The variation principle

There are many systems for which it is impossible to obtain explicit mathematical solutions of the Schrodinger equation. For many atoms and molecules it is therefore necessary to employ approximate methods. Exact solutions for molecular systems can be obtained only for the hydrogen molecule ion, which has a single electron. Approximate methods are required even for the hydrogen molecule, H2. 

The most useful and commonly used approximate method is the variation method. 

A derivation of the variation principle is outside the scope of this book, but it is important to have some understanding of how the method is applied, since some of its applications shed considerable light on the nature of the chemical bond. 

Suppose, first, that the Schrodinger equation can be solved exactly and that we have obtained an eigenfunction ij/ which need not be normalized. Then if H is the Hamiltonian operator, this eigenfunction satisfies the equation 

Suppose that we multiply both sides by the complex conjugate / of the eigenfunction  

For any trial wavefunction, the expectation value of the energy can never he less than the true ground state energy. 

This result comes about because, as mentioned in Section 8.2, the expectation value of the energy is a weighted mean of the true energies of the system, E, E, and this mean value cannot be less than E.  

9 Books on quantum mechanics are numerous. Understandably, the author inclines towards the Russian school of presenting quantum mechanics, which may be subdivided into three major subschools (one of Leningrad - Fock [19] and two of Moscow Landau [20] and Blokhintsev [21]). Classical works are also very useful [22-24]. 

10Literature on the variational principle is voluminous. In what relates to quantum mechanics the texts also contain necessary information on the variational principle. Here we add only [25] - a brilliant mathematical text on variations and [26] - a somewhat too casuistic, but useful, description of different aspects of the variational technique. I. Mayer [18] gives a good survey of what is necessary. 

An important special case in such an expectation value is the scalar product of two functions, where the operator A is the identity operator / /f.r. x-,.) = l.Then 

Similar relations hold for operators which will be explained below and widely used throughout the book. 

With these notation the proof evolves as follows The energy is the quantum mechanical expectation value E = (T // T) of the Hamiltonian calculated for any wave function k normalized to unity (( k k) = 1). The Schrodinger equation is a linear equation of the form  

We consider in this section the variation principle in molecular electronic-structure theory. Having established the particular relationship between the Schrddinger equation and the variational condition that constitutes the variation principle, we proceed to examine the variation method as a computational tool in quantum chemistry, paying special attention to the application of the variation method to linearly expanded wave functions. Next, we examine two important theorems of quantum chemistry - the Hellmann-Feynman theorem and the molecular virial theorem - both of which are closely associated with the variational condition for exact and approximate wave functions. We conclude this section by presenting a mathematical device for recasting any electronic energy function in a variational form so as to benefit to the greatest extent possible from the simplifications associated with the fulfilment of the variational condition. 

To establish the one-to-one relationship between the stationary points of the enogy functional [0] and the solutions to the Schrodinger equation, we first assume that 0) represents a solution to the Schrodinger equation (4.2.1) and that ) is an allowed variation 

Inserting this expression in the energy functional (4.2.2) and expanding in orders of S) around 0), we obtain 

The first-order variation in the energy functional [6] therefore vanishes whenever 0) corresponds to one of the eigenstates 0), thus demonstrating that the eigenstates of the Schrodinger equation represent stationary points of the energy functional. 

Conversely, to show that all stationary points of the energy functional represent eigenstates of the Schrodinger equation, let 0) be a stationary point of [0]. For the variation 5), we obtain by expanding the energy functional around the stationary point in the same way as in (4.2.4) 

Many of the calculations of quantum chemistry are based on the Rayleigh-Ritz variation principle which states For any normalized, acceptable function 4 , 

This statement is easily proved. We expand in terms of the complete, orthonormal set of eigenfunctions of H  

Now c Ci is never negative, and so Eq. (6-36) is merely a weighted average of the eigenvalues Ei. Such an average can never be lower than the lowest contributing member and the principle is proved. 

The variation principle is sometimes stated in an equivalent way by saying that the average value of H over / is an upper bound for the lowest eigenvalue of H. Following the approach of the example at the end of the previous section, if for a hydrogen atom happens to be a function equal to (1 /V2)t/ri + (1 the average energy for  

We are now in a position to state and prove an important theorem, called the variation principle Given a normalized wave function 0 that satisfies 

The equality holds only when ) is identical to o)-The proof of this theorem is simple. First we consider 

The variation principle for the ground state tells us that the energy of an approximate wave function is always too high. Thus one measure of the quality of a wave function is its energy The lower the energy, the better the wave function. This is the basis of the variation method in which we take a normalized trial function , which depends on certain parameters, and vary these parameters until the expectation value 0 reaches a minimum. This minimum value of 0 is then our variational estimate of the exact ground state energy. 

Exercise 1.18 The Schrodinger equation (in atomic units) of an electron moving in one dimension under the influence of the potential — (x) is 

Exercise 1.19 The Schrddinger equation (in atomic units) for the hydrogen atom is 

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In the quantum mechanics of atoms and molecules, both perturbation theory and the variational principle are widely used. For some problems, one of the two classes of approach is clearly best suited to the task, and is thus an established choice. Flowever, in many others, the situation is less clear cut, and calculations can be done with either of the methods or a combination of both. 

Another connnon approximation is to construct a specific fonn for the many-body waveftmction. If one can obtain an accurate estimate for the wavefiinction, then, via the variational principle, a more accurate estimate for the energy will emerge. The most difficult part of this exercise is to use physical intuition to define a trial wavefiinction. 

When the reciprocal relations are valid in accord with (A3.2.251 then R is also symmetric. The variational principle in this case may be stated as... 

C. Lanczos, The Variational Principles of Mechanics , University of Toronto Press, Toronto, 1970... 

Using the variation principle to optimize Cj and C2 we obtain dE/dc and dEjdc from Equation (7.44) and put them equal to zero, giving... 

The solvated Fock operator can be naturally derived from the variational principles [14] defining the Helmlioltz free energy of the system (fA) by... 

Note that this is also a functional of liaAr), Cas(r), and 4 ). Imposing constraints concerning the orthonormality of the configuration state function (C) and one-particle orbitals (pi) on the equation, one can derive the Eock operator from. A based on the variational principle ... 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

The variational principle leads to the following equations describing the molecular orbital expansion coefficients, c. , derived by Roothaan and by Hall ... 

In order to test the accuracy of the LCAO approximations, we use the variation principle if V lcao is an approximate solution then the variational integral... 

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. 

We now need to use the variation principle to seek the best possible values of the LCAO coefficients. To do this, I have to find Se as above, and set its first derivative to zero. I keep track of the requirement that the LCAO orbitals are... 

Then there is the question of quality. The variation principle only tells us about energies we might calculate the variational integral... 

It is also a common experience that traditional Cl calculations converge very poorly, because the virtual orbitals produced from an HF (or HF-LCAO) calculation are not determined by the variation principle and turn out to be very poor for representations of excited states. 

Sir William Hartree developed ingenious ways of solving the radial equation, and they are documented in Douglas R. Hartree s book (1957). By the time this book was published, the SCF method had been well developed, and its connection with the variation principle was finally understood. It is interesting to note that Chapter 2 of Douglas R. Hartree s book deals with the variation principle. 

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... 

The weight is the sum of coefficients at the given excitation level, eq. (4.2). The Cl method determines the coefficients from the variational principle, thus Table 4.2 shows that the doubly excited determinants are by far the most important in terms of energy. The singly excited determinants are the second most important, then follow the quadruples and triples. Excitations higher than 4 make only very small contributions, although there are actually many more of these highly excited determinants than the triples and quadruples, as illustrated in Table 4,1. 

The dissociation problem is solved in the case of a full Cl wave function. As seen from eq. (4.19), the ionic term can be made to disappear by setting ai = —no- The full Cl wave function generates the lowest possible energy (within the limitations of the chosen basis set) at all distances, with the optimum weights of the HF and doubly excited determinants determined by the variational principle. In the general case of a polyatomic molecule and a large basis set, correct dissociation of all bonds can be achieved if the Cl wave function contains all determinants generated by a full Cl in the valence orbital space. The latter corresponds to a full Cl if a minimum basis is employed, but is much smaller than a full Cl if an extended basis is used. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

Consider now making the variational coefficients in front of the inner basis functions constant, i.e. they are no longer parameters to be determined by the variational principle. The Is-orbital is thus described by a fixed linear combination of say six basis functions. Similarly the remaining four basis functions may be contracted into only two functions, for example by fixing the coefficient in front of the inner three functions. In doing this the number of basis functions to be handled by the variational procedure has been reduced from 10 to three. 

By including the doubly excited determinant, built from the antibonding MO, the amount of covalent and ionic terms may be varied, and be determined completely by the variational principle (eq. (4.19)). 

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