Legendre

The interaction energy can be written as an expansion employing Wigner rotation matrices and spherical hamionics of the angles [28, 130], As a simple example, the interaction between an atom and a diatomic molecule can be expanded hr Legendre polynomials as... 

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... 

By differentiating the defining equations for H, A and G and combining the results with equation (A2.T25) and equation (A2.T27) for dU and U (which are repeated here) one obtains general expressions for the differentials dH, dA, dG and others. One differentiates the defined quantities on the left-hand side of equation (A2.1.34), equation (A2.1.35), equation (A2.1.36), equation (A2.1.37), equation (A2.1.38) and equation (A2.1.39) and then substitutes die right-hand side of equation (A2.1.33) to obtain the appropriate differential. These are examples of Legendre transformations. ... 

In a canonical ensemble, the system is held at fixed (V, T, N). In a grand canonical ensemble the (V, T p) of the system are fixed. The change from to p as an independent variable is made by a Legendre transfomiation in which the dependent variable, the Flelmlioltz free energy, is replaced by the grand potential... 

It is convenient to expand7 (0) in a basis of Legendre polynomials / (cos 0) (as these define the natural... 

Figure Bl.12.11. Angular variation of the second- and fourth-rank Legendre polynomials.

Wlien expanded as a series of Legendre polynomials /Jj (cos 0), tire differential cross section has the following form... 

Soper S A and Legendre B L Jr 1998 Single-molecule detection in the near-IR using continuous-wave diode laser... 

The functions P " are associated Legendre polynomials of order m and degree I, and are associated Laguerre polynomials of degree (v — l)/2 in... 

These funetions are ealled Assoeiated Legendre polynomials, and they eonstitute the solutions to the 0 problem for non-zero m values. 

The functions are the associated Legendre polynomials of which a few are given in Table 1.1. They are independent of Z, the nuclear charge number, and therefore are the same for all one-electron atoms. 

M. A. Clarke and B. L. Legendre, Proceedings of the South African Sugar Technologists Association, 1996, in press. 

Equation 54 implies that U is a function of S and P, a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Hehnholt2 energy,, and Gibbs energy, G ... 

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