Sheridan, R.P. (2000) The centroid approximation for mixtures calculating similarity and deriving structure-activity relationships. J. Chem. Inf. Comput. Sci., 40, 1456—1469. [Pg.1169]

Details of the derivation for Kleinert s variational perturbation (KP) theory can be found elsewhere [21], The nth-order KP approximation W, (r) to the centroid potential W (f) is given by [Pg.90]

Figure 5. Plot of the centroid-constrained correlation function Q(t, q ) defined in Eq. (2.1) for the nonquadratic potential described in Section II.E [Eq. (2.66)]. The numerically exact results are shown by the solid circles. The correlation function obtained from the optimized LHO approximation in Eq. (2.39) is shown by the dashed line, while the solid line depicts the results obtained by including the two-line-loop correction from Eq. (2.42). The correlation functions are plotted as a function of m = r/fi/3 and for /3 = 10 and q = 0.0. |

The operator 4(5) is then represented in 2Af-dimensional Fourier space and the cumulant average over the phase-space path fluctuation variables 5(r) is carried out through second order [5]. After performing the inverse Fourier transformation, the final result for the operator average in the phase-space centroid picture is approximated by the classical-like form [5] [Pg.159]

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives [Pg.49]

Let us now consider A tt atomic orbital is essentially a pure p orbital. If there is any polarization (as will be discussed below), this will involve a very small displacement Ar,r/ of the centroid of the orbital. Then a monopole approximation of the 77 term of Eq. (11.9) suffices, giving [Pg.137]

The unknown cubic and quartic terms, which are small, enter with a small weight and may safely be neglected in this case. Even the second term in this series amounts to less than 1% of the leading term (sometimes referred to as the r-centroid approximation), except at small separations where R is comparable to the collision diameter ( 5.71 bohr) and corrections amount to +2%. [Pg.167]

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