Variation theory

The familiar variation theorem may be regarded as a particular instance of the inequalities referred to in the last section. Thus putting n = 1 in the n-term expansion (2.3.2) corresponds to using a single function (not necessarily normalized) as an approximate eigenfunction. The determinant (2.3.12) then reduces to a single element Hu-EMu, and on using W = Pi, 

E is said to be a functional of the functional E = E[W] takes a value that depends on the functional form of W, not merely on particular numerical values of a set of independent variables (adjustable parameters). The analysis of the behaviour of E[W] with respect to variation of W is functional analysis and parallels the study of the behaviour of f(x) with respect to variation of x (i.e. analysis in the usual sense) thus, for example, it is possible to discuss variations in terms of functional derivatives SEISfP. We shall, however, make little use of such concepts, in spite of their formal value (see e.g. Feynman and Hibbs, 1965), since variation functions are most commonly defined in terms of a finite number of parameters, to which the ordinary rules of differential calculus can be applied. The other problems we meet can also be solved by elementary methods. 

It is easily verified that if W in (2.4.1) is taken to be a linear variation function of the form Ci Pi -I- C2 2 + .. + then the condition that E be stationary against variation of Cj, Cz. , c leads back to the secular equation (2.3.4) in the truncated form corresponding to an n-term expansion. In other words, best approximate solutions at any level may be obtained by seeking stationary values of the corresponding energy functional. Thus the finite-basis form of (2.4.1) is 

Let us now consider a variation in which the coefficients change (c— c -1- 6c) to give a new value of E  

On e q anding the denominator using the binomial theorem and keeping only terms of first order in small quantities, we obtain easily 

The second major approximation theory used in quantum mechanics is called variation theory. Variation theory is based on the fact that any test wavefunction for a system has an average energy that is equal to or greater than the true ground-state 

Unless otherwise noted, all art on this page Is Cengage Learning 2014. 

One way of stating the basic idea behind variation theory is the following For a system having a Hamiltonian operator H, true wavefunctions I true nd some lowest-energy eigenvalue E, the variation theorem states that for any normalized trial wavefunction /  

Usually, the trial wavefunctions have some set of adjustable parameters (o, b,c.). The energy is calculated as an expression in terms of those parameters, and then the expression is minimized with respect to those parameters. In calculus terms, if the energy is some expression in terms of a single variable E a), then the minimum energy occurs when the slope of a plot of E versus a is zero  

This minimum energy is the best energy that this trial wavefunction can provide. When there are multiple variables in the trial wavefunction, then the absolute minimum with respect to all variables simultaneously is the lowest energy that such a trial wavefunction produces. Although variation theory does provide more complicated expressions for the energies of excited states, the above relatively simple expressions apply only to the ground state of a system. 

For the most realistic treatment of chemical problems, the Schrodinger equation does not tend to be separable and it is not usually a differential equation easily solved by analytical means. Or, more to the point, the problems of interest are not simple and are not strictly the same as ideal model problems. Powerful techniques have been formulated for dealing with realistic problems. Variation theory and perturbation theory are two such techniques that have been widely employed in understanding many quantum chemical systems. 

For the variational principle to hold, the chosen wavefunction must satisfy the same conditions we have considered necessary in interpreting the square of a true wave-function as a probability density. These are the smoothness and continuity and that the wavefunction be normalizable. The function and its first derivative must be continuous in all spatial variables of the system over the entire range of those variables the function must be single-valued and the integral of the square of the function over all space must be a finite value. Wavefunctions obtained from analytical solution of a Schrodinger equation and wavefunctions selected for variational treatment should satisfy these conditions. 

The process of adjusting chosen wavefunctions, or trial wavefunctions, to minimize the expectation value of the energy is the variational method. A useful first example is to return to a problem for which we have analytical solutions, the one-dimensional harmonic oscillator problem, and to see what application of the variational method yields. The Hamiltonian is that in Equation 7.28. The trial wavefunction, T, will be taken to be a Gaussian function with an undetermined parameter, a, that is adjustable. 

The constant in front of the exponential makes T normalized for any choice of a. Other functional forms could be used for the trial wavefunction, for instance a/(l + x ). Of course, the chosen trial function must satisfy the conditions of being smooth, continuous, and bounded. We know from the analytical treatment of the harmonic oscillator that T in Equation 8.67 is continuous, has a continuous first derivative, is single-valued, and yields a finite value if squared and integrated from -oo to The expectation value, designated W, is obtained by application of the Hamiltonian to the trial function followed by integration with the complex conjugate of the trial function. 

W turns out to be a function of the parameter a. Since the objective is to adjust the parameter so as to lower W as much as possible, the process needed is that of minimization of W(a), something that can be accomplished using the first derivative. 

The first step is to write the Hamiltonian for the problem. Then an educated guess is made at a reasonable wavefunction called formally the trial wavefunction, v /triai. The trial wavefunction will have one or more adjustable parameters, pi, that will be used for optimization. An energy expectation value in terms of the adjustable parameters, s, is obtained by using the same form as in Equation 2-23. 

The term in the denominator of Equation 4-1 is needed since the trial wavefunction is most likely not normalized. 

Variation theory states that the energy expectation value e is greater than or equal to the tme ground-state energy, Eo, of the system. The equality occurs only when the trial wavefunction is the tme ground-state wavefunction of the system. 

Since e is a function of the yet undetermined adjustable parameters Pi, the value of E can be optimized by taking the derivative of e with respect to each adjustable parameter and setting it equal to zero. A value for each parameter is then obtained for the optimized energy of the ground-state. 

Variation theory can be proven as follows. Take the trial wavefunction, H/triai, as a linear combination of the tme eigenfunctions of the Hamiltonian, 

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Equation (ASA. 110) represents the canonical fonn T= constant) of the variational theory. Minimization at constant energy yields the analogous microcanonical version. It is clear that, in general, this is only an approximation to the general theory, although this point has sometimes been overlooked. One may also define a free energy... 

Keck J 1960 Variational theory of chemical reaction rates applied to three-body recombinations J. Chem. Phys. 32 1035 Anderson J B 1973 Statistical theories of chemical reactions. Distributions in the transition region J. Chem. Phys. 58 4684... 

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

Now let us return to the Kolm variational theory that was introduced in section A3.11.2.8. Here we demonstrate how equation (A3.11.46) may be evaluated using basis set expansions and linear algebra. This discussion will be restricted to scattering in one dimension, but generalization to multidimensional problems is very similar. 

Several VTST techniques exist. Canonical variational theory (CVT), improved canonical variational theory (ICVT), and microcanonical variational theory (pVT) are the most frequently used. The microcanonical theory tends to be the most accurate, and canonical theory the least accurate. All these techniques tend to lose accuracy at higher temperatures. At higher temperatures, excited states, which are more difficult to compute accurately, play an increasingly important role, as do trajectories far from the transition structure. For very small molecules, errors at room temperature are often less than 10%. At high temperatures, computed reaction rates could be in error by an order of magnitude. 

CVT (canonical variational theory) a variational transition state theory technique... 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

Prager [302] examined diffusion in concentrated suspensions using the variational approach. (A discussion of the basic principles in variational theory is given in Ref. 6.) Prager s result is applicable to a very general class of isotropic porous media. Prager s solution for a limiting case of a dilute suspension of particles was... 

J. C. Keck, Variational theory of reaction rates, Adv. Chem. Phys. 13, 85 (1967). 

Variational Theory for a Nondegenerate Single Excited State. 125... 

Variational Theory for a Degenerate Single Excited State. 127... 

VARIATIONAL THEORY FOR A NONDEGENERATE SINGLE EXCITED STATE... 

We also adopt the above combination rule (Eq. [6]) for the general case of exp-6 mixtures that include polar species. Moreover, in this case, we calculate the polar free energy contribution Afj using the effective hard sphere diameter creff of the variational theory. 

THL.30. I. Prigogine, Sur la theorie variationnelle des phenomenes irreversibles (On the variational theory of irreversible phenomena), Bull. Cl. Sci. Acad. Roy. Belg. 40, 471 83 (1954). 

M. Nakata, M. Ehara, and H. Nakatsuji, Density matrix variational theory apphcation to the potential energy surfaces and strongly correlated systems. J. Chem. Phys. 116, 5432 (2002). 

Improving the foundation of manufacturing science in our current manufacturing practices should be the primary basis for moving away from the corrective action crisis to continuous improvement. Knowledge of the variation theory is, therefore, an essential element of manufacturing science. It requires an in-depth understanding of a process or system (15) ... 

Weiss-Marcus harmonic energy variation theory, 1513, 1519 Wenking, 1118... 

Correspondingly, a typical value for AG°/ES [cf Eq. (9.3)] is 0.5 so that (0r /3 In i) = (2RT/1.5F) = 1.3(RT/F). Although observed values of this coefficient vary from RT/4F to 2RT/F, and sometimes above this, the figure for the majority of electrochemical reactions is very near 2RT/F and thus the formation of the rate— overpotential relation to which this Weiss-Marcus harmonic energy variation theory gives rise is not consistent with experiment (Fig. 9.26). 

Over the years, several computational methods have been developed. The variational theory can be used either without using experimental data other than the fundamental constants (i.e., ab initio methods) or by using empirical data to reduce the needed amount of numerical work (i.e., semiempirical data methods). There are various levels of sophistication in both ab initio [HF(IGLO), DFT GIAO-MP2, GIAO-CCSD(T)] and semiempirical methods. In the ab initio methods, various kinds of basic sets can be employed, while in the semiempirical methods, different choices of empirical parameters and parametric functions exist. The reader is referred to reviews of the subject.18,77... 

A.D. Isaacson, D.G. Truhlar, Polyatomic canonical variational theory for chemical reaction rates. Separable-mode formalism with application to OH+I-p H2O+H, J. Chem. Phys. 76 (1982) 1380. 

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. 

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