Hamiltonian expression

The core Hamiltonian expressions, H)) and correspond to electrons moving in the... 

The autocorrelation function of the hydrogen bond within the effective Hamiltonian expression of the standard approach takes the form... 

Hamiltonians expressed in matrix forms have been extensively employed in the theory of radiationless transitions of electronically excited states of larger molecules (Bixon and Jortner, 1968 Schlag et al., 1971 Freed, 1972 Nitzan et al., 1972 Avouris et al., 1977 Jortner and Levine, 1981 Felker and Zewail, 1988 Seel and Domcke, 1991). 

Problem 11-16. Why do the results of the last two problems guarantee that the Hamiltonian, expressed in terms of the given molecular orbitals, will be diagonal ... 

Therefore, perturbation calculations and Huckel calculations are very similar (a) the Hamiltonian expression is not specified and (b) the required MOs are linear combinations of known orbitals. When Equations (3.1) and (3.3) are incorporated into the time-independent Schrodinger equation ... 

Wick s theorem (35) which gives us the prescription for treating a product of operators may of course be applied to the Hamiltonian, expressed in the second quantization formalism (29). This leads to the Hamiltonian in a form which is of primary importance in perturbation treatments. This form of the Hamiltonian which is called the normal product form is ... 

Recall that the canonical equations of classical mechanics can be used to derive the Hamiltonian expression for the total energy of a system from the momenta pk and positional coordinates qk ... 

We have derived the total Hamiltonian expressed in a space-fixed (i.e. non-rotating) coordinate system in (2.36), (2.37) and (2.75). We can now simplify the electronic Hamiltonian 3Q,i by transforming the electronic coordinates to the molecule-fixed axis system defined by (2.40) because the Coulombic potential term, when expressed as a function of these new coordinates, is independent of 0, ip and x From a physical standpoint it is obviously sensible to transform the electronic coordinates in this way because under the influence of the electrostatic interactions, the electrons rotate in space with the nuclei. We shall take the opportunity to refer the electron spins to the molecule-fixed axis system in this section also, and leave discussion of the alternative scheme of space quantisation to a later section. Since we assume the electron spin wave function to be completely separable from the spatial (i.e. orbital) wave function,... 

Since the canonical transformation is time independent, the new Hamiltonian equals the old Hamiltonian expressed in the new variables. We obtain... 

A new formulation of the theory of paramagnetic shifts particularly suited to shifts in liquid crystalline solvents has been presented. (29) The Hamiltonian expression used allows for the inclusion of effects of preferential orientational distribution of the solute molecules. 

The state of a classical system is specified in terms of the values of a set of coordinates q and conjugate momenta p at some time t, the coordinates and momenta satisfying Hamilton s equations of motion. It is possible to perform a coordinate transformation to a new set of ps and qs which again satisfy Hamilton s equation of motion with respect to a Hamiltonian expressed in the new coordinates. Such a coordinate transformation is called a canonical transformation and, while changing the functional form of the Hamiltonian and of the expressions for other properties, it leaves all of the numerical values of the properties unchanged. Thus, a canonical transformation offers an alternative but equivalent description of a classical system. One may ask whether the same freedom of choosing equivalent descriptions of a system exists in quantum mechanics. The answer is in the affirmative and it is a unitary transformation which is the quantum analogue of the classical canonical transformation. 

There are several versions of EH-CSD models. To make the exposition less cumbersome, in the next pages we shall only summarize one version, that was elaborated in Pisa and known with the acronym PCM (Polarizable Continuum Model) (Miertus et al., 1981 Miertus and Tomasi, 1982). We shall consider other versions later, and the differences with respect to PCM will be highlighted. Other approaches, based on effective Hamiltonians expressed in terms of discrete solvent distributions, EH-DSD, or not relying on effective Hamiltonians, will also be considered. 

It is straightforwardly shown that the spectrum and the eigenvalues of the effective spin Hamiltonian (expressed in H units)... 

The associated effective Hamiltonian expressed in terms of spin operators is ... 

The details of how this can be done are presented in some earlier work, "(5), (6) and (7) and the results can be summarised as yielding a new translation-free Hamiltonian expressed in terms of a set of H — 1 coordinates t that depend only on the original nuclear coordinates and L coordinates tf that are the original electronic coordinates referred to the centre of nuclear mass. 

The foregoing discussion deals with TDSCF as introduced for fixed, preselected coordinates. Very recently, Kucar et al.61 considered an optimal coordinates TDSCF approximation. For a two-mode model Hamiltonian expressed in terms of the Cartesian coordinates (x,y), these authors used an ansatz of the type... 

In the case of the Eckart barrier, when defining y = —exp(2Trx/L) for the Hamiltonian expression... 

Considering Equation 6.38 again, we need to transform the Hamiltonian expression. Thus, if cos(k) and ss(k) are the frequency and the polarization vector for the classic modes with polarization s and wave vector k, respectively, we can define the phonon creation (aks+) and annihilation ( /,s ) operators as... 

As shown in Ch. 3, the crystal Hamiltonian, expressed in terms of the operators Pj-f and P/ (in the following the index / will be omitted) when the lattice vibrations are not taken into account, has the following form (see also eqn 3.39)... 

An excellent example of non-accidental intramolecular resonance is provided by a polyatomic molecule with two chemically identical bonds. The spectrum and dynamics of such a molecule may equally well be described by an effective Hamiltonian expressed in basis sets corresponding to either of two opposite limiting cases normal mode (H -,RMAL) and local mode (Hlocal)-... 

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