Coupled Hartree-Fock perturbation generalizations

A simple example of the application of (11.9.5) is a derivation of the coupled Hartree-Fock (CHF) perturbation theory, first proposed by Peng (1941) and rediscovered, in various forms and with various generalizations, on many occasions. The essence of the approach is to start from a one-determinant wavefunction, optimized in the Hartree-Fock sense in the absence of the perturbation, and to seek the necessary first-order changes in the orbitals to maintain self-consistency when the perturbation is applied. The term coupled is used to indicate that, even if the perturbation contains only one-electron operators, the HF effective field must also change and will introduce a coupling , through the electron interactions, between the perturbation and the electron density. 

This is the basis of the so-called coupled perturbed Hartree-Fock (CPHF) theory, which, although generally traced to Ref. 233, actually is presented in essentially its complete form in Ref. 235. In Eq. (12), the Greek symbol p refers to an atomic basis function. 

In order to have a more complete picture of the many-body problem for more general or complicated cases that DFT could help to treat, it is necessary to make a correspondence with the use of many-body perturbation theory. Under this wider classification of perturbation theory are included all the methods that treat electron correlation beyond the Hartree-Fock level, including configuration interaction, coupled cluster, etc. This perturbational approach has traditionally been known as second quantization, and its power for some applications can be seen when dealing with problems beyond the standard quantum mechanics. 

We will always need to employ methods that can account for the electronic structure of our chosen model in some sense - either (1) totally explicitly by ab initio (Hartree-Fock (HF), MpUer-Plesset perturbation theory (MPx), coupled-cluster (CC), etc.) or by density functional theory (DFT) methods with a vast range, and continuously expanding number, of different functionals (B3LYP, M06, etc.), (2) partly explicitly like in semi-empirical (SE) methods (MNDO, AMI, PMx, etc.), or (3) we can even resort to a parameterization such as in classical MD simulations (being a much less prominent method though). All these are of course standard computational methods with generality across any elements and chemistry (for classical MD only if a proper parameter set exists). There is also ab initio MD (AIMD) emerging as a tool in the field. 

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