Perturbation theory, many-body

The idea in perturbation methods is that the problem at hand only differs slightly from a 3aoblem which has already been solved (exactly or approximately). The solution to the given problem should therefore in some sense be close to the solution of the already known system. This is described mathematically by defining a Hamilton operator which Consists of two part, a reference (Hn) and a perturbation The premise of 

Let us assume that the Sclu odinger equation for tlie reference Hamilton operator is solved. 

The solutions for the unperturhed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The penurbed Schrodinger equation is 

If A = 0, then H = Ho, S = o md W = Eq. As the perturbation is increased from zero to a finite value, the new energy and wave function must also change continuously, and they can be written as a Taylor expansion in powers of the perturbation parameter A. 

Once all the correction terms have been calculated, it is trivial to normalize the total 

It has been shown in the previous section that the exact total energy for a many-body system scales linearly with the number of electrons in the system. It has also been shown that a suitably chosen independent particle model leads to an approximation to the energy of a many-body system, which also scales linearly with the number of electrons in the system. 

In this section, we turn our attention to the development of approximations which are more accurate than the independent particle model and can take account of electron correlation effects. The many-body perturbation theory plays a pivotal role in the development of approximate treatments of correlation effects which scale linearly with the number of electrons. Indeed, many-body perturbation theory provides the foundation upon which almost all modern theories of electron correlation in molecules are constructed. 

A key feature of the many-body perturbation theory is the use of the method of second-quantization. We therefore open this section by introducing the second quantization formalism. We then discuss the Rayleigh-Schrddinger perturbation theory in its many-body form, that is, many-body Rayleigh-Schrodinger perturbation theory. We close this section by presenting the many-body perturbation theory with an emphasis on its diagrammatic formulation. 

Some general remarks on the perturbation theory and the many-body approach are now in order. Let us review first the essence of the nondegenerate Rayleigh-Schrodinger perturbation theory. Consider the time-independent Schrodinger equation for the ground state  

Finding solutions to this equation is, in most cases, a difficult task. Assume, however, that the Hamiltonian consists of two Hermitian parts  

Accordingly, the complete set of zeroth-order eigenfunctions is assumed to be known. The coefficient at the zeroth-order ground state in Eq. (12.4) is often chosen as Co = 1 which can be achieved by an appropriate renormalization of F. By this choice, Eq. (12.4) becomes  

This is the so-called intermediate normalization which can also be expressed as  

The validity of Eq. (12.6) can be seen from Eq. (12.5) by multiplying with from the right, integrating, and assuming the orthonormality of the zeroth-order 

In this section we examine the fundamental relationship between many-body perturbation theory (MBPT) and coupled cluster theory. As originally pointed out by Bartlett, this connection allows one to construct finite-order perturbation theory energies and wavefunctions via iterations of the coupled cluster equations. The essential aspects of MBPT have been discussed in Volume 5 of Reviews in Computational Chemistry,as well as in numerous other texts. We therefore only summarize the main points of MBPT and focus on its intimate link to coupled cluster theory, as well as how MBPT can be used to construct energy corrections for higher order cluster operators such as the popular (T) correction for connected triple excitations. 

The disconnected diagram of Eq. [193] is not unlinked because the inclusion of an additional fragment can connect its two components—Harris et al. (Ref. 80) have recommended that such terms be called linkable. With terms such as disconnected, connected, linked, and unlinked used to describe diagrams, it is not surprising that these techniques caused much confusion in the past. 

In this chapter, after a brief introduction to MBPT and Hedin s GW approximation, we will summarise some peculiar aspects of the Kohn-Sham xc energy functional, showing that some of them can be illuminated using MBPT. Then, we will discuss how to obtain ground-state total energies from GW. Finally, we will present a way to combine techniques from many-body and density functional theories within a generalised version of Kohn-Sham (KS) DFT. 

Our discussion focuses on the concepts from MBPT that will be useful in this chapter. We will also present a short overview of some current problems in ab-initio calculations of quasiparticle properties. We refer the reader to [3,4,5,6,7] and the review articles [9,10,11,12,13] for further information on theoretical foundations and applications to solid-state physics, respectively. 

For A = 0, it is seen that Fo = ho and Wo = Eq, and this is the unperturbed, or zeroth-order wave function and energy. The Ti, T2. and Wi, W2. are the first-order, second-order, etc., corrections. Tlie A parameter will eventually be set equal to 1, and the nth-order energy or wave function becomes a sum of all terms up to order n. 

Because the last term is too complex to handle directly, the Hamiltonian is broken up into two parts, H = Hq + with 

Magnetic interactions between electrons (the Breit interaction) can be treated perturbatively their effect on PNC is treated in Section 4.5.1. While in the previous section we chose U(r) to be one of a number of local potentials, for PNC we instead choose the Hartree-Fock potential, defined 

It is now trivial to solve HqtPq = Eotpo in terms of a Slater determinant of the occupied orbitals. These orbitals satisfy the Dirac equation 

The model of the atom provided by lowest order perturbation theory is rather inaccurate when the HF potential is used valence removal energies disagree with experiment by on the order of 10%, and matrix elements of the hyperfine operator by about 50%. Thus it is essential for accurate calculations to include the effects of Vc as fully as possible. MBPT proceeds by expanding the many-body wave function F(u) and the energy E v) in powers of Vc, 

The lowest-order wave function, which is an eigenfunction of Ho, is given 

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Wormer P E S and Hettema H 1992 Many-body perturbation theory of frequency-dependent... 

Bartlett R J and Silver D M 1975 Many-body perturbation theory applied to eleetron pair eorrelation energies I. Closed-shell first-row diatomie hydrides J. Chem. Rhys. 62 3258-68... 

Bartlett R J and Purvis G D 1978 Many-body perturbation theory coupled-pair... 

Bartlett R J and Purvis G D 1978 Many-body perturbation theory coupled-pair many-electron theory and the importance of quadruple excitations for the correlation problem int. J. Quantum Chem. 14 561-81... 

Highest occupied molecular orbital Intermediate neglect of differential overlap Linear combination of atomic orbitals Local density approximation Local spin density functional theory Lowest unoccupied molecular orbital Many-body perturbation theory Modified INDO version 3 Modified neglect of diatomic overlap Molecular orbital Moller-Plesset... 

Ei=i N F(i), perturbation theory (see Appendix D for an introduetion to time-independent perturbation theory) is used to determine the Ci amplitudes for the CSFs. The MPPT proeedure is also referred to as the many-body perturbation theory (MBPT) method. The two names arose beeause two different sehools of physies and ehemistry developed them for somewhat different applieations. Later, workers realized that they were identieal in their working equations when the UHF H is employed as the unperturbed Hamiltonian. In this text, we will therefore refer to this approaeh as MPPT/MBPT. 

Rayleigh-Schrodinger many-body perturbation theory — RSPT). In this approach, the total Hamiltonian of the system is divided or partitioned into two parts a zeroth-order part, Hq (which has... 

Another approach to electron correlation is Moller-Plesset perturbation theory. Qualitatively, Moller-Plesset perturbation theory adds higher excitations to Hartree-Fock theory as a non-iterative correction, drawing upon techniques from the area of mathematical physics known as many body perturbation theory. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

Hubbard, J., Proc. Roy. Soc. London) A240, 539, The description of collective motions in terms of many-body perturbation theory. ... 

Pipek J, Bogar F (1999) Many-Body Perturbation Theory with Localized Orbitals - Kapuy s Approach. 203 43-61... 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

I wish to stress that the meaning of the word Hole here is different and far more general than in Many Body Perturbation Theory. Indeed, no specific reference state is required in this definition and the difference between the RO s and the HRO"s follows exclusively from the different order of the creator operators with respect to the annihilator operators in E and in E respectively. 

In ab initio methods (which, by definiton, should not contain empirical parameters), the dynamic correlation energy must be recovered by a true extension of the (single configuration or small Cl) model. This can be done by using a very large basis of configurations, but there are more economical methods based on many-body perturbation theory which allow one to circumvent the expensive (and often impracticable) large variational Cl calculation. Due to their importance in calculations of polyene radical ion excited states, these will be briefly described in Section 4. 

Due to the size of the variational problem, a large Cl is usually not a practicable method for recovering dynamic correlation. Instead, one usually resorts to some form of treatment based on many-body perturbation theory where an explicit calculation of all off-diagonal Cl matrix elements (and the diagonalization of the matrix) are avoided. For a detailed description of such methods, which is beyond the scope of this review, the reader is referred to appropriate textbooks295. For the present purpose, it suffices to mention two important aspects. 

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