solution. 

This is the ground state, i.e. the lowest-energy solution, as will soon be clear. 

In order to find all solutions, we expand H(ξ) into a series (in principle an 

infinite one): 

∞∑ 

H(ξ) = A n ξ n , (1.10) 

n=0 

we derive the series to find H ′ and H ′′ , plug the results into Eq.(1.9) and regroup 

terms with the same power of ξ. We find an equation 

∞∑ 

[(n + 2)(n + 1)A n+2 + (2ε − 2n − 1)A n ] ξ n = 0 (1.11) 

n=0 

that can be satisfied for any value of ξ only if the coefficients of all the orders 

are zero: 

(n + 2)(n + 1)A n+2 + (2ε − 2n − 1)A n = 0. (1.12) 

Thus, once A 0 and A 1 are given, Eq.(1.12) allows to determine by recursion the 

solution under the form of a power series. 

Let us assume that the series contain an infinite number of terms. For large 

n, the coefficient of the series behave like 

A n+2 

A n 

→ 2 n , that is: A n+2 ∼ 1 

(n/2)! . (1.13) 

Remembering that exp(ξ 2 ) = ∑ n ξ2n /n!, whose coefficient also behave as in 

Eq.(1.13), we see that recursion relation Eq.(1.12) between coefficients produces 

a function H(ξ) that grows like exp(ξ 2 ), that is, produces unphysical diverging 

solutions. 

The only way to prevent this from happening is to have in Eq.(1.12) all 

coefficients beyond a given n vanish, so that the infinite series reduces to a 

finite-degree polynomial. This happens if and only if 

ε = n + 1 2 

(1.14) 

where n is a non-negative integer. 

Allowed energies for the harmonic oscillator are thus quantized: 

( 

E n = n + 1 ) 

¯hω n = 0, 1, 2, . . . (1.15) 

2 

The corresponding polynomials H n (ξ) are known as Hermite polynomials. H n (ξ) 

is of degree n in ξ, has n nodes, is even [H n (−ξ) = H n (ξ)] for even n, odd 

[H n (−ξ) = −H n (ξ)] for odd n. Since e −ξ2 /2 is node-less and even, the complete 

wave function corresponding to the energy E n : 

ψ n (ξ) = H n (ξ)e −ξ2 /2 

(1.16) 

7

Figure 1.1: Wave functions and probability density for the quantum harmonic 

oscillator. 

has n nodes and the same parity as n. The fact that all solutions of the 

Schrödinger equation are either odd or even functions is a consequence of the 

symmetry of the potential: V (−x) = V (x). 

The lowest-order Hermite polynomials are 

H 0 (ξ) = 1, H 1 (ξ) = 2ξ, H 2 (ξ) = 4ξ 2 − 2, H 3 (ξ) = 8ξ 3 − 12ξ. (1.17) 

A graph of the corresponding wave functions and probability density is shown 

in fig. 1.1. 

1.1.3 Comparison with classical probability density 

The probability density for wave functions ψ n (x) of the harmonic oscillator 

have in general n + 1 peaks, whose height increases while approaching the 

corresponding classical inversion points (i.e. points where V (x) = E). 

These probability density can be compared to that of the classical harmonic 

oscillator, in which the mass moves according to x(t) = x 0 sin(ωt). The probability 

ρ(x)dx to find the mass between x and x + dx is proportional to the time 

needed to cross such a region, i.e. it is inversely proportional to the speed as a 

function of x: 

ρ(x)dx ∝ 

√ 

Since v(t) = x 0 ω cos(ωt) = ω x 2 0 − x2 0 sin2 (ωt), we have 

ρ(x) ∝ 

dx 

v(x) . (1.18) 

1 

√x 2 0 − x2 . (1.19) 

This probability density has a minimum for x = 0, diverges at inversion points, 

is zero beyond inversion points. 

The quantum probability density for the ground state is completely different: 

has a maximum for x = 0, decreases for increasing x. At the classical inversion 

8

point its value is still ∼ 60% of the maximum value: the particle has a high 

probability to be in the classically forbidden region (for which V (x) > E). 

In the limit of large quantum numbers (i.e. large values of the index n), 

the quantum density tends however to look similar to the quantum one, but it 

still displays the oscillatory behavior in the allowed region, typical for quantum 

systems. 

1.2 Quantum mechanics and numerical codes: some 

observations 

1.2.1 Quantization 

A first aspect to be considered in the numerical solution of quantum problems 

is the presence of quantization of energy levels for bound states, such as for 

instance Eq.(1.15) for the harmonic oscillator. The acceptable energy values 

E n are not in general known a priori. Thus in the Schrödinger equation (1.1) 

the unknown is not just ψ(x) but also E. For each allowed energy level, or 

eigenvalue, E n , there will be a corresponding wave function, or eigenfunction, 

ψ n (x). 

What happens if we try to solve the Schrödinger equation for an energy E 

that does not correspond to an eigenvalue? In fact, a “solution” exists for any 

value of E. We have however seen while studying the harmonic oscillator that 

the quantization of energy originates from boundary conditions, requiring no 

unphysical divergence of the wave function in the forbidden regions. Thus, if 

E is not an eigenvalue, we will observe a divergence of ψ(x). Numerical codes 

searching for allowed energies must be able to recognize when the energy is not 

correct and search for a better energy, until it coincides – within numerical or 

predetermined accuracy – with an eigenvalue. The first code presented at the 

end of this chapter implements such a strategy. 

1.2.2 A pitfall: pathological asymptotic behavior 

An important aspect of quantum mechanics is the existence of “negative” kinetic 

energies: i.e., the wave function can be non zero (and thus the probability 

to find a particle can be finite) in regions for which V (x) > E, forbidden according 

to classical mechanics. Based on (1.1) and assuming the simple case in 

which V is (or can be considered) constant, this means 

d 2 ψ 

dx 2 = k2 ψ(x) (1.20) 

where k 2 is a positive quantity. This in turns implies an exponential behavior, 

with both ψ(x) ≃ exp(kx) and ψ(x) ≃ exp(−kx) satisfying (1.20). As a rule 

only one of these two possibilities has a physical meaning: the one that gives 

raise to a wave function that decreases exponentially at large |x|. 

It is very easy to distinguish between the “good” and the “bad” solution for 

a human. Numerical codes however are less good for such task: by their very 

nature, they accept both solutions, as long as they fulfill the equations. If even 

9

a tiny amount of the “bad” solution (due to numerical noise, for instance) is 

present, the integration algorithm will inexorably make it grow in the classically 

forbidden region. As the integration goes on, the “bad” solution will sooner or 

later dominate the “good” one and eventually produce crazy numbers (or crazy 

NaN’s: Not a Number). Thus a nice-looking wave function in the classically 

allowed region, smoothly decaying in the classically forbidden region, may suddenly 

start to diverge beyond some limit, unless some wise strategy is employed 

to prevent it. The second code presented at the end of this chapter implements 

such a strategy. 

1.3 Numerov’s method 

Let us consider now the numerical solution of the (time-independent) Schrödinger 

equation in one dimension. The basic assumption is that the equation 

can be discretized, i.e. written on a suitable finite grid of points, and integrated, 

i.e. solved, the solution being also given on the grid of points. 

There are many big thick books on this subject, describing old and new 

methods, from the very simple to the very sophisticated, for all kinds of differential 

equations and all kinds of discretizations and integration algorithms. 

In the following, we will consider Numerov’s method (named after Russian astronomer 

Boris Vasilyevich Numerov) as an example of a simple yet powerful 

and accurate algorithm. Numerov’s method is useful to integrate second-order 

differential equations of the general form 

d 2 y 

= −g(x)y(x) + s(x) (1.21) 

dx2 where g(x) and s(x) are known functions. Initial conditions for second-order 

differential equations are typically given as 

y(x 0 ) = y 0 , y ′ (x 0 ) = y ′ 0. (1.22) 

The Schrödinger equation (1.1) has this form, with g(x) ≡ 2m [E − V (x)] and 

¯h 2 

s(x) = 0. We will see in the next chapter that also the radial Schrödinger 

equations in three dimensions for systems having spherical symmetry belongs 

to such class. Another important equation falling into this category is Poisson’s 

equation of electromagnetism, 

d 2 φ 

= −4πρ(x) (1.23) 

dx2 where ρ(x) is the charge density. In this case g(x) = 0 and s(x) = −4πρ(x). 

Let us consider a finite box containing the system: for instance, −x max ≤ 

x ≤ x max , with x max large enough for our solutions to decay to negligibly small 

values. Let us divide our finite box into N small intervals of equal size, ∆x 

wide. We call x i the points of the grid so obtained, y i = y(x i ) the values of the 

unknown function y(x) on grid points. In the same way we indicate by g i and 

s i the values of the (known) functions g(x) and s(x) in the same grid points. In 

order to obtain a discretized version of the differential equation (i.e. to obtain 

10

an equation involving finite differences), we expand y(x) into a Taylor series 

around a point x n , up to fifth order: 

y n−1 = 

y n+1 = 

y n − y n∆x ′ + 1 2 y′′ n(∆x) 2 − 1 6 y′′′ n (∆x) 3 + 1 

+O[(∆x) 6 ] 

y n + y n∆x ′ + 1 2 y′′ n(∆x) 2 + 1 6 y′′′ n (∆x) 3 + 1 

+O[(∆x) 6 ]. 

If we sum the two equations, we obtain: 

24 y′′′′ 

24 y′′′′ 

n (∆x) 4 − 1 

n (∆x) 4 + 1 

120 y′′′′′ 

120 y′′′′′ 

n (∆x) 5 

n (∆x) 5 

(1.24) 

y n+1 + y n−1 = 2y n + y ′′ 

n(∆x) 2 + 1 

12 y′′′′ n (∆x) 4 + O[(∆x) 6 ]. (1.25) 

Eq.(1.21) tells us that 

y n ′′ = −g n y n + s n ≡ z n . (1.26) 

The quantity z n above is introduced to simplify the notations. The following 

relation holds: 

z n+1 + z n−1 = 2z n + z ′′ 

n(∆x) 2 + O[(∆x) 4 ] (1.27) 

(this is the simple formula for discretized second derivative, that can be obtained 

in a straightforward way by Taylor expansion up to third order) and thus 

y ′′′′ 

n ≡ z ′′ 

n = z n+1 + z n−1 − 2z n 

(∆x) 2 + O[(∆x) 2 ]. (1.28) 

By inserting back these results into Eq.(1.25) one finds 

y n+1 = 2y n − y n−1 + (−g n y n + s n )(∆x) 2 

+ 1 

12 (−g n+1y n+1 + s n+1 − g n−1 y n−1 + s n−1 + 2g n y n − 2s n )(∆x) 2 

+O[(∆x) 6 ] 

(1.29) 

and finally the Numerov’s formula 

[ 

] [ 

] ] 

(∆x) 

y n+1 1 + g 2 

(∆x) 

n+1 = 2y n 1 − 5g 2 

(∆x) 

n − y n−1 

[1 + g 2 

n−1 

12 

12 

+(s n+1 + 10s n + s n−1 ) (∆x)2 

12 

+ O[(∆x) 6 ] 

12 

(1.30) 

that allows to obtain y n+1 starting from y n and y n−1 , and recursively the function 

in the entire box, as long as the value of the function is known in the first 

two points (note the difference with “traditional” initial conditions, Eq.(1.22), 

in which the value at one point and the derivative in the same point is specified). 

It is of course possible to integrate both in the direction of positive x and 

in the direction of negative x. In the presence of inversion symmetry, it will be 

sufficient to integrate in just one direction. 

In our case—Schrödinger equation—the s n terms are absent. It is convenient 

to introduce an auxiliary array f n , defined as 

(∆x) 2 

f n ≡ 1 + g n 

12 , where g n = 2m 

¯h 2 [E − V (x n)]. (1.31) 

Within such assumption Numerov’s formula can be written as 

y n+1 = (12 − 10f n)y n − f n−1 y n−1 

f n+1 

. (1.32) 

11

1.3.1 Code: harmonic0 

Code harmonic0.f90 1 (or harmonic0.c 2 ) solves the Schrödinger equation for 

the quantum harmonic oscillator, using the Numerov’s algorithm above described 

for integration, and searching eigenvalues using the “shooting method”. 

The code uses the adimensional units introduced in (1.4). 

The shooting method is quite similar to the bisection procedure for the 

search of the zero of a function. The code looks for a solution ψ n (x) with a 

pre-determined number n of nodes, at an energy E equal to the mid-point of the 

energy range [E min , E max ], i.e. E = (E max + E min )/2. The energy range must 

contain the desired eigenvalue E n . The wave function is integrated starting 

from x = 0 in the direction of positive x; tt the same time, the number of nodes 

(i.e. of changes of sign of the function) is counted. If the number of nodes is 

larger than n, E is too high; if the number of nodes is smaller than n, E is too 

low. We then choose the lower half-interval [E min , E max = E], or the upper halfinterval 

[E min = E, E max ], respectively, select a new trial eigenvalue E in the 

mid-point of the new interval, iterate the procedure. When the energy interval 

is smaller than a pre-determined threshold, we assume that convergence has 

been reached. 

For negative x the function is constructed using symmetry, since ψ n (−x) = 

(−1) n ψ n (x). This is of course possible only because V (−x) = V (x), otherwise 

integration would have been performed on the entire interval. The parity of the 

wave function determines the choice of the starting points for the recursion. For 

n odd, the two first points can be chosen as y 0 = 0 and an arbitrary finite value 

for y 1 . For n even, y 0 is arbitrary and finite, y 1 is determined by Numerov’s 

formula, Eq.(1.32), with f 1 = f −1 and y 1 = y −1 : 

The code prompts for some input data: 

y 1 = (12 − 10f 0)y 0 

2f 1 

. (1.33) 

• the limit x max for integration (typical values: 5 ÷ 10); 

• the number N of grid points (typical values range from hundreds to a few 

thousand); note that the grid point index actually runs from 0 to N, so 

that ∆x = x max /N; 

• the name of the file where output data is written; 

• the required number n of nodes (the code will stop if you give a negative 

number). 

Finally the code prompts for a trial energy. You should answer 0 in order to 

search for an eigenvalue with n nodes. The code will start iterating on the 

energy, printing on standard output (i.e. at the terminal): iteration number, 

number of nodes found (on the positive x axis only), the current energy eigenvalue 

estimate. It is however possible to specify an energy (not necessarily an 

1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/harmonic0.f90 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/harmonic0.c 

12

eigenvalue) to force the code to perform an integration at fixed energy and see 

the resulting wave function. It is useful for testing purposes and to better understand 

how the eigenvalue search works (or doesn’t work). Note that in this 

case the required number of nodes will not be honored; however the integration 

will be different for odd or even number of nodes, because the parity of n 

determines how the first two grid points are chosen. 

The output file contains five columns: respectively, x, ψ(x), |ψ(x)| 2 , ρ cl (x) 

and V (x). ρ cl (x) is the classical probability density (normalized to 1) of the 

harmonic oscillator, given in Eq.(1.19). All these quantities can be plotted as a 

function of x using any plotting program, such as gnuplot, shortly described in 

the introduction. Note that the code will prompt for a new value of the number 

of nodes after each calculation of the wave function: answer -1 to stop the code. 

If you perform more than one calculation, the output file will contain the result 

for all of them in sequence. Also note that the wave function are written for 

the entire box, from −x max to x max . 

It will become quickly evident that the code “sort of” works: the results 

look good in the region where the wave function is not vanishingly small, but 

invariably, the pathological behavior described in Sec.(1.2.2) sets up and wave 

functions diverge at large |x|. As a consequence, it is impossible to normalize 

the ψ(x). The code definitely needs to be improved. The proper way to deal 

with such difficulty is to find an inherently stable algorithm. 

1.3.2 Code: harmonic1 

Code harmonic1.f90 3 (or harmonic1.c 4 ) is the improved version of harmonic0 

that does not suffer from the problem of divergence at large x. 

Two integrations are performed: a forward recursion, starting from x = 0, 

and a backward one, starting from x max . The eigenvalue is fixed by the condition 

that the two parts of the function match with continuous first derivative (as 

required for a physical wave function, if the potential is finite). The matching 

point is chosen in correspondence of the classical inversion point, x cl , i.e. where 

V (x cl ) = E. Such point depends upon the trial energy E. For a function 

defined on a finite grid, the matching point is defined with an accuracy that is 

limited by the interval between grid points. In practice, one finds the index icl 

of the first grid point x c = icl∆x such that V (x c ) > E; the classical inversion 

point will be located somewhere between x c − ∆x and x c . 

The outward integration is performed until grid point icl, yielding a function 

ψ L (x) defined in [0, x c ]; the number n of changes of sign is counted in the 

same way as in harmonic0. If n is not correct the energy is adjusted (lowered 

if n too high, raised if n too low) as in harmonic0. We note that it is not 

needed to look for changes of sign beyond x c : in fact we know a priori that in 

the classically forbidden region there cannot be any nodes (no oscillations, just 

decaying solution). 

If the number of nodes is the expected one, the code starts to integrate 

inward from the rightmost points. Note the statement y(mesh) = dx: its only 

3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/harmonic1.f90 

4 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/harmonic1.c 

13

goal is to force solutions to be positive, since the solution at the left of the 

matching point is also positive. The value dx is arbitrary: the solution is 

anyway rescaled in order to be continuous at the matching point. The code 

stopst the same index icl corresponding to x c . We thus get a function ψ R (x) 

defined in [x c , x max ]. 

In general, the two parts of the wave function have different values in x c : 

ψ L (x c ) and ψ R (x c ). We first of all re-scale ψ R (x) by a factor ψ L (x c )/ψ R (x c ), 

so that the two functions match continuously in x c . Then, the whole function 

ψ(x) is renormalized in such a way that ∫ |ψ(x)| 2 dx = 1. 

Now comes the crucial point: the two parts of the function will have in 

general a discontinuity at the matching point ψ ′ R (x c) − ψ ′ L (x c). This difference 

should be zero for a good solution, but it will not in practice, unless we are 

really close to the good energy E = E n . The sign of the difference allows us 

to understand whether E is too high or too low, and thus to apply again the 

bisection method to improve the estimate of the energy. 

In order to calculate the discontinuity with good accuracy, we write the 

Taylor expansions: 

yi−1 L = yL i − y i 

′L ∆x + 1 2 y′′L i (∆x) 2 + O[(∆x) 3 ] 

yi+1 R = yR i + y i 

′R ∆x + 1 2 y′′R i (∆x) 2 + O[(∆x) 3 ] 

(1.34) 

For clarity, in the above equation i indicates the index icl. We sum the two 

Taylor expansions and obtain, noting that yi L = yi R = y i , and that y i 

′′L = y i ′′R = 

y i ′′ = −g iy i as guaranteed by Numerov’s method: 

that is 

y L i−1 + y R i+1 = 2y i + (y ′R 

i − y ′L 

i )∆x − g i y i (∆x) 2 + O[(∆x) 3 ] (1.35) 

y i ′R − y i 

′L = yL i−1 + yR i+1 − [2 − g i(∆x) 2 ]y i 

+ O[(∆x) 2 ] (1.36) 

∆x 

or else, by using the notations as in (1.31), 

y i ′R − y i 

′L = yL i−1 + yR i+1 − (14 − 12f i)y i 

+ O[(∆x) 2 ] (1.37) 

∆x 

In this way the code calculated the discontinuity of the first derivative. If the 

sign of y i ′R −y i 

′L is positive, the energy is too high (can you give an argument for 

this?) and thus we move to the lower half-interval; if negative, the energy is too 

low and we move to the upper half interval. As usual, convergence is declared 

when the size of the energy range has been reduced, by successive bisection, to 

less than a pre-determined tolerance threshold. 

During the procedure, the code prints on standard output a line for each 

iteration, containing: the iteration number; the number of nodes found (on the 

positive x axis only); if the number of nodes is the correct one, the discontinuity 

of the derivative y i ′R − y i 

′L (zero if number of nodes not yet correct); the current 

estimate for the energy eigenvalue. At the end, the code writes the final wave 

function (this time, normalized to 1!) to the output file. 

14

1.3.3 Laboratory 

Here are a few hints for “numerical experiments” to be performed in the computer 

lab (or afterward), using both codes: 

• Calculate and plot eigenfunctions for various values of n. It may be 

useful to plot, together with eigenfunctions or eigenfunctions squared, the 

classical probability density, contained in the fourth column of the output 

file. It will clearly show the classical inversion points. With gnuplot, e.g.: 

plot "filename" u 1:3 w l, "filename" u 1:4 w l 

(u = using, 1:3 = plot column3 vs column 1, w l = with lines; the second 

”filename” can be replaced by ””). 

• Look at the wave functions obtained by specifying an energy value not 

corresponding to an eigenvalue. Notice the difference between the results 

of harmonic0 and harmonic1 in this case. 

• Look at what happens when the energy is close to but not exactly an 

eigenvalue. Again, compare the behavior of the two codes. 

• Examine the effects of the parameters xmax, mesh. For a given ∆x, how 

large can be the number of nodes? 

• Verify how close you go to the exact results (notice that there is a convergence 

threshold on the energy in the code). What are the factors that 

affect the accuracy of the results? 

Possible code modifications and extensions: 

• Modify the potential, keeping inversion symmetry. This will require very 

little changes to be done. You might for instance consider a “double-well” 

potential described by the form: 

[ (x ) 4 ( ) x 2 

V (x) = ɛ − 2 + 1] 

, ɛ, δ > 0. (1.38) 

δ δ 

• Modify the potential, breaking inversion symmetry. You might consider 

for instance the Morse potential: 

[ 

] 

V (x) = D e −2ax − 2e −ax + 1 , (1.39) 

widely used to model the potential energy of a diatomic molecule. Which 

changes are needed in order to adapt the algorithm to cover this case? 

15

Chapter 2 

Schrödinger equation for 

central potentials 

In this chapter we will extend the concepts and methods introduced in the previous 

chapter for a one-dimensional problem to a specific and very important 

class of three-dimensional problems: a particle of mass m under a central potential 

V (r), i.e. depending only upon the distance r from a fixed center. The 

Schrödinger equation we are going to study in this chapter is thus 

Hψ(r) ≡ 

[ 

− ¯h2 

2m ∇2 + V (r) 

] 

ψ(r) = Eψ(r). (2.1) 

The problem of two interacting particles via a potential depending only upon 

their distance, V (|r 1 − r 2 |), e.g. the Hydrogen atom, reduces to this case (see 

the Appendix), with m equal to the reduced mass of the two particles. 

The general solution proceeds via the separation of the Schrödinger equation 

into an angular and a radial part. In this chapter we will be consider the 

numerical solution of the radial Schrödinger equation. A non-uniform grid is 

introduced and the radial Schrödinger equation is transformed to an equation 

that can still be solved using Numerov’s method. 

2.1 Variable separation 

Let us introduce a polar coordinate system (r, θ, φ), where θ is the polar angle, 

φ the azimuthal one, and the polar axis coincides with the z Cartesian axis. 

After some algebra, one finds the expression of the Laplacian operator: 

∇ 2 = 1 r 2 ∂ 

∂r 

( 

r 2 ∂ ∂r 

) 

+ 

1 

r 2 sin θ 

∂ 

∂θ 

( 

sin θ ∂ ∂θ 

) 

+ 

1 

r 2 sin 2 θ 

∂ 2 

∂φ 2 (2.2) 

It is convenient to introduce the operator L 2 = L 2 x + L 2 y + L 2 z, the square of the 

angular momentum vector operator, L = −i¯hr × ∇. Both ⃗ L and L 2 act only 

on angular variables. In polar coordinates, the explicit representation of L 2 is 

L 2 = −¯h 2 [ 

1 

sin θ 

( 

∂ 

sin θ ∂ ) 

+ 1 

∂θ ∂θ sin 2 θ 

∂ 2 ] 

∂φ 2 . (2.3) 

16

The Hamiltonian can thus be written as 

( 

H = − ¯h2 1 ∂ 

2m r 2 r 2 ∂ ) 

+ L2 

+ V (r). (2.4) 

∂r ∂r 2mr2 The term L 2 /2mr 2 also appears in the analogous classical problem: the radial 

motion of a mass having classical angular momentum L cl can be described by an 

effective radial potential ˆV (r) = V (r) + L 2 cl /2mr2 , where the second term (the 

“centrifugal potential”) takes into account the effects of rotational motion. For 

high L cl the centrifugal potential “pushes” the equilibrium positions outwards. 

In the quantum case, both L 2 and one component of the angular momentum, 

for instance L z : 

L z = −i¯h ∂ 

(2.5) 

∂φ 

commute with the Hamiltonian, so L 2 and L z are conserved and H, L 2 , L z have 

a (complete) set of common eigenfunctions. We can thus use the eigenvalues of 

L 2 and L z to classify the states. Let us now proceed to the separation of radial 

and angular variables, as suggested by Eq.(2.4). Let us assume 

ψ(r, θ, φ) = R(r)Y (θ, φ). (2.6) 

After some algebra we find that the Schrödinger equation can be split into an 

angular and a radial equation. The solution of the angular equations are the 

spherical harmonics, known functions that are eigenstates of both L 2 and of 

L z : 

L z Y lm (θ, φ) = m¯hY lm (θ, φ), L 2 Y lm (θ, φ) = l(l + 1)¯h 2 Y lm (θ, φ) (2.7) 

(l ≥ 0 and m = −l, ..., l are integer numbers). 

The radial equation is 

( 

− ¯h2 1 ∂ 

2m r 2 ∂r 

r 2 ∂R nl 

∂r 

) [ 

] 

+ V (r) + ¯h2 l(l + 1) 

2mr 2 R nl (r) = E nl R nl (r). (2.8) 

In general, the energy will depend upon l because the effective potential does; 

moreover, for a given l, we expect a quantization of the bound states (if existing) 

and we have indicated with n the corresponding index. 

Finally, the complete wave function will be 

ψ nlm (r, θ, φ) = R nl (r)Y lm (θ, φ) (2.9) 

The energy does not depend upon m. As already observed, m classifies the 

projection of the angular momentum on an arbitrarily chosen axis. Due to 

spherical symmetry of the problem, the energy cannot depend upon the orientation 

of the vector L, but only upon his modulus. An energy level E nl will 

then have a degeneracy 2l + 1 (or larger, if there are other observables that 

commute with the Hamiltonian and that we haven’t considered). 

17

2.1.1 Radial equation 

The probability to find the particle at a distance between r and r + dr from the 

center is given by the integration over angular variables: 

∫ 

|ψ nlm (r, θ, φ)| 2 rdθ r sin θ dφdr = |R nl | 2 r 2 dr = |χ nl | 2 dr (2.10) 

where we have introduced the auxiliary function χ(r) as 

χ(r) = rR(r) (2.11) 

and exploited the normalization of the spherical harmonics: 

∫ 

|Y lm (θ, φ)| 2 dθ sin θ dφ = 1 (2.12) 

(the integration is extended to all possible angles). As a consequence the normalization 

condition for χ is 

∫ ∞ 

|χ nl (r)| 2 dr = 1 (2.13) 

0 

The function |χ(r)| 2 can thus be directly interpreted as a radial (probability) 

density. Let us re-write the radial equation for χ(r) instead of R(r). Its is 

straightforward to find that Eq.(2.8) becomes 

− ¯h2 d 2 [ 

] 

χ 

2m dr 2 + V (r) + ¯h2 l(l + 1) 

2mr 2 − E χ(r) = 0. (2.14) 

We note that this equation is completely analogous to the Schrödinger equation 

in one dimension, Eq.(1.1), for a particle under an effective potential: 

ˆV (r) = V (r) + ¯h2 l(l + 1) 

2mr 2 . (2.15) 

As already explained, the second term is the centrifugal potential. The same 

methods used to find the solution of Eq.(1.1), and in particular, Numerov’s 

method, can be used to find the radial part of the eigenfunctions of the energy. 

2.2 Coulomb potential 

The most important and famous case is when V (r) is the Coulomb potential: 

V (r) = − Ze2 

4πɛ 0 r , (2.16) 

where e = 1.6021 × 10 −19 C is the electron charge, Z is the atomic number 

(number of protons in the nucleus), ɛ 0 = 8.854187817 × 10 −12 in MKSA units. 

Physicists tend to prefer the CGS system, in which the Coulomb potential is 

written as: 

V (r) = −Zq 2 e/r. (2.17) 

18

In the following we will use q 2 e = e 2 /(4πɛ 0 ) so as to fall back into the simpler 

CGS form. 

It is often practical to work with atomic units (a.u.): units of length are 

expressed in Bohr radii (or simply, “bohr”), a 0 : 

a 0 = 

¯h2 

m e q 2 e 

= 0.529177 Å = 0.529177 × 10 −10 m, (2.18) 

while energies are expressed in Rydberg (Ry): 

1 Ry = m eq 4 e 

2¯h 2 = 13.6058 eV. (2.19) 

when m e = 9.11 × 10 −31 Kg is the electron mass, not the reduced mass of 

the electron and the nucleus. It is straightforward to verify that in such units, 

¯h = 1, m e = 1/2, q 2 e = 2. 

We may also take the Hartree (Ha) instead or Ry as unit of energy: 

1 Ha = 2 Ry = m eq 4 e 

¯h 2 = 27.212 eV (2.20) 

thus obtaining another set on atomic units, in which ¯h = 1, m e = 1, q e = 1. 

Beware! Never talk about ”atomic units” without first specifying which ones. 

In the following, the first set (”Rydberg” units) will be occasionally used. 

We note first of all that for small r the centrifugal potential is the dominant 

term in the potential. The behavior of the solutions for r → 0 will then be 

determined by 

d 2 χ l(l + 1) 

≃ 

dr2 r 2 χ(r) (2.21) 

yielding χ(r) ∼ r l+1 , or χ(r) ∼ r −l . The second possibility is not physical 

because χ(r) is not allowed to diverge. 

For large r instead we remark that bound states may be present only if 

E < 0: there will be a classical inversion point beyond which the kinetic energy 

becomes negative, the wave function decays exponentially, only some energies 

can yield valid solutions. The case E > 0 corresponds instead to a problem of 

electron-nucleus scattering, with propagating solutions and a continuum energy 

spectrum. In this chapter, the latter case will not be treated. 

The asymptotic behavior of the solutions for large r → ∞ will thus be 

determined by 

d 2 χ 

dr 2 ≃ −2m e 

2 

Eχ(r) (2.22) 

¯h 

yielding χ(r) ∼ exp(±kr), where k = √ −2m e E/¯h. The + sign must be discarded 

as unphysical. It is thus sensible to assume for the solution a form 

like 

χ(r) = r l+1 e −kr 

∞ ∑ 

n=0 

A n r n (2.23) 

which guarantees in both cases, small and large r, a correct behavior, as long 

as the series does not diverge exponentially. 

19

2.2.1 Energy levels 

The radial equation for the Coulomb potential can then be solved along the 

same lines as for the harmonic oscillator, Sect.1.1. The expansion (2.23) is 

introduced into (2.14), a recursion formula for coefficients A n is derived, one 

finds that the series in general diverges like exp(2kr) unless it is truncated to a 

finite number of terms, and this happens only for some particular values of E: 

E n = − Z2 

n 2 m e q 4 e 

2¯h 2 

= − Z2 

Ry (2.24) 

n2 where n ≥ l + 1 is an integer known as main quantum number. For a given l 

we find solutions for n = l + 1, l + 2, . . .; or, for a given n, possible values for l 

are l = 0, 1, . . . , n − 1. 

Although the effective potential appearing in Eq.(2.14) depend upon l, and 

the angular part of the wave functions also strongly depend upon l, the energies 

(2.24) depend only upon n. We have thus a degeneracy on the energy levels 

with the same n and different l, in addition to the one due to the 2l+1 possible 

values of the quantum number m (as implied in (2.8) where m does not appear). 

The total degeneracy (not considering spin) for a given n is thus 

n−1 ∑ 

l=0 

2.2.2 Radial wave functions 

(2l + 1) = n 2 . (2.25) 

Finally, the solution for the radial part of the wave functions is 

where 

χ nl (r) = 

√ 

(n − l − 1)!Z 

n 2 [(n + l)!] 3 a 3 0 

x ≡ 2Z n 

x l+1 e −x/2 L 2l+1 

n+1 (x) (2.26) 

√ 

r 

= 2 − 2m eE n 

a 0 ¯h 2 r (2.27) 

and L 2l+1 

n+1 (x) are Laguerre polynomials of degree n − l − 1. The coefficient has 

been chosen in such a way that the following orthonormality relations hold: 

∫ ∞ 

χ nl (r)χ n ′ l(r)dr = δ nn ′. (2.28) 

0 

The ground state has n = 1 and l = 0: 1s in “spectroscopic” notation (2p is 

n = 2, l = 1, 3d is n = 3, l = 2, 4f is n = 4, l = 3, and so on). For the Hydrogen 

atom (Z = 1) the ground state energy is −1Ry and the binding energy of the 

electron is 1 Ry (apart from the small correction due to the difference between 

electron mass and reduced mass). The wave function of the ground state is a 

simple exponential. With the correct normalization: 

ψ 100 (r, θ, φ) = Z3/2 

√ e −Zr/a 0 

. (2.29) 

π 

a 3/2 

0 

20

The dominating term close to the nucleus is the first term of the series, 

χ nl (r) ∼ r l+1 . The larger l, the quicker the wave function tends to zero when 

approaching the nucleus. This reflects the fact that the function is “pushed 

away” by the centrifugal potential. Thus radial wave functions with large l do 

not appreciably penetrate close to the nucleus. 

At large r the dominating term is χ(r) ∼ r n exp(−Zr/na 0 ). This means 

that, neglecting the other terms, |χ nl (r)| 2 has a maximum about r = n 2 a 0 /Z. 

This gives a rough estimate of the “size” of the wave function, which is mainly 

determined by n. 

In Eq.(2.26) the polynomial has n − l − 1 degree. This is also the number of 

nodes of the function. In particular, the eigenfunctions with l = 0 have n − 1 

nodes; those with l = n − 1 are node-less. The form of the radial functions can 

be seen for instance on the Wolfram Research site 1 or explored via the Java 

applet at Davidson College 2 

2.3 Code: hydrogen radial 

The code hydrogen radial.f90 3 or hydrogen radial.c 4 solves the radial equation 

for a one-electron atom. It is based on harmonic1, but solves a slightly 

different equation on a logarithmically spaced grid. Moreover it uses a more 

sophisticated approach to locate eigenvalues, based on a perturbative estimate 

of the needed correction. 

The code uses atomic (Rydberg) units, so lengths are in Bohr radii (a 0 = 1), 

energies in Ry, ¯h 2 /(2m e ) = 1, q 2 e = 2. 

2.3.1 Logarithmic grid 

The straightforward numerical solution of Eq.(2.14) runs into the problem of 

the singularity of the potential at r = 0. One way to circumvent this difficulty 

is to work with a variable-step grid instead of a constant-step one, as done until 

now. Such grid becomes denser and denser as we approach the origin. “Serious” 

solutions of the radial Schrödinger in atoms, especially in heavy atoms, invariably 

involve such kind of grids, since wave functions close to the nucleus vary on 

a much smaller length scale than far from the nucleus. A detailed description of 

the scheme presented here can be found in chap.6 of The Hartree-Fock method 

for atoms, C. Froese Fischer, Wiley, 1977. 

Let us introduce a new integration variable x and a constant-step grid in 

x, so as to be able to use Numerov’s method without changes. We define a 

mapping between r and x via 

x = x(r). (2.30) 

The relation between the constant-step grid spacing ∆x and the variable-step 

grid spacing is given by 

∆x = x ′ (r)∆r. (2.31) 

1 http://library.wolfram.com/webMathematica/Physics/Hydrogen.jsp 

2 http://webphysics.davidson.edu/physlet resources/cise qm/html/hydrogenic.html 

3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/hydrogen radial.f90 

4 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/hydrogen radial.c 

21

We make the specific choice 

(note that with this choice x is adimensional) yielding 

x(r) ≡ log Zr 

a 0 

(2.32) 

∆x = ∆r 

r . (2.33) 

The ∆r/r ratio remains thus constant on the grid of r, called logarithmic grid, 

so defined. 

There is however a problem: by transforming Eq.(2.14) in the new variable 

x, a term with first derivative appears, preventing the usage of Numerov’s 

method (and of other integration methods as well). The problem can be circumvented 

by transforming the unknown function as follows: 

y(x) = 1 √ r 

χ (r(x)) . (2.34) 

It is easy to verify that by transforming Eq.(2.14) so as to express it as a 

function of x and y, the terms containing first-order derivatives disappear, and 

by multiplying both sides of the equation by r 3/2 one finds 

d 2 [ ( 

y 

dx 2 + 2me 

¯h 2 r2 (E − V (r)) − l + 1 ) ] 2 

y(x) = 0 (2.35) 

2 

where V (r) = −Zqe/r 2 for the Coulomb potential. This equation no longer 

presents any singularity for r = 0, is in the form of Eq.(1.21), with 

g(x) = 2m ( 

e 

¯h 2 r(x)2 (E − V (r(x))) − l + 1 2 

(2.36) 

2) 

and can be directly solved using the numerical integration formulae Eqs.(1.31) 

and Eq.(1.32) and an algorithm very similar to the one of Sect.1.3.2. 

Subroutine do mesh defines at the beginning and once for all the values of 

r, √ r, r 2 for each grid point. The potential is also calculated once and for all 

in init pot. The grid is calculated starting from a minimum value x = −8, 

corresponding to Zr min ≃ 3.4 × 10 −3 Bohr radii. Note that the grid in r does 

not include r = 0: this would correspond to x = −∞. The known analytical 

behavior for r → 0 and r → ∞ are used to start the outward and inward 

recurrences, respectively. 

2.3.2 Improving convergence with perturbation theory 

A few words are in order to explain this section of the code: 

i = icl 

ycusp = (y(i-1)*f(i-1)+f(i+1)*y(i+1)+10.d0*f(i)*y(i)) / 12.d0 

dfcusp = f(i)*(y(i)/ycusp - 1.d0) 

! eigenvalue update using perturbation theory 

de = dfcusp/ddx12 * ycusp*ycusp * dx 

22

whose goal is to give an estimate, to first order in perturbation theory, of the 

difference δe between the current estimate of the eigenvalue and its final value. 

Reminder: icl is the index corresponding to the classical inversion point. 

Integration is made with forward recursion up to this index, with backward 

recursion down to this index. icl is thus the index of the matching point 

between the two functions. The function at the right is rescaled so that the 

total function is continuous, but the first derivative dy/dx will be in general 

discontinuous, unless we have reached a good eigenvalue. 

In the section of the code shown above, y(icl) is the value given by Numerov’s 

method using either icl-1 or icl+1 as central point; ycusp is the value 

predicted by the Numerov’s method using icl as central point. The problem 

is that ycusp≠y(icl). 

What about if our function is the exact solution, but for a different problem? 

It is easy to find what the different problem could be: one in which a delta function, 

v 0 δ(x−x c ), is superimposed at x c ≡x(icl) to the potential. The presence 

of a delta function causes a discontinuity (a ”cusp”) in the first derivative, as 

can be demonstrated by a limit procedure, and the size of the discontinuity is 

related to the coefficient of the delta function. Once the latter is known, we 

can give an estimate, based on perturbation theory, of the difference between 

the current eigenvalue (for the ”different” potential) and the eigenvalue for the 

”true” potential. 

One may wonder how to deal with a delta function in numerical integration. 

In practice, we assume the delta to have a value only in the interval ∆x centered 

on y(icl). The algorithm used to estimate its value is quite sophisticated. Let 

us look again at Numerov’s formula (1.32): note that the formula actually 

provides only the product y(icl)f(icl). From this we usually extract y(icl) 

since f(icl) is assumed to be known. Now we suppose that f(icl) has a 

different and unknown value fcusp, such that our function satisfies Numerov’s 

formula also in point icl. The following must hold: 

fcusp*ycusp = f(icl)*y(icl) 

since this product is provided by Numerov’s method (by integrating from icl-1 

to icl+1), and ycusp is that value that the function y must have in order to 

satisfy Numerov’s formula also in icl. As a consequence, the value of dfcusp 

calculated by the program is just fcusp-f(icl), or δf. 

The next step is to calculate the variation δV of the potential V (r) appearing 

in Eq.(2.35) corresponding to δf. By differentiating Eq.(2.36) one finds δg(x) = 

−(2m e /¯h 2 )r 2 δV . Since f(x) = 1 + g(x)(∆x) 2 /12, we have δg = 12/(∆x) 2 δf, 

and thus 

δV = − ¯h2 1 12 

2m e r 2 δf. (2.37) 

(∆x) 2 

First-order perturbation theory gives then the corresponding variation of the 

eigenvalue: 

∫ 

δe = 〈ψ|δV |ψ〉 = |y(x)| 2 r(x) 2 δV dx. (2.38) 

23

Note that the change of integration variable from dr to dx adds a factor r to 

the integral: 

∫ ∞ 

0 

f(r)dr = 

∫ ∞ 

−∞ 

f(r(x)) dr(x) 

dx dx = ∫ ∞ 

−∞ 

f(r(x))r(x)dx. (2.39) 

We write the above integral as a finite sum over grid points, with a single 

non-zero contribution coming from the region of width ∆x centered at point 

x c =x(icl). Finally: 

δe = |y(x c )| 2 r(x c ) 2 δV ∆x = − ¯h2 

2m e 

12 

(∆x) 2 |y(x c)| 2 ∆xδf (2.40) 

is the difference between the eigenvalue of the current potential (i.e. with a 

superimposed Delta function) and that of the true potential. This expression 

is used by the code to calculate the correction de to the eigenvalue. Since in 

the first step this estimate may have large errors, the line 

e = max(min(e+de,eup),elw) 

prevents the usage of a new energy estimate outside the bounds [elw,eip]. As 

the code proceeds towards convergence, the estimate becomes better and better 

and convergence is very fast in the final steps. 


• Examine solutions as a function of n and l; verify the presence of accidental 

degeneracy. 

• Examine solutions as a function of the nuclear charge Z. 

• Compare the numerical solution with the exact solution, Eq.(2.29), for 

the 1s case (or other cases if you know the analytic solution). 

• Slightly modify the potential as defined in subroutine init pot, verify 

that the accidental degeneracy disappears. Some suggestions: V (r) = 

−Zq 2 e/r 1+δ where δ is a small, positive or negative, number; or add an 

exponential damping (Yukawa) V (r) = −Zq 2 e exp(−Qr)/r where Q is a 

number of the order of 0.05 a.u.. 

• Calculate the expectation values of r and of 1/r, compare them with the 

known analytical results. 


• Consider a different mapping: r(x) = r 0 (exp(x) − 1), that unlike the one 

we have considered, includes r = 0. Which changes must be done in order 

to adapt the code to this mapping? 

24

Chapter 3 

Scattering from a potential 

Until now we have considered the discrete energy levels of simple, one-electron 

Hamiltonians, corresponding to bound, localized states. Unbound, delocalized 

states exist as well for any physical potential (with the exception of idealized 

models like the harmonic potential) at sufficiently high energies. These states 

are relevant in the description of elastic scattering from a potential, i.e. processes 

of diffusion of an incoming particle. Scattering is a really important 

subject in physics, since what many experiments measure is how a particle is 

deflected by another. The comparison of measurements with calculated results 

makes it possible to understand the form of the interaction potential between 

the particles. In the following a very short reminder of scattering theory is 

provided; then an application to a real problem (scattering of H atoms by rare 

gas atoms) is presented. This chapter is inspired to Ch.2 (pp.14-29) of the book 

of Thijssen. 

3.1 Short reminder of the theory of scattering 

The elastic scattering of a particle by another is first mapped onto the equivalent 

problem of elastic scattering from a fixed center, using the same coordinate 

change as described in the Appendix. In the typical geometry, a free particle, 

described as a plane wave with wave vector along the z axis, is incident on the 

center and is scattered as a spherical wave at large values of r (distance from the 

center). A typical measured quantity is the differential cross section dσ(Ω)/dΩ, 

i.e. the probability that in the unit of time a particle crosses the surface in the 

surface element dS = r 2 dΩ (where Ω is the solid angle, dΩ = sin θdθdφ, where 

θ is the polar angle and φ the azimuthal angle). Another useful quantity is the 

total cross section σ tot = ∫ (dσ(Ω)/dΩ)dΩ. For a central potential, the system 

is symmetric around the z axis and thus the differential cross section does not 

depend upon φ. The cross section depends upon the energy of the incident 

particle. 

Let us consider a solution having the form: 

ψ(r) = e ik·r + f(θ) 

r eikr (3.1) 

25

with k = (0, 0, k), parallel to the z axis. This solution represents an incident 

plane wave plus a diffused spherical wave. It can be shown that the cross section 

is related to the scattering amplitude f(θ): 

dσ(Ω) 

dΩ dΩ = |f(θ)|2 dΩ = |f(θ)| 2 sin θdθdφ. (3.2) 

Let us look for solutions of the form (3.1). The wave function is in general given 

by the following expression: 

ψ(r) = 

∞∑ 

l∑ 

l=0 m=−l 

χ l (r) 

A lm Y lm (θ, φ), (3.3) 

r 

which in our case, given the symmetry of the problem, can be simplified as 

ψ(r) = 

∞∑ 

l=0 

χ l (r) 

A l P l (cos θ). (3.4) 

r 

The functions χ l (r) are solutions for (positive) energy E = ¯h 2 k 2 /2m with angular 

momentum l of the radial Schrödinger equation: 

¯h 2 d 2 [ 

] 

χ l (r) 

2m dr 2 + E − V (r) − ¯h2 l(l + 1) 

2mr 2 χ l (r) = 0. (3.5) 

It can be shown that the asymptotic behavior at large r of χ l (r) is 

χ l (r) ≃ kr [j l (kr) cos δ l − n l (kr) sin δ l ] (3.6) 

where the j l and n l functions are the well-known spherical Bessel functions, 

respectively regular and divergent at the origin. These are the solutions of the 

radial equation for R l (r) = χ l (r)/r in absence of the potential. The quantities 

δ l are known as phase shifts. We then look for coefficients of Eq.(3.4) that yield 

the desired asymptotic behavior (3.1). It can be demonstrated that the cross 

section can be written in terms of the phase shifts. In particular, the differential 

cross section can be written as 

dσ 

dΩ = 1 ∣ ∣∣∣∣ ∞ ∑ 

2 

k 2 (2l + 1)e iδ l sin δ l P l (cos θ) 

, (3.7) 

∣ 

l=0 

while the total cross section is given by 

σ tot = 4π 

k 2 

∞ ∑ 

l=0 

(2l + 1) sin 2 δ l k = 

√ 

2mE 

¯h 2 . (3.8) 

Note that the phase shifts depend upon the energy. The phase shifts thus contain 

all the information needed to know the scattering properties of a potential. 

26

3.2 Scattering of H atoms from rare gases 

The total cross section σ tot (E) for the scattering 

of H atoms by rare gas atoms was measured by 

Toennies et al., J. Chem. Phys. 71, 614 (1979). 

At the right, the cross section for the H-Kr system 

as a function of the energy of the center of 

mass. One can notice “spikes” in the cross section, 

known as “resonances”. One can connect 

the different resonances to specific values of the 

angular momentum l. 

The H-Kr interaction potential can be modelled quite accurately as a Lennard- 

Jones (LJ) potential: 

[ (σ ) 12 ( ) ] 

σ 

6 

V (r) = ɛ − 2 

(3.9) 

r r 

where ɛ = 5.9meV, σ = 3.57Å. The LJ potential is much used in molecular and 

solid-state physics to model interatomic interaction forces. The attractive r −6 

term describes weak van der Waals (or “dispersive”, in chemical parlance) forces 

due to (dipole-induced dipole) interactions. The repulsive r −12 term models the 

repulsion between closed shells. While usually dominated by direct electrostatic 

interactions, the ubiquitous van der Waals forces are the dominant term for the 

interactions between closed-shell atoms and molecules. These play an important 

role in molecular crystals and in macro-molecules. The LJ potential is the first 

realistic interatomic potential for which a molecular-dynamics simulation was 

performed (Rahman, 1965, liquid Ar). 

It is straightforward to find that the LJ potential as written in (3.9) has a 

minimum V min = −ɛ for r = σ, is zero for r = σ/2 1/6 = 0.89σ and becomes 

strongly positive (i.e. repulsive) at small r. 

3.2.1 Derivation of Van der Waals interaction 

The Van der Waals attractive interaction can be described in semiclassical terms 

as a dipole-induced dipole interaction, where the dipole is produced by a charge 

fluctuation. A more quantitative and satisfying description requires a quantummechanical 

approach. Let us consider the simplest case: two nuclei, located 

in R A and R B , and two electrons described by coordinates r 1 and r 2 . The 

Hamiltonian for the system can be written as 

H = − ¯h2 

2m ∇2 1 − 

q 2 e 

|r 1 − R A | − ¯h2 

2m ∇2 2 − 

q 2 e 

|r 2 − R B | 

(3.10) 

− 

q 2 e 

|r 1 − R B | − q 2 e 

|r 2 − R A | + q 2 e 

|r 1 − r 2 | + q 2 e 

|R A − R B | , 

where ∇ i indicates derivation with respect to variable r i , i = 1, 2. Even this 

“simple” hamiltonian is a really complex object, whose general solution will be 

the subject of several chapters of these notes. We shall however concentrate on 

27

the limit case of two Hydrogen atoms at a large distance R, with R = R A −R B . 

Let us introduce the variables x 1 = r 1 −R A , x 2 = r 2 −R B . In terms of these new 

variables, we have H = H A +H B +∆H(R), where H A (H B ) is the Hamiltonian 

for a Hydrogen atom located in R A (R B ), and ∆H has the role of “perturbing 

potential”: 

∆H = − 

q2 e 

|x 1 + R| − q 2 e 

|x 2 − R| + q 2 e 

|x 1 − x 2 + R| + q2 e 

R . (3.11) 

Let us expand the perturbation in powers of 1/R. The following expansion, 

valid for R → ∞, is useful: 

1 

|x + R| ≃ 1 R − R · x 

R 3 + 3(R · x)2 − x 2 R 2 

R 5 . (3.12) 

Using such expansion, it can be shown that the lowest nonzero term is 

( 

∆H ≃ 2q2 e 

R 3 x 1 · x 2 − 3 (R · x ) 

1)(R · x 2 ) 

R 2 . (3.13) 

The problem can now be solved using perturbation theory. The unperturbed 

ground-state wavefunction can be written as a product of 1s states centered 

around each nucleus: Φ 0 (x 1 , x 2 ) = ψ 1s (x 1 )ψ 1s (x 2 ) (it should actually be antisymmetrized 

but in the limit of large separation it makes no difference.). It is 

straightforward to show that the first-order correction, ∆ (1) E = 〈Φ 0 |∆H|Φ 0 〉, 

to the energy, vanishes because the ground state is even with respect to both x 1 

and x 2 . The first nonzero contribution to the energy comes thus from secondorder 

perturbation theory: 

∆ (2) E = − ∑ i>0 

|〈Φ i |∆H|Φ 0 〉| 2 

E i − E 0 

(3.14) 

where Φ i are excited states and E i the corresponding energies for the unperturbed 

system. Since ∆H ∝ R −3 , it follows that ∆ (2) E = −C 6 /R 6 , the wellknown 

behavior of the Van der Waals interaction. 1 The value of the so-called 

C 6 coefficient can be shown, by inspection of Eq.(3.14), to be related to the 

product of the polarizabilities of the two atoms. 

3.3 Code: crossection 

Code crossection.f90 2 , or crossection.c 3 , calculates the total cross section 

σ tot (E) and its decomposition into contributions for the various values of the 

angular momentum for a scattering problem as the one described before. 

The code is composed of different parts. It is always a good habit to verify 

separately the correct behavior of each piece before assembling them into the 

final code (this is how professional software is tested, by the way). The various 

parts are: 

1 Note however that at very large distances a correct electrodynamical treatment yields a 

different behavior: ∆E ∝ −R −7 . 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/crossection.f90 

3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/crossection.c 

28

1. Solution of the radial Schrödinger equation, Eq.(3.5), with the Lennard- 

Jones potential (3.9) for scattering states (i.e. with positive energy). One 

can simply use Numerov’s method with a uniform grid and outwards 

integration only (there is no danger of numerical instabilities, since the 

solution is oscillating). One has however to be careful and to avoid the 

divergence (as ∼ r −12 ) for r → 0. A simple way to avoid running into 

trouble is to start the integration not from r min = 0 but from a small but 

nonzero value. A suitable value might be r min ∼ 0.5σ, where the wave 

function is very small but not too close to zero. The first two points can be 

calculated in the following way, by assuming the asymptotic (vanishing) 

form for r → 0: 

χ ′′ (r) ≃ 2mɛ 

¯h 2 σ 12 

r 12 χ(r) =⇒ χ(r) ≃ exp ⎛ 

⎝− 

√ 

2mɛσ 12 

25¯h 2 

r−5 ⎞ 

⎠ (3.15) 

(note that this procedure is not very accurate because it introduces and 

error of lower order, i.e. worse, than that of the Numerov’s algorithm. 

In fact by assuming such form for the first two steps of the recursion, 

we use a solution that is neither analytically exact, nor consistent with 

Numerov’s algorithm). 

The choice of the units in this code is (once again) different from that of 

the previous codes. It is convenient to choose units in which ¯h 2 /2m is a 

number of the order of 1. Two possible choices are meV/Å 2 , or meV/σ 2 . 

This code uses the former choice. Note that m here is not the electron 

mass! it is the reduced mass of the H-Kr system. As a first approximation, 

m is here the mass of H. 

2. Calculation of the phase shifts δ l (E). Phase shifts can be calculated by 

comparing the calculated wave functions with the asymptotic solution at 

two different values r 1 and r 2 , both larger than the distance r max beyond 

which the potential can be considered to be negligible. Let us write 

χ l (r 1 ) = Akr 1 [j l (kr 1 ) cos δ l − n l (kr 1 ) sin δ l ] (3.16) 

χ l (r 2 ) = Akr 2 [j l (kr 2 ) cos δ l − n l (kr 2 ) sin δ l ] , (3.17) 

from which, by dividing the two relations, we obtain an auxiliary quantity 

K 

K ≡ r 2χ l (r 1 ) 

r 1 χ l (r 2 ) = j l(kr 1 ) − n l (kr 1 ) tan δ l 

(3.18) 

j l (kr 2 ) − n l (kr 2 ) tan δ l 

from which we can deduce the phase shift: 

tan δ l = Kj l(kr 2 ) − j l (kr 1 ) 

Kn l (kr 2 ) − n l (kr 1 ) . (3.19) 

The choice of r 1 and r 2 is not very critical. A good choice for r 1 is about 

at r max . Since the LJ potential decays quite quickly, a good choice is 

r 1 = r max = 5σ. For r 2 , the choice is r 2 = r 1 + λ/2, where λ = 1/k is the 

wave length of the scattered wave function. 

29

3. Calculation of the spherical Bessel functions j l and n l . The analytical 

forms of these functions are known, but they become quickly unwieldy for 

high l. One can use recurrence relation. In the code the following simple 

recurrence is used: 

with the initial conditions 

z l+1 (x) = 2l + 1 

x z l(x) − z l−1 (x), z = j, n (3.20) 

j −1 (x) = cos x 

x , j 0(x) = sin x 

x ; n −1(x) = sin x 

x , n 0(x) = − cos x 

x . 

(3.21) 

Note that this recurrence is unstable for large values of l: in addition to 

the good solution, it has a ”bad” divergent solution. This is not a serious 

problem in practice: the above recurrence should be sufficiently stable up 

to at least l = 20 or so, but you may want to verify this. 

4. Sum the phase shifts as in Eq.(3.8) to obtain the total cross section and 

a graph of it as a function of the incident energy. The relevant range of 

energies is of the order of a few meV, something like 0.1 ≤ E ≤ 3 ÷ 4 

meV. If you go too close to E = 0, you will run into numerical difficulties 

(the wave length of the wave function diverges). The angular momentum 

ranges from 0 to a value of l max to be determined. 

The code writes a file containing the total cross section and each angular momentum 

contribution as a function of the energy (beware: up to l max = 13; for 

larger l lines will be wrapped). 


• Verify the effect of all the parameters of the calculation: grid step for integration, 

r min , r 1 = r max , r 2 , l max . In particular the latter is important: 

you should start to find good results when l max ≥ 6. 

• Print, for some selecvted values of the energy, the wave function and its 

asymptotic behavior. You should verify that they match beyond r max . 

• Observe the contributions of the various l and the peaks for l = 4, 5, 6 

(resonances). Make a plot of the effective potential as a function of l: can 

you see why there are resonances only for a few values of l? 


• Modify the code to calculate the cross section from a different potential, 

for instance, the following one: 

[ 

V (r) = −A exp −(r − r 0 ) 2] , r < r max ; V (r > r max ) = 0 

What changes do you need to apply to the algorithm, in addition to 

changing the form of the potential? 

30

• In the limit of a weak potential (such as e.g. the potential introduced 

above), the phase shifts δ l are well approximated by the Born approximation: 

δ l ≃ − 2m ∫ ∞ 

¯h 2 k r 2 jl 2 (kr)V (r)dr, 

0 

(k = √ 2mE/¯h). Write a simple routine to calculate this integral, compare 

with numerical results. 

31

Chapter 4 

The Variational Method 

The exact analytical solution of the Schrödinger equation is possible only in 

a few cases. Even the direct numerical solution by integration is often not 

feasible in practice, especially in systems with more than one particle. There are 

however extremely useful approximated methods that can in many cases reduce 

the complete problem to a much simpler one. In the following we will consider 

the variational principle and its consequences. This constitutes, together with 

suitable approximations for the electron-electron interactions, the basis for most 

practical approaches to the solution of the Schrödinger equation in condensedmatter 

physics. 

4.1 Variational Principle 

Let us consider a Hamiltonian H and a function ψ, that can be varied at will 

with the sole condition that it stays normalized. One can calculate the expectation 

value of the energy for such function (in general, not an eigenfunction of 

H): 

∫ 

〈H〉 = ψ ∗ Hψ dv (4.1) 

where v represents all the integration coordinates. 

The variational principle states that functions ψ for which 〈H〉 is stationary— 

i.e. does not vary to first order in small variations of ψ—are the eigenfunctions 

of the energy. In other words, the Schrödinger equation is equivalent to a stationarity 

condition. 

4.1.1 Demonstration of the variational principle 

Since an arbitrary variation δψ of a wave function in general destroys its normalization, 

it is convenient to use a more general definition of expectation value, 

valid also for non-normalized functions: 

∫ ψ ∗ Hψ dv 

〈H〉 = ∫ ψ ∗ (4.2) 

ψ dv 

32

By modifying ψ as ψ + δψ, the expectation value becomes 

〈H〉 + δ〈H〉 = 

= 

= 

∫ (ψ ∗ + δψ ∗ )H(ψ + δψ) dv 

∫ (ψ ∗ + δψ ∗ )(ψ + δψ) dv 

∫ ψ ∗ Hψ dv + ∫ δψ ∗ Hψ dv + ∫ ψ ∗ Hδψ dv 

∫ ψ ∗ ψ dv + ∫ δψ ∗ ψ dv + ∫ ψ ∗ δψ dv 

(∫ 

∫ 

ψ ∗ Hψ dv + 

( 

1 

∫ ψ ∗ 1 − 

ψ dv 

∫ 

δψ ∗ Hψ dv + 

) 

ψ ∗ Hδψ dv 

∫ δψ ∗ ∫ 

ψ dv ψ ∗ ) 

∫ ψ ∗ ψ dv − δψ dv 

∫ ψ ∗ ψ dv 

× 

(4.3) 

where second-order terms in δψ have been omitted and we have used the approximation 

1/(1+x) ≃ 1−x, valid for x

where λ k are constants to be determined (Lagrange multipliers). In our case we 

have 

∫ 

I 0 = ψ ∗ Hψ dv (4.11) 

∫ 

I 1 = ψ ∗ ψ dv (4.12) 

and thus we assume 

δ(I 0 + λI 1 ) = 0 (4.13) 

where λ must be determined. By proceeding like in the previous section, we 

find 

∫ 

δI 0 = δψ ∗ Hψ dv + c.c. (4.14) 

∫ 

δI 1 = δψ ∗ ψ dv + c.c. (4.15) 

and thus the condition to be satisfied is 

∫ 

δ(I 0 + λI 1 ) = δψ ∗ [H + λ]ψ dv + c.c. = 0 (4.16) 

that is 

Hψ = −λψ (4.17) 

i.e. the Lagrange multiplier is equal, apart from the sign, to the energy eigenvalue. 

Again we see that states whose expectation energy is stationary with 

respect to any variation in the wave function are the solutions of the Schrödinger 

equation. 

4.1.3 Ground state energy 

Let us consider the eigenfunctions ψ n of a Hamiltonian H, with associated 

eigenvalues (energies) E n : 

Hψ n = E n ψ n . (4.18) 

Let us label the ground state with n = 0 and the ground-state energy as E 0 . 

Let us demonstrate that for any different function ψ, we necessarily have 

〈H〉 = 

∫ ψ ∗ Hψ dv 

∫ ψ ∗ ψ dv ≥ E 0. (4.19) 

In order to demonstrate it, let us expand ψ using as basis energy eigenfunctions 

(this is always possible because energy eigenfunctions are a complete orthonormal 

set): 

ψ = ∑ c n ψ n (4.20) 

n 

Then one finds 

〈H〉 = 

∑ 

n |c n| 2 ∑ 

E n 

n 

∑n |c n| 2 = E 0 + 

|c n| 2 (E n − E 0 ) 

∑n |c n| 2 (4.21) 

34

This demonstrates Eq.(4.19), since the second term is either positive or zero, 

as E n ≥ E 0 by definition of ground state, 

This simple result is extremely important: it tells us that any function ψ, 

yields for the expectation energy an upper estimate of the energy of the ground 

state. If the ground state is unknown, an approximation to the ground state 

can be found by varying ψ inside a given set of functions and looking for the 

function that minimizes 〈H〉. This is the essence of the variational method. 

4.1.4 Variational method in practice 

One identifies a set of trial wave functions ψ(v; α 1 , . . . , α r ), where v are the 

variables of the problem (coordinates etc), α i , i = 1, . . . , r are parameters. The 

energy eigenvalue will be a function of the parameters: 

∫ 

E(α 1 , . . . , α r ) = ψ ∗ Hψ dv (4.22) 

The variational method consists in looking for the minimum of E with respect 

to a variation of the parameters, that is, by imposing 

∂E 

∂α 1 

= . . . = ∂E 

∂α r 

= 0 (4.23) 

The function ψ satisfying these conditions with the lowest E is the function 

that better approximates the ground state, among the considered set of trial 

functions. 

It is clear that a suitable choice of the trial functions plays a crucial role 

and must be carefully done. 

4.2 Secular problem 

The variational method can be reduced to an algebraic problem by expanding 

the wave function into a finite basis of functions, and applying the variational 

principle to find the optimal coefficients of the expansion. Based on Eq. (4.10), 

this means calculating the functional (i.e. a “function” of a function): 

G[ψ] = 〈ψ|H|ψ〉 − ɛ〈ψ|ψ〉 

∫ 

∫ 

= ψ ∗ Hψ dv − ɛ ψ ∗ ψ dv (4.24) 

and imposing the stationary condition on G[ψ]. Such procedure produces an 

equation for the expansion coefficients that we are going to determine. 

It is important to notice that our basis is formed by a finite number N of 

functions, and thus cannot be a complete system: in general, it is not possible 

to write any function ψ (including exact solutions of the Schrödinger equation) 

as a linear combination of the functions in this basis set. What we are going to 

do is to find the ψ function that better approaches the true ground state, among 

all functions that can be expressed as linear combinations of the N chosen basis 

functions. 

35

4.2.1 Expansion into a basis set of orthonormal functions 

Let us assume to have a basis of N functions b i , between which orthonormality 

relations hold: 

∫ 

〈b i |b j 〉 ≡ b ∗ i b j dv = δ ij (4.25) 

Let us expand the generic ψ in such basis: 

N∑ 

ψ = c i b i (4.26) 

i=1 

By replacing Eq.(4.26) into Eq.(4.24) one can immediately notice that the latter 

takes the form 

G(c 1 , . . . , c N ) = ∑ ij 

c ∗ i c j H ij − ɛ ∑ ij 

c ∗ i c j δ ij 

= ∑ ij 

c ∗ i c j (H ij − ɛδ ij ) (4.27) 

where we have introduced the matrix elements H ij : 

∫ 

H ij = 〈b i |H|b j 〉 = b ∗ i Hb j dv (4.28) 

Since both H and the basis functions are given, H ij is a known square matrix of 

numbers. The hermiticity of the Hamiltonian operator implies that such matrix 

is hermitian: 

H ji = H ∗ ij (4.29) 

(i.e. symmetric if all elements are real). According to the variational method, 

let us minimize Eq. (4.27) with respect to the coefficients: 

∂G 

∂c i 

= 0 (4.30) 

This produces the condition 

∑ 

(H ij − ɛδ ij )c j = 0 (4.31) 

j 

If the derivative with respect to complex quantities bothers you: write the 

complex coefficients as a real and an imaginary part c k = x k + iy k , require that 

derivatives with respect to both x k and y k are zero. By exploiting hermiticity 

you will find a system 

W k + W ∗ k = 0 

−iW k + iW ∗ k = 0 

where W k = ∑ j (H kj − ɛδ kj )c j , that allows as only solution W k = 0. 

We note that, if the basis were a complete (and thus infinite) system, this 

would be the form of the Schrödinger equation. We have finally demonstrated 

36

that the same equations, for a finite basis set, yield the best approximation to 

the true solution according to the variational principle. 

Eq.(4.31) is a system of N algebraic linear equations, homogeneous (there 

are no constant term) in the N unknown c j . In general, this system has only 

the trivial solution c j = 0 for all coefficients, obviously unphysical. A nonzero 

solution exists if and only if the following condition on the determinant is 

fulfilled: 

det |H ij − ɛδ ij | = 0 (4.32) 

Such condition implies that one of the equations is a linear combination of the 

others and the system has in reality N − 1 equations and N unknowns, thus 

admitting non-zero solutions. 

Eq.(4.32) is known as secular equation. It is an algebraic equation of degree 

N in ɛ (as it is evident from the definition of the determinant, with the 

main diagonal generating a term ɛ N , all other diagonals generating lower-order 

terms), that admits N roots, or eigenvalues. Eq.(4.31) can also be written in 

matrix form 

Hc = ɛc (4.33) 

where H is here the N ×N matrix whose matrix elements are H ij , c is the vector 

formed with c i components. The solutions c are also called eigenvectors. For 

each eigenvalue there will be a corresponding eigenvector (known within a multiplicative 

constant, fixed by the normalization). We have thus N eigenvectors 

and we can write that there are N solutions: 

ψ k = ∑ i 

C ik b i , k = 1, . . . , N (4.34) 

where C ik is a matrix formed by the N eigenvectors (written as columns and 

disposed side by side): 

Hψ k = ɛ k ψ k , (4.35) 

that is, in matrix form, taking the i−th component, 

(Hψ k ) i = ∑ j 

H ij C jk = ɛ k C ik . (4.36) 

Eq.(4.33) is a common equation in linear algebra and there are standard 

methods to solve it. Given a matrix H, it is possible to obtain, using standard 

library routines, the C matrix and a vector ɛ of eigenvalues. 

The solution process is usually known as diagonalization. This name comes 

from the following important property of C. Eq.(4.34) can be seen as a transformation 

of the N starting functions into another set of N functions, via a 

transformation matrix. It is possible to show that if the b i functions are orthonormal, 

the ψ k functions are orthonormal as well. Then the transformation 

is unitary, i.e. 

∑ 

CijC ∗ ik = δ jk (4.37) 

or, in matrix notations, 

i 

(C −1 ) ij = C ∗ ji ≡ C † ij (4.38) 

37

that is, the inverse matrix is equal to the conjugate of the transpose matrix, 

known as adjoint matrix. The matrix C having such property is also called a 

unitary matrix. 

Let us consider now the matrix product C −1 HC and let us calculate its 

elements: 

(C −1 HC) kn = ∑ ij 

= ∑ i 

(C −1 ) ki H ij C jn 

Cik 

∗ ∑ 

j 

H ij C jn 

= ∑ Cikɛ ∗ n C in 

i 

∑ 

= ɛ n 

i 

C ∗ ikC in 

= ɛ n δ kn (4.39) 

where the preceding results have been used. The transformation C reduces H 

to a diagonal matrix, whose non-zero N elements are the eigenvalues. We can 

thus see our eigenvalue problem as the search for a transformation that brings 

from the original basis to a new basis in which the H operator has a diagonal 

form, that is, it acts on the elements of the basis by simply multiplying them 

by a constant (as in Schrödinger equation). 

4.3 Plane-wave basis set 

A good example of orthonormal basis set, and one commonly employed in 

physics, is the plane-wave basis set. This basis set is closely related to Fourier 

transforms and it can be easily understood if concepts from Fourier analysis are 

known. 

A function f(x) defined on the entire real axis can be always expanded into 

Fourier components, ˜f(k): 

f(x) = 

˜f(k) = 

1 

√ 

2π 

∫ ∞ 

−∞ 

1 

√ 

2π 

∫ ∞ 

−∞ 

˜f(k)e ikx dk (4.40) 

f(x)e −ikx dx. (4.41) 

For a function defined on a finite interval [−a/2, a/2], we can instead write 

f(x) = 1 √ a 

∑ 

n 

˜f(k n ) = 1 √ a 

∫ a/2 

−a/2 

˜f(k n )e iknx (4.42) 

f(x)e −iknx dx (4.43) 

where k n = 2πn/a, n = 0, ±1, ±2, .... Note that the f(x) function of Eq.4.42 is 

by construction a periodic function, with period equal to a: f(x + a) = f(x), as 

can be verified immediately. This implies that f(−a/2) = f(+a/2) must hold 

(also known under the name of periodic boundary conditions). The expressions 

38

eported here are immediately generalized to three or more dimensions. In the 

following only a simple one-dimensional case will be shown. 

Let us define our plane-wave basis set b i (x) according to Eq.(4.42): 

b i (x) = √ 1 e ikix , k i = 2π i, i = 0, ±1, ±2, ..., ±N (4.44) 

a a 

and the corresponding coefficients c i for the wave function ψ(x) as 

c i = 

∫ a/2 

−a/2 

b ∗ i (x)ψ(x)dx = 〈b i |ψ〉, ψ(x) = ∑ i 

c i b i (x). (4.45) 

This base, composed of 2N + 1 functions, becomes a complete basis set in the 

limit N → ∞. This is a consequence of well-known properties of Fourier series. 

It is also straightforward to verify that the basis is orthonormal: S ij = 〈b i |b j 〉 = 

δ ij . The solution of the problem of a particle under a potential requires thus 

the diagonalization of the Hamiltonian matrix, whose matrix elements: 

H ij = 〈b i |H|b j 〉 = 〈b i | p2 

2m + V (x)|b j〉 (4.46) 

can be trivially calculated. The kinetic term is diagonal (i.e. it can be represented 

by a diagonal matrix): 

∫ 

〈b i | p2 

2m |b j〉 = − ¯h2 a/2 

b ∗ i (x) d2 b j 

2m −a/2 dx 2 (x)dx = δ ¯h 2 ki 

2 ij 

2m . (4.47) 

The potential term is nothing but the Fourier transform of the potential (apart 

from a multiplicative factor): 

〈b i |V (x)|b j 〉 = 1 a 

∫ a/2 

−a/2 

V (x)e −i(k i−k j )x dx = 1 √ aṼ (k i − k j ). (4.48) 

A known property of Fourier transform ensures that the matrix elements of the 

potential tend to zero for large values of k i − k j . The decay rate will depend 

upon the spatial variation of the potential: faster for slowly varying potentials, 

and vice versa. Potentials and wave functions varying on a typical length scale 

λ have a significant Fourier transform up to k max ∼ 2π/λ. In this way we can 

estimate the number of plane waves needed to solve a problem. 

4.4 Code: pwell 

Let us consider the simple problem of a potential well with finite depth V 0 : 

V (x) = 0 per x < − b 2 , x > b 2 

V (x) = −V 0 per − b 2 ≤ x ≤ b 2 

(4.49) 

(4.50) 

39

with V 0 > 0, b < a. The matrix elements of the Hamiltonian are given by 

Eq.(4.47) for the kinetic part. by Eq.(4.48) for the potential. The latter can be 

explicitly calculated: 

〈b i |V (x)|b j 〉 = − 1 a 

= − V 0 

a 

= V 0 

a 

∫ b/2 

V 0 e −i(k i−k j )x dx (4.51) 

−b/2 

e −i(k i−k j )x 

b/2 

−i(k i − k j ) ∣ 

−b/2 

The case k i = k j must be separately treated, yielding 

V 0 (4.52) 

sin (b(k i − k j )/2) 

, k i ≠ k j . (4.53) 

(k i − k j )/2 

Ṽ (0) = V 0b 

a . (4.54) 

Code pwell.f90 1 (or pwell.c 2 ) generates the k i , fills the matrix H ij and diagonalizes 

it. The code uses units in which ¯h 2 /2m = 1 (e.g. atomic Rydberg 

units). Input data are: width (b) and depth (V 0 ) of the potential well, width 

of the box (a), number of plane waves (2N + 1). On output, the code prints 

the three lowest energy levels; moreover it writes to file gs-wfc.out the wave 

function of the ground state. 

4.4.1 Diagonalization routines 

The practical solution of the secular equation, Eq.(4.32), is not done by naively 

calculating the determinant and finding its roots! Various well-established, 

robust and fast diagonalization algorithms are known. Typically they are based 

on the reduction of the original matrix to Hessenberg or tridiagonal form via 

successive transformations. All such algorithms require the entire matrix (or at 

least half, exploiting hermiticity) to be available in memory at the same time, 

plus some work arrays. The time spent in the diagonalization is practically 

independent on the content of the matrix and it is invariably of the order of 

O(N 3 ) floating-point operations for a N × N matrix, even if eigenvalues only 

and not eigenvectors are desired. Matrix diagonalization used to be a major 

bottleneck in computation, due to its memory and time requirements. With 

modern computers, diagonalization of 1000 × 1000 matrix is done in less than 

no time. Still, memory growing as N 2 and time as N 3 are still a serious obstacle 

towards larger N. At the end of these lecture notes alternative approaches will 

be mentioned. 

The computer implementation of diagonalization algorithms is also rather 

well-established. In our code we use subroutine dsyev.f 3 from the linear algebra 

library LAPACK 4 . Several subroutines from the basic linear algebra library 

1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/pwell.f90 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/pwell.c 

3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/Lapack/dsyev.f 

4 http://www.netlib.org/lapack/ 

40

BLAS 5 (collected here: dgemm.f 6 ) are also required. dsyev implements reduction 

to tri-diagonal form for a real symmetric matrix (d=double precision, 

sy=symmetric matrix, ev=calculate eigenvalues and eigenvectors). The usage 

of dsyev requires either linking to a pre-compiled LAPACK and BLAS libraries, 

or compilation of the Fortran version and subsequent linking. Instructions on 

the correct way to call dsyev are contained in the header of the subroutine. 

Beware: most diagonalization routines overwrite the matrix! 

For the C version of the code, it may be necessary to add an underscore (as in 

dsyev () ) in the calling program. Moreover, the tricks explained in Sec.0.1.4 

are used to define matrices and to pass arguments to BLAS and LAPACK 

routines. 


• Observe how the results converge with respect to the number of plane 

waves, verify the form of the wave function. Verify the energy you get for 

a known case. You may use for instance the following case: for V 0 = 1, 

b = 2, the exact result is E = −0.4538. You may (and should) also verify 

the limit V 0 → ∞ (what are the energy levels?). 

• Observe how the results converge with respect to a. Note that for values 

of a not so big with respect to b, the energy calculated with the variational 

method is lower than the exact value. Why is it so? 

• Plot the ground-state wavefunction. You can also modify the code to 

write excited states. Do they look like what you expect? 


• Modify the code, adapting it to a potential well having a Gaussian form 

(whose Fourier transform can be analytically calculated: what is the 

Fourier transform of a Gaussian function?) For the same ”width”, which 

problem converges more quickly: the square well or the Gaussian well? 

• We know that for a symmetric potential, i.e. V (−x) = V (x), the solutions 

have a well-defined parity, alternating even and odd parity (ground state 

even, first excited state odd, and so on). Exploit this property to reduce 

the problem into two subproblems, one for even states and one for odd 

states. Use sine and cosine functions, obtained by suitable combinations 

of plane waves as above. Beware the correct normalization and the k n = 0 

term! Why is this convenient? What is gained in computational terms? 

5 http://www.netlib.org/blas/ 

6 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/Blas/dgemm.f 

41

Chapter 5 

Non-orthonormal basis sets 

We consider here the extension of the variational method to the case of nonorthonormal 

basis sets. We consider in particular the cause of Gaussian functions 

as basis set. This kind of basis set is especially used and useful in molecular 

calculations using Quantum Chemistry approaches. 

5.1 Non-orthonormal basis set 

Linear-algebra methods allow to treat with no special difficulty also the case in 

which the basis is formed by functions that are not orthonormal, i.e. for which 

∫ 

S ij = 〈b i |b j 〉 = b ∗ i b j dv (5.1) 

is not simply equal to δ ij . The quantities S ij are known as overlap integrals. 

It is sometimes practical to work with basis of this kind, rather than with an 

orthonormal basis. 

In principle, one could always obtain an orthonormal basis set from a nonorthonormal 

one using the Gram-Schmid orthogonalization procedure. An orthogonal 

basis set ˜b i is obtained as follows: 

˜b1 = b 1 (5.2) 

˜b2 = b 2 − ˜b 1 〈˜b 1 |b 2 〉/〈˜b 1 |˜b 1 〉 (5.3) 

˜b3 = b 3 − ˜b 2 〈˜b 2 |b 3 〉/〈˜b 2 |˜b 2 〉 − ˜b 1 〈˜b 1 |b 3 〉/〈˜b 1 |˜b 1 〉 (5.4) 

and so on. In practice, this procedure is seldom used and it is more convenient 

to follow a similar approach to that of Sect.4.2. In this case, Eq.(4.27) is 

generalized as 

G(c 1 , . . . , c N ) = ∑ c ∗ i c j (H ij − ɛS ij ) (5.5) 

ij 

and the minimum condition (4.31) becomes 

∑ 

(H ij − ɛS ij )c j = 0 (5.6) 

or, in matrix form, 

j 

Hc = ɛSc (5.7) 

42

known as generalized eigenvalue problem. 

The solution of a generalized eigenvalue problem is in practice equivalent to 

the solution of two simple eigenvalue problems. Let us first solve the auxiliary 

problem: 

Sd = σd (5.8) 

completely analogous to the problem (4.33). We can thus find a unitary matrix 

D (obtained by putting eigenvectors as columns side by side), such that D −1 SD 

is diagonal (D −1 = D † ), and whose non-zero elements are the eigenvalues σ. 

We find an equation similar to Eq.(4.39): 

∑ 

D ∗ ∑ 

ik S ij D jn = σ n δ kn . (5.9) 

i j 

Note that all σ n > 0: an overlap matrix is positive definite. In fact, 

σ n = 〈˜b n |˜b n 〉, 

|˜b n 〉 = ∑ j 

D jn |b j 〉 (5.10) 

and |˜b〉 is the rotated basis set in which S is diagonal. Note that a zero eigenvalue 

σ means that the corresponding |˜b〉 has zero norm, i.e. one of the b functions is 

a linear combination of the other functions. In that case, the matrix is called 

singular and some matrix operations (e.g. inversion) are not well defined. 

Let us define now a second transformation matrix 

A ij ≡ D ij 

√ σj 

. (5.11) 

We can write 

∑ 

A ∗ ∑ 

ik S ij A jn = δ kn (5.12) 

i j 

(note that A is not unitary) or, in matrix form, A † SA = I. Let us now define 

With this definition, Eq.(5.7) becomes 

We multiply to the left by A † : 

c = Av (5.13) 

HA v = ɛSA v (5.14) 

A † HA v = ɛA † SA v = ɛ v (5.15) 

Thus, by solving the secular problem for operator A † HA, we find the desired 

eigenvalues for the energy. In order to obtain the eigenvectors in the starting 

base, it is sufficient, following Eq.(5.13), to apply operator A to each eigenvector. 

43

5.1.1 Gaussian functions 

Gaussian functions are frequently used as basis functions, especially for atomic 

and molecular calculations. They are known as GTO: Gaussian-Type Orbitals. 

An important feature of Gaussians is that the product of two Gaussian functions, 

centered at different centers, can be written as a single Gaussian: 

e −α(r−r 1) 2 e −β(r−r 2) 2 = e −(α+β)(r−r 0) 2 e − αβ 

α+β (r 1−r 2 ) 2 , r 0 = αr 1 + βr 2 

α + β . (5.16) 

Some useful integrals involving Gaussian functions: 

∫ ∞ 

0 

e −αx2 dx = 1 2 

( ) π 1/2 

, 

α 

∫ ∞ 

0 

xe −αx2 dx = 

] ∞ [− e−αx2 

2α 

0 

= 1 

2α , (5.17) 

from which one derives 

∫ ∞ 

e −αx2 x 2n dx = 

∫ 

(−1) n ∂n ∞ 

∂α n 

0 

∫ ∞ 

0 

∫ 

e −αx2 x 2n+1 dx = (−1) n ∂n ∞ 

∂α n 

5.1.2 Exponentials 

0 

0 

e −αx2 dx = 

(2n − 1)!!π1/2 

2 n+1 α n+1/2 (5.18) 

xe −αx2 dx = n! 

2α n+1 (5.19) 

Basis functions composed of Hydrogen-like wave functions (i.e. exponentials) 

are also used in Quantum Chemistry as alternatives to Gaussian functions. 

They are known as STO: Slater-Type Orbitals. Some useful orbitals involving 

STOs: 

∫ e 

−2Zr 

r 

∫ ∞ 

( 

d 3 r = 4π re −2Zr dr = 4π 

[e −2Zr − r 

0 

∫ e 

−2Z(r 1 +r 2 ) 

|r 1 − r 2 | 

5.2 Code: hydrogen gauss 

2Z − 1 )] ∞ 

4Z 2 0 

= π Z 2 (5.20) 

d 3 r 1 d 3 r 2 = 5π2 

8Z 5 . (5.21) 

Code hydrogen gauss.f90 1 (or hydrogen gauss.c 2 ) solves the secular problem 

for the hydrogen atom using two different non-orthonormal basis sets: 

1. a Gaussian, “S-wave” basis set: 

b i (r) = e −α ir 2 ; (5.22) 

2. a Gaussian “P-wave” basis set, existing in three different choices, corresponding 

to the different values m of the projection of the angular momentum 

L z : 

b i (r) = xe −α ir 2 , b i (r) = ye −α ir 2 , b i (r) = ze −α ir 2 . (5.23) 

(actually only the third choice corresponds to a well-defined value m = 0) 

1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/hydrogen gauss.f90 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/hydrogen gauss.c 

44

The Hamiltonian operator for this problem is obviously 

H = −¯h2 ∇ 2 

2m e 

− Zq2 e 

r 

(5.24) 

For the hydrogen atom, Z = 1. 

Calculations for S- and P-wave gaussians are completely independent. In 

fact, the two sets of basis functions are mutually orthogonal: S ij = 0 if i is a 

S-wave, j is a P-wave gaussian, as evident from the different parity of the two 

sets of functions. Moreover the matrix elements H ij of the Hamiltonian are 

also zero between states of different angular momentum, for obvious symmetry 

reasons. The S and H matrices are thus block matrices and the eigenvalue 

problem can be solved separately for each block. The P-wave basis is clearly 

unfit to describe the ground state, since it doesn’t have the correct symmetry, 

and it is included mainly as an example. 

The code reads from file a list of exponents, α i , and proceeds to evaluate all 

matrix elements H ij and S ij . The calculation is based upon analytical results 

for integrals of Gaussian functions (Sec.5.1.1). In particular, for S-wave one has 

∫ 

S ij = e −(α i+α j )r 2 d 3 r = 

while the kinetic and Coulomb terms in H ij are respectively 

∫ [ 

Hij K = e −α ir 2 −¯h2 ∇ 2 ] 

e −α jr 2 d 3 r = 

2m e 

∫ [ 

Hij V = e −α ir 2 − Zq2 e 

r 

] 

( 

π 

α i + α j 

) 3/2 

(5.25) 

( ) 3/2 ¯h2 6α i α j π 

(5.26) 

2m e α i + α j α i + α j 

e −α jr 2 d 3 r = − 2πZq2 e 

α i + α j 

(5.27) 

For the P-wave basis the procedure is analogous, using the corresponding (and 

more complex) analytical expressions for integrals. 

The code then calls subroutine diag that solves the generalized secular 

problem (i.e. it applies the variational principle). Subroutine diag returns a 

vector e containing eigenvalues (in order of increasing energy) and a matrix v 

containing the eigenvectors, i.e. the expansion coefficients of wave functions. 

Internally, diag performs the calculation described in the preceding section 

in two stages. The solution of the simple eigenvalue problem is performed by 

the subroutine dsyev we have already seen in Sect.4.4. 

In principle, one could use a single LAPACK routine, dsygv, that solves 

the generalized secular problem, Hψ = ɛSψ, with a single call. In practice, 

one has to be careful to avoid numerical instabilities related to the problem 

of linear dependencies among basis functions (see Eq.(5.10) and the following 

discussion). Inside routine diag, all eigenvectors of matrix S corresponding to 

very small eigenvectors, i.e. smaller than a pre-fixed threshold are thrown away, 

before proceeding with the second diagonalization. The number of linearly 

independent eigenvectors is reprinted in the output. 

The reason for such procedure is that it is not uncommon to discover that 

some basis functions can almost exactly be written as sums of some other basis 

45

functions. This does not happen if the basis set is well chosen, but it can happen 

if the basis set functions are too many or not well chosen (e.g. with too close 

exponents). A wise choice of the α j coefficients is needed in order to have a 

high accuracy without numerical instabilities. 

The code then proceeds and writes to files s-coeff.out (or p-coeff.out) 

the coefficients of the expansion of the wave function into Gaussians. The 

ground state wave function is written into file s-wfc.out (or p-wfc.out). 

Notice the usage of dgemm calls to perform matrix-matrix multiplication. 

The header of dgemm.f contains a detailed documentation on how to call it. 

Its usage may seem awkward at a first (and also at a second) sight. This 

is a consequence in part of the Fortran way to store matrices (requiring the 

knowledge of the first, or “leading”, dimension of matrices); in part, of the 

old-style Fortran way to pass variables to subroutines only under the form of 

pointers (also for vectors and arrays). Since the Fortran-90 syntax and MATMUL 

intrinsic are so much easier to use: why bother with dgemm and its obscure 

syntax? The reason is efficiency: very efficient implementations of dgemm exist 

for modern architectures. 

For the C version of the code, and how matrices are introduced and passed 

to Fortran routines, see Sec.0.1.4. 


• Verify the accuracy of the energy eigenvalues, starting with 1 Gaussian, 

then 2, then 3. Try to find the best values for the coefficients for the 1s 

state (i.e. the values that yield the lowest energy). 

• Compare with the the solutions obtained using code hydrogen radial. 

Plot the 1s numerical solution (calculated with high accuracy) and the 

“best” 1s solution for 1, 2, 3, Gaussians (you will need to multiply the 

latter by a factor √ 4π: why? where does it come from?). What do you 

observe? where is the most significant error concentrated? 

• Compare with the results for the following optimized basis sets (a.u.): 

– three Gaussians: α 1 = 0.109818, α 2 = 0.405771, α 3 = 2.22776 

(known as “STO-3G” in Quantum-Chemistry jargon) 

– four Gaussians: α 1 = 0.121949, α 2 = 0.444529, α 3 = 1.962079, 

α 4 = 13.00773 

• Observe and discuss the ground state obtained using the P-wave basis set 

• Observe the effects related to the number of basis functions, and to the 

choice of the parameters α. Try for instance to choose the characteristic 

Gaussian widths, λ = 1/ √ α, as uniformly distributed between suitably 

chosen λ min and λ max . 

• For Z > 1, how would you re-scale the coefficients of the optimized Gaussians 

above? 

46

Chapter 6 

Self-consistent Field 

A way to solve a system of many electrons is to consider each electron under the 

electrostatic field generated by all other electrons. The many-body problem is 

thus reduced to the solution of single-electron Schrödinger equations under an 

effective potential. The latter is generated by the charge distribution of all other 

electrons in a self-consistent way. This idea is formalized in a rigorous way in 

the Hartree-Fock method and in Density-Functional theory. In the following 

we will see an historical approach of this kind: the Hartree method. 

6.1 The Hartree Approximation 

The idea of the Hartree method is to try to approximate the wave functions, 

solution of the Schrödinger equation for N electrons, as a product of singleelectron 

wave functions, called atomic orbitals. As we have seen, the best 

possible approximation consists in applying the variational principle, by minimizing 

the expectation value of the energy E = 〈ψ|H|ψ〉 for state |ψ〉. 

The Hamiltonian of an atom having a nucleus with charge Z and N electrons 

is 

H = − ∑ ¯h 2 

∇ 2 i − ∑ Zqe 

2 + ∑ qe 

2 (6.1) 

2m 

i e r 

i i r 

〈ij〉 ij 

where the sum is over pairs of electrons i and j, i.e. each pair appears only 

once. Alternatively: 

∑ N−1 ∑ N∑ 

= 

(6.2) 

〈ij〉 

i=1 j=i+1 

It is convenient to introduce one-electron and two-electrons operators: 

With such notation, the Hamiltonian is written as 

f i ≡ − ¯h2 

2m e 

∇ 2 i − Zq2 e 

r i 

(6.3) 

g ij ≡ q2 e 

r ij 

(6.4) 

H = ∑ i 

f i + ∑ 〈ij〉 

g ij (6.5) 

47

provided that g act symmetrically on the two electrons (definitely true for the 

Coulomb interaction). 

6.2 Hartree Equations 

Let us assume that the total wave function can be expressed as a product of 

single-electron orbitals (assumed to be orthonormal): 

ψ(1, 2, . . . , N) = φ 1 (1)φ 2 (2) . . . φ N (N) (6.6) 

∫ 

φ i (1)φ j (1) dv 1 = δ ij . (6.7) 

Variables 1, 2, .., mean position and spin variables for electrons 1,2,...; ∫ dv i 

means integration on coordinates and sum over spin components. Index i labels 

instead the quantum numbers used to classify a given single-electron orbitals. 

All orbitals must be different: Pauli’s exclusion principle tells us that we cannot 

build a wave function for many electrons using twice the same single-electron 

orbital. In practice, orbitals for the case of atoms are classified using the main 

quantum number n, orbital angular momentum l and its projection m. 

The expectation value of the energy is 

⎡ 

⎤ 

∫ 

〈ψ|H|ψ〉 = φ ∗ 1(1) . . . φ ∗ ⎣ 

N(N) 

∑ f i + ∑ g ij ⎦ φ 1 (1) . . . φ N (N) dv 1 . . . dv N 

i 〈ij〉 

= ∑ i 

= ∑ i 

∫ 

∫ 

φ ∗ i (i)f i φ i (i) dv i + ∑ 〈ij〉 

φ ∗ i (1)f 1 φ i (1) dv 1 + ∑ ∫ 

〈ij〉 

∫ 

φ ∗ i (i)φ ∗ j(j)g ij φ i (i)φ j (j) dv i dv j 

φ ∗ i (1)φ ∗ j(2)g 12 φ i (1)φ j (2) dv 1 dv 2 

(6.8) 

In the first step, we made use of orthonormality (6.7); in the second we just 

renamed dummy integration variables with the ”standard” choice 1 and 2. 

Let us now apply the variational principle to formulation (4.10), with the 

constraints that all integrals 

∫ 

I k = φ ∗ k(1)φ k (1) dv 1 (6.9) 

are constant, i.e. the normalization of each orbital function is preserved: 

( 

δ 

〈ψ|H|ψ〉 − ∑ k 

ɛ k I k 

) 

= 0 (6.10) 

where ɛ k are Lagrange multipliers, to be determined. Let us vary only the 

orbital function φ k . We find 

∫ 

δI k = δφ ∗ k(1)φ k (1) dv 1 + c.c. (6.11) 

48

(the variations of all other normalization integers will be zero) and, using the 

hermiticity of H as in Sec.4.1.1, 

∫ 

δ〈ψ|H|ψ〉 = δφ ∗ k(1)f 1 φ k (1) dv 1 + c.c. 

+ ∑ ∫ 

δφ ∗ k(1)φ ∗ j(2)g 12 φ k (1)φ j (2) dv 1 dv 2 + c.c. (6.12) 

j≠k 

This result is obtained by noticing that the only involved terms of Eq.(6.8) are 

those with i = k or j = k, and that each pair is counted only once. For instance, 

for 4 electrons the pairs are 12, 13, 14, 23, 24, 34; if I choose k = 3 the only 

contributions come from 13, 23, 34, i.e. ∑ j≠k (since g is a symmetric operator, 

the order of indices in a pair is irrelevant) 

Thus the variational principle takes the form 

⎡ 

⎤ 

∫ 

∫ 

δφ ∗ k(1) 

φ ∗ j(2)g 12 φ k (1)φ j (2) dv 2 − ɛ k φ k (1) ⎦ dv 1 + c.c. = 0 

⎣f 1 φ k (1) + ∑ j≠k 

i.e. the term between square brackets (and its complex conjugate) must vanish 

so that the above equation is satisfied for any variation: 

f 1 φ k (1) + ∑ ∫ 

φ ∗ j(2)g 12 φ k (1)φ j (2) dv 2 = ɛ k φ k (1) (6.13) 

j≠k 

These are the Hartree equations (k = 1, . . . , N). It is useful to write down 

explicitly the operators: 

⎡ 

⎤ 

− ¯h2 ∇ 2 

2m 

1φ k (1) − Zq2 e 

φ k (1) + ⎣ ∑ ∫ 

φ ∗ 

e r 

j(2) q2 e 

φ j (2) dv 2 ⎦ φ k (1) = ɛ k φ k (1) 

1 r 

j≠k 

12 

(6.14) 

We remark that each of them looks like a Schrödinger equation in which in 

addition to the Coulomb potential there is a Hartree potential: 

where we have used 

∫ 

V H (r 1 ) = 

ρ k (2) q2 e 

r 12 

dv 2 (6.15) 

ρ k (2) = ∑ j≠k 

φ ∗ j(2)φ j (2) (6.16) 

ρ j is the charge density due to all electrons differing from the one for which we 

are writing the equation. 

6.2.1 Eigenvalues and Hartree energy 

Let us multiply Hartree equation, Eq(6.13), by φ ∗ k (1), integrate and sum over 

orbitals: we obtain 

∑ 

ɛ k = ∑ ∫ 

φ ∗ k(1)f 1 φ k (1)dv 1 + ∑ ∑ 

∫ 

φ ∗ k(1)φ k (1)g 12 φ ∗ j(2)φ j (2)dv 1 dv 2 . 

k k 

k j≠k 

(6.17) 

49

Let us compare this expression with the energy for the many-electron system, 

Eq.(6.8). The Coulomb repulsion energy is counted twice, since each < jk > 

pair is present twice in the sum. The energies thus given by the sum of eigenvalues 

of the Hartree equation, minus the Coulomb repulsive energy: 

E = ∑ ɛ k − ∑ ∫ 

φ ∗ k(1)φ k (1)g 12 φ ∗ j(2)φ j (2)dv 1 dv 2 . (6.18) 

k 

6.3 Self-consistent potential 

Eq.(6.15) represents the electrostatic potential at point r 1 generated by a charge 

distribution ρ k . This fact clarifies the meaning of the Hartree approximation. 

Assuming that ψ is factorizable into a product, we have formally assumed that 

the electrons are independent. This is of course not true at all: the electrons 

are strongly interacting particles. The approximation is however not so bad if 

the Coulomb repulsion between electrons is accounted for under the form of an 

average field V H , containing the combined repulsion from all other electrons on 

the electron that we are considering. Such effect adds to the Coulomb attraction 

exerted by the nucleus, and partially screens it. The electrons behave as if they 

were independent, but under a potential −Zq 2 e/r + V H (r) instead of −Zq 2 e/r of 

the nucleus alone. 

V H (r) is however not a “true” potential, since its definition depends upon 

the charge density distributions of the electrons, that depend in turn upon the 

solutions of our equations. The potential is thus not known a priori, but it is a 

function of the solution. This type of equations is known as integro-differential 

equations. 

The equations can be solved in an iterative way, after an initial guess of the 

orbitals is assumed. The procedure is as follows: 

1. calculate the charge density (sum of the square moduli of the orbitals) 

2. calculate the Hartree potential generated by such charge density (using 

classical electrostatics) 

3. solve the equations to get new orbitals. 

The solution of the equations can be found using the methods presented in 

Ch.2. The electron density is build by filling the orbitals in order of increasing 

energy (following Pauli’s principle) until all electrons are “placed”. 

In general, the final orbitals will differ from the starting ones. The procedure 

is then iterated – by using as new starting functions the final functions, or with 

more sophisticated methods – until the final and starting orbitals are the same 

(within some suitably defined numerical threshold). The resulting potential V H 

is then consistent with the orbitals that generate it, and it is for this reason 

called self-consistent field. 

6.3.1 Self-consistent potential in atoms 

For closed-shell atoms, a big simplification can be achieved: V H is a central 

field, i.e. it depends only on the distance r 1 between the electron and the 

50

nucleus. Even in open-shell atoms, this can be imposed as an approximation, 

by spherically averaging ρ k . The simplification is considerable, since we know 

a priori that the orbitals will be factorized as in Eq.(2.9). The angular part is 

given by spherical harmonics, labelled with quantum numbers l and m, while 

the radial part is characterized by quantum numbers n and l. Of course the 

accidental degeneracy for different l is no longer present. It turns out that even 

in open-shell atoms, this is an excellent approximation. 

Let us consider the case of two-electron atoms. The Hartree equations, 

Eq.(6.14), for orbital k = 1 reduces to 

− ¯h2 

2m e 

∇ 2 1φ 1 (1) − Zq2 e 

r 1 

φ 1 (1) + 

[ ∫ 

φ ∗ 2(2) q2 e 

r 12 

φ 2 (2) dv 2 

] 

φ 1 (1) = ɛ 1 φ 1 (1) (6.19) 

For the ground state of He, we can assume that φ 1 and φ 2 have the same spherically 

symmetric coordinate part, φ(r), and opposite spins: φ 1 = φ(r)v + (σ), 

φ 2 = φ(r)v − (σ). Eq.(6.19) further simplifies to: 

− ¯h2 

2m e 

∇ 2 1φ(r 1 ) − Zq2 e 

r 1 

φ(r 1 ) + 

6.4 Code: helium hf radial 

[ ∫ q 

2 

e 

r 12 

|φ(r 2 )| 2 d 3 r 2 

] 

φ(r 1 ) = ɛφ(r 1 ) (6.20) 

Code helium hf radial.f90 1 (or helium hf radial.c 2 ) solves Hartree equations 

for the ground state of He atom. helium hf radial is based on code 

hydrogen radial and uses the same integration algorithm based on Numerov’s 

method. The new part is the implementation of the method of self-consistent 

field for finding the orbitals. 

The calculation consists in solving the radial part of the Hartree equation 

(6.20). The effective potential V scf is the sum of the Coulomb potential of the 

nucleus, plus the (spherically symmetric) Hartree potential 

V scf (r) = − Zq2 e 

r 

+ V H (r), V H (r 1 ) = q 2 e 

∫ ρ(r2 ) 

r 12 

d 3 r 2 . (6.21) 

We start from an initial estimate for V H (r), calculated in routine init pot ( 

V (0) 

H 

(r) = 0, simply). With the ground state R(r) obtained from such potential, 

we calculate in routine rho of r the charge density ρ(r) = |R(r)| 2 /4π (note that 

ρ is here only the contribution of the other orbital, so half the total charge, and 

remember the presence of the angular part!). Routine v of rho re-calculates 

the new Hartree potential VH 

out (r) by integration, using the Gauss theorem: 

∫ r 

ṼH 

out (r) = V 0 + qe 

2 Q(s) 

r max s 

∫r

The Hartree potential is then re-introduced in the calculation not directly 

but as a linear combination of the old and the new potential. This is the 

simplest technique to ensure that the self-consistent procedure converges. It 

is not needed in this case, but in most cases it is: there is no guarantee that 

re-inserting the new potential in input will lead to convergence. We can write 

V in,new 

H 

(r) = βVH 

out (r) + (1 − β)VH in (r), (6.23) 

where 0 < β ≤ 1. The procedure is iterated (repeated) until convergence is 

achieved. The latter is verified on the norm, i.e. the integral of the square, of 

VH 

out (r) − V 

in 

H (r). square. 

In output the code prints the eigenvalue ɛ 1 of Eq.6.13, plus various terms of 

the energy, with rather obvious meaning except the term Variational correction. 

This is 

∫ 

δE = (VH in (r) − VH out (r))ρ(r)d 3 r (6.24) 

and it is useful to correct 3 the value of the energy obtained by summing the 

eigenvalues as in Eq.(6.18), so that it is the same as the one obtained using 

Eq.(6.8), where eigenvalues are not used. The two values of the energy are 

printed side by side. Also noticeable is the ”virial check”: for a Coulomb 

potential, the virial theorem states that 〈T 〉 = −〈V 〉/2, where the two terms 

are respectively the average values of the kinetic and the potential energy. It 

can be demonstrated that the Hartree-Fock solution obeys the virial theorem. 


• Observe the behavior of self-consistency, verify that the energy (but not 

the single terms of it!) decreases monotonically. 

• Compare the energy obtained with this and with other methods: perturbation 

theory with hydrogen-like wave functions (E = −5.5 Ry, Sect. 

D.1), variational theory with effective Z (E = −5.695 Ry, Sect. D.3), 

best numerical Hartree(-Fock) result (E = −5.72336 Ry, as found in the 

literature), experimental result (E = −5.8074 Ry). 

• Make a plot of orbitals (file wfc.out) for different n and l. Note that 

the orbitals and the corresponding eigenvalues become more and more 

hydrogen-like for higher n. Can you explain why? 

• If you do not know how to answer the previous question: make a plot 

of V scf (file pot.out) and compare its behavior with that of the −Zq 2 e/r 

term. What do you notice? 

• Plot the 1s orbital together with those calculated by hydrogen radial 

for Hydrogen (Z = 1), He + (Z = 2), and for a Z = 1.6875. See Sect. D.3 

if you cannot make sense of the results. 

3 Eigenvalues are calculated using the input potential; the other terms are calculated using 

output potential 

52

Chapter 7 

The Hartree-Fock 

approximation 

The Hartree method is useful as an introduction to the solution of many-particle 

system and to the concepts of self-consistency and of the self-consistent-field, 

but its importance is confined to the history of physics. In fact the Hartree 

method is not just approximate: it is wrong, by construction, since its wave 

function is not antisymmetric! A better approach, that correctly takes into 

account the antisymmetric character of the the wave functions is the Hartree- 

Fock approach. The price to pay is the presence in the equations of a non local, 

and thus more complex, exchange potential. 

7.1 Hartree-Fock method 

Let us re-consider the Hartree wave function. The simple product: 

ψ(1, 2, . . . , N) = φ 1 (1)φ 2 (2) . . . φ N (N) (7.1) 

does not satisfy the principle of indistinguishability, because it is not an eigenstate 

of permutation operators. It is however possible to build an antisymmetric 

solution by introducing the following Slater determinant: 

∣ ∣∣∣∣∣∣∣∣ φ 1 (1) . . . φ 1 (N) 

ψ(1, . . . , N) = √ 1 

. . . 

(7.2) 

N! . . . 

φ N (1) . . . φ N (N) ∣ 

The exchange of two particles is equivalent to the exchange of two columns, 

which induces, due to the known properties of determinants, a change of sign. 

Note that if two rows are equal, the determinant is zero: all φ i ’s must be 

different. This demonstrates Pauli’s exclusion principle. two (or more) identical 

fermions cannot occupy the same state. 

Note that the single-electron orbitals φ i are assumed to be orthonormal: 

∫ 

φ ∗ i (1)φ j (1)dv 1 = δ ij (7.3) 

53

where the “integral” on dv 1 means as usual “integration on coordinates, sum 

over spin components” . We follow the same path of Sec. (6.2) used to derive 

Hartree equations, Eq.(6.13). Since a determinant for N electrons has N! terms, 

we need a way to write matrix elements between determinants on a finite paper 

surface. The following property, valid for any (symmetric) operator F and 

determinantal functions ψ and ψ ′ , is very useful: 

〈ψ|F |ψ ′ 〉 = 1 ∫ ∣ ∣∣∣∣∣ φ ∗ 1 (1) . φ∗ 1 (N) 

∣ ∣∣∣∣∣∣ . . . F 

N! 

φ ∗ N (1) . φ∗ N (N) ∣ 

∫ 

= φ ∗ 1(1) . . . φ ∗ N(N)F 

∣ 

φ ′ 1 (1) . φ′ 1 (N) 

. . . 

φ ′ N (1) . φ′ N (N) ∣ ∣∣∣∣∣∣ 

dv 1 . . . dv N 

φ ′ 1 (1) . φ′ 1 (N) 

. . . 

φ ′ N (1) . φ′ N (N) ∣ ∣∣∣∣∣∣ 

dv 1 . . . dv N (7.4) 

(by expanding the first determinant, one gets N! terms that, once integrated, 

are identical). From the above property it is immediate (and boring) to obtain 

the matrix elements for one- and two-electron operators: 

〈ψ| ∑ f i |ψ〉 = ∑ ∫ 

φ ∗ i (1)f 1 φ i (1) dv 1 (7.5) 

i 

i 

(as in the Hartree approximation), and 

〈ψ| ∑ g ij |ψ〉 = ∑ ∫ 

φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − φ j (1)φ i (2)] dv 1 dv 2 (7.6) 

〈ij〉 

〈ij〉 

The integrals implicitly include summation over spin components. If we assume 

that g 12 depends only upon coordinates (as in Coulomb interaction) and not 

upon spin, the second term: 

∫ 

φ ∗ i (1)φ ∗ j(2)g 12 φ j (1)φ i (2) dv 1 dv 2 (7.7) 

is zero is i and j states are different (the spin parts are not affected by g 12 and 

they are orthogonal if relative to different spins). 

It is convenient to move to a scheme in which the spin variables are not 

explicitly included in the orbital index. Eq.(7.6) can then be written as 

〈ψ| ∑ g ij |ψ〉 = ∑ ∫ 

φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − δ(σ i , σ j )φ j (1)φ i (2)] dv 1 dv 2 

〈ij〉 

〈ij〉 

where σ i is the spin of electron i, and: 

(7.8) 

δ(σ i , σ j ) = 0 if σ i ≠ σ j 

= 1 if σ i = σ j 

In summary: 

〈ψ|H|ψ〉 = ∑ ∫ 

i 

+ ∑ ∫ 

〈ij〉 

φ ∗ i (1)f 1 φ i (1) dv 1 (7.9) 

φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − δ(σ i , σ j )φ j (1)φ i (2)] dv 1 dv 2 

54

Now that we have the expectation value of the energy, we can apply the variational 

principle. In principle we must impose normalization constraints such 

that not only all φ i stay normalized (as we did in the derivation of Hartree’s 

equation) but also all pairs φ i , φ j with same spin are orthogonal, i.e., a (triangular) 

matrix ɛ ij of Lagrange multipliers would be needed. It can be shown 

however (details e.g. on Slater’s book, Quantum theory of matter) that it is 

always possible to find a transformation to a solution in which the matrix of 

Lagrange multipliers is diagonal. We assume that we re dealing with such a 

case. 

Let us omit the details of the derivation and move to the final Hartree-Fock 

equations: 

f 1 φ k (1) + ∑ ∫ 

φ ∗ j(2)g 12 [φ k (1)φ j (2) − δ(σ k , σ j )φ j (1)φ k (2)] dv 2 = ɛ k φ k (1) 

j 

(7.10) 

or, in explicit form, 

− ¯h2 

2m e 

∇ 2 1φ k (1) − Zq2 e 

r 1 

φ k (1) + ∑ j 

∫ 

φ ∗ j(2) q2 e 

r 12 

[φ j (2)φ k (1) (7.11) 

− δ(σ k , σ j )φ k (2)φ j (1)] dv 2 = ɛ k φ k (1) 

The energy of the system, Eq. 7.9, can be expressed, analogously to the Hartree 

case, via the sum of eigenvectors of Eq.(7.11), minus a term compensating the 

double counting of Coulomb repulsion and of the exchange energy: 

E = ∑ ɛ k − ∑ ∫ 

φ ∗ k(1)φ ∗ j(2)g 12 [φ k (1)φ j (2) − δ(σ j , σ k )φ j (1)φ k (2)] dv 1 dv 2 . 

k 

(7.12) 

Eq.(7.11) has normally an infinite number of solutions, of which only the lowestenergy 

N will be occupied by electrons, the rest playing the role of excited 

states. The sum over index j runs only on occupied states. 

Let us carefully observe the differences with respect to Hartree equations, 

Eqs.(6.13): 

1. ∑ j 

also includes the case j = k. 

2. for electrons j with same spin as that of k, there is an additional exchange 

term 

3. due to the exchange term, the case j = k yields a nonzero contribution 

only if the spins are different. 

7.1.1 Coulomb and exchange potentials 

Let us analyze the physical meaning of the Hartree-Fock equations. We re-write 

them under the form 

− ¯h2 

2m e 

∇ 2 1φ k (1) − Zq2 e 

r 1 

φ k (1) + V H (1)φ k (1) + ( ˆV x φ k )(1) = ɛ k φ k (1), (7.13) 

55

where we have introduced a ”Hartree potential” V H (it is not the same as in the 

Hartree equations!) and an ”exchange potential” V x . The Hartree potential is 

the same for all orbitals: 

V H (1) = ∑ j 

∫ 

∫ 

φ ∗ j(2) q2 e 

φ j (2)dv 2 ≡ 

r 12 

where we have introduce the charge density 

ρ(2) q2 e 

r 12 

dv 2 , (7.14) 

ρ(2) = ∑ j 

φ ∗ j(2)φ j (2). (7.15) 

We can verify that ρ(1) is equal to the probability to find an electron in (1), 

that is, 

∫ 

ρ(1) = N |Ψ(1, 2, .., N)| 2 dv 2 ...dv N . (7.16) 

The exchange term: 

( ˆV x φ k )(1) = − ∑ j 

∫ 

δ(σ k , σ j ) 

φ ∗ j(2) q2 e 

r 12 

φ k (2)φ j (1) dv 2 (7.17) 

does not have the simple form of the Hartree potential: V H (1)φ k (1), where 

V H (1) comes from an integration over variable 2. It has instead a form like 

∫ 

( ˆV x φ)(1) ≡ V x (1, 2)φ k (2)dv 2 (7.18) 

typical of a non local interaction. 

7.1.2 Correlation energy 

The Hartree-Fock solution is not exact: it would be if the system under study 

were described by a wave function having the form of a Slater determinant. This 

is in general not true. The energy difference between the exact and Hartree- 

Fock solution is known as correlation energy. 1 The origin of the name comes 

from the fact that the Hartree-Fock approximation misses part of the ”electron 

correlation”: the effects of an electron on all others. This is present in Hartree- 

Fock via the exchange and electrostatic interactions; more subtle effects are not 

accounted for, because they require a more general form of the wave function. 

For instance, the probability P (r 1 , r 2 ) to find an electron at distance r 1 and 

one as distance r 2 from the origin is not simply equal to p(r 1 )p(r 2 ), because 

electrons try to ”avoid” each other. The correlation energy in the case of He 

atom is about 0.084 Ry: a small quantity relative to the energy (∼ 1.5%), but 

not negligible. 

An obvious way to improve upon the Hartree-Fock results consists in allowing 

contributions from other Slater determinants to the wave function. This is 

the essence of the “configuration interaction” (CI) method. Its practical application 

requires a sophisticated ”technology” to choose among the enormous 

1 Feynman called it stupidity energy, because the only physical quantity that it measures is 

our inability to find the exact solution! 

56

number of possible Slater determinants a subset of most significant ones. Such 

technique, computationally very heavy, is used in quantum chemistry to get 

high-precision results in small molecules. Other, less heavy methods (the socalled 

Møller-Plesset, MP, approaches) rely on perturbation theory to yield a 

rather good estimate of correlation energy. A completely different approach, 

which produces equations that are reminiscent of Hartree-Fock equations, is 

Density-Functional Theory (DFT), much used in condensed-matter physics. 

7.1.3 The Helium atom 

The solution of Hartree-Fock equations in atoms also commonly uses the central 

field approximation. This allows to factorize Eqs.(7.11) into a radial and 

an angular part, and to classify the solution with the ”traditional” quantum 

numbers n, l, m. 

For He, the Hartree-Fock equations, Eq.(7.11), reduce to 

∫ 

− ¯h2 ∇ 2 

2m 

1φ 1 (1) − Zq2 e 

φ 1 (1) + 

e r 1 

∫ 

+ 

φ ∗ 1(2) q2 e 

r 12 

[φ 1 (2)φ 1 (1) − φ 1 (2)φ 1 (1)] dv 2 

φ ∗ 2(2) q2 e 

r 12 

[φ 2 (2)φ 1 (1) − δ(σ 1 , σ 2 )φ 1 (2)φ 2 (1)] dv 2 = ɛ 1 φ 1 (1) (7.19) 

Since the integrand in the first integral is zero, 

∫ 

− ¯h2 ∇ 2 

2m 

1φ 1 (1) − Zq2 e 

φ 1 (1) + φ ∗ 

e r 

2(2) q2 e 

[φ 2 (2)φ 1 (1) 

1 r 12 

−δ(σ 1 , σ 2 )φ 1 (2)φ 2 (1)] dv 2 = ɛ 1 φ 1 (1). (7.20) 

In the ground state, the two electrons have opposite spin (δ(σ 1 , σ 2 ) = 0) and 

occupy the same spherically symmetric orbital (that is: φ 1 and φ 2 are the same 

function, φ). This means that the Hartree-Fock equation, Eq.(7.20), for the 

ground state is the same as the Hartree equation, Eq.(6.19). In fact, the two 

electrons have opposite spin and there is thus no exchange. 

In general, one speaks of Restricted Hartree-Fock (RHF) for the frequent 

case in which all orbitals are present in pairs, formed by a same function of r, 

multiplied by spin functions of opposite spin. In the following, this will always 

be the case. 

7.2 Code: helium hf gauss 

The radial solution of Hartree-Fock equations is possible only for atoms or in 

some model systems. In most cases the solution is found by expanding on a 

suitable basis set, in analogy with the variational method. 

Let us re-write the Hartree-Fock equation – for the restricted case, i.e. no 

total spin – in the following form: 

Fφ k = ɛ k φ k , k = 1, . . . , N/2 (7.21) 

The index k labels the coordinate parts of the orbitals; for each k there is 

a spin-up and a spin-down orbital. F is called the Fock operator. It is of 

57

course a non-local operator which depends upon the orbitals φ k . Let us look 

now for a solution under the form of an expansion on a basis of functions: 

φ k (r) = ∑ i c(k) i b i (r). We find the Rothaan-Hartree-Fock equations: 

F c (k) = ɛ k Sc (k) (7.22) 

where c (k) = (c (k) 

1 , c(k) 2 , . . . , c(k) N 

) is the vector of the expansion coefficients, S is 

the superposition matrix, F is the matrix of the Fock operator on the basis set 

functions: 

F ij = 〈b i |F|b j 〉, S ij = 〈b i |b j 〉. (7.23) 

that after some algebra can be written as 

⎛ 

F ij = f ij + ∑ N/2 

∑ ∑ 

⎝2 

l m k=1 

c (k)∗ 

l 

⎞ 

c (k) ⎠ 

m 

( 

g iljm − 1 ) 

2 g ijlm , (7.24) 

where, with the notations introduced in this chapter: 

∫ 

f ij = b ∗ i (r 1 )f 1 b j (r 1 )d 3 r 1 , (7.25) 

∫ 

g iljm = b ∗ i (r 1 )b j (r 1 )g 12 b ∗ l (r 2 )b m (r 2 )d 3 r 1 d 3 r 2 . (7.26) 

The sum over states between parentheses in Eq.(7.24) is called density matrix. 

The two terms in the second parentheses come respectively from the Hartree 

and the exchange potentials. 

The problem of Eq.(7.22) is more complex than a normal secular problem 

solvable by diagonalization, since the Fock matrix, Eq.(7.24), depends upon its 

own eigenvectors. It is however possible to reconduct the solution to a selfconsistent 

procedure, in which at each step a fixed matrix is diagonalized (or, 

for a non-orthonormal basis, a generalized diagonalization is performed at each 

step). 

Code helium hf gauss.f90 2 (or helium hf gauss.c 3 ) solves Hartree-Fock 

equations for the ground state of He atom, using a basis set of S Gaussians. The 

basic ingredients are the same as in code hydrogen gauss (for the calculation 

of single-electron matrix elements and for matrix diagonalization). Moreover 

we need an expression for the g iljm matrix elements introduced in Eq.(7.26). 

Using the properties of products of Gaussians, Eq.(5.16), these can be written 

in terms of the integral 

∫ 

I = 

e −αr2 1 e 

−βr 2 2 1 

r 12 

d 3 r 1 d 3 r 2 . (7.27) 

Let us look for a variable change that makes (r 1 −r 2 ) 2 to appear in the exponent 

of the gaussians: 

[ 

αr1 2 + βr2 2 = γ (r 1 − r 2 ) 2 + (ar 1 + br 2 ) 2] (7.28) 

⎡ 

⎛ √ ⎞ 

αβ ⎢ 

= ⎣(r 1 − r 2 ) 2 + ⎝√ α 

α + β 

β r β 

1 + 

α r 2⎠2 ⎤ ⎥ ⎦ . (7.29) 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/helium hf gauss.f90 

3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/helium hf gauss.c 

58

Let us now introduce a further variable change from (r 1 , r 2 ) to (r, s), where 

r = r 1 − r 2 , s = 

√ α 

β r 1 + 

√ 

β 

α r 2; (7.30) 

The integral becomes 

∫ 

I = 

e − αβ 1 ∣ 

α+β r2 αβ ∣∣∣ ∂(r 

e− α+β s2 1 , r 2 ) 

r ∂(r, s) 

∣ d3 rd 3 s, (7.31) 

where the Jacobian is easily calculated as the determinant of the transformation 

matrix, Eq.(7.30): 

( √ ) 3 ∂(r 1 , r 2 ) 

αβ 

∣ ∂(r, s) ∣ = . (7.32) 

α + β 

The calculation of the integral is trivial and provides the required result: 

g iljm = 

2π 5/2 

αβ(α + β) 1/2 (7.33) 

where α = α i + α j , β = α l + α m . 

The self-consistent procedure is even simpler than in code helium hf radial: 

at each step, the Fock matrix is re-calculated using the density matrix at the 

preceding step, with no special trick or algorithm, until energy converges within 

a given numerical threshold. 


• Observe how the ground-state energy changes as a function of the number 

of Gaussians and of their coefficients. You may take the energy given 

by helium hf radial as the reference. Start with the 3 or 4 Gaussian 

basis set used for Hydrogen, then the same set of Gaussians with rescaled 

coefficients, i.e. such that they fit a Slater 1s orbital for He + (Z = 2) and 

for a Hydrogen-like atom with Z = 1.6875 (see Sect. D.3). 

• Write to file the 1s orbital (borrow the relative code from hydrogen gauss), 

plot it. Compare it with 

– the 1s Slater orbital for Z = 1, Z = 1.6875, Z = 2, and 

– the 1s orbital from code helium hf radial. 

Beware: there is a factor between the orbitals calculated with radial integration 

and with a Gaussian basis set: which factor and why? 

• Try the following optimized basis set with four Gaussians, with coefficients: 

α 1 = 0.297104, α 2 = 1.236745, α 3 = 5.749982, α 4 = 38.216677 a.u.. 

59

Chapter 8 

Molecules 

8.1 Born-Oppenheimer approximation 

Let us consider a system of interacting nuclei and electrons. In general, the 

Hamiltonian of the system will depend upon all nuclear coordinates, R µ , and 

all electronic coordinates, r i . For a system of n electrons under the field of N 

nuclei with charge Z µ , in principle one has to solve the following Schrödinger 

equation: 

(T I + V II + V eI + T e + V ee ) Ψ(R µ , r i ) = EΨ(R µ , r i ) (8.1) 

where T I is the kinetic energy of nuclei, V II is the Coulomb repulsion between 

nuclei, V eI is the Coulomb attraction between nuclei and electrons, T e is the 

kinetic energy of electrons, V ee is the Coulomb repulsion between electrons: 

T I = − ∑ 

µ=1,N 

¯h 2 

2M µ 

∇ 2 µ, 

V ee = q2 e 

2 

T e = − ∑ 

∑ 

i≠j 

i=1,n 

¯h 2 

2m ∇2 i , 

1 

|r i − r j | , V eI = −q 2 e 

V II = q2 e 

2 

∑ 

µ≠ν 

∑ ∑ 

µ=1,N i=1,n 

Z µ Z ν 

|R µ − R ν | , 

Z µ 

|R µ − r i | . (8.2) 

This looks like an impressive problem. It is however possible to exploit the mass 

difference between electrons and nuclei to separate the global problem into an 

electronic problem for fixed nuclei and a nuclear problem under an effective 

potential generated by electrons. Such separation is known as adiabatic or Born- 

Oppenheimer approximation. The crucial point is that the electronic motion 

is much faster than the nuclear motion: while forces on nuclei and electrons 

have the same order of magnitude, an electron is at least ∼ 2000 times lighter 

than any nucleus. We can thus assume that at any time the electrons ”follow” 

the nuclear motion, while the nuclei at any time ”feel” an affective potential 

generated by electrons. Formally, we assume a wave function of the form 

Ψ(R µ , r i ) = Φ(R µ )ψ (l) 

R (r i) (8.3) 

where the electronic wave function ψ (l) 

R (r i) solves the following Schrödinger 

equation: 

(T e + V ee + V eI ) ψ (l) 

R (r i) = E (l) 

R ψ(l) R (r i). (8.4) 

60

The index R is a reminder that both the wave function and the energy depend 

upon the nuclear coordinates, via V eI ; the index l classifies electronic states. 

We now insert the wave function, Eq.(8.3), into Eq.(8.2) and notice that T e 

does not act on nuclear variables. We will get the following equation: 

( 

T I + V II + E (l) 

R 

) 

Ψ(R µ )ψ (l) 

R (r i) = EΨ(R µ )ψ (l) 

R (r i). (8.5) 

If we now neglect the dependency upon R of the electronic wave functions in 

the kinetic term: 

( 

T I Φ(R µ )ψ (l) ) 

R (r i) ≃ ψ (l) 

R (r i) (T I Φ(R µ )) . (8.6) 

we obtain a Schrödinger equation for nuclear coordinates only: 

( 

T I + V II + E (l) ) 

R 

Ψ(R µ ) = EΨ(R µ ), (8.7) 

where electrons have ”disappeared” into the eigenvalue E (l) 

R . The term V II +E (l) 

R 

plays the role of effective interaction potential between nuclei. Of course such 

potential, as well as eigenfunctions and eigenvalues of the nuclear problem, 

depends upon the particular electronic state. 

The Born-Oppenheimer approximation is very well verified, except the special 

cases of non-adiabatic phenomena (that are very important, though). The 

main neglected term in Eq.8.6 has the form 

∑ 

µ 

¯h 2 

( 

(∇ µ Φ(R µ )) ∇ µ ψ (l) ) 

R 

M (r i) 

µ 

and may if needed be added as a perturbation. 

8.2 Potential Energy Surface 

(8.8) 

The Born-Oppenheimer approximation allows us to separately solve a Schrödinger 

equation for electrons, Eq.(8.4), as a function of atomic positions, and 

a problem for nuclei only, Eq.(8.7). The latter is in fact a Schrödinger equation 

in which the nuclei interact via an effective interatomic potential, V (R µ ) ≡ 

V II +E (l) , a function of the atomic positions R µ and of the electronic state. The 

interatomic potential V (R µ ) is also known as potential energy surface (“potential” 

and “potential energy” are in this context synonyms), or PES. It is clear 

that the nuclear motion is completely determined by the PES (assuming that 

the electron states does not change with time) since forces acting on nuclei are 

nothing but the gradient of the PES: 

F µ = −∇ µ V (R µ ), (8.9) 

while equilibrium positions for nuclei, labeled with R (0) 

µ , are characterized by 

zero gradient of the PES (and thus of any force on nuclei): 

F µ = −∇ µ V (R (0) 

µ ) = 0. (8.10) 

61

In general, there can be many equilibrium points, either stable (a minimum: 

any displacement from the equilibrium point produces forces opposed to the 

displacement, i.e. the second derivative is positive everywhere) or unstable (a 

maximum or a saddle point: for at least some directions of displacement from 

equilibrium, there are forces in the direction of the displacement, i.e. there 

are negative second derivatives). Among the various minima, there will be a 

global minimum, the lowest-energy one, corresponding to the ground state of 

the nuclear system, for a given electronic state. If the electronic state is also 

the ground state, this will be the ground state of the atomic system. All other 

minima are local minima, that is, metastable states that the nuclear system can 

leave by overcoming a potential barrier. 

8.3 Diatomic molecules 

Let us consider now the simple case of diatomic molecules, and in particular, 

the molecule of H 2 . There are 6 nuclear coordinates, 3 for the center of mass 

and 3 relative to it, but just one, the distance R between the nuclei, determines 

the effective interatomic potential V (R): the PES is in fact invariant, both 

translationally and for rotations around the axis of the molecule. Given a 

distance R, we may solve Eq.(8.4) for electrons, find the l-th electronic energy 

level E (l) (R) and the corresponding interatomic potential V (R) = E II (R) + 

E (l) (R). Note that the nuclear repulsion energy E II (R) is simply given by 

E II (R) = q2 eZ 1 Z 2 

R 

(8.11) 

where Z 1 and Z 2 are nuclear charges for the two nuclei. 

Let us consider the electronic ground state only for H 2 molecule. At small 

R, repulsion between nuclei is dominant, V (R) becomes positive, diverging like 

q 2 e/R for R → 0. At large R, the ground state becomes that of two neutral 

H atoms, thus V (R) ≃ 2Ry. At intermediate R, the curve has a minimum at 

about R 0 = 0.74Å, with V (R 0 ) ≃ V (∞) − 4.5eV. Such value – the difference 

between the potential energy of atoms at large distances and at the minimum, 

is known as cohesive or binding energy. The form of the interatomic potential is 

reminiscent of that of model potentials like Morse (in fact such potentials were 

proposed to model binding). 

What is the electronic ground state for R ∼ R 0 ? We can get an idea by 

using the method of molecular orbitals: an approximate solution in which singleelectron 

states are formed as linear combinations of atomic states centered 

about the two nuclei. Combinations with the same phase are called ligand, as 

they tend to accumulate charge in the region of space between the two nuclei. 

Combinations with opposite phase are called antiligand, since they tend to 

deplete charge between the nuclei. Starting from molecular orbitals, one can 

build the wave function as a Slater determinant. Two ligand states of opposite 

spin, built as superpositions of 1s states centered on the two nuclei (σ orbitals), 

yield a good approximation to the ground state. By construction, the total spin 

of the ground state is zero. 

62

Molecular orbitals theory can explain qualitatively the characteristics of the 

ground (and also excited) states, for the homonuclear dimer series (i.e. formed 

by two equal nuclei). It is however no more than semi-quantitative; for better 

results, one has to resort to the variational method, typically in the framework 

of Hartree-Fock or similar approximations. Orbitals are expanded on a basis 

set of functions, often atom-centered Gaussians or atomic orbitals, and the 

Hartree-Fock or similar equations are solved on this basis set. 

8.4 Code: h2 hf gauss 

Code h2 hf gauss.f90 1 (or h2 hf gauss.c 2 ) solves the Hartree-Fock equations 

for the ground state of the H 2 molecule, using a basis set of S Gaussians. The 

basic ingredients are the same as in code helium hf gauss, but the basis set 

is now composed of two sets of Gaussians, one centered around nucleus 1 and 

one centered around nucleus 2. As a consequence, the overlap matrix and the 

matrix element of the Fock matrix contain terms (multi-centered integrals) in 

which the nuclear potential and the two basis functions are not centered on the 

same atom. 

The code requires in input a set of Gaussians coefficients (with the usual 

format); then it solves the SCF equations at interatomic distances d min , d min + 

δd, d min + 2δd, . . ., d max (the parameters d min , d max , δd have to be provided 

in input). It prints on output and on file h2.out the electronic energy (not 

to be confused with the Hartree-Fock eigenvalue), nuclear repulsive energy, the 

sum of the two (usually referred to as “total energy”), all in Ry; finally, the 

difference between the total energy and the energy of the isolated atoms at 

infinite distance, in eV. 

Note that h2 hf gauss.f90 also solves the H + 2 

is set to .false.. In this case the Schrödinger equation is solved, without any 

SCF procedure. 

8.4.1 Gaussian integrals 

case if the variable do scf 

A complete derivation of the Gaussian integrals appearing in the solution of the 

Hartree-Fock equations can be found at pages 77-81 of the book of Thijssen. 

The following is a quick summary. 

We use a basis set of gaussian functions, in which the index now labels not 

only the coefficient of the Gaussian but also the center (one of the two nuclei 

in practice): 

( 

b i (r) = exp −α i (r − R i ) 2) . (8.12) 

The following theorem for the product of two Gaussians: 

( 

exp −α i (r − R i ) 2) ( 

× exp −α j (r − R j ) 2) [ 

= K ij exp −(α i + α j )(r − R ij ) 2] , 

(8.13) 

1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/h2 hf gauss.f90 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/h2 hf gauss.c 

63

where 

K ij = exp 

[ 

− α iα j 

α i + α j 

|R i − R j | 2 ] 

, R ij = α iR i + α j R j 

α i + α j 

(8.14) 

allows to calculate the superposition integrals as follows: 

∫ 

S ij = 

b i (r)b j (r)d 3 r = 

( 

π 

α i + α j 

) 3/2 

K ij . (8.15) 

The kinetic contribution can be calculated using the Green’s theorem: 

∫ 

∫ 

T ij = − b i (r)∇ 2 b j (r)d 3 r = ∇b i (r)∇b j (r)d 3 r (8.16) 

and finally 

T ij = 

α [ 

iα j 

6 − 4 α ] 

iα j 

|R i − R j | 2 S ij . (8.17) 

α i + α j α i + α j 

The calculation of the Coulomb interaction term with a nucleus in R ′ ≠ R is 

more complex and requires to go through Fourier transforms. At the end one 

gets the following expression: 

∫ 

1 

V ij = − b i (r) 

|r − R ′ | b j(r)d 3 1 

r = −S ij 

|R ij − R ′ | erf (√ α i + α j |R ij − R ′ | ) . 

(8.18) 

In the case R ij − R ′ = 0 we use the limit erf(x) → 2x/ √ π to get the already 

known result: V ij = −2π/(α i + α j ). The bi-electronic integrals introduced in 

the previous chapter, Eq.(7.26), can be calculated using a similar technique: 

g iljm = 

∫ 

1 

b i (r)b j (r) 

|r − r ′ | b l(r ′ )b m (r ′ )d 3 rd 3 r ′ (8.19) 

= S ij S lm 

1 

|R ij − R lm | erf (√ 

(αi + α j )(α l + α m ) 

α i + α j + α l + α m 

|R ij − R lm | 

(beware indices!). 

Although symmetry is not used in the code, it can be used to reduce by a 

sizable amount the number of bi-electronic integrals g iljm . They are obviously 

invariant under exchange of i, j and l, m indices. This means that if we have 

N basis set functions, the number of independent matrix elements is not N 4 

but M 2 , where M = N(N + 1)/2 is the number of pairs of (i, j) and (l, m) 

indices. The symmetry of the integral under exchange of r and r ′ leads to 

another symmetry: g iljm is invariant under exchange of the (i, j) and (l, m) 

pairs. This further reduces the independent number of matrix elements by a 

factor 2, to M(M + 1)/2 ∼ N 4 /8. 


• Chosen a basis set that yields a good description of isolated atoms, find the 

equilibrium distance by minimizing the (electronic plus nuclear) energy. 

) 

64

Verify how sensitive the result is with respect to the dimension of the 

basis set. Note that the “binding energy” printed on output is calculated 

assuming that the isolated H atom has an energy of -1 Ry, but you should 

verify what the actual energy of the isolated H atom is for your basis set. 

• Make a plot of the ground state molecular orbital at the equilibrium 

distance along the axis of the molecule. For a better view of the binding, 

you may also try to make a two-dimensional contour plot on a plane 

containing the axis of the molecule. You need to write a matrix on a 

uniform two-dimensional N x M grid in the following format: 

x_0 y_0 \psi(x_0,y_0) 

x_1 y_0 \psi(x_1,y_0) 

... 

x_N y_0 \psi(x_N,y_0) 

(blank line) 

x_0 y_1 \psi(x_0,y_1) 

x_1 y_1 \psi(x_1,y_1) 

... 

x_N y_1 \psi(x_N,y_1) 

(blank line) 

... 

x_0 y_M \psi(x_0,y_M) 

x_1 y_M \psi(x_1,y_M) 

... 

x_N y_M \psi(x_N,y_M) 

and gnuplot commands set contour; unset surface; set view 0, 90 

followed by splot ”file name” u 1:2:3 w l 

• Plot the ground state molecular orbital, together with a ligand combination 

of 1s states centered on the two H nuclei (obtained from codes 

for hydrogen). You should find that slightly contracted Slater orbitals, 

corresponding to Z = 1.24, yield a better fit than the 1s of H. Try the 

same for the first excited state of H 2 and the antiligand combination of 

1s states. 

• Study the limit of superposed atoms (R → 0) and compare with the results 

of codes hydrogen gauss and helium hf gauss, with the equivalent basis 

set. The limit of isolated atoms (R → ∞) will instead yield strange 

results. Can you explain why? If not: what do you expect to be wrong 

in the Slater determinant in this limit? 

• Can you estimate the vibrational frequency of H 2 ? 

65

Chapter 9 

Electrons in a periodic 

potential 

The computation of electronic states in a solid is a nontrivial problem. A great 

simplification can be achieved if the solid is a crystal, i.e. if it can be described 

by a regular, periodic, infinite arrangement of atoms: a crystal lattice. In this 

case, under suitable assumptions, it is possible to re-conduct the solution of the 

many-electron problem (really many: O(10 23 )!) to the much simpler problem 

of an electron under a periodic potential. Periodicity can be mathematically 

formalized in a simple and general way in any number of dimensions, but in the 

following we will consider a one-dimensional model case, that still contains the 

main effects of periodicity on electrons. 

9.1 Crystalline solids 

Let us consider an infinite periodic system of “atoms”, that will be represented 

by a potential, centered on the atomic position. This potential will in some 

way – why and how being explained in solid-state physics books – the effective 

potential (or crystal potential) felt by an electron in the crystal. We will 

consider only valence electrons, i.e. those coming from outer atomic shells. 

Core electrons, i.e. those coming from inner atomic shells, are tightly bound 

to the atomic nucleus: their state is basically atomic-like and only marginally 

influenced by the presence of other atoms. We assume that the effects of core 

electrons can be safely included into the crystal potential. The pseudopotential 

approach formalizes the neglect of core electrons. 

The assumption that core electrons do not significantly contribute to the 

chemical binding and that their state does not significantly change with respect 

to the case of isolated atoms is known as frozen-core approximation. This is 

widely used for calculations in molecules as well and usually very well verified 

in practice. 

We also consider independent electrons, assuming implicitly that the crystal 

potential takes into account the Coulomb repulsion between electrons. The aim 

of such simplification is to obtain an easily solvable problem that still captures 

the essence of the physical problem. With a judicious choice of the crystal 

66

potential, we can hope to obtain a set of electronic levels that can describe the 

main features of the crystal. A rigorous basis for such approach can be provided 

by Hartree-Fock or Density-Functional theory. In the end, the basic step is to 

solve to the problem of calculating the energy levels in a periodic potential. 

We haven’t yet said anything about the composition and the periodicity of 

our system. Let us simplify further the problem and assume a one-dimensional 

array of atoms of the same kind, regularly spaced by a distance a. The atomic 

position of atom n will thus be given as a n = na, with n running on all integer 

numbers, positive and negative. In the jargon of solid-state physics, a is the 

lattice parameter, while the a n are the vectors of the crystal lattice. The system 

has a discrete translational invariance, that is: it is equal to itself if translated 

by a or multiples of a. Called V (x) the crystal potential, formed by the 

superposition of atomic-like potentials: V (x) = ∑ n V n(x − a n ), the following 

symmetry holds: V (x + a) = V (x). Such symmetry plays a very important role 

in determining the properties of crystalline solids. Our one-dimensional space 

(the infinite line) can be decomposed into finite regions (segments) of space, of 

length a, periodically repeated. A region having such property is called unit cell, 

and the smallest possible unit cell is called primitive cell. Its definition contains 

some degree of arbitrarity: for instance, both intervals [0, a[ and ] − a/2, +a/2] 

define a valid primitive cell in our case. 

9.1.1 Periodic Boundary Conditions 

Before starting to look for a solution, we must ask ourselves how sensible it is 

to apply such idealized modelling to a real crystal. The latter is formed by a 

macroscopically large (in the order of the Avogadro number or fractions of it) 

but finite number of atoms. We might consider instead a finite system containing 

N atoms with N → ∞, but this is not a convenient way: translational 

symmetry is lost, due to the presence of surfaces (in our specific 1D case, the 

two ends). A much more convenient and formally correct approach is to introduce 

periodic boundary conditions (PBC). Let us consider the system in a box 

with dimensions L = Na and let us consider solutions obeying to the condition 

ψ(x) = ψ(x + L), i.e. periodic solutions with period L >> a. We can imagine 

our wave function that arrives at one end “re-enters” from the other side. In the 

one-dimensional case there is a simple representation of the system: our atoms 

are distributed not on a straight line but on a ring, with atom N between atom 

N − 1 and atom 1. 

The advantage of PBC is that we can treat the system as finite (a segment 

of length L in the one-dimensional case) but macroscopically large (having N 

atoms, with N macroscopically large if a is a typical interatomic distance and 

L the typical size of a piece of crystal), still retaining the discrete translational 

invariance. Case N → ∞ describes the so-called thermodynamical limit. It is to 

be noticed that a crystal with PBC has no surface. As a consequence there is no 

“inside” and “outside” the crystal: the latter is not contemplated. This is the 

price to pay for the big advantage of being able to use translational symmetry. 

In spite of PBC and of translational symmetry, the solution of the Schrödinger 

equation for a periodic potential does not yet look like a simple problem. 

67

We will need to find a number of single-particle states equal to at least half the 

number of electrons in the system, assuming that the many-body wave function 

is build as an anti-symmetrized product of single-electron states taken as spinup 

and spin-down pairs (as in the case of He and H 2 ). Of course the resulting 

state will have zero magnetization (S = 0). The exact number of electrons in a 

crystal depends upon its atomic composition. Even if we assume the minimal 

case of one electron per atom, we still have N electrons and we need to calculate 

N/2 states, with N → ∞. How can we deal with such a macroscopic number 

of states? 

9.1.2 Bloch Theorem 

At this point symmetry theory comes to the rescue under the form of the Bloch 

theorem. Let us indicate with T the discrete translation operator: T ψ(x) = 

ψ(x + a). What is the form of the eigenvalues and eigenvectors of T ? It can be 

easily verified (and rigorously proven) that T ψ(x) = λψ(x) admits as solution 

ψ k (x) = exp(ikx)u k (x), where k is a real number, u k (x) is a periodic function 

of period a: u k (x + a) = u k (x). This result is easily generalized to three 

dimensions, where k is a vector: the Bloch vector. States ψ k are called Bloch 

states. It is easy to verify that for Bloch states the following relation hold: 

ψ k (x + a) = ψ k (x)e ika . (9.1) 

Let us classify our solution using the Bloch vector k (in our case, a onedimensional 

vector, i.e. a number). The Bloch vector is related to the eigenvalue 

of the translation operator (we remind that H and T are commuting operators). 

Eq.(9.1) suggests that all k differing by a multiple of 2π/a are equivalent 

(i.e. they correspond to the same eigenvalue of T ). It is thus convenient to 

restrict to the following interval of values for k: k: −π/a < k ≤ π/a. Values 

of k outside such interval are brought back into the interval by a translation 

G n = 2πn/a. 

We must moreover verify that our Bloch states are compatible with PBC. 

It is immediate to verify that only values of k such that exp(ikL) = 1 are 

compatible with PBC, that is, k must be an integer multiple of 2π/L. As 

a consequence, for a finite number N of atoms (i.e. for a finite dimension 

L = Na of the box), there are N admissible values of k: k n = 2πn/L, con 

n = −N/2, ..., N/2 (note that k −N/2 = −π/a is equivalent to k N/2 = π/a). In 

the thermodynamical limit, N → ∞, these N Bloch vectors will form a dense 

set between −π/a and π/a, in practice a continuum. 

9.1.3 The empty potential 

Before moving towards the solution, let us consider the case of the simplest 

potential one can think of: the non-existent potential, V (x) = 0. Our system 

will have plane waves as solutions: ψ k (x) = (1/ √ L)exp(ikx), where the factor 

ensure the normalization. k may take any value, as long as it is compatible 

with PBC, that is, k = 2πn/L, with n any integer. The energy of the solution 

68

with wave vector k will be purely kinetic, and thus: 

ψ k (x) = 1 √ 

L 

e ikx , ɛ(k) = ¯h2 k 2 

2m . (9.2) 

In order to obtain the same description as for a periodic potential, we simply 

“refold” the wave vectors k into the interval −π/a < k ≤ π/a, by applying 

the translations G n = 2πn/a. Let us observe the energies as a function of the 

“refolded” k, Eq.(9.2): for each value of k in the interval −π/a < k ≤ π/a there 

are many (actually infinite) states with energy given by ɛ n (k) = ¯h 2 (k+G n ) 2 /2m. 

The corresponding Bloch states have the form 

ψ k,n (x) = 1 √ 

L 

e ikx u k,n (x), u k,n (x) = e iGnx . (9.3) 

The function u k,n (x) is by construction periodic. Notice that we have moved 

from an “extended” description, in which the vector k covers the entire space, to 

a “reduced” description in which k is limited between −π/a and π/a. Also for 

the space of vectors k, we can introduce a “unit cell”, ] − π/a, π/a], periodically 

repeated with period 2π/a. Such cell is also called Brillouin Zone (BZ). It is 

immediately verified that the periodicity in k-space is given by the so-called 

reciprocal lattice: a lattice of vectors G n such that G n · a m = 2πp, where p is 

an integer. 

9.1.4 Solution for the crystal potential 

Let us now consider the case of a “true”, non-zero periodic potential: we can 

think at it as a sum of terms centered on our “atoms”‘: 

V (x) = ∑ n 

v(x − na), (9.4) 

but this is not stictly necessary. We observe first of all that the Bloch theorem 

allows the separation of the problem into independent sub-problems for each k. 

If we insert the Bloch form, Eq.(9.1), into the Schrödinger equation: 

(T + V (x))e ikx u k (x) = Ee ikx u k (x), (9.5) 

we get an equation for the periodic part u k (x): 

[ ( 

¯h 

2 

k 2 − 2ik d 

) 

] 

2m dx − d2 

dx 2 + V (x) − E u k (x) = 0 (9.6) 

that has in general an infinite discrete series of solutions, orthogonal between 

them: ∫ L/2 

∫ a/2 

u ∗ k,n(x)u k,m (x)dx = δ nm N u ∗ k,n(x)u k ′ ,m(x)dx, (9.7) 

−L/2 

−a/2 

where we have made usage of the periodicity of functions u(x) to re-conduct the 

integral on the entire crystal (from −L/2 to L/2) to an integration on the unit 

cell only (from −a/2 to a/2). In the following, however, we are not going to use 

69

such equations. Notice that the solutions having different k are by construction 

orthogonal. Let us write the superposition integral between Bloch states for 

different k: 

∫ L/2 

∫ L/2 

ψk,n(x)ψ ∗ k ′ ,m(x)dx = e i(k′ −k)x u ∗ k,n(x)u k ′ ,m(x)dx (9.8) 

−L/2 

−L/2 

( ) ∑ ∫ a/2 

= e ip(k′ −k)a 

e i(k′ −k)x u ∗ k,n(x)u k ′ ,m(x)dx, 

p 

−a/2 

where the sum over p runs over all the N vectors of the lattice. The purely 

geometric factor multiplying the integral differs from zero only if k and k ′ 

coincide: 

∑ 

e ip(k′ −k)a = Nδ k,k ′. (9.9) 

p 

we have used Kronecker’s delta, not Dirac’s delta, because the k form a dense 

but still finite set (there are N of them). We note that the latter orthogonality 

relation holds for whatever periodic part, u(x), the Bloch states may have. 

Nothing implies that the periodic parts of the Bloch states at different k are 

orthogonal: only those for different Bloch states at the same k are orthogonal 

(see Eq.(9.7)). 

9.1.5 Plane-wave basis set 

Let us come back to the numerical solution. We look for the solution using a 

plane-wave basis set. This is especially appropriate for problems in which the 

potential is periodic. We cannot choose any plane-wave set, though: the correct 

choice is restricted by the Bloch vector and by the periodicity of the system. 

Given the Bloch vector k, the “right” plane-wave basis set is the following: 

b n,k (x) = √ 1 e i(k+Gn)x , G n = 2π n. (9.10) 

L a 

The “right” basis must in fact have a exp(ikx) behavior, like the Bloch states 

with Bloch vector k; moreover the potential must have nonzero matrix elements 

between plane waves. For a periodic potential like the one in Eq.(9.4), matrix 

elements: 

∫ L/2 

〈b i,k |V |b j,k 〉 = 1 L 

= 1 ( ∑ 

L 

p 

−L/2 

V (x)e −iGx dx (9.11) 

) ∫ a/2 

e −ipGa V (x)e −iGx dx, (9.12) 

−a/2 

where G = G i − G j , are non-zero only for a discrete set of values of G. In fact, 

the factor ∑ p e−ipGa is zero except when Ga is a multiple of 2π, i.e. only on 

the reciprocal lattice vectors G n defined above. One finally finds 

〈b i,k |V |b j,k ) = 1 a 

∫ a/2 

−a/2 

V (x)e −i(G i−G j )x dx. (9.13) 

The integral is calculated in a single unit cell and, if expressed as a sum of 

atomic terms localized in each cell, for a single term in the potential. Note that 

the factor N cancels and thus the N → ∞ thermodynamic limit is well defined. 

70

Fast Fourier Transform 

In the simple case that will be presented, the matrix elements of the Hamiltonian, 

Eq.(9.13), can be analytically computed by straight integration. Another 

case in which an analytic solution is known is a crystal potential written as a 

sum of Gaussian functions: 

This yields 

V (x) = 

N−1 ∑ 

p=0 

〈b i,k |V |b j,k 〉 = 1 a 

v(x − pa), v(x) = Ae −αx2 . (9.14) 

∫ L/2 

−L/2 

Ae −αx2 e −iGx dx (9.15) 

The integral is known (it can be calculated using the tricks and formulae given 

in previous sections, extended to complex plane): 

∫ ∞ 

√ π 

e −αx2 e −iGx /4α 

dx = 

α e−G2 (9.16) 

−∞ 

(remember that in the thermodynamical limit, L → ∞). 

For a generic potential, one has to resort to numerical methods to calculate 

the integral. One advantage of the plane-wave basis set is the possibility to 

exploit the properties of Fourier Transforms (FT), in particular the Fast Fourier 

Transform (FFT) algorithm. 

Let us discretize our problem in real space, by introducing a grid of n points 

x i = ia/n, i = 0, n − 1 in the unit cell. Note that due to periodicity, grid points 

with index i ≥ n or i < 0 are “refolded” into grid points in the unit cell (that 

is, V (x i+n ) = V (x i ), and in particular, x n is equivalent to x 0 . Let us introduce 

the function f j defined as follows: 

f j = 1 ∫ 

V (x)e −iGjx dx, G j = j 2π , j = 0, n − 1. (9.17) 

L 

a 

Apart from factors, this is the usual definition of FT (and is nothing but matrix 

elements). We can exploit periodicity to show that 

f j = 1 a 

∫ a 

0 

V (x)e −iG jx dx. (9.18) 

Note that the integration limits can be translated at will, again due to periodicity. 

Let us write now such integrals as a finite sum over grid points (with 

∆x = a/n as finite integration step): 

f j = 1 a 

= 1 n 

= 1 n 

n−1 ∑ 

m=0 

n−1 ∑ 

m=0 

n−1 ∑ 

m=0 

V (x m )e −iG jx m 

∆x 

V (x m )e −iG jx m 

V m exp[−2π jm n i], V i ≡ V (x i). (9.19) 

71

Notice that the FT is now a periodic function in the variable G, with period 

G n = 2πn/a! This shouldn’t come as a surprise though: the FT of a periodic 

function is a discrete function, the FT of a discrete function is periodic. 

It is easy to verify that the potential in real space an be obtained back from 

its FT as follows: 

V (x) = 

n−1 ∑ 

j=0 

yielding the inverse FT in discretized form: 

f j e iG jx , (9.20) 

V j = 

n−1 ∑ 

m=0 

f m exp[2π jm n 

i]. (9.21) 

The two operations of Eq.(9.19) and (9.19) are called Discrete Fourier Transform, 

or DFT (not to be confused with Density-Functional Theory!). One may 

wonder where have all the G vectors with negative values gone: after all, we 

would like to calculate f j for all j such that |G j | 2 < E c (for some suitably 

chosen value if E c ), not for G j with j ranging from 0 to n − 1. The periodicity 

of DFT in both real and reciprocal space however allows to refold the G j on 

the “right-hand side of the box”, so to speak, to negative G j values, by making 

a translation of 2πn/a. 

9.2 Code: periodicwell 

Let us now move to the practical solution of a “true”, even if model, potential: 

the periodic potential well, known in solid-state physics since the thirties under 

the name of Kronig-Penney model: 

V (x) = ∑ n 

v(x − na), v(x) = −V 0 |x| ≤ b 2 , v(x) = 0 |x| > b 2 

(9.22) 

and of course a ≥ b. Such model is exactly soluble in the limit b → 0, V 0 → ∞, 

V 0 b →constant. 

The needed ingredients for the solution in a plane-wave basis set are almost 

all already found in Sec.(4.3) and (4.4), where we have shown the numerical 

solution on a plane-wave basis set of the problem of a single potential well. 

Code periodicwell.f90 1 (or periodicwell.c 2 ) is in fact a trivial extension 

of code pwell. Such code in fact uses a plane-wave basis set like the one 

in Eq.(9.10), which means that it actually solves the periodic Kronig-Penney 

model for k = 0. If we increase the size of the cell until this becomes large with 

respect to the dimension of the single well, then we solve the case of the isolate 

potential well. 

The generalization to the periodic model only requires the introduction of 

the Bloch vector k. Our base is given by Eq.(9.10). In order to choose when to 

1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/periodicwell.f90 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/periodicwell.c 

72

truncate it, it is convenient to consider plane waves up to a maximum (cutoff) 

kinetic energy: 

¯h 2 (k + G n ) 2 

≤ E cut . (9.23) 

2m 

Bloch wave functions are expanded into plane waves: 

ψ k (x) = ∑ n 

c n b n,k (x) (9.24) 

and are automatically normalized if ∑ n |c n| 2 = 1. The matrix elements of the 

Hamiltonian are very simple: 

H ij = 〈b i,k |H|b j,k 〉 = δ ij 

¯h 2 (k + G i ) 2 

2m 

+ 1 √ aṼ (G i − G j ), (9.25) 

where Ṽ (G) is the Fourier transform of the crystal potential. Code pwell may 

be entirely recycled and generalized to the solution for Bloch vector k. It is 

convenient to introduce a cutoff parameter E cut for the truncation of the basis 

set. This is preferable to setting a maximum number of plane waves, because 

the convergence depends only upon the modulus of k +G. The number of plane 

waves, instead, also depends upon the dimension a of the unit cell. 

Code periodicwell requires in input the well depth, V 0 , the well width, 

b, the unit cell length, a. Internally, a loop over k points covers the entire BZ 

(that is, the interval [−π/a, π/a] in this specific case), calculates E(k), writes 

the lowest E(k) values to files bands.out in an easily plottable format. 


• Plot E(k), that goes under the name of band structure, or also dispersion. 

Note that if the potential is weak (the so-called quasi-free electrons case), 

its main effect is to induce the appearance of intervals of forbidden energy 

(i.e.: of energy values to which no state corresponds) at the boundaries 

of the BZ. In the jargon of solid-state physics, the potential opens a gap. 

This effect can be predicted on the basis of perturbation theory. 

• Observe how E(k) varies as a function of the periodicity and of the well 

depth and width. As a rule, a band becomes wider (more dispersed, in 

the jargon of solid-state physics) for increasing superposition of the atomic 

states. 

• Plot for a few low-lying bands the Bloch states in real space (borrow and 

adapt the code from pwell). Remember that Bloch states are complex 

for a general value of k. Look in particular at the behavior of states for 

k = 0 and k = ±π/a (the “zone boundary”). Can you understand their 

form? 

73

Chapter 10 

Pseudopotentials 

In general, the band structure of a solid will be composed both of more or less 

extended (valence) states, coming from outer atomic orbitals, and of strongly 

localized (core) states, coming from deep atomic levels. Extended states are the 

interesting part, since they determine the (structural, transport, etc.) properties 

of the solid. The idea arises naturally to get rid of core states by replacing 

the true Coulomb potential and core electrons with a pseudopotential (or effective 

core potential in Quantum Chemistry parlance): an effective potential that 

“mimics” the effects of the nucleus and the core electrons on valence electrons. 

A big advantage of the pseudopotential approach is to allow the usage of a 

plane-wave basis set in realistic calculations. 

10.1 Three-dimensional crystals 

Let us consider now a more realistic (or slightly less unrealistic) model of a 

crystal. The description of periodicity in three dimensions is a straightforward 

generalization of the one-dimensional case, although the resulting geometries 

may look awkward to an untrained eye. The lattice vectors, R n , can be written 

as a sum with integer coefficients, n i : 

R n = n 1 a 1 + n 2 a 2 + n 3 a 3 (10.1) 

of three primitive vectors, a i . There are 14 different types of lattices, known 

as Bravais lattices. The nuclei can be found at all sites d µ + R n , where d µ 

runs on all atoms in the unit cell (that may contain from 1 to thousands of 

atoms!). It can be demonstrated that the volume Ω of the unit cell is given by 

Ω = a 1 · (a 2 × a 3 ), i.e. the volume contained in the parallelepiped formed by 

the three primitive vectors. We remark that the primitive vectors are in general 

linearly independent (i.e. they do not lye on a plane) but not orthogonal. 

The crystal is assumed to be contained into a box containing a macroscopic 

number N of unit cells, with PBC imposed as follows: 

ψ(r + N 1 a 1 + N 2 a 2 + N 3 a 3 ) = ψ(r). (10.2) 

Of course, N = N 1 · N 2 · N 3 and the volume of the crystal is V = NΩ. 

74

A reciprocal lattice of vectors G m such that G m · R n = 2πp, with p integer, 

is introduced. It can be shown that 

G m = m 1 b 1 + m 2 b 2 + m 3 b 3 (10.3) 

with m i integers and the three vectors b j given by 

b 1 = 2π Ω a 2 × a 3 , b 2 = 2π Ω a 3 × a 2 , b 3 = 2π Ω a 1 × a 2 (10.4) 

(note that a i · b j = 2πδ ij ). The Bloch theorem generalizes to 

ψ(r + R) = e ik·R ψ(r) (10.5) 

where the Bloch vector k is any vector obeying the PBC. Bloch vectors are 

usually taken into the three-dimensional Brillouin Zone (BZ), that is, the unit 

cell of the reciprocal lattice. 

It can be shown that there are N Bloch vectors; in the thermodynamical 

limit N → ∞, the Bloch vector becomes a continuous variable as in the onedimensional 

case. We remark that this means that at each k-point we have 

to “accommodate” ν electrons, where ν is the number of electrons in the unit 

cell. For a nonmagnetic, spin-unpolarized insulator, this means ν/2 filled states 

(usually called “valence” states, while empty states are called “conduction” 

states). We write the electronic states as ψ k,i where k is the Bloch vector and 

i is the band index. 

10.2 Plane waves, core states, pseudopotentials 

For a given lattice, the plane wave basis set for Bloch states of vector k is 

b n,k (r) = 1 √ 

V 

e i(k+Gn)·r (10.6) 

where G n are reciprocal lattice vector. A finite basis set can be obtained, as 

seen in the previous section, by truncating the basis set up to some cutoff on 

the kinetic energy: 

¯h 2 (k + G n ) 2 

≤ E cut . (10.7) 

2m 

In realistic crystals, however, E cut must be very large in order to get a good 

description of the electronic states. The reason is the very localized character 

of the core, atomic-like orbitals, and the extended character of plane waves. 

Let us consider core states in a crystal: their orbitals will be very close to the 

corresponding states in the atoms and will exhibit the same strong oscillations. 

Moreover, these strong oscillations will be present in valence (i.e. outer) states 

as well, because of orthogonality (for this reason these strong oscillations are 

referred to as “orthogonality wiggles”). Reproducing highly localized functions 

that vary strongly in a small region of space requires a large number of delocalized 

functions such as plane waves. 

75

Let us estimate how large is this large number using Fourier analysis. In 

order to describe features which vary on a length scale δ, one needs Fourier 

components up to q max ∼ 2π/δ. In a crystal, our wave vectors q = k + G have 

discrete values. There will be a number N P W of plane waves approximately 

equal to the volume of the sphere of radius q max , divided by the volume Ω BZ 

of the BZ: 

N P W ≃ 4πq3 max 

, Ω BZ = 8π3 

3Ω BZ Ω . (10.8) 

The second equality follows from the definition of the reciprocal lattice. 

A simple estimate for diamond is instructive. The 1s orbital of the carbon 

atom has its maximum around 0.3 a.u., so δ ≃ 0.1 a.u. is a reasonable value. 

Diamond has a ”face-centered-cubic” lattice with lattice parameter a 0 = 6.74 

a.u. and primitive vectors: 

( 1 

a 1 = a 0 

2 , 1 ) 

( 1 

2 , 0 , a 2 = a 0 

2 , 0, 1 ) 

( 

, a 3 = a 0 0, 1 2 

2 , 1 ) 

. (10.9) 

2 

The unit cell has a volume Ω = a 3 0 /4, the BZ s a volume Ω BZ = (2π) 3 /(a 3 0 /4). 

Inserting the data, one finds N P W ∼ 250, 000 plane wave, clearly too much for 

practical use. 

It is however possible to use a plane wave basis set in conjunction with 

pseudopotentials: an effective potential that “mimics” the effects of the nucleus 

and the core electrons on valence electrons. The true electronic valence orbitals 

are replaced by “pseudo-orbitals” that do not have the orthogonality wiggles 

typical of true orbitals. As a consequence, they are well described by a much 

smaller number of plane waves. 

Pseudopotentials have a long history, going back to the 30’s. Born as a 

rough and approximate way to get decent band structures, they have evolved 

into a sophisticated and exceedingly useful tool for accurate and predictive 

calculations in condensed-matter physics. 

10.3 Code: cohenbergstresser 

Code cohenbergstresser.f90 1 (or cohenbergstresser.c 2 ) implements the 

calculation of the band structure in Si using the pseudopotentials published by 

M. L. Cohen and T. K. Bergstresser, Phys. Rev. 141, 789 (1966). These are 

“empirical” pseudopotentials, i.e. devised to reproduce available experimental 

data, and not derived from first principles. 

Si has the same crystal structure as Diamond: 

a face-centered cubic lattice with two atoms in 

the unit cell. In the figure, the black and red 

dots identify the two sublattices. The side of 

the cube is the lattice parameter a 0 . In the Diamond 

structure, the two sublattices have the 

same composition; in the zincblende structure 

(e.g. GaAs), they have different composition. 

1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/cohenbergstresser.f90 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/cohenbergstresser.c 

76

The origin of the coordinate system is arbitrary; typical choices are one of 

the atoms, or the middle point between two neighboring atoms. We use the 

latter choice because it yields inversion symmetry. The two atoms in the unit 

cell are thus at positions d 1 = −d, d 2 = +d, where 

( 1 

d = a 0 

8 , 1 8 , 1 ) 

. (10.10) 

8 

The lattice parameter for Si is a 0 = 10.26 a.u.. 3 

Let us re-examine the matrix elements between plane waves of a potential 

V , given by a sum of spherical symmetric potentials V µ centered around atomic 

positions: 

V (r) = ∑ ∑ 

V µ (|r − d µ − R n |) (10.13) 

n µ 

With some algebra, one finds: 

〈b i,k |V |b j,k 〉 = 1 ∫ 

V (r)e −iG·r dr = V Si (G) cos(G · d), (10.14) 

Ω 

Ω 

where G = G i − G j . The cosine term is a special case of a geometrical factor 

known as structure factor, while V Si (G) is known as the atomic form factor: 

V Si (G) = 1 ∫ 

V Si (r)e −iG·r dr. (10.15) 

Ω 

Ω 

Cohen-Bergstresser pseudopotentials are given as atomic form factors for a few 

values of |G|, corresponding to the smallest allowed modules: G 2 = 0, 3, 4, 8, 11, ..., 

in units of (2π/a 0 ) 2 . 

The code requires on input the cutoff (in Ry) for the kinetic energy of plane 

waves, and a list of vectors k in the Brillouin Zone. Traditionally these points 

are chosen along high-symmetry lines, joining high-symmetry points shown in 

figure and listed below: 

Γ = (0, 0, 0), 

X = 2π 

a 0 

(1, 0, 0), 

W = 2π 

a 0 

(1, 1 2 , 0), 

K = 2π 

a 0 

( 3 4 , 3 4 , 0), 

L = 2π 

a 0 

( 1 2 , 1 2 , 1 2 ), 

3 We remark that the face-centered cubic lattice can also be described as a simple-cubic 

lattice: 

a 1 = a 0 (1, 0, 0) , a 2 = a 0 (0, 1, 0) , a 3 = a 0 (0, 0, 1) (10.11) 

with four atoms in the unit cell, at positions: 

( 

d 1 = a 0 0, 1 2 , 1 ) ( 1 

, d 2 = a 0 

2 

2 , 0, 1 ) ( 1 

, d 3 = a 0 

2 

2 , 1 ) 

2 , 0 , d 4 = (0, 0, 0) . (10.12) 

77


• Verify which cutoff is needed to achieve converged results for E(k) 

• Reproduce the results in the original paper for Si, Ge, Sn. 

• Try to figure out what the charge density would be by plotting the sum 

of the four lowest wavefunctions squared at the Γ point. It is convenient 

to plot along the (110) plane (that is: one axis along (1,1,0), the other 

along (0,0,1) ). 

• In the zincblende lattice, the two atoms are not identical. Cohen and 

Bergstresser introduce a “symmetric” and an “antisymmetric” contribution, 

corresponding respectively to a cosine and a sine times the imaginary 

unit in the structure factor: 

〈b i,k |V |b j,k 〉 = V s (G) cos(G · d) + iV a (G) sin(G · d). (10.16) 

What do you think is needed in order to extend the code to cover the case 

of Zincblende lattices? 

78

Chapter 11 

Exact diagonalization of 

quantum spin models 

Systems of interacting spins are used since many years to model magnetic phenomena. 

Their usefulness extends well beyond the field of magnetism, since 

many different physical phenomena can be mapped, exactly or approximately, 

onto spin systems. Many models exists and many techniques can be used to 

determine their properties under various conditions. In this chapter we will deal 

with the exact solution (i.e. finding the ground state) for the Heisenberg model, 

i.e. a quantum spin model in which spin centered at lattice sites interact via 

the exchange interaction. The hyper-simplified model we are going to study is 

sufficient to give an idea of the kind of problems one encounters when trying to 

solve many-body systems without resorting to mean-field approximations (i.e. 

reducing the many-body problem to that of a single spin under an effective field 

generated by the other spins). Moreover it allows to introduce two very important 

concepts in numerical analysis: iterative diagonalization and sparseness of 

a matrix. 

11.1 The Heisenberg model 

Let us consider a set of atoms in a crystal, each atom having a magnetic moment, 

typically due to localized, partially populated states such as 3d states in 

transition metals and 4f states in rare earths. The energy of the crystal may in 

general depend upon the orientation of the magnetic moments. In many cases 1 

these magnetic moments tend to spontaneously orient (at sufficiently low temperatures) 

along a given axis, in the same direction. This phenomenon is known 

as ferromagnetism. Other kinds of ordered structures are also known, and in 

particular antiferromagnetism: two or more sublattices of atoms are formed, 

having opposite magnetization. Well before the advent of quantum mechanics, 

it was realized that these phenomena could be quite well modeled by a system 

of interacting magnetic moments. The origin of the interaction was however 

mysterious, since direct dipole-dipole interaction is way too small to justify the 

1 but not for our model: it can be demonstrated that the magnetization vanishes at T ≠ 0, 

for all 1-d models with short-range interactions only 

79

observed behavior. The microscopic origin of the interaction was later found 

in the antisymmetry of the wave functions and in the constraints it imposes on 

the electronic structure (this is why it is known as exchange interaction). 

One of the phenomenological models used to study magnetism is the Heisenberg 

model. This consists in a system of quantum spins S i , localized at lattice 

sites i, described by a spin Hamiltonian: 

H = − ∑ 

(J x (ij)S x (i)S x (j) + J y (ij)S y (i)S y (j) + J z (ij)S z (i)S z (j)) (11.1) 

The sum runs over all pairs of spins. 

In the following, we will restrict to the simpler case of a single isotropic 

interaction energy J between nearest neighbors only: 

H = −J ∑ 

S(i) · S(j). (11.2) 

The restriction to nearest-neighbor interactions only makes physical sense, since 

in most physically relevant cases the exchange interaction is short-ranged. We 

will also restrict ourselves to the case S = 1/2. 

11.2 Hilbert space in spin systems 

The ground state of a spin system can be exactly found in principle, since the 

Hilbert space is finite: it is sufficient to diagonalize the Hamiltonian over a 

suitable basis set of finite dimension. The Hilbert space of spin systems is in 

fact formed by all possible linear combinations of products: 

|µ〉 = |σ µ (1)〉 ⊗ |σ µ (2)〉 ⊗ . . . ⊗ |σ µ (N)〉 (11.3) 

where N is the number of spins and the σ µ (i) labels the two possible spin states 

(σ = −1/2 or σ = +1/2) for the i−th spin. The Hilbert space has dimension 

N h = 2 N (or N h = (2S + 1) N for spin S), thus becoming quickly intractable for 

N as small as a few tens (e.g. for N = 30, N h ∼ 1 billion). It is however possible 

to reduce the dimension of the Hilbert space by exploiting some symmetries of 

the system, or by restricting to states of given total magnetization. For a 

system of N spins, n up and N − n down, it can be easily demonstrated that 

N h = N!/n!/(N − n)!. For 30 spins, this reduces the dimension of the Hilbert 

space to ”only” 155 millions at most. The solution “reduces” (so to speak) to 

the diagonalization of the N h × N h Hamiltonian matrix H µ,ν = 〈µ|H|ν〉, where 

µ and ν run on all possible N h states. 

For a small number of spins, up to 12-13, the size of the problem may still 

tractable with today’s computers. For a larger number of spin, one has to resort 

to techniques exploiting the sparseness of the Hamiltonian matrix. The number 

of nonzero matrix elements is in fact much smaller than the total number of 

matrix elements. Let us re-write the spin Hamiltonian under the following form: 

H = − J 2 

∑ 

 

(S + (i)S − (j) + S − (i)S + (j) + 2S z (i)S z (j)) . (11.4) 

80

The only nonzero matrix elements for the two first terms are between states |µ〉 

and |ν〉 states such that σ µ (k) = σ ν (k) for all k ≠ i, j, while for k = i, j: 

〈α(i)| ⊗ 〈β(j)|S + (i)S − (j)|β(i)〉 ⊗ |α(j)〉 (11.5) 

〈β(i)| ⊗ 〈α(j)|S − (i)S + (j)|α(i)〉 ⊗ |β(j)〉 (11.6) 

where α(i), β(i) mean i−th spin up and down, respectively. The term S z (i)S z (j) 

is diagonal, i.e. nonzero only for µ = ν. 

Sparseness, in conjunction with symmetry, can be used to reduce the Hamiltonian 

matrix into blocks of much smaller dimensions that can be diagonalized 

with a much reduced computational effort. 

11.3 Iterative diagonalization 

In addition to sparseness, there is another aspect that can be exploited to make 

the calculation more tractable. Typically one is interested in the ground state 

and in a few low-lying excited states, not in the entire spectrum. Calculating 

just a few eigenstates, however, is just marginally cheaper than calculating all 

of them, with conventional (LAPACK) diagonalization algorithms: an expensive 

tridiagonal (or equivalent) step, costing O(Nh 3 ) floating-point operations, 

has to be performed anyway. It is possible to take advantage of the smallness 

of the number of desired eigenvalues, by using iterative diagonalization 

algorithms. Unlike conventional algorithms, they are based on successive refinement 

steps of a trial solution, until the required accuracy is reached. If an 

approximate solution is known, the convergence may be very quick. Iterative 

diagonalization algorithms typically use as basic ingredients Hψ, where ψ is 

the trial solution. Such operations, in practice matrix-vector products, require 

O(Nh 2 ) floating-point operations. Sparseness can however be exploited to speed 

up the calculation of Hψ products. In some cases, the special structure of the 

matrix can also be exploited (this is the case for one-electron Hamiltonians in 

a plane-wave basis set). It is not just a problem of speed but of storage: even 

if we manage to store into memory vectors of length N h , storage of a N h × N h 

matrix is impossible. 

Among the many algorithms and variants, described in many thick books, 

a time-honored one that stands for its simplicity is the Lanczos algorithm. 

Starting from |v 0 〉 = 0 and from some initial guess |v 1 〉, we generate the following 

chain of vectors: 

where 

|w j+1 〉 = H|v j 〉 − α j |v j 〉 − β j |v j−1 〉, |v j+1 〉 = 1 

β j+1 

|w j+1 〉, (11.7) 

α j = 〈v j |H|v j 〉, β j+1 = (〈w j+1 |w j+1 〉) 1/2 . (11.8) 

It can be shown that all vectors |v j 〉 are orthogonal: 〈v i |v j 〉 = 0 ∀i ≠ j, and 

that in the basis of the |v j 〉 vectors, the Hamiltonian has a tridiagonal form, 

81

with α j elements on the diagonal, β j on the subdiagonal. After n steps: 

⎛ 

⎞ 

α 1 β 2 0 . . . 0 

β 2 α 2 β 3 0 . 

H t = 

. 0 β 3 α .. 3 0 

. (11.9) 

⎜ . 

⎝ . 0 .. . .. ⎟ βn ⎠ 

0 . . . 0 β n α n 

If n = N h , this transformation becomes exact: H t = H, and constitutes an alternative 

tridiagonalization algorithm. In practice, the Lanczos recursion tends 

to be unstable and may lead to loss of orthogonality between states. If however 

we limit to a few steps, we observe that the lowest eigenvalues, and especially 

the lowest one, of matrix H t converge very quickly to the corresponding ones 

of H. Since the diagonalization of a tridiagonal matrix is a very quick and 

easy operation, this procedure gives us a convenient numerical framework for 

finding a few low-lying states of large matrices. If moreover it is possible to 

exploit sparseness (or other properties of the matrix) to quickly calculate H|v〉 

products without storing the entire matrix, the advantage over conventional 

diagonalization becomes immense. 

11.4 Code: heisenberg exact 

Code heisenberg exact.f90 2 (or heisenberg exact.c 3 ) finds the ground state 

energy of the 1-dimensional Heisenberg model, using Periodic Boundary Conditions: 

N∑ 

H = −J S(i) · S(i + 1), S(N + 1) = S(1). (11.10) 

i=1 

In the code, energies are in units of |J|, spins are adimensional. If J > 0 a 

ferromagnetic ground state, with all spins oriented along the same direction, 

will be favored, while the J < 0 case will favor an antiferromagnetic ordering. 

The sign of J is set in the code (to change it, edit the code and recompile). 

For the totally magnetized (ferromagnetic) case, the solution is trivial: there 

is just one state with all spins up (let us call it |F 〉), yielding E 0 = 〈F |H|F 〉 = 

−NJ/4. Also the case with N − 1 spins up can be exactly solved. We have 

N states with N − 1 spins up, that we label as |n〉 = S − (n)|F 〉. Exploiting 

translational symmetry, one introduces Bloch-like states 

|k〉 = 1 √ 

N 

N ∑ 

n=1 

e ikn |n〉, k = 2πm/N, m = 0, ..., N − 1. (11.11) 

It can then be shown that these are the eigenvectors of H with eigenvalues 

E(k) = E 0 + J(1 − cos k). Careful readers will recognize spin waves in this 

solution. A general exact solution (for other similar spin problems as well) can 

be found by means of the Bethe Ansatz, a highly nontrivial technique. 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/heisenberg exact.f90 

3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/heisenberg exact.c 

82

The code requires the number N of spins and the number nup of up spins, 

computes the dimension nhil of the Hilbert space. It then proceeds to the 

labeling of the states, using a trick often employed for spin-1/2 systems: an 

integer index k, running from 1 to 2 N −1, contains in its i−th bit the information 

(0=down, 1=up) for the i−th spin. Of course this works only up to 32 spins, 

for default integers (or 64 spins for INTEGER(8)). The integer k is stored into 

array states for the states in the required Hilbert space. 

The Hamiltonian matrix is then filled (the code does not takes advantage of 

sparseness) and the number of nonzero matrix elements counted. For the S + S − 

and S − S + terms in the Hamiltonian, only matrix elements as in 11.5 and 11.6, 

respectively, are calculated. We remark that the line 

k = states(ii)+2**(j-1)-2**(i-1) 

is a quick-and-dirty way to calculate the index for the state obtained by flipping 

down spin i and flipping up spin j in state states(ii). 4 

We then proceed to the generation of the Lanczos chain. The number nl of 

chain steps (should not exceed nhil) is prompted for and read from terminal. 

The starting vector is filled with random numbers. Note the new BLAS routines 

dnrm2 and dgemv: the former calculates the module of a vector, the latter a 

matrix-vector product and is used to calculate H|v〉. 

The Hamiltonian in tridiagonal form (contained in the two arrays d and e) 

is then diagonalized by the LAPACK routine dsterf, that actually finds only 

the eigenvalues. The lowest eigenvalues is then printed for increasing values of 

the dimension of the tridiagonal matrix, up to nl, so that the convergence of 

the Lanczos chain can be estimated. You can modify the code to print more 

eigenvalues. 

As a final check, the matrix is diagonalized using the conventional algorithm 

(routine dspev). Note how much slower this final step is than the rest of the 

calculation! Once you are confident that the Lanczos chain works, you can 

speed up the calculations by commenting out the exact diagonalization step. 

The limiting factor will become the size of the Hamiltonian matrix. 

11.4.1 Computer Laboratory 

• Examine the convergence of the Lanczos procedure to the exact energy 

• For the antiferromagnetic case, verify that the ground state has zero magnetization. 

Plot the ground-state energy E 0 and the first excited state E 1 

as a function of N. Try to verify if the gap E − E 0 has a 1/N dependence. 

• For the ferromagnetic case, verify that the ground state has all spins 

aligned. Note that the ground state will have the same energy no matter 

what total magnetization you choose! This is a consequence of the 

rotational invariance of the Heisenberg Hamiltonian. Verify that the case 

with N − 1 spins up corresponds to the spin-wave solution, Eq.(11.11). 

You will need to print all eigenvalues. 

4 A more elegant but hardly more transparent way would be to directly manipulate the 

corresponding bits. 

83

Possible code extensions: 

• Modify the code in such a way that open boundary conditions (that is: the 

system is a finite chain) are used instead of periodic boundary conditions. 

You may find the following data useful to verify your results: E/N = 

−0.375 for N = 2, E/N = −1/3 for N = 3, E/N = −0.404 per N = 4, 

E/N → −0.44325 per N → ∞ 

• Modify the code in such a way that the entire Hamiltonian matrix is no 

longer stored. There are two possible ways to do this: 

– Calculate the Hψ product “on the fly”, without ever storing the 

matrix; 

– Store only nonzero matrix elements, plus an index keeping track of 

their position. 

Of course, you cannot any longer use dspev to diagonalize the Hamiltonian. 

Note that diagonalization packages for sparse matrices, such as 

ARPACK, exist. 

84

Appendix A 

From two-body to one-body 

problem 

Let us consider a quantum system formed by two interacting particles of mass 

m 1 and m 2 , with no external fields. The interaction potential V (r) depends 

only upon the distance r = |r 2 −r 1 | between the two particles. We do not make 

any further assumption on the form V (r). For the case of the Hydrogen atom, 

V will be of course the Coulomb potential. The Hamiltonian is 

H = p2 1 

2m 1 

+ p2 2 

2m 2 

+ V (|r 2 − r 1 |) 

(A.1) 

As in the case of classical mechanics, one can make a variable change to the 

two new variables R and r: 

R = m 1r 1 + m 2 r 2 

(A.2) 

m 1 + m 2 

r = r 2 − r 1 (A.3) 

corresponding to the position of the center of mass and to the relative position. 

It is also convenient to introduce 

M = m 1 + m 2 (A.4) 

m = 

m 1 m 2 

m 1 + m 2 

(A.5) 

where m is known as the reduced mass. 

By introducing the new momentum operators: P = −i¯h∇ R and p = −i¯h∇ r , 

conjugate to R and r respectively, the Hamiltonian becomes 

H = P 2 

2M + p2 

2m + V (r) 

(A.6) 

i.e. we have achieved separation of the variables. The center of mass behaves 

like a free particle of mass M; the solutions are plane waves. The interesting 

part is however the relative motion. The Schrödinger for the relative motion is 

the same as for a particle of mass m under a central force field V (r), having 

spherical symmetry with respect to the origin. 

85

For Hydrogen (and all one-electron atoms: He + , Li 2+ , etc.,) the two particles 

are a proton (or a heavier nucleus) and an electron, with a mass ratio equal 

to or larger than 1836. The reduced mass will be just a little bit smaller than 

the electron mass, m e = 0.911 × 10 −30 Kg. 

86

Appendix B 

Accidental degeneracy and 

dynamical symmetry 

The energy levels of the Coulomb potential depend only upon the main quantum 

number n. We have thus a degeneracy on the energy levels with the same n 

and different l, in addition to the one due to the 2l + 1 possible values of the 

quantum number m The total degeneracy (not considering spin) for a given n 

is thus 

n−1 ∑ 

(2l + 1) = n 2 . (B.1) 

l=0 

The degeneracy of the energies having different l and same n is present only 

if the interaction potential is Coulombian. Although it is known as accidental 

degeneracy, it is not really “accidental”. A degeneracy usually indicates the 

presence of a symmetry and thus of a conserved quantity. For instance, the 

degeneracy in m is related to spherical symmetry and to conservation of angular 

momentum. 

In classical mechanics, the equivalent of the accidental degeneracy is the 

conservation of the Runge-Lenz vector 

verified for a classical Hamiltonian 

M = p × L 

m − α r r (B.2) 

H = p2 

2m − α r . 

(B.3) 

This is the case of the relative motion of two bodies interacting via gravitational 

forces. In this case, the orbits are ellipses, and they are always closed: the 

orientation of the ellipses does not change with time. The Runge-Lenz vector 

lies along the major ellipses axis. The corresponding quantum vector has a 

slightly different expression: 

M = 1 

2m (p × L − L × p) − Zq2 e 

r r. 

(B.4) 

It is possible to show that M is orthogonal to L, and that [M, H] = 0, i.e. it is 

a conserved quantity. 

87

Appendix C 

Composition of angular 

momenta: the coupled 

representation 

Let us consider a system in which J 1 and J 2 are two mutually commuting angular 

momentum operators. This may happen when they refer to independent 

physical systems: e.g. angular momenta of two different particles, or orbital 

and spin angular momentum of the same particle. Let us also assume that 

there are no interactions coupling them. We have four commuting observables 

that describe the system: J1 2, J 1z, J2 

2 and J 2z. The common eigenstates are 

characterized by the set of quantum numbers: j 1 , m 1 , j 2 and m 2 . For fixed j 1 

and j 2 , we have thus (2j 1 + 1)(2j 2 + 1) distinct states. 

There is also another useful set of observables that describes the same system. 

We define the operator total angular momentum 

J = J 1 + J 2 

(C.1) 

It is immediate to verify that also J satisfies to the commutation algebra of the 

angular momentum: 

[J x , J y ] = [J 1x + J 2x , J 1y + J 2y ] 

= [J 1x , J 1y ] + [J 2x , J 2y ] 

= i¯hJ 1z + i¯hJ 2z 

= i¯hJ z 

(C.2) 

since we have assumed that the commutators between components relative to 

different systems are zero. Thus [J z , J 2 ] = 0. 

We can then describe the system using the four operators J 2 1 , J 2 2 , J 2 and J z . 

This is known as the coupled representation. In order to demonstrate that all 

these four operators commute, we just need to show that [J 2 1 , J 2 ] = [J 2 2 , J 2 ] = 0 

88

and [J 2 1 , J z] = [J 2 2 , J z] = 0: 

[J1 2, J 2 ] = [J1 2, J xJ x + J y J y + J z J z ] 

= J x [J1 2, J x] + [J1 2, J x]J x + (. . .) 

= J x [J1 2, J 1x + J 2x ] + [J1 2, J 1x + J 2x ]J x + (. . .) 

= J x [J1 2, J 1x] + [J1 2, J 1x]J x + (. . .) 

i = 0 

(C.3) 

and 

[J 2 1 , J z ] = [J 2 1 , J 1z + J 2z ] = [J 2 1 , J 1z ] = 0 (C.4) 

Let us label with j 1 , j 2 , j and m the quantum numbers characterizing the 

eigenvalues of our operators. For fixed j 1 and j 2 , j will assume values between 

j min and j max . By definition, m = m 1 + m 2 , where m is the projection of 

J z = J 1z + J 2z . Thus, we deduce that j max = j 1 + j 2 . j min can be obtained 

by the condition that the total number of states is the same as the one for the 

original representation, i.e. 

j 1 

∑+j 2 

j=j min 

(2j + 1) = (2j 1 + 1)(2j 2 + 1) 

(C.5) 

It can be verified that this conditions yields j min = |j 1 − j 2 |. o In summary: 

j = |j 1 −j 2 |, . . . , j 1 +j 2 . One can use a ”vector” picture: for j = |j 1 −j 2 | the two 

vectors have same direction and opposite sign, for j = j 1 +j 2 same direction and 

same sign, while intermediate cases correspond to angular momenta pointing 

in different directions. 

As an important example, let us consider the case j 1 = 1/2 and j 2 = 1/2. 

There are four independent states, m 1 = ±1/2 and m 2 = ±1/2. Let us move 

to the coupled representation. The allowed values for j are j = 0 and j = 1. 

For j = 0 only m = 0 is possible (“singlet state”). For j = 1 one has m = 0, ±1 

(“triplet states”). In total, there are always four states. 

The reason to introduce the coupled representation, apparently equivalent 

to the “ordinary” one, is that in many cases there are terms in the Hamiltonian 

that couple angular momenta. Quite common for instance is the case of a 

“torque” that tends to align the two vectors: 

H = . . . − AJ 1 · J 2 

(C.6) 

If such term is present, J 1z and J 2z are no longer conserved and their eigenvalues 

are no longer good quantum numbers. In fact 

[J 1z , −AJ 1 · J 2 ] = −A[J 1z , J 1x J 2x + J 1y J 2y + J 1z J 2z ] (C.7) 

= −A[J 1z , J 1x ]J 2x − A[J 1z , J 1y ]J 2y (C.8) 

= −i¯hA(J 1y J 2x − J 1x J 2y ) (C.9) 

and this operator is in general not zero. 

It is however immediate to verify that J 1z +J 2z is conserved, since [J 2z , −AJ 1· 

J 2 ] is equal to the term calculated above, but with opposite sign. 

89

Appendix D 

Two-electron atoms 

Let us assume that the spin and the coordinates are separable variables (this 

is surely true if the Hamiltonian does not contain spin-dependent terms): i.e., 

one can write 

ψ(1, 2) = Φ(r 1 , r 2 )χ(σ 1 , σ 2 ) 

(D.1) 

where Φ is a function of coordinates r alone, χ of the spins σ alone. ψ(1, 2) 

is always antisymmetric because electrons are fermions. It is however clear 

that this can be achieved by an antisymmetric Φ and a symmetric χ, or by a 

symmetric Φ and an antisymmetric χ. The spin eigenfunctions of the single 

electron have two possible values that we indicate by v + and v − . We can build 

three symmetric functions for the spin: 

χ 1,1 = v + (σ 1 )v + (σ 2 ) (D.2) 

χ 1,0 = √ 1 [v + (σ 1 )v − (σ 2 ) + v − (σ 1 )v + (σ 2 )] (D.3) 

2 

χ 1,−1 = v − (σ 1 )v − (σ 2 ) (D.4) 

and an antisymmetric one: 

χ 0,0 = 1 √ 

2 

[v + (σ 1 )v − (σ 2 ) − v − (σ 1 )v + (σ 2 )] 

(D.5) 

The symmetric functions constitute a triplet and correspond to a state of 

the two-electron system with total spin equal to 1 and three possible values: 

−1, 0, +1, for the projection of spin along z. The antisymmetric function constitutes 

a singlet and corresponds to a state with total spin equal to 0. 

The value of total spin thus determines the symmetry of the spin part, and 

as of consequence, of the coordinate part of the wave function. An antisymmetric 

coordinate part tends to “push apart” the two electrons, since Φ(r, r) = 0, 

i.e. the wave function tends to zero when the two electrons move into the 

same position. The presence of electrostatic repulsion lowers the energy for 

the antisymmetric coordinate wave function with respect to the corresponding 

symmetric case, in which there is a finite probability that electrons are close 

together (no reason for Φ(r, r) to vanish). This is why excited states in He are 

labeled as ortho-helium (triplet state with symmetric spin part and antisymmetric 

coordinate part) and para-helium (singlet state with antisymmetric spin 

90

part and symmetric coordinate part), with the former lower in energy than the 

latter for the same single electron levels. The ground state of He turns out to 

be a singlet, so the coordinate wave function must be symmetric. 

D.1 Perturbative Treatment for Helium atom 

The Helium atom is characterized by a Hamiltonian operator 

H = −¯h2 ∇ 2 1 

2m e 

− Zq2 e 

− ¯h2 ∇ 2 2 

− Zq2 e 

r 1 2m e r 2 

+ q2 e 

r 12 

(D.6) 

where r 12 = |r 2 − r 1 | is the distance between the two electrons. The last term 

corresponds to the Coulomb repulsion between the two electrons and makes the 

problem non separable. 

As a first approximation, let us consider the interaction between electrons: 

V = q2 e 

r 12 

as a perturbation to the problem described by 

(D.7) 

H 0 = −¯h2 ∇ 2 1 

2m e 

− Zq2 e 

− ¯h2 ∇ 2 2 

− Zq2 e 

r 1 2m e r 2 

(D.8) 

which is easy to solve since it is separable into two independent problems of 

a single electron under a central Coulomb field, i.e. a Hydrogen-like problem 

with nucleus charge Z = 2. The ground state for this system is given by the 

wave function described in Eq.(2.29) (1s orbital): 

φ 0 (r i ) = Z3/2 

√ π 

e −Zr i 

(D.9) 

in a.u.. We note that we can assign to both electrons the same wave function, as 

long as their spin is opposite. The total unperturbed wave function (coordinate 

part) is simply the product 

ψ 0 (r 1 , r 2 ) = Z3 

π e−Z(r 1+r 2 ) 

(D.10) 

which is a symmetric function (antisymmetry being provided by the spin part). 

The energy of the corresponding ground state is the sum of the energies of the 

two Hydrogen-like atoms: 

E 0 = −2Z 2 Ry = −8Ry 

(D.11) 

since Z = 2. The electron repulsion will necessarily raise this energy, i.e. make 

it less negative. In first-order perturbation theory, 

E − E 0 = 〈ψ 0 |V |ψ 0 〉 (D.12) 

∫ 

= Z6 2 

π 2 e −2Z(r 1+r 2 ) d 3 r 1 d 3 r 2 

(D.13) 

r 12 

= 5 ZRy. (D.14) 

4 

91

For Z = 2 the correction is equal to 2.5 Ry and yields E = −8 + 2.5 = −5.5 Ry. 

The experimental value is −5.8074 Ry. The perturbative approximation is not 

accurate but provides a reasonable estimate of the correction, even if the “perturbation”, 

i.e. the Coulomb repulsion between electrons, is of the same order 

of magnitude of all other interactions. Moreover, he ground state assumed in 

perturbation theory is usually qualitatively correct: the exact wave function for 

He will be close to a product of two 1s functions. 

D.2 Variational treatment for Helium atom 

The Helium atom provides a simple example of application of the variational 

method. The independent-electron solution, Eq.(D.10), is missing the phenomenon 

of screening: each electron will ”feel” a nucleus with partially screened 

charge, due to the presence of the other electron. In order to account for this 

phenomenon, we may take as our trial wave function an expression like the 

one of Eq.(D.10), with the true charge of the nucleus Z replaced by an “effective 

charge” Z e , presumably smaller than Z. Let us find the optimal Z e 

variationally, i.e. by minimizing the energy. We assume 

ψ(r 1 , r 2 ; Z e ) = Z3 e 

π e−Ze(r 1+r 2 ) 

and we re-write the Hamiltonian as: 

[ 

H = −¯h2 ∇ 2 1 

− Zq2 e 

− ¯h2 ∇ 2 ] 

2 

− Zq2 e 

+ 

2m e r 1 2m e r 2 

[ 

− (Z − Z e)q 2 e 

r 1 

− (Z − Z e)q 2 e 

r 2 

We now calculate 

∫ 

E(Z e ) = ψ ∗ (r 1 , r 2 ; Z e )Hψ(r 1 , r 2 ; Z e ) d 3 r 1 d 3 r 2 

(D.15) 

] 

+ q2 e 

r 12 

(D.16) 

(D.17) 

The contribution to the energy due to the first square bracket in Eq.(D.16) is 

−2Ze 2 a.u.: this is in fact a Hydrogen-like problem for a nucleus with charge Z e , 

for two non-interacting electrons. By expanding the remaining integrals and 

using symmetry we find 

∫ 

E(Z e ) = −2Ze 2 − 

(in a.u.) with 

|ψ| 2 4(Z − Z ∫ 

e) 

d 3 r 1 d 3 r 2 + 

r 1 

|ψ| 2 = Z6 e 

π 2 e−2Ze(r 1+r 2 ) 

Integrals can be easily calculated and the result is 

|ψ| 2 2 

r 12 

d 3 r 1 d 3 r 2 

(D.18) 

(D.19) 

E(Z e ) = −2Z 2 e − 4(Z − Z e )Z e + 2 5 8 Z e = 2Z 2 e − 27 

4 Z e (D.20) 

where we explicitly set Z = 2. 

immediately leads to 

Minimization of E(Z e ) with respect to Z e 

Z e = 27 = 1.6875 (D.21) 

16 

92

and the corresponding energy is 

E = − 729 = −5.695 Ry (D.22) 

128 

This result is definitely better that the perturbative result E = −5.50 Ry, 

even if there is still a non-negligible distance with the experimental result 

E = −5.8074 Ry. 

It is possible to improve the variational result by extending the set of trial 

wave functions. Sect.(7.1) shows how to produce the best single-electron functions 

using the Hartree-Fock method. Even better results can be obtained 

using trial wave functions that are more complex than a simple product of 

single-electron functions. For instance, let us consider trial wave functions like 

ψ(r 1 , r 2 ) = [f(r 1 )g(r 2 ) + g(r 1 )f(r 2 )] , 

(D.23) 

where the two single-electron functions, f and g, are Hydrogen-like wave function 

as in Eq.(D.9) with different values of Z, that we label Z f and Z g . By 

minimizing with respect to the two parameters Z f and Z g , one finds Z f = 2.183, 

Z g = 1.188, and an energy E = −5.751 Ry, much closer to the experimental 

result than for a single effective Z. Note that the two functions are far from 

being similar! 

D.3 ”Exact” treatment for Helium atom 

Let us made no explicit assumption on the form of the ground-state wave function 

of He. We assume however that the total spin is zero and thus the coordinate 

part of the wave function is symmetric. The wave function is expanded over 

a suitable basis set, in this case a symmetrized product of two single-electron 

gaussians. The lower-energy wave function is found by diagonalization. Such 

approach is of course possible only for a very small number of electrons. 

Code helium gauss.f90 1 (or helium gauss.c 2 ) looks for the ground state 

of the He atom, using an expansion into Gaussian functions, already introduced 

in the code hydrogen gauss. We assume that the solution is the product of a 

symmetric coordinate part and of an antisymmetric spin part, with total spin 

S = 0. The coordinate part is expanded into a basis of symmetrized products 

of gaussians, B k : 

ψ(r 1 , r 2 ) = ∑ c k B k (r 1 , r 2 ). 

(D.24) 

k 

If the b i functions are S-like gaussians as in Eq.(5.22), we have: 

B k (r 1 , r 2 ) = √ 1 

) 

(b i(k) (r 1 )b j(k) (r 2 ) + b i(k) (r 2 )b j(k) (r 1 ) 

2 

(D.25) 

where k is an index running over n(n+1)/2 pairs i(k), j(k) of gaussian functions. 

The overlap matrix ˜S kk ′ may be written in terms of the S ij overlap matrices, 

Eq.(5.25), of the hydrogen-like case: 

˜S kk ′ = 〈B k |B k ′〉 = ( S ii ′S jj ′ + S ij ′S ji ′) . (D.26) 

1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/helium gauss.f90 

2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/helium gauss.c 

93

The matrix elements, ˜H kk ′, of the Hamiltonian: 

˜H kk ′ = 〈B k |H|B k ′〉, H = −¯h2 ∇ 2 1 

2m e 

− Zq2 e 

− ¯h2 ∇ 2 2 

− Zq2 e 

r 1 2m e r 2 

+ q2 e 

r 12 

(D.27) 

can be written using matrix elements H ij = Hij K + HV ij , obtained for the 

hydrogen-like case with Z = 2, Eq.(5.26) and (5.27): 

˜H kk ′ = ( H ii ′S jj ′ + H ij ′S ji ′ + S ii ′H jj ′ + H ij ′S ji ′) + 〈Bk |V ee |B k ′〉, (D.28) 

and the matrix element of the Coulomb electron-electron interaction V ee : 

〈B k |V ee |B k ′〉 = 

∫ 

∫ 

+ 

b i(k) (r 1 )b j(k) (r 2 ) q2 e 

b 

r i(k ′ )(r 1 )b j(k ′ )(r 2 )d 3 r 1 d 3 r 2 

12 

b i(k) (r 1 )b j(k) (r 2 ) q2 e 

r 12 

b j(k ′ )(r 1 )b i(k ′ )(r 2 )d 3 r 1 d 3 r 2 . 

These matrix elements can be written, using Eq.(7.33), as 

(D.29) 

〈B k |V ee |B k ′〉 = 

qeπ 2 5/2 

αβ(α + β) 1/2 + qeπ 2 5/2 

α ′ β ′ (α ′ + β ′ ) 1/2 , 

(D.30) 

where 

α = α i(k) + α i(k ′ ), β = α j(k) + α j(k ′ ), α ′ = α i(k) + α j(k ′ ), β ′ = α j(k) + α i(k ′ ). 

(D.31) 

In an analogous way one can calculate the matrix elements between symmetrized 

products of gaussians formed with P-type gaussian functions (those 

defined in Eq.5.23). The combination of P-type gaussians with L = 0 has the 

form: 

B k (r 1 , r 2 ) = √ 1 ( 

) 

(r 1 · r 2 ) b i(k) (r 1 )b j(k) (r 2 ) + b i(k) (r 2 )b j(k) (r 1 ) 

2 

(D.32) 

It is immediately verified that the product of a S-type and a P-type gaussian 

yields an odd function that does not contribute to the ground state. 

In the case with S-type gaussians only, the code writes to file ”gs-wfc.out” 

the function: 

P (r 1 , r 2 ) = (4πr 1 r 2 ) 2 |ψ(r 1 , r 2 )| 2 , 

(D.33) 

where P (r 1 , r 2 )dr 1 dr 2 is the joint probability to find an electron between r 1 and 

r 1 + dr 1 , and an electron between r 2 and r 2 + dr 2 . The probability to find an 

electron between r and r + dr is given by p(r)dr, with 

∫ 

∫ 

p(r) = 4πr 2 |ψ(r, r 2 )| 2 4πr2dr 2 2 = P (r, r 2 )dr 2 . 

(D.34) 

It is easy to verify that for a wave function composed by a product of two 

identical functions, like the one in (D.10), the joint probability is the product 

of single-electron probabilities: P (r 1 , r 2 ) = p(r 1 )p(r 2 ). This is not true in 

general for the exact wave function. 

94

D.3.1 

Laboratory 

• observe the effect of the number of basis functions, to the choice of coefficients 

λ of the gaussians, to the inclusion of P-type gaussians 

• compare the obtained energy with the one obtained by other methods: 

perturbation theory with hydrogen-like wave functions, (Sec.D.1), variational 

theory with effective Z (Sec.D.3), exact result (-5.8074 Ry). 

• Make a plot of the probability P (r 1 , r 2 ) and of the difference P (r 1 , r 2 ) − 

p(r 1 )p(r 2 ), using for instance gnuplot and the following commands: 

set view 0, 90 

unset surface 

set contour 

set cntrparam levels auto 10 

splot [0:4][0:4] "gs-wfc.out" u 1:2:3 w l 

Note that the probability P (r 1 , r 2 ) (column 3 in ”splot”) is not exactly 

equal to the product p(r 1 )p(r 2 ) (column 4; column 5 is the difference 

between the two). 

95

Appendix E 

Useful algorithms 

E.1 Search of the zeros of a function 

Short notes on the problem of numerical determination of the zeros of a function, 

i.e. the values of x that solve f(x) = 0 for a given function. 

There are two different types of zeros: 

• odd – when f(x) changes sign at x 0 

• even – when f(x) does not change sign at x 0 

The latter case is more difficult from a numerical point of view. Imagine to 

add a small numerical noise to f(x): the zero disappears, or else it splits into 

two odd zeros. It is usually convenient to resort to methods to localize extrema 

(minima or maxima) rather than trying to pinpoint the zero. In the following 

we will deal with odd zeros only. 

E.1.1 

Bisection method 

We first have to find an interval [a, b] that includes a single zero, in such a way 

that f(a)f(b) < 0. The bisection algorithm halves the interval at each iteration, 

by refining the estimate of x 0 : 

1. c = (a + b)/2 

2. if f(a)f(c) < 0 we re-define b = c; else if f(b)f(c) < 0 we re-define a = c. 

3. In this way we get a new interval [a, b] of half the width at the previous 

step, and we repeat the procedure. 

Convergence is guaranteed (it is impossible to ”miss” the zero), and the logarithm 

of the uncertainty on the solution linearly decreases with the number of 

iterations. Some difficulty may arise in the choice of the stopping criterion. For 

instance: 

• Absolute error: |a − b| < ɛ. This may run into trouble if x 0 is very large: 

rounding errors |a − b| might be larger then ɛ. 

96

• Relative error: |a − b| < ɛa. This may run into trouble close to x = 0. 

E.1.2 

• If the slope of f(x) close to zero is very small, there might be a finite 

interval in which f(x) is indistinguishable from zero in machine representation. 

Newton-Raphson method 

The function is linearly approximated at each iteration in order to obtain a 

better estimate of the zero. Let us assume that we know both f(x) and f ′ (x). 

Then, close to x, 

f(x + δ) ≃ f(x) + f ′ (x)δ 

(E.1) 

and thus, to first order, 

δ = − f(x) 

f ′ (x) 

(E.2) 

would yield f(x + δ) = 0. We proceed by iterating in this way. It is possible to 

show that the rate of convergence is quadratic, i.e. the number of significant 

figures approximately doubles at each iteration (while with bisection method it 

grows linearly). 

The problem of this method is that convergence is not guaranteed, notably 

when f ′ (x) varies a lot close to the zero. Moreover, the method assumes that 

f ′ (x) can be directly calculated for any given x. In cases in which this snot 

true and the derivative must be calculated via finite differences, it is preferable 

to use the secant method described below. 

E.1.3 

Secant method 

This is based on a linear expansion of f(x) between two successive points, x n e 

x n+1 , of an iteration: 

f(x) = f(x n−1 ) + x − x n−1 

x n − x n−1 

[f(x n ) − f(x n−1 )] 

(E.3) 

that provides as an estimate for the zero 

x n − x n−1 

x n+1 = x n − f(x n ) 

f(x n ) − f(x n−1 ) 

(E.4) 

The procedure consists in iterating the above step. It is not necessary that 

the zero is contained inside the considered interval. This however may lead to 

non-convergence in some pathological cases. In regular cases, the speed of convergence 

is much better than for the bisection method, although slightly slower 

that the Newton-Raphson method (which however requires the knowledge of 

the derivative). 

In most difficult cases, it is convenient to split the search into two step: first 

bisection, with a sure and safe bracketing of the zero; then the secant method 

that quickly stabilizes the searched value up to the required precision. 

97

Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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