Three-Dimensional Addition Theorem for Arbitrary Functions Involving Expansions in Spherical Harmonics

Sack, R. A.

doi:10.1063/1.1704115

Cited by 77 publications

(30 citation statements)

References 9 publications

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“…For the special choice of the function f(r), f(r) = r n , the explicit form of the coefficients in the multipole expansion (17) may be easily derived from (18), (19) where 2 F 1 (α, β; γ ; x) is the Gauss hypergeometric function. These results coincide with those derived by Sack [13]. As we mentioned above in Section 2, the multipole expansion simplifies for tensor functions F jm (r) obeying the Laplace equation, Ñ 2 F jm (r) = 0.…”

Section: Multipole Expansion Of Two-center Functions Of the Form F (Rsupporting

confidence: 78%

“…Unlike the results in [13], our operator expression in (18) for the coefficients B does not depend on the relation between the indices l, l′ and j . In addition, by dealing initially with an arbitrary f(r), we have obtained the nontrivial relation (A.8) for the differential operators dˆ = (1/r)(d/dr).…”

Section: Multipole Expansion Of Two-center Functions Of the Form F (Rmentioning

confidence: 66%

“…In that section we treat the multipole expansion (in terms of bipolar harmonics) of a tensor function that is dependent on the distance between two points specified by the vectors r 1 and r 2 . This type of function is important in many two-center problems, and its multipole expansion, which is similar to that in our equations (17) and (18), was obtained earlier [13] by a noninvariant differential method which requires more cumbersome calculations. In section 4 we derive the multipole expansion (25) for our invariant representation for FRMs in vector form.…”

Section: Introductionmentioning

confidence: 93%

“…However, the projection method allows one to obtain analytic results only for a limited set of functions having simple functional forms so that necessary integrals can be calculated analytically. We present here an alternative, differential approach, which generalizes earlier results by Sack [13] (who dealt with multipole expansions for the special case of two-center functions) as well as by Avery [14] and Wen and Avery [15] (who dealt with bicenter expansions using the vector differentiation technique in three dimensions [14] and m dimensions [15] for the special case of scalar functions). Our general results for the multipole expansion of T jm (a 1 , a 2 , .…”

Section: Introductionmentioning

confidence: 93%

“…Such functions of the distance between two points specified by the vectors r 1 and r 2 are important in many twocenter problems. A number of results for concrete choices of f(r) are presented in [9], and an analysis of the general form of f(r) was performed by Sack [13]. Below we shall show that for the case considered here the expressions for the coefficients in the general multipole expansion (8) may be reduced to a simpler form, not involving gradient-operators.…”

Section: Multipole Expansion Of Two-center Functions Of the Form F (Rmentioning

confidence: 99%

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Multipole expansions of irreducible tensor sets and some applications

Manakov

Meremianin

Starace

2001

J. Phys. B: At. Mol. Opt. Phys.

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A formal multipole expansion of tensor functions dependent on several vector parameters is obtained in an invariant differential form. We apply this result to derive multipole expansions of finite rotation matrices in terms of finite sums of bipolar harmonics. The multipole expansion in terms of bipolar harmonics of unit vectors r 1 and r 2 are analyzed for functions of the type f (r)Y lm (r) (with r = r 1 − r 2 ), which are important in two-center problems. As another example of the application of the multipole expansion technique, reduction formulae are obtained for tensor constructions which appear in analyses of angular distributions in photoprocesses which take exact account of non-dipole (or retardation) effects. As a result, the "photon factors" in the angular distribution for an arbitrary photoprocess are expressed in an invariant form involving only the photon polarization vector and spherical harmonics of the photon wavevector.

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Section: Multipole Expansion Of Two-center Functions Of the Form F (Rsupporting

confidence: 78%

Section: Multipole Expansion Of Two-center Functions Of the Form F (Rmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 93%

Section: Multipole Expansion Of Two-center Functions Of the Form F (Rmentioning

confidence: 99%

See 3 more Smart Citations

Multipole expansions of irreducible tensor sets and some applications

Manakov

Meremianin

Starace

2001

J. Phys. B: At. Mol. Opt. Phys.

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Addition theorems as three-dimensional Taylor expansions

Weniger

2000

Int. J. Quant. Chem.

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A principal tool for the construction of the addition theorem of a r Ји١Ž . function f is the translation operator e , which generates f r q rЈ by doing a threedimensional Taylor expansion of f around r. In atomic and molecular quantum mechanics, one is usually interested in irreducible spherical tensors. In such a case, the application of the translation operator in its Cartesian form e xЈѨ rѨ x e yЈѨ rѨ y e zЈѨ rѨ z leads to serious technical problems. A much more promising approach consists of the use of an operator expansion for e r Ји١ which contains exclusively irreducible spherical tensors. This expansion contains as differential operators the Laplacian ٌ 2 and the spherical tensor m Ž . gradient operator Y Y ٌ , which is an irreducible spherical tensor of rank l. Thus, if l m Ž . Y Y ٌ is applied to another spherical tensor, the angular part of the resulting expression l is completely determined by angular momentum coupling, whereas the radial part is obtained by differentiating the radial part of the spherical tensor with respect to r. Consequently, the systematic use of the spherical tensor gradient operator leads to a considerable technical simplification. In this way, the originally three-dimensional expansion problem in x, y, and z is reduced to an essentially one-dimensional expansion problem in r. The practical usefulness of this approach is demonstrated by constructing the Laplace expansion of the Coulomb potential as well as the addition theorems of the regular and irregular solid harmonics and of the Yukawa potential.

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Accurate modeling of the intramolecular electrostatic energy of proteins

1995

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The (4, +) energy surface of blocked alanine (N-acetyl-N'-methyl alanineamide) was calculated at the Hartree-Fock (HF)/6-31G* level using ab initio molecular orbital theory. A collection of six electrostatic models was constructed, and the term electrostatic model was used to refer to (1) a set of atomic charge densities, each unable to deform with conformation; and (2) a rule for estimating the electrostatic interaction energy between a pair of atomic charge densities. In addition to two partial charge and three multipole electrostatic models, this collection includes one extremely detailed model, which we refer to as nonspherical CPK. For each of these six electrostatic models, parameters-in the form of partial charges, atomic multipoles, or generalized atomic densities-were calculated from the HF/6-31G* wave functions whose energies define the ab initio energy surface. This calculation of parameters was complicated by a problem that was found to originate from the locking in of a set of atomic charge densities, each of which contains a small polarization-induced deformation from its idealized unpolarized state. It was observed that the collective contribution of these small polarization-induced deformations to electrostatic energy differences between conformations can become large relative to ab initio energy differences between conformations. For each of the six electrostatic models, this contribution was reduced by an averaging of atomic charge densities (or electrostatic energy surfaces) over a large collection of conformations. The ab initio energy surface was used as a target with respect to which relative accuracies were determined for the six electrostatic models. A collection of 42 more complete molecular mechanics models was created by combining each of our six electrostatic models with a collection of seven models of repulsion + dispersion + intrinsic torsional energy, chosen to provide a representative sample of functional forms and parameter sets. A measure of distance was defined between model and ab initio energy surfaces; and distances were calculated for each of our 42 molecular mechanics models. For most of our 12 standard molecular mechanics models, the average error between model and ab initio energy surfaces is greater than 1.5 kcal/mol. This error is decreased by (1) careful treatment of the nonspherical nature of atomic charge densities, and (2) *~uthor to whom all correspondence should be addressed. Journal of Computational

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Three-Dimensional Addition Theorem for Arbitrary Functions Involving Expansions in Spherical Harmonics

Cited by 77 publications

References 9 publications

Multipole expansions of irreducible tensor sets and some applications

Multipole expansions of irreducible tensor sets and some applications

Addition theorems as three-dimensional Taylor expansions

Accurate modeling of the intramolecular electrostatic energy of proteins

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