Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

where
K ij = exp
[
− α iα j
α i + α j
|R i − R j | 2 ]
, R ij = α iR i + α j R j
α i + α j
(8.14)
allows to calculate the superposition integrals as follows:
∫
S ij =
b i (r)b j (r)d 3 r =
(
π
α i + α j
) 3/2
K ij . (8.15)
The kinetic contribution can be calculated using the Green’s theorem:
∫
∫
T ij = − b i (r)∇ 2 b j (r)d 3 r = ∇b i (r)∇b j (r)d 3 r (8.16)
and finally
T ij =
α [
iα j
6 − 4 α ]
iα j
|R i − R j | 2 S ij . (8.17)
α i + α j α i + α j
The calculation of the Coulomb interaction term with a nucleus in R ′ ≠ R is
more complex and requires to go through Fourier transforms. At the end one
gets the following expression:
∫
1
V ij = − b i (r)
|r − R ′ | b j(r)d 3 1
r = −S ij
|R ij − R ′ | erf (√ α i + α j |R ij − R ′ | ) .
(8.18)
In the case R ij − R ′ = 0 we use the limit erf(x) → 2x/ √ π to get the already
known result: V ij = −2π/(α i + α j ). The bi-electronic integrals introduced in
the previous chapter, Eq.(7.26), can be calculated using a similar technique:
g iljm =
∫
1
b i (r)b j (r)
|r − r ′ | b l(r ′ )b m (r ′ )d 3 rd 3 r ′ (8.19)
= S ij S lm
1
|R ij − R lm | erf (√
(αi + α j )(α l + α m )
α i + α j + α l + α m
|R ij − R lm |
(beware indices!).
Although symmetry is not used in the code, it can be used to reduce by a
sizable amount the number of bi-electronic integrals g iljm . They are obviously
invariant under exchange of i, j and l, m indices. This means that if we have
N basis set functions, the number of independent matrix elements is not N 4
but M 2 , where M = N(N + 1)/2 is the number of pairs of (i, j) and (l, m)
indices. The symmetry of the integral under exchange of r and r ′ leads to
another symmetry: g iljm is invariant under exchange of the (i, j) and (l, m)
pairs. This further reduces the independent number of matrix elements by a
factor 2, to M(M + 1)/2 ∼ N 4 /8.
8.4.2 Laboratory
• Chosen a basis set that yields a good description of isolated atoms, find the
equilibrium distance by minimizing the (electronic plus nuclear) energy.
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