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Some comments on the electrostatic potential of a molecule
Author(s) -
Larson Everett G.,
Li Mingsheng,
Larson Guy C.
Publication year - 1992
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560440817
Subject(s) - multipole expansion , scalar (mathematics) , physics , charge density , operator (biology) , charge (physics) , atomic orbital , quantum mechanics , field (mathematics) , point particle , fast multipole method , classical mechanics , chemistry , electron , mathematics , biochemistry , geometry , repressor , transcription factor , pure mathematics , gene
In this article we discuss several principles and tools which should expedite description of the electrostatic potentials and electrostatic interactions of molecules, and show that these also lead to some rather remarkable results in the theory of the irreducible representations of the full rotation group SO (3). First, by representing a molecule's charge‐density matrix over a basis of atomic‐like orbitals (on the various atoms), we observe that outside its charge distribution the molecule's electrostatic potential is exactly the same as if that charge distribution were merely a sum (and in the case of a finite orbital basis, this is a finite sum) of point multipoles on each of the atomic centers and line multipoles on the line segments joining each of those atomic centers. Possible methods of approximating the field of these line charges and line multipoles, as if they were due to point charges and point multipoles, are discussed. The calculation of the interaction of point multipoles of high order, as is necessary for this procedure to successfully calculate the interaction of arbitrarily oriented molecules, motivates our second topic. Here we present a differential operator which, when acting on the 3‐dimensional delta function, produces the source density for a scalar field that is exactly an ( l , m ) multipole field. Using the Hermitian adjoint of this operator, we express the interaction of this ( l , m ) multipole with an external scalar field as the result of this differential operator acting on that external field at the location of this multipole source. Irreducible representation matrices of the full rotation group are then used, together with these relations, to simplify the interaction of two arbitrarily oriented multipoles of any orders. Finally, we use the representation of the Condon and Shortley “raising and lowering” relations on eigenstates of the z ‐component of angular momentum, in an orientation that is not aligned with its fundamental basis states, to generate recursion relations that allow simple calculations of the irreducible representation matrices of the full rotation group, SO (3), and the special unitary group, SU (2). From these recursion relations we display some useful symmetry properties of our parameterization of these matrices, that allow the entire matrix to be very simply generated from an explicit calculation of only about 1/8 of its elements. © 1992 John Wiley & Sons, Inc.