Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

The matrix elements, ˜H kk ′, of the Hamiltonian:
˜H kk ′ = 〈B k |H|B k ′〉, H = −¯h2 ∇ 2 1
2m e
− Zq2 e
− ¯h2 ∇ 2 2
− Zq2 e
r 1 2m e r 2
+ q2 e
r 12
(D.27)
can be written using matrix elements H ij = Hij K + HV ij , obtained for the
hydrogen-like case with Z = 2, Eq.(5.26) and (5.27):
˜H kk ′ = ( H ii ′S jj ′ + H ij ′S ji ′ + S ii ′H jj ′ + H ij ′S ji ′) + 〈Bk |V ee |B k ′〉, (D.28)
and the matrix element of the Coulomb electron-electron interaction V ee :
〈B k |V ee |B k ′〉 =
∫
∫
+
b i(k) (r 1 )b j(k) (r 2 ) q2 e
b
r i(k ′ )(r 1 )b j(k ′ )(r 2 )d 3 r 1 d 3 r 2
12
b i(k) (r 1 )b j(k) (r 2 ) q2 e
r 12
b j(k ′ )(r 1 )b i(k ′ )(r 2 )d 3 r 1 d 3 r 2 .
These matrix elements can be written, using Eq.(7.33), as
(D.29)
〈B k |V ee |B k ′〉 =
qeπ 2 5/2
αβ(α + β) 1/2 + qeπ 2 5/2
α ′ β ′ (α ′ + β ′ ) 1/2 ,
(D.30)
where
α = α i(k) + α i(k ′ ), β = α j(k) + α j(k ′ ), α ′ = α i(k) + α j(k ′ ), β ′ = α j(k) + α i(k ′ ).
(D.31)
In an analogous way one can calculate the matrix elements between symmetrized
products of gaussians formed with P-type gaussian functions (those
defined in Eq.5.23). The combination of P-type gaussians with L = 0 has the
form:
B k (r 1 , r 2 ) = √ 1 (
)
(r 1 · r 2 ) b i(k) (r 1 )b j(k) (r 2 ) + b i(k) (r 2 )b j(k) (r 1 )
2
(D.32)
It is immediately verified that the product of a S-type and a P-type gaussian
yields an odd function that does not contribute to the ground state.
In the case with S-type gaussians only, the code writes to file ”gs-wfc.out”
the function:
P (r 1 , r 2 ) = (4πr 1 r 2 ) 2 |ψ(r 1 , r 2 )| 2 ,
(D.33)
where P (r 1 , r 2 )dr 1 dr 2 is the joint probability to find an electron between r 1 and
r 1 + dr 1 , and an electron between r 2 and r 2 + dr 2 . The probability to find an
electron between r and r + dr is given by p(r)dr, with
∫
∫
p(r) = 4πr 2 |ψ(r, r 2 )| 2 4πr2dr 2 2 = P (r, r 2 )dr 2 .
(D.34)
It is easy to verify that for a wave function composed by a product of two
identical functions, like the one in (D.10), the joint probability is the product
of single-electron probabilities: P (r 1 , r 2 ) = p(r 1 )p(r 2 ). This is not true in
general for the exact wave function.
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