Chemical Physics Letters 459 (2008) 60–64

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Chemical Physics Letters

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Effects of monomer flexibility on spectra of N2–HF

Piotr Jankowski a,*, Krzysztof Szalewicz b

a Department of Quantum Chemistry, Institute of Chemistry, Nicolaus Copernicus University, Gagarina 7, PL-87-100 Torun, Polandb Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 March 2008In final form 15 May 2008Available online 21 May 2008

0009-2614/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.cplett.2008.05.047

* Corresponding author.E-mail address: teojan@chem.uni.torun.pl (P. Jank

The interaction energies of N2–HF were computed ab initio on a five-dimensional grid, including thedependence on the H–F separation. The coupled-cluster method with up to noniterative triple excitationswas employed and the interaction energies were extrapolated to the complete basis set limit. These ener-gies were then averaged over the vibrational wave functions of HF corresponding to the ground and thethird excited states and two four-dimensional potential energy surfaces were fitted to these values. Rovi-brational calculations performed using these surfaces gave dissociation energies, fundamental frequen-cies, rotational constants, and distortion constants in excellent agreement with experimental values.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

Spectroscopy of the N2–HF complex has been the subject ofextensive experimental research in the past two decades [1–6].Compared to other van der Waals complexes, the dependence ofN2–HF spectra on the vibrational excitations of one of the mono-mers, HF, has been particularly well characterized. Only one ab ini-tio study [7] developed potential energy surfaces for N2–HF (thesesurfaces will be denoted as V01). These four-dimensional surfacescorrespond to the HF molecule in the ground and third excitedvibrational states. The dependence on the monomer vibration wasintroduced in the simplest possible way, by fixing the H–F distanceat the value averaged over the particular vibration. The surfaceswere used to calculate rovibrational spectra [7] and a reasonableagreement with experiment was achieved. However, substantialdiscrepancies have remained. It is not clear whether these discrep-ancies are due to inaccuracies of ab initio interaction energies or tothe neglected monomer-flexibility effects. In the present Paper, wedevelop new potential energy surfaces for N2–HF with improve-ments upon both sources of the discrepancies. The interaction ener-gies are calculated by the coupled-cluster method with single,double, and noniterative triple excitations [CCSD(T)] [8] withinthe supermolecular approach. The CCSD(T) method includes asomewhat higher level of theory than the symmetry-adapted per-turbation theory (SAPT) [9] method at the level used to calculateV01. In practice, however, the two methods give very similar inter-action energies and for some systems SAPT energies may be closerto the exact interaction energies. This is the case in particular forHe2 (except at very small interatomic separations), the only systemfor which nearly exact interaction energies are known [10–12]. Amore important difference compared to the previous work is that

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owski).

we use larger basis sets than those of Ref. [7] and basis set extrap-olation techniques. To take into account the monomer-flexibility ef-fects due to the vibrations of HF, we employ the recently developedmethod [13] of approximately accounting for intramonomer de-grees of freedom. The interaction energies of N2–HF have been com-puted on a five-dimensional grid, with the N–N distance frozen.These energies have then been averaged over v ¼ 0 and v ¼ 3 vibra-tional states of HF and two four-dimensional surfaces have been fit-ted to these results. Moreover, two surfaces have been obtainedanalogously as in Ref. [7], but using the current level of ab initiomethods. Rovibrational calculations have been performed for allthe surfaces and the results compared to experimental data.

2. Outline of the method

2.1. Construction of the interaction energy surface

The geometry of the N2–HF system with the N2 and HF mole-cules treated as rigid can be characterized by the Jacobi coordi-nates: three angles h1, h2, and / and the parameter R denotingthe separation between the center of the mass of N2, CM(N2), andthe center of the mass of HF, CM(HF). To define the angles h1, h2,and / let us choose the z axis pointing from CM(N2) to CM(HF).The parameter / denotes the dihedral angle between two half-planes extending from the z axis to the chosen N atom of N2 (N1)and to the H atom. The angle between the z axis and the vectorpointing from CM(N2) (CM(HF)) to the N1 (H) atom is denoted h1

ðh2Þ. The set of these four coordinates will be denoted further asX ¼ fR; h1; h2;/g. To discuss the dependence of the interaction en-ergy on the intramolecular degrees of freedom, two more parame-ters, the N2 and HF bond lengths hereafter referred to as rN2 and r,respectively, have to be taken into account. The set of six parame-ters fR; h1; h2;/; rN2 ; rg defines a general geometry of the complex.

P. Jankowski, K. Szalewicz / Chemical Physics Letters 459 (2008) 60–64 61

In the present work, only the dependence of the interaction en-ergy on the H–F distance is taken into account explicitly, whereasthe N2 molecule is treated as rigid. This choice is due to two rea-sons: first, as already mentioned, spectra of N2–HF with vibration-ally excited HF monomer have been measured; second, even if bothmonomers are in their ground states, one expects the spectra to bemuch more sensitive to the monomer-flexibility effects in HF thanin N2. The former effects have been extensively investigated for Ar–HF in Ref. [14]. This study has shown that if HF is excited by a fewvibrational quanta, to get quantitative agreement with three-dimensional spectra, monomer-flexibility effects have to includedin a more sophisticated way than by calculations of potentials athriv. This is due to the relatively large amplitude and anharmonicityof the HF vibration and the strong and nonlinear dependence of theAr–HF interaction energy on the elongation of the H–F distance. Incomparison, the amplitude of the vibrational motion in N2 is rela-tively small and the motion is fairly harmonic, at least in the groundstate, the only state included in measurements. The interaction en-ergy of the N2–HF complex with N2 treated as a rigid molecule,VðX; rÞ ¼ VðR; h1; h2;/; rÞ, can be represented for a given value ofthe intermolecular coordinates X as a truncated Taylor expansionaround some reference intramolecular configuration rc withnumerically computed derivatives [13]

VðX; rÞ ¼ f0ðX; rcÞ þ f1ðX; rcÞðr � rcÞ þ12

f2ðX; rcÞðr � rcÞ2; ð1Þ

where

f0ðX; rcÞ ¼ VðX; rcÞ; ð2Þ

f1ðX; rcÞ ¼oVor

����

r¼rc

’ VðX; rc þ hÞ � VðX; rc � hÞ2h

; ð3Þ

and

f2ðX; rcÞ ¼o2Vor2

�����r¼rc

’ VðX; rc þ hÞ � 2VðX; rcÞ þ VðX; rc � hÞh2 : ð4Þ

The right-hand sides of these equations correspond to the sim-plest finite-difference approximations to the derivatives at the ref-erence point rc with a step h. In our calculations, the value of rc hasbeen set to 1.85 bohr and h to 0.025 bohr [13].

The form (1) of VðX; rÞ could in principle be used in five-dimen-sional rovibrational calculations for N2–HF, but such calculationswould be fairly difficult. However, it has been demonstrated inRef. [14] (on Ar–HF) that the accuracy of spectra is decreased onlymarginally if a properly averged, reduced-dimensionality potentialis used. Such potential is obtained simply by averaging VðX; rÞ overthe vibrational function vvðrÞ of HF, at each point X. If VðX; rÞ is gi-ven by Eq. (1), such an average reduces to

hVivðXÞ ¼ f0ðX; rcÞ þ f1ðX; rcÞhðr � rcÞiv þ12

f2ðX; rcÞhðr � rcÞ2iv: ð5Þ

To obtain the vibrationally averaged distances

hðr � rcÞaiv ¼ hvvðrÞjðr � rcÞavvðrÞi;

the empirical HF potential by Ogilvie [15] has been applied. We will usebelow a short-hand notation Vav for the averaged energies of Eq. (5)regardless of the value of v. Using Eq. (1), one can also define a surfacecorresponding to the rigid-monomer approximation applied for HF

V hriv ðXÞ ¼ f0ðX; rcÞ þ f1ðX; rcÞðhriv � rcÞ þ12

f2ðX; rcÞðhriv � rcÞ2: ð6Þ

Such surface for any value of v will hereafter be referred to asV hri.

Four interaction energy surfaces, Vav and V hri for v ¼ 0 andv ¼ 3, have been fitted to the functional form developed in Refs.[16,7] which consists of two terms

VðXÞ ¼ V shðXÞ þ VasðXÞ; ð7Þ

where VasðXÞ, representing the long-range part of the potential, wastaken from Ref. [7] and kept constant in all fits. The form of theshort-range component, V shðXÞ, was also taken from Ref. [7], butits parameters were separately optimized for each surface underconsideration.

2.2. Calculations of the interaction energy

The values of the interaction energies of the N2–HF complexhave been obtained within the supermolecular approach usingthe CCSD(T) method. The Boys and Bernardi counterpoise scheme[17,18] has been used in all calculations. The details of these com-putations are very similar to those in a recent study of the H2–COcomplex [19]. The main reason of the insufficient accuracy of theearlier V01 surface was a limited quality of the basis set used inthe calculations. Thus, a special care has been taken in selectingthe basis sets of the present study, following Ref. [19] where vari-ous such choices and the performance of different complete basisset (CBS) extrapolation schemes have been discussed. A compre-hensive description of the tests performed for the N2–HF complex,using various basis sets and levels of theory, can also be found inRef. [20].

The total interaction energy Eint has been obtained as the sumEint ¼ EHF þ Ecorr of the Hartree–Fock ðEHFÞ and correlation ðEcorrÞcomponents. The Hartree–Fock part of the interaction energy hasbeen calculated without any extrapolation procedure using theaug-cc-pVQZ basis sets [21]. The correlation part of the interactionenergy has been obtained from the two-point 1=X3 extrapolation[22] based on the aug-cc-pVTZ and aug-cc-pVQZ basis sets. Nomidbond functions have been used since the importance of suchfunctions is diminished if CBS extrapolations are used.

The electronic structure calculations have been performedusing the MOLPRO [23] package. The regular grid of ðR; h1; h2;/Þ forwhich interaction energies were computed is a combination ofthe values of R = 5.5, 6.0, 6.5, 7.0, 8.0, 10.0, 12.0 bohr and the irre-ducible set of 33 combinations of angles [7], with h1, h2, and / ta-ken in increments of 45�. For R ¼ 6:5 and 7.5 bohr, the angular gridhas been extended by 47 configurations. Finally, a few points havebeen added for short intermolecular distances, leading to 337 gridpoints for ðR; h1; h2;/Þ. Each of these points has been combinedwith three values of r, thus the total number of grid points amountsto 1011.

To fit the Vav and V hri potentials, we have used the grid pointsfor which the calculated interaction energies were less than1000 cm�1. The fitted surfaces reproduce the computed pointswith an error smaller than 2%, except in the region where the po-tential crosses zero. However, in the range of energies probed incalculations of the rovibrational states published in this Letter,the accuracy of the fits is much better than 1%. The average rootmean square error deviations of the fits for points with negativeinteraction energy were about 0.6 cm�1 and 1 cm�1 for v ¼ 0 andv ¼ 3 surfaces, respectively.

3. Results and discussion

3.1. General features of the potential surface

In Table 1, the characteristic features of the Vav and V hri poten-tial energy surfaces are shown and compared with those of the ear-lier V01 potential. The new surfaces are much deeper than their V01

counterparts. The V01 and V hri surfaces were both obtained withinthe rigid-monomer approximation, thus, the differences betweenthem demonstrate the variation in the performance of the methodsand basis sets used to calculate the interaction energies in both

Table 3Dissociation energies ðD0Þ and fundamental frequencies m3, m4, and m5 of N2–HF for thev ¼ 0 and v ¼ 3 vibrational states of HF

v D0 m3 m4 m5 RS

0 V01 352.2 79.6 230.2 63.3V hri 399.1 87.0 260.8 59.1

Table 1The position ðReÞ and the depth ðDeÞ of the global minimum for various ab initiointeraction energy surfaces of N–N–H–F in the linear configuration

Surface v ¼ 0 v ¼ 3

Re De Re De

V01 6.73 762.4 6.71 897.9V hri 6.627 832.7 6.603 1004.9Vav 6.616 843.3 6.529 1090.0

Distances are in bohr, energies in cm�1.

62 P. Jankowski, K. Szalewicz / Chemical Physics Letters 459 (2008) 60–64

cases. A comparison of Re and De for the Vav surfaces with those ob-tained from V hri indicates that the vibrationally averaged surfacediffers only slightly from the rigid-HF surface for the v ¼ 0 case,whereas for v ¼ 3 the differences become significant. This observa-tion parallels earlier findings for Ar–HF [14]; the performance ofthe rigid-monomer approximation deteriorates if the vibrationalexcitation of the monomer increases.

With CBS extrapolations applied by us, the residual errors dueto the basis set incompleteness should be of the order of 1%. Thus,the main source of error of our potential is probably the approxi-mation in the description of the electron correlation introducedby the CCSD(T) method. Although no direct information is availablefor N2–HF, such effects have been studied for the H2–CO complexby Noga et al. [24]. These authors computed H2–CO interactionenergies using the CCSDT, CCSDT(Q), and CCSDTQ methods which,respectively, take into account the complete contribution of tripleexcitations, the noniterative contribution of quadruple excitations,and the complete contribution of quadruple excitations. These con-tributions amounted to about 3% of the interaction energy at theglobal minimum and to 1.4% at the secondary minimum. Thus,the neglect of these contributions may change not only the magni-tude of interaction energies, but also the anisotropy of the poten-tial energy surface. We will assume that the errors due to theCCSD(T) theory level truncation will also be about 3% for N2–HF.This estimate is probably an upper limit for the post-CCSD(T) com-ponent of the interaction energy since the relative correlation en-ergy contribution to the total interaction energy is much smallerin the case of N2–HF than H2–CO (see Table 2). The additional argu-ment is the fact that the dipole moment of the HF molecule ismuch less sensitive to the high-order electron correlation effectsthan the dipole moment of the CO molecule [25]. Thus, the leadingelectrostatic component of the interaction energy should be less af-fected by the truncation of higher order excitations in the CCSD(T)method in the case of N2–HF than for H2–CO.

3.2. Bound states calculations

The ultimate test of the accuracy of the potential energy sur-faces obtained in this work follows from the evaluation of the spec-troscopic properties for the N2–HF complex and comparisons ofsuch properties with the experimental data.

Table 2Comparison of the Hartree–Fock ðEHFÞ and correlation ðEcorrÞ components of the totalinteraction energy ðEintÞ for the N2–HF and H2–CO complexes, obtained with the aug-cc-pVQZ basis sets

EHF Ecorr Eint

N2–HF �310.1 �559.9 �870.0H2–CO 39.3 �127.6 �88.3

The complexes are in the linear configurations N–N–H–F and H–H–C–O withintermolecular separation R equal to 6.5 bohr and 7.8 bohr, respectively. The fol-lowing molecular bond lengths were used in the calculations: 2.081458 bohr (N2),1.85 bohr (HF), 1.448736 bohr (H2), and 2.13992 bohr (CO). Energies are in cm�1.

The rovibrational calculations reported in this Letter have beenperformed with the BOUND program of Hutson [26]. The values ofthe rotational constants of the monomers and the reduced massof the dimer were the same as those given in Ref. [7]. The angularbasis set used in calculations has been significantly extended incomparison with that used in Ref. [7] and is defined by jmax

HF ¼ 10and jmax

N2¼ 16. However, even for such large basis set, inaccuracies

of 1–2 cm�1 can be expected for some of the calculated rovibra-tional energies, especially those corresponding to the states involv-ing higher excitations in N2 bending.

We followed the literature and used the linear semirigid mole-cule model [27,3] to label rovibrational states of the N2–HF com-plex. According to this model, the quantum numbers areðv3vl1

4 vl25 Þ where vi, i ¼ 3;4;5, are the vibrational quantum numbers

for intermolecular oscillations: the van der Waals stretch ði ¼ 3Þand the HF and N2 bends (i ¼ 4 and i ¼ 5, respectively). The sym-bols l1 and l2 indicate the vibrational angular momenta of thedegenerate modes. The transition frequencies from the groundstate of the complex to the rovibrational states vi will be denotsas mi.

The dissociation energies and the frequencies mi are listed inTable 3. For v ¼ 0, the results obtained from both the V hri and Vav

potentials are very close to the experimental values, with the dis-crepancies below 1% and 2%, respectively. The unexpectedly betteragreement in the case of the former potential is likely accidentaland in the case of the transition frequencies it is partly related touncertainties of experimental data, see below. The value ofD0 ¼ 399:1 cm�1 predicted by V hri is well within the error bars ofthe experimental result of 398� 2 cm�1 [5], whereas Vav predicted405.5 cm�1, 2% off experiment. The reason that the experimentalvalues of the fundamental frequencies have relatively large uncer-tainties is that none of them was actually measured in molecularbeams. These frequencies were either obtained by scaling the val-ues measured for v ¼ 3 with factors derived from the Ar–HF com-plex, or estimated from the diatomic approximation (m3 ¼ 91cm�1) [2], or measured in matrices (m4 ¼ 262 cm�1) [28]. The dif-ferences between the experimental frequencies obtained by differ-ent methods are relatively large and therefore these data cannot beused to judge which of the surfaces, V hri or Vav, is better. In fact, thetheoretical frequencies may be the best estimates of the truevalues.

For v ¼ 3, the experimental data are much more precise thanthose for v ¼ 0. Table 3 shows that the values of D0 and mi calcu-lated from Vav are in an excellent agreement with experiment.

Vav 405.5 88.4 266.1 58.6Exp. 398a 91b, 82c 259d, 262e 59.2d

3 V01 438.3 86.2 256.1 74.4 86.1V hri 516.9 98.5 300.0 64.7 117.8Vav 553.5 100.7 325.0 68.5 148.0Exp. 551f 98.6g 328.6g 68.5g 152.6g

The red shifts (RS) of the HF v ¼ 3 0 transition, defined asRS ¼ D0ðv ¼ 3Þ � D0ðv ¼ 0Þ, are also given. All values are in cm�1.

a Ref. [5].b Estimated, from Ref. [2].c Scaled, from Ref. [7],d Scaled, from Ref. [6].e Ref. [28].f Ref. [6].g Ref. [3].

Table 6Values of the l-doubling parameter ql (in 10�4 cm�1) for selected rovibrational levelsof N2–HF and intramonomer v ¼ 0 and v ¼ 3 states

v ðv3vl14 vl2

5 Þ Vav Exp.

0 (00011) 4.47 4.61(8)a

(01100) 1.33

3 (00011) 4.01 3.93b

(01100) 0.88 9.33b

a Ref. [2].b Ref. [6].

P. Jankowski, K. Szalewicz / Chemical Physics Letters 459 (2008) 60–64 63

The value of D0 ¼ 553:5 cm�1 deviates by only 0.5% from theexperimental value of 551 cm�1. The largest discrepancy in transi-tion frequencies is 2% (for m3). The V hri surface performs, as ex-pected, significantly worse: D0 is equal to 516.9 cm�1, whichamounts to 6% error. Similar discrepancies are found for transitionfrequencies, up to 9% (for m5). One exception is the value of m3

which almost perfectly matches the experimental result, althoughin view of much lower accuracies of the other frequencies, thisagreement should be regarded as accidental.

One more quantity that can be compared to experiment is thered shift of the HF v ¼ 3 0 transition, defined as RS ¼ D0

ðv ¼ 3Þ � D0ðv ¼ 0Þ. The value calculated from Vav, 148.0 cm�1, isvery close to the measured one of 152.6 cm�1. It is also nearlythe same as an earlier theoretical estimation of this quantity[29], amounting to 148.9 cm�1, obtained from a one-dimensionalinteraction potential. The Vav potential reproduces the experimen-tal red shift much better than the other potentials. This is likelydue to the fact that the five-dimensional potential from whichVav has been obtained includes terms proportional to the secondderivative of the interaction energy with respect to the HF intra-monomer coordinate. It has been argued in Ref. [3] that to explainthe quadratic increase of the magnitude of the redshift treated as afunction of the vibrational quantum number v, one has to includesuch terms. Apparently, the subsequent averaging of the five-dimensional potential, leading to Vav, preserves this property.

The calculated and measured rotational constants B and distor-tion constants D for the N2–HF complex are collected in Tables 4and 5, respectively. Discussing these results one should keep inmind that there are some methodological differences between the-ory and experiment in deriving the values of B and D, which maycontribute to discrepancies, especially for the D constant. Theexperimental B and D were obtained from fitting of the standardpolynomial expansion in JðJ þ 1Þ to all the measured transitionenergies. The calculated values of B (D) were obtained by us from

Table 4Rotational constants (in cm�1) for selected rovibrational levels of N2–HF andintramonomer v ¼ 0 and v ¼ 3 states

v ðv3vl14 vl2

5 Þ V01 V hri Vav Exp.

0 (00000) 0.10247 0.10629 0.10652 0.10659a

(00011) 0.10267 0.10686 0.10707 0.10733b

(10000) 0.09825 0.10226 0.10248(01100) 0.09875 0.10203 0.10221

3 (00000) 0.10268 0.10700 0.10854 0.10921b

(00011) 0.10259 0.10747 0.10892 0.10992b

(10000) 0.09893 0.10333 0.10491 0.10584b

(01100) 0.09915 0.10340 0.10405 0.10345b

a Ref. [1].b Ref. [3].

Table 5Centrifugal distortion constants (in 10�7 cm�1) for selected rovibrational levels of N2–HF and intramonomer v ¼ 0 and v ¼ 3 states

v ðv3vl14 vl2

5 Þ V01 V hri Vav Exp.

0 (00000) 5.95 5.54 5.42 5.78(5)a

(00011) 5.92 6.6b

(10000) 7.29 7.63 7.38(01100) 7.84

3 (00000) 5.08 4.38 4.38 3.98(44)b

(00011) 4.75 5.88(40)b

(10000) 5.75 5.75 5.42(01100) 6.57

a Ref. [4].b Ref. [3].

the difference of rovibrational energies for only two (three) consec-utive, smallest values of J. We have considered four intermolecularrovibrational states, for the HF monomer in the v ¼ 0 and v ¼ 3vibrational states. As can be seen from Table 4 the values of B cal-culated from Vav are again in a very good agreement with the ob-served constants. For v ¼ 3, the largest discrepancy is less than 1%,whereas for v ¼ 0 the agreement is almost perfect and the discrep-ancies are smaller than 0.2%. For the v ¼ 0 mode, for which the fun-damental frequencies computed from V hri were closer to theexperimental ones than those from Vav, the rotational constantsare less accurate in the former cases, as expected.

In Table 5 the distortion constants calculated for selected vibra-tional states are given and compared to the available experimentaldata. The values calculated from both Vav and V hri are in goodagreement with the experimental ones. This agreement has beenachieved despite the earlier-mentioned methodological differencesin deriving the values of D.

Finally, in Table 6 the values of the l-doubling parameter ql cal-culated for the (00011) and (01100) rovibrational states and the int-ramonomer v ¼ 0 and v ¼ 3 states are presented. The ql parameteris obtained as the difference Bf � Be, where Be and Bf are the rota-tional constants calculated for the e and f symmetry components ofthe P state, respectively [2]. The agreement between the theoret-ical and experimental values of ql is very good for the (00011) state(N2 bending) for both v ¼ 0 and v ¼ 3 cases. For the (01100) state(HF bending) in the v ¼ 3 case a significant difference, amountingto one order of magnitude, between calculated and observed val-ues of ql has been found. Most likely this discrepancy is causedby perturbations of the (01100) state that have been observed inthe experimental study [30]. Thus, to obtain a meaningful compar-ison of the ab initio and experimental values of the l-doublingparameter for the (01100) state, a deperturbation procedure shouldbe applied in both the theoretical and experimental studies.

4. Conclusions

New ab initio four-dimensional potential energy surfaces for theN2–HF complex, with the HF molecule in the ground and third ex-cited vibrational states, have been developed. The surfaces havebeen constructed according to the method of Ref. [13]. The interac-tion energies have been computed on a five-dimensional gridincluding the dependence on the H–F separation. The more accu-rate Vav surfaces have been obtained by averaging over the intramo-lecular vibrations of HF in v ¼ 0 and v ¼ 3 states. In addition,different four-dimensional surfaces, V hri, have been constructedby assuming a rigid-HF monomer with the H–F separation corre-sponding to the average H–F distance in the v ¼ 0 and v ¼ 3 states.For both sets of surfaces, the rovibrational calculations for the N2–HF complex have been performed. The dissociation energies, funda-mental frequencies, rotational constants, and distortion constantscalculated from the Vav surfaces are in an excellent agreement withthe experimental data. For v ¼ 0, the theoretical transition frequen-cies may be more accurate than the experimental data. The surfaces

64 P. Jankowski, K. Szalewicz / Chemical Physics Letters 459 (2008) 60–64

calculated within the rigid-monomer approximation perform verywell for v ¼ 0 and much worse for v ¼ 3. This shows that to makean ab initio surface reliable for higher vibrational excitations ofmonomers, one needs to go beyond the rigid-monomer approxima-tions. Our results demonstrate that the simple method of construc-tion the full-dimensional and vibrationally averaged surfacesproposed in Ref. [13], combined with the high-accuracy ab initiomethod and an appropriate choice of basis sets, gives the four-dimensional surfaces which are able to recover most of the nonrig-idity effects. Since typical discrepancies of the predictions from theVav surfaces with experiment are of the order of 1%, this level ofagreement shows, first, that our intrinsic estimate of the errors ofthe ab initio interaction energies to be about 3–4% is reasonable(although spectra are more sensitive to relative rather than abso-lute errors in potential energy surfaces). Second, it shows that theapproximation of averaging over the H–F distance introduces onlysmall uncertainties of the order of 1%.

Acknowledgments

We are grateful to Professor William Klemperer for his com-ments on the manuscript. This work was supported by the NSFGrant CHE-0555979.

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