The General Hamiltonian

The kinetic energy for a molecule in a center-of-mass coordinate system may be written 

The first two terms are pure rotational terms, the third a pure vibrational term, and the last a Coriolis term. The partitioning of the energy between the pure rotational terms and the Coriolis term depends on the choice of the rotating axis system used to describe the problem18-22.  

In order to obtain a quantum mechanical kinetic energy operator, a momentum representation of the kinetic energy is required. If coordinates are chosen having conjugate momenta 

The 3 N-3 x 3 N-3 matrix in Eq. (3.9) is hereafter referred to as the rotation-vibration G matrix. Transformation to the correct quantum mechanical from according to Kemble23 yields 

As pointed out by Gwinu and Gaylord19), the solutions of the eigenvalue problems associated with a vibration-rotation problem do not and must not depend on the choice of the rotating axis system as long as an adequate Hamiltonian is used. What do depend on the axis system used are the numerical values of elements of the inertial tensor or vibration-rotation interaction constants determined from analysing the data. 

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The above formulation can be generalized to a general multidimensional case in the form invariant under any coordinate transformation, as was done before for the ground-state case. We consider the general Hamiltonian given by Eq. (32). The formulation can be carried out in the same way as before. The equation for the additional term w is given by... 

The energy over i electrons and /i nuclei is computed by evaluating the integrals defined by the operation of the general Hamiltonian... 

For simplicity, let us illustrate the calculations of entanglement for two spin- particles. The general Hamiltonian, in atomic units, for such a system is given by [57]... 

The general Hamiltonian including magnetic as well as electric interactions can be written... 

The general Hamiltonian is given by Eq. (6.3), where the sum is over all the j-k pairs. The treatment is in fact based on bimetallic coupling. The S levels are obtained and the hyperfine coupling for each S level is given by... 

The general Hamiltonian formalism was developed principally by Dirac [11-13] ... 

To take into account both discrete energy levels of a system and the electron-electron interaction, it is convenient to start from the general Hamiltonian... 

To establish the Hamiltonian describing the interaction of electrons with vibrons in nanosystems, we can start from the generalized Hamiltonian... 

Marginally stable equilibrium points are such that d V/dq = 0. The stability is then determined by the nonlinearity of the force —dV/dq. In the 1-DOF case, this is easily extended to the general Hamiltonian (8). An equilibrium point is such that dH/dp = dH/dq = 0. The linear stability proceeds as follows. 

The general Hamiltonian for A charged point particles under Coulomb interactions is represented in Cartesian coordinates by [11]... 

Finally we must examine the property of additivity of the exchange Hamiltonian. Let us consider, for instance, four interacting electrons and let us specifically examine the exchange coupling between electrons 1 and 2. The effect of electrons 3 and 4 is treated through the effective (mean) potentials SV n) and SV r2) seen by electrons 1 and 2 (Hartree-Fock approximation). Under these conditions the general Hamiltonian for electrons 1 and 2 may be written ... 

Of course this argument holds for the six distinct electron pairs and, in the present Hartree-Fock approach, the general Hamiltonian will be ... 

At first, the general Hamiltonian can be described from products of spherical tensors expressed in terms of spin and orbital components. Since no spin-orbit coupling has been explicitly introduced in the general expression of H , the spin operators must appear in a spherical way. As they occur linearly, the Hamiltonian will take the form ... 

The mass of atom a is given by m and its displacement vector by r,. The general Hamiltonian was used for the treatment of the multi vibrational problem,... 

The PES arises naturally upon application of the BO approximation to the solution of the Schrodinger equation. We begin by considering the general Hamiltonian... 

We have applied NESGET to study the charge conductivity of a molecular wire attached to two perfectly conducting leads. In the simplest approach the leads a and b are treated as two free electron reservoirs. Nuclear motions in the molecular region are described as harmonic phonons which interact with the surrounding electronic structure and the environment (secondary phonons) [26]. We first recast the general Hamiltonian, Eq. (1), in a single electron local basis and partition it as... 

A Hamiltonian is an operator which operates upon a wave function. When it is applied to the wave function of a particular system, it gives the permitted energy levels of that system. A simple form of the general Hamiltonian for an ion in a crystalline environment and with an applied magnetic field may be VTitten as... 

The part of the general Hamiltonian incorporating (in particular) the g and A factors needs to be explained in more detail, i.e.,... 

For an excellent brief discussion of the general Hamiltonian, including external electric and magnetic held effects, see Reference 40. 

Two mechanisms have been proposed to explain the appearance of an asymmetric doublet in randomly oriented substances with no magnetic ordering. One mechanism is based on the combination of the directional quantities — the angular distribution function of the magnetic dipole radiation and the Debye-Waller factor which becomes anisotropic in systems of lower than cubic symmetry. This mechanism predicts an asymmetry which should decrease as the temperature is lowered, in contradiction to the experimental observations in hemoglobin. The second mechanism is based on magnetic interactions described by the general Hamiltonian Eq. (234). 

The general Hamiltonian of a molecule interacting with an external field in second quantization form reads "... 

For a quadrupolar nucleus such as Zn in a diamagnetic material, the general Hamiltonian operator describing the SSNMR spectrum is given by the following ... 

Here H and H are the generalized Hamiltonians (Wu and Moszkowski 1965) and the Q-s are the coupling strengths. The H, operators have different shapes for S, V( T A, and P classes of weak interaction. They are complicated combinations of the so-called y matrices of the... 

This field turns out to he Hamiltonian on the common level surface of two integrals /2 = 2 = const, and /s = C3 = const, with respect to a certain natural Poisson bracket. This makes it possible to apply to this system the rich techniques used for the study of the general Hamiltonian systems. 

Within the framework of two-level approximation at excited nucleus consideration, the general Hamiltonian operator of... 

We now consider the general Hamiltonian, which is the same as Equation (6.82) in Subsection 6.1.3,... 

For a system with a doublet (S = i) non-degenerate electronic ground state, the interaction with the external magnetic field can be expressed in terms of a perturbation of the general Hamiltonian by the following three terms... 

Let us proceed to the fermion part of the general Hamiltonian, particularly the fermion part difference AHp between the general Hamiltonian and the original crude... 

If there are several magnetic nuclei in a molecule, the high-field energy level expression that follows from a first-order perturbative treatment of the general Hamiltonian in Equation 12.19 is... 

The general Hamiltonian for the Klein-Gordon equation for order parameter iy at nth site is written as ... 

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