Multipole expansions of irreducible tensor sets and some applications

Abstract

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.

Decay of a negative molecular ion in a constant electric field

“…The explicit form of matrix elements (⑀) is given by the relations (10) (11) in deriving these expressions, we used formula (6) from [37] for the expansion of the irreducible rank J tensor depending on vector r j + R j into a series in spherical harmonics (r j ) taking into account only the zero order terms in r j in the expansion coefficients. Formulas (10) and (11) obviously lead to the following symmetry relations:…”

“…where Taking into account only the principal (first) terms in expansions (36) and (37) and replacing exact energy …”

“…(13) for ⑀ will be taken into account in perturbation theory based on the solu tion of system (38). For this purpose, we write ⑀ in the form and confine the solution of transcendental equation (13) to the term linear in Δ⑀ and the terms on the order of F and F 2 , which appear from expansions (36) and (37) for matrix elements in Eq. (13).…”

“…(62) using the substitution k 0 k = in the exchange term Q(k 0 ), in (k 0 ), and in (k 0 ). Writing now ⑀ in formula (67) in the form ⑀ ± = E ± (F) + Δ⑀ ± and taking into account the terms on the order of F appearing in expansions of type (36), (37) for matrix elements in (67), we obtain complex correction Δ⑀ ± to E ± (F):…”

Multipole expansions in four-dimensional hyperspherical harmonics

“…In the three-dimensional space such a technique is already known [4]. In the present paper, the multipole expansions in four-dimensional space are considered.…”