Computational Physics

82 The Hartree–Fock method

We construct the functionsandχfrom the orthonormal spatial orbitalsφ 1 (r),

φ 2 (r)and the spin-up and -down functionsα(s)andβ(s)respectively.

(a) Write down the antisymmetric wave functions that can be constructed in this

way (there are six of them).

(b) Write down all possible Slater determinants that can be built from the

one-electron spin-orbitals consisting of a product of one of the orbitalsφ 1 and

φ 2 and a spin-up or -down spinor (you will find six of these determinants too).

(c) Express the wave functions of part (a) of this problem in those of (b).

4.3 Consider a Slater determinant

AS(x 1 ,...,xN)=

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)

∣∣

∣∣

∣∣

∣∣

∣

=

1

√

N!

∑

P

PPψ 1 (x 1 )...ψN(xN).

The spin-orbitalsψk(x)are orthonormal.

(a) Show that the Slater determinant is normalised, by considering the inner product

of two arbitrary terms occurring in the sum of the Slater determinant and then

summing over all possible pairs of such terms.

(b) Show in the same way that the density of electrons with coordinatesx, given by:

n(x)=N

∫

dx 2 ...dxN|AS(x,x 2 ,...,xN)|^2 ,

can be written in terms of theψkas:

n(x)=

∑

k

|ψk(x)|^2.

Suppose all spin-orbitals can be written as the product of a normalised orbital

and a normalised one-particle spinor, what is then the spatial charge density of

the electrons (i.e. regardless of the spin)?

(c) DeriveEqs. (4.35)and(4.36)using the methods employed in (a) and (b).

4.4 Consider the helium atom with two electrons having thesamespin, represented by

the spinorα(s).

(a) Give the form of the two-electron wave function, expressed in orthonormal

spatial orbitalsφ 1 andφ 2.

(b) Write down the Schrödinger equation for this system.

(c) Give an expression for the expectation value of the energy in the orbitalsφi.

4.5 The Hartree–Fock analysis can be performed not only for fermions, but also for

bosons. Consider a system consisting ofNspin-0 particles in one dimension having

spin-orbital coordinatesxi(for spin-0 particles, only the orbital coordinate matters).