Scaled external potentials

It should be noticed that in general, due to different effective contributions from the kinetic energy functional [Eq.(15)], the two scaled external potentials for subsystems differ from each other for 0 s X < 1. The... 

The above definition of the scaled external potentials of subsystems immediately implies their shapes f or the limiting values of X (see Fig. 2) ... 

For the (X = 0, X = 1) point in Fig. 2 the scaled external potential for electrons in B corresponds to the f ully interacting electrons of this subsystem moving in the the effective potential due to nuclei of M and electrons in s4 ... 

The same reasoning identifies the following scaled external potential coupled to in the (X ... 

Examples of a non-uniform scaling of the subsystem electronic charges ame represented by the paths 2-6 in Fig. 2 we have summarized in the figure Eqs. (44) and (53)-(55) identifying the subsystem scaled external potentials corresponding to the four points defining these scaling trajectories. 

Consider now a special case of = 0 (path 2 in Fig. 2). The relevant scaled external potentials for the limiting points (0, 0) and (2, o) of this integration are given by Eqs. (54b) and (55b) they are also explicitly listed in Fig. 2. Using these expressions in the above Hellmann-Feynman theorem for the path 2 in Fig. 2 gives ... 

The relativistic adiabatic connection formula is based on a modified Hamiltonian H g) in which not only the electron-photon coupling strength is multiplied by the dimensionless scaling parameter g but also a g-dependent, multiplicative, external potential is introduced. 

Now suppose that an external potential U(x) is added, which varies on a macroscopic scale and is therefore written as a function of x. The number of sites per macroscopic unit of length is our parameter Q. The presence of U(x) alters the heights of potential barriers. The new jump probabilities are... 

The corresponding energy shift can be evaluated by a coupling constant integration with respect to the external potential. Scaling the external potential Hamiltonian by A,... 

For a given (( )dl), there are three parameters that determine the dynamic behavior e, which describes the ratio of the time scales of potential and concentration changes U, the dimensionless external voltage and p, the dimensionless external resistance. If / ((t)DL) 1 diffusion is the rate-limiting step while (< )dl) 1 indicates reaction control. 

Note that here and later on r denotes the single-particle coordinate whereas R is still used as abbreviation for all nuclear positions as in Eq. (1). The potential (5) consists, on one hand, of an external potential V(r,R), which in our case is time-dependent owing to the atomic motion R( ). On the other hand, there are electron-electron interaction terms, namely the Hartree and the exchange-correlation term, which depend both via the density p on the functions tpj. The exchange-correlation potential VIC is defined within the so-called adiabatic local density approximation [25] which is the natural extension of the lda from stationary dpt. It is assumed to give reliable results for problems where the time scale of the external potential (in our case typical collision times) is larger than the electronic time scale. 

The present review has been very selective, stressing the rationale behind density-functional methods above their applications and excluding many important topics (both theoretical and computational). The interested reader may refer to anyone of the many books [91-93] or review articles [94-101] on density-functional theory for more details. Of special importance is the extension of density-functional theory to time-dependent external potentials [102-105], as this enables the dynamical behavior of molecules, including electronic excitation, to be addressed in the context of DFT [106-108]. As they are particularly relevant to the present discussion, we cite several articles related to the formal foundations of density-functional theory [85,100,109-111], linear-scaling methods [63,112-116], exchange-correlation energy functionals [25, 117-122], and qualitative tools for describing chemical reactions [123-126,126-132]. 

It goes beyond the scope of this chapter to discuss the matter of time scale where such a density function appears justified nevertheless, one should recognize that the effects of zero-point energy and the associated vibrations modify the role of the effective external potential experienced by the actual electron density, which also involves the role of time. Whereas electron density changes and fluctuations can carry information, all such effects on the molecular information are also dependent on the ground-state electron density. 

The operator (175) measures the energy of a given state with respect to the vacuum IO5) in the presence of the external potential. In the noninteracting situation these energy differences correspond directly to the observable ionization potentials. However, the operator (175) does not yet reflect the fact that the vacuum energies resulting from different external potentials are not identical (Casimir effect). The differences between the vacua are most easily seen on a local scale The vacuum expectation value of the current density operator (7) reads... 

G. C. Lynch, R. Steckler, D. W. Schwenke, A. J. C. Varandas, D. G. Truhlar, and B. C. Garrett, Use of scaled external correlation, a double many-body expansion, and variational transition state theory to calibrate a potential energy surface for FH2, J. Chem. Phys. 94 7136 (1991). 

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