Modern Quantum Chemistry

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Exercise. This is the solution to exercise 3.1 in the book.

Solution. We have that:

fχj(1) = h(1)χj(1) + bJb(1)χj(1) bKb(1)χj(1)

Multiply by χi(1) and integrate over x1.

χi|f|j = χi(1)h(1)χ j(1)dx1 + + b bj χi(1)J b(1)χj(1)dx1 b bj χi(1)K b(1)χj(1)dx1

Replace definitions for

Jb(1)χj(1) =χb(2) 1 r12χb(2)χj(1)dx2

Kb(1)χj(1) =χb(2) 1 r12χj(2)χb(1)dx2

Hence,

χi|f|χj = χi|h|χj+ b bj χi(1)χ j(1) 1 r12χb(2)χ b(2)dx1dx2 b bj χi(1)χ b(1) 1 r12χb(2)χ j(2)dx1dx2

or

χi|f|χj = χi|h|χj+ b bj [ij|bb] b bj [ib|bj]

Since

[ib|bb] [ib|bb] = 0

we can also add index j = b, therefore

χi|f|χj = χi|h|χj + b[ij|bb] b[ib|bj]

or

χi|f|χj = χi|h|χj + b ib|jbib|bj

χi|f|χj = χi|h|χj + bib||jb