Mpller-Plesset MP Perturbation Theory

Physicists and chemists have developed various perturbation-theory methods to deal with systems of many interacting particles (nucleons in a nucleus, atoms in a solid, electrons in an atom or molecule), and these methods constitute many-body perturbation theory (MBPT). In 1934, Mpller and Plesset proposed a perturbation treatment of atoms and molecules in which the unperturbed wave function is the Hartree-Fock function, and this form of MBPT is called Moller-Plesset (MP) perturbation theory. Actual molecular applications of MP perturbation theory began only in 1975 with the work of Pople and co-workers and Bartlett and co-workers [R. J. Bartlett, Ann. Rev. Phys. Chem.,31,359 (1981) Hehre et al.]. 

The treatment of this section will be restricted to closed-shell, ground-state molecules. Also, the development will use spin-orbitals m rather than spatial orbitals For spin-orbitals, the Hartree-Fock equations (13.148) and (13.149) for electron m in an n-electron molecule have the forms Szabo and Ostlund, Section 3.1) 

The MP unperturbed Hamiltonian is taken as the sum of the one-electron Fock operators f m) in (15.81)  

The ground-state Hartree-Fock wave function I o is the Slater determinant mi 2 m of spin-orbitals. This Slater determinant is an antisymmetrized product of the spin-orbitals [for example, see Eq. (10.36)] and, when expanded, is the sum of n terms, where each term involves a different permutation of the electrqns among the spin-orbitals. Each term in the expansion of s an eigenfunction of the MP H for example, for a four-electron system, application of H to a typical term in the I o expansion gives 

The perturbation H is the difference between the true interelectronic repulsions and the Hartree-Fock interelectronic potential (which is an average potential). 

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Perturbative approximation methods are usually based on the Mpller-Plesset (MP) perturbation theory for correcting the HF wavefunction. Energetic corrections may be calculated to second (MP2), third (MP3), or higher order. As usual, the open- versus closed-shell character of the wavefunction can be specified by an appropriate prefix, such as ROMP2 or UMP2 for restricted open-shell or unrestricted MP2, respectively. 

The type of correlated method that has enjoyed the most widespread application to H-bonded systems is many-body perturbation theory, also commonly referred to as Mpller-Plesset (MP) perturbation theory This approach considers the true Hamiltonian as a sum of its Hartree-Fock part plus an operator corresponding to electron correlation. In other words, the unperturbed Hamiltonian consists of the interaction of the electrons with the nuclei, plus their kinetic energy, to which is added the Hartree-Fock potential the interaction of each electron with the time-averaged field generated by the others. The perturbation thus becomes the difference between the correct interelectronic repulsion operator, with its instantaneous correlation between electrons, and the latter Hartree-Fock potential. In this formalism, the Hartree-Fock energy is equed to the sum of the zeroth and first-order perturbation energy corrections. 

A similar approach with the following modifications was used in the present paper (1) The more rigorous Mpller—Plesset (MP) perturbation theory was selected instead of the Hartree—Fock method. (2) Clusters of two molecules were employed for the geometry... 

A year or so after our initial MBPT studies, Pople s group joined in this MBPT effort, but chose to call their approach Mpller-Plesset (MP) perturbation theory [44,45], and they scruptulously avoided any use of second-quantization or diagrammatic techniques in its implementation. This was much preferred by some In 1978 we [13] and Pople et al. 

One of us [1] reviewed the situation of electron correlation a quarter of a century ago in a paper with the title electron correlation in the seventies [2]. At that time most quantum chemists did not care about electron correlation, and standard methods for the large scale treatment of electron correlation, like Mpller-Plesset (MP) perturbation theory or coupled-cluster (CC) theory were not yet available. However precursors of these methods such as lEPA (independent electron pair approximation) and CEPA (coupled-electron-pair approximation) had already been developped and were being used, mainly in research groups in Germany [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. 

The most convenient choice for Hq is the Fock operator since the Hartree-Fock wavefunction is an eigenfunction of the Fock operator, as are the excited configurations derived from it by replacing occupied orbitals by virtual orbitals. This choice of Hq yields Mpller-Plesset (MP) perturbation theory or many-body perturbation theory (MBPT)26,27 Through second order, the energy is ... 

A powerful technique in quantum chemical manipulations is called perturbation theory. In many cases one has to deal with a hamiltonian operator for which the quantum chemical equations are too difficult or impossible to solve. A simpler hamiltonian may then be used to provide a zero-order solution, and then a perturbation operator is introduced, whose effect on the final results of the calculation can be obtained as a separate correction to the zero-order approximation. In Mpller-Plesset (MP) perturbation theory, the Fock operator is the zero-order hamiltonian (equation (c) in Box 3.1) and a Slater determinant is the zero-order wavefunction. The zero-order energy... 

This inclnsion of electron correlation can be accomplished in several ways. One method has been to nse Mpller-Plesset (MP) pertnrbation theory. This theorized that the electron correlation was a perturbation of the wavefnnction, so the MP perturbation theory conld be applied to the HF wavefnnction to inclnde the electron correlation. As more perturbations are made to the system, more electron correlation is inclnded (these methods are denoted MP2, MP3, and MP4). Another method is to calculate the energy of the system when electrons are moved into vacant orbitals. These methods move electrons either one at a time (single), two at a time (double, such as the QCISD method), or three at a time (triple, such as the QCISDT method). These methods calculate energy values more accurately but at greater computational cost. 

One of the most widely used methods for treating the electron correlation missing in the Hartree-Fock wavefunction is MpUer Plesset (MP) perturbation theory (Mpller... 

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 Mpller-Plesset (MP) treatment of electron correlation [84] is based on perturbation theory, a very general approach used in physics to treat complex systems [85] this particular approach was described by M0ller and Plesset in 1934 [86] and developed into a practical molecular computational method by Binkley and Pople [87] in 1975. The basic idea behind perturbation theory is that if we know how to treat a simple (often idealized) system then a more complex (and often more realistic) version of this system, if it is not too different, can be treated mathematically as an altered (perturbed) version of the simple one. Mpller-Plesset calculations are denoted as MP, MPPT (M0ller-Plesset perturbation theory) or MBPT (many-body perturbation theory) calculations. The derivation of the Mpller-Plesset method [88] is somewhat involved, and only the flavor of the approach will be given here. There is a hierarchy of MP energy levels MPO, MP1 (these first two designations are not actually used), MP2, etc., which successively account more thoroughly for interelectronic repulsion. 

SM calculations are broadly based on either the (i) Hartree-Fock method (ii) Post-Hartree-Fock methods like the Mpller-Plesset level of theory (MP), configuration interaction (Cl), complete active space self-consistent field (CASSCF), coupled cluster singles and doubles (CCSD) or (iii) methods based on DFT [24-27]. Since the inclusion of electron correlation is vital to obtain an accurate description of nearly all the calculated properties, it is desirable that SM calculations are carried out at either the second-order Mpller-Plesset (MP2) or the coupled cluster with single, double, and perturbative triple substitutions (CCSD(T)) levels using basis sets composed of both diffuse and polarization functions. 

The Mpller-Plesset (MP) method [26] constitutes a relatively less expensive alternative and is conceptually related to Rayleigh-Schrddinger perturbation theory. The lowest term in the MPn series is MP2. The MP2 energy can be efficiently calculated as... 

The least expensive and conceptually simplest correlation treatment that can be applied to medium-size molecules is the second-order Mpller-Plesset perturbation theory (MP2), which is the most popular single-reference approach for the low-level treatment of electron correlation [21]. Higher order MP perturbation theory such as MP3 and MP4 are typically less useful in particular, results coming from MP3 level are inferior to MP2 because of the characteristically oscillatory convergence of perturbation theory. MP4, MP5, and MP6 offer some improvements, but the high computational costs required prevent their routine application [22]. 

The other area in which projection of an unrestricted result has received attention is projected Mpller-Plesset perturbation theory, the PUMPn methods [31], where n is the order of the perturbation theory. In cases in which the UHF approximation is a poor starting point (considerable spin contamination, for example), the convergence of the MP perturbation expansion can be slow and/or erratic. The PUMP methods apply projection operators to the perturbation expansion, although usually not full projection but simply annihilation of the leading contaminants. This approach has met with mixed success again, it represents a rather expensive modification to a technique that was originally chosen partly for its economy — seldom a recipe for success. 

The coupled cluster (CC) method is actually related to both the perturbation (Section 5.4.2) and the Cl approaches (Section 5.4.3). Like perturbation theory, CC theory is connected to the linked cluster theorem (linked diagram theorem) [101], which proves that MP calculations are size-consistent (see below). Like standard Cl it expresses the correlated wavefunction as a sum of the HF ground state determinant and determinants representing the promotion of electrons from this into virtual MOs. As with the Mpller-Plesset equations, the derivation of the CC equations is complicated. The basic idea is to express the correlated wave-function Tasa sum of determinants by allowing a series of operators 7), 73,... to act on the HF wavefunction ... 

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