Second-order many-body perturbation approaches

All the above mentioned functionals generally provide atomic or molecular properties with a reasonable precision, so that conventional DF methods are claimed to deliver results comparable to those obtained by second-order many body perturbation approaches (MP2). From a purely formal point of view, the reasons for such good performances are not yet evident. 

The incorporation of correlation effects in calculations for periodic solids requires the use of a many-body formalism. Second order many-body perturbation theory, in its MP2 form, should provide the basis of an efficient computational approach to this problem. In particular, the local MP2 methods originally developed for large molecules can be adapted for the treatment of periodic solids. 

Solution of the matrix equations associated with an independent particle model gives rise to a representation of the spectrum which is an essential ingredient of any correlation treatment. Finite order many-order perturbation theory(82) forms the basis of a method for treating correlation effects which remains tractable even when the large basis sets required to achieve high accuracy are employed. Second-order many-body perturbation theory is a particularly simple and effective approach especially when a direct implementation is employed. The total correlation energy is written... 

The applications of many-body perturbation theory in contemporary research in the molecular sciences are manifold and it is certainly not possible to describe more than a mere fraction of the enormous number of publications which have exploited this approach to the molecular structure problem over recent years. Calculations based on second order many-body perturbation theory or MP2 theory are particularly prevalent offering unique advantages in terms of efficiency and accuracy over many other theoretical and computational approaches. Here, we shall briefly describe the use of graphical user interfaces and then concentrate on two recent applications of the many-body perturbation theory which have established new levels of precision. 

Often, the HF approximation provides an accurate description of the system and the effects of the inclusion of correlations with Cl or MCSCF methods are of secondary importance. In this case, the correlation effects may be considered as a smaller perturbation and as such treated using the perturbation theory. This is the approach of Moller-Plesset [104] or many-body perturbation theory for the inclusion of correlation effects. In the MP2 approximation only the second-order many-body perturbations are taken into account. 

The most widely used ab initio approach to the correlation energy in molecular species is the mp2 theory - second-order many-body perturbation theory. This approach is available in many quantum chemistry packages, such as the well-known Gaussian suite of computer programs [50] for which the British theoretical chemist Professor Sir John Pople frs was awarded a half share of the 1998 Nobel Prize in Chemistry. mp2 is essentially a robust, black-box method which is today the most... 

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

The original ab initio approach to calculating electronic properties of molecules was the Hartree-Fock method [31,32,33,34]. Its appeal is that it preserves the concept of atomic orbitals, one-electron functions, describing the movement of the electron in the mean field of all other electrons. Although there are some inherent deficiencies in the method, especially those referred to the absence of correlation effects. Improvements have included the introduction of many-body perturbation theory by Mollet and Plesset (MP) [35] (MP2 to second-order MP4 to fourth order). The computer power required for Hartree-Fock methods makes their use prohibitive for molecules containing more than very few atoms. 

In order to have a more complete picture of the many-body problem for more general or complicated cases that DFT could help to treat, it is necessary to make a correspondence with the use of many-body perturbation theory. Under this wider classification of perturbation theory are included all the methods that treat electron correlation beyond the Hartree-Fock level, including configuration interaction, coupled cluster, etc. This perturbational approach has traditionally been known as second quantization, and its power for some applications can be seen when dealing with problems beyond the standard quantum mechanics. 

A many-body perturbation theory (MBPT) approach has been combined with the polarizable continuum model (PCM) of the electrostatic solvation. The first approximation called by authors the perturbation theory at energy level (PTE) consists of the solution of the PCM problem at the Hartree-Fock level to find the solvent reaction potential and the wavefunction for the calculation of the MBPT correction to the energy. In the second approximation, called the perturbation theory at the density matrix level only (PTD), the calculation of the reaction potential and electrostatic free energy is based on the MBPT corrected wavefunction for the isolated molecule. At the next approximation (perturbation theory at the energy and density matrix level, PTED), both the energy and the wave function are solvent reaction field and MBPT corrected. The self-consistent reaction field model has been also applied within the complete active space self-consistent field (CAS SCF) theory and the eomplete aetive space second-order perturbation theory. ... 

Moller-Plesset Perturbation Theory. - Many-body perturbation theory with a Moller-Plesset reference hamiltonian is the most widely used approach to the correlation problem in atomic and molecular systems. Second-order theory, which is often designated MP2 and which was the order of theory originally presented by Moller and Plesset, is computationally efficient and facilitates the use of very large basis sets which allows basis set truncation errors to be reduced to a level where other effects, such as relativity, are often more significant... 

Many body perturbation theory is one of the fundamental tools in Quantum Chemistry. It takes a central place both in the calculation of accurate energies and wave functions, and in the analysis of results for reaching a better understanding of the sometimes complicated physics contained in the system. There are basically two flavors of many-body perturbation theory. The first is what one calls the diagonalize-and-then-perturb method, and the second one inverts this order, it follows a perturb-and-then-diagonalize approach. 

The first approach is Moller-Plesset (MP) many-body perturbation theory. To the Hartree-Fock wavefunction is added a correction corresponding to exciting two electrons to higher energy Hartree-Fock MOs. Second-order, third-order, and fourth-order corrections to the Hartree-Fock total energy are designated MP2, MP3, and MP4, respectively. For double substitutions, i,j (occupied) into m,n (virtual),... 

It is known that within the framework of the Roothaan coupling operator approach, there is no unique way of choosing a reference Hamiltonian, with respect to which a perturbation expansion for correlation effects can be developed. Several proposals have been made for open-shell many-body perturbation theory expansions based on a reference from the ROHE formalism [41, 42] or the umestricted Hartree-Fock formalism [43, 44]. We follow our papers [10, 45] where an alternative technique for the open-shell systems has been developed. In our method, the second-order correction to the ground state energy can be presented by [45] ... 

Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141 134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented. 

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