Ab Initio Calculations of Vibrational Spectra and Their Use in the

  Chem. Rev. 1988, 86, 709-730 709 Contents Ab Initio Calculations of Vibrational Spectra and Their Use in theIdentification of Unusual Molecules B. ANDES HESS, JR.' and LAWRENCE J. SCHAAD Department of Chemistry, Vanderbilt Universiv, Nashville, Tennessee 37235 PETR CARSKY and RUDOLF ZAHRADNiK J. Heyrovskp Institute of Physical Chemistrv and Eiectrochemistry, Czechoslovak Academy of Sciences, Michova 7, 12 1 38 Prague 2, CzechoslovakiaReceived December 12, 1985 (Revised Manuscript Received March 27, 1986) I. Introduction I I. Theoretical Approaches to Vibrational Spectra A. Harmonic Approximation 8. Perturbation Treatment of Anharmonicky C. Classification of ab Initio Calculations A. Force Constants and Frequencies B. Intensities C. Use ofEmpirical ParametersIV. A Test of the Method V. Examples ofAppiicatlons A. HNO B. Cyciobutadiene C. [l.l.l]Propeilane D. Methylenecyciopropene E. Cyciopropenyiidene F. ThiireneMoleculesCalculations 111. Computation of IR Spectra VI. Some Predictions of IR Spectra ofUnknown VII. Compilation ofab Initio Vibrational VIII. Summary I. Introductlon 709 71 1 71 1 712 713 714 714 715 716 717 718718 719 722 722 723 723 723726 726 The synthesis of new compounds and the determi-nation of their structures is one of the main goals inchemistry. In recent decades various kinds of spec-troscopy have played a major role in structural deter-mination. Apart from the direct determination ofstructure by means of X-ray analysis, the most usefulamong these spectroscopies are electronic (ultravioletand visible), vibrational (infrared and Raman), nuclearmagnetic resonance, and mass spectrometry. A full use of experimental spectroscopy for the pur-pose of structure determination requires the availabilityof a theoretical tool that can provide computed spectraldata in a completely independent way and with anaccuracy sufficient for a meaningful comparison withthe experimental data. An deal procedure is the fol-lowing. First record the spectrum of a compound of unknown structure. Assign to it one or more plausiblestructures, and compute theoretical spectra for all thesestructures. Compare experimental and theoreticalspectra, and accept or reject the assumed structure onthe basis of agreement or disagreement between thetwo.Historically, quantum chemistry had an impact first in the field of electronic spectroscopy. In the late fiiies and early sixties the quantum chemistry of all but thesimplest systems was dominated by semiempiricaltechniques such as the Pariser-Parr-Pople (PPP) me-thod.' Calculations of this type were applied mostly inthe field of electronic spectroscopy to interpret theelectronic spectra of conjugated a-electron systems. An example,2 he main goal of which was structure deter-mination, concerns the benzenoid hydrocarbon zethrene (I). A hydrocarbon to which the structure of zethrenewas srcinally assigned was actually either a derivativeof acepleiadiene (11) or the hydrocarbon (111). Com- 883 00 00 00 I II Ill parison of theoretical spectral data for 1-111 with twoavailable absorption curves showed that the assumedzethrene was actually I11 and that a newly synthesizedhydrocarbon was really zethrene.Since the 1960s he development of larger and fastercomputers together with elaborate ab initio programshas allowed the routine computation of theoretical vi- brational spectra for molecules of up to, at present,about ten first-row atoms. During the same period theavailability of low-temperature matrix isolation hasmade it possible to isolate highly reactive molecules andto obtain their spectra. This technique coupled withthe advent of Fourier-transform IR spectroscopy hasprovided a powerful new tool to the experimentalist. 0 1986 American Chemical Society  710 Chemical Reviews. 1966. Vol. 86. No. 4 B. Andes Hess. Jr.. is Professor and Chahman of the Deprmwmt of Chemistry a1 Vanderbill University. He was bom h WMngton. DE, in 1940. received a B.A. (Wllllams College) In 1982. an M.S. (Yale) h 1963. and a W.D. (Yale) n 1966 under Professor Kenneth B. Wberg. After two years of postdoctoral studles wiih ProfessorVkgil BoekelheMe at the University of Oregon. he plned the De-partment of Chemistry at Vanderbin University as an assistant professor in 1968. He spent a year (1973-1974) at the Heyow Institute. Czechoslovak Academy of Sciences. Prague wnh Dr. Rudolf Zahradnik as a National Academy of Sclences ExchangeScientist. His research nterests include the thmry of aromaticity.ab initio wmputatbn of vibratonal spectra and tkaetkal shdies of organic reaction mechanisms. Q \ -. A 1 LJ. schaad. WhosereJearch hieresbareh~chemlsby. b ~ofessor f Chemistry. varderbin university. ashv vine. TN. ~e was ban (1930) h cokmbus. OH. W A.B. deqee is I arvard h 1952. and his W.D. was done at the Massachusetts 1 of Technology under the direction of C. G. Swain. After postdoaoral study wiih C. A. Coulson (Oxford) and Harrison Shuil (Indiana). he joined Vanderbin University in 1961. MEBrsky was tan h 1942 h slavakla and he hk RNX from Charles University in Prague in 1964. He then !dmdDr. Zakadnik's gwp t the CzechoslovakAcademy of sdenees and received his CSc. (equivalent to Ph.D.) in 1968. He was a post-doctoral fellow at the University of Winburg (1968) and visningprofessor at Vanderbin University in Nashville (1982) end PasteuUniversity in Strasboug (1984). His research interests Center on chmkal amtiom of molecu$r Uway and computational chemistry. Hess et al. Rudolf Zahradnk was barn 1928 and graduated from th9 Prague Institute of Technology (1952). In 1956 he received his C.Sc. wee (appoximately equivalent to Ph.D.). and in 1968. the D.Sc. At present he is the Head of the Group of Theay of the ChemicalReactivity in J. Heyrovskg Institute of Physical Chemistry andElectrochemistry. Czechoslovak Academy of Sciences. Prague.From 1965 to 1984 he spent several periods as visiting professorat universities n Wuzburg. Darmstadt. Groningen. Giessen. Basb. Sandal. Osaka. Erlangen. and Leipzig. In 1970 he received a NSF(Washington) senior fellowship. In 1981 he was elected in theInternational Academy of Quantum Molecular Sciences. He actsas a member of ediorial boardsofJournal of Molecular slrucfure (THEOCHEM). Reactiviiy and Structure (Springer). and ChemicalReviews, and is a member pf the advisoly board of TheoreficaChemica Acta. Dr. Zahradnik has authored or coauthored about 300 papers, a number of textbooks. and 7 books. These worksdeal mainly wiih molecular orbital theory. theory of chemical re- activity. and weak intermolecular interactions in chemistry andbiology. For example the highly reactive thiirene was preparedby the following ~cheme.~ &gee; he then became Assodate Professor O Charles vntvenny. L Also present in the product mixture were ethynylmercaptan and thioketene. IR spectroscopy was thetool by which the presence of thiirene was identified. A spectrum of a reaction mixture obtained by the de-composition of thiadiazol is shown in Figure l. It is to be expected that the spectra so obtained are of lowrmlution and that the assignment is uncertain because of side product formation.The aim of this review is to show the utility of the-oretical vibrational spectra in such experiments. It tums out that the accuracy of these spectra is sufficient to give correct overall patterns, but fine details are notyet to be trusted. The next sections outline the theo-retical background of vibrational calculations. A practical chemist, whose interest is seeing whethertheoretical vibrational spectra can be of use to him,might profitably skip these sections at first reading and turn o the examples of sections IV and V. Such readersshould however keep in mind the distinction betweentwo ways of calculating vibrational spectra. In both a force constant matrix is diagonalized to give molecularvibrational frequencies. In the ab initio method of thisreview, these force constanta are derived by as rigorous as possible a solution of the electronic SchrBdingerequation of the molecule in the fixed-nucleus approx-imation. In the other method, which is older and more  Calculations of Vibrational SpectraChemicalReviews, 1986, Vol. 86, No. 4 711 systems such as liquids, clusters, or floppy molecules,for which normal mode analysis is inappropriate. Thesubject of the review by Schrader, Bougeard, and Nig-gemann6 is most closely related to our paper. It alsodeals with structure determination by infrared and Raman spectra. Their review is however more general,and the results of only a few ab initio calculations arepresented as illustrative examples. Recent reviews byFogarasi and Pulay7 summarize the progress made inthe last several years in ab initio methods and compu-tational techniques (analytical computation of secondand higher derivatives of energy, basis set and corre-lations effects), and are recommended as an extensionof our sections IIC and IIIA. 0 I I I1 3000 2000 1600 I200 800 c m-1 Figure 1. Infrared spectrum of the photolyzed thiadiazole? Thearrows mark bands assigned to thiirene. bond length - Figure 2. Schematic representation of a potential curve fromthe fixed nucleus calculation (solid line) and ita harmonic ap-proximation (dashed line). usual among experimental spectroscopists, the forceconstants are simply guessed. This older method canbe useful if the molecule studied is similar enough toothers with known force constants, but since there areusually more force constants than vibration frequencies,there are many ways to choose force constants to pro-duce exact agreement with experimental frequencies. It is common to find in the literature a molecule studiedby two groups who have produced different, but equallyaccurate and equally plausible, sets of force constants.One application of ab initio vibrational calculations isin the choice between these sets of experimental forceconstants.However it is in the case of unusual mole-cules, such as reactive intermediates, where analogiesare poor and guessed force constants particularlydoubtful, that we suggest ab initio calculations will beof greatest use.There are several review articles in the literature with titles similar to ours though their scope is different. Asstated by Schutte4 he purpose of his paper is to ex-amine the progress which has been made since 1926 inthe ab initio calculation of both the vibrational fre-quencies of molecules as well as the forces acting uponindividual atoms when the equilibrium of the moleculehas been disturbed . Molecular orbital calculations aredealt with only very briefly, and the paper concernsmostly diatomic molecules. A review by Fredkin, Ko- mornicki, White, and Wilson5 focuses on treatment of II. Theoretlcal Approaches to VlbratlonalSpectra A. Harmonic Approximatlon molecule with M nuclei and N electronsConsider the Schrodinger equation for a polyatomic c c- @>a a=i rab j>i i=irjj where ma nd 2, are the mass and charge (in units ofelectron charge) of nucleus a, m, and e are the electronicmass and charge, ria is the distance between electron i and nucleus a, nd rij and ra6 are defined analogously.The terms on the left of eq 1 give, in order, the kineticenergy of the nuclei, that of the electrons, the elec-tron-nuclear attraction, the nuclear-nuclear repulsion,and the electron-electron repulsion. The wavefunctiondepends upon the 3(M + N) coordinates of theelectrons and nuclei. Since the nuclei are heavier andmove more slowly than the electrons, it is usual toneglect the nuclear kinetic energy term in eq 1 byholding the nuclei stationary. The resulting equationdescribes the motion of the N electrons in the field ofthe fured nuclei, and the energy E of these electrons will depend upon the chosen relative nuclear positions. Inthe case of a diatomic molecule there is only a singlerelative position, the bond length Ri2; and the plot of E(R12) w. R12 will be as shown by the solid line in Figure 2. For a polyatomic molecule the electronic energy inthis fixed-nuclear approximation will depend on allnuclear positions so that E = E(R ...RM), hough E isconstant for those combinations of nuclear displace-ments that give translation or rotation of the molecule as a whole.Born and Oppenheimer8 showed that if the nuclearkinetic energy is treated as a perturbation on the fix- ed-nuclear Hamiltonian, the first-order energy correc-tion vanishes at all positions of nuclear equilibrium (i.e.,for stable molecules). The second-order energy cor-rection is given by an equation describing the nuclearmotion  712 Chemical Reviews,1980, Vol. 80, No. 4 Hess et al. where the first term in eq 2 is the kinetic energy op-erator for the nuclei and E,, is the second-order energycorrection and gives the energy of the nuclear motion.The potential V(R, .. RM) onsists of the quadratic termsin a power series expansion of E(R ...RM) about theequilibrium position, 1 3M a2E - - AXiAXj + ... (3) 2 ij=ldxiaxj where X1, 2..Xm are the Cartesian coordinates of the A4 nuclei. The first term E(X10...X3Mo) dds only aconstant to all allowed E,, and can be ignored for thepurposes of mqst vibrational calculations. Since E is a minimum at the equilibrium geometry, alland the second term in eq 3 vanishes. If all terms oforder higher than second are neglected, the potential is a quadratic function of the nuclear displacements AXi 1 3M 2 lJ'1 E(X1 .. X3M) - Fii AXiAXj V(R1 .. M) (4) where the force constants Fij are given by F.. = - ~ dxiaxj 2E I (5) Equation 4 is known as the harmonic approximation.In the diatomic case this corresponds to replacing thesolid curve of Figure 2 by the parabola shown by thedashed line. As can be seen, this approximation maybe satisfactory for small displacements from the equi-librium position R120, ut it cannot be accurate for largedistortions of the molecule. A transformation to mass-weighted coordinates, pi = (mi)*/2AXi, ollowed by rotation of the coordinates tocoincide with the principal axes of the quadratic formin eq 4, gives an expression for V containing onlysquared terms 1 3M where the Q, are coordinates relative to the principalaxes and are called normal coordinates . As a resultof these transformations eq 2, which depends on 3M variables, separates into 3M equations, each dependingupon a single Q,. Further, each of these is a harmonicoscillator equation with force constant hi. For a nonlinear polyatomic molecule there turn outto be six of the A, with value zero. These correspondto the three translational motions and three rotationsof the entire molecule. The six zero-frequency motions can be removed by working in a coordinate system withthe srcin at the center of mass and rotating with themolecule. The remaining 3M - 6 degrees of freedomare usually specified by internal coordinates such asbond lengths and bond angles. As a result, fewer forceconstants need be evaluated, and those that are evalu-ated have chemical interpretation. Minor difficultiesarise because the transforrqation to these coordinatesis not linear except in the limit of infinitesimal dis-placements, and because in particular cases there may TABLE . Harmonic and Anharmonic VibrationalFrequencies (cm-') of the HCOOH Molecule Given by abInitio SCF 4-31G Calculations obsd frequency, calcd frequency (cm-')mode cm-' harmonic anharmonic OH 35703769 3629 CH 29443160 3047 c=o 17761909 1882 co 11051150 1140 LHCO 13871510 1490 LOCO 625659 653 LCOH 12231366 1347Reference 13. be no nonredundant set of simple internal coordinates.All of this is taken care of in the commonly used GF matrix formulation of the vibrational problem byWil~on.~ 6. Perturbation Treatment of Anharmonicity Most of the vibrational calculations discussed insections IV-VI and listed in Tables IV and V have usedthe harmonic approximation of eq 4, but in recent yearsthere has been progress in the ab initio calculation ofcubic and quartic force coestants. These are the thirdand fourth order terms that were dropped from eq 3, and their availability allows the possibility of includinganharmonicity in ab initio treatments of the vibrationalproblem. Prior to 1980 his was rarely done, but onemay anticipate it being done more commonly in thenear future. Perturbation theory yields the followingformula for the anharm0nicity:'O 1 1 xrr = 16 4ww - 16 C4,2[(8 4 3 d)/~s(4 41 (7) Equation 7 contains the quadratic, diagonal and sem-idiagonal cubic, and diagonal quartic force constants.The wk are the harmonic frequencies. The quarticconstant +rm may be estimated easily1' (for stretchingvibrations) or it may be calculated numerically.Equation 7 does not yield anharmonicities of high ac-curacy, but it represents the only method suitable forpractical application to polyatomic molecules of arbi-trary structure. Equation 7 is relatively simple. Theonly complication met in the calculation of xW is thatthe force constants 4- and 4- are derivatives of energywith respect to normal coordinates. They are nottherefore force constants obtained by the differentiationof energy with respect to internal coordinates (J nd F,, in the notation of the next sections). The conver-sion F - may be performed by a rather complicatednonlinear transformation.12 Alternatively, 4w8 may beobtained from the changes of the energy gradient alongnormal coordinates. This way of obtaining 4rrs s con-ceptually simple, but computationally it is not eco-nomic.We have selected two examples of the ab initio cal-culation of anharmonicity by means of eq 7. The fiist13 is for the formic acid molecule. The level of the ab initiocalculations was modest, so only a rather approximatepotential was obtained. From the results presented inTable I it is seen that the inclusion of anharmonicityby means of eq 7 leads to better agreement betweentheory and experiment, though the error in the com- 8