Variational methods. Coupled Hartree-Fock theory

9 VARIATIONAL METHODS. COUPLED HARTREE-FOCK THEORY 

In Section 2.4 it was shown that with a wavefunction W, containing numerical parameters p = (pi, p2.), the variation of the usual energy functional around any point po in parameter space could be expressed as 

Let us now suppose that a variational approximation is available for the case H = Ho, and that the perturbation H is added. The energy functional will become 

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. 

Let US start from a one-determinant function Wq of spin-orbitals V l assumed orthonormal, and introduce parameters by considering variations of the form as in Section 8.2, A being an anti-Hermitian 

---

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

By ab initio we refer to quantum chemical methods in which all the integrals of the theory, be it variational or perturbative, are exactly evaluated. The level of theory then refers to the type of theory employed. Common levels of theory would include Hartree-Fock, or molecular orbital theory, configuration interaction (Cl) theory, perturbation theory (PT), coupled-cluster theory (CC, or coupled-perturbed many-electron theory, CPMET), etc. - We will use the word model to designate approximations to the Hamiltonian. For example, the zero differential overlap models can be applied at any level of theory. The distinction between semiempirical and ab initio quantum chemistry is often not clean. Basis sets, for example, are empirical in nature, as are effective core potentials. The search for basis set parameters is not usually considered to render a model empirical, whereas the search for parameters in effective core potentials is so considered. 

The second main group of methods analyse the response to successive powers of the perturbation separately using analytical re-arrangements of the perturbed equations. The procedure is typified in the Coupled Perturbed Hartree-Fock39,40,41 method (CPHF). which produces variationally optimized solutions in each order. Since the results represent a solution of the variational Hartree-Fock equations to each order they satisfy the energy derivative equations for the polarisabilities and the (2n+ 1) rule for the derivatives can be used to simplify the calculations. Corrections to the perturbed HF solutions can be made through MP2 or MP4 perturbation theory. 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

---