Computational Chemistry 4.13 - Spin Integration

Short lecture on integrating spin in electron integrals to obtain spatial orbital integrals.
Two-electron integrals in chemists's and physicist's notation use spin orbitals, which contain spin coordinates. One-electron integrals similarly contain spin coordinates, but are always the same electron on each side, thus the spin always match, and due to the normalization of spin functions integrates out to 1. The result is a one-electron spatial integral, indicated by parenthesis rather than brackets. Two electron integrals fall either into the Coulomb or exchange category. Coulomb integrals always match between the spin of spin orbitals and their complex conjugate, thus all electron pairs have a spatial Coulomb integral (in chemist's notation). Exchange integrals, however, have also exchanged spin functions, thus the spins of each electron must match for the integral to be non-zero, indicating exchange can only occur between same-spin electrons.
Notes Slide: i.imgur.com/77...
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[@mennaemam1971]

Hello, what is the i/2 subscript notation refer to? Why is not it just i?

[@qammaralmas8915]

As I understand, i/2 represents half of the i terms because half of the i electrons have spin up (alpha) and the other half have spin down (beta).

[@b6kf367]

I think there's a missing (i+1)/2 in the first row

[@seitdrs2467]

Hi, can it be assumed that the spatial orbitals are always represented by the chemist's notation or does a physicists notation exist as well. What is the convention?

[@TMPChem]

As far as I know, a two-electron integral within parentheses always indicates spatial orbitals in chemist's notation.

[@apburner1]

First.