JOURNAL OF CHEMICAL PHYSICS
VOLUME 109, NUMBER 6
8 AUGUST 1998
Benchmark calculations with correlated molecular wave functions. IX. The weakly bound complexes Ar–H2 and Ar–HCl David E. Woon,a) Kirk A. Peterson,b) and Thom H. Dunning, Jr.c) Theory, Modeling and Simulation, Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352
~Received 6 March 1998; accepted 7 May 1998! The interaction of Ar with H2 and HCl has been studied using Mo” ller–Plesset perturbation theory ~MP2, MP3, MP4! and coupled-cluster @CCSD, CCSD~T!# methods with augmented correlation consistent basis sets. Basis sets as large as triply augmented quadruple zeta quality were used to investigate the convergence trends. Interaction energies were determined using the supermolecule approach with the counterpoise correction to account for basis set superposition error. Comparison with the available empirical potentials finds excellent agreement for both binding energies and transition state. For Ar–H2, the estimated complete basis set ~CBS! limits for the binding energies of the two equivalent minima and the connecting transition state ~TS! are, respectively, 55 and 47 cm21 at the MP4 level and 54 and 46 cm21 at the CCSD~T! level, respectively @the XC~fit! empirical potential of Bissonnette et al. @J. Chem. Phys. 105, 2639 ~1996!# yields 56.6 and 47.8 cm21 for H2 ( v 50)#. The estimated CBS limits for the binding energies of the two minima and transition state of Ar–HCl are 185, 155, and 109 cm21 at the MP4 level and 176, 147, and 105 cm21 at the CCSD~T! level, respectively @the H6(4,3,0) empirical potential of Hutson @J. Phys. Chem. 96, 4237 ~1992!# yields 176.0, 148.3, and 103.3 cm21 for HCl ( v 50)#. Basis sets containing diffuse functions of (d f g) symmetries were found to be essential for accurately modeling these two complexes, which are largely bound by dispersion and induction forces. Highly correlated wave functions were also required for accurate results. This was found to be particularly true for ArHCl, where significant differences in calculated binding energies were observed between MP2, MP4, and CCSD~T!. © 1998 American Institute of Physics. @S0021-9606~98!31030-2# I. INTRODUCTION
N2 –HF and even in more strongly bound complexes, but purely electrostatic interactions account for most of the binding energy in interactions between two species with permanent electric multipole moments. The Ar–H2 and Ar–HCl complexes fall into the intermediate regime. The long-range attraction depends upon the dipole and higher polarizabilities of Ar, but induction and dispersion contributions are roughly comparable. Whenever dispersive forces are important in describing intermolecular interactions, electron correlation must be included in the treatment. Our rare gas dimer and N2 –HF studies2–4 found that the MP4 and CCSD~T! methods are well suited to the task of recovering correlation energy in weakly bound complexes. As important as it is to extend the n-electron expansion as far as practical, it is also critical to extend the one-electron basis sets in order to satisfactorily describe molecular properties that require functions of high angular momentum. For rare-gas dimers, the force arising from the dipole polarizabilities of the two atoms dominates the attractive terms, but contributions from at least the quadrupole and octupole polarizabilities must be included in order to achieve high accuracy. Consequently, suitable (d f g) functions, including diffuse functions, must be present in the basis set. Similar demands upon both basis set and electron correlation methodology can be anticipated for Ar–H2 and Ar–HCl. The goal of the present work is to determine the requisite level of theory and one-electron basis sets necessary to ac-
The last several years have seen rapid advances in the application of high resolution molecular spectroscopy to the study of weakly interacting molecular complexes, with a number of groups presenting results of unprecedented detail and accuracy. Concomitant developments in the description of the potential energy functions of these systems have accompanied the elegant new experimental techniques. At the same time, although somewhat overshadowed by the success of the experimental studies, advances in the technology of computational chemistry have led to breakthroughs in the ability to make ab initio predictions for weakly bound complexes that rival the accuracy of the experimental results.1 The present work builds upon previous benchmark studies of rare gas dimers2,3 that established a strategy for describing interactions dominated by exceptionally weak dispersion forces as well as related studies on stronger interactions such as N2 –HF, 4 (HF) 2 , 5 (H2O) n , 6–10 and an extensive compendium of ion-water and related investigations.11–14 Rare-gas dimers are bound by dispersive forces dominated by induced dipole–induced dipole interactions. Polarizability based contributions remain important in a!
Present address: Molecular Research Institute, 845 Page Mill Rd, Palo Alto, CA 94304. b! Department of Chemistry, Washington State University, Richland, WA 99352. c! E-mail addresses:
[email protected],
[email protected], th –
[email protected] 0021-9606/98/109(6)/2233/9/$15.00
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© 1998 American Institute of Physics
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J. Chem. Phys., Vol. 109, No. 6, 8 August 1998
Woon, Peterson, and Dunning
TABLE I. Bond lengths ~Å! of H2 and HCl monomers at the MP2, MP3, MP4, CCSD, and CCSD~T! levels with various basis sets.a Species H2
HCl
a
Basis set
MP2
MP3
MP4
aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ
0.7549 0.7375 0.7363 0.7359 1.2880 1.2747 1.2734 1.2721 1.2877 1.2747 1.2733
0.7584 0.7399 0.7388 0.7384 1.2891 1.2754 1.2744 1.2730 1.2888 1.2754 1.2743
0.7602 0.7416 0.7406 0.7402 1.2913 1.2783 1.2773 1.2759 1.2911 1.2783 1.2772
CCSD CCSD~T! 0.7617 0.7430 0.7420 0.7416 1.2906 1.2767 1.2753 1.2739 1.2903 1.2766 1.2753
¯ ¯ ¯ ¯ 1.2922 1.2789 1.2776 1.2762 1.2919 1.2788 1.2776
The experimental values are 0.7414 Å for H2 and 1.2746 Å for HCl ~Ref. 39!.
curately characterize the Ar–H2 and Ar–HCl interactions. As in previous studies, monomer properties will be reviewed ~Sec. III! before discussing the dimer behavior ~Secs. IV and V for Ar–H2 and Ar–HCl, respectively!. Comparison with the empirical potentials of Le Roy and Hutson15 and Bissonnette et al.16 for Ar–H2 and Hutson17 for Ar–HCl, as well as with selected previous theoretical results is incorporated into the discussion of the present results. II. METHODOLOGY
Augmented correlation consistent basis sets18–22 (aug-cc-pVnZ) were employed in this work. In addition, multiply augmented21 sets @x-aug-cc-pVnZ, x5d(doubly) or t(triply)# were used on Ar and for HCl ~but not on H2). The MOLPRO program suite23 was utilized to perform the Mo” ller–Plesset perturbation theory ~MP2, MP3, MP4! and coupled cluster24–27 @CCSD, CCSD~T!# calculations. On the basis of previous work2–4 and for the sake of brevity, RHF ~restricted Hartree–Fock! results are not reported here for the complexes. Only valence electrons were correlated and pure spherical harmonic components were employed for the (d f gh) polarization functions. Binding energies were determined using the supermolecule approach with the usual Boys and Bernardi counterpoise ~CP! correction28 applied to account for basis set superposition error ~BSSE!. Uncorrected results will not be presented in this study. The rare-gas dimer studies2,3 demonstrated that uncorrected results for very weakly bound complexes converge irregularly, though toward very similar limits as the CP-corrected calculations. The bond distances of H2 and HCl ~see Table I! were frozen at aug-cc-pVnZ CCSD and MP4 values, respectively. This is valid for two reasons: MP2, MP3, MP4, CCSD, and CCSD~T! bond lengths are comparable, particularly for MP4 and CCSD~T!, and the extra sets of diffuse functions on H and Cl do not change the HCl bond lengths significantly. Also, since monomer relaxation in the complexes is negligible, the intermolecular binding energy depends predominately on the intermolecular coordinate r m ~the distance between Ar and the H2 or HCl centers of mass! and the angle u between the intermolecular axis and the H2 or HCl bond axis.
With u fixed appropriately, from five to eleven points were run at various intermolecular separations around each minima or transition state ~TS! and the calculated ~counterpoise corrected! interaction energies were then fit with either a four-parameter modified Morse potential2
H FA FA
U ~ r m ! 5D e Q exp 2
2 ~ Q11 ! exp 2
~ Q11 ! k e ~ r m 2r e ! QD e
G
Qk e ~ r 2r ! ~ Q11 ! D e m e
GJ
, ~1!
where D e is the binding energy, r e is the equilibrium value of r m , k e is the force constant, and Q is an adjustable parameter, or as in the case of Ar–HCl, a polynomial expansion in the displacement coordinates, Dr m 5r m 2r e . The greatest advantage to using the family of correlation consistent basis sets is their regular convergence characteristics. As previously observed in a large number of benchmark calculations,3,5,29–37 many molecular properties calculated using these sets converge systematically toward the apparent complete basis set ~CBS! limit. Simple functions, such as exponentials6 or polynomials in inverse powers of l max ~the highest angular momenta in a given set, see, e.g., Ref. 38!, have been shown to yield accurate estimates of the CBS limit for total energies, dissociation energies, bond lengths, etc. Throughout the present work, we have chosen to use the function32 2
A ~ n ! 5A ` 1Be 2 ~ n21 ! 1Ce 2 ~ n21 ! ,
~2!
where n is equal to 2–5 for DZ25Z, respectively, A ` is the estimated CBS limit for property A, and A ` , B, and C are adjustable parameters. Other functional forms were also investigated and these generally yielded very similar results. III. RESULTS AND DISCUSSION A. Monomer properties
The basis set dependence and convergence of the polarizabilities of Ar has been discussed in previous work.21 The dipole, quadrupole, and octupole polarizabilities require 2 to 3 diffuse functions of d, f, and g symmetries, respectively. Here we shall examine the basis set and electron correlation requirements of the lowest order permanent moments and dipole polarizability components of H2 and HCl. All properties were determined using the finite field approximation based upon the energy changes due to adding a weak ~0.002 a.u.! electric field F a or field gradient F ab 8 to the oneelectron Hamiltonian. The calculated quadrupole moment Q zz of H2 for the aug-cc-pVnZ sets and the dipole moment m z of HCl for the aug-cc-pVnZ and d-aug-cc-pVnZ sets are given in Table II. These were computed at their respective experimental bond distances39 @r e (H2)50.7414 Å and r e (HCl)51.2746 Å#. The measured values for comparison are Q zz (H2)50.637 60.046 D Å ~in v 50) ~Ref. 40! and m z (HCl)51.093 D ~at r e ) ~Ref. 41!. The five correlated methods used later @MP2, MP3, MP4, CCSD, and CCSD~T!# are compared with RHF values. For the quadrupole moment of H2, the correlated
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J. Chem. Phys., Vol. 109, No. 6, 8 August 1998
Woon, Peterson, and Dunning
TABLE II. Equilibrium quadrupole moment Q zz ~D Å! of H2 and dipole moment m z ~D! of HCl as a function of basis set and level of theory computed at their experimental bond distances (1.401 30a o and 2.408 55a o for H2 and HCl, respectively!. Basis set
RHF
MP2
MP3
MP4
CCSD
CCSD~T!
Q zz (H2) @exp’t: 0.63760.046 D Å ~Ref. 40! # aug-cc-pVDZ 0.609 0.570 0.557 0.551 0.546 aug-cc-pVTZ 0.672 0.641 0.630 0.624 0.620 aug-cc-pVQZ 0.667 0.637 0.626 0.620 0.616 aug-cc-pV5Z 0.665 0.636 0.625 0.619 0.614 a
aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ d-aug-cc-pV5Z a
m z (HCl) 1.241 1.199 1.189 1.181
@exp’t: 1.171 1.115 1.118 1.116
1.093 D 1.158 1.107 1.118 1.117
1.232 1.198 1.188 1.181
1.159 1.113 1.118 1.117
1.146 1.106 1.118 1.117
¯ ¯ ¯ ¯
~Ref. 41!# 1.133 1.146 1.086 1.097 1.099 1.110 1.098 1.109
1.132 1.085 1.099 1.098
1.121 1.085 1.099 1.098
1.120 1.084 1.098 1.098
1.135 1.096 1.110 1.109
Refers to v 50.
methods yield similar values that fall within the experimental uncertainty; they are largely converged at the aug-cc-pVTZ level. The CCSD ~equivalent to a full CI for H2) and MP4 values are very similar, while MP2 is slightly larger. The RHF result overshoots the correlated values by about 10%. The trends for the dipole moment of HCl are completely analogous, with CCSD~T! and MP4 yielding accurate dipole moments and MP2 yielding a value somewhat too large. Adding extra diffuse functions ~i.e., the d-aug-cc-pVnZ sets! has essentially no effect. Table III gives the computed dipole polarizability components perpendicular ( a xx 5 a y y ) and parallel ( a zz ) to the molecular axes of H2 and HCl. Adding extra diffuse functions ~d-aug sets! improves the convergence of the polarizability components of HCl, while they are not necessary for H2 as indicated by the fast convergence of the standard augmented sets. For H2, there is little dependence on correlation method, although electron correlation reduces the magnitude
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slightly compared to RHF. There is greater variation for HCl, however, where correlation raises the values by several percent. MP2 yields somewhat larger values than MP4 and CCSD~T!, which again are very similar. It is also observed that the difference between the doubly and singly augmented basis set results is nearly negligible at the quadruple zeta level. The polarizability components calculated for HCl can be compared to the accurate work of Hammond and Rice,42 whose CCSD~T! calculations in a large atomic natural orbital ~ANO! basis set yielded a xx 516.65 and a zz 518.42 a.u. Our CCSD~T! results with the d-aug-cc-pV5Z basis set, a xx 516.73 and a zz 518.37 a.u., are probably somewhat more accurate due to the more complete d-aug-cc-pV5Z basis set. On the basis of these properties, it is possible to anticipate trends in the convergence of the Ar–H2 and Ar–HCl binding energies. If the intermolecular attraction is solely dependent upon the lowest order induction contribution, one would expect little improvement beyond the aug-cc-pVTZ level for H and Cl ~and d-aug-cc-pVDZ on Ar!. However, the results will demonstrate that higher angular momentum functions and extra diffuse exponents are needed to saturate the basis function space and recover the full binding energies of the two complexes. This reflects the importance of dispersion and higher order induction contributions that depend on the additional functions. B. Ar–H2
Computed results at the counterpoise-corrected MP2, MP4, and CCSD~T! levels of theory for the two equivalent weakly bound ~linear! minima and the interposed ~T-shaped! transition state ~TS! on the Ar–H2 surface are shown in Table IV. The equilibrium dissociation energy D e ~relative to the Ar1H2 asymptote! and intermolecular separation r e ~equilibrium distance from Ar to the H2 center of mass! are given. Eleven different basis sets were employed for ArH2. Nine of these made up a 333 grid of combinations between the double, triple, and quadruple zeta standard cc-pVnZ base sets with one, two, and three additional even-tempered diffuse functions for each angular symmetry present in the base set.
TABLE III. Dipole polarizability components of H2 and HCl as a function of basis set and level of theory ~all values in atomic units! computed at their respective experimental bond lengths ~see Table II!.
a xx 5 a y y
a zz
Species
Basis set
RHF
MP2
MP3
MP4
CCSD
CCSD~T!
RHF
MP2
MP3
MP4
CCSD
CCSD~T!
H2
aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ d-aug-cc-pV5Z
4.406 4.635 4.617 4.607 14.712 15.806 16.121 16.120 16.008 16.114 16.145 16.177
4.364 4.612 4.592 4.582 15.164 16.476 16.818 16.786 16.734 16.902 16.874 16.860
4.360 4.615 4.593 4.582 15.131 16.335 16.601 16.565 16.644 16.710 16.636 16.630
4.358 4.613 4.592 4.581 15.231 16.449 16.720 16.686 16.800 16.853 16.764 16.758
4.352 4.607 4.586 4.575 15.163 16.293 16.525 16.490 16.670 16.657 16.559 16.554
¯ ¯ ¯ ¯ 15.247 16.437 16.693 16.657 16.813 16.833 16.735 16.727
6.537 6.455 6.457 6.459 17.056 17.739 17.916 17.938 17.788 17.958 17.960 17.969
6.553 6.433 6.426 6.423 17.451 18.228 18.392 18.397 18.343 18.488 18.432 18.401
6.556 6.423 6.414 6.411 17.399 18.103 18.225 18.226 18.259 18.361 18.278 18.256
6.554 6.417 6.409 6.406 17.529 18.230 18.352 18.354 18.423 18.501 18.410 18.386
6.548 6.410 6.402 6.399 17.434 18.089 18.196 18.197 18.290 18.346 18.254 18.232
¯ ¯ ¯ ¯ 17.531 18.213 18.330 18.331 18.422 18.484 18.391 18.367
HCl
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J. Chem. Phys., Vol. 109, No. 6, 8 August 1998
Woon, Peterson, and Dunning
TABLE IV. Dissociation energy (D e , cm21) and equilibrium separation (r e , Å! of the minima and transition state ~TS! of Ar–H2 at the counterpoise-corrected MP2, MP4, and CCSD~T! levels of theory. The estimated x-aug limits for each level of basis set quality as well as estimated complete basis set ~CBS! limits are also given. Minimum
TS
Property
Basis set
MP2
MP4
CCSD~T!
MP2
MP4
CCSD~T!
De
aug-cc-pVDZ d-aug-cc-pVDZ t-aug-cc-pVDZ x-aug limit
34.55 35.16 35.36 35.47
34.96 36.08 36.32 36.44
34.26 35.42 35.66 35.78
21.73 23.29 23.59 23.74
22.76 24.87 25.24 25.42
22.32 24.45 24.82 25.00
aug-cc-pVTZ d-aug-cc-pVTZ t-aug-cc-pVTZ x-aug limit
43.85 45.91 46.22 46.36
46.07 48.77 49.12 49.26
44.84 47.54 47.85 47.97
33.34 35.86 36.15 36.26
36.59 39.94 40.30 40.43
35.71 38.98 39.33 39.45
aug-cc-pVQZ d-aug-cc-pVQZ t-aug-cc-pVQZ x-aug limit
48.99 50.13 50.22 50.24
51.69 53.05 53.16 53.19
50.00 51.36 51.47 51.50
38.43 40.21 40.32 40.33
42.58 44.79 44.90 44.90
41.28 43.43 43.54 43.54
aug-cc-pV5Z d-aug-cc-pV5Za
51.11 51.61
53.75 54.31
51.99 52.56
40.83 41.47
45.21 46.00
43.80 44.58
est. CBS ~x-aug! est. CBS ~D-5! est. CBS ~dD-d5!
52.5 52.3 52.6
55.4 55.0 55.4
53.5 53.1 53.5
42.3 41.9 42.6
46.8 46.5 47.3
45.2 44.9 45.7
re
aug-cc-pVDZ d-aug-cc-pVDZ t-aug-cc-pVDZ
3.814 3.811 3.810
3.812 3.808 3.807
3.819 3.815 3.814
3.888 3.878 3.874
3.871 3.858 3.853
3.877 3.864 3.859
aug-cc-pVTZ d-aug-cc-pVTZ t-aug-cc-pVTZ
3.692 3.684 3.681
3.678 3.669 3.667
3.688 3.678 3.675
3.698 3.677 3.672
3.668 3.642 3.637
3.675 3.650 3.646
aug-cc-pVQZ d-aug-cc-pVQZ t-aug-cc-pVQZ
3.641 3.640 3.640
3.628 3.627 3.627
3.639 3.637 3.637
3.637 3.630 3.629
3.604 3.596 3.596
3.613 3.605 3.605
aug-cc-pV5Z
3.629
3.618
3.628
3.618
3.587
3.596
est. CBS ~D-5!
3.617
3.606
3.616
3.605
3.573
3.582
a
Calculated at the CCSD~T!/aug-cc-pV5Z geometries.
The 2D ~two-dimensional! nature of the expansion over angular functions and the diffuse radial region suggests performing two independent extrapolations to predict estimated complete basis set ~CBS! limits for D e and r e . This strategy was first used in a previous study of rare-gas dimers.3 Since the convergence is more rapid when additional diffuse functions are added to a given base set ~e.g., the x-aug-cc-pVDZ series! than the expansions in angular space ~e.g., the aug-cc-pVnZ series!, these multiply augmented extrapolations are performed first to yield ‘‘x-aug’’ limits. These intermediate limits are then extrapolated to yield an estimate of the CBS limit @labeled ‘‘CBS(x-aug)’’#. To illustrate this, consider the convergence behavior of any of the methods in Table IV. The changes in D e due to adding a second diffuse function are on the order of a few percent, while the changes due to adding a third function are less than 1%. In Table IV these CBS(x-aug) limits are compared to values obtained from single extrapolations of the aug-cc-pVnZ (D25) and
d-aug-cc-pVnZ (dD2d5) series. Since the convergence of r e as a function of the number of diffuse functions is very rapid, only the single extrapolations (nZ) have performed for these quantities. As can be seen in Table IV, the DZ x-aug limits represent only a fraction ~65%–68%! of the well depths, so it is critical to use the TZ and QZ sets. As in Ar2, contributions from the quadrupole and octupole polarizabilities are again implicated. The nZ extrapolation indicates that contributions higher than the QZ set are quite small, only about 2 cm21 in D e and 0.02 Å in r e with respect to the x-aug-cc-pVQZ limits. Our CBS predictions for the binding energy of the Ar–H2 minima are 53, 55, and 54 cm21 for MP2, MP4, and CCSD~T!, respectively, at r e values of 3.62, 3.61, and 3.62 Å. The TT 3 (6,8) potential of Le Roy and Hutson15 for H2 ( v 50) has a well depth of 56.6 cm21 at an equilibrium sepa-
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J. Chem. Phys., Vol. 109, No. 6, 8 August 1998
Woon, Peterson, and Dunning
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TABLE V. Dissociation energies (D e , cm21) and equilibrium separations (r e , Å! for the two minima and saddle point on the ArHCl surface at the counter-poise-corrected MP2, MP4, and CCSD~T! levels of theory. The estimated CBS limiting values are also given. Ar¯HCl minimum
HCl¯Ar minimum
Transition state
Property
Basis Set
MP2
MP4
CCSD~T!
MP2
MP4
CCSD~T!
MP2
MP4
CCSD~T!
De
aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z Est. CBS ~D-5!
132.56 176.08 197.44 202.88 208.1
116.02 156.53 176.55 180.65 185.7
112.94 150.15 168.13 172.09 176.6
105.50 143.78 165.80 175.34 180.2
80.76 118.49 139.05 147.73 152.3
78.72 113.79 132.43 140.70 144.7
61.94 94.73 108.31 116.03 118.6
49.80 82.62 96.25 103.49 106.1
49.16 80.23 92.57 99.49 101.8
d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ d-aug-cc-pV5Za Est. CBS ~dD-d5!b
138.02 184.16 199.81 203.97 207.5
121.92 165.82 179.33 181.86 185.0
118.81 159.31 170.89 173.27 175.9
110.97 157.59 173.14 179.68 182.8
88.07 133.55 147.15 152.14 154.9
86.21 128.54 140.33 144.93 147.2
67.78 104.74 115.30
57.12 94.16 104.02
56.65 91.61 100.12
121.2
109.4
104.7
re
a
a
aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z Est. CBS ~D-5!
4.169 4.011 3.970 3.960 3.951
4.216 4.053 4.009 4.001 3.991
4.225 4.067 4.024 4.016 4.006
3.795 3.685 3.618 3.601 3.584
3.870 3.743 3.667 3.649 3.630
3.878 3.755 3.682 3.664 3.646
4.132 3.976 3.917 3.892 3.880
4.207 4.019 3.953 3.928 3.914
4.212 4.029 3.966 3.940 3.927
d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ Est. CBS ~dD-dQ!
4.151 4.005 3.970 3.951
4.198 4.045 4.009 3.990
4.206 4.058 4.024 4.006
3.790 3.663 3.613 3.584
3.863 3.713 3.660 3.629
3.870 3.725 3.674 3.645
4.121 3.951 3.904 3.878
4.187 3.988 3.937 3.909
4.190 3.996 3.949 3.924
Calculated at the respective aug-cc-pV5Z equilibrium geometries for each method. Only d-aug-cc-pVDZ through d-aug-cc-pVQZ were fit for the transition state.
b
ration of 3.609 Å. The recent XC~fit! potential of Bissonnette et al.16 yields nearly identical values (56.6 cm21 and 3.610 Å!. Our best estimates of the binding energies ~relative to Ar1H2) for the transition state are 43, 47, and 46 cm21 at r e values of 3.61, 3.57, and 3.58 Å. The empirical values from the TT 3 (6,8) potential are D e 548.4 cm21 and r e 53.567, while those from the more accurate XC~fit! potential are 47.8 cm21 and 3.578 Å, respectively. Converting the above empirical binding energies to those appropriate for H2 at r e is expected to lower these values by about 0.7 cm21. Both MP4 and CCSD~T! at the CBS limit yield results very close to these empirical values, i.e., within 1 to 2 cm21 in D e and 0.01 Å in r e . MP2 is the least accurate method, but it still does remarkably well with differences of ;3 cm21 for the minimum and ;4 cm21 for the TS compared to the empirical potentials. The barrier heights for MP2, MP4, and CCSD~T! at the estimated CBS limit are 10, 8, and 8 cm21, respectively, where that of the XC~fit! potential is 8.73 cm21 ~for H2 in v 50). The most accurate previous ab initio work for Ar–H2 are the symmetry-adapted perturbation theory ~SAPT! calculations of Williams and co-workers.43–45 They also employed basis sets with (spd f g) functions and reported D e 557.6 cm21, r e 53.59 Å for the minima and D e 548.1 cm21, r e 53.56 Å for the TS. Also, Schwenke, Walch, and Taylor46 have published a global ArH2 potential-energy surface using averaged coupled pair functional ~ACPF! multireference wave functions and empirical corrections to better reproduce the TT 3 (6,8) potential of Le Roy and Hutson.15
C. Ar–HCl
Counterpoise-corrected MP2, MP4, and CCSD~T! results for the three critical points on the Ar–HCl surface with various basis sets are summarized in Table V. The two linear minima are designated Ar¯HCl ~global minimum! and HCl¯Ar ~local minimum!. As in Ar–H2, the intervening transition state is bound with respect to the Ar1HCl asymptote. We have adopted u 594.02° ~measured as Ar–CMHCl –Cl, where CMHCl is the center of mass of HCl! for the angular displacement of the TS as derived from Hutson’s H6(3) potential.47 The barrier height was found to be very insensitive to changes of several degrees in u. Exchanging one H for Cl in moving from Ar–H2 to Ar– HCl limits the largest x-aug calculations to the d-aug-cc-pVnZ series. Hence, for Ar–HCl we were unable to perform the double extrapolation that was applied to the Ar–H2 data. Furthermore, only single point calculations were run with the d-aug-cc-pV5Z basis set using the CCSD~T! center-of-mass separations, and the largest d-aug set used for the transition state was d-aug-cc-pVQZ due to computer resource limitations. Table V, therefore, includes only ‘‘nZ’’ extrapolations for the aug-cc-pVnZ and d-augcc-pVnZ series. The calculated binding energies for all three critical points at the MP2, MP3, MP4, CCSD, and CCSD~T! levels of theory are plotted in Fig. 1 as a function of the aug-cc-pVnZ basis sets. Well behaved, nearly exponential convergence is observed in each case. However, the convergence of D e for the Ar–ClH minimum is clearly slower than
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FIG. 1. The calculated binding energies ~in cm21) at the MP2 ~m!, MP3 ~j!, MP4 ~d!, CCSD ~L!, and CCSD~T! ~l! levels of theory for ~a! Ar–HCl, ~b! Ar–ClH, and ~c! the intervening transition state are plotted as a function of the aug-cc-pVnZ basis set used. The binding energies obtained from the H6(4,3,0) empirical potential of Hutson are shown by the dotted lines.
that of D e for the more strongly bound Ar–HCl minimum. Similar behavior was also observed recently in the closely related Ar–HF system.48 Furthermore, as shown in Fig. 1 and explicitly tabulated in Table V, there are relatively large
differences among binding energies obtained with the different correlation methods. For example, with the aug-cc-pV5Z basis set, which is very near the CBS limit, D e (Ar–HCl) from MP2 calculations is over 20 cm21 more strongly bound
FIG. 2. The calculated CCSD~T! binding energies ~in cm21) calculated with the aug-cc-pVnZ ~j! and d-aug-cc-pVnZ ~d! basis sets are compared for ~a! Ar–HCl, ~b! Ar–ClH, and ~c! the connecting transition state. The binding energies obtained from the H6(4,3,0) empirical potential of Hutson are shown by the dotted lines.
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FIG. 3. The calculated equilibrium intermolecular center of mass separations ~in Å! at the MP2 ~m!, MP3 ~j!, MP4 ~d!, CCSD ~L!, and CCSD~T! ~l! levels of theory for ~a! Ar–HCl, ~b! Ar–ClH, and ~c! the intervening transition state are plotted as a function of the aug-cc-pVnZ basis set used. The values obtained from the H6(4,3,0) empirical potential of Hutson are shown by the dotted lines.
than from MP4 calculations, which is nearly 10 cm21 larger than that calculated with the CCSD~T! method. This is in contrast to recent results for Ar–HF,48 as well as the Ar–H2 values shown above, and indicates the increased importance of dispersion contributions in HCl compared to HF ~and H2). The importance of dispersion-type interactions, especially in the case of the Ar–ClH minimum, is also indicated by the relative convergence of the aug-cc-pVnZ and d-aug-ccpVnZ results. Figure 2 compares these corresponding binding energies at the CCSD~T! level of theory and demonstrates the strong effect of the additional diffuse functions for Ar–ClH, as well as the transition state; the effects are not inconsequential even at the 5Z level. As in Ar2 and Ar–H2, the importance of higher angular momentum functions can also be seen from the significant improvements in D e that are evident when the sets are enlarged from, e.g., d-aug-ccpVDZ to d-aug-cc-pVQZ. But once again, there are only relatively small gains when going beyond quadruple zeta quality sets ~especially beyond d-aug-cc-pVQZ!—only 5 – 9 cm21 in D e and ;0.01 Å in r e . Using a CBS extrapolation is obviously a cost-effective means to approximate these small higher order effects. In the case of the equilibrium center of mass separations, smaller differences are observed between the aug and d-aug results. The differences, however, among the correlation methods are not insignificant, as shown in Fig. 3 for the MPn and CCSD methods. In each case, the r e ’s from the MP2 calculations are too short by 0.04–0.06 Å, while those from the MP4 and CCSD~T! calculations agree to within ;0.01 Å. The best estimates ~CBS limits from the d-aug-cc-pVnZ and aug-cc-pVnZ series for D e and r e , respectively! for the
Ar¯HCl global minimum at the MP2, MP4, and CCSD~T! levels of theory are 208, 185, and 176 cm21 for the binding energy and 3.95, 3.99, and 4.01 Å for the equilibrium center of mass separation, respectively. The values from the H6(4,3,0) potential of Hutson17 are D e 5174.3 cm21 ~HCl at its r e ) and r e 54.004 Å, with quoted uncertainties of ;63 cm21 and 0.01 Å, respectively. The CCSD~T! method yields a binding energy within this estimated uncertainty, while the MP4 method overestimates D e by about 8 – 11 cm21. The agreement for the other minimum and the barrier height is similarly good. Specifically, for the HCl¯Ar minimum, the MP4 and CCSD~T! predictions for D e are 155 and 147 cm21 ~Hutson: 148.365.0 cm21 for HCl in v 50) at r e values of 3.63 and 3.65 Å, respectively ~Hutson: 3.61060.02 Å). In this case it would appear that the equilibrium separation for the HCl¯Ar minimum may be somewhat underestimated by the H6(4,3,0) potential. The estimated CBS limit barrier heights at the MP4 and CCSD~T! levels are 76 and 71 cm21 ~binding energies with respect to Ar1HCl of 109 and 105 cm21), while the H6(4,3,0) value is 71.065.0 cm21. Chalasinski et al.49 has reported supermolecular MP4 ~with BSSE corrections! results for ArHCl with a (spd f ) basis, yielding 151.8 and 122.6 cm21 for the binding energies of the two minima, which are consistent with our augcc-pVTZ results. Further unpublished results by Chalasinski and co-workers1 also employed a set of bond functions, yielding improved values of 174.6 and 139.3 cm21 for the two minima, respectively ~bond distances of 4.18 and 3.61 Å!.
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TABLE VI. Estimated complete basis set ~CBS! limits obtained in the present work compared to results from empirical, as well as other ab initio, potentials. Ar–H2 21
D e (cm )
Method
Transition state D e (cm21)
r e (Å)
r e (Å)
MP2 MP4 CCSD~T!
53 55 54
3.62 3.61 3.62
43 47 46
3.61 3.57 3.58
TT3 ~6,8!a,b XC~fit!a,c SAPTd
56.6 56.6 57.6
3.567 3.610 3.59
48.4 47.8 48.1
3.567 3.578 3.56
Ar–HCl
Ar–ClH
Transition state
D e (cm21)
r e (Å)
D e (cm21)
r e (Å)
D e (cm21)
r e (Å)
MP2 MP4 CCSD~T!
208 185 176
3.95 3.99 4.01
183 155 147
3.58 3.63 3.65
121 109 105
3.88 3.91 3.93
H6~4,3,0!e MP4f MP41b f g
176.063.0 151.8 174.6
4.0060.01
148.365.0 122.6 139.3
3.6160.02
105.065.0
3.8960.02
Method
4.18
3.61
These correspond to values for H2 in v 50. The binding energies should be decreased by about 0.7 cm21 for H2 at r e . b Reference 15. c Reference 16. d References 43–45. e Reference 17. These correspond to values for HCl in v 50. For Ar–HCl with HCl at its r e , D e 5174.3 cm21. The values for D e ~Ar–ClH! and D e ~T.S.! should also probably be decreased by about 1.7 cm21 for HCl at r e . f Reference 49. g MP41bond function calculations reported in Ref. 1. a
IV. CONCLUSIONS
The weakly bound Ar–H2 and Ar–HCl complexes have been studied with large basis sets and correlated methods including MPn and CCSD~T!. Basis sets with diffuse (d f gh) polarization functions are essential for predicting an accurate value for the interaction energy, while an extrapolated estimate of the complete basis set limits can be used to account for the remaining higher order contributions. As in previous benchmark studies, the MP4 method, and particularly the CCSD~T! method, emerge as the most reliable methods for describing weak complexes. Agreement between the computed binding energies and intermolecular separations with values derived from empirical potentials is very good, especially for the global minima. The empirical potentials are known to be less reliable for other critical points where less experimental data exists,17 but agreement is still quite good. Table VI compares our best calculated estimates for the binding energies and equilibrium separations for ArH2 and ArHCl with the available empirical potentials and some previous ab initio results. For Ar–H2, MP4 and CCSD~T! calculations predict D e ’s of 55 and 54 cm21 ~estimated complete basis set limits! respectively, at center of mass r e separations of 3.61 and 3.62 Å. Even the MP2 method predicts reasonable results: D e 553 cm21 at a separation of 3.62 Å. Our best estimates of the binding energies for the intervening T-shaped transition state are 47 and 46 cm21 at r e values of 3.57 and 3.58 Å for the MP4 and CCSD~T! meth-
ods, respectively. The predictions are to be compared to values of 55.9 ~minimum! and 47.1 ~TS! cm21 at separations of 3.61 Å ~minimum! and 3.58 Å ~TS! from the XC~fit! potential of Bissonnette et al.16 The best estimates for the Ar¯HCl global minimum at the MP2, MP4, and CCSD~T! levels of theory yield D e ’s of 208, 185, and 176 cm21 and r e ’s of 3.95, 3.99, and 4.01 Å, respectively. For the HCl¯Ar minimum, the MP4 and CCSD~T! methods predict D e ’s of 155 and 147 cm21 at r e values of 3.63 and 3.65 Å. The estimated CBS limit for D e at the barrier are 109 ~MP4! and 105 @CCSD~T!# cm21. The corresponding empirical D e ’s for comparison are 174.3, 146.6, and 103.3 cm21, respectively ~including approximate corrections for HCl v 50→r e ). For the Ar–HCl complex, there is a significant variation in the D e ’s predicted by the three correlation methods considered here—the CCSD~T! method appears to yield the most accurate description, while the MP2 method yields much less reliable results. Similar behavior has also been observed for the HCl dimer.50 This is in strong contrast to Ar–H2 and recent results on Ar–HF ~Ref. 48! where the MP2 and MP4 methods yielded results of very similar accuracy to the CCSD~T! method. ACKNOWLEDGMENTS
The authors wish to acknowledge the support of the Chemical Sciences Division in the Office of Basic Energy Sciences of the U.S. Department of Energy at Pacific Northwest Laboratory, a multiprogram national laboratory oper-
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J. Chem. Phys., Vol. 109, No. 6, 8 August 1998
ated by Battelle Memorial Institute for the U.S. Department of Energy, under Contract No. DE-AC06-76RLO 1830. This research was also supported by the Associated Western Universities, Inc., Northwest Division under Grant DE-FG0689ER-75522 with the U.S. Department of Energy. K.A.P. would also like to thank Dr. R. J. Le Roy for helpful correspondence on the Ar–H2 empirical potentials. Cf., Chem. Rev. 94~7! ~1994!. D. E. Woon, Chem. Phys. Lett. 204, 29 ~1993!. 3 D. E. Woon, J. Chem. Phys. 100, 2838 ~1994!. 4 D. E. Woon, T. H. Dunning, Jr., and K. A. Peterson, J. Chem. Phys. 104, 5883 ~1996!. 5 K. A. Peterson and T. H. Dunning, Jr., J. Chem. Phys. 102, 2032 ~1995!. 6 D. Feller, J. Chem. Phys. 96, 6104 ~1992!. 7 S. S. Xantheas and T. H. Dunning, Jr., J. Chem. Phys. 98, 8037 ~1993!. 8 S. S. Xantheas and T. H. Dunning, Jr., J. Chem. Phys. 99, 8774 ~1993!. 9 S. S. Xantheas, J. Chem. Phys. 100, 7523 ~1994!. 10 M. W. Feyereisen, D. Feller, and D. A. Dixon, J. Phys. Chem. 100, 2993 ~1996!. 11 S. S. Xantheas and T. H. Dunning, Jr., J. Phys. Chem. 96, 7505 ~1992!. 12 S. S. Xantheas and T. H. Dunning, Jr., J. Phys. Chem. 98, 13489 ~1994!. 13 D. Feller, E. D. Glendening, R. A. Kendall, and K. A. Peterson, J. Chem. Phys. 100, 4981 ~1994!. 14 D. E. Woon and T. H. Dunning, Jr., J. Am. Chem. Soc. 117, 1090 ~1995!. 15 R. J. Le Roy and J. M. Hutson, J. Chem. Phys. 86, 837 ~1987!. 16 C. Bissonnette, C. E. Chuaqui, K. G. Crowell, R. J. Le Roy, R. J. Wheatley, and W. J. Meath, J. Chem. Phys. 105, 2639 ~1996!. 17 J. M. Hutson, J. Phys. Chem. 96, 4237 ~1992!. 18 T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 ~1989!. 19 R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys. 96, 6796 ~1992!. 20 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 98, 1358 ~1993!. 21 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 100, 2975 ~1994!. 22 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 ~1995!. 23 MOLPRO is a package of ab initio programs written by H.-J. Werner and P. J. Knowles with contributions from J. Almlo¨f, R. D. Amos, M. J. O. Deegan, S. T. Elbert, C. Hampel, W. Meyer, K. A. Peterson, R. M. Pitzer, A. J. Stone, P. R. Taylor, and R. Lindh. 1 2
Woon, Peterson, and Dunning
2241
24
K. Ragavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 157, 479 ~1989!. 25 C. Hampel, K. A. Peterson, and H.-J. Werner, Chem. Phys. Lett. 190, 1 ~1992!. 26 M. J. O. Deegan and P. J. Knowles, Chem. Phys. Lett. 227, 321 ~1994!. 27 C. Mo” ller and M. S. Plesset, Phys. Rev. 46, 618 ~1934!. 28 S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 ~1970!. 29 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 99, 1914 ~1993!. 30 K. A. Peterson, R. A. Kendall, and T. H. Dunning, Jr., J. Chem. Phys. 99, 1930 ~1993!. 31 K. A. Peterson, R. A. Kendall, and T. H. Dunning, Jr., J. Chem. Phys. 99, 9790 ~1993!. 32 K. A. Peterson, D. E. Woon, and T. H. Dunning, Jr., J. Chem. Phys. 100, 7410 ~1994!. 33 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 101, 8877 ~1994!. 34 K. A. Peterson and T. H. Dunning, Jr., J. Chem. Phys. 106, 4119 ~1997!. 35 A. K. Wilson and T. H. Dunning, Jr., J. Chem. Phys. 106, 8718 ~1997!. 36 K. A. Peterson and T. H. Dunning, Jr., J. Phys. Chem. 101, 6280 ~1997!. 37 K. A. Peterson, A. K. Wilson, D. E. Woon, and T. H. Dunning, Jr., Theor. Chem. Acc. 97, 251 ~1997!. 38 J. M. L. Martin, Chem. Phys. Lett. 259, 669 ~1996!. 39 K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules ~Van Nostrand, Princeton, 1979!. 40 W. H. Flygare and R. C. Benson, Mol. Phys. 20, 225 ~1971!. 41 E. W. Kaiser, J. Chem. Phys. 53, 1686 ~1970!. 42 B. L. Hammond and J. E. Rice, J. Chem. Phys. 97, 1138 ~1992!. 43 H. L. Williams, K. Szalewicz, B. Jeziorski, R. Moszynski, and S. Rybak, J. Chem. Phys. 98, 1279 ~1993!. 44 R. Moszynski, B. Jeziorski, S. Rybak, K. Szalewicz, and H. L. Williams, J. Chem. Phys. 100, 5080 ~1994!. 45 R. Moszynski, B. Jeziorski, P. E. S. Wormer, and A. van der Avoird, Chem. Phys. Lett. 221, 161 ~1994!. 46 D. W. Schwenke, S. P. Walch, and P. R. Taylor, J. Chem. Phys. 98, 4738 ~1993!. 47 J. M. Hutson, J. Chem. Phys. 107, 2451 ~1988!. 48 T. van Mourik and T. H. Dunning, Jr., J. Chem. Phys. 107, 2451 ~1997!. 49 G. Chalasinski, M. M. Szczesniak, and B. Kukawska-Tarnawska, J. Chem. Phys. 94, 6677 ~1991!. 50 K. A. Peterson ~in preparation!.
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