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This 60 page Class Notes was uploaded by Alexandria Bauch on Monday October 26, 2015. The Class Notes belongs to CHEM1410 at University of Pittsburgh taught by Staff in Fall. Since its upload, it has received 103 views. For similar materials see /class/229441/chem1410-university-of-pittsburgh in Chemistry at University of Pittsburgh.
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Date Created: 10/26/15
Chapter 2 When do we need to use QM 1 A dimensions of the system 2 Energy level spacing gtgt kT kT n g quantum classical populations I I Degeneracies two or more levels with the same energy quoti g gi gjkT e Boltzmann eq Can treat the system classically if energy spectrum continuous Classical waves 52 1 52 8x2 Pxt A sinkx oat show this satisfies the wave eq V 2 52 wave equation 7 k wave vector on 27m angular freq V velocity Add two travelling waves of same freq and amplitude opposite direction 1 Asinkx wt sinkx wt 2 2A sin or cos at ix cos wt Complex representation LP Aeikx at 39 Euler 6quot 00505 isina L standing wave fixed nodes Derivation of the Schr dinger eq 6sz 3qu 6x2 v2 61 2 l a 2 x wz C fxi v 2ltx0 dzw 472 0 dx2 12 l dzw 47r2p2 d2 2 0 x h l h2 dzw V E 2m dx2 W W time independent S E Substitute L11x t 2 Mac cos at for classical standing wave vv Substitute xi E p h Substitute h and 27 2 p Vx E Classical expression for total energy timedependent S E 6 t Form of wavefunction for a stationary state Energy is constant over time Q is a soln of the timeindep SE Px l WXeiE h ln QM all observables are associated with operators Ow anw Eigenvalue eq I eigenfunction operator eigenvalue 2 x iy i 1 ln QM the eigenvalues correspond to the Exam le ofa observables and are real 390 complex number 22 w W Ew 2m dxz A l H is the Hamiltonian HM Ely operator 39 ikx ikx d 1s sze 36 an ef of dx d z39kx z39kx lkAe lkBe 5 const 1 2 is it an ef of 2 dx d2 Ely k2Ae x kZBe 39kx k2 AdoC Se m No Yes Kronecker Orthogonality delta vector space function space fundlon x 0 0 a x 2 0 w mgrj matix fry V 39 Z 0 gt E 339 3 where x y z are vectors in the x y 2 directions i G 339 3 The different eigenfunctions of a QM operator are orthogonal degenerate eigenfunctions are a special case If r ijidx 1 the functions are normalized Normalize aa x on O 5 x 5 a let l Naa x JOaN2a2a x2dx N2c12ja2 2axx2dx 3 a 3 2 5 x Na N2a2a2x ax2 a 0 5 set NZa zl gt Nais 3 a 3 1 yam x is normalized on 0 5 X 5 a Orthonormal set of functions orthogonal and normalized The EF s of a QM operator form a complete set gt any function in that space can be written in terms of the eigenfunctions f x lenbc n1 1 f xwmx Wm 902 19W x n1 2 Integrate over both sides bn I fxlnxdx bn is the projection of fonto LJn The analogue in vector spaces is v ai bj ck where i j k are unit vectors in the X y 2 directions Fourier 1 mix mrx x 2 17 b cos a sun f0 L En L 22296 for a function periodic over L 5 x 5 L teXt Key ideas time independent and time dependent Schrodinger equations operators eigenvalue equations orthogonal functions and complete basis sets Chapter 9 The HAtom o last example we will solve analytically o foundation of electronic structure theory 0 fixed nucleus of charge 1e dynamic electron charge 1e A hZ 62 62 62 62 spherical polar coordinates r 6 p xzrsin cosw yrsin65imp zzrcose A h2 1 d d Hr 2 r2 p 2mer2 dr d1 Mr 6 p RrYZ 6 p 2 2 E e Zmer2 4722901 9 4722901 r c2yzz2 Radial equation for each E hz d rzijh2 l e2 Zmer2 E dr Zmer2 4721801quot This equation gives a new quantum number n in addition to and m n1 10m10 n21omo Enavgy ieV i i n2ilmf0 n2lmil sawqu in ma Penman Emmaquot In pubiiihing as Dunlmnm Cummings Rr ERr Vcentripeta r 1 o1 rm il 1 3 2 wiwir7 ZJ equotquot 1 I 32 H L quotr a ZWr 4mao a0e 312 r r2 e access 0 2mir99 3 l 1 4 a 2IIT0 5J i 112 8 Lay an 63331 0 n 3 levels of the H atom n310WL0 n311m0 n3llmil n3l2m0 n3l2mli39l n312mr2 w r is 27 xxL2i 43 30 8134r 110 a0 a3 1 2 V2 1 3 2 r r2 r30 9 wmwm 739 an 6 a3 2 cos 9 i M 5Li eAVJ osinegtw 339 quot31 an an ag 32 1 r2 A 391 31039y9v 815 g2 393quotn3cosZB l 72 zer3au sinecosthm a j l 3mr0 m a 32 w no 4 L evaa Sinz g 322139 W 1625 no 13 4 Energy levels E mee i 136eV 1 Rydberg 05 au n 880h2 n2 2 13605 eV n Energy is independent of 720000 i i Paschen n13 E m f Balinet n12 5740000 n2132333m f 760000 g 2 E 0n 1 LL 750000 7100000 1 m Z EE n1 120000 Lmn 1S2S2px2py32pz o Wavefunctions Atomic orbitals yr6go at Y 6gooe quot polynomial in r Note X y 2 have same angular dependence as px py and p2 This is Am 0 i1 why s gt p is allowed 0 Spectroscopy A any A6 i1 WW WWW Normalization N2 sin sdsj d jequotZquot Zrzdr 1 magi 11 probability density r2Rr2dr radial distribution function probability of being found in a shell of radius r of thickness dr We have integrated out and H Max in radial distribution function ofthe 25 orbital 3 2 PrL i r2 2 L e m o 327239 a0 a0 dP r 8513 16a2r8a r2 r3 e ra0 dr 327m lt 0 0 0 PU dPrdr was We see that the principal maximum in Pr is a 524 a This corresponds to he most probable distance ofa 2 electron from the nucleus The subsndiary maximum is at 076 aai The minimum is at Z 110 1s 2p 25 3d 3p as Shell Model The orbital plots provide some justification to the use of shell models Radial distribution lunclion cuwmlvmznns Penman Edualion inc mime as mnumm Cummlnv l 25 as 39 l l u I v 339 1s m 2 39 5 t 105 j 1 39 gt o x 0 gt o 7 1 r A 5t 39 2 quotCF T Z 1 l 3 0 l l l l l l l l l u y I I l 3 2 1 0 l 2 3 710 5 0 5 IO 20 7 0 0 ll 20 x x x H atom 4 mee ZZZ W3 11212333 0 hZ a080 205291 Emee l mee4 1 247r502n2h2 2r quot 339 39 atomic units 71 1 e 1 energy 1 au 27211 eV distance 1 au 0529 A Note We really should be using y instead of me H D T have slightly different y and thus can be distinguished spectroscopically Treatment of H atom applies to He Li U9 Radial distribution function integrate over angular degrees of freedom Prdr r2 Rr2 dr 6 E u m a 10 Distance 50 Chapter 10 Many e39 atoms 1 1 2 2 1 He atom Vf V yzEy In au u 2 2 r1 72 712 1 cannot separate dueto r term 2 2 r However it is still useful to use an ner approximate wavefunction that does separate quot2 y 1 am orbital approximation 1 Simplest approach neglectthe Z term poor approximation each e experiences a potential from the average charge distribution ofthe other e39 0 Better approach 6 a 2 a a Vlf r1 7J39 2 r2r 2 may Hartree model 1 12 N electrons gt N oneelectron hamiltonians gt 5 To proceed further we must consider e39 spin spin of e39 12 two components mS 12 mS 12 spin wavefunctions 2a h2ss 1 2 05722 A2 3 2 s 72 4 A h A h sazmhaz a s Z S 2 Z Mada Ema 1 Ia d039 o Indistinguishability of electrons wavefunction must be antisymmetric wrt exchange of two e Pauli He w 151 1 ls2 061 2 la2 exclusion I I f f principle r r2 1 0391 0392 1 W Z i 131a1 131 1 3151152a1 2 la2 xE 1s2a2 1s2 2 1 Elsl a at 1 511a1 MIMI ml 1 In general wl2n 12a2 415126 2 m282 1nan 1n6 n m n6 n mn2 ifneven 1 Slater determinant n 1fn odd 2 Two electrons cannot have all quantum s the same V 2L1 25393 1 150 258 eXCited 23 singlet 1 ISO 2S0 states W2 Elsa 2505 T 1 1S triplet 1 ls 233 W3 J5 156 256 1 156 250 W4 J5 156 250 Variational method approximate wavefunction q HQD EQD jquchr 2 lt Ifqb has a parameter b Id M7 solve 6E6b20 3 5 7 9 example q x3ax5lx7x9 a a a 2 a a for particleinbox problem with O lt X lt a Relative amplitude The variational parameter is a Plot of the exact and approximate wavefunctions 2 ifaO gt E20203 2 ma 2 05720 2 E0gt0 6345 gtE0127 hz 2a ma hZ EeszJZS 2 ma The energy can never fall below the exact energy HartreeFock SelfConsistent Field method cDis taken to be a Slater determinant parameters in orbitals are varied 12 Vle n in 814 depends on orbitals that we are trying to solve for Guess a set of orbitals construct Vier 6 solve for orbitals energies lterate until energies and orbitals are converged Etotal 228i Z2Jij ITltij jgtl exchange I Coulomb I E empty to remove double counting in 22 81 O 53 H i 52 4 filled 81 4 IP g for filled orbitals Koopmans theorem EA 3 for empty orbitals many electron atoms H atom ens lt snp lt8nd ens snp 8nd deshielding Radxal distribution function Dwstance ac Cuwuyhmzuns Mm Edmslmn m pubthlvwg n Emu1mm Cummhvgs s has more weight near nucleus than does p which has more weight near nucleus than d He Is2 2 Ne 1s22s22p6 2810 Ar 1s22s221753s23176 28818 ScAr4s2 3d M1391Ars2 3515 CuArs3d1 T139Ar4s23d2 FeAr4sz3d6 ZnArsz3d1 C VAr4sz3d3 C0Ar4s2 3517 CrAr4s3d5 N139Ar4sz3d8 2 Q s 2P hs Cuwuulvmzana Penman Edumlmn m pubhslvhvg as 9mm Cumrnng tnergy Cr 4s3dS vs 493514 Cu 453d vs 493519 d5 Especially stable Electronegativity X P EA de nition due to Mulliken Does Na transfer an e39 to CI AE 1PM iEAC 5136158V quot Cawughmzans Pemsou Edumhan MK uUbHshmg a Bnmumm Cummwga Ho Nu m u u l v x w 20 an m 5 m mmzalmn mmth mm mam Hams m x x an au up so mum numhlr Many electron atoms n t m0 m5 not good quantum s L 8 ML MS are the good quantum s Actually for heavy elements one needs to add a spinorbit coupling term to H L2 L2 2 52 no longer commute with Fl with spinorbit coupling need to use j i 5 z 2 40 the good quantum s are J M Electron lectron No electron electron repulsion and Spinorbit interaction External repulsion indistinguishability included magnetic field included Conligu ration gt Terms gt Levels gt States n LS JMJ MAJ cowwmezuns Psalmquot Educalion lnc publishing as Emmmni Cummings Terms States H He Ne Ti 3 1s L 039 S 21 gt 2S 2 Term symbol 237L 1s2 L0S0 gt1S 1s22s22p L 1 S gt 2P wu39t Clt 1s22s22p2 L2130 5210 gt 3P 1D1S 1s22s22p3 L210 S22l gt 4S 2P 2D 2 2 1s22s22p4 L 2210 5210 gt 3P1D1S 1s22s22p5 L21 Sz gt 2P 1s22s22p6 L20 S20 gt 1S Ar4s23d2 L 43210 S 01 gt 1S 11 16 3P 3F A closer look at spin 1 2 a B M8 0 There must be a S 1 state B 0L Ms0 91825 S1gtMS101 CL CL MS 1 There must also be an B B Ms7 SOstategtMs0 1 WT E 1s2s 2s1saa T T T 1S ls2s 2s1sa6 Ba H STlt l T w ls2s 2slsa a 1 l T 1 WT E 1s2s 2s1s6 8 In the absence of a magnetic field the three triplet components are degenerate The Tand 8 states are different energy ET lt ES term 237L a 2812L1 degeneracy C 1s22p3d L321 S 210 filled shells gt 7S 3F71F73D 1D73P71P 2171559360states 2p 6 chorces 60 States 3d 10 choices C 1322322p3p L210 SL0 3D ID 3P IP 35 IS C 1322p2 3P1S1D what happens to 3D 7P 3S violate the Pauli exclusion principle States and Terms for the Ilpl Configuration m mZ MLmn m2 m mr2 Ms n m Term 71 71 72 12 712 0 39D 712 712 71 P 71 2 12 1 0 1 1 l 39D 31gt 12 712 0 12 12 1 P o o 0 12 712 0 712 12 0 3911 P s 12 712 0 1 71 o 717 712 71 P 12 12 1 P 712 712 71 P 712 12 0 1 o 1 3911 P 12 12 0 12 12 1 P ffH f H 71Hf f Possible Terms for Indicated Configurations Electron Con guration Term Symbol 3 3 S P lt p5 P pipquot s 39D 31gt 1 1 R D 45 d d 1 D dadx S 39D 39G 7P F d Jl7 quot F quotR 3H JG 1F ZD21P didquot i D 1H YCL F 2 JD 3132 I 39G 2 39F 39D 2L Is 2 di quot S G quotF quotD quotP 1I 2H 1G 2 1F 2 2D 3 1P ZS Note p 75 172174 dz d8 d3 a7 etc give the same states for each pair Hund s Rules 1 The lowest energy term is that with the highest spin 2 For terms that have the same spin that with the greatest L value lies lowest in energy C lszlpz 3P lt1D lt15 S O Coupling 11454cmquot 3 3 3 3 P gt P2 P1 P0 3 1D 02 9 V I I I E J 10194cmquot aP aPz Energy level diagram for O atom Spinorbitcoupling addsaterm jog t0 1 generally can ignore for light atoms Hund s rule 3 If a subshell is a half full 3quot32 the level with the highest J is lowest in 3P 3P energy If it is lt half full the level with 7 the lowest J is lowest in energy 3P0 3P1 9 splits into 3 levels upon application of magnetic field Return to the 1323 singlettriplet problem E singlet 1523 2319190523 2S1Sdl391dl392 E15 EZS 1523 2Slsils2s 2S1Sdl391dl392 r12 E1sE2s 12 K12 J12 J1s2si1s2sa z391d2392 quot12 1 exchange K12 J1s2s 2s1sd2391a 2392 quot12 coulomb I1s2s 2s1sH1s2s 2s1sa 2391d2392 triplet 2 EU E2S le K12 J K are positive 3 K12 arises from the I 2K12 T antisymmetrlzatlon of w 1323 J Chapter 1 Late 1800 s Several failures of classical Newtonian physics discovered 1905 1925 Development of QM resolved discrepancies between expt and classical theory QM Essential for understanding many phenomena in Chemistry Biology Physics photosynthesis vision magnetic resonance imaging radioactivity operation of transistors lasers CD DVD players van der Waals interactions Examples where classical Physics inadequate rzaelo lliomj 1 Blackbody Radiation K 4 90mm heated objects gt light 0000K Classical theory 5009 pdv 8 Emdv 2 U 87w 2 cs kT 70 u frequency p density oscillators EOSC average energyosc Emits energy at all T Planck 87ZhU3 1 on mail b fIttIn ex erlment 1900 0 0 c3 elmH4611 g y y g p Planck39s constant k 6626x10 34 J S httpenwikipediaorgwikiPlanck39sconstant Planck later showed this is consistent with the energies of the oscillators making up the blackbody object taking on discrete values Eznhu n012 hU T70 thkl 1 T kT Classical result Taylor series of eX for small x ex 1 x cquot l 2 Photoelectric effect jh observed expected behavior e emitted oc I light is a wave so each e39 absorbs small fraction of the energy I I I I e39 emitted at all u if intensity 1 939 em39ttEd 39f D gt Lo critical great enough freq KE gt With I KE gt With U and independent ofI Explained by Einstein in 1905 light has energy ho and acts particlelike enabling its energy to be focused on one e39 Eel ho 4 work function of metal 3 Heat capacity of solids Spectra of atoms molecules discrete lines spectrumHatom I I I I l U 1 1 U 2 RH 7 7 Rydberg series n1 1 n1 n integers n n11 n12 n13 We will return to this RH 109677581 cm1 Waveparticle duality photoelectric effect 2 light can behave as a particle diffraction of light 2 light can behave as a wave h de Broglie 1924 particles have a wavelength 1 Demonstrated by diffraction of e39 He H2 from crystalline surfaces e39 with KE 17 eV has A 3 A a typical lattice spacing in a crystal gt interference diffraction large objects baseballs cars etc have de Broglie wavelengths too small to be detected Diffraction experiments light incident on a 7 single slit 396 minima Sing 2 n i1 i2 i3 a well separated peaks when 9 z a X a can t see diffraction doubleslit expt with e39 the e39 goes through both slitsll Ii In 1977 the expt was done with the He atoms gt Each atom goes through both slitsll Summary energy oscillators are quantized waveparticle duality de Broglie relationship these ideas paved the way for QM NOTE Frequencies of a guitar string are quantized and guitars are clearly Classical Quantization comes from boundary conditions Fourier transforms frequency time position momentum are conjugate variables We will come back to these considerations Chapter 7 Vibrations and Rotations Vx translation particle in box rotation rigid rotor vibration harmonicoscillator 1 harmonic potential Vx Eloc2k force constant 39 true potential actually we generally use the of variable x39 x xe so Vx39kx392 x390 3 xxe Diatomic molecule imam x24 center of mass coordinates for Vibration what W0 matters is the m1 m2 separation between the atoms dV true potential V06 V069 5 X X Xe can be written e as a Taylor 2 series l d V x Xe2 2 cix2 Xe dV I O x xe3 xxe l 0131 Xe dx 6 dx3 2 choose Vxe to be the zero of energy Vx l 2 x x 2 m 2 dx 3 1 kx x 2 2 e 2 d2 1 h 12 2W 2 EV Schrodinger Eq for 1D 2 dx 2 harmonic oscHIator I Note 6 2 Isasolutlon d Zx2 Zx2 d axe 2 aa2x2e 2 x Do you see why this solves the equation 3x2 e 2 also solves the differential equation But we reject it VVhy The general form of the wavefuction is 1 a 2 ynzAana2xe 2 n012 HAM236 w Hermite polynomials a 14 Zx2 W09 W29 W49 even W2 20m2 1e 2 47 3 14 W19W39W5 Odd a 2ax3 3xe 2 2 W3 72 even function fx fx odd function fx fx Enzh k nl 2 nl hu nl n012 u 2 2 2 quantization due to requiring w gt 0 as x gt 00 EKEEPEgtJU 1 2 2 4 As n becomes large there is a high probability of nding the oscillator nearthe classical turning points E in units of hvo Q 2 L velocity gt 0 Classical lt maximum velocity situation Similar situation for the Classical oscillator 0x00 hhho no0 1x0gt 2 0 shorthand nomenclature lt11 mgt Jw121tmdx The integral HMO is the transition moment for going from state l0 to wn A Transition probability 2 0C x 0gt integral non zero only if n 1 Later we will see that it is also essential that the dipole moment is Changing Chapter 15 Electronic Spectroscopy Diatomic molecules ignoring spinorbit coupling the good quantum s are ML 8 MS ML ZMeia MS ZMsi Term symbols ZS1A A 2 IM l a L A0123 symbol 2 H A CD 9 u subscripts if there is an inversion center H2 10 12g 10gb gt 32 12L excited states 2 129 327 1A symmetry 2 states only depends on whether y changes sign upon reflection through a plane through the molecular axis Selection rules AA0i1 AS0 ult gtg gt gt Electronic states of 02 First two transitions of 02 are forbidden If allowed the earth s atmosphere would not be transparent Photodissociation ht 20 Requires 51 eV 02 002M gt03M L Bagu 03PO1D AU 03P 03P Energy Bond length WWWmam Penman Eduualiun u publishing r Bmuamm bummiligs filters UV radiation If ground excited states had the same potential energy curves would get a single line Exclled stale If excited state potential is displaced can get a very long progression Intensities of vibrational peaks m vfl lva leY Ground sta vertical adiabatic e ll421 WA lllrbllr blz Energy 5 FrankCondon Factor Note vibrational structure is seen in the electronic Distance transitions of molecules such as H2 N2 02 nuns Mm Edmzlion w publishing as Bmiwmml 0 Formaldeh de H 2 2 2 2 2 2 2 y c o lsO lsc UCH OCH0CO7Z39COI ZO 7rCO H using localized orbitals 039 7r not really valid symmetries for a nonlinear molecule Excited states C O n gt7r triplet 80232 n gt7r singlet 80232 7 gt 72 triplet BO 1 7 gt 7 singlet BO 1 internal conversion internal conversion Inte rsyslem Tr Energy Absorption Fluorescence Fhosphorescence so Cuwwlvmzans Pearson Educalinn lipc mum as Bummln Cummings W5 NonRadiative Transitions internal conversion absorption intersystem crossing fluorescence phosphorescence collisions usually important Internal conversion is generally fast compared to uorescence intersystem crossing Sax Tl z fluorescence phosphorescence SO Energy E 77 fluorescence T H 10 S phosphorescence T 2103S Distance H UV Photoelectron Spectroscopy Removal ofan electron from this orbital has 39 little impact on the bonding gt not a very wide FC envelope Removal of an e from this orbital will cause a long progression in the bending vibration Removal of an e39 from this orbital causes excitation of both stretch and bending vibrations
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