Concept of Effective Hamiltonians

Often one wants to solve an operator eigenvalue equation in two steps. Rather than in transforming the matrix representation H of the Hamiltonian H in an orthonormal basis by a unitary transformation 

Effective Hamiltonians play a key role in quasidegenerate perturbation theory. If an eigenvalue is degenerate or near-degenerate, the standard perturbation cannot be applied, because zero or small energy denominators would arise. However the first block diagonalization can often be done by perturbation theory, the final diagonalization of Hg/f must be done nonperturbatively. 

Often one starts from a given approximate (model) Hamiltonian Hq, the eigenfunctions p of which are known, and expands H in this basis. The model space is then spanned by some eigenfunctions of Hq. 

The transformation W need not be unitary, a non-unitary similarity transformation 

The block Lpp is an effective Hamiltonian in either case. It is recommended to multiply (579) from the left by W  

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To follow the scale of complexity, the review is divided into three parts. The first two parts deal with the key concept of effective Hamiltonians which describe the dynamical and spectroscopic properties of interfering resonances (Section 2) and resonant scattering (Section 3). The third part. Section 4, is devoted to the resolution of the Liouville equation and to the introduction of the concept of effective Liouvillian which generalizes the concept of effective Hamiltonian. The link between the theory of quantum resonances and statistical physics and thermodynamics is thus established. Throughout this work we have tried to keep a balance between the theory and the examples based on simple solvable models. 

We outline briefly in this section how to link the theory of quantum resonances to statistical physics and thermodynamics by extending the concept of effective Hamiltonian as recently discussed in Ref. [60]. The quantum Liouville-von Neumann equation is written in the form... 

The concept of model space plays a central role in the theory of effective Hamiltonians. It is a finite Af -dimensional subspace S of the entire Hilbert... 

First one can build up other effective Hamiltonians based on hierarchized orthogonalization procedures. The Gram-Schmidt procedure is recommended if one starts from the best projected wavefunctions of the bottom of the spectrum. Thus one can obtain a quite reliable effective Hamiltonian with well behaved wavefunctions and good transferability properties (see Section III.D.2). The main drawback of this approach is that the Gram-Schmidt method, which involves triangular matrices, does not lead to simple analytical expressions for perturbation expansions. A partial solution to these limitations is brought about by the new concept of intermediate Hamiltonian,... 

We suppose that the effective Hamiltonian is known. Let us first recall how it is directly related to the spectroscopical and dynamical observables [24,25]. Since the role of effective Hamiltonians in both line profiles and dynamics is already well documented the reader is referred to some review on this wide subject (see, e.g.. Refs. [16-18,26]). The reports [18] and [26] contain numerous references inside and outside chemical physics. Reference [17] is a review of time-dependent effective Hamiltonians. Earlier application can be found in references [27] and [28]. Memory kernels are discussed in references [14,15,18] with references to irreversible statistical mechanics. Here we briefly review the subject for introducing the basic concepts and the notations. In the second part of this section we will present corrections to the dynamics for taking into account the dependence on energy of the effective Hamiltonian. 

Abstract In this chapter we examine some basic concepts of quantum chemistry to give a solid foundation for the other chapters. We do not pretend to review all the basics of quantum mechanics but rather focus on some specific topics that are central in the theoretical description of magnetic phenomena in molecules and extended systems. First, we will shortly review the Slater-Condon rules for the matrix elements between Slater determinants, then we will extensively discuss the generation of spin functions. Perturbation theory and effective Hamiltonians are fundamental tools for understanding and to capture the complex physics of open shell systems in simpler concepts. Therefore, the last three sections of this introductory chapter are dedicated to standard Rayleigh-Schrddinger perturbation theory, quasi-degenerate perturbation theory and the construction of effective Hamiltonians. 

The early attempts to develop MR perturbation methods focused on the use of an effective Hamiltonian determined from the Bloch equation [7, 18, 24, 27]. The main drawback of these theories is the intrader state problem, i.e., the appearance of close to zero denominators which give nonphysical contribution to the effective Hamiltonian matrix elements, especially for large CAS spaces where the high lying model functions are energetically not separated from the outer space determinants. To tackle with this problem the application of incomplete model spaces [18, 27], various level shift-based techniques [14, 23, 30, 45] and the concept of intermediate Hamiltonian [26] were intensively studied, but the most common solution is to use a state-specific description, where only a single target state is described [3-5, 8,16,17, 29,44]. 

In the early years of quantum theory, Hiickel developed a remarkably simple form of MO theory that retains great influence on the concepts of organic chemistry to this day. The Hiickel molecular orbital (HMO) picture for a planar conjugated pi network is based on the assumption of a minimal basis of orthonormal p-type AOs pr and an effective pi-Hamiltonian h(ctT) with matrix elements... 

Before investigating the qualitative concepts of the VSEPR model it is worth noting that the details of the interactions between the electron pairs have been ascribed to a size-Pauli exclusion principle result . But objects do not repel each other simply because of their sizes (i.e. interpenetrations) only if the constituents of the objects interact is any interaction possible10). If we are to use the idea of orbital size at all we must avoid the danger of contrasting a phenomenon (electron repulsion) with one of its manifestations (steric effects). The only quantitative tests which we can apply to the VSEPR model are ones based on the terms in the molecular Hamiltonian specifically, electron repulsion. 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

Prior to considering semiempirical methods designed on the basis of HF theory, it is instructive to revisit one-electron effective Hamiltonian methods like the Huckel model described in Section 4.4. Such models tend to involve the most drastic approximations, but as a result their rationale is tied closely to experimental concepts and they tend to be inmitive. One such model that continues to see extensive use today is the so-called extended Huckel theory (EHT). Recall that the key step in finding the MOs for an effective Hamiltonian is the formation of the secular determinant for the secular equation... 

The examination takes place in two stages, one corresponding to the formal interelectronic repulsion component of the Hamiltonian HER and the second to the spin-orbit coupling term Hes. As will be pointed out, in principle, and in certain cases in practice, it is not proper to separate the two components. However, the conventional procedure is to develop HLS as a perturbation following the application of Her. That suffices for most purposes, and simplifies the procedures. Any interaction between the d- or/-electron set and any other set is ignored. It is assumed that it is negligible or can be taken up within the concept of an effective d-orbital set. 

Obviously the electrons most affected in the process of formation of the condensed state are the valence electrons and in the lighter elements the primary physical effects of the remaining (core) electrons is often included through the concept of a pseudo potential, generally non-local. With this understanding Hamiltonian (1) is then modified to reproduce simply the valence electron spectrum. To within density dependent constants (1) is therefore replaced by... 

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form ... 

Formally it applies to any atom, but it is nontrivial only for the frontier ones. The condition which specifies the distribution of the core charge ZA between the R- and M-systems is that the cores of the R-system must be as much as possible screened by the electrons of the R-system i.e. the effective Hamiltonian must be as close as possible to the Hamiltonian of the free M-system H°M. This reduces to the electron counting rules based on the concept of the formal oxidation state (see [60] for details). With this we arrive at the possibility of distributing not only the electronic density, but also the total effective charges between the R- and M-systems. This is done by the formulae ... 

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