CHM 3411 - Physical Chemistry II Chapter 12 - Supplementary Material 1. Einstein A and B coefficients Consider two singly degenerate states in an atom, molecule, or ion, with wavefunctions 1 (for the lower energy state) and 2 (for the higher energy state). The following processes involving the interaction of light with these states of the molecule can take place. Absorption M 1 + h M 2 (12.1.1) Stimulated emission M 2 + h M 1 + 2 h (12.1.2) Spontaneous emission M 2 M 1 + h (12.1.3) Note that the photon produced by stimulated emission (12.1.2) will travel in the same direction and have the same phase as the incident photon. Spontaneous emission (12.1.3), in contrast, will be isotropic (probability of light emission is the same in all directions). If we look at the change in the number of molecules in state 1 with time, we get dn 1 /dt = - B 12 ( ) N 1 + B 21 ( ) N 2 + A 21 N 2 (12.1.4) where N 1 and N 2 are the number of molecules in state 1 and state 2, B 12, B 21, and A 21 are the rate constants for absorption, stimulated emission, and spontaneous emission, and ( ) is the density of electromagnetic radiation at frequency. For a system at equilibrium dn 1 /dt = 0. By setting dn 1 /dt = 0 in eq 12.1.4, the following expression for N 2 /N 1 can be found. N 2 = B 12 ( ) (12.1.5) N 1 B 21 ( ) + A 21 However, there is a second expression for N 2 /N 1 (from the Boltzmann distribution law) N 2 = exp( - E 21 /kt) (12.1.6) N 1
where E 21 = E 2 E 1 is the difference in energy between the two states, and k is the Boltzmann constant. If we look eq 12.1.5 and 12.1.6 in the limit T (which corresponds to ( ) ) we get Lim T N 2 = B 12 = 1 (12.1.7) N 1 B 21 where the first term in eq 12.1.7 comes from eq 12.1.5 and the second term comes from 12.1.6. From this, we can conclude that B 12 = B 21 = B, meaning that the rate constant for absorption and stimulated emission are equal. If we again assume equilibrium, and solve eq 12.1.5 for A 21, we get (after some algebra) A 21 = A = B ( ) [ (N 1 /N 2 ) 1 ] = B ( ) [ exp(h /kt) 1 ] (12.1.8) where we have used the fact that E 21 = h and N 1 /N 2 = exp(h /kt). But ( ) is given by the Planck distribution law ( ) = 8 h 3 1 (12.1.9) c 3 [ exp(h /kt) 1 ] Substitution of eq 12.1.9 into eq 12.1.8 gives as a final result A = 8 h 3 B (12.1.10) c 3 Finally, it can be shown in the dipole approximation that B is given by the expression where B = 8 2 2 * 1 d 2 (12.1.11) 3h 2 = i=1 N q i r i (12.1.12) is the dipole moment operator. Note that the sum in eq 12.1.12 is over all of the charges present in the molecule and r i is the vector representing the location of the i th charge. What is the significance of the above? 1) The rate constants for absorption, stimulated emission, and spontaneous emission are related to one another by eq 12.1.10 and the fact that B 12 = B 21 = B. 2) The rate constant for spontaneous emission is proportional to 3 (eq 12.1.10). This means that spontaneous emission (fluorescence) is easily detected only for high frequency light (visible and UV light). 3) While the integral in eq 12.1.11 is generally not easy to evaluate, we can often use symmetry based arguments to decide whether the integral is or is not equal to zero. This is the origin of selection rules, which classify transitions as either allowed (integral is not equal to zero) or forbidden (integral is equal to zero). Since selection rules are based on the dipole approximation and usually also make use of approximations for 1 and 2, forbidden transitions may be observed, but are generally much weaker than allowed transitions.
2. Notation in spectroscopy (general considerations) Chemists are not always consistent in the notation they use in rotational. vibrational, and electronic spectroscopy. The following summarizes a consistent presentation of spectroscopy that is commonly used by spectroscopists. Energies are usually given in wavenumbers (cm -1 ), where E(cm -1 ) = 1/, where is the wavelength (in cm) of the photon that is absorbed or admitted when a transition is made between two states a molecule. This is related to the energy in MKS units (J = Joule) by E(cm -1 ) = E(J)/hc (12.2.1) where E(cm -1 ) is energy in wavenumbers and E(J) is energy in Joule. The conversion factor between the two is 1 cm -1 = 1.986 x 10-23 J (12.2.2) Often the symbol Ẽ is used when energy is given in wavenumbers. Note that in the material given below we will always use wavenumbers as our energy unit (which means that rotational and vibrational constants will also be given in units of wavenumbers). In labeling states the lower energy state is usually given a " (double prime) and the higher energy state is given a ' (single prime). This is done for all types of spectroscopy - absorption, emission (fluorescence) and Raman. Diatomics 3. Pure rotational spectroscopy In the rigid rotor (RR) approximation the rotational energy of a diatomic molecule is given by the expression E = J(J+1) B (12.3.1) where B, the rotational constant, is B = h/8 2 Ic = h/8 2 c r 2, I = r 2, = m A m B /(m A +m B ) (12.3.2) In the above expressions I is the moment of inertia, which for a diatomic molecule is equal to r 2, where is the reduced mass and r is the bond length for the molecule. with For high precision work the rotational energy can be written in terms of a power series expansion in J(J+1), E = J(J+1) B v - J 2 (J+1) 2 D v (12.3.3) In the above expression B v takes into account that the rotational constant is different for different vibrational states of the molecule (because the average bond distance increases as v ncreases), and D v takes into account the stretching of the bond as the molecule rotates (due to the centrifugal effect). B v is itself often written as B v = B e - (v + 1 / 2 ) e (12.3.4) since for small values of v the changes in the value of the rotational constant are approximately proportional to v. Note that B e is the value of the rotational constant (eq 12.3.2) at r = r e (equilibrium bond distance), the bond distance at the minimum in the molecular potential energy curve for the molecule.
Polyatomics In discussing rotational energy for polyatomic molecules the molecules are classified by their principal moments of inertia (I aa, I bb, I cc ) as follows (note that I must be greater than or equal to zero). 1) Linear molecules (including diatomic molecules) I aa = I bb = 0, I cc 0 2) Spherical top molecules (pure tetrahedral or octahedral) I aa = I bb = I cc 0 3) Symmetric top molecules (molecule has a C 3 or higher I aa = I bb < I cc or I aa < I bb = I cc proper axis of rotation 4) Asymmetric top molecule I aa < I bb < I cc In the rigid rotor approximation the energy levels for a linear polyatomic molecule or a spherical top molecule are the same as for a diatomic molecule (eq 12.3.1). If the moment of inertia for the molecule is known then the rotational constant can be found using eq 12.3.2. Note that for a spherical top molecule there are now three rotational quantum numbers J = 0, 1, 2,... M J = 0, 1, 2,... J K = 0, 1, 2,... J (12.3.5) M J and K have no effect on the energy, but the presence of a third rotational quantum number does have an effect on the degeneracy of the rotational energy levels. For a symmetric top molecules the rotational energy levels are given by the expression E = J(J+1) B + K 2 (A - B) (12.3.6) where A = h/8 2 I c B = h/8 2 I c (12.3.7) In the above expressions I is the moment of inertia parallel to the principal axis (and so is the the principal moment of inertia that is different than the others) and I is the moment of inertia perpendicular to the principal axis (and so corresponds to the two equal principal moments of inertia). The quantum numbers are the same as for a spherical rotor (eq 12.3.5) but the degeneracies of the energy levels are in general smaller than those for a spherical rotor due to the fact that the rotational energy depends on both J and K. For an asymmetic top molecule there is no general expression for the rotational energy levels. Approximate values for these levels (in the rigid rotor approximation) can be found using perturbation theory. Selection rules (pure rotational spectroscopy) 1) Molecule must have a permanent dipole moment. This means that homonuclear diatomic molecules, some linear polyatomic molecules (such as CO 2 and C 2 H 2 ) spherical top molecules, and some other polyatomic molecules do not exhibit a pure rotational spectrum. 2) J = 1 3) M J = 0, 1 4) K = 0 (for nonlinear molecules) Note that selection rule on K means that the location of allowed transitions for symmetric top molecules (in the rigid rotor approximation) is the same as for linear molecules, even though the rotational energy depends on K.
4. Vibrational spectroscopy Diatomics - Vibrational energy In the harmonic oscillator (HO) approximation the vibrational energy of a diatomic molecule is given by the expression E = (v + 1 / 2 ) (12.4.1) where, the vibrational constant, is = [1/(2 c)] (k/ ) 1/2, = m A m B /(m A +m B ) (12.4.2) In the above expressions k is the force constant for the molecule and is the reduced mass of the molecule. Note that k is related to the curvature of the molecular potential at the equilibrium bond distance, r e, with k = (d 2 V/dr 2 ) e. Real molecules deviate from the harmonic oscillator model even at low energies. By analogy to rotation, the vibrational energy levels for a real diatomic molecule can be written in terms of a power series expansion in (v + 1 / 2 ) E = (v + 1 / 2 ) e - (v + 1 / 2 ) 2 e x e + (v + 1 / 2 ) 3 e y e +... (12.4.3) where e is the value for the vibrational constant at r = r e, e x e (a single constant, despite how it is written) is the leading order correction term for the vibrational energy, and e y e,... are higher order correction terms. Real molecules, unlike the harmonic oscillator, will also have a highest value for vibrational quantum number, usually written as v max. It is interesting to note that eq 12.4.3, cut off after the leading order correction term, gives the correct expression for the energy levels of a Morse oscillator. A Morse potential has the form V(r) = D e { 1 - exp[-a(r - r e )] } 2 (12.4.4) where D e is the dissociation energy of the molecule (measured from the minimum in the potential energy curve) and a is a constant. Solution of the corresponding TISE for this potential gives E MORSE = (v + 1 / 2 ) e - (v + 1 / 2 ) 2 e x e (12.4.5) Since a Morse potential is close in appearance to the potential for a real diatomic molecule, eq 12.4.5 is expected to be a good approximation for the vibrational energy levels of a real diatomic molecule. In this case, the following relationship, which may be derived for a Morse oscillator, will be approximately correct (it is of course exact for a Morse potential) D e = e 2 /(4 e x e ) (12.4.6) where D e is the dissociation energy of the molecule as measured from the minimum of the potential energy curve. This is closely related to D 0, the dissociation energy measured from the v = 0 vibrational energy level. These two dissociation energies are related by the expression D 0 = D e - ( e /2 - e x e /4 ) (12.4.7)
Diatomics - Rotational structure and selection rules If we have sufficient energy to excite a vibrational transition then we can also change the rotational state of a diatomic molecule. Vibrational spectra will therefore exhibit rotational structure, which can provide additional information about the molecule. In the RRHO (rigid rotor-harmonic oscillator) model the combined vibrational + rotational energy of a diatomic molecule is E = (v + 1 / 2 ) + J(J+1)B (12.4.8) The combined selection rules for an allowed transition are 1) Dipole moment of the molecule changes as the molecule vibrates (which for diatomic molecules means transitions are allowed for heteronuclear diatomics and forbidden for homonuclear diatomics) 2) v = 1 3) J = 1 For absorption spectroscopy v = +1, while for emission spectroscopy (IR fluorescence) v = -1. Since at room temperature at equilibrium most diatomic molecules are in the v = 0 vibrational state, the only allowed transition commonly seen is v" = 0 v' = 1. However, since the selection rule for changes in v is obtained in the harmonic oscillator approximation, one can occasionally observe weak transitions where v = 2, 3,... These are called overtone transitions. Finally, while for most diatomic molecules J = 1, there are a few diatomics where the selection rule is J = 1, 0. There can also be additional complicating factors for some diatomics, which we will ignore. For molecules with initial quantum numbers v" = 0, J", there will be two types of allowed transitions v" = 0, J" v' = 1, J' = J"+1 R-branch transitions v" = 0, J" v' = 1, J' = J"-1 P-branch transitions The transitions where J increases by 1 are called R-branch transitions, while those that occur when J decreases by 1 are called P-branch transitons. If there were transitions where J did not change, as is true for some diatomic molecules, they are called Q-branch transitions. Transitions are labeled by the value of J in the lower energy state, so, for example, R(3) (for the transition where J increases from 3 to 4), or P(5) (for the transition where J decreases from 5 to 4). There are R-branch transitions for J" = 0, 1, 2,..., and P-branch transitions for J" = 1, 2, 3,... Note that there is no P(0) transition because the final state required for this (J' = -1) does not exist. The energy at which a particular allowed vibrational-rotational transition will occur can be found in the harmonic oscillator approximation using eq 12.4.8 E = E(v',J') - E(v",J") (12.4.9) Using eq 12.4.8 and 12.4.9 we can derive the following expressions for the R-branch and P-branch transitions R-branch E = + 2(J"+1) B (12.4.10) P-branch E = - 2J"B (12.3.11) The above equations predicts two families of transitions, one at energies larger than (the R-branch transitions) and the other at energies smaller than (the P-branch transitions). Adjacent peaks within a family of transitions are
separated by an energy equal to 2B, and there is a central gap between the R(0) and P(1) peaks separated by an energy 4B. The intensity of the observed peaks is determined by a combination of the degeneracy of the initial rotational state and the Boltzmann factor. For real molecules, where anharmonicity in the vibrational energy and the change in the rotational constant with vibration are included, one can use a modified form of eq 12.4.8 along with eq 12.4.9 to find an improved expression for the location of R-branch and P-branch transitions. The main effect of including anharmonicity is to cause the spacing between adjacent peaks in the spectrum to change slightly as J" changes, and to only be approximately equal to 2B. Polyatomics For a polyatomic molecule there will be several different fundamental, or normal mode, vibrations. The total number is related to the size of the molecule # normal mode vibrations = 3N - 5 linear molecule # normal mode vibrations = 3N - 6 nonlinear molecule where N is the number of atoms per molecule. For small polyatomic molecules it is usually possible to write down the normal mode vibrations by inspection (as done for CO 2 below). For larger polyatomic molecules the normal mode vibrations can be classified by their symmetry properties, which can help in determining the form of each vibration. In the harmonic oscillator approximation the total vibrational energy for a polyatomic molecule is given by the expression E(v 1,v 2,...v k ) = (v 1 + 1 / 2 ) 1 + (v 2 + 1 / 2 ) 2 +... + (v k + 1 / 2 ) k = i=1 k (v i + 1 / 2 ) i (12.4.12) where k is the total number of normal mode vibrations, and v i and i are the vibrational quantum number and vibrational constant for each normal mode vibration. In the harmonic oscillator approximation v i = 0, 1, 2,... for each normal mode vibration. The selection rules for allowed vibrational transitions are 1) Dipole moment of the molecule changes as the molecule vibrates 2) v i = 0, 1 3) At least one normal mode vibration has a change in its quantum number
For molecules where there are degenerate vibrations (such as in CO 2, where the bending vibration has a degeneracy of 2) the vibrations are usually grouped together and described by a single quantum number. In this case, the selection rule becomes v i = 0, 1, 2,..., g, where g is the degeneracy of the vibration. The types of vibrations observed in absorption polyatomic molecules are classified as follows 1) Fundamental transitions. All vibrational quantum numbers are initially zero. One quantum number increases by 1, while the other vibrational quantum numbers remain zero. (Example: 0,0,0 0,0,1) 2) Overtone transitions. All vibrational quantum numbers are initially zero. One quantum number increases by more than 1, while the other vibrational quantum numbers remain zero. (Example: 0,0,0 2,0,0) 3) Combination transition. All vibrational quantum numbers are initially zero. More than one quantum number changes. (Examples: 0,0,0 0,1,1 ; 0,0,0 2,1,0) 4) Hot band transitions. Transitions where the initial state does not have all of the vibrational quantum numbers equal to zero. (Example: 0,1,0 0,2,0). Note that hot band transitions occur from vibrationally excited initial states. The temperature dependence of a vibrational transition can be used to identify hot band transitions. Since the fraction of molecules in vibrationally excited states decreases as temperature decreases, the intensity of a hot band transition should decrease if the sample is cooled (and increase if the sample is heated). The intensity for the other types of transitions will be relatively unaffected by changes in temperature. Generally speaking, allowed fundamental transitions will be very strong, while allowed overtone and combination transitions will be moderately strong. Forbidden transitions, when they are observed, are usually weak. For real molecules, where the harmonic oscillator approximation breaks down, the location of overtone and combination bands will be close to the values predicted by the harmonic oscillator model, but lower in energy due to the effects of anharmonicity. The intensity and location of vibrational transitions can often be used to find the vibrational constants for each normal mode vibration. Each vibrational transition will have associated rotational structure. However, since this structure is usually not resolved, we will not discuss rotational structure any further.