Coupled and Time-Dependent Hartree-Fock

In the coupled Hartree-Fock method (CHF), which was probably derived the first time by Peng (1941) and rederived many times (Stevens et al, 1963 Gerratt and Mills, 1968), second- and higher-order static properties are obtained by solving the 

Contrary to the unperturbed Hartree-Fock theory, where the molecular orbitals are expanded in atomic one-electron basis functions Eq. (9.4), one normally expands the perturbed occupied spin orbitals in the set of orthonormalized unperturbed molecular spin orbitals V g  

One of the consequences of this choice is that the unperturbed Fock matrix becomes diagonal in this basis. 

Inserting this ansatz in the perturbed Hartree-Fock equations, (11.1) and (11.2), 

Multiplying the perturbed Hartree-Fock equations, Eq. (11.4), from the left with another basis function, an unperturbed molecular orbital V r, followed by integration one obtains a matrix form of the perturbed Hartree-Fock equations 

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