Computational Physics

84 The Hartree–Fock method

4.7 Consider a Slater determinant constructed from spin-orbitalsψk,k=1,...,N.A

unitary transformation transforms the spin-orbitalsψkinto new ones, which we

denote byψk′:

ψk′=

∑N

l= 1

Uklψl.

Uklare the elements of the unitary matrixU.

(a) Show that the new basis is orthonormal.

(b) Show that the Slater determinant constructed from the new spin-orbitals can be

written as the determinant of the product of the matrix

1

√

N!







ψ 1 (x 1 )ψ 2 (x 1 ) ··· ψN(x 1 )

ψ 1 (x 2 )ψ 2 (x 2 ) ··· ψN(x 2 )

..

.

..

.

..

.

ψ 1 (xN)ψ 2 (xN) ··· ψN(xN)







and the matrixU.

(c) Show that the Slater determinant built fromψkand that built fromψk′are equal

up to a complex constant of absolute value unity.

4.8 [C] In this problem, the program for calculating the electronic structure of the

hydrogen atom (seeSection 3.2.2), is extended to the H 2 +ion. The H+ 2 ion contains

only one electron and the problem is therefore essentially the same as that of

Section 3.2.2, the difference being a second nucleus at a distanceRfrom the first

one. The global structure of the program is therefore the same, the main difference

being that the basis now consists of eight functions: four functions centred around

each nucleus. Therefore, all matrices now have dimension 8.

It is important to note that for basis functions centred around one of the two

nuclei, the Coulomb attraction of the other nucleus is still important, as is

immediately clear from the expression of the Coulomb matrixA:

Apq=

∫

d^3 rχp(r)

(

1

|r−RA|

+

1

|r−RB|

)

χq(r)

whereRAandRBdenote the positions of the two nuclei. The integrals for

calculating the matrix elements of the various operators can be found inSection 4.8.

Write a program to determine the ground state of the H+ 2 ion.

For a distance 1a 0 between the nuclei, the program should yield an energy

(without the Coulomb repulsion between the nuclei) equal to−1.442 455 a.u.

4.9 [C] In this problem, the program developed in the previous problem is extended

along the lines of the helium ground state calculation ofSection 4.3.2in order to

calculate the electronic structure of the hydrogen molecule. This means that a

second electron is added to the ionic hydrogen system and we must solve the Hartree

equation for a finite basis,Eq. (4.14), self-consistently analogous to the helium

calculation.