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# Fundamental Concepts of Molecular Spectroscopy

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## Book Description

This practical and unique textbook explains the core areas of molecular spectroscopy as a classical teacher would. The author carefully explores and explains each concept, walking side by side with the student through carefully constructed text, pedagogy, and derivations to ensure comprehension of the basics before approaching higher level topics. The author incorporates both electric resonance and magnetic resonance in the textbook.

- Uses boxes to explain more difficult topics and provides derivations to demonstrate "how and why".
- Includes coverage of electronic and NMR spectroscopy, both in sufficient detail.
- Discusses the density matrix method and its use in electronic spectroscopy before addressing it in NMR.
- Includes a chapter on Vibrational and Rotational Coherence Spectroscopy.
- Each chapter ends with problems with varying level of difficulty.

## Table of Contents

Electromagnetic Wave Nature of Light

Gauss’s Law of Electrostatics

Gauss’s Law of Magnetism

Faraday’s Law of Induced Electric Field

Ampere’s Law of Induced Magnetic Field

Maxwell’s Equations

Wave Equation

Homogeneous Travelling Plane Wave

Wave Packet

Reference

Problems

2. Postulates of Quantum Mechanics

Stern-Gerlach Experiment

Postulates of Quantum Mechanics

Perturbation Theory

Perturbation of a Nondegenerate System

Perturbation of a Degenerate State

References

Problems

**3. Semiclassical Theory of Spectroscopic Transition**

Two-Level System

System-Radiation Interaction

Time Development of Eigenstate Probabilities

Probability Expressions

Rabi Oscillations

Transition Probability and Absorption Coefficient

Limitations of the Theory

Collisional Line Broadening

Line Broadening from Excited State Lifetime

Spectral Line Shape and Line Width

Homogeneous or Lorentzian Line Shape

Inhomogeneous or Gaussian Line Shape

Doppler Interpretation of Inhomogeneous Line Shape

General Reading

Problems

4. Hydrogen Atom Spectra

Free Hydrogen Atom

Eigenvalues, Quantum Numbers, Spectra, and Selection Rules

Hydrogen Atom in External Magnetic Field: Zeeman Effect and Spectral Multiplets

Magnetic Moment in External Magnetic Field

Larmor Precession

Eigenstate, Operator, and Eigenvalue in External Magnetic Field

Anomalous Zeeman Effect and Further Splitting of Spectra

Electron Spin and Spin Magnetic Moment

Lande *-factor*

Spin-Orbit Coupling

Spin-Orbit Coupling Energy

Spectroscopic Notation

Fine Structure of Atomic Spectra

Splitting of *Degeneracy: Anomalous Zeeman Effect*

Zeeman Effect in Weak Magnetic Field

Zeeman Splitting Changeover from Weak to Strong Magnetic Field

Electron-Nuclear Hyperfine Interaction

Zeeman Splitting of Hyperfine Energy Levels

Zeeman Splitting of Hyperfine States in Weak Magnetic Field

Hyperfine States of Hydrogen Atom in Strong Magnetic Field

Stark Effect

Hydrogen Atom in External Electric Field

Effect on the *Level*

Effect on the *Level*

References and Text

Problems

5. Molecular Eigenstates

Born-Oppenheimer Approximation

Solution of the Total Schrödinger Equation

States of Nuclear Motion

Adiabatic and Non-Adiabatic Processes

Molecular Potential Energy States

One-Electron Hydrogen-Like Atom States

Molecular Electronic States Derived from Atom States

LCAO-MO

Molecular Eigenstates of H_{2}^{+}

Molecular Eigenstates of H_{2}

Singlet and Triplet Excited States of H_{2}

Electric Dipole Transition in H_{2}

Molecular Orbital Energy and Electronic Configuration

Molecular Orbitals of Heteronuclear Diatomic Molecule

Molecular Orbitals of Large Systems

LCAO-MO of Porphyrins

Free-Electron Orbitals of Porphyrin

References

Problems

6. Elementary Group Theory

Symmetry Operations

Rotation

Reflection

Improper Rotation

Inversion

Point Group

Properties of Point Groups

Representation of Symmetry Operators of a Group

Group Representations

Labels of Irreducible Representations

Reduction of Representations to Irreducible Representations

Direct Product of Irreducible Representations

Applications

Energy Eigenvalues of Molecular Orbitals

Removal of Energy Degeneracy by Perturbation

General Selection Rules for Electronic Transitions

Specific Transition Rules

References

Problems

7. Rotational Spectra

Rotational Spectra of Diatomic Molecules

Schrödinger Equation for Diatomic Rotation

Rotational Energy of Rigid Rotor

Rotational Energy of Non-Rigid Rotor

Stationary State Eigenfunctions and Rotational Transitions

Energy Levels and Representation of Pure Rotational Spectra

Rotational Spectra of Polyatomic Molecules

Rotational Inertia

Energy of Rigid Rotors

Wavefunctions of Symmetric Tops

Commutation of Rotational Angular Momentum Operators

Eigenvalues for Tops

Selection Rules for Polyatomic Rotational Transition

General Reading

Problems

8. Diatomic Vibrations, Energy, and Spectra

Classical Description of an Oscillator

Schrödinger Equation for Nuclear Vibration

Selection Rules for Vibrational Transitions

Rotational-Vibrational Combined Structure

General Reading

Problems

**9. Polyatomic Vibrations and Spectra**

A Simple Classical Model to Define a Normal Mode

Vibrational Energy from Classical Mechanics

Solution of Lagrange’s Equation

Vibrational Hamiltonian and Wavefunction

Symmetry of Normal Modes

Finding the Vibrational Frequencies

Activity of Normal Modes of Vibration

Secondary Band Manifold in Infrared Spectra

Overtone Band

Hot Band

Combination Band

Fermi Resonance Band

Vibrational Angular Momentum and Coriolis-Perturbed Band Structure

Rotational Band Structure in Vibrational Bands

Selection Rules for Vibrational Transition

References

Problems

10. Raman Spectroscopy

Light Scattering

Frequencies of Rayleigh and Raman-Scattered Light

Limitation of the Classical Theory of Raman Scattering

Brillouin Scattering

Raman Tensor

Polarizability Tensor Ellipsoid

Nomenclature of the Polarizability Tensor

Anisotropy of Polarizability

Isotropic Average of Scattered Intensity

Semi-Classical Theory of Raman Scattering

Rotational Raman Spectra

Vibration-Rotation Raman Spectra

Raman Tensor and Vibrational Symmetry

Secondary or Coupled Bands in Raman Spectra

Solution Phase Raman Scattering

Resonance Raman Scattering

Sundries and Outlook

References

Problems

11. Electronic Spectra

Energy Term-Value Formulas for Molecular States

Dipole Transitions in the Electronic-Vibrational-Rotational Spectra

Electronic Transition Dipole with Nuclear Configurations

Franck-Condon Factor

Progression of Vibrational Absorption in an Electronic Band

Analysis of Vibrational Bands

Analysis Rotational Bands

Electron-Nuclear Rotational Coupling and Splitting of Rotational Energy Levels

Hund’s Cases

-type Doubling

Selection Rules for Electronic Transitions in Diatomic Molecules

Symmetry-Based General Rules for Electronic Transitions

Selection Rules

Selection Rules Pertaining to Hund’s Coupling Cases

Perturbation Manifests in Vibronic Spectra

Rotational Perturbation and Kronig’s Selection Rules

Frequency Shift and *-doubling in Rotational Perturbation*

Vibrational Perturbation

Predissociation

Diffused Molecular Spectra

Stark Effect in Rotational Transitions: Observation and Selection Rules

Zeeman Effect on Rotational Energy Levels and Selection Rules

Magnetooptic Rotational Effect

References

Problems

**12. Vibrational and Rotational Coherence Spectroscopy**

Ultrashort Time of Spectroscopy

Wave Packet

Coherence

Linear Superposition and Interference

Vibrational Coherence

Rotational Coherence

Coherence Decay

Wave Packet Oscillation

Frequency Spectrum of Time-Domain Coherence

Assignment of Vibrational Bands

Pure Rotational Coherence

Density Operator, Coherence, and Coherence Transfer

Homogeneous and Statistical Mixture of States of a System

Density Operator

Time Evolution of the Density Operator

Matrix Representation of the Unitary Transformation Superoperator

Matrix Representation of the Commutator Superoperator

Partial Density Matrix

Density Operator Expression Using Irreducible Tensor Operator

Density Matrix Treatment of an Optical Experiment

References

Problems

13. Nuclear Magnetic Resonance Spectroscopy

Nuclear Spin of Different Elements

Excited-State Nuclear Spin

Nuclear Spin Angular Momentum and Magnetic Moment

Zeeman Splitting of Nuclear Energy Levels

Larmor Precession of Angular Momentum

Transition Torque Mechanics

Spin Population and NMR Transition

Static Field Dependence of Signal Intensity

Nuclear Receptivity

Macroscopic Magnetization

Bloch Equations and Relaxation Times

The Rotating Frame

Bloch Equations in the Rotating Frame

RF Pulse and Signal Generation

Origin of Chemical Shift: Local Shielding

Long-Range Shielding

Ring Current Effect,

Electric Field Effect,

Bond Magnetic Anisotropy,

Shielding by Hydrogen Bonding,

Hyperfine Shielding,

Shielding from Solvent Effect,

Chemical Shift Scale

Spin-Spin Coupling

Basic Theory of the Origin of Nuclear Spin Relaxation

Mechanism of Spin Relaxation

Shielding Anisotropy

Spin-Rotation Interaction

Scalar Interaction

Paramagnetic Effect

Dipole-Dipole Interaction

Dipolar Interaction and Cross Relaxation

Effect of Dipolar Interaction on Nuclear Relaxation

Spin Cross Relaxation: Solomon Equations

Nuclear Overhauser Effect (NOE)

Positive and Negative NOE

Direct and Indirect NOE Transfer

Rotating Frame Overhauser Effect

Transient NOE

Chemical Exchange

Effect of Chemical Exchange on Line Shape

One-Sided Chemical Reaction

Hahn Echo and Double Resonance

Echo Modulation and *-spectroscopy*

Heteronuclear *-spectroscopy*

Polarization Transfer (INEPT and Refocused INEPT)

Two-Dimensional *-resolved Spectroscopy*

Absence of Coherence Transfer in 2D *-spectroscopy*

2D *-spectroscopy in Strong Coupling Limit*

Density Matrix Method in NMR

Outline of the Density Matrix Apparatus in NMR

Expression of Nuclear Spin Density Operators

Transformations of Product Operators

Homonuclear Correlation Spectroscopy (COSY)

Relayed Correlation Spectroscopy (Relay COSY)

Total Correlation Spectroscopy (TOCSY)

2D Nuclear Overhauser Enhancement Spectroscopy (NOESY)

Pure Exchange Spectroscopy (EXSY)

Phase Cycling, Spurious Signals, and Coherence Transfer

Coherence Transfer Pathways

Magnetic Field Gradient Pulse

Heteronuclear Correlation Spectroscopy

3D NMR

Dissection of a 3D Spectrum

NOESY-[^{1}H-^{15}N]HSQC

Triple-Resonance 3D Spectroscopy

Calculation of 3D Molecular Structure

References

Problems

## Author(s)

### Biography

**Abani K. Bhuyan** has been in the Chemistry faculty at the University of Hyderabad since 2000, and is currently a Senior Professor of Physical Chemistry. He received his Ph.D. in Molecular Biophysics from the University of Pennsylvania in 1995 and was a Visiting Fellow at Tata Institute of Fundamental Research from 1995 to 2000.