# Moller Plessett Perturbation Theory

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Supplementary subject: Quantum Chemistry

Perturbation theory6 lectures, (Tuesday and Friday, weeks 4-6 of Hilary term)

Chris-Kriton Skylaris(chris-kriton.skylaris @ chem.ox.ac.uk)

Physical & Theoretical Chemistry LaboratorySouth Parks Road, Oxford

February 24, 2006

Bibliography

All the material required is covered in Molecular Quantum Mechanics fourth editionby Peter Atkins and Ronald Friedman (OUP 2005). Specically, Chapter 6, rst half ofChapter 12 and Section 9.11.

Further reading:Quantum Chemistry fourth edition by Ira N. Levine (Prentice Hall 1991).Quantum Mechanics by F. Mandl (Wiley 1992).Quantum Physics third edition by Stephen Gasiorowicz (Wiley 2003).Modern Quantum Mechanics revised edition by J. J. Sakurai (Addison Wesley Long-man 1994).Modern Quantum Chemistry by A. Szabo and N. S. Ostlund (Dover 1996).

1Contents

1 Introduction 2

2 Time-independent perturbation theory 22.1 Non-degenerate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 The rst order correction to the energy . . . . . . . . . . . . . . . 42.1.2 The rst order correction to the wavefunction . . . . . . . . . . . 52.1.3 The second order correction to the energy . . . . . . . . . . . . . 62.1.4 The closure approximation . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Perturbation theory for degenerate states . . . . . . . . . . . . . . . . . . 8

3 Time-dependent perturbation theory 123.1 Revision: The time-dependent Schrodinger equation with a time-independent

Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Time-independent Hamiltonian with a time-dependent perturbation . . . 133.3 Two level time-dependent system - Rabi oscillations . . . . . . . . . . . . 173.4 Perturbation varying slowly with time . . . . . . . . . . . . . . . . . . 193.5 Perturbation oscillating with time . . . . . . . . . . . . . . . . . . . . . . 20

3.5.1 Transition to a single level . . . . . . . . . . . . . . . . . . . . . . 203.5.2 Transition to a continuum of levels . . . . . . . . . . . . . . . . . 21

3.6 Emission and absorption of radiation by atoms . . . . . . . . . . . . . . . 23

4 Applications of perturbation theory 284.1 Perturbation caused by uniform electric eld . . . . . . . . . . . . . . . . 284.2 Dipole moment in uniform electric eld . . . . . . . . . . . . . . . . . . . 284.3 Calculation of the static polarizability . . . . . . . . . . . . . . . . . . . . 294.4 Polarizability and electronic molecular spectroscopy . . . . . . . . . . . . 304.5 Dispersion forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6 Revision: Antisymmetry, Slater determinants and the Hartree-Fock method 354.7 Mller-Plesset many-body perturbation theory . . . . . . . . . . . . . . . 36

Lecture 1 2

1 Introduction

In these lectures we will study perturbation theory, which along with the variation theorypresented in previous lectures, are the main techniques of approximation in quantummechanics. Perturbation theory is often more complicated than variation theory butalso its scope is broader as it applies to any excited state of a system while variationtheory is usually restricted to the ground state.

We will begin by developing perturbation theory for stationary states resulting fromHamiltonians with potentials that are independent of time and then we will expandthe theory to Hamiltonians with time-dependent potentials to describe processes suchas the interaction of matter with light. Finally, we will apply perturbation theory tothe study of electric properties of molecules and to develop Mller-Plesset many-bodyperturbation theory which is often a reliable computational procedure for obtaining mostof the correlation energy that is missing from Hartree-Fock calculations.

2 Time-independent perturbation theory

2.1 Non-degenerate systems

The approach that we describe in this section is also known as Rayleigh-Schrodingerperturbation theory. We wish to nd approximate solutions of the time-independentShrodinger equation (TISE) for a system with Hamiltonian H for which it is dicult tond exact solutions.

Hn = Enn (1)

We assume however that we know the exact solutions (0)n of a simpler system

with Hamiltionian H(0), i.e.H(0)(0)n = E

(0)n

(0)n (2)

which is not too dierent from H . We further assume that the states (0)n are non-

degenerate or in other words E(0)n = E(0)k if n = k.

The small dierence between H and H(0) is seen as merely a perturbation onH(0) and all quantities of the system described by H (the perturbed system) can beexpanded as a Taylor series starting from the unperturbed quantities (those of H(0)).The expansion is done in terms of a parameter .

We have:

H = H(0) + H(1) + 2H(2) + (3)n =

(0)n +

(1)n +

2(2)n + (4)En = E

(0)n + E

(1)n +

2E(2)n + (5)

Lecture 1 3

The terms (1)n and E

(1)n are called the rst order corrections to the wavefunction and

energy respectively, the (2)n and E

(2)n are the second order corrections and so on. The

task of perturbation theory is to approximate the energies and wavefunctions of theperturbed system by calculating corrections up to a given order.

Note 2.1 In perturbation theory we are assuming that all perturbed quantities are func-tions of the parameter , i.e. H(), En() and n(r;) and that when 0 we haveH(0) = H(0), En(0) = E

(0)n and n(r; 0) =

(0)n (r). You will remember from your maths

course that the Taylor series expansion of say En() around = 0 is

En = En(0) +dEnd

=0

+1

2!

d2End2

=0

2 +1

3!

d3End3

=0

3 + (6)

By comparing this expression with (5) we see that the perturbation theory corrections to

the energy level En are related to the terms of Taylor series expansion by: E(0)n = En(0),

E(1)n =

dEnd|=0, E(2)n = 12! d

2End2

|=0, E(3)n = 13! d3End3

|=0, etc. Similar relations hold for theexpressions (3) and (4) for the Hamiltonian and wavefunction respectively.

Note 2.2 In many textbooks the expansion of the Hamiltonian is terminated after therst order term, i.e. H = H(0) + H(1) as this is sucient for many physical problems.

Note 2.3 What is the signicance of the parameter ?In some cases is a physical quantity: For example, if we have a single electron

placed in a uniform electric eld along the z-axis the total perturbed Hamiltonian is just

H = H(0) + Ez(ez) where H(0) is the Hamiltonian in the absence of the eld. The eectof the eld is described by the term ez H(1) and the strength of the eld Ez plays therole of the parameter .

In other cases is just a ctitious parameter which we introduce in order to solvea problem using the formalism of perturbation theory: For example, to describe the twoelectrons of a helium atom we may construct the zeroth order Hamiltonian as that oftwo non-interacting electrons 1 and 2, H(0) = 1/221 1/222 2/r1 2/r2 which istrivial to solve as it is the sum of two single-particle Hamiltonians, one for each electron.The entire Hamiltonian for this system however is H = H(0) + 1/|r1 r2| which is nolonger separable, so we may use perturbation theory to nd an approximate solution forH() = H(0) + /|r1 r2| = H(0) + H(1) using the ctitious parameter as a dialwhich is varied continuously from 0 to its nal value 1 and takes us from the modelproblem to the real problem.

To calculate the perturbation corrections we substitute the series expansions of equa-tions (3), (4) and (5) into the TISE (1) for the perturbed system, and rearrange and

Lecture 1 4

group terms according to powers of in order to get

{H(0)(0)n E(0)n (0)n }+ {H(0)(1)n + H(1)(0)n E(0)n (1)n E(1)n (0)n } (7)+ 2{H(0)(2)n + H(1)(1)n + H(2)(0)n E(0)n (2)n E(1)n (1)n E(2)n (0)n }+ = 0

Notice how in each bracket terms of the same order are grouped (for example H(1)(1)n

is a second order term because the sum of the orders of H(1) and (1)n is 2). The powers

of are linearly independent functions, so the only way that the above equation can besatised for all (arbitrary) values of is if the coecient of each power of is zero. Bysetting each such term to zero we obtain the following sets of equations

H(0)(0)n = E(0)n

(0)n (8)

(H(0) E(0)n )(1)n = (E(1)n H(1))(0)n (9)(H(0) E(0)n )(2)n = (E(2)n H(2))(0)n + (E(1)n H(1))(1)n (10)

To simplify the expressions from now on we will use bra-ket notation, representingwavefunction corrections by their state number, so

(0)n |n(0), (1)n |n(1), etc.

2.1.1 The first order correction to the energy

To derive an expression for calculating the rst order correction to the energy E(1), takeequation (9) in ket notation

(H(0) E(0)n )|n(1) = (E(1)n H(1))|n(0) (11)

and multiply from the left by n(0)| to obtain

n(0)|(H(0) E(0)n )|n(1) = n(0)|(E(1)n H(1))|n(0) (12)n(0)|H(0)|n(1) E(0)n n(0)|n(1) = E(1)n n(0)|n(0) n(0)|H(1)|n(0) (13)E(0)n n(0)|n(1) E(0)n n(0)|n(1) = E(1)n n(0)|H(1)|n(0) (14)

0 = E(1)n n(0)|H(1)|n(0) (15)

where in order to go from (13) to (14) we have used the fact that the eigenfunctions ofthe unperturbed Hamiltonian H(0) are normalised and the Hermiticity property of H(0)

which allows it to operate to its eigenket on its left

n(0)|H(0)|n(1) = (H(0)n(0))|n(1) = (E(0)n n(0))|n(1) = E(0)n n(0)|n(1) (16)

Lecture 1 5

So, according to our result (15), the rst order correction to the energy is

E(1)n = n(0)|H(1)|n(0) (17)which is simply the expectation value of the rst order Hamiltonian in the state |n(0)

(0)n of the unperturbed system.

Example 1 Calculate the rst order c