Solid spherical harmonics

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

The importance of this formula is that each spherical harmonic is defined by specific values of the quantum numbers l and mi. As an example, the solid spherical harmonics... 

Molecular Integrals by Solid Spherical Harmonic Expansions. 

Table 7. The two sets of solid spherical harmonics (/ = 2 and 1 = 4), appearing in the integrals d>, are normalized to 4nj(21+1) over the unit sphere...

If we now examine solutions of Laplace s equation (9-128) for do, we find that there is only one term that exhibits an r l dependence, namely, C/r, where C is an arbitrary constant. Furthermore, Brenner12 has shown that the next term in an expression for 60 for large r 1 will be a solid spherical harmonic of 0(r 2), provided only that the origin of coordinates is chosen at the proper point inside the body, and this is always true independent of the body shape. Thus, in general,... 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

E. O. Steinbom, K. Ruedenberg, Rotation and Translation of Regular and Irregular Solid Spherical Harmonics, Adv. Quantum Chem. 7 (1973) 1-81. 

Even-tempered basis sets. - Large basis sets can be efficiently generated by utilizing the concept of an even-tempered basis set. Such a basis set consists of pure exponential or pure Gaussian functions multiplied by a solid spherical harmonic, that is a spherical harmonic multiplied by rl. Thus an even-tempered basis set consists of Is, 2p, 3d, 4/,. .. functions. A set of even-tempered basis functions is thus defined by... 

P6rez-Jorda, J., and Yang, W. (1996). A concise redefinition of the solid spherical harmonics and Its use In fast multipole methods,/. Chem. Phys. 104,8003-8006. 

They vanish at the origin and are finite at large distances r. This is in contrast to irregular solid spherical harmonics ... 

External electric fields transform as the regular solid spherical harmonics, (p), for which the Laplacian vanishes, V r Yemi, v) = 0. There is therefore no spin-free correction of 0(c ) to the property operator, and the operator can be written as... 

Point nuclear electric fields transform as the irregular solid spherical harmonics for which the Laplacian is a generalized delta function. For finite... 

The potential within the volume of the solute molecule is a smooth function and can be expanded in an appropriate set of expansion functions gifr). This is a generalization of the usual multipole expansion. The latter uses Cartesian monomials 1, x, y, z, xy, y, xz, yz, x, ..., or the corresponding solid spherical harmonics to expand the potential. However, for reasons explained in [17], the origin-centered multipole expansion is unsuitable for most systems. We experimented with several expansion sets. One important requirement is that the expansion functions should not diverge to infinity like the solid harmonics do. We finally settled on a sine function expansion of the potential. It is conveniently defined in an outer box that is larger than the extent of the molecular electronic density, to avoid problems with the periodic nature of the sine expansion. Other... 

Fig. 1 Projected oscillator orbitals (b-d) generated by projection of the products of first-order solid spherical harmonics polynomitils with the O lone pair orbital of H2C = O seen in a. The harmonics are aligned with the local frame axes, i.e., the principeil axes of the...

Higher order oscillator orbitals, not used in the present work, can be generated in an analogous manner, using higher order solid spherical harmonic functions. 

The quantities r Yij 9,) and Yi,m(9,)/r + appearing in equations (67) and (68) are known as solid spherical harmonics. Because of their central role in multipole approximations, it is important to have optimized procedures for their generation and manipulation. An equivalent and efficient reformulation of the solid spherical harmonics has been published recently, with the interesting property of having very simple derivatives with respect to Cartesian coordinates, which are required for the computation of forces, or for obtaining useful recurrence relations for the integrals of solid spherical harmonic with Gaussian basis functions. 

The evenly-loaded solid spherical harmonics have a terminating two-range ADT given by ... 

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