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Lecture 14

The variation method

An exact solution of the Schrodinger equation is possible only for a one electron system. In largersystems, approximate methods have to be used. There are two kinds of approximate methods, the

perturbation method and the variation method. The latter method is more useful in chemicalbonding.

The wave equation is,

2+ 82m/h2 (E V) = 0

We can rewrite this in the familiar form,

(-h2/82m2 + V) = E

The quantity in the parentheses is an operator, H.

Thus,

H= E

Let us integrate within the space of the wave function.

*Hd = E * d

or

E = *Hd / * d

Thus, if the wave function is known, we can get the energy. Although it is possible to get the exactform of H, it is difficult to get the exact wave function. Still, the method is useful in getting theapproximate wave equation.

If we have the correct wave function we have the correct energy. But, often we do not have it.

Variation theorem says that if we have a trial wave function i, the energy Ei we get from the

equation above will be higher than the true energy Eo. This principle can be used in determiningthe best trial function and can be used further to improve the function. One can actually write thewave function in terms of certain parameters and can optimize the parameters to get the minimumenergy.

The variation principle can be demonstrated using the hydrogen atom. One can take the trial wave

function of the form, = e-ar. Using the right H, we can show that we get the right energy.

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The Secular Equations

The trial wave function can be complex. It may be in the form of functions, i, the coefficients, aiare arbitrary.

= a11 + a22 + . . . + ann

We can take a simple function,

= a11 + a22

The energy,

E = (a11* + a22*) H (a11 + a22) d / (a11* + a22*) (a11 + a22) d

Or

E (a11* + a22*) (a11 + a22) d = (a11* + a22*) H (a11 + a22) d

Which leads to,

E ( a121*1 d + 2a1a2 1*2 d + a22 2*2 d ) = a121*H1d

+ 2a1a21*H2d + a222*H2d

The equation, 1*2 d = 21* d is not always right for all operator, . This will be validonly for Hermetian operators. The eigen values are always real.

We need to have minimum value for E. Thus, energy should be minimized with respect to a1 anda2.

Differentiate with respect to a1.

E [ 2a11*1d + 2a21*2d ] + E/a1 [ a121*1d + 2a1a21*2d + a221*2d ] = 2a11*H1d + 2a21*H2d --- Eq (1)

In order to minimize energy, we need

(E/a1)a2 = (E/a2)a1 = 0

Let us use,

Hij = i*Hjd and Sij = i*j d

Let us assume E/a1 = 0, in eq. 1. Then we can write,

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(H11 ES11) a1 + (H12 ES12) a2 = 0

For a similar condition, E/a2 = 0, we can get,

(H21 ES21) a1 + (H22 ES22) a2 = 0

These two equations are called secular equations.The equations are of the form,ax + by = 0

cx + dy = 0

If we solve this set of linear, homogeneous equations, we get:

(ad bc) y = 0

For this to be valid, either y has to be zero or the coefficient has to be zero. The non-trivial solutionis the later which can be expressed in the determinental form.

a b= 0

c d

These conditions are the same for secular equations. Then,

H11 ES11 H12 ES12

= 0H21 ES21 H22 ES22

For a wave function represented as n independent terms,

H11 ES11 H12 ES12 Hln - ESln

H21 ES21 H22 ES22 H2n ES2n

. . . = 0

. . .Hn1 ESn1 Hn2 ESn2 Hnn ESnn