Computational Physics - Department of Physics

50215 Many-body approaches to studies of electronic systems: Hartree-Fock theory and Density Functional Theory

15.4 Derivation of the Hartree-Fock equations

Having obtained the functionalE[Φ], we now proceed to the second step of the calculation.
With the given functional, we can embark on at least two typesof variational strategies:

• We can vary the Slater determinant by changing the spatial part of the single-particle
  wave functions themselves.


• We can expand the single-particle functions in a known basis and vary the coefficients,
  that is, the new single-particle wave function|a〉is written as a linear expansion in terms
  of a fixed chosen orthogonal basis (for the example harmonic oscillator, or Laguerre poly-
  nomials etc)
  ψa=∑
  λ

Caλψλ.

In this case we vary the coefficientsCaλ.

We will derive the pertinent Hartree-Fock equations and discuss the pros and cons of the two
methods. Both cases lead to a new Slater determinant which isrelated to the previous one
via a unitary transformation.
Before we proceed we need however to repeat some aspects of the calculus of variations.
For more details see for example the text of Arfken [53].
We have already met the variational principle in chapter 14.We give here a brief reminder
on the calculus of variations.

15.4.1Reminder on calculus of variations

The calculus of variations involves problems where the quantity to be minimized or maximized
is an integral.
In the general case we have an integral of the type

E[Φ] =

∫b

a

f(Φ(x),

∂ Φ

∂r

,r)dr,

whereEis the quantity which is sought minimized or maximized. The problem is that although
fis a function of the variablesΦ,∂ Φ/∂randr, the exact dependence ofΦonris not known.
This means again that even though the integral has fixed limitsaandb, the path of integration
is not known. In our case the unknown quantities are the single-particle wave functions and
we wish to choose an integration path which makes the functionalE[Φ]stationary. This means
that we want to find minima, or maxima or saddle points. In physics we search normally for
minima.
Our task is therefore to find the minimum ofE[Φ]so that its variationδEis zero subject
to specific constraints. In our case the constraints appear as the integral which expresses
the orthogonality of the single-particle wave functions. The constraints can be treated via
the technique of Lagrangian multipliers. We assume the existence of an optimum path, that
is a path for whichE[Φ]is stationary. There are infinitely many such paths. The difference
between two pathsδ Φis called the variation ofΦ.
The condition for a stationary value is given by a partial differential equation, which we
here write in terms of one variablex

∂f

∂ Φ

−

d

dx

∂f

∂ Φx

= 0 ,