Variational principles stationary

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

For a family of trajectories all starting at the value X(to) and at t=t all arriving at X(t), there is one trajectory that renders the action stationary. The classical mechanical trajectory of a given dynamical system is the one for which 5S=0, i.e. the action becomes stationary. The equation of motion is obtained from this variational principle [59], The corresponding Euler-Lagrange equations are obtained d(3L/3vk)/dt = 9L/dXk. In Cartesian coordinates these equations become Newton s equations of motion for each nucleus of mass Mk ... 

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. 

In principle both approaches may differ in the index of the stationary geometries sensed by the eigenvalues of the Hessian matrix. The variational principle implies, in the present electronuclear approach, that any stationary conformation is a minimum since geometric variations must conserve symmetry to do actual computing one should use variations in the an-space. On the other hand, in the... 

This part is concerned with variational theory prior to modem quantum mechanics. The exception, saved for Chapter 10, is electromagnetic theory as formulated by Maxwell, which was relativistic before Einstein, and remains as fundamental as it was a century ago, the first example of a Lorentz and gauge covariant field theory. Chapter lisa brief survey of the history of variational principles, from Greek philosophers and a religious faith in God as the perfect engineer to a set of mathematical principles that could solve practical problems of optimization and rationalize the laws of dynamics. Chapter 2 traces these ideas in classical mechanics, while Chapter 3 discusses selected topics in applied mathematics concerned with optimization and stationary principles. 

The calculus of variations [5,322] is concerned with problems in which a function is determined by a stationary variational principle. In its simplest form, the problem is to find a function v(x) with specified values at end-points xo, x such that the integral J = /(x, y, y )dx is stationary. The variational solution is derived... 

Variational principles for classical mechanics originated in modem times with the principle of least action, formulated first imprecisely by Maupertuis and then as an example of the new calculus of variations by Euler (1744) [436], Although not stated explicitly by either Maupertuis or Euler, stationary action is valid only for motion in which energy is conserved. With this proviso, in modem notation for generalized coordinates,... 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

Defining the functional V = (i/ v f). where u(r) is a local potential for which E = T + V is bounded below, the Schrodinger variational principle requires E to be stationary subject to normalization =. The variation SE induced... 

The Schrodinger variational principle requires (T //1T) to be stationary subject to constant normalization ( ). Introducing a Lagrange multiplier, the variational condition is... 

In bound-state calculations, the Rayleigh-Ritz or Schrodinger variational principle provides both an upper bound to an exact energy and a stationary property that determines free parameters in the wave function. In scattering theory, the energy is specified in advance. Variational principles are used to determine the wave function but do not generally provide variational bounds. A variational functional is made stationary by choice of variational parameters, but the sign of the residual error is not determined. Because there is no well-defined bounded quantity, there is no simple absolute standard of comparison between different variational trial functions. The present discussion will develop a stationary estimate of the multichannel A -matrix. Because this matrix is real and symmetric for open channels, it provides the most... 

The derivation by Kohn [202] can readily be extended to the multichannel 7 -matrix. The underlying logic depends on the variational principle that the multichannel Schrodinger functional is stationary for variations of a trial function that satisfies the correct boundary conditions if and only if that function satisfies the Schrodinger equation. In a matrix notation, suppressing summations and indices, the variational functional of Schrodinger is... 

At the risk of being redundant, we may state here the salient features of the TD-functional formalism. The first requirement is a variational principle, and for a time-dependent quantum description only a stationary action principle is available. With this a mapping theorem is established which turns the action functional into a functional of relevant physical quantities (which are the expectation values), and the condition of stationarity is now in terms of these variables instead of the entire density matrix. Thus the stationary property with respect to the density matrix now becomes one with respect to all the variables... 

Variational principles are widely applied to calculations of bound states in quantum mechanics [18,231]. One usually considers the expectation value of the energy of the system, FJ[ ] = ( ]ff ), and looks for stationary points (i.e., 5E = 0) of this functional with respect to arbitrary variations... 

Stationary points of the functional [c] should be calculated through variation of the coefficients c. Kohn s variational principle requires the wave function on dS to remain fixed during the variation, 6fa = 0. In view of Eq. (27), this means that variation of the Ck is subject to the additional condition Y.k Tak Sck — 0. The standard way to solve a variational problem with constraints is to use undetermined Lagrange multipliers [234]. A technical realization of this method, which we do not describe here, is given in Ref. 60. Using it, one obtains a compact expression for a set of coefficients c which render [c] stationary, namely... 

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. 

The Schrodinger variational principle can be applied directly for fixed occupation numbers nt, E is required to be stationary subject to (i j) = 8y. To simplify the derivation, trial orbitals can be required to be orthogonal, but diagonal Lagrange multipliers e, are used for the normalization constraint. The variational condition is... 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

It will be recalled that the Variation Principle asks us to choose those values of cr which make the Rayleigh Ratio, y, stationary. We thus seek a situation in which a change in cr (r = 1,2,..., , in turn) produces no change in y. In other words, we shall require n conditions of the sort... 

Instead of applying tail cancellation as in Sect.2.1 where we derived the KKR-ASA equations, one may use the linear combination of muffin-tin orbitals (5.27) directly in a variational procedure. This has the advantages that it leads to an eigenvalue problem and that it is possible to include non-muffin-tin perturbations to the potential. According to the Rayleigh-Ritz variational principle, one varies y to make the energy functional stationary, i.e. 

Here L ig a certain function of the co-ordinates and velocities of all the particles, and, in certain circumstances, also an explicit function of the time, and equation (2) as an expression of Hamilton s Principle is to be interpreted as follows the configuration (co-ordinates) of the system of particles is given at the times l1 and t2 and the motion is sought (i.e. the co-ordinates as function of the time) which will take the system from the first configuration to the second in such a way that the integral will have a stationary value.2 The chief advantage of such a variation principle is its independence of the system of coordinates. 

The first three methods we will discuss are based on variational principles— not minimum principles for the energy, but stationary principles for the scattering amplitude or some related quantity. While these methods are the most elaborate and computationally demanding, they are also potentially the most flexible and the most accurate, in that they make the fewest simplifications and approximations. More approximate methods are also in use, and descriptions can be found elsewhere (e.g., Huo and Gianturco, 1995). Because of its extraordinary utility, we will also briefly consider the method of Kim and Rudd (Kim and Rudd, 1994 Hwang etal, 1996) for obtaining electron impact ionization cross sections, which is based on a very simple model of the electron-target interaction. 

The results of the variation principle calculation are shown in Figure 5.2. The stationary value of the total energy of the helium atom within the variation principle calculation occurs for the parameter k of equation 5.22 equal to 3.375 a.u. (54/16) of inverse of length squared. Hartree pointed out that this result... 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

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