Variational derivatives

The variational derivative of this with respect to ((> yields the continuity equation... 

Variationally deriving with respect to a leads to the Hamilton-Jacobi equation... 

We thus obtain a Lagrangean density, whieh is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic Lagrangian density, Eq. (140). Its variational derivations with respect to v / and v / lead to the solutions shown in Eq. (152), as well as to other solutions. 

The variational derivatives of with respect to the moduli ai give the following equations ... 

To obtain the last equality one uses a partial integration so that the derivative by e is now taken on d y/de instead of on S. Consequently, the variational derivative gives the free energy variation as... 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

Function F (t r ) (64) represents the density of probability for the particle ancestor at the moment t to give birth to i descendants (whose colors are 0,l,...,z-l) at the instants r0[Pg.200]

When one takes its variational derivative with respect to the phases <[), , one obtains the continuity equation in the form... 

The physical properties of atoms and molecules embedded in polar liquids have usually been described in the frame of the effective medium approximation. Within this model, the solute-solvent interactions are accounted for by means of the RF theory [1-3], The basic quantity of this formalism is the RF potential. It is usually variationally derived from a model energy functional describing the effective energy of the solute in the field of an external electrostatic perturbation. For instance, if a singly negative or positive charged atomic system is considered, the RF potential is simply given by... 

The constraints at Eqs. (65) and (68) are important checks on the accuracy of (non-variationally derived) approximate potentials, as they are usually not fulfilled by approximate potentials except in those cases where the fulfillment of these constraint is caused for symmetry reasons such as the spherical symmetry in atoms with a nondegenerate ground state. In the case of molecules these constraints will in general not be equal to zero for non-variationally derived potentials. 

VARIATIONAL DERIVATION OF HARTREE-FOCK EQUATIONS FOR EXCITED STATES... 

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. 

For proteins that are members of gene families, such as cytochrome P450, available structural models derived from X-ray crystallography and NMR data allow inference of the structure of newly discovered family members. The inference is based on the sequence variations derived from comparing the reference and new DNA sequences. While a high degree of computational power is needed, predictions based on this approach provide preliminary estimates of protein structure without a huge investment in expression, purification, and crystallization. A full structural elucidation requires several years for each protein, even if it can be crystallized. 

Now we turn to the function /2(r) Using the definition of the Green function (38) one can take the variation derivative in (42)) obtaining... 

Taking the variation derivative of the action A 2 with respect to A and A we obtain J2k f k an<4 Sfc fik respectively. Therefore up to the boundary terms the functional (99) should coincide with 2fcTrln[<9r + TLk]. Moreover, the possible boundary terms should vanish because the Green functions of initial t = —00 and final r = 00 coincide. Writing the action (99) in terms of the original functions g, /, /t and using Eqs. (47-49) to bring it into symmetric form for A2 we finally get... 

Levenston, M.E., Frank, E.H., and Grodzinsky, A.J. (1998) Variationally derived 3-field finite element formulations for quasistatic poroelastic analysis of hydrated biological tissues. Comput. Methods Appl. Mech. Eng. 156, 231-246... 

Physics described by the model with so many parameters is very rich and the model is able particularly to treat heavy fermion systems. To study the model many approaches were suggested (see reviews [2-5]). They are successful for particular regions of the parameter space but no one is totally universal. In this paper we apply to PAM the generating functional approach (GFA) developed first by Kadanoff and Baym [6] for conventional systems and generalized for strongly correlated electron systems [7-10]. In particular it has been applied to the Hubbard model with arbitrary U in the X-operators formalism [10]. The approach makes it possible to derive equations for the electron Green s function (GF) in terms of variational derivatives with respect to fluctuating fields. 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

Flavones. Flavone structural variation derives from hydroxylation, O-methylation and O-glycosylation. In addition, there can be C6- and C8-linked C-glycosides, isoprenyl (isopen-tenyl, Cj) substituents and G G or C-O—C links to form biflavones. Methylation of the phenolic OHs decreases polarity to permit an external location such as in the waxy leaf or fruit surface. 

The two association reactions have been examined theoretically by Marcus, Wardlaw and co-workers [47-49, 69]. They treated these reactions using Flexible Transition State Theory (FTST), a variational derivative of transition state theory. The difficulty with association reactions such as reactions (31) and (32) is that there is no barrier to association and so there is no obvious location on the reaction coordinate for the transition state. Recent developments of TST place more emphasis in locating the molecular geometry for which the reactive flux is a minimum, and the transition state is associated with this geometry. 

The atomic statement of the principle of stationary action, eqn (6.3), yields a variational derivation of the hypervirial theorem for any observable G, a derivation which applies only to a region of space H bounded by a surface satisfying the condition of zero flux in the gradient vector field of the charge density,... 

The variational derivation of the integral atomic force law, eqn (8.175), is applicable only to a region of space bounded by a zero-flux surface in Vp(r), i.e. to an open system whose Lagrangian integral vanishes at the point of variation. Thus the variational derivation of the atomic force. 

---