Density functional derivatives notations

The mixed, v t — % notation here has historic causes.) The Schrodinger equation is obtained from the nuclear Lagrangean by functionally deriving the latter with respect to t /. To get the exact form of the Schrodinger equation, we must let N in Eq. (95) to be equal to the dimension of the electronic Hilbert space (viz., 00), but we shall soon come to study approximations in which N is finite and even small (e.g., 2 or 3). The appropriate nuclear Lagrangean density is for an arbitrary electronic states... 

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of averaged quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

Let both the emission and absorption line be centered at Eq and have the same natural line width F = Ff, = F. The distribution of the ground-state Mbssbauer atoms in question (e.g., Fe) is taken to be uniform in both the source and the absorber. The distribution of the parent atoms (e.g., Co) producing the excited Mossbauer atoms ( Fig. 25.7) (and thus the y photons) is described by the density function p x) along the x-axis (see Fig. 25.6 for the notation). In the derivation of the peak shape of the Mossbauer spectrum, nonresonant and resonant absorption processes are to be considered in both the source and the absorber as illustrated schematically in Fig. 25.6. 

The moments of Y are obtained from similar expressions simply by changing the signs of all g /, g f and g f that appear in Eqs. 6.78 through 6.80, so that we need not repeat those expressions here. We note that the reduced mass is m, B is short for Bj.cJ, and the Vv, Vv> are the vibrational averages of the interaction potential. Superscripted Roman numericals I. .. IV mean the first. .. fourth derivatives with respect to R. The radial distribution functions g = g(R) depend on the interaction potentials, Vv, Vv>, and are thus subscripted like the potentials the low-density limit of the distribution function will be sufficient for our purposes. The functions g and g M are defined in Eq. 6.23. The notation f f R)d2R stands for 4n /0°° / (R) R2 dR as usual. 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

Here / = 1/7 in the standard notation. From our general statements in Section in. A, the spinodal criterion derived from the exact free energy (38) must be identical to this this is shown explicitly in Appendix C. Note that the spinodal condition depends only on the (first-order) moment densities p, and the second-order moment densities py of the distribution p(cr) [given by Eqs. (40) and (41)] it is independent of any other of its properties. This simplification, which has been pointed out by a number of authors [11, 12], is particularly useful for the case of power-law moments (defined by weight functions vt>f(excess free energy only depends on the moments of order 0, 1... K — 1 of the density distribution, the spinodal condition involves only 2K— moments [up to order 2(K — 1)]. 

Hereafter, the expressions of h, c and p are matrices. In Equation (1) the notation f ), denotes the derivatives of quantity / with respect to a under constant value of b. The integral equation of the number density derivatives of the radial distribution function is obtained in a similar way. The differential form of the closure relation is obtained in a similar way. In what follows a rather extended description is given for calculation of the heat capacity. Detailed procedure for calculation of these thermodynamic quantities will be published elsewhere. ) The functions S, and calculated from... 

The second derivative term, shorthand notation, has rather exotic behavior. Figure 5 is a qualitative sketch of the function plotted against density. An empirical relationship which can predict such behavior is... 

Correlation is a techniccil term borrowed from probability theory, and it is appropriate to analyze electron correlation in this framework. All the information about electron correlation is contained in two functions that can be derived from the wave functions, namely electron density q t) and pair density n(ri, r ). We use this notation proposed by Ruedenberg instead of the original notation Pi r) and P iri, of McWeeny n), as it cuts out some of the subscripts. 

Atomic basis functions in quantum chemistry transform like covariant tensors. Matrices of molecular integrals are therefore fully covariant tensors e.g., the matrix elements of the Fock matrix are F v = (Xn F Xv)- In contrast, the density matrix is a fully contravariant tensor, P = (x IpIx )- This representation is called the covariant integral representation. The derivation of working equations in AO-based quantum chemistry can therefore be divided into two steps (1) formulation of the basic equations in natural tensor representation, and (2) conversion to covariant integral representation by applying the metric tensors. The first step yields equations that are similar to the underlying operator or orthonormal-basis equations and are therefore simple to derive. The second step automatically yields tensorially correct equations for nonorthogonal basis functions, whose derivation may become unwieldy without tensor notation because of the frequent occurrence of the overlap matrix and its inverse. 

---