209 Lecture 34 We now turn to our final quantum mechanical topic, molecular spectroscopy. Molecular spectroscopy is intimately linked to quantum mechanics. Before the mid 1920's when quantum mechanics was developed, people had begun to develop the science of molecular spectroscopy. However, its early use was purely qualitative, based solely on the observation that molecules and atoms, when exposed to light, absorbed and emitted it in patterns that were unique to each species. The interpretation of these spectra in terms of the physical parameters that describe the geometries and bonding of the molecules that produce them had to wait until the development of quantum mechanics. The close interrelation between spectroscopy and quantum mechanics is shown by the fact that Robert Mulliken, who was the foremost practitioner and proponent of Molecular Orbital theory in its early years, was spurred on by an attempt to understand the details of molecular spectra. It is further shown by the fact that Gerhard Herzberg, the foremost practitioner of molecular spectroscopy from the middle of the '30s to the 70's, used to keep a bound copy of Mulliken's papers by his desk. In fact, I keep a bound copy of Mulliken's major papers, edited by Norman Ramsey, Herzberg's collaborator, by my desk. Molecular spectroscopy, which is the study of the interaction of light with atoms and molecules, is one of the richest probes into molecular structure. Radiation from various regions of the electromagnetic spectrum yields different information about molecules. For example, microwave radiation is used to investigate the rotation of molecules and yields moments of inertia and therefore bond lengths. Infrared radiation is used to study the vibrations of molecules, which yields information about the stiffness or rigidity of chemical bonds. Visible and ultraviolet radiation is used to investigate electronic states of molecules, and yields 209 210 information about ground and excited state vibrations, electronic energy levels, and bond strengths. We will do a fairly detailed treatment of rotational and vibrational spectroscopy and a somewhat briefer treatment of electronic spectroscopy. In our treatment we will begin with the basic interpretation of molecular spectra, and finish with the quantum mechanical basis of these interpretations. The features of the electromagnetic spectrum that are of interest to us are summarized below. The absorption of microwave radiation is due to transition between rotational energy levels; the absorption of infrared radiation is due to transitions between vibrational levels, Region Frequency/Hz Wavelength/m Wave Energy/ Molecular number/cm-1 J *molecule-1 Process Microwave 109-1011 3x10-1-3x10-3 0.033-3.3 6.6x10-25- Rotation of 6.6x10-23 polyatomic molecules Far Infrared 1011-1013 3x10-3-3x10-5 3.3-330 6.6x10-23- Rotation of 6.6x10-21 small molecules Infrared 1013-1014 3x10-5-3x10-6 330-4000 6.6x10-21- Vibration of 8.0x10-20 flexible bonds Visible and 1015-1016 9 x 10-7-3x10-8 11,000-3.3x105 2.2x10-19- Electronic Ultraviolet 6.6x10-18 transitions accompanied by transitions between rotational levels; and the absorption of visible and ultraviolet radiation is due to transitions between electronic energy levels, accompanied by simultaneous transitions between vibrational and rotational levels. The frequency of radiation absorbed is calculated from the energy difference between the upper and lower states in the transition, given by ∆=EEul − E = hν where Eu and El are the energies of the upper and lower states respectively. 210 211 For example, if we have an absorption with wavenumber, ν = 1.00 cm-1, we calculate ∆E as follows. Remember that νλ=1/ (cm ). Therefore ∆E is related to ν by hc ∆=E hνν = = hc λ and so ∆E = hcν = (6.626 x 10-34 Js)(3.00 x 1010 cm/s)(1.00cm-1) = 1.99 x 10-23 J. TO WHAT TYPE OF MOLECULAR PROCESS WILL THIS RADIATION CORRESPOND? We will begin with molecular rotation. A rigid rotor is the simplest model of molecular rotation. We discussed the quantum-mechanical properties of a rigid rotor before, but will review the pertinent results here. The energy of rotation of a rigid rotor is all kinetic energy. For a motion with circular symmetry, it is convenient to express this energy in terms of the angular variables L, angular momentum, and I, moment of inertia. The moment of inertia for a diatomic molecule is given by 2 IR= µ 0 where µ is the reduced mass of the molecule and R0 is the bond length of the molecule. The angular momentum L is given by LI= ω , where ω is the angular frequency. When the Schrödinger equation for this problem is solved, it is found that the angular momentum is quantized according to the equation J 2 = J(J +1)2 where J is the spectroscopic notation for the total angular momentum of a molecule, and that the angular momentum in the z direction is quantized according to the equation 211 212 JMz = , where M = J, J-1, ..., -J. The energy of the rigid rotor, in the absence of an external field, depends on J only, is also quantized and is given by 2 E = J(J +1), J = 0,1,2,... J 2I Since there are 2J+1 values of M for each energy EJ, the energy states are 2J+1 fold degenerate. When microwave or far infrared radiation shines on a rotating molecule, the J value can either increase or decrease, and the energy of the light absorbed is given by ∆=EEul − E. An important characteristic of all molecular spectra is that the choice of upper and lower states is not completely free but is in fact extremely limited. The rules that govern the choices of upper and lower states are called selection rules. These selection rules are determined by time dependent perturbation theory. For the pure rotational transitions of a diatomic rigid rotor the selection rule is ∆J = ±1. In other words, when a diatomic rigid rotor absorbs light, its rotational quantum number can increase by 1 or decrease by 1. These are the only possible transitions that occur. (The reason that I specify diatomic rigid rotors is that the selection rules depend in part on symmetry, and molecules with other symmetries have different selection rules.) In addition, for pure rotational transitions the molecule must have a permanent dipole moment. Thus molecules like H2 and I2 will have no pure rotational spectra, while HI or CO, which have permanent dipole moments, will exhibit rotational spectra. For molecules that do have permanent dipole moments, the larger the dipole moment, the more strongly the molecule will absorb light. 212 213 The consequence of this selection rule is that the rotational absorption spectrum of a rigid rotor is a series of evenly spaced lines in the microwave or far infrared region. To see this we begin with the energy eigenvalues, 2 E = J(J +1), J = 0,1,2,... J 2I According to our selection rule, for absorption ∆J = +1. The energy change for this transition is 2 2 ∆E = E - E = (J +1)(J + 2)- J(J +1) J +1 J 2I 2I 2 = (J +1), J = 0,1,2,.... 2I The wavenumber change for this absorption is given by ∆E h ν = = (J +1), J = 0,1,2 hc 4π 2cI It is typical in spectroscopy to write the energy in the form E J = hcBJ(J +1) where B , the rotational constant of the molecule, has units of wavenumbers and is given by h B = 8π 2cI In this form the wavenumber of the rotational absorptions is given by ν = 2B(J +1), J = 0,1,2,... Thus for the lowest energy rotational transition, J = 0 to J = 1, the absorption occurs at a wavenumber of 2B . The second transition, J = 1 to J = 2, occurs at a wavenumber of 4 B . The third transition occurs at a wavenumber of 6 B and so on. Thus the rotational spectrum consists of a series of equally spaced lines separated by 2 B . [Illustrate] Lets check on our assertion that 213 214 the rotational spectrum is in the microwave region, by calculating the value of B for the rotation of HI. h B = 8π 2cI 2 The moment of intertia I is given by µR0 , where µ is the reduced mass in kg and R0 is the bond length in meters. For HI, the reduced mass is given by mH mI 1AMUx127AMU −27 kg −27 µ HI = = x1.66x10 = 1.65x10 kg mH + mI 1AMU +127AMU AMU and the bond length is 160.4 pm, so the moment of intertia is 2 -27 -12 2 -47 2 I = µR0 = 1.65 x 10 kg (160.4 x 10 m) = 4.25 x 10 kg m , 6.6262x10−34 Js and B = = 6.58cm−1 (8π 2 )(3.00x1010 cm / s)(4.25x10-47 kg m2 ) Our first rotational transition will occur at a wavenumber of 2 B = 13.16 cm-1, which is in the far infrared region, as predicted. A more typical application is to determine the value of B from the spacing of a rotational spectrum, and use this to determine the moment of inertia and therefore the bond length. For example, the microwave spectrum of 39K127I consists of a series of lines whose spacing is almost constant at 3634 MHz. Calculate the bond length of 39K127I. The rotational spacing for KI is given in Hz, a unit of frequency. Our equation for the rotational wavenumber is ν = 2B(J +1) The relation between wavenumbers and frequency is ν = cν where c is the speed of light in cm/s.