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48 CHAPTER 5. MANY ELECTRONS
Note that the factor of 2 in the electron-electron repulsion is due to the sum running over all
pairs, e.g., including (1,2) and (2,1). The Schr¨odinger equation is then
> ?
ˆ
ˆ
ˆ
T + V ee + V ext Ψ(x 1 , . . . , x N ) = EΨ(x 1 , . . . , x N ). (5.8)
Chapter 5 The ground-state energy can be extracted from the variational principle:
ˆ
ˆ
ˆ
E = min%Ψ|T + V ee + V ext |Ψ&, (5.9)
Ψ
Many electrons
once the minimization is performed over all normalized antisymmetic wavefunctions.
5.2 Hartree-Fock
In this chapter, we generalize the concepts and ideas introduced for two electrons to the
N electron case. In the special case of non-interacting particles, we denote the wavefunction by Φ instead of
Ψ, and this will usually be a single Slater determinant of occupied orbitals, i.e.,
5.1 Ground state @ @
@ @
@ φ 1 (x 1 ) · · · φ N (x 1 ) @
@ . . @
@ . . @ (5.10)
We define carefully our notation for electronic systems with N electrons. First, we note Φ(x 1 , . . . , x N ) = @ @ . . @ @
@ @
that the wavefunction for N electrons is a function of 3N spatial coordinates and N spin @ φ 1 (x N ) · · · φ N (x N ) @
coordinates. Writing x i = (r i , σ i ) to incorporate both, we normalize our wavefunction by: For systems with equal numbers of up and down particles in a spin-independent external
1 1 2 potential, the full orbitals can be written as a product, φ i (x) = φ i (r)|σ&, and each spatial
dx 1 . . . dx N |Ψ(x 1 , . . . , x N )| = 1, (5.1)
orbital appears twice. The Hartree-Fock energy is
where ! dx denotes the integral over all space and sum over both spins. Note that the ˆ ˆ ˆ
antisymmetry principle implies E = min%Φ|T + V ee + V ext |Φ&, (5.11)
Φ
Ψ(x 1 , . . . , x j , . . . , x i , . . .) = −Ψ(x 1 , . . . , x i , . . . , x j , . . .). (5.2) The first contribution is just the non-interacting kinetic energy of the many orbitals, and the
last is their external potential energy.
The electronic density is defined by
The Coulomb interaction for a single Slater determinant yields, due to the antisymmetric
1 1
- 2
n(r) = N dx 2 . . . dx N |Ψ(r, σ, x 2 , . . . , x N )| , (5.3) nature of the determinant, two contributions:
σ
ˆ
3
and retains the interpretation that n(r)d r is the probability density for finding any electron %Φ|V ee |Φ& = U[Φ] + E X [Φ]. (5.12)
3
in a region d r around r. The density is normalized to the number of electrons The first of these is called the direct or Coulomb or electrostatic or classical or Hartree
1 3
d r n(r) = N. (5.4) contribution, given in Eq. F.4, with the density being the sum of the squares of the occupied
orbitals. This is the electrostatic energy of the charge density in electromagnetic theory,
Our favorite operators become sums over one- and two-particle operators: ignoring its quantum origin. The second is the Fock or exchange integral, being
N
∗
∗
#
#
ˆ 1 - 2 1 1 φ (r) φ (r ) φ iσ (r ) φ jσ (r)
3
T = − ∇ , (5.5) 1 - - d r d r iσ jσ (5.13)
3 #
i
2 i=1 E X [φ i ] = −
#
2 σ i,j |r − r |
occ
N
ˆ
-
V ext = v ext (r i ), (5.6) for a determinant of doubly-occupied orbitals. This is a purely Pauli-exclusion principle effect.
i=1
and Exercise 20 Exchange energies for one and two electrons
N 1
ˆ
V ee = 1 - (5.7) Argue that, for one electron, E X = −U, and show that, for two electrons in a singlet,
2 i!=j |r i − r j | E X = −U/2.
47