Molecular structure: Diatomic molecules in the rigid rotor and harmonic oscillator approximations Notes on Quantum Mechanics

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1 Molecular structure: Datomc molecules n the rgd rotor and harmonc oscllator approxmatons Notes on Quantum Mechancs Last updated Thursday, November 30, 006 8:0:55-05:00 Copyrght 003 Dan Dll (dan@bu.edu) Department of Chemstry, Boston Unversty, Boston MA 05 We have seen how the adabatc approxmaton and the Born-Oppenhemer approxmaton allow us to get separate Schrödnger equatons for the electronc and the nuclear coordnates. The nuclear Schrödnger equaton descrbes both the moton of the molecule as a whole through space and relatve moton vbratons and rotatons of the atoms that mae up the molecule. The way to separate the motons through space from the nternal moton s to reexpress the coordnates of each atom wth respect to the laboratory n terms of () the three coordnates of the center of mass of the molecule and () the coordnates of each atom wth respect to the center of mass. Ths change of coordnates s llustrated most smply for the example of a datomc molecule. The result s that the nuclear wave functon factors nto the product of () the wave functon of the molecule movng freely through space, as a pont mass centered at the center of mass of the molecule and wth mass equal to the mass of the molecule, and () the wave functon of the relatve moton of the atoms wth respect to the center of mass. The Schrödnger equaton for the relatve moton of the atoms wth respect to the center of mass depends on the detals of the structure of the molecule. The smplest example, whch we wll explore here, s a datomc molecule AB. à Transformaton to center of mass and nternal coordnates In ths case of a datomc molecule AB the center of mass s at the pont R cm = M A R A + M B R B ÅÅÅÅ ÅÅ M between the atoms, where M = M A + M B s the total mass of the molecule, and the relatve moton s descrbed by the R = R A - R B. That s, the sx coordnates, consstng of the three coordnates, R A = HX A, Y A, Z A L, of atom A and the three coordnates, R B = HX B, Y B, Z B L, of atom B, are replaced by the three coordnates, R cm = HX cm, Y cm, Z cm L, of the center of mass, and the three coordnates, R rel = HX, Y, ZL, of the relatve moton wth respect to the center of mass. If we assume for smplcty that the molecule s confned n a cubcal volume L 3, then the Schrödnger equaton for the center of mass s just the free partcle Schrödnger equaton. Ths means that the wave functon for the moton of the molecule through space just the free partcle wave functon 3 y cm = j $%%%%%% y ÅÅÅÅÅ snj n x p X cm N sn j n y p Y cm y ÅÅÅÅÅÅÅ Å z snj n z p Z cm N, L z L L L and ts contrbuton to the total energy s just

2 Molecular structure: Datomc molecules n the rgd rotor and harmonc oscllator approxmatons h E cm = ÅÅÅÅÅÅ 8 ML Hn x + n y + n z L. The Schrödnger equaton for the relatve moton about the center of mass s Ñ j Å m X + Y + ÅÅÅÅÅÅÅÅÅÅ y Å Z z + E HX, Y, ZL Ö Ñ f HX, Y, ZL = E nt f HX, Y, ZL, where E nt = - E cm n the contrbuton of the nternal moton to the total energy, and m= M A M B ÅÅÅÅ M A + M B s the reduced mass of the molecule. We can actually smplfy ths Schrödnger equaton somewhat by realzng that for a datomc molecule the electronc egenvalue, E, changes only when the separaton between the two atoms changes. Ths means that E depends only on the nternuclear dstance, E HX, Y, ZL = E HRL, and not at all on the ortentaton q, f of the axs n space. Note that now we (re-)use the symbol R to be R = è!!!!!!!!!!!!!!!! X + Y!!!!!!!!!! + Z. Ths smplfcaton n turn means that we can separate the moton along the nternuclear axs the stretchng and shrnng of R from the rotaton of the molecule about the center of mass by transformng to sphercal polar nternal coordnates, HX, Y, ZLöHR, q, fl. From our study of rotatonal moton and the one-electron atom we now that ths means we can wrte the nuclear wave functon as f HX, Y, ZL = S HRL Y JM Hq, fl, n terms of sphercal harmoncs and a wave functon S HRL determned by the vbratonal Schrödnger equaton Ñ m ÅÅÅÅÅ R R R + Ñ JHJ + L + E m R HRL ÖÑ S HRL = E nt S HRL. The quantum number ndexes the number of loops n the wave functon S HRL and the ndex ndcates the parametrc dependence of wave functon S HRL on the electronc potental curve, E HRL. In a way analogous to what we dd n the one-electron atom, we can defne a related vbratonal wave functon c HRL = RS HRL to transform the vbratonal Schrödnger equaton nto pure curvature form Ñ m R + Ñ JHJ + L + E m R HRL ÖÑ c HRL = E nt c HRL. Copyrght 003 Dan Dll (dan@bu.edu). All rghts reserved

3 Molecular structure: Datomc molecules n the rgd rotor and harmonc oscllator approxmatons 3 à The rgd rotor and harmonc oscllator approxmatons A smple pcture of a nuclear moton n a datomc molecule s two masses connected by a sprng. The masses can vbrate, stretchng and compressng the sprng wth respect to the equlbrum sprng length (the bond length), The masses can also rotate about the fxed pont at the center of mass. The queston arses whether the rotaton can affect the vbraton, say by stretchng the sprng. To answer ths queston, we can compare the expected frequences of vbratonal moton and rotatonal moton. Now, we now that snce molecules n an egenstate do not move, we need to dscuss moton n terms of wave pacets. For vbratons, wave pacets oscllate wth at the harmonc (angular) frequency w vb = è!!!!!!!!! ê m. For rotatons, the wave pacets rotate wth (angular) frequency roughly correspondng to the frequency angular momentum w rot = ÅÅÅÅ ÅÅÅÅÅÅÅ Å moment of nerta ö Ñ è!!!!!!!!!!!!!!!! JHJ + L Ñ ÅÅÅÅ ÅÅÅÅÅÅÅ º ÅÅÅ m R e m R. e As example, for N ground state, w è vb = cm -, R e = Þ, and for N-4 the reduced mass s amu. These values gve w vb = μ 0 3 Hz and w rot = μ 0 Hz. Verfy these values for w vb and w rot. The rato of these values, w vb ê w rot = 93.99, means that the ntrogen molecule undergoes about 00 vbratons for each full rotaton. Ths means that on average the molecule rotates at the mdpont of ts vbratonal excruson, namely at the nternuclear dstance R e. For ths reason, t s a very good approxmaton s to assume that the rotatonal netc energy can be replaced by ts value at the equlbrum nternuclear dstance, R e. Ñ JHJ + L ö Ñ JHJ + L. m R m R e Ths s nown as the rgd rotor approxmaton. In ths approxmaton the vbratonal Schrödnger equaton becomes Ñ m R + E HRL ÖÑ c HRL = je nt - Ñ JHJ + L y z c HRL. m R e The equlbrum nternuclear dstance s determned by the balance of attractve electron-nuclear forces, repulsve nuclear-nuclear forces, and repulsve electron-electron forces. We can smplfy thngs further by expandng the electronc potental energy n a Taylor seres wth respect to the equlbrum separaton, E HRL = E HR e L + J E HRL ÅÅÅÅÅÅ R N HR - R e L + ÅÅÅÅÅ j E HRL y HR - R R=R e R e L zr=r + e Snce, by defnton, E HRL s a mnmum at R = R e, the second term n the seres vanshes snce the slope of E HRL s zero at the mnmum. If we also gnore the cubc and hgher terms n the expanson, then we can wrte the electronc potental energy as E HRL = E HR e L + ÅÅÅÅÅ j E HRL y HR - R R e L zr=r. e Copyrght 003 Dan Dll (dan@bu.edu). All rghts reserved

4 4 Molecular structure: Datomc molecules n the rgd rotor and harmonc oscllator approxmatons Ths s nown as the harmonc oscllator approxmaton. In ths approxmaton the vbratonal Schrödnger equaton smplfes further to Ñ m R + ÅÅÅÅÅ j E HRL y R z HR - R e L R=R e ÖÑ c HRL = je nt - E HR e L - Ñ JHJ + L y z c HRL. m R e Ths smplfed equaton loos a lttle untdy. We can neaten t up n two ways. Frst, the nternal energy s the sum of contrbutons from electronc moton, vbratonal moton, and electronc moton, E nt = E elec + E vb + E rot. If we mae the dentfcatons E elec = E HR e L E rot = Ñ JHJ + L, m R e ths means that the egenvalue of the smplfed vbratonal Schrödnger equaton s just E vb, snce E nt - je HR e L + Ñ JHJ + L y z = HE elec + E vb + E rot L - HE elec + E rot L = E vb. m R e Second, harmonc potental energy s expressed most naturally as x ê, n terms of a force constant (Gree letter appa) and dsplacement from equlbrum x (Gree letter x). We can transform from the coordnate R to the dsplacement x = R - R e usng R = Hx + R e L = x. The result s the vbratonal Schrödnger equaton ÅÅÅÅÅÅÅÅ m j- Ñ x + ÅÅÅÅÅ where the force constant s = j E HRL y R xy z c HxL = E vb c HxL, zr=r e. à Grand summary of datomc molecule molecular structure Usng the adabatc and Born-Oppenhemer approxmatons, we express the molecular wave functon and total energy as Y HR, rl ºy HR, rl f HRL =y HR, rl y cm HX cm, Y cm, Z cm L f HX, Y, ZL. = E cm + E nt = E cm + E elec + E vb + E rot The electronc wave functon and energy are determned from H y HR, rl = E HRL y HR, rl, Copyrght 003 Dan Dll (dan@bu.edu). All rghts reserved

5 Molecular structure: Datomc molecules n the rgd rotor and harmonc oscllator approxmatons 5 usng the nfnte nuclear mass hamltonan H =- Ñ ÅÅÅÅÅÅÅÅÅÅÅ m e e +VHR, rl. The hamltonan assumes the molecular center of mass s nfntely heavy. The electronc energy, E HRL, contrbutes to the potental energy experenced by the nuclear moton, as descrbed below. The moton of the molecule through space, assumed to be bounded by a cubcal volume L 3, has wave functon and energy 3 y cm HX cm, Y cm, Z cm L = j $%%%%%% y ÅÅÅÅÅ snj n x p X cm N sn j n y p Y cm y ÅÅÅÅÅÅÅ Å z snj n z p Z cm N, L z L L L h E cm = ÅÅÅÅÅÅ 8 ML Hn x + n y + n z L, n x, n y, n z =,, 3, Usng the rgd rotor and harmonc oscllator approxmatons, we express the wave functon of nternal nuclear moton (wth respect to the center of mass) as f HX, Y, ZL = c HxL ÅÅÅÅÅÅ Y JM Hq, fl, x where the harmonc oscllator wave functon s determned from the Schrödnger equaton ÅÅÅÅÅÅÅÅ m j- Ñ The nternal energy, x + ÅÅÅÅÅ E nt = E elec + E vb + E rot, xy z c HxL = E vb c HxL. s the sum of the electronc, vbratonal and rotatonal contrbutons, E vb = h ÅÅÅÅÅÅÅÅ p $%%%%%% ÅÅÅÅÅ m J - ÅÅÅÅÅ N, = j E HRL y, =,, 3, R zr=r e E elec = E HR e L, E rot = Ñ JHJ + L, J = 0,,,. m R e These equatons are a complete, approxmate descrpton of the quantum aspects of datomc molecules. To test ther correctness, the next step s to see how to use them to account for the structure and spectra of datomc molecules. Copyrght 003 Dan Dll (dan@bu.edu). All rghts reserved