Coupled pair functional study on the hydrogen fluoride dimer. I. Energy surface and characterization of stationary points

Chemical Physics 121 (1988) North Holland, Amsterdam

137-153

COUPLED PAIR FUNCTIONAL STUDY ON THE HYDROGEN FLUORIDE DIMER. I. ENERGY SURFACE AND CHARACTERIZATION OF STATIONARY POINTS Manfred

KOFRANEK,

Hans LISCHKA

and Alfred KARPFEN

Institut ftir Theoretische Chemie und Strahlenchemie der Universitiit Wien, Wdhringerstrasse 17, A-1090 Vienna, Austria Received

I2 October

1987

The energy surface of the hydrogen fluoride dimer has been investigated within the framework of the coupled pair functional (CPF) approach applying an extended, polarized basis set. More than 1000 individual points on the 6D energy surface have been evaluated at this level. The 4D intermolecular part of the energy surface is presented with the aid of contour plots for various selected 2D cuts through the surface. The C, energy minimum, the cyclic CZh saddle point and the linear C,, configuration were optimized separately and are characterized by harmonic vibrational analysis including the calculation of infrared intensities.

contributed significantly to a more detailed understanding of the spectroscopic properties of the (HF) z complex. Pine, Lafferty and Howard [ 5-71 recorded rotationally resolved infrared spectra in the HF stretching region and thus determined accurately the shifts of HF-stretching frequencies occurring upon complex formation. From these data Pine and Howard [ 71 extracted an improved value for the dissociation energy of ( HF)2. The region of low-lying intermolecular vibrations could not yet be studied in the gas phase. The only source of experimental information on this important frequency domain thus stems from matrix spectroscopy [ 8-111. Dynamic phenomena, such as vibrational relaxation, vibrational predissociation and the lifetime of intramolecularly excited (HF), complexes were addressed to in a large number of experimental investigations [5,6,12-l 91. Similar to the great efforts undertaken from the experimental side to determine the static and dynamic properties of ( HF)2 this complex has attracted many theoreticians as well. An impressive number of quantum chemical studies has been performed on (HF)*. The majority of these investigations was carried out at the SCF level and aimed at an elucidation of the equilibrium structure of the ( HF)z complex [ 20-331. More sophisticated treatments including electron correlation have, however, also been reported [ 23,34-441. Harmonic vibrational analysis at the re-

1. Introduction The hydrogen fluoride dimer, ( HF) 2, has been one of the first weakly bound intermolecular complexes studied by molecular beam electric resonan$e spectroscopy (MBERS) [ 11. In their pioneering work, Dyke, Howard and Klemperer determined for the first time a number of structural parameters of the equilibrium geometry of ( HF)2, ( DF)z and HFDF in the gas phase. Moreover they revealed the existence of a large-amplitude hydrogen tunnelling motion between energetically equivalent minima of ( HF)z and ( DF)2. In the course of this motion “H-bonded” and “free” H-atoms exchange their roles. The corresponding tunnelling-induced splitting of rotational levels was observed for (HF) z and (DF) z but not for HFDF. The alternative unsymmetric complex DFHF could not be detected. Quite recently, the microwave spectra of (HF)* and of the various other deuterated complexes have been reinvestigated by several groups [ 2-41. In addition to the K=O rotational level, Howard, Dyke and Klemperer [ 21 reported also microwave and radiofrequency transitions for the K= 1 rotational level. Gutowsky et al. [ 31 detected also the DFHF species. Finally Lafferty, Suenram and Lovas [4] reported K= 0, 1, 2 transitions for all four isotopic species. Apart from these microwave investigations, further vapor phase studies have been performed which 030 1-O104/88/$ 03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

138

A4. Kofranek et al. /Coupled pair functional study on (HF) z

spective equilibrium configurations has been performed, both at the SCF [25,30] and at correlated level [ 34,36-381. In most of these studies the cyclic transition state, relevant for the dynamics of (HF)*, has been dealt with too. The first attempt to scan a considerable portion of the intermolecular energy surface of (HF) z has been undertaken by Yarkony et al. [24]. These authors computed 294 points on the four-dimensional intermolecular energy surface. HF monomers were kept frozen at their equilibrium distance and all calculations were done at the SCF level applying a basis set of DZ+P quality. Similar investigations using several smaller basis sets were later carried out by Jorgensen [ 28,291. Quite recently, Michael et al. [ 361 reported a methodically more advanced study of the energy surface of ( HF)2. Applying a TZ+ P basis set, more than 200 geometries were computed at the SCF level. For 73 points electron correlation has been taken into account within the framework of the ACCD approach. The region of the energy surface considered was, however, much more restricted than in ref. [24]. In the course of our study a paper by Redmon and Binkley appeared who carried out a thorough and extensive investigation of the energy surface of ( HF)2 at the MP4 level applying a 63 11 G** basis set. A large number of points, similar in magnitude to ours, obtained at a correlated level has been presented in their work. They scanned the energy surface in a somewhat wider range in terms of the intermolecular distance and therefore used also a less dense grid for this particular internal coordinate than we do. Redmon and Binkley thus emphasized the global behavior of the interaction between two rigid HF molecules and did not attempt to describe the detailed equilibrium and saddle point geometries or the vibrational spectroscopic properties of (HF)*. To a certain extent the information that can be gained from the Redmon-Binkley investigation and from our study is therefore complementary. Particularly the energy surface of Yarkony et al. [ 241 and the subsequent lit of this surface by Alexander and DePristo [ 451 has been the basis for many dynamical studies [ 46-5 11. In a notable effort Barton and Howard [ 521 used the spectroscopic data derived in ref. [ 1 ] together with the tit of ref. [ 451 to desigh an improved analytic representation of the energy surface of (HF) *. A further theoretical energy

surface has been constructed by Brobjer and Murrell [ 531 based on the results of SCF energy partitioning. The latter two surfaces and that of Jorgensen were also used in a few dynamical calculations [ 54-571. The Redmon-Binkley surface was also already used for close-coupling and quasiclassical trajectory calculations [ 5 8 1. In view of this considerable interest in the (HF)2 system exemplified by the increasing number of highresolution spectroscopic investigations and improved theoretical studies which appeared in the last years it seemed timely to us to undertake a major computational effort and to reinvestigate the energy surface of (HF) 2. In this work we aim at a more accurate and more complete quantum chemical description of the energy surface of ( HF)2 than has been hitherto achieved. Applying large basis sets and using the coupled pair functional (CPF) technique [ 591 to take account of electron correlation contributions a systematic scan of the intermolecular energy surface has been carried out. Additionally, sufficient points have been computed to locate the global C, energy minimum, the cyclic C2,, saddle point and the linear C,, configuration and to characterize these stationary points within the framework of the harmonic approximation. In case of the C, minimum infrared intensities were evaluated too.

2. Method of calculation 2.1. Quantum chemical methodology The weak intermolecular forces governing the interaction of polar subunits in hydrogen bonded systems require a careful choice of the computational approach from the pool of the quantum chemical methods available. Experience over the years with applications of the conventional SCF (HartreeFock) approximation to hydrogen bonded complexes indicates the need for sufficiently large atomic basis sets in order to both minimize the well known superposition error in the computations on the complexes and to obtain at the same time reasonable monomer charge distributions (multipole moments) and polarizabilities which determine the asymptotic longrange behavior between interacting, polar molecules. “Sufficiently large” in this context means s and p ba-

M. Kofranek et al. / Coupledpairfunctional

sis sets of triple zeta (TZ) quality for second- and third-row atoms and the addition of at least one, preferably two, d sets as polarization functions. Similarly, large s basis sets and at least one polarizing p set has turned out to be necessary for hydrogen atoms. SCF computations performed with basis sets as just described, in general, lead to qualitatively satisfactory results for equilibrium geometries, hydrogen bond energies, shifts of vibrational frequencies induced upon complex formation and for other ground state properties. In case more quantitative results are desired electron correlation must be taken into account explicitly in order to correct for the neglect of dispersion energy within the SCF approach and simultaneously for remaining errors originating from inaccurate SCF electrostatic moments and polarizabilities of the constituent monomers. For our computations on (HF)* we chose an 11 s7p/6s Huzinaga basis set [ 60,6 1 ] and augmented it with diffuse s and p functions with exponents 0.08 and 0.06 and with 2 d sets (exponents 1.6 and 0.15) on fluorine and one p set (0.75) on hydrogen. The resulting primitive 12s8p2d/6slp basis was contracted to [ 8, 6, 2/4, 1 ] with the contraction_scheme [5, 1, 1, 1, 1, 1, 1, 1;3, 1, 1, 1, 1, 1; 1, l/3, 1, 1, 1; 11. This basis set is, as far as the s and p parts are concerned, significantly more flexible than the usual triple-zeta basis sets and is therefore in line with the abovementioned principles. Moreover is is also considerably larger than the basis sets applied in the previous potential energy surface computations on ( HF)z [ 24,28,29,36,44]. Application of this extended basis set (ext. +2P) leads to a close to Hartree-Fock limit description of expectation values relevant for our study. Electron correlation effects were incorporated within the framework of the CPF formalism [ 591 which consists of an explicit treatment of all single and double substitutions with respect to the HartreeFock determinant and takes also account of higherorder excitations in a way closely related to CEPA- 1 [ 621 in order to assure size extensivity. In particular the size extensivity is of great importance for a correct description of intermolecular interactions. All calculations on HF and (HF)* reported in this study were performed with the COLUMBUS program system [ 63-661. For all points on the energy surface we obtained not only the total energy but also a variety

study on (HF),

Fig. I. Definition of the coordinate ergy surface of ( HF)2.

139

system used to scan the en-

of one-electron expectation values such as electrostatic moments and electric field gradients at the nuclei at the CPF level. As will be demonstrated below this CPF energy surface as obtained with the ext. + 2P basis set is capable to describe accurately many structural and spectroscopic features of the (HF), complex. 2.2. Scanning the energy surface The coordinate system chosen for the (HF) z complex is shown in fig. 1. Denoting the two intramolecular HF stretching coordinates as r, and r2 we used the following intermolecular coordinates: R, the distance between the two fluorine atoms, and (Yand b, the angles between R and r,, and r2, respectively. R is oriented along the z-axis, planar arrangements are supposed to be located in the xz-plane and non-planar geometries are additionally characterized by the rotational angle y of r, around an axis through F2 parallel to the x-direction. For planar geometries y = 0’. The symmetry properties of the intermolecular ( HF)2 energy surface for fixed R, r, and rz have already been discussed in detail by Yarkony et al. [ 241. Using a grid of 45 o for (Y,p and y (I!),, e2 and @ in their notation) they showed that instead of the 512 ( 83) points only 49 symmetry unique points must be computed. Since we used a grid of 30” the total number of 123 = 1728 geometries is reduced to 170 symmetry unique points. 43 configurations of these correspond to planar arrangements. The grid we chose is the same as in ref. [ 441. In our systematic scan of the intermolecular energy surface (r, and r2 frozen at the monomer equilibrium

140

M. Kofranek et al. /Coupled pair functional study on (HF) 2

distance) we computed in a first step five series of 2D planar cuts for different R values centered around the optimized intermolecular distance at the (HF) z energy minimum. Since it is our primary aim in this work to provide an energy surface useful for spectroscopic and dynamic properties of (HF) Zin the vapor phase the range of R values considered was rather narrow extending from about 4.67 to 5.87 bohr. For smaller R only a fraction of the points will be still attractive or moderately repulsive and there is no need for a systematic scan. For larger R values it is very probable that a significant amount of computer time can be saved without significant loss of accuracy by taking advantage of more approximate methods based on perturbation theoretical treatments of intermolecular interaction in combination with a limited number of carefully selected CPF calculations. In the case of non-planar configurations. three series of 3D cuts (127 points each) at R values of about 4.67, 5.21 and 5.87 bohr have been computed. Apart from these systematic scans of the rigid monomer intermolecular energy surface we additionally located the C, global energy minimum, the cyclic CZh saddle point and the linear C,, configuration. The latter may be viewed as a second-order saddle point. Sufficient points have been computed in each case to find the optimal geometries of these stationary points and to perform standard harmonic vibrational analysis. Dipole moment derivatives with respect to internal coordinates were obtained numerically. A presentation of our potential energy and oneelectron property surfaces for the large number of individual points treated in this study would require undue space. Files containing complete lists of all computed energies and of the various one-electron expectation values such as dipole, quadrupole and octupole moments and fields gradients at the nuclei with geometries specified either by the internal coordinate system shown in fig. 1 or simply by the Cartesian coordinates of the four atoms involved may, however, be obtained upon request from the authors.

3. Results and discussion 3.1. HF monomer An accurate description of the ground state potential energy and property surfaces of the monomer HF in the vicinity of the energy minimum is a necessary prerequisite for a correct evalution of the intermolecular interaction in (HF) *. Spectroscopic constants of the isolated HF molecule as obtained within the framework of the CPF method are compiled in table 1 and confronted with previous accurate theoretical results [ 67,681 and with available experimental data. In general, we observe excellent agreement between our results and those of the CEPA investigations of Werner and Meyer [ 681 and Werner and Rosmus [ 671. Equilibrium bond length, rotational constant and harmonic vibrational frequency are also in close agreement with the corresponding experimental data. Our computed HF bond length is only about 0.003 8, larger than the experimental value. The harmonic vibrational frequency differs by only 3 cm- ’ from the corresponding experimental quantity. Electrostatic properties such as dipole moment, polarizabilities and dipole moment derivatives are also satisfactorily described. The computed dipole moment deviates by less than 0.03 D from the experimental value. In case of polarizabilities the discrepancy between theoretical and experimental values is somewhat larger. Polarizabilities evaluated via dipole moment derivatives are in better agreement with experimental data than those obtained via energy derivatives. 3.2. (HF)> dimer

3.2.1. Structure, energetics and dipole moment In this section we characterize in detail the stationary points on the energy surface of the hydrogen fluoride dimer. Of particular interest are the global energy minimum of the complex possessing C, symmetry and the cyclic transition state with a C2,, configuration. We also discuss the fully linear C,, structure (see fig. 2). Although the latter is a secondorder saddle point (two degenerate imaginary frequencies) only it constitutes an energetically low-lying stationary point and is therefore of relevance for the dynamics of the ( HF)2 complex. Table 2 provides a comparison between computed

hf. Kojiianek et al. /Coupled pair functional study on (HF) z Table 1 Ground state spectroscopic

constants

of the HF monomer

method reference

CPF this work

CEPA

E(h) r,(A) B, (cm-‘) w, (cm-‘)

- 100.30386 0.9194 20.84(10.95) a) 4135(2998) 1.7728 1.43 90.27(47.45) 6.29 “, 6.43 d’ 5.01 “, 5.24 ‘)

- 100.34744 0.919 20.8 4131 1.786 I .46

p(D)

aplar (D/A) I ‘) (km/mol) (Y, (bohr3) (Y, (bohr3)

141

experiment

[67,681

6.44 5.17

0.9168 [69] 20.9(11.01) [69] 4138(2998)[69] 1.797, 1.803 [70-721 _ 6.59 [ 731 5.10 [73,74]

a) Values in parentheses refer to DF. b, Absolute infrared intensity. ‘) Obtained via energy derivative. d, Obtained via dipole moment derivative.

FyH--------F

a

\H b

F-

H__------F-

Fig. 2. C, (a), CZh (b) and C,,

H (c) stationary

C

points of ( HF)2.

structural parameters at the C, energy minimum of (HF)* with previous theoretical and experimental data. Intramolecular geometry relaxations taking place upon complex formation could up to now not be extracted from experimental data. In the course of analyzing experimental microwave spectra [ l-41 unperturbed monomer bond lengths have invariably been assumed. We obtained bond length elongations of 0.0042 and 0.0026 A for hydrogen bonded and free HF bonds, respectively. Very similar elongations have been observed previously at the SCF level [ 301 applying a comparable basis set and by the ACCD study of Michael et al. [ 361.

The three intermolecular coordinates R, a and j? have been determined experimentally [ 1,2]. One should, however, keep in mind that these are vibrationally averaged quantities. Moreover, transformation of these data with the aim to establish the equilibrium structure depends on a number of model assumptions [ 1,2]. For R,a and p we obtain 2.79 19 A, 6.8” and - 65.6”, respectively. These values are well inside the range as obtained by microwave spectroscopy. In previous geometry optimizations of (HF) z using correlated wavefunctions significantly smaller intermolecular distances were obtained, Gaw et al. [ 341 obtained 2.724 8, for R applying a DZ+ P basis set, whereas Michael et al. [ 361 reported 2.7675 8, using a TZ + P basis set. Our value of 2.79 19 A must be attributed to the use of a doubly polarized basis set. Basis set extension from DZ to TZ and the addition of a second d set usually leads to an elongation of intermolecular distances because of a further reduction of the basis set superposition error and a smaller monomer dipole moment. These trends are well known from other studies on hydrogen bonded dimers and are already visible at the SCF level [ 301. Compared to the SCF result [ 301 the incorporation of electron correlation contributions leads to a shortening of the intermolecular distance by about 0.04 A; and to a decrease of j? to more negative values by about 10 O. The latter effect is very probably a consequence of the smaller dipole moment of HF once electron correlation is taken into account. The bent equilibrium structure of (HF)* is the result of a subtle interplay between dipole-dipole, dipole-

M. Kofranek

142

Table 2 Geometry

of (HF),

method basis set reference

at the C, equilibrium

structure.

CI

0.9236 0.0042 0.9220 0.0026 2.7919 6.81 -65.55

0.9255 0.0037 0.9242 0.0024 2.7615 6.42 - 59.92

0.935 0.002 0.934 0.001 2.724 9.3 - 74.9

of (HF)2.

HFDF,

(DF):

pairfunctional study on (HF) z

in A. bond angles in degree a)

MP2 Tz+P 1381

ACCD TZ+P 1361

DZ+P 1341

0.923 0.006 _ 2.759 5.5 -68.0

SCF

TZ+2P 1301 0.904 0.004 0.902 0.002 2.83 6.0 -56.8

SCF DZ

experiment

~251

[ 1921

0.927 0.005 0.925 0.003 2.687 8.1 -55.9

2.79 &0.05 lo*6 -63f6

see fig. 1. HF monomers.

and ((DF)?

complexes.

Experiment (B+C)/2

CPF

(HF)z HFDF DFHF

Bond distances

CPF ext. + 2P this work

‘) For definition of geometrical parameters h’ Elongations with respect to unperturbed

Table 3 Rotational constants All values in MHz

et al. /Coupled

B

c

(B+C)/2

6528.9 6507.1 6280.5 6256.6

6472.1 6449.5 6183.4 6159.0

6500.5 6418.3 6232.0 6207.8

6495.0 6500.4 6246.4 6252.3

[ 21 [ 21 [4] [2]

quadrupole and quadrupole-quadrupole interactions between the two subunits. The pure dipole-dipole term would favor a strictly linear arrangement, whereas an approximate 90” angle for j3 would be optimal for pure quadrupoles. The equilibrium structure of ( HCl)2 with a corresponding /? of about 98 o is a good example for this behavior [ 751. Table 3 confronts experimentally determined rotational constants (B + C)/2 as obtained for the four isotopomers ( HF)2, HFDF, DFHF and ( DF)2 with our computed values. Agreement is seemingly best for ( HF)z with a deviation of 5 MHz only and worst for (DF) z where our value is about 45 MHz lower than the corresponding experimental number. In case of the isotopically mixed dimers the deviations amount to 22 and 14 MHz for HFDF and DFHF, respectively. Again averaging over zero-point vibrational motions is included in the experimental numbers but not in ours which correspond to the static equilibrium structure. Energetic quantities and dipole moments of ( HF)z

are collected in table 4. We obtain -4.32 kcal/mole for the stabilization energy of (HF),. Comparison with previous computations again reveals the well known trends due to basis set improvements. The CI (DZ + P) calculations of Gaw et al. [ 341 gave a value of - 5.72 kcal/mole which is significantly too large. Similarly the - 5.0 kcal/mole obtained by Frisch et al. [ 381 represent an overestimation of the intermolecular interaction energy. Our value is rather close to the one reported by Michael et al. [36] who arrived at -4.55 kcahmole. compared to the SCF reelectron correlation sult [30] enhances the stabilization energy of (HF) z by about - 0.5 kcal /mole. Our computed stabilization energy AE is in close agreement with the experimental estimate reported recently by Pine and Howard [ 71. This point will be discussed in greater detail in connection with our vibrational analysis results. The actually measured quantity corresponds to our AH0 which contains the contributions due to zero point energy motions. For the total dipole moment of the complex we obtainp(tota1) = 3.283 D with the componentspL,=2.94 and pi-= - 1.46 D (see fig. 1 for the definition of the coordinate system). Assuming unperturbed monomer dipole moments simple vectorial addition leads to p( total) = 2.988 D with pL,= 2.676 and pL,= - 1.330 D. Complex formation thus leads to a total induced moment of 0.607 D and p=( ind) =0.264 and pFc-;( ind) = -0.13 D. From the analysis of Stark effect splittings Dyke et al. [ 1,2] reported ,u== 2.989 D in close agreement with our value of 2.97 D. The perpendicular component ,u~ was estimated to be

M. Kofranek Table 4 Stabilization

energy and dipole moment

et al. /Coupled pair functional

of ( HF)2. Energies in kcal/mol,

in D experiment -

CPF ext. + 2P this work

ACCD TZ+P

CI DZ+P

MP2 TZ+ZP

SCF TZ+ZP

SCF DZ

t361

I341

I381

[301

[251

-

AE AH0 p

-4.32 -2.62 3.283

-4.55 3.707

-5.12 -3.86 -

- 5.0 -3.1 . -

-3.8 -2.2 _

-8.0 -5.7

-4.56 -2.97

& PL

2.91 -1.40

and stabilization

-

_ _

energy of the CZh transition

state of (HF),

about 0.1 8, if compared to the C, structure. The energies of C,, and CZh structures are almost identical with a difference of -0.05 kcal/mol in favor of the linear C,, configuration. Both saddle points are therefore of importance for the dynamics of ( HF)2. The dipole moment enhancement is with 0.039 D smaller than in the C, structure. 3.2.2. Vibrational spectra and infrared intensities Vibrational analysis of intermolecular complexes is complicated by the fact that the intermolecular potential is generally flat in the vicinity of the equilibrium structure. Large-amplitude motions are therefore expected to occur and consequently anharmanic effects will give rise to significant shifts of the harmonic frequencies of intermolecular vibrations. The peculiar C, equilibrium structure of ( HF)2 with a slightly non-linear hydrogen bond (small value of (Y) and the existence of a second, energetically equivalent minimum in which “free” and “H-bonded” HF

a’

CPF ext. + 2P this work

ACCD TZ+P

CI DZ+P

SCF TZ+ZP

SCF DZ

I361

I341

[301

I251

rl (A) Ar, b’ (A) R (A) AR <’ (A) LY(deg) AE (kcal/mol) A.!& (kcal/mol

0.9223 0.0029 2.796 0.004 54.23 -3.30 - 1.02

0.9248 0.0030 2.728 - 0.040 54.42 -3.55 - 1.00

2.721 - 0.003 53.4 -4.92 -0.8

2.80 -0.03 57.0 -2.7 - I.1

0.926 0.004 2.551 -0.136 55.2 -6.9 - 1.1

)

[7] [ 71

2.989 [ 1,2] - 1.2kO.O [ 1,2]

method basis set reference

‘) For definition of geometrical parameters see fig. I. b, Elongation with respect to unperturbed HF monomer ‘) R(L)-R(G).

dipole moments

method basis set reference

- 1.2 ? 0.1 D by these authors whereas we arrive at - 1.40 D. As before vibrational averaging effects make a direct comparison somewhat difficult. Energy and optimized structure of the cyclic transition state are given in table 5. In agreement with all previous theoretical studies the C,, configuration is destabilized with respect ot the C, minimum by about 1 kcal/mole. The modification of the optimal intermolecular distance going from C, to CZh structure seems, however, to be quite sensitive to the quality of the calculations. We find negligible relaxation of R along this tunnelling path. The elongation of HF bond lengths in the cyclic transition state lies midway between hydrogen bonded and free HF bonds in the C, structure. Structure, stabilization energy and dipole moment of the linear C,, configuration of (HF)* are shown in table 6. Intramolecular geometry relaxations are weaker than in the C, minimum and the CZh saddle point. The intermolecular distance is widened by

Table 5 Structure

143

study on (HF) I

M. Kofianek et al. /Coupled

144

Table 6 Structure, stabilization energy and dipole moment C,, configuration of (HF), Bond lengths (A)

either be described as the torsion r( HIF1FZH2) or as an out-of-plane bending q of the nearly linar FI H IFZ entity. While within the framework of the harmonic approximation this choice has no influence on the computed harmonic frequency inspection of the energy surface for torsional motion reveals that the outof-plane degree of freedom should rather be interpreted as an out-of-plane bending since the barrier for torsional motion is too low. Thus at the equilibrium structure rotation of the non-bonded H2 along the torsional coordinate is almost free and should be treated as a hindered rotation. Indeed for cy= 0 this torsional motion degenerates into a true rotation of the complex. From table 7 we infer that all HF stretching force constants at the three stationary points are reduced compared to the HF monomer value of 9.643 mdynlA. The variations of these force constants directly mimic the trends in HF bond lengths discussed in the previous section. The intermolecular force constant fRR of 0.14 1 mdyn/A may be compared with the experimental value of Dyke et al. [ 1,2] who reported a value of 0.136 ? 0.004 mdynl.& Computed harmonic vibrational frequencies of (HF) 2 and of deuterated forms as obtained at the C,

of the linear

Energies and dipole moment

zrz a) R AR b’

0.9214 0.0020 0.9204 0.010 2.8907 0.988

a) Elongation b, Elongation

with respect to unperturbed HF monomer. with respect to C, minimum.

zr, a’

pair functional study on (HF) z

AE (kcallmol) AIf,, (kcahmol) AEbarr (kcal/mol )

-3.35 - 2.40 0.97 3.937 0.391

P(D) 4u (D)

molecules exchange their roles lead to additional complexity. Double minimum potentials with low barriers exist in several directions along internal coordinates or combinations thereof. Because of these difficulties our harmonic frequencies will in the case of intermolecular motions only provide a very crude picture of the true fundamentals. Harmonic force constants as obtained at C,, CZh and C,, stationary points are compiled in table 7. For inplane distortions r,, r2, R, Q!and /I have been used as internal coordinates (see fig. 1). In the case of the C, minimum the out-of-plane degree of freedom could Table 7 Harmonic

force constants

C,

of (HF)* at C,, Czh and C,,

rl r2

R s” Q(o)

C

rn”

rl r2

R s”

stationary

points a)

9.272 b’ -0.053 -0.018 0.025 0.025 0

9.487 0.029 0.022 0.023 0

0.141 -0.011 - 0.009 0

0.104 0.043 0

9.423 -0.043 0.012 0.039 0.03 1 0

9.423 0.012 0.031 0.039 0

0.103 - 0.004 - 0.004 0

0.056 0.076 0

0.056 0

9.522 -0.027 -0.067 0 0

9.603 0.014 0 0

0.097 0 0

0.037 0.014

-0.028

‘) Force constants for stretch-stretch, stretch-bend b’ The stretching force constant of the unperturbed

0.044 0

and bend-bend in mdyn/& mdymrad HF monomer is 9.643 mdyn/&

0.0012

(0.039)

0.021

and mdyn&rad’,

respectively.

M. Kofranek et al. / Coupled pair functional study on (HF) z Table 8 Harmonic vibrational frequencies and absolute (0) in cm-’ and intensities (I) in km/mol Type of mode

infrared

intensities

w(r2) w(r,) w(o+P) w(e) No--8) w(R)

of (HF)*, HFDF, DFHF and (DF)* at the C, minimum.

DFHF

HFDF

(HF),

vibrational

Type of mode

w(r2) w(r,) w(o+P) w(r) w(R) Na--P)

frequencies

I

0

I

0

I

w

I

4103 4052 510 413 216 150

98 379 159 133 144 1.4

4101 2939 409 297 193 148

131 139 126 71 109 1.6

2913 4054 485 413 179 138

71 344 140 126 73 15

2975 2937 370 297 169 136

81 196 83 68 60 18

of (HF)* and deuterated

shifts of -93.1 and -30.5 cm-‘, respectively. The good agreement between our shifts of harmonic frequencies and the experimentally determined shifts of HF stretching fundamentals suggests that anharmonicity effects in the complex are comparable to the monomer case. The additional anharmonicity induced by hydrogen bonding is therefore of the order of lo- 15 cm- ’ only. In case of the fully deuterated

forms as obtained

Cl,,

Type of mode

(HF),

HFDF

(DF),

4097 4078 520 361 132 203i

4087 2963 451 320 139 169i

2970 2956 376 266 128 147i

w(r2) w(r,) w((Y+fi) w(R) w(cu-p)

a) =)

at CZh and C,,

structures.

All values in cm-’

C,, (HF),

HFDF

DFHF

(DF),

4128 4106 286 128 2591

4126 2977 208 127 2551

4108 2992 282 126 194i

2992 2977 206 125 191i

‘I Doubly degenerate.

Table 10 Analysis of zero-point

energy contributions

to the stabilization

energy of ( HF)2 and deuterated

(HF)z

DC AZPEin,ra 09 ZPL., 00

CPF

exp. [71

1513 58 1571 644 927

1595( 62 1657( 619( 1038(

Frequencies

(DF)z

w

minimum are shown in table 8 together with the corresponding infrared intensities. Turning first to intramolecular modes we observe shifts of -83 and - 32 cm- ’ of “H-bonded” and “free” HF stretching frequencies in the complex relative to the monomer harmonic frequency. From their rotationally resolved vapor phase infrared investigations Pine, Lafferty and Howard [5-71 obtained corresponding

Table 9 Harmonic

145

+ 103, -93) + 103, -93) +60) +43, -34)

forms. All values in cm-’

HFDF CPF

DFHF CPF

(DF), CPF

1513 53 1560 526 1034

1513 53 1566 610 956

1513 42 1555 439 1116

hf. Kofranek et al. /Coupled pair functional study on (HF) z

146

R(Wl-4.6761

RtFF)-5.4761

bohr

100

P

100

a

O

-100

P

d

O

-100

-100

100

0

a

-100

bohr

RIFFI-5.0761

0

a

R(FFl-5.8761

100

bohr

100

100

P

bohr

b

O

P

e:

0

-100

-100

-100

0

a

RIFFI-5.2761

100

-.lOO

0

a

100

bohr

100

P

0

C

-100

-100

0

a

100

Fig. 3. 2D planar energy surfaces of (HF)* for different molecular separations R.

inter-

141

M. Kofranek et al. /Coupled pair functional study on (HF)>

complex ( DF) z we obtain similar red-shifts of - 6 1 and - 23 cm-’ again in good agreement with the experimental values [ 61 of - 72.1 and -24 cm-‘. Experimental data for isotopically mixed dimers are not yet available. Infrared intensities of “H-bonded” HF stretches are significantly enhanced over the monomer intensity. For ( HF)z and (DF) z these vibrations are predicted to be more intense by factors of 4.2 and 4.1, whereas the intensities of “free” HF and DF stretching vibrations are only slightly modified and exceed the monomer intensities by less than ten percent only.

To date none of the intermolecular fundamentals of ( HF)z could be determined with the aid of vapor phase investigations. Although this frequency region has been scanned by matrix isolation spectroscopy [ 8- 111 for isotopically “pure” as well as for mixed dimers interpretation of these data in terms of assignments to particular normal modes turned out to be difficult. By the very nature of the intermolecular energy surface of ( HF)z it is to be expected that vapor phase fundamentals and vibrational frequencies as observed in inert matrices will differ substantially.

a=0

a=60

100

a

Y O

C

-100

-100

0

100

-100

P

100

0

P

a=30

a= 90

100

Y

b

0

-100

Fig. 4. 2D cuts through obtained for R=5.2761

the energy surface of (HF)* bohr.

for different

fixed values of cr with /I and y as variable

parameters.

All plots were

hf. Kofranek et al. /Coupled pair functional study on (HF) z

148

a= 180

a= 120

e

-100

0

100

-100

100

P

a= 150 100

f

-100

0

100

P

Vibrational frequencies as obtained for C,, and C,, structures of (HF) 2 and their deuterated forms are collected in table 9. Contrary to the C, case the torsional coordinate t is more suitable in the C,, configuration since the corresponding potential surface is sufficiently steep and the barrier relates to a geometric arrangement where one of the hydrogen bonds is broken. One imaginary frequency corresponding to an in-plane o-j.3 motion is obtained for the CZh saddle point. Splitting of the two intramolecular vibrational frequencies is considerably smaller than in the C, case since it originates mainly from the intermolecular coupling fr, R. Because of the weaker hydro-

Fig. 4 (continued).

gen bond in the linear C,, arrangement this splitting is smaller too. Two imaginary frequencies stemming from a pair of degenerate intermolecular vibrations are obtained for the C,, case indicating a second-order saddle point. The harmonic vibrational analyses performed at C, and CZh configurations allow for the approximate determination of the hydrogen bond enthalpy and also of the effective barrier height of the CZh saddle point within the framework of the harmonic approximation. Of particular interest is the hydrogen bond enthalpy AH0 which may be compared to the experimentally determined Do [ 71. Using the notation of

M. Kofranek et al. / Coupledpair.finctional

ref. [ 71 intra- and intermolecular zero-point energy contributions are compiled in table 10. From their rotationally resolved infrared spectra and from the infrared intensities of HF stretching bands Pine and Howard obtained 1038 ( +43, - 34) cm-’ for D,. In order to enable a comparison with the quantum chemically computed AE (D,) intra- and intermolecular zero-point energy contributions must be taken into account. The intramolecular zero-point energy contribution has been measured accurately [ 5-71 and amounts to 62 cm- ‘. From our computations we obtain 58 cm-’ in good agreement with the experimental value. The intermolecular zero-point energy contributions pose a more difficult problem both from the theoretical as well from the experimental side. As already mentioned no experimental values are available for the intermolecular fundamentals. Therefore Pine and Howard had to estimate this contribution on the basis of previous theoretical investigations of the Barton-Howard surface [ 621. They reported an estimate of 6 19 ? 60 cm- ‘. We obtain 644 cm- ‘. Evidently, our value as originating from harmonic vibrational analysis must represent an overestimate. With the values of ZPEinter=644 cm-‘, AZPEi”l,a ~58 cm-’ and De=1513 cm-’ we obtain DO=927 cm-‘. The discrepancy of 111 cm-’ (x0.3 kcal/mol) to the corresponding experimental value will certainly be further reduced by the eventual inclusion of anharmonicity contributions. However, already at the current stage the agreement seems acceptable. The DOvalues for the other isotopomers of ( HF)z explain also nicely the relative instability of DFHF compared to HFDF with a difference of 76 cm-’ in favor of the “D-bonded” species. This trend is in agreement with the earlier experimental difficulties to detect the “H-bonded” mixed dimer. From zero-point energy corrections at the CZh saddle point and at the C, minimum an effective barrier height of 294 cm- ’ is obtained for the tunneling motion Cs-*C2h+ C,. In order to arrive at this value the contribution of the lowest intermolecular vibration at the C, minimum has been omitted since it corresponds approximately to the reaction coordinate. 3.3. Intermolecular energy surface In the following

we qualitatively

discuss some of

study on (HF) z

149

the features of the 4D intermolecular energy surface of ( HF) 2 as it originates from the interaction of two rigid HF molecules frozen at the monomer equilibrium distance. With the aid of contour plots we present a series of 2D cuts through the 4D energy surface. The contour plots were obtained using bicubic spline interpolation. We emphasize the attractive and moderately repulsive region of the energy surface. Therefore a cutoff at + 5 mh has been applied for all plots. Successive contour lines differ by 1 mh. Fully drawn lines are used for the attractive region. The repulsive part is represented with dashed lines. 3.3.1. Planar energy surface Five contour plots of the planar part of the energy surface ( y = 0) as obtained for five different intermolecular separations (R=4.6761, 5.0761, 5.2761, 5.4761 and 5.8761 bohr) are shown in figs. 3a-3e. The most important qualitative feature, namely the existence of various energetically equivalent C, minima separated by CZh and C,, saddle points, is in agreement with the previous investigations ofthe energy surface of ( HF)2, in particular with the most recent study by Redmon and Binkley [ 441. Close to the equilibrium intermolecular separation (figs. 3b-3d) this structure is most clearly visible since both saddle point configurations are about equally destabilized with respect to the C, minima. With decreasing R (see fig. 3a) the cyclic CZh saddle point approaches the energy of the C, configuration whereas the linear saddle point becomes relatively destabilized. The opposite trend occurs for larger intermolecular distances (see fig. 3e). This behavior is a consequence of the relative weight of dipole-dipole versus quadrupole-quadrupole interaction. For larger R the dipole-dipole term dominates and the energy gain obtained by off-line bending out of the linear C,, configuration becomes smaller. Evidently, for shorter R the attractive valley becomes narrower. Restricting to the attractive part of the planar energy surface of (HF) 2 transitions between equivalent C, minima are thus only possible via the C,, and the CZh saddle point. Only in case of the latter “Hbonded” and “free” HF molecules exchange their role as proton donor. This “trans-path” and a trans-path modulated by transitions via the C,, saddle point are therefore relevant for the interpretation of the microwave data [ 1,2]. The alternative “cis-path” which

.M. Kofranek et al. / Coupied pai~,~unc[ional srzcd.yon (HF) z

150

inevitably proceeds via configurations where the two hydrogen atoms have close contact is energetically exceedingly unfavorable. 3.3.2. ~~~-pf~~a~ energy surface In order to give an impression of the topology of the non-planar parts of the energy surface we selected several 2D cuts fixing one of the three intermolecular angular coordinates (Y,p or y in each case. As representative examples we provide a series of contour plots as obtained for R=5.2761 bohr, the equilibrium value. Plots of (Yfixed at 0, 30, 60,90, 120, 150

f3=0

and 180” are shown in figs. 4a-4g. Similar scans for fixing /? or y, respectively, are given in figs. 5 and 6. Since plots for fixed p(y) are equivalent to those for 180°-p (180”--y) fixedonlytheplotsforj?=O,30, 60 and 90” and y= 30.60 and 90” are shown. y=O corresponds to fig. 4~. A detailed inte~retation of the various contour plots of figs. 4-6 requires some contemplation and probably the use of geometrical models. In the following only a few general features are therefore discussed. We first turn to the series with fixed N values (fig. 4). Fixing CYmeans to conserve a definite geometry

13=60

a

0

a

100

b

a Fig. 5. 2D cuts through the energy surface of (HF), for different fixed values of@ with (r and y as variable parameters. All plots were obtained for R = 5.276 bohr.

M. Kofranek et al. / Coupledpalrfunctional

for the triangle F,H,F2. All the plots of fig. 4 are therefore determined by the angular motion of Hz alone. For (Y= 0 and 180 o (see figs. 4a and 4g) the 2D surfaces have therefore a fourfold rotational symmetry and all geometries with either /I = f 90” or y = f. 90” are energetically equivalent. Fig. 4a gives also an impression of the flatness of the energy surface at the energy minimum with respect to in-plane and out-of-plane motions of Hz for a fixed linear hydrogen bond F,H, Fz. The energetical equivalence of points with /? = + 90” for all values of y is obviously also valid for figs. 4b-4f. We observe that for (Yvalues in the region between 0 and 100” non-planar attractive pathways between equivalent minima are

O

151

available. These correspond, however, always to transitions without exchange of donor and acceptor roles. Fig. 5a (/3 = 0 O, a and y as variables) displays again the double minimum structure with the C,, saddle point in the center. The just barely attractive saddle points in the (cYy)-plane situated at cr= -t 72” and y = k 108” may be understood as arising from a strongly distorted CZh saddle. For /? = 30 and 60” (see figs. 5b and SC) this apparent saddle point approaches progressively the planar CZh configuration. The pattern for p fixed at 90” is particularly simple. Comparison of figs. 3c and 6a again reveals the torsional flexibility at the C, minimum whereas torsion

100

P

study on (HF),

100

a

-100

P

C

O

-100

-100

0

100

a

100

P o

b

-100

Fig. 6. 2D cuts through the energy surface of (HF), for different fixed values of y with (Yand p as variable parameters. All plots were obtained for R= 5.276 I bohr.

152

M. Kofranek et al. /Coupled pair functional study on (HF)

the CZh saddle results in much greater energetic destabilization. Although these 2D plots cannot give a full representation of the complicated 4D energy surface we are led to the conclusion that transitions in which “H-bonded” and “free” H-atoms exchange their role proceed necessarily in planar or near-planar orientations passing the cyclic CZh saddle point, in agreement with the interpretation of Dyke et al. [ 1,2].

at

4. Conclusions Many of the spectroscopically known properties of (HF) z in the gas phase are well reproduced by our CPF-potential energy surface. Equilibrium geometry, dipole moment, the shift of intramolecular HF stretching frequencies and their infrared intensities are satisfactorily described. The main weakness of our approach lies in the use of the harmonic approximation when analyzing the energy surface. In particular, at the present stage we cannot estimate the anharmanic contributions to the four intermolecular funcamentals. Analysis of the property surfaces of ( HF)2 which have been obtained in this study and attempts to fit the energy surface to various analytic formulas will be the subject of forthcoming publications in this series.

Note added in proof

z

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[ 19I E.L. Knuth, H.-G. Rubahn, J.P. Toennies and J. Wanner, J. a recent high-resolution interferometric FTIR spectroscopic study by von Puttkamer and Quack [ 761 one of the foJr intermolecular vibrational frequencies of (HF)2 has been observed at 400 cm-’ and was tentatively assigned to the torsional vibration. This value is in close agreement with the present value of 413 cm-’ for the out-of-plane vibration of (HF), (see table 8). In

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