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Determination of the intermolecular potential energy surface for (HCl) 2
from vibration–rotation–tunneling spectra
M. J. Elroda)^ and R. J. Saykally Department of Chemistry, University of California, Berkeley, California 94720 ~Received 26 October 1994; accepted 5 April 1995! An accurate and detailed semiempirical intermolecular potential energy surface for ~HCl! 2 has been determined by a direct nonlinear least-squares fit to 33 microwave, far-infrared and near-infrared spectroscopic quantities using the analytical potential model of Bunker et al. @J. Mol. Spectrosc. 146 , 200 ~ 1991 !# and a rigorous four-dimensional dynamical method ~described in the accompanying paper!. The global minimum ~ D (^) e 52 692 cm^21! is located near the hydrogen-bonded L-shaped geometry ~ R 5 3.746 Å, u 15 9°, u 25 89.8°, and f 5 180°!. The marked influence of anisotropic repulsive forces is evidenced in the radial dependence of the donor–acceptor interchange tunneling pathway. The minimum energy pathway in this low barrier ~48 cm^21! process involves a contraction of 0.1 Å in the center of mass distance ( R ) at the C 2 h symmetry barrier position. The new surface is much more accurate than either the ab initio formulation of Bunker et al. or a previous semiempirical surface @J. Chem. Phys. 78 , 6841 ~ 1983 !#. © 1995 American Institute of Physics.
I. INTRODUCTION
During the past decade, the investigation of intermolecu- lar forces via the study of van der Waals complexes has been greatly advanced by progress in both high resolution molecu- lar spectroscopy techniques and the complementary theoreti- cal methods required to calculate the spectrum from a trial intermolecular potential energy surface. In favorable situa- tions, it has been possible to determine experimental poten- tial energy surfaces by directly fitting spectroscopic data to detailed analytical models of the intermolecular interactions. 1 In order to invert spectroscopic data to obtain a potential surface in this fashion, it is necessary to solve the associated multidimensional intermolecular dynamics problem very ac- curately and efficiently such that the calculation of the rel- evant eigenvalues and eigenvectors can be placed inside of a least-squares fitting loop. The various methods currently in use for this purpose are summarized in recent reviews.2, Principally due to limitations in these computational tech- niques ~and available computers!, the systems for which such calculations have been used in conjunction with least-squares fitting of high precision spectroscopic data to determine im- proved intermolecular potential surfaces ~IPS! have been limited to ~ 1! rare gas ( R (^) g ) pairs^4 ~one intermolecular degree of freedom!, ~ 2! Rg–H 2 systems,5–7^ and Rg–hydrogen halides 8 ~two intermolecular degrees of freedom!, and ~ 3! Ar–H 2 O,^9 and Ar–NH 3 ~Ref. 10! ~three intermolecular de- grees of freedom!. Quack and Suhm 11 have obtained several six-dimensional ~including chemical bond stretch! semi- empirical IPS for the HF dimer system through fitting of ab initio points to an analytical model, then adjusting various terms in the model to force agreement with several known properties of the dimer ~dissociation energy, F—F bond dis- tance, rotational constant! with the use of quantum Monte
Carlo ~QMC! methods. Most recently, an empirical six- dimensional intermolecular potential for ~NH 3! 2 has been constructed by van der Avoird et al. which impressively re- produces many of the measured spectroscopic properties. 12 However, since the latter two studies did not employ direct least-squares fitting of the data, the reliability of these poten- tial surfaces for ~NH 3! 2 and ~HF! 2 is not really comparable to the simpler IPS listed above. The new direction addressed in the present work is the extension of the direct least-squares fitting methods to the determination of interaction potentials that are of genuine chemical significance. All of the existing rigorously deter- mined potential surfaces involve rare gas atoms and, al- though these systems provide important benchmarks and paradigms, these results are not generalizable to the more strongly interacting molecular systems. The simplest mo- lecular system from the point of view of intermolecular dy- namics is that of a pair of linear molecules, which constitutes a four-dimensional intermolecular dynamics problem ~the coordinate system is depicted in Fig. 1!. In particular, we address appropriate theoretical techniques for precisely cal- culating the vibration–rotation–tunneling ~VRT! spectra of the ~HCl! 2 complex in the accompanying paper ~which will be referred to as I!, 13 and we turn here to a general discus- sion of this system. Because hydrogen chloride is a prototypical hydrogen- bonding system, its condensed phases have been the subject of extensive experimental investigation through dielectric properties, 14 and nuclear magnetic resonance, 15,16^ infrared and Raman, 17–23^ spectroscopies, and x-ray and neutron dif- fraction techniques.24 –27^ More recently, the HCl intermolecu- lar interaction has been investigated with gas phase experi- mental techniques. The ~HCl! 2 complex was initially characterized by unresolved vibrational structure in the infra- red spectrum of the HCl monomer by Rank and co-workers. 28,29^ In a related development, Dyke et al.^30 used high resolution molecular beam electric resonance tech- niques in a study of ~HF! 2 , and found that although the
a!Present address: Department of Earth, Atmospheric, and Planetary Sci- ences, Massachusetts Institute of Technology, 54-1312, Cambridge, MA
J. Chem. Phys. Downloaded¬30¬Jun¬2005¬to¬132.162.177.17.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp 103 (3), 15 July 1995 0021-9606/95/103(3)/933/17/$6.00 © 1995 American Institute of Physics 933
hydrogen-bonded equilibrium structure was an L-shaped configuration, quantum tunneling rapidly interchanges the roles of the hydrogen bond donor and acceptor. This tunnel- ing motion results in the splitting of each rotational level into a symmetric and antisymmetric component, with the selec- tion rules requiring that the transitions result in a change in the tunneling state symmetry. Since the tunneling splitting provides a very sensitive characterization of the height and shape of the barrier to donor–acceptor interchange, these data provide crucial information on the intermolecular poten- tial energy surface, which is the feature of ultimate interest. Although extensive high resolution infrared studies of ~HCl! 2 were later carried out 31 as well as coherent anti-Stokes Ra- man spectroscopy ~CARS!,^32 similar measurements of the tunneling splitting for ~HCl! 2 remained elusive. In the late 1980s, far-infrared laser spectroscopy measurements of the K (^) a 50 →1 tunneling subband 33 and near-infrared mea- surements of the tunneling splitting for the mixed dimer ~H^35 Cl–H^37 Cl!^34 finally led to the direct measurement of the ground state tunneling splitting by far-infrared laser techniques.^35 Since that time, Fourier-transform far-infrared techniques have been used to measure several rotation– tunneling subbands associated with the intermolecular out- of-plane bending vibration,^36 and Schuder et al. recently re- ported extensive near-infrared measurements for ~HCl! 2 ~Ref. 37! and ~DCl! 2.^38 The extensive theoretical efforts in the study of the HCl–HCl interaction can roughly be divided into ab initio investigations, semiempirical methods which emphasize con- densed phase properties, and the calculation of spectra de- signed to test proposed potential energy surfaces ~including ab initio models!. Although most of the previous ab initio calculations were performed at the self-consistent field ~SCF! level,39– 42^ there have been several recent studies which in- cluded the influence of electron correlation.43– 45^ Other ab initio calculations have also been carried out at lower levels of theory to facilitate a global mapping of the four- dimensional IPS ~see Fig. 1!. 46 – 48^ These potentials were then used in molecular dynamics simulations of liquid HCl and subsequently refined to yield more accurate semiempirical surfaces. In particular, the work of Votava et al.^48 is notable in that both radial distribution functions obtained from neu- tron scattering from liquid HCl^27 and experimental second virial coefficients^49 were used to refine the potential. In a study vital to the present work, Karpfen et al.^50 determined the ~HCl! 2 potential surface at 1654 geometries via relatively high level ab initio calculations. Bunker et al.^51 subsequently fit a six-dimensional analytical function ~cast in a single cen-
ter spherical expansion! through these points. Jensen et al.^52 used this analytical function in a full four-dimensional close- coupling calculation ~only the high frequency intramolecular modes neglected! of the lowest intermolecular vibrational levels of ~HCl! 2 in an attempt to evaluate the quality of the ab initio analytical potential by comparing to existing experi- mental spectra and to predict unobserved spectra. Because these calculations only included states with total angular mo- mentum of zero, much of the existing experimental data could not actually be directly compared with the calculated results. In order to address this deficiency, Althorpe et al.^53 performed approximate three-dimensional calculations ~ra- dial coordinate fixed—‘‘reversed adiabatic approximation’’! for total angular momentum values of up to two using the analytical ab initio potential, as well as an electrostatic po- tential surface based on only the dipole and quadrupole mo- ment of HCl. Although the agreement between these poten- tials and the experimental spectra was not quantitative, the relative success of the simple electrostatic potential sug- gested that all of the essential anisotropy in the total IPS was indeed contained in the long-range attractive terms. We show in this paper, that such is not the case. In an early attempt to deduce an experimental potential for ~HF! 2 , Barton and Howard 54 used the BOARS approximation^55 ~separation of the radial and angular degrees of freedom! to calculate ~HF! 2 spectra and to directly fit ~via least-squares methods! a very simple low-order anisotropic repulsion 1 electrostatic potential energy surface. This approximation—which also reduces the maximum dimen- sionality of the problem to three—is also not sufficiently quantitative for the fitting of high resolution spectra for ~HF! 2. However, the nature of the experimental data avail- able at the time—which included only eigenstates located very near the bottom of the potential well—is the main limi- tation in the reliability of this IPS for ~HF! 2. It should be emphasized, however, that this work represents the only pre- vious attempt to directly fit a set of spectroscopic data to a flexible potential surface for a four-dimensional system— albeit with an approximate dynamical method and a quite crude potential model. In contrast, the ~HF! 2 potential deter- mination of Quack and Suhm 11 did not involve such a least- squares fit of the extensive spectroscopic data that exist ~see references given in Ref. 11!. In this paper, we report the measurement of new far- infrared VRT spectra for ~HCl! 2 and ~DCl! 2 which access excited states in the tunneling coordinate, thus providing im- portant new constraints on the form of the IPS—particularly with respect to the barrier to donor–acceptor interchange. Using a rigorous four-dimensional quantum dynamics tech- nique to precisely calculate the spectroscopic observables ~described in the accompanying paper!, the best ab initio and semiempirical potential energy surfaces are evaluated by comparison to all new and existing experimental spectra. Fi- nally, we describe the determination of a highly accurate experimental potential surface from a least-squares fit of the complete spectroscopic data set for ~HCl! 2 and ~DCl! 2 using a fully coupled four-dimensional dynamics method and a de- tailed intermolecular potential model.
FIG. 1. Coordinate system for ~HCl! 2. The ‘‘L-shaped’’ equilibrium struc- ture determined in this work occurs at u 15 9°, u 25 89.8°, f 5 180°, and R 5 3.746 Å. The monomer bond lengths ~ r 1 and r 2! were fixed at their equilibrium values ~2.412 a.u.!.
states of K (^) a 5 0 and K (^) a 5 1 for a given vibrational level! for ~DCl! 2 because both parallel and perpendicular transitions were measured connecting the K (^) a 5 0 and K (^) a 5 1 levels in both vibrational states. The spectroscopic constants for ~HCl! 2 are contained in Table I and for ~DCl! 2 in Table II. The fitted rotational constants confirm n 551 ~the first excited ‘‘tunneling’’ state! as the lower state for all of the new tran- sitions reported here for ~HCl! 2 and ~DCl! 2. The nuclear quadrupole hyperfine structure was fit to energies obtained from a Hamiltonian appropriate for a two- spin system. The following angular momentum coupling scheme was used:
I 1 1 I 2 5 I , I 1 J 5 F. ~ 3!
The matrix elements are evaluated in a symmetric top basis, 59
^ I 1 I 2 IJKFM F u H Q u I 1 I 2 I 8 J 8 K 8 F 8 M F 8 &
5 d FF 8 d M F M F 8
~ 21! ~ I^81 I^11 I^21 F^1 K^!
@~ 2 I 11 !~ 2 I 811!
3 ~ 2 J 11 !~ 2 J 811 !#1/2^ H
I I 8 2
J 8 J F J
(^3) S
J 2 J 8
2 K 2 q K 8 D^ F~
21! I^8
H
I I 8 2
I 1 I 1 I 2 J
S
I 1 2 I 1
2 I 1 0 I 1 D
3 ~x 22 q! 11 ~ 21! I
H
I I 8 2
I 2 I 2 I 1 J
S
I 2 2 I 2
2 I 2 0 I 2 D^
~x 22 q! 2 G
where the quantities in parentheses and braces are 3- j and 6- j symbols, respectively, and the nuclear quadrupole cou- pling constants, x 22 q , can be related to the usual Cartesian components by
x 205 x aa , ~ 5!
x 2615 ~2/3 !1/2 x ab , ~ 6!
and
x 2625 ~1/6 !1/2~x bb 2 x cc !. ~ 7!
Only first-order energies ~D J 50 and D K 50! were evaluated for the purposes of fitting the experimental spectra. The value of ^ P 2 ~cos u!& is reported rather than x aa , since that quantity @^ P 2 ~cos u!& 5 x aa / xHCl# is more directly related to the angular orientation of the HCl monomers within the complex. These quantities are reported in Tables I and II for ~HCl! 2 and ~DCl! 2 , respectively.
IV. ANALYTICAL INTERMOLECULAR POTENTIAL ENERGY SURFACES FOR (HCl) 2
In 1983, Votava et al.^48 performed ab initio calculations ~electron correlation included at the CEPA-1-SD level! on ~HCl! 2 in order to construct an analytical potential energy surface for the purposes of performing molecular dynamics calculations on liquid HCl. The ab initio points were fit to an effective three point-charge model @at the positions of the hydrogen atom ~charge 1 0.403 a.u.!, chlorine atom ~charge 2 0.909 a.u.!, and a dummy center ~charge 0.506 a.u.! posi- tioned outside the molecule on the side of the chlorine atom# with exp-6 functions between H–Cl and Cl–Cl sites and angle-dependent H–H exponential repulsion. This model was refined so that agreement with experimental second virial coefficients and experimental liquid properties ~radial distribution functions! was improved. Since this semiempir- ical potential energy surface continues to be employed in modern studies of HCl, 60 it is important to test the quality of this potential with respect to the results from spectroscopy. This is described later in the paper.
TABLE I. Spectroscopic constants a^ ~ 1 s uncertainties in parentheses! for ~HCl! 2.
u n 3 n 4 n 5 n 6 & K (^) a Parameter ~H 35 Cl! 2 ~H 35 Cl!~H^37 Cl!
u0 0 1 0& 0 ( B 1 C )/2 ~MHz! 1915.071~ 18! 1864.56~ 3! D (^) J ~kHz! 8.2~ 3! 6.9~ 7! ^ P 2 ~cos u!&^35 0.187~ 6! 0.187 b ^ P 2 ~cos u!&^37 0.191~ 6! n 552 ← 1 ~cm^21! 37.645 395~ 3! 37.298 817~ 6!
u0 0 2 0& 0 ( B 1 C )/2 ~MHz! 1923.984~ 18! 1872.60~ 2! D (^) J ~kHz! 10.1~ 2! 9.0~ 7! ^ P 2 ~cos u!&^35 0.179~ 8! 0.179 b ^ P 2 ~cos u!&^37 0.186~ 17!
a (^) Observed transition frequencies available on request from authors. bFixed to value for ~H 35 Cl! 2.
TABLE II. Spectroscopic constantsa^ ~ 1 s uncertainties in parentheses! for ~DCl! 2.
u n 3 n 4 n 5 n 6 & K (^) a Parameter ~D 35 Cl! 2 ~D^35 Cl!~D 37 Cl!
u0 0 1 0& 0 ( B 1 C )/2 ~MHz! 1896.27~ 3! 1846.98~ 4! D (^) J ~kHz! 8.2~ 6! 6.7~ 10! ^ P 2 ~cos u!&^35 0.189~ 5! 0.189 b ^ P 2 ~cos u!&^37 0.20~ 2! n 552 ← 1 ~cm^21! 31.012 263~ 10! 30.955 251~ 13!
u0 0 2 0& 0 ( B 1 C )/2 ~MHz! 1929.85~ 4! 1879.48~ 4! D (^) J ~kHz! 7.6~ 5! 6.4~ 9! ^ P 2 ~cos u!&^35 0.193~ 8! 0.193 b ^ P 2 ~cos u!&^37 0.19~ 3!
u0 0 1 0& 1 ( B 1 C )/2 ~MHz! 1896.03~ 3! 1846.86~ 4! ( B 2 C ) ~MHz! 27.77~ 5! 26.37~ 7! A eff ~cm^21! 5.386 162~ 13! 5.380 828~ 15! D (^) J ~kHz! 7.8~ 5! 5.2~ 11!
u0 0 2 0& 1 ( B 1 C )/2 ~MHz! 1931.74~ 3! 1881.32~ 3! ( B 2 C ) ~MHz! 35.81~ 4! 34.00~ 6! A eff ~cm^21! 5.291 146~ 13! 5.287 334~ 15! D (^) J ~kHz! 7.3~ 4! 6.4~ 9!
a (^) Observed transition frequencies available on request from authors. bFixed to value for ~D 35 Cl! 2.
In 1991, Bunker et al.^51 determined an ab initio six- dimensional ~including both HCl monomer stretching mo- tions! analytical potential energy surface for ~HCl! 2 from the earlier calculations of Karpfen et al.^50 by fitting the potential points to an elaborate, accurate single-center spherical ex- pansion form
V ~ r 1 , r 2 , R , u 1 , u 2 , f (^)!
(^5) ( l 1 l 2 l
A (^) l 1 l 2 l ~ R , r 1 , r 2! g (^) l 1 l 2 l ~u 1 , u 2 , f! 1 V short range. ~ 8!
Here the angular basis set for the expansion is defined as
g (^) l 1 l 2 l ~u 1 , u 2 , f! 5 ~ 21! l^12 l^2
2 l 11
2 p1/2^ H
S
l 1 l 2 l
0 0 0 D^ P^ l^1 0 ~u^1! P^ l^2 0 ~u^2!
(^1) ( m 51
l (^) m
~ 21! m 2 S
l 1 l 2 l
m 2 m 0 D^
P (^) l 1 m ~u 1! P (^) l 2 m ~u 2 !cos~ m f!
J
and where, in turn, the symbols in parentheses are 3 2 j sym- bols, l (^) m is the smaller of l 1 and l 2 , and P (^) lm ~u! is an unnor- malized associated Legendre function related to the spherical harmonic functions by
Y (^) lm ~u, f! 5 P (^) lm ~u !exp~ im f !. ~ 10!
The very short-range repulsion ~for geometries where the atoms begin to overlap! is separately parametrized,
V short range 5 @ a exp~ 2 2.5 R 12 !h^2 ~2.0,4.0; R 12!
1 b exp~ 23 R 14 !h^2 ~2.0,4.0; R 14!
1 b exp~ 23 R 23 !h^2 ~2.0,4.0; R 23 !#
3 h^2 ~2.0,3.5; r 1 !h^2 ~2.0,3.5; r 2 !, ~ 11!
and damped out at long range by the following functions:
h^6 ~ t , u ; R! 5 0.5$ 16 tanh@ t ~ R 2 u !# %. ~ 12!
The very short-range repulsion can be entirely neglected in calculations of bound intermolecular states since the wave functions for those states do not possess significant ampli- tude in the region of the potential where the short-range term is significant. Because of the invariance of the potential un- der the Eq. ~ 12! permutation operation, the A (^) l 1 l 2 l ( R , r 1 , r 2 )
coefficients are related by
A (^) l 1 l 2 l ~ R , r 1 , r 2! 5 ~ 21! l^11 l^2 A (^) l 2 l 1 l ~ R , r 2 , r 1 !. ~ 13!
In addition, the invariance of the potential under the E * operation requires
~ 21! l^11 l^21 l 5 1. ~ 14!
The expression for A 000 is given by
A 0005 k 00001 @ k 1 sr^ exp~ 2 d 3! 1 k 2 sr^ exp~ 22 d 3 !#
3 h^2 ~2.0,6.0; R! 1 @ k 1000 exp~ 2 d 3! 1 k 2000
3 exp~ 22 d 3 !#h^1 ~2.0,6.0; R !h^2 ~0.5,8.5; R!
k 6000 R^6
h^1 ~2.0,6.0; R! 1 k 11000 ~ y 121 y 22 !, ~ 15!
where
y 1512 exp~ 2 c 3 d 1 !, ~ 16!
y 2512 exp~ 2 c 3 d 2 !, ~ 17!
d 15 r 12 c 1 , ~ 18!
d 25 r 22 c 1 , ~ 19!
and
d 35 R 2 c 2 , ~ 20!
and where c 1 , c 2 , and c 3 are adjustable parameters. The ex- pression for other A (^) l 1 l 2 l ( R , r 1 , r 2 ) is
A (^) l 1 l 2 l 5 @ k 1 l 1 l 2 l exp~ 2 d 3! 1 k 2 l 1 l 2 l exp~ 22 d 3!
1 k 3 l 1 l 2 l exp~ 23 d 3! 1 k 4 l 1 l 2 l exp~ 24 d 3!
1 k 5^ l^1 l^2 l^ exp~ 25 d 3! 1 k 9^ l^1 l^2 l^ y 1
1 k 10^ l^1 l^2 l^ y 2 #h^1 ~2.0,6.0; R !h^2 ~0.5,8.5; R!
1 F
k 6^ l^1 l^2 l R^6
k 7^ l^1 l^2 l R^7
k 8^ l^1 l^2 l R^8
G h^1 ~2.0,6.0; R !. ~ 21!
In addition, the electrostatic contribution to each of the l 11 l 25 l terms was added to give the potential correct long- range behavior
A l^ elec 1 l 2 l 5 ~ 21! l^2 8 p3/2 d ~ l 11 l 2 !, l
3 S
~^2 l 111 !!
~ 2 l 11 !~ 2 l 111 !!~ 2 l 211 !! D
1/2 (^) Q (^) l 1 Q^ l 2 R l^11
where Q l are the HCl multipole moments. The dipole, quad- rupole, and octupole moments were held fixed at the ab initio values and the l 5 4 and l 5 5 moments were adjustable pa-
rameters. The damping functions ~h! were used to damp out
~DCl! 2 from near-infrared spectroscopy34,38^ constitute the previously reported data set for this coordinate. The n 552 spectra are particularly important as these eigenstates are lo- cated near the top of the barrier to donor–acceptor inter- change. The antigeared in-plane bending fundamental ~n 3! has yet to be observed, probably due to its small predicted band intensity. In spite of this, the dependence of the poten- tial on the in-plane coordinates is probably constrained better
by experiment than any other region of the potential. The FTIR far-infrared measurement of the out-of-plane bend for ~HCl! 2 ~Ref. 36! provides a good constraint on the torsional coordinate ~f! and the measurement of a similar combina- tion out-of-plane bend/ n 55 1 band provides a constraint on the extent of the coupling of f to the in-plane angles ~u 1 and u 2 !. Unfortunately, no spectra accessing excited stretching states have been reported; such data would provide the most exacting constraint on the radial coordinate. However, the J 50 →1 spacings are largely determined by ^1/ R^2 & and there- fore provide a good constraint on the average value of R for each bending state. In addition, the dissociation energy ~ D 0! for ~HCl! 2 has been measured from a consideration of abso- lute infrared intensities,^61 and this information helps to con- strain the isotropically averaged well depth. In order to com- pare the experimental dissociation energy directly with the results from the intermolecular four-dimensional dynamics calculations, it is necessary to properly account for the change in zero-point energy upon complexation due to shifts in the intramolecular frequencies. The n 1 ‘‘free’’ and n 2 ‘‘bonded’’ HCl stretches have been measured at 2880. and 2854.0593 cm^21 , respectively.^37 Since the HCl monomer stretching frequency occurs at 2885.9777 cm^21 , this amounts to a total redshift of 38 cm^21 , or a decrease in the zero-point energy of 19 cm^21. Therefore, the experimentally determined D 0 of 431 cm^21 is shifted to 412 cm^21 so as to include contributions from the intermolecular modes only. It should also be emphasized that the ~DCl! 2 spectra essentially con-
TABLE IV. Comparison of calculations using the ab initio ~Ref. 51! and the fitted ES1 potentials with experi- mentally determined values for ~H^35 Cl! 2.
Observable Ab initio Experimental ES u0 0 0 0& D 0 2 373.0 2 412.0 2 411. J 50 → 1 K (^) a 50 → 0 0.121 3 0.129 7 0.129 6 J 51 → 1 K (^) a 512 → 11 0.000 570 0.000 624 0.000 638 J 50 → 1 K (^) a 50 → 12 11.076 92 11.085 64 11.085 50 ^ P 2 ~cos u!& 0.145 0 0.170 2 0.160 1 u0 0 1 0& Ground state J 50 → n 551 J 50 12.432 81 15.476 68 15.667 07 J 50 → 1 K (^) a 50 → 0 0.121 0 0.127 8 0.127 8 J 51 → 1 K (^) a 512 → 11 0.000 450 0.000 503 0.000 510 J 50 → 1 K (^) a 50 → 12 11.101 55 10.883 49 11.062 66 ^ P 2 ~cos u!& 0.187 9 0.186 8 0.196 3 u0 0 2 0& n 551 J 50 → n 552 J 50 46.660 43 37.645 40 37.616 12 J 50 → 1 K (^) a 50 → 0 0.120 6 0.128 4 0.128 6 ^ P 2 ~cos u!& 0.180 6 0.178 8 0.195 5 u0 0 0 1& Ground state J 50 → n 651 J 50 136.055 73 160.778 160.601 52 J 50 → 1 K (^) a 50 → 0 0.118 3 0.128 6 0.127 7 J 51 → 1 K (^) a 512 → 11 0.000 520 0.000 615 0.000 583 J 50 → 1 K (^) a 50 → 12 11.515 20 11.198 10.925 70 u0 0 1 1& Ground state J 50 → n 651 n 551 J 50 157.061 64 184.862 185.059 77 J 50 → 1 K (^) a 50 → 0 0.118 7 0.126 8 0.125 7 J 51 → 1 K (^) a 512 → 11 0.000 290 0.000 438 0.003 55 J 50 → 1 K (^) a 50 → 12 11.149 30 8.381 8.661 30
TABLE V. Comparison of calculations using the ab initio ~Ref. 51! and the fitted ES1 potentials with experimentally determined values for ~D 35 Cl! 2.
Observable Ab initio Experimental ES
u0 0 0 0& J 50 → 1 K (^) a 50 → 0 0.120 4 0.128 1 0.128 1 J 51 → 1 K (^) a 512 → 11 0.000 942 0.001 2 0.001 06
u0 0 1 0& Ground state J 50 → n 551 J 50 3.633 62 5.96 5.734 94 J 50 → 1 K (^) a 50 → 0 0.120 1 0.126 5 0.126 8 J 51 → 1 K (^) a 512 → 11 0.000 828 0.000 926 0.000 906 J 50 → 1 K (^) a 50 → 12 5.608 63 5.512 18 5.539 59 ^ P 2 ~cos u!& 0.193 1 0.188 7 0.193 1
u0 0 2 0& n 551 J 50 → n 552 J 50 35.858 79 31.012 25 30.984 97 K 50 → 1 K (^) a 50 → 0 0.120 5 0.128 9 0.128 9 J 51 → 1 K (^) a 512 → 11 0.001 06 0.001 20 0.001 18 J 50 → 1 K (^) a 50 → 12 5.223 48 5.419 45 5.275 01 ^ P 2 ~cos u!& 0.165 5 0.192 8 0.186 1
stitute an independent data set due to the fact that DCl pos- sesses a much smaller rotational constant than HCl. This results in the ~DCl! 2 eigenstates being localized in quite dif- ferent regions of the potential surface than the corresponding ~HCl! 2 eigenstates.
VII. LEAST-SQUARES FITS
The fully coupled four-dimensional dynamics calcula- tion was imbedded in a standard nonlinear least-squares fit- ting routine ~Levenberg–Marquardt algorithm!^63 using a ba- sis set of j max 58 n 5 4 for both ~HCl! 2 and ~DCl! 2. The largest matrices ~for J 51 B^2 symmetry! were 1620 basis functions square. In order to expedite the fit, only ~DCl! 2 potential basis functions with coefficients larger than 0. cm^21 were included in the energy calculation. The effects of this approximation were found to be less than the inaccura- cies due to the lack of total basis set convergence. Basis set convergence was investigated for the final fitted potential, but since the results closely paralleled the findings reported in Paper I for the ab initio potential, we do not explicitly report these properties. One-sided numerical derivatives were used in the least-squares procedure such that each it- eration required n parameters 1 1 calls to the energy subroutine. The radial basis set was precontracted using the V effcut^ ( R ) po- tential method at each least-squares iteration ~also described in Paper I!. This was an important consideration—especially for the initial fits—since the potential could shift substan- tially between iterations. Because some energy levels crossed during the course of the fitting process @in particular for ~HCl! 2 , u0 0 2 0& and u0 1 0 0&#, a simple eigenvalue number assignment was not sufficient to identify the desired energy differences. To distinguish between the bending and stretch- ing states, the eigenvectors were used to sum up the total excited state radial contribution to the wave function, thus allowing identification of the stretching states. In a typical fit of eight parameters requiring five iterations for least-squares convergence, about 50 calls to the energy subroutine are nec- essary. The total CPU time ~on our IBM RISC 6000 model 560 computer! required for a typical fit was usually about 50 h. Based mainly on the basis set convergence properties and in some cases on experimental uncertainties, the following uncertainties were assigned to each type of experimental data: ~ 1! vibrational band origins—0.3 cm^21 , ~ 2! J 50 → 1 K (^) a 50 → 0 spacings—0.0002 cm^21 , ~ 3! J 50 → 1 K (^) a 50 →1—spacings—0.1 cm^21 , ~ 4! J 51 → 1 K (^) a 512 → 11 spacings—0.0001 cm^21 , and ~ 5! angular expectation values ~^ P 2 ~cos u!&! 2 0.01. The uncertainty for the dissociation en- ergy was set at the experimental estimate^61 of 22 cm^21 and the rotational term value uncertainties for the u0 0 0 1& and u 0 0 1 1& states were doubled due to slower basis set conver- gence for these levels. In their analytical fit of the ab initio potential, Bunker et al.^51 used 27 variable intermolecular parameters. Given the current experimental data set, it is not practical to attempt to vary all 27 of these parameters in a least-squares fit. Ini- tially, we intended to use a simpler model for the potential energy surface by fixing the electrostatic, induction, and dis- persion forces according to their known asymptotic long- range form and basically determining the sum of the repul-
sive terms and errors in the long-range terms by direct fitting to the experimental data. This approach has been success- fully applied to the ArHF,^9 ArHCl,^8 ArH 2 O,^10 and ArNH (^3) ~Ref. 11! systems. However, despite the fact that the electro- static forces are indeed the source of most of the anisotropy at the radial potential minimum, trial potentials containing estimates ~from SCF ab initio calculations!^46 for the aniso- tropic repulsive terms did not lead to a convergent least- squares fitting result. Because of the limited data set and the complexity of the IPS, it is crucial to possess a reasonably accurate initial potential which can then be iteratively refined by nonlinear least-squares regression. In particular, the repul- sive anisotropy simply cannot be determined completely by the available data and must be adequately constrained by other ~theoretical! means. The apparent difficulty in obtain- ing a good initial potential is the splicing of the short- and long-range contributions together at an intermolecular dis- tance where the two contributions nearly cancel. If the two contributions are not very closely distance consistent ~i.e., the breakdown of the long-range expressions at the radial minimum!, the resulting potential could be substantially in error. Despite many trials, no potentials with a fixed long- range form and variable short-range terms led to convergent fitting results. Of course, the ab initio potential is not plagued by the above problems which occur when the potential is parti- tioned into short- and long-range terms, as is common in the theory of intermolecular forces. On the other hand, it is dif- ficult to perform ab initio calculations at high enough levels of theory to reproduce the correct long-range behavior, par- ticularly for systems wherein dispersion is the dominant force. In their fitting of the analytical potential model to the ab initio points, Bunker et al. fixed the dipole, quadrupole, and octupole moments at their ab initio values so the ana- lytical function would approximately retain the correct long- range electrostatic behavior.^51 The ab initio values for the dipole and quadrupole moments are in very good agreement with experiment, such that this constraint ensures an ‘‘ex- perimentally’’ accurate long-range electrostatic form. How- ever, Bunker et al. did not explicitly include induction terms, nor did they assess the quantitative accuracy of the terms attributed to dispersion. In order to explore the total long- range validity of the ab initio potential, we used the canoni- cal long-range expressions 64 to estimate the electrostatic ~from the ab initio dipole, quadrupole, and octupole mo- ments!, induction ~from the ab initio moments and the ex- perimental polarizabilities!,^65 and dispersion ~from estimates for the C 6 coefficient!^66 contributions at the potential mini- mum. The partitioning of the long-range attractive forces was as follows: the electrostatic forces ~of which 93% arise from the dipole–quadrupole and quadrupole–quadrupole in- teractions! accounted for 470 cm^21 , the induction forces ac- counted for only 40 cm^21 , and the dispersion forces contrib- uted 250 cm^21. Since this estimate predicts the dispersion forces to be dominated by the electrostatic forces at the mini- mum, the ab initio potential should provide a reasonable— but not quantitative—model at both short- and long-range. Although the induction effects are perhaps safely neglected in light of the uncertainty in the larger and similarly aniso-
tential for ArNH 3 has recently been shown to accurately re- produce the results of state-selected scattering experiments which probe much higher energy regions of the potential than were accessed by the spectroscopy experiments.^67 Be- cause of the lack of experimental information on the excited van der Waals stretching states, the primary weakness in the ES1 potential is probably in the radial degree of freedom. It should be noted, however, that the fitted potential produces the J 50 →1 energy differences ~proportional to ^1/ R^2 &! for all states that are in much better agreement with experiment than does the ab initio potential. These observables also serve to constrain the degree of angular–radial coupling, an important point which will discussed in a later section. The in-plane coordinates, although extensively characterized near the minimum and the donor–acceptor interchange barrier, are only constrained by experiment at relatively low ener- gies, as discussed earlier. Measurements of both the n 3 anti- geared in-plane bending fundamental and the n 4 intermolecu- lar stretching fundamental will be necessary to extend the reliability of the ES1 surface to higher energies for these coordinates. As a further test of the reliability of the ES1 potential surface ~with respect to the radial coordinate, in particular!, the temperature dependence of the second virial coefficients were calculated and compared to the experimental results^49 as well as with the results calculated from the ab initio po- tential ~the Votava et al. potential was already adjusted to fit these coefficients!. In order to calculate second virial coeffi- cients for ~HCl! 2 , the following integral must be evaluated:
B ~ T! 5
N av 4 E 0
2 p E 0
p E 0
p E 0
` ~ 12 exp@ 2 V ~ R , u 1 , u 2 , f !/ kT #!
3 sin u 1 sin u 2 R^2 d f d u 1 d u 2 dR , ~ 24!
where N Av is Avogadro’s number and B ( T ) is expressed in cm 3 /mol. At very short range, the exponential term is zero and the integral can be evaluated analytically. For ~HCl! 2 , the integral is evaluated analytically for R 5 0–2 Å and nu- merically evaluated ~by Simpson’s rule! for 2 Å , R ,30 Å. Because no Gaussian quadratures are applicable to this inte- gral, the numerical evaluation requires 100 points in R and 6 points in each angular coordinate ~to achieve accuracies of ;10 cm^3 /mol! and is therefore too expensive to include in the least-squares fitting process. In Fig. 3 we present a com- parison of the experimental and calculated second virial co- efficients. It is clear that the ES1 potential satisfactorily re- produces the experimental second virial coefficients while the ab initio potential does not. These results seem to indi- cate that the radial dependence of the ES1 potential is at least as accurate as the Votava et al. potential for the regions of the potential probed by the second virial coefficients. The ES1 potential is, of course, much more accurate than the Votava et al. potential in the bound region of the potential, particularly with respect to the angular coordinates. Additional spectroscopic measurements will provide a precise test of the accuracy of the new potential described here, and will permit further details of the intermolecular forces in this system to be elucidated. To aid in this en- deavor, the ES1 potential was used to calculate energy levels
for ~HCl! 2 and ~DCl! 2 using a j max 5 8 and n 5 4 basis and the results are presented in Table VIII. In addition, calculations for ~HCl!~DCl! were also performed ~in a manner identical to calculations using the ab initio potential! using the ES1 sur- face and are presented in Table IX. We find that the ~HCl!~DCl! isomer with the DCl monomer acting as the hy- drogen bond donor is 22 cm^21 more stable than the corre- sponding isomer with HCl subunit as the donor, which is in good agreement with the recent experimental estimate of 166 4 cm^21 obtained by a consideration of near-infrared band intensities. 68 Although the energy levels are reported to many digits, the reader should refer to the basis set conver- gence section in Paper I for an estimate of the actual number of significant figures. There are many VRT states of interest that have not yet been measured. In Table X we list several frequencies and intensities for transitions which access states that would be of particular utility in a testing of the new ES potential. The intensities were calculated according to the method outlined in the accompanying paper; however, the Boltzmann factors were left out so that the intensities could be easily weighted either for the 5 K molecular beam or the 140 K cooled cell experiments. Intensities for transitions already measured are presented for comparison. For example, the ~HCl! 2 u0 0 0 0& J 50 K (^) a 50 →u0 0 1 0& J 51 K (^) a 5 0 transition ~to which the other transitions are normalized! was measured using the tunable far-infrared laser technique with a signal-to-noise ratio in excess of 1000:1. 35 The n 6 out-of-plane bending fundamental has not yet been reported for ~DCl! 2 , although such work is in progress.^69 The measurement of this state ~in addition to the combination state u0 0 1 1&! would provide an indication of the reliability of the out-of-plane dependence of the ES potential. Of course, measurements of excited van der Waals stretching states would provide important information on the radial dependence of the potential. Intensities calculated us- ing the ab initio potential predicted that transitions involving the van der Waals stretching states would be too weak to observe with current experimental techniques. However, the results from the ES1 potential indicate that the ~HCl! 2 u0 0 1 0 & J 50 K (^) a 50 →u0 1 0 0& J 51 K (^) a 5 0 or 1 transitions may indeed be detectable, although the Boltzmann factor for the 5 K molecular beam experiment is quite unfavorable. The
FIG. 3. Calculated and experimental second virial coefficients for HCl.
TABLE VIII. Calculated energy levels from the ES1 potential surface ~in cm^21 !.
- ~H^35 Cl! 2 ~D 35 Cl! Assignment
- J 50 J 51 J 50 J
- u0 0 0 0& K a 50 2 411.783 68 2 411.654 04 2 469.412 94 2 469.284 A^1 symmetry B^2 symmetry A^1 symmetry B^2 symmetry
- u0 0 0 0& K a 512 2 400.698 19 2 463.852
- u0 1 0 0& K a 50 2 339.716 63 2 339.591 86 2 398.672 77 2 398.549
- u0 0 2 0& K a 50 2 358.500 52 2 358.371 93 2 432.673 40 2 432.544
- u0 1 0 0& K a 512 2 328.907 39 2 393.268
- u0 0 2 0& K a 512 2 347.278 03 2 427.399
- u0 2 0 0& K a 50 2 275.897 62 2 275.774 89 2 335.428 55 2 335.309
- u0 2 0 0& K a 512 2 265.005 79 2 330.094
- u0 1 2 0& K a 50 2 300.612 89 2 300.487 89 2 359.444 98 2 359.321
- u0 1 2 0& K a 512 2 288.994 89 2 353.787
- u0 0 4 0& K a 50 2 264.037 45 2 263.910 83 2 379.403 41 2 379.277
- u0 0 4 0& K a 512 2 250.046 50 2 373.402
- u0 2 2 0& K a 50 2 234.530 10 2 234.402 43 2 301.174 21 2 301.050
- u0 2 2 0& K a 512 2 222.680 02 2 296.076
- u0 0 1 1& K a 511 2 218.060 84 2 330.898
- u0 1 4 0& K a 50 2 203.356 09 2 203.233 64 2 322.996 96 2 322.874
- u0 1 4 0& K a 512 2 188.399 15 2 317.048
- u0 0 6 0& K a 50 2 184.741 73 2 184.614 27 2 289.892 14 2 289.770
- u1 0 0 0& K a 50 2 171.032 60 2 170.909 30 2 283.079 61 2 282.954
- u0 0 6 0& K a 511 2 168.016 87 2 284.377
- u0 1 1 1& K a 511 2 163.790 72 2 278.863
- u1 0 0 0& K a 512 2 155.362 52 2 275.765
- u0 0 1 0& K a 511 2 385.053 29 2 458.076 A^2 symmetry B^1 symmetry A^2 symmetry B^1 symmetry
- u0 1 1 0& K a 511 2 320.948 64 2 384.042
- u0 0 3 0& K a 511 2 295.914 87 2 402.787
- u0 2 1 0& K a 511 2 258.721 93 2 326.694
- u0 0 0 1& K a 50 2 251.182 21 2 251.054 50 2 347.155 05 2 347.027
- u0 0 0 1& K a 512 2 240.256 83 2 341.352
- u0 1 3 0& K a 511 2 231.814 40 2 345.201
- u0 1 0 1& K a 50 2 199.862 07 2 199.737 52 2 308.891 40 2 308.764
- u0 1 0 1& K a 512 2 189.873 83 2 277.833
- u0 0 5 0& K a 512 2 207.017 16 2 316.087
- u0 2 3 0& K a 511 2 176.274 64 2 286.558
- u0 0 2 1& K a 50 2 179.911 07 2 179.786 19 2 283.180 37 2 283.058
- u0 0 1 0& K a 50 2 396.116 46 2 395.988 61 2 463.615 86 2 463.489 B^1 symmetry A^2 symmetry B^1 symmetry A^2 symmetry
- u0 0 1 0& K a 512 2 385.053 80 2 458.077
- u0 1 1 0& K a 50 2 332.254 25 2 332.130 81 2 389.374 30 2 389.251
- u0 1 1 0& K a 512 2 320.949 14 2 384.042
- u0 0 3 0& K a 50 2 307.861 79 2 307.735 01 2 408.332 41 2 408.206
- u0 0 3 0& K a 512 2 295.915 39 2 402.788
- u0 2 1 0& K a 50 2 270.248 17 2 270.123 70 2 332.498 11 2 332.375
- u0 2 1 0& K a 512 2 258.722 46 2 326.695
- u0 1 3 0& K a 50 2 244.289 88 2 244.167 45 2 350.878 28 2 350.754
- u0 0 0 1& K a 511 2 240.256 24 2 341.351
- u0 1 3 0& K a 512 2 231.814 84 2 345.202
- u0 0 5 0& K a 50 2 223.791 56 2 223.664 39 2 321.414 63 2 321.294
- u0 2 3 0& K a 50 2 190.476 89 2 190.350 69 2 292.893 96 2 292.771
- u0 1 0 1& K a 511 2 189.873 10 2 277.832
- u0 0 5 0& K a 512 2 207.017 07 2 316.088
- u0 2 3 0& K a 512 2 176.274 64 2 286.559
- u0 0 0 0& K a 511 2 400.697 55 2 463.851 B^2 symmetry A^1 symmetry B^2 symmetry A^1 symmetry
- u0 1 0 0& K a 511 2 328.906 78 2 393.267
- u0 0 2 0& K a 511 2 347.277 27 2 427.398
- u0 2 0 0& K a 511 2 265.005 13 2 330.093
- u0 1 2 0& K a 511 2 288.994 12 2 353.786
- u0 0 4 0& K a 511 2 250.046 17 2 373.401
- u0 0 1 1& K a 50 2 226.723 78 2 226.598 09 2 335.920 02 2 335.794
- u0 2 2 0& K a 511 2 222.679 38 2 296.076
- u0 0 1 1& K a 512 2 218.061 19 2 330.899
- u0 1 4 0& K a 511 2 188.399 06 2 317.047
- u0 1 1 1& K a 50 2 171.514 40 2 171.393 00 2 283.259 72 2 283.135
- u0 0 6 0& K a 512 2 168.017
- u0 1 1 1& K a 512 2 163.790 72 2 278.864
- u1 0 0 0& K a 511 2 155.361
analytical ab initio barrier of Bunker et al.^51 and the center of mass separation only contracts by about 0.1 Å as ~HCl! 2 moves from the global minimum to the C 2 h barrier geometry, these features are crucial in order to accurately reproduce the measured spectroscopic observables. The Bunker et al. ana- lytical ab initio potential shows only a 0.013 Å radial con- traction along this coordinate, but it should be noted that the directly obtained ab initio points 50 ~which were fitted to the analytical form! at the minimum and the C 2 h barrier indicate a larger contraction of 0.056 Å. This suggests that the accu- racy of the analytical model itself could be a limiting factor
in obtaining purely theoretical quantitatively correct poten- tial surfaces for systems such as ~HCl! 2 , in addition to the usual problems associated with the limited accuracy of the ab initio methods. It is also interesting to note that the ab initio potential of Latajka et al.^44 precisely predicts a 0.1 Å contraction in R along the donor–acceptor interchange coor- dinate. None of the ab initio calculations, however, correctly predict the absolute magnitude of these center of mass dis- tances. The large variations in the experimental J 50 →1 energy differences for each vibrational state directly indicate the ex- istence of strong angular–radial coupling. Indeed, this angular–radial coupling is responsible for the breakdown of the semirigid bender model ~described in Paper I!, which predicted a barrier of 20 cm^21 or less based on the experi-
TABLE XI. Comparison of potential surfaces ~f fixed at 180°!.
E ~cm^21 !a^ R ~Å! u 1 ~deg! u 2 ~deg!
Latajka et al. ab initio b minimum 2555 3.838 11.3 89. barrier 45 3.740 50.6 129. Karpfen et al. ab initio c minimum 2602 3.912 6.60 88. barrier 71 3.856 47.23 132. Bunker et al. d analytical fit to Karpfen et al. ab initio minimum 2626 3.820 7.4 87. barrier 64 3.807 47.2 132. Votava et al. expanded semiempirical c minimum 2740 3.700 5 80. barrier 188 3.725 52 128. ES minimum 2692 3.746 9 89. barrier 48 3.650 47 133
a (^) Barrier energies relative to minimum. bReference 31. c (^) Reference 37. dReference 38. e (^) Reference 35.
FIG. 4. Contour plot of ES1 potential. Contours from 2 690 to 2 590 cm^21 at 10 cm^21 intervals ~other coordinates fixed at their equilibrium values!.
FIG. 5. Contour plot of ES1 potential showing radial dependence of donor– acceptor interchange tunneling pathway ~other coordinates determined by energy minimization!.
FIG. 6. Contour plot of ab initio potential showing radial dependence of donor–acceptor interchange tunneling pathway ~other coordinates deter- mined by energy minimization!.
mental n 5 energy levels. Moreover, this finding quite graphi- cally illustrates the inaccuracy of approximate angular ~fixed R! calculations such as those utilized by Althorpe et al.^53 and Schuder et al. 37,38^ which totally neglect angular–radial cou- pling. These results further suggest that as systems with less spherical symmetry than HCl are investigated, the errors due to the neglect of angular–radial coupling in approximate an- gular calculations are likely to render the information to be gained from such approximate approaches of questionable validity. The two remaining intermolecular coordinates to be ex- amined are the radial ( R ) and torsional ~f, out-of-plane! degrees of freedom. Because of the large amplitude donor– acceptor interchange dynamics occurring in the in-plane co- ordinates ~u 1 and u 2 !, it is difficult to find a suitable graphical method to demonstrate the radial and torsional dependence of the potential. Therefore, we present one-dimensional plots for both R and f with u 1 and u 2 fixed at both the global minimum geometry and the C 2 h barrier geometry ~Figs.
8 –11! in order to provide a lower and upper limit for the effects due to the ‘‘graphical’’ in-plane averaging. For geom- etries with the in-plane angles fixed at the global minimum, the fitted radial potential closely matches the repulsive wall of the ab initio potential ~as intended! while having a deeper well and shorter equilibrium intermolecular distance. The ra- dial dependence of the ES1 potential more closely follows the Votava et al. potential at long range, which is encourag- ing since the latter potential was adjusted to reproduce the experimental low temperature second virial coefficients, which are sensitive to this region of the potential. The tor- sional dependence ~f! of the ES1 potential is steeper than the Bunker et al. ab initio potential, which is not surprising, considering that the out-of-plane bending frequency calcu- lated from the ab initio surface was almost 25 cm^21 lower than the experimental value. The ES1 potential also repro- duces the experimental combination frequency, u0 0 1 1&, better than the ab initio surface indicating that the coupling between the in- and out-of-plane angular coordinates is much better described by the new potential.
FIG. 8. Comparison of radial potentials with angles fixed at potential mini- mum.
FIG. 9. Comparison of radial potentials with angles fixed at C 2 h geometry.
FIG. 7. Contour plot of Votava et al. potential showing radial dependence of donor–acceptor interchange tunneling pathway ~other coordinates deter- mined by energy minimization!.
FIG. 10. Comparison of out-of-plane torsional ~f! potentials with angles fixed at potential minimum.
~HF! 2 is found to be 332 cm^21 , while the simple electrostatic model depicted in Fig. 12 predicts a barrier of only about 200 cm^21. However, it is also apparent that there is a fairly pronounced breakdown of the long-range electrostatic ex- pressions at these short intermolecular distances, which com- plicates the ~HF! 2 interpretation. Despite these difficulties, the new fitted ~HCl! 2 potential surface does provide convinc- ing evidence for the importance of anisotropic repulsive forces. The angular–radial coupling evident in the donor– acceptor interchange tunneling pathway cannot be repro- duced by the mathematical form of the electrostatic expres- sions alone. Indeed, intuition would suggest that the reduced repulsive forces present at the C 2 h geometry ~where the hy- drogen atom on one HCl molecule is as far as possible from both the hydrogen and chlorine atoms on the other HCl mol- ecule! are actually responsible for the observed contraction of the intermolecular bond. These effects will undoubtedly be even more important in less spherical systems such as NH 3 , H 2 O, and HF. Therefore, fully rigorous dynamical treatments of these systems ~with no assumptions involving the separability of the radial and angular degrees of freedom! will also be necessary in the prediction and fitting of high precision VRT data for the associated dimers. The dominant influence of the electrostatic forces also ensures a stiff out-of-plane torsional barrier since both the dipole–quadrupole and quadrupole–quadrupole interactions demand planarity in order to maximize these attractive forces. The internal axis method ~IAM! promoted by Hougen^72 to describe the donor–acceptor interchange tunnel- ing problem in ~HF! 2 holds only for planar tunneling paths. For ~HCl! 2 this description should be quite accurate since the torsional barrier is much greater than the donor–acceptor interchange barrier. However, the increased effects of aniso- tropic repulsion expected for ~HF! 2 could compromise this planar dynamical model.
XI. SUMMARY
We have described the first rigorous experimental deter- mination of an intermolecular potential energy surface for a hydrogen bonded system. The ~HCl! 2 surface was obtained by a direct least-squares fit of 8 parameters in a detailed analytical model to 33 spectroscopic observables for ~HCl! 2 and ~DCl! 2. We have employed a rigorous four-dimensional variational method to calculate the spectroscopic observables from the surface. Although the fitted surface confirms the preeminent role of the electrostatic forces in determining the topology of the potential surface, we find that anisotropic repulsive forces are crucial in rationalizing the radial depen- dence of the donor–acceptor interchange tunneling pathway. This potential surface will serve as a benchmark for testing simplified potential forms appropriate for condensed phase simulations, as well as for testing ab initio calculations. This work should provide useful insights for potential surface de- terminations for the other classical hydrogen bonded sys- tems.
ACKNOWLEDGMENTS
The authors wish to thank Ron Cohen and Kun Liu for useful conversations, and Al van der Avoird for constructure criticism of the manuscript. Jeff Cruzan for help with graph- ics. This work was supported by the Experimental Physical Chemistry Program of the National Science Foundation ~Grant No. CHE-9123335!.
(^1) R. J. Saykally and G. A. Blake, Science 259 , 1570 ~ 1993 !. (^2) R. C. Cohen and R. J. Saykally, Annu. Rev. Phys. Chem. 42 , 381 ~ 1991 !. (^3) J. M. Hutson, Annu. Rev. Phys. Chem. 41 , 123 ~ 1990 !. (^4) Inert Gases , edited by R. A. Aziz ~Springer, New York, 1984!. (^5) S. Green, J. Chem. Phys. 96 , 4679 ~ 1992 !. (^6) R. J. LeRoy and J. M. Hutson, J. Chem. Phys. 86 , 837 ~ 1987 !. (^7) A. R. W. McKellar, J. Chem. Phys. 92 , 3261 ~ 1990 !. (^8) J. M. Hutson, J. Phys. Chem. 96 , 4237 ~ 1992 !; J. Chem. Phys. 96 , 6752 ~ 1992 !. (^9) R. C. Cohen and R. J. Saykally, J. Chem. Phys. 98 , 6007 ~ 1993 !. (^10) C. A. Schmuttenmaer, R. C. Cohen, and R. J. Saykally, J. Chem. Phys. 101 , 146 ~ 1994 !. (^11) M. Quack and M. Suhm, J. Chem. Phys. 95 , 28 ~ 1991 !. (^12) E. H. T. Olthof, A. van der Avoird, and P. E. S. Wormer, J. Chem. Phys. 101 , 8430 ~ 1994 !. (^13) M. J. Elrod and R. J. Saykally, J. Chem. Phys. 103 , 921 ~ 1995 !. (^14) R. H. Cole and S. Havriliak, Discuss. Faraday Soc. 23 , 13 ~ 1957 !. (^15) J. G. Powles and M. Rhodes, Phys. Lett. A 24 , 523 ~ 1967 !. (^16) H. Okama, N. Nakamura, and H. Chihara, J. Phys. Soc. Jpn. 24 , 452 ~ 1967 !. (^17) D. F. Hornig and W. E. Osberg, J. Chem. Phys. 23 , 662 ~ 1965 !. (^18) R. Savoie and A. Anderson, J. Chem. Phys. 44 , 548 ~ 1966 !. (^19) T. S. Sun and A. Anderson, Spectrosc. Lett. 4 , 377 ~ 1971 !. (^20) R. Savoie and M. Pezolet, J. Chem. Phys. 50 , 2781 ~ 1969 !. (^21) H. B. Friedrich and R. E. Carlson, J. Chem. Phys. 53 , 4441 ~ 1970 !. (^22) M. Ito, M. Suzuki, and T. Yokoyama, J. Chem. Phys. 50 , 2949 ~ 1969 !. (^23) C. H. Wang and P. A. Fleury, J. Chem. Phys. 53 , 2243 ~ 1970 !. (^24) E. Sandor and R. F. C. Farrow, Nature 213 , 171 ~ 1967 !. (^25) E. Sandor and R. F. C. Farrow, Nature 215 , 1265 ~ 1967 !. (^26) E. Sandor and M. W. Johnson, Nature 217 , 541 ~ 1968 !. (^27) A. K. Soper and P. A. Egelstaff, Mol. Phys. 42 , 399 ~ 1981 !. (^28) D. H. Rank, B. S. Rao, and T. A. Wiggins, J. Chem. Phys. 37 , 2511 ~ 1962 !. (^29) D. H. Rank, P. Sitaram, W. A. Glickman, and T. A. Wiggins, J. Chem. Phys. 39 , 2673 ~ 1963 !. (^30) T. R. Dyke, B. J. Howard, and W. Klemperer, J. Chem. Phys. 56 , 2442 ~ 1972 !. (^31) N. Ohashi and A. S. Pine, J. Chem. Phys. 81 , 73 ~ 1984 !. (^32) A. Furlan, S. Wulfert, and S. Leutwyler, Chem. Phys. Lett. 153 , 291 ~ 1988 !. (^33) G. A. Blake, K. L. Busarow, R. C. Cohen, K. B. Laughlin, Y. T. Lee, and R. J. Saykally, J. Chem. Phys. 89 , 6577 ~ 1988 !. (^34) M. D. Schuder, C. M. Lovejoy, D. D. Nelson, Jr., and D. J. Nesbitt, J. Chem. Phys. 91 , 4418 ~ 1989 !. (^35) G. A. Blake and R. E. Bumgarner, J. Chem. Phys. 91 , 7300 ~ 1989 !. (^36) N. Moazzen-Ahmadi, A. R. W. McKellar, and J. W. C. Johns, J. Mol. Spectrosc. 138 , 282 ~ 1989 !. (^37) M. D. Schuder, C. M. Lovejoy, R. Lascola, and D. J. Nesbitt, J. Chem. Phys. 99 , 4346 ~ 1993 !. (^38) M. D. Schuder, D. D. Nelson, Jr., and D. J. Nesbitt, J. Chem. Phys. 99 , 5045 ~ 1993 !. (^39) M. Allavena, B. Silvi, and J. Cipriani, J. Chem. Phys. 76 , 4573 ~ 1982 !. (^40) P. Hobza and R. Zahradnik, Chem. Phys. Lett. 82 , 473 ~ 1981 !. (^41) P. Hobza and J. Sauer, Theor. Chim. Acta 65 , 279 ~ 1984 !. (^42) Y. Hannachi and B. Silvi, J. Mol. Struct. ~Theochem! 200 , 483 ~ 1989 !. (^43) M. J. Frisch, J. A. Pople, and J. DelBene, J. Phys. Chem. 89 , 3664 ~ 1985 !. (^44) Z. Latajka and S. Scheiner, Chem. Phys. 122 , 413 ~ 1988 !. (^45) J. DelBene and I. Shavitt, Intern. J. Quantum Chem. Symp. 23 , 445 ~ 1989 !. (^46) I. R. McDonald, S. F. O’Shea, D. G. Bounds, and M. L. Klein, J. Chem. Phys. 72 , 5710 ~ 1980 !. (^47) M. L. Klein and I. R. McDonald, Mol. Phys. 42 , 243 ~ 1981 !.
(^48) R. Votava, R. Ahlrichs, and A. Geiger, J. Chem. Phys. 78 , 6841 ~ 1983 !. (^49) B. Schramm and U. Leuchs, Ber. Bunsenges. Phys. Chem. 83 , 847 ~ 1979 !. (^50) A. Karpfen, P. R. Bunker, and P. Jensen, Chem. Phys. 149 , 299 ~ 1991 !. (^51) P. R. Bunker, V. C. Epa, P. Jensen, and A. Karpfen, J. Mol. Spectrosc. 146 , 200 ~ 1991 !. (^52) P. Jensen, M. D. Marshall, P. R. Bunker, and A. Karpfen, Chem. Phys. Lett. 187 , 594 ~ 1991 !. (^53) S. C. Althorpe, D. C. Clary, and P. R. Bunker, Chem. Phys. Lett. 187 , 345 ~ 1991 !. (^54) A. E. Barton and B. J. Howard, Faraday Discuss. Chem. Soc. 73 , 45 ~ 1982 !. (^55) S. L. Holmgren, M. Waldman, and W. Klemperer, J. Chem. Phys. 67 , 4414 ~ 1977 !. (^56) G. A. Blake, K. B. Laughlin, K. L. Busarow, R. C. Cohen, D. Gwo, C. A. Schmuttenmaer, D. W. Steyert, and R. J. Saykally, Rev. Sci. Instrum. 62 , 1693 ~ 1991 !. (^57) G. A. Blake, K. B. Laughlin, K. L. Busarow, R. C. Cohen, D. Gwo, C. A. Schmuttenmaer, D. W. Steyert, and R. J. Saykally, Rev. Sci. Instrum. 62 , 1701 ~ 1991 !. (^58) D. P. Shoemaker, C. W. Garland, and J. W. Nibler, Experiments in Physi- cal Chemistry , 5th ed. ~McGraw-Hill, New York, 1989!.
(^59) M. R. Keenan, D. B. Wozniak, and W. H. Flygare, J. Chem. Phys. 75 , 631 ~ 1981 !. (^60) A. B. McCoy, Y. Hurwitz, and R. B. Gerber, J. Phys. Chem. 97 , 12516 ~ 1993 !. (^61) A. S. Pine and B. J. Howard, J. Chem. Phys. 84 , 590 ~ 1986 !. (^62) This work. (^63) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Nu- merical Recipes , 2nd ed. ~Cambridge University Press, Cambridge, 1992!. (^64) A. D. Buckingham, Adv. Chem. Phys. 12 , 107 ~ 1967 !. (^65) J. O. Hirschfelder, C. F. Curtis, and R. B. Bird, Molecular Theory of Gases and Liquids ~Wiley, New York, 1954!. (^66) A. Kumar and W. J. Meath, Mol. Phys. 54 , 823 ~ 1985 !. (^67) G. C. M. van der Sanden, P. E. S. Wormer, A. van der Avoird, C. A. Schmuttenmaer, and R. J. Saykally, Chem. Phys. Lett. 226 , 22 ~ 1994 !. (^68) M. D. Schuder and D. J. Nesbitt, J. Chem. Phys. 100 , 7250 ~ 1994 !. (^69) A. R. W. McKellar ~private communication! (^70) A. van der Avoird, Faraday Discuss. Chem. Soc. 97 , ~ 1994 !. (^71) P. R. Bunker, P. Jensen, A. Karpfen, M. Kofranek, and H. Lischka, J. Chem. Phys. 92 , 7432 ~ 1990 !. (^72) J. T. Hougen and N. Ohashi, J. Mol. Spectrosc. 109 , 134 ~ 1985 !.
