Permutational symmetry potential energy surfaces

HCCS radical, Renner-Teller effect, tetraatomic molecules, II electronic states, 633-640 H2D molecule, non-adiabatic coupling, two-state molecular system, 107-109 HD2 molecule, permutational symmetry isotopomers, 713-717 potential energy surfaces, 692-694 Heaviside function ... 

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

In the case of potentials having a lower permutation symmetry, the coordinates (i = 1,2,3) may still be used, but other terms besides those of equation (15) are obviously allowed for the representation. For example, for AB2-type systems, there are two linear terms, two quadratic terms, and one cubic term with C2 symmetry which may be used for the representation of the potential-energy surface, namely29... 

Abstract When considering the work of Carl Ballhausen on vibrational spectra, it is suggested that his use of the Born-Oppenheimer approximation is capable of some refinement and extension in the light of later developments. A consideration of the potential energy surface in the context of a full Coulomb Schrodinger Hamiltonian in which translational and rotational motions are explicitly considered would seem to require a reformulation of the Born-Oppenheimer approach. The resulting potential surface for vibrational motion should be treated, allowing for the rotational motion and the nuclear permutational symmetry of the molecule. 

Keywords Born-Oppenheimer Rotation-vibration spectra Potential energy surface Self-adjoint Permutational symmetry... 

The local permutational symmetry [Aa ] [AB ] is restricted such that the total permutational symmetry [A] is contained in T a 1 [V . When [Aa] [Ab] and [Aa ] [AB ] are not equal the corresponding separated molecule energies are different. Then for [Aa] [AB] / [Aa ] [AB ], the [Aa] [Ab] and [Aa ] [AB ] states are on different potential surfaces, and the process (5-9) is nonadiabatic. Thus the nonadiabatic reaction (5-9) might be expected to be most probably when the spin-free adiabatic potentials approach close to one another, since this is just the condition for the breakdown of the adiabatic approximation (see Sect. IV). 

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