Power series expansions

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

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 ... 

We then write the solution of equation B1.5.7 as a power series expansion hi temis of the strength X of the perturbation ... 

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... 

Assuming now that the power series expansion in F(p) can be terminated to keep /f(p) well behaved at large p values, it may be shown [95] that... 

Expressing p(R) in a power series expansion about the equilibrium bond length position (denoted Rg collectively and Ra,g individually) ... 

The desired physieal property must be extraeted from the power series expansion of AE in powers of V. 

Using the faet that k is an eigenfunetion of HD and employing the power series expansion of /k allows one to generate the fundamental relationships among the energies Ek ) and the wavefunetions /kl ) ... 

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. 

Generating functions are used in calculating moments of distributions for power series expansions. In general, the nth moment of a distribution,/fxj is E x ") = lx" f x) dx, where the integration is over the domain of x. (If the distribution is discrete, integration is replaced by summation.)... 

USC may be modeled as a power-series expansion of non-CCF component failure nates. No a priori physical information is introduced, so the methods are ultimately dependent on the accuracy of data to support such an expansion. A fundamental problem with this method is that if the system failure rate were known such as is required for the fitting process then it would not be neces.sary to construct a model. In practice information on common cause coupling in systems cannot be determined directly. NUREG/CR-2300 calls this "Type 3" CCF. 

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... 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

An essential improvement of the entire approach has recently been given by Kinoshita (1957), who pointed out that, instead of using Eq. III. 113, one could just as well use a power series expansion in the three variables... 

We note that the power series expansion III. 119is a direct generalization of the Hylleraas form III. 114 to which it should go over in the limiting case Rab = 0. James and Coolidge obtained a value of the electronic energy, —1.17347 at.u., in excellent agreement with the experimental results available, and their work forms even today the best basis for our understanding of the electronic structure of the chemical bond. 

In table VII, the values for Eexact are estimated from experiments or taken from highly accurate calculations based on correlated power series expansions. The energy values in the column for km x = 1 are associated with wave functions of the type III. 120, but, since the a-values are actually optimized for km x — 3, they could probably all be slightly improved. For comparison, we have included the energy values for the best functions (u)2(l- -atr12), where u is expressed in the form III. 121. 

The moments of this-distribution can now be obtained by making use of the known power series expansion for e ... 

To integrate the line integral in equation (2.18), heat capacity is often expressed as a power series expansion in T of the type... 

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 ... 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

The error function erfx has a power series expansion for small x and an asymptotic expansion for large x... 

The approximate expression (10.50) for the nuclear energy levels E j is observed to contain the initial terms of a power series expansion in (n -I- ) and J J + 1). Only terms up to (n + ff and J(J + )f and the cross term in (n + )J(J 4-1) are included. Higher-order terms in the expansion may be found from higher-order perturbation corrections. 

The ratio of consecutive terms in the power series expansion F is given by equation (G.48) as... 

This equation is of the form of Eq. (15) and hence can be solved by the power-series expansion p) = J2k akPi- The resulting recursion formula is... 

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

It is only natural to consider ways that would allow us to use our knowledge of the whole distribution P0(AU), rather than its lew-AU tail only. The simplest strategy is to represent the probability distribution as an analytical function or a power-series expansion. This would necessarily involve adjustable parameters that could be determined primarily from our knowledge of the function in the well-sampled region. Once these parameters are known, we can evaluate the function over the whole domain of interest. In a way, this approach to modeling P0(AU) constitutes an extrapolation strategy. 

In all cases we need to take at least a brief look at what happens when A j. is close to the asymptote at —ET basically, it goes off to -(-infinity when approaching from above (as we saw), and to -infinity when approaching from below. If we do not try to compute values when we are too close to —Er, therefore, using either approach there will be a tendency toward cancellation of the positive and negative terms, leaving a finite result. In the case of a power series expansion, the closer we come to unity, the more terms we would need to include in the series. 

---