The variation theorem

The Variation Theorem. Given a system whose Hamiltonian operator H is time independent and whose lowest-energy eigenvalue is if is any normalized, well-behaved function of the coordinates of the system s particles that satisfies the boundary conditions of the problem, then 

The variation theorem allows us to calculate an upper bound for the system s ground-state energy. 

To prove (8.1), we expand f in terms of the complete, orthonormal set of eigenfunctions of H, the stationary-state eigenfunctions 

Note that the expansion (8.2) requires that f obey the same boundary conditions as the i/r/t s. Substitution of (8.2) into the left side of (8.1) gives 

Using the eigenvalue equation (8.3) and assuming the validity of interchanging the integration and the infinite summations, we get 

To deal with the time-independent Schrodinger equation for systems (such as atoms or molecules) that contain interacting particles, we must use approximation methods. This chapter discusses the variation method, which allows us to approximate the ground-state energy of a system without solving the Schrodinger equation. The variation method is based on the following theorem  

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Wlien first proposed, density llinctional theory was not widely accepted in the chemistry conununity. The theory is not rigorous in the sense that it is not clear how to improve the estimates for the ground-state energies. For wavefiinction-based methods, one can include more Slater detenuinants as in a configuration interaction approach. As the wavellmctions improve via the variational theorem, the energy is lowered. In density fiinctional theory, there is no... 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

In accordance with the variation theorem we require the set of coefficients that gives the lowest-energy wavefunction, and some scheme for changing the coefficients to derive that wavefunction. For a given basis set and a given functional form of the wavefunction (i.e. a Slater determinant) the best set of coefficients is that for which the energy is a minimum, at which point... 

Suppose the ground state solution to this problem were unknown, and that you wish to approximate it using the variational theorem. Choose as your trial wavefunetion,... 

The variational theorem which has been initially proved in 1907 (24), before the birthday of the Quantum Mechanics, has given rise to a method widely employed in Qnantnm calculations. The finite-field method, developed by Cohen andRoothan (25), is coimected to this method. The Stark Hamiltonian —fi.S explicitly appears in the Fock monoelectronic operator. The polarizability is derived from the second derivative of the energy with respect to the electric field. The finite-field method has been developed at the SCF and Cl levels but the difficulty of such a method is the well known loss in the numerical precision in the limit of small or strong fields. The latter case poses several interconnected problems in the calculation of polarizability at a given order, n ... 

In many applications of quantum mechanics to chemical systems, a knowledge of the ground-state energy is sufficient. The method is based on the variation theorem-, if 0 is any normalized, well-behaved function of the same variables as and satisfies the same boundary conditions as then the quantity = (p H (l)) is always greater than or equal to the ground-state energy Eq... 

To prove the variation theorem, we assume that the eigenfunctions 0 form a complete, orthonormal set and expand the trial function 0 in terms of that set... 

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... 

Following the same procedure used to prove the variation theorem, we obtain... 

According to the variation theorem, the lowest root g o is an upper bound to the ground-state energy Eq Eo So- The other roots may be shown to be upper bounds for the excited-state energy levels... 

The numerical value of S is listed in Table 9.1. The simple variation function (9.88) gives an upper bound to the energy with a 1.9% error in comparison with the exact value. Thus, the variation theorem leads to a more accurate result than the perturbation treatment. Moreover, a more complex trial function with more parameters should be expected to give an even more accurate estimate. 

Although the variational theorem was expressed in Eq. (111) with respfect to the ground state of the system, it is possible to apply it to higher, so-called excited, states. As an example, consider again the particle in a box. In Section 5.4.2 a change in coordinate was made in order to apply symmetry considerations. Thus, fire DOtential function was written as... 

As a simple proof of the variational theorem, consider the case in which 4> tJ/0. The variational function can be expanded in terms of die complete... 

The variation theorem has been an extremely powerful tool in quantum chemistry. One important technique made possible by the variation theorem is the expression of a wave function in terms of variables, the values of which are selected by minimizing the expectation value of the energy. 

One of the remarkable results of quantum mechanics is the variation theorem, which states that... 

Thus the singlet spatial function is symmetric and the triplet one antisymmetric. If we use the variation theorem to obtain an approximate solution to the ESE requiring symmetry as a subsidiary condition, we are dealing with the singlet state for two electrons. Alternatively, antisymmetry, as a subsidiary condition, yields the triplet state. 

The relative values of the coefficients indicate that the variation theorem thinks better of the covalent function, but the other appears fairly high at first glance. If, however, we apply the EGSO process described in Section 1.4.2, we obtain 0.996 50 Vc + 0.083 54 where, of course, the covalent function is unchanged,... 

We define a ten-function AO basis for the H2 molecule that has two different s -type orbitals and one />-type set on each H atom. It will be recalled that Weinbaum allowed the scale factor of the s orbital to adjust at each intemuclear distance. Using two different sized 5-type orbitals on each center accomplishes a similar effect by allowing the variation theorem to choose the amount of each in the mixture. Our orbitals are shown in Table 2.2. The 5-type orbitals are a split version of the Huzinaga 6-Gaussian H function and the p-type orbitals are adjusted to optimize the energy at the minimirm. It will be observed that the p and scale factors are different. We will present an interpretation of this below. 

If we wish to apply the variation theorem to ir, we still need the condition it must satisfy. Reflecting back upon the two-electron systems, we see that the requirement of symmetry for singlet functions could have been written... 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

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