Trial eigenfunctions

The procedure used to solve second-order differential equations of the form of equation (7.50) is essentially the same as that described in Worked Problem 7.4 and involves the construction of trial eigenfunctions from some of the functions introduced in Chapter 2. 

The usefulness of this method is that we can calculate energies from a variety of functions, and we know that the lowest energy obtained is closest to the truth. More systematically, we can express a trial eigenfunction as a function of certain variables and calculate the energy from equation (1.32) as a function of these variables. The energy can be minimized with respect to these variables, and we will then have obtained the best energy value from this type of function. 

In 1937, John Slater proposed to partition a crystal into regions of atomic nucleus and an interstitial space. He used different functions for approximation of these two areas of a many-body system. He replaced the trial eigenfunction for many particles by a determinant (3.26). 

The method of applying this theorem is equally simple in principle. A trial eigenfunction (Xi, X2, ), normalized to unity, is chosen, this trial eigenfunction being a function of a number of parameters Xi,... 

In simple cases, and with a judiciously chosen trial eigenfunction, the results of the variation method are identical with the results obtained by the solution of the Schrodinger equation. As an example, let us consider the harmonic oscillator, for which the Hamiltonian operator is... 

The fact that these results are identical with those of section 5e is, of course, due to the well-chosen form which we took for our trial eigenfunction. 

By introducing more parameters into the trial eigenfunction experimental result. Particularly good results are obtained, even with quite simple trial eigenfunctions, if the variable R12 is explicitly introduced into the trial eigenfunction. For example, Hylleras, using the trial eigenfunction... 

Our next problem is then to combine the determinantal eigenfunctions, the D functions, into new trial eigenfunctions which are eigenfunctions of M, S, Mg, and Sg. We shall first show that these trial eigenfunctions are already eigenfunctions of Mg and Sg. Let us take a general determinantal eigenfunction... 

Having found a set of A s which are eigenfunctions of Mg, and Sg, we repeat this same process with taking the place of The resulting linear combinations of A s are then eigenfunctions of S, Mg, and Sg. We thus obtain a set of trial eigenfunctions B which satisfy the relations... 

Let us therefore suppose that we have set up an approximate potential field V for the molecule, in which an electron is to move. This field might, for example, be that obtained by the superposition of the Hartree fields of the component atoms. Let us also suppose that we are given a set of functions atomic orbitals of the various atoms of the molecule, although any set of independent functions could be used. The approximate orbitals can then be found by the method of trial eigenfunctions, using zero-order functions. The approximate energies of the molecular orbitals will therefore be the roots of the secular equation... 

The variation principle then says that the energy E0 of the ground state is the lower bound of the quantity Eq. II.6 for arbitrary normalized trial wave functions W and that further all eigenfunctions satisfy the relation... 

It seems as if an energy value of sufficiently high accuracy has now been found for the helium problem, but we still do not know the actual form of the corresponding exact eigenfunction. In this connection, the mean square deviation e = J — W 2 (dx) and criteria of the Eckart type (Eq. III.27) are not very informative, since s may turn out to be exceedingly small, even if trial function... 

Except for the restrictions stated above, the function 0, called the trial function, is completely arbitrary. If 0 is identical with the ground-state eigenfunction 00, then of course the quantity S equals Eq. If 0 is one of the excited-state eigenfunctions, then is equal to the corresponding excited-state energy and is obviously greater than Eq. However, no matter what trial function 0 is selected, the quantity W is never less than 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... 

When the quantity W is not identieal to Eq, we assume that the trial funetion 0 which minimizes W is an approximation to the ground-state eigenfunction ipQ. However, in general, is a closer approximation to Eq than 0 is to 0o-... 

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... 

The reason why we obtain the exact ground-state energy in this simple example is that the trial function 0 has the same mathematical form as the exact ground-state eigenfunction, given by equation (4.39). When the parameter c is evaluated to give a minimum value for S , the function 0 becomes identical to the exact eigenfunction. 

When the exact eigenfunctions ipa, ip, . .., tpk- are not known, they may be approximated by trial functions 0o, 0i,. .., (pk-i which successively give upper bounds for Eq, E, , Ek-, respectively. In this case, the function 0i... 

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 ... 

Of these three classes (ii) is most easily disposed of clearly if a co-ordinate q does not appear in H then we can anticipate that the variation process will be completely indifferent to symmetry classifications involving q. Unless, of course, the form of the trial function is chosen with these variationally phantom degrees of freedom in mind. In the case of electron spin the unrestricted solution of Eq. (22) would not therefore lead to a total wave function which is an eigenfunction of operators depending on spin co-ordinates. 

The overlap of ip with the true ground state eigenfunction ipo is greater than or equal to 1 — e that is, the spatial distribution of the trial wave function is a very good approximation to the true wave function, and... 

These permutational symmetries are not only characteristics of the exact eigenfunctions of H belonging to any atom or molecule containing more than a single electron but they are also conditions which must be placed on any acceptable model or trial wavefunction (e.g., in a variational sense) which one constructs. 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

A useful trial variational function is the eigenfunction of the operator L for the parabolic barrier which has the form of an error function. The variational parameters are the location of the barrier top and the barrier frequency. The parabolic barrierpotential corresponds to an infinite barrier height. The derivation of finite barrier corrections for cubic and quartic potentials may be found in Refs. 44,45,100. Finite barrier corrections for two dimensional systems have been derived with the aid of the Rayleigh quotient in Ref 101. Thus far though, the... 

The vectors corresponding to the zero eiegenvalues of the matrix (M-o l) are the eigenfunctions of the matrix M corresponding to eigenvalue u. These eigenfunctions will be available from the TDDFT calculation and can be projected out of the trial solution of the system of equations to improve convergence. 

Because (4> ff S) is not in itself a variational expression, its unconstrained minimum value is not simply related to an eigenstate of the Hamiltonian Hv defined by v in Eq.(3), whereas Eq.(2) defines F[p only for such eigenstates. Any arbitrary trial function J —> can be expressed in the form + Aca with ca = 1. If the minimizing trial function in Eq.(3) were not an eigenfunction of Hv, then for some subset of trial functions, using the Brueckner-Brenig condition,... 

Even when confining the variation of the trial wavefunction to the LCAO-MO coefficients c U, the respective approximate solution of the Schrodinger equation is still quite complex and may be computationally very demanding. The major reason is that the third term of the electronic Hamiltonian, Hel (Equation 6.12), the electron-electron repulsion, depends on the coordinates of two electrons at a time, and thus cannot be broken down into a sum of one-electron functions. This contrasts with both the kinetic energy and the electron-nucleus attraction, each of which are functions of the coordinates of single electrons (and thus are written as sums of n one-electron terms). At the same time, orbitals are one-electron functions, and molecular orbitals can be more easily generated as eigenfunctions of an operator that can also be separated into one-electron terms. 

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