Hartree-Fock model perturbation theory

As usual, the Hartree-Fock model can be corrected with perturbation theory (e.g., the Mpller-Plesset [MP] method29) and/or variational techniques (e.g., the configuration-interaction [Cl] method30) to account for electron-correlation effects. The electron density p(r) = N f P 2 d3 2... d3r can generally be expressed as... 

The use of the Hartree-Fock model allows the perturbation-theory equations (1.2)-(1.5) to be conveniently recast in terms of underlying orbitals (,), orbital energies (e,), and orbital occupancies (n,). Such orbital perturbation equations will allow us to treat the complex electronic interactions of the actual many-electron system (described by Fock operator F) in terms of a simpler non-interacting system (described by unperturbed Fock operator We shall make use of such one-electron perturbation expressions throughout this book to elucidate the origin of chemical bonding effects within the Hartree-Fock model (which can be further refined with post-HF perturbative procedures, if desired). 

Many-body calculations which go beyond the Hartree-Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects superposition-of-configurations (or configuration interaction (Cl)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree-Fock (MCHF) approach, etc. However, to date only Cl and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions. 

In order to improve on the Hartree-Fock model, the use of perturbation theory is common. The first energy correction is obtained at second order and the corresponding method is calles second-order Moller-Plesset perturbation theory (MP2). MP2 calculations provide a first estimate for the correlation energy SE, which turned out to be also useful for estimates of the interaction energy in cases dominated by dispersive interactions (see the next section for an overview on how interaction energies can be calculated from total electronic energy estimates). 

Before we do so it is worth-while to establish some conventions and terminology in this area. The obvious name for a model of electronic structure which has a time-dependent Hamiltonian and consists of a single determinant of orbit s and remains a single determinant at all times is the Time-Dependent Hartree-Fock (TDHF) model, and this is the terminology which will be used here. However, there is, particularly in the theoretical physics literature, another related usage. Because the use of perturbation theory is so much their stock-in-trade, many theoretical physicists use the term time-dependent Hartree-Fock to mean the first-order (in the sense of perturbation theory) approximation to what we will call the time-dependent Hartree-Fock model. 

In 1934, Mailer and Plesset applied the Rayleigh-Schrddinger perturbation theory taken through second-order in the energy to the electronic structure problem in which the Hartree-Fock model is employed as a zero-order approximation. The Hartree-Fock wavefunction is a single determinant of the form... 

An efficient approach to improve on the Hartree-Fock Slater determinant is to employ Moller-Plesset perturbation theory, which works satisfactorily well for all molecules in which the Dirac-Hartree-Fock model provides a good approximation (i.e., in typical closed-shell single-determinantal cases). The four-component Moller-Plesset perturbation theory has been implemented by various groups [519,584,595]. A major bottleneck for these calculations is the fact that the molecular spinor optimization in the SCF procedure is carried out in the atomic-orbital basis set, while the perturbation expressions are given in terms of molecular spinors. Hence, all two-electron integrals required for the second-order Moller-Plesset energy expression must be calculated from the integrals over atomic-orbital basis functions like... 

During the 1960s, Kelly [37-43] pioneered the application of what is today the most widely used approach to the description of correlation effects in atomic and molecular systems namely, the many-body perturbation theory [1,2,43 8]. The second-order theory using the Hartree-Fock model to provide a reference Hamiltonian is particularly widely used. This Mpller-Plesset (mp2) formalism combines an accuracy, which is adequate for many purposes, with computational efficiency allowing both the use of basis sets of the quality required for correlated studies and applications to larger molecules than higher order methods. 

In the M0ller-Plesset formalism, a single-reference function is employed and the partition of the Hamiltonian into a reference or zero-order operator and a perturbation uses the Hartree-Fock model to define the reference. Third-order theory (mp3) and fourth-order theory (mp4) are computationally tractable. 

The use of M0ller-Plesset or Hartree-Fock model to label particular choices of zero-order Hamiltonian in many-body perturbation theory dates from the work of Pople et al. [2] and of Wilson and Silver [3]. In their original publication of 1934, MpUer and Plesset [4] did not recognize the many-body character of the theory in the modern (post-Brueckner) sense. 

Ab initio methods are applicable to the widest variety of property calculations. Many typical organic molecules can now be modeled with ab initio methods, such as Hartree Fock, density functional theory, and Moller Plesset perturbation theory. Organic molecule calculations are made easier by the fact that most organic molecules have singlet spin ground states. Organics are the systems for which sophisticated properties, such as NMR chemical shifts and nonlinear optical properties, can be calculated most accurately. 

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