THEORETICAL : MOLECULAR AND DYNAMICS 276 CHIMIA 2004, 58, No. 5

Chimia 58 (2004) 276–280 © Schweizerische Chemische Gesellschaft ISSN 0009–4293

Ab initio Vibration-Rotation Spectroscopy

Walter Thiel*

Abstract: This review surveys recent theoretical work from our group in the area of vibration-rotation spectroscopy. It addresses the computation of anharmonic force fields and spectroscopic constants in the context of rovibrational per- turbation theory as well as variational calculations of vibrational levels, on the basis of highly accurate ab initio po- tential energy surfaces. Results are presented for three case studies involving difluorovinylidene, bismuthine, and am- monia, to illustrate current contributions from ab initio to spectroscopic studies.

Keywords: Ammonia · · Force fields · Reactive molecules · Spectroscopy

1. Introduction 2. Anharmonic Force Fields and diagonal quartic force constants which are Spectroscopic Parameters needed in the equations for the spectroscop- High-level ab initio quantum chemistry ic constants [1–3]. In this procedure, the nu- is a natural partner of high-resolution spec- The harmonic-oscillator rigid-rotor ap- merical precision of the results must be con- troscopy. On the one hand, the quality of ab proximation provides the simplest textbook trolled carefully, through an appropriate initio calculations can be gauged by compar- model for vibration-rotation spectroscopy: choice of step sizes and tight convergence ison with accurate spectroscopic data. On the each vibration is independent and represent- criteria for the quantum-chemical calcula- other hand, after such validation, ab initio ed by a normal mode, and the molecule ro- tions and for geometry optimization. Suit- theory can provide realistic predictions tates as a rigid body. Second-order perturba- able guidelines are available for this purpose which may guide the spectroscopic identifi- tion theory can then be employed to account [3][4]. cation of unknown reactive molecules and for the anharmonic coupling between the vi- As illustrated in Fig. 1, the spectroscopic assist in the analysis and assignment of high- brations and the vibration-rotation coupling constants provide the meeting ground be- resolution spectra. due to nonrigid rotation. This treatment tween theory and experiment in the standard This article reviews some of our recent yields an effective Hamiltonian and expres- approach outlined above. The next question theoretical work on vibration-rotation spec- sions for the rovibrational levels in terms of is obvious: How accurate are current ab ini- troscopy which addresses both of these ob- quantum numbers and spectroscopic con- tio calculations for these spectroscopic pa- jectives. The prediction of theoretical vibra- stants. The experimental high-resolution rameters? This can be tested in systematic tion-rotation spectra requires knowledge of spectra are interpreted on the basis of such convergence studies using increasingly ac- the potential energy and dipole moment sur- effective Hamiltonians, and the spectroscop- curate levels of theory and basis sets. Fig. 2 faces which can be computed using elec- ic constants are determined through appro- and 3 show corresponding results for the tronic structure theory. These quantum- priate fits of the observed spectral lines. geometry and the harmonic vibrational chemical data can then be converted into In the context of second-order perturba- wavenumbers of water, respectively. Three spectroscopic information using either rovi- tion theory [1][2], there are explicit formulas levels of theory have been applied (SCF: brational perturbation theory or variational that relate the spectroscopic constants of a self-consistent-field Hartree-Fock theory, treatments of nuclear motion. These two ap- given molecule to its equilibrium geometry MP2: second-order Møller-Plesset perturba- proaches will be discussed in sections 2 and and its quartic force field (i.e. the derivatives tion theory; CCSD(T): coupled cluster theo- 3, respectively, followed by a brief summa- of the potential energy with respect to the nu- ry with single and double excitations aug- ry in section 4. clear coordinates, evaluated at the equilibri- mented by a perturbational treatment of con- um geometry up to fourth order). Ab initio nected triple excitations), in combination methods can be applied to compute these with five correlation-consistent polarized molecular properties. Due to the availability valence basis sets of increasing size (from of analytic second derivatives for most of the cc-pVDZ to cc-pV6Z, with 24 to 322 basis established methods, it is nowadays routine functions). The results are seen to converge to calculate the equilibrium geometry and at each level as the size of the basis ap- *Correspondence: Prof. Dr. W. Thiel Max-Planck-Institut für Kohlenforschung the quadratic force field. The required cubic proaches the complete basis set (CBS) limit. Kaiser-Wilhelm-Platz 1 and quartic force constants are best deter- The converged Hartree-Fock results deviate D-45470 Mülheim mined from analytic second derivatives by a considerably from the experimental data, Tel.: +49 208 306 2150 which are however well reproduced upon in- Fax: +49 208 306 2996 numerical finite-difference procedure [3] E-Mail: [email protected] that involves displacements along normal clusion of electron correlation at the simple www.kofo.mpg.de/thiel coordinates and yields all cubic and semi- MP2 level and particularly at the more accu- : MOLECULAR SPECTROSCOPY AND DYNAMICS 277 CHIMIA 2004, 58, No. 5

rate CCSD(T) level. The superiority of CCSD(T) over MP2 becomes more obvious for electronically more demanding mole- cules such as HOF and F2O [4]. Based on many such convergence studies, we find that CCSD(T) calculations with the cc-pVQZ ba- sis are usually adequate for molecular geometries and harmonic force fields: bond Perturbation lengths are normally given within 0.2 pm, bond angles within 0.2°, and harmonic wavenumbers within 10 cm–1 (or typically 1%). Concerning the quartic force fields, MP2/cc-pVTZ yields reasonable estimates of anharmonic spectroscopic constants such as the vibration-rotation interaction con- X stants αi and the anharmonicity constants xij (often within 20%), whereas CCSD(T)/cc-pVTZ is recommended for higher accuracy (often within 5%). Larger Fig. 1. Spectroscopic constants: theory meets experiment basis sets such as cc-pVQZ are normally not needed for anharmonic force fields. Having established an efficient and reli-

bond length Re[Å] bond angle Θe[deg] able computational approach, it is possible to predict the vibration-rotation spectra and oth- er properties of yet unknown molecules in or- der to assist in their spectroscopic identifica- tion. This will now be illustrated for the ex- ample of difluorovinylidene F2C=C. Vinylidenes are elusive reactive intermedi- ates. The parent compound (singlet H2C=C) and 1-H-vinylidenes rearrange easily into alkynes, by a very facile 1,2-H-shift with a barrier of ca. 1–4 kcal/mol. F2C=C is a more promising target because the barrier to a 1,2- F-shift is much higher. Consequently, there have been several attempts to detect F2C=C, and a vibrational progression in the photo- electron spectrum of the corresponding anion has indeed been attributed to F2C=C [5]. Our CCSD(T) calculations show that F2C=C has to overcome a barrier of 36 kcal/mol to re- Fig. 2. Equilibrium geometry of water: convergence of ab initio results. Color code: green SCF, blue arrange to difluoroethyne FCCF which is MP2, red CCSD(T), black experiment. The cc-pVnZ basis sets are denoted as VnZ (n=D,T,Q,5,6). more stable than F2C=C by 29 kcal/mol. It is therefore most unlikely that F2C=C can be prepared from FCCF by a thermal reaction, but when generated photochemically from FCCF it should live long enough to be de- tectable. Irradiation of matrix-isolated FCCF in Ar at 10 K with a pulsed 193 nm ArF laser yields a photolysis product [6] that exhibits several sharp infrared lines (Fig. 4). Through comparison with the computed ab initio gas- phase spectrum (red in Fig. 4) these lines can unambiguously be assigned to F2C=C. Their positions are predicted by the ab initio calcu- lations within the expected accuracy of about 10 cm–1, both for the six fundamental bands νi and for the combination band ν2+ν5 which is in anharmonic resonance with ν4. More- over, their intensities are also predicted well from the calculated dipole moment deriva- tives (double harmonic approximation) in- cluding those for the very weak bending modes ν3 and ν6. Hence, there is no doubt that F2C=C can indeed be made photochem- Fig. 3. Harmonic wavenumbers of water: convergence of ab initio results. See caption of Fig. 2 for ically in an Ar matrix [6]. F2C=C behaves as notation. a superelectrophilic carbene and reacts with THEORETICAL CHEMISTRY: MOLECULAR SPECTROSCOPY AND DYNAMICS 278 CHIMIA 2004, 58, No. 5

example is permanganyl fluoride MnO3F [29]. Another obvious advantage of DFT is

F2C=C its rather low computational cost compared with high-level ab initio methods, and it can therefore be used to calculate anharmonic force fields for medium-sized molecules such as benzene [30] and azabenzenes [31]. While the validation studies on DFT anhar- monic force fields [28][32][33] generally show reasonable performance of the estab- lished functionals, the DFT results cannot be improved systematically (unlike ab initio re- sults) which is a major disadvantage. When- ever feasible in practice, it is therefore ad- visable to employ high-level ab initio meth- ods in connection with high-resolution spectroscopic studies on small molecules. DFT will be the method of choice for larger molecules, and especially for transition met- Fig. 4. Matrix infrared spectrum (black) of a photostationary mixture of difluorovinylidene and di- fluoroethyne in Ar at 10 K, and computed gas-phase infrared spectrum of difluorovinylidene (red) al compounds, when the emphasis is on qualitatively reliable results: these can often already be obtained from harmonic DFT many nucleophiles added to the Ar matrix sues [24–26] has led to the conclusion that force fields [34][35]. [7]. Even with xenon, it forms a charge- outer-core correlation in the 5s5p5d shell is transfer complex F2C=CXe which has again important in BiH3 and strongly affects the been identified and characterized with the bond length (by almost 2 pm) and the 3. Potential Energy Surfaces aid of ab initio calculations [8]. Experimen- stretching wavenumbers (by about 20 cm–1). and Vibrational Levels tally, it has unfortunately not been possible, Therefore it is necessary to employ small- in spite of many attempts, to measure gas- core pseudopotentials and explicitly corre- The standard approach outlined in the phase high-resolution spectra of photochem- late the outer-core electrons. This treatment preceding section is usually appropriate for ically generated F2C=C. The ab initio results requires very large basis sets at Bi to avoid describing the fundamental vibrations of for the equilibrium geometry and various basis set superposition errors, and the rec- semirigid molecules. It is less suitable for vi- harmonic and anharmonic spectroscopic ommended optimized [8s8p7d5f3g2h] basis brationally excited polyatomic molecules constants of F2C=C thus serve as predictions at Bi contains 151 basis functions (aug-cc- with high vibrational energy and for floppy [9] that may hopefully be verified in future pVQZ at H). CCSD(T) calculations with this molecules with large amplitude motions. In experimental work. extended basis and a relativistic pseudo- such cases, it is necessary to resort to a vari- Using the combined theoretical and ex- potential yield satisfactory results for the ational treatment of molecular motion [36]. perimental approach outlined above, we equilibrium geometry and the fundamental The variational approach requires an accu- have characterized a number of reactive and wavenumbers for the bending modes, while rate potential energy surface (PES) for the short-lived molecules over the past years, those for the stretching modes are overesti- configurational space that is sampled during mostly in close cooperation with the group mated more strongly than usually found at the vibrations, along with a corresponding of Hans Bürger [10]. This includes joint this level of theory. This is rectified by in- dipole moment surface (DMS) if intensities work on difluorophosphorane PH3F2 [11], cluding relativistic spin-orbit corrections are of interest. These ab initio surfaces are difluoroethyne FCCF [10][12–14], fluo- from an appropriate spin-orbit configuration computed at a large number of grid points rochloroethyne FCCCl [10][15], silaethene interaction (CI) treatment which lowers the and then represented by a suitable analytic –1 H2C=SiH2 [16–18], difluorosilanethione stretching wavenumbers by up to 12 cm . function through least-squares fitting. The F2Si=S [19], phosphenous fluoride FP=O After these measures, the ab initio results for actual variational treatment proceeds as fol- [20], thiophosphenous fluoride FP=S BiH3 reproduce the experimental data well lows: After choosing appropriate basis func- [21][22], the fluorocarboxyl radical FCO2 [24–26] and satisfy our target accuracy (see tions (e.g. Morse oscillators for stretching [23], and bismuthine BiH3 [24–26]. Here we above). Hence, the available ab initio meth- and harmonic oscillators for bending) and briefly comment only on the most recent ods can describe heavy closed-shell mole- setting up the Hamiltonian (explicit kinetic study of bismuthine, the heaviest and least cules with similar accuracy as compounds energy expressions plus the analytic poten- stable XH3 species of group 15 consisting only of lighter elements. tial function), the Hamiltonian matrix is cal- (X=N,P,As,Sb,Bi). Experimentally, the chal- A final point in this section concerns culated and diagonalized to obtain the vibra- lenging synthesis [27] of BiH3 could be re- analogous applications of density functional tional levels and eigenfunctions. The exact peated successfully such that it became pos- theory (DFT). The procedures for computing solution for a given input potential can be sible to record high-resolution vibrational anharmonic force fields can also be em- reached by converging the variational results and rotational spectra [24][25]. From a theo- ployed at the DFT level, provided that nu- with regard to the chosen basis set for nu- retical point of view, the presence of a heavy merical issues are handled with proper care clear motion. For this reason, the variational atom (Bi) poses two problems that are not [28]. DFT results for spectroscopic constants approach is superior to second-order pertur- encountered with lighter atoms and may lim- are normally of similar accuracy to MP2 re- bation theory (section 2) with regard to the it the accuracy of our calculations: it is prac- sults, except for occasional outliers [28], and accuracy that can be achieved, but it is also tically inevitable to replace the inner-shell DFT has the advantage that it can often still considerably more costly. core electrons by a suitable pseudopotential, be used in electronically demanding situa- A recent benchmark study on water [37] and it may be necessary to explicitly account tions, e.g. for transition metal compounds, illustrates the state of the art in exact varia- for valence-shell relativistic effects. A sys- where single-reference ab initio methods tional calculations on triatomic molecules. tematic investigation of these and other is- such as MP2 or CCSD(T) may fail. One such The underlying PES has been determined by THEORETICAL CHEMISTRY: MOLECULAR SPECTROSCOPY AND DYNAMICS 279 CHIMIA 2004, 58, No. 5 multi-reference MRCI calculations with very large basis sets (up to aug-cc-pV6Z) and extrapolation to the CBS limit, aug- mented by corrections for core-valence cor- relation, relativistic effects, quantum elec- trodynamics, and diagonal Born-Oppen- heimer terms. The quality of the computed vibrational levels has been evaluated Fig. 5. Inversion poten- through comparisons with the 104 experi- tial and inversion levels mentally known vibrational band origins up (cm-1) in ammonia. The –1 to about 25000 cm . The corresponding experimental value of root-mean-square (rms) deviations decrease 4055 cm-1 is uncertain systematically with improvements in the (stated error bar of 5 PES, typical values [37] being 16.56 cm–1 cm-1). The other experi- for MRCI/CBS, 4.23 cm–1 for MRCI/CBS mental levels and inver- plus core-valence correlation and relativistic sion splittings are well corrections, and 1.90 cm–1 after including all reproduced by the cal- corrections considered. This accuracy is tru- culations. ly remarkable, especially when taking into account that the standard deviations are CBS** PES, respectively, which are of sim- ambiguous assignments: 56 bands in NH3 up much smaller in the low-energy region, by ilar quality to the target CBS+ PES. Our vi- to 15500 cm–1 have been chosen for this pur- more than a factor of 2. Individual positions brational energy calculations are based on pose as well as 50 bands in its isotopomers of rovibrational lines (i.e. differences be- the Hougen-Bunker-Johns (HBJ) formalism up to 5100 cm–1. The rms deviation between tween rovibrational levels connected by an [52] involving a nonrigid reference configu- experiment and theory for all bands in NH3 allowed transition) are typically accurate to ration that follows the large amplitude inver- is 34 cm–1 for the CCSD(T)/aug-cc-pVTZ 0.2 cm–1 due to the cancellation of systemat- sion motion of ammonia. The latter is de- surface, and ranges from 13–16 cm–1 for the ic errors [37]. scribed by a trigonometrically defined inver- more accurate surfaces such as CBS+ and Our own variational calculations have sion coordinate (corresponding to the angle CBS* [50]. Lower-energy levels are repro- focused on ammonia which exhibits a low between the C3 axis and an N–H bond in the duced more accurately, e.g. for the 28 funda- –1 inversion barrier of less than 1800 cm and C3v case), while the five small amplitude vi- mental bands of all isotopomers considered is therefore the prototype of a molecule with brations are treated as displacements from (rms deviations less than 5 cm–1 for CBS+- large amplitude inversion motion. It is obvi- this nonrigid reference configuration and ex- type surfaces). Inversion splittings are com- ous that a variational approach is needed for pressed in terms of Morse coordinates for the puted even more accurately, typically within a proper understanding of the vibration-rota- three N–H stretches and internal symmetry 2 cm–1 of experiment, which is illustrated in tion-inversion spectrum of ammonia. Theo- coordinates for the remaining two bends. Fig. 5 for the umbrella mode ν2 and its over- retical challenges associated with this en- The analytic representations of the ab initio tones. deavor include the computation of accurate PES employ fourth-order polynomials of While these results for ammonia are six-dimensional (6D) PES and DMS, the these coordinates, with up to 78 prefactors quite encouraging, they are certainly less ac- best choice for the 6D variational treatment that are determined by least-squares fits. In curate than the benchmark results for water of nuclear motion, the description of rota- the HBJ formalism, the Hamiltonian con- [37] (see above). We have therefore tried to tionally excited states, the evaluation of line tains an approximate kinetic energy operator improve them by further refining our theo- intensities, and the simulation of tunneling that is derived by a Sorensen procedure [53] retical approach [51]. The regular grid has dynamics. Given these many challenges, it is with Eckart-Sayvetz constraints and in- been extended from 14400 to 45760 points, not surprising that several groups have re- volves expansions in linearized internal co- and another 6056 points with energies up to cently been attracted to ammonia and are ordinates. For consistency, the analytic po- about 20000 cm–1 have been added outside currently studying these topics [38–51]. For tential function is also reexpanded in terms this grid to cover all low-energy regions in the sake of brevity, we shall only survey our of these linearized coordinates up to fourth 6D space. CCSD(T)/aug-cc-pVTZ and own investigations in this area [50][51]. order. The Hamiltonian matrix is then con- CBS** energies are available at all these In our published work on ammonia [50] structed in a basis set whose functions are points. The analytic PES representations we have generated 6D PES and DMS at the products of Morse oscillator functions for have been upgraded from fourth-order to CCSD(T)/aug-cc-pVTZ level using a regu- stretching, two-dimensional isotropic har- sixth-order polynomials, which reduces the lar grid of 14400 points. For a subset of 1244 monic oscillator functions for bending, and rms error of the PES fits typically from 4–6 points, more accurate energies (labelled numerical inversion functions obtained by cm–1 to 2–3 cm–1. The reexpansions of the CBS+) have been computed by extrapolat- numerically solving the one-dimensional in- kinetic and potential energy terms in the ing the CCSD(T) energies to the CBS limit version problem. Diagonalization of the Hamiltonian have been extended to eighth and adding core-valence correlation and rel- Hamiltonian matrix finally yields the vibra- order and sixth order, respectively, and con- ativistic corrections; another 389 CBS+ data tional energy levels. vergence to better than 0.1 cm–1 has been are available on a denser two-dimensional Concerning our published results on am- demonstrated for the approximate kinetic monia [50], the CBS+ inversion barrier is energy operator. Each of these improve- (2D) grid for C3v symmetric geometries. The differences between CCSD(T)/aug-cc- 1790 cm–1 which is lowered to 1766 cm–1 by ments in our computational approach leads pVTZ and CBS+ energies form a rather including an improved core-valence correla- to some improvement in the computed vi- smooth surface (EDS) which can be repre- tion and a diagonal Born-Oppenheimer cor- brational energies. Their combined effect is sented by an interpolated or fitted function to rection. These values are close to the best a general shift upwards, by about 10 cm–1, provide corrections for all those points available theoretical estimate of 1777±13 and a reduction of the rms deviations be- where only CCSD(T)/aug-cc-pVTZ data are cm–1 [54]. The computed vibrational energy tween experiment and theory to the range of available. Adding these corrections from levels can be compared to experimentally 5–10 cm–1, for CBS** and related surfaces. EDS interpolation or fitting yields CBS* or observed vibrational band centers with un- These new results [51] are more satisfactory THEORETICAL CHEMISTRY: MOLECULAR SPECTROSCOPY AND DYNAMICS 280 CHIMIA 2004, 58, No. 5 than the published ones [50], but they are [4] J. Breidung, W. Thiel, J. Gauss, J.F. Stanton, [26] J. Breidung, W. Thiel, D. Figgen, H. Stoll, J. still less accurate than the corresponding J. Chem. Phys. 1999, 110, 3687. Chem. Phys., in press. benchmark results for water [37] with rms [5] M.K. Gilles, W.C. Lineberger, K.M. 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