1. Topic III: Spectroscopy Chapter 12 Rotational and vibrational spectra

2. Spectroscopy
The analysis of the EM radiations emitted, absorbed or scattered by atoms, molecules or matter
Photons of the radiation bring information to us about the atom, molecule or matter.
The difference between molecular and atomic spectroscopy: a molecule can make a transition between its electronic, rotational and vibrational states.
The rotational and vibrational spectroscopy of a molecule can provide information about the bond lengths, bond angles and bond strength in the molecule.

3. Electromagnetic spectrum

4. General features of spectroscopy
is called the wavenumber of the photon and gives the number of complete wavelengths per centimeter. It is in the unit of cm-1.
Emission spectrum: A molecule returns to a state of lower energy E1 from an excited state of energy E2 by emitting a photon.
Absorption spectrum: A molecule is excited from a lower energy state to a higher energy state by absorbing a photon as the frequency of the incident radiation is swept over a range

5. Raman spectrum
A monochromatic radiation is incident and scattered by the molecule. The frequency of a scattered radiation is different from the frequency of the incident radiation. The spectrum of the scattered radiation is called the Raman spectrum.
Stokes Raman spectrum: The frequency of the scattered radiation is lower than the frequency of the incident radiation.
Anti-Stokes Raman spectrum: The frequency of the scattered radiation is higher than the frequency of the incident radiation.

6. Stimulated and Spontaneous emissions
A molecule in an excited state will decay to a lower energy state without any stimulus from the outside.
Stimulated emission:
As an EM radiation incident upon a molecule in an excited state can cause the molecule to decay to a lower energy state. If the incoming photon has the same frequency as the emitted photon, the incident and emitted photons have the same wavelength and phase and travel in the same direction. The incoming photon is not absorbed by the molecule but triggers emission of a second photon. The two photons are said to be coherent.
The stimulated emission is the fundamental physical process of the operation of the laser.

7. Experimental techniques
Absorption spectrometer
Raman spectrometer
Emission spectrometer
Sample
Source of radiation
Dispersion elements
Detector

8. Source of radiation
To produce radiation, either monochromatic or polychromatic spanning a range of frequencies.
Tungsten-Lamps (for visible), LASER (almost monochomatic), a discharge through deuterium gas or Xe in quartz (for near ultraviolet), a mercury arc in a guartz (for far infrared), a heated ceramic filament containing rare-earth oxides (for mid infrared)
Synchrotron radiation emitted from an electron beam traveling in a storage ring

9. Dispersion element and Detector
Dispersing element: To separate the radiation into different frequencies
Glass, quartz prism, diffraction grating and Fourier transform spectrometer (Michelson interferometer)
Detector: To convert the scattered radiation into an electric current for the appropriate signal processing by radiation-sensitive semiconductor devices, photodiode or Si (for the visible region).
A charge-coupled device (CCD) is a two-dimensional array of photodiode detectors.
The mercury cadmium telluride (MCT) is a photovoltaic device for mid infrared.
Diffraction grating
Michelson interferometer

10. Samples
For rotational spectroscopy, the samples are in a gas phase in which molecules are almost in free rotation and infrequently collide.
For infrared spectroscopy, the samples are embedded in a condensed phase (liquids or solids).
The measured spectroscopy depends on the number of molecules in the initial state. For samples at thermal equilibrium, the populations of the initial states follow the equilibrium statistics. For samples at non-equilibrium thermodynamic state , the populations of the initial states do not follow the statistics.

11. The Beer-Lambert law for absorption
The intensity of radiation transmitted by an absorbing sample decreases exponentially with the path length through the sample.
I0 and I : Intensities of the incident and transmitted radiations
A : Absorbance of the sample, T: Transmittance
L : Length of the sample
[J] : Concentration of the absorbing species in the sample
e : Molar absorption coefficient or the extinction coefficient
e depends on wavelength of the incident radiation.

12. Transition dipole moment and selection rules
The strength with which individual molecules are able to interact with the EM radiation and generate or absorb photons. The transition dipole moment depends on the initial and the final states of the molecules
: Electric dipole moment operator
Selection rules: Rules for non-zero transition dipole moments.
Two kinds of selection rules:
Gross selection rules: the general features that a molecule must have to cause the spectrum of a given kind. Example: A molecule gives a rotational spectrum only if it has a permanent electric dipole moment.
Specific selection rules: statements about which changes in quantum numbers may occur in a transition.
An allowed transition (1s→2p) is permitted by a specific selection rule
A forbidden transition (1s→2s) is disallowed by a specific selection rule
Exception: A forbidden transition sometimes may occur weakly.

13. The absorption spectrum
For a Gaussian spectrum,

14. The spectral linewidths
I. Doppler broadening: the frequency of a radiation is shifted when the source of the radiation is moving toward or away from the observer. This effect is important for samples in a gas phase.
In a gas phase, the velocities of the molecules follow the Maxwell-Boltzmann distribution. Some move toward the detector and some move away. The linewidth of the detected spectrum is resulted from all of the Doppler shifts. The spectrum is generally a Gaussian. The full width of the spectrum is defined at half of its maximum, denoted as FWHM.
T: temperature of the sample
m : Mass of the molecule
Decreasing temperature of the sample reduces the Doppler broadening

15. II. Lifetime broadening: line broadening due to the lifetime of the states involved in the transition.
According to the uncertainty principle, the state of a system that is changing with time do not have precisely defined energies. If the system in an excited state that exponentially decays with a time constant t, which is called the lifetime of the state, the energy level of the excited state is blurred by dE.
Two processes are responsible for the finite lifetime of the excited states
I. Collision deactivation: Collisions between molecules or with the wall of the container (In gas phase, this effect can be minimized by working at low pressures.)
II. Spontaneous emission: The emission of radiation as the system in the excited state collapses into a lower state. The rate of the emission depends on the wavefunctions of the excited and lower states. The linewidth of spontaneous emission increases as n3 (n is the frequency). This broadening is the nature linewidth of the transition and is not avoidable by modifying T or P of the system.

16. Rigid-rotor models of molecules
Rigid-rotor model of molecules: The relative distances and orientations of atoms in a molecule do not change during the rotational motion of the molecule
A rigid rotor has three orthogonal axes passing through the center of mass of the rotor, which are called the principal axes of the rotor.
Corresponding to each principal axis, a moment of inertia of the molecule is defined. A rigid rotor has three principal moments of inertia, Ix, Iy and Iz.
The moment of inertia of a molecule about a rotational axis is a sum of the mass of each atom multiplied by the square of its distance from the axis.

17. Three principal moments of inertia of a rigid molecule
An asymmetric rotor has three different moments of inertia.
Ix = Iy= Iz
Ix = Iy  Iz
Ix  Iy  Iz

19. Rotational energy levels of a rigid rotor
The kinetic energy of a rigid rotor with angular momentum Jx, Jy and Jz. with respect to the three principal axes of the rotor.
Classical description
For a linear or spherical rigid rotor with a moment of inertia I
Quantum
description

20. Rotational energy levels of a symmetric rotor
I|| > I⊥, for a oblate rotor
like a pancake.
I|| < I⊥, for a prolate rotor like a cigar.
I|| : The moment of inertia about the axis parallel to the molecular axis
I⊥ : The moment of inertia about the axis perpendicular to the molecular axis
K: Quantum number to specify the component of angular momentum parallel to the symmetric axis. There are two special cases
Case I: As the rotational axis is parallel to the molecular axis, K = ± J.
Case II: As the rotational axis is perpendicular to the molecular axis, K = 0.

21. Centrifugal distortion
In real molecules, the atoms of rotating molecules are subject to centrifugal forces due to the rotation. The centrifugal forces tend to distort the molecule geometry by stretching the bonds between atoms and increase the moment of inertia about the rotational axis. Hence, the molecular distortion due to the centrifugal effect reduces the separations of energy levels predicted by the rigid-rotor models.
The centrifugal distortion constant

22. Rotational spectroscopy
Gross selection rule: The molecules must have a permanent electric dipole moment so that the molecules are polar.
Classical description
Rotational-inactive molecules: Molecules without rotational spectrum
Homonuclear diatomic molecules: N2, O2
Symmetric linear molecules: CO2
Tetrahedral molecules: CH4
Octahedral molecules: SF6, C6H6
Rotational-active molecules: Molecules with rotational spectrum
Heteronuclear diatomic molecules: HCl
Less symmetric polar molecules: NH3, H2O
To an observer, a rotating polar molecule has an electric dipole that appears to oscillate. This oscillating dipole can interact with the EM field.

23. Specific selection rules for rotational transition
DJ = 1 and DK = 0
Conservation of angular momentum for DJ = 1
A photon is a spin-1 particle. When the molecule absorbs one photon, the angular momentum of the molecule must increase to conserve the total angular momentum, so J increases by one. When the molecule emits a photon, the angular momentum of the molecule must decreases, so J decreases by one.
For symmetric rotors, DK = 0
The dipole moment of a polar molecule does not move when a molecule rotates around its symmetric axis and, therefore, there is no absorption or emission of the EM radiation by the rotation of the molecule about the axis.
Quantum description

24. Absorption of allowed rotational transitions
A rigid molecule with K=0 makes a transition from J to J+1
Rigid-rotor model
The rotational spectrum consists of a series of lines at frequencies separated by
The rotational spectra of gas-phase samples are microwave spectroscopy.
We can use the value of obtained from the rotational spectrum to estimate the bond length of a heteronuclear diatomic molecule.

25. Intensity of rotational spectrum: Populations of rotational states
Intensity of absorption spectrum is proportional to the population of molecules in the absorbing state.
The temperature effect arises from Boltzmann statistics.
The degeneracy of a rotational state gives a factor gJ = 2J+1.
The maximum intensity for linear rotors is at

26. Electric dipole moment
2. Unit of dipole moments
1. Definition
1 D (debye) = × C·m
The electric dipole moment of two charges e and –e separated by 100 pm is 4.8D.
An electric dipole is composed of two charges, Q and –Q, separated by a distance R. The electric dipole moment is defined as Q times , which is a vector pointing from the negative charge to the positive one.
3. Addition
The electric dipole moment is a vector.
The addition of two dipole moments follows the law of vector addition.

27. Polar and nonpolar molecules
A polar molecule has a permanent dipole moment arising from the partial charges of its atoms.
A nonpolar molecules has no permanent dipole moment.
All homonuclear diatomic molecules are nonpolar molecules.
All heteronuclear diatomic molecules are polar molecules.
Whether a polyatomic molecule is polar or not is strongly related to the geometric symmetry of the molecule.
O3 is polar.
CO2 is nonpolar.

28. Electric dipole moments of dichlorobenzene isomers
Benzene is a nonpolar molecule, due to its molecular symmetry.

29. Electric dipole moments of molecules without geometric symmetry
Qj is the partial charge of atom j.
xj, yj and zj are the Cartesian coordinates of atom j.

30. Multipoles of point charges
An n-pole is an array of point charges with an n-pole moment but no lower moment.

31. Induced dipole moment Physical reason:
Without an external electric field, the centers of the negative and positive charge distributions of a nonploar molecule are at the same place, so there is no permanent dipole moment. But, in an external electric field, the centers of the negative and positive charge distributions are separated by a distance and give rise to an induced dipole moment. The proportional constant between the induced dipole mement and the external field is called the electric polarizability.
A nonpolar molecule may acquire a temporary induced dipole moment m* by the electric field E generated by a nearby ion or polar molecule.

32. Electric polarizability
The polariability depends on the orientation of the molecule with respect to the electric field.
In general, the polarizability is a 3 × 3 matrix. Except for atoms, tetrahedral, octahedral and icosohedral molecules, the polarizabilities of all other molecules have anisotropic polarizabilities, corresponding to the off-dagonal element of the matrix.
A large polarizability means the electronic density can undergo relatively large fluctuation. The polarizability is inversely proportional to the ionization energy.
Polarizability tensor of a nonpolar molecule
Polarizability volume (in unit of volume)

33. Rotational Raman spectroscopy
Visible or ultraviolet Lasers
Rayleigh line
Stokes lines: the scattered lines shifted to lower frequency than the incident radiation (nscat < ninc)
Anti-Stokes lines: the scattered lines shifted to higher frequency than the incident radiation (nscat > ninc)
Rayleigh lines: the scattered lines in the forward direction and with the same frequency as the incident radiation (nscat = ninc)

34. Gross selection rule
The molecules must have anisotropic polarizability.
The polarizability of a molecule is a measure of the extent to which an applied electric field can induce an electric dipole moment m in addition to any permanent dipole moment. The anisotropy of the polarizability is its variation with the orientation of the molecule.
All spherical rotors, like tretrahedral (CH4), octahedral (SF6) and icosahedral molecules (C60), are both rotationally and rotationally Raman inactive.
All homonuclear diatomic molecules and linear molecules are rotationally inactive but rotationally Raman active.

35. Specific selection rules of rotational Raman spectroscopy
The distortion induced in a molecule by an applied electric field returns to its initial value after a rotation of only 180°.
Rotational Raman scattering is associated with the variation of molecular polarizability due to rotational motions of the molecule.
wB: Angular frequency of molecular rotation
DJ =  2 for linear rotors
Positive for Stokes lines
Negative for Anti-Stokes lines

36. Stokes and anti-Stokes lines of rotational Raman spectrum
For Stokes lines (DJ= +2)
For anti-Stokes lines (DJ= -2)
Stokes lines
Anti-Stokes lines
Dn : Frequency shift relative to the incident radiation.
The rotational Raman spectrum consists of a series of lines at frequencies of
6 , and … , which are separated by 4 .
We can use the value of B obtained from the rotational Raman spectrum to estimate the bond length of a homonuclear diatomic molecule.
The intensities of the Stokes lines are generally stronger than those of the Anti-Stokes lines.

37. Rotational Raman spectrum of a diatomic molecule with two identical nuclei of spin ½
For H2 molecules with nonzero nuclear spins, the intensities of the odd-J lines are three times more than those of the even-J lines.
Under rotation through 180°,
Wavefunctions with even J do not change sign.
Wavefunctions with odd J do change sign.
Nuclear statistics must be taken into account whenever a rotation interchanges equivalent nuclei.

38. Ortho- and Para-hydrogen molecules
If the nuclear spin I is a half-integer, the total wavefunction should be antisymmetric. The even-J rotational states are associated with the antisymmetric nuclear wavefucntions and the odd-J rotational states are with the symmetric nuclear wavefunctions.
If the nuclear spin I is an integer, the total wavefunction should be symmetric. The even-J rotational states are associated with the symmetric nuclear wavefucntions and the odd-J rotational states are with the anti-symmetric nuclear wavefunctions.
Ortho-hydrogen molecule (Ortho-H2): Molecule with parallel nuclear spin
Para-hydrogen molecule (Para-H2): Molecule with antiparallel nuclear spin
For H2, I=1/2
For D2, I=1

39. Vibrations of diatomic molecules
Harmonic approximation around the equilibrium
A diatomic molecule has three translational modes, two rotational modes and one vibrational mode.
kf : force constant of the bond,
the curvature of the potential at Re

40. Infrared spectroscopy: Vibrational transitions
Gross selection rule
The molecule need not to have a permanent dipole moment but the electric dipole moment of the molecule must change during the vibration. The rule only requires a change in electric dipole moment, possibly from zero.
The typical vibrational frequency of a molecule is about 1013~1014 Hz. The vibrational spectroscopy of molecules is in the infrared region, which normally lies in 300 ~ 3000 cm-1.
Homonuclear diatomic molecules are infrared inactive, because their dipole moments remain zero as they vibrate.
Heteronuclear diatomic molecules are infrared active, because their dipole moments change as they vibrate.
Bending motion of CO2

41. Specific selection rules for infrared spectroscopy
Electric dipole moment
Molecules with stiff bonds joining atoms with low masses have high vibrational wavenumbers.
The bending modes of a linear molecule are usually less stiff than stretching modes. So, the bending modes usually occur at lower wavenumbers than the stretching modes in a spectrum.
At room temperatures, almost all molecules are in their vibrational ground states (n = 0). So, the most probable transition is from n = 0 to n = 1.
For Dn = 1, the molecule absorbs one photon and, for Dn = -1, the molecule emits one photon.
Transition dipole element

42. Anharmonicity
The harmonic approximation is only good for the potential in the region near Re but is poor as the molecule is deviated far from Re.
Morse potential is an anharmonic potential but the Schrodinger’s equation of the model is exactly solvable.
The transition moves to lower wavenumbers as n increases.
Anharmonicity also accounts for the appearance of weak absorption lines corresponding to n = 0 to n = 2 and n = 0 to n = 3, 
De: Depth of the potential minimum
ce : Anharmonicity constant

43. Vibration-rotation spectrum of HCl
Each line of the high-resolution vibrational spectrum of a gas-phase heteronuclear diatomic molecule is found to consist of a large number of closely spaced components in the order of 10 cm-1, which suggests that the structure is due to the rotational transitions accompanying the vibrational transitions.
There is no Q branch.
The lines appear in pair, because both H35Cl and H37Cl contribute in their abundance ratio 3:1.

44. Vibration-rotation spectra
In reality, the vibrational spectra of gas molecules are complicated as the excitation of a vibration also results in the excitation of rotation.
The rotational and vibrational energy levels of a linear molecule
Selection rules
DJ = 0 or ±1
Dn = ±1
Two quantum numbers n and J
n : quantum number for vibration
J : quantum number for rotation
Transition wavenumber

45. Vibration-rotation absorption spectrum
P branch: DJ = -1
Q branch: DJ = 0
R branch: DJ = +1

46. Vibrational Raman spectroscopy
The incident photon leaves some of its energy in the vibrational modes of the molecule it strikes (Stokes lines), or collects additional energy from a vibration that has already been excited (Anti-Stokes lines).
Gross selection rule: The molecular polarizability must change as the molecule vibrates.
Both homonuclear and heteronuclear diatomic molecules are vibrationally Raman active.
Specific selection rules:
The Stokes lines are more intense than the Anti-Stokes lines, because very few molecules are in an excited vibrational state initially.
Induced dipole moment
DJ = 0 or ±2, Dn =  1

48. Vibration-rotation Raman spectrum of a linear rotor
O branch: DJ = -2
Q branch: DJ = 0
S branch: DJ = +2

49. Vibration-rotation Raman spectrum of CO
DJ=-2
DJ=0
DJ=+2

50. Normal modes of polyatomic molecules
The motions of a non-rigid polyatomic molecule of N atoms have 3N degrees of freedom, including three translational and three rotational modes of the center of mass and 3N-6 vibrational modes among atoms.
Translational motions:
Motion of center of mass of the molecule.
Rotational motions:
Motion of the molecule with all bond lengths and bond angles among atoms unchanged.
Vibrational motions:
Internal relative motions among atoms of the molecule with changing bond lengths (stretching) or bond angles (bending).
Harmonic approximation: Deviated not too far away from the equilibrium of a molecule, the vibrational motions of the molecule can be described as a linear combination of normal modes.
The number of normal modes should equal the number of degrees of freedom.
Number of vibrational modes:
3N-6 for nonlinear molecules
3N-5 for linear molecules

51. Features of normal modes
A vibrational normal mode describes a specific collective motion of atoms, with each atom in a harmonic oscillation of the same frequency. The collective motion of a normal mode is called a vibrational excitation.
In the harmonic approximation, all normal modes of a molecule are independent from one another. Each normal mode behaves like an independent harmonic oscillator. The energies of the vibrational levels of the i-th normal mode with frequency ni are
Symmetric atretching and antisymmetric stretching modes of CO2 molecule 
Symmetric
Stretching mode
Antisymmetric stretching mode

52. Four vibrational modes of CO2
Symmetric stretching and antisymmetri stretching modes
The frequency of symmetric stretching mode is higher than that of the antisymmetric stretching mode.
Two bending modes
The frequencies of bending modes are double degeneracy.
Typically, the frequencies of bending modes are smaller than those of stretching modes.

53. Three vibrational modes of H2O
Symmetric stretch
Bending mode
Antisymmetric stretch

54. Typical vibrational modes of a tetrahedral molecule

55. Typical vibrational wavenumbers

56. Gross selection rules of vibrational normal modes
The collective motion of a vibrational normal mode must give rise to a change in dipole moment of the molecule.
The symmetric stretching mode of CO2 leaves the dipole moment unchanged, so this mode is infrared inactive.
The antisymmetric stretching mode of CO2 makes the dipole moment changed, so this mode is infrared active.
The two bending modes of CO2 are infrared active.
The solar energy strikes the top of the Earth’s atmosphere and 30 percent of the energy is reflected back into space. The Earth atmosphere absorbs the remaining energy, with most of the intensity in the infrared range. The trapping of the infrared radiation by certain gases in the atmosphere is the greenhouse effect. O2 and N2 in the atmosphere do not contribute to the greenhouse effect. H2O and CO2 do absorb infrared radiation and are responsible for the greenhouse effect.

57. Fingerprint of infrared absorption spectrum
The infrared absorption spectrum of a molecule is a characteristic of the molecule and can be used as a fingerprint to identify the presence of the molecule in a particular substance.
A sample (O2N-C6H4-CC-COOH)

58. Vibrational Raman spectra of polyatomic molecules
Gross selection rule: The vibrational normal mode is accompanied by a change in the polarizability of the molecule.
The symmetric stretch of CO2 is Raman active, and the other are Raman inactive.
A general exclusion rule: If the molecule has a center of inversion, then no modes can be both infrared and Raman active.

59. Resonant Raman spectroscopy
In the conventional Raman spectroscopy, the incident radiation does not match an electronic absorption frequency of the molecule and there is only a virtual transition to an excited state. In the resonant Raman spectroscopy, the incident radiation has a frequency that nearly coincides with a molecular electronic transition. This resonant Raman spectroscopy gives a much greater intensity in the scattered radiation and is used to study biologic molecules that absorb strongly in the ultraviolet and visible regions.

60. Exercises of Chapter 12 12A.2(a), 12A.9(b) 12B.1(a), 12B.4(a)
12C.2(a), 12C.8(a)
12D.2(a), 12D.3(a)
12E.1(a), 12E.5(a)