Molecular potential symmetry coordinates

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

The symmetry coordinates show themselves to be particularly useful for the functional representation of the molecular potential. For example, the potential function of a X3-type molecule must be invariant with respect to the interchange of any internal coordinate ft, (/ = 1, 2, 3) hence it must be totally symmetric in relation to those coordinates. Thus, in terms of the coordinates Qi (/ =1,2, 3), such a function can only be written in terms of or totally symmetric combinations of Q2 and Q3. Such combinations may in fact be obtained by using the projection-operator technique.16"27 In fact, one can demonstrate16 27 that any totally symmetric function of three variables is representable in terms of the integrity basis,28... 

The second symmetry requirement that the expression for the inter-molecular potential has to meet is that it must be invariant under any rotation of the global coordinate frame. The transformation properties of the symmetry-adapted functions Gj Hw) under such a rotation are easily obtained from Eqs. (10) and (5) ... 

In summary, symmetry is a useful tool in structure correlation for at least three purposes (1) to enumerate all equivalent ways of labeling atoms in a given molecule or molecular fragment and hence to eliminate the element of arbitrariness inherent in any particular labeling scheme (2) to quantify the notion of approximate symmetry with the help of symmetry coordinates (3) to analyze the symmetry properties of the molecular potential energy surface. 

One use of the symmetry coordinate classification is that it can tell us which types of distortion are expected to be coupled to other types. Each representative point can be associated with a value of the molecular potential energy, a structural invariant B that must be independent of such matters as the choice of coordinate system or the way the labels of the various atoms or bonds have been chosen. Consider, for the spiro-ketal example, the general quadratic energy expression for the S2 and Sg coordinates, which both transform as B2, symmetric with respect to the (yz) plane, antisymmetric with respect to the (xz) plane ... 

The occurrence of electron degeneracy brings two effects. First, the molecular Hamiltonian is invariant with respect to arbitrary symmetry operations, so that the adiabatic potential belongs to the totally symmetric representation (A-type or 2-type). Therefore, only certain combinations of QjQj or QjQjQk are allowed for symmetry coordinates and they must span the totally symmetric representation. Secondly, the interaction matrix U... 

Thus, for a transition between any two vibrational levels of the proton, the fluctuation of the molecular surrounding provides the activation. For each such transition, the motion along the proton coordinate is of quantum (sub-barrier) character. Possible intramolecular activation of the H—O chemical bond is taken into account in the theory by means of the summation of the probabilities of transitions between all the excited vibrational states of the proton with a weighting function corresponding to the thermal distribution.3,36 Incorporation in the theory of the contribution of the excited states enabled us in particular to improve the agreement between the theory and experiment with respect to the independence of the symmetry factor of the potential in a wide region of 8[Pg.135]

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

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