External molecular rotations

9.4 External molecular rotations

All non-linear polyatomics have principal moments of inertia I A ,I B ,I C and corresponding rotational constants A = ħ 2 /2I

A , etc. When two of the rotational constants are equal to each other, the molecule is termed a symmetric top. In the following, we will assume that A ≠ B = C. The rotational energy of a symmetric top is given by

E r ( JK

) 2 = BJ J + ( ) + (A ! B)K

where quantum number J refers to a two-dimensional 2-D adiabatic rotor (i.e. one that conserves angular momentum J) and K refers to a one-dimensional rotation about the top axis (projection of J on the top axis). For a rigid rotor, the K quantum number is constrained to take integer values from -J to +J. Thus each value of |K| is doubly degenerate except for |K|=0, which is singly degenerate.

The rotational states of asymmetric tops are not easily calculated, but sufficiently accurate partition functions and densities of states (but not for high resolution spectroscopy) are obtained by averaging the two rotational constants that are most similar to each other and treating the asymmetric top approximately as a symmetric top. For use in DenSum one can use

either of two reasonable approximations for the effective rotational constant B 2D for the J-Rotor (when B ≈ C, for example):

B 1/ 2 16,18

2D = ( C ) [Ref. ] or B 2D ! ( B +C 64 ) [Ref. ]

The effective rotational constant B K for the K-rotor is given by

B K =A!B ( ) .

(This is often pragmatically approximated as B K ≈ A.) For use in Thermo , the best approximate forms are B 1/2

2D = (BC) and B K = A since these expressions give the correct partition functions for all non-spherical, non-linear molecules. The corresponding rotational energy becomes

E r 2 ( J ,K ) =B 2D JJ ( +1 ) +B K K

Note that entering I A ,I B , and I C (or A, B, and C) in the data file for DenSum or Thermo is NOT correct, since the state densities and partition functions will not be computed correctly. In any non-linear polyatomic, the three external rotations are constrained by the total angular momentum: the three rotors are not independent. Thus inputting the three rotors as independent 1-D rotations leads to an incorrect result. The approximate symmetric top treatment, which is the convention adopted in MultiWell, includes the constraint and thus gives the right answer (with quite high accuracy).

Active Energy and the K-rotor

In a free molecule, only the total angular momentum is conserved. Thus the J quantum number is assumed to be adiabatic (conserved), but the K quantum number can vary within its constraints, as do the vibrational quantum numbers, subject to conservation of energy in the "active" degrees of freedom. The "active energy" is the energy associated with the vibrational quantum numbers and K, collectively. Thus it is said that the energy in the active degrees of freedom randomizes, or the K-rotor energy "mixes" with the vibrational energy.

For a prolate symmetric top (e.g. a cigar shape), A>B and therefore B K >0. For prolate symmetric tops, the rigorous constraint -J≤K≤+J is often relaxed in the interest of more convenient computation of densities of states. This popular approximation is reasonably accurate except at the lowest total energies and is simple to implement because the rotational energy

E r (J,K) ≥0 for all values of J and K. For an oblate symmetric top (e.g. disk-shaped), A<B and hence B K <0; this can give

E r (J,K) <0 when the unconstrained K >> J. Thus the approximate treatment of the K-rotor may fail seriously for oblate tops in Densum. Moreover, densities of states are usually computed by

treating the J-rotor and the K-rotors as separable. Thus the term B 2 K K in the expression for rotational energy is <0, confounding the usual methods for computing densities of states, which assume only positive energies.

To utilize the simple approximation in DenSum, the K-rotor is declared as a simple 1-

D free rotation (type "rot" or "qro") and included with the vibrational degrees of freedom when computing the density of states. The kro degree of freedom type employs the correct constraints on K for a single user- specified value of J. Because it considers only a single value of J, it is not used for general applications.

The rotational degrees of freedom in a MultiWell master equation calculation are distributed in two input files: the K-ROTOR properties are listed in densum.dat and included in density and sums of states calculations. The 2-D ADIABATIC ROTOR moment of inertia is listed in multiwell.dat on Line 8 (for wells) or Line 14 (for transition states).