Hamiltonian operator force fields

For these vibrations, the quantization scheme of Section 4.2 can be carried over without any modification (Iachello and Oss, 1991a). The potentials in each stretching coordinate 5 are in an anharmonic force field approximation represented by Morse potentials. The boson operators (Ot,xt) correspond to the quantization of anharmonic Morse oscillators, with classical Hamiltonian... 

Because we will use rectilinear coordinates / , and normal coordinates Q, to formulate the Hamiltonian operator, it is necessary to convert the input force field in curvilinear coordinates to a force field in rectilinear coordinates. After using the transfer-... 

Despite being quite different from one another, the QM/discrete models are ah. characterized by maintaining the information on the atomic structure of the environment. The most popular formulation of these models is to use the MM force fields to describe the interaction within the ENV part of the system as weU as the nonelectrostatic interactions between the QM subsystem and the ENV. The electrostatic interactions between the two parts are instead kept in the effective QM Hamiltonian as an additional operator which contributes to determine the MS wavefrmction. Such an operator is generahy written in terms of a set of fixed multipoles usuaUy placed on the atoms of the environment molecules. In most cases just the partial charges are considered, but there are instances where multipoles up to the quadrupoles are included. The resulting MS/ENV interaction term thus becomes ... 

Now consider the quantum-mechanical definition of the electric dipole moment. Suppose we apply a uniform external electric field E to an atom or molecule and ask for the effect on the energy of the system. To form the Hamiltonian operator, we first need the classical expression for the energy. The electric field strength E is defined as E = F/Q, where F is the force the field exerts on a charge Q. We take the z direction as the direction of the applied field E = The potential energy V is [Eq. (4.24)]... 

There are two types of situation to distinguish. One is situations where the potential is changing as a function of time, and hence the hamiltonian operator is time dependent. An example is a molecule or atom in a time-varying electromagnetic field. The other is situations where the potential and hamiltonian operator do not change with time, but the particle is nonetheless in a nonstationary state. An example is a particle that is known to have been forced into a nonstationary state by a measurement of its position. We deal here with the second category. 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

This theorem is also valid for many variational wavefunctions, e.g. for the Hartree-Fock one, if complete basis sets are used. As only the one-electron part of the Hamiltonian depends on the nuclear coordinates, H is a one-electron operator, and the evaluation of the Hellmann-Feynman forces is simple. Because of this simplicity, there have been a number of early suggestions to use the Hellmann-Feynman forces for the study of potential surfaces. These attempts met with little success, and the discussion below will show the reason for this. It is perhaps fair to say that the main value of the Hellmann-Feynman theorem for geometrical derivatives is in the insight it provides, and that numerical applications do not appear promising. For other types of perturbations, e.g. for weak external fields, the theorem is widely used, however. For a survey, see a recent book (Deb, 1981). 

From our experience to date with the transformations, we can immediately foresee a problem. If the transformation is some complicated function of the momentum, we might not be able to separate out the perturbation from the zeroth-order Hamiltonian. This would be unfortunate, because magnetic operators break Kramers symmetry and we would be forced to perform calculations without spin (or time-reversal) symmetry. We might also be forced to perform finite-field calculations. We will address this problem as it arises. 

Searching along the lines of selective modification of the radial wavefunctions, Newman (1970) and Judd (1977b) found another source of correlated crystal field in the exchange forces between 4f electrons with similarly directed spins. They introduce a correction of the type s, S (where s,- is the zth electron spin and S the total spin) as a multiplicative factor to apply to the operators. The effective crystal field Hamiltonian therefore becomes ... 

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