Power series expansion of the

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

Performing power series expansion of the exponents and using formulae (5.30) and (5.28) respectively we obtain... 

To derive working expressions for the dispersion coefficients Dabcd we need the power series expansion of the first-order and second-order responses of the cluster amplitudes and the Lagrangian multipliers in their frequency arguments. In Refs. [22,29] we have introduced the coupled cluster Cauchy vectors ... 

Table 1. Numerical results for the first four terms in the power series expansions of the velocity profile near a rotating sphere [2].

The two-point correlation function has been worked out explicitly by Berkolaiko et.al. (2001) and has been shown to coincide with the statistics of so-called Seba billiards, that is, rectangular billiards with a single flux line. The first few terms in a power series expansion of the form factor have been derived by Kottos and Smilansky (1999) and Berkolaiko and Keating (1999) and yield... 

The power series expansion of the generalised James-Coolidge firnction, given in elliptic coordinates, has been, therefore, developed specially for two-electron systems and, moreover, cannot be used for nonlinear molecules. 

Following the standard series expansion of Bessel functions, the power-series expansion of the function i (z) near z = 0 has the following form ... 

A power series expansion of the state energy E, computed in a manner consistent with how P is determined (i.e., as an expectation value for SCF, MCSCF, and Cl wavefunctions or as <IHIvP> for MPPT/MBPT or as <lexp(-T)Hexp(T)l> for CC wavefunctions), is carried out in powers of the perturbation V ... 

This equivalence can be shown most easily by carrying out a power series expansion of the function of M (e.g., of exp(M)) and allowing each term in the series to act on an eigenvector. 

The power series expansions of the spherical Bessel functions are (Antosiewicz, 1964)... 

In order to estimate the transcendental number e, we will expand the exponential function ex in a power series using a simple iterative procedure starting from its definition Eq. (25) together with Eq. (12). As a prelude, we first find the power series expansion of the geometric series y — 1/(1 + x), iterating the equivalent expression ... 

This is termed a memoryless non-linearity since the output is a function of only the present value of the input s[n The expression may be regarded as a power series expansion of the non-linear input-output relationship of the non-linearity. In fact this representation is awkward from an analytical point of view and it is more convenient to work in terms of the inverse function. Conditions for invertibility are discussed in Mercer [Mercer, 1993],... 

The coefficients of the power series expansion of the extinction efficiency, to a 5th order approximation, were derived by Penndorf ( 17) and independently at our laboratories using Macsyma (18). The coefficients of the series are ... 

Appendix 3.1. Power series expansion of the transformed Hamiltonian... 

Later, Landau and Lifshitz (1959) obtained the same result by averaging the stress tensor over the entire space, thereby initiating one of the first dynamical (i.e., nonenergetic) approaches to calculating the rheological properties of suspensions. Attempts to extend Eq. (4.1) to higher concentrations are legion. Most propose a power series expansion of the form... 

The numerator and the denominator in this equation are the leading terms in the power-series expansions of the logarithms in the numerator and the denominator in Eq. (13). Under the conditions of the experiment, the difference between the values of 7 given by Eqs. (13) and (23) is only about 1 or 2 percent. 

By utilizing the fact that H n> n> and using the power series expansion of the exponential operator, we get... 

The reduced partition functions of isotopic molecules determine the isotope separation factors in all equilibrium and many non-equilibrium processes. Power series expansion of the function in terms of even powers of the molecular vibrations has given explicit relationships between the separation factor and molecular structure and molecular forces. A significant extension to the Bernoulli expansion, developed previously, which has the restriction u = hv/kT < 2n, is developed through truncated series, derived from the hyper-geometric function. The finite expansion can be written in the Bernoulli form with determinable modulating coefficients for each term. They are convergent for all values of u and yield better approximations to the reduced partition function than the Bernoulli expansion. The utility of the present method is illustrated through calcidations on numerous molecular systems. 

It is possible to differentiate the quantum-mechanical electronic energy beyond first order, and means for doing this are discussed in Section III. The second derivatives are the usual polarizabilities, the third derivatives are the hyperpolarizabilities, and so on. These properties are associated with a power series expansion of the energy in terms of the elements of V. A second-degree polytensor is introduced for handling all the polarizabilities [7]. It is a square matrix whose rows and columns are labeled, in anticanonical order, by the same indices that label the elements of the column array M. For example. 

The power series expansion of the second derivative with respect to R [from Eqn. (95)] is truncated at the first term to give an expression for... 

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