Invariant expansion. IV. The exponentials of tensorial expressions

Abstract

The expansion of the exponential of a tensorial expression such as the interaction or pair correlation function between two nonspherical molecules 1, 2 is of the form ∑mnl λmnlΦmnl(12), where Φmnl(12) are invariant tensorial expressions that depend only on the orientation of 1 and 2. The generating function e−∑mnl λmnlΦmnl =∑pqt ipqt(λ) Φpqt defines a generalized Bessel function (GBF). We discuss integral representations and recurrence relations for the GBF. The first GBFs for dipolar and linear quadrupolar exponents, which are of interest in the theory of ionic solutions are computed explicitly.

Orientation-Averaged Pair Potentials between Dipolar Proteins or Colloids

“…Colloidal forces often represent a combination of charge−charge, charge−dipole, and dipole−dipole interactions. In a general scenario, integration in eq 8 requires an expansion of the integrand using techniques such as the cumulant expansion, , or expansion in terms of rotational invariants recently exemplified in the analysis of pure dipolar interaction. A standard procedure applicable in systems with weak dipole−dipole interactions (ν μμ small compared to 1/β), is based on the expansion of the Boltzmann factor in eq 8 leading to For a particular type of interaction (charge−dipole or dipole−dipole), it is easy to show that odd-order cumulants vanish; the series is usually truncated beyond the third-order term giving Clearly, eq 11 captures the exact weak-coupling (high-temperature) limit of the rigorous result given by eq 9.…”

An exponential approximation for continuum percolation in dipolar hard-sphere fluids

“…The replacement of h † (1, 2) in equation ( 9) by equation (24) gives the mean cluster size. The involved integrations are greatly simplified if use is made of the invariant expansion for the exponentials of tensorial expressions reported by Blum and Torruella [38]. We obtain…”

“…In equation (27) the generalized Bessel functions i 00 0 {f 0 (r 12 ); f 1 (r 12 )} are [38] i 00 0 f 0 r 12 ;…”

“…In this last equation i 1 (x) = cosh(x)/x − sinh(x)/x 2 is the modified spherical Bessel function of first order and functions h †0 MSA (r 12 ) and h †1 MSA (r 12 ) are given by equation (28). Equation (38) follows from equation (37) considering the expansion of Blum and Torruella again [38]. The GMSA-based EXP approximation compares with the results obtained from Monte Carlo simulations better than the other exponential approximations, particularly the connectedness y expansion [29].…”

Modelo microscópico de agua líquida: aproximación esférica media generalizada

“…A partir del conocimiento de las funciones de correlación, y considerando que la energía potencial total es una suma de términos de interacciones de pares, es posible calcular la energía interna como integral sobre }Ec 2. Para el caso de la energía interna de exceso, está dada por [64]: En general, utilizando los desarrollos invariantes (3.3.4) para el potencial de interacción y la función de correlación total, y considerando además la ortogonalidad de los invariantes rotacionales [87]:…”