MP perturbation theory

So far the theory has been completely general. In order to apply perturbation theory to the calculation of correlation energy, the unperturbed Hamilton operator must be selected. The most common choice is to take this as a sum over Fock operators, leading to Mdller-Plesset (MP) perturbation theory. The sum of Fock operators counts the (average) electron-electron repulsion twice (eq. (3.43)), and the perturbation becomes... 

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. 

CC) methods, which have largely superseded Cl methods, in the limit can also be used to give exact solutions but again with same prohibitive cost as full Cl. As with Cl, CC methods are often truncated, most commonly to CCSD (N cost), but as before these can still only be applied to systems of modest size. Finally, Moller-Plesset (MP) perturbation theory, which is usually used to second order (MP2 has a cost), is more computationally accessible but does not provide as robust results. 

Unlike the Cl case, where for a long time there has been general agreement as to what treatment to use (although not necessarily as to how it should be implemented computationally), until recently there were several different approaches to the use of perturbation theory in treating dynamical correlation. For the last few years, however, one approach, M0ller-Plesset (MP) perturbation theory, has been dominant [7]. It employs the MP partitioning of the Hamiltonian... 

PTE and PTD describe, respectively, the effects of the solvation on the electron correlation on the solvent polarization and vice versa the PTED scheme leads instead to a comprehensive description of these two separate effects, revealing coupling between them. However, the PTDE scheme is not suitable for the calculation of analytical derivatives, even at the lowest order of the MP perturbation theory. 

Another approach to the problem of computing the electron correlation energy is the M0ller54-Plesset55 (MP) perturbation theory (which is philosophically akin to the many-body perturbation theory of solid-state physics). The mechanics are the conventional Rayleigh-Schrodinger perturbation theory One introduces a generalized electronic Hamiltonian Hi, where... 

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. 

The set of atomic orbitals Xk is called a basis set, and the quality of the basis set will usually dictate the accuracy of the calculations. For example, the interaction energy between an active site and an adsorbate molecule might be seriously overestimated because of excessive basis set superposition error (BSSE) if the number of atomic orbitals taken in Eq. [4] is too small. Note that Hartree-Fock theory does not describe correlated electron motion. Models that go beyond the FiF approximation and take electron correlation into account are termed post-Flartree-Fock models. Extensive reviews of post-HF models based on configurational interaction (Cl) theory, Moller-Plesset (MP) perturbation theory, and coupled-cluster theory can be found in other chapters of this series. ... 

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... 

Non-variational wavefunctions, in which there is no optimisation of parameters, also exist. One approach is Moller-Plesset (MP) perturbation theory which, like Cl methods, is employed most often to improve upon a previously determined HF wavefunction. Fuller details of all these approaches may be found in the monograph by Szabo and Ostlund [40]. 

Ab initio MO methods based on HF or small multiconfigurational wave-functions have been the method of choice, up to the present, for studies of organic systems and other molecules with light nuclei. The properties of stable species on the PES are often reproduced very well by calculations with just HF wavefunctions. Studies of reactions usually require the more sophisticated and expensive techniques, such as Cl or MP perturbation theory, that take into account the effects of the correlation between the electrons that is omitted from the HF approximation. The additional energy lowering computed with these methods with respect to that obtained with an HF calculation is called the correlation energy. A detailed and up-to-date discussion of the accuracy of state-of-the-art MO methods when applied to a variety of problems may be found in the book by Hehre et al. 

The form of the SCEP treatment will vary in certain aspects depending upon whether it is employed to carry out a Cl, CC or Moller-Plesset (MP) perturbation theory calculation. However, the differences are modest and the same quantities appear in one place or another. For convenience we utilize here the MP perturbation theory version of SCEP as formulated by Pulay and Saebo [30, 31] for their local correlation treatment. The (Hylleraas) variation condition on the first-order coefficient matrix, C = CP, may be written in the form... 

This completes our treatment of MP perturbation theory through fourth-order. We have shown elsewhere [6] that the SCEP approach is readily adapted for CC and Cl calculations within the LSA as well. As already noted, essentially the same quantities appear in all cases. Thus, our formalism is now sufficient to include the CCSD(T) procedure, which is the current method of choice where feasible. 

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. 

CC theory is inherently better than an equivalent level of Cl because it eliminates unlinked diagrams and as a consequence, is size-extensive [13]. It is also inherently better than an equivalent level of MBPT because it is not hmited to finite-orders, or potential difficulties encountered in the convergence of perturbation theory. It is well known, e.g. that ordinary MP perturbation theory does not converge for the electron gas, and this has also been emphasized recently for molecules [47], though resummations (including CC theory) work fine [48]. But, the operable word is equivalent level . For Cl, that meant at least single and double excitations, and frequently some more, perhaps even from a multi-reference space. MBPT had been done with single excitations in fourth-order SDQ-MBPT(4) in the above two papers [13,46]. 

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 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]. 

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