Legendre expansion

This Legendre expansion converges rapidly only for weakly anisotropic potentials. Nonetheless, truncated expansions of this sort are used more often than justified because of their computational advantages. 

A more natural way to account for the anisotropy is to treat tire parairreters in an interatomic potential, such as equation (A 1.5.64). as fiurctioirs of the relative orientation of the interacting molecules. Comer [131] was perhaps the first to use such an approach. Pack [132] pointed out that Legendre expansions of the well depth e and equilibrium location of the interaction potential converge more rapidly tirair Legendre expansions of the potential itself... 

Lead, excess entropy of solution of noble metals in, 133 Lead-thalium, solid solution, 126 Lead-tin, system, energy of solution, 143 solution, enthalpy of formation, 143 Lead-zinc, alloy (Pb8Zn2), calculation of thermodynamic quantities, 136 Legendre expansion in total ground state wave function of helium, 294 Lennard-Jones 6-12 potential, in analy-... 

The Raman measurements provide values directly for P)mn, the coefficients of the Legendre expansion related to coordinates axes chosen with respect to the principal axes of the differential polarizability tensor, hence the superscript r. The coefficients Pimn for the orientation of the units of structure must then be obtained by further calculation from the P)mn. 

Figure 1. Spin-dependent terms Cq, —7, in Legendre expansion coefficients as functions of the interfragment distance R dashed curves from Asymptotic Theory (AT) solid curves from AT corrected to fit ah initio data.

Legendre expansion with two terms (2) and elliptic-sine profile (3)... 

G.D. JOANOU and H. FENECH, Fast Neutron Cross Sections and Legendre Expansion Coefficients tor Oxygen-16, GA-3564, General Atomic (1962). 

Higher-order moments P2, Pe of the Legendre expansion can also play a role as the order parameters of the nematic phase. 

In Fig. 2 we report the spherical average potential Vq(R) for the Ne-N2/ O2 and NO systems compared with the corresponding isotropic Ne-Ar case. The Vq(R) term, i.e., thefirst term in the Legendre expansion or the potential surface, is easily obtained from the full PES by a Gauss-Legendre quadrature, according to the formula ... 

A second type of simplification present in systems having symmetry is that the diatom in at least one arrangement will be homonuclear. This can be exploited in several places. First of all, the work in the quadratures to determine the potential expansion coefficients of Eq. (28) can be cut in half because the interaction potential is an even function. This also means that only even values of A occur in the Legendre expansion. Thus even and odd rotational quantum numbers are not coupled by the distortion potential and so the size of the distortion potential blocks decreases as does also the time to solve for the radial fimctions and the resources required to save them. 

This causes a change in sign of the terms in the Legendre expansion of the potential which correspond to odd angular functions [see Eq. (28)]. This is... 

Art expression for the fluidity might be developed a number of ways. Here we collate the fluidity with the temperatures at the Lobatto points that is, at the walls and at two interior points. The Legendre expansion for the fluidity is then developed in a manner completely analogous to that for the temperature. For example,... 

The expression for the sur ce energy can also be predicted for other shapes of the sur ce energy which are different from the Rapini-Papoular energy type, for example the elliptic type, Legendre expansion [14], and so oa Thwe is presently a widespread intwest in the inclusion of the surface elastic moduli Kb and K24 in the theory to allow for novel contributions to the surfrice anchoring energy to be taken into account [23] (see Datareview 5.3). 

Recently Gislason and Kosmas have suggested expanding 1(0) sin 0 in a Fourier sine series. Their procedure is more convenient to use than the Legendre expansion method. The uncertainty (noise) in the Fourier coefficients is known exactly, so it is relatively easy to determine where to truncate the series. In addition, they have... 

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