CHEMISTRY 158A, FALL, 2008

CHEMISTRY 158a, FALL, 2008

PERTURBATION THEORY

The student of quantum mechanics quickly discovers that very few problems in quantum mechanics can be solved exactly. The small set of soluble problems includes the particle in the box, the harmonic oscillator, the rigid rotor, and the hydrogen atom. In most cases, we must apply approximations which if we are clever will yield useful results. This document will develop perturbation theory, an approach for the case where the problem differs slightly from one that can be solved exactly.

Notation. This document will use the Dirac notation. The index in the ket or bra is the quantum number that specifies the state. A superscript 0 will label the operators and results of problems that can be solved exactly. Operators will be designated with bold, Italic font, e.g. H. For example, |i0> represents the i’th energy state with an energy Ei0 of the system with Hamiltonian is H0.

Consider first a problem such as the particle in the box that can be solved analytically. This is called the unperturbed problem. Given the Hamiltonian energy operator H0, one can solve the energy eigenvalue problem

H0 |i0> = Ei0|i0> (1)

And thereby obtains the energy eigenvalues Ei0 and the corresponding energy eigenfunctions |i0> . Two properties of these eigenfunctions, orthogonality and completeness, are mathematical consequences of the form of equation (1) and are very useful for solving general problems. Completeness allows us to construct the solution, i.e. the energy eigenfunctions, for any problem, from a linear combination (weighted superposition) of the |i0>. That is, the eigenfunction of state i for a system with a Hamiltonian H can be written as

|i> = S aij|j0> (2)

j

where the aij is the contribution of the unperturbed basis function |j0> to |i>.

In solving a problem related to an exactly soluble problem, we can divide its Hamiltonian operator H into two pieces, H0 and H’.

H = H0 + H' (3)

H0 is the Hamiltonian for the system whose operator-eigenfunction equation, e.g. equation (1), can be solved exactly. In perturbation theory, the approximation method developed here, one modifies the soluble Hamiltonian H0 by adding a term H' which is assumed to make a small contribution to the total energy. H' is called the perturbation. Planetary astronomy provides a straightforward example. Consider a solar system consisting of a massive sun and two planets. The gravitational potential energy of the three-body system is dominated by the gravitational attraction of each planet for the sun; the gravitational attraction between the two, much less massive planets constitutes the small perturbation is this example.

The perturbation H' is the term that prevents an exact solution of the problem. Our method of attack is guided by its magnitude and the small contributions that it makes to the solution. We shall employ the method of successive approximations, sometimes called a bootstrap method, in our approach to the problem. This strategy is useful in a broad range of problems including chemical equilibrium. As a first step, we identify the largest term in equation (2), the linear combination of basis functions. For state i, this is |i0>; that is, |i> @ |i0>. For example, if we modify the particle in the box potential where V is zero inside the box by adding a small symmetric barrier in the center of the box, the ground state for the system with the small barrier will very closely resemble the ground state for the soluble problem without the barrier. With this first estimate for the energy eigenfunction, we can now calculate the energy. Since |i0> is only an approximate solution, it is not a solution of the operator-eigenfunction equation for the full operator H. Hence, we employ Postulate 4 (page 122 of McQuarrie & Simon) by which an average value is calculated.

Ei = <i|H|i>/<i|i> @ <i0|H|i0>/<i0|i0> (4)

If we use normalized basis functions |i0>, the expression can be simplified by setting the denominator to one. Also, note that H0|i0> = Ei0|i0>.

Ei = <i0|H|i0> = <i0|H0|i0> + <i0|H'|i0>

= Ei0<i0|i0> + <i0|H'|i0> = Ei0 + <i0|H'|i0> (5)

The final result given in equation (5) yields an estimate of the energy to first order. It employs the exact operator and an approximate energy eigenfunction. The resulting estimate of the energy is an average of the correct operator over the approximate energy eigenfunction. This approach is a feature of many quantum mechanical methods. If the perturbation is small, the results are quantitatively very accurate. This is the case in NMR when the scalar spin-spin coupling constants are small compared with the difference in chemical shifts. First order perturbation theory works quite well and yields the results on splitting patterns that are found in every organic textbook.

In the event that one requires a more accurate energy or properties requiring a better eigenfunction, one takes the next step and improves the energy eigenfunction. The improved eigenfunction can then be substituted into equation (4), thus generating the energy to second order. The simple example of the low, symmetric barrier in the middle of a one-dimensional box is instructive. The ground state wave function has mostly the shape of a half sine wave. This is the basis of the first-level approximation that is the source of first-order perturbation theory. The barrier in the center of the well raises the potential energy in the center of the well and decreases the probability density and the wave function. This result, a small dip in the half wave, can be seen as adding a bit of the function (2/L)0.5sin[3px/L] which is |30> to (2/L)0.5sin[px/L] which is |10>. Nota bene; the ground state is even and only even functions such as (2/L)0.5sin[3px/L] can be included in the superposition.

This section provides the bootstrap derivation of the improvement of the energy eigenfunction. An alternate derivation, Rayleigh-Schroedinger perturbation theory, can be found in many quantum mechanics texts such as Pauling & Wilson. The derivation begins by substituting equation (2), which is an exact expression for |i> until we make some approximations later, into the equation H|i> = Ei|i>. One obtains the result

S aij(H0 + H')|j0> = S aijEi|j0> (6)

j j

Keeping our eye on the target, an expression for aij, one multiplies both sides by <k0| and takes the inner product, the generalization of the dot product. This is done to select out the desired component of the weighted sum of eigenfunctions. One obtains

S aij<k0|(H0 + H')|j0> = S aij<k0|Ei|j0> (7)

j j

The right hand side of equation (7) simplifies to aikEi since <k0|j0> is zero (orthogonality!) for all values of j except k. Orthogonality selects one term in the sum. Now we shall make the first of several simplifying approximations to equation (7) which to this point is exact. We shall replace the exact energy eigenvalue Ei with its approximate value derived to first order, namely Ei0 + <i0|H'|i0>. These manipulations yield

S aij<k0|(H0 + H')|j0> = aik(Ei + <i0|H'|i0) (7)

j

The derivation proceeds with the simplification of the left hand side of the equation. The left side has two terms. In the first, one uses the fact that |j0> is an energy eigenfunction of H0 so H0|j0> = Ej0|j0>. Each term of the sum is aij Ej0<k0|j0>. Orthogonality of the eigenfunctions eliminates all terms of the sum except for the term where j equals k. One now obtains.

aik Ek0 + S aij<k0|H'|j0> = aik(Ei0 + <i0|H'|i0> (8)

j

The next step involves the simplification of the remaining sum. Each term in the sum is the product of a coefficient and a matrix element. Recall that for a small perturbation, the leading term in the expansion given by equation (2) is the largest. That is, aii @ 1 and aii > |aik|. Therefore, the largest term in the sum is the one in which j equals i. We discard the remaining smaller terms and end up with

aik Ek0 + aii<k0|H'|i0> = aik(Ei0 + <i0|H'|i0>) (9)

Having discarded many small terms, one discards one final small term, <i0|H'|i0> on the right hand side and uses the result aii @ 1. If this last step bothers you, you can always renormalize the final eigenfunction. After making these final changes, one obtains after a bit of algebra the final result.

aik = <k0|H'|i0>/( Ei0 - Ek0) (10)

Equation (10) and the result aii @ 1 yield the energy eigenfunction to second order. Setting |i> equal to |i0> is the first order result. Equation (10) is an important result and will guide our qualitative discussion of molecular orbital theory. Several features of equation (10) are noteworthy. The magnitude of the contribution that eigenfunction |k0> make to |i> is inversely proportional to the energy difference between |k0> and |i0>. Please note that we have not considered the case where the two states are degenerate. This case requires a separate derivation and will not be considered here. Also note that |k0> will only make a contribution to |i> if the numerator is non-zero. In the case of our simple example of the symmetric barrier, H' is an even function. Therefore, |k0> and |i0> must have the same symmetry. The numerator is the basis of a very important result in MO theory. One only mixes orbitals of the same symmetry. Finally note that the contribution of |k0> to |i> is equal in magnitude but opposite in sign of the contribution of |i0> to |k>.

With an improved expression, now to second order, for the energy eigenfunction, one can substitute this into equation (4), Ei = <i|H|i>/<i|i>, and obtain a second-order expression for the energy eigenvalue. Equation (11) presents the result.

Ei = Ei0 + <i0|H'|i0> + S<i0|H'|k0<k0|H'|i0>/( Ei0 - Ej0) (11)

The running index in the sum is j and in equation (11) j does not equal i. Note that the second term in equation (11) can become very large for small energy differences. For functions of a real variable, the two matrix elements in the numerator are equal. This Hermitian property follows from the representation of all observeables as real numbers.

We shall stop here. If the perturbation is so large as to require further stages of iteration, it is better to employ other methods. These other methods are usually numerical. They are more accurate, especially if the complete set of eigenfunctions has a finite dimension. Numerical methods, however accurate, do not provide closed-form algebraic expressions, a source of insight into the interaction. Perturbation theory does provide algebraic results and allows one to examine directly the relationship between parameters and variables. For this reason alone, perturbation theory is a technique that should be in the toolbox of every quantum mechanic. Perturbation theory is also useful for cases in which there is not a finite number of basis functions.

WES, 17 May 2008