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

Lectures on Modern Physics

Jiunn-Ren Roan

12 Oct. 2007

Chemical PhysicsWhat Is Chemical Physics?

Methods of Computer SimulationMolecular DynamicsMonte Carlo Method

Gas Phase Dynamics and StructureAtomic SpectraMolecular Spectra: General FeaturesMolecular Spectra: Separation of Electronic and Nuclear

MotionMolecular Spectra: Rotational SpectraMolecular Spectra: Vibrational SpectraMolecular Spectra: Rotation-Vibration SpectraMolecular Spectra: Electronic Transitions

Chemical PhysicsAppendix A

Solving the Harmonic Schrödinger Equation

Appendix BVibration-Rotation Interaction

References

Chemical physics, according to Wikipedia’s definition, is

“a subdiscipline of physics that investigates physicochemical phenomenausing techniques from atomic and molecular physics and condensed matterphysics”

and is

“the branch of physics that studies chemical processes from the point ofview of physics.”

It is

“distinct from physical chemistry in that it focuses more on the characteristicelements and theories of physics.”

but

“the distinction between the two fields is vague, and workers often practicein each field during the course of their research.”

For this course, Wikipedia’s definition is too broad to land us on specific topicsin chemical physics. An alternative way to find an answer to the question posedabove is to see what kind of research chemical physicists do.

What Is Chemical Physics?

http://en.wikipedia.org/wiki/Chemical_physics

http://en.wikipedia.org/wiki/Chemical_physics

The Journal of Chemical Physics, a major journal in this field, divides its contentsinto six sections:

1. Theoretical methods and algorithms2. Gas phase dynamics and structure: Spectroscopy, molecular

interactions, scattering, and photochemistry3. Condensed phase dynamics, structure, and thermodynamics:

Spectroscopy, reactions, and relaxation4. Surfaces, interfaces, and materials5. Polymers, biopolymers, and complex systems6. Biological molecules, biopolymers, and biological systems

This list better outlines the major themes of chemical physics. We shall discussin this lecture some basic knowledge and fundamental issues in the first fourtopics and leave the last two topics to the lecture on soft condensed matter andbiophysics.

What Is Chemical Physics?

http://scitation.aip.org/dbt/dbt.jsp?KEY=JCPSA6&Volume=CURVOL&Issue=CURISS

Methods of Computer SimulationComputer simulation can directly connect the microscopic details of a systemto macroscopic properties of experimental interest. It is not only academicallyinteresting but also technologically useful: When it is difficult or impossible tocarry out experiments under extreme conditions, computer simulation wouldbe perfectly feasible.

Even when experiments are feasible, results of computer simulations may becompared with those of real experiments, offering tests for the underlyingmodel used in the simulations. Computer simulations also constitute tests forapproximate theories, because simulations provide essentially exact results forproblems that would otherwise only be soluble by approximate methods. Beingable to test models or theories, computer simulations are thus often called“computer experiments”.

Two popular simulation methods are molecular dynamics (MD) and MonteCarlo (MC) method. A molecular dynamics simulation is in principle entirelydeterministic in nature. By contrast, an essential part of any MC simulation isa probability element. MD has the great advantage that it allows the study oftime-dependent phenomena, but if static properties alone are required, MCmethod is often more efficient and useful.

Methods of Computer Simulation

RealSystem

ModelSystem

ExperimentalResults

TheoreticalPredictions

Exact Resultsfor Model

Tests ofModels

Tests ofTheories

MakeModels

PerformExperiments

Carry outComputer

Simulations

ConstructApproximate

Theories

Compare Compare

Adapted from M. P. Allen and D. J. Tildesley., Computer Simulation of Liquids, Oxford University Press, New York (1987).

Molecular DynamicsMD was first devised in the late 1950s by Alder and Wainwright to studysystems of particles with hard cores and was extended in 1960s by Rahman tosystems 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 inthese equations make the way by which the solution is obtained qualitativelydifferent, depending on whether or not the potential energy V is a continuousfunction of particle positions.

The potential energy V in the equations of motion may be divided into termsdepending 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 verytime-consuming, these (and higher-order) terms are only rarely included.


The effects of the three-body and higher-order terms can be partially included bydefining an effective pair potential:

(rij is the distance between particles i and j). A consequence of this approximationis that the effective pair potential needed to reproduce experimental data maydepend 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) andsquare-well potentials:

Although these potentials are unrealistic, they are very simple and convenient touse in simulation.


r

To simulate atoms in liquid argon, Rahman used the famous Lennard-Jones12-6 potential:

which provides a reasonable description of the properties of argon ( /kB= 120 Kand = 0.34 nm). It has a minimum – at r = 21/6.


Experimental data

Lennard-Jones potential

From M. P. Allen and D. J. Tildesley., Computer Simulation of Liquids, Oxford University Press, New York (1987).

MD simulation of molecules interacting via hard potentials (i.e., discontinuousfunctions of distance) proceeds according to the following scheme:

(a) locate next collision;(b) move all particles forward until collision occurs;(c) implement collision dynamics for the colliding pair;(d) calculate any properties of interest, ready for averaging, before

returning to (a).Whenever the distance between two particles becomes equal to a point ofdiscontinuity in the potential, a “collision” (in a broad sense) occurs and theparticle velocities change suddenly according to the particular model under study.

For hard spheres, steps (a) and (b) require the solution of the equation for thetime t+tij when the next collision occurs:

where and are known at time t. If this equation hastwo real roots, then the smaller positive root,

gives the time when the next collision will occur.


The collision dynamics needed in (c) is determined by conservation of energyand 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 alongthe vector joining the centers, , the change in velocity at a collision is

For the square-well potential, there are two distances where collisions occur. Atthe inner distance 1, collisions obey normal hard-core dynamics. At the outerdistance 2, if the two particles are approaching one another, the potential energydrops and the kinetic energy increases a corresponding amount; if the particlesare separating, then either the particles suffer a loss in the kinetic energy tocompensate the rise in the potential energy, or the kinetic energy is insufficientfor the particles to cross the boundary and remain bound.


Instead of evolving the system on a collision-by-collision basis, MD simulationfor 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 earliertimes) one obtains the positions, velocities etc. at a later time t+t, to a sufficientdegree of accuracy. The detailed scheme will depend on how the positions andvelocities at a later time are determined, namely, on which algorithm is used.The choice of t depends on the algorithm as well, but in general it should bemuch 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 thecentral-difference algorithm) developed by Verlet in 1967, the equations ofmotion to be solved are written in the form

Expanding about , we have

This equation allows us to compute the trajectories without using velocities, whichcan be readily obtained when needed:


http://en.wikipedia.org/wiki/Loup_Verlet

Desirable qualities for a successful algorithm include(a) It should be fast and require little memory.(b) It should permit the use of a long time step t.(c) It should duplicate the classical trajectory as closely as possible.(d) It should satisfy the known conservation laws and be time-reversible.(e) It should be simple in form and easy to program.

The calculation of forces usually is very time-consuming, so it is very importantto 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 t is, the less accuratelywill (c) and (d) be satisfied. All simulations involve a trade-off between economyand accuracy: a good algorithm permits a large time step, while preserving, to anacceptable 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 excellentenergy-conserving properties even with long time steps. Its main weakness liesin the handling of velocities: whereas positions are subject to errors of order (t)4,velocities suffer from errors of order (t)2.


Monte Carlo MethodThe Maxwell-Boltzmann velocity distribution at equilibrium temperature T,

gives the probability density of finding a particle with velocity . Its formsuggests (and it can be shown) that for a system of N particles underthermodynamic equilibrium, the probability density of finding the system ina specific state ≡ (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 experimentallyobserved 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 computethe 6N-dimensional integration if we want to compare experimental results andtheoretical predictions.


To efficiently and accurately compute the 6N-dimensional integration for largeN, the only practical method is the method developed by von Neumann, Ulam,and Metropolis for the Manhattan Project and named after Ulam’s uncle, whowas 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 coloredregion: Randomly generate a number of trial shots in thesquare and count the number of shots that hit the coloredregion 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, sumup all the sampled values, and do a proper normalization.


http://en.wikipedia.org/wiki/John_von_Neumann

http://en.wikipedia.org/wiki/Stanislaw_Marcin_Ulam

http://en.wikipedia.org/wiki/Nicholas_Metropolis

For the 6N-dimensional integral

the simple “hit and miss” approach leads to

where US = uniformly sampled points. Unfortunately, this approach is boundto fail, because the factor

decays very quickly as moves away from the state that minimizes the energyand, therefore, most shots will only make an infinitesimal contribution. To solvethis problem, the shots cannot be random, instead they should be distributed according to the probability density P(), so that regions where P() is large aresampled more frequently while those where P() is small are less sampled.When this is achieved, the integral has a very simple form:

where is the number of sampled points.


There are different schemes to make the sampling consistent with the probabilitydensity. The most popular scheme is the following Metropolis algorithm:

(a) Randomly choose a new state n from the neighboring states of thepresent state m.

(b) Compute the energy difference between them: E = E(n)-E(m).(c) If E ≤ 0, accept the new state and return to (a).(d) Otherwise, accept the new state with a probability exp[-E/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 macroscopicproperty X depends only on particle positions,

where rN ≡ (r1, r2, ..., rN), then the consecutive states rNi and rN

i+1 in

are “neighboring states” in the sense that the maximum displacement rmax in the“move” rN

i+1-rNi = (r1, r2, ..., rN) 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.


Gas Phase Dynamics and Structure

When bombarding a piece of metal with electrons of sufficient energy to producex-rays, an inner-shell electron of a metal atom is knocked out. By falling intothe vacancy left by the knocked-out electron, the remaining electrons in the atomemit radiation that has wavelengths on the order of 10-11 to 10-8 m (x-ray) andenergies of 102 to 105 eV.

Since the inner-shell energies change from metal to metal, the wavelengths ofthe 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 onthe continuous spectrum (which is nearly independent ofthe target element) generated by the bombardments belongto this characteristic x-ray spectrum. The designation ofthese peaks are

Atomic Spectra


Moseley found that the frequencies of the most energetic lines versus atomicnumber is nearly linear:

where R = 109677.57 cm-1 is the Rydberg constant and is an empirical constantthat can be interpreted as a screening constant: its value is the amount of nuclearcharge screened or shielded from theelectron involved in the transition by theother electrons in the atom. For the K line, experimental data give = 1.13. Forconvenience, = 1 may be used quitesatisfactorily.

From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).


Unlike the inner-shell spectra, the spectra arising from transitions of outer-shellelectrons are similar for elements of the same group of the periodic table. Thesimplest are those of the alkali metals. In ordinary atomic spectra, the inner-shellremains 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 levelsof different n :


The spectra of alkali metals follow the same selection rules. The transitionsamong lowest terms of the sodium atom are


Na D-linedoublet

From M. Karplus and R. N. Porter, Atoms andMolecules, Benjamin/Cummings, Menlo Park (1970).



Depending on the spectrometer’s resolution, the observed molecular spectrumis 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 ofmolecular spectra appears to comprise continuums.


Molecular Spectra: General Features

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 (thefirst excited state), B (the second excited state), and so on. All the above featurescan be explained by considering the potential-energy curves for the ground stateX1g

+ and for the second excited state B1u+:






Transitions A, B, and Care predicted by theFranck-Condon principleto have high intensities. Transition C is a continuouselectronic spectrum froma H2 molecule to twounbound H atoms.

Molecular Spectra: Separation of Electronic and Nuclear MotionThe 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 twonuclei, Ki the kinetic energy of the ith electron, and the coulombic potentialenergy V contains three parts: the electron-electron repulsion energy, theelectron-nucleus attraction energy, and the nucleus-nucleus repulsion energy.

The Born-Oppenheimer approximation considers the massive nuclei to bestationary relative the motion of electrons and focuses on the electronicSchrödinger equation

This equation is solved by taking the inter-nuclear distance R as a parameterand gives, for each R, the electronic wave function (rN; R) and energy E(R).

Under this approximation, the molecular wave function should be the productof the electronic wave function and the nuclear wave function (R):


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 hydrogenatom, so its solution has the same form as the H-atom wave function:

and three quantum numbers will emerge and characterize the wave function:


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


Molecular Spectra: Rotational SpectraThe quantized rotational energy,

where Re is the equilibrium inter-nuclear distance, gives spectral lines thatcorrespond to transitions between different rotational states within the sameelectronic 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

0 ← 1 1 ← 2 2 ← 3 3 ← 4 4 ← 5

2Be 4Be 6Be 8Be 10Be


Order-of-magnitude estimation gives

so the rotational spectrum typically is found in the far infrared or microwaveregion:

Measurement of the line spacing can be used to determine the equilibriumdistance Re.

Molecules without permanent dipole moments can have collision-induced dipolemoments, 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 pressureitself as does the rotational spectrum intensity of polar molecules.


From W. J. Moore, Physical Chemistry,Prentice-Hall, Englewood Cliffs (1972).


The quantized rotational energy can be written in the form

where Ie is the equilibrium moment of inertia of the rotor. This form can bedirectly applied to polyatomic molecules.

Consider as an example the linear polyatomic molecule OCS. The moment ofinertia about the rotational axis that passes through the center of mass is givenby

The molecule has a permanent dipole moment, so its rotational energy levels areobservable in its rotational spectrum.The selection rule is the same as thatof diatomic molecules:

The rotational lines can be used to findthe moment of inertia. However, thereare two inter-nuclear distances, rCO andrCS, so a single value of the moment ofinertia is insufficient.


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 inertiafor two different isotopic compositions, we can solve the two unknowns rCO andrCS:

For nonlinear polyatomic molecules, molecular symmetry plays an importantrole. The classical form of the rotational energy is given by

where Ii and Mi are the moment of inertia and component of angular momentumwith respect to the i-axis of a specific coordinate system set up by the so-calledprincipal axes associated with the rotating object.

From W. J. Moore, Physical Chemistry, Prentice-Hall, Englewood Cliffs (1972).



For nonlinear planar molecules such as H2O and non-planar molecules withouta three-fold or higher symmetry axis, such as CH2F2, all three moments of inertiaare different and the molecule is called an asymmetric top.

For nonlinear molecules that possess a three-fold or higher symmetry axis suchas NH3 and CH4, the symmetry axis is a principal axis, usually taken to be the zaxis, 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.



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 rotationalenergy levels are given by

If the molecule has a permanent electric dipole moment inthe z axis, it exhibits a rotational spectrum. It can be shownthat the selection rule for the spectrum is

The CH4 molecule has no permanent dipole moment, so itcannot exhibit pure rotational spectrum. On the other hand,the NH3 molecule has a strong dipole moment and is capableof giving a rich microwave spectrum.

From

P. H

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. Gordy, P

hys. Rev. 188, 100 (1969).



Some asymmetric-top molecules are so simple that they can be regarded as a“pseudo-symmetric top”. An example is the molecule CH3CH2CH2Cl (n-propylchloride).



Molecular Spectra: Vibrational SpectraConsider the radial part of the Schrödinger equation for nuclear motion andwork 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-Re, and

so the Schrödinger equation becomes



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 = e. Thebond strength of a typical covalent bond is about102 kcal/mol, i.e. 4 eV. Assuming that breakingsuch a bond requires stretching it to twice as muchits equilibrium length, which is about 0.15 nm, wecan give an order-of-magnitude estimation:


Thus, vibrational spectra are normally found in the infrared region and havewavelengths much shorter than those of rotational spectra.

Beyond the harmonic approximation, higher-order (anharmonic) terms must beincluded. 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 xee/c is called the first anharmonicity constant. Therefore, the energy levelsare no longer equally spaced as they are for a harmonic oscillator. As for thethe selection rule, it is much more relaxed–no “selection” at all:



Morsepotential

Harmonicpotential

Thus, the vibrational absorption lines from the ground state to the excited statesare

The “fundamental” line 0→1 is the most intense, and the intensity decreases veryrapidly 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 thelowest vibration level is

This is the bond dissociation energy at 0 K.



A molecule containing N atoms requires 3N coordinates to completely specifythe atoms’ positions in space. Three of them can be taken as the coordinates ofthe 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 remainingcoordinates for specifying the atoms’ relative positions is associated with anoscillation (mode) in the relative positions. Thus, the number of vibrational modesis 3N-6 for nonlinear molecules or 3N-5 for linear molecules. This number isthe molecule’s vibrational degrees of freedom.

For a molecule with s vibrational degrees of freedom, the kinetic and potentialenergies written in terms of the remaining coordinates (1, 2, ..., s) have thefollowing 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 positionsV(0, 0, ..., 0) = 0.


It is possible to linearly combining the coordinates (1, 2, ..., s) to form a newset of coordinates (Q1, Q2, ..., Qs), so that both the kinetic and potential energieswhen written in terms of the new coordinates are free of cross terms:

When this is achieved, the new coordinates are called normal coordinates andthe 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 sumof the individual vibrational energies. In the harmonic approximation, theselection 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 frequentlyobserved in polyatomic spectra, although their intensities are relatively weak.


From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).

1 = 3651.7 cm-1 2 = 1595.0 cm-1 3 = 3755.8 cm-1

H2O

1 = 1388.3 cm-1 2 = 2349.3 cm-1 3 = 667.3 cm-1

From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).

CO2

For linear symmetric triatomic molecules such as CO2, there are four modes:

Since the symmetric stretching mode does not produce an oscillation in the dipolemoment 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 otherhand, produce an oscillating dipole (even though CO2 in static equilibrium doesnot have a permanent dipole moment), so there are two strong bands centered on2 and 3.

Nonlinear symmetric triatomic molecules such as H2O have three modes:


All the three modes produce an oscillating dipole moment, three fundamentalbands centered on 1, 2, and 3 appear in the infrared region. In addition tothese bands, overtones due to anharmonicity and combination bands due tosimultaneous change of two or more modes.


From G. W. Castellan, Physical Chemistry, 3rd ed. Addison-Wesley, Reading (1983)

It is found that the vibrational frequencies for agiven chemical bond are relatively invariant frommolecule to molecule. Thus, the characteristicfrequencies of various chemical groups found inthe IR spectrum of an unknown molecule can beused to identify the molecule.

Because the exact position depends on the typeof compound, it requires some experience. It isalso possible to use infrared spectroscopy todetermine quantitatively the amount of varioussubstances present in mixtures.

Gas Phase Dynamics and StructureFr

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http://en.wikipedia.org/wiki/Acetone

Acetone, CH3-CO-CH3

The energy level for nuclear motion is given by

For simultaneous vibrational and rotational transitions,

the same selection rules hold

and since e is generally larger than Be, the vibrational part determines whethera transition is absorption,

or emission,

whereas the rotational part determines to which group of lines, the so-called Pand R branches associated with a single vibrational transition, the transitionbelongs.


Molecular Spectra: Rotation-Vibration Spectra

The intensity of a line is proportional to the number of molecules in the initialstate, which in turn is proportional to the Boltzmann factor

where gvJ is the degeneracy of the state. Each rotational state contains 2J+1degenerate states, corresponding to M = -J, -J+1, …, J, so gvJ = 2J+1. Everybranch therefore has a most intense line somewhere away from the band centerJ = 0.


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For some polyatomic molecules and in the rotation-vibration spectra of diatomicmolecules with nonzero electronic momentum about the molecular axis, transitionswith J = 0 are allowed. The associated spectral lines are called Q branch. Thebending vibration of CO2 offers an example of Q branch.


Because a rotating molecule experiences a centrifugal force which stretches themolecule so that the effective equilibrium inter-nuclear distance increases with J,rotation indeed interacts with vibration. Appendix B shows that after includingthe lowest-order correction due to vibration-rotation interaction, we have (Morsepotential is assumed)

The effect of vibration-rotation interaction on rotation is to narrow the spacingbetween rotational energy levels. This is a natural consequence of increasing themoment of inertia by increasing the equilibrium inter-nuclear distance. As forits effect on vibration, the interaction reduces the vibrational frequency when theoscillation is sufficiently anharmonic (xee > Be), which is the case for mostmolecules. This is because the centrifugal force drives the molecule into the“softer” region of the potential. Also note that xee e for most molecules,so the effect of vibration-rotation interaction normally is masked by the effectof anharmonicity.



Without vibration-rotation interaction

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Because electrons are much lessmassive than nuclei, the motionof electrons is much faster thanthat of nuclei, and electronictransitions occur in times too shortfor any appreciable nuclear motionto take place. Consequently, theinter-nuclear distance in the finalstate is the same as it was in theinitial state and the transition isrepresented by a vertical linebetween the initial and finalpotential-energy curves. Thisis called the Franck-Condonprinciple. Accordingly, amongthe possible transitions, the mostintense are those in which theinstantaneous inter-nucleardistance is a highly probable onefor both states.


Molecular Spectra: Electronic Transitions


Transitions can originate in any of the multitude of rotational states associatedwith the initial vibrational level and can end up in either of the two rotationalstates determined by the selection rule, J = ±1 in the final electronic andvibrational state. Thus, a group of closely spaced lines are observed. Thisgroup of lines is called an electronic band.

Owing to the anharmonicity of the potential-energy curve, there is no selectionrule for vibration, so transitions may occur between any two vibrational statesthat 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.


Appendix ASolving the Harmonic Schrödinger Equation

To solve the equation

first consider its asymptotic behavior.

For large , the equation becomes

It can be readily seen by setting () = exp[f()] that

Thus, the solution should be in the form

and the equation for the unknown function H() is

Appendix A

We can see immediately that the power series method will give

which must terminate somewhere; otherwise the power series will lead to aasymptotically divergent solution. Therefore, there exists a specific n such that

This requirement gives quantized energy:

where

The equation to be solved now becomes

or

Appendix A

Writing the equation in the following form

leads us to set

where the new known function satisfies

To eliminate the term outside the square bracket, we find that has to be writtenas

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 0 is a constant, so

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 0 is a proper normalization constant.

Appendix BVibration-Rotation Interaction

In the rigid rotor model, since the inter-nuclear distance R is fixed at Re, aparameter also taken as the equilibrium distance of electronic energy E(R),vibration and rotation become decoupled. This is unrealistic, as the rotoris expected to respond to the influence of centrifugal force by stretching itself,thus increasing the equilibrium inter-nuclear distance. This Appendix derivesthe lowest-order correction to the energy of nuclear motion.

Without taking R = Re in the rotor term, EJ, in the radial Schrödinger equationfor nuclear motion,

we have

Thus, the effective potential is E(R) augmented by a centrifugal term:

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 potentialcan be written as

Appendix B

where

Substitution of the formula for Re' into Eeff(Re') gives

So we obtain, after a little algebra,

where

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. Thenthe vibrational energy levels are given by

where

Substituting the expressions for ke', e', and Re' into e' and xe', we get

Appendix B

So the vibrational energy is

where

We finally have the expression that includes vibration-rotation interaction andlowest-order anharmonicity:

Appendix B

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.

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