Computational Physics

60 The Hartree–Fock method

The resulting statesψ′kthen form an orthonormal set, satisfying(4.50)with

(^) kl=

∑

lm

UkmmUml†. (4.53)

In fact, a unitary transformation of the set{ψk}leaves the Slater determinant
unchanged (see Problem 4.7).
Equation (4.51) has the form of an ordinary Schrödinger equation although the
eigenvalueskare identified as Lagrange multipliers rather than as energies; nev-
ertheless they are often called ‘orbital energies’. From(4.51)and(4.38)it can be
seen that the energy is related to the parameterskby

E=

1

2

∑

k

[k+〈ψk|h|ψk〉] =

∑

k

[

k−

1

2

〈k|J−K|k〉

]

. (4.54)

The second form shows how the Coulomb and exchange contribution must be
subtracted from the sum of the Fock levels to avoid counting the two-electron
integrals twice.
In the previous section we have already seen how the self-consistency procedure
for solving the resulting equations is carried out.

4.5.3 Koopman’s theorem

If we were to calculate an excited state, we would have to take the lowestN− 1
spin-orbitals from the Fock spectrum and one excited spin-orbital for example (see
Fig. 4.1c), and carry out the self-consistency procedure for this configuration. The
resulting eigenstates will differ from the corresponding eigenstates in the ground
state. If we assume that the states do not vary appreciably when constructing the
Slater determinant from excited spin-orbitals instead of the ground state ones, we
can predict excitation energies from a ground state calculation. It turns out that –
within the approximation that the spin-orbitals are those of the ground state –
the difference between the sums of the eigenvalues,k, of the ground state and
excited state configuration, is equal to the real energy difference; see Problem 4.6.
This is known asKoopman’s theorem. This is not really a theorem but a way of
approximating excitation energies which turns out to work well for many systems.
Forfurtherreading,seeRefs.[ 5 , 6 , 8 ].

4.6 Basis functions

In the derivation leading to (4.51) (or (4.31)), the possible solutions of the
Schrödinger equation were restricted to the space of single Slater determinants. To
solve the resulting eigenvalue equation, another variational principle in the same