Perturbed many-body wave function

Now we should specify the response Hamiltonian hres r, t). For this aim, we use the scaling transformation and define the perturbed many-body wave function of the system as... 

Collective modes can be viewed as superpositions of Iph configurations. It is convenient to define this relation by using the Thouless theorem which establishes the connection between two arbitrary Slater determinants [25]. Then, the perturbed many-body wave function reads... 

SRPA equations are very general and can be applied to diverse systems (atomic nuclei, atomic clusters, etc.) described by density and current-dependent functionals. Even Bose systems can be covered if to redefine the many-body wave function (25) exhibiting the perturbation through the elementary excitations. In this case, the Slater deterninant for Iph excitations should be replaced by a perturbed many-body function in terms of elementary bosonic excitations. 

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,... 

Density-functional theory (DFT) is one of the most widely used quantum mechanical approaches for calculating the structure and properties of matter on an atomic scale. It is nowadays routinely applied for calculating physical and chemical properties of molecules that are too large to be treatable by wave-function-based methods. The problem of determining the many-body wave function of a real system rapidly becomes prohibitively complex. Methods such as configuration interaction (Cl) expansions, coupled cluster (CC) techniques or Moller Plesset (MP) perturbation theory thus become harder and harder to apply. Computational complexity here is related to questions such as how many atoms there are in the molecule, how many electrons each atom contributes, how many basis functions are... 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

The multiphonon problem involves a complex many-body (both many-ion and many-electron) Hamiltonian and, in order to induce transitions, some perturbation. Essentially, headway on this problem has been possible only by (1) approximating the unperturbed Hamiltonian, (2) assuming some sensible wave functions for this unperturbed Hamiltonian, and (3) assuming some perturbation. 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

Another possibility to improve the Martree-Fock wave function is to estimate electron correlation effects by many-body perturbation theory. The division of the... 

A general approach to the intramonomer correlation problem is known as the many-electron (or many-body) SAPT method88,141 213-215. In this method the zeroth-order Hamiltonian H0 is decomposed as H0 = F + W, where F = FA + FB is the sum of the Fock operators, FA and FB, of monomer A and B, respectively, and W is the intramonomer correlation operator. The correlation operator can be written as W = WA + WB, where Wx = Hx — Fx, X = A or B. The total Hamiltonian can be now be represented as H = F + V + W. This partitioning of H defines a double perturbation expansion of the wave function and interaction energy. In the SRS theory the wave function is obtained by expanding the parametrized Schrodinger equation as a power series in and A,... 

In the section that follows this introduction, the fundamentals of the quantum mechanics of molecules are presented first that is, the localized side of Fig. 1.1 is examined, basing the discussion on that of Levine (1983), a standard quantum-chemistry text. Details of the calculation of molecular wave functions using the standard Hartree-Fock methods are then discussed, drawing upon Schaefer (1972), Szabo and Ostlund (1989), and Hehre et al. (1986), particularly in the discussion of the agreement between calculated versus experimental properties as a function of the size of the expansion basis set. Improvements on the Hartree-Fock wave function using configuration-interaction (Cl) or many-body perturbation theory (MBPT), evaluation of properties from Hartree-Fock wave functions, and approximate Hartree-Fock methods are then discussed. 

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. 

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