Generalized moment approximation

To our knowledge, the first paper devoted to obtaining characteristic time scales of different observables governed by the Fokker-Planck equation in systems having steady states was written by Nadler and Schulten [30]. Their approach is based on the generalized moment expansion of observables and, thus, called the generalized moment approximation (GMA). 

The starting point of the generalized moment approximation (GMA) is the Laplace transformation of an observable ... 

It should also be noted that in general, the magnetic density is a vector field, and this would introduce some problems with computational efficiency, since the sizes of matrices would increase. In both the EMTO method and the BGFM, an approximation called the Atomic Moment Approximation is made. This is an approximation where you assume that the spin direction is the same within each atomic sphere. This can be motivated by noticing that the spin density often is centered around the nuclei, and that it is almost collinear around each nuclei [128]. In this way, one can coarse grain the problem, from a big mesh of the whole vector field to a much smaller mesh consisting of only the nuclei. 

AC electric fields and includes dielectrophoresis (DEP), travelling wave dielectrophoresis (twDEP) and electrorotation (ROT). Generally, non-uniform electric fields are used in AC electrokinetics. The assumption that the uniform field solution for the dipole moment is valid, is referred to as the dipole moment approximation, and is sufficient if the size of the particle is small compared to the scale of the electric field non-uniformity, which is true for most cases. In this chapter, we describe the forces on particles due to the action of AC fields, and discuss applications for manipulation of particles. We finish with a discussion of scaling effects. 

Finer details of magnetic properties such as orbital contributions, unusual temperature-dependencies of magnetic moments and magnetic anisotropies can also be calculated by using CFT or, better, ACFT, and the results are generally good approximations. They are not perfect because the basic... 

Dipole Moments While dipole moments are not generally useful with organic unknowns, occasionally they provide the best means of distinguishing among a few possible structures which would have differing dipole moments. Approximate moments can be calculated by vector addition of bond moments from the literature.K For example, rra is-l,4-dibro-... 

The functional form (11.15) was originally derived as a generalization of the effective medium theory and the second moment approximation to tight-binding theory. Later, however, it lost its close ties with original physical meaning and came to be treated as a semiempirical expression with adjustable parameters. 

The Cn-kij A) terms are easy to determine. The zero-body term, Cq a)i equals 1 the one-body term, C mA)-> equals Ti the two-body term, C2(mA), equals T2 + if rriA > 2 the three-body term, Cz rnA) equals riT2 + if rriA = 2 and Ts + T1T2 + if rriA > 3, etc. The coefficients, Eq. (27), define the generalized moments of CC equations (for a discussion of the relationship between the method of moments of Krylov [99] and the single-reference CC theory, see Ref. [100]). They are readily available for the basic CC approximations, such as CCSD (the rriA = 2 case). As pointed out in our original work [11,30], the generalized moments represent the most fundamental quantities... 

Quantum mechanical and electromagnetic theory provide an additional extremely important restriction upon the occurrence of rotational transitions, namely, to a first and generally adequate approximation they can occur only for molecules having non-zero electric dipole moments. Thus, microwave spectra occur for the polar molecules of water, carbon monoxide and acetone but not for the non-polar moleculess of methane, carbon dioxide and benzene. It is worth stressing also that rotational spectra are produced only by gaseous molecules, not by liquids or solids. While this seems at first a serious limitation it should be noted that it is possible to vaporize even very refractory materials at elevated temperatures. Thus, the microwave spectrum of gaseous sodium chloride (Na-Cl) molecules is perfectly well known. 

Equation (22), with Cn-j mA) defined by eq (8) and defined by eqs (24) and (26), defines the excited-state MMCC theory. In analogy to the groimd-state case, the main elements of eq (22) are the generalized moments of the EOMCC equations defining approximate method A, i.e., the EOMCC equations, in which T is approximated by and Rk is approximated by projected onto the excited configurations not included in method A. For example, if we want to correct the exdted-state energies obtained in EOMCCSD calculations (the mA — 2 case) and recover the full Cl energies Ek, we must calculate the EOMCCSD equations projected on triples, quadruples, etc., or... 

The general nature of the MMCC theory, on which all renormalized and completely renormalized CC methods described here are based, allows us to proposed many other potentially useful approximations. We can, for example, introduce the MMCC(2,6) method, in which the CCSD results are corrected by considering all nonzero moments of the CCSD equations, including those corresponding to projections on pentuply and hextuply excited configurations. We can also introduce the active-space variants of the renormalized and completely renormalized CC approaches, in which we consider small subsets of the generalized moments of CC equations defined... 

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