Symmetry eigenvectors

Rigid Molecule Group theory will be given in the main part of this paper. For example, synunetry adapted potential energy function for internal molecular large amplitude motions will be deduced. Symmetry eigenvectors which factorize the Hamiltonian matrix in boxes will be derived. In the last section, applications to problems of physical interest will be forwarded. For example, conformational dependencies of molecular parameters as a function of temperature will be determined. Selection rules, as wdl as, torsional far infrared spectrum band structure calculations will be predicted. Finally, the torsional band structures of electronic spectra of flexible molecules will be presented. 

From this table, the symmetry eigenvectors which factorize the matrix Hamiltonian corresponding to operator (22) into boxes are easily deduced in the base of the solutions of the planar free rotor equation. There are ... 

From this character table, the symmetry eigenvectors for the double internal rotation in planar pyrocatechin are easily deduced on the basis of the double free rotor equation solutions ... 

The symmetry eigenvectors corresponding to the G36 acetone-like group are not so easily deduced by tryings and errors, from the character table, as in the case of the one-fold rotor molecules. The projector technique has to be used for [21,34] ... 

Applying this projector on each of the terms of a rotational configuration interaction expansion, built up on the basis of the double free rotor solutions, the symmetry eigenvectors are obtained [21,22,34]. 

FVom this character table, and the symmetry eigenvectors of planar acetone (54-56), the symmetry eigenvectors of pyramidal acetone are easily deducible. For this purpose, linear combinations of the eigenvectors, which exhibit the same behavior for all the operations except for WU and VU, are built up. In addition, to the coefBcients of which are trigonometric functions of the wagging angle, a. The coefficients are chosen in such a way that the linear combinations fulfill the characters corresponding to operators WU and VU,... 

The non-degenerate symmetry eigenvectors (49) are identical to those of non-planar pyrocatechine(40) except for the threefold multiplicity. The twofold... 

The energy potential energy function of the Hamiltonian operator for the double Csv internal rotation and wagging may be written in terms of Ai symmetry eigenvectors [37] ... 

The same symmetry eigenvectors as in the case of benzaldehyde with cogwheel effect are encountered (29), but in the present case their group properties are different. 

Using these symmetry eigenvectors the Hamiltonian matrix is factorized in eight boxes instead of four. This feature introduces a new simplification in the solution of the double rotor Schrodinger equation, paying a very small loss of accuracy. 

They are the same symmetry eigenvectors as those of pyrocathechin with the cog-wheel effect (34), but the group properties are different. 

The symmetry eigenvectors of such a group are easily deduced multiplying those of planar pyrocatechin (34) by cos Ma or sin A/a. 

From this character table the symmetry eigenvectors are easily derivable on the basis of the triple free rotor solution. It is found ... 

The symmetry eigenvectors of any local NRG may be advantageously used as basis functions in a pre-diagonalization of the complete Hamiltonian matrix. For this purpose, the local NRG has only to be a larger group than the complete NRG. In the present case we have for example ... 

The potential energy function for acetone without cog-wheel effect between the rotors is then written in terms of symmetry eigenvectors corresponding to the completely symmetric representation ... 

Using the symmetry eigenvectors the Hamiltonian matrix is factorable into 32 boxes instead of 16, with a very slight loss of accuracy. 

The symmetry eigenvectors may be constructed from those of the former group, by multiplying by cos Ma or sin Ma, and gathering the products by pairs according to their symmetry properties with respect to operations of the subgroup WUy A [ UU y x (Cft/ ) ] of (98), as in the case of non-planar pyrocatechine (89). 

As in the previous case of pyrocatechin, the symmetry eigenvectors of such a group may be deduced from those of the preceding group (98) by taking into account in the present case the products by cos Ma and sin Ma belonging to different irreducible representations. 

In the present paper, symmetry eigenvectors which factorize the Hamiltonian matrix into boxes are given for the single rotation in phenol (24) for double rotations in benzaldehyde (29), pyrocatechin (34) and acetone (44-46), for double rotation and inversion in non-planar pyrocatechin (40) and pyramidal acetone (49-51). In the same way, symmetry eigenvectors deduced in the local approach are deduced for some of these non-rigid systems (79), (83), and (89). Symmetry eigenvectors for the double internal Czv rotation in molecules with frame of any symmetry are given in reference [36]. 

In the quantum mechanical approach. Group Theory permits us to handle synunetry species, especially by developing the solution on the base of the symmetry eigenvectors. This produre would shorten appreciably the sums on t and j in (118). 

Throughout this paper, we give potential energy functions, symmetry eigenvectors, as examples, for systems of one, two and three internal degrees of freedom, in the formalism of the restricted Hamiltonian operator, as well as in the local one. A generalization of these ideas can be found in the scientific literature [21,22] and [30-37]. 

This technique assumes previous knowledge of the symmetry eigenvectors which may be deduced by using Group Theory for Non-Rigid Molecules as well as some information about the necessary basis length [2,3]. 

Furthermore, we have to remark that Group Theory for Non-Rigid Molecules may be advantageously used to deduce a symmetry adapted analytical form for the potential, as well as the symmetry eigenvectors for simplifying the Hamiltonian matrix solution. In the same way. Group Theory permits to label and classify the energy levels and the vibrational functions. Finally, it may be also used to deduce selection rules for the infrared transitions. 

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