Variation theorem extended

Prove an extended variational theorem, which is that if the trial function (p is orthogonal to the correct ground-state wave function, the variational enCTgy calculated with

correct energy of the first excited state. 

The extended variation theorem states that if a variational trial function is orthogonal to the exact ground-state wave function, it provides an upper limit to the energy of the... 

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

The variation theorem may be extended in some cases to estimate the energies of excited states. Under special circumstances it may be possible to select a trial function 0 for which the first few coefficients in the expansion (9.3) vanish ao = a = = = 0, in which case we have... 

It is easy to extend proofs of the variational theorem to the case of states which are the lowest roots of a given spatial and spin symmetry.70 Since the... 

Use an approach similar to that in Problem 11.1 to discuss the effect of scaling in which all the interparticle distances in the wavefunction are multiplied by a parameter p. Hence establish the virial theorem For a system of particles with inverse distance interactions, (V) = —2(7 ) and (E) = V), either for a variationally optimized value of p or for an exact wavefunction. Then extend the theorem to admit external forces applied to the nuclei. [Hint Show, by change of variables in the integrations, how the expectation values depend on p. Use a stationary-value condition and suppose that p = l for the exact wavefunction. Note that the nuclei may be held fixed by applying forces opposite to the (Hellmann-Feynman) forces exerted by the electrons. These forces must be included in forming the expectation value of the classical virial.]... 

The nonrelativistic Schrodinger theory is readily extended to systems of N interacting electrons. The variational theory of finite A-electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived. 

Note that all these theorems can be easily understood by relying on intuition. We can assume that the frequency dependence of the field (electromagnetic skin-effect) provides information about the vertical variations of conductivity, while the spatial dependence of the data on the surface allow us to reconstruct the horizontal changes in conductivity. Thus, one can expect that these theorems can be extended to 3-D cases as well. 

A totally different point of view is proposed by Time-Dependent Density Functional Theory [211-215] (TD-DFT). This important extension of DFT is based on the Runge-Gross theorem [216]. It extends the Hohenberg-Kohn theorem to time-dependent situations and states that there is a one to one map between the time-dependent external potential t>ea t(r, t) and the time-dependent charge density n(r, t) (provided we know the system wavefunction at t = —oo). Although it is linked to a stationary principle for the system action, its demonstration does not rely on any variational principle but on a step by step construction of the charge current. 

Within a variational framework, a simple, sufficient condition of applicability of the Hellmann-Feynman theorem has been proposed by HurleyIf the variation extends to a family of trial functions where the family is invariant to changes in a parameter o, then the optimum trial function fulfills the Hellmann-Feynman theorem. In other words, if the variation in parameter c merely interconverts the trial functions within the same family, then the Hellmann-Feynman theorem applies. One trivial case is a family of trial functions where each function is independent of ct. Hurley s condition is fulfilled in variational approaches involving Lagrange multipliers, such as the Hartree-Fock and multiconfigurational self consistent field (MCSCF) meth-ods." ... 

We find that a layer model analysis can adequately describe the Pt NMR spectrum of nanoscale electrode materials. The shifts of the surface and sub-surface peaks of Pt NMR spectra correlate well with the electronegativity of various adsorbates, while the Knight shift of the adsorbate varies linearly with the f-LDOS of the clean metal surface. The Pt NMR response of Pt atoms from the innermost layers of the nanoparticles does not show any influence of the adsorbate present on the surface. This provides experimental evidence, which extends the applicability of the Friedel-Heine invariance theorem to the case of metal nanoparticles. Further, a spatially-resolved oscillation in the s-like E( -LDOS was observed via Pt NMR of a carbon-supported Pt catalyst sample. The data indicate that much of the observed broadening of the bulk-like peak in Pt NMR spectra of such systems can be attributed to spatial variations of the A( f). The oscillatory variation in A(A) beyond 0.4 nm indicates that the influence of the metal surface goes at least three layers inside the particles, in contrast to the predictions based on the Tellium model. 

The reason why the Heitler-London method gives such a bad Hellmann-Feynman force-constant is thus that R is "attached to the nuclei so that 9 / R is not zero the same applies to the Weinbaum function 94), to the Wang function (95), and to the Coulson function 96) for Hg. To yield better force results, the variable parameters ) must be "detached from the nuclei and their values determined at each internuclear configuration. A wave function in which the parameters are determined by the variational procedure is called a floating function by Hurley 93, 97, 30, 31, 32) (this is eqtuvalent to Hall s stable wave functions 88)). This procedure can be extended to the scale factors, as discussed by McLean 81) and Lowdin 83). The vibrational frequency of H2 determined by Ross and Phillipson using the differentiation of the virial theorem (which assumed that all the variable parameters are variationally... 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

That works fine and can be extended to N unknowns as long as the right sides of the equation are nonzero. In the variational molecular orbital problem, all the right sides of the equation are zero and if we use Cramer s rule we only get the trivial solution with all values equal to zero. The Cayley-Hamilton theorem implies that if all the equations equal zero, you can still get a nonzero solution by forcing the denominator determinant to be zero, that is, by solving for the roots of the corresponding... 

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