5815
J. Phys. Chem. 1992,96, 5815-5816
Valldlty of the Function Counterpoise Method: Results from the Complete Fourth-Order MBPT Calculations Fu-Ming Taot and Yuh-Kang Pan* Department of Chemistry, Boston College, Chestnut Hill, Massachusetts 02167 (Received: December 13, 1991)
The validity of the function counterpoise (CP) correction method for the basis set superposition error (BSSE) is tested by ab initio calculation of the Ne-Ne interaction energy at the level of the complete fourth-order M~rller-Plesset perturbation theory. The scheme of the test procedure is briefly introduced. The results of this calculation support the CP method. The related issues such as BSSE and the basis set for the calculation of intermolecular potentials are discussed.
I. Introduction A scheme to test the validity of the function counterpoise (CP) correction method for the basis set superposition error (BSSE) was recently presented by us1 with ab initio calculations of the Ne-Ne interaction energy at a given distance (4.0 A) using a series of extended basis sets. The Mailer-Plesset perturbation theory (MBPT) was employed for the calculation of electron correlation energy, based on the Hartree-Fock SCF wavefunctions. The highest theoretical level used was the fourth-order approximation (MP4) including the single, double, and quadruple excitation configurations. Our more recent study2and several other reviews3s4 have however shown that the contribution of correlation energy from the triple excitations is nonnegligible for the accurate determination of van der Waals interaction energy. The triple excitations describe the influence of intramolecular correlation on the intermolecular correlation, or the intra-inter coupling, which often accounts for -20% of the total correlation interaction energy (depending on the interatomic distance). The lowest level to incorporate the triple excitations in MBPT is the complete fourth-order approximation with all the single, double, triple, and quadruple excitations (MP4SDTQ).S*6 In this work, we report the results on the validity of the CP method from the MP4SDTQ calculations.
II. Method In this section, we introduce the outlines of the scheme for the test of the validity of the CP method for the BSSE correction. For details, including the choice of the basis sets, the readers may refer to our original paper.’ In the supermolecular approach, the calculated interaction energy between two neon atoms is given by AE = E(Ne2) - 2E(Ne) (1) where E(Ne) and ,?(Ne2) are the total energies of the neon atom and the dimer, respectively. With the CP correction,’ it becomes
hEcp= E(Ne2) - 2E(NeG)
(2) where E(NeG) is the total energy of the neon atom calculated with the dimer basis set. The difference between E(Ne) and E(Ne,) is defined as gCp and we have
hEc’ = AE
+2
P
(3)
The quantity 2aCpis the correction of the calculated interaction energy for the BSSE in the CP method. This implies the following assumption 26” - BSSE = 0 (4)
Thus the verification of the CP method is to verify eq 4. The major difficulty is that the actual BSSE value could never be surely known despite that it may be trivially defined as BSSE = AE, - AE (5) ‘Current address: Department of Chemistry, Brown University, Providence, RI 02912.
where A,??,refers to the physical interaction energy assuming that the BSSE is completely removed. combining eqs 3 and 5 , we have
@’ = AE, + 26”
- BSSE
(6) For any basis set x, we can always find an interatomic distance beyond which 2aCp(x) = 0 and BSSE(x) = 0 so that AECP(X)= MAX) (7) When a basis function X is added so that the new basis set, x’ = x A, becomes more diffuse, we have hEc‘(x’) AE,(x’) + 2FP(x’) - BSSE(x’) (8)
+
Combining eqs 7 and 8, it becomes h E c P ( X ’ ) - AECP(X) = Ur(x’) - AEr(x) + 26“(~’) - BSSE(x’) (9) Equation 4 holds immediately if hEc‘(x’) - @‘(x) = AE,(x’) - AE,(x). We assume that AE,(x’) - AE,(x) has a definite value for a given function X and is independent of the original basis set that X is added to. We may tentatively expect that this quantity is the limit of hEcp(x’) - hEcp(x) in a very large basis set. Therefore we can estimate the value of AE ( ’) - AE,(x) by extrapolating the corresponding value of A,#$&’) - hEcp(x). In other words, the verification of eq 4 is equivalent to observing how hEcp(x’) - hEcp(x) for a given function X varies with the increase of the basis set x. One of three possible results may be expected: (1) it stays at a same value for different basis sets; (2) it increases in magnitude with the increase of basis set and converges to a limiting value for a reasonably large basis set; (3) it shows no tendency of convergence within the basis sets considered. The first result means that hEcp(x’) - hEcp(x) = AE,(x’) AE,(x) and thus leads to the validity of the CP method. The second result indicates the overcorrection of BSSE by the CP method in a small basis set but it also confirms the validity of the CP method for practical calculations. The third result leads to the rejection of the CP method. Our previous work arrived at the second result, that is, at the MP2, MP3, and MP4SDQ levels the CP method overcorrects the BSSE if a deficient basis set is used but the term of the overcorrection tends to fall off if the basis set is improved within the practical limit. Furthermore, it also showed that even with the deficient basis set the overcorrection of the BSSE by the CP method is a negligible error as compared to other errors such as those caused by the deficiency of the calculational method and basis set. We use the same basis sets (extended from 6-31 1G) and basis functions (A = sp and d with the same exponent of 0.1) as those for the previous work. We omit some of the basis sets which are irrelevant to the main objective of this work. The interatomic distance remains at R = 4.0 A.
III. Results and Discussions The total energies of the monomer (Ne and NeG) and of the dimer (Ne2) from the MP4SDTQ calculations in the various extended basis sets are presented in Table I. The calculated interaction energies after the CP corrections, hEcp, and the
0022-3654/92/2096-58 1S$03.00/0 0 1992 American Chemical Society
5816
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992
Tao and Pan
TABLE I: Total Energies of Ne, NeC, and Ne2 (in hartrees) and tbe Corresponding Interaction Energies AECPand the Values of microhartrees) from the MPlSDTQ Calculations Using the Various Extended Basis Sets basis E(Ne) E(NeG) E(Ne,) 26CP 6-31 1G(d2,d) -128.764679 15 -128.764679 47 -257.529 383 93 0.64 -128.768 949 98 -128.768 950 52 -257.537 929 33 6-3 1 1G(d2,d0 1.08 -128.773 161 62 -128.773 205 11 6-3 1 1G(d2,d,sp) -257.546441 04 86.98 -128.765 067 03 6-3 1 1G(d2,d,d) -128.764 75078 632.50 -257.530 161 72 -128.777 475 27 6-31 1G(d2,df,sp) -128.117 51964 -257.555 07423 88.74 -128.769021 78 6-31 lG(d2,df,d) -128.769 34025 -257.53871 170 636.94 -128.773 27284 6-31 1G(d2,d,spd) -128.773 385 35 -257.546 806 30 225.02 6-3 1 1G(d2,df,spd) -128.777 702 27 -257.555 444 52 -128.777 58697 230.60 -128.77791631 -128.777 69099 6-3 1 1G(d2,df,spdf) -257.555 87495 450.64 TABLE 11: Improvement of the Interaction Energy with the Counterpoise Correction, AECP(x’)- AEcp(x), by the Addition of the sp or d Diffuse Function (in microhartrees) addition of sp addition of d
-
(d2,d) (d2,d,sp) (d2,df) -+ (d2,df,sp) (d2,d,d) (d2,d,spd) (d2,df,d) (d2,df,spd) -+
-+
-5.83 -6.66 -7.94 -8.78
-
(d2,d) (d2,d,d) (d2,df) --+ (d2,df,d) (d2,d,sp) + (d2,d,spd) (d2,df,sp) (d2,df,spd) +
-2.67 -2.91 -4.78 -5.03
corresponding values of 26cp are also listed in the same table. It is seen that the values of 26cp varies violently, from almost zero for the relatively compact basis sets (Le. 6-31 1G(d2,d))to more than 20 times the corrected interaction energies for the diffuse basis sets (Le. 6-31 1G(d2,df,d)). However, the calculated interaction energies after the C P correction for BSSE fall just into the correct range expected with the quality of the basis set. This may indicate that the magnitude of 2aCpdoes not have much to do with the quality of the basis set used in the practical calculations. A common expectation about 26cp is that it approaches to zero in the limit of a complete basis set.8 In practice, however, it could possibly never reach that limit, and in fact it seems that all of the basis sets considered so far in the literature are probably still far from reaching such a limit. For instance, the value of 26cp for the basis set of 6-31 lG(d2,df,spdf), 230.60 phartrees, is almost doubled by the addition of only a diffuse f function. Moreover, it seems that the addition of a high polarization function (e.g., d, f, etc.) increases the value of 2 F P much more severely than the addition of a lower polarization function (e.g., s or p) with the same exponent. Wells and Wilson9 studied the behavior of gCp and found that it may increase and pass through a maximum before decreasing to its limiting value of zero. Although this conclusion can be generally true, the question about the completeness of a basis set still remains, since the authors studied the problem only with the Hartree-Fock calculations with the basis sets consisting just of the sp functions, whose exponents were not very diffuse. The completeness of a basis set means that no effect (for example, on 26cp) could be produced by further addition of any other basis functions to the basis set. Our result and the argument in this paragraph is trying to explain that in a practical calculation the magnitude of 2aCp should not be so much concerned, causing the unnecessary doubt about the adequacy of the basis set and the accuracy of the result, as long as an appropriate correction such as C P is followed. The effort to find the complete basis set with negligible 26cp is a trivial attempt and it may be practically impossible without introducing arbitrary restrictions on the basis set. Compared to the results of lower theoretical levels as given in our previous paper,] the MP4SDTQ interaction energies are all improved considerably. The addition of the triple excitation configurations at the MP4 level improves the interaction energy by more than 10% for all the basis sets considered. The best value of this work, AEcp = -42.33 phartrees, should be very close to the estimated experimental value. For instance, the HFD-C 1 potential for Ne2 of Aziz et al. at the same distance gives a value
BCp(in
ALP -24.99 -28.29 -30.82 -27.66 -34.95 -3 1.20 -35.60 -39.98 -42.33
of -41.07 phartrees while the HFD-C2 potential gives -43.93 phartrees.’O Our calculation with a much larger basis set gives -42.38 phartrees.* Obviously, the converged value of the calculated interaction energy is consistent with our previous argument and with the conclusion in the following paragraph. The values of the improvement of the calculated interaction energy by the addition of the sp or d diffuse function, hEcp(x’) - AEcp(x), are summarized in Table 11. It is seen that the magnitudes of the improvement are smaller with the smaller basis sets and tend to converge fairly fast with the increase of basis set. If we take the highest values as the estimates for the improvement of the calculated interaction energy due to the addition of the specific diffuse functions that we consider, according to the analysis in the previous section, the overcorrections of BSSE by the C P method could be determined. For the addition of the diffuse sp function, the consecutive values of the overcorrections are 2.95, 2.12, and 0.84 phartrees for the increasing basis set of x = 631 1G(d2,d), 6-31 1G(d2,df), and 6-31 lG(dz,d,d), respectively. Similarly, the values for the addition of the diffuse d function are 2.36, 2.12, and 0.25 phartrees. This indicates that the overcorrection of BSSE by the C P method does occur, but it tends to vanish with the saturation of basis set. All these results are quite similar to those from the lower theoretical levels as we found previously’ except that the decreasing rates of the overcorrection are slower. The slower decreasing rates of the overcorrection are also in agreement with our previous result’,* that the calculation at a higher theoretical level demands more for the requirement of basis set. It is very likely that the overcorrection of BSSE is not even nearly completely vanished for the basis set of x = 6-31 1G(d2,df,d) or 6-31 1G(d2,df,sp) for which we have taken for the estimation of aE,(x’) - aE,(x). It should however be noted that the quality of most basis sets that we use for practical calculations can be much superior to the quality of such a basis set, so that the resulting overcorrection of BSSE for the larger basis sets could be really negligible. On the other hand, a basis set inferior to the adequate basis set cannot produce the correct interaction energy anyway, even if the overcorrection of the BSSE is recovered. For example, the value of hEcpfor 6-31 1G(d2,df,d) is -31.20 phartrees. If the overcorrection of the BSSE, 2.12 phartrees, is recovered, it would be -33.32 phartrees, still much below the anticipated value. As a result, we can be safe to say that the overcorrection of BSSE by the C P method is negligible, for the use of either a sufficient or insufficient basis set. The validity of the C P method should therefore be confirmed. References and Notes ( I ) Tao, F.-M.; Pan, Y.-K. J . Phys. Chem. 1991, 95, 3582. (2) Tao, F.-M.; Pan, Y.-K. J . Phys. Chem. 1991, 95, 9811. (3) Hobza, P.; Zahradnik, R. Chem. Reu. 1988, 88, 871. (4) Chalasinski, G.;Gutowski, M. Chem. Reu. 1988, 88, 943. ( 5 ) Merller, C.; Plesset, M. S.Phys. Reu. 1934, 46, 618. (6) Pople, J . A. Faraday Discuss. Chem. SOC.1982, 73, 7 . (7) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553. (8) van Lenthe, J. H.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Duijneveldt, F. B. Ab initio Methods in Quantum Chemisfry-11;Lawley, K. P., Ed.; Wiley: New York, 1987; p 521. (9) Wells, B. H.; Wilson, S.Mol. Phys. 1983, 50, 1295. ( I O ) Aziz, R. A,; Meath, W. J.; Allnatt, A. R. Chem. Phys. 1983, 78, 295.
