J. Phys. Chem. 1993,97, 1011-1018
1011
Ab Initio Study of the Potential Energy Surface for the Interaction of Na+ with Hz and the Geometries and Energies of Na+(Ht),,, n = 2-4 M. F. Falcetta, J. L. Pazun, M. J. Dorko, D. Kitchen, and P. E. Siska’ Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 Received: August 5, I992
A near-Hartree-Fock-limit calculation of the dependence of the interaction between Na+ and H2 on distance, orientation, and H2 bond length has been carried out a t 291 geometries and fit to an analytic, nonpiecewise potential function with correct long-range behavior. The H atom basis set was chosen to give accurate polarizability components and quadrupole moment for Hz, and each point on the surface was corrected for a small basis-set superposition error using the counterpoise method. With the large basis sets employed, electron correlation effects are shown to be nearly negligible, changing the well depth a t the minimum point on the surface by less than 1%. Equilibrium geometry, energy, and vibrational frequencies are compared with recent ab initio studies, and the repulsion a t short range with experimental estimates derived from scattering studies. In the cluster calculations, the additional H i s are arranged in the classic electron-pair-repulsion geometries with the H-H bond axes perpendicular to the Na+-HZ axes; a less constrained optimization for n = 2 shows this to be realistic. In the complex, the H i s are too distant to repel each other by overlap even for n = 4; the determining factor appears to be repulsion between H2 dipoles induced by Na+. Due to this repulsion, the (classical) energy released when an HZis added declines slightly as n grows. The structures are nonrigid with respect to translational or rotational motion of Hz, so long as its internal bond axis remains within the solvation shell.
point in relating the nonbonded forces to the ionizationdynamics. The Penning system itself is presently at or beyond the frontier of what is possible in ab initio studies, being a quasi-discretestate embedded in the ionization continuum. Semiempiricalelectronic structure calculations9 have established, however, that much insight into the features of the excited-state PES may be gained by neglecting the coupling to the continuum and replacing all the core electrons by model potentials. The use of this method requires a core ion-molecule (Ne+H2) PES as input, which the present study was undertaken to provide. Because of the difficulties in performing accurate calculationson Ne+H2,an open-shellchargetransfer excited state, we chose to use Na+H2 as a model for Ne+H2. Although Ne+ and Na+ are not isoelectronic, the 2p hole in Ne+ is much lower in energy than the ug orbital of H2, and only a small splitting among the 2A” and two 2A’ states (in C,symmetry) is anticipated, with all expected to be similar to the IA’ state of Na+H2, which as a closed-shell ground state is much easier to treat theoretically. A point of current interest in Penning ionization is the effect of preionization bond stretching or compression on the excited-state dynamics and ionic product distributions;I0 we have therefore included r values for H2 that span its zero-point vibrational motion. In the study of ion-solvent forces, there may still be much to learn by comparing homologous systems; here the well-studied Li+H2 system can serve as a standard. The scattering of Li+ by H2 and its isotopes has been studied extensively in ion-molecule crossed-beam experiments,’ I yielding rotationally and vibrationally resolved inelastic cross sections and angle-resolved inelastic-to-elastic ratios. PES calculations have been carried out by Lester12at the HartrebFock level and by Kutzelnigg et al.13 including electron correlation. Either PES yielded excellent agreement with the beam results when used as input in scattering calculation^.^^ Again, cluster studies have not been done. We present here near-HartrebFock-limit (hereafter referred to simply as HartrebFock or HF) calculations of the monomer Na+H2 PES over a dense grid of R ranging from 1.8 to 20 A, y = 0, 45, and 90°, and at four values of r spanning the turning points of the v = 0 state of H2, with special emphasis on r = re, using the GAUSSIAN 90 program package.15 Electron correlation effects are examined at a few key geometries and found to be
Introduction Dtspite the ubiquity of the sodium cation and the hydrogen molecule in chemistry, their mutual interaction has received relatively little attention. Experimental probing of the Na+H2 potential energy surface (PES) has been limited to early ionmolecule inelastic backscattering studies;l-3 over the range of collision energies employed (E 1-40 eV), the observed scattering is mainly sensitive to the short-range repulsive forces. Recent ab initio theoretical studiese of the equilibriumgeometry of Na+H2have shown the C, form to be most stable, with a well depth of -3 kcal/mol located at a distance R between Na+ and the center of mass of H2 of -2.5 A, and a bond length r of H2 verycloseto its value for isolated H2. These studies also indicated a relatively small contribution from electron correlation effects. A more global ab initio study of the Na+H2 PES over a wide range of R , r, and y (the angle between R and r), as well as trajectory-calculations on this surface, has been carried out by Jeziorek, Zurawski, and co-workers;’ their PES results, however, are not well-suited to our purposes, as will be discussed later. The respectable binding found theoretically suggests that H2 should readdysolvateNa+,yet toourknowledgeNa+(H2).clusters have not yet been examined experimentally. Castleman and Keesees have reviewed experimental studies of the stepwise solvation of Na+ by several diatomic and triatomic molecules; the smallest diatomic solvent they report is Nz, with a monomeric ( n = 1) binding energy of 8.0 kcal/mol. Shortly after the completion of our ab initio computational effort, Bauschlicher et a1.6 reported an ab initio study of the energies and structures of Na+(H2), clusters for n = 2 and 3, finding binding energies and bond lengths in good agreement with the present work, as given later. Despite its simplicity, Na+(H2), presents an interaction with substantial anisotropy and both electrostatic (chargequadrupole) and inductiveattraction as well as exchange repulsion, features typical of more complex ion-solvent systems. Our own interest in this system stems from our (as yet unpublished) molecular beam and theoretical studies of Ne”(3s ~Pz,o) + Hz Penning ionization,
-
Ne*
+ H 2 + N e + H: + e-
for which the similarity between Ne* and Na is a fruitful starting 0022-3654/58/2097-1011$04.00/0
-
1993 American Chemical Society
Falcetta et al.
1012 The Journal of Physical Chemistry, Vol. 97, No. 5, 1993
TABLE I: Varirtion of Quadruople Moment and Polarizability Compoeenb of H2 with Basis Set and Bond Length at the Hartree-Fock Level' ~H-HA , H atom basis set all a1 e 6-31G(2~) 6-31GS*(2slp) 3s2p 4s2p 5s3pld 6s4p2d accurateb 5s3pld 5s3pld 5s3pld
0.7412 0.7412 0.7412 0.7412 0.7412 0.7412 0.7412 0.633 0.833 0.883
0.316 0.401 0.451 0.486 0.496 0.488 0.4577 0.374 0.607 0.671
6.50 6.42 5.42 6.43 6.43 6.48 6.385 5.09 7.75 8.55
0.00 0.57 2.78 4.42 4.54 4.51 4.580 3.86 5.20 5.56
a 8 and a values are given in atomic units (ao3 for CY, euo2 for e). Conversions are .(AI) = 0 . 1 4 8 1 8 ~(au) ~ and e ( D A) = 1.34508 (au). The first two bases are those used by Curtiss and Pople, ref 4, and the third by Dixon et al., ref 5. Kdos and Wolniewicz, ref 18, interpolated at 1.40066~70from values given at 1.40 and 1.45~0.A value of r slightly smaller than the best recommended re (0.74144 A, 1.40112ao (Huber, K. P.; Herzberg, G. Consranrs of Diatomic Molecules; Van Nostrand: New York, 1977)] was inadvertently used in the calculations reported here. Based on the calculations of Kolos and Wolniewicz, the properties in this table vary only by one or two in the fourth significant figure due to this0.0002-Adiscrepancy;likewise, theenergies presented in subsequent tables change by at most 0.002 kcal/mol, well within the accuracy of the calculations and fits.
unimportant. The H atom basis set is chosen to give accurate dipole polarizability components as well as a moderatelyaccurate quadrupole moment for Hz, A detailed analysis of basis-set superposition error (BSSE) reveals that in this system, and perhaps generally in ionic interactions, the BSSE at large R is accurately obtained from the difference between the ab initio potential points and the known long-rangeform of the interaction using moments and polarizabilities given by the same basis set. The calculated points are used to construct an analytic potential energy surface based on a realistic nonpiecewise,differentiablepotential function, derived from the Tang-Toennies mode1,16 containing only four adjustable parameters (for fixed r ) and yielding an rms fitting error of 0.05 kcal/mol, with a maximum error, in the highly repulsive short-range regime, of 0.50 kcal. The variable parametersand long-range coefficientsare well-represented by linear functions in r - re. This potential function should be suitable for both low-energy scattering and molecular dynamicscalculations. Calculationsoptimized in Rand r on n = 1-4 clusters of assumed symmetry are also presented; these show a slight decline in the classical binding energy for each added HZligand. The HzNa+-H2 bond angle was optimized for n = 2 and found to be 1BOo as assumed. Harmonic vibrational frequencies, zero-point energies, and rotational constants are given for n = 1.
Theoretical Metbods Choice of Basis Set and Level of Treatment. Obtaining a potential surface for an ion-molecule system that is accurate at long range requires a wave function for the neutral molecule of sufficient quality to yield an accurate lowest-order permanent electric moment and dipole polarizability tensor. Dykstra" has shown that accurate polarizabilities may be obtained for H2 at the Hartrce-Fock level, provided the atomic hydrogen basis set is not overbalanced in favor of high angular momentum orbitals. In calculations using 4s2p, 5s3pld, and 6s4p2d contracted Gaussian basis sets on each hydrogen atom, it was found that the 5s3pld and 6s4p2d bases agreed as to both the polarizability components al and aL and the quadrupole moment 8 within 2%. while the 5s3pld basis reproduced the accurate ab initio results of KO)" and Wolniewiczl*for the polarizabilities within 1% and the quadrupole moment to 8%. as given in Table I. Also given there are the results for basis sets employed in recent HFstudies.4.5 Inclusion of d functions was found to be essential to obtaining an accurate al. Even with a well-balanced basis, convergence
t I
J
t 6
9
12
15
18
1
R (4 Figure 1. Interaction energy of H2 with a positive point charge for y = 90' as a function of R as predicted by the E1 model using the accurate values of the parameters from Table I (solid curve) and Hartrce-Fock calculations (points) using various basis sets.
on the true properties of HZas the basis is augmented at the H F level is of course not guaranteed, as indicated by the results for the larger bases. Moreover, we have not been able to establish whether a H F wave function can yield both permanent moments and polarizabilities of high accuracy. Bauschlicher et al.'j have been able to achieve this with a correlated wave function derived from a basis set comparable to ours. At large R, the Na+H2interaction should be well-approximated by that of a point positive charge with HI,i.e., by the H+Hz interaction. The leading terms in this potential are c h a r g e quadrupole and charge-induced-dipole attractions, given by (in au)l9 v ( R , ~ ) = BR-~P,(COS 7) - (.,,/2)~~[1
+ aP,(cos y)] -
o(R-s) (1) where 8 is the quadrupole moment, Pz(x) the second-degree Legendre polynomial, a0 = (all + 2a1)/3, and a = (2/3)(0( al)/ao. We will refer to eq 1 as the electrostatic-inductive (EI) model, in which higher-order terms, mainly involving the hexadecapole moment and quadrupole polarizability of Hf, and dispersion forces are neglected. Replacing Na+ by H+afforded a quick, inexpensive way to check the actual behavior of the surface at long range, as well as its sensitivity to the H atom basis set and inclusion of electron correlation. Figure 1 compares the results from point-charge H F calculations for each of the three bases to the E1 model, for an orientation angle y = 90' (C, symmetry), while Figure 2 illustrates the influence of electron correlation (within Hz)for the 5s3pld basis using Morlbr-Plesset perturbation theory to second order (MP2) and to fourth order including all single and double excitations (Le., all excitations in Hz)(MP4SD). The adequacy of the 5s3pld basis is confirmed by the good agreement with EI, while the small changes in the energies upon including electron correlation suggest that the H F approximation should yield an accurate attractive part of the PES and, moreover, that the higher-order terms neglected in cq 1 contribute at the most a few tens of calories to the energy even
Interaction of Na+ with H2
The Journal of Physical Chemistry, Vol. 97, No. 5, 1993 1013
3
s
A
I
I
t
n
1
I
I
I
I
I
2
3
I
4
5
6
€3
6 ~
9
1
MP4SD
1
y3-
I
12
15
18
21
R (4 Figure 2. Interaction energy of H2 with a positive point charge for y = 90' as a function of R as predicted by the E1 model (solid curve) and ab initio calculations (points) employing the HF, MP2,and MP4SD approximations and using the 5s3pld basis set. at R = 4 A. Similar calculations for 7 = Oo (Lo) also confirmed that the long-range anisotropy is well-described by H F in the chosen basis. On the basis of these calculations and the well-documented finding that for closed-shell interactions H F does very well at predicting the repulsive part of the PES even for modest-sized basis sets, we expect that, aside from weak dispersion attraction between Na+ and H2 and basis-set superposition error (BSSE), the H F model should yield an accurate PES over the entire range of(R,y). Figure3 showsaHFcomparisonofthey = 90° energies, for the three H atom bases combined with a 6-31 1G* basis set20 on Na+, along with an estimate of the BSSE for the 5s3pld basis using the standard functional counterpoise method. The small discrepancies among the energy curves are about the same as would be predicted from point-charge calculations using a's and 8's from each basis (given in Table I), even at smaller distances, while the two larger bases are in close agreement, again indicating the adequacy of the 5s3pld H atom basis. Our definition of the superposition error is BSSE = PA- E A + EB- EB,where PAis an energy for species A computed with the basis functions of species B present at the intermolecular geometry of interest but without B's nuclei or electrons and EA is the energy without B's functions and similarly for B. This definition makes the BSSE always negative, and the corrected interaction energy is obtained by subtracting the BSSE from the H F interaction energy. For our basis sets, the BSSE is fortunately small, only4.2 kcal/mol at the potential minimum, and at larger R is accurately given by the difference between the ab initio points and the E1 model. Roughly two-thirds of the BSSE is due to the relatively poorer Na+ basis, causing Na+ to borrow the H atom functions. The latter finding is probably a more general phenomenonoccurringwhen electrostatic and/or induction forces dominate the long-range behavior; though it should not be used uncritically,we suggest the difference between the ab initio energy and its expected long-range form, using the moments and polarizabilities predicted by the same basis set, as an inexpensive
(4
Figure 3. V(R,90°)for Na+H2predicted by Hartree-Fock calculations using the 6-3 11G* basis set on the Na atom and various basis sets on the H atoms. Also shown is the counterpoise approximation to the basis-set superposition error obtained using the 5s3pld basis set on the H atoms. estimate of BSSE in larger systems, where direct counterpoise calculations may be prohibitive. The relatively sharp change in the slope of the BSSE curve near the energy minimum indicates an additional contribution arising from orbital overlap. To assess the influence of correlation in the full system, H F and MP2 energies using the 5s3pld basis, counterpoise-corrected for BSSE, were compared and found to agree again within a few tens of calories near the 90° minimum, with a maximum error of 0.1 kcal/mol well up on the repulsive wall of the surface. A few calculations with the 6s4p2d basis at the H F and MP2 levels showed negligible change from the smaller basis. Based on a dispersion C6constant of 6.5 au derived from combination rules,21 the undamped dispersion energy for Na+H2is estimated as 4 . 3 6 kcal/mol at R = 2.5 A, far larger than any energy differences calculated ab initio (but comparable to the BSSE). Weconclude that the Na+ basis is inadequate to give an appreciable dispersion contribution when correlation is included via perturbation theory; a configuration interaction calculation in a larger basis would be required. As the addition of the 10 Na+ electrons, even with a modest basis, greatly increases computational effort, and we wished to extend the calculations to clusters, the use of a pseudopotential to represent Na+ was also investigated. Keeping the 5s3pld H atom basis, Na+ was represented by theGoddard/Smedly effective core potential;22 the PES was found to be less attractive than that from the full calculation, with a y = 90' well 0.5 kcal/mol shallower. This isprobablydue to theinabilityofthecorepotential to contract when no valence electron is present, and a H F calculation on Na+ itself, rather than Na, would probably yield an improved pseudopotential for the present use. Based on these trial calculations, the 5s3pld H atom basis and 6-31 1G* Na+ basis were employed in an extensive exploration of the Na+H2PES and in geometry-optimizedcalculations of the energetics and structure of Na+(H2),, n = 2-4, at the HartrecFock level. An Analytic Potential Energy Surface for Na+H2. The total H F energy for Na+H2, corrected for BSSE, was com uted at 291 geometries chosen as follows. For r = re (0.7412 ), a grid of 69 R values, stepped by 0.1 A for R = 1.8-2.0 A,by 0.05 A for 2.04.1 A,byO.lAfor4.1-6.0A,andbyl.OAfor6.&2O.OA, was selected for each of the y values 0,45, and 90°. To obtain
1
Falcetta et al.
1014 The Journul of Physical Chemistry, Vol. 97, No. 5, 1993
TABLE II: Ab initio Potential Energy Surface Values for Na+H2,with Radii Given in A and Energies in kcil/mol Relative to Separated Na+ and Hl, at the Hartree-Fock Level with a 5s3pld H Atom Basis and 6-311G* Na Basis, Including BSSE Correction' R r
Y9 deg
1.8
0.633
0 90 0 45 90 0 90 0 90
20.639 7.244 26.776 16.199 7.676 32.906 7.934 36.628 8.028
0.7412 0.833 0.883
2.2 3.136 -1.741 1 1.627 1.022 -2.086 6.139 -2.390 7.068 -2.560
2.6 -0.001 -2.428 0.284 -1.379 -2.903 0.559 -3.316 0.724 -3.543
3.0 -0.278 -1.792 -0.240 -1.238 -2.180 -0.226 -2.524 -0.226 -2.7 15
3.4 -0.145 -1.203 -0.132 -0.821 -1.481 -0.143 -1.731 -0.158 -1.870
40
6.0
10.0
0.006 -0.673 0.034 -0.408 -0.836 0.047 -0.983 0.047 -1.066
0.061 -0.156 0.087 -0.055 -0.196 0.109 4232 0.120 -0.253
0.022 -0.027 0.030 -0.021 -0.034 0.037 -0.041 0.041 -0.045
~
a R is the distance between the Na nucleus and the midpoint of the H-H bond, r is the H-H bond length, and cos y = k?. The total HF energy of isolated Na+ is -161.664 3289 au, and those of H2 at the four bond lengths r are -1.123 421 295, -1.132 805 33 1, -1 .I27 156 878, and -1 .I21 050 100 au at r = 0.633,0.7412,0.833,and 0.883 A, respectively. The energy and length conversions 1 au (hartree) = 627.510 kcal/mol and 00 = 0.529 177 A were used. A complete set of ab initio points is available from the authors upon request.
the dependenceon r, rvalues of 0.633,0.833, and 0.883 A, which span the classical zero-point vibrational motion of H2,were each chosen for calculations at 14 R values, stepped by 0.2 A for R = 1.8-4.0 A and at R = 6.0, 10.0, and 14.0 A for the y values 0 and 90'. To get suitable values for the PES, the asymptotic energies of separated Na+ and H2 were subtracted from each point. Table I1 presents a small selection of calculated points; the entire set is available from the authors upon request. To cast the large number of PES values into a useful form, the calculated points for a fixed r were transformed into Legendre expansions
V(R,W') = A=0,2,4,
...
VA(&"' p ~ ( c o 'Ys)
(2)
where the P i s are Legendre polynomials. For a set of fixed 7's and the same set of R's for each y, eq 2 yields independent sets of simultaneous equations that can be solved for the VAat each R. In this collocation method, the number of Legendre terms included is dictated by the number of 7's. Thus, for r = re, the calculated points allow three terms to be obtained, and for other r two. The pointwise V i s were then fit to analytic functions; a representation of the complete surface retaining X = 0 and 2 and a more detailed three-term fit for r = re were obtained. For the full surface, the two leading Legendre terms were fit to a modified form23of the Tang-Toennies modell6 for nonbonded potentials, given by
Vo(R) = V,(R) + VL(R) Vz(R) = asVs(R) + aLVL(R) + f'Q(R)
(3)
where VSand VLare the short-range and long-range potentials, respectively, US and U L are anisotropy coefficients, VQ is the ionquadrupoleinteraction, and the r dependence has been suppressed. These functions are specified by
Vs(R) = A exp(-bR) - B exp(-bR/2) V,(R) = -f4(R)C4R-4
where the damping function is the Tang-Toennies form " ( b R ) k / k ! exp(-bR) ]
(5)
The functional form specified by eqs 3-5 contains the E1 model, q 1, in VLand VQ,which are suitably damped as the V, repulsion sets in at intermediate R. By comparison with eq 1, the long-
range parameters may be identified as U L = u = (2/3)(all- aL)/ ao,C4= 4 2 = (all + 2a1)/6,and C3 = -8, while the parameters A, B, b, and US remain to be specified by fitting the ab initio points. For r = re, where y = 45' was included, along with a more extensive set of R's, we augmented eq 3 by adding a V4 term, given by
V,(R) = u,V,(R) -fs(R)c,R-s (6) where Cs may be identified with the negative of the hexadecapole moment of Hz but was treated, along with 04, as an additional fitting parameter. The gradient-expansion least-squares method of M a r q ~ a r d t 2 ~ was used to optimize the values of the free parameters. Two separate fits were carried out, one for the "rigid rotor" surface V(R,y;r,)including the V4term of eq 6 and the other for the full surface, neglecting V4. In our potential-function routine, the VO parameters c, the well depth, R,, the well position, and u, the zero-crossing radius, were specified and varied, with the repulsion parameters A, B, and b of eq 4 determined from these algebraically using the three conditions Vo(R,) -e, VO(U)= 0, and dVo/dR = 0 at R = R,. For the full surface, separate fits were first made for each r using the r-dependent ab initio values of the long-range coefficients (see Table I); the resulting parameters were found to be well-represented by a linear r dependence; for example, c = e(re)+ mc(r-re). This linear assumptionwas then incorporated into a final parameter adjustment, fitting the ab initio data for the four r values simultaneously. In the full-surface fit, only those ( R , y )points for r = rethat were alsocomputed for theother r's were included in the input data set, to ensure a self-consistent set of optimized parameters. The rigid rotor surface is thus the result of a six-parameter variation (e, R,, u, US, 04, C,)and the full surface of an eight-parameter one (e, R,, u, US, and their r slopes). Table I11 collects the resulting parameters. The fullsurface rms fitting error was 0.08 kcal/mol, dominated by the larger absolute errors in the repulsive region, with a maximum error of 0.50 kcal/mol at R = 1.8 A, y = 90°, and r = 0.883 A, while for the r = re surface these figures are 0.04 kcal rms and 0.3 1 kcal at R = 1.8 A and y = 90'. These errors are within the estimated error in the HF calculation itself, making the fits useful as representations of the functionality implied by the calculated points. A detailed comparison will be made in the discussion below. The short-range behavior of the ab initio surface at large r is more difficult to represent by two Legendre terms; thus, extrapolation of the analytic surface function to larger r is not recommended. Results and Uscussion Monomer Potential Surface: Equilibrium Properties. Table V compares the computed equilibrium properties of Na+H2 derived from recent ab initio studies of this systemM with the present findings. To obtain an equilibrium geometry, Curtiss
Interaction of Na+ with Hz
The Journal of Physical Chemistry, Vol. 97, No. 5, 1993 1015
TABLE III: Potential Energy Surface Parameters for Na+ + Hf thrte,tcrm
fit for
r = 0.7412 A (re)
u
1 A835 2.6418 2.2014 0.3197 0.2400 -46.09 127.57 0.0291 -6.547 13635.6 78.517 3.4755
f
Rm U 0s UL
c3 c 4
04
cs
A
E b
two-term fit for 0.633 A < r < 0.883 A u(rA ma 1 A461 2.1871 2.65 1 1 0.3338 2.2122 0.2 153 0.3380 0.7754 0.2400 0.4434 46.09 -108.06 127.57 225.94
+
TABLE I V Numerical Comparison of ab Initio Points and Analytic Fit for the Global Na+H2Potential Eoergy Surface Using the Parameters in Columas 2 and 3 of Table III, in the Format of Table 111 R r
y,deg
1.8
2.6
3.4
6.0
0.633
0
20.639 20.442 7.244 7.391 26.776 26.889 7.676 7.954 32.906 32.850 7.934 8.350 36.628 36.288 8.028 8.533
-0.001 0.043 -2.428 -2.434 0.284 0.376 -2.903 -2.939 0.559 0.667 -3.316 -3.352 0.724 0.831 -3.543 -3.569
-0.145 -0.156 -1.203 -1.211 -0.132 -0.139 -1.481 -1.502 -0.143 -0.156 -1.731 -1.750 -0.158 -0.177 -1.870 -1.885
0.061 0.063 -0.156 -0,153 0.087 0.089 -0.196 -0,195 0.109 0.109 -0.232 -0.230 0.120 0.118 -0.253 -0.250
0
90 0.833
0 90
0.883
0
90
rc(H2),A rmin, A &in,
A
V,in, kcal/mol
cm-I cm-I u2, cm-1 v3, cm-1 AZPE, kcal/mol* DO,kcal/mole u(H2), VI,
a
Units are kcal/mol and A. Parameters refer to eqs 4-6. c, R,, and respectively refer to the well depth, well position, and zero-crossing radius of Vo(R);these parameters, along with US and 114 and C5 for the three-term fit, have been determined by least-squares fits to the ab initio points. The parameters A, E, and b are determined algebraically from c, R,, and u. The r dependence in the two-term fit is represented by linear dependence of each parameter u on r, as u(r) = u(re) m,(r - re). The error in this linear approximation to u(r) is alwap
