Many-body Hamiltonians methods

DFT is the modern alternative to the wave-function based ab initio methods and allows to obtain accurate results at low computational cost, that also helps to understand the chemical origin of the effect. DFT, like Hartree-Fock (HF) methods, exploit molecular symmetry which is crucial in the case of computational studies of the JT effect. It also includes correlation effects into the Hamiltonian via the exchange-correlation functional. HF and many-body perturbation methods are found to perform poorly in the analysis of JT systems for obvious reasons, at contrast to the methods based on DFT, or multiconfigurational SCF and coupled cluster based methods [73]. The later are very accurate but have some drawbacks, mainly the very high computational cost that limits the applications to the smaller systems only. Another drawback is the choice of the active space which involves arbitrariness. 

Methods for Solving Model Many-Body Hamiltonians... 

For our purposes, there are three important aspects of the density functional method. First, it is in principle exact and provides an alternative to the direct treatment of the full many-body Hamiltonian discussed above. It is therefore relevant to establish rigorous expressions for other physical quantities, such as the stress, in the density functional formalism. Second, for any functional, variational solutions of the equations satisfy all the properties required to derive the requisite theorems for force, stress, and other derivatives. Third, there are local approximations to the exact... 

The HF equations solve the exact Hamiltonian (i.e., the molecular Hamiltonian) with approximate many-body wavefunctions (i.e.. Slater determinants). As discussed earlier, HF-based methods converge to the exact solution through systematic improvements in the form of many-body wavefunctions such as Cl wavefunctions. Approximations in DF theory are introduced only in the exchange-correlation operator xcCf)- Thus, the DF equations solve an approximate many-body Hamiltonian with exact wavefunctions. DF theory approaches the exact solution... 

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. 

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. 

The mean-field approximation has been extensively applied in many-body physics. Its application to molecular algebraic Hamiltonians and the connection with the coherent-states expectation method was begun by van Roosmalen (1982). See also, van Roosmalen and Dieperink (1982), and van Roosmalen, Levine, and Dieperink (1983). For applications in the geometrical context see Bowman (1986) and Gerber and Ratner (1988). 

A. Landan, E. Ehav, and U. Kaldor, Intermediate Hamiltonian Fock-Space Coupled-Cluster Method and Applications. In R. F. Bishop, T. Brandes, K. A. Gernoth, N. R. Walet, and Y. Xian (Eds.) Recent Progress in Many-Body Theories, Advances in Quantum Many-Body Theories, Vol. 6. (World Scientific, Singapore, 2002), pp. 355-364 and references therein. 

Much interest has developed on approximate techniques of solving quantum mechanical problems because exact solutions of the Schrodinger equation can not be obtained for many-body problems. One of the most convenient of such approximations for the solution of many-body problems is the application of the variational method. For instance, with approximate eigen-functions p , the eigen-values of the Hamiltonian H are En... 

Two methods appear to be very powerful for the study of critical phenomena field theory as a description of many-body systems, and cell methods grouping together sets of neighboring sites and describing them by an effective Hamiltonian. Both methods are based on the old idea that the relevant scale of critical phenomena is much larger than the interatomic distance and this leads to the notion of scale invariance and to the statistical applications of the renormalization group technique.93... 

As with the solution of other many-body electronic structure problems, determination of the unperturbed eigenvalues is numerically challenging and involves compromises in the following areas (1) approximations to the hamiltonian to simplify the problem (e.g., use of semi-empirical molecular orbital methods) (2) use of incomplete basis sets (3) neglect of highly excited states (4) neglect of screening effects due to other molecules in the condensed phase. 

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