Validity of the function counterpoise method and ... - ACS Publications

May 1, 1991 - Validity of the function counterpoise method and ab initio calculations of van der Waals interaction energy. Fu Ming Tao, Yuh Kang Pan. ...
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3582

J . Phys. Chem. 1991, 95, 3582-3588

Validity of the Function Counterpoise Method and ab Initio Calculations of van der Waals Interactlon Energy Fu-Ming Tao and Yuh-Kang Pan* Department of Chemistry, Boston College, Chestnut Hill, Massachusetts 021 67 (Received: September 12, 1990) A scheme to test the validity of the function counterpoise correction method for the basis set superposition error has been designed and presented with ab initio calculations of the Ne-Ne interaction energy at the distance of 4.0 A using a series of extended basis sets. The efficiency of the extended basis functions and the convergence of the calculated interaction energy with respect to basis set and to the Mdler-Plesset approximation level have also been studied. The calculations and careful analysis have confirmed the validity of the function counterpoise method; namely, it produces negligible overcorrections for the basis set superposition error compared to other errors caused by the deficiency of basis set and calculation method. The diffuse functions (including the diffuse polarization functions) have been found to be more effective than the regular polarization functions in the description of the interaction energy. The higher M~ller-Plesset approximation levels (MP3 or MP4) have been determined to improve the calculated interaction energy significantly over lower levels of the method (MP2 or MP3). This is only when the basis set in calculations exceeds a certain quality level. The higher Maller-Plesset levels are also subject to relatively larger overcorrections for the basis set superposition error.

Introduction Recent reviews and publications'v2 have shown that, although ab initio calculations of van der Waals interactions are now routinely being carried out, there still remain major difficulties in achieving very high precision and accuracy of the calculational results except for the smallest systems such as H e H e or H e H P f S One of the main reasons behind this is the fact that in the conventional supermolecular calculation of interaction energy the basis set superposition error (BSSE) is inevitable, and furthermore, it cannot be fully corrected in any practical calculation using a truncated basis set.613 The BSSE is explained as a nonphysical energy contribution to the calculated interaction energy resulting from the lowering of the monomer energy in the dimer calculation due to the improvement of the monomer basis set caused by the presence of the partner basis set. The interaction energy itself is usually so small (10-1000 rhartree) that it could be totally overshadowed by the BSSE. Other major problems that also prevent accurate calculation of van der Waals potentials are the tremendous size of the requisite basis set with highly flexible polarization functions and the demanding level of the calculational method. The scheme most widely used to correct for the BSSE is the full function counterpoise method (or full counterpoise or function counterpoise method) proposed by Boys and Bernardi.I4 This method uses the same basis functions in the monomer calculations as in the dimer calculation; namely, the monomer energies are calculated by using the full basis set employed in the dimer calculation rather than by using separate monomer basis sets. The validity of this recipe has been questioned frequently on the basis of the Pauli exclusion principle, which will prevent one component from fully utilizing the basis set of the other component in the dimer calculation. The full counterpoise method is believed by ( I ) Chem. Reo. 1988, 88 (van der Waals interactions). (2) van Lenthe. J. H.; van Duijneveldt-van de Rijdt, J. G. C. M.; van

Duijneveldt, F. B. Ab initio Merhods in Quantum Chemistry-II; Lawley, K. P., Ed.; Wiley: New York, 1987; p 521. (3) van Lenthe, J. H.; Vos, R. J.; van Duijneveldt-van de Rijdt, J. G. C. M. Chem. Phys. Lerr. 1988, 143,435. (4) Liu, B.; McLean, A. D. J . Chem. Phys. 1989, 91, 2348. (5) Meyer, W.; Hariharan, P. C.; Kutzelnigg, W. J . Chem. Phys. 1980, 73, 1880. (6) Liu, B.; McLean. A. D. J . Chem. Phys. 1973,59,4557. (7) Kestner, N. R. J . Chem. Phys. 1968, 48, 252. (8) Clementi, E.J . Chem. Phys. 1967, 46, 3851. (9) Urban, M.; Hobza, P. Theor. Chim. Acto 1975, 36, 215. (IO) Ostlund, N. S.; Merrifield, D. L. Chem. Phys. Lerr. 1976, 39, 612. (11) Dacre, P. D. Chem. Phys. Lerr. 1977, 50, 147. (12) Price, S. L.; Stone, A. J. S . Chem. Phys. Lerr. 1979, 65. 127. (1 3) Johansson, A.; Kollman, P.;Rothenberg, S . Theor. Chim. Acta 1973, 29, 167. (14) Boys, S . F.; Bernardi, F. Mol. Phys. 1970, 19, 553.

0022-3654191 12095-3582302.50/0

the majority of a u t h o r ~ ' ~ Jto~ overestimate -~~ the real BSSE because of the restriction on the orbital space by the Pauli principle is not considered in the monomer calculations. This leads to the proposal of various alternative counterpoise schemes such as virtuals-only counterpoise method.25-28 None of them can be rigorously justified nor can they give more satisfactory and consistent results in practical calculation^.^^^^^^^^ Up to this time the problem of how to determine and completely remove the basis set superposition error is considered unsolvable by many, and no general agreement has been reached in the l i t e r a t ~ r e . ~More ~-~ recently, formal arguments as well as numerical results of calculation have been supplied in favor of the recipe of the full counterpoise scheme both at the Hartree-Fock and CI levels?'-33 However, the problem of whether and how much the scheme overcorrects the real BSSE remains unsolved. As a result, widespread hesitation exists in accepting it as correction for the BSSE because of the conflict with the Pauli (15) Maggiora, G. M.; Williams, I. H.J . Mol. Strucr. 1982. 88, 23. (16) Pettersson, L.; Wahlgren, U. Chem. Phys. 1982, 69, 185. (17) Newton, M. D.; Kestner, N. R. Chem. Phys. Lerr. 1983, 94, 198. (18) Fowler, P. W.; Madden, P. A. Mol. Phys. 1983, 49, 913. Wilson, S. Mol. Phys. 1983, 50, 1295. (19) Wells, B. H.; (20) Sqiegelmann, F.; Malrieu, J. P. Mol. Phys. 1980, 40, 1273. (21) Miyoshi, E.; Tatewaki, H.; Nakamura, T. J . Chem. Phys. 1983,78, 815. (22) Hayes, I. C.; Hurst, G. J. B.; Stone, A. J. Mol. Phys. 1984, 53, 107. (23) Senff, U. E.; Burton, P. G. J . Phys. Chem. 1985, 89, 797. (24) Olivares del Valle, F. J.; Tolosa, S.;Ojalvo, E. A.; Espinosa, J. Chem. Phys. 1988, 127, 343. (25) Morokuma, K.; Kitaura, K. In Chemicol Applications of Atomic and

Molecular Electronic Potentials; Politzer, P., Truhlar, D. G., Eds.; Plenum: New York, 1981; p 215. (26) van Lenthe, J. H.; van Dam, T.; van Duijneveldt, F. B.; Kroon-Batenbuurg, L. M. J. Faraday Symp. Chem. SOC.1984, 19, 125. (27) Pullman, A.; Sklenar, H.; Ranganathan, S. Chem. Phys. Len. 1984,

110, 346. (28) Tomonari, M.; Tatewaki, H.; Nakamura, T. J . Chem. Phys. 1984,80, 344. (29) Schwenke, D. W.; Truhlar, D. G. J . Chem. Phys. 1985.82, 2418. (30) Cybulski, S. M.; Chalasinski, G.; Moszynski, R. J. Chem. Phys. 1990, 92, 4357. (31) Gutowski, M.; van Lenthe, J. H.; Verbeek, J.; van Duijneveldt, F. B. Chem. Phys. Lerr. 1986, 124, 370. (32) Gatowski, M.; van Duijneveldt, F. B.; Chalasinski, G.; Piela, L. Mol. Phys. 1987, 61, 233. (33) Szczesniak, M. M.; Scheiner, S. J . Chem. Phys. 1986, 84, 6328. (34) Schwenke, D.; Truhlar, D. G. J. Chem. Phys. 1985,82, 2418; 1986, 84, 41 13. (35) Collins, J. R.; Gallup, G. A. Chem. Phys. Lert. 1986, 123, 56. (36) Loushin, S . K.; Liu, S.-Y.; Dykstra, C. E. J . Chem. Phys. 1986,84, 2720. (37) Olivares del Valle, F. J.; Tolosa, S.; Ojalvo, E. A.; Espinosa, J. 1988, 127, 343.

0 199 1 American Chemical Society

Validity of the Function Counterpoise Method As the next major problem in accurate calculation of van der Waals potential, the basis set is generally demanded to accurately describe not only the short-range regions of orbitals near the nucleus but also the long-range polarizabilities over the entire intermolecular regions. The most popular strategy is to use an energy-o timized even-tempered basis set as advocated by Wilson et al.19*3g But, the size of such basis has to be so large that it is practically unmanageable at the present time. This is because most of the added functions are used up to improve the description of the short-range orbitals while the long-range behavior of electrons improves slowly. A more practical alternative is to use a given energy-optimized basis set such as those used for regular molecular calculations and extend it with certain polarization functions and diffuse functions. The number of the extended functions should be controlled to a manageable range and therefore the problem becomes: what are the most effective basis functions as the extended functions? The last problem of the same importance is which calculational method is efficient and adequate in the description of the electron correlation energy, which dominates the contribution to the interaction energy of van der Waals molecules. A major requirement for a calculational method is that it is ~ i z e - c o n s i s t e n t ,which ~ ~ - ~ implies that a calculation on the dimeric system at infinite distance between the monomers should yield the sum of the monomer energies. The HartreeFock method is size-consistent. The Mailer-Plesset perturbation theory is a widely used correlation calculation beyond the HartreeFock level and it also provides a size-consistent description of electron correlation effects.41 However, it was stated by many authors that its convergence is slow, and the third and fourth orders of Maller-Plesset (MP3 and MP4) contributions cancel each other appr~ximately.'~~~~ Therefore, it is not clear whether a few higher orders of Maller-Plesset perturbation theory (e.g., MP3 and MP4) will show any advantage over MP2 in producing a reasonable van der Waals interaction energy. This work is to study the above problems by calculating the interaction energy of the neon dimer at a distance of 4.0 A in order to provide some general information for ab initio calculations of van der Waals potentials. The Ne-Ne system is a typical van der Waals molecule which resembles a large class of van der Waals molecules of the first row atoms, such as CH4-CH4,CH4-NH3, NH3-NH3, CH4-H20, and NH3-H20. Therefore, a b initio calculation of the neon dimer potential is of very great interest. Numerous attempts have been made to determine the Ne-Ne potential by a b initio appr0ach,2~*"yet, no accurate determination has been achieved because of the limitation of the current computer resources. Before the ab initio Ne-Ne potential or other ab initio van der Waals potentials can be accurately calculated with the improvement of computer technology, however, the attempt to search for a better method of calculation is always necessary. This work is not intended to calculate an accurate a b initio Ne-Ne potential. Instead, the primary goal of this work is to provide evidence and detailed analysis which could determine the validity of the full counterpoise scheme through a special design of our calculations. At the same time, it will carefully examine the efficiency of various extended basis functions and the performance of the second, third, and fourth orders of Maller-Plesset approximation. Several empirical neon dimer potentials have been reported in the literature, based on experimental results and empirical (38) Wells, 9. H.; Wilson, S. Mol. Phys. 1985, 54, 787. (39) March, N . H.; Young, W. H.; Sampanther, S. The Muny-Body Problem in Quuntum Mechunics: Cambridge University Prws: Cambridge, UK, 1967. (40) Pople, J. A.; Binkley, J. S.; Seeger, R. In?.J. Quuntum Chem. 1976,

SIO, 1.

(41) Frisch, M. J.; Pople, J. A.; Binkley, J. S. J . Chem. Phys. 1984, 80, 3265. (42) Knowlw. P. J.; Somasundram, K.; Handy, N. C.; Hirao, K. Chem. Phys. Leu. 1985, 113, 8. (43) Frisch, M. J.; Pople, J. A,; Del Bene, J. E. J . Phys. Chem. 1985, 89, 3664. (44) Bulski, M.; Chalasinski, G.; Jtseziorski, 9. Theor. Chim. Acru 1979, 52, 93.

The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 3583 TABLE I: Internuclear Distance ( R &) Dependence of the Function Counterpoise Correctiom &Calculated by Hartree-Fock Method and the Second-, Third-, and Fourth-Order Mailer-Plesset Perturbation Theory Using the 6-311G and 6-311G(d) Basis Setso RN-N.. A 2.0 2.5 3.0904 3.5 4.0 6-311G HF 496.45 378.06 66.03 6.51 0.14 MP2 848.28 557.18 84.66 7.97 0.17 MP3 756.06 502.38 78.91 7.52 0.16 MP4 843.03 540.40 82.14 7.75 0.15 6-311G(d) HF 498.66 379.33 66.04 6.51 0.14 MP2 950.09 558.10 83.57 7.86 0.16 MP3 861.26 504.26 77.87 7.41 0.15 MP4 930.68 533.73 80.31 7.59 0.15 'Energies are in fihartree. 6cp = E(Ne)

- E(NeG).

Considerable discrepancy exists among these empirical potentials and no final agreement has been reached to date. One of the most satisfactory empirical potentials is HFD-C1 reported by Aziz et a1.,4s which has a well depth of 136.17 phartree at the equilibrium distance of 3.06 A. According to the HFD-C1 potential, the Ne-Ne interaction energy at the distance of 4.0 A is -41.07 phartree.

Method The starting basis set used in this work is the energy-optimized basis set 6-31 1G" (roughly equivalent to 9s5p) which is usually large enough to adequately describe the valence-electron density around the nuclei and most of the short-range behaviors of the individual Ne atoms. Preliminary calculations are made with the 6-31 1G basis set to find the dependence of the function counterpoise correction gCp with the internuclear distance RNeNc. The function counterpoise correction hcp, which is a measure of the BSSE,is given by

bcp = E(Ne) - E(NeG)

(1)

where E(Ne) and E(NeG) are both the total energies of the neon atom but E(NeG) is different in that it is calculated by using a ghost nucleus G (a set of the same N e basis set with its nuclear charge of zero) at R N - N ~ . The hcp - RNtNe dependence from the calculations is given in Table I. The same table also displays the bCp - R N - N ~ dependence from calculations using the 631 lG(d),51a basis set of the 6-31 1G extended with a regular d function. It is seen that ticp decreases sharply with the increase of RNr and is practically negligible (-0.15 phartree) at R = 4.0 for both of the basis sets. This means that at 4.0 larger distance the 6-3 1 1G or 6-3 1 1(d) basis set does not cause measurable gCp and that the calculated interaction energy should be the real one reflecting the quality of the basis set. As seen in Table I, the addition of the regular d function to the 6-31 1G basis set does not increase the ticp because of the relative tightness (or nondiffuseness) of the regular polarization functions. In our first stage of calculation, two groups of extended basis sets are used. The 6-31 1G basis set is first extended step by step to contain more of the regular polarization functions such as d, d2, and d2f and the resulting basis sets are grouped in group I and denoted as 6-31 1G(d2) or 6-31 lG(d2f).52 The basis sets are further extended by adding d and f polarization functions with the maximal diffuseness, or with the minimal exponents, under the restriction that no BSSSE arises (maximal tolerance hcp = 0.6 phartree). The minimal exponents of these d and f functions

f

le:;

(45) Aziz, R. A.; Meath, W. J.; Allnatt, A. R. Chem. Phys. 1983,78,295. (46) Aziz, R. A. High Temp. High Press 1980, 12, 565. (47) Aziz, R. A. Chem. Phys. Lett. 1976, 41, 5767. (48) LeRoy, R. J.; Klein, M. L.; McGee, I. Mol. Phys. 1974, 28, 587. (49) Tanaka, Y.; Yoshino, K. J . Chem. Phys. 1972,57, 2964. (50) Frisch, M. J.; Pople, J. A.; Azieman, A. Mol. Strucr. 1976, 34, 145. (51) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Pbys. 1980, 72, 650. (52) Hariharan, P. C.; Pople, J. A. Chem. Phys. Lett. 1972, 16, 217.

3584 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991

4

are both found to be 0.48. Basis sets containing these d and or f functions are grouped in group I1 and denoted as 6-31 1G(d ,d) or 6-3 1 1G(d2,df). In our next stage of calculation, the restriction on the extended functions is removed and the sp, d, and/or f diffuse functions are added to the basis sets. The exponents of all the sp, d, and f diffuse functions are chosen to be 0.0.1, which is typical of the sp diffuse functions incorporated for the first-row at0ms.4~3~~ Basis sets containing the diffuse functions are grouped in group I11 and denoted as 6-31 1G(d2,d,sp) or 6-31 1G(d2,d,spd). All the extended basis functions are symbolically represented in the parentheses of the basis sets in the order of the regular polarization functions, the polarization functions with the exponent of 0.48, and the diffuse functions with the exponent of 0.1. For example, 6-31 1G(d2,d,sp)contains two regular d function, one d function with the exponent of 0.48, and a set of sp functions with the exponent of 0.1. It should be pointed out that not all the combinations of the extended functions will be considered in this work because not all of them would produce interesting results. The interatomic distance of 4.0 A, which is slightly beyond the potential well, is the minimal distance that satisfies the following condition as required by this work: at this distance the starting basis set 6-31 1G does not cause the BSSE in calculation. At the first stage of calculation with the group I and I1 basis sets, basis functions are added to the starting basis set while they are restricted not to cause the BSSE (Le., bCp does not exceed the tolerance level of 0.6 phartree). Thus the calculated interaction energy needs no correction for the BSSE. At the following stage of calculations with the group 111 basis set, the restriction is removed and diffuse functions are further added to the extended basis sets. Interaction energies calculated by using such basis sets are then corrected with the function counterpoise scheme. If the calculated interaction energy gets improved consistently as the basis set changes from the first to the second stage of calculation, it could possibly mean that no overcorrection for the BSSE is made by the function counterpoise method. But this is not sufficient to accept the function counterpoise method because the overcorrection by the method may still exist when the improved interaction energy insufficiently reflects the real improvement by the addition of the diffuse function. On the other hand, if the calculated interaction energy gets worse as the basis set changes from the first to the second stage of calculation, evidence of overcorrections would be observed and a negative conclusion about the validity of the function counterpoise method should be reached. In order to explain further the complete scheme of testing the validity of the full counterpoise method, we present the following formulation and analysis. The interaction energy calculated without any correction is given by AE = E(Ne2) - 2E(Ne) (2) where E(Ne2) is the total energy of the neon dimer. With the counterpoise correction, it becomes A E p = E(Ne2) - 2E(NeG)

(3)

Equation 3 is then combined with ( I ) and (2) to give A E p = AE 26cp

+

(4) It is commonly known that, due to the basis set extension (BSE) effect, AE is usually an overestimation of the real interaction energy, AE,, which reflects the quality of the basis set used in calculation (the real interaction energy refers to the physical interaction energy assuming that the BSSE is completely removed; it is not the true or the experimental interaction energy). The difference of AE and AE, is defined as the basis set superposition error, BSSE, namely,

+

AE, = AE BSSE, BSSE > 0 Combining (4) and ( 5 ) , we then have AECP = AE, 2~ - BSSE

+

(5)

(6)

Tao and Pan

So far, we have independently defined the two different quantities: 2aCp is the correction for the interaction energy by the counterpoise method (eq 4), and BSSE is the real error incurred by the basis set extension (BSE) effect (eq 5 ) . In the function counterpoise scheme, they are assumed to be the same quantities, namely, 2bCP- BSSE = 0 (7) and therefore eq 6 becomes A@' = AE,, namely, the function counterpoised interaction energy is the estimation of the real interaction energy. In order to verify the validity of the function Counterpoise method, we should verify eq 7. The quantity of 2 p can be readily determined according to eq 1; however, that of BSSE cannot be directly determined likewise. As a result, the direct verification of the validity of the function counterpoise method is considered impossible at the present time. The scheme designed in this work is to provide a possible indirect way to the verification of eq 7. In our first stage of calculation as stated earlier in this section, a basis set x, of group I or group I1 satisfies GCp(xI) = 0 and BSSE(xl) = 0 with a certain tolerance level (-0.6 phartree), and therefore A@'(xI) = AE(xI) = AEr(X1) (8) In our second stage of calculation, a diffuse basis set x2of group 111 is used; Le., 6cp(x2) # 0, BSSE(x2) # 0, and therefore AECP(x2) = AEr(x2) + 26"(~2) - BSSE(x2) (9) It is possible that x2is extended slightly from x1by adding a single sp or d diffuse function to cause large GCp(x2)or large BSSE(x2) while the improvement of the real interaction energy, AE,(x2) AE,(xl), is expected in a small range. If the same expected improvement of the interaction energy with the function counterpoise method is observed, namely AE'(x2) - A@'(xI) = AEr(x2) - AEr(xl), we then have (by subtracting eq 8 from eq 9)

26''(~2) - BSSE(x2) = 0

(10) Equation 7 is thus verified, and therefore the validity of the function counterpoise method is confirmed. Otherwise, if no expected improvement of the interaction energy is observed, or the interaction energy is even worse after the sp or d diffuse function is added to the basis set, namely IhEC'(x2) - A@'(xI)I < Imr(x2) - Mr(X1)l or IAF'(X~)I < l@'(xI)I, we then have 26"(~2) - BSSE(x2) > 0 (11) This means that the function counterpoise method overcorrects for the BSSE and therefore the method may not be valid. The problem now is how to get the improvement of the real interaction energy, AE,(x2) - AEr(xl), as the sp or d diffuse function is added to an original basis set xI. Two assumptions are made in order to solve this problem: (1) AE,(xJ - mr(XI) is independent of the original basis set x l , to which the sp or d diffuse function is added; and (2) when the original basis set is increased to a certain large limit (for example, a group I11 basis set x i ) , we approximately have A E p ( x F ) - AEcp(xi) = AE,(x?) - AE,(xi), where x; is a group 111 basis set and XF is also a group I11 basis set extended from x2 by the sp or d diffuse function. Combining the two assumptions, we then have AEr(x2) - PEAXI) = AEP(x2") - AE'(x2') (12) One may suspect that the second assumption is based on the validity of the function counterpoise method and therefore the entire logic of our scheme above is circular. This is not the case because the second assumption itself does not directly lead to eq 7 or eq 10. The Hartree-Fock S C F calculations followed by the second, third, and fourth orders of Mailer-Plesset perturbation (MP2, MP3, and MP4) calculation^^^^^^ have been carried out with the ~

(53) Latajka, Z.; Scheiner, S. Chem. Phys. Lerr. 1984, 105, 435.

~

_

_

_

(54) Maller, C.; Plesset, M. S. Phys. Reu. 1934, 46, 618. ( 5 5 ) Pople, J. A. Faraday Discuss. Chem. SOC.1982, 73, 7.

_

_

The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 3585

Validity of the Function Counterpoise Method

TABLE II: Totrl h t g i e s of Ne, NeC, and Ne2 (in bartree) a d tbe Corresponding Interaction Energies AE and AECP(in pbartree) Calculated Usinn the GWD I Basis Sets"

basis 6-311G

6-31 IG(d)

6-31 IG(d2)

6-31 IG(d2f)

method

EO")

E(NeG)

me21

AE

ALP

HF MP2 MP3 MP4 HF MP2 MP3 MP4 HF MP2 MP3 MP4 HF MP2 MP3 MP4

-128.522 55305 -128.65506722 -1 28.650 779 08 -128.654 328 79 -128.52255305 -128.731 571 73 -128.731 32069 -128.733 241 73 -128.522 55305 -128.756 344 36 -128.756 17477 -128.75766078 -128.522 55305 -128.77969433 -128.781 48847 -128.782617 39

-128.522553 19 -128.65506740 -1 28.650 779 24 -128.654 328 95 -128.522553 19 -128.731 571 89 -128.731 32084 -128.733 241 88 -128.522553 19 -128.756 344 52 -128.756 17493 -128.75766094 -128.522553 19 -128.77969449 -128.781 48862 -128.782 617 54

-257.045 106 30 -257.310 13538 -257.301 559 26 -257.308 658 70 -257.045 106 30 -257.463 147 69 -257.462 645 79 -257.466 487 82 -257.045 106 30 -257.51270055 -257.512 36088 -257.51533267 -257.045 106 30 -257.559 40062 -257.562 988 42 -257.565 249 57

-0.19 -0.94 -1.10 -1.12 -0.19 -4.23 -4.41 -4.36 -0.19 -1 1.83 -1 1.34 -11.11 -0.19 -1 1.96 -1 1.48 -14.79

+0.08 -0.58 -0.78 -0.80 +0.08 -3.91 -4.1 1 -4.06 +0.08 -11.51 -1 1.02 -10.79

"AE = E(Ne2) - 2E(Ne). A@'

+0.08 -1 1.64

-11.18 -14.49

= E(Ne2) - 2E(NeG).

TABLE 111: Total Energies of Ne, NeC, and Ne2 (in hartree) and the Corresponding Interaction Energies AE and AECP(in phartree) Calculated Using the Group 11 Basis Sets" basis method E(Ne) E(NeG) We21 AE AECP 6-31 1G(d2,d) HF -128.522 553 05 -128.522 553 33 -257.045 106 59 -0.49 +0.07 -24.05 -23.41 -128.759 14627 -257.51 8 3 15 95 MP2 -128.759 14595 -23.68 -23.06 -128.758951 69 -257.51792644 MP3 -128.758951 38 -23.59 -22.97 -128.760505 70 -257.521 034 37 MP4 -128.760505 39 6-31 IG(d2f,d) HF -128.522 553 09 -128.522553 33 -257.045 106 59 -0.49 +0.09 MP2 -128.782 676 62 -128.78267694 -257.565 377 47 -24.23 -23.59 MP3 -128.784441 59 -128.784441 90 -257.568 907 03 -23.85 -23.23 MP4 -128.785601 78 -128.785 60209 -257.571 227 16 -23.60 -22.98 6-3 1 1G(d2,df) HF -1 28.522 553 05 -128.522553 52 -257.045 10699 -0.89 +0.06 MP2 -128.762931 44 -128.762931 98 -257.525 88991 -27.03 -25.95 MP3 -128.762 874 68 -128.762875 20 -257.525 776 16 -26.80 -25.76 MP4 -128.764470 54 -128.764471 07 -257.528 967 90 -26.82 -25.76 6-3 1 1G(d2f,df) HF -1 28.522 553 05 -128.522553 53 -257.045 10699 -0.89 +0.06 MP2 -128.785 11 179 -128.785 112 34 -257.57025066 -27.08 -25.98 MP3 -128.786792 17 -128.78679270 -257.573 61 1 17 -26.83 -25.77 MP4 -128.787 965 40 -128.787 965 94 -257.575 957 47 -26.67 -25.59

"See Table 11.

computer program GAUSSIAN 82.56757 The single, double, and quadrupole substitutions are considered in the MP4 calculations. It should be noted that the MP2, MP3, and MP4 methods, like other approximate CI methods, incompletely describe the correlation effects and therefore could cause underestimation of the interaction energy. For example, in the case of He2, the MP4 method underestimates the interaction energy at the equilibrium distance by 1-22 K,%or 9-l8% of the true interaction energy upon saturation of the basis set.

Results and Discussion A. The First Stage of Calculation: Group I and Group II Basis Sets. The basis sets of group I include the starting basis set, 6-3 1 IG, the extended basis sets with the regular polarization functions, 6-31 lG(d), 6-31 1G(d2), 6-311G(d2f). The basis sets of group I1 include 6-31 1G(d2,d),6-31 1G(d2f,d),6-31 1G(d2,df), and 6-31 1G(d2f,df), of which the extended functions are the d and/or f functions with the exponent of 0.48 in addition to the regular polarization functions. The results of calculations using the basis sets of group I and group I1 are shown in Table I1 and Table 111, respectively. The calculated interaction energies are given b AE (without any corrections for the BSSE) as well as by AE 5 (with full function counterpoise correction for the BSSE). As seen in both tables, the AE and GP values are very close to each other (within 1.2 phartree) as a result of the restriction (56) Seeger, R.; Pople. J. A. J . Chem. Phys. 1977, 66, 3045. (57) GAUSSIAN 82; Binkley, J. S.;Frisch, M.J.; DeFrees, D.J.; Rahgavachari, K.; Whiteside, R. A,; Schlegel, H.B.; Fluder, E. M.; Pople, J. A. Department of Chemistry, Carnegie-Mellon University, Pittsburgh, PA. (58) Saver, J.; Hobza, P.;Carsky, P.; Zahradnik, R. Chem. Phys. Left. 1987, 134, 553.

of the basis sets in these two groups. This indicates that no serious influences of the BSSE effect have resulted in the calculated interaction energies, AE, and therefore corrections for the BSSE are unnecessary (with the tolerance of aCp = 0.6 phartree). It is seen that with the increase of the number of the polarization functions the calculated interaction energies, AE or A@p, are improved progressively and consistently and the rate of convergence of the interaction energy to basis set is moderate. However, with the increase of the Maller-Plesset approximation level, the improvement of the interaction energy is very poor and in most cases the results are even worse at the higher Maller-Plesset approximation levels (e.g., MP3 or MP4) than MP2, even though the individual energies, E(Ne2), E(Ne), or E(NeG), are all improved significantly. This seems in agreement with the statement that the Maller-Plesset theory converges slowly and the contributions of MP3 and MP4 tend to cancel each If the two tables are compared, it is seen that major improvement of the interaction energy is achieved by introducing only the d function or the d and f functions with the exponent of 0.48 to the basis sets of group I. For example, the AE of -1 1.1 1 phartree at MP4/6-31 1G(d2) is improved to -23.59 phartree at MP4/631 1G(d2,d),or to -26.82 phartree at MP4/6-31 1G(d2,df). On the other hand, no noticeable improvement of the interaction energy is made by the inclusion of the regular f function (with the exponent of 2.5) as is seen by comparison of 6-3 11G(d2,d) with 6-31 1G(d2f,d), or 6-31 1G(d2,df) with 6-31 lG(d2f,df),or even 6-31 1G(d2) with 6-31 1G(d2f) (except for MP4/6-31 1G(d2f)). This means that polarization functions with smaller exponents, or the extended basis functions that are more diffuse, are more effective in the description of the interaction energy. Of course, this conclusion is based on our special case of the relatively large

3586 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991

Tao and Pan

TABLE I V Total Energies of Ne, NeG, and Ne2 (in bartree) n d tbe Corresponding Interaction Energies AE n d AECP(in fibtree) Calculated Using the Group 111 Basis Setsa basis method EW) E(NeG) EW2) hE LLP -257.054 394 24 -34.59 +1.17 6-31 IG(d2,d,sp) HF -128.527 17982 MP2 -128.767367 15 -257.534 76069 -104.83 -26.39 -128.767 32793 -101.52 -128.765 97805 -257.531 983 16 -27.06 -128.76594082 MP3 -105.76 -27.80 MP4 -128.768 32071 -257.536 669 22 -128.768281 73 -257.045 525 79 -419.69 +0.64 6-31 IG(d2,d,d) HF - I 28.522 553 05 -128.522 762 57 -257.519084 19 -6 57.09 -25.57 MP2 -128.75921355 -128.759 529 31 -128.759 30072 -257.518 62664 -583.12 -25.20 MP3 -128.759021 76 -624.58 -25.34 -128.76087205 -257.521 76944 M P4 -128.76057243 -34.57 +1.19 6-31 IG(d2,df,sp) HF -128.527 17982 -128.527 197 70 -257.054 394 21 -257.541 161 60 -103. I4 -29.86 M P2 -128.770565 87 -128.77052923 -99.76 -30.50 -128.768 82078 -257.537 672 06 MP3 -128.768 786 15 -31.53 -128.771 30902 -257.542 649 57 -105.45 -128.771 27206 M P4 -0.68 -257.045 515 83 -409.74 6-31 1G(d2,df,d) HF -128.522 55305 -128.522757 58 -28.66 -257.525441 42 -654.76 -128.76270638 MP2 -128.762 393 33 -28.38 -257.524371 16 -582.32 -128.761 89442 -128.762 171 39 MP3 -28.63 -257.527 659 11 -624.47 MP4 -128.76351732 -128.763815 24 +0.85 -257.054 462 41 -128.527231 63 -102.77 6-31 1G(d2,d,spd) HF -128.527 17982 -29.38 -257.53508282 -128.767 526 72 -235.28 MP2 -128.767 423 77 -30.43 -257.532 291 83 -23 1.45 MP3 -128.766030 19 -128.766 13070 -3 1.89 -257.536 996 53 -238.75 MP4 -128.768 378 89 -128.768482 32 +0.85 -257.054463 16 -103.52 6-31 IG(d2,df,spd) HF -128.527 17982 -128.527 23201 -32.52 -257.542 728 56 -243.76 -128.771 24240 MP2 -128.771 34802 -33.78 -239.74 -I 28.769 96467 -257.540 16908 -128.770067 65 MP3 -35.58 -247.64 -128.772 372 53 -257.544 992 70 -128.772478 56 MP4 +0.23 -257.054 573 48 -213.84 6-31 IG(d2,df,spdf) HF -128.527 17982 -128.527 286 85 -34.24 -450.32 M P2 -128.771 33865 -257.543 127 62 -128.771 54669 -35.65 MP3 -128.770068 19 -257.540 583 01 -128.770 273 68 -446.63 -37.68 -457.40 M P4 -128.772473 82 -1 28.772 683 68 -257.545 405 04 “See Table 11.

interacting distance (4.0 A). Diffuse (polarization) functions with smaller exponents will be further included in the basis sets of group I11 but the regular f function (with the exponent of 2.5) will be excluded in order to save the space and CPU time of the computer. The best value of the interaction energy achieved at this stage of calculations is AE = -27.08 phartree, or A E P = -25.98 phartree, by MP2/6-31 1G(d2f,df). This is still far below (or only 60% of) the estimated experimental value of -41.07 phartree. B. The Second Stage of Calculation: Group III Basis Sets. The group 111 basis sets and the corresponding results of calculations are shown in Table IV. Because at least one of the basis functions with the small exponent of 0.1 is included in each of the basis sets, large influences of the BSSE effects are no longer evitable as in the previous calculations. This can be seen by the notable energy differences between E(Ne) and E(NeG) and also by the large overestimates of the interaction energy without corrections, AE. However, with the full function counterpoise correction scheme the interaction energies, A S p , are corrected to the expected range and great improvement over the results of the group I1 basis sets is also achieved. As the size of the basis set increases in this group, the calculated interaction energy is improved progressively and consistently from -25.35 phartree by MP4/6-31 1G(d2,d,d) to -37.68 phartree by MP4/6-311G(d2,df,spdf). More interestingly, unlike in the previous calculations, higher Moiler-Plesset level apparently shows advantage in producing better interaction energies. Furthermore, the improvement of the results is progressively larger with the increase of the basis size and with the increase of the Mdler-Plesset level. For example, the changes of hEcpfrom MP2 to MP3 are -0.67 phartree at 6-31 1G(d2,d,sp),-1.05 phartree at 6-31 IG(d2,d,spd), and -1.41 phartree at 6-31 lG(d2,df,spdf), and from MP3 to MP4, -0.74, -1.80, and -2.03 phartree, respectively. If the results of 631 1G(d2,df,spdf) are compared, the interaction energy, hECp, is improved from MP2 to MP4 by 3.44 phartree, almost 10% of the interaction energy calculated. All of these could bring us a fresh look at the Moller-Plesset perturbation theory: the convergence of the theory is moderately fast and the higher Maller-Plesset level does show considerable advantage over the lower level of the method in the description of the van der Waals interaction energies; Le., MP4 shows advantage over MP3, and MP3 shows advantage over MP2. It should be stressed that the above con-

clusion is based on the condition that the basis set used in the calculation should reach a certain quality level; otherwise the results of higher Mdler-Plesset level are approximately the same as or even worse than that of MP2, the lowest Mdler-Plesset level, as demonstrated in the first stage of calculations. The conclusion reached here may also provide us a criterion for the sufficiency of a basis set in calculation of van der Waals interaction energy: a basis set is sufficient only when the results of higher MallerPlesset level are improved considerably compared to that of lower level of the method. The best calculated interaction energy is now hEcp= -37.68 phartree by MP4/6-31 IG(d2,df,spdf). It recovers 91.7% of the total interaction energy as compared to the estimated experimental value of the HFD-Cl potential. Considering that the MP4 method itself may underestimate the interaction energy by as much as 9-18% in the case of He2, this result is very encouraging: it may mean that the basis set, 6-31 IG(d*,df,spdf), could be already as effective as the saturated basis set in the description of the interaction energy of the neon dimer at the distance of 4.0 A. The drastic improvement of the calculated interaction energy by introducing the diffuse functions could give us an indication that the orbital overlap between the interacting components may play an important role in the description of the van der Waals interaction energy. This means that the basis set used in calculation should be diffuse enough to have considerable orbital overlap between the two components in order to completely describe the long-range interaction. Therefore, as the distance of the two components increases, more and more diffuse basis set should be used in the calculation of the interaction energy. In usual sense, the van der Waals interaction energy at a large distance is mainly the dispersion energy which is contributed from the correlation effects instead of the Coulombic and exchange effects, and therefore only polarization functions would be important in the description of the interaction energy. It turns out to be not fully true from our calculations. Diffuse functions, including diffuse polarization functions, should also be needed to ensure the large enough orbital overlap between the components in order to thoroughly describe the interaction energy. Consider the hEcp results from the two basis sets, 6-311G(d2,d,sp) and 6-311G(d2,d,d), as given in Table 1% we could see that even the sp diffuse function (which is not a polarization function in our sense) could

The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 3587

Validity of the Function Counterpoise Method

TABLE V Improvement of tbe Interaction Energy with Function Counterpoise Correction, AECP(xJ - AE"(xl) and AE"(x,') Addition of the sp or d Diffuse Function (in phartree)

(d2& A S P ( x 2 ) - AFP(xi)

-2.98 -3.91

AflP(xz') - A@'(xz)

-

MP3

-4.00 (d2,fd,sp) -4.74

(d2,df)

(d2,d,d) -3.81

-

(d2,d,spd)

-5.23

(d2,df,d) -3.86

(d2,d,sp)

MP4

MP2

-4.83

-2.16

-5.77

-2.7 1

-6.55

-2.99

greatly improve the interaction energy and the improvement by this sp function is even greater than that by the d diffuse function with the same exponent. The reason may be from the fact that a set of sp functions potentially produce larger interatomic orbital overlap than does a d function. This again demonstrates the importance of the orbital overlap between the components in the description of the van der Waals interaction energy. The above analysis enables us to think about a possible and more economical way to find the efficient basis sets for the calculations of van der Waals interaction potentials: the exponents of some extended basis functions in a basis set, especially of the polarization functions, are varied according to the distance of interaction in order to,reach the maximal efficiency in the calculation of the interaction energy at each specific distance. This means, at a large distance, extended functions mainly with small exponents should be considered in a basis set, while at a short distance, regular extended functions (with relatively large exponents) should be considered. This can be realized possibly by employment of the continuous variables for the exponents of those extended basis functions. C. Proof of the Validity of the Counterpoise Scheme: Comparison of the Two Stages of Calculations. The basis sets 631 1G(d2,d,sp)and 6-31 1G(d2,d,d)of group 111 in the second stage of calculations are both extended from the basis set 6-31 1G(d2,d) of group I1 in the first stage of calculations by adding the sp diffuse function and the d diffuse function, respectively. Similarly, 631 1G(d2,df,sp) and 6-31 1G(d2,df,d) are from 6-31 1G(d2,df). It is seen in Table I11 and IV that the interaction energies calculated with the function counterpoise correction for the BSSE, A E p , are consistently improved by these basis set extensions in spite of the considerably large overestimations of the interaction energies without correction, AE, caused from the BSSE through the use of the diffuse functions. For example, from MP4/6-31 1G(d2,d) to MP4/6-3 1 1 G(d2,d,sp) and to MP4/6-3 1 1G(d2,d,d),the AEcp is improved from -22.97 to -27.80 phartree and to -25.34 phartree, respectively, or from MP4/6-31 1G(d2,df) to MP4/631 1G(d2,df,sp) and to MP4/6-31 1G(d2,df,d),hEcpis from -25.76 phartree to -31.53 phartree and to -28.63 phartree, respectively. Similar improvement is also found for the corresponding MP2 and MP3 calculations. Thus, there is no apparent overcorrection for the BSSE made by the function counterpoise method. However, solid proof of the validity of the function counterpoise method still needs further detailed analysis for the calculational results. Table V lists quantitatively the improvement of the interaction energy with the function counterpoise correction by the addition of the sp or d diffuse function, AEp(x2)- AEp(xI) and AEP(x?) - AEP(xi),where xI is a basis set of group I1 and x i and x2/1 are of group 111. Because both x i and x2/1are subject to the BSSE effect. The overcorrections for the BSSE made by the function counterpoise method, if any, may be a uniform constant with these two basis sets and therefore is approximately Up(x2'[)- A E p ( x i ) = AEr(X?) - AEr(xi) which is the expected improvement of the real interaction energy achieved by the addition of the sp or d diffuse function. This speculation is nothing but one of the assumptions made in the earlier section (the second assumption above eq 12). As seen in the table, for the addition of the sp diffuse function, the expected improvement at MP2 is -3.81 phartree when x i = 6-31 1G(d2,d,d) and x? = 6-31 1G(d2,d.spd), and -3.86 phartree when x i = 6-31 1G(d2,df,d)

MP3 (d2& (d2,d,d) -2.14 (d2,df) (d2,df,d) -2.62

-

-

(d2,d,sp)

(d2,df,spd)

-5.40

by

d

SP

MP2

- AE"(x&,

-6.95

-2.66

(d*,d,spd)

-3.37

(d2,df,sp)

(d2,df,spd)

-3.28

MP4 -2.37 -2.87 -4.09 -4.05

and x? = 6-311G(d2,df,spd), at MP3, -5.23 phartree and -5.40 phartree; and at MP4, -6.55 and -6.95 phartree, respectively. This indicates that the expected improvement achieved by the addition of the sp diffuse function is approximately independent of the basis set to which it is added (the maximal variance is only 0.40 phartree from 6-31 1G(d2,d,d) to 6-31 1G(d2,df,d)). Similarly, for the addition of the d diffuse function, the expected improvement at MP2 is -2.99 phartree when x i = 6.31 1G(d2,d,sp) and -2.66 phartree when x i = 6-31 1G(d2,df,sp), a t MP3 -3.37 and -3.28 phartree, and at MP4 -4.09 and -4.05 phartree, respectively. This again demonstrates the independence of the expected improvement on the basis set to which the d diffuse function is added (the maximal variance is 0.33 phartree). If we assume that the same independence still remains true for the basis extension from group I1 to group 111 by the addition of the sp or d diffuse function, i.e., AEr(x2)- AEr(xl)= AEr(x2/1)- AEr(x2)),the other assumption in the earlier section (first assumption above eq 12), we have then established the expected improvement of the real interaction energy as a standard to compare with the corresponding results from the function counterpoise calculations in order to determine the validity of the function counterpoise method. If the results of hEcp(x2)- AlFP(xI)in Table V are compared with that of AEp(xT)- AEp(xi), which are taken to be of AEr(x2) - AE,(xI) according to above analysis, noticeable overcorrections are seen to have been incurred by the function counterpoise method. In the case of the sp diffuse function, for example, the improvement of the interaction energy calculated with the function counterpoise method at MP4 is AEP(x2)AEp(xI)= -4.83 phartree when xi = 6-31 1G(d2,d) and x2 = 6-31 1G(d2,d,sp)while it is expected to be AEr(x2)- AEr(xi)= -6.55 or -6.95 phartree and therefore an overcorrection of 26cp(x2) - BSSE(x2) = 1.72 or 2.12 phartree has been incurred by the function counterpoise method with the basis set of x2 = 63 11G(d2,d,sp). Similarly an overcorrection of 1.68 or 1.72 phartree has been incurred with MP4/6-31 1G(d2,d,d). However, two aspects of evidence can be seen for the overcorrections. First, the overcorrections are negligibly small compared to the overall errors which may result from the deficiency of the basis set and calculational method. For example, the calculated interaction energy by MP4/6-31 1G(d2,d,sp) is -27.80 phartree and therefore the overall error is still at -13.27 phartree. Similarly, the overall error with MP4/6-31 1G(d2,d,d) is -15.73 phartree. Second, the overcorrections diminish rapidly with the saturation of the basis set. If xi = 6-31 1G(d2,df) and x2 = 6-31 1G(d2,df,sp) are considered, A E p ( x 2 ) - AEp(xl) = -5.77 phartree at MP4, and therefore the overcorrection for the BSSE decreases to 0.78 or 1.18 phartree with the basis set of 6-31 1G(d2,df,sp). A similar trend is also found in the case of the d diffuse function; for example, the overcorrection of 1.68 or 1.72 phartree with 631 1G(d2,d,d) at MP4 decrease to 1.18 or 1.22 phartree with 6-31 1G(d2,df,d). It is easy to imagine that with further increase of the basis size the overcorrection for the BSSE would decrease well under 1.Ophartree, which is negligible compared to the error caused by the deficienciesof basis set and the calculational method. For example, an overcorrection of 0.1-0.4 phartree for the BSSE is estimated for the MP4/6-3 11G(d2,df,spdf) calculation while the deficiencies of the basis set and the MP4 method contribute the underestimation of the total interaction energy by more than 3.0 phartree.

3588 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 So far, following conclusive remarks about the validity of the function counterpoise method have been drawn: (1) the overcorrection for the BSSE by the function counterpoise method is negligibly small compared to other calculational error caused by the deficiencies of basis set and of the calculational method, namely,

26cp(x2) - BSSE(x2) = O(4XZ)) (13) where c(x2) denotes the other calculational error associated with the basis set x2 and O(c(x2)) represents the infinitesimal quantity of t(x2); and (2) the overcorrection vanishes more rapidly than does the other calculational error with the increase of basis set and therefore when a basis set used is reasonably large (e.g., xz = 6-31 1G(d2,df,spdf) in this work) we have Equation 7 or 10 is thus valid and the validity of the full counterpoise method has been demonstrated and verified. Note that the problem of the contradiction between the Pauli principle and the validity of the full counterpoise method has now been well solved. This means that when a small basis set is used 26cp will be an overcorrection for the BSSE due to the Pauli principle, but the overcorrection is negligibly small compared to other error and, furthermore, as the size of the basis set increases, it converges to zero much faster than does the underestimation of the calculated interaction energy incurred by the deficiencies of basis set and of the calculational method. Another interesting results can be seen in Table V that the overcorrections for the BSSE by the full counterpoise method are always larger at higher Maller-Plesset approximation levels. For example, the overcorrections with the basis set of 6-3 11G(d2,d,sp) are 0.83, 1.23, and 1.72 phartree or 0.88, 1.40, and 2.12 phartree at MP2, MP3, and MP4, respectively. This indicates a strong coupling between the effect of basis set and that of the calculational method in calculation of van der Waals potential with the full counterpoise method: a high level of calculational method should be always incorporated with the use of large basis set in order to avoid its potentially large overcorrections for the BSSE; on the other hand, a low level of calculational method should be enough or even preferred if a small basis set is used. This discovery is consistent with the results found in the first stage of calculations where the interaction energies of MP2 are better than those of MP3 and MP4.

Conclusions The scheme and calculation presented in this work is a new approach to the study of the full counterpoise correction for the BSSE and other methodological strategies for a b initio determination of van der Waals potentials. The following major conclusions can be drawn from the results of calculation and from the analysis. 1. The full counterpoise method is a valid approach to the correction for the BSSE. When a small basis set is used, which is not expected to be capable of correctly describing the interaction energy, overcorrection for the BSSE will be made by the full counterpoise method. However, the overcorrection is negligibly small compared to other calculational error incurred by the de-

Tao and Pan ficiencies of basis set and of the calculational method. Furthermore, as the basis set is improved, the overcorrection vanishes more rapidly than the other calculational error. When a reasonably large basis set is used, the validity of the full counterpoise method is fully established. By this explanation, the contradiction of the Pauli principle and the validity of the full counterpoise method is well resolved. It is interesting to note that, as a result of this conclusion, one should focus the attempt on the improvement of basis set and calculational method, rather than the recovery of the overcorrection for the BSSE by the function counterpoise method, in order to achieve the accurate calculational interaction energy. The fact is that even an ideal method (with no overcorrection for the BSSE at all) could not yield any good interaction energy if insufficient basis set and inadequate calculational method are employed. Therefore, the effort in searching for any better correction scheme other than the full counterpoise method is senseless. 2. Diffuse functions as well as polarization functions should be added to the regular energy-optimized basis set in order to calculate the complete interaction energy. Diffuse functions (including diffuse polarization functions) with small exponents tend to be more effective in the description of the interaction energy than those with large exponents. The diffuse functions are needed possibly to ensure the large orbital overlap between the interacting components that may play an important role in the determination of the van der Waals interaction energy. 3. A few higher orders of Maller-Plesset approximation method (MP3 and MP4) yield better interaction energy than does the second order of the method (MP2). This is true only on the condition that the basis set used in a calculation exceeds a certain quality level. Otherwise, the MP3 or MP4 calculation produces nearly the same or even worse results than does the MP2 calculation. Furthermore, the better the basis set used, the greater will be the improvement achieved at the higher Maller-Plesset level. In addition, the higher Mdler-Plesset level tends to produce the larger overcorrection for the BSSE by the full counterpoise method and therefore it should always employed with the use of the larger basis in order to ensure the advantage of the method. Finally, we have realized that the verification of the validity of the counterpoise method should be supported by results for other geometries, even other van der Waals systems. The primary interest to us is of course at the equilibrium distance or nearby. The verification at this distance with similar strategies as stated in this work is also possible, but systematic modifications are needed and the procedure is more complicated because the BSSE cannot be neglected for the starting basis set 6-31 1G. We will be working on this as a separate work in the near future. Nevertheless, our results at R = 4.0 A have reflected most of the general features of the BSSE problem. Basically, at other distances there is no essential difference in the features of the BSSE behavior except the different numerical magnitudes. Acknowledgment. We thank Professor Y. N . Chiu of the Catholic University of America for reading the manuscript and for helpful suggestions. We also thank John Gary and Edmund Greene of Boston College Computer Center for their enthusiastic assistance.