Quadratic expansion coefficients

By equating the individual terms, the dependence on the time interval of the coefficients in the quadratic expansion of the second entropy may be obtained. Consider the expansions... 

The scattering function can be expanded for low values of q. The expansion coefficients correspond to high moments involving summations of powers of different intramolecular distances (it is well known that the mean quadratic ra-... 

Here we give explicit expressions for the coefficients in the power series expansions of the potential, the azimuthal angle, and the vector potential. The coefficients in the quadratic expansion, (3.13), of the potential are... 

The subscript / has been omitted from A fc(r) for notational simplicity. The action S becomes a quadratic function of the expansion coefficients (p<, Qi) when the expansions (118) are substituted into the action integral. The variational principle then leads to a system of linear equations for the expansion coefficients,... 

McCormick developed two complementary sets of equations involving quadratic integrals for determining the albedo and an arbitrary number of scattering expansion coefficients, provided unpolarized intensity measurements are dependent on both the polar and azimuthal angles [7]. Numerical tests were performed by Oelund and McCormick [11] that demonstrated the sensitivity of the estimated parameters to simulated errors in the measurements, and provided insight into which of the two independent sets of equations was better. 

Note that Eq. (26) assumes a quadratic expansion of the energy as a function of the Cl parameters E(C) about a minimum. This approximation results in some error in the gradient. In practice, we also neglect the derivative due to the complex phase of the Cl expansion coefficients, i.e., we calculate the matrix C by expressing the real vector M(t ) by a rotation between M(r -]) and its orthogonal complements. As we show in Sect. 4.2, these errors are corrected via numerical fitting of the hypersurface along the trajectory. 

NORDIO - According to the Rotational Isomeric State approximation, the geometry dependent interactions are simply expressed by additional terms in the total energy. On the other hand, the hydrodynamic interactions give rise to configuration dependent friction coefficients. Under these assumptions, the stationary distribution function is uniquely defined, but the construction of the site functions requires identification of a reactive path. This is done by quadratic expansion of the multivariate diffusion equation about the saddle point connecting two stable conformers, followed by a normal mode analysis. 

A quadratic form of the kinetic energy can be obtained similarly. However, the expansion coefficient corresponding to the mass is now related through... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The form of the symmetric matrix of coefficients in Eq. 3-20 for the normal equations of the quadratic is very regular, suggesting a simple expansion to higher-degree equations. The coefficient matrix for a cubic fitting equation is a 4 x 4... 

To prove this let us make more precise the short-time behaviour of the orientational relaxation, estimating it in the next order of tfg. The estimate of U given in (2.65b) involves terms of first and second order in Jtfg but the accuracy of the latter was not guaranteed by the simplest perturbation theory. The exact value of I4 presented in Eq. (2.66) involves numerical coefficient which is correct only in the next level of approximation. The latter keeps in Eq. (2.86) the terms quadratic to emerging from the expansion of M(Jf ). Taking into account this correction calculated in Appendix 2, one may readily reproduce the exact... 

In a recent publication [22] we reported the implementation of dispersion coefficients for first hyperpolarizabiiities based on the coupled cluster quadratic response approach. In the present publication we extend the work of Refs. [22-24] to the analytic calculation of dispersion coefficients for cubic response properties, i.e. second hyperpolarizabiiities. We define the dispersion coefficients by a Taylor expansion of the cubic response function in its frequency arguments. Hence, this approach is... 

The following notation has been introduced in Eq. (4.92) As denote coefficients of terms linear in the Casimir operators, A.s denote coefficients of terms linear in the Majorana operators, Xs denote coefficients of terms quadratic in the Casimir operators, Ks denote coefficients of terms containing the product of one Casimir and one Majorana operator, and Zs denote coefficients of terms quadratic in the Majorana operators. This notation is introduced here to establish a uniform notation that is similar to that of the Dunham expansion, where (Os denote terms linear in the vibrational quantum numbers, jcs denote terms that are quadratic in the vibrational quantum numbers and y s terms which are cubic in the quantum numbers (see Table 0.1). Results showing the improved fit using terms bilinear in the Casimir operators are given in Table 4.8. Terms quadratic in the Majorana operators, Z coefficients, have not been used so far. A computer code, prepared by Oss, Manini, and Lemus Casillas (1993), for diagonalizing the Hamiltonian is available.2... 

The hydrogen atom in a linear external field (charmonium) and in a quadratic external field (harmonium) has already been studied by Vrscay (1983, 1984, 1985). In both cases, Lie algebraic methods provided a considerable number of coefficients in the perturbation series expansion for small values of the coupling constant. However, these results were obtained for a... 

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