Expansion series

Altschuler M.D and Herman G. Fully three-dimensional image reconstruction using series expansion methods., A Review of Information Processing in Medical Imaging, Oak Ride National Lab., Oak Ride, TN, 1977, p.124-142. 

Dalgarno A and Lewis J T 1956 The representation of long-range forces by series expansions. I. The divergence of the series Proc. Phys. Soc. A 69 57... 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

At the critical pohit (and anywhere in the two-phase region because of the horizontal tie-line) the compressibility is infinite. However the compressibility of each conjugate phase can be obtained as a series expansion by evaluating the derivative (as a fiuictioii of p. ) for a particular value of T, and then substituting the values of p. for the ends of the coexistence curve. The final result is... 

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 ... 

Evidently, this fomuila is not exact if fand vdo not connnute. However for short times it is a good approximation, as can be verified by comparing temis in Taylor series expansions of the middle and right-hand expressions in (A3,11,125). This approximation is intrinsically unitary, which means that scattering infomiation obtained from this calculation automatically conserves flux. 

The molecular dipole moment (not the transition dipole moment) is given as a Taylor series expansion about the equilibrium position... 

Raman scattering has been discussed by many authors. As in the case of IR vibrational spectroscopy, the interaction is between the electromagnetic field and a dipole moment, however in this case the dipole moment is induced by the field itself The induced dipole is pj j = a E, where a is the polarizability. It can be expressed in a Taylor series expansion in coordinate isplacement... 

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... 

The basic idea of NMA is to expand the potential energy function U(x) in a Taylor series expansion around a point Xq where the gradient of the potential vanishes ([Case 1996]). If third and higher-order derivatives are ignored, the dynamics of the system can be described in terms of the normal mode directions and frequencies Qj and Ui which satisfy ... 

Series expansion Smith and van Gunsteren [4] investigated the first approach expanding the free energy as a function of the coupling parameter A into a T ylor series around a given reference state, A = 0,... 

A potential advantage of methods based on a series expansion of the free energy is that the convergence of the series is determined by the A dependence of the potential energy function meaning that the efficiency of the approach could be enhanced by a judicious choice of coupling scheme. 

The three moments higher than the quadrupole are the octopole, hexapole and decapoli. Methane is an example of a molecule whose lowest non-zero multipole moment is the octopole. The entire set of electric moments is required to completely and exactly describe the distribution of charge in a molecule. However, the series expansion is often truncated after the dipole or quadrupole as these are often the most significant. 

Many of the torsional terms in the AMBER force field contain just one term from the cosine series expansion, but for some bonds it was found necessary to include more than one term. For example, to correctly model the tendency of O-C—C-O bonds to adopt a gauche conformation, a torsional potential with two terms was used for the O—C—C—O contribution ... 

Irude model only considers the dipole-dipole interaction if higher-order terms, due to e-quadrupole, quadrupole-quadrupole, etc., interactions are included as well as other i in the binomial expansion, then the energy of the Drude model is more properly an as a series expansion ... 

This is similar in spirit to the arithmetic-mean rule but with each individual r,) being weighted according to the square of its value. The well depth in this function starts with a formula proposed by Slater and Kirkwood for the Cg coefficient of the dispersion series expansion ... 

When discussing derivative methods it is useful to write the function as a Taylor series expansion about the point jc. ... 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

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