Abstract
The spherical tensor gradient operator Y-l(m)(del), which is obtained by replacing the Cartesian components of r by the Cartesian components of del in the regular solid harmonic Y-l(m)(r), is an irreducible spherical tensor of rank l. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank l. Thus, it is in principle sufficient to consider only ...
Abstract
The spherical tensor gradient operator Y-l(m)(del), which is obtained by replacing the Cartesian components of r by the Cartesian components of del in the regular solid harmonic Y-l(m)(r), is an irreducible spherical tensor of rank l. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank l. Thus, it is in principle sufficient to consider only multicenter integrals of scalar functions: Higher angular momentum states can be generated by differentiation with respect to the nuclear coordinates. Many of the properties of Y-l(m)(del) can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. Math. London Soc. 24, 54 ( 1892)]. It follows from Hobson's theorem that some scalar functions of considerable relevance as for example the Coulomb potential, Gaussian functions, or reduced Bessel functions produce particularly compact results if Y-l(m)(del) is applied to them. Fourier transformation is very helpful in understanding the properties of Y-l(m)(del) since it produces Y-l(m)(-ip). It is also possible to apply Y-l(m)(del) to generalized functions, yielding for instance the spherical delta function delta(m)(l)(r). The differential operator Y-l(m)(del) can also be used for the derivation of pointwise convergent addition theorems. The feasibility of this approach is demonstrated by deriving the addition theorem of r(v)Y(l)(m)(r) with v is an element of R.
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