Transformed Hamiltonian eigenfunctions, calculation

F. Calculation of Approximate Eigenvalues and Eigenfunctions to the Transformed Hamiltonian H... 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

The perturbation calculation may also be described as a contact transformation. The original hamiltonian is transformed to a new effective hamiltonian which has the same eigenvalues but different eigenfunctions, to some carefully chosen order of magnitude. This contact transformation of the vibration-rotation hamiltonian was originally studied by Nielsen and co-workers. >33... 

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

Each top experiences the same on-site potential 1 3 and the coupling depends only on the phase difference of the two rotors. In contrast to the textbook case of coupled harmonic oscillators, this Hamiltonian cannot be diagonalized by simple transformation into normal coordinates [86], On the other hand, numerical calculation of the eigenstates and eigenfunctions cannot be carried out with standard methods, for it requires very large basis sets with dimension 10, and the analysis of the eigenfunctions would be cumbersome. 

Two successive contact transformations remove from the expression of the Hamiltonian the first and second degree terms. Thus, the wave functions of die zero order term which, in our case, are the standard linear harmonic oscillator wave functions, are also eigenfunctions of the Hamiltonian H up to the second order. If a standard perturbation dieory is applied, there will be an extensive number of off-diagonal matrix elements of the first-order perturbation Hamiltonian appearing in the expressions for any molecular quantity estimated from second order matrix elements. By the contact transformations the matrix elements will be diagonal through second order which greatly simplifies the calculations. If the linear harmonic oscillator wave function is denoted by ( n i, die matrix element < n I H" I m) may be expressed as [Eq. (6.10)]... 

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