Session 7. Spectroscopy and Thermochemistry. Session 7 Overview: Part A I. Prediction of Vibrational Frequencies (IR)

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1 Sesson 7 Spectroscopy and Thermochemstry CCCE Sesson 7 Overvew: Part A I. Predcton of Vbratonal Frequences (IR) II. Thermochemstry Part B III. Predcton of Electronc Transtons (UV-Vs) IV. NMR Predctons CCCE I. Predcton of Vbratonal Frequences Purposes: 1. IR data helps determne molecular structure and envronment Compare expermental vs. computed spectra Fngerprnt regon assgnments dffcult 2. Identfyng transton state structures Nature of statonary ponts on the PES 3. Compute force constants for a geometry optmzaton CCCE

2 Predcton of Vbratonal Frequences Purposes: 4. Compute zero pont vbraton and thermal energy correctons to the total energes, as well as other thermodynamc quanttes of nterest Enthalpy (H), Entropy (S), Free energy (G), etc. CCCE Revew Normal Modes: Nonlnear: 3N-6 normal modes Lnear: 3N-5 Bond stretches: Hghest n energy Bends: Somewhat lower n energy Torsonal motons: Lower stll Breathng modes (very large molecules): Lowest energy Only modes whch cause a change n dpole moment wll be IR actve CCCE Types of Moton - Anmatons Stretchng Symmetrc Asymmetrc In Plane Bendng Out of Plane Scssorng Rockng Waggng Twstng CCCE

3 Harmonc Oscllator Approxmaton Based on Hooke s Law: 1 2 ~ 1 ν = k 2π c MaMb Ma + Mb ~ ν = vbratonal frequency (cm -1 ) Atomc masses (g) are Ma and Mb c = velocty of lght (cm / sec) k = bond force constant (dynes / cm) CCCE A Better Descrpton? When stretched enough, a bond wll break Bond stretchng s more accurately descrbed usng a Morse potental: ab ( rab r ) EMorse( rab ) = Dab{ e } α 0 1 D ab = bond dssocaton energy 2 k α = ab k = 2 Dab ab force constant r ab = nternuclear dstance r o = "equlbrum" bond length Morse potental: Approxmaton to an anharmonc oscllator CCCE Dfferences Harmonc Oscllator Approxmaton gves nfnte number of evenly spaced energy levels gven by: E = hν ( v ) ν = frequency of the vbraton v = vbratonal quantum number (0, 1, 2, etc.) hν When v = 0, E s the zero- pont energy E o = 2 Morse Potental Vbratonal energy levels more closely spaced at hgher quantum number (due to anharmoncty) and are fnte n number (See next slde) CCCE

4 Harmonc Oscllator vs. Morse Comparson CCCE Whch Model to Use Under expermental condtons, vbratonal transtons observed are between the (v = 0) (v = 1) states Both models are nearly the same for ths fundamental vbraton (See prevous slde) Computatonally: Easer to deal wth polynomal expressons (HOA) than wth exponental expressons (MP) HOA s most commonly used for computng molecular vbratonal frequences More accurate methods could be used, at the expense of ncreased CPU tme CCCE Calculatons Normal modes of vbraton are centered at the equlbrum geometry of the molecule Structure must be optmzed pror to the start of a frequency calculaton Frequency calculatons are only vald at statonary ponts on the PES Frequency calculatons should be computed at the same level of theory used to optmze the molecule Frequences should all be postve for an optmzed structure (statonary pont on PES) If magnary (negatve) frequences are found, the geometry represents a saddle pont on the PES CCCE

5 Calculatons - contnued 2nd dervatve of the energy (E) w.r.t. the Cartesan nuclear co-ordnates gves the curvature at the bottom of the PES (Slde #9) Snce the real (Morse) PES s shallower, frequences calculated n the above manner are always greater than the actual (expermental) frequences (more on ths later) CCCE Method Comparson MM force felds are emprcally created to descrbe atomc motons Can get usable results usng the HOA and f the compound of nterest s smlar n structure to those used to create the force feld Lmtaton: Many molecules of nterest wll not have an adequate MM force feld avalable Molecular vbratons are therefore best nvestgated usng a quantum mechancal approach CCCE Method Comparson - contnued Sememprcal Depends on the parameters If the molecule of nterest s smlar to the tranng set, qualtatve results can be obtaned Calculated frequency values are often erratc In general: PM3 s better than AM1 To compensate for systematc errors and get better agreement wth expermental results, some authors wll multply PM3 and AM1 frequences by a scalng factor (See Slde #18) CCCE

6 Method Comparson - contnued HF Calc. frequences are ~10% too hgh Ths s due to: (1) The HOA, and (2) the lack of electron correlaton n the calculaton These (known) systematc errors can be compensated for Much better results can be obtaned by scalng the calculated frequences by a factor of ~0.9 The best scalng factor depends on the bass set used (See Slde #18) 6-31G(d) s the smallest bass set that gves decent results for a varety of molecules CCCE Method Comparson - contnued DFT Erratc behavor (sometmes), but wth smaller devatons than sememprcal results Overall systematc errors wth the better DFT functonals are less than those obtaned usng Hartree-Fock The pure DFT functonal BLYP requres lttle scalng Errors are random about the expermental values Hybrd DFT functonals whch nclude HF character need to be scaled, snce they agan gve consstently hgh results CCCE Scalng factors (pg. 340, Cramer, 2 nd Ed.) (More extensve lst at: Level of Theory Scale factor RMS error (cm -1 ) Outlers (%) 1 AM PM HF/3-21G HF/6-31G(d) BLYP/6-31G(d) B3LYP/6-31G(d) B3PW91/6-31G(d) ) Number of frequences stll n error by more than 20% of the expermental value after applcaton of the scalng factor CCCE

7 Exp. vs. Calc. frequences (cm -1 ) for formamde All results scaled usng factors from Slde #18 Expermental PM3 HF/6-31G(d) B3LYP CCCE Peak Intenstes Intenstes can help n peak assgnments, Can get relatve ntenstes usng the wavefuncton to compute the transton dpole moments Ab nto: Preferred way of dong ths (sememprcal methods often gve poor results) HF results are often scaled Scaled HF, DFT, MP2 all gve smlar accuraces Hybrd DFT functonals gve best results Hgher-level correlated methods (CISD, CCSD) gve mproved values, but the computatonal cost s hgh CCCE Vbratonal Frequences: Summary Computatonal methods are a powerful tool to gan nsght nto molecular vbratonal moton Dfferent methods wll produce dfferent degrees of agreement wth expermental results Best: Use HF or DFT wth a scalng factor If HF or DFT can t be used, try MM or sememprcal methods, but beware of lmtatons If workng wth new compounds, try to calbrate your frequency calculatons usng smlar, known compounds that have expermental data avalable CCCE

8 II. Thermochemstry Once vbratonal frequences are calculated, t takes a small amount of CPU tme to compute varous thermochemcal parameters Results are usually ncluded n program output As wth IR calculatons, a bad startng geometry can gve ncorrect thermodynamc results Optmze geometry, THEN calculate the vbratonal frequences usng the same (or hgher) level of theory To relate calculated molecular propertes to macroscopc thermodynamc propertes, statstcal mechancs s used CCCE Statstcal Mechancs Quantum mechancs depends on Ψ Approprate operator Property of nterest Statstcal mechancs depends on the partton functon, whch allows calculaton of macroscopc values Partton functon (q) for a sngle molecule s a sum of exponental terms nvolvng all possble quantum energy states (ε ): all states q = e - ε / k T B CCCE Statstcal Mechancs - contnued Partton functon (Q) for N dentcal molecules: Q q N = N! The molecular partton functon (q) may also be wrtten as a sum over all dstnct energy levels multpled by a degeneracy factor (g ): q = all levels g e Once the partton functon s determned, a number of thermochemcal and macroscopc observables can be calculated (Next two sldes) ε / k T B CCCE

9 Relatonshps Internal energy (U): Helmholtz free energy (A): 2 ln Q U = k T B Pressure (P): T Constant volume heat capacty (C v ): U Q Cv = kt B kt B ln = 2 + T T V V A= kbtln Q A Q P = kt B ln = V V T V CCCE T 2 ln Q 2 T V Other Relatonshps Enthalpy (H): 2 ln Q ln Q H = U + PV = kbt + ktv B T V Entropy (S): U A ln Q S = = kt B + k T T Gbbs free energy (G): ln Q G = H TS = kbtv V V T CCCE V B T ln Q ktln Q B Partton functons Molecular energy can be approxmated as a sum of the varous contrbutons: εtot = εtrans + εrot + εvb + εelec The partton functon then becomes a product of terms: qtot = qtransqrotqvbqelec Enthalpy and Entropy nvolve ln(q), so: H = H + H + H + H S = S + S + S + S tot trans rot vb elec tot trans rot vb elec CCCE

10 Partton Functon Contrbutons Translatonal: Only need MW MkBT qtrans = 3 V 2π 2 2 h M = MW; V = Molar gas volume Rotatonal: q rot = π σ π kt B h III Where I = Moment of nerta; σ = # of dstnct proper rotatonal operatons plus the dentty operaton CCCE Partton Functon Contrbutons Vbratonal: Total molecular vbratonal energy = sum of energes for each vbraton. Total partton functon s a product of partton functons for each vbraton h n kt B qvb = exp ν = 1 h Electronc: ν 1 exp kt B Only ground electronc state s consdered Excted states typcally le much hgher n energy Most molecules have a nondegenerate ground state, whch means: q elec = 1 CCCE Zero Pont Energy (ZPE) In comparng theoretcal energes of ndvdual molecules (calc. at 0K, fxed nucle) to expermental results (done at ~298K wth vbratng nucle), two correctons are normally requred: 1. Zero-pont energy (ε o ): At 0K, a molecule wll have vbratonal energy. Summaton of energy over all vbratonal modes gves: normal modes 1 ε0 = H ( 0 ) vb = h ν 2 May need to add zero-pont energy to total energy CCCE

11 Thermal Energy 2. At temperature (T) above 0K, ΔH s gven by: ΔHT ( ) = Utrans( T) + Urot ( T) + ΔUvb( T) + RT 3 3 Utrans( T) = RT U rot( T) = RT ( = RT f lnear) 2 2 ΔU ( T) = U ( T) U ( 0) = Nh vb vb vb normal modes hν kbt e 1 Ths calculaton takes nto account the effects of molecular translaton, rotaton, and vbraton at the temperature of nterest Note that the thermal energy correcton ncludes the ZPE automatcally ν CCCE Scalng Factors As wth frequences, ZPE and thermal energy values are scaled to elmnate systematc errors. Same value used for frequences may be used, or specfc factors for energes may be used Level of Theory Frequency Scale factor ZPE/Thermal energy scale factor HF/3-21G HF/6-31G(d) BLYP/6-31G(d) B3LYP/6-31G(d) Adapted from Foresman & Frsch, Explorng Chemstry wth Electronc Structure Methods, 2 nd. Ed., p. 64 CCCE Program Output - Sememprcal Output vares by program and by method used MOPAC (CAChe) default values lsted n mopac.out: Δ f HE, I, MW, Pont group, ZPE, enthalpy, C p, and entropy at varety of temperatures. Includes ndvdual partton functon values for vbraton, rotaton, and translaton (1) Electronc energy = Sum of potental energes for all electrons n the molecule (negatve number) (2) Core-Core repulson energy = Nuclear repulson (3) Total energy (not prnted) = (1) + (2) N (4) Atomzaton energy = () 3 + E () (5) Heat of Formaton = Δ H = Δ H() ( 4) f N atomzaton, exp f,exp CCCE

12 Program Output DFT DGauss (CAChe) default values lsted n DGauss.log: Quanttes from IR frequency calculatons ZPE, total energy, and the rotatonal symmetry number (σ), heat capacty, enthalpy, entropy, and free energy values at a varety of temperatures (1) Energy (a.u.) = electronc energy + core-core repulson energy (2) Zero pont energy (postve number) (3) Total energy = (1) + (2) The total energy s not converted nto Δ f H, as the errors would be large Total energes can be compared to calculate reacton energes, relatve stabltes of somers, etc. CCCE Representatve Results Calc. of Δ r H o for: CO 2 + CH 4 2 H 2 CO (Exp. Value: 59.9 ± 0.2 kcal/mole) Method Result (kcal/mole) AM PM PM B88-LYP 58.4 CCCE Sesson 7: Part B III. Predcton of Electronc Transtons (UV-Vs) IV. NMR Predctons CCCE

13 Predcton of Electronc Transtons In order to obtan energes of electronc excted states, the followng steps are taken: 1. A geometry optmzaton s performed for the ground state molecule Could use MM, Sememprcal, HF, or DFT methods to do ths 2. Ground state wavefuncton s calculated, generatng occuped and vrtual (unoccuped) orbtals Could use Sememprcal, HF or DFT methods CCCE Steps - contnued 3. Typcally, a CIS (Confguraton Interacton, Sngles) calculaton s performed Vrtual orbtals (Ψ ) are mxed nto the ground state wave-functon (Ψ o ) (.e. electrons are swapped between occuped and vrtual orbtals obtaned from the ground state geometry) The geometry s held constant To keep a small number of excted states, only orbtals near the HOMO and LUMO are used (restrcted actve space) Ψ = coψo + c1ψ1+ c2ψ c = mxng coeffcents CCCE Steps - contnued 4. Ground state molecular electronc Hamltonan s used to fnd the coeffcents of mxng Ths gves an approxmaton to the energy of the excted electronc states at the fxed molecular geometry chosen to begn wth (.e. the ground state energy does not change) Eex Eg 5. Transton frequency found by: ν = Note ths gves a vertcal exctaton energy, snce E ex wll not be n ts equlbrum geometry O.K. for short-lved excted states (as n UV- Vs) h CCCE

14 Steps - contnued 6. Transton ntensty depends on the energy and the oscllator strength Oscllator strength depends on the transton dpole moment between any two states (selecton rules) μmn = Ψ $ μψ 1 2 CCCE Methods Ground state geometry MM, Sememprcal, HF, or DFT CIS Use sememprcal or ab nto methods for ths Can use Tme-dependent DFT (TDDFT) Works well for lower energy exctatons Ablty to do ths not ncluded n all programs CCCE UV-Vs usng ZINDO INDO/S (also called ZINDO/S, part of ZINDO) Sememprcal method specfcally parameterzed to reproduce UV-Vs spectra Other MOPAC methods parameterzed for ground-state heats of formaton (better suted for geometry optmzatons) ZINDO output s n atomc unts CCCE

15 ZINDO Zerner s Intermedate Neglect of Dfferental Overlap Only uses valence electrons Parameterzaton s theoretcally-based, and more elements have parameters than n MOPAC Can use d-orbtals wth many transton metals (ths s very lmted n MOPAC) Used to calculate electronc spectra CCCE ZINDO - contnued Lmtatons of ZINDO: Handles molecules up to ~200 atoms Stran energy of small rng systems not handled well May have to calbrate for partcular systems of nterest If studyng new compounds, do calculatons on known compounds of smlar structure whose electronc spectra have been measured See how well the calculated spectra match CCCE Representatve Results Calc. gas phase (ZINDO CI at MM/PM3 Geometry) Compound Exp.(nm) Calc.(nm) Assgnment 1,3-butadene π π* 1,3,6-hexatrene π π* 1,3-cyclohexadene π π* Napthalene 221, 286, 312 Acetophenone Benzophenone 252 Lqud Phase , 268, π π* π π* n π* π π* n π* CCCE

16 NMR Spectroscopy Chemcal shft s the most mportant magnetc property Most wdely appled spectroscopc technque for structure determnaton In addton to 1 H and 13 C, many other nucle are ncreasngly mportant ( 15 N, 29 S, 31 P, etc.) All are equally amenable to computatonal nvestgaton Need to know e - densty at the nucleus of an atom CCCE NMR - contnued Computed magnetc propertes are very senstve to the geometry used Optmze the geometry frst! Complex problem More dffcult to model the nteracton of a wavefuncton wth a magnetc feld (B) than an electrc feld (E) Electrc feld (E) perturbs the potental energy term of the Hamltonan CCCE NMR Calculaton Problems Magnetc feld (B) perturbs the knetc energy term Electron moton produces electronc magnetc moments Angular momentum operator (L) s magnary An orgn must be specfed defnng the coordnate system for the calculaton; The operators used depend on ths orgn Use of the exact Ψ gves orgn ndependent results Ψ HF wll also gve orgn ndependent results f a complete bass set s used Snce nether of these are lkely, the calculated results wll depend on the orgn used CCCE

17 Gauge Orgn The orgn of the coordnate system used for the calculaton s called the gauge orgn One way to elmnate the gauge dependence s to construct bass functons that are dependent on a magnetc feld Include a complex phase factor that refers to the poston of the bass functon (the nucleus) Makes all calculated propertes ndependent of the gauge orgn Older versons of ths approach called Gauge Invarant Atomc Orbtals (GIAO) CCCE Gauge Orgn - contnued More recent versons keep the same acronym, that now stands for Gauge Includng Atomc Orbtals Most popular technque, probably the most robust Based on perturbaton theory Uses HF or DFT wavefuncton to calculate sheldng tensors Programs lke Gaussan use ths method CCCE NMR Calculatons There are two magnetc felds to worry about: NMR Chemcal Shft α 2 E B I B = External magnetc feld I = Nuclear magnetc moment Computng absolute chemcal shfts s dffcult Shfts are therefore calculated relatve to a standard (TMS for 1 H and 13 C) Gas phase results, often reasonably close to expermental CCCE

18 NMR Calculatons cont. Heavy atom chemcal shfts for frst row elements can be computed wth a far degree of accuracy In general: CCSD(T) > MP2 > DFT > HF CCSD(T) & MP2 usually not feasble due to hgh cost CCCE H NMR Calculatons Small chemcal shft range (~15ppm) Rovbratonal and solvent effects gve errors comparable to the range of chemcal shfts DFT method shows best results: 80 modest-sze organcs: B3LYP rated best GIAO scheme used wth a G(2df,p) bass set Lnear scalng mproved results (factor = ) CCCE C NMR Calculatons Calculatons much more common than for 1 H Much larger chemcal shft range Large bass sets gve the best results It s largely the tals of the valence orbtals at the nucleus that affect chemcal shfts, not the core orbtals Mnmum recommended combnaton: B3LYP/6-31G(d) geometry followed by HF/6-31G(d) calc. of the chemcal shfts CCCE

19 Spn-Spn Couplng Calculatons Less routne than chemcal shft calculatons Addtonal complcaton assocated wth 2 local magnetc moments Expermentally, 1 H/ 1 H couplngs are usually reported These are the most dffcult to calculate Tend to be small n magntude, so absolute errors are magnfed Have to use very flexble bass sets, makng the cost qute hgh Gaussan does do these calculatons CCCE The End Questons? CCCE