Solid harmonics

The Slater-Roothaan method uses solid-harmonic Gaussians [18] to fit the molecular orbitals and five other nonnegative quantities, namely the total density, the cube root of the partitioned density for both spins, and the 2/3 power of the partitioned density for both spins. They are treated as five additional orbitals of the totally symmetric irreducible representation. Changing them slightly does not affect the robust energy at all. 

Solid-harmonic solutions J1/ and N j of the homogeneous Helmholtz equation in cell pi are products of spherical Bessel functions and spherical harmonics. Specific functional forms for the regular and irregular solid harmonics, respectively, are [188]... 

The function x can be represented in x by a sum of regular solid harmonic functions,... 

If the matrix C is not singular, which requires the number of basis functions to match the number of solid harmonics used to expand the Green function, a local r-matrix is defined by t = —SC l. The consistency condition expressed above in terms of C and S matrices then reduces to the simple matrix expression... 

Matrices C, S, C and S here are to be considered as rectangular matrices. The internal sums over solid-harmonic L-indices should be carried to convergence. The L, L indices of the square matrix co are basis function indices and may have a smaller range. 

The complementary potential Ad or generalized Madelung term is expanded in regular solid harmonics as... 

Vi = Y so that Ytm is the value of the solid harmonic at points on the surface of the unit sphere defined by the coordinates 8 and (p, and hence Y is called a surface harmonic of degree l. Surface harmonics are orthogonal on the surface of the unit sphere and not at r = 0, as commonly assumed in the definition of atomic orbitals. 

Here Vr is an appropriate permutational symmetry projection operator for the desired state, T, and YfcM is a product of coupled solid harmonics labeled by the total angular momentum quantum numbers L and M. Permutational symmetry is handled using projection methods in the same manner as described for the potential expansion in the previous section. Again, the reader is referred to the references for details[9,10,12],... 

YkM is a vector coupled product of solid harmonics[69] given by the Clebsch-Gordon expansion,... 

We are now able to perform an integration over the unit sphere of the product of three solid harmonics and with... 

In the Cartesian scheme (Eq. (19)), there are (/+1)(/+ 2)/2 components of a given /, whereas the number of independent spherical harmonics is only 21+ 1. Usually, therefore, the Cartesian GTOs are not used individually but instead are combined linearly to give real solid harmonics (see Ref. 1). In addition, for a more compact and accurate description of the electronic structure, the GTOs (Eq. (19)) are not used individually as primitive GTOs but mostly as contracted GTOs (i.e., as fixed, linear combinations of primitive GTOs with different exponents a). 

Solid Harmonic Bases and Their Symmetry Properties. 206... 

Here we use as basis functions the spherical harmonics on a real solid harmonic form. They are eigenfunctions of the orbital angular momentum operator,... 

Phase-fixed 3-jT Symbols and Coupling Coefficients for the Point Groups When fi is a solid harmonic hi, it is subject to the condition... 

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