Frenkel principle

In the variational multi-configuration Gaussian wavepacket (vMCG) method [10, 11, 59-63] the basis functions follow coupled quantum trajectories whereby the mean positions and momenta are treated as variational basis-function parameters that evolve according to the Dirac-Frenkel principle applied to the TDSE. Each basis function directly simulates quantum phenomena in a rigorous way, and the method thus promises much faster convergence than classical-trajectory-based methods due to a better sampling of the phase space. 

The second approach is based on a Hamiltonian with explicit time dependence, provided with an opportune form of the of the time-dependent variation principle (the Frenkel principle is generally used (Cammi et al. 1996)). From this starting point linear and non linear response functions are derived. 

The variational equations formulated in Section 12.2 may be used to obtain a /nu/time-dependent case in particular, spedal care is needed in using the Frenkel principle (McWeeny, 1983). We therefore proceed cautiously, writing the principle in the fiiller form (12.2.5a), namely... 

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

Principles and Applications, Longman, 1996 D. Frenkel and B. Smith, Understanding Molecular Simulations, Academic Press, 1996. 

In essentially all of the prior formulations of TDDFT a complex Lagrangian is used, which would amount to using the full expectation value in Eq. (2.9), not just the real part as in our presentation. The form we use is natural for conservative systems and, if not invoked explicitly at the outset, emerges in some fashion when considering such systems. A discussion of the different forms of Frenkel s variational principle, although not in the context of DFT, can be found in (39). 

Computational strategies can be based on variational procedures using the Dirac-Frenkel time-dependent variational principle (TDVP). Introducing a shorthand notation so that... 

Beginning with values of ]r) and ]v) at time 0, one calculates the new positions and then the new velocities. This method is second-order in At, too. For additional details, see Allen, M. R, and D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford (1989) Frenkel, D., and B. Smit, Understanding Molecular Simulation, Academic Press (2002) Haile, J. M., Molecular Dynamics Simulation, Wiley (1992) Leach, A. R., Molecular Modelling Principles and Applications, Prentice-Hall (2001) Schlick, T., Molecular Modeling and Simulations, Springer, New York (2002). 

Stein GS, Lian JB, Stein JL, van Wijnen AJ, Frenkel B, Montecino M (1996) Mechanism regulating osteoblast and proliferation. In Bilezikian JP, Raisz LG, Rodan GA (eds) Principles of bone biology. Academic, New York, pp 69-86... 

Equations of motion for the time-dependent coefficients Aj time-dependent single particle functions, and time-dependent Gaussian parameters A K s) = aj c s), f r]jK s can be derived via the Dirac-Frenkel variational principle [1], leading to... 

Let us start with an analogy. An ideal crystal, in which all the atoms are exactly located at the nodes of a geometrically perfect space lattice, can be conceived only on classical grounds and at absolute zero. However, it is impossible to accept this somewhat naive concept because of the uncertainty principle and thermal agitation at T 0°K. This does not, however, mean that the idea of crystallinity loses all definiteness or that, for instance, a crystal can melt in a continuous process, as Frenkel [1] seems to suggest. 

A cation vacancy may be paired with a nearby cation interstitial. This is called a Frenkel pair. An example is the formation of Zn+2 vacancies and Zn+2 interstitials in ZnO. This is illustrated in Figure 5.2B. In principle, paired anion vacancies and interstitials are possible, but this is less likely because of the larger size of the anions. 

Approximate solutions of the time-dependent Schrodinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree-Fock or Kohn-Sham equation ... 

Dirac-Frenkel stationary action principle involving the functional W determines the exact system density matrix. 

Reuter, K., Frenkel, D., Scheffler, M. (2004). The steady state of heterogeneous catalysis, studied by first-principle statistical mechanics. Phys. Rev. Lett. 93,116105. 

The change in / to the value below the critical (fo = 0.63) may result in the liquid crystalline state appearing in a polymeric system which in the absence of external field or without a change in the nature of a solvent is in the isotropic state. This principle of equivalence of inherent and induced rigidity was additionally discussed in the review article by Frenkel ... 

Molecular dynamics uses classical mechanics to study the evolution of a system in time. At each point in time the classical equations of motion are solved for a system of particles (atoms), interacting via a set of predefined potential functions (force field), after which the solution obtained is applied to predict positions and velocities of the particles for a (short) step in time. This step-by-step process moves the system along a trajectory in phase space. Assuming that the trajectory has sampled a sufficiently large part of phase space and the ergodicity principle is obeyed, all properties of interest can then be computed by averaging along the trajectory. In contrast to the Monte Carlo method (see below), the MD method allows one to calculate both the structural and time-dependent characteristics of the system. An interested reader can find a comprehensive description of the MD method in the books by Allen and Tildesley or Frenkel and Smit. ... 

The examples discussed here are the simplest possible and real systems are usually more complicated. Moreover, we have used simplified models for example, v is the characteristic frequency of the interstitial ion or an ion adjacent to a vacant lattice position it will be diflerent from the characteristic lattice frequency and will not be the same for Schottky and Frenkel defects. However, the general principles of the transfer of matter through a crystalline solid are as they have been given here. 

To obtain the set of coupled equations of motion for the coefficients and SPFs, the Dirac-Frenkel variational principle is used. Dividing the Hamiltonian into parts that act only on a given particle (separable or correlated term). 

As is often the case in any new development in quantum chemistry, the fundamental formalism has been around for some time, in this case TDCPHF based on Frenkel s variational principle and density matrices [7]. However, the detailed theory will not be our immediate concern, but rather how it has been put to use. 

We return to the self-trappping (ST) of Frenkel excitons in 3D organic structures. As a result of the Franck-Condon principle, photo-exciting a crystal from its ground state, with a regular lattice, leads to the initial creation of a coherent... 

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