Gaunt coefficient

Since the Gaunt Coefficient (2 k CfJ- > 2 k) may be expressed in terms of Clebsch-Gordon coefficients it is easy to show that the parameters Ejm and 22jm are proportional so that the reduction of the matrix elements of the static potential to combinations of radial integrals over the r/-orbital basis (potential parameters) has the same structure for both local and non-local potentials which can therefore be treated on the same footing. 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

Table 1.5 The unique nonzero real-valued spherical harmonic Gaunt coefficients for expanding atomic orbital products to quadrupole. where...

Real-Valued Spherical Harmonic Gaunt Coefficients... 

A real-valued spherical harmonic Gaunt coefficient corresponds to the integral... 

Fig. 18.7. The integral over the bound-free Gaunt factor which enters the expression for the radiative recombination coefficient. Results are for the k distribution family for capture into the ground n = 1 shell of hydrogen. The curves are generated by numerical quadrature over the distribution function. The limiting Maxwellian curve is the analytic expression elH kT Ei(Ib /ET)/T 2 and corresponds to k —> oo. The x-axis coordinate is Tefr = 2E/Z...

Gaunt and coworkers [77, 78] devised a more technical method. They expanded the coefficient as a function of... 

D, S. Gaunt and A. J. Guttmann, Asymptotic Analysis of Coefficients, in Phase Transitions and Critical Phenomena, ed. C. Domb and M. S. Green (London Academic Press, 1974), p. 181. 

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