Coordinates molecular Hamiltonians

The tliree protons in PH are identical aud indistinguishable. Therefore the molecular Hamiltonian will conmuite with any operation that pemuites them, where such a pemiutation interchanges the space and spin coordinates of the protons. Although this is a rather obvious syimnetry, and a proof is hardly necessary, it can be proved by fomial algebra as done in chapter 6 of [1]. 

The traditional treatment of molecules relies upon a molecular Hamiltonian that is invariant under inversion of all particle coordinates through the center of mass. For such a molecular Hamiltonian, the energy levels possess a well-defined parity. Time-dependent states conserve their parity in time provided that the parity is well defined initially. Such states cannot be chiral. Nevertheless, chiral states can be defined as time-dependent states that change so slowly, owing to tunneling processes, that they are stationary on the time scale of normal chemical events. [22] The discovery of parity violation in weak nuclear interactions drastically changes this simple picture, [14, 23-28] For a recent review, see Bouchiat and Bouchiat. [29]... 

The coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiahatic (diabatic) Hamiltonian. This leads to renormalized fermions and renotmahzed diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

In previous part we developed canonical transformation (through normal coordinates) by which we were able to pass from crude adiabatic to adiabatic Hamiltonian. We started with crude adiabatic molecular Hamiltonian on which we applied canonical transformation on second quantized operators... 

Let us focus on molecular systems for which we know molecular Hamiltonian models, H(q,Q). Electronic and nuclear configuration coordinates are designated with the vectors q and Q, respectively x = (q,Q) = (qi,..., qn, Qi,---, Qm- For an n-electron system, q has dimension 3n Q has dimension 3m for an m-nuclei system. The wave function is the projection in configuration space of a particular abstract quantum state, namely P(x) P(q,Q), and base state func-... 

In this chapter, we are dealing only with spin wave functions and Hamiltonians. In the complete molecular Hamiltonian, which involves electronic spatial coordinates and momenta in addition to spin coordinates, the contact term in the Hamiltonian would have to be written in a form such that the integral over the electronic spatial wave function J/,... 

The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical A-particle molecular Hamiltonian,... 

The Born-Oppenheimer approximation permits the molecular Hamiltonian H to be separated into a component H, that depends only on the coordinates of the electrons relative to the nuclei, plus a component depending upon the nuclear coordinates. This in turn can be wriuen as a sum Hr + H, of terms for vibrational and rotational motion of ihe nuclei. 

Using primed coordinates in the laboratory-centered frame to help comparisons with Pack-Hirschfelder s work [8,15], the molecular hamiltonian... 

In order to illustrate electronic transitions we discuss the simple two-dimensional model of a linear triatomic molecule ABC as depicted in Figure 2.1. R and r are the appropriate Jacobi coordinates to describe the nuclear motion and the vector q comprises all electronic coordinates. The total molecular Hamiltonian Hmoi, including all nuclear and electronic degrees of freedom, is given by Equation (2.28) with Hei and Tnu being the electronic Hamiltonian and the kinetic energy of the nuclei, respectively. 

As an example of application of the method we have considered the case of the acrolein molecule in aqueous solution. We have shown how ASEP/MD permits a unified treatment of the absorption, fluorescence, phosphorescence, internal conversion and intersystem crossing processes. Although, in principle, electrostatic, polarization, dispersion and exchange components of the solute-solvent interaction energy are taken into account, only the firsts two terms are included into the molecular Hamiltonian and, hence, affect the solute wavefunction. Dispersion and exchange components are represented through a Lennard-Jones potential that depends only on the nuclear coordinates. The inclusion of the effect of these components on the solute wavefunction is important in order to understand the solvent effect on the red shift of the bands of absorption spectra of non-polar molecules or the disappearance of... 

We dealt with the effects of applied static fields on the electronic Hamiltonian in section 3.7. In this section we first give the relevant terms for the nuclear Zeeman and Stark Hamiltonians and then perform the same coordinate transformations that proved to be convenient for the field-free molecular Hamiltonian. 

Thus we see that the operator g is not strictly an angular momentum operator in the quantum mechanical sense, which is why we have assigned it a different symbol. More importantly for the present purposes, we cannot use the armoury of angular momentum theory and spherical tensor methods to construct representations of the molecular Hamiltonian. In addition, the rotational kinetic energy operator, equation (7.89), takes a more complicated form than it has for a nonlinear molecule where there are three Euler angles (rotational coordinates). 

Here <5FS signifies the fluctuating solvent force on the coordinate qs, while < qs (t) is the Heisenberg time-dependent operator, with dynamics governed by the full internal anharmonic molecular Hamiltonian, associated with the fluctuation <5qs = qs — (i qs f). Finally, the prefactor yi( is (2)... 

For quantum-mechanical systems, the moment definitions in Eqn. (8) can be made into definitions of moment operators by replacing position coordinates with their corresponding position operators. The molecular Hamiltonian for a molecule experiencing an external electrical potential following the convention of Eqn. (7) is... 

A. Nauts and X. Chapuisat, Momentum, quasi-momentum and Hamiltonian operators in terms of arbitrary curvilinear coordinates, with special emphasis on molecular Hamiltonians. Mol. 

The essential step in the construction[of the molecular Hamiltonian]is the removal of the ignorable coordinates corresponding to the overall translation and overall rotation of the molecule. 

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