Scaled coordinates

The new box length is computed, and all the particle coordinates scaled by an appropriate factor ... 

In practice modifications are made to incorporate thermostats or barostats that may destroy the time-reversible and symplectic properties. While extended-system algorithms such as Nose dynamics [41] can be designed on the principles of the reversible operators, methods that use proportional velocity or coordinate scaling [42] cannot. Such methods arc very... 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

Exchange identities utilizing the principle of adiabatic connection and coordinate scaling and a generalized Koopmans theorem were derived and the excited-state effective potential was constructed [65]. The differential virial theorem was also derived for a single excited state [66]. 

Response surface plot a graphical representation of the response surface as a contour map of the dependent variable on a coordinate scale of two of the independent variables. 

Using the same method as for the first excited electronic state, we select a level shift in the region Xi < p < X2. This procedure may indeed lead to a transition state but in this way we always increase the function along the lowest mode. However, if we wish to increase it along a higher mode this can only be accomplished in a somewhat unsatisfactory manner by coordinate scaling. Nevertheless, this method has been used by several authors with considerable success.14 The problem of several first-order saddle points does not arise in electronic structure calculations since there is only one first excited state.15... 

Nagy, A. (1995). Coordinate scaling and adiabatic connection formula for ensembles of fractionally occupied excited states, Ini. J. Quantum Chem. 56, 225-228. 

Wilson LC, Levy M (1990) Nonlocal Wigner-like correlation-energy density functional through coordinate scaling, Phys Rev B, 41 12930-12932... 

Use coordinate scales that give good proportionment of the curve over the entire plot, but do not distort the apparent accuracy of the results. 

It is possible to write eqn.(ll) in a form that makes scaling and perturbation corrections to the infinite nuclear mass approximation more obvious [13]. Since for the cases being considered p = p = p2, coordinate scaling pi = (p/me) rt gives a Hamiltonian,... 

Among the known properties of this functional are the coordinate scaling conditions first obtained by Levy and Perdew [55]... 

In analogy to the coordinate scaling of Eqs. (59) - (61), this property is often called spin-scaling , and it can be used to construct an SDFT exchange functional from a given DFT exchange functional. In the context of the LSDA, von Barth and Hedin [89] wrote the exchange functional in terms of an interpolation between the unpolarized and fully polarized electron gas which by construction satisfies Eq. (99). Alternative interpolation procedures... 

A large number of known exact properties of density functionals involve coordinate scaling transformations of the density. Most of such relations have been derived by Levy and coworkers [78-84]. The uniform scaling of the density is defined by... 

The effect of this transformation is to contract or thin the density while preserving its normalization. Coordinate scaling constraints for exchange and correlation functionals are reviewed by Levy [85]. The most important among these constraints are the following ... 

Fig. 13a. Predicted variation of X,ff/(Axo) vs (N0/N) i Eq. (134). Dot-dash line shows the asymptotic behavior 3Urf/(Axo)oc(N/N) 1/2> while the marks indicate the scale for (N/Nc) itself, b Variation of the observed normalized critical temperature k Tc/(eN) with N1/2, from a simulation of the bond fluctuation lattice model (see Sect. 4). Marks indicate the values of N chosen. Note that for large enough N the integral equation theory of Schweizer and Curro [44-49] implies that this plot could be mapped on Fig. 13a), by multiplying the coordinate scales with suitable constants, without any other adjustable parameters being available. From Deutsch and Binder [92]...

Figure 10 Degeneracy-summed integral cross sections for the H + H, (v = 0, j = 0) - H (v =0, j = 1) + H as a function of total energy. The open circles (squares) depict the GP (NGP) results. The open triangles represent the difference between the GP and NGP cross sections, multiplied by a factor of 25 before plotting. The full circles represent their ratio, and correspond to the coordinate scale at the right of the figure. (From Ref. 36.)...

For any well-behaved density functional Q[p] whose homogeneity in coordinate scaling is degree k, one has the identity [35] ... 

The exchange functional E fp] is well known to be homogeneous of degree one in coordinate scaling [1,2], that is, the Levy-Perdew relation [34],... 

The coordinate scaling of the exchange energy should be linear, i.e. multiplying the electron coordinates with a constant factor should result in a similar linear scaling of the exchange energy. ... 

Levy, M. (1991). Density-functional exchange correlation through coordinate scaling in adiabatic connection and correlation hole. Phys. Rev. A 43,4637-4645. 

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