On the helium pair potential

Volume 155, number

CHEMICAL

4,5

ON THE HELIUM

PHYSICS LETTERS

10 March 1989

PAIR POTENTIAL

J.A. MONTGOMERY

Jr.

UnlnrtedTechnologies Research Center, East Hartford, CT061 08, USA

G.A. PETERSSON

and N. MATSUNAGA

Hall-Atwater Laboratories of Chemistry, Wesleyan University, Middletown. CT 06457, USA Received 4 November

1988; in final form 26 December

1988

Ab initio calculations of the helium pair potential including extrapolation to the complete basis set (CBS) Iimit are presented. It is shown that CBS extrapolations of Moller-Plesset calculations using small atomic basis sets supplemented with bond functions give excellent agreement with previously reported empirical and ab initio potentials at very modest computational expense. The CBS extrapolation reduces the basis set superposition correction by an order of magnitude. Our calculated CBS binding energies of 34.66 u&, (TZDP +bond basis set) and 34.09 u&, (TZDP+ 2dfbond basis set) are very close to the recent empirical estimate of 34.67 pE,, by Aziz, McCourt, and Wong.

1. Introduction In recent years, the He pair potential has been the subject of a number of studies, both experimental and theoretical [ I-61. Although the He2 system has only four electrons, accurate ab initio calculations of the interaction potential over the entire range of experimentally accessible internuclear separations have proven difficult, due to the slow convergence of the one-electron basis set expansion and the effects of the basis set superposition error (BSSE). In this paper, we present a new ab initio calculation of the He pair potential including extrapolation to the complete basis set (CBS) limit [ 7-121. The CBS extrapolation will be shown to give an accurate estimate of the He pair potential with relatively small computational effort and well-controlled BSSE.

2. Method and results The He* interaction potential can be expanded in a Moller-Plesset perturbation series [ 131. An accurate calculation of the interaction potential requires convergence with both the one-electron basis set and the Mraller-Plesset series. In our calculations, 0 009-26 14/89/$ 03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division )

this is accomplished by the CBS”,‘) method [ 121. Using the known convergence properties of pair natural orbital expansions [7-l I], direct calculations of the pair correlation energies (to infinite order by pair CI) and the pair coupling terms (by third-order Moller-Plesset theory ) with different size molecular pair natural orbital (MPNO) basis sets are extrapolated to give the CBS(m,3) results. Calculations with the less accurate, but faster, CBS2’ method (based on extrapolation of the second-order Moller-Plesset pair energies) will also be reported. The basis sets used are the double zeta plus polarization (DZP), triple zeta plus double polarization (TZDP), and quadruple zeta plus triple polarization (QZTP ) atomic pair natural orbital ( APNO ) basis sets developed by Petersson et al. [ 121. Calculations were also performed with the TZDP basis set augmented by two atom-optimized Gaussian d functions (exponents = 2.1, 0.875 ). Because the expansion in atom-centered one-electron basis functions is so slowly convergent [ 141, in some of the calculations these basis sets are augmented by optimized s (exponentz0.325 ) and p (exponent ~0.258) bond functions located at the midpoint of the internuclear axis. In table 1, the convergence of the energy of atomic helium is shown relative to the B.V.

413

Volume

155,number 45

Table 1 Convergence

CHEMICAL

of the calculated

MPr’r

MPl”‘1

-2S61617035 -2.861623462 -2.861623482 -2.861624279

-0.028699614 -0.030150501 -0.031386292 -0.033341912

-0.035920056 -0.037042852 -0.037992026 -0.039360977

-2.861617035 -2.861623462 -2.861623482 -2.861624279

-0.037934836 -0.037587292 -0.037535360 -0.037580202

-0.042239369 -0.042008799 -0.041974577 -0.042032988

-2.861679996@

-0.03725

-0.042044379

Basis set

SCF

direct

DZP TZDP QZTP TZDP+2d

CBS

DZP TZDP QZTP TZDPfZd

exact [15].

b)Ref.[16].

“Ref.

Table 2 Convergence of the calculated separation (in u.&)

He, interaction

Method

Basis set

SCF

MP”’

MP’“.j’

direct

DZP TZDP TZDP + bond TZDPfZd +bond

28.77 28.93 29.04

19.15 - 2.22 - 15.95

18.75 -11.45 -29.32

29.09

- 18.43

-32.14

DZP TZDP TZDPfbond TZDPfZd fbond

28.77 28.93 29.04

16.23 -7.07 - 20.65

15.20 - 16.81 - 34.66

29.09

-19.76

- 34.09

exact a) Ref. [20]. ‘D)Ref. [ 121, Hartree-Fock

414

1989

b,

c)

(171.

known exact results [ 15- 171. It is essential to make corrections for the effects of BSSE, especially if a reliable value for the He, van der Waals minimum is to be obtained. Recent numerical calculations [ 181 indicate this is most reliably done through the counterpoise method of Boys and Bernardi [ 19 1, which has therefore been used in this work. Somewhat to our surprise, the BSSE from the bond functions is very strongly coupled to the BSSE from the second helium atom. The convergence of the interaction energy at 5.6 bohr with basis set and order of Moller-Plesset perturbation theory is shown in table 2. A parabolic lit to the TZDP+bond CBS(“13) results at 5.5, 5.4, and 5.7 bohr gave a minimum at 5.58 bohr with a D, of

CBS

lOMarch

energy of the helium atom (in Es)

Method

“‘Ref.

PHYSICS LETTERS

29.2 ‘!

damped dispersion,

energy at 5.6 bohr

-34.67

HFD-B form.

b,

34.68 &, ( 10.95 K). Complete potential curves obtained with the two largest basis sets are shown in fig. 1. All of these results include counterpoise corrections. The magnitude of these corrections is shown in fig. 2 for the TZDP + bond basis set. Note that the counterpoise corrections obtained without the use of bond functions are much smaller ( < 5 @,, direct and x 1 @,, extrapolated at 5.6 bohr with the TZDP basis) than those shown in fig. 2. A numerical comparison of the CBSf”z3) potential curves with previously reported He2 potentials is presented in table 3. The close agreement between the CBS’“*3’ TZDP + bond and the HFD-B potential curves is apparent in fig. 1. Near the van der Waals minimum, the SCF interaction (table 2) has converged to within 1 p&. Included in table 2 is the recent large basis counterpoise corrected SCF result of Gutowski, van Duijneveldt, Chalasiriski, and Piela [ 201, which is in very good agreement with our calculations. A comparison of our second-order calculations with the MP2 results for the He* interaction obtained by Sauer, Hobza, Carsky, and Zahradnfk [ 2 1] using much larger basis sets demonstrates the effectiveness of our extrapolation to the MP2 CBS limit. From their calculated MP2 interaction energy ( - 17 u&,), Sauer et al. estimate (by a different procedure from ours) the complete basis set MP2 interaction energy to be -22 &, at an internuclear separation of 5.6 bohr (ref. [21], table 2). Our TZDP+bond and TZDP + 2d + bond CBS(‘) results from table 2 are in good agreement with this value. As this accounts for only about 2/3 of the observed van der Waals well

Volume

155.number 4,5

CHEMICAL

10 March 1989

PHYSICS LETTERS

a

‘:O

1

2 I/

3

4

(N+d)

4

6

8

10

R( bohr) Fig. 2. The basis set superposition error with the TZDP+bond basis set as determined by the counterpoise method of Boys and Bemardi.

-

-13.12 I

;'O

0.1

0.2 1/

0.3

0.4

(N+d)

Fig. I. The helium pair potential. The curves are the direct and extrapolated MPt2’ and MP’“,3J results obtained with the fZDP+ bond basis set. The points are the HFD-B empirical potential (A&, McCourt and Wong [ 21).

depth [ 21, it is clear that higher orders of perturbation theory are necessary to accurately describe the He* interaction. Both the direct and extrapolated MPfmp3) results in table 2 show the slow convergence of the binding energy toward the observed value. Calculations at 5.6

bohr using the QZTP basis gave an insignificant improvement (0.029 +!$,) over the TZDP basis and consequently no further calculations were performed with the QZTP basis. As found by other workers [ 4,5,14,21], the use of bond functions is an effective way of overcoming this difficulty. Our TZDP + bond CBS(“z3’ binding energy of 34.66 JLE,, agrees well with the 34.670 @,, (10.948 K) value of Aziz, McCourt, and Wong [ 21, as does the TZDP +2d+bond CBS’a*3’ binding energy of 34.09 @,,_ It might seem surprising that the addition of d functions to the basis set decreases the interaction energy. The reason this occurs has to do with the details of the extrapolation procedure. In this case, the CBS{“13) extrapolation using the TZDP+ bond basis set (as described in ref. [ 121) is based on direct calculations with one and four natural orbitals (fig. 3a). However, with the inclusion of d functions in the TZDP + 2d+ bond basis set the more accurate fourand six-natural-orbital extrapolation is used to obtain the CBS(“x’j results (fig. 3b). The 0.5 flh difference between the TZDP +bond and TZDP +2d + bond results can be seen as a crude measure of the absolute error in the CBS’“v3) extrapolation procedure. 415

Volume 155, number 4,5

CHEMICAL PHYSICS LETTERS

10 March 1989

Table 3 Comparison of Hez interaction potentials (in K) ‘) Ab initio

Empirical R (bohr)

HFD-B b,

HFIMD =’

QMC d,

CEPAi =’ 11ls, 5~73d, 2f+2s, 2p

(bond11 1.0 2.0 3.0 4.0 4.5 5.0 5.3

291.3(3) 362.4(2) 381.9(l) 291.4 57.18 -0.82 - 9.24

5.6

- 10.95

5.8 6.0 7.0 8.0 9.0 10.0

- 10.52 -9.60 -4.61 -2.07 -0.99 -0.51

369.6(l) 291.1 0.19 - 10.74 -9.56 -4.60 -2.06

291.9(3) 358(2) 380( 1)

MRCI ‘I ]7s, 4~9 4d, 4f, lg, lh+2s, 2p, 1d (bond)]

288.3(3) 363.2(2) 381*0(l) 302.5

360.9(2) 379.5(l) 269.1

2.38

0.0

CBS (rn.3) [Js, 3p+ Is, 1P (bond)1 291.9(3) 366.9(2) 383.9(l) 292.8 56:9 1 -1.25 -9.46

[3s, 2p, 2d+ Is, 1~ (bond)1 289.8(3) 363.2(2) 381.4(l) 294.9 58.89 -0.26 -8.95

-9.54

- 10.86

- 10.95

-10.76

-8.69 -4.24 -1.90

-9.61 -4.61 -2.06

- 10.40 -9.38 -4.29 -1.86 -0.87 -0.44

-10.36 - 9.43 -4.41 - 1.92 -0.90 -0.46

a) 1 p& = 0.3 1578 K. b, Hartree-Fock plus damped dispersion, HFD-B form, ref. [2]. <) Hartree-Fock plus intra-atomic correlation plus model dispersion, ref. [ 11. d, Quantum Monte Carlo, ref. 151. e, Coupled electron-pair approximation, ref. 141. ‘) Multi-reference CI, ref. [6].

Fig. 3. The convergence of the MPNO expansion of the Her a/? interatomic pair energy at 5.6 bohr separation. The CBS pair energy corresponds to the intercept in this graph using the inverse of the number, iV, of MPNOs as the abscissa. The filled circles indicate closed shells. (a) The extrapolation of the N= 1 and N= 4 calculations using the TZDP + bond basis set. (b) The extrapolation of the N=4 and N=6 calculations using the TZDP + 2d +bond basis set.

416

If we had used the exact APNOs as basis functions, the one-natural-orbital energy of the He atom would be converged completely, as would the fivenatural-orbital energy, and the CBS atomic extrapolation. Therefore, additions to the basis set could not have any effect on the CBS atomic energy and the CBS counterpoise correction would vanish. One indication of the effectiveness of our APNO basis sets is the reduction of the CBS counterpoise corrections (fig. 2 ) relative to those for the direct calculations by an order of magnitude. Indeed, the direct BSSE is larger than the binding energy, giving one little confidence in the direct binding energy obtained with these small basis sets. As it has been found previously [ 121 that excellent results for the correlation energy of first-row hydrides may be obtained from APNO expansions without the use of bond functions, calculations were performed where the bond functions were replaced with additional s, p, and d atom centered functions chosen to have maximum overlap with the bond functions. The resulting l-2 @, increases in the van der Waals well depth relative to the TZDP calcula-

CHEMICAL

Volume 155, number 4,s Table 4 Partitioning of the He, CBS@‘,‘1 TZDP+2d+bond energy at 5.6 bohr separation (in @,,)

interaction

Partition

He

Hcl

Interaction

SCF

-2861624.83

-5723220.56

29.09

-42038.22

-84073.44

3.01

PHYSICS LETTERS

intra-atomic pair energy, Fe,, interatomic pair energy,

-71.55

Ze,, pair couplings,

5.37

&..k, total correlation energy total energy

-71.55 5.37

-42038.22

-84139.62

-63.18

-2903663.05

-5807360.19

-34.09

tion suggest that there is no practical alternative to the use of bond functions in this particular instance.

3. Discussion As shown in table 4, the helium dimer interaction potential is the net result of a series of small differences between large numbers. At 5.6 bohr separation, the Hez SCF energy ( -5.723220565 Ed is only 29 @,, above the separated atom limit while the correlation energy ( -84.13962 E,,) is only 66 p& below that of the separated atoms. Note also that if we localize the occupied orbitals, the bulk of the correlation energy is intra-atomic ( - 84 mJ& versus - 70 pE,, interatomic). It is interesting to note that the intra-atomic pair energies show a small exclusion effect [ 221. As atoms A and B approach, the electrons on atom B are excluded from the region occupied by those on atom A, and vice versa. This effect manifests itself in the 3.01 p& reduction of the intraatomic pair energies at 5.6 bohr relative to those at infinite separation.

10 March 1989

calculation of very weak atomic interactions. The specific CBS(“,3)‘{KK, KL, LL, LL’} model chemistry that we have recently proposed [ 121 recovers 77.8% of the interatomic correlation energy, but only 58.8% of the binding energy. Ifwe augment the basis with s and p bond functions and use counterpoise corrections for BSSE we no longer have a model chemistry, since the energy of a single helium atom is no longer uniquely defined, but is instead a function of the interaction being studied. However, we now recover 100 + 0.5% of the interatomic correlation energy and 1OO? l.OI of the binding energy. Because the effects of higher angular momentum basis functions are included in the CBS extrapolation, the basis sets used in the direct calculations are not large and the machine time required to perform the calculations is quite modest. Simplifications of the CBS program and the basis sets used are under investigation. The success of the CBS method in the calculation of the Hez interaction suggests the method offers an opportunity to accurately calculate the (relatively much larger) dispersion forces between polyatomic molecules.

Acknowledgement We would like to thank H.H. Michels for suggesting this problem and for useful discussions. This work was supported in part by the National Science Foundation under Grant No. CHE-8419381. Acknowledgement is made to the donors of the Petroleum Research Fund, administered by the ACS, for partial support of this research under Grant No. 16587-AC6. The authors are also grateful to Wesleyan University, Merch Sharp and Dohme Research Laboratories, and Digital Equipment Corporation for providing funds toward the purchase of computer equipment used in the research.

References 4. Conclusions The purpose of this study was to determine the effectiveness of the CBS extrapolation procedure in the

[ 1 ] R. Feltgen, H. Kirst, K.A. Kiihler, H. Pauly and F. Torello, J. Chem. Phys. 76 (1982) 2360. [2] R.A. Aziz, F.R.W. McCourt and C.C.K. Wong, Mol. Phys. 61 (1987) 1487.

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{31 B. Liu and A.D. McLean, as cited in ref. [ 11. [4] U.E. Scnff and P.G. Burton, Mol. Phys. 58 ( 1986) 637. [ 51D.M. Ceperly and H. Partridge, J. Chem. Phys. 84 ( 1986) 820. [ 61 J.H. van Lenthe, R.J. Vos, J.G.C.M. van Duijneveldt-van de Rijdt and F.B. van Duijneveldt, Chem. Phys. Letters 143 (1988) 435. [ 71 M.R. Nyden and G.A. Petersson, J. Chem. Phys. 75 ( 1981) 1843. [ 81 G.A. Petersson and M.R. Nyden, J. Chem. Phys. 75 ( I981 ) 3423. [9] G.A. Petersson and S.L. Licht, J. Chem. Phys. 75 ( 1981) 4556. [ IO] G.A. Petersson, AK. Yee and A. Bennett, J. Chem. Phys. 83 (1985) 5105. [ I I ] G.A. Petersson and M. Braunstein, J. Chem. Phys. 83 (1985) 5129. [ 121G.A. Petersson, A. Bennett, T.G. Tensfeldt, M.A. Al-Labam, W.A. Shirley and J. Mantzaris, J. Chem. Phys.. 89 ( 1988) 2193.

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[ 131C. Mdler and M.S. Plesset, Phys. Rev. 46 (1934) 618.

[ 14]M. Gutowski, J. Verbeek, J.H. van Lenthe and G. Chalasinski, Chem. Phys. 11I (1987) 271.

[ 151K. Sza1ewiczandH.J. Monkhorst, J. Chem. Phys. 75 (1981) 5785.

[ 161F.W. Byron Jr. and C.J. Joachain, Phys. Rev. 157 (1967) [ 17] Y. Accad, CL. Pekeris and B. Schiff, Phys. Rev. A I I ( 1975) 1474. [ 18 ] M. Gutowski, J.H. van Lenthe, J. Verbeek, F.B. van Duijneveldt and G. Chalasinski, Chem. Phys. Letters 124 (1986) 370. [ 191SF. Boys and F. Bernardi, Mol. Phys. 19 ( 1970) 553. [20] M. Gutowski, F.B. van Dunijneveldt, G. Chalasidski and L. Piela, Mol. Phys. 61 (1987) 233. [2 1 ] J. Sauer, P. Hobza, P. C&sky and R. Zahradnik, Chem. Phys. Letters 134 (1987) 553. [22] 0. Sinano&, Advan. Chem. Phys. 6 (1964) 315.