Lectures on Modern Physics

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

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. s s1 r s2 e 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.

10. Lennard-Jones potential Experimental data From M. P. Allen and D. J. Tildesley., Computer Simulation of Liquids, Oxford University Press, New York (1987). 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.

11. 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. 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. 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. 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. 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. 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. 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 10-11 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. Gas Phase Dynamics and Structure Moseley found that the frequencies of the most energetic lines versus atomic number is nearly linear: where R = 109677.57 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 = 1.13. For convenience, s = 1 may be used quite satisfactorily.

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

22. Na D-line doublet From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970). 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

23. 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). 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.

24. 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). 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+:

25. 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. 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. 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. M R 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,

29. 3 ← 4 3 → 4 4 → 5 4 ← 5 0 → 1 0 ← 1 1 → 2 1 ← 2 2 ← 3 2 → 3 2Be 2Be n = n = 4Be 4Be 6Be 6Be 8Be 8Be 10Be 10Be 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 moleculeswith permanent dipole moments (e.g., HF, HCl, etc.) Thus, the allowed frequencies are where is the rotational constant. Either case gives uniformly spaced lines:

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

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

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

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

34. From P. Helminger and W. Gordy, Phys. Rev.188, 100 (1969). From W. J. Moore, Physical Chemistry, Prentice-Hall, Englewood Cliffs (1972). 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.

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

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

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

39. Harmonic potential Morse potential From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970). 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:

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

41. 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. 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. CO2 n1 = 1388.3 cm-1n2 = 2349.3 cm-1n3 = 667.3 cm-1 From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983). H2O n1 = 3651.7 cm-1n2 = 1595.0 cm-1n3 = 3755.8 cm-1 From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983). 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:

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

45. 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. Acetone, CH3-CO-CH3 http://en.wikipedia.org/wiki/Acetone From G. W. Castellan, Physical Chemistry, 3rd ed. Addison-Wesley, Reading (1983) Gas Phase Dynamics and Structure

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

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

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.