MBPT

Although the MBPT and RPAE approaches differ, they suffer from the same basic difficulty in that neither method is complete. To some extent, the methods complement each other. In practice, each one has different advantages for different situations. 

The problem of N identical fermions interacting with each other through a two-body potential V j and with a centre of charge through a one-body potential Ti leads to the Hamiltonian 

In order to define the basis states, the interaction potential V, - is, as usual in atomic physics, approximated by a sum over individual potentials Vi, where Vi must be Hermitian, but can be chosen arbitrarily. We can thus define an unperturbed or independent particle Hamiltonian 

The unperturbed wavefunction of the ground state is 4 o- It is made up from a determinant containing the N single particle orbitals 4 n which are solutions of the equation 

Those of 4 n which are occupied in 4 o are called unexcited states, and those of (f n which are empty are called excited states. Unoccupied unexcited states are referred to as holes and occupied excited states are referred to as particles the words hole and particle, being part of the language of many-body theory, describe properties of the complete system rather than those of actual holes and particles. 

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In the MPPT/MBPT method, once the reference CSF is chosen and the SCF orbitals belonging to this CSF are detennined, the wavefiinction T and energy E are detennined in an order-by-order maimer. The perturbation equations determine what CSFs to include and their particular order. This is one of the primary strengdis of this technique it does not require one to make fiirtlier choices, in contrast to the MCSCF and Cl treatments where one needs to choose which CSFs to include. 

As Bartlett [ ] and Pople have both demonstrated [M], there is a close relationship between the MPPT/MBPT and CC methods when the CC equations are solved iteratively starting with such an MPPT/MBPT-like initial guess for these double-excitation amplitudes. 

These approaches provide alternatives to the conventional tools of quantum chemistry. The Cl, MCSCF, MPPT/MBPT, and CC methods move beyond the single-configuration picture by adding to the wavefimction more configurations whose amplitudes they each detennine in their own way. This can lead to a very large number of CSFs in the correlated wavefimction and, as a result, a need for extraordinary computer resources. 

The essential features of the MPPT/MBPT approaeh are deseribed in the following artieles ... 

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. 

The amplitude for the so-ealled referenee CSF used in the SCF proeess is taken as unity and the other CSFs amplitudes are determined, relative to this one, by Rayleigh-Sehrodinger perturbation theory using the full N-eleetron Hamiltonian minus the sum of Foek operators H-H as the perturbation. The Slater-Condon rules are used for evaluating matrix elements of (H-H ) among these CSFs. The essential features of the MPPT/MBPT approaeh are deseribed in the following artieles J. A. Pople, R. Krishnan, H. B. Sehlegel, and J. S. Binkley, Int. J. Quantum Chem. 14, 545 (1978) R. J. Bartlett and D. M. Silver, J. Chem. Phys. 3258 (1975) R. Krishnan and J. A. Pople, Int. J. Quantum Chem. 

B. Non-Variational Methods Sueh as MPPT/MBPT and CC do not Produee Upper Bounds, but Yield Size-Extensive Energies... 

P I H I P >. It ean be shown (H. P. Kelly, Phys. Rev. 131, 684 (1963)) that this differenee allows non-variational teehniques to yield size-extensive energies. This ean be seen in the MPPT/MBPT ease by eonsidering the energy of two non-interaeting Be atoms. The referenee CSF is = Isa 2sa Isb 2sb the Slater-Condon rules limit the CSFs in P whieh ean eontribute to... 

The additivity of E and the separability of the equations determining the Cj eoeffieients make the MPPT/MBPT energy size-extensive. This property ean also be demonstrated for the Coupled-Cluster energy (see the referenees given above in Chapter 19.1.4). However, size-extensive methods have at least one serious weakness their energies do not provide upper bounds to the true energies of the system (beeause their energy funetional is not of the expeetation-value form for whieh the upper bound property has been proven). 

The implementation of the CC method begins mueh as in the MPPT/MBPT ease one seleets a referenee CSF that is used in the SCF proeess to generate a set of spin-orbitals to be used in the subsequent eorrelated ealeulation. The set of working equations of the CC teehnique given above in Chapter 19.1.4 ean be written explieitly by introdueing the form of the so-ealled eluster operator T,... 

The CC method, as presented here, suffers from the same drawbaeks as the MPPT/MBPT approaeh its energy is not an upper bound and it may not be able to aeeurately deseribe waveflinetions whieh have two or more CSFs with approximately equal amplitude. Moreover, solution of the non-linear CC equations may be diffieult and slowly (if at all) eonvergent. It has the same advantages as the MPPT/MBPT method its energy is... 

Within the CC, Cl, and MPPT/MBPT methods, one must evaluate the so-ealled responses of the Ci and Ca,i eoeffieients (3Cj/3)i)o and (3Ca,i/3/i)o that appear in the full energy response as (see above)... 

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). 

Just as single reference Cl can be extended to MRCI, it is also possible to use perturbation methods with a multi-detenninant reference wave function. Formulating MR-MBPT methods, however, is not straightforward. The main problem here is similar to that of ROMP methods, the choice of the unperturbed Hamilton operator. Several different choices are possible, which will give different answers when the tlieory is carried out only to low order. Nevertheless, there are now several different implementations of MP2 type expansions based on a CASSCF reference, denoted CASMP2 or CASPT2. Experience of their performance is still somewhat limited. 

The exaet definition is slightly more eomplieated, sinee the wave funetion has to be properly antisymmetrized and projected onto the acmal basis, but for illustration the above form is sufficient. Such R12 wave funetions may then be used in eonnection with the Cl, MBPT or CC methods described above. Consider for example a Cl calculation with an R12 type wave funetion. The energy is given as... 

Approximating a many-electron wave function by a finite sum of Slater determinants, e.g. truncating the Cl, CC or MBPT wave function to include only certain excitation types. 

SCF + MBPT + ZPE calculations [92THE(277)313] and MP2/6-31+G calculations [97JPC(B)9199] of the three possible dimers (oxo-oxo, oxo-hydroxy, and hydroxy-hydroxy). 

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. 

It is seen that neither the MBPT nor the Cl approaches are the panacea. 

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