Effective Hamiltonian, concept

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

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... 

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 ... 

It is important to note that the Hamiltonian (2.120) contains the terms which produce both the adiabatic and non-adiabatic effects. In chapter 7 we shall show how the total Hamiltonian can be reduced to an effective Hamiltonian which operates only in the rotational subspace of a single vibronic state, the non-adiabatic effects being treated by perturbation theory and incorporated into the molecular parameters which define the effective Hamiltonian. Almost for the first time in this book, this introduces an extremely important concept and tool, outlined in chapter 1, the effective Hamiltonian. Observed spectra are analysed in terms of an appropriate effective Hamiltonian, and this process leads to the determination of the values of what are best called molecular parameters . An alternative terminology of molecular constants , often used, seems less appropriate. The quantitative interpretation of the molecular parameters is the link between experiment and electronic structure. 

Kutzelnigg/76/ and Kutzelnigg and Koch/77/ emphasized that the classification of the operators into the categories C, A, B and 0 is essentially a Fock space concept, so that the condition that L, vanishes will automativally generate operator equivalent of the eq. (7.1.3). L, is then the Fock-space effective hamiltonian H, and appropriate... 

The effective Hamiltonian approach clearly shows the important role of intramolecular energy flow in the quantum dynamics of unimolecular dissociation. It suggests that unless intramolecular energy flow is dominantly rapid, there exist two drastically different time scales in the reaction dynamics. This is consistent with the classical concept that nonstatistical behavior in intramolecular energy flow, such as bottleneck effects, can dramatically alter the kinetics of unimolecular reaction. 

For very small field amplitudes, the multiphoton resonances can be treated by time-dependent perturbation theory combined with the rotating wave approximation (RWA) [10]. In a strong field, all types of resonances can be treated by the concept of the rotating wave transformation, combined with an additional stationary perturbation theory (such as the KAM techniques explained above). It will allow us to construct an effective Hamiltonian in a subspace spanned by the resonant dressed states, degenerate at zero field. 

Notice that the definitions (17) and (18) do not necessary lead to real values of and jg. This is not critical, however, because, due to the hermiticy of the Hamiltonian, imaginary parts will vanish after integration over r. Also, both variables become real when the classical limit is approached and, in case of significant quanmm effects, the concept of local densities itself becomes inconsistent. Thus, without serious consequences, the imaginary parts of and jg can be excluded from consideration. 

Under the second topic of Ligand-field Theory and its Extensions we describe the basic concepts behind the various versions of LFT - the angular-overlap model (AOM) and its extensions. In the section named The physical background conditions for the applicability of the ligand-field approach we sketch briefly the theoretical foundation and limits of applicability of the effective-hamiltonian approach with special attention to electronic multiplets. In the theory section, we describe various approaches in current calculations of electronic structure, such as LFDFT, SORCI and TDDFT, with the various applications detailed in the following section, before an outlook for further developments. 

The SRTS sequence consists of a preparatory pulse and an arbitrary long train of the phase-coherent RF pulses of the same flip angle applied with a constant short-repetition time. As was noted above, the "short time" in this case should be interpreted as the pulse spacing T within the sequence that meets the condition T T2 Hd. The state that is established in the spin system after the time, T2, is traditionally defined as the "steady-state free precession" (SSFP), ° and includes two other states (or sub-states) quasi-stationary, that exists at times T2effective relaxation time) and stationary, that is established after the time " 3Tie after the start of the sequence.The SSFP is a very particular state which requires a specific mechanism for its description. This mechanism was devised in articles on the basis of the effective field concept and canonical transformations. Later approaches on the basis of the average-Hamiltonian theory were developed. ... 

The power of the coupled equation method is that, in principle, if the electronic interaction parameters are either calculated ab initio or taken as adjustable parameters, the microscopic vibrational Hamiltonian can be used without introducing a finite-basis effective Hamiltonian matrix. Scattering concepts have penetrated into the field of molecular spectroscopy (Mies, 1980). The coupled equations and MQDT methods are both borrowed from scattering theory and are extensively used to treat perturbations and problems of dissociation and ionization (see Chapters 7 and 8). 

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 have shown in the previous sections that an exactly solvable model can provide generic results concerning line profiles and dynamics. The physics was discussed in terms of resonances and effective Hamiltonians. These concepts are also of fundamental importance for real systems. Here we recall the simplest one a hydrogen atom in its ground state exposed to a static electric field described in Refs. [10, 35]. 

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... 

Recently the same problem has been reanalyzed by Dicus et al. [86], and indeed they confirmed that the survival probability deviates from exponential at long times. This model and its variants have been applied to study the effect of a distant detector (by adding an absorptive potential) [87], anomalous decay from a flat initial state [44], resonant state expansions [3], initial state reconstruction (ISR) [58], or the relevance of the non-Hermitian Hamiltonian concept (associated with a projector formalism for internal and external regions of space) in potential scattering [88]. In Ref. [88] the model was extended to a chain of delta functions to study overlapping resonances. 

The particular aspect we shall treat here is the generalization of the basic theory of these methods to QM descriptions of the solute. To do that, one has to introduce the concept of the Effective Hamiltonian (EH) for the isolated system, the fundamental equation to solve is the standard Schrddinger equation... 

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