1. Lectures on Modern Physics
Jiunn-Ren Roan
12 Oct. 2007

2. Chemical Physics What Is Chemical Physics?
Methods of Computer Simulation
Molecular Dynamics
Monte Carlo Method
Gas Phase Dynamics and Structure
Atomic Spectra
Molecular Spectra: General Features
Molecular Spectra: Separation of Electronic and Nuclear
Motion
Molecular Spectra: Rotational Spectra
Molecular Spectra: Vibrational Spectra
Molecular Spectra: Rotation-Vibration Spectra
Molecular Spectra: Electronic Transitions

3. Chemical Physics Appendix A Appendix B References
Solving the Harmonic Schrödinger Equation
Appendix B
Vibration-Rotation Interaction
References

4. What Is Chemical Physics?
Chemical physics, according to Wikipedia’s definition, is
“a subdiscipline of physics that investigates physicochemical phenomena
using techniques from atomic and molecular physics and condensed matter
physics”
and is
“the branch of physics that studies chemical processes from the point of
view of physics.”
It is
“distinct from physical chemistry in that it focuses more on the characteristic
elements and theories of physics.”
but
“the distinction between the two fields is vague, and workers often practice
in each field during the course of their research.”
For this course, Wikipedia’s definition is too broad to land us on specific topics
in chemical physics. An alternative way to find an answer to the question posed
above is to see what kind of research chemical physicists do.

5. Theoretical methods and algorithms
What Is Chemical Physics?
The Journal of Chemical Physics, a major journal in this field, divides its contents
into six sections:
Theoretical methods and algorithms
Gas phase dynamics and structure: Spectroscopy, molecular
interactions, scattering, and photochemistry
Condensed phase dynamics, structure, and thermodynamics:
Spectroscopy, reactions, and relaxation
Surfaces, interfaces, and materials
Polymers, biopolymers, and complex systems
Biological molecules, biopolymers, and biological systems
This list better outlines the major themes of chemical physics. We shall discuss
in this lecture some basic knowledge and fundamental issues in the first four
topics and leave the last two topics to the lecture on soft condensed matter and
biophysics.

6. Methods of Computer Simulation
Computer simulation can directly connect the microscopic details of a system
to macroscopic properties of experimental interest. It is not only academically
interesting but also technologically useful: When it is difficult or impossible to
carry out experiments under extreme conditions, computer simulation would
be perfectly feasible.
Even when experiments are feasible, results of computer simulations may be
compared with those of real experiments, offering tests for the underlying
model used in the simulations. Computer simulations also constitute tests for
approximate theories, because simulations provide essentially exact results for
problems that would otherwise only be soluble by approximate methods. Being
able to test models or theories, computer simulations are thus often called
“computer experiments”.
Two popular simulation methods are molecular dynamics (MD) and Monte
Carlo (MC) method. A molecular dynamics simulation is in principle entirely
deterministic in nature. By contrast, an essential part of any MC simulation is
a probability element. MD has the great advantage that it allows the study of
time-dependent phenomena, but if static properties alone are required, MC
method is often more efficient and useful.

7. Real System Model Experimental Results Theoretical Predictions
Methods of Computer Simulation
Real
System
Model
Experimental
Results
Theoretical
Predictions
Exact Results
for Model
Tests of
Models
Theories
Make
Perform
Experiments
Carry out
Computer
Simulations
Construct
Approximate
Compare
Adapted from M. P. Allen and D. J. Tildesley., Computer Simulation of Liquids, Oxford University Press, New York (1987).

8. Methods of Computer Simulation
Molecular Dynamics
MD was first devised in the late 1950s by Alder and Wainwright to study
systems of particles with hard cores and was extended in 1960s by Rahman to
systems of particles that interact through continuous potentials. In a nutshell,
what MD does is to solve the Newtonian equations of motion,
for a system of particles and then compute from the solution (i.e., particles’
trajectories) various thermodynamic quantities. The spatial derivatives in
these equations make the way by which the solution is obtained qualitatively
different, depending on whether or not the potential energy V is a continuous
function of particle positions.
The potential energy V in the equations of motion may be divided into terms
depending on the coordinates of individual atoms (effect of an external field),
pairs, triples etc.:
Because computation of the N(N-1)(N-2)/6 three-body terms will be very
time-consuming, these (and higher-order) terms are only rarely included.

9. defining an effective pair potential:
Methods of Computer Simulation
The effects of the three-body and higher-order terms can be partially included by
defining an effective pair potential:
(rij is the distance between particles i and j). A consequence of this approximation
is that the effective pair potential needed to reproduce experimental data may
depend on the density, temperature etc. while the true two-body potential does not.
Alder and Wainwright used hard-core (hard-disc in 2D, hard-sphere in 3D) and
square-well potentials:
Although these potentials are unrealistic, they are very simple and convenient to
use in simulation. s s1 s2 r e

10. and s = 0.34 nm). It has a minimum –e at r = 21/6s.
Methods of Computer Simulation
To simulate atoms in liquid argon, Rahman used the famous Lennard-Jones
12-6 potential:
which provides a reasonable description of the properties of argon (e /kB= 120 K
and s = 0.34 nm). It has a minimum –e at r = 21/6s.
Experimental data
Lennard-Jones potential
From M. P. Allen and D. J. Tildesley., Computer Simulation of Liquids, Oxford University Press, New York (1987).

11. functions of distance) proceeds according to the following scheme:
Methods of Computer Simulation
MD simulation of molecules interacting via hard potentials (i.e., discontinuous
functions of distance) proceeds according to the following scheme:
locate next collision;
move all particles forward until collision occurs;
implement collision dynamics for the colliding pair;
calculate any properties of interest, ready for averaging, before
returning to (a).
Whenever the distance between two particles becomes equal to a point of
discontinuity in the potential, a “collision” (in a broad sense) occurs and the
particle velocities change suddenly according to the particular model under study.
For hard spheres, steps (a) and (b) require the solution of the equation for the
time t+tij when the next collision occurs:
where and are known at time t. If this equation has
two real roots, then the smaller positive root,
gives the time when the next collision will occur.

12. for the particles to cross the boundary and remain bound.
Methods of Computer Simulation
The collision dynamics needed in (c) is determined by conservation of energy
and linear momentum (assuming that the colliding particles are of equal mass):
It is straightforward to derive from the conservation laws the following identities:
Because the velocity change is equal to impulse and impulse must be along
the vector joining the centers, , the change in velocity at a collision is
For the square-well potential, there are two distances where collisions occur. At
the inner distance s1, collisions obey normal hard-core dynamics. At the outer
distance s2, if the two particles are approaching one another, the potential energy
drops and the kinetic energy increases a corresponding amount; if the particles
are separating, then either the particles suffer a loss in the kinetic energy to
compensate the rise in the potential energy, or the kinetic energy is insufficient
for the particles to cross the boundary and remain bound.

13. motion to be solved are written in the form
Methods of Computer Simulation
Instead of evolving the system on a collision-by-collision basis, MD simulation
for continuous potentials is carried out on a step-by-step basis: From positions,
velocities and other dynamic information at time t (and, if necessary, at earlier
times) one obtains the positions, velocities etc. at a later time t+dt, to a sufficient
degree of accuracy. The detailed scheme will depend on how the positions and
velocities at a later time are determined, namely, on which algorithm is used.
The choice of dt depends on the algorithm as well, but in general it should be
much smaller than the typical time taken for a particle to travel its own length.
In the algorithm of the commonest choice, the Verlet algorithm (also called the
central-difference algorithm) developed by Verlet in 1967, the equations of
motion to be solved are written in the form
Expanding about , we have
This equation allows us to compute the trajectories without using velocities, which
can be readily obtained when needed:

14. Desirable qualities for a successful algorithm include
Methods of Computer Simulation
Desirable qualities for a successful algorithm include
It should be fast and require little memory.
It should permit the use of a long time step dt.
It should duplicate the classical trajectory as closely as possible.
It should satisfy the known conservation laws and be time-reversible.
It should be simple in form and easy to program.
The calculation of forces usually is very time-consuming, so it is very important
to cover a given period of simulation time by as few steps as possible, that is, (b)
should be given a very high priority. However, the larger dt is, the less accurately
will (c) and (d) be satisfied. All simulations involve a trade-off between economy
and accuracy: a good algorithm permits a large time step, while preserving, to an
acceptable accuracy, conservation laws.
The Verlet algorithm is exactly reversible in time and, given conservative forces,
is guaranteed to conserve linear momentum. It has been shown to have excellent
energy-conserving properties even with long time steps. Its main weakness lies
in the handling of velocities: whereas positions are subject to errors of order (dt)4,
velocities suffer from errors of order (dt)2.

15. Methods of Computer Simulation
Monte Carlo Method
The Maxwell-Boltzmann velocity distribution at equilibrium temperature T,
gives the probability density of finding a particle with velocity . Its form
suggests (and it can be shown) that for a system of N particles under
thermodynamic equilibrium, the probability density of finding the system in
a specific state G ≡ (r1, r2, …, rN, p1, p2, …, pN) is given by
where
is a normalization factor.
Knowing the probability density, it is natural to suppose that the experimentally
observed value of a macroscopic property X is equal to its expected value:
Since this supposition is correct in many cases, we must know how to compute
the 6N-dimensional integration if we want to compare experimental results and
theoretical predictions.

16. was a gambler, as “Monte Carlo” method.
Methods of Computer Simulation
To efficiently and accurately compute the 6N-dimensional integration for large
N, the only practical method is the method developed by von Neumann, Ulam,
and Metropolis for the Manhattan Project and named after Ulam’s uncle, who
was a gambler, as “Monte Carlo” method.
The spirit of the MC method is the “hit and miss integration”,
illustrated here by its use in finding the area of the colored
region: Randomly generate a number of trial shots in the
square and count the number of shots that hit the colored
region whose area is being computed; then
Thus, to evaluate the two-dimensional integral
for the function
all one has to do is to “sample” the function randomly over the entire square, sum
up all the sampled values, and do a proper normalization.

17. For the 6N-dimensional integral
Methods of Computer Simulation
For the 6N-dimensional integral
the simple “hit and miss” approach leads to
where US = uniformly sampled points. Unfortunately, this approach is bound
to fail, because the factor
decays very quickly as G moves away from the state that minimizes the energy
and, therefore, most shots will only make an infinitesimal contribution. To solve
this problem, the shots cannot be random, instead they should be distributed
according to the probability density P(G), so that regions where P(G) is large are
sampled more frequently while those where P(G) is small are less sampled.
When this is achieved, the integral has a very simple form:
where is the number of sampled points.

18. Randomly choose a new state Gn from the neighboring states of the
Methods of Computer Simulation
There are different schemes to make the sampling consistent with the probability
density. The most popular scheme is the following Metropolis algorithm:
Randomly choose a new state Gn from the neighboring states of the
present state Gm.
Compute the energy difference between them: DE = E(Gn)-E(Gm).
If DE ≤ 0, accept the new state and return to (a).
Otherwise, accept the new state with a probability exp[-DE/kBT] and
return to (a)
Here the “neighboring states” of a given state can be understood as states that are
“not very different from” the given state. More specifically, if the macroscopic
property X depends only on particle positions,
where rN ≡ (r1, r2, ..., rN), then the consecutive states rNi and rNi+1 in
are “neighboring states” in the sense that the maximum displacement drmax in the
“move” rNi+1-rNi = (dr1, dr2, ..., drN) is not too small (so that a large fraction of
moves are acceptable) or too large (so that nearly all the trial moves are rejected).
Note that although in principle an MC move can involve more than one particle,
most MC simulations choose to displace only one particle in each move.

19. Gas Phase Dynamics and Structure
Atomic Spectra
When bombarding a piece of metal with electrons of sufficient energy to produce
x-rays, an inner-shell electron of a metal atom is knocked out. By falling into
the vacancy left by the knocked-out electron, the remaining electrons in the atom
emit radiation that has wavelengths on the order of to 10-8 m (x-ray) and
energies of 102 to 105 eV.
Since the inner-shell energies change from metal to metal, the wavelengths of
the x-ray thus produced vary with the target substance,
forming what is called the characteristic x-ray spectrum
of the target element. The sharp peaks superimposed on
the continuous spectrum (which is nearly independent of
the target element) generated by the bombardments belong
to this characteristic x-ray spectrum. The designation of
these peaks are

20. number is nearly linear:
Gas Phase Dynamics and Structure
Moseley found that the frequencies of the most energetic lines versus atomic
number is nearly linear:
where R = cm-1 is the Rydberg constant and s is an empirical constant
that can be interpreted as a screening
constant: its value is the amount of nuclear
charge screened or shielded from the
electron involved in the transition by the
other electrons in the atom. For the Ka
line, experimental data give s = For
convenience, s = 1 may be used quite
satisfactorily.

21. Because of the selection rules,
Gas Phase Dynamics and Structure
Unlike the inner-shell spectra, the spectra arising from transitions of outer-shell
electrons are similar for elements of the same group of the periodic table. The
simplest are those of the alkali metals. In ordinary atomic spectra, the inner-shell
remains intact, so the spectra of alkali metals are similar to that of hydrogen.
Because of the selection rules,
the spectrum of the hydrogen atom lacks certain transitions between energy levels
of different n :
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

22. among lowest terms of the sodium atom are
Gas Phase Dynamics and Structure
The spectra of alkali metals follow the same selection rules. The transitions
among lowest terms of the sodium atom are
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
Na D-line
doublet

23. Molecular Spectra: General Features
Gas Phase Dynamics and Structure
Molecular Spectra: General Features
Depending on the spectrometer’s resolution, the observed molecular spectrum
is composed of
At low resolution: Continuous bands, each for a pair of electron states.
At high resolution: Many individual lines for each electronic transition.
However, even at the highest resolving power, the short-wavelength part of
molecular spectra appears to comprise continuums.
From M. Karplus and R. N. Porter, Atoms and
Molecules, Benjamin/Cummings, Menlo Park (1970).
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
From M. Karplus and R. N. Porter, Atoms and
Molecules, Benjamin/Cummings, Menlo Park (1970).

24. X1Sg+ and for the second excited state B1Su+:
Gas Phase Dynamics and Structure
On the other hand, the long-wavelength part of molecular spectra consists of
At low resolution:
At high resolution:
The standard symbols for electronic potentials are X (the ground state), A (the
first excited state), B (the second excited state), and so on. All the above features
can be explained by considering the potential-energy curves for the ground state
X1Sg+ and for the second excited state B1Su+:
From M. Karplus and R. N. Porter, Atoms and
Molecules, Benjamin/Cummings, Menlo Park (1970).
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

25. Franck-Condon principle to have high intensities.
From M. Karplus and R. N. Porter, Atoms and
Molecules, Benjamin/Cummings, Menlo Park (1970).
Gas Phase Dynamics and Structure
Transitions A, B, and C
are predicted by the
Franck-Condon principle
to have high intensities.
Transition C is a continuous
electronic spectrum from
a H2 molecule to two
unbound H atoms.

26. Molecular Spectra: Separation of Electronic and Nuclear Motion
Gas Phase Dynamics and Structure
Molecular Spectra: Separation of Electronic and Nuclear Motion
The Schrödinger equation for a diatomic molecule with N electrons is
where rN = r1, r2, …, rN, K is the kinetic energy of relative motion of the two
nuclei, Ki the kinetic energy of the ith electron, and the coulombic potential
energy V contains three parts: the electron-electron repulsion energy, the
electron-nucleus attraction energy, and the nucleus-nucleus repulsion energy.
The Born-Oppenheimer approximation considers the massive nuclei to be
stationary relative the motion of electrons and focuses on the electronic
Schrödinger equation
This equation is solved by taking the inter-nuclear distance R as a parameter
and gives, for each R, the electronic wave function y(rN; R) and energy E(R).
Under this approximation, the molecular wave function should be the product
of the electronic wave function and the nuclear wave function c(R):

27. Substituting it into the original Schrödinger equation, we obtain
Gas Phase Dynamics and Structure
Substituting it into the original Schrödinger equation, we obtain
The electronic wave function varies slowly with the inter-nuclear distance R, i.e., or
so the Schrödinger equation for the nuclear wave function is
This equation has the same form as the Schrödinger equation for the hydrogen
atom, so its solution has the same form as the H-atom wave function:
and three quantum numbers will emerge and characterize the wave function:

28. To understand the physical meaning of these quantum number, define
Gas Phase Dynamics and Structure
To understand the physical meaning of these quantum number, define
so that
and the nuclear kinetic energy is split into two parts:
The angular part has a classical analog – the kinetic energy of a rigid rotor:
(M: angular momentum) Because
the rotational energy of the rotor is quantized, M R

29. Molecular Spectra: Rotational Spectra
Gas Phase Dynamics and Structure
Molecular Spectra: Rotational Spectra
The quantized rotational energy,
where Re is the equilibrium inter-nuclear distance, gives spectral lines that
correspond to transitions between different rotational states within the same
electronic state. The selection rule for rotational spectra can be shown to be
for molecules with permanent dipole moments (e.g., HF, HCl, etc.) Thus,
the allowed frequencies are
where
is the rotational constant. Either case gives uniformly spaced lines:
0 → 1
1 → 2
2 → 3
3 → 4
4 → 5
2Be
4Be
6Be
8Be
10Be
n =
0 ← 1
1 ← 2
2 ← 3
3 ← 4
4 ← 5

30. Order-of-magnitude estimation gives
Gas Phase Dynamics and Structure
Order-of-magnitude estimation gives
so the rotational spectrum typically is found in the far infrared or microwave
region:
Measurement of the line spacing can be used to determine the equilibrium
distance Re.
Molecules without permanent dipole moments can have collision-induced dipole
moments, which allow the observation of rotational lines under right conditions.
The intensity of these collision-induced lines is proportional to the collision rate,
which in turn depends on the square of the pressure, rather than the pressure
itself as does the rotational spectrum intensity of polar molecules.
From M. Karplus and R. N. Porter, Atoms and
Molecules, Benjamin/Cummings, Menlo Park (1970).

31. The quantized rotational energy can be written in the form
Gas Phase Dynamics and Structure
The quantized rotational energy can be written in the form
where Ie is the equilibrium moment of inertia of the rotor. This form can be
directly applied to polyatomic molecules.
Consider as an example the linear polyatomic molecule OCS. The moment of
inertia about the rotational axis that passes through the center of mass is given by
The molecule has a permanent dipole moment, so its rotational energy levels are
observable in its rotational spectrum.
The selection rule is the same as that
of diatomic molecules:
The rotational lines can be used to find
the moment of inertia. However, there
are two inter-nuclear distances, rCO and
rCS, so a single value of the moment of
inertia is insufficient.
From W. J. Moore, Physical Chemistry,
Prentice-Hall, Englewood Cliffs (1972).

32. role. The classical form of the rotational energy is given by
Gas Phase Dynamics and Structure
Since inter-nuclear distances are mainly determined by electrostatic interactions,
isotope substitution will change the moment of inertia while keeping the inter-
nuclear distances nearly unchanged. Thus, by measuring the moment of inertia
for two different isotopic compositions, we can solve the two unknowns rCO and
rCS:
For nonlinear polyatomic molecules, molecular symmetry plays an important
role. The classical form of the rotational energy is given by
where Ii and Mi are the moment of inertia and component of angular momentum
with respect to the i-axis of a specific coordinate system set up by the so-called
principal axes associated with the rotating object.
From W. J. Moore, Physical Chemistry, Prentice-Hall, Englewood Cliffs (1972).

33. are different and the molecule is called an asymmetric top.
Gas Phase Dynamics and Structure
For nonlinear planar molecules such as H2O and non-planar molecules without
a three-fold or higher symmetry axis, such as CH2F2, all three moments of inertia
are different and the molecule is called an asymmetric top.
For nonlinear molecules that possess a three-fold or higher symmetry axis such
as NH3 and CH4, the symmetry axis is a principal axis, usually taken to be the z
axis, and the moments of inertia with respect to the other two principal axes, Ix
and Iy, are equal. The molecule is called a symmetric top. Conventionally, IA = Iz
and IB = Ix = Iy are used instead and the classical rotational energy becomes
The special case IA = IB occurs when the molecule has a high degree of symmetry,
e.g., CH4. Such a molecule is called a spherical top.
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

34. energy levels are given by
Gas Phase Dynamics and Structure
In the quantum world, angular momentum and its z-component are quantized
where J = 0, 1, 2, ... and K = 0, ±1, ±2, ..., ±J. Thus, the symmetric-top rotational
energy levels are given by
If the molecule has a permanent electric dipole moment in
the z axis, it exhibits a rotational spectrum. It can be shown
that the selection rule for the spectrum is
The CH4 molecule has no permanent dipole moment, so it
cannot exhibit pure rotational spectrum. On the other hand,
the NH3 molecule has a strong dipole moment and is capable
of giving a rich microwave spectrum.
From P. Helminger and W. Gordy, Phys. Rev. 188, 100 (1969).
From W. J. Moore, Physical Chemistry, Prentice-Hall, Englewood Cliffs (1972).

35. Gas Phase Dynamics and Structure
Some asymmetric-top molecules are so simple that they can be regarded as a
“pseudo-symmetric top”. An example is the molecule CH3CH2CH2Cl (n-propyl
chloride).
From W. J. Moore, Physical Chemistry, Prentice-Hall, Englewood Cliffs (1972).

36. Gas Phase Dynamics and Structure
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

37. Molecular Spectra: Vibrational Spectra
Gas Phase Dynamics and Structure
Molecular Spectra: Vibrational Spectra
Consider the radial part of the Schrödinger equation for nuclear motion and
work around Re, i.e., taking R = Re in the expression of EJ,
where
By writing
we can simplify the equation and obtain
Since R ≈ Re,
where the zero of energy is chosen at the minimum, E(Re) = 0, r = R-Re, and
so the Schrödinger equation becomes

38. In Appendix A it is shown that the energy is quantized according to
Gas Phase Dynamics and Structure
The so-called harmonic approximation neglects all but the second order term:
In Appendix A it is shown that the energy is quantized according to
where
The selection rule for vibrational spectra is
so the allowed frequency is simply n = ne. The
bond strength of a typical covalent bond is about
102 kcal/mol, i.e. 4 eV. Assuming that breaking
such a bond requires stretching it to twice as much
its equilibrium length, which is about 0.15 nm, we
can give an order-of-magnitude estimation:
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

39. wavelengths much shorter than those of rotational spectra.
Gas Phase Dynamics and Structure
Thus, vibrational spectra are normally found in the infrared region and have
wavelengths much shorter than those of rotational spectra.
Beyond the harmonic approximation, higher-order (anharmonic) terms must be
included. A potential called the Morse potential is often used for this purpose:
where De is the equilibrium dissociation energy:
De = E(∞) – E(0). Its second order term gives
It can be shown that for the Morse oscillator,
in which
and xene/c is called the first anharmonicity constant. Therefore, the energy levels
are no longer equally spaced as they are for a harmonic oscillator. As for the
the selection rule, it is much more relaxed–no “selection” at all:
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
Morse
potential
Harmonic

40. lowest vibration level is
Gas Phase Dynamics and Structure
Thus, the vibrational absorption lines from the ground state to the excited states
are
The “fundamental” line 0→1 is the most intense, and the intensity decreases very
rapidly as the order of overtone (0→2: first overtone; 0→3: second overtone, etc.)
increases.
The difference between the energy for infinite separation of the atoms and the
lowest vibration level is
This is the bond dissociation energy at 0 K.
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

41. the molecule’s vibrational degrees of freedom.
Gas Phase Dynamics and Structure
A molecule containing N atoms requires 3N coordinates to completely specify
the atoms’ positions in space. Three of them can be taken as the coordinates of
the center of the mass of the molecule. Another 3 (if the molecule is nonlinear)
or 2 (if it is linear) coordinates are needed to specify the molecule’s orientation.
Since it takes only one coordinate to define an oscillator, each of the remaining
coordinates for specifying the atoms’ relative positions is associated with an
oscillation (mode) in the relative positions. Thus, the number of vibrational modes
is 3N-6 for nonlinear molecules or 3N-5 for linear molecules. This number is
the molecule’s vibrational degrees of freedom.
For a molecule with s vibrational degrees of freedom, the kinetic and potential
energies written in terms of the remaining coordinates (x1, x2, ..., xs) have the
following forms (with harmonic approximation):
where the potential energy has been expanded around the equilibrium positions
(0, 0, ..., 0) and the zero of energy is chosen to be at the equilibrium positions
V(0, 0, ..., 0) = 0.

42. when written in terms of the new coordinates are free of cross terms:
Gas Phase Dynamics and Structure
It is possible to linearly combining the coordinates (x1, x2, ..., xs) to form a new
set of coordinates (Q1, Q2, ..., Qs), so that both the kinetic and potential energies
when written in terms of the new coordinates are free of cross terms:
When this is achieved, the new coordinates are called normal coordinates and
the vibrating molecule can be regarded as a collection of s independent oscillators,
called normal modes of vibration:
Each oscillator has its own quantized energy levels,
and because the modes are independent, the total vibrational energy is the sum
of the individual vibrational energies. In the harmonic approximation, the
selection rule for the i-th normal mode is
As in the diatomic case, anharmonic terms introduce overtones for a given mode,
or allow simultaneous change of several modes. These transitions are frequently
observed in polyatomic spectra, although their intensities are relatively weak.

43. Nonlinear symmetric triatomic molecules such as H2O have three modes:
Gas Phase Dynamics and Structure
For linear symmetric triatomic molecules such as CO2, there are four modes:
Since the symmetric stretching mode does not produce an oscillation in the dipole
moment of the molecule, it does not produce a band in the infrared spectrum.
This mode is said to be infrared inactive. The other three modes, on the other
hand, produce an oscillating dipole (even though CO2 in static equilibrium does
not have a permanent dipole moment), so there are two strong bands centered on
n2 and n3.
Nonlinear symmetric triatomic molecules such as H2O have three modes:
n1 = cm n2 = cm n3 = cm-1
From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).
CO2
From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).
n1 = cm n2 = cm n3 = cm-1
H2O

44. simultaneous change of two or more modes.
Gas Phase Dynamics and Structure
All the three modes produce an oscillating dipole moment, three fundamental
bands centered on n1, n2, and n3 appear in the infrared region. In addition to
these bands, overtones due to anharmonicity and combination bands due to
simultaneous change of two or more modes.
From G. W. Castellan, Physical Chemistry, 3rd ed. Addison-Wesley, Reading (1983)

45. It is found that the vibrational frequencies for a
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
Gas Phase Dynamics and Structure
It is found that the vibrational frequencies for a
given chemical bond are relatively invariant from
molecule to molecule. Thus, the characteristic
frequencies of various chemical groups found in
the IR spectrum of an unknown molecule can be
used to identify the molecule.
Because the exact position depends on the type
of compound, it requires some experience. It is
also possible to use infrared spectroscopy to
determine quantitatively the amount of various
substances present in mixtures.

46. Gas Phase Dynamics and Structure
From G. W. Castellan, Physical Chemistry, 3rd ed. Addison-Wesley, Reading (1983)
Acetone, CH3-CO-CH3

47. Molecular Spectra: Rotation-Vibration Spectra
Gas Phase Dynamics and Structure
Molecular Spectra: Rotation-Vibration Spectra
The energy level for nuclear motion is given by
For simultaneous vibrational and rotational transitions,
the same selection rules hold
and since ne is generally larger than Be, the vibrational part determines whether
a transition is absorption,
or emission,
whereas the rotational part determines to which group of lines, the so-called P
and R branches associated with a single vibrational transition, the transition
belongs.

48. state, which in turn is proportional to the Boltzmann factor
Gas Phase Dynamics and Structure
The intensity of a line is proportional to the number of molecules in the initial
state, which in turn is proportional to the Boltzmann factor
where gvJ is the degeneracy of the state. Each rotational state contains 2J+1
degenerate states, corresponding to M = -J, -J+1, …, J, so gvJ = 2J+1. Every
branch therefore has a most intense line somewhere away from the band center
J = 0.
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
Most intense

49. bending vibration of CO2 offers an example of Q branch.
Gas Phase Dynamics and Structure
For some polyatomic molecules and in the rotation-vibration spectra of diatomic
molecules with nonzero electronic momentum about the molecular axis, transitions
with DJ = 0 are allowed. The associated spectral lines are called Q branch. The
bending vibration of CO2 offers an example of Q branch.
From G. W. Castellan, Physical Chemistry, 3rd ed. Addison-Wesley, Reading (1983)

50. Gas Phase Dynamics and Structure
Because a rotating molecule experiences a centrifugal force which stretches the
molecule so that the effective equilibrium inter-nuclear distance increases with J,
rotation indeed interacts with vibration. Appendix B shows that after including
the lowest-order correction due to vibration-rotation interaction, we have (Morse
potential is assumed)
The effect of vibration-rotation interaction on rotation is to narrow the spacing
between rotational energy levels. This is a natural consequence of increasing the
moment of inertia by increasing the equilibrium inter-nuclear distance. As for
its effect on vibration, the interaction reduces the vibrational frequency when the
oscillation is sufficiently anharmonic (xene > Be), which is the case for most
molecules. This is because the centrifugal force drives the molecule into the
“softer” region of the potential. Also note that xene  ae for most molecules,
so the effect of vibration-rotation interaction normally is masked by the effect
of anharmonicity.

51. Gas Phase Dynamics and Structure
Without vibration-
rotation interaction
With vibration-
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

52. Molecular Spectra: Electronic Transitions
Gas Phase Dynamics and Structure
Molecular Spectra: Electronic Transitions
Because electrons are much less
massive than nuclei, the motion
of electrons is much faster than
that of nuclei, and electronic
transitions occur in times too short
for any appreciable nuclear motion
to take place. Consequently, the
inter-nuclear distance in the final
state is the same as it was in the
initial state and the transition is
represented by a vertical line
between the initial and final
potential-energy curves. This
is called the Franck-Condon
principle. Accordingly, among
the possible transitions, the most
intense are those in which the
instantaneous inter-nuclear
distance is a highly probable one
for both states.
From G. W. Castellan, Physical Chemistry, 3rd ed. Addison-Wesley, Reading (1983)

53. group of lines is called an electronic band.
Gas Phase Dynamics and Structure
Transitions can originate in any of the multitude of rotational states associated
with the initial vibrational level and can end up in either of the two rotational
states determined by the selection rule, DJ = ±1 in the final electronic and
vibrational state. Thus, a group of closely spaced lines are observed. This
group of lines is called an electronic band.
Owing to the anharmonicity of the potential-energy curve, there is no selection
rule for vibration, so transitions may occur between any two vibrational states
that match properly and each one of these possible transitions produce a band.
The collection of the bands is called a band system.
Since there are several electronic states between which transitions may occur,
the electronic spectrum of a molecule is composed of several band systems.

54. Appendix A Solving the Harmonic Schrödinger Equation
To solve the equation
first consider its asymptotic behavior.
For large r, the equation becomes
It can be readily seen by setting z(r) = exp[f(r)] that
Thus, the solution should be in the form
and the equation for the unknown function H(r) is

55. We can see immediately that the power series method will give
Appendix A
We can see immediately that the power series method will give
which must terminate somewhere; otherwise the power series will lead to a
asymptotically divergent solution. Therefore, there exists a specific n such that
This requirement gives quantized energy:
where
The equation to be solved now becomes or

56. Writing the equation in the following form
Appendix A
Writing the equation in the following form
leads us to set
where the new known function h satisfies
To eliminate the term outside the square bracket, we find that h has to be written as
so that the equation can be cast into a very simple form:
It is now rather easy to obtain the solution. From the last equation, we get
where g0 is a constant, so

57. The convention is to define the so-called n-th Hermite polynomial as
Appendix A
The convention is to define the so-called n-th Hermite polynomial as
so the solution should be written
Finally, the solution to the radial Schrödinger equation is
where z0 is a proper normalization constant.

58. Appendix B Vibration-Rotation Interaction
In the rigid rotor model, since the inter-nuclear distance R is fixed at Re, a
parameter also taken as the equilibrium distance of electronic energy E(R),
vibration and rotation become decoupled. This is unrealistic, as the rotor
is expected to respond to the influence of centrifugal force by stretching itself,
thus increasing the equilibrium inter-nuclear distance. This Appendix derives
the lowest-order correction to the energy of nuclear motion.
Without taking R = Re in the rotor term, EJ, in the radial Schrödinger equation
for nuclear motion,
we have
Thus, the effective potential is E(R) augmented by a centrifugal term:

59. It gives to the lowest order
Appendix B
The new equilibrium distance Re' that minimizes Eeff(R) is given by the condition
It gives to the lowest order so
where
When the inter-nuclear distance oscillates around R = Re', the effective potential
can be written as

60. Substitution of the formula for Re' into Eeff(Re') gives
Appendix B
where
Substitution of the formula for Re' into Eeff(Re') gives
So we obtain, after a little algebra,

61. Now the radial Schrödinger equation can be written as
Appendix B
Now the radial Schrödinger equation can be written as
where
is the potential well in which the diatomic molecule vibrates.
Assume that this potential well is well approximated by a Morse potential. Then
the vibrational energy levels are given by
Substituting the expressions for ke', he', and Re' into ne' and xe', we get

62. So the vibrational energy is
Appendix B
So the vibrational energy is
where
We finally have the expression that includes vibration-rotation interaction and
lowest-order anharmonicity:

63. References
1. H. D. Young and R. A. Freedman, Sears and Zemansky’s University Physics
(Pearson, 2008) 12th ed.
2. M. Karplus and R. N. Porter, Atoms and Molecules (Benjamin, 1970).
3. D. A. McQuarrie, Quantum Chemistry (Oxford University Press, 1983).
4. I. N. Levine, Quantum Chemistry (Prentice Hall, 1991) 4th ed.
5. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University
Press, 1987)
6. J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, 1986)
2nd ed.
7. W. J. Moore, Physical Chemistry (Prentice-Hall, 1972) 4th ed.
8. G. W. Castellan, Physical Chemistry (Addison-Wesley, 1983) 3rd ed.