Power series expansion convergence

McWeeny then regarded the elements of A as determining the minimization problem and clearly the steepest descent is along a negative multiple (—A, say) of the quantity in square brackets in equation (32). Given that there is a convergent power-series expansion for the inverse in equation (29), it then... 

For small displacements, of the order of vibrational amplitudes at room temperature, the terms in the power series expansion (1) converge fairly rapidly, and higher-order terms are related to successively smaller-order effects in the spectrum, so that they become more and more difficult to determine. Almost all calculations to this date have been restricted to determining quadratic, cubic, and quartic force constants only [the first three terms in equation (1)], and in this Report we shall not consider higher-than-quartic terms in the force field. The paper by Cihla and Chedin11 is one of the few exceptions in which force constants involving up to the sixth power have been determined for a polyatomic molecule, namely COa. 

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

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. 

Unfortunately, the above procedure cannot be used when [] - [/]] is singular and cannot be inverted. Hence, the matrix function [3], though finite, cannot be obtained in this way. A power series expansion of [3] is not convergent for all [< >]. On the other hand, an expansion of the inverse, [3] is convergent and can be calculated even when [] is singular. The series [3] may be expressed as (Taylor and Webb, 1981)... 

By using the general power series expansion for U all the infinitely many parametrizations of a unitary transformation are treated on an equal footing. However, the question about the equivalence of these parametrizations for application in decoupling Dirac-like one-electron operators needs to be studied. It is furthermore not clear a priori whether the antihermitean matrix W can always be chosen in the appropriate way the mandatory properties of W, i.e., its oddness, antihermiticity and behavior as a certain power in the chosen expansion parameter, have to be checked for every single transformation U applied to the untransformed or any pre-transformed Hamiltonian. Since the even expansion coefficients follow from the odd coefficients, the radius of convergence Rc of the power series depends strongly on the choice of the odd coefficients. 

The (monochromatic) electric fields are characterized by Cartesian directions indicated by the Greek letters and by circular optical frequencies, ( i, ( 2, and 0)3. The induced dipole moment oscillates at (0 = 2i cOi. and are such that the p and y values associated with different NLO processes converge towards the same static value. The 0 superscript indicates that the properties are evaluated at zero electric fields. Eq. (2) is not the unique phenomenological expression defining the (hyper)polarizabilities. Another often-applied expression is the analogous power series expansion where the 1/2 and 1 /6 factors in front of the second- and third-order terms are absent. 

Approximate many-electron wave functions are then constructed from the Hartree-Fock reference and the excited-state configurations via some sort of expansion (e.g., a linear expansion in Cl theory, an exponential expansion in CC theory, or a perturbative power series expansion in MBPT). When all possible excitations have been incorporated (S, D, T,. .., for an -electron system), one obtains the exact solution to the nonrelativistic electronic Schrodinger equation for a given AO basis set. This -particle limit is typically referred to as the full Cl (FCI) limit (which is equivalent to the full CC limit). As Figure 5 illustrates, several WFT methods can, at least in principle, converge to the FCI limit by systematically increasing the excitation level (or perturbation order) included in the expansion technique. 

For this purpose, we consider the functions of momentum to be defined by their power series expansion. The expansion is not convergent for p > me, but provided we do not truncate the expansion we may make transformations on the series outside its radius of convergence as a representation of the operator and re-sum the series in the... 

Still Strong, must be moderated. We note, first, that it tells us the appropriate form of the equation of state, namely that the compression factor Z = pV/nRT, has a power series expansion in the molar density n/V, where n is the amount of substance. This expansion is convergent at the low densities and divergent at high. We do not know the limit of convergence but it must be below the critical density at temperatures at and below T. Perhaps fortunately, the mathematical range of convergence is unimportant in practice since the expansion is useless when it needs to be taken to more than two or three terms. 

The two important convergence parameters introduced by the power series expansion of the ABC Green s function are the total propagation time T, and the time step At. The total time T represents the time required for reaction and absorption by e(q). The time step At is the duration for which the STP is a faithful representation of the propagator. These are both a function of the dynamics and the choice of absorbing potential. We measure At in units of a fundamental small time given by... 

This can be achieved most easily with a careful choice of internal coordinates. Since anharmonic force fields are usually expanded only up to fourth order, it is of considerable importance that cubic and quartic interaction terms in internal coordinates are minimized due to the curvilinear nature of these coordinates. However, this does not mean that a truncated power series expansion will behave correctly farther away from the reference geometry usual expansions suffer from convergence problems. 

Using equations (28) and (29) we obtain a set of equations for the new expansion coefficients B . The new expansion has better convergence properties than the power series expansion. The functions (x) are harmonic oscillator wave-functions. Thus the hermite expansion involves an expansion around the classical path in a harmonic oscillator basis set. The hermite-corrected GWP can be used in a basis set expansion so as to approach the exact quantum theory fi"om the classical path limit in a systematic fashion. Hence in this sense the method is complementary to the multitrajectory approach discussed below. The advantage of the hermit basis is however that it is an orthorgonal basis and therefore somewhat simpler to work with in practical computations. ... 

The coefficients are found from the orthonormal behavior of sine and cosine functions, which we will discuss later in the chapter. Fourier series differ from power series expansions in at least two important ways. First, the interval of convergence of a power series is different for different functions the Fourier series, on the other hand, always converges between —n and -l- r. Second, many functions cannot be expanded in a power series, whereas it is rare to find a function that cannot be expanded in a Fourier series. 

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

The values are supposed to be finite. There are no assumptions on the convergence of the power series (I.I) formal expansions... 

The generality of a simple power series ansatz and an open-ended formulation of the dispersion formulas facilitate an alternative approach to the calculation of dispersion curves for hyperpolarizabilities complementary to the point-wise calculation of the frequency-dependent property. In particular, if dispersion curves are needed over a wide range of frequencies and for several optical proccesses, the calculation of the dispersion coefficients can provide a cost-efficient alternative to repeated calculations for different optical proccesses and different frequencies. The open-ended formulation allows to investigate the convergence of the dispersion expansion and to reduce the truncation error to what is considered tolerable. 

In the limit as 2 —> oo, this ratio becomes p/k, which approaches zero for finite p. Thus, the series converges for all finite values of p. To test the behavior of the power series as p oo, we consider the Taylor series expansion of... 

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