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: https://i.imgur.com/77FjxyU.png
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