Body Perturbation Theory

Now let us turn to the many-body perturbation theory treatment of atoms with more than one electron. As discussed in the introduction, our approach is the no-pair Hamiltonian, which is given by 

For simplicity, we write only the Coulomb interaction in (71) however, to obtain correct two-electron fine structure intervals it is also necessary to include the Breit interaction bij, discussed in the following subsection. 

It should be noted that the projection operator A+ and, consequently, the no-pair Hamiltonian depends on the background potential U. One finds however that energies obtained from the no-pair Hamiltonian are only weakly dependent on the potential and that small differences between calculations starting from different potentials can be accounted for in terms of omitted negative-energy corrections. We elaborate on this point in Sec. 4. 

To bypass the complicated issue of constructing A+ in configuration space, we work with the second-quantized version of the no-pair Hamiltonian, 

In this expression, a and a] designate electron annihilation and creation operators, respectively. The projection operators in Eq. (71) are implemented by simply restricting the indices i, j, k, I to bound states and positive-energy continuum states, omitting contributions from negative-energy states entirely. In Eq. (74), the quantity Cj is the eigenvalue 

The previous section considered the derivation of second quantized Hamiltonians that can be used in post-DHF calculations. From now on we will regard the matrix elements of h and g as (complex) numbers and direct the attention to the associated operators. By applying the no-pair approximation we retained only particle conserving operators in the Hamiltonian. Such operators can concisely be expressed using the replacement operators Eq = a p Q and 

The first step is to define the diagonal Fock operator as the zeroth order Hamiltonian. We have Mocc. occupied spinors that generate a mean field potential 

Second order perturbation theory gives then expressions for the first 

The first order correction simply corrects the double counting of electron interactions in the zero order energy expression. The second order correction gives the correlation energy that we are interested in. For convenience we defined a shorthand notation for spinor energy differences 

Simple open shell cases may also be treated via this kind of perturbation theory. The high spin case with one electron outside a closed shell is of course easy when an unrestricted formalism is used. Dyall also worked out equations for the restricted HE formalism and the more complicated case of two electrons in two Kramers pairs outside a closed shell [32]. Also in this method the crucial step remains the efficient formation of two-electron integrals in the molecular spinor basis. 

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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. 

The idea in perturbation methods is that the problem at hand only differs slightly from a problem 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 (Hq) and a perturbation (H )- The premise of perturbation methods is that the H operator in some sense is small compared to Hq. In quantum mechanics, perturbational methods can be used for adding corrections to solutions which employ an independent particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory (MBPT). 

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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