Chem 44 Review of Spectroscopy General spectroscopy Wavelength (nm), frequency (s -1 ), wavenumber (cm -1 ) Frequency (s -1 ): n= c l Wavenumbers (cm -1 ): n =1 l Chart of photon energies and spectroscopies Degrees of freedom 3n+3N Born-Oppenheimer separation: electronic (3n) + nuclear (3N) Nuclear (3N): translation (3) exactly separable; rotations ( or 3) approximately separable; vibrational (3N-5 or 3N-6) Sources of radiation, dispersing elements, detectors (do not need to know details) Einstein s theory of three modes of optical transitions: stimulated absorption (B), stimulated emission (B ), spontaneous emission (A) Equilibrium condition for the rates of these three transitions: W = W NBr= NB r+ NA 3 3 8pn h c Blackbody radiation formula for photon density: r = h kt e n -1 Solve for relation between A, B, and B Lasers: high power, monochromatic and polarized, coherent, low divergence and long path lengths; population inversion Linewidths: lifetime broadening, collisional deactivation, natural linewidth, Doppler broadening Absorption/emission spectroscopy: microwave, IR, UV/Vis, fluorescence 1-photon process, first-order perturbation theory Related to dipole moment and its changes (changes in molecular length) Dipole moment is a vector This spectroscopy occurs when molecular vibration or rotation alters dipole moment Tends to transform as x, y, z Scattering spectroscopy: Raman (excite to virtual state) -photon process, second-order perturbation theory Stokes Raman: n0-n ; Rayleigh:n 1 0 ; anti-stokes Raman: n + 0 n1 Related to polarizability and its changes (changes in molecular volume) Polarizability: softness of wavefunction; the larger the polarizability, the more easily the wavefunction is distorted by external electric field to create an induced dipole Polarizability is a tensor Classical theory of Rayleigh and Raman scattering: oscillating electric field causes molecule to have induced dipole; polarizability and therefore induced dipole vary with molecular vibration This spectroscopy occurs when molecular vibration or rotation alters polarizability Tends to transform as xx, yy, zz, xy, yz, zx 1
Rotational spectroscopy Microwave Rigid rotor in 3D (particle on a sphere, diatomic molecule with fixed bond length R): J( J + 1) =, mm, A B I = mr m= I ma+ mb Moments of inertia: tensor; eigenvector is principal axis of rotation, eigenvalue is principal moment of inertia mm A B Linear rotors: I = A I = B R ; IC ma+ m = 0 ; degeneracy J+1 B Spherical rotors: I A = IB = IC; degeneracy (J+1) Symmetric rotors: I A = IB ¹ IC; degeneracy (J+1) Degeneracies determined by considering both K and M J (K refers to rotation about principal axis, and M J refers to rotation about externally fixed z-axis) = J( J + 1) = hcbj( J + 1) I Rotational constant (given in cm -1 ): B = 4 pci Note: = BJ( J +1 ) if energy also given in cm -1 Energy levels at 0B, B, 6B, 1B, ; transitions from J to J+1 are at B(J+1): B, 4B, 6B; peaks separated by B Zeeman effect: lifts M J degeneracy by applying static magnetic field along z-axis; 1 st -order perturbation theory; even spacing of perturbed levels Stark effect: lifts M J degeneracy by applying static electric field along z-axis; nd -order perturbation theory; uneven spacing of perturbed levels Selection rules: Transition dipole moment: m * ˆ x = ò yfxyid t Intensity of transition: I µ m Selection rules for pure rotational absorption: Gross selection rules: nonzero permanent dipole, m ev Specific selection rules: D J = D M J = 0, Centrifugal distortion: Rotation slightly increases bond length R Larger R larger I smaller B smaller spacing = BJ( J + 1) - DJJ ( J + 1 ) Centrifugal distortion leads to smaller spacing at higher J Rotational spectrum of diatomic molecule: Many lines because many states populated, B<<kT
Approximately even spacing: B (see energy expression above) Centrifugal distortion line spacing slightly narrower at higher energies Intensities initially increase with energy and then decrease, creating a bell-shaped envelope Balance between degeneracy and Boltzmann populations: (1) Degeneracy leads to greater intensities at higher energies (from above, see that degeneracies increase with J) () Boltzmann population decreases at higher J and therefore leads to lower intensities at higher energies (less population of higher states to excite) - kt - hcbj ( J + 1) kt Boltzmann distribution: e = e Selection rules for pure rotational Raman scattering Gross selection rules: polarizability changes upon rotation (non-spherical volume); spherical rotors are inactive Specific selection rules: D J = Rotational spectrum of diatomic molecule: Stokes wing, Rayleigh peak, anti-stokes wing Each wing s envelope explained by competing effects of degeneracy and Boltzmann distribution Separation between peaks in Stokes and anti-stokes wings is 4B Anti-Stokes wing slightly less intense than Stokes wing because of Boltzmann distribution (temperature effect) H rotational Raman spectra: 3:1 intensity alternation Odd J levels are triply degenerate and even J levels are singly degenerate Nuclear spin statistics: protons are fermions Rotational wavefunction must be antisymmetric wrt exchange of two protons para-h : singlet; symmetric spatial part, antisymmetric spin part; even J ortho-h : triplet; antisymmetric spatial part, symmetric spin part; odd J Vibrational spectroscopy Infrared (IR) Harmonic approximation harmonic oscillator wavefunctions Selection rules for infrared absorption Gross selection rule: dipole varies with vibration Specific selection rule: n f = ni Selection rules for Raman scattering Gross selection rule: polarizability varies with vibration Specific selection rule: n f = ni Splittings between vibrational energy levels usually larger than kt, so only ground state is significantly populated Harmonic potential: single peak =1 0 Hot bands: =, =3 Weak because not much population 3
Same frequency because harmonic so evenly spaced Anharmonicity: Hot bands shifted to lower frequencies because energy levels not evenly spaced, splitting decreases as energy levels get higher Overtones (specific selection rules not exactly true): = 0, =3 0 occur at higher frequencies with decreasing spacing and intensity Normal modes: 3N-5 for linear molecules, 3N-6 for other molecules Normal mode: classical motion of nuclei with well-defined frequency System can be decomposed into independent 1D harmonic oscillators with welldefined frequencies, called normal modes (uncoupled) vibrational wavefunction is product of 1D harmonic oscillator wavefunctions for nomal mode coordinates energy is sum of 1D harmonic oscillators energies Electronic spectroscopy UV/vis Total dipole moment of a molecule involves electronic and nuclear coordinates Born-Oppenheimer approximation allows separation of electronic/nuclear coordinates Intensity proportional to square of transition dipole moment: I e ˆ fxeidte n fnidt n (Note: Greek symbols in integrals are electronic and vibrational wavefunctions, and the integrals are over electronic and nuclear coordinates, respectively) Franck-Condon factor: overlap of initial and final vibrational wavefunctions UV/vis spectroscopy gives information on electronic excitation energies and vibrational frequencies in excited electronic states Vertical transition vibrational progression with intensities depending on Franck- Condon factors Short progression dominated by 0 0 transition: PES of electronic states have similar equilibrium structures and similar vibrational frequencies; excited electron from non-bonding or weaker orbital Long progression with weak 0 0 transition: PES of electronic states have very different equilibrium structures; excited electron from bonding orbital; vertical excitation leads to maximum at classical turning point of vertically accessible state and decreasing transitions on either side; molecule is far from new equilibrium structure and vibrates toward it; vibration is along totally-symmetric structure change from ground to excited state Long progression with alternating intensities: PES of electronic states have similar equilibrium structures but very different vibrational frequencies; may occur for non-totally-symmetric vibrations Progression with blurred structure: upper PES (electronic state) becomes dissociative where blurring starts Fluorescence (emission) spectroscopy: Kasha s rule: molecule loses energy to surroundings non-radiatively (collisions in = ò ò 4
the gas phase dispense of a quantum of vibrational energy) and goes to ground vibrational state of excited electronic state; fluorescence occurs from the lowest vibrational/rotational state Fluorescence quenching: a quantum of electronic energy can be dispensed nonradiatively by more frequent collisions in solution, leading to quenching of fluorescence in solution Vibronic coupling and vibronic transition: breakdown of Born-Oppenheimer separation, cannot separate electrons and nuclei, cannot write overall wavefunction as product of electronic and vibrational wavefunction Examples: d-d transitions in metal complexes, Franck-Condon-forbidden transitions in benzene 5