Moller-Plesset perturbation theory applications

R. A. DiStasio, Jr. R. P. Steele, Y. M. Rhee, Y. Shao, and M. Head-Gordon, J. Comput. Chem., 28, 839-859 (2007). An Improved Algorithm for Analytical Gradient Evaluation in Resolution-of-the-Identity Second-Order Moller-Plesset Perturbation Theory Application to Alanine Tetrapeptide Conformational Analysis. 

Ah initio methods are applicable to the widest variety of property calculations. Many typical organic molecules can now be modeled with ah initio methods, such as Flartree-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. 

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 ... 

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). 

In many cases electronic properties calculated at the Hartree-Fock level do not have the accuracy sufficient to make them useful in chemical predictions. For example, as revealed in a recent study,the stability of the cage isomer of the C20 carbon cluster relative to that of the cyclic isomer is underestimated at the Flartree-Fock level by as much as 200 kcal/mol. In such systems, the electron correlation effects have to be taken into account in quantum chemical calculations through application of approximate methods. One such approximate electron correlation methods that has gained a widespread popularity is the second-order Moller-Plesset perturbation theory (MP2). Until recently calculations involving the MP2 approach have used a traditional formulation in which the MP2 energy is evaluated as the sum... 

Problem was solved by application of intrinsic reaction coordinate (IRC) procedure in combination with Density Functional Theory and second order Moller-Plesset perturbation theory methods [35]. It was demonstrated that potential energy surface around boat conformation is extremely fiat (Fig. 19.2) and contains wide plateau where energy of molecule remains almost the same. Conformation of cyclohexene ring is changed from one twist-boat to symmetric another twist-boat via boat conformation representing a central point of plateau. Width of plateau in terms of value of the C3-C4-C5-C6 torsion angle slightly depends on method of calculations (Table 19.2) and is varied within 30 0°. 

One of the most dramatic changes in the standard theoretical model used most widely in quantum chemistry occurred in the early 1990s. Until then, ab initio quantum chemical applications [1] typically used a Hartree-Fock (HF) starting point, followed in many cases by second-order Moller-Plesset perturbation theory. For small molecules requiring more accuracy, additional calculations were performed with coupled-cluster theory, quadratic configuration interaction, or related methods. While these techniques are still used widely, a substantial majority of the papers being published today are based on applications of density functional theory (DFT) [2]. Almost universally, the researchers use a functional due to Becke, whose papers in 1992 and 1993 contributed to this remarkable transformation that changed the entire landscape of quantum chemistry. 

The most recent of the publications to question the applicability of the low order theory studies is authored by Schaefer and his co-workers who ask the question Is Moller-Plesset perturbation theory a convergent ab initio method These authors write... 

Olsen et al. and others rests on the assumption that the utihty of lower-order Moller-Plesset perturbation theory can be inferred from the behaviour of the higher-order terms in the perturbation series. It is widely appreciated that MoUer-Plesset perturbation theory and the equivalent many-body perturbation theory for a single determinantal reference function are not as robust as, for example, configuration interaction. Some care must therefore be exercised in applications to ensure that an appropriate reference function is employed. 

Moller-Plesset many-body perturbation theory taken through second order in the energy is the most commonly used ab initio molecular electronic structure method in contemporary quantum chemistry. For this report on many-body perturbation theory and its application to the molecular electronic structure problem we restricted our survey of applications to second-order Moller-Plesset perturbation theory. Even with this restriction, the nmnber of pubhcations appearing in the period covered by our review - namely, June 1999 to May 2001 - is sizeable. We recorded in the introduction that 883 publications containing the string MP2 in the title or keywords appeared in the year 2000 alone. However, rather than review just a small subset of these publications we decided to try to convey the... 

Figrire 8 An overview of quantum chemical methods for excited states. Bold-italic entries indicate methods that are currently applicable to large molecules. Important abbreviations used Cl (configuration Interaction), TD (time-dependent), CC (coupled-cluster), HF (Hartree-Fock), CAS (complete active space), RAS (restricted active space), MP (Moller-Plesset perturbation theory), S (singles excitation), SD (singles and doubles excitation), MR (multireference). Geometry optimizations of excited states for larger molecules are now possible with CIS, CASSCF, CC2, and TDDFT methods. 

Computational methods such as density functional theory (DFT) [1-3], Moller-Plesset perturbation theory (MP) [4, 5], and coupled-cluster (CC) theory [6-8] are common computational methods used to calculate ground state properties with quantum chemistry. These methods are black-box in the sense that the system can be analysed solely in terms of giving an input structure and a level of theory. When a problem is formulated, the choice of which method to choose comes down to a balance between levels of accuracy required versus computational expense. Even for ground state problems black-box methods may not be applicable in all cases. For example, in cases of near or actual degeneracy then any method based on a single determinant as a reference will be invalid. Such things are far more common when transition metals are involved. 

Many-body methods, based on the linked-cluster expansion (LCE), were first developed by Brueckner [1] and Goldstone [2] in the 1950s for nuclear physics problems. Perturbation-theory applications to atomic and molecular systems (in a numerical, one-center frame) were pioneered by Kelly [3] in the early 1960s. Basis sets were later introduced, first in second-order [4] and then in third-order [5]. The 1970s saw a proliferation of molecular applications with basis sets, under the names of many-body perturbation theory (MBPT) [6] or the Moller-Plesset method [7]. Nowadays, many-body methods offer some of the most powerful tools in the quantum chemistry arsenal, in particular the coupled-cluster (CC) method, and are available in many widely used quantum chemistry program packages. 

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. 

In this chapter we employ a special version of the second-order Moller-Plesset [13] perturbation theory (MP2/CA) which is especially useful in applications to closed-shell atoms. In this approach the first-order wave function consists of spinor-bitals and symmetry-adapted pair functions (SAFE). A detailed presentation of the MP2/CA method can be found in Ref. [14]. For sake of describing the nomenclature used, we repeat here the basic equations. 

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.]. 

Two relevant topics have been ignored completely in this short chapter the treatment of electron correlation with more sophisticated methods than DFT (that remains unsatisfactory from many points of view) and the related subject of excited states. Wave function-based methods for the calculation of electron correlation, like the perturbative Moller-Plesset (MP) expansion or the coupled cluster approximation, have registered an impressive advancement in the molecular context. The computational cost increases with the molecular size (as the fifth power in the most favorable cases), especially for molecules with low symmetry. That increase was the main disadvantage of these electron correlation methods, and it limited their application to tiny molecules. This scaling problem has been improved dramatically by modern reformulation of the theory by localized molecular orbitals, and now a much more favorable scaling is possible with the appropriate approximations. Linear scaling with such low prefactors has been achieved with MP schemes that the... 

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