Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

where the “integral” on dv 1 means as usual “integration on coordinates, sum
over spin components” . We follow the same path of Sec. (6.2) used to derive
Hartree equations, Eq.(6.13). Since a determinant for N electrons has N! terms,
we need a way to write matrix elements between determinants on a finite paper
surface. The following property, valid for any (symmetric) operator F and
determinantal functions ψ and ψ ′ , is very useful:
〈ψ|F |ψ ′ 〉 = 1 ∫ ∣ ∣∣∣∣∣ φ ∗ 1 (1) . φ∗ 1 (N)
∣ ∣∣∣∣∣∣ . . . F
N!
φ ∗ N (1) . φ∗ N (N) ∣
∫
= φ ∗ 1(1) . . . φ ∗ N(N)F
∣
φ ′ 1 (1) . φ′ 1 (N)
. . .
φ ′ N (1) . φ′ N (N) ∣ ∣∣∣∣∣∣
dv 1 . . . dv N
φ ′ 1 (1) . φ′ 1 (N)
. . .
φ ′ N (1) . φ′ N (N) ∣ ∣∣∣∣∣∣
dv 1 . . . dv N (7.4)
(by expanding the first determinant, one gets N! terms that, once integrated,
are identical). From the above property it is immediate (and boring) to obtain
the matrix elements for one- and two-electron operators:
〈ψ| ∑ f i |ψ〉 = ∑ ∫
φ ∗ i (1)f 1 φ i (1) dv 1 (7.5)
i
i
(as in the Hartree approximation), and
〈ψ| ∑ g ij |ψ〉 = ∑ ∫
φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − φ j (1)φ i (2)] dv 1 dv 2 (7.6)
〈ij〉
〈ij〉
The integrals implicitly include summation over spin components. If we assume
that g 12 depends only upon coordinates (as in Coulomb interaction) and not
upon spin, the second term:
∫
φ ∗ i (1)φ ∗ j(2)g 12 φ j (1)φ i (2) dv 1 dv 2 (7.7)
is zero is i and j states are different (the spin parts are not affected by g 12 and
they are orthogonal if relative to different spins).
It is convenient to move to a scheme in which the spin variables are not
explicitly included in the orbital index. Eq.(7.6) can then be written as
〈ψ| ∑ g ij |ψ〉 = ∑ ∫
φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − δ(σ i , σ j )φ j (1)φ i (2)] dv 1 dv 2
〈ij〉
〈ij〉
where σ i is the spin of electron i, and:
(7.8)
δ(σ i , σ j ) = 0 if σ i ≠ σ j
= 1 if σ i = σ j
In summary:
〈ψ|H|ψ〉 = ∑ ∫
i
+ ∑ ∫
〈ij〉
φ ∗ i (1)f 1 φ i (1) dv 1 (7.9)
φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − δ(σ i , σ j )φ j (1)φ i (2)] dv 1 dv 2
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