IISER Pune
PH4224 Atomic & Molecular Physics (Jan 2025)
Course Notes by T S Mahesh
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appreciated.
• Updates: The notes are periodically updated, so always refer to
the latest version.
April 17, 2025
2
Contents
1 Introduction
5
1.1
Origin of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
The origin of atomic & molecular concepts . . . . . . . . . . . . . . . . . . .
9
1.2.1
Time-line of atomic concepts (Source: Wikipedia) . . . . . . . . . . .
9
Planned contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.3
2 Hydrogenic atoms
15
2.1
Quantum theory of single electron atoms . . . . . . . . . . . . . . . . . . . .
16
2.2
Hydrogenic energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3
Hydrogenic eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4
Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.5
Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.6
Virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3 Atom in an EM field
3.1
27
EM field in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.1.1
Energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.1.2
Coherent & incoherent radiation . . . . . . . . . . . . . . . . . . . . .
29
A hydrogenic atom in EM field . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2.1
Absorption (ω ≈ +ωba ) . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.2.2
Stimulated Emission (ω ≈ −ωba = ωab ) . . . . . . . . . . . . . . . . .
33
3.2.3
Spontaneous emission
. . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.3
The dipole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.4
The Einstein Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.5
Selection rules for one-electron atoms
. . . . . . . . . . . . . . . . . . . . .
37
3.6
Spectrum of hydrogenic atoms . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.6.1
Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.6.2
Line intensities and oscillator strengths . . . . . . . . . . . . . . . . .
40
3.6.3
Lifetimes of excited states . . . . . . . . . . . . . . . . . . . . . . . .
42
3.6.4
Lineshapes and linewidths . . . . . . . . . . . . . . . . . . . . . . . .
42
3.2
3
4
CONTENTS
3.7
3.8
The photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.7.1
The quantum theory of photoionization . . . . . . . . . . . . . . . . .
49
Scattering of radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4 Hydrogenic atom: Higher order corrections
55
4.1
Relativistic correction to the kinetic energy . . . . . . . . . . . . . . . . . . .
56
4.2
Spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
4.3
Darwin term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.4
Total relativistic correction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.4.1
Fine structure of spectral lines . . . . . . . . . . . . . . . . . . . . . .
61
Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.5.1
Lamb-Retherford Experiment . . . . . . . . . . . . . . . . . . . . . .
62
4.5.2
Lamb-shift as a radiative correction in QED . . . . . . . . . . . . . .
63
Nuclear effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.6.1
Nuclear spin and Hyperfine structure . . . . . . . . . . . . . . . . . .
67
4.6.2
Electric Quadrupole Hyperfine structure . . . . . . . . . . . . . . . .
71
4.6.3
Volume effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
4.5
4.6
5 Two electrons atoms
73
5.1
Exchange symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5.2
Independent electron-model . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
1
5.3
First-order perturbation of ground state 1 S . . . . . . . . . . . . . . . . . .
81
5.4
Variational method for ground state 11 S . . . . . . . . . . . . . . . . . . . .
82
5.5
Singly Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.6
Doubly excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.7
Ionization energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
5.7.1
87
Franck-Hertz Experiment
. . . . . . . . . . . . . . . . . . . . . . . .
6 Many electrons atoms
89
6.1
Central field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.2
Electronic configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
6.3
The Thomas-Fermi model of multielectron atom/ion . . . . . . . . . . . . . .
94
6.4
The Hartree-Fock method or the self-consistent field approach . . . . . . . .
96
6.5
Correction to the central field approximation: Spin orbit interaction . . . . .
98
6.5.1
LS coupling or Russel-Saunders coupling . . . . . . . . . . . . . . . .
98
6.5.2
jj-Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7 Atoms in E, M, or EM Fields
7.1
107
Hydrogenic atom in an Electric field . . . . . . . . . . . . . . . . . . . . . . . 107
CONTENTS
5
7.1.1
Linear Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.2
Quadratic Stark effect . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2
The Stark effect of multi-electron atom . . . . . . . . . . . . . . . . . . . . . 112
7.2.1
7.3
Ionization by a Static Electric Field . . . . . . . . . . . . . . . . . . . 113
Hydrogen atom in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 115
7.3.1
|HLS | |HZ | < |H0 |: The Normal Zeeman effect . . . . . . . . . . . 116
7.3.2
|HLS | . |HZ | < |H0 |: The Paschen-Back effect . . . . . . . . . . . . . 119
7.3.3
|HZ | |HLS | < |H0 |: The anomalous Zeeman effect . . . . . . . . . . 120
7.3.4
HZ > HD H0 : Ultra-strong field . . . . . . . . . . . . . . . . . . . 121
7.4
The Zeeman effect of multi-electron atoms . . . . . . . . . . . . . . . . . . . 121
7.5
Interaction of many-electron atoms with EM fields . . . . . . . . . . . . . . . 122
7.5.1
Absorption & Emission . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.5.2
Dipole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.5.3
Selection rules for Electric Dipole transitions . . . . . . . . . . . . . . 123
7.5.4
Spectra of Alkali Elements . . . . . . . . . . . . . . . . . . . . . . . . 124
8 Molecular Structure
129
8.1
A diatomic molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2
Born–Oppenheimer adiabatic approximation (1927): Clamped nuclear model 130
8.3
Symmetry in a diatomic molecule . . . . . . . . . . . . . . . . . . . . . . . . 133
8.4
The Hydrogen molecule ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.5
H2 molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.6
The rotation and vibration of diatomic molecules . . . . . . . . . . . . . . . 143
9 Spectra of Diatomic Molecules
147
9.1
Electric Dipole of a Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.2
Molecular transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.2.1
Pure rotational transitions in heteronuclear systems . . . . . . . . . . 149
9.2.2
Vibrational rotational spectrum . . . . . . . . . . . . . . . . . . . . . 150
9.3
Scattering by molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.4
Electronic spectra of diatomic molecules . . . . . . . . . . . . . . . . . . . . 155
9.5
Franck-Condon Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.6
Dissociation and predissociation . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.7
Fluorescence and Phosphorescence
10 Electron Spin Resonance (ESR)
. . . . . . . . . . . . . . . . . . . . . . . 158
161
10.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.1.1 Zeeman Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.1.2 Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6
CONTENTS
10.1.3 Hyperfine interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.2 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
10.3 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11 Nuclear Magnetic Resonance (NMR)
167
11.1 Nuclear Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
11.2 The Zeeman energy gap and resonance . . . . . . . . . . . . . . . . . . . . . 168
11.3 Magnetization, RF pulse, FID, and Chemical Shift . . . . . . . . . . . . . . 170
11.4 NMR qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11.5 Magnetic Resonance Imaging (MRI) . . . . . . . . . . . . . . . . . . . . . . 172
11.6 Two interacting spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Chapter 1
Introduction
1.1
Origin of matter
Atoms and molecules make up the matter around us. Where did all these come from?
According to the current cosmological model, the universe originated from singularity about
13.8 billion years ago, in an explosion known as ‘Big-Bang’. Fig. 1.1 illustrates the timeline
of universe as we understand today.
Figure 1.1: Elementary particles (Source: Wikipedia)
In the beginning, it was only an energy field, with quantum fluctuations. At 10−37 s, the
rapid expansion of the initial hot primordial soup resulted in the production of elementary
particles including quark-gluon plasma (see Fig. 1.2).
According to the standard model 1 , the fermionic particles, lead to matter as well as
1
Wikipedia: "Elementary particle"
7
8
CHAPTER 1. INTRODUCTION
Figure 1.2: Elementary particles (Source: Wikipedia)
antimatter that have mass and occupy space (latter due to Pauli exclusion principle). The
bosonic particles constitute interaction carriers. Within the first microsecond after Big Bang,
gluons combined quarks to form protons, antiprotons, neutrons, and antineutrons. Most of
the matter annihilated with antimatter, but a small fraction of matter particles survived due
to their slight excess, 1 part in 108 . One second into the expansion, a similar annihilation
of leptons, like electrons and positrons resulted the survival of 1 part in 108 electrons. The
reason for the excess matter over antimatter is one of the biggest questions in cosmology known as the matter-antimatter asymmetry or baryon asymmetry problem.
Big Bang nucleosynthesis took place in the first three minutes of expansion, in which a
small fraction of neutrons and protons combined to form nuclei such as 2 H (D), He ion (α
particle), and a small fraction of Li and Be ions. Almost all of 2 H (D) was produced during
the Big Bang only, since stars use 2 H (D) as a fuel along with protons. After three minutes,
the universe got cooled enough that the nucleosynthesis got shut down.
Nothing much happened for the next 100,000 years, except the initial quantum fluctuations were frozen resulting in nonuniform distribution of matter density. At about 100,000
years, when the temperature is over 200,000 K and the atomic Hydrogen (ionization energy
13.6 eV) was yet to form, He nuclei captured pairs of electrons to form the first atoms. A
1.1. ORIGIN OF MATTER
9
He atom, which has the first and second ionization energies of 24.6 eV and 54.4 eV, could
hold on to its pair of electrons even at such high temperatures, and became the first atom
to born. Another 200,000 years passed by before atomic hydrogen was formed.
Exercise: The first and second ionization energies of He are 24.6 eV and 54.4 eV. Determine the corresponding temperatures.
At about 400,000 years, the temperature dropped to about 3000 K and proton-electron
pairs could form atomic Hydrogen. Now, He atom could also capture a proton and form
the first molecule Helium hydride ion HeH+ . The primordial thermal radiation of 3000 K
eventually red-shifted to microwave, is presently observed as the omnipresent 2.725 K blackbody radiation, known as the cosmic microwave background (CMB). Now is the dawn of
chemistry. Neutral atoms and ions combine to form H2 , HD, LiH, etc.
The universe would have expanded uniformly, without any further interesting stuff, but
for the density fluctuations (resulted from initial quantum fluctuations), which started forming clumps of matter attracted by gravity. These clumps, which are seen now in the cosmic
microwave background (CMB) patterns, later formed the large scale structures of universe.
In about 100 million years after Big Bang, the cold gases made up of neutral atoms
formed gas clouds, called nebulae, which started to collapse under gravity to form stars.
Our Sun fuses 600 million tons of H into He every second, in which 0.7% mass (4 million
tons) is converted into energy. Sun is 4.5 billion years old and has consumed about half of
H fuel. Fig. 1.3 shows the relative abundances of various elements in the solar system. The
fact that even atomic numbers are more abundant indicates that He is the building block in
most cases.
Figure 1.3: Abundances of elements in the solar system (Source: Wikipedia)
10
CHAPTER 1. INTRODUCTION
Nuclei heavier than Lithium are largely created by stars via nuclear fusion in their cores.
However, stars can only produce elements up to Fe, which is the most stable element. Fusion
or fission with Fe is no longer exothermic, and therefore the star’s engine shuts down when
there is enough Fe. Elements heavier than Fe are produced when giant stars explode as
supernovae. The following table is for a star with 25 times mass of Sun.
Fuel
Core Temp (Million K)
Consuming Time (Years)
H → He
He → C
C → Ne
Ne → O
O → Si
Si → Fe
40
200
600
1200
1500
2700
7 million
500 thousand
600
1
0.5
1 day
The next generation stars, when they exploded released clouds of heavy atoms such as Si,
which combined with O, resulted in SiO2 , which clumped together to form interstellar dust.
The dusty clouds provided platforms for formation of molecules such as CH, OH, CO, etc.
Explosions of successive stars released heavier elements into space, which ultimately resulted
in H2 O, SO2 , NO, CH3 OCH3 , CH3 COOH, H2 CO, etc. Hundreds of chemicals, organic as well
as inorganic, have been detected in Milky Way nebulae. Interestingly, some chemicals were
detected in space before synthesizing in laboratory (eg. Cyclopropenylidene). Astrochemists
have observed water, ammonia, formaldehyde, fullerene, and even amino acids in galactic
nebulae.
Things around us and we ourselves are made up of atoms. By mass, we are 65% O,
18.5% C, 9.5% H, 3.3% N, and 3.7% of other atoms. Consider I atoms which are essential for
thyorid hormones. I was formed in a second or third generation star, which later ended up
in an exploding star’s gas ejections, then became part of Solar system, ended up on Earth’s
surface while it was forming, finally consumed by us. So, we are indeed star dust. After we
die, I goes back to Earth’s surface and may end up in another life form. Eventually, when
our Sun explodes as a red giant, Earth’s outer surface gets evaporated, and these atoms will
end up space again.
What we have discussed so far is about the matter that is observable through its interaction with the electromagnetic waves. However, one of the biggest mysteries of the universe
is that we don’t know what 96% of it is made up of, which we call as dark energy and
dark matter. Dark matter is observed through its gravitational effects, while dark energy is
believed to be accelerating the expansion of universe. The remaining 4% of the observable
universe constitutes the ordinary matter, most of which is free hydrogen and helium atoms
pervading the universe. Less than 1% exists in the bulk in the form of stars and planetary
systems (see Fig. 1.4).
1.2. THE ORIGIN OF ATOMIC & MOLECULAR CONCEPTS
11
Figure 1.4: Dark energy and dark matter.
1.2
The origin of atomic & molecular concepts
1.2.1
Time-line of atomic concepts (Source: Wikipedia)
• 6th Century BCE Kanada (philosopher) proposes that anu is an indestructible particle
of matter, an "atom"; anu is an abstraction and not observable.[1]
• 4th Century BCE Democritus speculates about fundamental indivisible particles—calls
them "atomos" ("a"+"tomos" means not divisible)
• Next two millennia: Religion takes over science
• 1766 Henry Cavendish discovers and studies hydrogen
• 1778 Carl Scheele and Antoine Lavoisier discover that air is composed mostly of nitrogen and oxygen
• 1781 Joseph Priestley creates water by igniting hydrogen and oxygen
• 1789 Antoine Lavoisier, after carefully observing several chemical reactions, propounded
the law of conservation of mass in chemical reactions.
• 1799 J. L. Proust, found in a chemical reaction A+B → C, the relative masses were
always in a definite ratio (Eg. 70g of Cl consumed exactly 2g of H2 , to give 72g of
HCl); proposed the law of definite proportions.
• 1800 William Nicholson and Anthony Carlisle use electrolysis to separate water into
hydrogen and oxygen
• 1803 John Dalton came up with the law of multiple proportions in multicomponent
chemical reactions: Eg., In A+B → C + D, mA : mB = mC : mD . Simplest way to
12
CHAPTER 1. INTRODUCTION
explain the discreteness of masses is via discreteness of du concept of atoms. Proposes
that elements are made up atoms of different weights.
• 1805 Thomas Young conducts the double-slit experiment with light
• 1811 Amedeo Avogadro claims that equal volumes of gases should contain equal numbers of molecules
• 1827 Robert Brown observed random zig-zag motion of pollen particles suspended in
water
• 1832 Michael Faraday states his laws of electrolysis; led to the notion of fundamental
electric charge.
• 1871 Dmitri Mendeleev systematically examines the periodic table and predicts the
existence of gallium, scandium, and germanium
• 1885 Johann Balmer finds a mathematical expression for observed hydrogen line wavelengths
• 1887 Heinrich Hertz discovers the photoelectric effect
• 1897 J. J. Thomson discovered the electron;
• 1899 Ernest Rutherford discovered the alpha and beta particles emitted by uranium;
• 1905 Albert Einstein explains the Brownian motion, photoelectric effect; formulates
the special theory of relativity.
• 1909 Hans Geiger and Ernest Marsden discover large angle deflections of alpha particles
by thin metal foils
• 1911 Ernest Rutherford explains the Geiger–Marsden experiment by invoking a nuclear
atom model and derives the Rutherford cross section; The atomic model similar to the
planetary model. Can not explain the stability of atoms.
• 1913 Niels Bohr presents his quantum model of the atom
• 1913 Robert Millikan measures the fundamental unit of electric charge
• 1917 Albert Einstein introduces the idea of stimulated radiation emission
• 1924 Satyendra Nath Bose and Albert Einstein introduce Bose–Einstein statistics
• 1925 Werner Heisenberg, Max Born, and Pascual Jordan formulate quantum matrix
mechanics
1.2. THE ORIGIN OF ATOMIC & MOLECULAR CONCEPTS
13
• 1925 Wolfgang Pauli states the quantum exclusion principle for electrons
• 1926 Erwin Schrödinger proves that the wave and matrix formulations of quantum
theory are mathematically equivalent
• 1927 Werner Heisenberg states the quantum uncertainty principle
• 1932 James Chadwick discovers the neutron
The concept of atoms dates back at least to 6 BCE, when Indian philosopher, Kanada,
suggested that everything can be subdivided until one finally gets the smallest indivisible
particle, ‘paramaanu’. In 4 BCE, Greek philosophers Leucippus, Democritus, and Plato
proposed the idea of indivisible particles, and they named it ‘atoms’. They postulated that
the macroscopic properties of bulk matter emerge from the properties of constituent atoms.
The philosophical and logical notions were largely suppressed by religious beliefs over the
next two thousand years, until about 18th century, when several experimental observations
compelled the revisit of atomic concept. For example, Daniel Bernoulli explained the experimentally observed Boyle Gas Law, P · V = constant (at constant temperature) as due to
the collisions of atoms against one another and against the container walls. He also related
the temperature of the gas to average kinetic energy of atoms.
In the beginning of 19th century came the Dalton’s law of multiple proportions. He
found that when two elements combine, their mass ratios were always integers. For example,
water was formed when hydrogen and oxygen were combined with a respective mass ratio
of one to eight; Oxygen to Manganese mass ratios in oxides happened to be 2, 3, 4, 6, or 7.
Chemists also found that when two or more gasses combined to form a new gas, the ratios of
volumes of the reactants were simple integers, under equal pressure and temperature. These
observations led Avagadro to propose that the smallest particle that determines the property
of a substance is ‘molecule’ and it consists of two or more atoms. Examining the ratios in
multiple reactions, Dalton proposed relative atomic weights with respect to Hydrogen atom,
which was set to a value 1. The law of multiple proportions then implied that if we take
two substances in the ratio of their atomic or molecular weights, they have same number of
atoms. For example, 2 g of Hydrogen gas have twice the number of atoms as 16 g of Oxygen,
which when combined resulted in 18 g of water.
In 1827, Robert Brown observed random zig-zag motion of pollen particles suspended in
water. After about eight decades, in 1905, Albert Einstein explained the origin of Brownian motion as due to collisions of thermally driven water molecules against the suspended
particles.
Meanwhile in 1833, Faraday’s electrolysis experiments led to the notion of unit electric
charge. By determining the quantity of charge required to liberate one mole of a monovalent
substance in an electrolysis experiment, Stoney, in 1874, determined the unit electric charge
14
CHAPTER 1. INTRODUCTION
value to be around 10−19 C, and termed it as ‘electron’. W. Crookes and P. Lenard observed a
faint green glow on the glass wall of a vacuum tube containing a gas under very low pressure
and two electrodes with high potential difference. In around 1895, J. J. Thomson postulated
Figure 1.5: Crookes tube (Source: Wikipedia).
‘cathode ray’ as the beam of invisible particles emerging from one electrode, passing towards
the other, and finally impinging on the wall producing the glow. Using an electrometer, the
charge of the beam was found to be negative. By measuring deflection of the cathode beam
by a transverse magnetic field, the charge to mass ratio e/m was estimated, and found to
be 1840 higher than that of hydrogen ion. Thomson postulated the cathode particles to be
electrons of charge e, same in magnitude but opposite sign as that of hydrogen ion.
In 1909, Millikan performed his famous ‘oil-drop’ experiment, wherein charged oil droplets
were balanced against gravity by applying an electric field, while looking through a microscope. He found the value of e ∼ 1.6 × 10−19 C. Using the known values of e and e/m, one
obtained the mass of electron m = 9.11 × 10−30 kg. For several elemental ions, the maximum
charge observed was +Ze, where Z is a characteristic integer came to be known as atomic
number.
By measuring the density of one mole of various solid elemental substances, it was noticed
that atomic sizes did not vary much. For example Lead (atomic weight 207) atoms are only
10 % larger than Lithium (atomic weight 7) atoms. So, how the atomic mass distributed
inside the atom was an intriguing question.
1.3. PLANNED CONTENTS
15
Figure 1.6: Millikan’s oil-drop experiment (Source: Wikipedia).
Around 1910, Rutherford and coworkers reported the famous experiment of α particle
(with atomic weight 4, and charge +2e) scattering by a gold-foil. Most of the α particles
passed without significant deflection (< 1◦ ), while only about 0.01 % got deflected by more
than 90◦ . This established that most of the atomic mass is concentrated in a very small and
dense nucleus. So, Rutherford postulated that electrons in an atom move around a central
dense nuclei like a planetary system.
Atomic spectra: The presence of characteristic dark lines in atomic spectra was known
since 18th century. In 1885, Balmer found that the frequencies of the dark lines of an atom
followed the empirical relation νm,n ∝ m12 − n12 , where m, n ∈ {1, 2, 3, · · · }.
Niels Bohr, working with Rutherford, was trying to fix the instability of Rutherford
atomic model. An electron circling its nucleus must radiate electromagnetic energy and
should ultimately collapse to the nucleus. Combining the spectroscopic observation, Bohr
postulated the quantization of orbital angular momentum L = n~, where ~ = h/2π is
the scaled Planck’s constant. This in turn, quantized energy, and explained the discrete
absorption spectra. However, a general basis for Bohr’s postulate was missing, which was
subsequently provided by the framework of Quantum Theory.
Today, we can see atoms. For example, transmission electron microscopy allows to image
at atomic resolution 2 .
1.3
Planned contents
• 1. How did matter particles, atoms, and molecules originate? (1 hr)
• 2. How to understand hydrogenic atoms? How do they interact with EM fields? (6
hrs) Revisiting Bohr atom* fine and hyperfine corrections* isotope effects* Stark &
Zeeman effects* Bohr atom in EM fields* scattering* Raman effect* absorption* emission* dipole approximation* selection rules* Einstein coefficients* hydrogenic spec2
One can even trap a single atom and control its dynamics; Eg. Kim et al, Nature Comm.
16
CHAPTER 1. INTRODUCTION
Figure 1.7: Transmission electron microscopy (Source: Wikipedia).
trum* photoelectric effect* lasers/masers/X- rays.
• 3. How to understand multi-electron atoms? How do they interact with EM fields?
(12 hrs) Approximate methods* excitations, ionizations* Franck- Hertz Experiment*
central field approximation* electronic configuration and periodic table* Thomas-Fermi
model* Hartree-Fock method* LS/JJ corrections* EM absorption, emission* selection
rules* lifetimes of excited states* spectra of alkali elements* atomic collisions and
spectral line shapes* Doppler-free techniques.
• – MID SEM –
• 4. How to understand molecules? How do they interact with EM fields? (12 hrs)
Born-Oppenheimer approximation* diatomic molecules* electric dipole* rotational, vibrational, electronic transitions* scattering* Franck-Condon principle* dissociation*
fluorescence & phosphorescence* spectroscopy: infrared, Raman, microwave, magnetic
resonance, X-ray, electron spectroscopy* spectroscopy with synchrotron radiation.
• 5. Modern atomic physics: (8 hrs) Atom optics* cooling & trapping atoms, ions* laser
cooling* BEC* atomic clocks* optical tweezers* quantum memory with atoms, ions,
spins.
Chapter 2
Hydrogenic atoms
A Hydrogenic atom is a bound state of a positively and a negatively charged particles. Below
are some examples.
Figure 2.1: Examples for Hydrogenic atoms.
17
18
CHAPTER 2. HYDROGENIC ATOMS
2.1
Quantum theory of single electron atoms
We consider a bound state of an electron of mass m with a nucleus of atomic number Z and
mass M . In the center of mass frame, the Hamiltonian is
P2
Ze2
~2
Ze2
−
= − ∇2 −
2µ 4π0 r
2µ
4π0 r
"
!
#
2
2
∂
1 ∂
L
~
Ze2
2
r
−
,
=−
−
2m r2 ∂r
∂r
~2 r2
4π0 r
H=
(2.1)
mM
≈ m, if m M is the reduced mass. Now we
where r is the position variable and µ = m+M
have to solve time-independent Schrödinger equation HΨ(r) = EΨ(r) to obtain the energy
eigenvalue E and the corresponding wave function Ψ(r).
First we invoke the symmetry: here, the Coulomb potential is a central potential (recall from the 2nd chapter of QM-II notes). Each component Lx/y/z of the orbital angular
momentum operator L, as well as L2 are conserved. That is
[Lx/y/z , H] = [L2 , H] = 0.
(2.2)
This implies, there exists a common eigenbasis |E, l, mi → Ψ(r, θ, φ) = RE,l (r)Yl,m (θ, φ) of
H, L2 , and one of the components, say Lz (since Lx/y/z don’t commute with one another).
Since Hamiltonian is central, it allows separation of radial and angular variables: Ψ(r, θ, φ) =
RE,l (r)Yl,m (θ, φ), where, the angular part is Ylm (θ, φ) the spherical harmonic function, the
simultaneous eigenfunction of L2 and Lz . The radial part of the Schrödinger equation can
be written as
2µ
d
uE,l + 2 [E − Veff (r)] uE,l (r) = 0,
2
dr
~
(2.3)
Ze2
l(l + 1)~2
+
.
Veff = −
4π0 r
2µr2
(2.4)
where uE,l = rRE,l (r) and
Here the repulsive term |L2 |/2µr2 is the ‘centrifugal barrier’ (See Fig. 2.2).
Solving the radial equation in terms of the dimensionless quantities
8µE
ρ= − 2
~
2
µc2
r & λ = Zα −
2E
with fine structure constant α =
!1/2
1
e2
≃
,
4π0 ~c
137
(2.5)
2.1. QUANTUM THEORY OF SINGLE ELECTRON ATOMS
19
Figure 2.2: The effective potential.
leads to the quantization of λ = 1, 2, · · · , which leads to the quantization of energy,
En = −
e2 Z 2
1 2 (Zα)2
Z2
=
−
µc
≃
−R
.
y
4π0 aµ 2n2
2
n2
n2
(2.6)
Here Ry = 21 mc2 α2 ≈ 13.6 eV is the Rydberg unit of energy and n = 1, 2, · · · is known
as the principal quantum number, which also bounds the orbital quantum number l =
0, 1, 2, · · · , n − 1, which in turn fixes the azimuthal quantum number m = −l, −l + 1, · · · , l −
1, l. The corresponding radial wave function is of the form
Rnl (r) = Nnl e−ρ/2 ρl L2l+1
n+l (ρ), where ρ =
2Z
r.
naµ
(2.7)
20
CHAPTER 2. HYDROGENIC ATOMS
Here Nnl is the normalization factor, L2l+1
n+l is the Laguerre polynomial, and
4π0 ~2
aµ =
µe2
(2.8)
is the called Bohr radius.
Figure 2.3: Laguerre polynomials.
For Hydrogen atom, µ ≈ m, and therefore,
aµ ≈
4π0 ~2
= a0 ≈ 0.529 × 10−10 m.
me2
(2.9)
Thus, the total wave function is of the form
ψnlm (r, θ, φ) = Rnl (r)Ylm (θ, φ).
2.2
(2.10)
Hydrogenic energy levels
The energy levels En for Hydrogen atom (Z = 1) are shown in Fig. 2.4.
• There are infinite bound states.
• The ground state energy is E1 = −13.6 eV and the subsequent levels are at En =
−13.6/n2 eV.
• The energy levels are degenerate w.r.t. the azimuthal quantum number m because of
the rotational symmetry of the central potential (see section 2.3 of QM-II notes).
2.2. HYDROGENIC ENERGY LEVELS
21
Figure 2.4: Energy level diagram of Hydrogen atom (Source: Young & Freedman, 2012).
Exercise: Prove the degeneracy w.r.t. the azimuthal quantum number in Hydrogenic atoms.
This degeneracy can be lifted by introducing a magnetic field, i.e., the Zeeman effect.
• The energy levels are degenerate w.r.t to l quantum number, because of Symmetry
w.r.t. Runge-Lenz vector
N=
1
(P × L − L × P ),
2m
(2.11)
which commutes with the Hydrogen atom Hamiltonian. This is also known as the
accidental symmetry of (−1/r) potentials.
Exercise: Explain the degeneracy w.r.t. the orbital quantum number in Hydrogenic
atoms using ladder operators of N (you may refer to R. Shankar).
In multi-electron systems, when the screening effect of inner electrons modifies the
potential away from −1/r form, this degeneracy will be lifted.
• The various l levels are given spectroscopic notations, s, p, d, f, g, h,· · · respectively
for l = 0, 1, 2, 3, 4, 5, · · · . Levels are labeled by the principal quantum number as well
22
CHAPTER 2. HYDROGENIC ATOMS
as the spectroscopic label. In this notation, ground state is labeled 1s, and the various
excited states are labeled 2s, 2p, etc.
Figure 2.5: State labels (Source: Demtroder).
• The total degeneracy of the nth level is
n−1
X
(2l + 1) = n2 .
(2.12)
l=0
• Transitions: The most common transitions (called the electric-dipole transitions) between levels (n, l, m) and (n0 , l0 , m0 ) obey the selection rules
∆l = l − l0 = ±1
∆m = m − m0 = 0, ±1.
(2.13)
∆n can be arbitrary. The corresponding spectral energy is
2
hν = ±Z Ry
1
1
− 02 .
2
n
n
(2.14)
This leads to various spectral series as shown in Fig. 2.4.
2.3
Hydrogenic eigenfunctions
• The radial part Rnl is purely real. Note that only l = 0 has no centrifugal term, and
therefore has nonzero wave function at r = 0.
• Radial density per unit length: Dnl = r2 |Rnl (r)|2 is the probability per unit length at
a distance r from the nucleus averaged over all angles.
• The density function Dnl has n−l maxima. It follows from the property of the Laguerre
polynomials. Accordingly, Dn,n−1 has only one maximum.
23
2.4. PROBABILITY DENSITY
Figure 2.6: Hydrogen eigenfunctions (Source: Demtroder).
∗
(θ, φ)
• Conjugation of angular part: Yl,±m (θ, φ) = (−1)m Yl,∓m
• Parity transformation (~r → −~r): (r, θ, φ) → (r, π − θ, π + φ).
Under parity transformation,
Ylm (θ, φ) → (−1)l Ylm (θ, φ).
(2.15)
∴ Even and odd l correspond to even and odd parities.
2.4
Probability Density
Probability distribution |ψnlm |2 for various wave functions of Hydrogen atom are shown in
Fig. 2.9.
24
CHAPTER 2. HYDROGENIC ATOMS
Figure 2.7: Radial wave functions Rnl (Source: Demtroder).
2.5
Expectation values
Consider the ground state eigenfunction of a Hydrogenic atom
ψ100 =
Z3
πa3µ
!1/2
!
−Zr
, where aµ is the Bohr radius.
exp
aµ
(2.16)
Now we ask how far is the electron from the nucleus. This is answered by the expectation
value
hri100 = hψ100 |r| ψ100 i =
Z ∞
0
∗
dr ψ100
rψ100 =
Z ∞
Z3
−2Zr
= 3 4π
dr r3 exp
πaµ
aµ
0
3
=
aµ .
2Z
Z 2π
0
dφ
Z π
dθ sin θ
0
Z ∞
0
dr r2 |ψ100 |2 r
!
(2.17)
Thus, for Hydrogen atom, the average distance of the electron from the nucleus is 1.5a0 ≈
0.8 × 10−10 m.
For a general eigenfunction ψnlm , the expectation value is
(
"
n2
1
l(l + 1)
hrinlm = aµ
1+
1−
Z
2
n2
#)
.
(2.18)
25
2.6. VIRIAL THEOREM
Figure 2.8: Radial density functions Dnl (Source: Demtroder).
2.6
Virial theorem
One can show that h1/rinlm = Z/(aµ n2 ). Now, the average value of the Coulomb potential
is
Ze2
hV inlm = −
4π0
1
r
=−
nlm
Ze2 Z
(compare with Eq. 2.6)
4π0 aµ n2
= 2En = 2(hV inlm + hT inlm ).
∴ hV inlm = −2 hT inlm ,
which is the virial theorem.
(2.19)
26
CHAPTER 2. HYDROGENIC ATOMS
Exercises:
(1) Prove virial theorem
hT i =
1
hr · ∇V i
2
for a time-independent Hamiltonian.
(2) Show that for Coulomb potential, r · ∇V = −V , and thereby arrive at Eq. 2.19.
(3) Determine hpi1s , hpi2s , hpi2p , hp2 i1s , hp2 i2s , and hp2 i2p .
(4) For the Hydrogen atom in ground state find the classically forbidden region. Find
the probability of finding the electron in the forbidden region.
(5) Consider a Hydrogen atom in state |Ψi = √114 (2 |1, 0, 0i − 3 |2, 0, 0i + |3, 2, 2i)
(a) Does |Ψi have a definite parity? If yes, what is its parity?
(b) Find the energy of the atom, hL2 i, and hLz i.
(6) Estimate the size of a Rydberg atom in n = 100 state.
2.6. VIRIAL THEOREM
27
Figure 2.9: Probability distribution of various wave functions of Hydrogen atom (source:
internet).
28
CHAPTER 2. HYDROGENIC ATOMS
Chapter 3
Atom in an EM field
In this chapter, we shall study the interaction of an atom with electromagnetic field. As
we have seen in QM-II class (see Sec. 7.5 in QM-II notes), broadly there are three types of
atom-EM interactions: Absorption, Stimulated Emission, Spontaneous Emission (See Fig.
3.1).
Figure 3.1: Types of atom-EM interactions.
We shall study these in further details and try to understand the origin of selection rules.
We make the following assumptions:
• Semi-classical approximation: We consider a classical EM field interacting with a quantum atom.
• Long-wave length approximation: We consider the wavelength of the radiation much
longer than atomic sizes. It holds well for visible/UV light (λ ∼ 102 − 104 nm) interacting with normal atoms (of size ∼ 10−1 nm).
• Heavy nucleus approximation: Nucleus is three orders of magnitude heavier than electron, and therefore it oscillates relatively weakly in the EM field.
• Low density approximation: We assume that atoms are rarely dispersed, and therefore
radiation can be treated as EM waves in vacuum.
29
30
CHAPTER 3. ATOM IN AN EM FIELD
• Weak-field approximation: If atom-EM interaction is weak compared to the Coulomb
interaction, we can use perturbation theory.
3.1
EM field in vacuum
The E and M components of the EM field are governed by the Maxwell’s equations in vacuum,
∇·E =0
∇×E =−
∂B
∂t
∇·B =0
∇ × B = µ0 0
1
∂E
, with µ0 0 = 2 .
∂t
c
(3.1)
The electric and magnetic fields of an EM field can be generated from a vector potential A
∂
A(r, t)
∂t
B(r, t) = ∇ × A(r, t),
E(r, t) = −
with, ∇ · A(r, t) = 0 (called Coulomb gauge)
(3.2)
(3.3)
Using the above in Maxwell’s equations leads to,
∇2 A −
1 ∂2
A = 0, which has the solution,
c2 ∂t2
A(r, t) = A0 (ω)ε̂ cos (k · r − ωt + δω ) .
(3.4)
Exercise: Prove Eqs. 3.4.
Here k is the wave vector, ω = kc is the angular frequency, ε̂ is the polarization vector
that satisfies k · ε̂ = 0, and δω is the phase. By solving the wave equation we obtain (see
Fig. 3.2)
E(r, t) = E0 (ω)ε̂ sin (k · r − ωt + δω ) with E0 (ω) = −ωA0 (ω), and
B(r, t) = B0 (ω)b̂ sin (k · r − ωt + δω ) , where,
|E0 (ω)|
ω
= = c & b̂ = k̂ × ε̂.
|B0 (ω)|
k
Note that all three directions b̂, k̂, and ε̂ are mutually orthogonal.
(3.5)
31
3.1. EM FIELD IN VACUUM
Figure 3.2: EM field.
Exercise: Workout the dimensions of the vector potential.
3.1.1
Energy density
The average energy density ρ(ω) is defined as the average energy or N photons each of energy
~ω in a given volume V .
"
#
N ~ω
1ZT 1
1
1
1
I(ω)
dt ε0 |E|2 + |B|2 = ε0 E02 = ε0 ω 2 A20 =
ρ(ω) =
=
,
V
T 0
2
µ0
2
2
c
(3.6)
where T = 2π/ω is the period of the wave and I(ω) is the intensity of EM, or the average
rate of energy flow through unit area.
Exercise: Prove Eqs. 3.6.
3.1.2
Coherent & incoherent radiation
A general radiation can be a mixture of several components with random directions and
random polarization axes. A linearly polarized radiation, propagating along a particular
direction, is described by the vector potential
A(r, t) = ε̂
Z ∞
0
dωA0 (ω) cos (k · r − ωt + δω )
h
i
1 Z∞
= ε̂
dωA0 (ω) ei(k·r−ωt+δω ) + e−i(k·r−ωt+δω ) .
2 0
(3.7)
32
CHAPTER 3. ATOM IN AN EM FIELD
If the frequencies and phases are random, the radiation is incoherent. For a monochromatic
but incoherent radiation, the frequency distribution A0 (ω) is a delta function at some frequency ω0 , but in general the phases δ are random. A coherent radiation is linearly polarized,
monochromatic ω = ω0 , as well as of same phase δω0 .
3.2
A hydrogenic atom in EM field
Consider a Hydrogenic atom with Hamiltonian H0 , eigenvalues Ek , and normalized eigenfunctions ψk (r) s.t.,
H0 = −
~2 2
Ze2
∇ −
2m
4π0 r
and
H0 ψk (r) = Ek ψk (r).
(3.8)
We place the atom in a weak EM field described by the vector potential A(r, t). The resulting
Hamiltonian is
H = H0 + H 0 (t), where,
−i~e
e
A(r, t) · ∇ is the time-dependent EM perturbation. (3.9)
H 0 (r, t) = A(r, t) · p =
m
m
Exercise: Check Eqs. 3.9 by dimensional analysis.
We shall now use the time-dependent perturbation theory (see Sec. 7.1 of QM-II notes).
The time-dependent Schrödinger equation is now
−i~
∂
Ψ(r, t) = [H0 + H 0 (r, t)] Ψ(r, t), with the general solution
∂t
X
Ψ(r, t) =
ck (t)Ψk (r)e−iEk t/~ .
(3.10)
k
wherein the coefficients satisfy
.
i~cb (t) =
X
0
ck (t)Hbk
(r, t)eiωbk t ,
with matrix elements
(3.11)
k
0
Hbk
(r, t) = hΨb |H 0 (r, t)| Ψk i ,
Eb − Ek
ωbk =
.
~
and
(3.12)
Suppose, the atom is initially prepared in an eigenstate, i.e., Ψ(t = 0) = Ψa . In this case,
33
3.2. A HYDROGENIC ATOM IN EM FIELD
we can solve Eq. 3.11 to first order in the perturbation to obtain
1 Zt 0 0
Eb − Ea
0
dt Hba (r, t0 )eiωba t , with ωba =
;
i~ 0
~
Z t
e
0
dt0 hΨb |A(r, t) · ∇| Ψa i eiωba t , using Eq. 3.7 we get
=−
m 0
EZ t
D
e Z∞
0
dt0 ei(ωba −ω)t
=−
dωA0 (ω) eiδω Ψb eik·r ε̂ · ∇ Ψa
2m −∞
0
(1)
cb (t) =
D
+ e−iδω Ψb e−ik·r ε̂ · ∇ Ψa
EZ t
0
(3.13)
dt0 ei(ωba +ω)t .
(3.14)
0
In practice, the duration t of the radiation is much longer than the period 2π/ωba . Therefore,
the time-integrals vanish unless ω ≈ ±ωba . If ω ≈ +ωba , the first time-integral survives,
Eb ≈ Ea + ~ωba , which corresponds to absorption. Else, if ω ≈ −ωba , the second timeintegral survives, Eb ≈ Ea − ~ωba , which corresponds to stimulated emission. An extreme
case of absorption is ionization, wherein the electron goes from a bound state to a scattering
(free) state. We shall consider these cases in the following.
3.2.1
Absorption (ω ≈ +ωba )
We have defined (see Eq. 3.12)
~ωba = Eb − Ea , is positive & ∴ Eb = Ea + ~ωba is the higher state.
(3.15)
This corresponds to absorption.
Figure 3.3: Absorption of EM radiation leading to a transition from a lower level a to a
higher level b.
In this case, only the first term in Eq. 3.14 survives. The transition probability from a
level a to a level b is given by
Pb←a = |cba (t)|2 .
(3.16)
To simplify the analysis, we assume the radiation to be incoherent, so that we can ignore
34
CHAPTER 3. ATOM IN AN EM FIELD
interference terms. From the time-integral in Eq. 3.14, we obtain
Z t
0 i(ωba −ω)t0
dt e
0
2
1 − ei(ωba −ω)t
=
ωba − ω
2
1 − cos (ωba − ω) t
(ωba − ω)2
2 sin2 ((ωba − ω)t/2)
= 2F (t, ωba − ω) (say).
=2
(ωba − ω)2
t2
∴ F (t, ωba − ω) = sinc2 [(ωba − ω)t/2].
(3.17)
2
=2
Figure 3.4: Function F (t, ωba − ω) plotted versus ω with ωba = 1 and for various values of t.
The function F is plotted in Fig. 3.4. Note that it peaks at ω = ωba , and the peak gets
sharper with time t. It means that for long irradiation, we can simply ignore the off-resonance
components. Thus using Eqs. 3.14, 3.16, and 3.17, we obtain
E
D
e2 Z ∞
ik·r
2
2
e
ε̂
·
∇
Ψ
dωA
(ω)|M
|
F
(t,
ω
−
ω),
where
M
=
Ψ
a ,
ba
ba
ba
b
0
2m2 −∞
Z ∞
e2 2
2
=
A
(ω
)|M
|
dωF (t, ωba − ω), since F samples A0 and Mba at ω = ωba ,
ba
ba
2m2 0
−∞
πe2 2
=
A (ωba )|Mba |2 t, since value of the integral is πt.
(3.18)
2m2 0
Pb←a =
Exercise: Compare Eq. 3.18 with Fermi’s Golden Rule.
Thus, the absorption probability Pb←a increases linearly with time t. The transition rate
in the semiclassical theory is given by
.
πe2 2
4π 2
2
Wb←a = Pb←a =
A
(ω
)|M
|
=
ba
ba
2m2 0
m2 c
e2
4π0
!
I(ωba )
|Mba |2 .
2
ωba
(3.19)
This matches with the QED rate given in terms of photons as
4π 2
QED
Wb←a
= 2
m
e2
4π0
!
N (ωba )~
|Mba |2 δ(ω − ωba ).
V ωba
(3.20)
35
3.2. A HYDROGENIC ATOM IN EM FIELD
The absorption cross-section is defined as
~ωba Wb←a
I(ωba )
4π 2 α~2
e2
= 2
|Mba |2 where, α =
is the fine-structure constant.
m ωba
4π0 ~c
σb←a =
(3.21)
Exercise:
• Prove that α is dimensionless.
• Prove that σb←a has the dimension of area per time.
3.2.2
Stimulated Emission (ω ≈ −ωba = ωab )
Since the sign of the frequency is reversed, this corresponds to the release of energy from an
atom in a higher level b going to a lower level a. In this case, only the second term in Eq.
Figure 3.5: Stimulated emission causing a transition from a higher level b to a lower level a.
3.14 survives and the transition probability is given by
Pa←b = |cab (t)|2 =
πe2 2
A0 (ωba )|Mba |2 t, that matches exactly with Eq. 3.18,
2
2m
∴ Pa←b = Pb←a
& Wa←b = Wb←a .
& also, σa←b = σb←a .
(3.22)
Thus the probability (as well as rate & cross-section) for the absorption and stimulated
emission processes are the same. This is known as the principle of detailed balancing. In the
case of stimulated emission, the emerging radiation is coherent with the incident radiation.
36
CHAPTER 3. ATOM IN AN EM FIELD
3.2.3
Spontaneous emission
Interestingly, in 1927, Dirac came up with QED and predicted a slightly different expression
for the emission rate, i.e.,
4π 2
QED
Wa←b
= 2
m
e2
4π0
!
[N (ωba ) + 1] ~
|Mba |2 δ(ω − ωba )
V ωba
QED,stimu
QED,spont
= Wa←b
+ Wa←b
,
(3.23)
where the first term,
4π 2
QED,stimu
Wa←b
= 2
m
e2
4π0
!
N (ωba )~
|Mba |2 δ(ω − ωba )
V ωba
(3.24)
QED,stimu
matched with the semiclassical expression for the rate of stimulated emission, i.e., Wa←b
=
QED
Wb←a = Wa←b = Wb←a . On the other hand, the second term in Eq. 3.23, i.e.,
4π 2
QED,spont
Wa←b
= 2
m
e2
4π0
!
~
|Mba |2 δ(ω − ωba )
V ωba
(3.25)
exists independent of external radiation, i.e., even when N (ωba ) = 0. This process of natural
emission of light by atoms is known as spontaneous emission.
The spontaneous emission is the reason for much of the light around us. It is attributed
to the zero-point (vacuum/cavity) fluctuations of the radiation field interacting with the
atom.
Unlike the stimulated emission, which is coherent with the incident radiation, the spontaneous emission is completely random. The emerging photon can take a random polarization
(linear combination of any two independent polarization vectors ε̂1 & ε̂2 ) and arbitrary direction along an infinitesimal solid angle dΩ. Thus the total rate of spontaneous emission is
given by
QED,spont
Wa←b
=
~
2πm2 c3
!
2
D
E
X
e2 Z
λ 2
λ
dΩ
ωba |Mba
| , where Mba
= Ψb eik·r ε̂λ · ∇ Ψa .
4π0
λ=1
(3.26)
3.3
The dipole approximation
D
E
The exponential in the matrix element Mba = Ψb eik·r ε̂ · ∇ Ψa can be expanded as
eik·r = 1 + (ik · r) +
(ik · r)2
+ ··· .
2!
(3.27)
37
3.4. THE EINSTEIN COEFFICIENTS
As we shall see now, the first term unity corresponds to the interaction of EM with the
electric dipole moment (E1) of the atom. Similarly the second term corresponds to both
electric quadrupole moment (E2) and the magnetic dipole moment (M1). The other terms
correspond to higher moments. If the atomic size r is negligible compared to the wavelength
λ = 2π/k, then kr = 2πr/λ is small, and we can approximate eik·r ≈ 1. The matrix element
under the dipole approximation is
D
Mba
= ε̂ · hψb |∇| ψa i =
−1
im
.
ε̂ · hψb |p| ψa i =
ε̂ · hψb |r| ψa i
i~
~
.
im
ε̂ · hψb |[H0 , r]| ψa i , using Heisenberg Eq. i~O = −[H0 , O] for any observable O,
~(−i~)
−m
−m(Eb − Ea )
−mωba
= 2 ε̂ · hψb |(H0 r − rH0 )| ψa i =
ε̂ · hψb |r| ψa i
ε̂ · hψb |r| ψa i =
2
~
~
~
−mωba
=
ε̂ · rba
~
mωba
=
ε̂ · Dba , where D = −er is the electric dipole moment operator.
(3.28)
e~
=
Thus primarily, emission and absorption are the results of interaction between the electric
component of EM with the atomic dipole! If Dba = 0, we say that electric-dipole transition
(E1) is forbidden. In this case, higher order moments (E2 or M1) may still lead to a small
transition rate.
For an unpolarized radiation, ε̂ has random orientation and so the average of |ε̂ · Dba |2 =
cos2 Θ has an average value of 1/3. Now using Eqs. 3.19, 3.22, and 3.26, we obtain the rates
of
1
4π 2
I(ωba ) |Dba |2 , and
3c~2 4π0
4
4α 3
1
D,spont
3
=
spontaneous emission: Wa←b
ωba
|Dba |2 = 2 ωba
|rba |2 , (3.29)
3
3~c 4π0
3c
D
D
absorption/stimulated emission: Wb←a
= Wa←b
=
where α is the fine-structure constant.
3.4
The Einstein Coefficients
Einstein in 1916 came up with an empirical approach to understand the absorption and
emission processes. Consider an ensemble of atoms undergoing transitions between a lower
level a and a higher level b. Let Na and Nb be the populations of the two levels.
The number of atoms undergoing upward transitions from a to b via absorption in the
presence of a radiation of energy density ρ(ω) is
.
Nb←a = Bb←a Na ρ(ωba ).
(3.30)
38
CHAPTER 3. ATOM IN AN EM FIELD
Figure 3.6: Einstein coefficients.
Here Bb←a is the Einstein coefficient for absorption, which can be written as
.
Nb←a /Na
WD
4π 2
Bb←a =
= b←a = 2
ρ(ωba )
ρ(ωba )
3~
1
|Dba |2 , using Eq. 3.29 and ρ = I/c.
4π0
(3.31)
Now, the number of atoms undergoing downward transitions from b to a depends on both
stimulated emission as well as spontaneous emission, i.e.,
.
Na←b = Ba←b Nb ρ(ωba ) + Aa←b Nb .
(3.32)
Here Ba←b is the Einstein coefficient for the stimulated emission and Aa←b is the Einstein
coefficient for the spontaneous emission. Note that
.
spont
Na←b
spont
= Wa←b
.
Ab←a =
Nb
(3.33)
We assume the gas of atoms to be in thermal equilibrium at a temperature T . The equilibrium is usually established by inter atomic collisions. In thermal equilibrium, the upward
and downward transitions must be equal, i.e.,
.
.
Nb←a = Na←b , now from Eqs. 3.30 & 3.32,
Na Bb←a ρ(ωba ) = Nb (Ba←b ρ(ωba ) + Aa←b ) ,
Na
Ba←b ρ(ωba ) + Aa←b
=
= e~ωba /kT , as per the Boltzmann distribution.
Nb
Bb←a ρ(ωba )
Aa←b
=⇒ ρ(ωba ) =
.
(3.34)
~ω
Bb←a e ba /kT − Ba←b
=⇒
A body in thermal equilibrium is associated with a thermal (black-body) radiation described
by the Planck’s distribution law, according to which,
ρ(ωba ) =
3
~ωba
1
.
2
3
~ω
/kT
π c e ba
−1
(3.35)
39
3.5. SELECTION RULES FOR ONE-ELECTRON ATOMS
From Eqs. 3.34 and 3.35, we obtain
Bb←a = Ba←b
D
3
3
Wb←a
~ωba
~ωba
using Eq. 3.31
B
=
b←a
π 2 c3
π 2 c3 ρ(ωba )
3
D
~ωba
Wb←a
D,spont
i.e., Wb←a
= 2 3
using Eq.3.29.
π c ρ(ωba )
Aa←b =
(3.36)
Thus, Einstein coefficient Aa←b directly relates to the rate of the spontaneous emission.
Exercise: Show that if ga and gb are the degeneracies of levels a and b, then
ga Bb←a = gb Ba←b .
3.5
Selection rules for one-electron atoms
Under the dipole approximation, the transition rate depends on the perturbative term |ε̂·rba |2
(see Eq. 3.28). The electric polarization vector ε̂ can be expressed in terms of orthonormal
spherical components (see Fig. 3.7)
1
ε1 = − √ (ε̂x + iε̂y , )
2
1
ε−1 = √ (ε̂x − iε̂y , ) , and
2
ε0 = ε̂z
(3.37)
If the radiation is along ẑ, then the 0 = 0 since the electric component should be
transverse to the k̂ vector. In this case, ε±1 represent right and left circular polarization
components.
In the same way, the position coordinate r can also be expressed in orthonormal spherical
components:
1
1
4π 1/2
r1 = − √ (x + iy, ) = − √ r sin θeiφ = r
Y1,1 (θ, φ)
3
2
2
1
1
4π 1/2
−iφ
r−1 = √ (x − iy, ) = √ r sin θe
=r
Y1,−1 (θ, φ), and
3
2
2
4π 1/2
r0 = z = r cos θ = r
Y1,0 (θ, φ).
3
(3.38)
40
CHAPTER 3. ATOM IN AN EM FIELD
Figure 3.7: Optical polarizations.
Thus, using the orthonormal spherical components, the perturbation term is defined as
X
ε̂ · r̂ =
ε∗q (rab )q =
Inq 0 l0 m0 ←nlm =
4π
3
ε∗q Inq 0 l0 m0 ←nlm , which using Eq. 2.10 gives
q=1,0,−1
q=1,0,−1
X
1/2 Z ∞
0
Z
3
drr Rn0 l0 (r)Rnl (r)
dΩYl∗0 m0 (θ, φ)Y1,q (θ, φ)Ylm (θ, φ)
(3.39)
The radial integral is always nonzero. However, the angular integral vanishes, unless certain
special conditions are met, thus resulting in the selection rules.
Magnetic-quantum m-selection rule
In Eq. 3.39, the integral over φ leads to
Jm,m0 ,q =
Z 2π
0
dφei(m−m +q)φ .
(3.40)
0
Photon is a spin-1 particle, but owing to the transverse nature it is described by two helicities - right and left circular polarizations. A general state is a superposition of these two
polarizations (see Fig. 3.8). Now we consider the following cases:
• Polarization vector ε̂ along ẑ axis:
From Eq. 3.37, this implies q = 0. Now, Eq. 3.40 and therefore Eq. 3.39 vanish, unless
∆m = m0 − m = 0.
• Propagation vector k̂ along ẑ axis:
This implies q = ±1. Again, Eq. 3.40 and therefore Eq. 3.39 vanish, unless ∆m =
m0 − m = ±1.
3.6. SPECTRUM OF HYDROGENIC ATOMS
41
Figure 3.8: Linear and circular polarizations of EM. (source: Demtroder).
Figure 3.9: Magnetic-quantum m-selection rule (source: Demtroder).
Orbital-quantum l-selection rule (Parity-selection rule)
The total parity of the angular integrand is l + l0 + 1. Since integral of odd-parity functions
vanish, the angular integral is nonvanishing only if l + l0 + 1 is even, i.e., l + l0 is odd, or in
other words, only if the levels a and b have opposite parity. Conservation angular momentum
requires that ∆l = ±1, since it is caused by the photon, a spin-1 particle. A rigorous way
to understand this selection rule is via Clebsch-Gordan coefficients (see Chapter 3 of QMII notes), which vanish unless ∆l = ±1. Thus the selection rules for the electric-dipole
transitions are
∆l = ±1 and
∆m = 0, ±1.
3.6
(3.41)
Spectrum of hydrogenic atoms
Considering the selection rules discussed above, the allowed transitions from the first four
levels of a Bohr atom are shown in Fig. 3.10.
42
CHAPTER 3. ATOM IN AN EM FIELD
Figure 3.10: Allowed first-order transitions in lowest four Bohr levels.
2s matastable state
From Fig. 3.10, it is clear that there is no lower allowed state from the 2s state, despite
it being an excited state. The electric-dipole transition 2s→1s is forbidden, since both are
even-parity levels. As a result, 2s has a very long lifetime of ∼ 1/7 s, almost hundred million
times longer lived than 2p state, and accordingly it is known as a metastable state.
3.6.1
Frequencies
Previously, we had discussed frequencies of a Hydrogenic atom (see Eq. 2.14)
ωn0 l0 m0 ←nlm =
Z 2 Ry
~
1
1
−
n2 n02
(3.42)
corresponding to various spectral series as shown in Fig. 2.4.
3.6.2
Line intensities and oscillator strengths
We have seen that, under the dipole approximation, the transition rate depends on |ε̂ · rba |2
(see Eq. 3.28), leading to selection rules given in Eq. 3.41. Now, what about the relative lineintensities of various transitions? As seen from Eq. 3.29, the rate of spontaneous emission
from an excited level a to an arbitrary lower level k depends on |Dka |2 ∝ |rka |2 . One therefore
defines oscillator strength, a dimensionless quantity,
fka =
X
2mωka
|rka |2 , which satisfies
fka = 1, on summing over complete basis. (3.43)
3~
k
43
3.6. SPECTRUM OF HYDROGENIC ATOMS
To prove the sum-rule, we note |rka |2 = |xka |2 + |yka |2 + |zka |2 and first consider sum of the
x-components over the complete basis of atomic wave functions.
2m X
mX
ωka |xka |2 =
ωka [x∗ka xka + xka x∗ka ]
3~
3~
k
k
k
mX
ωka [ha |x| ki hk |x| ai + ha |x| ki hk |x| ai] .
=
3~ k
m
m~(ωk − ωa )
.
Consider (px )ka = mxka = − hk |[H0 , x]| ai = −
xka = imωka hk |x| ai .
i~
i~
Similarly, (px )ak = imωak xak = −imωka ha |x| ki .
X
X m
ωka
x
=
∴
fka
[ha |x| ki hk |px | ai − ha |px | ki hk |x| ai]
3~ k i
m
ωka
k
X
x
fka
=
"
!
!
X
X
1
=
ha|x
|ki hk| px |ai − ha|px
|ki hk| x|ai
3i~
k
k
X
1
=
ha |[x, px ]| ai , ∵
|ki hk| = 1,
3i~
k
1
1
i~ = .
=
3i~
3
#
(3.44)
Since the atom has no bias for the x, y, and z components, summing over all three components
yields
X
fka = 1.
(3.45)
k
Using Eqs. 3.19 and 3.43, we obtain the rate equation
D,spont
Wk←a
=
2~α 2
ω |fka |.
mc2 ka
(3.46)
In general, higher the atomic level, lower are the oscillator strengths as well as the rates
of spontaneous transition (see Fig. 3.11).
Figure 3.11: Oscillator strengths (source: Bransden & Joachain).
44
CHAPTER 3. ATOM IN AN EM FIELD
3.6.3
Lifetimes of excited states
Suppose Nb (0) atoms are present in an excited state b at time t = 0. Under spontaneous
emission, the atoms can make transitions to various lower levels allowed by the selection rules.
Therefore at any time t, the instantaneous population of the level b is changing according to
.
Nb (t) = −Nb (t)
X
D,spont
Wk←b
, where the sum is over allowed lower levels.
(3.47)
k
The solution for Nb (t) is
1
D,spont is the lifetime or half-life of level b. (3.48)
k Wk←b
Nb (t) = Nb (0)e−t/τb , where τb = P
Figure 3.12: Calculated life-times of various levels of Hydrogen atom (source: Bransden &
Joachain).
Since oscillator strengths decrease for higher atomic levels, the life times increase (see
Fig. 3.12). For hydrogenic ions one observes a scaling law,
τ (Z) =
3.6.4
τ (Z = 1)
.
Z4
(3.49)
Lineshapes and linewidths
According to the energy-time uncertainty principle, longer the lifetime of a level, smaller is
the uncertainty in its energy. Thus, ~/τa is known as the natural width of a level a. Thus,
for a transition b ↔ a, between levels a and b of respective lifetimes τa and τb , the quantity
Γa←b =
~
~
+
τa τb
(3.50)
is called the natural linewidth. For example, for the 2p → 1s transition, τ2p = 0.16 × 10−8 s,
for the ground state 1s, the lifetime τ1s is taken to be infinity, and the corresponding natural
linewidth Γ1s←2p = 4.11 × 10−7 eV, which is negligible compared to the energy 3.4 eV of the
transition.
Since the decay of excited states is normally described by an exponential decay function,
3.6. SPECTRUM OF HYDROGENIC ATOMS
45
the spectral lineshape is described its Fourier transform, which is Lorentzian,
f (ω) =
Γ2a←b /(4~2 )
(ω − ωba )2 + Γ2a←b /(4~2 )
(3.51)
Figure 3.13: Natural line shape (source: Bransden & Joachain).
The spontaneous decay of an atom in an excited state to a lower level state discussed
above is also known as fluorescence. The spectral line shape of a pure fluorescence process
should be Lorentzian. However, in practice, the spectral lines undergo broadening due to
multiple factors. Two main factors are described below
Pressure broadening or Collision broadening
If atoms are undergoing collisions, the lifetime τb of an excited level b depends not only on the
rate of spontaneous decays, but also on the rate of transitions Wc induced by elastic/inelastic
collisions with other atoms (see Fig. 3.14). The effective rate of transitions from the level b
can be expressed by modifying Eq. 3.48,
X
1
D,spont
=
Wk←b
+ Wc .
τb k(lowerlevels)
(3.52)
Thus the overall life-time of the excited state is reduced resulting in additional Lorentzian
broadening. Generally, inelastic collisions result in broadening, while elastic collisions result
46
CHAPTER 3. ATOM IN AN EM FIELD
in broadening as well as a shift of the spectral line.
Figure 3.14: Pressure/collision broadening (source: Demtroder).
Doppler broadening
Suppose an observer picks a radiation with propagation vector k emitted by an atom moving
with a velocity v (see Fig. 3.15). The observed frequency is given by
Figure 3.15: Pressure/collision broadening (source: Demtroder).
0
ωba
= ωba + k · v.
(3.53)
47
3.6. SPECTRUM OF HYDROGENIC ATOMS
Thus the observed frequency increases or decreases depending on whether the atom is approaching or receding respectively. Let the observer picking the radiation along ẑ direction.
Then, using k = ω/c, we find the observed frequency to be
ωba
0
ωba
= ωba +
vz = ωba
c
vz
1+
, where vz = ẑ · v.
c
(3.54)
In thermal equilibrium at temperature T , the random motion of atoms of atomic mass M is
such that the number nb (vz ) of atoms in level b with a particular velocity component vz is
given by the Gaussian distribution,
s
−(vz /v0 )2
nb (vz ) ∝ e
, where v0 =
2kB T
is the most probable velocity component.
M
(3.55)
Here it is essentially a Gaussian process. Fourier transform of Gaussian is also Gaussian.
Accordingly, Doppler broadening leads to a Gaussian line broadening function (see Fig. 3.16)
"
M c2
J(ω) ∝ exp −
2kB T
ω − ωba
ωba
2 #
.
(3.56)
Figure 3.16: Lorentzian vs Gaussian broadening (source: Demtroder).
Total line shape
The combination of natural line shape, collision broadening, and Doppler broadening leads
to a resultant line shape called Voigt line shape as shown in Fig. 3.17.
48
CHAPTER 3. ATOM IN AN EM FIELD
Figure 3.17: Lorentzian vs Gaussian broadening (source: Demtroder).
Doppler-free spectroscopy
A direct method of measuring fine structures is by dopper-free spectroscopy or saturation
spectroscopy (see Fig. 3.18). It involves a probe beam whose transmission is measured
through a gas chamber. We also send a counter-propagating pump beam, always with same
frequency as the probe, but ten times intense. Typically, both probe and pump are derived
from a common source, and then divided into different optical paths. The mismatch between
the atomic resonance frequency and the probe/pump frequency is called the detuning. In
Fig. 3.18, the probe goes right and the pump goes left.
First suppose pump is off (see Fig. 3.18 (left upper graph)). Note that the if the probe
has a positive detuning, it must be red-shifted to cause resonance, and that happens to only
right-going atoms. Similarly, if the probe has a negative detuning, it must be blue-shifted
to cause resonance, and that happens to only left-going atoms. At resonance, the probe
is absorbed by transverse going atoms, which are of higher probability. So, we see more
absorption at zero-detuning.
Now let us switch-ON the pump (see Fig. 3.18 (left lower graph)). At positive detuning,
probe is red-shifted and absorbed by the right-going atoms, but they see blue-shifted pump
and therefore remain unaffected by the pump. Similarly, at negative detuning, the probe
is blue-shifted and absorbed by the left-going atoms, but they see red-shifted pump and
therefore again remain unaffected by the pump. However, at zero detuning, the transverse
going atoms get saturated by absorbing the strong pump and they no longer can absorb
the probe. This results in a sharp peak, called Lamb peak, at zero-detuning arising from
saturated transverse atoms which are free from doppler broadening w.r.t. the longitudinal
direction.
3.7. THE PHOTOELECTRIC EFFECT
49
Figure 3.18: Doppler-free or saturation spectroscopy: the transmission signal without (left
upper plot) and with intense pump beam (left lower plot), and experimental spectra (right)
(source: https://www.mpq.mpg.de/4992695/saturation_spectroscopy.pdf).
3.7
The photoelectric effect
Towards the late 19th century, it was observed (by Heinrich Hertz and others) that charges
were produced when EM is shined on metals. It was later established that electrons can be
ejected from the metal upon shining a EM radiation (usually visible, UV, X-Rays, or γ-rays).
A simple setup to study photoelectric effect is shown in Fig. 3.19. Here, the photoelectrons
ejected from a metal using light of frequency ν are collected by an electrode and the resulting
current is measured.
Systematic experiments with vacuum tubes (see Fig. 3.19 and 3.20) established the
following facts:
• For a given metal, there exists a minimum or threshold frequency ν0 of EM below
which no photoelectric current Iph is found irrespective of the intensity.
• Above the threshold frequency ν ≥ ν0 :
50
CHAPTER 3. ATOM IN AN EM FIELD
Figure 3.19:
Setup for studying photoelectric effect (source:
https://www.toppr.com).
Demtroder and
– The photoelectric current Iph gradually increases and saturates for positive potential U . This indicates only so many electrons are released with a given intensity
of light.
– The saturation value of photoelectric current Iph (or total number of ejected electrons) increased linearly with the intensity of EM radiation.
– The photoelectric current vanishes completely with a negative potential −U0 .
This indicates ejected electrons do have certain maximum kinetic energy.
– The stopping potential increases linearly with frequency of EM. This indicates
electron kinetic energy increases with EM frequency.
– The release of electrons from the metal appeared almost instantaneous, without
any observable delay after exposure to light.
These observations are inconsistent with the classical electromagnetic theory, according to
which, there should be sufficient energy to release electrons with sufficient intensity and
sufficiently long duration of exposure for any frequency of EM.
In 1905, Einstein explained photoelectric effect using the quantum nature of light, which
fetched him the Nobel prize in 1921. Let W be the work-function or the binding energy
3.7. THE PHOTOELECTRIC EFFECT
51
of electron within the metal. Then, part of the EM energy hν is utilized against the workfunction and remaining energy is carried by the photoelectron as the kinetic energy T . Thus,
hν = W + T.
(3.57)
The kinetic energy is balanced by the stopping potential, i.e., T = eU0 . Writing W = hν0 ,
we obtain
T = eU0 = hν − W.
(3.58)
Thus, by plotting the stopping potential U0 against the frequency ν, we can determine the
work-function as the negative y-intercept or positive x-intercept (note that W = hν0 when
T = 0). The slope can be used to determine h/e value (see Fig. 3.20).
Figure 3.20: Determining work-function of Zinc with UV irradiation of varying frequencies
and measuring the stopping potential in each case (source: Wikipedia).
3.7.1
The quantum theory of photoionization
Photoionization is the extreme case of absorption in which an electron transitions from a
bound state a to a free-particle state b with energy Ef = ~2 kf2 /(2m) and wave function
√
ψb = (1/ 3 2π)eikf ·r (see Fig. 3.21).
Using Eq. 3.21 and integrating over kf , one obtains the differential cross section for
ejecting electron within a solid angle dΩ as
dσ
4π 2 α~kf
=
|Mba (ω)|2 , where,
2
dΩ
m
D
E
1 Z
ik·r
√
Mba = b e ε̂ · ∇ a = 3
dr eiK·r ε̂ · ∇ψa , with K = k − kf .
2π
(3.59)
52
CHAPTER 3. ATOM IN AN EM FIELD
Figure 3.21: Photoionization as the extreme case of absorption.
The total cross-section for ejecting 1s electron happens to be
√
!7/2
16π 2 8 5 mc2
σ=
α Z
a20 .
3
~ω
(3.60)
In bulk materials, σ ∝ Z (4 to 5) . That is the reason why heavier elements such as lead
(Z = 82) are used for stopping high-energy radiation such as γ-rays.
3.8
Scattering of radiation
So far we had discussed the atom-EM interaction under weak-field approximation using
first-order perturbation theory. Unlike the photoelectric effect, the scattering of EM by
atoms is generally a second-order process, and accordingly one has to apply the secondorder perturbation theory. The process involves two steps:
• Step 1: A photon of energy ~ω and wave vector k is absorbed exciting the atomic
system from state a to an intermediate state n.
• Step 2: The atomic system emits a photon of energy ~ω and wave vector k0 and gets
de-excited from state n to state b.
The different possibilities of light scattering are shown in Fig. 3.22.
• Rayleigh scattering is elastic scattering:
Here a and b are same, and accordingly, incident and scattered photons have the same
energy, but different wave vectors. The emitted radiation is called resonance radiation.
Rayleigh scattering by the Earth’s atmosphere is responsible for the diffused sky light
as well as blue sky.
• The Raman scattering is inelastic scattering. In 1928, Raman observed that 1 in 108
photons are scattered inelastically. He won the Physics Nobel Prize in 1930.
Here a and b are different. The scattered light is called resonance fluorescence. So,
there are two types of Raman scattering.
53
3.8. SCATTERING OF RADIATION
Figure 3.22: Types of scattering processes (Source: Wikipedia).
– Stokes Raman scattering:
The atom (or molecule) absorbs energy, while the scattered photon has less energy
than the incident photon. In molecules, the remaining energy is usually absorbed
by vibrational or rotational modes. The resulting Raman spectrum consisting of
all the Stokes and Anti-Stokes lines leaves a finger-print of molecular structure.
We shall study more about the vibrational and rotational spectroscopy later in
the course.
– Anti-Stokes Raman scattering:
The atom (or molecule) loses energy, while the scattered photon has more energy
than the incident photon. If the number of prior excited atoms is low, the antiStokes Raman lines are weaker than Stokes Raman lines. In molecules the extra
energy is usually provided by the vibrational or rotational modes.
Under dipole-approximation, levels a and n must have different parities, i.e., la −ln = ±1.
Similarly, n and b, must also have different parities and lb − ln = ±1. This means, levels a
and b must have same parity or la − lb = 0, ±2.
Assuming resonance region, i.e., incident frequency ω is nearly equal to ωna , and using
2nd order perturbation theory, we find that the cross-section for scattering depends on
2
(0 · Dbn )( · Dna )
dσba
∝
,
dΩ
ωna − ω − iΓn /(2~)
(3.61)
54
CHAPTER 3. ATOM IN AN EM FIELD
where and 0 are the polarizations of the incident and emergent waves and Γn = ~/τn is
the life-time of the intermediate state. In general Γn ≥ Γna + Γbn . Integrating and taking
the maximum, one obtains the following:
• Resonant (ω ≈ ωna ) case:
!2
c 2 Γan
Rayleigh scattering:
, and
ωna
Γn
!
c 2 Γan Γbn
max
Raman scattering: σRaman = 2πgn
,
ωna
Γ2n
max
= 2πgn
σRayleigh
(3.62)
where gn is the degeneracy of the intermediate state.
• Non resonant, low-frequency (ω ωna ) case:
r0 mᾱ 2 4
ω , where,
e2
X |(Dz )an |2
ᾱ = 2
,
E
−
E
n
a
n6=a
8π
Total scattering: σtot =
3
(3.63)
is the static polarizability of the atom, Dz is the z-component of the electric dipole
moment, and r0 =∼ 10−15 m. Note that the scattering cross-section increases rapidly
with the frequency. This is the reason for blue sky of Earth (see Fig. 3.23). However,
the color of sky may vary in different planets. On Mars for example, the scattering by
particles (with size comparable to EM wavelength), called Mie scattering dominates
over the Rayleigh scattering resulting in a reddish sky. The vibrant colors of twilight
sky on Earth is also a result of multiple types (Rayleigh, Raman, and Mie) of scatterings
occurring simultaneously.
Figure 3.23: On Earth, blue light is scattered more efficiently than red (Source: Wikipedia).
3.8. SCATTERING OF RADIATION
55
Figure 3.24: Scattering profiles of Rayleigh vs Mie. Right: Human eye sensitivity to colors
(source: http://hyperphysics.phy-astr.gsu.edu/)
Raman spectrometer
56
CHAPTER 3. ATOM IN AN EM FIELD
Figure 3.25: Raman spectrometer (Source: Wikipedia).
Chapter 4
Hydrogenic atom: Higher order
corrections
The simplest Bohr model for a Hydrogenic atom considers only the Coulomb potential, which
in the center of mass frame, has the Hamiltonian
Ze2
~2
Ze2
P2
−
= − ∇2 −
2µ 4π0 r
2µ
4π0 r
"
!
#
2
2
~
1 ∂
∂
L
Ze2
2
=−
r
−
.
−
2m r2 ∂r
∂r
~2 r2
4π0 r
H0 =
(4.1)
We had obtained the energy levels of a Hydrogenic atom as (see Eq. 2.6)
1
Z2
Z2
En = − µc2 α2 2 ≃ −Ry 2 ,
2
n
n
(4.2)
which depends only on the principal quantum number n but independent of other quantum
numbers. The energy levels with same n, but different l and m quantum numbers are
supposed to be degenerate. The corresponding spectral lines are of frequency (see Eq. 2.14)
2
hν = ±Z Ry
1
1
− 02 .
2
n
n
(4.3)
The energy gaps can thus be confirmed experimentally by measuring the frequencies of
Hydrogen spectral lines. Normally the spectral lines broadened by pressure broadening as
well as Doppler broadening. However, very careful experiments revealed a fine structure,
which not only involved splitting spectral lines but also small shifts in the spectral lines from
the above expression. Fine structure confirming non degeneracy w.r.t. l and m quantum
numbers. For example, the splitting of Balmer-α line: (n = 2) ← (n0 = 3) is shown in Fig.
4.1. These corrections to energy En , which are of the order of α2 or less, remained puzzling
over several decades even after quantum mechanics was well formulated. In this chapter, we
57
58
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
Figure 4.1: Fine-structure of Balmer-α line (Source: Ingolf V. HertelClaus-Peter Schulz,
Atoms, Molecules and Optical Physics).
shall study these corrections and try to appreciate the depth of science behind the simplest
atom. While the rigorous approach would be to solve the hydrogenic atom using Dirac’s
equation, the relativistic version of Schrödinger equation, this is beyond the scope of this
course. Even solving Dirac’s equation is not enough to account for all the observed effects.
So, we shall take certain important effects individually and perturb the Bohr atom to obtain
solutions to get as close to experimental observations as possible.
4.1
Relativistic correction to the kinetic energy
The relativistic form of kinetic energy is total energy minus rest mass energy, i.e.,
Trel =
q
p2 c2 + m2 c4 − mc2

p2
= mc2  1 + 2 2
mc
"
!1/2

− 1 , now using binomial expansion,
!
#
1 p2
1 p4
= mc
1 +
−
+ · · · − 1
2 m2 c2 8 m4 c4
p4
1
p2
p2
2
=
−
=
T
−
T
,
where
T
=
is the nonrelativistic kinetic energy,
2m 8m3 c2
2mc2
2m
= T + H1 .
(4.4)
2
59
4.2. SPIN-ORBIT INTERACTION
We shall treat the extra term H1 = −T 2 /(2mc2 ), that was missing earlier, as a perturbation
to the Bohr atom:
1
1
1
ze 2
2
2
H1 = −
, where ze =
T
=
−
[H
−
V
]
=
−
H
−
0
0
2mc2
2mc2
2mc2
r
1
1
2 1
2
=−
H0 − 2ze H0 + ze 2 .
2mc2
r
r
−Ze2
4π0
!
(4.5)
Note that H10 is a function of r only. In polar coordinates, all components of L are independent of r, and are dependent on only θ and φ coordinates. So, for any general function
f (r), [L, f (r)] = 0 = [L2 , f (r)]. By the same argument, [L, H10 ] = 0 = [L2 , H10 ]. Thus, for
any given n quantum number, H10 is diagonal in the n2 -degenerate basis of l, ml . Now using
the time-independent perturbation theory, the energy correction is
∆E1 = hψnlml |H10 | ψnlml i
D
E
1
H0
1
2
2
=−
ψnlml + ze ψnlml 2 ψnlml
ψnlml H0 ψnlml − 2ze ψnlml
2mc2
r
r
using standard expectation values,
"
#
1
mc2 α2 Z 2
Z2
ze Z 2
Z
2
2
=−
+
z
=
E
,
where
E
=
−
2E
z
n
n
e
e 2 3
n
2mc2
an2
a n (l + 1/2)
2an2
2n2
"
#
n
(Zα)2 3
= −En
−
.
(4.6)
n2
4 l + 1/2
Note that the correction is by a factor ∝ Z 2 α2 , which is ∼ 10−4 for hydrogen and ∼ 10−3
for Li++ .
Exercise: Complete the calculations of Eq. 4.6.
4.2
Spin-orbit interaction
Let us now invoke the electron spin angular momentum S, which had been ignored so far.
The electron is a spin-1/2 particle. Let {|αi , |βi} be the eigenbasis for the electron spin, i.e.,
~
|αi and
2
~
S |βi = − |βi .
2
S |αi =
(4.7)
Accordingly, we associate the spin quantum number ms = ±1/2. We had studied the spinorbit interaction in QM-II (3rd chapter of QM-II notes).
60
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
The Hamiltonian for the spin-orbit interaction is
H20 = ξ(r) L · S, where ξ(r) =
1 Ze2 1
.
2m2 c2 4π0 r3
(4.8)
The Bohr atom is generally solved in the simultaneous eigenbasis of H0 , L2 , and Lz . The
spin operator S commutes with all these operators and therefore it is tempting to use
{|n, l, ml , ms i} as the common eigenbasis. However, not all components of L simultaneously commute with L2 , and therefore H20 does not commute with L2 and Lz .
Recall the commutation relations:
[Li , Rj ] = i~ijk Rk
[Ri , Rj ] = 0
[Ri , Pj ] = i~δij
[Li , Pj ] = i~ijk Pk
[Li , Lj ] = i~ijk Lk
[Pi , Pj ] = 0
[Li , R2 ] = 0
[Li , P 2 ] = 0
[Li , L2 ] = 0
Therefore {|n, l, ml , ms i} can’t be the simultaneous eigenbasis for H0 and H20 . In QM-II,
we had learned that, given two angular momenta J1 and J2 , and the total angular momentum,
J = J1 + J2 , then {J2 , Jz , J21 , J22 } form a simultaneous set of commuting operators. We
therefore define the total angular momentum
J = L + S, with eigenstates |j, mj ; l, si s.t.
J2 |j, mj ; l, si = j(j + 1)~2 |j, mj ; l, si ,
Jz |j, mj ; l, si = mj ~ |j, mj ; l, si ,
L2 |j, mj ; l, si = l(l + 1)~2 |j, mj ; l, si , and
S2 |j, mj ; l, si = s(s + 1)~2 |j, mj ; l, si , where s = 1/2 for electron.
(4.9)
Here the quantum number j can take integral values from |l − 1/2| to l + 1/2, which follows
from the triangle inequality. Also, mj = ml + ms ∈ [−j, −j + 1, · · · , j − 1, j]. We can now
choose the common eigenbasis {|n, j, mj i} of {H0 , J2 , Jz , L2 , S2 } as the common eigenbasis.
Now, expressing
1
H20 = ξ(r) L · S = ξ(r) J2 − L2 − S2 ,
2
(4.10)
we find that H20 also commutes with all of {H0 , J2 , Jz , L2 , S2 } and thus we can continue to
work in the {|n, j, mj i} eigenbasis.
Again using the first-order perturbation theory in the degenerate subspace, we obtain
61
4.3. DARWIN TERM
the correction
E
D
1
hn, j, mj |ξ(r)| n, j, mj i n, j, mj J2 − L2 − S2 n, j, mj
2
1 Ze2 1
3
=
j(j + 1) − l(l + 1) −
,
2m2 c2 4π0 r3
4
∆E2 =
(4.11)
which vanishes for l = 0 since j = 1/2 in this case. This is expected since spin-orbit
interaction survives only for l > 0. For l > 0, again using standard expectation values,


l
2
∆E2 = −En
(Zα)
×
2nl(l + 1/2)(l + 1) 
(−l − 1)
for j = l + 1/2 and
(4.12)
for j = l − 1/2.
Again the correction is by a factor ∝ (Zα)2 .
4.3
Darwin term
In QED, the spontaneous creation and annihilation of particles leads to an interference
between particles and antiparticles that appears as the fluctuations in particle position.
This is often referred to as Zitterbewegung. For electron, this fluctuation is roughly over
Compton wavelength, λc = ~/(mc). Note that, we can ignore nuclear oscillations, which are
three orders of magnitude smaller. Effectively, it adds a small delta-function contribution,
called the Darwin term,
H30 =
π~2 Ze2
δ(r).
2m2 c2 4π0
(4.13)
Since only s-orbitals have wave functions at r = 0, it applies only for l = 0. Moreover,
H30 commutes with the observables {H0 , L2 , Lz }. Therefore, the energy correction for l = 0
states is
π~2 Ze2
hψ(r)n,l=0,ml =0 |δ(r)| ψ(r)n,l=0,ml =0 i
2m2 c2 4π0
π~2 Ze2
=
|ψ(0)n,l=0,ml =0 |2
2m2 c2 4π0
(Zα)2
= −En
, for l = 0.
n
∆E3 =
(4.14)
62
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
4.4
Total relativistic correction
Adding the contributions of ∆E1 , ∆E2 , and ∆E3 , we obtain (for any l value)
!
(Zα)2
n
3
∆Erel = ∆E1 + ∆E2 + ∆E3 = En
−
, and so the revised energies
2
n
j + 1/2 4
!#
"
n
3
(Zα)2
−
, where,
Enj = En + ∆Erel = En 1 +
n2
j + 1/2 4
j = 1/2, if n = 1
j ∈ {l ± 1/2}l=1,··· ,n−1 = {1/2, 3/2, · · · , n − 1/2}, if n > 1.
(4.15)
Note:
• ∆Erel depends uniquely on j but not on l. This is interesting from the parity of states,
Parity
+1
−1
+1
−1
+1
l
0(s) 1(p) 2(d) 3(f ) 4(g)
j = l − 1/2
1/2
3/2
5/2
7/2
Symbol
np1/2 nd3/2 nf5/2 ng7/2
j = l + 1/2 1/2
3/2
5/2
7/2
9/2
Symbol
ns1/2 np3/2 nd5/2 nf7/2 ng9/2
···
···
···
···
···
···
Table 4.1: States, parities, and energies. Spectroscopic notation (symbol) nlj is used to
indicate the levels.
that is given by (−1)l . Look at Table 4.1. Here, same j and hence same energy Enj
appears for two states of opposite parity. This degeneracy is further removed by Lamb
shift correction.
• The relativistic correction ∆Erel has the following dependencies:
– ∝ (Zα)2 ,
– ∝ 1/n, and
– for a given n, decreases with j.
• For a given principal quantum number n, with Bohr energy En , now there are n values
of j, each corresponding to a nondegenerate level Enj . This splitting of levels is known
as fine structure. Since the splitting is controlled by α (splitting is ∝ α2 ), α is known
as the fine structure constant.
n
n
• For n = 1, we have j = 1/2. So, j+1/2
− 34 = 1/4. For n > 1, the value of j+1/2
− 34
ranges from (n − 3/4) (for j = 1/2) to 1/4 (for j = n − 1/2). Thus, in all the cases,
4.4. TOTAL RELATIVISTIC CORRECTION
n
− 34
j+1/2
63
is positive. Since En is negative, Enj is negative for all values of j. Thus,
relativistic corrections increase the binding energy for all levels.
• Fig. 4.2 shows relativistic corrections for n = 1, 2, 3 levels of Hydrogen atom.
Figure 4.2: Relativistic corrections to low levels of Hydrogen atom (Bransden & Joachain).
4.4.1
Fine structure of spectral lines
We have learned that the electric dipole transition has the selection rule ∆l = ±1. This
translates, following Table 4.1, to selection rule in j as
∆j = 0, ±1.
(4.16)
The spectral lines nlj → n0 l0 j 0 for a given pair (n → n0 , l → l0 ), is known as a multiplet.
Some multiplets are shown in Fig. 4.3.
Doublet of Lyman-α is shown in Fig. 4.4 and the multiplet of Balmer-α is shown in Fig.
4.5.
64
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
Figure 4.3: Fine structure multiplets (source: Bransden & Joachain).
4.5
Lamb Shift
As explained in Table 4.1, the two sets of levels corresponding to same j values, but different
l values are degenerate up to relativistic corrections. However, some careful experiments by
Lamb and Retherford in late 1947 and 1953 revealed an energy gap. We shall first discuss
this interesting experiment.
4.5.1
Lamb-Retherford Experiment
The focus here is to see if there is an energy gap between 2s1/2 and 2p1/2 levels. The plan is
to see the frequency of the EM stimulating absorption between these two levels.
The Lamb-Retherford experimental setup is illustrated in Fig. 4.6. It starts with dissociating molecular hydrogen into a beam of atomic hydrogen using a tungsten oven maintained
at 2500 K. The atomic beam is now bombarded with electrons with kinetic energy ∼ 10 eV.
This is sufficient to excite the hydrogen atoms to 2s metastable state, which has a lifetime
of over 10−1 s. The average velocity of the beam is ∼ 106 cm/s. Therefore, the metastable
atoms can easily travel up to a detector kept about 10 cm away, with negligible decay. The
detector consists of a tungsten plate, that emits electrons when impinged by the excited 2s
atoms, but not when impinged by ground state 1s atoms. The emitted electrons are captured
by an anode and are measured as current in the external circuit. Now, the metastable hydrogen atoms are irradiated by EM radiation on-resonant with 2s1/2 -2p1/2 transition, in a EM
resonator kept before the detector. Note that the lifetime of 2p1/2 is just 10−9 s and therefore
survives only for a distance of about 10−3 cm before spontaneously decaying to ground state
4.5. LAMB SHIFT
65
Figure 4.4: Lyman-α doublet (source: Bransden & Joachain).
by emitting fluorescence. Thus, an on-resonant EM causes a rise in the population of ground
state hydrogen atoms and therefore a dip in the detector current. Between 1947 and 1953,
Lamb and coworkers measured the resonance frequency with increasing precision and arrived
at a value of 1057 ± 0.1 MHz ≡ 4.4 µeV ≡ 0.035 cm−1 . This splitting of levels 2s1/2 and
2p1/2 could not be explained by the standard form of Dirac’s theory, and provided the basis
for formulating Quantum Electrodynamics (QED) by Feynman and other over next several
decades.
4.5.2
Lamb-shift as a radiative correction in QED
In QED, vacuum is filled with zero-point fluctuation of radiation field, which acts on electron
causing position oscillations and smearing of charges. Notice the similarties with the Darwin
term. This effect is also more pronounced for s-orbital which raises in energy and less so for
p-orbital that lowers in energy (see Fig. 4.7). Effect on higher orbitals is even smaller and
is usually ignored.
Direct detection of fine structure multiplets via Doppler free spectroscopy is shown in
Fig. 4.8.
66
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
Figure 4.5: Balmer-α multiplet. The degeneracy between (f) and (g) are lifted by Lamb shift
(source: Bransden & Joachain).
4.5. LAMB SHIFT
67
Figure 4.6: Lamb - Retherford experimental setup (Source: Demtroder)
Figure 4.7: Lamb shift is explained as a radiative correction in QED (source: Bransden &
Joachain).
68
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
Figure 4.8: Fine structure of Balmer-α line (source: Demtroder).
69
4.6. NUCLEAR EFFECTS
4.6
Nuclear effects
Bohr model assumes nucleus as a spin-less point electric monopole. But nucleus has a finite
volume, may have magnetic dipole moment (spin), and may also possess electric quadrupole
moment. Now we shall introduce the properties of nucleus one by one and see how the
atomic states are affected.
4.6.1
Nuclear spin and Hyperfine structure
Both proton and neutron are spin-1/2 particles. Thus, for every nuclear isotope, addition
of these spins results in a specific net spin angular momentum I (in the ground state) with
eigenvalue equation
I |I, MI i = I(I + 1)~2 |I, MI i , and
Iz |I, MI i = MI ~ |I, MI i .
(4.17)
where I is the nuclear spin quantum number and MI ∈ {−I, −I + 1, · · · , I − 1, I} are the
magnetic quantum numbers. Following table summaries the empirical rule (from standard
model) for nuclear spin quantum number.
No. of protons
No. of neutrons
Nuclear Spin
Type
Eg. (I)
Even
Even
Zero
Boson
Even
Odd
Half integral
Fermion
13
Odd
Even
Half integral
Fermion
1
Odd
Odd
Integral
Boson
12
C (0), 16 O (0)
C (1/2), 17 O (5/2)
H (1/2), 7 Li (3/2)
2
H (1), 14 N (1)
For I 6= 0, the magnetic dipole moment associated with the nuclear spin is given by,
MN = gI µN I/~.
(4.18)
Here gI is called the nuclear g factor or Landé g factor and the nuclear magneton
µN =
m
e~
=
µB (with proton mass Mp )
2Mp
Mp
(4.19)
is much smaller than the Bohr magneton µB = e~/(2m).
The nuclear spin I interacts with both electronic orbital angular momentum L and electronic spin angular momentum S. The nuclear magnetic-dipole interaction Hamiltonian is
given by (for r 6= 0)
0
HM
D =
µ0 2
1
(S · r)r
gI µB µN 3 G · I, where G = L − S + 3
.
2
4π ~
r
r2
(4.20)
70
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
We consider this as a weak perturbation to the Dirac Hamiltonian H0 , with eigenbasis
{|lsjmj i}. Like we have done in the case of electronic spin-orbit coupling, we introduce the
total angular momentum operator
F = I + J, whose eigenvalues are given by
F ∈ {|I − j|, |I − j| + 1, · · · , I + j − 1, I + j}, and magnetic quantum numbers
MF ∈ {−F, −F + 1, · · · , F − 1, F }.
(4.21)
Note that the total number nF of eigenvalues F is smaller of the two numbers (2j + 1)
and (2I + 1). For example, for hydrogen, I = 1/2, so nF = 2. But for deuterium, I = 1, and
so nF = 1 for j = 1/2 and nF = 3 for j > 1/2.
We can calculate the energy shifts in the simultaneous eigenbasis eigenbasis |lsjIF MF i
using 1st order perturbation theory, i.e.,
1
µ0 2
lsjIF
M
g
µ
µ
G · I lsjIF MF
F
I
B
N
4π ~2
r3
C(j)
=
[F (F + 1) − I(I + 1) − j(j + 1)] , where
2
µ0
1
Z3
C(j) =
4gI µB µN
4π
j(j + 1)(2j + 1) a3 n3
∆EjF =
(4.22)
Thus, each of the Dirac energy level Enj are split into nF levels, depending on the number
of F values. This is known as hyperfine structure. For a given j, the separation follows the
interval rule
∆EjF − ∆Ej(F −1) =
C(j)
[F (F + 1) − F (F − 1)] = C(j)F.
2
(4.23)
The selection rule for the electric dipole transition |nlsjIF MF i → |n0 l0 s0 j 0 I 0 F 0 MF0 i are now
∆l = ±1
∆j = 0, ±1, and
∆F = 0, ±1 (F = 0 = F 0 not allowed).
(4.24)
Certain electric dipole transitions of hydrogen hyperfine levels are shown in Fig. 4.10.
4.6. NUCLEAR EFFECTS
Figure 4.9: Dirac atom with hyperfine splittings (source: Bransden & Joachain).
71
72
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
Figure 4.10: Some ED transitions of hydrogen atom hyperfine levels (source: Bransden &
Joachain).
73
4.6. NUCLEAR EFFECTS
4.6.2
Electric Quadrupole Hyperfine structure
Spin > 1/2 nuclei are associated with nonspherical, specifically, ellipsoidal electric charge
distribution, and therefore possess electric quadrupole moment Q. The corresponding Hamiltonian is
3
0
HEQ
=B2
I · J(2I · J + 1) − I2 J2
,
2I(2I − 1)j(2j − 1)
(4.25)
where B is the quadrupole coupling constant. The first-order energy correction is
B 32 K(K + 1) − 2I(I + 1)j(j + 1)
,
4
I(2I − 1)j(2j − 1)
E
D
0
nlsjIF MF =
∆E = nlsjIF MF HEQ
where K = F (F + 1) − I(I + 1) − j(j + 1).
(4.26)
The quadrupole correction renders the levels to deviate from the interval rule of Eq. 4.23.
4.6.3
Volume effect
While the Coulomb potential has 1/r form, we know that nucleus is not a point charge, but
has a finite size R, and therefore 1/r form can’t hold near the nucleus, i.e., for r ≤ R. In
one particular model, the corrected potential is piece-wise continuous function
VR (r) =

2

 Ze
h 2
4π0 2R
i
r
−3
R2
r≤R

− Ze2
(4.27)
r > R.
4π0 r
This potential is plotted in Fig. 7.1. Note that the singularity at r = 0 of Coulomb potential
(dashed line) is now avoided with a finite potential (orange line)
VR (0) = −
Ze2
.
4π0 R
(4.28)
This corresponds a perturbative correction
2



Ze2
Ze
H0 = +
+ VR (r) = 4π0 (2R)

4π0 r
0
h 2
r
+ 2R
−3
R2
r
i
for r ≤ R,
and
(4.29)
for r > R.
This perturbation introduces correction only to l = 0 states
∆E ≈
e2 2 2 Z 4
R
.
4π0 5 a3 n3
(4.30)
74
CHAPTER 4. HYDROGENIC ATOM: HIGHER ORDER CORRECTIONS
Figure 4.11: The form of potential VR (r) for r ≤ R (orange line) and for r > R (solid blue
line) vs r/R. The dashed line shows the diverging behavior of Coulomb potential. Here
y-scale is arbitrary.
Summary
In this chapter, we have seen how intricate is the theory of the simplest atomic system. The
entire Hamiltonian for a hydrogenic atom is of the form 1
(101 eV ≡ 1015 Hz )
H = HCoulomb
+ Hfinestr










HRel.KE









+ Hhyperfine











(10−4 eV ≡ 1010 Hz )
+HEl.spin−orbit
(10−4 eV ≡ 1010 Hz )
+HDarwin−term
(10−4 eV ≡ 1010 Hz )
+HLamb−shift
(10−5 eV ≡ 109 Hz )
(10−6 eV ≡ 108 Hz )
HNu.spin−El.spin−orbit
+HNu.elec.quad
(10−8 eV ≡ 106 Hz )
+HNu.volume
(10−9 eV ≡ 105 Hz ).
It is clear that there is no simple analytical treatment to solve the Hamiltonian in its entirety
and obtain an exact solution - even for the simplest atom!
1
https://arxiv.org/ftp/arxiv/papers/1612/1612.08042.pdf
Chapter 5
Two electrons atoms
In this chapter we shall study the bound states of two electrons with a nucleus of charge Z.
Examples include Hydrogen anion H− (Z = 1), He (Z = 2), Li+ (Z = 3), etc.
There is a dramatic difference between Hydrogenic atoms and two-electron atoms. In
Hydrogenic atoms, electronic charge played the central role, but not electronic spin. But in
the two-electron case, in addition to electronic charge, the spin also plays the central role
via Pauli exclusion principle.
We had studied some aspects of two electron atoms in QM-II (chapter 5). In the following,
we shall study them in further details. Let us first recall the atomic units:
Physical quantity
Atomic Unit
Angular momentum
~
Electric charge
Electronic charge e−
Mass
Electronic mass me
Length
Bohr radius a0
Permittivity
4π0
Speed
αc
Energy
me (αc)2
Result
~→1
e→1
me → 1
a0 → 1
4π0 → 1
αc → 1
2Ry = 2 21 me (αc)2 → 1
Hamiltonian
Figure 5.1: Position coordinates for electrons in a Helium-like atom/ion.
75
76
CHAPTER 5. TWO ELECTRONS ATOMS
The Hamiltonian is
1 Ze2
~2 2
H = −
∇1 −
2me
4π0 r1
= H1 + H2 + H12
!
1 Ze2
~2 2
+ −
∇2 −
2me
4π0 r2
!
+
1 e2
,
4π0 r12
(5.1)
where, r12 = |r12 | = |r2 − r1 |, H1 , H2 are Hydrogen-atom like single-electron terms, while
the last term H12 is due to the repulsion between the electrons (see Fig. 5.1).
In atomic units, the Hamiltonian appears cleaner
∇2
Z
H = H1 + H2 + H12 = − 1 −
2
r1
!
∇2
Z
+ − 2−
2
r2
!
+
1
,
r12
(5.2)
and the corresponding Schrödinger equation is
Hψ(r1 , r2 ) = Eψ(r1 , r2 ).
(5.3)
The above Hamiltonian is deceptively simple, and we don’t know an exact analytical solution!
But there are powerful approximation methods which can be precise up to fifteen digits or
more.
Here ψ(r1 , r2 ) is not the full wave function, but only the space part of the wave function.
Including the spin part, the full wave function can be written as
Ψ(q1 , q2 ) = ψ(r1 , r2 )χ(1, 2),
(5.4)
where qi form the space-spin collective variables and χ is the spin part of the wave function.
5.1
Exchange symmetry
Indistinguishable particles
Unlike classical particles, which are tractable, at least in principle, the wave nature of quantum particles renders them intractable (recall from QM-II, chapter 1). So, if we have a system
of two identical particles, we can no longer label them separately. In other words, for a pair
of identical particles the state vectors P12 |Ψ(q1 , q2 )i is indistinguishable from |Ψ(q2 , q1 )i,
where P12 is the exchange operator that exchanges particles 1 and 2. If two state vectors are
indistinguishable, they must be equal up to a global phase, i.e.,
P12 |Ψ(q1 , q2 )i = |Ψ(q2 , q1 )i = eiα |Ψ(q1 , q2 )i .
(5.5)
77
5.1. EXCHANGE SYMMETRY
†
−1
2
= P12
= P12 , i.e., P12 is unitary as well as Hermitian. An
Also, P12
= 1, which means P12
operator that is both unitary and Hermitian can have only two eigenvalues, viz., ±1. To see
this explicitly, we apply Pij again,
2
P12
|Ψ(q1 , q2 )i = |Ψ(q1 , q2 )i = e2iα |Ψ(q1 , q2 )i .
(5.6)
We obtain e2iα = 1, i.e., α = 0 or α = π. Thus we have
P12 |Ψ(q1 , q2 )i = + |Ψ(q1 , q2 )i or P12 |Ψ(q1 , q2 )i = − |Ψ(q1 , q2 )i .
(5.7)
Thus the eigenvalues of the exchange operator are ±1, depending on the type of particles.
Pauli Exclusion Principle
According to the Pauli exclusion principle (QM-II notes Sec. 1.4.2), for a set of N indistinguishable fermions, the wave function Ψ(q1 , q2 , · · · , qN ) must be antisymmetric w.r.t.
exchange of any two particles. Therefore, in the case of the two-electron atom,
P12 Ψ(q1 , q2 ) = Ψ(q2 , q1 ) = −Ψ(q1 , q2 ).
(5.8)
This implies that either the space part or the spin part must be antisymmetric, i.e.,
Ψ(q1 , q2 ) = ψ+ (r1 , r2 )χ− (1, 2), or
Ψ(q1 , q2 ) = ψ− (r1 , r2 )χ+ (1, 2), where
P12 ψ± (r1 , r2 ) = ψ± (r2 , r1 ) = ±ψ± (r1 , r2 ), and
P12 χ± (1, 2) = χ± (2, 1) = ±χ± (1, 2).
(5.9)
Spin wave function
Let {α, β} be the spin wave function of a single electron with spin operator Si and its
z-component Sz , i.e.,
3~
S2i α = ~2 si (si + 1)α =
2
α,
4
3~2
S2i β = ~2 si (si + 1)β =
β,
4
~
S2iz α = ~msi α = + α, and
2
~
S2iz β = ~msi β = − β.
2
since si = 1/2.
(5.10)
78
CHAPTER 5. TWO ELECTRONS ATOMS
The resultant spin angular momentum of two electrons is S = S1 + S2 . The basis
{α(1)α(2), α(1)β(2), β(1)α(2), β(1)β(2)}
(5.11)
is known as the product basis. These form eigenbasis of S1z + S2z , but not of S2 . In terms of
the product states. the eigenstates of S2 can be expressed as χS,mS where the total quantum
numbers S ∈ {|s1 − s2 |, · · · , s1 + s2 } = {0, 1} and mS = {−S, −S + 1, · · · , S}. Specifically,
1
χ0,0 (1, 2) = √ (α(1)β(2) − β(1)α(2)) ,
2
χ1,1 (1, 2) = α(1)α(2),
1
χ1,0 (1, 2) = √ (α(1)β(2) + β(1)α(2)) ,
2
χ1,−1 (1, 2) = β(1)β(2),
(5.12)
You may verify that these spin wave functions χS,mS satisfy the eigenvalue equations
S2 χS,mS = ~2 S(S + 1)χS,mS and
Sz χS,mS = ~mS χS,mS .
(5.13)
Note that the singlet state χ0,0 is antisymmetric w.r.t. particle exchange, while the three
triplet states are symmetric w.r.t. particle exchange:
P12 χ0,0 (1, 2) = χ0,0 (2, 1) = −χ0,0 (1, 2),
P12 χ1,1 (1, 2) = χ1,1 (2, 1) = χ1,1 (1, 2),
P12 χ1,0 (1, 2) = χ1,0 (2, 1) = χ1,0 (1, 2), and
P12 χ1,−1 (1, 2) = χ1,−1 (2, 1) = χ1,−1 (1, 2),
(5.14)
where P12 is the exchange operator.
Space-Spin wave function
From the above discussions, we can now set up the antisymmetric space-spin wave function
consistent with the Pauli exclusion principle as follows:
Ψpara (q1 , q2 ) = ψ+ (r1 , r2 ) × χ0,0 (1, 2), or
Ψortho (q1 , q2 ) = ψ− (r1 , r2 ) ×











χ1,1 (1, 2), or
χ1,0 (1, 2), or
χ1,−1 (1, 2),
5.1. EXCHANGE SYMMETRY
79
Thus, there are two species of Helium atom: parahelium with total spin quantum number
S = 0 and orthohelium with total spin quantum number S = 1.
Recall that the electric dipole transitions do not change the spin quantum number. Thus,
the ortho to para transitions are forbidden. Therefore, spectroscopically, one observes two
different sets of lines corresponding to each of these.
Spectroscopic Term Symbol
Let L1 and L2 be the orbital angular momentum operators of individual electrons and L =
L1 + L2 be the total angular momentum operator. The eigenvalues of L are of the form
L(L+1)~2 , with the total orbital angular momentum quantum numbers L. The spectroscopic
term symbol 2S+1 L is used to denote the states.
Hydrogen anion (H− ; Z = 1)
Figure 5.2: The energy levels (experimental) of Hydrogen anion (source: Bransden and
Joachain).
Hydrogen anion has only one bound state as shown in Fig. 5.2. The only bound state
corresponds to the S = 0, i.e., the singlet state. It is a gently held system by only 0.75 eV.
Shining near-infra-red light suffices to decompose it into hydrogen atom and an electron.
Helium atom (Z = 2)
We can consider a Helium atom as a bound state of (i) He++ and two electrons or (ii) He+
and one electron.
(i) Various energy levels of helium atom w.r.t. double ionization are shown in Fig. 5.3.
Note that the double excitation of helium atom is more unstable than the helium ion.
(ii) The discrete energy levels of the singly excited Helium atom (100; nlm) w.r.t. the
first ionization are shown in Fig. 5.4. In this case l1 = 0; l2 = l (say), so the total orbital
angular momentum quantum number L essentially indicates the quantum number of the
excited electron.
80
CHAPTER 5. TWO ELECTRONS ATOMS
Figure 5.3: Full energy levels (experimental) of Helium (singly/doubly excited atom,
singly/doubly ionized) w.r.t. double ionization (source: Bransden and Joachain).
5.1. EXCHANGE SYMMETRY
81
Figure 5.4: Discrete energy levels (experimental) of singly-excited Helium atom w.r.t. the
first ionization level (source: Bransden and Joachain).
82
CHAPTER 5. TWO ELECTRONS ATOMS
5.2
Independent electron-model
Here we ignore the troublesome term H12 , so that each electron does not see the other. The
Hamiltonian is then
∇21
Z
H = H1 + H2 = −
−
2
r1
!
∇22
Z
+ −
−
2
r2
!
(5.15)
In this case, we can immediately apply separation of variables. Let ψni ,li ,mi (ri ) be the
Hydrogenic wave function for the ith electron. We now set up the unperturbed total wave
function
(0)
ψ1,2 (r1 , r2 ) = ψn1 ,l1 ,m1 (r1 )ψn2 ,l2 ,m2 (r2 ), which yield (anti)symmetric wave functions
1
(0)
ψ± (r1 , r2 ) = √ (ψn1 ,l1 ,m1 (r1 )ψn2 ,l2 ,m2 (r2 ) ± ψn2 ,l2 ,m2 (r1 )ψn1 ,l1 ,m1 (r2 )) .
(5.16)
2
The corresponding energy eigenvalue is
"
E
(0)
#
Z2 1
1
= En1 + En2 = −
+ 2 .
2
2 n1 n2
(5.17)
As we have seen earlier, the doubly excited states of Helium are unstable and lead to
ionization. The general singly excited states are of the form,
1
(0)
ψ± (r1 , r2 ) = √ (ψ100 (r1 )ψnlm (r2 ) ± ψnlm (r1 )ψ100 (r2 )) , with energy
2
1
Z2
1+ 2 .
E (0) = E1 + En = −
2
n
(5.18)
For the ground state n = 1,
(0)
ψ11 S (r1 , r2 ) = ψ100 (r1 )ψ100 (r2 )
s
 s

Z 3 −Zr1   Z 3 −Zr2 
=
e
e
π
π
=
Z 3 −Z(r1 +r2 )
e
,
π
(5.19)
which is already symmetric w.r.t. particle exchange. Therefore,
(0)
(0)
(0)
ψ+ = ψ11 S (r1 , r2 ) & ψ− = 0,
(0)
(0)
and thus the total wave function Ψ11 S (q1 , q2 ) = ψ+ χ0,0 (1, 2).
(5.20)
Accordingly, we have only parahelium in the ground state. The ground state energy of the
5.3. FIRST-ORDER PERTURBATION OF GROUND STATE 11 S
83
Helium-like atom becomes twice that of a Hydrogen-like atom with i.e.,
E (0) = −Z 2 = −4 a.u.,
(5.21)
which is off by 38% from the experimental value −2.904 a.u.
Effective screening
One way to fix the deviation of the ground state energy of the independent electron model
from the experimental value is by finding an effective atomic number Zeff so that
2
−2.904 = −Zeff
=⇒ Zeff =
√
2.904 = 1.704,
(5.22)
which is ≈ 0.3 less than Z = 2 assumed earlier. This observation hints that the repulsion
between the two electrons is screening the effective nuclear charge as seen by either electron.
Here the screening constant is ≈ 0.3.
5.3
First-order perturbation of ground state 11S
We shall now introduce the ignored term H12 = 1/r12 as a perturbation to the independent
electron model, and try to refine the ground state 11 S. The correction in energy is
E
(1)
1
(0)
=
ψ0 , now using Eq. 5.19,
r12
1
Z6 Z
= 2 dr1 dr2 e−2Z(r1 +r2 )
π
r12
5
= Z.
8
(0)
ψ0
(5.23)
Thus, 1s orbital of each electron shields the nucleus thereby reducing the effective nuclear
charge as seen by the other electron. The overall reduction in the Coulomb attractive
potential is roughly 5Z/8. Therefore, up to first-order, the ground state energy of Helium
(Z = 2) is
E = E (0) + E (1) = −4 +
10
= −2.75 a.u.,
8
which is only 5% away from the experimental value.
(5.24)
84
CHAPTER 5. TWO ELECTRONS ATOMS
5.4
Variational method for ground state 11S
This method (see QM-II notes, Chapter 5) is based on the variational principle according to
which, given a trial wave function φ (normalized), the ground state energy E0 of a Hamiltonian is upper bounded by the expectation value hφ |H| φi. If φ is a function of a parameter λ,
called variational parameter, one obtains the best estimate for the ground state by minimizing the expectation hφ(λ) |H| φ(λ)i w.r.t. λ. Closer the trial function to the actual ground
state wave function, better would be the estimate for the ground state energy. For estimating
the energy of nth excited state, we need to choose a trial function that is orthogonal to all
the n − 1 lower energy wave functions, and then use the variational method.
We are going to use the ground state wave function of independent electrons as the trial
wave function.
φ(Zeff , r1 , r2 ) = ψ (0) (Zeff , r1 , r2 ) = ψ1s (Zeff , r1 )ψ1s (Zeff , r2 ).
(5.25)
Here, Zeff is the effective nuclear charge, after the screening effect. We now rewrite the
Hamiltonian in Eq. 5.2 in terms of the unperturbed Hamiltonian consisting of effective Zeff
and rest as the perturbation term:
!
!
Z
∇2
Z
1
∇2
+ − 2−
+
,
H = − 1−
2
r1
2
r2
r12
!
! ∇22 Zeff
Zeff − Z Zeff − Z
1
∇21 Zeff
−
+ −
−
+
+
+
= −
,
2
r1
2
r2
r1
r2
r12
= H1,eff + H2,eff + H12,eff
= H0,eff + H12,eff ,
(5.26)
where H0,eff is the independent electron Hamiltonian with nuclear charge Zeff . The eigenfunctions of H1,eff and H2,eff are the hydrogenic wave functions ψ1s (Zeff , r1 ) and ψ1s (Zeff , r2 )
respectively. The expectation value of the trial function is
Eφ = hφ(Zeff ) |H0,eff | φ(Zeff )i + hφ(Zeff ) |H12,eff | φ(Zeff )i
1
Zeff − Z Zeff − Z
2
+
+
φ(Zeff ) using Eq. 5.26
= −Zeff + φ(Zeff )
r1
r2
r12
1
1
2
= −Zeff
+ 2(Zeff − Z)
+
r ψ1s (Zeff )
r12 φ(Zeff )
5Zeff
1
Z
1
2
= −Zeff
+ 2(Zeff − Z)Zeff +
, ∵
=
=⇒
= Zeff a.u.
2
8
r nlm a0 n
r ψ1s (Zeff )
5Zeff
2
= Zeff
− 2ZZeff +
,
(5.27)
8
which is plotted in Fig. 5.5.
5.4. VARIATIONAL METHOD FOR GROUND STATE 11 S
85
Figure 5.5: Eφ (Z = 2, Zeff ) versus Zeff .
We have taken φ(0) (Zeff , r1 , r2 ) as the parametric trial function in terms of the effective
atomic number Zeff , whose energy is Eφ (Z, Zeff ). To estimate the ground state energy, we
minimize the energy w.r.t. Zeff , by setting
0=
5
∂Eφ (Z, Zeff )
= 2Zeff − 2Z +
∂Zeff
8
5
=⇒ Zeff = Z − ,
16
(5.28)
which for Helium atom (Z = 2), becomes 1.688, as can be seen in Fig. 5.5. This corresponds
to the screening constant 5/16 = 0.3125 slightly higher than the value obtained in the
1st order perturbation theory. Substituting this value back in Eq. 5.27, we get the best
variational estimate
5 2
5
5
5
− 2Z Z −
+
Z−
Eφ (Z) = Z −
16
16
8
16
2
(5 − 16Z)
,
= −
256
(5.29)
For the ground state of helium we obtain Eφ (2) = −2.848 a.u. that is within 2% of the
experimental value −2.904 a.u. Table 5.6 compares perturbation, variation, and experimental
values of ground state energies of various two-electronic systems.
For more precise calculations, one can use more sophisticated trial functions. Higher order
corrections as well as the contribution from the spin degree of freedom are also considered.
For example, one estimate (Drake 1996) reads E0 = −2.903 724 377 034 119 598 13 a.u.,
that is accurate up to one part in 1019 !
86
CHAPTER 5. TWO ELECTRONS ATOMS
Figure 5.6: Perturbation, variation, and experimental values of ground state energies of
various two-electronic systems (source: Bransden and Joachain).
5.5
Singly Excited States
As seen in Eq. 5.18, the singly excited states can have both space-symmetric (para) or spacei
h
2
antisymmetric (ortho) forms ψ± both with same zeroth order energy E (0) = − Z2 1 + n12 .
This degeneracy of para and ortho states in the independent electron model is called exchange
degeneracy. There is also degeneracy w.r.t. l and m quantum numbers. Thus, the first order
correction to the singly excited state is of the form
(1)
E± = hψ± |H12 | ψ± i = Jnl ± Knl (say; no m-dependence!).
(1)
(1)
Now, 2Jnl = E+ + E− = hψ+ |H12 | ψ+ i + hψ− |H12 | ψ− i
= hψ100 (r1 )ψnlm (r2 ) |H12 | ψ100 (r1 )ψnlm (r2 )i .
1Z
1
dr1 dr2 |ψ100 (r1 )|2
|ψnlm (r2 )|2 ,
∴ The direct (Coulomb) integral Jnl =
2
r12
1Z
1
∗
∗
& similarly, exchange integral Knl =
(r2 ) ψnlm (r1 )ψ100
(r2 ),
dr1 dr2 ψ100 (r1 )ψnlm
2
r12
(0)
and the total integral Enl,± = Enl + Jnl ± Knl
1
Z2
1 + 2 + Jnl ± Knl .
2
n
Therefore, Enl,+ − Enl,− = 2Knl .
=−
(5.30)
In general, Jnl and Knl are positive integrals.
Note that Jnl represents the Coulomb repulsion between the electrons, which is expected
87
5.5. SINGLY EXCITED STATES
from the form of the Hamiltonian. However, Knl is entirely due to exchange interaction. To
see this, we can write
(0)
Enl,± = Enl + Jnl ± Knl
Z2
1
1
=−
1 + 2 + Jnl − h1 + σ1 · σ2 i Knl ,
2
n
2
(5.31)
since for the Pauli operators σi = ~2 Si ,
h1 + σ1 · σ2 ipara = −3 and
h1 + σ1 · σ2 iortho = 1.
(5.32)
Therefore, the space symmetric (para) states have higher energy Enl,+ than the energy
Enl,− of the space antisymmetric (ortho) states. Thus the exchange degeneracy is lifted by
the electron-electron repulsion. The degeneracy in l quantum number is also lifted, but m
degeneracy is retained. Some lowest singly excited states of helium atom is shown in Fig.
5.7.
Figure 5.7: Some lowest singly excited states of Helium atom (source: Bransden and
Joachain).
88
5.6
CHAPTER 5. TWO ELECTRONS ATOMS
Doubly excited states
As shown in Fig. 5.3, the doubly excited states of two-electron systems are unstable, and
spontaneously undergo autoionization resulting in a radiationless transition to an ion and
an electron that carries most of the extra energy. This is known as Auger effect. It takes
place when an atom is doubly excited either by a radiation or by collision:
hν + He(11 S) → He∗∗ → e− + He+ ,
e− + He(11 S) → He∗∗ → e− + He+ .
or
(5.33)
In general, Auger effect can occur when an inner shell electron is knocked out creating
a hole, which gets filled by an electron from an outer shell. Another electron from an outer
shell takes the extra energy and goes away leaving behind an ion (see Fig. 5.8).
Figure 5.8: Auger effect: Two views of the Auger process. (a) illustrates sequentially the
steps involved in Auger deexcitation. An incident electron (or photon) creates a core hole
in the 1s level. An electron from the 2s level fills in the 1s hole and the transition energy is
imparted to a 2p electron which is emitted. The final atomic state thus has two holes, one
in the 2s orbital and the other in the 2p orbital. (b) illustrates the same process using X-ray
notation, KL1L2,3. (source: Wikipedia) .
5.7. IONIZATION ENERGY
5.7
89
Ionization energy
Ionization energy (or first ionization potential) is the energy required to remove an outermost
electron of an atom and thereby convert it into an ion. The ionization energies for various
atoms are plotted versus atomic number in Fig. 5.9. Note that the alkali metals (Li, Na, K,
etc) have the least ionization energies while the nobel gases (He, Ne, Ar, etc) have the most
ionization energies. Among all the atoms, Helium has the highest ionization energy of 24.6
eV.
Figure 5.9: Ionization energy trends plotted against the atomic number, in units eV. The
ionization energy gradually increases from the alkali metals to the noble gases. The maximum
ionization energy also decreases from the first to the last row in a given column, due to the
increasing distance of the valence electron shell from the nucleus. Predicted values are used
for elements beyond 104 (source: Wikipedia).
5.7.1
Franck-Hertz Experiment
Ionization energy can be measured using one variant of the Franck-Hertz experiment. FranckHertz (Noble prize in 1925) experiment was carried out around 1914, and played an important
role in understanding the electronic structure of atoms.
The setup is shown in Fig. 5.10. It consists of a vacuum tube filled with Mercury vapor
at a low pressure. The vacuum tube has two parts. In the first part, the electrons emitted
from a hot filament K are accelerated from grid S1 up to a grid S2 by a positive potential.
In the second part, the electrons are prevented from reaching the anode A by maintaining
a strong negative potential. The accelerated electrons can ionize atoms by spending their
kinetic energy. The ions so formed reach the anode and contribute to the anode current. For
mercury, the anode current is observed only after 10.4 eV, which is its ionization potential.
90
CHAPTER 5. TWO ELECTRONS ATOMS
Figure 5.10: One variant of the Franck-Hertz experiment to measure ionization potential of
mercury (source: internet).
Chapter 6
Many electrons atoms
The Hamiltonian for a many electron atom, in atomic units, is of the form
N
X
1
1 2 Z
,
+
H=
− ∇i −
2
ri
i=1,j>i rij
i=1
N X
& Schrödinger eqn. HΨ(q1 , q2 , · · · , qN ) = EΨ(q1 , q2 , · · · , qN ),
(6.1)
According to Pauli exclusion principle, the wave function Ψ(q1 , q2 , · · · , qN ) is antisymmetric
w.r.t. particle exchange, i.e.,
Pij Ψ(q1 , q2 , · · · , qN ) = −Ψ(q1 , q2 , · · · , qN ),
(6.2)
where Pij is the exchange operator. In the last chapter, we had considered the two-electron
case of the above Hamiltonian, and found that the repulsive term prevents an exact solution.
We had also studied the approximations involved in solving it. We shall now extend the
approximation methods to solve multi-electron systems.
6.1
Central field approximation
In this approximation, we assume that each electron experiences a central potential that is
an average effect of attraction due to nucleus and repulsion due to all other electrons. Eq.
6.1 can be cast as
H=
N X
N
N
X
X
1
1
Z
− ∇2ri + V (ri ) +
−
+ V (ri )
2
i=1
i=1,j<i rij
i=1 ri
(N X
)
1
=
− ∇2ri + V (ri )
2
i=1

N
 X

N

X
1
Z
+
−
S(ri ) , where S(ri ) = + V (ri )
ri
i=1,j<i rij
i=1
= Hc + H1 ,
(6.3)
91
92
CHAPTER 6. MANY ELECTRONS ATOMS
with Hc is the Hamiltonian corresponding to the central field approximation, H1 is the
perturbation, and S(ri ) is the screening term. The central potential V (r) for an N electron
atom/ion system should have the following behavior:
V (ri ) = −


− Z
Z
+ S(ri ) →  ri
ri
− Z−(N −1)
ri
ri → 0 (S(ri ) = 0 for innermost electron i)
ri → ∞ (S(ri ) = Nr−1
for outermost electron i).
i
(6.4)
Of course, for a neutral atom, Z − (N − 1) = 1, and therefore the effective central potential
for the outermost electron is,
V (r) =
N
X
1
V (ri ) → − , for r → ∞.
r
i=1
(6.5)
Now, the Schrödinger equation with the central field approximation is separable into that
of individual electron, i.e.,
Hc ψc = Ec ψc , where ψc = u1 (r1 )u2 (r2 ) · · · uN (rN ) s.t.
N
X
1
− ∇2r + V (r) unlml (r) = Enl unlml (r) & Ec =
Enl ,
2
i=1
where, unlml (r) = Rnl (r)Ylml (θ, φ).
(6.6)
Here ui (r) is the central field orbital for an individual electron. Note that since the potential
V (r) may deviate from the 1/r form, the accidental symmetry due to the conservation of
Runge-Lenz vector is broken, and therefore the eigenvalues are no longer degenerate w.r.t.
the l quantum number. However, the central potential retains the degeneracy w.r.t the
magnetic quantum number (see Sec. 2.3 of QM-II notes). It turns out that the eigenvalues
Enl depend on n + l value. For the same reason, the radial wave function Rnl depends on the
form of the central potential V (r) and hence is not same as the hydrogenic wave function.
Let the individual spin wave functions be χ1/2,ms . The total individual wave function is
unlml ms (q) = ψnlml (r)χ1/2,ms , and
1
− ∇2r + V (r) unlml ms (q) = Enl unlml ms (q).
2
(6.7)
Thus the individual orbitals unlml ms (q) with same n and l quantum numbers have 2l + 1
distinct ml quantum numbers, each having 2 ms quantum numbers (±1/2). Thus the central
field approximation for each (n, l) allows 2(2l + 1)-fold degeneracy and form what is known
as a subshell. All the orbitals with same n quantum number together form a shell.
According to the Pauli exclusion principle, the total wave function must be antisymmetric.
93
6.2. ELECTRONIC CONFIGURATION
Such a wave function can be constructed using the Slater determinant
1
Ψ(q1 , q2 , · · · , qN ) = √
N!
u2 (q1 ) · · ·
u2 (q2 ) · · ·
uN (q1 )
uN (q2 )
u1 (qN ) u2 (qN ) · · ·
uN (qN )
u1 (q1 )
u1 (q2 )
..
.
.
(6.8)
Note that the determinant changes sign if any two rows (or columns) are interchanged. Also,
the determinant vanishes if any two rows (or columns) are identical.
In general, we may need to form linear combinations of the Slater determinants to obtain
P
common eigenfunctions |γLSML MS i of {Hc , L2 , S2 , Lz , Sz }, where L = N
i=1 Li and S =
PN
i=1 Si .
6.2
Electronic configuration
This order of filling is often termed as aufbau (building) principle or Janet-Madelung rule:
• The electrons are filled into various subshells in the order of increasing energy, i.e.,
increasing n + l value, starting from the ground state.
• For subshells with the same value of n + l, the subshell with lower value of n is filled
first.
• There can be exceptions, or anomalous configurations, due to higher stability of filled
or half filled subshells.
94
CHAPTER 6. MANY ELECTRONS ATOMS
Figure 6.1: Order for building-up or filling up the atomic orbitals (source: Demtroder). The
n + 1 value is shown in the left.
6.2. ELECTRONIC CONFIGURATION
95
Figure 6.2: Electron Configurations of the Elements. The electron configurations of elements
indicated in red are exceptions due to the added stability associated with half-filled and filled
subshells. The electron configurations of the elements indicated in blue are also anomalous,
but the reasons for the observed configurations are more complex. For elements after No, the
electron configurations are tentative. (source: 6.9: Electron Configurations and the Periodic
Table in https://chem.libretexts.org/Courses/University...).
96
CHAPTER 6. MANY ELECTRONS ATOMS
6.3
The Thomas-Fermi model of multielectron atom/ion
Consider N electrons confined in a cube of length L. Let the potential be zero inside the
cube, and infinite outside. The eigenfunctions and energies are of the form
nx πx
ny πy
nz πz
sin
sin
L
L
L
2 2
π 2 ~2 2
π
~ 2
2
2
E=
(n
+
n
+
n
)
=
n,
x
y
z
2
2mL
2mL2
ψnx ,ny ,nz = C sin
(6.9)
where nx , ny , and nz are the positive integers, and n2 = n2x + n2y + n2z . In this 3-dimensional
grid of quantum-number space, two electrons can occupy in each coordinate point (nx , ny , nz ).
Centered about this point (nx , ny , nz ), we can imagine a cube of unit volume, in which there
q
are only two electrons. If the maximum principal quantum number is n = n2x + n2y + n2z ,
it is the radius of a sphere in the coordinate space. Then, the total number of electrons
occupied in states up to n is twice the volume of one octant (because nx , ny , and nz are all
positive integers) of the sphere of radius n, i.e.,
1
1
14
2mEL2
Ns = 2 πn3 = πn3 = π
83
3
3
π 2 ~2
1
= 2
3π
2m
~2
3/2
!3/2
, using Eq. 6.9
L3 E 3/2 .
(6.10)
At T = 0, energy E is up to EF , and therefore the electron density
Ns
1
ρ= 3 = 2
L
3π
2m
~2
3/2
3/2
(6.11)
EF .
So far we assumed zero potential. Now we assume the Fermi electron gas in an attractive central potential V (r). We take semiclassical approximation by assuming that V (r) is
roughly constant over the de Broglie wavelengths of electrons. For bound electrons at T = 0,
Emax = EF + V (r) ≤ 0,
(6.12)
which using Eq. 6.11 gives
ρ(r) =

3/2
3/2

 1 2 2m
[Emax − V (r)]3/2 = 1 2 2m2
[eΦ(r)]3/2 ,
2
Emax − V (r) ≥ 0

0,
Emax − V (r) < 0.
3π
~
3π
~
(6.13)
where Φ(r) is the effective electrostatic potential. Similarly,
φ0 = −Emax /e, φ(r) = −V (r)/e, ∴ Φ(r) = φ(r) − φ0 .
(6.14)
97
6.3. THE THOMAS-FERMI MODEL OF MULTIELECTRON ATOM/ION
Also, Emax < V (r) is the classically forbidden region, and therefore Φ(r) = 0 can be treated
as the electrostatic boundary of the atom. For a neutral atom (N = Z), V (r) and φ(r) falls
as 1/r, and therefore we set φ0 = 0, i.e., Emax = 0. For a cation N < Z, Emax < 0, and
therefore φ0 > 0. Anions N > Z are not explained Thomas-Fermi model.
According to Poisson’s equation
∇2 Φ(r) =
1 d2
e
[rΦ(r)] = ρ(r); now, using Eq. 6.13
2
r dr
0
=

3/2

 e 1 2 2m
[eΦ(r)]3/2 ,
2
Φ(r) ≥ 0

0
Φ(r) < 0
0 3π
~
(6.15)
Now we make a substitution in terms of the dimensionless variable x, i.e., r = xb,
rΦ(r) =
Ze
(3π)2/3
χ(x), with b =
a0 Z −1/3 .
4π0
27/3
(6.16)
With this substitution, Eq. 6.15 yields a 2nd order nonlinear differential equation
2


x−1/2 χ3/2
dχ
=
dx2 
0
χ≥0
(6.17)
χ < 0,
known as the Thomas-Fermi equation. The graphical solutions of the Thomas-Fermi
equation are illustrated in Fig. 6.3.
Figure 6.3: Solutions of Thomas Fermi equation (source: Bransden and Joachain).
98
CHAPTER 6. MANY ELECTRONS ATOMS
For a neutral atom and for intermediate values of r
Ze2
χ
4π0 r
Z
≈ − + 1.794Z 4/3 + · · ·
r
V (r) = −
6.4
(in a.u.)
(6.18)
The Hartree-Fock method or the self-consistent field
approach
Here start with the Hamiltonian
H = H1 + H2 , where
H1 =
H2 =
N
X
Z
1
and
hi , with hi = − ∇2i −
2
ri
i=1
N
X
1
.
i=1,j<i rij
(6.19)
Clearly, H1 alone is separable in terms of individual electronic Hamiltonians and hence is
easily solvable. However, as we have seen earlier, H2 renders the Hamiltonian non-separable
and hence unsolvable. Now we make the mean field approximation or Hartree-Fock approximation that each electron i sees an average potential or Hartree-Fock potential ViHF (ri ) that
depends on the overall position distribution of all other electrons. How to find the mean
field? It is based on the variational method. We start with guess wave function uλ (qi )
for each electron and setup the antisymmetric wave function ΦHF (q1 , q2 , · · · , qN ) using their
Slater determinant.
Normally in QM, given a potential, we determine the wave function. Here we do the reverse. Given the wave function ΦHF (q1 , q2 , · · · , qN ), we construct the Hartree-Fock potential
as
*
ViHF (ri ) =
ΦHF
X 1
j6=i rij
+
ΦHF .
(6.20)
This method of constructing the potential from the wave function is called the self consistent field (SCF) approach. Note that if at all ΦHF (q1 , q2 , · · · , qN ) happens to be an exact
eigenfunction of H2 , then this would have yielded the exact potential energy. We can now
setup the new individual Hamiltonian, called Fock operator
HiHF (ri ) = hi (ri ) + ViHF (ri ).
(6.21)
6.4. THE HARTREE-FOCK METHOD OR THE SELF-CONSISTENT FIELD APPROACH99
Now we define the pseudo-eigenvalue equation, called the Hartree-Fock equation
HiHF uλ (qi ) = Ei uλ (qi ).
(6.22)
It is not a simple eigenvalue equation since the potential ViHF (ri ) depends on all other
eigenfunctions. So it is actually a coupled set of eigenfunctions. Therefore the Hartree-Fock
involves an iterative method. Diagonalizing HiHF gives the new wave function u(qi ). Using
these new individual wave functions, we can setup the new Slater determinant ΦH and with
this trial function, we find the Hartree-Fock energy
E[ΦHF ] = hΦHF |H| ΦHF i =
X
Iλ +
λ
Iλ =
X
1X
Jλ,µ − Kλ,µ ,
2 λ,µ
where
huλ (qi ) |hi | uλ (qi )i ,
i
Jλ,µ =
X
Kλ,µ =
X
1
uλ (qi )uµ (qj ) , the direct integral, and
uλ (qi )uµ (qj )
rij
*
1
uλ (qi )uµ (qj )
uµ (qi )uλ (qj ) , the indirect integral.
rij
i<j
i<j
+
*
+
(6.23)
According to the variational principle, E[Φ] ≥ E0 , the ground state energy of the atom. Now
one employs an iterative procedure to improve the wave function and minimize the energy
as described in Fig. 6.4.
Figure 6.4: Iterations of Hatree method.
100
CHAPTER 6. MANY ELECTRONS ATOMS
6.5
Correction to the central field approximation: Spin
orbit interaction
We have seen that the Hartree-Fock method is based on the variational principle, and therefore it gives an upperbound to the ground state, i.e., EHF ≥ Eexact . The difference
Ecorr = Eexact − EHF
(6.24)
is called the correlation energy and is always negative. This energy is associated with the
electrostatic correction term H1 in Eq. 6.3. Now we also include the spin orbit interaction
HLS . Thus the total Hamiltonian,
(N X
1
H=
− ∇2ri + V (ri )
2
i=1
)

N
 X

N

X
1
Z
+
−
+ V (ri ) + HLS


i=1,j<i rij
i=1 ri
= Hc + H1 + HLS .
(6.25)
As usual, the nature of eigenfunctions depends on the relative strengths of H1 and HLS .
We have seen that HLS varies as (Zα)2 . Therefore, for small atoms and ions (up to Z = 20)
|H1 | |HLS | and therefore we chose the eigenbasis of Hc + H1 and treat HLS as the
perturbation. That leads to LS coupling or Russel-Saunders coupling. For large atoms
(Z > 60), |HLS | |H1 |, so that Hc + HLS eigenbasis is to be used and H1 is treated as
perturbation, which leads to the jj-coupling. For intermediate atoms and ions, one has to
take both H1 and HLS together and it is complicated.
6.5.1
LS coupling or Russel-Saunders coupling
For small atoms (Z . 20), we can work in the eigenbasis of Hc + H1 and treat HLS as
the perturbation. The main Hamiltonian Hc + H1 has no spin-dependence, so it commutes
P
with total spin angular momentum S = i Si . Moreover, it also commutes with L2 , where
P
L = i Li , the total orbital angular momentum. Therefore, we can work in the simultaneous
eigenbasis |γLSML MS i of {Hc + H1 , L2 , Lz , S2 , Sz }. For two electrons with l1 and l2 , the
L quantum number takes values |l1 − l2 |, |l1 − l2 | + 1, · · · , l1 + l2 . Similarly, two electrons
s1 = s2 = 1/2, the S quantum number takes values 0, 1. If there is a third electron with l3
and s3 = 1/2, it adds to L and S of the first two. Since, the total angular momentum is
a conserved quantity, there exists degeneracy w.r.t. ML and MS . Thus, the atomic states
characterized by (L, S) have a degeneracy of (2L + 1)(2S + 1). However, this degeneracy
is broken by the perturbation HLS which does not commute with L and S, but only with
J. The J quantum number can take values |L − S|, |L − S| + 1, · · · , L + S. The number of
possible J values is 2S + 1 if L ≥ S and 2L + 1 if L < S. The resulting levels are denoted
6.5. CORRECTION TO THE CENTRAL FIELD APPROXIMATION: SPIN ORBIT INTERACTION101
by the term symbol 2S+1 LJ . Here 2S + 1 is called the multiplicity (recall singlet and triplet
states). The term 2S+1 LJ represents a level that has a degeneracy of 2J + 1 w.r.t. MJ
quantum number. This degeneracy can be broken by Zeeman effect.
Determining the possible states of a multielectron system
• Possible values of L, S, and J are calculated according to the rules of (i) addition of
angular momenta and (ii) Pauli exclusion principle.
• Closed (completely filled) subshells (levels of same n and l) have L = 0, S = 0, and
therefore only possible term is 1 S0 :
P
Note that ML = i mli . But for closed subshells, mli takes all values from −l to +l.
P
Therefore, ML = 0, which implies L = 0. Similary, MS = i msi = 0, since msi takes
both ±1/2. Thus, when unclosed subshells are present, we can ignore all the closed
shells for adding the angular momenta.
• Electrons in an unclosed shell are called optically active, since they decide the angular
momentum of the atom, and thereby decide the optical properties of the atom.
• Equivalent electrons: Two electrons in the same subshell are termed equivalent electrons. Since they are indistinguishable, the Pauli exclusion principle applies.
• Nonequivalent electrons: Two electrons in different subshells are termed nonequivalent
electrons. Here Pauli exclusion principle does not apply since they can’t have same
quantum numbers.
While one may rigorously apply the perturbation analysis to find the energy eigenvalues,
the empirical rules are given by the Hund’s rules.
Hund’s empirical rules
1. (Bus-seat rule) For a given electron configuration, the term with maximum multiplicity
(2S + 1) has the lowest energy. Eg. ↑ ↑ ↑ has lesser energy than ↑↓ ↑ ↑ .
2. For a given multiplicity, the term with largest L has the lowest energy.
3. (i) For subshell half-filled or lesser, the term 2S+1 L with lowest value of J has the least
energy.
(ii) For subshell more than half-filled, the term 2S+1 L with highest value of J has the
least energy.
102
CHAPTER 6. MANY ELECTRONS ATOMS
Non equivalent electrons: Eg. Configuration np n’p (n’ 6= n) (say, Li∗ )
Here, Pauli exclusion principle is naturally satisfied. l1 = l2 = 1 and s1 = s2 = 1/2. So,
L = 0, 1, 2 and S = 0, 1. Here l1 = 1, l2 = 2, s1 = s2 = 1/2. Thus, L = 1, 2, 3 and S = 0, 1.
So, possible terms are 1 S, 1 P, 1 D, 3 S, 3 P, and 3 D.
Their relative energies are shown in Fig. 6.5.
Figure 6.5: Terms and levels for the configuration np n0 p. Here H2 = HLS (source: Bransden
and Joachain).
6.5. CORRECTION TO THE CENTRAL FIELD APPROXIMATION: SPIN ORBIT INTERACTION103
Landé interval rule
The perturbative correction due to HLS is
E(J) = hγLSJMJ |HLS | γLSJMJ i
= A hγLSJMJ |L · S| γLSJMJ i ,
E
1 D
∵ 2L · S = (L + S)2 − L2 − S2 ,
= A γLSJMJ J2 − L2 − S2 γLSJMJ
2
1
= A~2 [J(J + 1) − L(L + 1) − S(S + 1)] .
(6.26)
2
Thus the energy difference between the adjacent levels is
E(J) − E(J − 1) = A~2 J,
(6.27)
which is known as the Landé interval rule. Empirically, it has been found that
• A > 0 (leads to normal multiplets) for less than half-filled subshell
• A < 0 (leads to inverted multiplets) for more than half-filled subshell,
• A = 0 for exactly half-filled subshell.
Non equivalent electrons: Eg. Configuration np n’d
Here l1 = 1, l2 = 2, s1 = s2 = 1/2. Thus, L = 1, 2, 3 and S = 0, 1. So, possible terms are 1 P,
1
D, 1 F, 3 P, 3 D, and 3 F.
Equivalent electrons: Eg. 1s2 2s2
The first shell is closed. The second subshell is also closed. So, L=0, S=0. Therefore, only
possible term is 1 S.
Equivalent electrons: Eg. Configuration of Carbon: 1s2 2s2 2p2
The first two subshells are closed, and can be ignored. So, we need to consider only two
equivalent electrons 2p2 , with n1 = n2 = 2, l1 = l2 = 1 and s1 = s2 = 1/2. So, L = 0, 1, 2
and S = 0, 1. Here, there are totally δ = 2(2l + 1) levels for ν = 2 equivalent electrons.
However, according to the Pauli exclusion principle,
• All four quantum numbers of the two electrons can’t be identical, which implies
(ml1 , ms1 ) 6= (ml2 , ms2 ).
• Since the equivalent electrons are indistinguishable, (ml , ms ; m0l , m0s ) ≡ (m0l , m0s ; ml , ms ),
so they should be counted as a single term.
104
CHAPTER 6. MANY ELECTRONS ATOMS
The number of distinct states (ml , ms ; m0l , m0s ) with ν electrons is given by
d = δ Cν =
δ!
.
(δ − ν)!ν!
(6.28)
6!
= 15. The possible quantum numbers are shown
For np2 , δ = 6 and ν = 2, so d = 6 C2 = 4!·2!
in the table of Fig. 6.6.
Figure 6.6: Possible quantum numbers of np2 configuration (source: Bransden and Joachain).
Note:
• Rows 1 to 5 correspond to 1 D: ML = 2, MS = 1 is missing. That could have come
from (1, 1/2; 1, 1/2) which is prohibited by the Pauli exclusion principle. So, the term
3
D corresponding to L = 2 and S = 1 is missing.
• Rows 6 to 14 correspond to 3 P . So, 1 P is missing.
• Row 15 corresponds to 1 S. So, 3 S is missing.
Thus, the possible terms are 1 S, 1 D, and 3 P . The various possible levels of the carbon atom
are shown in Fig. 6.7.
6.5. CORRECTION TO THE CENTRAL FIELD APPROXIMATION: SPIN ORBIT INTERACTION105
Figure 6.7: Energy levels of np2 configuration (source: Bransden and Joachain).
6.5.2
jj-Coupling
For large atoms (Z & 60) the spin-orbit term HLS is much larger than the correlation
Hamiltonian H1 (in Eq. 6.25). So, the main Hamiltonian is
Hc + HLS =
N
X
e , where h
e = − 1 ∇2 + V (r ) + ξ(r )L · S .
h
i
i
i
i
i
i
2
i=1
(6.29)
Thus, without the correlation term H1 , the main Hamiltonian can be separated for individual
electrons, each with eigenbasis |γi , li , si , ji , mji i, where si = 1/2, ji = li ± 1/2, and mji =
−j, −j + 1, · · · , j. The corresponding term symbol is written as (j1 , j2 , · · · )J , where J is the
total angular momentum quantum number.
Configuration ns n’p
Nonequivalent electrons. So, Pauli exclusion principle is anyway satisfied.
For ns: l = 0, s = 1/2, so j = 1/2.
For np: l = 1, s = 1/2, so j = 1/2, 3/2.
106
CHAPTER 6. MANY ELECTRONS ATOMS
So, possible terms are:
(1/2, 1/2)0 , (1/2, 1/2)0 , (1/2, 3/2)1 and (1/2, 1/2)2 .
Equivalent electrons
In this case, the Pauli exclusion principle needs to be carefully implemented. Table in Fig.
6.8 shows possible terms for j k configuration.
Figure 6.8: Possible terms for j k configuration (source: Bransden and Joachain).
Connecting the two extreme cases
One can roughly estimate the levels of intermediate atoms by interpolating the extreme
cases of small and large atoms. This is illustrated in Fig. 6.9. Note that Hund’s rules
are essentially for LS coupling, and qualitatively extrapolated to larger atoms. Here Pb
electronic configuration is [ ]6p2 .
6.5. CORRECTION TO THE CENTRAL FIELD APPROXIMATION: SPIN ORBIT INTERACTION107
Figure 6.9: Connecting the extreme cases (source: Bransden and Joachain).
108
CHAPTER 6. MANY ELECTRONS ATOMS
Chapter 7
Atoms in E, M, or EM Fields
7.1
Hydrogenic atom in an Electric field
7.1.1
Linear Stark Effect
The shifting and splitting of spectral lines by static electric field was observed by the German
physicist Johannes Stark in 1913, and hence the phenomenon goes by the name Stark effect.
Stark got Nobel prize in 1919.
Figure 7.1: An experimental setup to study Stark effect (Source:
arXiv:physics/0512111).
Theory:
We make the following assumptions:
• We assume a static electric field which is uniform over space.
109
Stalnaker et al,
110
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
• Without loss of generality, we can assume the electric field to be along ẑ axis, i.e.,
~E = E ẑ same as the quantization axis of hydrogen atom. We again ignore the effect on
the nucleus and write the corresponding Hamiltonian as
~ · ~E = −(−e~z) · ~E = eEz,
H 0 = −D
(7.1)
~ = −e~z is the electric dipole moment of the electron.
where D
• We assume the electric perturbation H 0 to be weak compared to the Coulomb interaction, but strong compared to the fine-structure corrections. We may thus treat H 0 as
Ze2
~2
∇2 − 4π
.
a perturbation to the Bohr Hamiltonian H0 = − 2m
0r
Thus under the above assumptions, the first order Hamiltonian is
H = H0 + H 0 = −
~2 2
Ze2
∇ −
+ eEz.
2m
4π0 r
(7.2)
n = 1 level
For the nondegenerate ground state, the 1st order energy correction is
(1)
∆E100 = eE hψ100 |z| ψ100 i = eE
Z
∗
dr ψ100
(r) z ψ100 (r) = 0,
(7.3)
since ψ100 (r) has a definite parity (even ∵ l = 0) while z has odd parity. In other words, the
s-orbital has no electric dipole moment, and therefore does not interact with the external
electric field to first-order. In fact, we had a similar situation when we discussed the selection
rule for electric dipole-transitions with linearly polarized EM: in 3rd chapter: see Eqs. 3.38,
3.39, 3.40, and Table 3.9 in Sec. 3.5. The selection rules are
∆l = ±1 and ∆m = 0.
(7.4)
n = 2 levels
Now consider the four degenerate states of n = 2: (2, 0, 0), (2, 1, 1), (2, 1, 0), (2, 1, −1). We
had studied this case in QM-II (see Sec. 4.2 and 4.2.1). The eigenbasis within the degenerate subspace is not uniquely defined because of degeneracy. So, we find the good basis
E
(0)
li
formed by the linear combination of unperturbed eigenstates that also diagonalizes the
0
corresponding subspace HD
of the perturbation H 0 . However, from the selection rule Eq.
7.4, we find only states that mix are (2, 0, 0) and (2, 1, 0).
Therefore only non-zero matrix elements happen to be the only non-zero matrix elements
happen to be
h2, 0, 0 |H 0 | 2, 1, 0i = eE h2, 0, 0 |z| 2, 1, 0i = −3eEa0 /Z and its adjoint.
(7.5)
111
7.1. HYDROGENIC ATOM IN AN ELECTRIC FIELD
0
Here a0 is the Bohr radius. Thus in the basis {ψnlm }, HD
is of the form






0
HD = 




|2, 0, 0i

|2, 1, 0i
|2, 1, 1i |2, 1, −1i 
|2, 0, 0i
0
−3eEa0 /Z
|2, 1, 0i −3eEa0 /Z
0
0
0
|2, 1, 1i
|2, 1, −1i
0
0
0
0
0
0
0
0
0
0




.




(7.6)
0
Now, we solve the eigenvectors and corresponding eigenvalues of HD
and obtain the zerothorder good-basis and first-order corrections:
|2, 0, 0i − |2, 1, 0i
√
2
E
(0)
|ψ211 i = l2 = |2, 1, 1i
(0)
|ψ2 i = l1
E
E
= |2, 1, −1i
E
=
(0)
|ψ21−1 i = l3
(0)
|ψ1 i = l4
=
(0)
The above state vectors { li
|2, 0, 0i + |2, 1, 0i
√
2
El1 = 3eEa0 /Z,
with
El2 = 0,
with
El3 = 0,
with
El4 = −3eEa0 /Z.
E
(1)
(1)
(0)
(0)
(0)
(1)
(7.7)
(1)
} form the good basis and E2 + Eli are the correspond-
ing energies (see Fig. 7.2). Note that although l2
E
(1)
with
(0)
E
(0)
and l3
E
are of odd parity, the
E
electrically-mixed states l1 and l4 have no definite parity, which allows them to have
finite expectation value of z, therefore possess a net electric dipole-moment, and hence interact with the electric field. Thus, in the first-order, the degeneracy is only partially lifted
by the perturbative electric field. Effectively, the n = 2 state of Hydrogen is behaving as if
it has an electric dipole of magnitude 3ea0 which can be oriented parallel, or antiparallel, or
perpendicular to the field.
The electric dipole transition selection rule ∆m = 0, ±1 still holds, but the orbital rule
∆l = ±1 is relaxed for mixed states with no definite parity (but it applies to at least one
component of the a mixed state is considered). The Lyman alpha line (see Fig. 7.2) is now
split into a triplet with frequencies ν+ , ν0 , ν− s.t.,
hν0 = E (2) − E (1) , hν± = hν0 ± 3eEa0 /Z,
& splitting h∆ν± = h(ν± − ν0 ) ∼ ±10−10 (E/Z) eV.
(7.8)
To have an effect stronger than fine-structure (10−4 eV), we need |E| ∼ 107 V/m or in
practice ∼ 104 V/mm. Thus the splitting is proportional to E and therefore this scenario is
known as the linear Stark effect.
112
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
Figure 7.2: Left: Energy levels of unperturbed hydrogen atom (in Bohr’s atomic model).
Right: Partial lifting of degeneracy by the first-order electric perturbation, and the Stark
triplet of the Lyman-α line.
Quenching of metastable state
From Fig. 7.2, it is clear that the electric dipole transitions ν± now connect the states ψ1(2)
obtained by mixing 2s and 2p orbitals to ground state ψ000 . If an atom is prepared in the 2s
orbital at t =, it can be expressed in terms of the new eigenstates ψ1(2) as
1
1
Ψ(0) = ψ200 = √ ψ1 + √ ψ2 ,
2
2

1
Ψ(t) = √ ψ1 exp it
2
∴ at time t,

(0)
E2 − ∆E 
~
"
1 (0)
t∆E
= eiE2 t/~ ψ200 cos
2
~
!

1
+ √ ψ2 exp it
2

(0)
E2 + ∆E 
t∆E
+ iψ210 sin
~
~
, with ∆E = 3eEa0 /Z,
!#
,
(7.9)
which oscillates with a period
T =
π~
10−33
∼ −3
∼ 10−11 s.
∆E
10 × 10−19
(7.10)
This period is much smaller than the life time 10−9 s of 2p state. Thus, if an atom was
initially prepared in the 2s metastable orbital, under the electric field, it undergoes a rapid
time-dependent oscillation with 2p orbital, and therefore can de-excite rapidly to 1s ground
state, thereby loosing the metastable property. This is known as quenching of metastable
state.
7.1. HYDROGENIC ATOM IN AN ELECTRIC FIELD
113
n = 3 levels
The Stark splittings n = 3 levels and corresponding allowed transitions are shown in Fig.
7.3.
Figure 7.3: Stark splittings and allowed electric-dipole transitions for n = 3 levels (Source:
Bransden and Joachain).
7.1.2
Quadratic Stark effect
We have seen that up to first order perturbation, there is no Stark effect for the 1s state, which
lacks a permanent electric dipole moment. Now let’s estimate the second order perturbation
114
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
(see Sec. 4.1 of QM-II notes):
(2)
∆E100 =
X |H100,nlm |2
= e2 E 2
(0)
(0)
n6=1 E1 − En
2 2
X
X | h100 |z| nlmi |2
(0)
n6=1
(0)
E1 − En
= −e2 E 2
X | h100 |z| nlmi |2
(0)
n6=1
(0)
En − E1
eE
h100 |z| nlmi hnlm |z| 100i ∵ h100 |z| 100i = 0
E2 − E1 n
!
X
e2 E 2
e2 Z 2
h100|
z
≃−
|nlmi
hnlm|
z
|100i
,
∵
E
=
−
n
e2 Z 2
1
4π0 a0 2n2
− (4π
−1
n
4
0 )2a0
≃−
E
8(4π0 )a0 E 2 D
2
100
∵ {|nlmi} is a complete basis
z
100
3Z 2
D E
D E
D E
8(4π0 )a0 E 2 a20
1 D 2E
a20
2
2
2
≃−
,
∵
x
=
y
=
z
=
r
=
100
100
100
100
3Z 2
Z2
3
Z2
3
1
8 (4π0 )a0
= − ᾱE 2 , where ᾱ ≃
, is called electric dipole polarizability.
2
3 Z4
≃−
(7.11)
Thus, the Stark effect of 1s orbital is due to the induced electric dipole moment. As a result,
it is an order of magnitude weaker than the first order energy corrections for 2p orbital 1 .
7.2
The Stark effect of multi-electron atom
The interaction of an electric field E ẑ with a multielectron atom is given by
H 0 = eE
N
X
zi = −EDz ,
(7.12)
i=1
where
D = −eR, and R =
N
X
ri .
(7.13)
i=1
Here D is the total electric dipole moment operator of the atom. For an atomic orbital with
a definite parity,
(1)
EγJMJ = −E hγJMJ |Dz | γJMJ i = 0,
(7.14)
and therefore the first order level shift vanishes. The second order shift
(2)
∆E = EγJMJ = E 2
| hγ 0 J 0 MJ0 |Dz | γJMJ i |2
= E 2 (A + BMJ2 ).
0J 0
E
−
E
0
γJ
γ
0
0
γJ M
X
(7.15)
J
1
Bethe, H.A. and Salpeter, E.E. (1957) Quantum Mechanics of One and Two Electron Atoms. Springer,
Berlin. https://doi.org/10.1007/978-3-662-12869-5
7.2. THE STARK EFFECT OF MULTI-ELECTRON ATOM
115
Here A and B are functions of γ and J. For hydrogenic atoms, we have seen that the
an electric field mixes states with same n, but ∆l = ±1 and ∆m = 0 (see Eq. 7.4), and
effectively, the degeneracy in l is partially or completely lifted. The J quantum number of
a multielectron atom exhibits (2J + 1)-fold degeneracy w.r.t MJ quantum number, since
there is no preferred Z axis in free space. This symmetry is broken by the external electric
field, and the MJ degeneracy is partially lifted. Note that levels of ±MJ have same shift. In
addition, the levels of opposite parity may get mixed.
7.2.1
Ionization by a Static Electric Field
Now we shall consider an electric field that is strong enough to ionize it by knocking out the
electron. The total potential is
V =−
Ze2
+ eEz.
4π0 r
(7.16)
To visualize this potential, we shall fix x = x0 , y = y0 , but vary z, as plotted in Fig. 7.4.
Note that in the presence of the external electric field, the potential is tilted such that the
potential minimum is no longer at the nucleus, but closer to anode. In such a case, a bound
state may become a scattering or free state, which results in the escape of electron towards
the anode and ionization of atom. Even a bound state can have a finite tunneling probability
of escape and thus becomes a metastable state. The effect is seen in the gradual broadening
spectral lines - called Stark broadening.
116
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
Figure 7.4: Ionization or tunneling of electrons in the presence of a strong electric field.
(Source: Bransden and Joachain).
7.3. HYDROGEN ATOM IN A MAGNETIC FIELD
7.3
117
Hydrogen atom in a Magnetic Field
In 1896, Pieter Zeeman found splitting of atomic spectral lines in the presence of magnetic
field, and he won Nobel prize in 1902. This phenomenon is known as Zeeman effect. Fig 7.5
show the UG lab setup to observe Zeeman effect.
Figure 7.5: Top two: Zeeman effect setup. Bottom: Zeeman splitting. The spectral lines
of mercury vapor lamp at wavelength 546.1 nm, showing anomalous Zeeman effect. (A)
Without magnetic field. (B) With magnetic field, spectral lines split as transverse Zeeman
effect. (C) With magnetic field, split as longitudinal Zeeman effect. The spectral lines were
obtained using a Fabry–Pérot interferometer. (Source: Wikipedia and other sources from
the internet).
118
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
Theory
Assumptions:
~ = Bẑ.
• We again assume uniform magnetic field along ẑ, i.e., B
• The magnetic field interacts with electronic orbital and spin magnetic moments:
µB
~ = µB L z B z ,
ML = − L, ∴ HL0 = −ML · B
~
~
µB
gS µB
0
~
S, ∴ HS = −MS · B =
2Sz Bz ,
MS = −
~
~
(7.17)
where electronic spin gyromagnetic ratio gS = 2. The interaction with the nuclear spin
leads to Nuclear Magnetic Resonance, which is observed at the radio-wave frequencies
and it will be considered later.
• The total Hamiltonian is of the form
H=−
~2 ∇2
Ze2
−
2m
4π0 r
+ ξ(r)L · S
µB
(Lz + 2Sz )Bz
+
~
e2 2 2 2
+
B r sin θ
8m
(H0 Bohr atom)
(HLS spin-orbit coupling)
(HZ paramagnetic term)
(HD diamagnetic term),
(7.18)
where θ is polar angle of ~r. The diamagnetic term is rather small and is ignored in the
following.
7.3.1
|HLS | |HZ | < |H0 |: The Normal Zeeman effect
The simplest case is when the paramagnetic term HZ is smaller than H0 , but much larger than
spin-orbit term HLS . In this case, we can ignore HLS , and treat the HZ as the perturbation
to H0 . Moreover, HZ commutes with H0 , so the normal Bohr-basis {|nlmi} is also the
simultaneous eigenbasis, but extended with spin quantum numbers as {|nlmsms i} with
s = 1/2 and ms = ±1/2. Therefore, the corrected energies are
Enlmsms = En + µB B(ml + 2ms ).
(7.19)
7.3. HYDROGEN ATOM IN A MAGNETIC FIELD
119
The Zeeman energy gaps are
En,l,m0l ,s,m0s − En,l,ml ,s,ms = µB B(∆ml + 2∆ms ),



µB B(∆ml ± 1) if ∆ms = ±1/2

µB B∆ml if ∆ms = 0.
=
(7.20)
Thus, the n-fold degeneracy in l are not lifted, but the 2(2l + 1)-fold degeneracy in ml &
ms is partially lifted. Fig. 7.6 shows the Zeeman splitting of l = 1 level.
Figure 7.6: Zeeman splitting of p orbital (Source: Bransden and Joachain).
Zeeman spectrum
Note that the electric dipole transitions are due to the interaction of the electric component
of EM wave with the electric-dipole moment of the atom. Thus, the spin magnetic quantum
number ms remains unaffected. Therefore, the selection rules are
∆l = ±1,
∆ml = 0, ±1,
and ∆ms = 0.
(7.21)
Accordingly, the optical transitions between the two subspaces of Zeeman levels corresponding to ms = +1/2 and ms = −1/2 are not allowed. Therefore there are two sets of transitions,
from the ms = ±1/2 subspaces, and their frequencies overlap. For example, the transitions
from n’d to np for ms = +1/2 subspace is shown in Fig. 7.7.
The central three transitions with ∆ml = 0 are linearly polarized π-transitions and have
the same frequency ν0 = (En0 − En )/h. The right three transitions with ∆ml = +1 and
the left three transitions with ∆ml = −1 form the circularly polarized σ-transitions with
120
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
Figure 7.7: Zeeman spectral lines between d and p orbitals (Source: Bransden and Joachain).
frequencies ν0 ± νL , where
µL =
µB B
h
(7.22)
is known as the Larmor frequency. These three spectral lines are together known as Lorentz
triplet.
One can see the Lorentz triplet from arbitrary directions w.r.t. the magnetic field. But
when observed along the longitudinal direction, the π-transition will be missing. This can
be reasoned as follows. Earlier in Sec. 3.3 we have seen that the electric-dipole spontaneous
emissions are proportional to |rba |2 . Also, from Sec. 3.5 we found for π-transitions, the
electric component of EM ε̂ is along ẑ axis. So, when viewed along ẑ, for the π-transition,
7.3. HYDROGEN ATOM IN A MAGNETIC FIELD
121
the wave vector k̂ is in the xy plane, so it will be missed by the observer focusing along the
ẑ direction (see Fig. 7.8).
Figure 7.8: Longitudinal and transverse observation of Lorentz triplet (Source: Bransden
and Joachain).
7.3.2
|HLS | . |HZ | < |H0 |: The Paschen-Back effect
Now HLS acts a perturbation to H0 + HZ . The resulting correction is (for m0s = ms )
En0 l0 m0l ms − En0 l0 m0l ms = En0 − En + µB B(m0l − ml ) + (λn0 l0 m0l − λnl ml ) ms ,
where λnl =
−α2 Z 2
1
En .
1
n
l l + (l + 1)
(7.23)
2
Thus, unlike the strong field case discussed earlier, now the degeneracy in l is also removed.
This is known as Paschen-Back effect.
122
7.3.3
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
|HZ | |HLS | < |H0 |: The anomalous Zeeman effect
Now, HZ is the perturbation to H0 + HLS , whose eigenbasis is {n, l, s, j, mj }. The first-order
correction is
∆E = hn, l, s, j, mj |HZ | n, l, s, j, mj i
µB B
=
hn, l, s, j, mj |Lz + 2Sz | n, l, s, j, mj i
~
µB B
=
hn, l, s, j, mj |Jz + Sz | n, l, s, j, mj i
~
µB B
= µB Bmj +
hn, l, s, j, mj |Sz | n, l, s, j, mj i
~
µB B
1
= µB Bmj +
hn, l, s, j, mj |(S · J)Jz | n, l, s, j, mj i
~ j(j + 1)~2
(follows from certain commutation relations)
E
µB B
mj ~ 1 D
2
2
2
= µB Bmj +
n,
l,
s,
j,
m
J
+
S
−
L
n,
l,
s,
j,
m
j
j
~ j(j + 1)~2 2
µB B
mj ~ j(j + 1) + s(s + 1) − l(l + 1) 2
= µB Bmj +
~
~ j(j + 1)~2
2
j(j + 1) + s(s + 1) − l(l + 1)
is the Landé g-factor.
= gµB Bmj , where g = 1 +
2j(j + 1)
(7.24)
For the single electron case, s = 1/2, and therefore
∆E =


 2l+2 µB Bmj ,
j = l + 1/2

 2l µB Bmj ,
j = l − 1/2
2l+1
2l+1
.
(7.25)
The total energy is therefore
Enlsjmj = En + En,j + ∆Ej,l,mj .
(7.26)
Thus the splitting is dependent on l & mj , and not simply on ml as was in the strongfield case. Since the concept of spin was not yet introduced in the early 20th century, the
splittings were not well understood at that time, and hence the name “anomalous”. Examples
for splitting of energy levels and for the corresponding transitions are shown in Fig. 7.9.
Although we have studied different regimes of Zeeman effect separately, there is a smooth
variation in energy levels w.r.t. the strength of the magnetic field as shown in Fig. 7.10.
7.4. THE ZEEMAN EFFECT OF MULTI-ELECTRON ATOMS
7.3.4
123
HZ > HD H0 : Ultra-strong field
In ultra-strong fields such as those near neutron stars (millions of Tesla), the Coulomb
interaction can be neglected and Hamiltonian can be written as
1
~2 ∇2
+ HZ + mωL2 (x2 + y 2 ), where last term is the diamagnetic term
2m
2
!
!
~2 ∂ 2
~2 ∂ 2
1
1
2 2
2 2
= −
(7.27)
+ mωL x + −
+ mωL y + HZ ,
2m ∂x2 2
2m ∂y 2 2
H=
which is the 2D harmonic oscillator plus the Zeeman term. In this case, the eigenvalues
happen to be
~2 k 2
E(k, ms , r) =
+ ~ωL (2r + 2ms + 1),
2m
r = 0, 1, 2, · · · ; ms = ±1/2.
(7.28)
Thus the electrons are no longer bound by the nucleus and can have any energy defined by
k. They also undergo precession about the magnetic field with Larmor frequency ωL . For a
given value of k and ms , they have discrete Landau levels labeled by the quantum number
r.
7.4
The Zeeman effect of multi-electron atoms
The interaction of a multielectron atom with magnetic field Bẑ is given by (see Eq. 7.18)
N
N
X
X
µB
Siz .
Liz & Sz =
(Lz + 2Sz )B, where Lz =
H =
~
i=1
i=1
0
(7.29)
• Strong-field case or Normal Zeeman effect:
Here we ignore L·S interaction and consider the simultaneous eigenbasis of L2 , S2 , Lz , Sz .
∆E = µB B(ML + 2MS ).
(7.30)
• Intermediate case or Paschen-Back effect:
Here L · S is added to the perturbation.
∆E = µB B(ML + 2MS ) + AML MS ,
(7.31)
where A is a constant.
• Weak field case or anomalous case:
Here L·S is part of the main Hamiltonian and Zeeman Hamiltonian is the perturbation.
124
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
So, we should work in the simultaneous eigenbasis of J 2 , Jz , L2 , S2 .
µB
B hJLSMJ |L + 2Sz | JLSMJ i
~
= gµB BMJ ,
∆E =
where Landé g-factor g = 1 +
J(J + 1) + S(S + 1) − L(L + 1)
.
2J(J + 1)
(7.32)
7.5
Interaction of many-electron atoms with EM fields
7.5.1
Absorption & Emission
We shall again make the following assumptions as in the Hydrogenic case.
• Semi-classical approximation: We consider a classical EM field interacting with a quantum atom.
• Long-wave length approximation: We consider the wavelength of the radiation much
longer than atomic sizes.
• Heavy nucleus approximation.
• Low density approximation: radiation can be treated as EM waves in vacuum.
• Weak-field approximation: (compared to Coulomb interaction) we can use perturbation
theory.
Using the same steps as in Chapter 3, we obtain the transition probability for absorption
e2
4π0
4π 2
Wb←a = 2
mc
Mba =
N D
X
!
I(ωba )
|Mba |2 where,
2
ωba
E
Ψb eik·ri ε̂ · ∇ri Ψa , since terms are same for all electrons,
i=1
D
E
= N Ψb eik·r1 ε̂ · ∇r1 Ψa .
(7.33)
Since |Mab |2 = |Mba |2 , the principle of detailed balancing holds again, i.e., the rates of
stimulated emission is same as that of absorption
Wa←b = Wb←a .
(7.34)
The rate of spontaneous emission also has the same expression as in 3.26, which for a solid
7.5. INTERACTION OF MANY-ELECTRON ATOMS WITH EM FIELDS
125
angle dΩ in the direction (θ, φ) is
QED,spont
Wa←b
(θ, φ)dΩ =
7.5.2
~
2πm2 c3
!
e2
ωba |Mba (ωba )|2 dΩ.
4π0
(7.35)
Dipole approximation
Under the dipole approximation in the long wavelength limit, ek·r ≈ 1, and therefore from
7.33,
D
= −N hΨb |ε̂ · ∇r1 | Ψa i , now following Eq. 3.28,
Mba
mωba
=
ε̂ · Dba , where
~e
D = −eR = −e
N
X
ri is atom’s electric dipole operator,
(7.36)
i=1
Therefore, in terms of the atom’s electric dipole operator, the transition rates have the same
form as that of Hydrogen atom:
D
D
absorption/stim. emission by upolarized EM: Wb←a
= Wa←b
1
4π 2
I(ωba ) |Dba |2 ,
2
3c~ 4π0
4
1
D,spont
3
=
ωba
spontaneous emission: Wa←b
|Dba |2
3
3~c 4π0
4α 3
= 2 ωba
|rba |2 ,
(7.37)
3c
=
where α is the fine-structure constant.
7.5.3
Selection rules for Electric Dipole transitions
The matrix element
Dba = hψb |D| ψa i = hγ 0 JMJ0 |D| γJMJ i
(7.38)
of the electric dipole operator vanishes, unless following selection rules are obeyed:
(a) ∆MJ = 0, ±1
(b) ∆J = 0, ±1 (J = 0 ↔ J 0 = 0 forbidden)
(c) Laporte’s rule: The atomic states a and b must have opposite parity (w.r.t. inversion about origin: ri → −ri ). The even/odd parity states are are also termed as
gerade(g)/ungerade(u) (German). Therefore, g↔g and u↔u transitions are not allowed.
126
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
In addition, small atoms following LS coupling follow:
(d) ∆ML = 0, ±1
(e) ∆L = 0, ±1 (L = 0 ↔ L0 = 0 forbidden)
(f) ∆S = 0
In most cases, only one electron, say ith electron, makes the transition. In that case,
∆mi = 0, ±1 and ∆li = ±1.
The selection rules for electric-dipole (E1) transitions, magnetic dipole (M1) transitions,
as well as the electric quadrupole (E2) transitions are shown in Fig. 7.11.
7.5.4
Spectra of Alkali Elements
Alkali metals have one valance electron. Ignoring the closed shells, the ground state corresponds to S = 1/2, L = 0, J = 1/2, and therefore has the term (n0 s)2 S1/2 . For large
r, the inner electrons effectively shield the nucleus and therefore the valance electron sees
effectively one protonic charge. Thus, the highly excited atomic levels are roughly hydrogenic and show approximate degeneracy w.r.t. l and m quantum numbers. For small r, the
screening is not very effective and the attraction is stronger than a single protonic charge.
Thus, lower atomic states exhibit dependence on both l and m quantum numbers, and can
be expressed as
Enl = −
1 1
, with
2 (n∗ )2
the effective quantum number n∗ = n − α(l) − β(l)/n2 ,
(7.39)
where α(l) and β(l) are constants.
As an example, we shall consider the spectrum of Sodium, with electronic configuration
(1s)2 (2s)2 (2p)6 (3s)1 (see Fig. 7.12). The famous sodium doublet 589.0 nm and 589.6 nm
are due to fine-structure (spin-orbit coupling). Here, the 3p (l = 1) level is split into
j = l ± 1/2 = {1/2, 3/2} with terms 2 P1/2 and 2 P3/2 , and the corresponding energy gap is
(similar to Eq. 6.27)
1
∆E = ~2 hξ(r)i [j(j + 1) − l(l + 1) − 3/4] .
2
(7.40)
7.5. INTERACTION OF MANY-ELECTRON ATOMS WITH EM FIELDS
127
Figure 7.9: Top: Anomalous Zeeman splitting of np levels. Bottom: Anomalous Zeeman
transitions from n = 2 to n = 1 levels. (Source: Bransden and Joachain).
128
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
Figure 7.10: (Source: Bransden and Joachain).
Figure 7.11: Selection rules (source: Tennyson, J. 2005).
7.5. INTERACTION OF MANY-ELECTRON ATOMS WITH EM FIELDS
129
Figure 7.12: The energy level diagram and some important transitions of Sodium (source:
Bransden and Joachain). Levels under EH are hydrogen levels shown for reference.
130
CHAPTER 7. ATOMS IN E, M, OR EM FIELDS
Chapter 8
Molecular Structure
8.1
A diatomic molecule
A molecule is a bound state of two or more nuclei and one or more electrons. Let us consider
the center of mass frame coordinate system of a molecule with two nuclei of masses MA and
MB as well as N electrons as illustrated in Fig. 8.1. Let
Figure 8.1: Molecular coordinates.
µ=
MA MB
MA + MB
131
(8.1)
132
CHAPTER 8. MOLECULAR STRUCTURE
be the reduced mass of nuclei and R = RA −RB be the internuclear vector. The Hamiltonian
is
H = TN + Te + V where
"
!
#
~2 1 ∂
~2
N2
2 ∂
R
−
TN = − ∇2R = −
,
2µ
2µ R2 ∂R
∂R
~2 R2
where N is the nuclear orbital angular momentum operator,
V = VN N + VN e + Vee ,
N
N
∂
~2 X
1 ∂
~2 X
ri2
Te = −
∇2ri = −
2
2m i=1
2m i=1 ri ∂ri
∂ri
"
!
#
L2
− 2i2 ,
~ ri
1 ZA ZB e2
4π0
R
N
N
1 X ZA e2
1 X
ZB e2
VN e = −
−
4π0 i=1 |ri − RA | 4π0 i=1 |ri − RB |
VN N =
Vee =
N
1 X
e2
.
4π0 i,j<i |ri − rj |
(8.2)
The Schrödinger equation for the molecule is
HΨ(QA , QB , q1 , q2 , · · · , qN ) = EΨ(QA , QB , q1 , q2 , · · · , qN ),
(8.3)
where Ψ(QA , QB , q1 , q2 , · · · , qN ) is the joint eigenfunction for nuclei and electrons together.
This equation does not allow a simple analytical solution. This is the motivation for the
approximation discussed below.
8.2
Born–Oppenheimer adiabatic approximation (1927):
Clamped nuclear model
Born-Oppenheimer approximation (BOA) or adiabatic approximation of a molecular analysis
hinges on the order of magnitude differences in the electronic, vibration, rotational, and spininteraction energies. The same is summarized in the following table for the simple case of
two nuclei of mass M held by a chemical bond.
8.2. BORN–OPPENHEIMER ADIABATIC APPROXIMATION (1927): CLAMPED NUCLEAR MODEL1
Type of energy
Coulomb
(Ee )
Energy Estimate
attraction
EM-wave
range
~2
∼ Ry =
2me a20
Vibrational (EV )
UV/Visible
Infrared
s
s
spring-constant
binding energy 1
·
∼~
∼~
nuclear mass (M)
bond-length2 M
v
u
u
∼t
Rotational (ER )
Spin-Orbit or SpinSpin (ES )
~4 me
me
∼ Ry
∼ 10−2 Ry
4
2
m e a0 M
M
r
L2
~2
~2
me
∼
∼
∼
·
∼ 10−4 Ry
2
2
2I
2M a0
2me a0 M
µ0 µ2B
∼
·
∼ 10−5 Ry
4π a30
Microwave
Microwave/
Radiowave
134
CHAPTER 8. MOLECULAR STRUCTURE
Writing the energies as hν, we can see that the frequencies associated with the electronic
Coulomb energy is 102 times that associated with vibrational motions, which is 102 times
that associated with rotational motions, which is 101 to 102 times than that with spin contributions. Therefore, we may assume that while nuclei are changing their positions slowly,
due to vibrations/rotations, rapid-paced electrons can adiabatically (without undergoing
any transitions) readjust themselves to continuously remain in the eigenstates of the instantaneous Hamiltonian. Note that this approximation owes itself to the lucky separation of
electron-nuclear mass-scales by three orders of magnitudes or more, i.e., me /M ∼ 10−3 . Extending this approximation, is the clamped nuclei model, wherein we treat nuclei stationary
at some internuclear distance R and solve the electronic eigenstates and eigenvalues. The
electronic Schrödinger equation is
He (R)Φq (R; r1 , r2 , · · · , rN ) = Eq (R)Φq (R; r1 , r2 , · · · , rN ), where He (R) = Te + V (R).
(8.4)
Using {Φq } as the basis functions, we now express the molecular eigenfunction as
Ψ(R; r1 , r2 , · · · , rN ) =
X
Fq (R)Φq (R; r1 , r2 , · · · , rN ),
(8.5)
q
where Fq (R) represents nuclear motion corresponding to vibration/rotation with quantum
number q. Using this in the full Schrödingereqn (TN + He − E)Ψ = 0, we obtain
X
hΦq0 |TN + He − E| Φq i Fq = 0.
(8.6)
q
Since electronic motions are faster compared to nuclear motions, the associated wave functions vary slowly w.r.t. molecular coordinates R, Θ, and Φ. Therefore, BOA amounts to
neglecting the off-diagonal elements in the above, by saying that for each Φq , there exists a
corresponding Fq so that we can setup a nuclear wave equation
"
!
#
~2 1 ∂
hΦq |N2 | Φq i
2 ∂
R
+
+ Eq (R) Fq (R) = EFq (R),
−
2µ R2 ∂R
∂R
2µR2
(8.7)
where the electronic energy eigenvalue Eq adds to the potential. Thus, the total wave function
is approximated to the form
Ψq = Fq (R)Φq (R; r1 , r2 , · · · , rN ).
(8.8)
It is basically saying that the electronic states are not very much mixed if electrons can
adiabatically readjust to changing nuclear positions. Compare this with the quantum particle
in a box problem. If the box suddenly widens, the states get mixed, and particle originally
135
8.3. SYMMETRY IN A DIATOMIC MOLECULE
in the ground state gets excited. But slowly changing the box width avoids excitation and
the particle remains in the ground state.
Thus, we first solve the electronic wave equation Eq. 8.4 at fixed values of R and find the
energies Eq (R). Now, plotting an eigenvalue as a function of R gives us the potential energy
curve, whose minimum gives us the equilibrium bond-length. If there exists no minimum, it
suggests that a bound state is not possible.
8.3
Symmetry in a diatomic molecule
Rotation
In the case of a central potential, all components of angular momentum Lx , Ly , Lz , as well
as L2 commute with the Hamiltonian. For a diatomic molecule, the potential is not central,
but axial - about the axis of the molecule, which we choose to be ẑ. The potential is invariant
under rotation about the ẑ axis and therefore Lz commutes with the Hamiltonian, but Lx ,
Ly , and L2 do not. Thus, the electronic wave function Φs is a simultaneous eigenfunction of
He and Lz , i.e.,
Lz Φs = ML ~Φs ,
= ±Λ~Φs ,
ML = 0, ±1, ±2, · · ·
where Λ = |ML | =


0(σ), 1(π), 2(δ), · · ·
for individual electron

0(Σ), 1(Π), 2(∆), · · ·
for molecule.
Thus, all eigenfunctions with Λ > 0 are doubly degenerate.
Reflection
He is invariant under reflection about any plane containing the axis of diatomic molecule.
Eg. reflection about XZ plane is defined by
Ay : yi → −yi .
(8.9)
So, the reflection operator Ay commutes with He , i.e., [Ay , He ] = 0. Also, we have Ay Lz =
−Lz Ay . Now consider
Lz (Ay Φs ) = −Ay (Lz Φs ) = −Ay (±Λ~Φs ) = ∓Λ~(Ay Φs ).
(8.10)
Thus Ay Φs is an eigenfunction of Lz , but has an opposite sign compared to Φs . But, Ay Σ
and Σ both have same eigenvalue of Lz , namely, Λ = 0. Since A2y = 1, eigenvalues of Ay are
±1, and therefore Ay Σ± = ±Σ. Thus the Σ± states are simultaneous eigenstates of He , Lz ,
and Ay .
136
CHAPTER 8. MOLECULAR STRUCTURE
Figure 8.2: Symmetries in a diatomic molecule (Source: Demtroder).
Inversion symmetry or Parity symmetry in homonuclear diatomic molecules
In homonuclear molecules such as H2 , N2 , O2 , etc., the Hamiltonian is invariant under the
inversion P : ri → −ri about the molecular center. Note that the parity symmetry commutes
with Lz . So, we can think of simultaneous eigenbasis of He , Lz , and P, the parity operator.
Again P 2 = 1, so its eigenvalues are ±1. The eigenstates with +1 eigenvalue, i.e., even parity
are known as gerade states (Σg , Πg , ∆g , · · · ), and those with −1 eigenvalue, i.e., odd parity
are known as ungerade states (Σu , Πu , ∆u , · · · ). Thus the homonuclear diatomic states have
±
four degenerate Σ states: Σ±
g , Σu .
Molecular Term
The molecular term is of the form 2S+1 Λ.
8.4
The Hydrogen molecule ion
It turns out that one electron can bond two protons! Thus the simplest molecule, Hydrogen
molecule ion or H+
2 , is a bound state of two protons with one electron (Fig. 8.3) (apparently,
one electron is insufficient, but at least two electrons are needed, to hold three protons!).
Figure 8.3: The hydrogen molecule ion H+
2.
137
8.4. THE HYDROGEN MOLECULE ION
As described above, we shall use the Born-Oppenheimer approximation, and clamp the
protons with an inter-proton distance R, so that the Hamiltonian is
He = −
∇2
1
1
1
−
−
+ .
2
rA rB R
(8.11)
Here rA = r − R/2 and rB = r + R/2 are the electronic position vectors from the protons A
and B respectively. Since HN N = 1/R simply shifts the energy, we shall first consider only
the first three terms. Now we are going to setup a trial wave function and use variational
method to estimate the ground state energy.
Since we know the ground state eigenfunction of Hydrogen atom ψ1s (r) = √1π e−r , we may
+
think of H2+ as H + e+
A or H + eB , i.e., an unperturbed Hydrogen atom and the remaining
bare proton. Respecting the symmetry of the molecule, we shall consider two orthogonal the
trial functions,
i
Ag (R) h −rA
Φg (R; r) = Ag (R) [ψ1s (rA ) + ψ1s (rB )] = √
e
+ e−rB ,
π
which is symmetric w.r.t exchange & even w.r.t inversion, i.e., gerade, and
Φu (R; r) = Au (R) [ψ1s (rA ) − ψ1s (rB )] =
i
Au (R) h −rA
√
e
− e−rB ,
π
which is antisymmetric w.r.t exchange & odd w.r.t inversion, i.e., ungerade.
(8.12)
The above trial function is called LCAO, for Linear Combination of Atomic Orbitals. The
normalization constant is obtained by setting,
Z ∞
−∞
2
dτ |Φg (R; r)| =
and we obtain, Ag (R) = q
1
2(1 + S)
,
Z ∞
−∞
dτ |Φu (R; r)|2 = 1,
1
Au (R) = q
,
2(1 − S)
"
−R
where the overlap integral S = hψ1s (rA )|ψ1s (rB )i = e
R2
1+R+
3
#
a.u..
(8.13)
Note that as R → ∞, the overlap integral S → 0, as expected for an isolated hydrogen and
an isolated proton.
Using the variational method, an upper estimate for the ground state is found by evaluating the expectation value of He with the trial wave functions. Before we do that, let us
138
CHAPTER 8. MOLECULAR STRUCTURE
apply He on φg/u (R) (for now we shall ignore the proton-proton repulsion)
"
He Φg (R; r) =
=
=
=
#
∇2
1
1
−
−
−
Ag (R) [ψ1s (rA ) + ψ1s (rB )] , rearraging six terms on left,
2
rA rB
!
!
#
"
∇2
1
∇2
1
1
1
−
ψ1s (rA ) + −
−
ψ1s (rB ) − ψ1s (rB ) − ψ1s (rA )
Ag (R) −
2
rA
2
rB
rA
rB
1
1
1
1
Ag (R) − ψ1s (rA ) − ψ1s (rB ) − ψ1s (rB ) − ψ1s (rA ) ,
2
2
rA
rB
∵ first two terms are simply of H-atoms & Ry ≡ 1/2 in atomic units.
1
1
1
ψ1s (rB ) + ψ1s (rA ) .
(8.14)
− Φg (R; r) − Ag (R)
2
rA
rB
Similarly,
He Φu (R; r) = −Au (R) −
1
1
ψ1s (rB ) + ψ1s (rA ) .
rA
rB
(8.15)
Now let us evaluate the expectation values
Ee,Φg = hΦg (R; r) |He | Φg (R; r)i
Z ∞
1
1
1Z ∞
2
dτ Ag (R) (ψ1s (rA ) + ψ1s (rB ))
dτ Φg (R)Φg (R) −
ψ1s (rB ) + ψ1s (rA )
= −
2 −∞
rA
rB
−∞
here first integral is unity because φg (R) is normalized
also, by symmetry of the molecule 2nd integral collapses into two terms
Z ∞
Z ∞
1
1
1
= − − 2A2g (R)
dτ ψ1s (rB ) ψ1s (rB ) − 2A2g (R)
dτ ψ1s (rA ) ψ1s (rB )
2
rA
rA
−∞
−∞
here 2nd and 3rd terms are called direct (D) and exchange (X) terms
1
1 D+X
= − − 2A2g (R)(D + X) = − −
.
2
2
1+S
(8.16)
Similarly,
1 D−X
.
Ee,Φu = hΦg (R; r) |He | Φg (R; r)i = − −
2
1−S
(8.17)
The evaluation of direct and exchange integrals result in
1
1 −2R
D = − 1+
e
and X = (1 + R)e−R .
R
R
(8.18)
Now, including the proton-proton repulsion the total energy becomes,
1 D+X
1
EΦg = hΦg (R; r) |He | Φg (R; r)i = − −
+
and,
2
1+S
R
1
1 D−X
EΦu = hΦu (R; r) |He | Φu (R; r)i = − −
+ .
2
1−S
R
(8.19)
8.4. THE HYDROGEN MOLECULE ION
139
Figure 8.4: The variational analysis of hydrogen molecule ion showing estimates Eφg and
Eφu plotted vs bond-length R.
The above variational estimates are plotted versus bond-length in Fig. 8.4. The function
EΦg (R) shows a minimum at bond-length R = 2.4 a.u indicating the existence of a bound
state. Accordingly, Φg is called the bonding wave function. However, the function EΦu (R)
has no minimum at a finite bond-length, and therefore has no bound state, and is called
anti-bonding wave function.
We notice that both bonding and anti-bonding wave functions diverge at R = 0, and
converge asymptotically to 0.5 a.u., the same value as that of the ground state of Hydrogen
for R → ∞. This is expected from LCAO, since as we stretch the bond, it separates into a
H-atom and a proton.
The variational equilibrium bond-length R0 = 2.4 a.u. differs from the experimental
value, which is Rexp = 2.00 a.u.
The difference between the energies of bound state and that of dissociated state is called
the binding energy or the dissociation energy, which in this case is H + e+ . Notice that since
the variational estimate gives an upper bound for the ground state, we get a lower bound
for the binding energy or dissociation energy. The energy minimum is −0.565 a.u., which is
0.065 a.u. below the ground state energy (−0.5 a.u.) of Hydrogen. So, the binding energy
of H2+ is 0.065 a.u., or about 1.8 eV. Experimentally, the binding energy of H2+ is 0.104
a.u. or about 2.8 eV. While improved trial functions can lead to better agreement with the
experimental values, the simplest trial function that we have used captures all the essential
features of the molecular bond.
140
CHAPTER 8. MOLECULAR STRUCTURE
Now to better understand the nature of the two wave functions, let us fix the bondlength to the minimum energy configuration, i.e., R = R0 = 2.4 a.u., and analyze the wave
functions along the line joining the two protons. In this case, rA + rB = R0 , and therefore,
rB = R0 − rA . Substituting this in the wave functions of Eq. 8.12 we get,
i
Ag (R0 ) h −|rA |
√
e
+ e−|R−rA | and
π
i
Au (R0 ) h −|rA |
φu (R0 , rA ) = Au (R0 ) [ψ1s (R0 , rA ) − ψ1s (R − rA )] = √
e
− e−|R−rA | .(8.20)
π
φg (R0 , rA ) = Ag (R0 ) [ψ1s (rA ) + ψ1s (R − rA )] =
Figure 8.5: Solid lines are the axial profiles of optimized trial wave functions φg (R0 , rA ) and
φu (R0 , rA ) plotted versus the axial distance rA from proton A. The bonding wave function
φg (R0 , rA ) shows significant electronic density in between the two protons, thus contributing
to the bound state. On the other hand, the anti-bonding wave function has a central node,
so it provides little central electronic density to hold the protons together. The dashed lines
are for the unperturbed Hydrogen atom wave functions, with electron at either proton A or
at proton B.
These axial profiles of the wave functions φg (R0 , rA ) and φu (R0 , rA ) are plotted in Fig. 8.5.
For comparison, also shown in dotted lines are the axial profiles of the unperturbed wave
functions of Hydrogen atom with electron being either entirely with proton A or entirely
with proton B. The bonding wave function φg (R0 , rA ) can be thought of as a constructive
interference between the two unperturbed Hydrogen wave functions, while the anti-bonding
wave function φu (R0 , rA ) can be thought of as a destructive interference.
Note that the bonding wave function is symmetric and has no nodes, as expected for
8.4. THE HYDROGEN MOLECULE ION
141
ground state wave function, where as the anti-bonding wave function has a node exactly in
between the two protons. Therefore, the bonding wave function places a significant electron
density in between the two repelling protons thereby binding them together. On the other
hand, the anti-bonding wave function has got no central electron density to hold the two
protons together. Fig. 8.6 illustrates the electron densities in bonding and anti-bonding
∗
cases. Often, one denotes φg ≡ σ1s and φu ≡ σ1s
, as indicated on the right side of Fig. 8.6.
+
In H2 , there is only one bonding electron, where as in Hydrogen molecule, there are two
electrons of opposite spins participating in the covalent bond as illustrated in Fig. 8.7.
Figure 8.6: (Left) Illustrating bonding and anti-bonding wave functions formed by the constructive and destructive interference of Hydrogen atom wave functions. (Right) Energy
level diagram describing the formation of the covalent bond.
142
CHAPTER 8. MOLECULAR STRUCTURE
Figure 8.7: LCAO of Hydrogen molecule showing two electrons of opposite spins in the
bonding orbital forming a covalent bond.
143
8.5. H2 MOLECULE
8.5
H2 molecule
It is a bound state of two protons and two electrons. The coordinate system for H2 molecule
is shown in Fig. 8.8.
Figure 8.8: Coordinate system for H2 molecule (source: Bransden and Joachain).
Like before, we shall focus on electronic wave function. The electronic Hamiltonian
consists of electronic kinetic energies, nuclear-electron attractions, electronic repulsion, and
nuclear repulsion. It is of the form,
1
1
1
1
1
1
1
1
−
−
−
+
+
He = − ∇2r1 − ∇2r2 −
2
2
rA1 rA2 rB1 rB2 r12 R
1
1
= H0 (1) + H0 (2) +
+ , where,
r12 R
1
1
1 2
−
, i = 1, 2.
H0 (i) = − ∇ri −
2
rAi rBi
(8.21)
Spin symmetry
Like in the Helium atom, in many electronic molecules, spin part plays a key role. The two
electrons are indistinguishable, so the spin part is either singlet or triplet (χS,MS ):
1
χ0,0 = √ [α(1)β(2) − β(1)α(2)]
2
χ1,1 = α(1)α(2)
1
χ1,0 = √ [α(1)β(2) + β(1)α(2)]
2
χ1,−1 = β(1)β(2).
(8.22)
144
CHAPTER 8. MOLECULAR STRUCTURE
Space symmetry
The lowest energy state of the diatomic molecule is a Σ state (Λ = 0 state). We shall consider
even symmetry states under reflection Ay , i.e., Σ+ state. Further, since it is a homonuclear
diatomic molecule, it can have even or odd symmetry w.r.t. inversion about center P.
The overall wave function must be antisymmetric.
Hund-Mulliken or Molecular orbital (MO) method
In MO method, we shall use the orbitals from a smaller molecular system to build a larger
molecular system. In this case, we can use H+
2 wave functions
i
Ag (R) h −rA
Φg (R; r) = Ag (R) [ψ1s (rA ) + ψ1s (rB )] = √
e
+ e−rB ,
π
i
Au (R) h −rA
Φu (R; r) = Au (R) [ψ1s (rA ) − ψ1s (rB )] = √
e
− e−rB .
π
(8.23)
Since these are trial wave functions of Hydrogen molecular ion Hamiltonian (Eq. 8.11), they
satisfy,
1
Φg,u .
H0 (i)Φg,u = Eg,u −
R
(8.24)
Let us consider the following states (with symmetry identified):
ΦA (1, 2) = Φg (1)Φg (2)χ0,0 (1, 2);
(1 Σ+
g)
ΦB (1, 2) = Φu (1)Φu (2)χ0,0 (1, 2) = ΦB (2, 1); (1 Σ+
g)
1
ΦC (1, 2) = √ [Φg (1)Φu (2) + Φu (1)Φg (2)] χ0,0 (1, 2); (1 Σ+
u)
2
1
ΦD (1, 2) = √ [Φg (1)Φu (2) − Φu (1)Φg (2)] χ1,MS (1, 2), MS = 0, ±1;
2
(3 Σ+
u ).
(8.25)
Note that the first two wave functions are even and the last two are odd w.r.t. inversion.
However, all are antisymmetric w.r.t particle exchange. By analogy with the Helium atom,
we expect ΦA to be the ground state:
ΦA (1, 2) = Φg (1)Φg (2)χ0,0 (1, 2), expanding via Eq. 8.12 & normalizing,
1
= (ψ1s (rA1 ) + ψ1s (rB1 )) (ψ1s (rA2 ) + ψ1s (rB2 )) χ0,0 (1, 2)
2
ion
= Φcov
where,
A + ΦA
1
Φcov
[ψ1s (rA1 )ψ1s (rB2 ) + ψ1s (rA2 )ψ1s (rB1 )] χ0,0 (1, 2) and,
A =
2
1
Φion
[ψ1s (rA1 )ψ1s (rA2 ) + ψ1s (rB1 )ψ1s (rB2 )] χ0,0 (1, 2).
A =
2
(8.26)
8.6. THE ROTATION AND VIBRATION OF DIATOMIC MOLECULES
145
In the component Φcov
A , called covalent bonding orbital, the two electrons are associated
with different protons - more like two hydrogen atoms. On the other hand, in Φion
A , called
ionic bonding orbital, both electrons are associated with one of the protons.
The energy of this state is
EA (R) = hΦA |He | ΦA i .
(8.27)
Minimizing this w.r.t R gives the dissociation energy De = 2E1s − EA (R0 ) = 0.098 a.u.
at R0 = 1.5 a.u. However, the experimental values of binding energy is 0.175 a.u and
equilibrium bond length is 1.4 a.u. Thus, the above orbital needs further fine tuning. We
may envisage that the exact wave function may not be an equal superposition of covalent
and ionic orbitals as in the above. So, we setup a variational trial function,
ion
ΦT = Φcov
A + qΦA
≡ ΦA + λΦB , s.t.
1+λ
= q,
1−λ
(8.28)
and determine the trial parameters q, λ by variational method. It yields, q = 0.2, λ = −2/3,
R0 = 1.42 a.u., and De = 0.147 a.u., which is improved over the unoptimized wave function
ΦA .
Correlation effects
As discussed in the case of Helium atom, we can use Hartree-Fock method to introduce the
correlation effects neglected in the above model. This yields even better agreement (up to
90%) with the experimental value.
8.6
The rotation and vibration of diatomic molecules
So far we had been clamping the nuclei in fixed positions using Born-Oppenheimer approximation, and studied the electronic part Φs (R) of the wave function. Now we shall unclamp
the nuclei. A diatomic molecule can rotate or vibrate about its center of mass. We have
seen that the q
rotational energy is of the order of (m/M )Ry and the vibrational energy is of
the order of ( m/M )Ry . Recall the form of the total wave function from Eq. 8.8
Ψs = Fs (R)Φs (R; r1 , r2 , · · · , rN ).
(8.29)
146
CHAPTER 8. MOLECULAR STRUCTURE
Also recall the nuclear wave equation from Eq. 8.7:
"
!
#
hΦs |N2 | Φs i
~2 1 ∂
2 ∂
R
+
−
+ Es (R) Fs (R) = EFs (R),
2µ R2 ∂R
∂R
2µR2
(8.30)
The term hΦq |N2 | Φq i/(2µR2 ) is the rotational kinetic energy due to rotation of the internuclear vector. The total molecular orbital angular momentum operator is (see Fig. 8.9)
K = N + L.
(8.31)
Figure 8.9: Various orbital angular momentum vectors (source: Bransden and Joachain).
For an isolated molecule the total orbital angular momentum is a conserved quantity. Its
component along the inter-nuclear axis is also conserved. So, Ψs is a simultaneous eigenstate
of K2 and Kz ,
K2 Ψs = ~2 K(K + 1)Ψs
Kz Ψs = ~MK Ψs
K = 0, 1, 2, · · ·
MK = −K, −K + 1, · · · , K.
(8.32)
We also have
Lz Ψs = Fs (R)Lz Φs = ±Fs (R)~ΛΦs = ±~ΛΨs .
(8.33)
From Fig. 8.9 it is clear that Kz and Lz have equal magnitude. So, K must be at least Λ.
Therefore,
K = Λ, Λ + 1, Λ + 2, · · · .
(8.34)
8.6. THE ROTATION AND VIBRATION OF DIATOMIC MOLECULES
147
Since
hΦq |N2 | Φq i
~2 K(K + 1)
≈
,
2µR2
2µR2
(8.35)
the nuclear wave equation becomes
~2
−
2µ
!
!
1 ∂
∂
K(K + 1)
R2
−
Fs (R) = (E − Es (R))Fs (R).
2
R ∂R
∂R
R2
(8.36)
We now express
Fs (R) =
1 s
Fv,K (R)Λ HK,MK (Θ, Φ),
R
(8.37)
with rotational & vibrational quantum numbers K and v of the rovibronic states. Substituting this form in Eq. 8.36, we obtain the radial equation
"
~2
−
2µ
!
#
K(K + 1)
d2
s
−
+ Es (R) − Es,v,K Fv,K
(R) = 0.
2
dR
R2
(8.38)
Now we shall assume small oscillations about the equilibrium bond length R0 . At low
amplitudes it can be viewed as a Harmonic potential, but a better approximation is given
by the Morse potential (see Fig. 8.10)
h
i2
V (R) = De 1 − ea(R−R0 ) .
(8.39)
The energy levels are now described by
Es,v,K = Es,v,r = Es (R0 ) + Ev + Er , where
1
electronic levels : Es (R) = Es (R0 ) + ks (R − R0 )2 with spring constant ks
2
s
"
#
1
1 2
ks
vibrational levels : Ev = ~ω0 v +
−β v+
, v = 0, 1, 2, · · · , ω0 =
, &
2
2
µ
~2
rotational levels : Er = BK(K + 1), B =
.
(8.40)
2I0
Here ω0 is the vibrational constant, β is the anharmonicity parameter, B is the rotational
constant, and I0 = µR02 is the molecular moment of inertia.
Thus, the overall energy level diagram consists of electronic, vibrational, and rotational
levels as shown in Fig. 8.11. Note that the equilibrium bond-length R0 itself is a function
of the electronic quantum number s. Moreover, the bond-length stretches with rotational
quantum number due to centrifugal effect, R(K) − R0 ∝ K(K + 1), and can even lead to
dissociation.
148
CHAPTER 8. MOLECULAR STRUCTURE
Figure 8.10: Vibrational energy (source: Wikipedia). Here re = R0 , r = R.
Figure 8.11: Electronic, Vibrational, and Rotational States (source: Demtroder).
Chapter 9
Spectra of Diatomic Molecules
9.1
Electric Dipole of a Molecule
Consider a molecule in state Ψa , with a ≡ (s, v, K, MK , Λ), The energy levels are now
described by
Es,v,K = Es,v,r = Es (R0 ) + Ev + Er , where
1
electronic levels : Es (R) = Es (R0 ) + ks (R − R0 )2 with spring constant ks
2
s
"
#
1
ks
1 2
, v = 0, 1, 2, · · · , ω0 =
vibrational levels : Ev = ~ω0 v +
−β v+
,
2
2
µ
~2
rotational levels : Er = BK(K + 1), B =
.
(9.1)
2I0
EM interacts with the electric dipole of the molecule, which is the sum of the nuclear and
electronic dipole moments. The permanent electric dipole moment of the molecule is
Daa = hΨa |D| Ψa i , where


X
X
D = e  Zi Ri −
rj 
i
(9.2)
j
is the molecular electric dipole moment operator. Homonuclear molecules H2 , N2 , O2 , etc.
have no permanent electric dipole moment in their ground state. Heteronuclear molecules
such as HCl have permanent electric dipole.
The electric-dipole transitions between levels a and a0 are determined by the matrix
element
Daa0 = hΨa |D| Ψ0a i .
149
(9.3)
150
9.2
CHAPTER 9. SPECTRA OF DIATOMIC MOLECULES
Molecular transitions
The molecular transitions are illustrated in Fig. 9.1. For homonuclear molecules, which
have no permanent electric dipole, the matrix element vanishes unless there is an electronic
transition, i.e., s 6= s0 . However, heteronuclear molecules can have vibrational or rotational
transitions without electronic transition.
Figure 9.1: Electronic transitions, vibrational-rotational transitions, and pure rotational
transitions (source: Demtroöder).
151
9.2. MOLECULAR TRANSITIONS
9.2.1
Pure rotational transitions in heteronuclear systems
The pure rotational spectrum lie in far-infra-red to microwave region. The selection rules
for rotational transitions are
•


∆K
= ±1
∆MK
= 0, ±1
∆K
= 0, ±1
∆MK
=0
if Λ = 0 (Σ state)

if Λ = 1, 2, · · · ,

• ∆Λ = 0.
(9.4)
Figure 9.2: Pure rotation spectrum of HCl (source: Bransden and Joachain).
In a pure rotational spectrum, the transition frequencies are linearly dependent on the
rotational quantum number K, i.e.,
hνK+1,K = B(K + 1)(K + 2) − BK(K + 1)
= B(K + 1)(K + 2 − K) = 2B(K + 1).
(9.5)
Therefore, the separation of adjacent absorption lines of the rotation spectrum is constant,
namely,
νK+1,K − νK,K−1 =
2B(K + 1) − 2BK
2B
=
.
h
h
(9.6)
The rotation spectrum of HCl is shown in Fig. 9.2. The rotational spectrum allows very
precise determination of the rotational constant B = ~2 /2I0 = ~2 /(2µR02 ), from which we
152
CHAPTER 9. SPECTRA OF DIATOMIC MOLECULES
can estimate the equilibrium bond-length R0 .
9.2.2
Vibrational rotational spectrum
For a pure harmonic oscillator, the electric-dipole vibration transition selection rule is v =
±1. However, because of anharmonic nature, v = ±2, ±3, · · · can also occur with low
probabilities, an order of magnitude lower or even weaker. Since photons carry spin angular
momentum, the absorption of a photon causing vibrational transition is associated with a
change in the rotational quantum number also. Together they are referred as a rovibronic
transition. They appear in the infrared region. Based on the rotational selection rules in
Eqs. 9.4, we consider the two cases.
Λ = 0 (Σ state): As per Eqs. 9.4, ∆K = ±1
R branch (∆K = 1) :
R
hνv+1,K+1;v,K
= E(v + 1, K + 1) − E(v, K) ≈ hν0 + 2B(K + 1),
P branch (∆K = −1) :
P
hνv+1,K−1;v,K
= E(v + 1, K − 1) − E(v, K) ≈ hν0 − 2BK.
(9.7)
The approximate expressions are due to the fact that the rotational constant B slightly
change with the vibrational quantum number v. The lowest transitions in R branch is
R
= E(v + 1, 1) − E(v, 0) ≈ hν0 + 2B and the lowest transitions in R branch is
hνv+1,1;v,0
P
hνv+1,0;v,1 = E(v + 1, 0) − E(v, 1) ≈ hν0 − 2B. Therefore the central gap between the two
branches is 4B. These two branches of transitions are illustrated in Fig. 9.3.
9.2. MOLECULAR TRANSITIONS
153
Figure 9.3: The R and P branches of rovibronic transitions (source: Bransden and Joachain).
154
CHAPTER 9. SPECTRA OF DIATOMIC MOLECULES
The rovibronic spectra of HCl is shown in Fig. 9.4. Note that unlike in the pure rotational
spectrum of Fig. 9.2, here the spacings are not uniform (∵ bond-length R0 and hence the
rotational constant B change with v).
Figure 9.4: The R and P branches of in the fundamental rovibronic band of HCl (source:
Bransden and Joachain).
At room temperatures, small molecules generally exist in v = 0, the vibrational ground
state. Therefore, most commonly observed vibration band is from v = 0 to v = 1. This is
called the fundamental band.
Λ ≥ 1: As per Eqs. 9.4, ∆K = 0, ±1
Now, in addition to P and R branches, we see Q branch corresponding to ∆K = 0,
Q
= E(v + 1, K) − E(v, K) ≈ hν0 .
hνv+1,K;v,K
9.3
(9.8)
Scattering by molecules
As discussed in Sec. 3.8, scattering is essentially a second order process, often involving an
intermediate state of the molecule. Unlike the absorption or emission, scattering can take
place with the induced dipole moment, and therefore even homonuclear diatomic molecules
like H2 , N2 , O2 , can also participate in scattering. The selection rule for scattering is
∆K = 0, ±2.
(9.9)
9.3. SCATTERING BY MOLECULES
155
If a molecule in energy Ea gets scattered by a photon of energy ~ω, the energy conservation equation is given by
Ea + ~ω = Eb + ~ω 0 ,
(9.10)
where Eb is the final state of the molecule and ~ω 0 is the energy of the scattered photon. If
ω 0 = ω and Eb = Ea , then it is an elastic scattering or Rayleigh scattering. Otherwise, it
is inelastic scattering or Raman scattering (see Fig. 3.22). A molecule in a lower state Ea
can get scattered into higher energy state Eb while the scattered photon looses energy, i.e.,
ω 0 < ω. The observed spectral line is then called Stokes line. On the other hand, scattering
of an initially excited atom may bring it to a lower energy state Eb < Ea and the extra energy
will be carried away by the photon ω 0 > ω. The resulting spectral line is called anti-Stokes
line.



K 0 = K







K0 = K + 2



h∆νK,K 0 = 





K0 = K − 2







0, K = 0, 1, 2, · · · (K ≥ Λ) (Rayleigh)
B(K + 2)(K + 3) − BK(K + 1)
= B(4K + 6), K = 0, 1, 2, · · · (K ≥ Λ) (Stokes)
B(K − 2)(K − 1) − BK(K + 1)
= −B(4K − 2), K = 2, 3, 4, · · · (K ≥ Λ) (anti-Stokes)
(9.11)
156
CHAPTER 9. SPECTRA OF DIATOMIC MOLECULES
Figure 9.5: Molecular scattering (source: Bransden and Joachain).
157
9.4. ELECTRONIC SPECTRA OF DIATOMIC MOLECULES
9.4
Electronic spectra of diatomic molecules
Electronic transitions of molecules generally involve changes in the vibrational and rotational
quantum numbers also. They appear in visible to ultraviolet region. The frequencies are
given by
0
hνs0 ,v0 ,K 0 ;s,v,K = (Es0 − Es ) + (Ev0 − Ev ) + (EK
− EK ).
(9.12)
As a result we observe a band structure with numerous lines instead of a single sharp line
as in the atomic case.
Vibrational structure of electronic spectra
Ignoring the rotational substructure the frequencies are given by Deslandres formula,
hνs0 ,v0 ;s,v = (Es0 − Es ) + (Ev0 − Ev )
= hνs0 s + hν00 v 0 +
1
1
− hβ 0 ν00 v 0 +
2
2
2
− hν0 v +
1
1
+ hβν0 v +
2
2
2
.
(9.13)
Rotational band of electronic spectra
Selection rules: There is no selection rule for the vibrational quantum number since s and
s0 have different vibrational potential wells. Other selection rules are:
∆K


= 0, ±1



∆Λ
= 0, ±1 Λ = 0=Λ0 = 0.
∆S
=0
∆K
= ±1
Σ+
↔ Σ+ or
+
Σ
−
=Σ
∆S
=0










−
−

Σ ↔Σ 








Λ = 0(Σ)↔Λ0 = 0(Σ).
(9.14)
158
CHAPTER 9. SPECTRA OF DIATOMIC MOLECULES
The different branches of the rotational band are:
P-branch: ∆K = −1
hνhP νs0 ,v0 ,K−1;s,v,K = hνs0 ,v0 ;s,v + B 0 K(K − 1) − BK(K + 1)
Q-branch: ∆K = 0
hνhP νs0 ,v0 ,K;s,v,K = hνs0 ,v0 ;s,v + B 0 K(K + 1) − BK(K + 1)
R-branch: ∆K = 1
hνhP νs0 ,v0 ,K+1;s,v,K = hνs0 ,v0 ;s,v + B(K + 1)(K + 2) − B 0 K(K + 1).
9.5
(9.15)
Franck-Condon Principle
Now consider an electronic transition in relation to the nuclear oscillations. As we discussed
before, the electronic dynamics in a molecule is much faster than the nuclear dynamics.
Therefore, during the transitions, we may assume the nuclei to be stationary, and the bondlength R0 to be a constant. Therefore, the transitions can be represented by vertical lines
as shown in Fig. 9.6.
Figure 9.6: Electronic transitions in molecules and the Franck-Condon principle (source:
Wikipedia).
Since there is no selection rule on the vibrational quantum number, we may ask, which
159
9.6. DISSOCIATION AND PREDISSOCIATION
would be the most probable final vibrational state. To answer this, let us consider the
transition
[Ψs,v = Φs ψv ] → [Ψs0 ,v0 = Φ0s ψv0 ]
ignoring other degrees of freedom. The transition probability is proportional to

*
Ps0 ,v0 ←s,v = Ψs0 ,v0 e

X
i
=e
X
Zi Ri − e
X
rj  Ψs,v
j
Zi hΦs0 ψv0 |Ri | Φs ψv i − e
i
+
X
hΦs0 ψv0 |rj | Φs ψv i .
(9.16)
j
Franck-Condon principle amounts to (i) fixing the nuclear coordinates and dropping the first
term and (ii) simplifying the second term such that

Ps0 ,v0 ←s,v = fv0 v −e

X
hΦ0s |rj | Φs i , where the overlap of the vibrational wave functions
j
fv0 v = hψv0 |ψv i ,
so that
Ps0 ,v0 ←s,v ∝ fv0 v , which is called the Franck-Condon factor.
(9.17)
According to the Franck-Condon principle, the transition with maximum overlap between
the vibrational wave functions is the strongest. As illustrated in Fig. 9.6, the low level
vibrational wave functions are approximately the eigenfunctions of harmonic oscillator, i.e.,
approximate Hermite functions. The ground state is a Gaussian centered at the equilibrium
bond length R0 . For higher vibrational quantum numbers, the probability density gets
more and more concentrated at the classical turning points, i.e., the edges of the harmonic
potential. Thus the vertical line connecting the two maxima (as indicated by arrows in Fig.
9.6) correspond to the strongest transitions.
9.6
Dissociation and predissociation
As illustrated in Fig. 9.7, an electronic transition may lead to excitation, dissociation,
ioniziation, or dissociative ionization. The excitation to a repulsive state leads to dissociation.
In some cases, before dissociation, the molecule may pass through an intermediate step
state called predissociation (see Fig. 9.7). Here the electronic transition first leads to an
excitation, which decays to a repulsive state thus causing dissociation.
160
CHAPTER 9. SPECTRA OF DIATOMIC MOLECULES
Figure 9.7: (Left) Predissociation (source: Internet).
Demtroder).
9.7
(Right) Predissociation (source:
Fluorescence and Phosphorescence
Often electronic excitation to a high vibrational state may be followed by a sequence of
vibrational decays ultimately reaching the vibrational ground state, from which an electronic
decay takes place emitting a light of lower frequency (see Fig. 9.8 (a)). This process, which
was historically observed in CaF2 , is known as fluorescence. Since the electronic selection rule
requires ∆S = 0, the fluorescence takes place between levels of same multiplicity. Typically,
the lifetimes associated with vibrational states are nanoseconds and so the whole process
may last up to a microsecond.
If the electronic excitation gets to a level that is coupled to another level of different
multiplicity, the molecule may undergo an intersystem crossing, which is a rather slow process
(see Fig. 9.8 (b)). Subsequently, vibrational decays bring it to the vibrational ground state,
from which an electronic decay results in the emission of a lower frequency light. The overall
process may last seconds to hours, and the substance undergoing this process keeps shining
during this time. This process, that was historically observed in phosphorous, is known as
phosphorescence.
Fluorescence and phosphorescence are different from inelastic scattering, which is based
on virtual intermediate levels and leads to absorption and emission of a wide range of frequencies. Fluorescent and phosphorescent materials have a number applications such as
traffic signals that glow in night, emergency signals, etc.
9.7. FLUORESCENCE AND PHOSPHORESCENCE
Figure 9.8: Predissociation (source: Demtroder).
161
162
CHAPTER 9. SPECTRA OF DIATOMIC MOLECULES
Chapter 10
Electron Spin Resonance (ESR)
ESR is essentially due to the interaction of the electron spin with both static and oscillating magnetic fields. First discovered by Soviet Physicist Yevgeny Zavoisky in 1944, it is also
known as Electron Paramagnetic Resonance (EPR) or Electron Magnetic Resonance (EMR).
Since typically the oscillating magnetic fields are the magnetic component of the microwave
radiation, it is also a type of microwave spectroscopy. ESR is very useful in studying paramagnetic substances, such as free radicals and metal complexes. In this chapter, we discuss
the essential elements of ESR.
10.1
Hamiltonian
10.1.1
Zeeman Hamiltonian
Atoms or molecules of a paramagnetic substance have unpaired electrons. Each unpaired
electron is associated with a net spin angular momentum ~S and associated magnetic moment,
µe = −ge µB ~S,
(10.1)
where ge = 2.0023193043617 is the free-electron Zeeman factor or free-electron g-factor,
|e|~
is the Bohr magneton, e is the electronic charge, and me is the mass of electron.
µ
~B =
2me
~ = B0 ẑ, the electronic
When the paramagnetic sample is placed in a strong magnetic field B
magnetic moment interacts with the magnetic field via the Zeeman Hamiltonian,
~ = ge µB B0 Sz .
H = −~µe · B
(10.2)
Since the eigenvalues of Sz are ms = ±1/2, the corresponding energy eigenvalues are
E±1/2 = ±ge µB B0 /2,
163
(10.3)
164
CHAPTER 10. ELECTRON SPIN RESONANCE (ESR)
and the energy gap
∆E = ge µB B0 = ~ω0 ,
(10.4)
where ω0 = ∆E/~ is the called the Larmor frequency.
Figure 10.1: ESR energy levels (source: Wikipedia).
~ = B1 cos(ωt) incident
Now consider an electromagnetic wave with magnetic component B
on the sample. Resonance occurs when frequency ω electromagnetic wave matches with the
Larmor frequency ω0 , i.e.,
~ω = ge µB B0 .
(10.5)
Under resonance, maximum absorption of the electromagnetic wave occurs resulting in maximum transitions.
10.1.2
Local fields
It must be noted that the above relation is for an isolated electron. An atomic/molecular
~ loc such that the effective magnetic field
electron is subjected to a net local magnetic field B
experienced by it is
~ eff = B
~ +B
~ loc = (1 − σ)B,
~
B
(10.6)
165
10.1. HAMILTONIAN
where σ is the chemical shift tensor. For an isotropic system, like a liquid, wherein molecules
undergo rapid isotropic motions, only the scalar part of σ survives, and we can write
~ eff = (1 − σ)B
~ = gB
~ = g B0 ẑ.
B
ge
ge
(10.7)
Thus, the Hamiltonian and the ESR energy gap formulae of Eqs. 10.2 and 10.4 get modified
to
H = gµB B0 Sz
∆E = gµB B0 = hν.
(10.8)
Thus the chemical environment information information is encoded in the resonance frequency, which is what makes ESR more interesting and useful.
10.1.3
Hyperfine interaction
The unpaired electronic spin of a free radical can interact with surrounding nuclear spins via
the hyperfine interaction. In a strong external magnetic field B0 , the net Hamiltonian under
secular approximation is,
H = gµB B0 Sz + ASz Iz ,
(10.9)
EmS ,mI = gµB B0 mS + AmS mI ,
(10.10)
and the energy eigenvalues
where ~I is the nuclear spin operator and A is the Hyperfine coupling parameter. The effect
of the Hyperfine interaction is splitting of levels as shown in Fig. 10.2. Here, note the
splitting of levels according to Eq. 10.10 and the eigenvalues (mS , mI ) and the selection
rules ∆mS = ±1 and ∆mI = 0. Thus, it results in two peaks, as in Fig. 10.2. Also, note
that the ESR spectral lines are usually analyzed in the 1st derivative form for ease of finding
the peak frequency.
In general, if there are M number of spin-I nuclear spins, then there will be (2M I + 1)
splittings - therefore (2M I + 1) is the multiplicity of the spectrum. The example of CH∗3 free
radical is shown in Fig. 10.3. Here the electronic spin ESR line is hyperfine split by three
H spins, each of spin 1/2. The intensities of these lines are according to the Pascal triangle
as shown.
In the case of electronic spin coupling to a spin-1 nucleus, such as a 14 N spin, each
electronic energy level is split into three levels as shown in Fig. 10.4. Note that, in this case,
all three peaks are of each height, just as in Fig. 10.2.
166
CHAPTER 10. ELECTRON SPIN RESONANCE (ESR)
Figure 10.2: ESR doublet (source: internet).
Figure 10.3: ESR multiplicty (source: internet).
10.2
Hardware
Typically, ESR is carried out at the microwave range. Following table shows different ranges
of the electromagnetic waves. A block-diagram of ESR spectrometer is shown in Fig. 10.6.
167
10.3. SENSITIVITY
Figure 10.4: ESR multiplicty (source: internet).
10.3
Sensitivity
The magnetization and hence the sensitivity of ESR depends on the population difference
∆N = N−1/2 − N1/2 . According to the Boltzmann distribution,
N1/2
∆E
= exp(−∆E/kT ) ≈ 1 −
N−1/2
kT
N ge µB B0
∴ ∆N = N−1/2 − N+1/2 ≈
,
2kT
(10.11)
where N = N−1/2 + N+1/2 . Thus, sensitivity increases with B0 and decreases with temperature T .
168
CHAPTER 10. ELECTRON SPIN RESONANCE (ESR)
Figure 10.5: ESR microwave bands (source: Wikipedia).
Figure 10.6: ESR microwave bands (source: Wikipedia).
Chapter 11
Nuclear Magnetic Resonance (NMR)
11.1
Nuclear Spin
Many fundamental particles have spin angular momentum I~ with associated eigenvalue equations
I~2 |I, mi = ~2 I(I + 1) |I, mi
Iz |I, mi = ~m |I, mi ,
(11.1)
where m ∈ {−I, −I +1, · · · , I}. We refer to the number I as the spin number. Like electrons,
protons and neutrons are also spin-1/2 particles. A given nuclear isotope with multiple protons and neutrons will have a resultant spin which may be zero, or integral, or half integral.
In general, following empirical rules apply:
No. of protons
No. of neutrons
Nuclear Spin (I)
Even
Even
Odd
Odd
Even
Odd
Even
Odd
zero (eg. 12 C, 16 O, 32 S)
half-integral (eg. 13 C, 17 O)
half-integral (eg. 7 Li, 15 N)
integral (eg. 6 Li, 14 N)
The nuclear magnetic moment mu
~ is related to the spin angular momentum I~ via
µ
~ = ~γ I~
where γ is the gyromagnetic ratio which varies from isotope to isotope.
169
(11.2)
170
11.2
CHAPTER 11. NUCLEAR MAGNETIC RESONANCE (NMR)
The Zeeman energy gap and resonance
~ = B0 ẑ via the
The nuclear magnetic moment µ
~ interacts with an external magnetic field B
Zeeman effect
~ = −µz B0 = ~(−γB0 )Iz = ~ω0 Iz ,
H = −~µ · B
(11.3)
where ω0 is called the Larmor frequency, i.e., the frequency with which nuclear spins precess
about the magnetic field. The corresponding energy eigenvalues are
Em = ~ω0 m.
(11.4)
In the rest of this chapter, for simplicity, we consider only spin 1/2 nuclei, i.e., I = 1/2,
m = ±1/2, and
E±1/2 = ∓~γB0 /2,
(11.5)
∆E = E−1/2 − E1/2 = ~γB0 = ~|ω0 |.
(11.6)
with the Zeeman energy gap
Typically, B0 ∼ 1 − 20 Tesla and ω0 ∼ 106 − 108 Hz (depending on the isotope), which is
in the radio-frequency (RF) range, and the corresponding Zeeman energy gap is ∼ 10−6 eV,
which is much less than the room temperature thermal energy kT (∼ 10−3 eV). The strong
magnetic fields beyond 2 Tesla are usually achieved by superconducting coils.
Although, the energy gaps are tiny, the nuclear spins are sufficiently isolated from external
noise in order to retain quantum coherences for long durations to allow intricate control on
their quantum dynamics, which makes it attractive not only for spectroscopy, but also for
quantum simulations and other quantum information studies (see Fig. 11.1).
Suppose we shine an electromagnetic (EM) wave with the same energy as the Zeeman
energy gap, i.e., a radiowave with energy ~ω = ~ω0 matching the Zeeman energy gap.
In this case, the nuclear magnetic moment interacts with the magnetic component of the
EM and resonantly absorb the electromagentic energy and undergo transitions. By sending
AC current through a pair of Helmholtz coils, we can generate a linearly polarized EM,
171
11.2. THE ZEEMAN ENERGY GAP AND RESONANCE
Figure 11.1: NMR and the NMR Research Center @ IISER Pune.
~ ( t) = 2B1 cos(ωt)x̂. Thus, we achieve a time-dependent lab-frame Hamiltonian,
B
~ −µ
~1
H(t) = −~µ · B
~ ·B
= ~(−γB0 )Iz + ~(−γB1 )Ix 2 cos(ωt)
= ~ω0 Iz + ~ω1 Ix 2 cos(ωt),
h
i
= ~ω0 Iz + ~ω1 eiωtIz Ix e−iωtIz + e−iωtIz Ix eiωtIz ,
≈ ~ω0 Iz + ~ω1 eiωtIz Ix e−iωtIz ,
(11.7)
where we have dropped the nonresonant circularly polarized component. The time-dependence
can be avoided by moving to a rotating frame along ẑ with frequency ω,
H R = ~∆ωIz + ~ω1 Ix ,
(11.8)
where ∆ω = ω0 − ω is called the offset frequency. If the resonance condition is perfectly
achieved, ∆ω = 0, and the rotating frame Hamiltonian
H R = ~ω1 Ix ,
(11.9)
172
CHAPTER 11. NUCLEAR MAGNETIC RESONANCE (NMR)
looks exactly like the Zeeman Hamiltonian in the Lab frame. Thus, in the rotating frame,
under resonant condition, the nuclear spin precesses about the direction of RF with frequency
ω1 , called the Rabi frequency.
11.3
Magnetization, RF pulse, FID, and Chemical Shift
In practice, one usually takes a bulk sample of diamagnetic atoms / molecules having no
net electronic spin. The thermal magnetization M = N µ(p(1/2) − p(−1/2)), where N is the
total number of spins in the sample and p(±1/2) are the relative probabilities in the two
levels, is provided by the Boltzman distribution,
∆E
p(−1/2)
= e−∆E/kT ≈ 1 −
p(1/2)
kT
N µ∆E
N ~γB0
∴ M≈
=
.
2kT
2kT
(11.10)
Thus, strong magnetic fields and low sample temperatures lead to high magnetization along
ẑ, the direction of the Zeeman field. This thermal magnetization undergoes Rabi precession
along x̂ the direction of the resonant RF field. For an RF pulse, the RF field is switched on
only for a duration the Rabi precession makes an angle θ. Thus, by controlling the duration,
frequency, and phase of the RF fields, we can control spin dynamics (see Fig. 11.2).
Figure 11.2: RF pulses for spin rotations.
Using a π/2 pulse, we can tilt the magnetization from ẑ to x̂ direction. The magnetization
11.4. NMR QUBITS
173
then keeps precessing about the Zeeman field along ẑ inside the pair of Helmholtz coils
mentioned before. Like in a dynamo, wherein a magnet spins inside a copper coil, the
precessing nuclear magnetization produces emf in the Helmoholtz coil, which is detected as
an oscillating and decaying signal - called free induction decay (FID) (see Fig. 11.3). Fourier
transform of FID reveals all the Larmor frequencies in the signal.
Figure 11.3: NMR signal: the free induction decay (FID) and its Fourier transform.
For two nuclei of same isotope but in different chemical environment, the local field at the
site of nuclei differ due to different electron densities surrounding them, resulting in different
Larmor frequencies. Thus, Larmor frequencies are different even for nuclei of same isotope
in different chemical groups. This effect is known as chemical shift (see Fig. 11.4).
11.4
NMR qubits
The efficient quantum control of nuclear spins and their long coherence times allow encoding
quantum information on nuclear spins and performing quantum gates and quantum algorithms (see Fig. 11.5). Thus, nuclear spin systems offer a convenient quantum registers for
experimental investigations of Physics of quantum computing and quantum simulations. For
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CHAPTER 11. NUCLEAR MAGNETIC RESONANCE (NMR)
Figure 11.4: Chemical shift) (source: internet).
example, the Hadamard gate can be realized via a (π/2)y pulse followed by a πx pulse, i.e.,


1 1 
1
e−iIx π e−iIy π/2 = √ 
.
2 1 −1
11.5
(11.11)
Magnetic Resonance Imaging (MRI)
Another way to spread the Larmor frequencies is by using magnetic field gradients. For
example, consider a sample as in Fig. 11.6, where shaded squares have higher nuclear
densities of certain isotope, say 1 H spins. If we apply uniform magnetic field B ẑ, all the
nuclei will have same Larmor frequency, and the NMR signal will have a single peak as in
Fig. 11.6 (a). On the other hand, if we apply B(x)ẑ, where B(x) ∝ x, then the Larmor
frequencies ω0 ∝ x, so different columns in the sample will have different frequencies, yielding
the spectrum as in Fig. 11.6 (b). Similarly, if we apply B(y)ẑ, where B(y) ∝ y, then the
Larmor frequencies ω0 ∝ y, so different rows in the sample will have different frequencies,
yielding the spectrum as in Fig. 11.6 (c). In effect, we got projections along different
directions, which together can reconstruct the overall nuclear density profile of the sample.
This is the principle of MRI (see Fig. 11.7).
11.6. TWO INTERACTING SPINS
175
Figure 11.5: NMR qubits.
11.6
Two interacting spins
Two or more nuclear spins interact with one another via (i) direct dipole-dipole interaction
known as D coupling or (ii) via indirect interaction known as J coupling. In liquids, due to
isotropic motion of molecules D coupling averages out to zero, while the J coupling survives.
The Hamiltonian, under secular approximation, for a two J coupled spins can be written as
H = ~ωA IzA + ~ωB IzB + ~2πJIzA IzB ,
(11.12)
EmA ,mB = ~ωA mA + ~ωB mB + ~2πJmA mB .
(11.13)
with eigenvalues
Note that this leads to four nondegenerate levels and four transitions following the selection
rule ∆mA + ∆mB = ±1 as illustrated in Fig. 11.8. Using the interaction, we can also realize
CNOT gate as shown in Fig. 11.9.
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CHAPTER 11. NUCLEAR MAGNETIC RESONANCE (NMR)
Figure 11.6: Uniform magnetic field (a), and spreading the Larmor frequencies using magnetic field gradients along x (b) or along y (c).
Figure 11.7: Magnetic Resonance Imaging (MRI) (source: internet).
11.6. TWO INTERACTING SPINS
Figure 11.8: Energy levels and spectral lines from two J-coupled nuclear spins.
Figure 11.9: NMR CNOT gate (source: internet).
177