The Spherical Tensor Gradient Operator

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Title:The Spherical Tensor Gradient Operator
Creators:
Weniger, Ernst Joachim
Journal or Publication Title:
Collection of Czechoslovak Chemical Communications, 70, 8, pp. 1225-1271
Uncontrolled Keywords:Bossel functions, Spherical delta functions, Hobson's theorem, Spherical tensor, Cartesian components, Quantum chemistry

Abstract

The spherical tensor gradient operator <i>Y</i><sub>l</sub><sup>m</sup>(∇), which is obtained by replacing the Cartesian components of <strong><i>r</i></strong> by the Cartesian components of ∇ in the regular solid harmonic <i>Y</i><sub>l</sub><sup>m</sup>(<strong><i>r</i></strong>), is an irreducible spherical tensor of rank <i>l</i>. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank <i>l</i>. 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 <i>Y</i><sub>l</sub><sup>m</sup>(∇) can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. Math. London Soc. <i>24</i>, 54 (<strong>1892</strong>)]. 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 <i>Y</i><sub>l</sub><sup>m</sup>(∇) is applied to them. Fourier transformation is very helpful in understanding the properties of <i>Y</i><sub>l</sub><sup>m</sup>(∇) since it produces <i>Y</i><sub>l</sub><sup>m</sup>(-i<strong><i>p</i></strong>). It is also possible to apply <i>Y</i><sub>l</sub><sup>m</sup>(∇) to generalized functions, yielding for instance the spherical delta function δ<sub>l</sub><sup>m</sup>(<strong><i>r</i></strong>). The differential operator <i>Y</i><sub>l</sub><sup>m</sup>(∇) 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 <i>r<sup>v</sup>Y</i><sub>l</sub><sup>m</sup>(<strong><i>r</i></strong>) with <i>v</i> ∈ πR. <p>

Title:The Spherical Tensor Gradient Operator
Creators:
Weniger, Ernst Joachim
Uncontrolled Keywords:Bossel functions, Spherical delta functions, Hobson's theorem, Spherical tensor, Cartesian components, Quantum chemistry
Divisions:Life and Chemical Sciences > Institute of Organic Chemistry and Biochemistry > Collection of Czechoslovak Chemical Communications
Journal or Publication Title:Collection of Czechoslovak Chemical Communications
Volume:70
Number:8
Page Range:pp. 1225-1271
ISSN:0010-0765
E-ISSN:1212-6950
Publisher:Institute of Organic Chemistry and Biochemistry
Related URLs:
URLURL Type
http://dx.doi.org/10.1135/cccc20051225UNSPECIFIED
ID Code:2412
Item Type:Article
Deposited On:06 Feb 2009 17:18
Last Modified:06 Feb 2009 16:18

Citation

Weniger, Ernst Joachim (2005) The Spherical Tensor Gradient Operator. Collection of Czechoslovak Chemical Communications, 70 (8). pp. 1225-1271. ISSN 0010-0765

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