THE VARIATION METHOD

In this section we discuss an important approach to finding approximate solutions to the eigenvalue problem 

We are interested in eigenvalue problems because the time-independent Schrodinger equation is an eigenvalue equation  

Given the operator, there ei ists a set of exact solutions to the Schrodinger equation, infinite in number, labeled by the index a 

We have assumed for the sake of simplicity that the set of eigenvalues / is discrete. Since is a Hermitian operator, the eigenvalues are real and the corresponding eigenfunctions are orthonormal 

Furthermore, we assume that the eigenfunctions of form a complete set and hence any function 0 that satisfies the same boundary conditions as the set oe can be written as a linear combination of the 

The variation principle provides us with a simple and powerful procedure for generating approximate wave functions the variation method. For some proposed model or ansatz for the wave function, we express the electronic state C) in terms of a finite set of numerical parameters C, and the best values of C are deemed to be those that correspond to the stationary points of the energy function 

The stationary points of (C) represent the approximate electronic states C and the values of (C) at the stationary points are the approximate energies. We shall discuss the meaning of the term best approximation later. 

The exact wave function 0) corresponds to a stationary point of the expectation value of the Hamiltonian. The expectation value of the Hamiltonian for any approximate wave function 0) + 5) is therefore correct to second order in the error 3 and it follows that the energy calculated as an expectation value is more accurate than the wave function itself (for sufficiently small errors). This result is important as it shows that small contributions to the wave function may be neglected without affecting the calculated energy significantly. In Exercise 4.1, we show how the error in the energy can be estimated from the norm of the gradient. 

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Chemisoq)tion bonding to metal and metal oxide surfaces has been treated extensively by quantum-mechanical methods. Somoijai and Bent [153] give a general discussion of the surface chemical bond, and some specific theoretical treatments are found in Refs. 154-157 see also a review by Hoffman [158]. One approach uses the variation method (see physical chemistry textbooks) ... 

The complexity of molecular systems precludes exact solution for the properties of their orbitals, including their energy levels, except in the very simplest cases. We can, however, approximate the energies of molecular orbitals by the variational method that finds their least upper bounds in the ground state as Eq. (6-16)... 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

The relative strengths and weaknesses of perturbation theory and the variational method, as applied to studies of the electronic structure of atoms and molecules, are discussed in Section 6. 

This Introductory Section was intended to provide the reader with an overview of the structure of quantum mechanics and to illustrate its application to several exactly solvable model problems. The model problems analyzed play especially important roles in chemistry because they form the basis upon which more sophisticated descriptions of the electronic structure and rotational-vibrational motions of molecules are built. The variational method and perturbation theory constitute the tools needed to make use of solutions of... 

The variational method ean be used to optimize the above expeetation value expression for the eleetronie energy (i.e., to make the funetional stationary) as a funetion of the Cl eoeffieients Cj and the ECAO-MO eoeffieients Cv,i that eharaeterize the spin-orbitals. However, in doing so the set of Cv,i ean not be treated as entirely independent variables. The faet that the spin-orbitals ([ti are assumed to be orthonormal imposes a set of eonstraints on the Cv,i ... 

This characteristic is commonly referred to as the bracketing theorem (E. A. Hylleraas and B. Undheim, Z. Phys. 759 (1930) J. K. E. MacDonald, Phys. Rev. 43, 830 (1933)). These are strong attributes of the variational methods, as is the long and rich history of developments of analytical and computational tools for efficiently implementing such methods (see the discussions of the CI and MCSCF methods in MTC and ACP). 

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

This relationship is often used for computing electrostatic properties. Not all approximation methods obey the Hellmann-Feynman theorem. Only variational methods obey the Hellmann-Feynman theorem. Some of the variational methods that will be discussed in this book are denoted HF, MCSCF, Cl, and CC. 

A more accurate quantum mechanical approach makes use of the variational method. - The goal is the solution of the basic wave equation... 

Berencz, F., Acta Phys. Hung. 6, 423, Calculation of the ground state of H2 on the basis of the variation method. ... 

We return now to those molecular orbitals, variation method shows that the best values of the coefficients a are those which satisfy the equations... 

In the unrestricted treatment, the eigenvalue problem formulated by Pople and Nesbet (25) resembles closely that of closed-shell treatments.-On the other hand, the variation method in restricted open-shell treatments leads to two systems of SCF equations which have to be connected in one eigenvalue problem (26). This task is not a simple one the solution was done in different ways by Longuet-Higgins and Pople (27), Lefebvre (28), Roothaan (29), McWeeny (30), Huzinaga (31,32), Birss and Fraga (33), and Dewar with co-workers (34). 

In all the variational methods, the choice of trial function is the basic problem. Here we are concerned with the choice of the trial function for the polarization orbitals in the calculation of polarizabilities or hyperpolarizabilities. Basis sets are usually energy optimized but recently we can find in literature a growing interest in the research of adequate polarization functions (27). 

S. T. Epstein, The Variation Method in Quantum Chemistry, Academic, New... 

The variation method gives an approximation to the ground-state energy Eq (the lowest eigenvalue of the Hamiltonian operator H) for a system whose time-independent Schrodinger equation is... 

As a simple application of the variation method to determine the ground-state energy, we consider a particle in a one-dimensional box. The Schrodinger equation for this system and its exact solution are presented in Section 2.5. The ground-state eigenfunction is shown in Figure 2.2 and is observed to have no nodes and to vanish at x = 0 and x = a. As a trial function 0 we select 0 = x(a — x), 0 X a... 

If we use the wave function for the unperturbed ground state as a trial function 0 in the variation method of Section 9.1 and set H equal to then we have from equations (9.2), (9.18), and (9.24)... 

In this section we examine the ground-state energy of the helium atom by means of both perturbation theory and the variation method. We may then compare the accuracy of the two procedures. 

As a normalized trial function 0 for the determination of the ground-state energy by the variation method, we select the unperturbed eigenfunction r2) of the perturbation treatment, except that we replace the atomic number Zby a parameter Z ... 

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the ffee-electron gas and Bose-Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. 

Several examples of the application of quantum mechanics to relatively simple problems have been presented in earlier chapters. In these cases it was possible to find solutions to the Schrtidinger wave equation. Unfortunately, there are few others. In virtually all problems of interest in physics and chemistry, there is no hope of finding analytical solutions, so it is essential to develop approximate methods. The two most important of them are certainly perturbation theory and the variation method. The basic mathematics of these two approaches will be presented here, along with some simple applications. 

A different approach to obtaining approximation solutions to quantum mechanical problems is provided by the variation method. It is particularly useful when there is no closely related problem that yields exact solutions. The perturbation method is not applicable in such a case. 

The variation method is usually employed to determine an approximate value of the lowest eneigy state (the ground state) of a given atomic or molecular system. It can, furthermore, be extended to the calculation of energy levels of excited stales. It forms the basis of molecular orbital theory and that which is often referred to (incorrectly) as theoretical chemistry". 

T ie determinant in Eq. (59) is of course a secular determinant, a description that refers to its application to the temporal evolution of a mechanical system, historically in astronomy. It will re-appear later in this chapter in the development of the variation method. 

One of the most important techniques in quantum mechanics is known as the variation method. That method provides a way of starting with a wave function and calculating a value for a property (dynamical... 

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