Construction of a model Hamiltonian

An important simplification can be made at this point by noting that the amplitude q of a molecular vibration is usually much smaller than intermolecular and intramolecular distances. Therefore the interaction potential may be expanded according to 

The resulting system-bath Hamiltonian can therefore be written in the form 

The bath consists of all solvent atoms and the spherical impurity with its internal motion frozen at q = 0, 

These normal modes evolve independently of each other. Their classical equations of motion are Uk = —co Uk, whose general solution is given by Eqs (6.81). This bath is assumed to remain in thermal equilibrimn at all times, implying the phase space probability distribution (6.77), the thermal averages (6.78), and equilibrium time correlation functions such as (6.82). The quantum analogs of these relationships were discussed in Section 6.5.3. 

Equations (13.6), (13.7), (13.9), and (13.10) provide the general stracture of our model. The relaxation process depends on details of this model mainly through the form and the magnitude of T/sb- We will consider in particular two fonns for this interaction. One, which leads to an exactly soluble model, is the bilinear interaction model in which the force F( ry ) is expanded up to first order in the deviations rj of the solvent atoms from their equilibrium positions, F( ry ) = F ( rJ )5ry, 

These normal modes evolve independently of each other. Their classical equations of motion are Uk = whose general solution is given by Eqs (6.81). This bath 

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We refer to Chapter 4 for a detailed discussion on the definition and explicit construction of diabatic states. The diabatic representation is generally advantageous for the computational treatment of the nuclear dynamics if the adiabatic potential-energy surfaces exhibit degeneracies such as conical intersections. Moreover, the diabatic representation often reflects more clearly than the Born ppenheimer adiabatic representation the essential physics of curve crossing problems and is thus very useful for the construction of appropriate model Hamiltonians for polyatomic systems. 

A model Hamiltonian that describes the excitation spectrum of the crystal in the energy region E = 2MI can be readily constructed on the basis of the qualitative considerations presented above. As a matter of fact, the Hamiltonian of the crystal, describing the effect of intermolecular interaction on the spectrum, for example, of nondegenerate molecular vibrations can be written in the harmonic approximation as follows ... 

The construction of a Hamiltonian is normally an easy problem. The solution of the Schrodinger equation, on the contrary, represents a serious problem. It can be solved exactly for several model cases a particle in a box (one-, two- or three-dimensional), harmonic oscillator, rigid rotor, a particle passing through a potential barrier, hydrogen atom, etc. In most applications only an approximate solution of the Schrodinger equation is attainable. 

Throughout this paper, we have seen that algebraic techniques often provide extremely simple numerical results with small computational effort. This is particularly true in the preliminary phases of one-dimensional calculations, where almost trivial relations can be found for the initial guesses for the algebraic parameters, as shown in Sections II.C.l and III.C.2. However, it is also true that as soon as real calculations of more complex vibrational spectra are requested, the problem of adapting the various algebraic Hamiltonian and transition operators to suitable computer routines must be resolved. The construction of a computer interface between theoretical models and numerical results is absolutely necessary. Nonetheless, it is rather atypical to discuss these problems explicitly in a theoretical paper such as this one. However, the novelty of these methods itself justifies further explanation and comment on the computational procedures required in practical applications. In this section we present only a brief outline of the development of algebraic software in the last few years, as well as the most peculiar situations one expects to encounter. 

This is the start of the construction of multichannel models of laser-driven molecules. In practice, one restricts to a finite number, N h, of electronic states, selected on the basis of physical relevance and Eq. (7) defines an Nch-channel molecular model system. The electronic" Hamiltonian is, in this model, described by the operator... 

When the effective Hamiltonian is constructed from ab initio wave functions the resulting matrix is numerical in nature. This matrix can be used to determine the values of the parameters of a phenomenological model Hamiltonian, but also to check the validity of the model. In most cases the structure of the effective Hamiltonian matrix coincides with the structure of the model Hamiltonian, but when significant deviations are observed, it should not be discarded that important interactions are missing in the model. For instance, when non-zero matrix elements appear in the... 

Because of the simplicity, Eq. (3.5) has gained fairly wide acceptance for the interpretation of experimental facts. Furthermore, it served as a basis for the construction of a number of model Hamiltonians (see below). 

A considerable shortcoming of Eq. (3.2) consists in the assumption of invariability of the electron structure of the solute molecule when it passes from free space into solution, i.e., in the assumption of constancy of Eq. However, Eq can change significantly, particularly for the systems of high polarizability. This limitation of the calculational scheme may be overcome by constructing a model Hamiltonian of the solute molecule. The general form of this Hamiltonian H is ... 

Strictly speaking, the Born equation of Eq. (3.5) can be used only for calculating the solvaton energy of an ion. However with some additional assumptions, it is possible to construct from Eq. (3.5) a model Hamiltonian for neutral molecules. Such a scheme was first suggested by Klopman [19], while its nonselfconsistent version had already been employed in Ref. [20]. 

In this section we shall marginally discuss two selected topics of quantum chemistry emphasizing the points where second quantization is useful. First in Sect. 17.1 the second quantized representation of spin operators will be introduced and they will be applied to construct a model Hamiltonian. Section 17.2 will explain the connection between second quantization and the unitary group approach. 

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