Page 6.1

SECTION 6: MULTI-ELECTRON ATOMS

1. Introduction 2

2. Central-field approximation 2

2.1 The central field and electron screening 3 2.2 The single-electron states 4

2.3 The atomic states 5

3. Perturbative correction to the central-field approximation 7

3.1 Spectral terms 7

3.2 Example: the ground state of carbon 8

4. Spin-Orbit Coupling 10

4.1 Weak spin-orbit coupling 11

4.2 Example: the ground state of carbon 12 4.3 Strong spin-orbit coupling 13

4.4 Angular momentum coupling schemes for composite systems 14

References

The original version of these notes was based on various sections of the textbook by Mandl, subsequently

augmented with material from Quantum Mechanics Vol II by A Messiah, published by North Holland (1965).

For specific references, see the footnotes to the text.

Original Version: 1995; Last Major Revision: May 2009

Revision Date: 26 October 2010

Printed: 26 October 2010

Page 6.2

1. INTRODUCTION

It is possible to do very simple calculations for the helium atom and get reasonable quantitative agreement

with experiment, but for more complicated atoms this is not possible quantitative agreement requires the use

of complex numerical calculations. In this course we restrict our attention to simple models that allow us to

gain a qualitative understanding of observed atomic spectra, in particular focussing on the quantum numbers

that are used to label atomic states.

Ignoring for now the spin-orbit interaction, the Hamiltonian for a neutral atom with Z electrons is

2 2 2

0 01 1 , 1

1

2 4 4

Z Z Zi

i i ji i i ji j

p Ze eH

m r r r = = = 1, so that approximate techniques must be used.

As was the case with the helium atom, it is the last term that causes the difficulties, because it couples the

coordinates of the individual electrons. Treating this term as a perturbation is not possible since even for

helium it is not sufficiently small; for Z > 2 its effect becomes progressively larger as Z increases (it grows

approximately as Z2 rather than linearly with Z).

2. CENTRAL-FIELD APPROXIMATION1

In the usual approach, the first step is to approximate the electron-electron interaction using the central-field

approximation, in which it is assumed that:

The main effect on a particular electron of the remaining Z 1 electrons is to produce a spherically

symmetric central potential due to the charge distribution that they create. The central field is taken to

be the same for all electrons.

The residual electron-electron interaction is assumed to be sufficiently small that it can be either ignored

completely or, if necessary, estimated using first-order perturbation theory.

So in this approximation the Hamiltonian is written:

1centH H V= +

where the central-field Hamiltonian contains the kinetic energy of the electron and the central field seen by

each electron:

2

1

( ) ,2

Zi

cent cent ii

pH V r

m=

= +

(2)

1 See Mandl section 4.4.1 for a concise discussion.

Page 6.3

and V1 contains the remainder of the full Hamiltonian:

2 2

10 0, 1 1

1( ) .

4 4

Z Z

cent iii ji j i

i j

e ZeV V r

rr r = =