Hamiltonian operators, algebraic models

To reiterate, we prefer to describe the one-dimensional model first because of its mathematical simplicity in comparison to the three-dimensional model. From a strictly historical point of view, the situation is slightly more involved. The vibron model was officially introduced in 1981 by lachello [26]. In his work one can find the fundamental idea of the dynamical symmetry, based on U(4), for realizing an algebraic version of the three-dimensional Hamiltonian operator of a single diatomic molecule. After this work, many other realizations followed (see specific... 

In the one-dimensional algebraic model, the study of molecular vibrations requires proper treatment of bending modes as well. As alluded to earlier, the one-dimensional Hamiltonian operator is equally well suited for the description of both stretches and bends, by virtue of... 

The three-dimensional algebraic model can reproduce, in detail, the aforementioned classification of the rovibrational Hamiltonian operator in partials fi22- To achieve this goal, we start by writing a compact form of this operator in its usual (normal coordinate) notation ... 

This means that the point symmetry of H changes from to D2h ( L and H stand for low- and high -symmetry site, respectively, as should now be evident). The first step in the construction of the benzene dimer is to modify force constants of the two sites according to Eqs. (5.1) and (5.2). A similar study can be easily and systematically implemented within the one-dimensional algebraic model by taking the Hamiltonian operator for CH stretching modes of the benzene molecule (Section III.C.2),... 

In this book we shall write the Hamiltonian as an (algebraic) operator using the appropriate Lie algebra. We intend to illustrate by many applications what we mean by this cryptic statement. It is important to emphasize that one way to represent such a Hamiltonian is as a matrix. In this connection we draw attention to one important area of spectroscopy, that of electronically excited states of larger molecules,4 which is traditionally discussed in terms of matrix Hamiltonians, the simplest of which is the so-called picket fence model (Bixon and Jortner, 1968). A central issue in this area of spectroscopy is the time evolution of an initially prepared nonstationary state. We defer a detailed discussion of such topics to a subsequent volume, which deals with the algebraic approach to dynamics. 

As emphasized, one of the advantages of this model is that it provides explicit wavefunctions which can be used in the computation of expectation values for various operators of interest. Due to limitations of space, we cannot reproduce here the complete set of vibrational wavefunctions obtained in the HCN calculation [76]. However, the typical outcome of the algebraic procedure can be outlined. We obtain a polyad of levels labeled by the numbers Vj and Ig of Eq. (4.56). Each polyad contains a number of local states, such as those listed in Eq. (4.57). The numerical diagonalization of the Hamiltonian matrix is performed separately for each polyad. Thus the eigenvectors derived represent the vibrational wavefunctions in the local basis. A possible outcome of the analysis of the HCN molecule could therefore be given by the following sequence of numbers ... 

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. 

Just as the use of a finite basis set in independent-electron models restricts the domain of the relevant one-electron operator, h, so the algebraic approximation results in the restriction of the domain of the total Hamiltonian to a finite-dimensional subspace of the Hilbert space. In most applications of quantum mechanics to atoms and molecules which go beyond the independent-electron models, the JV-electron wavefunction is expressed in terms of the fVth-rank direct product space M generated by a finitedimensional single-particle space MlS that is... 

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