Key Concepts for Final

Heterogeneous Broadening of Lines - different molecules in the sample may have instantaneously different environments (or be subject to different perturbations) leading to a distribution of values for molecular eigenenergies.

Lifetime broadening - If on average a system survives in a state for a time  , then its energy h levels are blurred by E , where E  and  is the lifetime of the state. The uncertainty of 2 molecular eigenenergy is due to the Heisenberg Uncertainty Principle. Short-lived states have a large E , while long-lived states have a small E . Excited-state lifetimes are determined by the intrinsic transition frequency (energy), intrinsic transition strength (probability), extrinsic deactivation processes (e.g. non-radiative energy transfers such as intermolecular collisions).

Doppler Line Broadening - Radiation is shifted in frequency when the source is moving towards or away from the observer. This causes different sub-populations of molecules in the sample to "see" different radiation frequencies even though the radiation entering the sample is monochromatic. For molecules moving parallel to (towards) the direction of light propagation, f f   v . For molecules moving anti-parallel to (away from) the direction of light 1 c f f   propagation, v where f is the actual frequency of light, f ' is the apparent frequency of 1 c light, v is the molecular speed, and c is the speed of light.

Electric Dipole Selection Rules for radiative transitions in molecules The electric dipole transition moment must be non-zero. * Note: The electric dipole transition moment = μ   μˆ d where μˆ is the electric dipole if  f i operator and μˆ  erˆ .

Magnetic Dipole Selection Rules for radiative transitions in molecules The magnetic dipole transition moment must be non-zero. * Note: The magnetic dipole transition moment = m   mˆ  d where ˆ is the magnetic if  f i m dipole operator.

Absorption Spectroscopy

I is proportional to Niif Dif where I is the integrated absorption-line intensity, Ni is the number of molecules in the initial state, if is the transition frequency, and Dif is the transition dipole strength.

Luminescence Spectra (spontaneous emission) 4 I is proportional to Niif Dif where I is the integrated absorption-line intensity, Ni is the number of molecules in the initial state, if is the transition frequency, Dif is the transition dipole strength, and the initial state (i) is the emitting state.

 Absorption: the transition raises the energy of the atom or molecule  Stimulated (or induced) emission: the transition lowers the energy of the atom or molecule. The emission is stimulated by the radiation field. The radiation that is emitted has the same wavelength as the incident radiation, moves in the same direction, and is in phase with the incident radiation. This type of radiation is also called coherent radiation and is emitted by lasers.  Spontaneous emission: occurs in the absence of stimulating radiation. It is emitted in all directions and is not coherent.

Hydrogen Atom Selection Rules m  0,1 l  1 n  no restrictions

Multielectron Atom Selection Rules L  1 S  0 J  0,1 (0 to 0 forbidden)

M J  0,1 (0 to 0 forbidden for J  0)

Translation Motion of Molecules in a Confined Volume  In a big box, the spacings between energy levels are small enough that we can approximate a  3  continuum of energy. Classically, the average translational energy E   k BT where kB is  2  Boltzman's constant.  In a confined volume (e.g. electrons in the  system of benzene), the energy levels must be treated as quantized because the spacings between energy levels are large. 2 2 2 2  h  n n y n  E    x   z  where n ,n ,n  1,2,3,...   2 2 2  x y z  8m  a b c 

Rigid Rotor  The object rotates, but does not vibrate J J 1h2  EJ  where J = 0,1,2… 82 I

 The degeneracy of the energy level corresponding to each J is given by g J  2J 1  Rotational energy levels are NOT equally spaced  Selection Rules: The molecule must have a permanent electric-dipole moment and J  1;

M J  0,1; K J  0 .

Types of Rigid Rotor Linear ……………………………………………. Ic = Ib and Ia = 0 Spherical Top …………………………………… Ia = Ib = Ic Symmetric Top (oblate)………………………….. Ia = Ib < Ic Symmetric Top (prolate)………………………….. Ia < Ib = Ic Asymmetric Top …………………………………. Ia < Ib < Ic

Where Ia, Ib, and Ic are the moments of inertia about the principal axes of inertial a, b, and c. These axes are mutually perpendicular. The unique principal axis (or figure axis) of a symmetric top is always an axis of symmetry of order greater than or equal to 3.

Moment of Inertia of a Diatomic

 m1m2  2 I   R  m1  m2  m m where I is the moment of inertia, R is the bond length, and the reduced mass =   1 2 . m1  m2

Rotational Spectra of Diatomic Molecules h h  Rotational constant = B  when B is expressed in cm1; and B  when B is 8 2cI 8 2 I expressed in Hz.

Rotational Energy levels

J J 1h2  Linear: EJ  with a degeneracy of g J  2J 1, where J = 0,1,2,3,… and 82 I

M J  0,1,2,...  Linear; Spherical Top; Symmetric Top … expressions for energy, total angular momentum, and angular momentum about lab z-axis for each  Eigenfunctions are spherical harmonic functions

Vibrational Motion  Solutions are Hermite Polynomials  1   1  k  Ev   v  h0 where the fundamental frequency  0    and v = 0,1,2,…  2   2  

 Vibrational energy levels are equally spaced (  h 0 )  Anharmonicity: deviation from harmonic behavior  Selection Rules: v  1 for harmonic behavior; and v  1,2,3... for anharmonic behavior. There must also be a change in dipole moment when the molecule is distorted along the normal vibration coordinate.

Vibration-Rotation Absorption Spectra

 Selection rules: v  1, J  0,1; M J  0,1; K J  0 . There must also be a change in dipole moment when the molecule is distorted along the normal vibration coordinate; and the molecule must have a permanent electric-dipole moment. ( J  0 only in special cases. Typicaly, J  1).

Raman Spectroscopy  Selection Rules: Molecule must have anisotropic electric polarizablity; J  0,2 for linear molecules, J  0,1,2 for non-linear molecules; and v  0,1.  A molecule has anisotropic polarizability if its polarizability is different in different directions  Anti-Stokes lines correspond to transitions from higher to lower energy levels. In this case, the molecule makes a transition with J  2 and the scattered photon emerges with increased energy (and therefore higher frequency than the incident radiation).  Spectral lines corresponding to transitions from a lower to a higher molecular energy levels are Stokes lines. The molecule makes a transition with J  2 and the lines appear at lower frequency than the incident radiation.

Lasers - Light Amplification by Stimulated Emission  Must disturb an equilibrium population to obtain an inverted population  Stimulated vs. spontaneous emission  3-level laser, 4-level laser  Types of lasers: solid-state lasers, liquid lasers, gas lasers  General characteristics of lasers: high photon flux, beam collimation, phase coherence (spatial and temporal), monochromatic light output

Other Concepts  Band Theory: insulators vs. semiconductors (p-type, n-type semiconductors)  Classical vs. quantum descriptions of light  Degrees of freedom: translational (3 per molecule); rotational (2 per linear molecule, 3 per non-linear molecule); vibrational (3N  5 per linear molecule, 3N  6 per non-linear molecule) where N is the number of atoms in the molecule  Einstein transitions probabilities; einstein corefficients  Rotational Stark Effect

 Resonance condition (also known as the Bohr frequency rule): h  E'J 'K ' EJK  Centrifugal Distortion  Normal modes of vibration: specify the atomic trajectories associated with distinct (and independent) modes of vibrational motion  Franck-Condon principle states that for an electronic transition (optical excitation), the nuclei do not respond immediately, and a non-equilibrium state results. This is followed by nuclear relaxation.  Circular Dichroism  Fluorescence  Phosphorescence  Internal Conversion  Intersystem Crossing  Chromophore: a molecule or part of a molecule that exhibits a characteristic absorption  P, Q, and R branches

Note: This list is only a guide to help you study. It is NOT comprehensive, and the exam may cover any topics discussed in class.