Power-series expansions generalized

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.)... 

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 1901, H. Kamerlingh Onnes introduced a fundamentally new and improved description of real gas PVT properties in terms of the virial equation of state. [The word virial, deriving from the Latin word viris ( force ) was introduced into physics by R. Clausius, whom we shall meet later.] This equation expresses the compressibility factor Z(Vm, T) in terms of a general power series expansion in inverse molar volume Vm. The starting point for the virial expansion is the ideal limiting behavior (2.12), which can also be expressed as... 

The determination of these normal frequencies, and the forms of the normal vibrations, thus becomes the primary problem in correlating the structure and internal forces of the molecule with the observed vibrational spectrum. It is the complexity of this problem for large molecules which has hindered the kind of detailed solution that can be achieved with small molecules. In the general case, a solution of the equations of motion in normal coordinates is required. Let the Cartesian displacement coordinates of the N nuclei of a molecule be designated by qlt q2,... qsN. The potential energy of the oscillating system is not accurately known in the absence of a solution to the quantum mechanical problem of the electronic energies, but for small displacements it can be quite well approximated by a power series expansion in the displacements ... 

Constitutive Relations. A more general representation of the nonlinear polarization is that of a power series expansion in the electric field. For molecules this expansion is given by... 

In this review, the focus is on electro-optic materials. Such materials are members of the more general class of second-order nonlinear optical materials, which also includes materials used for second harmonic generation (frequency doubling). The term second-order derives from the fact that the magnitude of these effects is defined by the second term of the power series expansion of optical polarization as a function of applied electric fields. The power series expansion of polarization with electric field can be expressed either in terms of molecular polarization (p Eq. 1) or macroscopic polarization (P Eq. 2)... 

Higher order terms (e.g., Tl) do not appear, of course, because our example system contains only four electrons. If we remember that Ti and T2 commute, then all the terms from the equation above match those from the power series expansion of an exponential function Thus, the general expression for Eq. [30]... 

Numerical calculations using Kapuy s partitioning scheme have shown that for covalent systems the role of one-particle localization corrections in many-body perturbation theory is extremely important. For good quality results several orders of one-particle perturbations have to be taken into account, although the additional computational power requirement is much less in these cases than for the two-electron perturbative corrections. Another alternative for increasing the precision of the calculations is to estimate of the asymptotic behavior of the double power series expansion (24) from the first few terms by applying Canterbury approximants [31], which is a two-variable generalization of the well-known Pade approximation method. It has also been found [6, 7] that in more metallic-like systems the relative importance of the localization corrections decreases, at least in PPP approximation. 

In order to prepare the discussion of the relativistic generalization of the HK-theorem in Section 3 we finally consider the renormalization procedure for inhomogeneous systems. As the underlying renormalization program of vacuum QED is formulated within a perturbative framework (see Appendix B) we assume the perturbing potential to be sufficiently weak to allow a power series expansion of all relevant quantities with respect to V. In particular, this allows an explicit derivation of the counterterms required for the field theoretical version of the KS equations, i.e. for the four current and kinetic energy of noninteracting particles. 

It should be mentioned that there are only a few restrictions on the choice of the matrices C/j. Firstly they have to be unitary and analytical (holomorphic) functions on a suitable domain of, and secondly they have to permit a decomposition of Hm in even terms of definite order in the external potential according to Eq. (73). It is thus possible to parametrise them without loss of generality by a power series expansion in an odd and antihermitean operator Wi of ith order in the external potential. In the following, the physical consequences of this freedom in the choice of the unitary transformations will be investigated. Therefore we shall start with a discussion of all possible parametrisations in terms of such power series expansions. Afterwards the most general parametrisation of Ui is applied to the Dirac Hamiltonian in order to derive the fourth-order... 

The most general ansatz to construct a unitary transformation U = fiW) as an analytical function of an antihermitean operator W is a power series expansion,... 

By using the general power series expansion for U all the infinitely many parametrisations of a unitary transformation are treated on equal footing. However, the question about the equivalence of these parametrisations for application in the Douglas-Kroll method, which represents a crucial point, is more subtle and will be analysed in the next section. It is especially not clear a priori, if the antihermitean matrix W can always be chosen in the appropriate way the mandatory properties of W, i.e., its oddness, antihermiticity and behaviour as a correct power in the external potential, have to be checked for every single transformation Ui of Eq. (73). 

For regular singular points, a series solution of the differential equation can be obtained by the method of Frobenius. This is based on the following generalization of the power series expansion ... 

In general the polymer-solvent Interaction depends upon the polymer volume fraction. This is taken into account by a power series expansion... 

For calculating the spinodal curve and the critical point, there are two possible ways in the framework of continuous thermodynamics. The most general one is the application of the stability theory of continuous thermodynamics [45-47]. The other way is based on a power series expansion of the phase equilibrium conditions at the critical point. Following the second procedure. Sole et al. [48] studied multiple critical points in homopolymer solutions. However, in the case of divariate distribution functions the method by Sole has to be modified as outlined in the text below. 

Our expression for [Eq. (105)] is thus far still exact. The first term in (105) is seen to be the contribution from local two-body types of interaction including static correlation effects. This term represents a generalized Boltzmann-Enskog term, which controls the short-time behavior of We note that the first two terms in a power series expansion of (l/z) both come from YG V. The second term in (105) contains recollision and mode-coupling " effects. The T-matrices on the ends have a simple form in the case of hard spheres, and they describe close collisions between two particles. The quantity R describes the motion of these two particles after their collision. 

The general method for solving Eqs. (11) consists of transforming the partial differential equations with the help of Fourier-Laplace transformations into a set of linear algebraic equations that can be solved by the standard techniques of matrix algebra. The roots of the secular equation are the normal modes. They yield the laws for the decays in time of all perturbations and fluctuations which conserve the stability of the system. The power-series expansion in the reciprocal space variables of the normal modes permits identification of relaxation, migration, and diffusion contributions. The basic information provided by the normal modes is that the system escapes the perturbation by any means at its disposal, regardless of the particular physical or chemical reason for the decay. 

According to Eq. (11.93), the decoupled Hamiltonian within the Foldy-Wouthuysen framework is formally given as a series of even terms of well-defined order in 1/c. In most presentations of the Foldy-Wouthuysen transformation the exponential function parametrization Hjj] = exp(W[j]) is applied for each transformation step. However, in the light of the discussion in section 11.4 the specific choice of this parametrization does not matter at all, since one necessarily has to expand Ui into a power series in order to evaluate the Hamiltonian. Consequently, in order to guarantee a most general analysis, the most general parametrization for the Foldy-Wouthuysen transformation should be employed [610]. Thus, li is parametrized as a power series expansion in an odd and antihermitean operator W, , which is of (2/+l)-th order in 1/c, (cf. section 11.4). After n transformation steps, the intermediate, partially transformed Hamiltonian f has the following structure. 

The function (1,2,3..) is a much more complicated object, and in general does not admit a simple diagram expansion. The understanding of the meaning of this function is clarified using a functional series expansion Consider the functional power series expansion of In j(l) around the uniform density [67, 68] p,... 

Appendix A includes a sechon on curve fitting that outlines a means to find the cs for a truncated power series expansion. We have, then, a general approach for finding an expression for X in ferms of P, V, and T. When that is obtained, we return to Equation 2.28 and write X = PV - nRT, now using the fitting expression in P, V, and T in place of X. Such a result is a real gas law based on the PVT laboratory data that are available. 

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