Hamiltonian equation zero-point energy

Of course, in view of Eq. (292) the effective non-Hermitean Hamiltonians are given by the following equations, where, as usual, for the non-Hermitean Hamiltonians, the zero-point energies have been ignored... 

Note that in the eigenvalue equation (204) the zero-point energy was not taken into account, and so the corresponding factor 0.5 in the expression of the Hamiltonians in terms of Bosons. 

Then, using the eigenvalue equations dealing with the two Hamiltonians of interest and neglecting the zero-point energy involved in the Boltzmann operator,... 

The critical points of the equivalent classical Hamiltonian occur at stationary state energies of the quantum Hamiltonian H and correspond to stationary states in both the quantum and generalized classical pictures. These points are characterized by the constrained generalized eigenvalue equation obtained by setting the time variation to zero in Eq. (4.17)... 

The Hiickel Hamiltonian-matrix for an n-annulene (Fig. 5-1) is the matrix in the determinant found in equation (5-1), with the quantities ( — x), along the diagonal, replaced by zero (i.e. a). As pointed out in Appendix A, this matrix is isomorphic with the adjacency matrix, A(C ), of the correspondingly-labelled molecular graph furthermore, the eigenvalues of the matrix, A(C ), are the annulene energy-levels we require (expressed in units of fi and... 

These simple numerical exercises illustrate the point that the total Hamiltonian of the system in the presence of the electric field must not only include the well-known electrostatic energy term but also the hitherto ignored strain energy term. The small strains that are encountered in the polarization orocess also justify the use of the electrostatic equations of the sphere. It is also quite clear that for very unsymmetrical species where equilibrium compliance approaches infinity, such as the case of bulky polar liquids, the strain energy term may also be zero and the Onsager-Kirkwood equations are once more applicable. It should be pointed out that the strain energy is also zero for finite values of Do but symmetrical species such as spherical dipoles. 

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