Spherical Harmonics

    The implementation of ISPHER in $CONTRL does not rely 
on using a spherical harmonic basis set, in fact the atomic 
basis remains the Cartesian Gaussians.  Instead, certain 
MOs formed from particular combinations of the Cartesian 
Gaussians (for example, xx+yy+zz) are deleted from the MO 
space.  Thus a run with ISPHER=1 will have fewer MOs than 
AOs.  Since neither the occupied nor virtual MOs contain 
any admixture of xx+yy+zz, the resulting energy and wave- 
function is exactly equivalent to the use of a spherical 
harmonic basis.

    The log file output will contain expansions of each MO 
in terms of 6 d's, 10 f's, and 15 g's, and the $VEC also 
contains the same expansion over Cartesian Gaussians.  Both 
the matrix in your log file and in $VEC will contain fewer 
MOs than AOs, the exact number of MOs used is printed in 
the initial guess section of the log file.  It should be 
possible to read such $VEC groups into runs with different 
settings of ISPHER, should you choose to do so.

    The advantage of this approach is that intelligence in 
the generation of symmetry orbitals combined with the 
capability to drop linearly dependent MO combinations means 
that the details of ISPHER are located only in the orbital 
optimization code, where the variational spaces are simply 
reduced in size to eliminate the undesired contaminant 
functions.  This means that none of the integral routines 
need be modified, as the atomic basis remains the Cartesian 
Gaussians.  The disadvantage is that AO integral files run 
over the Cartesian Gaussians, and thus are not reduced in 
size.  Of course transformed MO integrals and various 
computations in correlated calculations are reduced in 
size, since the number of MOs may be greatly reduced.

    Computationally, the advantages of ISPHER=1 are not 
limited to the reduced CPU time associated with fewer total 
MOs.  Questions about d orbital participation as measured 
by Mulliken populations are cleanly addressed when the d's 
usage in the MOs does not contain any contamination from 
the s shape xx+yy+zz.  Less obviously, the use of spherical 
harmonics frequently greatly reduces problems with linear 
dependency, that exhibit as poor SCF convergence.

created on 7/7/2017