Effective-operator Hamiltonian

The Hamiltonian is an effective electronic Hamiltonian that operates on h ... 

A simple example of an effective operator with which the reader will be familiar is the use of Zeff e r as the effective nuclear potential experienced by an electron outside of a closed inner shell. Thus, we may compute the energies and wavefunctions for a 2s or 2p electron outside a shell, using the hydrogen-like Hamiltonian,... 

Any set of energetically well-isolated levels can be described by an effective spin Hamiltonian operator by choosing S to match the corresponding number of levels. This can be just one isolated Kramers doublet of a high-spin multiplet if the... 

With these data, we calculated numerically the (mi, m2 He[t m u m 2) matrix elements of the effective exchange Hamiltonian, which are presented in Table 1. Each group of matrix elements can be associated with a definite equivalent spin operator, which has the same matrix elements in the basis set as those... 

Table 1. Non -zero (ni],m2 Haflm3,m4) matrix elements of the effective exchange Hamiltonian and the equivalent spin operators (all parameters are in cm-1)...

Summing up, the structure of the effective Hamiltonian of Equation (1.107) makes explicit the nonlinear nature of the QM problem, due to the solute-solvent interaction operator depending on the wavefunction, via the expectation value of the electronic density operator. The consequences of the nonlinearity of the QM problem may be essentially reduced to two aspects (i) the necessity of an iterative solution of the Schrodinger Equation (1.107) and (ii) the necessity to introduce a new fundamental energetic quantity, not described by the effective molecular Hamiltonian. The contrast with the corresponding QM problem for an isolated molecule is evident. 

Averaging the interaction operators in eq. (1.246) - they are both two-electronic ones - over the ground states of each subsystem does not touch the fermi-operators of the other subsystem. The averaging of the two-electron operators PWCP and PwrrP yields the one-electron corrections to the bare subsystem Hamiltonians. The wave functions and d, )7 are calculated in the presence of each other. The effective operator iTff describes the electronic structure of the R-system in the presence of the medium, whereas HIf describes the medium in the presence of the R-system. 

The effective bond Hamiltonians also can be rewritten in terms of the pseudospin operators. Indeed, the effective Hamiltonian eq. (3.1) for each of the bond geminals can be presented in the form ... 

As mentioned previously, the density ESVs must be obtained from the effective bond Hamiltonian eq. (4.1). In terms of the geminal amplitudes, the ESVs are given by eq. (2.78). To get the required direct estimates of the ESVs, we use again the projection operator technique. In terms of the geminal amplitudes (subject to the normalization condition) the projection operator upon die ground state of a geminal has the form ... 

For this purpose, consider selective canonical transformations leading to a new quantum representation that we name /// in order to diagonalize the effective Hamiltonian corresponding to k = 1, without affecting that corresponding to k = 0. This may be performed on the different effective operators B dealing with k = 0,1 with the aid of... 

It took several decades for the effective Hamiltonian to evolve to its modem form. It will come as no surprise to learn that Van Vleck played an important part in this development for example, he was the first to describe the form of the operator for a polyatomic molecule with quantised orbital angular momentum [2], The present formulation owes much to the derivation of the effective spin Hamiltonian by Pryce [3] and Griffith [4], Miller published a pivotal paper in 1969 [5] in which he built on these ideas to show how a general effective Hamiltonian for a diatomic molecule can be constructed. He has applied his approach in a number of specific situations, for example, to the description of N2 in its A 3 + state [6], described in chapter 8. In this book, we follow the treatment of Brown, Colbourn, Watson and Wayne [7], except that we incorporate spherical tensor methods where advantageous. It is a strange fact that the standard form of the effective Hamiltonian for a polyatomic molecule [2] was established many years before that for a diatomic molecule [7]. 

In the present treatment, we retain essentially all the diagonal matrix elements of X these are the first-order contributions to the effective electronic Hamiltonian. There are many possible off-diagonal matrix elements but we shall consider only those due to the terms in Xrot and X o here since these are the largest and provide readily observable effects. The appropriate part of the rotational Hamiltonian is —2hcB(R)(NxLx + NyLy). The matrix elements of this operator are comparatively sparse because they are subject to the selection rules AA = 1, A,Y=0 and AF=0. The spin-orbit coupling term, on the other hand, has a much more extensive set of matrix elements allowed... 

The simplest example of the way in which the various terms arise in the effective electronic Hamiltonian involves the rotational kinetic energy operator, 3Qot ... 

We see from the way in which the effective rotational Hamiltonian is constructed that it is naturally expressed in terms of the angular momentum operator N. In the scientific literature, however, it is frequently written in terms of the vector R (which represents the rotational angular momentum of the nuclei) rather than N. While R = N — L occurs in the fundamental Hamiltonian (7.71), its use in the effective Hamiltonian is not satisfactory because R has matrix elements (due to L) which connect different electronic states and so is not block diagonal in the electronic states. In practice, authors who claim to be using R in their formulations usually ignore any matrix elements which they find inconvenient such as those of Lx and Ly. We shall return to this point in more detail later in this chapter. 

The first-order contribution of these hyperfine interactions to the effective electronic Hamiltonian involves the diagonal matrix elements of the individual operator terms over the electronic wave function, see equation (7.43). As before, we factorise out those terms which involve the electronic spin and spatial coordinates. For example, for the Fermi contact term we need to evaluate matrix elements of the type ... 

We use the operator on the left-hand side of equation (7.169) as the zeroth-order vibrational Hamiltonian. The remaining terms in the effective electronic Hamiltonian, given for example in equations (7.124) and (7.137), are treated as perturbations. In a similar vein to the electronic problem, we consider only first- and second-order corrections as given in equations (7.68) and (7.69) to produce an effective Hamiltonian 3Q, which is confined to act within a single vibronic state rj, v) only. Once again, the condition for the validity of this approximation is that the perturbation matrix elements should be small compared with the vibrational intervals. It will therefore tend to fail for loosely bound states with low vibrational frequencies. 

The right member of this equation contains an operator which depends only on the nucler coordinates, X, and on the quantum number n. As a result, the following approximation may be written for an effective nuclear Hamiltonian operator ... 

The Group Theory for Non-Rigid Molecules considers isoenergetic isomers, and the interconversion motions between them. Because of the discernability between the identical nuclei, each isomer possesses a different electronic Hamiltonian operator in (3), with different eigenfunctions, but the same eigenvalue. In contrast, a non-rigid molecule has an unique effective nuclear Hamiltonian operator (7). 

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