A variation principle

The original equation of Wilemski and Fixman [51], eqn, (272) can be re-written in terms of the operator S as 

By expanding the function 0 in terms of (( )) and exp (— ejjf) (i.e. effectively separation of variables), the probability, 0(t), that the reactants remain is 

The smallest eigen-value e is associated with the slowest decay of (t), consequently, k i =  

Doi [485] then drew attention to an arbitrary function of the form sp( r ) and from it suggested that a functional /[ ] be considered such that (note the similarity to eqn. (276) if p = 1) 

Subject to certain restrictions, a variational principle can be given where 

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One drawback is that, as a result of the time-dependent potential due to the LHA, the energy is not conserved. Approaches to correct for this approximation, which is valid when the Gaussian wavepacket is narrow with respect to the width of the potential, include that of Coalson and Karplus [149], who use a variational principle to derive the equations of motion. This results in replacing the function values and derivatives at the central point, V, V, and V" in Eq. (41), by values averaged over the wavepacket. 

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

If classical Coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and the term for the Cl coefficients (C) is omitted, the solvated Eock operator is reduced to Eq. (6). The significance of this definition of the Eock operator from a variational principle is that it enables us to express the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule R ,... 

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

Frieden s theory is that any physical measurement induces a transformation of Fisher information J I connecting the phenomenon being measured to intrinsic data. What we call physics - i.e. our objective description of phenomenologically observed behavior - thus derives from the Extreme Physical Information (EPI) principle, which is a variational principle. EPI asserts that, if we define K = I — J as the net physical information, K is an extremum. If one accepts this EPI principle as the foundation, the status of a Lagrangian is immediately elevated from that of a largely ad-hoc construction that yields a desired differential equation to a measure of physical information density that has a definite prior significance. 

The theoretical results provided by the large basis sets II-V are much smaller than those from previous references [15-18] the present findings confirm that the second-hyperpolarizability is largely affected by the basis set characteristics. It is very difficult to assess the accuracy of a given CHF calculation of 2(ap iS, and it may well happen that smaller basis sets provide theoretical values of apparently better quality. Whereas the diagonal eomponents of the eleetrie dipole polarizability are quadratic properties for which the Hartree-Fock limit can be estimated with relative accuracy a posteriori, e.g., via extended calculations [38], it does not seem possible to establish a variational principle for, and/or upper and lower bounds to, either and atris-... 

Still another useful approximation is introduced by reformulating Eq. (1.1) as a variational principle,16... 

Among various theories of electronic structure, density functional theory (DFT) [1,2] has been the most successful one. This is because of its richness of concepts and at the same time simplicity of its implementation. The new concept that the theory introduces is that the ground-state density of an electronic system contains all the information about the Hamiltonian and therefore all the properties of the system. Further, the theory introduces a variational principle in terms of the ground-state density that leads to an equation to determine this density. Consider the expectation value (H) of the Hamiltonian (atomic units are used)... 

This functional satisfy a variational principle [3,5] EVo o[p0, m0 < EVo>Bo[p, m. EVoJ3o [p0, hi0] denote the ground state energy with density p0(r), and magnetization m0(r) of a particular system characterized by the external fields (v0(r), Bt>(r)). One of the main differences between the spin-restricted and spin-polarized cases is that the one-to-one relation between the external potential and the density cannot be extrapolated to the set of quantities (v0(r), B0(r)) and (p0(r), m0(r)) [3]. 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

The exact approach to the problem of dynamic (linear) stability is based on the solution of the equations for small perturbations, and finding eigenvalues and eigenfunctions of these equations. In a conservative system a variational principle may be derived, which determines the exact value of eigenfrequency... 

Models of hot isentropic neutron stars have been calculated by Bisnovatyi-Kogan (1968), where equilibrium between iron, protons and neutrons was calculated, and the ratio of protons and neutrons was taken in the approximation of zero chemical potential of neutrino. The stability was checked using a variational principle in full GR (Chandrasekhar, 1964) with a linear trial function. The results of calculations, showing the stability region of hot neutron stars are given in Fig. 7. Such stars may be called neutron only by convention, because they consist mainly of nucleons with almost equal number of neutrons and protons. The maximum of the mass is about 70M , but from comparison of the total energies of hot neutron stars with presupemova cores we may conclude, that only collapsing cores with masses less that 15 M have... 

In an important paper (TNC.l), they offered for the first time an extension of nonequilibrium thermodynamics to nonlinear transport laws. As could be expected, the situation was by no means as simple as in the linear domain. The authors were hoping to find a variational principle generalizing the principle of minimum entropy production. It soon became obvious that such a principle cannot exist in the nonlinear domain. They succeeded, however, to derive a half-principle They decomposed the differential of the entropy production (1) as follows ... 

In this work, the electronic kinetic energy is expressed in terms of the potential energy and derivatives of the potential energy with respect to nuclear coordinates, by use of the virial theorem (5-5). Thus, the results are valid for ail bound electronic states. However, the functional derived for E does not obey a variational principle with respect to (Pg ( )), even though in... 

It is highly desirable to formulate a variational principle valid for the Dirac Hamiltonian. The first attempts are due to Drake and Goldman [9], Wood etal. [10], Talman [4] and Datta and Deviah [5], Very recently the subject has been discussed in detail by Griesemer and Siedentop [6] and also by Kutzelnigg [7] andby Quiney efa/. [11],... 

Our object of interest is a many electron finite system (such as an atom, molecule, cluster etc.), having, by assumption, a nondegenerate ground state (GS) (this assumption will be removed in Sects. 4.4 and 5). The numter of electrons N and the electron-nuclei potential energy v(r) = Ve (r) (the so-called external potential) are given and common for all schemes to be discussed. The GS energy qs aod the GS wave function Vqs of the system can be found from a variational principle as... 

Though having a variational principle that works is all that is technically required in a useful theory, this condition is actually necessary and sufficient for the A-representability of the Q-matrix. That is,... 

The second law of thermodynamics states that an isolated system in equilibrium has maximum entropy. This is the basis for a variational principle often used in determining the equilibrium state of a system. When the system contains several elements which are allowed to exchange mass with each other, the variational principle yields the condition that all elements must have equal chemical potential once equilibrium is established. 

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

The overestimation of the TST rate constant leads to a variational principle for the optimization of the position of the dividing surface constituting the transition state. In general, one can write ... 

The first theorem of Hohenberg and Kohn is an existence theorem. As such, it is provocative with potential, but altogether unhelpful in providing any indication of how to predict the density of a system. Just as with MO theory, we need a means to optimize our fundamental quantity. Hohenberg and Kohn showed in a second theorem that, also just as with MO theory, the density obeys a variational principle. 

The behavior of a mixture is determined by a system of ordinary differential equations, while the required state, either equilibrium or stationary, is determined by a time-independent system of algebraic equations. Therefore, at first glance one would not expect any qualitative difference between the equilibrium and stationary states. Ya.B. shows that in the equilibrium case, even for an ideal system, a variational principle exists which guarantees uniqueness. Such a principle cannot be formulated for the case of an open system with influx of matter and/or energy. 

These concepts stem from a variational principle applied to the action... 

We can now construct a topological model of electromagnetism in empty space, which can be formalized by means of a variational principle as follows. Let us take two pairs of dual scalars k, 0, where k 1,2 as fundamental fields and define an electromagnetic field by the equations... 

The limitations of the SH formalism can be overcome when we omit the perturbation theory and involve the spin-orbit coupling in a variational principle. The straightforward way is to diagonalize the (complex) interaction... 

Another important aspect of the theory is that it has the distinction of being a variational principle. Sato et al. showed that the solvation Fock operator can be naturally derived from the variational principle when starting from the Helmholtz-type free energy of the system (A) [7]. 

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