Moller-Plesset form

The spin free electronic Hamiltonian of the stem,, is partitioned according to the usual Moller-Plesset form (129),... 

Moller-Plesset form, 1,10, 48-50 Moller-Plesset perturbation theory, 6, 31,123 MOLPRINT 2D, 2,145 Monte Carlo methods, 1, 216-218, 239, 242,... 

Its first term, Emp2, has exactly the same functional form as the conventional second order Moller-Plesset correlation contribution (MP2). However, in Eqs (4 7 4.8) the ( )9 represent KS orbitals, which experience the multiplicative KS potential (2 7), rather... 

One approach is to construct a more flexible description of electron motions in terms of a combination of Hartree-Fock descriptions for ground and excited states. Configuration interaction (Cl) and Moller-Plesset (MP) models are two of the most commonly used models of this type. The so-called second-order Moller-Plesset model (MP2) is the most practical and widely employed. It generally provides excellent descriptions of equilibrium geometries and conformations, as well as thermochemistry, including the thermochemistry of reactions where bonds are broken and formed. Discussion is provided in Section n. 

Basis sets for use in practical Hartree-Fock, density functional, Moller-Plesset and configuration interaction calculations make use of Gaussian-type functions. Gaussian functions are closely related to exponential functions, which are of the form of exact solutions to the one-electron hydrogen atom, and comprise a polynomial in the Cartesian coordinates (x, y, z) followed by an exponential in r. Several series of Gaussian basis sets now have received widespread use and are thoroughly documented. A summary of all electron basis sets available in Spartan is provided in Table 3-1. Except for STO-3G and 3 -21G, any of these basis sets can be supplemented with additional polarization functions and/or with diffuse functions. It should be noted that minimal (STO-3G) and split-valence (3-2IG) basis sets, which lack polarization functions, are unsuitable for use with correlated models, in particular density functional, configuration interaction and Moller-Plesset models. Discussion is provided in Section II. 

Intramolecular nucleophilic substitution to form thiiranes was studied by means of ab initio MO computations based on the 6-31G basis set <1997JCC1773>. Systems studied included the anions SCH2CH2F and CH2C(=S)CH2F which would afford thiirane and 2-methylenethiirane, respectively (Equations Z and 3). It was important to include electron correlation which was done with the frozen-core approximation at the second-order Moller-Plesset perturbation level. Optimized structures were confirmed by means of vibrational frequency calculations. The main conclusions were that electron correlation is important in lowering AG and AG°, that the displacements are enthalpy controlled, and that reaction energies are strongly dependent on reactant stabilities. 

Li et studied the monochalcogenocarboxylic acids CH3COXH with X being S, Se, or Te (Fig. 10) both in vacuum and in a THF (tetrahydrofuran) solution. They used the polarizable continuum model and calculated the electronic properties of the solute by using Hartree-Fock calculations with a second order Moller-Plesset treatment of correlation effects. They compared the enol and the keto forms and calculated also the transition state between the two. Finally, they considered both... 

Published in 2002, the second report covered the period June 1999 to May 2001 and provided an opportunity to review the wide range of applications to which many-body perturbation theory in its simplest form, that is Moller-Plesset perturbation theory through second order, was being put at the turn of the millennium. The main development considered in this second report were ... 

As in my three most recent reviews, "" I have attempted to provide a snapshot of the many applications of many-body perturbation theory in its simplest form, i.e. Moller-Plesset theory designated MP2, during the period under review by performing a literature search for publications with the string MP2 in the title only. During the period covered by the present report, a total of 80 papers were discovered satisfying this criterion. 

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... 

Bach et al. have calculated the energies and structures of parent dioxirane (lc) as well as the open diradical form (10) and the isomeric carbonyl oxide (11) <92JA7207>. The molecular orbital calculations were carried out using the GAUSSIAN 90 program and the second order Moller-Plesset (MP2) optimization was used to obtain the structure of (lc). The C—O and O—O bond distances were calculated to be 1.397 and 1.531 A, respectively, which compare well to the distances... 

It has been shown that the reaction path from the pyran to the most stable open form involved a two-step mechanism through a cis open form (Figure 1). The first step of this mechanism (transition state TS1) is the rate-limiting step, with a barrier of 138 kJmol-1 at the Hartree-Fock 6-31G(d) level, corrected to 92kJmol-1 with Moller-Plesset second-order perturbation theory. This value is in agreement with experimental measurements of the activation energy of the ring... 

More than 100 years ago Thiele and Lapworth put forward the hypothesis that the exclusive substitution of phenol at the ortho- and para-positions might be attributed to rapid equihbration of phenol 3 with the transient keto forms 4 and 5. Since that time, the keto-enol equilibrium ratio in phenol itself has been estimated repeatedly and by application of various research methods. Thus, ab initio 6-31G basis set calculations were recently carried out on the structures of phenol 3, and its keto tautomers 2,4-cyclohexadienone 4 and 2,5-cyclohexadienone 5 . Energy calculations were carried out by using the all-electron ab initio Hartree-Fock formalism (RHF) as well as 2nd-order Moller-Plesset formalism (MP2) on the RHF-optimized geometries. It was shown that phenol 3 is significantly more stable than dienones 4 and 5 by 47.4 and 42.5 kJ mor (RHF) as well as 72.5 and 70.6 kJ moH (MP2), respectively. An equilibrium constant 3 4 was estimated as 1.98 x 10, i.e. in excellent agreement with experimental results as shown below. 

In practical applications of the sapt approach to interactions of many-elect ron systems, one has to use the many-body version of sapt, which includes order-by-order the intramonomer correlation effects. The many-body SAPT is based on the partitioning of the total Hamiltonian as H = F+V+W, where the zeroth-order operator F = Fa + Fb is the sum of the Fock operators for the monomers A and B. The intermolecular interaction operator V = H — Ha — Hb is the difference between the Hamiltonians of interacting and noninteracting systems, and the intramonomer correlation operator W = Wa + Wb is the sum of the Moller-Plesset fluctuation potentials of the monomers Wx — Hx — Fx, X — A or B. The interaction operator V is taken in the non-expanded form, i.e., it is not approximated by the multipole expansion. The interaction energy components of Eq. (1) are now given in the form of a double perturbation series,... 

The theoretical description of any many body system is usually approached in two distinct stages. First, the solution of some independent particle model yielding a set of quasi-particles, or dressed particles, which are then used to formulate a systematic scheme for describing the corrections to the model. Perturbation theory, when developed with respect to a suitable reference model, affords the most systematic approach to the correlation problem which today, because it is non-iterative and, therefore, computationally very efficient, forms the basis of the most widely used approaches in contemporary electronic structure calculations, particularly when developed with respect to a Moller-Plesset zero order Hamiltonian. 

The theoretical treatment of the Van der Waals interaction, on the other hand, definitely requires the application of more sophisticated, correlated methods such as perturbation theory (performed mostly in the form of Moller-Plesset perturbation theory, MP), coiffiguration interaction (Cl) or coupled cluster (CC) approaches (see below). 

The question for a more systematic inclusion of electronic correlation brings us back to the realm of molecular quantum chemistry [51,182]. Recall that (see Section 2.11.3) the exact solution (configuration interaction. Cl) is found on the basis of the self-consistent Hartree-Fock wave function, namely by the excitation of the electrons into the virtual, unoccupied molecular orbitals. Unfortunately, the ultimate goal oi full Cl is obtainable for very small systems only, and restricted Cl is size-inconsistent the amount of electron correlation depends on the size of the system (Section 2.11.3). Thus, size-consistent but perturbative approaches (Moller-Plesset theory) are often used, and the simplest practical procedure (of second order, thus dubbed MP2 [129]) already scales with the fifth order of the system s size N, in contrast to Hartree-Fock theory ( N ). The accuracy of these methods may be systematically improved by going up to higher orders but this makes the calculations even more expensive and slow (MP3 N, MP4 N ). Fortunately, restricted Cl can be mathematically rephrased in the form of the so-called coupled clus-... 

The Moller-Plesset (MP) perturbation theory [13] is based on the zero-order Hamiltonian of the form ... 

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.]. 

Some of the above reasons for preferring the energy derivative over the expectation value will only hold for variational wavefunctions. However, Diercksen et have argued that this technique is more suitable with MBPT methods as well. The techniques developed in analytic derivative methods can also be applied to the calculation of MBPT properties. In Moller-Plesset theory (the simplest form of MBPT), the zeroth-order wavefunction is SCF and the Hamiltonian is partitioned so that... 

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