Power series coefficient

This, combined with the more complex integration of dG over a temperature interval at one bar pressure, allowed us to calculate the position of phase boundaries at high pressures and temperatures. The next question is how to evaluate the pressure integral (11.1) when a fluid such as H2O or CO2 is involved, either in the pure form, mixed with other fluid components, or reacting with solid phases Obviously, assuming that the molar volume of a fluid is a constant is not even approximately true, and is unacceptable. A possible way to proceed would be to express U as a function of P in some sort of power series, just as we did for Cp as a function of T (equation 7.12). V dP could then be integrated, and we could determine the values of the power series coefficients for each gas or fluid and tabulate them as we do for the Maier-Kelley coefficients. 

EjAi where Emix — ATpianc E/N above which becomes essentially hard-shell E/N maximizing Kq Convergence radius for K E/N) expansion in a power series Coefficient that sets Eft) within a certain class of profiles Optimum / value Profile of Eft)... 

We obtain the following relation for the power series coefficients of 5 ... 

The lowest energy solution corresponds to = 0. This means that co is the highest nonzero coefficient in the power series expansion for /(y). Hence, we must set C2 = 0. (The odd-powered series coefficients are all zero for this case.) Thus, for n = 0, we can write the unnonnalized wavefunction as [from Eq. (3-24)]... 

Coefficients representing T= f V) are often needed and are also given by Fowdl etal. for types E, K, and T in reduced temperature ranges. These power series coefficients are given for second-, third-, and fourth-order equations with maximum uncertainties indicated for each order and temperature range. 

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... 

Actually the assumptions can be made even more general. The energy as a function of the reaction coordinate can always be decomposed into an intrinsic term, which is symmetric with respect to jc = 1 /2, and a thermodynamic contribution, which is antisymmetric. Denoting these two energy functions h2 and /zi, it can be shown that the Marcus equation can be derived from the square condition, /z2 = h. The intrinsic and thermodynamic parts do not have to be parabolas and linear functions, as in Figure 15.28 they can be any type of function. As long as the intrinsic part is the square of the thermodynamic part, the Marcus equation is recovered. The idea can be taken one step further. The /i2 function can always be expanded in a power series of even powers of hi, i.e. /z2 = C2h + C4/z. The exact values of the c-coefficients only influence the... 

The combinatorial interpretation (or the computations of Sec. 14 regardless of combinatorial considerations) implies the following useful result Substituting a power series with non-negative integer coefficients in the difference of the cycle indices of A, and we get a power series with non-negative integer coefficients. 

Consider, as an example, the functional equation (4) which is satisfied by the power series (3). Comparing the coefficients we find the equations... 

Each of the functional equations (4.1) - (4.10) determines a unique power series. The constant term of the first five solutions is equal to, of the second five it is zero. The other coefficients are positive moreover, they are integers except in the case of equation (4.8). The sequence of coefficients does not decrease. 

These statements are a consequence of the recursion relations obtained by identifying the coefficients of the power series expansion on the right- and left-hand side of the equation. For example, in (4.6), the coefficient of x" is (n > 1) on the left-hand side, and on the right-hand side a polynomial in R, . [cf. (2.56)], which implies the uniqueness. The coefficients of the polynomial mentioned are non-negative the term occurs, coming from x/, thus Rj > n-i statements that the coefficients are... 

Since the coefficients of the power series r(x) are all positive, the point X = p must be singular thus, equation (4.25) holds for x = p. The coefficients of the power series on the left-hand side of (4.25) are all positive, except for the constant, which is equal to 0. On the circle of convergence x = p, the absolute value of the series assumes, therefore, its maximum at x = p. We conclude that x p is the only solution of (4.25) on the circle of convergence the point... 

Asymptotic Values of the Coefficients of Certain Power Series... 

In the preceding section the analytical behavior of the power series q x) r(x), s(x), t(x) on the circle of convergence has been examined. An easily derived and well-known relation between the singularities and the coefficients of a power series is given below it allows for several inferences. ... 

The function III. 120 with more general forms of the functions u and g has also been studied in greater detail by Baber and Hasse (1937) and by Pluvinage (1950). The latter expanded g(rl2) in a power series in r12 and, by studying the formal properties of the wave equation itself, Pluvinage could derive certain general relations for the coefficients. At the Paris molecular symposium in 1957, Roothaan reported that, by expressing u and g in the form... 

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. 

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 ... 

Coefficients in the virial expansion of the osmotic pressure as a power series in the concentration c (Chap. XII et seq.). 

The variable s is a dummy variable in the sense that it does not enter die final result. Thus, if the exponential function in Eq. (94) is expanded in a power series in s, the coefficients of successive powers of s are just the Hermite polynomials divided by u . It is not too difficult to show that Eqs. (93) and (94) are equivalent definitions of the Hermite polynomials. 

The function y(x) can now be developed in a power series following the method presented in Section 5.2.1. The recursion formula for the coefficients is then of the form... 

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