Chemistry 3502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2003 Christopher J. Cramer. Lecture 12, October 1, 2003

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1 Cemistry 350 Pysica Cemistry II (Quantum Mecanics) 3 Credits Fa Semester 003 Cristoper J. Cramer Lecture 1, October 1, 003 Soved Homework We are asked to demonstrate te ortogonaity of te functions Φ(φ) tat are te φ-dependent components of te sperica armonics. We know tat tey are eigenfunctions of L z, and tus, since tey are non-degenerate eigenfunctions of a Hermitian operator, tat tey must be ortogona, but demonstrating tis point expicity serves as a ceck on our accuracy, if you ike. So, te question is, given two eigenfunctions Φ caracterized by different eigenvaues m and m, is it true tat Φm ( φ) Φm ( φ) 0? Te variabe φ ranges from 0 to π. So, we evauate te above integra as Φ π im φ * im φ m m 0 π imφ im φ 0 π im m φ e dφ 0 φ Φ φ e e dφ e e dφ 1 im m im m e φ ( ) ( ) 0 πim m e 1 im m i m m π ( ) We may use te reationsip πim e ( m ) [ π ( m m ) ]+ i π m m cos sin[ ] to evauate te first part of te soved integra. Since m and m are integers, teir difference is aso an integer. Te cosine of an integra mutipe of π is 1 and te sine of an integra mutipe of π is 0. Substituting tis simpification into te eqs. above provides

2 1- Φm πim m e ( ) ( φ) Φm ( φ) im ( m) 1 im m 1 im ( m) 1 im ( m) 0 Tus proving te ortogonaity of te eigenfunctions. Note tat if m and m ad been equa to one anoter, te origina integrand woud ave simpified to e 0 dφ dφ, te integra for wic woud be π, and as we've aready seen, tis gives rise to te normaization constant (π) 1/ incuded in Φ. Te Rigid Rotator We ve previousy considered two masses connected by a spring in soving te vibrationa Scrödinger equation. Te soutions were te quantum mecanica armonic osciator wave functions. Now, if we repace te spring wit a soid rod (no vibration) and permit te system to rotate about an axis perpendicuar to te rod, it wi rotate about its center of mass. As we discussed in ecture, te kinetic energy for a rotating system is T I (1-1) were is te anguar momentum and I is te moment of inertia. Atoug our origina discussion considered ony a singe partice orbiting a fixed position, te moment of inertia generaizes to mutipe partices as N I mr ii i 1 (1-) were tere are N tota partices eac aving a distinct mass m and distance from te center of mass r. Wen tere are ony two partices, one can sow wit some straigtforward agebra tat were µ is te reduced mass defined by I µ R (1-3) µ mm 1 m1+ m (1-4)

3 1-3 and R is te engt of te rigid rod connecting tem. For a rigid rotator in free space, tere is no potentia energy affecting te system, and te Hamitonian operator is simpy te kinetic energy operator. Tus, te timeindependent rigid rotator Scrödinger equation for a diatomic moecue is HΨ ( T + V) Ψ TΨ L Ψ EΨ I (1-5) were L is te tota anguar momentum squared operator wit wic we've just spent muc time. We aready know its eigenfunctions and eigenvaues, so we may rewrite eq. 1-5 as LI Y ( + 1) m, Ym EY, m, I (1-6) Tus, we see tat te aowed energies of te rigid rotator are quantized by te tota anguar momentum quantum number, and depend inversey on te moment of inertia. Te energies are independent of m. We aready know tat m can take on + 1 different vaues depending on, so eac energy eve of te rigid rotator is + 1 degenerate. In discussing te rigid rotator for moecuar rotation, it is traditiona to repace te notation wit J for te tota anguar momentum quantum number. It is aso traditiona to define a rotationa constant B according to B I (1-7) In tat case, te aowed energy eves for te rigid rotator may be compacty written as EJ J J + 1 B (1-8) Note tat te separation between aowed energy eves depends on B, wic is itsef a function of te atomic masses and R, te distance between tem. If we know te identity of our moecue, say CO (carbon monoxide), we know te masses and te ony unknown is te bond distance. So, if we can measure te separation between rotationa energy eves (and know wic eves are wic), we can determine te bond engt. Tis is indeed a standard protoco for determining moecuar structure.

4 1-4 Microwave Spectroscopy of te Rigid Rotator Consider te energy eves of te rigid rotator. Te ground state, for wic J 0, as zero rotationa energy according to eq Tus, tere is no zero-point energy. Tis does not vioate te uncertainty principe because, atoug we know te anguar momentum exacty, we don't know anyting about te anguar "position" (ow te moecue is oriented in space, if you wi). Te ground state as no degeneracy, since te ony aowed vaue for te z component of te anguar momentum is zero. Te next iger energy eves correspond to J 1,, 3, etc., and ave energies according to eq. 1-8 of E J B, 6B, 1B, etc. wit degeneracies of 3, 5, 7, etc. Te degeneracies may be tougt of as refecting te different anges tat are aowed between te axis of rotation and te z axis of te system. Wen te two axes are ortogona, te z component of te anguar momentum is zero. Wit iger tota anguar momenta, tere are more defections aowed from m J 0, but it can never get to te point were te z axis corresponds to te rotation axis, because ten te z component of te anguar momentum woud be te tota anguar momentum, we woud know te oter two components must be zero, and tis woud vioate te uncertainty principe. In any case, we now know te separation in energy between eves, but we do not know te seection rues for wat transitions are aowed, and may terefore be observed spectroscopicay. Remember tat to observe a transition, it must be true tat te integra * Ψ Ψ Ψ r e r Ψ r d r (1-9) µ () () m m m n is not zero. Some paying around wit te reevant integras in sperica poar coordinates aows one to prove tat spectroscopic transitions wi ony be observed between two rotationa states m and n if J ± 1 and 0or ± 1 (1-10) Tus, we can ony observe transitions between adjacent rotationa states, and tose transitions cannot cange te z component by more tan one quantum number (one -bar), atoug tey are aowed to eave it uncanged. Te restrictions of transitions to ony te next iger or ower quantum number is a deigtfu simpification for spectroscopic purposes. Let's consider absorption spectroscopy. For any state J being excited to state J + 1, we ave m J EJ J+ 1 EJ+ 1 EJ ( J + 1) ( J + 1)+ 1 B J( J 1) B J + 1 B [ ] + (1-11)

5 1-5 Te Bor ypotesis says tat potons of energy E ν wi be absorbed at tese caracteristic energies, so we expect to see absorptions at frequencies ν ( J + 1) B (1-1) B 4B 6B,,, etc. Moecuar rotationa frequencies are in te microwave region of te spectrum, so a typica rotationa spectrum for a diatomic wi appear very simpe, namey J 1 J 3 Absorbance J 0 1 J 3 4 J 4 5 B B B B B Frequency Note tat te first absorption corresponds to te ground state to first excited state transition and occurs at B/. Typicay, owever, it may be rater ard to decide exacty wat ine in a spectrum corresponds to te 0 1 transition. A muc better way to determine B/ is to notice tat te separation between every pair of adjacent ines is aso equa to tis vaue. So, if we ave a nice spectrum ike tat above, and we know te masses of our two atoms, it is trivia to measure B/ and sove for R, te interatomic distance (Piar does an exampe for HF). Note tat te intensities of te absorptions depend on ow many moecues are in te starting state, and tat depends on temperature (tis is one of te reasons wy it may be ard to see te 0 1 transition; te Botzmann distribution may ave very few

6 1-6 moecues in te ground state). So, tis kind of spectroscopy can aso be used to determine te temperature of a sampe! Wie we've considered tus far ony a diatomic system, te approac outined above works for any inear moecue. Non-inear moecues are more compicated tan inear ones because tey are caracterized by tree separate moments of inertia (one about eac cartesian axis). In igy symmetric cases, owever, reativey simpe soutions of te correct rotationa Scrödinger equation continue to exist. For instance, in moecues possessing an axis of rotation tat is 3-fod or iger in symmetry, te two moments of inertia for rotation about te two axes perpendicuar to te ig-symmetry axis wi be equa. For exampe, in fuorometane, wic is C 3v, tere is one moment of inertia, I A, about te symmetry axis A, and tere are two equa moments of inertia, I B and I C, about te axes perpendicuar to axis A. In tis particuar case, te magnitude of te atter two moments is arger tan tat of te former moment because te eavy atoms C and F ave dispacements of 0 from axis A but not from te oter two, and suc a moecue is caed a proate top. In te case of a proate top, te rotationa eigenvaues are given by EJ K JJ ( + ) K IB I A I B (1-13) were K is te quantum number corresponding to m, running over J, J+1,..., J 1, J, and expressing te component of te anguar momentum aong te igest symmetry axis. Te seection rues for a rotationa transition in tis case are J±1 and K0, and tus eqs and 1-1 continue to be vaid for absorption frequencies using I I B in te rotationa constant B. Less symmetric moecues require a consideraby more compicated treatment, wic we wi not go troug ere, but in te end teir spectra transitions are functions of teir 3 moments of inertia. From a spectroscopy standpoint, ten, prediction of rotationa spectra ines depends ony on te moments of inertia, and ence ony on te moecuar geometry. Tus, rotationa spectra provide a good way to measure moecuar structure witin te regime were te rigid-rotor approximation is vaid. To furter nai down structures, one often carries out te spectroscopy on many different isotopomers of te moecue in wic one is interested, in order to be abe to vary te different moments of inertia differenty and tus narrow down te possibe ways in wic te atoms may be arranged reative to one anoter and sti give te observed spectra. Te Diffuse Interstear Bands Let's eave te word of te tiny and cange engt scaes by, o, 5 orders of magnitude or so. Wat constitutes te matter wic is found in space? It's a fairy cear point tat sending a spacesip out to coect te contents of a nebua 000 igt years away is probaby not te best experimenta option for answering tis question.

7 1-7 So, astropysicists use spectroscopy (it's sti sort of a ong experiment, since te igt being observed takes 000 years to reac Eart in te above exampe, but we' sette for knowing about te composition of te matter 000 years ago...) One approac is to find a nebua (a coud of gas and dust) in wic you are interested, ook at te spectrum of a star sining troug it, and observe wat frequencies of igt tat you expect to see from te star (based on aving observed simiar stars tat aren't stuck beind nebuae) are depeted. Someting must be absorbing tose frequencies, and given teir magnitudes you may be abe to decide wat tat someting is. Suc spectroscopy is usuay done in te UV/Vis region, and corresponds to eectronic absorptions, ike tose we've aready considered for te H atom in te Bor mode for tat atom. But wat if te nebua is so tick tat no starigt comes troug? In tat case, you can't ook at absorptions, you ave to ook for emissions. Wy migt tere be emissions? We, one way in wic a moecue can be in an excited rotationa state is if it excanged energy wit anoter moecue tat just based into it. In tat case, if it takes some energy from te oter moecue in te coision, it can be in an excited rotationa state. Given a sufficient amount of time, it wi utimatey radiate a poton (in te microwave region) to come back into terma equiibrium. Space is fantasticay diffuse, so moecuar coisions, even in nebuae, occur on te time scae of days to years (!). Neverteess, if you consider tat a receiver on Eart gets to sampe a pat engt of te widt of te universe, you wi peraps not be surprised to earn tat tis microwave radiation can be detected by enormous radio antennae, and tis is one function of radioastronomy: to identify moecuar components of interstear gas couds based on teir rotationa emission spectra, wic form te microwave region of wat are caed te diffuse interstear bands (DIBs) in te universa spectrum. By observing canges in te DIBs as a function of wat sice of sky is samped, one can assign variations in signa to particuar interstear objects. None of tis woud be possibe if te rotationa spectra were not to be quantized! If a vaues of anguar momenta were aowed, we'd see no ines in spectra, ony broad burs tat woud be competey uninformative. Homework Piar 4-7. Note tat tere is an error in Piar s probem. Te correct frequency is 115,71,000,000 s 1, not 115,71 s 1.