Russell A. Bonham

## Some Integrals Involving Symmetric-Top Eigenfunctions NB   CDF   PDF

Using closure over a complete set of rotational states for methane to evaluate the intensity for quasi-elastic electron scattering in the first Born approximation, a number of integrals were encountered that appear not to have been evaluated previously. Mathematica was employed to evaluate these and similar integrals, and it was discovered that in all the cases studied the results could be represented by simple formulas.

### Details of the Integral Evaluations

The integrals discussed here are related to averages over the spherical matrix elements [1, 2] defined by

which are symmetric-top eigenfunctions with eigenvalues and a degeneracy . The function is defined [1, 2] as

where the sum is over all integer values of from 0 to the first negative factorial that occurs in the denominator. In the impulse approximation for quasi-elastic electron scattering [3], averages over the rotational motion of a spherical top molecule such as methane are integrals of the following form:

where

is the Hamiltonian operator whose eigenfunctions are the Legendre polynomials with eigenvalues . Equation (4) is also a part of the Hamiltonian for which the are solutions, but the are not eigenfunctions of (). The occurrence of arises in the treatment of the scattering problem in the impulse approximation [3]. By use of Mathematica it was possible to prove that

where the normalization factor is given by

for all values of and . For integer the results are

where

The use of the word “prove” is based on the results from a Mathematica program. All of these expressions occur as ratios of positive integers and were found to be in exact agreement with the results on the right-hand sides in equation (7) for values of from 0 through 18.

Note that the same result can be obtained for the case where is replaced by if is replaced by

Except for the factor of for and for in the definitions of the , the remaining parts are equivalent to the integrals

and

Further investigation using found the same result with

where it is noted that

Define the double sum over the integral

and

These results suggest the conjecture that , where is a constant. For and , the constant factor was found to be for , as shown in the next section.

### Implementation

The function d is .

The functions d1 and d2 are the first and second derivatives of with respect to .

The function h is the result of the Hamiltonian operating on d.

Define the integral a.

The function int1 is the integral in equation (14).

The function int2 is the normalization integral of the square of .

The function s is the sum of the ratios of the integrals int1 and int2 over and .

For various choices of , , and , the table compares the exact evaluation of (equation (14)) with the proposed result (equation (15)) and shows that the ratio of the two is a constant, which in this case is . The evaluation of takes some time.

### References

 [1] D. M. Brink and G. R. Satchler, Angular Momentum, Oxford: Clarendon Press, 1979 pp. 21-25. [2] R. A. Bonham, G. Cooper, and A. P. Hitchcock, “Electron Compton-like Quasielastic Scattering from , , and ,” Journal of Chemical Physics, 130 144303, 2009. dx.doi.org/10.1063/1.3108490. [3] G. I. Watson, “Neutron Compton Scattering,” Journal of Physics: Condensed Matter, 8(33) 5955, 1996. iopscience.iop.org/0953-8984/8/33/005. R. A. Bonham, “Some Integrals Involving Symmetric-Top Eigenfunctions,” The Mathematica Journal, 2012. dx.doi.org/doi:10.3888/tmj.14-12.