J . Phys. Chem. 1988, 92,4329-4333
4329
Rare-Gas Interaction Energy Curves M. Krauss,* National Bureau of Standards, Gaithersburg, Maryland 20899
Rebecca M. Regan, and D. D. Konowalow Department of Chemistry, State University of New York at Binghamton, Binghamton, New York 13901 (Received: October 15, 1987)
Interaction energy curves for all the heteronuclear rare-gas dimers including He through Xe are calculated as the sum of the exchange repulsion and the attractive interatomic correlation energy between the atoms. The self-consistentfield (SCF) energy represents the repulsion, and a perturbation method is used to calculate a dispersion energy that takes account of the overlap of the atoms. The effect of overlap is to reduce or damp the dispersion energy. This hybrid model yields binding energies that are in the range of 75-90% of experimental values. The differences are due to slow convergence of the damped dispersion energy and to the difficulty in obtaining Hartree-Fock limit SCF energy curves within 1-cm-' accuracy. The comparable accuracy for the binding energies over a range of atoms from He through Xe suggests that the model correctly represents the dominant energy terms. The model is presently widely applicable since it depends only on the polarization behavior of the fragments for the dispersion calculation and on SCF energies, which can be more accurately calculated in routine calculations than can correlated energies.
Introduction The fragments in a van der Waals complex are sufficiently weakly interacting that the energy curve can be approximated as the sum of the Hartree-Fock (HF) or single-configuration selfconsistent field (SCF) energy, E l , and the interfragment correlation energy, D.' The interfragment correlation energy can be expanded as a superposition of contributions from configurations that are ordered by the number of excitations from the dominant HF configuration. Interfragment doubly excited configurations are categorized by single excitations from the occupied localized orbitals on each "atom". Since the virtual orbitals are difficult to localize in an orthogonal set, the correlation energy consists of both the damped dispersion and a charge-transfer correlation energy. Although accurate calculations on small systems2g3have shown that correlation of the fragments cannot be ignored for quantitative results, the difficulty of treating even modest sized systems forces the continued exploration of an additive model such as the present one. In one version, the damped dispersion energy is obtained by a second-order perturbation theory that accounts for the charge overlap of the interacting fragments! A convergent multipole expansion series is found to represent the dispersion in the form
D(R) =
-x
C(LA,LB)x ( L , , L ~ ; R ) R ~ ( ~ + ' ) LAJB where C(LA,LB) are the usual van der Waals or dispersion coefficients and LA+ LB = L. The reduction of the Coulombic dispersion energy by the charge overlap is measured by the damping function, x . The damping functions and dispersion coefficients are calculated by using the atomic or separated fragment polarizabilities of imaginary argument, al(iw),which can be approximated by al(iw)
= a1(0)m2/(m2
+ 0')
where a(0)is the static polarizability and q , is an energy parameter that is determined variati~nally.~The calculation of the S C F energy curves for any pair of rare-gas atoms is now simplified by the use of effective core potentiah6 Application of this model (1) Karo, A.; Krauss, M.; Wahl, A. C. Inr. J . Quantum Chem. Quantum Chem. Symp. 1973, 7 , 143. (2) Liu, B.;McLean, A. D. J. Chem. Phys. 1973, 59,4557. (3) Meyer, W.; Frommhold, L. Phys. Reu. A 1986, 33, 3807. (4) Koide, A. J. Phys. 1976, B9, 3173. Koide, A,; Meath, W. J.; Allnatt, A. R. Chem. Phys. 1981,58, 105. (5) Krauss, M.; Neumann, D. B. J . Chem. Phys. 1979, 71, 107.
0022-3654/88/2092-4329$01.50/0
to the calculation of the energy curves of He2,' Ar2,*and Xe29 has shown it to be applicable to heavy as well as light systems. Other analogous models for calculating van der Waals energy curves using semiempirical damped dispersion parameters have also been used successfully.1° In this paper SCF, damped dispersion, and total energy curves will be presented for all heteronuclear rare-gas interactions for He through Xe. The earlier calculations have already shown that the S C F energy curves can be obtained with sufficient accuracy both by comparison to published curves and by selective addition to the basis set used to represent the molecular orbitals. In addition, the recent results of Toennies and Tang" on the Born potentials for mixed rare gases are found to agree well with the Born fits to the S C F energy curves. We carry the dispersion expansion through the Eloterms including both quadrupolequadrupole and dipoleoctupole contributions. The results support earlier observations12on simpler systems that the convergence of the dispersion energy as a function of angular momentum is slow. Within the present model, the calculated bond energies agree with experimentally deduced values within 10-25%. There is a long list of effects not included in the present model, including the charge-transfer correlation, the effect of exchange on the dispersion, and the intraatomic correlation effect on the repulsion and its coupling into the dispersion. The magnitude of these effects cannot be quantified at this time for interactions that involve heavier atoms, but configuration interaction calculations for HeNe show that the model does approximate the comparable configuration interaction calculation, at least in this case.
Results
SCF Energy Curves. The SCF calculations used both allelectron and effective core potential (ECP) basis sets. All-electron bases for He, Ne, Ar, and Kr were chosen from the largest sets developed by Bagus et al.,13 Le., He 3s, N e 5s,4p, Ar 6s,5p, and (6) Krauss, M.; Stevens, W. J. Annu. Rev. Phys. Chem. 1984, 35, 357. (7) Krauss, M.; Neumann, D. B.; Stevens, W. J. Chem. Phys. Lett. 1979, 66, 29. (8) Krauss, M.; Stevens, W. J. Chem. Phys. Lett. 1982, 85, 423. (9) Krauss, M.; Stevens, W. J.; Neumann, D. B. Chem. Phys. Lett. 1980, 71, 500. (10) Ahlrichs, R.;Penco, R.; Scoles, G. Chem. Phys. 1977,19,119. Tang, K. T.; Toennies, J. P. J . Chem. Phys. 1977, 66, 1496. Douketis, C . ; Scoles, G.; Mordetti. S.;Zen, M.; Thakkar, A. J. J . Chem. Phys. 1982, 76, 3037. (1 1) Tang, K. T.; Toennies, J. P. Z . Phys. D At., Mol. Clusters 1986, 1 , 91.
(12) Dacre, P. D. Chem. Phys. Lett. 1977, 50, 147.
0 1988 American Chemical Society
4330 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
Krauss et al.
TABLE I: SCF STF Polarization Bases
all-electron bases He"
nl 2P 2p 3d 4f
Neb
r 1.624 2.812 2.273 5.062
Arb
f
nl 3s 3p 3d 3d 4f
ECP bases
1.550 1.200 1.200 2.100 1.850
Krb
r
nl 4s 4p 3d 3d 4f
1.336 1.200 1.605 1.120 1.850
Kr
f
nl 5s 5p 4d 4f
1.650 1.104 1.514 1.850
nl 4s 4s 4s 5s 4P 4P 4P 4d
Xe
r
nl 55 55 55 5p 5P 5d 4f
6.29291 3.01652 1.68381 1.48259 16.85011 2.65893 1.44650 1.46659
r 3.93474 2.64807 1.26278 2.69358 1.56798 1.43413 1.85000
dThese functions were added to the basis of the following: Tatewaki, et al., Mol. Phys. 1984,53, 233. bThese functions were added to the Bagus setI3 as S C F polarization functions. TABLE 11: Exponential Fit to SCF Energy Curves
TABLE I V SCF Atomic Polarizabilities
energy = AdbR(au) Alb
He Ne
He 23.8 2.430
Ne 53.1 2.422 192.1 2.493
Ar
Ar 88.9 2.107 199.2 2.117 300.2 1.896
Kr 140.7 2.051 217.0 2.015 398.9 1.849 999.7 1.889
Kr Xe
Xe 132.7 1.878 290.0 1.929 387.6 1.734 676.6 1.730 805.1 1.650
TABLE 111: Atomic Polarization Bases"
He Ne nl f nl f 2p 1.156d 3s 1.150d 3d 1.188q 3p 2.550 4f 1.2030 3d 1.525 4d 2.141 3p 1.250q 3d 1.634 4f 1.437 3p 0.6200 3d 0.996 4f 1.709 5g 1.475
Ar
nl 4s 3d 4d 4p 3d 4f 4p 3d 4f 5g
f 1.300d 1.093 2.211 0.866q 1.142 1.129 0,8660 0.855 1.362 1.131
Kr
nl 5s 5p 5p 4d 5d 5p 4d 4f 5p 4d 4d 4f 5g
Xe
nl 1.483d 5d 0.990 6d 0.200 5p 1.387 5d 0.900 5f 1.OOOq 5d 1.453 5d 1.002 5f 1.5670 5g 2.500 0.961 1.029 0.994 f
f 1.537d 1.037 1.131q 1.522 1.237 1.4340 0.887 1.300 0.976
"The designations d, q, or o denote the first function for the dipole, quadrupole, or octupole polarization basis.
Kr 8s,7p,4d. The added polarization functions are shown in Table I. T h e Xe atom was represented with an ECP in both t h e SCF a n d polarization calculations, but the ECP was used for Kr only for t h e polarizability calculation. A double-l basis was energy optimized for the valence electrons in the ECP calculations. T h e SCF interaction energy curves were fit to the Born expression Ae-bRover a region around the energy minimum for the total energy, a n d these parameters a r e given in T a b l e 11. T h e fit is not accurate enough t o be used for evaluating t h e total energy, and t h e acutal computed energy points a r e given below in Table VI. Comparison of our results with published SCF energy curves near the HF limit show a good agreement. T w o examples will illustrate the quantitative accuracy. Wells and Wilson14examined the variation of the SCF energy with basis for Ar?. A t a distance of 7.5 au, our SCF interaction energy agrees with the Wells a n d Wilson value to within 1 cm-l. Present results also agree to within 1 cm-' for t h e interaction energy reported for H e A r at a distance of 5.5 a ~ Comparable . ~ SCF energy curves h a d been reported ~~
(13) Bagus, P. S.;Gilbert, T. L.; Roothaan, C. C. J. J . Chem. Phys. 1972, 56, 5195; Argonne National Laboratory report, Feb 1972. (14) Wells, B. H.; Wilson, S. Mol. Phys. 1985, 54, 787.
present work numerical CHF' exptlb Ne present work numerical CHF" exptl Ar present work numerical CHFc exptl Kr present work numerical CHFc exptl Xe present work numerical CHFC exptl
1.30 1.332 1.38
2.27 2.33
9.50 10.01
2.36 2.374 2.66
6.36 6.42
33.9 34.3
10.75 10.76 11.1
48.0 50.2
515.0 531.0
16.30 16.46 16.74
94.1 95.6
1407.0 1260.0
27.55 27.06 27.29
213.7 212.6
3670.0 3602.0
"Sitter, R. E., Jr.; Hurst, R. P. Phys. Rev.A 1972, 5, 5. bTeachout, R. R.; Pack, R. T. At. Data 1971, 3, 195. CReference21. TABLE V SCF van der Waals Coefficients
DD He2 HeNe HeAr HeKr HeXe Ne2 NeAr NeKr NeXe Ar2 ArKr ArXe Kr2 KrXe Xe2
1.40 2 70 9.66 13.5 20.6 5.23 18.4 25.7 38.9 68.3 96.7 149 137 214 334
DO
OD
00
DO
OD
6.48 6.48 56.1 52.7 52.7 17.4 12.5 151 176 102 107 44.2 917 2016 357 191 61.7 1625 4847 497 405 93.5 3432 11241 752 32.4 32.4 407 341 34 1 205 120 3833 1209 2489 362 168 4423 9150 1689 765 255 9361 21113 2567 761 761 15802 14407 14407 1371 1077 28504 35201 20468 2930 1664 60921 82723 31766 1952 1952 51749 50340 50340 4185 3036 11 1073 118960 78790 6539 6539 239074 187538 187538
earlier for H e A r , NeAr, NeKr, a n d ArKr" in good agreement with those reported earlier by Ahlrichs et al.IOfor HeAr and NeAr. For the Xe2 energy curve, the SCF interaction energy curve obtained with the ECP a n d the valence basis set is more accurate than are published curves for a n all-electron SCF.I6 W e d o not claim, however, that the energy curves are a t the HF limit by any means. Estimates of the basis set superposition energy (BSSE) were made with the counterpoise method" that indicate significant shifts m a y be required in the SCF curves. However, when SCF (15) Birnbaum, G.; Krauss, M.; Frommhold, L. J . Chem. Phys. 1984,80, 2669. (16) Wadt, W. R. J. Chem. Phys. 1978, 68, 402. (17) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553.
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4331
Rare-Gas Interaction Energy Curves TABLE VI: Damped Dispersion, SCF, and Total Energy (cm-')
R
DD
DQ
-D QD DO
OD
QQ
SCF
R
total
DD
DQ
-D QD DO
OD
QQ
SCF
total
HeNe 5.50 5.65 5.80 6.00
20.6 17.7 15.2 12.5
4.0 3.3 2.7 2.1
2.8 2.3 1.9 1.5
1.1 0.8 0.7 0.5
0.5 0.4 0.4 0.3
0.9 0.7 0.6 0.4
19 13 9 6
-10.8 -12.1 -12.3 -11.6
6.25 6.50 6.75 7.00
33.5 26.9 21.7 17.6
8.2 6.3 4.8 3.7
3.4 2.6 2.0 1.5
3.3 2.4 1.7 1.3
0.5 0.4 0.3 0.2
1.3 1 .o 0.7 0.5
37 20 13 7
-13.1 -17.8 -18.7 -17.8
6.75 7.00 7.25 7.50
29.8 24.3 19.9 16.4
2.6 2.0 1.6 1.2
8.0 6.3 4.9 3.8
0.3 0.3 0.2 0.2
3.5 2.6 2.0 1.5
1.1
0.9 0.7 0.5
30 18 6
-15.1 -18.4 -18.7 -17.3
7.25 7.50 7.75 8.00
29.4 10.0 24.4 7.9 20.2 6.3 16.9 5.0
2.2 1.8 1.4 1.1
4.4 3.4 2.5 1.9
0.2 0.2 0.2 0.1
1.3 1 .o 0.7 0.6
36 22 14 8
-11.4 -16.0 -17.5 -17.1
6.25 6.50 6.75 7.00
63.0 15.0 50.8 11.6 41.1 8.9 33.4 6.9
9.0 6.9 5.4 4.1
5.9 4.3 3.2 2.3
1.7 1.3 1.0 0.7
3.5 2.6 1.9 1.5
77 45 25 14
-20.6 -33.3 -36.0 -34.7
6.75 7.00 7.25 7.50
55.8 45.7 37.5 30.9
14.7 11.7 9.1 7.2
7.0 5.5 4.3 3.4
6.1 4.7 3.6 2.7
1.2 0.9 0.7 0.5
3.0 2.3 1.7 1.3
58 34 20 11
-29.6 -36.0 -36.9 -34.5
7.25 7.50 7.75 8.00
54.9 45.6 37.9 31.7
18.3 14.6 11.6 9.2
5.9 4.7 3.8 3.0
0.8 0.7 0.5 0.4
7.8 6.0 4.6 3.5
3.3 2.6 2.0 1.5
53 33 20 12
-38.0 -41.5 -40.2 -37.1
7.25 134.0 29.8 23.5 10.6 7.50 111.7 24.1 18.9 8.3 7.75 93.2 19.4 15.2 6.6 8.00 78.0 15.6 12.2 5.1
6.6 5.1 4.0 3.1
8.3 6.6 5.2 4.1
130 80 48 29
-82.5 -94.6 -95.2 -89.8
7.50 164.5 48.6 26.0 7.75 138.7 44.5 21.3 8.00 117.4 32.1 17.4 8.25 98.9 26.1 14.2
6.5 5.2 4.1 3.3
18.6 14.7 11.6 9.1
12.6 10.1 8.1 6.4
192 123 79 59
-85.1 -106.4 -111.5 -108.0
35.8 29.7 24.5 20.2
18.7 15.0 12.0 9.5
10.6 8.6 7.0 5.7
16.0 13.0 10.5 8.4
226 144 91 55
-99.6 -127.9 -136.9 -135.2
6.25 7.00 8.00
9.8 5.0 2.3
1.6 0.7 0.2
1.1 0.5 0.2
0.4 0.1
0.2 0.1
0.3 0.1
3
-10.3
7.25 8.00 9.00
14.4 8.0 4.0
2.9 1.4 0.5
1.2 0.6 0.2
0.9 0.4 0.1
0.2 0.1 0.0
0.4 0.2 0.1
4
-15.8
7.75 8.00 9.00
13.5 11.2 5.6
1.0 0.8 0.3
3.0 2.4 1.0
0.1 0.1 0.0
1.1
0.4 0.3 0.1
4
-15.6
0.9 0.3
8.25 8.50 10.0
14.1 11.8 4.5
3.9 3.1 0.9
0.9 0.7 0.2
1.5 1.1 0.2
0.1 0.1 0.0
0.4 0.3 0.1
5
-15.8
7.50 8.50 10.0
22.4 10.7 4.0
4.2 1.6 0.4
2.5 1.0 0.3
1.3 0.4 0.1
0.4 0.1 0.0
0.8 0.3 0.1
4
-27.9
7.75 8.50 10.0
25.5 14.8 5.6
5.6 2.8 0.8
2.6 1.3 0.4
2.1 0.9 0.2
0.4 0.2 0.0
1.0 0.4 0.1
6
-31.1
8.25 8.50 10.0
26.5 22.3 8.5
7.4 5.9 1.7
2.4 1.9 0.6
0.3 0.2 0.1
2.7 2.1 0.5
1.2 0.9 0.2
7
-33.1
8.25 8.50 10.0
65.5 55.2 21.2
12.6 10.2 3.0
9.9 7.9 2.3
4.0 3.1 0.7
2.4 1.8 0.4
3.2 2.9 0.6
16
-81.2
8.50 8.75 10.0
83.6 21.2 11.6 71.0 17.2 9.5 32.6 6.3 3.5
2.6 2.0 0.6
7.2 5.6 1.7
5.1 4.0 1.2
31
-100.1
7.6 3.8 2.4
4.6 2.4 1.5
6.8 3.4 2.2
32
-127.3
HeAr
HeKr 11
HeXe
NeAr
NeKr
NeXe
ArKr
ArXe
KrXe 7.75 8.00 8.25 8.50
191.9 162.9 138.3 117.6
52.5 43.1 35.4 29.0
energies with larger bases are calculated, the shifts suggested by the BSSE are not consistent. At this time no BSSE corrections will be made. Polarizabilities and Dispersion Energies. SCF dipole (a,), quadrupole (a2), and octupole (a3) polarizabilities were calculated with the finite field method.'* Since the S C F polarizabilities are bounded, orbital exponents for the polarization basis can be optimized and are shown in Table 111. All-electron bases were used for He, Ne, and Ar. Kr and Xe were studied with the ECP bases. The resulting polarizabilities are compared with the literature values in Table IV. As is well-known,19 the S C F dipole polarizabilities for H e and N e are substantially smaller than the experimental or correlated values. On the other hand, the S C F (18) Cohen, H.D.;Roothaan, C. C. J. J. Chem. Phys. 1965, 43, S34. (19)Standard,J. M.; Certain, P. R. J . Chem. Phys. 1985,83,3002. Tang, K. T.; Norbeck, J. M.; Certain, P. R. J. Chem. Phys. 1976, 64, 3063.
8.75 100.1 23.7 9.50 62.6 13.1 10.0 46.4 8.9
16.7 9.3 6.4
polarizabilities for Ar, Kr, and Xe are much closer to the accurate values. This is related to the differing effect of correlation on the radial charge density of the rare-gas atoms as pointed out by Werner and Meyef.zo Comparisons of the finite-field values with numerical coupled HartreeFock (CHF) results are, on the whole, good. Only the a3 for Kr differs substantially from the value Since the dipole and reported for the CHF calculation.2' quadrupole polarizabilities for the C H F and the present calculations are in good agreement, it is unlikely that the ECP for Kr is at fault. Our present use of larger polarization bases for Xe has resulted in considerably larger values of a2and a3than were reported earlier: and the present values are in good agreement (20) Werner, H. J.; Meyer, W. Phys. Rev. A 1976, 13, 13. (21)McEachran, R.P.;Ryman, A. G.; Stauffcr, A. D. J . Phys. B 1977, 10, L681. McEachran, R.P.;Stauffer, A. D.; Greita, S.J. Phys. B 1979, 12,
3119.
4332 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 TABLE VII: Characteristic Constants for the Rare-Gas Pairs molecule ref Re3 a0 De, cm-' He2 present 5.62 6.81 a 5.60 7.65 HeNe present 5.76 12.3 b 5.67 16.1 HeAr present 6.71 19.1 C 6.53 20.5 HeKr present 7.17 18.9 b 6.97 21.5 HeXe present 7.83 17.7 C 7.43 19.7 Ne2 present 5.95 21.6 d 5.83 29.3 NeAr present 6.78 37.3 b 6.55 42.9 NeKr present 7.14 38.0 b 6.81 45.5 NeXe present 7.54 42.6 e 7.08 52.1 AT2 present 7.32 79.6 7.10 99.6 f ArKr present 7.65 97.1 b 7.33 116.3 ArXe present 7.73 119.6 g 7.69 131.1 Kr2 present 7.94 112.9 h 7.60 139.3 KrXe present 8.32 138.9 g 7.89 162.3 Xe2 present 8.69 168.7 i 8.25 196.2
"Farrar, J. H.; Lee, Y. T. J . Chem. Phys. 1972, 56, 5801. *Watanabe, K.; Allnatt, A. R.; Meath, W. J. Chem. Phys. 1982, 68, 423. CSmith,K. M.; Rulis, A. M.; Scoles, G.; Aziz, R. A,; Nain, V. J . Chem. Phys. 1977, 67, 152. dAziz, R. A,; Meath, W. J.; Allnatt, A. R. Chem. Phys. 1983, 78, 295. eNg, C. Y.; Lee, Y. T.; Barker, J. A. J . Chem. Phys. 1974, 61, 1996. fKoide, A,; Meath, W. J.; Allnatt, A. R. Mol. Phys. 1980, 39, 895. gAziz, R. A.; van Dalen, A. J . Chem. Phys. 1983, 78, 2402. hNg, K. C.; Meath, W. J.; Allnatt, A. R. Mol. Phys. 1979, 37, 237. 'Barker, J. A,; Klein, M. L.; Bobetic, M. V. IBM J . Res. Deu. 1976, 20, 222. with the C H F values. N o attempt was made to saturate the polarization basis for any of the atoms. The dispersion or van der Waals coefficients and damping coefficients were obtained by methods described by Krauss and N e ~ m a n n .The ~ van der Waals coefficients, summarized in Table V, are determined from the perturbation of S C F wave functions. The coefficients for the homonuclear rare-gas molecules are included for completeness. For systems containing H e and Ne, which have relatively large correlation effects, our coefficients often do not fit within the bounds established by Standard and Certain.lg These van der Waals coefficients satisfy the combining rules presented by Tang and Toennies" with the average excitation energies determined variationally. The damping and dispersion energy components are given in Table VI. They confirm the earlier conclusiongregarding the slow convergence of the dispersion energy with L, especially for atoms heavier than Ne. For example, the DO, OD, and QQ terms for KrXe at R = 8.00 au, which is near Re, contribute 36.6 cm-l out of a total of 272 cm-' of the attractive dispersion energy. The total binding energy curves are fit with a Morse function. They are compared to the best available experimental data in Table VII. Our De fall in a range of 75-90% of the experimental values. For several of the He-containing molecules, He2, HeAr, HeKr, and HeXe, our De values are about 90% of experimental. However, our Re values are determined to lie at substantially larger R than the experimental even when the absolute error in D, is about 2 cm-I, since the energy curves are so shallow and the percentage error is still high. For NeAr, NeKr, NeXe, Arz, ArKr, ArXe, Krz, KrXe, and Xe2, our calculated De range between 80% and 85% of experimental. For the present model, the errors reflect the slow convergence of the damped dispersion energy with angular momentum and also the uncertainties in the S C F energy curves.
Krauss et al. CI Calculation of HeNe The configuration interaction (CI) calculation, which is emulated by this model, was performed for HeNe at two distances, 5.0 and 6.0 ao. The dipole and quadrupole van der Waals polarization bases were added to atomic bases optimized for both S C F and atomic correlation. However, only one of the two d polarization functions for Ne was included. The S C F wave functions were localized for the C I calculation. At 6.0 a. the van der Waals CI yields a binding energy of 13.9 cm-', which compares to 11.2 cm-I in the model calculation. Even though the S C F orbitals are localized, there is still a small charge-transfer correlation and intraatomic correlation that results from the orbital overlap and exchange. The intraatomic contribution represents a correlation superposition error, while the charge-transfer contribution is a molecular effect. Unfortunately, the two terms cannot be separated unless the virtual space is localized, which proved difficult to do with the large and diffuse basis. The superposition error cannot be large since the intraatomic correlation C I was also calculated and the counterpoise BSSE" was only 6.0 cm-' out of a total intraatomic correlation energy of 0.32 au. The charge transfer and intraatomic BSSE, although small in absolute terms, have a significant percent effect on the very shallow energy curve. The dipole moment at 6.0 a. is essentially unchanged from the S C F value by the van der Waals CI, suggesting that the dominant effect is the BSSE and not charge transfer in the small overlap region. Considering heavier atoms, these large C I calculations are less practical and must be replaced by balanced multiconfiguration (MC) S C F methods.'*22 Preliminary results with the M C wave functions for HeNe also show that there are small shifts in the dipole moment when van der Waals configurations are added to a CI that reproduces the M C result. For example, at a distance of 6 a. the dipole moment increases by less than au when the van der Waals configurations are included.
Discussion The results in Table VI1 are interpreted as a successful application of the S C F plus damped dispersion potential model. There are two main problems in applying the model. The first is the calculation of accurate S C F energy curves, and the second is the convergence of the damped dispersion. With the ability to use large basis sets in the available codes, the calculation of accurate S C F energy curves should not remain a problem. We recommend the use of the ECP for all atoms since they accurately simulate the all-electron curves in the small overlap region and tend to avoid the BSSE since the cores are replaced by an effective potential. The counterpoise energies do not seem to be useful in shifting the SCF curves,23but they are indicative of the need to improve the S C F calculation. S C F curves for the heavier atoms (>Ar) still need refinement. Because of BSSE, it seems likely the accurate S C F curves are more repulsive than tabulated. The slow convergence of the damped dispersion multipolar expansion is more evident for the heavier systems, and cancellation of these errors may have contributed to the apparent accuracy of the present results. Estimates of the higher multipole terms can be made by using the various methods available to extend the van der Waals coefficients and a general expression for the damping.1° Comparison of the present damping curves with the general one shows differences in detail, but the overall agreement is sufficient to approximate the small remainder in the series. The C I analysis of HeNe shows how difficult it is to calculate accurate energies of van der Waals systems because of the well-known intraatomic correlation BSSE. Correlated fragment energy curves can be estimated or experimentally derived for only a few systems. Application of the present model to more general problems is of interest. However, to be useful the method has to be based on the use of S C F fragment interaction energy curves. The results for the rare-gas molecules suggest that this method will represent the van der Waals contribution with good accuracy. (22) Liu, B.; McLean, A. D. J. Chem. Phys. 1973,59,4557. Lengsfield, B. H., 111; McLean, A. D.; Yoshimine, M.; Liu, B. J . Chem. Phys. 1983, 79, 1891. (23) Schwenke, D. W.; Truhlar, D. C. J . Chem. Phys. 1985, 82, 2418.
J . Phys. Chem. 1988, 92,4333-4339 The slow convergence for atoms may not be as severe for molecules where the centers of fluctuating potential may be distributed over many bond orbitals. Charge-transfer correlation is not significant for HeNe. This conclusion has to be checked for the heavier systems. The relatively constant percentage error in the binding energy suggests that the terms such as the charge transfer contribute proportionately to all the rare-gas molecules. This method of analysis depends on the dominance of a single configuration in the wave function representation. Separate calculation of the van der Waals
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contribution is then possible. The present method also avoids BSSE errors for the calculation of the van der Waals contribution, and the SCF BSSE can be reduced to small values by using the ECP and near-saturated basis sets. Calculation of the van der Waals contribution for a correlated atom is also possible, and the energy curve for correlated rare-gas atoms can be determined by using effective potentials in limited M C S C F calculations. Registry No. He, 7440-59-7; Ne, 7440-01-9; Ar, 7440-37-1; Kr, 7439-90-9; Xe, 7440-63-3.
On the Distinction between Weak and Strong Collisions in the Theory of Unimolecular Reactions H.0. Pritchard Centre for Research in Experimental Space Science, York University, Downsview, Ontario, Canada M3J 1P3 (Received: May 7, 1987; In Final Form: February I , 1988)
Beginning with the conjecture that the principal difference between weak- and strong-collision reactions is intrinsic to the reactant molecule, rather than in the nature of the collisions between molecules, a reanalysis of the unimolecular falloff phenomenon is given. The rate at which the molecule acquires energy to react is identified with the bulk vibrational relaxation rate, and the rate at which reactive molecules lose their energy of activation is taken to be the collisional deactivation rate, as used in existing theory; certain choices must be made for the rates of intramolecular vibrational relaxation (randomization) among the reactive states, with the result that the same analytic expression for the unimolecular rate can be made to reproduce the falloff shapes for such diverse reactant molecules as nitrous oxide, methyl isocyanide, and cyclopropane, as well as their correct positions on the pressure axis.
Introduction At the present time, there are two principal flavors of unimolecular reaction theory to correspond with the principal classifications of reaction type: weak-collision and strong-collision reactions. Strong-collision theory itself comes in several versions, which differ mainly in the methods used to construct the specific rate function k(E): one begins with an attempt to formulate a transition state for the reaction in question and derives a k(E) function by the well-known RRKM prescription,’v2 whereas another derives the k(E) function as the inverse Laplace transform of the Arrhenius rate la^.^-^ The “strongness” quality of the collisions has taken a much less central position in these theories, but it turns out that the usual steady-state expression used for calculating the falloff of the rate with p r e s s ~ r e ~ *implies ~ - ’ that the collisional deactivation of reactive molecules must be a single pureexponential process;g1othis had been suspected for some time earlier”J* but not proved. It was also found that if this rate, often denoted by w , was assumed to be equal to the collision rate, then the predicted position of the falloff on the pressure axis was usually (1) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley: London, 1972. (2) Forst, W. Theory of Unimolecular Reactions; Academic: New York, 1973. (3) Slater, N. B. Theory of Unimolecular Reactions; Methuen: London, 1959. (4) Forst, W. J . Phys. Chem. 1972, 76, 342. (5) Pritchard, H. 0. Quantum Theory of Unimolecular Reactions; Cambridge University Press: Cambridge, England, 1984. (6) Rice, 0. K.; Ramsperger, H. C. J . Am. Chem. SOC.1927,49, 1617. (7) Kassel, L. S. Kinetics of Homogeneous Gas Reactions; The Chemical Catalog Co.: New York, 1932. (8) Nordholm, S. Chem. Phys. 1978, 29, 55. (9) Yau, A. W.; Pritchard, H. 0. Can. J . Chem. 1978, 56, 1389. (IO) Singh, S. R.; Pritchard, H. 0. Chem. Phys. Lett. 1980, 73, 191. ( I 1) Bunker, D. L. Theory ofElementary Gas Reaction Rares; Pergamon: Oxford, England, 1966. (12) Tardy, D. C.; Rabinovitch, B. S. Chem. Rev. 1977, 77, 369.
well within an order of magnitude of that observed, and that the falloff shape was in virtually perfect agreement with experiment. Unfortunately, this dual quality of strong collisions, their pure exponential character and their near-unit efficiency, has given rise to some confusion in their d e f i n i t i ~ n . ~ Moreover, we need to be careful about the use of the term “exponential decay” in collisional relaxation. One may have, in theory, the situation that all populations decay exponentially with a single time constant, whence the total energy decays exponentially, too: this is the “strong collision”, strictly definede8-12One can also have the less restrictive situation in which the total energy decays exponentially, but the populations do not? the condition for this to occur is that ( h E ) / E = constant;I4 such behavior has been observed in large molecules at low energies,IsJ6 but there is insufficient information content in the results to be able to say whether or not, strictly, these are strong collisions. In the remainder of this article, the terms single exponential, pure exponential, or strong collision will refer to the former kind of process. Likewise, there are variations in the treatment of weak-collision reactions, but the dominant feature of weak-collision theory is the collision efficiency factor &, which is the amount by which the collision rate must be scaled to cause the computed strongcollision low-pressure limiting rate constant to coincide with the observed ne.^**'^ Again, little is said about the nature of the collision process, but it is never conceived as being a pure-exponential relaxation; in fact, for triatomic moIecuIes,’5 ( A E ) / Evaries with E, and the relaxation clearly is not exponential. It is not necessary to go into further details of strong- and weak-collision unimolecular reaction theory, as they are so well-known: all we have done here is to select a few points for (13) Troe, J. J . Chem. Phys. 1977, 66,4745, 4758. (14) Forst, W.; Xu, G.-Y.; Gidiotis, G. Can. J . Chem. 1987, 65, 1639. (15) Troe, J. 2.Phys. Chem. (Neue Folge) 1987, 154, 73. (16) Wallington, T. J.; Scheer, M. D.;Braun, W. Chem. Phys. t e t t . 1987, 138, 538.
0022-3654/88/2092-4333$01.50/0 0 1988 American Chemical Society
