Advanced Physical Chemistry Chemistry 5350 ROTATIONAL AND VIBRATIONAL SPECTROSCOPY

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1 Advanced Physical Chemistry Chemistry 5350 ROTATIONAL AND VIBRATIONAL SPECTROSCOPY Professor Angelo R. Rossi Department of Chemistry, Room CHMT215 The University of Connecuticut Spectroscopy Molecular spectroscopy is a powerful tool for learning about molecular structure and molecular energy levels. The study of rotational spectra gives us information about moments of inertia, interatomic distances, and bond angles. Vibrational spectra yield fundamental vibrational frequencies and force constants. Electronic spectra provide electronic energy levels and dissociation energies. The Basic Ideas of Spectroscopy Fall Semester 2013 angelo.rossi@uconn.edu When an isolated molecule undergoes a transition from one quantum eigenstate with energy E 1 to another with energy E 2, energy is conserved by the absorption or emission of a photon. The frequency ν of the photon is related to the difference in energy of the two states by the relation hν = hc ν = E 1 E 2 where ν = 1 and is the transition energy in wave numbers (the λ number of waves per unit length). The SI unit is m 1, but usually cm 1 is used. If E 1 > E 2. the process is photon emission. If E 1 < E 2. the process is photon absortion. 2 Spectroscopic Regions of the Electromagnetic Spectrum The energy eigenvalues of a molecule can be written as E = E rot + E vib + E elec where E rot is the rotational energy, E vib is the vibrational energy, and E elec is the electronic energy. When a molecule undergoes a transition to another state with the emission or absorption of a single photon of frequency, ν, then hν = (E rot E rot) + (E vib E vib) + (E elec E elec) where the primes refer to the state of higher energy and double primes to the state of lower energy. 3 4

2 Spectroscopic Regions of the Electromagnetic Spectrum The classification of the various regions of the electromagnetic by the type of transition involved because E rot E rot << E vib E vib << E elec E elec Electronic energy level differences are much greater that vibrational energy differences, which in turn are much greater than rotational energy level differences. Electronic transitions are often in the visible and ultraviolet part of the spectrum. Vibrational transitions are in the infrared, and rotational transitions are in the far infrared and microwave regions. Spectroscopic Regions of the Electromagnetic Spectrum The frequency range of photons, or the electromagnetic spectrum, is classified into different regions according to custom and experimental methods outlined in the table below. Wavelength, λ Wave Frequency, ν Photon Molar Number, ν Energy, hν Energy, N A hν γ rays 10 pm 1 x 10 9 cm EHz 19.9 x J 12.0 GJ/mol X-rays 10 pm 1 x 10 6 cm PHz 19.9 x J 12.0 MJ/mol Vacuum UV 200 nm 50.0 x 10 3 cm PHz 993 x J 598 kj/mol Near UV 380 nm 26.3 x 10 3 cm THz 523 x J 315 kj/mol Visible 780 nm 12.8 x 10 3 cm THz 255 x J 153 kj/mol Near IR 2.5 µm 4.00 x 10 3 cm THz 79.5 x J 47.9 kj/mol Mid IR 50 µm 200 cm THz 3.98 x J 2.40 kj/mol Far IR 1 mm 10 cm GHz 199 x J 120 J/mol Microwaves 100 mm 0.1 cm GHz 1.99 x J 12.0 J/mol Radio Waves 1000 mm 0.01 cm MHz x J 1.2 J/mol 5 Spectroscopic Regions of the Electromagnetic Spectrum The frequency of the photon in the absorption or emission often indicates the kinds of molecular that are involved: 6 The Electromagnetic Spectrum Depicted on a Logarithmic Wavelength Scale The electromagnetic spectrum is shown schematically below: In the radio-frequency region (very low energy), transitions between nuclear spin states can occur. In the microwave region, transitions in molecules with unpaired electrons and transitions between rotational states take place. In the infrared region, transitions between vibrational states take place with and without transitons between rotational states. In the visible and ultraviolet regions, the transitons occur between electronic states accompanied by vibrational and rotational changes. Finally, in the far ultraviolet and X-ray regions, transitions occur that can ionize or dissociate molecules. 7 The above figure shows that visible light is a very small part of the electromagnetic spectrum. 8

3 Electric and Magnetic Fields Associated with a Traveling Light Wave Light is an electromagnetic traveling wave that has perpendicular magnetic and electic field components: Spectroscopic Selection Rules Atoms and molecules possess a set of discrete energy levels which is an essential feature of all spectroscopies. If all molecules had a continuous energy spectrum, it would be very difficult to distinguish one from another on the basis of their absorption spectra. Selection rules indicate which transitions will be experimentally observed. 9 Spectroscopic Selection Rules 10 Energy Transfer from the Electromagnetic Field to a Molecule Leading to Vibrational Excitation Because spectroscopies involve transitions between quantum states, it is important to describe how electromagnetic radiation interacts with molecules. The principal interactions of molecules with electromagnetic radiation are of the electric dipole type and will be the focus of study. Magnetic dipole transitions are about 10 5 weaker than electric dipole transitions, and electric quadrupole transitions are about 10 8 weaker

4 Energy Transfer from the Electromagnetic Field to a Molecule Leading to Vibrational Excitation Consider the effect of a time-dependent electric field on a classical polar molecule constrained to move in one dimension: The spring allows the two masses to oscillate about their equilibrium position generating a a periodically varying dipole moment. If the electric field and oscillation of the dipole moment have the same frequency and if they are in phase, the molecule can absorb energy from the field. For a classical molecule, any amount of energy can be absorbed, and the spectrum is continuous. For a quantum mechanical molecule, the interaction with the electromagnetic field is similar but only discrete amounts of energy can be absorbed. Interaction of a Rigid Rotor with an Electromagnetic Field Imagine a sinusoidally varying electric field applied between a pair of capacitor places. The arrows indicate the direction of force on each of the two charged masses. If the frequencies of the field and rotation are equal, the rotor will absorb from the electric field. The amount of energy absorbed from the electromagnetic field will be controlled the quantum mechanical energy for a rigid rotor Spectroscopic Selection Rules The types of transitions that can occur are limited by selection rules. As with atoms, the principle interactions of molecules with electronmagnetic radiation are of the dipole type, and will be our principle focus. Although the selection rules limit radiative transitions that can occur, molecular collisions can cause many additional kinds of transitions. Because of molecular collisions, the populations of the various molecular energy levels are in thermal equilibrium. Absorption, Spontaneous Emission, and Stimulated Emission The basic processes by which photon-assisted transitions between energy levels occur are by absorption, spontaneous emission, an stimulated emission as shown in the figure below for a two level-system. In absorption, the incident photon induces a transition to a higher level. In emmision, a photon is emitted as an excited state relaxes to one of lower energy. Spontaneous emission is a random event, and its rate is related to the lifetime of the excited state

5 Absorption, Spontaneous Emission, and Stimulated Emission Schrödinger Equation for Nuclear Motion These three processes are not independent in a system at equilibrium where the overall rate for level 1 to 2 must be the same as that for 2 to 1. B 12ρ(ν)N 1 = B 21ρ(ν)N 2 + A 21N 2 Spontaneous emission is independent of the radiation density at a given frequency, ρ(ν), but the rates of stimulated absorption and emission are directly proportional to ρ(ν). Each of these rates is directly proportional to the number of molecules (N 1 or N 2) in the state from which the transition originates. This means that unless the lower state is populated, a signal will not be seen in the absorption experiment, and unless the upper state is populated, a signal will not be observed in the emission experiment. ρ(ν) is the blackbody spectral density which provides the distribution of frequencies at equilibrium for a given temperature. Previously we showed how the Schrödinger equation can be treated in the Born-Oppenheimer approximation so that the electronic Hamiltonian is for fixed nuclei. The Hamiltonian for nuclear motion contains the kinetic energy operator and the electronic energy as a function of the nuclear coordinates as the potential energy operator Ĥ = 2 2µ 2 R + E(R) The potential energy term E(R) depends only on the relative positions and not where the molecule is placed or on the orientation of the molecule in space. B 12 = B 21 and A21 = 16π2 ν 3 B 21 c Schrödinger Equation for Nuclear Motion The kinetic energy operator consists of the kinetic energy of the center of mass (translational energy), kinetic energy associated with rotational motion, and the kinetic energy of the vibrational motion Ĥ = Ĥtrans + Ĥrot + Ĥvib where the translational and rotatational Hamiltonians contain only kinetic energy terms, and the vibrational Hamiltonian contains E(R) depending on the internuclear distances. These internuclear distances are the vibrational coordinates of the molecule. Schrödinger Equation for Nuclear Motion If the Hamiltonian is the sum of three terms, one for each kind of motion, then the wave function ψ can be written as a product of wave functions: ψ = ψ trans ψ rot ψ vib The Schrödinger equations for the three terms are Ĥ trans ψ trans = E trans ψ trans Ĥ rot ψ rot = E rot ψ rot Ĥ vib ψ vib = E vib ψ vib The translational wavefunction is that for a free particle with a mass equal to the mass of a molecule. The translational eigenvalues are very closely spaced and cannot be probed in molecular spectroscopy

6 Schrödinger Equation for Nuclear Motion The total number of coordinates required to describe a polyatomic molecule with N atoms in a molecule is 3N. However, to describe the internal motions in a molecule, we are not interested in its location in space. So three coordinates required to specify the position of the center of mass of a molecule can be subtraced leaving 3N 3 coordinates. To describe rotational motions in a molecule, we are interested in its orientation in a coordinate system. The orientation of a linear molecule with respect to a coordinate system requires two angles leaving 3N 5 coordinates to describe the internal motions whereas a nonlinear polyatomic molecule requires three angles leaving 3N 6 coordinates. These 3N 3 or 3N 6 internal motions are referred to as vibrational degrees of freedom. Schrödinger Equation for Nuclear Motion For a diatomic molecule, Ĥ rot depends only on two angles, θ and ψ. Ĥ vib depends only on R, the internuclear separation. For polyatomic molecules, Ĥ vib is more complex, depending on 3N 6 coordinates for nonlinear molecules and 3N 5 coordinates for linear molecules Introduction to Vibrational Spectroscopy Spectroscopy is an important chemical tool. Two features have enabled vibrational spectroscopy to achieve the importance that it has as a tool in Chemistry. 1. The vibrational frequency depends primarily on the identity of the two vibrating atoms on either end of the bond and to a much lesser degree on the presence of atoms farther away from the bond. 2. This property generates characteristic frequencies for atoms joined by a bond known as group frequencies. Introduction to Vibrational Spectroscopy The table below shows the number of diatomic molecules in the first vibrational state (N 1 ) relative to those in the ground state (N 0 ) at 300 K and 1000 K calculated using the Boltzmann distribution. Vibrational State Populations for Selected Diatomic Molecules Molecule ν(cm 1 ) ν(s 1 N ) 1 N 0 for 300 K N1 N 1 N 0 for 100 K H-H x x x 10 3 H-F x x x 10 3 H-Br x x x 10 2 N-N x x x 10 2 C-O x x x 10 2 Br-Br x

7 Introduction to Vibrational Spectroscopy For absorption by a quantum mechanical harmonic oscillator, v = v final v initial = +1. Nearly all molecules are in the ground state even at 1000 K because N 1 /N 0 << 1 except for Br 2. Vibrational Spectra of Diatomic Molecules The potential energy curve for diatomic molecules are not exactly parabolic but is approximately parabolic in the vicinity of the equilibrium internuclear distance R e as indicated by the dashed line. This means that absorption of light will occur from molecules in the v = 0 state. It will be shown that in most cases only the v = 0 v = 1 transition is observed in vibrational spectroscopy Vibrational Spectra of Diatomic Molecules The potential energy indicated by the dashed line is given by the parabola (harmonic potential) where k is the force constant. E(R) = 1 2 k(r R e) 2 The energy levels for a simple harmonic oscillator are given by where and µ is the reduced mass. E v = (v + 1 )hν v = 0, 1, 2,... 2 ν = ( ) ( ) 1 1 k 2 2π µ Vibrational Spectra of Diatomic Molecules For larger internuclear separations, the potential curve becomes anharmonic. It is difficult to solve for the exact form of E(R), but to a good approximation, a realisitc anharmonic potential can be described in analytical form by the Morse potenital: V (R) = D e [1 e α(r Re) ] 2 D e is the dissociation energy relative to the bottom of the well. α = k 2D e The force constant, k is defined by k = ( d2 V (R) dr 2 ) R=Re and ν = k µ

8 Vibrational Spectra of Diatomic Molecules The spectroscopic dissociation energy D 0 is defined with respect to the lowest allowed vibrational energy level, rather than to the bottom of the potential as shown in the figure below Vibrational Spectra of Diatomic Molecules The energy levels for the Morse potential are given by ( E v = hν v + 1 ) ( (hν)2 v + 1 ) 2 2 4D e 2 The second term gives the anharmonic correction to the energy levels. Measurements of the frequencies of the overtone vibrations allow the parameter D e in the Morse potential to be determined for a specific molecule Values of Molecular Constants for Selected Diatomic Molecules The Origin of Selection Rules Not all diatomic molecules have an infrared (vibrational) absorption spectrum. To determine which transitions are possible, an equation for a transition dipole momemt is used. According to the Born-Oppenheimer approximation, the wavefunction for a molecule in the electronic state ψ elec, the vibrational state ψ vib, and having a particular set of rotational quantum numbers, can be written as a product of ψ elec ψ vib ψ rot. µ elec The permanent dipole moment µ elec 0 of a molecule in this electronic state is equal to the expectation value of the operator ˆµˆµˆµ over the wavefunction of the electronic state: µ elec ψelec ˆµˆµˆµ ψ elec dτ elec 0 = 31 32

9 The Origin of Selection Rules Since the dipole moment for a diatomic molecule depends on the internuclear distance, the electronic dipole moment can be expanded in a Taylor series about R = R e : ( ) µ µ elec 0 = µ e + (R R e ) + 1 ( ) 2 µ (R R R R e 2 R 2 e ) 2 + R e For a molecule in a given electronic state, the transition dipole moment for a vibrational transition is given by ψ n µψ n = µ e ψ n ψ n dτ + ( µ e ) R R e ψ n (R R e ) ψ n dτ+ ( 2 µ e R 2 )R e ψ n (R R e ) 2 ψ n dτ + The Origin of Selection Rules ψ n µψ n = µ e ψ n ψ n dτ + ( µ e ) R R e ψ n (R R e ) ψ n dτ+ ( 2 µ e R 2 )R e ψ n (R R e ) 2 ψ n dτ + The first term is zero because the vibrational wavefunctions for different v are orthogonal. The second term is nonzero if the dipole moment depends on the internuclear distance R. The integral in the second term vanishes unless v = v + 1 for harmonic oscillator wavefunctions. The second and higher derivatives of the dipole moment with respect to internuclear distance are small but do give rise to overtone transitions with v = ±2, ±3,, with rapidly diminishing intensities The Origin of Selection Rules The selection rule for a diatomic is that a molecule will show a vibrational spectrum only if the dipole moment changes with internuclear distance. Vibrational Overtones The vibrational absorption spectrum of HCl is shown schematically below in a stick representation: Homonuclear diatomic molecules such as H 2 and N 2 have zero dipole moments for all lengths and therefore do not show vibrational spectra. In general, heteronuclear diatomic molecules do have dipole moments that depend on internuclear distance, so they exhibit vibrational spectra. The strongest absorption band is at 3.46 µm. There is a much weaker band at 1.76 µm and a very much weaker one at µm. These are overtone transitions v = 0 v = 2 and v = 0 v =

10 Group Frequencies Group Frequencies For a molecule such as R the vibrational frequency of the C and O atoms is determined by the force constant for the C=O bond. This force constant is primarily determined by the chemical bond between the C and O atoms and to a much lesser degree by the adjacent R and R groups. O C For this reason, the carbonyl group (C=O) has a characteristic frequency (group frequency) at which it absorbs infrared radiation in a narrow range for different molecules. R Group frequencies are very valuable in determining the structure of molecules, and a few values are given in the table below: Selected Group Frequencies Group Frequency (cm 1 Group Frequency (cm 1 ) O-H stretch 3600 C=O stretch 1700 N-H stretch 3350 C=C stretch 1650 C-H stretch 2900 C-C stretch 1200 C-H bend 1400 C-Cl stretch Experimental Vibrational Spectra The T d Character Table Vibrational spectra for gas-phase CO and CH 4 are shown in the figure below: E 8C 3 3C 2 6S 4 6σ d A x 2 + y 2 + z 2 A E (2z 2 x 2 y 2, x 2 y 2 ) T (R x, R y, R z ) T (x, y, z) (xy, xz, yz) Because CO and CH 4 are linear and nonlinear molecules, we expect one (3 2-5) and nine (3 5-6) vibrational modes, respectively. However, the spectrum for CH 4 shows two rather than nine peaks as well as several unexpected broad peaks. Γ reducible = A 1 + E + 2T

11 Normal Mode Vibrations of Molecules Normal mode characteristics of molecules. During a vibrational period, the center of mass of the molecule remains fixed, and all atoms undergo in-phase periodic motion about their equilibrium positions. All atoms in a molecule reach their minimum and maximum amplitudes at the same time. These collective motions are called normal modes, and the frequencies are called the normal mode frequencies. Normal Mode Vibrations of Molecules Normal mode characteristics of molecules. The frequencies measured in vibrational spectroscopy are the normal mode frequencies. All normal modes are independent in the harmonic approximation; the excitation of one normal mode does not transfer vibrational energy into another normal mode. All motions of the atoms in a molecule can be expressed as a linear combination of the normal modes of that molecule Normal Mode Vibrations of Molecules To see what normal modes of vibration are, we first consider the vibration of polyatomic molecules from a classical mechanical point of view. The kinetic energy T of a polyatomic molecule is given by [ T = 1 N (dxk ) 2 ( ) 2 ( ) ] 2 dyk dzk m k dt dt dt k=1 The equation can be simplified by introducing mass-weighted Cartesian displacement coordinates q 1,..., q 3N : q 1 = m (x 1 x 1e ) q 2 = m (y 1 y 1e ) q 3 = m (z 1 z 1e ) q 4 = m (x 2 x 2e )... q 3N = m 1 2 N (z N z Ne ) where x ie are the values of the coordinates at the equilibrium geometry of the molecule. 43 Normal Mode Vibrations of Molecules The q i are independent of time, and the kinetic energy becomes T = 1 2 3N i=1 ( ) 2 dqi dt For a molecule with N vibrational degrees of freedom, the potential energy is given by V (q 1, q 2,..., q N ) = 1 2 Q 2 i=1 i 44 N N i=1 j=1 2 V q i q j q i q j A new set of vibrational coordinates can be found Q j (q 1, q 2,..., q N ) that simplify the above equation to V (Q 1, Q 1,..., Q N ) = 1 2 N ( 2 V ) Q 2 i

12 Normal Mode Vibrations of Molecules The Transformation Coordinate System for the Atoms in H 2 O Under the C 2v Point Group The Q j (q 1, q 2,..., q N ) are known as the normal coordinates of the molecule. Because there are no cross terms of the type Q i Q j in the potential energy, the vibrational modes are independent in the harmonic approximation ψ vibrational (Q 1, Q 2,... Q N ) = ψ 1 (Q 1 )ψ 1 (Q 1 )ψ 2 (Q 2 )... ψ N (Q N ) E vibrational = N ( ) vj + 1 hνj 2 Because of the transformation to normal coordinates, each of the normal modes contributes independently to the energy, and the vibrational motions of different normal coordinates are coupled. i=1 Ĉ 2 = C 2 Transformation Matrix for H 2O Molecule Transformations of the Coordinate Systems on the Atoms in H 2 O Under Symmetry Operations of the C 2v Group Ĉ 2 x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 = x 3 y 3 z 3 x 2 y 2 z 2 x 1 y 1 z 1 46 Guidelines for Calculating Characters in Reducible Representations for Coordinate Transformations Only the diagonal elements contribute to the character. Therefore, only atoms that are not shifted by an operation contribute to the character. If the atoms remains in the same position under the transformation, and the sign of x, y, or z is not changed, the value of +1 is associated with each unchanged coordinate. If the sign of x, y, or z is changed, the value of -1 is associated with each changed coordinate. If the coordinate system is exchanged with the position of another coordinate system, the value of 0 is associated with each of the coordinates

13 The C 2v Character Table Reducible Representation into an Irreducible Representation C 2 σ v σ v E C 2 σ v σ v A z x 2, y 2, z 2 A R z xy B x, R y xz B y, R x yz The leftmost column shows the symbol for each irreducible representation. By convention, a representation that is symmetric (+1) with respect to rotation about the principle axis (Ĉ2 in this case) is given the symbol A. A representation that is antisymmetric (-1) with respect to rotation about the principle axis is given the symbol B. C 2 σ v σ v E C 2 Γ reducible = The number of times, a p, any irreducible representation, p occurs in any reducible representation is given by ( ) 1 a p = n R χ(r)χ p (R) h sum over classes Remove translations and rotations Γ reduciible = 3A 1 + A 2 + 2B 1 + 3B 2 Γ reduciible = 3A 1 + A 2 + 2B 1 + 3B 2 (B 1 + B 2 + A 1 ) (B 2 + B 1 + A 2 ) then Γ reduciible = 2A 1 + B The Normal Modes of H 2 O Rotational Spectroscopy To a first approximation, the rotational spectrum of a diatomic molecule can be understood in terms of the Schrödinger equation for rotational motion of a rigid rotor. The normal modes of H 2 O: bond bending O-H symmetric stretch O-H antisymmetric stretch. The vectors indicate atomic displacements. The wavefunctions are the spherical harmonics YJ M (θ, φ), and there are two quantum numbers J and M for molecular rotation. The energy eigenvalues are given by 2 h2 E = J(J + 1) = J(J + 1) = hcbj(j + 1) 2µR0 2 8π 2 µr0 2 h where B =, the rotational constant, has units of cm 1 and 8π 2 cµr0 2 constants specific to the molecule. Since the energy does not depend on M = J,..., 0,..., +J, the rotational levels are (2J + 1)-fold degenerate

14 Rotational Spectroscopy The energies levels and transitions allowed are given by the selection rule J = J final J initial = ±1 The energies corresponding to rotational transitions of J = +1 correspond to absorption E + = hcb(j + 1)(J + 2) hcbj(j + 1) = hcbj(j + 1) while those transitions of J = 1 correspond to emission. E = hcb(j 1)(J) hcbj(j + 1) = hcbj The E + E because the rotational energy levels are not equally spaced. Rotational Spectroscopy The larger J the value of the orginating energy level, the more energetic the photon must be to promote excitation to the next highest level. For successive initial values of J, the E associated with the transitions increases in such a way that the difference between these frequencies ( ν), i.e. ( ν) is constant. J J ( ν) 0 1 2cB 2cB 1 2 4cB 2cB 2 3 6cB 2cB 3 4 8cB 2cB cB 2cB This means that the spectrum will show a series of equally spaced lines separated in frequency by 2cB Rotational Spectroscopy Vibration-Rotation Spectra of Diatomic Molecules Up to this point, rotation and vibration have been considered separately. In the microwave region of the electromagnetic spectrum, the photon energy is sufficient to excite rotational transitions but not to excite rotational transitions. For infrared radiation, diatomic molecules can make transitions in which both v and J change according to the selection rules v = +1 and J = ±1. The energy levels for the rigid rotor are shown on the left with the allowed transitions between levels shown as vertical bars. The spectrum observed through absorption of microwave radiation is shown on the right. Because of this structure, molecular spectra are often referred to a band spectra

15 Vibration-Rotation Spectra of Diatomic Molecules Schematic Representation of Rotational and Vibrational Levels Vibration-Rotation Spectra of Diatomic Molecules The fundamental vibration band for HCl (v = 0 1) is shown below: The double peaks are due to the presence of H 35 Cl (75% abundance) and H 37 Cl (25% abundance) Vibration-Rotation Spectra of Diatomic Molecules Vibrational Spectroscopy For vibrational spectroscopy, one intense peak is expected because the energy level spacing is the same for all quantum numbers so that for v = ±1, all transitions have the same frequency. Only the v = 0 energy level has significant population so that even taking anharmonicity into account will not generate additional peaks originating from v > 0. Vibration-Rotation Spectra of Diatomic Molecules Rotational Spectroscopy Because the rotational energy levels are not equally spaced in energy, different transitions give rise to separate peaks. E rotation << kt under most conditions so that many rotational energy levels will be populated. Many peaks are generated in the rotational spectrum. Although the overtone frequencies differ from the fundamental frequency, these peeks will have a low intensity

16 Relative Intensities of Peaks in Vibration-Rotation Spectra Relative Intensities of Peaks in Vibration-Rotation Spectra The intensity of a spectral line is determined by the mumber of molecules in the energy level from which it originates. Relative Number of Molecules The ratio of the number of molecules in a state for a given value of J relative to the number in the ground state (J = 0) can be calculated using the Boltzmann distribution: n J = g J e (e J e 0 ) kt n 0 g 0 = (2J + 1)e 2 J(J+1) 2IkT The 2J + 1 term in front of the exponential gives the degeneracy of the energery level of J and generally dominates nj for small J and large T. n0 The exponential term causes nj n0 to decrease rapidly with increasing J For a molecule such as HD with a small momentum of intertia, the rotational energy levels can be far enough apart that few rotational states are populated. For molecules with a large moment of inertia such as CO, the exponential term does not dominate until J is quite large yielding many occupied rotational energy levels. 61 Vibrational and Rotational energy Levels for a Diatomic Molecule 62 Simulated Absorption Spectrum for HCl at 300K The transitions with J = +1 give rise to the lines in the R Branch, and the transitions with J = 1 give rise to the lines in the P Branch of the spectrum. The intensities of the lines in these branches reflect the thermal populations of the initial rotational states. The Q Branch, when it occurs, consists of lines corresponding to J = 0. Generally, these transitions are forbidden, except for molecules such as NO which have orbital angular momentum about their axis. 63 P Branch R Branch 64

17 Return to Experimental Vibrational Spectra for CO and CH 4 High-Resolution Spectrum for CO In Which P and R Branches are resolved into the individual rotational transitions The broad resolved peaks seen for CO between 2000 cm cm 1 are the P and R branches corresponding to rotational-vibrational transitions. The minimum near 2200 cm 1 corresponds to the forbidden J = 0 transition. The broad and only partially resolved peaks for CH 4 seen around the sharp peaks centered near 1300 cm 1 and 3000 cm 1 are again the P and R branches Raman Spectra When a sample is irradiated with monochromatic light, the incident radiation may be absorbed, may stimulate emission, or may be scattered. A part of the scattered radiation radiation is referred to as the Raman spectrum. It is found that some photons lose energy in scattering from a molecule and emerge with a lower frequency. These photons produce Stokes lines in the spectrum of scattered radiation. A smaller fraction of the scattered photons gains energy in striking a molecule and emerges with higher frequency. These photons produce anti-stokes lines in the spectrum of scattered radiation. Raman Spectra The intepretation of Raman spectra is based on the conservation of energy. This requires that when a photon of frequency ν is scattered by a molecule in a quantum state with energy E i and the outgoing photon has a frequency ν, the molecule ends up in quantum state f with energy E f : hν + E i = hν + E f or h(ν ν) = E i E f = h ν R = hc ν R where the shift in frequency is labeled ν R, and the shift in wave number is labeled ν R

18 Raman Spectra Raman spectroscopy is different from absorption or emission spectroscopy in that the incident light need not coincide with the quantized energy difference in the molecule. Any frequency of light can be used. Since many final states consisting of both higher and lower energy than the initial state are possible, many Raman spectral lines can be observed. The frequency shifts seen in Raman experiments correspond to vibrational or rotational energy differences, so this kind of frequency gives us information on the vibrational and rotational states of molecules. Raman Spectra The Raman effect arises from the induced polarization of scattering molecules that is caused by the electric vector of the electromagnetic radiation. An isotropic molecule is one that has the same optical properties in all directions, e.g. the CH 4 molecule. A dipole moment µ is induced in the molecule by an electric field E. µ = αe where α is the polarizability which has units of dipole moment divided C m by electric field strength, i.e. = C2 m 2. V m 1 J For an isotropic molecule the vectors µ and E point in the same direction, and the polarizability is a scalar Raman Spectra The polarizability α of a rotating or vibrating molecule is not constant but varies with some frequency, ν vib or ν rot. Consider a molecule with a characteristic vibrational frequency ν vib in a time-dependent electromagnetic field: E = E 0 cos 2πνt The electric field distorts the molecule slightly because the negative valence electrons and positive nuclei experience forces in opposite directions. This induces a time-dependent dipole moment of magnitude µ induced of the same frequency as the electric field. Raman Spectra As shown earlier, the dipole moment is related to the polarizability and the magnitude of the electric field. The polarizability is an anisotropic quantity, and its value depends on the direction of the electric field relative to the molecular axes: µ induced (t) = αe 0 cos πνt The polarizability depends on bond length x e + x(t), where x e is the equilibrium value. The polarizability can be expanded ( ) dα α(x) = α(x e ) + x + dx x=x e The vibration of a molecule is time dependent and is given by x(t) = x max cos 2πν vib t 71 72

19 Raman Spectra Combining the equations from the previous slide gives Raleigh {}}{ µ induced = α(x e )E 0 cos eπνt ] ( +[ dα ) dx x=x e x max E 0 cos(2πν + 2πν vib )t + cos(2πν 2πν }{{} vib )t }{{} anti-stokes Stokes The time-varying dipole moment radiates light at the same frequency (Raleigh), at a higher frequency (anti-stokes), or at lower frequency (Stokes). In addition to scattered light at the incident frequency, light will also be scattered at frequencies corresponding to vibrational excitation and de-excitation. Raman Spectra The equation on the previous slide demonstrates that the intensity of the Stokes and anti-stokes peaks is zero unless dα dx 0. The polarizability of a molecule must change as it vibrates. This condition is satisfied for many vibrational modes including the stretching vibration of a homonuclear diatomic molecule. Thus, the stretching vibration of a homonuclear molecule is infrared inactive but Raman active. Not all vibrational modes that are active for the absorption of light are Raman active and vice versa demonstrating that infrared and Raman spectroscopies complement each other Raman Spectra Are the intensities of the Stokes and anti-stokes peaks equal? The relative intensitiy is governed by the relative number of molecules in the originating states For the Stokes line, the transition originates from v = 0 state, whereas for the anti-stokes line, the transition originates from the v = 1 state. The relative intensities of the Stokes and anti-stokes peaks can be calculated from the Boltzmann distribution: I anti Stokes I Stokes = n excited = e n ground 3hν 2kT e hν 2kT = e hν kt For vibrations for which ν is in the range cm 1, this ratio ranges between 8 x 10 3 and 5 x 10 7 at 300K demonstrating that the intensities of the Stokes and anti-stokes lines will be different. Raman Spectra In order for a molecular motion to be Raman active, the polarizability α must change when that motion occurs ( dα dx 0 ). In order for a vibrational mode to be active, the polarizability α must change during a vibration, and for a rotation to be Raman active, the polarizability must change as the molecule rotates in an electric field. The polarizability of both homonuclear and heteronuclear diatomic molecules changes as the distance changes because this alters the electronic structure. The polarizability of a spherical molecule does not change in a rotation. Thus, spherical rotors do not have a rotational Raman effect. All other molecules are anisotropically polarizable which means that the polarization is dependent on the orientation of the molecule in the electric field. The mutual exclusion rule states that for molecules with a center of symmetry, fundamental transitions that are active in the infrared, are forbidden in Raman scattering and vice versa

20 Raman Active Modes Rotational Raman Spectrum for a Linear Molecule To be Raman active, a vibrational mode must produce a change in molecular polarizability of the molecule. The Raman modes are those that have the same symmetry as the molecular polarizability (α) as the binary functions (x 2, y 2, z 2, xy, xz, yz). In addition, Raman active modes will leave plane polarized light polarized, if they are totally symmetric, otherwise the light will be depolarized. The theoretical rotation-vibration Raman spectrum for v = +1, 0, 1 and J = +2, 0, 1 is shown below for a linear molecule. The lines which appear at higher frequencies are referred to as the O branch. In addition, there is a Q branch for J = 0. The S, Q, and O branches correspond to the P, Q, and R branches of infrared spectroscopy. Example: H 2 O For H 2 O, the Raman active modes must be a 1, a 2, b 1, b 2 symmetry. The Raman activity of a 1 modes is identified by adding pol in the brackets after the listed modes. and the activity of the other modes is identified by adding depol. Γ vib (H 2 O) = 2A 1 (IR, pol) + B 2 (IR, depol) Raman Spectra and Selection Rules Symmetry Selection Rules for Infrared Spectra The specific selection rule for the vibrational Raman effect is v = ±1. The vibrational transitions are accompanied by rotational Raman transitions with the specific selection rules J = 0, ±2. A Fundamental Transition consists of a transition from a molecule in a vibrational ground state (initial vibrational state wave function, ψ i) to a vibrationally excited state (final vibrational state wave function, ψ f ) where the molecule absorbs one quantum of energy in one vibrational mode. A vibrational transition in the infrared occurs when the molecular dipole moment (µ) interacts with incident radiation which occurs with a probability which is proportional to the transition moment: ψ iµψ f dτ A transition is said to be forbidden in the infrared if the value of this integral is zero because the probability of that transition is zero and no absorption will be observed. The integral will be zero unless the direct product of ψ iµψ f contains the totally symmetric representation which has the character +1 for all symmetry operations for the molecule under consideration. The vector µ can be split into three components, µ x, µ y, and µ z along the Cartesian coordinate axes, and only one of the three integrals needs to be non-zero: ψ i µx µ y ψ f dτ µ z The pure rotational Raman spectrum of CO 2 The intense peak at 488 nm is due to elastic scattering

21 Symmetry Selection Rules for Infrared Spectra of RuO 4 Symmetry Selection Rules for Raman Spectra Consider the vibrations of the tetrahedral molecule, ruthenium tetroxide (T d symmetry), The probability of a vibrational transition occurring in Raman scattering is proportional to: ψ iαψ f dτ where there are vibrations of A 1, E, and T 2, and deduce the infrared activity of each of them. 1. ψ i has A 1 symmetry. 2. The character table for T d shows that T x, T y, T z together have T 2 symmetry. The direct products are then A 1 vibration: A 1 T 2 A 1 = T 2 E vibration: A 1 T 2 E = T 2 E = T 1 + T 2 T 2 vibration: A 1 T 2 T 2 = T 2 T 2 = A 1 + E + T 1 + T 2 Thus, the T 2 vibrations are infrared active because the direct products produce an A 1 representation, but the E and T 2 vibrations will not appear in the infrared spectrum. An important result of the above analysis is that if an excited vibrational mode has the same symmetry as the translation vectors, T x, T y, T z, for that point group, then the totally symmetric irreducible representation is present and a transition from the vibrational ground state to that excited vibrational mode will be infrared active. where α is the polarizability of the molecule. The Raman effect depends on a molecular dipole induced by the electromagnetic field of the incident radiation and is proportional to the polarizability of the molecule which is a measure of the ease with which the molecular electron distribution can be distorted. α is a tensor, i.e. a 3 x 3 array of components α x 2 α xy α xz α yx α y 2 α yz α zx α zy α z 2 so there will be six distinct components α x 2 α y 2 α ψ i z 2 ψ α xy f dτ α yz α zx where one non-zero integral is needed to have an allowed Raman transition Symmetry Selection Rules for Raman Spectra of RuO 4 For the ruthenium tetroxide molecule with T d symmetry, the components of polarizability have the following symmetries: A1 x 2 + y 2 + z 2 E 2z 2 x 2 y 2, x 2 y 2 T2 xy, yz, zx The vibrations are A1, E, and T2, and it is possible to deduce the infrared activity of each of them. The direct products are then A1 A1 E A1 = A1, E, T2; A1 vibration is possible. T2 A1 A1 E E = E, (A1 + A2 + E), (T1 + T2); E vibrations are possible. T2 A1 A1 E T2 = T2, (T1 + T2), (A1 + E + T1 + T2); T2 vibrations are possible. T2 Thus, all of the vibrations in the RuO4 molecule are Raman active. A summary of the infrared and Raman activity is given as A1: x 2 + y 2 + z 2 Raman Only E: 2z 2 x 2 y 2, x 2 y 2 Raman Only T2: (Tx, Ty, Tz), (xy, yz, zx) Infrared and Raman Active 83