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Comment on “A possible definition of basis set superposition error”
14 July 1995
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 241 (1995) 140-145
Comment
Comment on “A possible definition of basis set superposition error’ ’ Maciej Gutowski a1*, Matgorzata M. Szcz@niak b, Grzegorz Chalasiiiski
’
a Department of Chemistry, Unioersity of Gdarisk, 80-952 Gdarisk, Poland h Department of Chemistry, Oakland University, Rochester, MI 48309, USA ’ Department of Chemistry, Uniuersity of Warsaw, 02-093 Warsaw, Poland Received
15 September
1994; in final form 15 May 1995
Abstract In their recent Letter Davidson and Chakravorty proposed a framework for analysis of basis set effects which trouble calculations of potential energy surfaces for weakly interacting species. Their analysis suggests that the complete basis set interaction energy differs by some monomer- and dimer-type non-additive corrections from the conventional counterpoised interaction energy obtained in a finite basis set. In this Comment we point out that the dimer-type correction does not fit the definition of non-additivity applied for the monomer terms and that the sum of the non-additive corrections reduces to a component of the interaction energy. This component disappears if the finite basis set is flexible enough to describe all important terms of the interaction energy. Thus the discrepancy between the counterpoise-corrected and the complete basis set interaction energy is exclusively determined by the suitability of the basis set used to reproduce the interaction energy terms.
1. End of the deadlock? Recent studies on van der Waals interactions indicate that the discussion on the basis set superposition error (BSSE) has reached a deadlock. On the one hand, a few research groups insist that the Boys and Bernardi function counterpoise procedure (CP) [l] is the correct approach to eliminate the lowering of monomer energies which takes place in the dimer calculation [2-161, but other researchers remain sceptical. First, there is a lingering doubt whether occupied orbitals of monomer A should be available
??
Corresponding
author.
0009-2614/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0009-2614(95)00611-7
for monomer B in the course of counterpoise calculations [17,18]. Second, other studies emphasize that the CP-corrected interaction energies differ from the expected full-basis set results [19-281. Some of the most persuasive arguments in favor of the CP correction come from the analysis of the CP-corrected interaction energies in terms of the perturbation theory of intermolecular forces [29,30]. These arguments demonstrate that the sum of the perturbation theory corrections, which are free from BSSE, amounts to the CP-corrected interaction energy [7-161. Unfortunately, these arguments remain largely unnoticed. A breakthrough in this impasse is welcome and we believe that the recent series of papers from Davidson’s group [31-331 offers a convenient start-
141
M. Gutowski et al. /Chemical Physics Letters 241 (1995) 140-145
ing point to revisit the issue. The recent contribution by Davidson and Chakravorty (D&Ch) [33] is of particular relevance as the authors discussed the efficacy of the counterpoise method. For this purpose, they studied the HF and H,O dimers at the SCF and MP2 levels of theory. In their work, the dimer basis set JZ!L%’of the 6-31G * * (HF) and 6-31G * (H,O) quality was augmented with an orthogonal complement E’. For convenience, it is assumed that the resulting &LZ?Z??basis set is complete. D&Ch invoked some non-additive terms of the monomer and dimer type to interpret the disrepancy between the CP-corrected interaction energy obtained in the tiL&’ basis set and the complete basis set (CBS) interaction energy [33]. The presence of the non-additive terms, especially of the monomer type, might mislead readers that something is fundamentally wrong with CP. Moreover, the authors did not offer any practical guide as to how to minimize the discrepancy between the CBS and CP-corrected interaction energies. In this Comment we discuss the nature of the CP-corrected interaction energy within the framework proposed by D & Ch. We demonstrate that their approach leads, after minor modifications, to essentially the same conclusions as those reached in earlier studies, in which the supermolecular interaction energy was analyzed in terms of the perturbation theory of intermolecular forces [7-141. It should be mentioned that D&Ch also addressed other important issues which will not be discussed in this Comment. In particular, we note their conclusion that the occupied orbitals of the partner should be included in the monomer energy calculation, a point strongly advocated by us for a long time [7-91.
In the D& Ch analysis, the CBS interaction ergy between A and B, denoted -D,
-D=E(AB(dLZ?~) -E(Al&‘L&Z’) -E(B\
JZ’L%?‘),
was split into the following -D
+ NAC,
+ NAC,
First, the nominal
We will follow the notation from Ref. [33] where E(X 1$V) stands for the energy of the system X calculated in basis set 9. The analysis presented below applies to any size-consistent method such as SCF, Moller-Plesset perturbation theory, coupled cluster theory etc.
terms [33]:
.
+ NAC,,
(counterpoise
(2)
uncorrected)
energy
-Q,,, -D,,,
=E(ABl&~)
-E(AId)
-E(BIL%‘), (3)
has to be corrected by the counterpoise and CPC,, CPC,
= +(A(
ML&‘) +E(AI
terms CPC,
JZ’)
(4)
(and similarly for CPC,), to deliver the counterpoise corrected interaction energy - Dcpc,
-D cpc= -On,,+ CPC, + CPC a.
(5)
Obtaining the CBS interaction energy -D requires further inclusion of some NAC,, NAC,, and NAC,, corrections. The monomer non-additive correction NAC,, NAC,
= -E(A +E(AI
1_w’c%W’)+ E(A 1~~‘58’) ti’Z’> -E(AI
.@‘)
(6)
(and similarly for NAC,), describes the non-additive effect resulting from extension of the monomercentered basis set ti by the orthogonal complement 55’ and the partner basis set L@. The correction NAC,, has the form =E(ABI~&?%)-E(AB(ti.%J) -E(A).&Z)
Discrepancy between the counterpoise-corrected and complete basis set interaction energies
(1)
= -On,,,, + CPC, + CPC,
NAC,,
2.
en-
+E(Aj
-E(B~L%‘%Y) +E(B(s)
&‘) (7)
and was dubbed the dimer non-additive correction [33]. We should point out that the expression for NAC, does not fit the definition of non-additivity applied for the monomer NAC, and NAC, terms. The decomposition of -D given by Eq. (2) resulted from a specific sequence in which +@, L&‘,and E’ were combined to form LZ?L&“Z?? [33]. First, monomer basis sets & and L@ were supplemented
hf. Gutowski et al. / Chemical Physics Letters 241 (1995) 140-145
142
with the orthogonal complement ‘&’(&-t&V’, L8’ +L%‘%). In the next step, &L%? and L&Z’ were supplemented with z&’and &‘, respectively, to form the complete basis set &&5”Z. The partitioning given by Eqs. (2)--(7) shows that the counterpoise-corrected interaction energy - DCpc would be equal to -D if all NAC terms vanished or cancelled each other. D & Ch demonstrated, however, that separate NAC terms are significant for the 631G* * and 6-31G * basis sets and that a partial cancellation of the dimer and monomer NAC terms occurs at the MP2 level only. What is the meaning of the NAC terms present in Eq. (2)? Do their nonzero values disqualify the counterpoise procedure? An alternative splitting of -D will be presented below to provide a physically straightforward interpretation of the sum of non-additive terms. Let us change the order in which the complete basis set &L%%? is approached from a monomer perspective. For monomer A, for instance, we first supplement & with .%’ to allow for CPC,. In the second step, we supplement &L%’with the orthogonal complement $? which allows us to reproduce the dimer basis set incompleteness (DBSI) terms. We define the DBSI term for a quantity E(X) (which may be a monomer or dimer energy as well as a component of the interaction energy) as DBSI( E(X))
= E(X 1&L&Z’) - E(X 1ML%‘). (8)
Then, the CBS energy of monomer
E(A ) s’B!F)
A is expressed as
= E(A 1s’) - CPC,
+ DBSI( E(A))
(9)
and similarly for E(B I ~.GZW). For the dimer, CBS energy may be written as E(AB 1d.B?)
the
energy,
defined by Eq.
+ DBSI( -D),
(11)
where the term denoted DBSI( - D) has this form: DBSI( -D)
which is close to
= E(AEl 1MB’) + DBSI( E(AE3)).
Finally, the CBS interaction (l), takes the form = -Dcgc
3. Guidelines to obtain -D,,,
the complete basis set result (10)
-D
Eq. (11) offers an alternative to Eq. (2). The sum of the non-additive NAC,, NAC,, and NAC,, terms simplifies to the component of the interaction energy, DBSI(-D). Due to the reduction of the monomer E(X 1~) and E(X I%5?) terms (X= A, B) present in Eqs. (6) and (71, DBSI( -D) can no longer be interpreted as non-additive with respect to different basis set extensions. Eq. (11) follows the decomposition of the complete basis set .@‘L%Z’into ~Y.58’and the orthogonal complement %Yand offers a natural separation of the and CBS interaction energy -D into -Dcpc DBSI( - 0). - DCPC is the interaction energy which can be reproduced within the finite dimer basis set _&L%‘. Since the monomer energy artifacts were eradicated by the CPC corrections, -DC,, is a meaningful interaction energy for a finite d&5’ basis set. However, it is not equal to the CBS energy -D due to the missing DBSI( -D) part. DBSI( -D) is the part of the interaction energy which cannot be reproduced within the finite dimer basis &LlY, as it requires the orthogonal complement 5?. The nonzero values of DBSI( -D) are often used as an argument against the counterpoise method [19-281. The counterpoise method, however, cannot make a finite basis set any larger (not to mention complete!) and thus is not intended to compensate for this kind of deficiency. As concluded elsewhere interaction energy is as close to [6-141, the -DC,, the CBS limit as the finite dimer basis set allows it to be.
= DBSI( E(AB)) - DBSI( E(B))
- DBSI( E(A))
.
(12)
Eqs. (11) and (12) indicate that -DC,, would be equal to -D if the monomer and dimer DBSI terms vanished or cancelled each other. The first possibility is impractical as it would require the d.58’ basis set to be complete. The second possibility is much more attractive. In fact, this is the way the supermolecular approach works in the case of both weak intermolecular interactions and strong chemical bonds. The monomer and dimer DBSI terms would cancel each other providing the interaction energy terms
M. Gutowski et al. /Chemical
Physics Letters 241 (1995) 140-145
were already saturated within the .@A%’basis set. Then, inclusion of the orthogonal complement %? affects monomer energies only; the DBSI( -D) term becomes zero (see Eq. (12)), and -D = -DC,,. This ideal is hard to achieve in practice, but recent studies indicate that the goal is feasible [15]. A remarkable agreement with experimental data was achieved in recent ab initio studies in which the counterpoise method was employed [4-6,15,34,35]. The theoretical results were proved to be saturated with respect to further extensions of one-electron basis sets and the level of electron correlation treatment. A common feature of these studies is that much effort has been spent to design basis sets which accurately reproduce dominant components of the interaction energy and hence minimize IDBSI(-D) ( rather than the monomer energies. Interaction energies thus obtained are quite accurate even though the monomer DBSI terms may remain large. It should be mentioned that the basis sets optimized for atomic energies (or energies of atomic anions) were also used in a number of recent studies [21-231. By their design, these basis sets are suitable to make CPCs and the monomer DBSI terms relatively small. Unfortunately, their small values do not guarantee that the interaction energy is accurately reproduced within ti.9. In fact, the quality of separate interaction energy components has never been tested for these basis sets. This seems to be an urgent
Table 1 The HF dimer interaction energies CD,,,,,, Dc,,, and Dl, counterpoise corrections (CPC) and the dimer basis set incompleteness terms (DBSI) for the basis sets and geometry from Ref. [33]. FC stands for the ‘frozen core’ results. All entries in kcal/mol SCF - D,,, CPC, CPC, - DCPC DBSI(E@B)l - DBSI(E(A)) - DBSI( E(B)) DBSI( - D) -D
AMP2(FC)
AMP2
- 6.04
- 0.94
- 1.02
1.30 0.13
0.77 0.13
0.82 0.16
- 4.61
- 0.04
- 0.04
- 72.14 35.88 37.04 0.78
- 124.55 61.91 62.45 - 0.20
- 153.47 76.35 76.90 - 0.22
- 3.83
- 0.24
- 0.26
143
Table 2 The Ha0 dimer interaction energies CD,,,,,, D,,c, and D), counterpoise corrections (CPC) and the dimer basis set incompleteness terms (DBSI) for the basis sets and geometry from Ref. [33]. FC stands for the ‘frozen core’ results. All entries in kcal/mol SCF - D,,, CPC, CPC, - DCPC DBSI(E(AB)l - DBSI(E(Al) - DBSI( E(B)) DBSI( - D) -D
-5.65 0.23 0.67
A MP2(FCl -1.37 0.16 0.75
-4.74
-0.45
- 69.33 35.45 35.01 1.13
- 111.88 56.00 55.41 - 0.47
-3.61
-0.92
Ak4P2 - 1.45 0.20 0.79 - 0.46 - 163.77 81.93 81.34 - 0.50 - 0.96
task since the convergence of the - Dcpc interaction energies was reported to be far from smooth 121-231.
4. Numerical
results
Numerical aspects of the splitting of -D, given by Eqs. (ll)-(121, are presented in Tables 1 and 2 for the HF and H,O dimers, respectively. Geometries, basis sets and other details are d&cussed in Ref. [33]. The counterpoise corrections CPC, and CPC, are significant at both the SCF and MP2’ level. The CP-corrected interaction energy, - DcRc, is the meaningful interaction energy for the LX?& basis set and the theoretical model used. The separate monomer and dimer DBSI terms are huge, some two orders of magnitude larger than the CPC terms. These numbers reflect the fact that the 6-31G * * type basis sets leave a lot of room for the improvement of monomer energies. The huge DBSI terms cancel each other to a large extent but the residue DBSI ( -D) is still significant. This is expected, since the 6-31G * * basis set is not suitable to deliver accurate interaction energies [2,12]. It is worth noting that for both dimers the DBSI(-D) terms are positive at the SCF level and negative at the MP2 level, see Tabled 1 and 2, showing that even the sign of the discmpancy between -Dcpc and the CBS result cannot1 be a priori
M. Gutowski et al. /Chemical
144
Table 3 Perturbation components (for definitions see Ref. [15]) of the interaction energy in the HF dimer. All entries in kcal/mol
SCF decomposition (10) %s %xh
*& MP2 decomposition (20)
?? ;$ Edis
A&h
6-31G * ’
WTdf(b-ext)
DBSJ
-7.51 4.46 - 1.60
- 6.22 4.13 - 1.85
1.29 - 0.33 - 0.25
- 0.91 -0.19 0.99
- 1.55 0.27 0.90
-0.64 0.46 - 0.09
taken for granted. Such a behavior, however, can easily be rationalized using arguments based on the perturbation theory of intermolecular forces. Let us briefly discuss the HF dimer, for which the basis set dependence of the interaction energy components was recently analyzed in Refs. [12,15] and the results are presented in Table 3. For convenience, the results obtained with the WTdRb-ext) basis set from Ref. [15] are considered as the CBS limits. Due to the differences in dimer geometries and basis sets employed in Refs. [15,33], the perturbation interpretation of the DBSI(-D) terms from Table 1 is semiquantitative. The SCF/6-31G * * counterpoise-corrected interaction energy is too attractive primarily due to overestimation of the first-order electrostatic interaction (lo), by 1.29 kcal/mol, see Table 3. Two es energy, ?? other components of the SCF interaction energy, ,tki, and AEdS,r, are also affected by the dimer basis set incompleteness, and the corresponding DBSI terms are - 0.33 and - 0.25 kcal/mol, respectively. The sum of the above DBSI terms of 0.71 kcal/mol reproduces well the SCF DBSI(-D) term of 0.78 kcal/mol reported in Table 1. The MP2/6-31G * * counterpoise-corrected interaction energy is too repulsive primarily due to underestimation of the second-order dispersion energy, , by 0.64 kcal/mol. Another component of the MP2 interaction energy which is inaccurate in the 6-31G * * basis set is the second-order intramonomer correlation correction to the electrostatic term, earn). It differs from the WTdf(b-ext) result by 0.46 kcal/mol. The sum of the DBSI terms of -0.27 kcal/mol reproduces satisfactorily the MP2
??
??
Physics Letters 241 (1995) 140-145
DBSI( -D) term of - 0.22 kcal/mol, reported in Table 1. To conclude, the NAC terms (Eq. (2)) are shown to sum up to DBSI(-D), a part of the interaction energy. The perturbation theory results demonstrate that the DBSI(-D) term can be made to vanish by selecting the d.58’ basis set so as to saturate the leading components of the interaction energy.
5. Conclusions The current analysis of the counterpoise issue within the framework proposed by Davidson and Chakravorty [33] and the previous discussions based on the perturbation theory of intermolecular forces [7-141 lead to essentially the same conclusion that the counterpoise-corrected interaction energy is the meaningful interaction energy for a given finite basis set. It differs from the complete basis set result by the dimer basis set incompleteness term, DBSI( -D), defined by Eq. (12). Our analysis demonstrates that DBSI( -0) should be interpreted as a genuine component of the interaction energy. It may be reduced to a small magnitude if the dimer basis set accurately reproduces the interaction energy terms. Conversely, the large values of DBSI( -D) are caused by a poor choice of the dimer basis set and not by a failure of the counterpoise procedure.
Acknowledgement This work was supported by the National Science Foundation (Grant No. CHE-9215082) and by the Polish Committee for Scientific Research KBN (Grant No. 2 0556 91 01).
References [l] SF. Boys and F. Bernardi, Mol. Phys. 19 (1970) 553. [2] J.H. van Lenthe, J.G.C.M. van Duijneveldt-van de Rijdt and F.B. van Duijneveldt, Adv. Chem. Phys. 69 (1987) 521. [3] G. Chalasihski and M. Gutowski, Chem. Rev. 88 (1988) 943. [4] R.J. Vos, J.H. van Lenthe and F.B. van Duijneveldt, J. Chem. Phys. 93 (1990) 643. [5] B. Liu and A.D. McLean, J. Chem. Phys. 91 (1989) 2348.
M. Gutowski et al. /Chemical [6] J.G.C.M. van Duijneveldt-van de Rijdt and F.B. van Duijneveldt, J. Chem. Phys. 97 (1992) 5019. [7] M. Gutowski, F.B. van Duijneveldt, G. Chalasitiski and L. Piela, Mol. Phys. 61 (1987) 233. [8] M. Gutowski, J.H. van Lenthe, J. Verbeek, F.B. van Duijneveldt and G. Chalasibski, Chem. Phys. Letters 124 (1986) 370. ]9] M.M. SzcqLniak and S. Scheiner, J. Chem. Phys. 84 (1986) 6328. [lo] M. Gutowski, J.G.C.M. van Duijneveldt-van de Rijdt, J.H. van Lenthe and F.B. van Duijneveldt, J. Chem. Phys. 98 (1993) 4728. [11] M. Gutowski and G. Chalasihski, J. Chem. Phys. 98 (19931 5.540. 1121 SM. Cybulski, G. Chalasihski and R. Moszyhski, J. Chem. Phys. 92 (1990) 4357. ]13] Q. Zhang, L. Chenyang, Y. Ma, F. Fish, M.M. Szcz&tiak and V. Buch, J. Chem. Phys. 96 (1992) 6039. [14] S.M. Cybulski and G. ChaIasihski, Chem. Phys. Letters 197 (19921.591. 1151 G. Chaiasinski and M.M. Szcz@tiak, Chem. Rev. 94 (1994) 1723. [16] F.B. van Duijneveldt, J.G.C.M. van Duijneveldt-van de Rijdt and J.H. van I-enthe, Chem. Rev. 94 (1994) 1873. 1171 A. Johansson, P. Kollman and S. Rothenberg, Theoret. Chim. Acta 29 (1973) 167. 1181 J.P. Daudey, P. Claverie and J.P. Malrieu, Intern. J. Quantum Chem. 8 (1974) 1. [19] D.W. Schwenke and D.G. Truhlar, J. Chem. Phys. 82 (1985) 2418;84(1986)4113;86 (1987) 3760.
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[20] M.J. Frisch, J.E. Del Bene, J.S. Binkley and H.F. Schaefer III, J. Chem. Phys. 84 (1986) 2279. [21] D. Feller, J. Chem. Phys. 96 (1992) 6104. [22] J.E. Del Bene, Intern. J. Quantum Chem. Symp. 26 (19921 527. [23] D. Feller, E.D. Glendening, R.A. Kendall and K.A. Peterson, J. Chem. Phys. 100 (1994) 4981. [24] K.A. Franken and C.E. Dykstra, J. Chem. Phys. 100 (1994) 2865. [25] F. Tao and Y. Pan, J. Phys. Chem. 95 (1991) 3582, 9811; 96 (1991) 5815. [26] J. Yang and N.R. Kestner, J. Phys. Chem. 95 (1991) 9214, 9221. [27] S. Saebo, W. Tong and P. Pulay, J. Chem. PHys. 98 (1993) 2170. [28] D.B. Cook, J.A. Sordo and T.L. Sordo, Intern. J. Quantum Chem. 48 (1993) 375. [29] B. Jeziorski and W. Kolos, in: Molecular interactions, Vol. 3, eds. H. Ratajczak and W.J. Orville-Thomas (Wiley, New York, 1982) p. 1. [30] B. Jeziorski, R. Moszyhski and K. Szalewicz, Chem. Rev. 94 (1994) 1887. [31] S.C. Racine and E.R. Davidson, J. Phys. Chem. 97 (1993) 6367. [32] S.J. Chakravorty and E.R. Davidson, J. Phys. Chem. 97 (19931 6373. [33] E.R. Davidson and S.J. Chakravorty, Chem. Phys. Letters 217 (1994) 48. 1341 D.E. Woon, J. Chem. Phys. 100 (1994) 2838. [35] F. Tao and W. Klemperer, J. Chem. Phys. lOi (1994) 1129.
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 241 (1995) 140-145
Comment
Comment on “A possible definition of basis set superposition error’ ’ Maciej Gutowski a1*, Matgorzata M. Szcz@niak b, Grzegorz Chalasiiiski
’
a Department of Chemistry, Unioersity of Gdarisk, 80-952 Gdarisk, Poland h Department of Chemistry, Oakland University, Rochester, MI 48309, USA ’ Department of Chemistry, Uniuersity of Warsaw, 02-093 Warsaw, Poland Received
15 September
1994; in final form 15 May 1995
Abstract In their recent Letter Davidson and Chakravorty proposed a framework for analysis of basis set effects which trouble calculations of potential energy surfaces for weakly interacting species. Their analysis suggests that the complete basis set interaction energy differs by some monomer- and dimer-type non-additive corrections from the conventional counterpoised interaction energy obtained in a finite basis set. In this Comment we point out that the dimer-type correction does not fit the definition of non-additivity applied for the monomer terms and that the sum of the non-additive corrections reduces to a component of the interaction energy. This component disappears if the finite basis set is flexible enough to describe all important terms of the interaction energy. Thus the discrepancy between the counterpoise-corrected and the complete basis set interaction energy is exclusively determined by the suitability of the basis set used to reproduce the interaction energy terms.
1. End of the deadlock? Recent studies on van der Waals interactions indicate that the discussion on the basis set superposition error (BSSE) has reached a deadlock. On the one hand, a few research groups insist that the Boys and Bernardi function counterpoise procedure (CP) [l] is the correct approach to eliminate the lowering of monomer energies which takes place in the dimer calculation [2-161, but other researchers remain sceptical. First, there is a lingering doubt whether occupied orbitals of monomer A should be available
??
Corresponding
author.
0009-2614/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0009-2614(95)00611-7
for monomer B in the course of counterpoise calculations [17,18]. Second, other studies emphasize that the CP-corrected interaction energies differ from the expected full-basis set results [19-281. Some of the most persuasive arguments in favor of the CP correction come from the analysis of the CP-corrected interaction energies in terms of the perturbation theory of intermolecular forces [29,30]. These arguments demonstrate that the sum of the perturbation theory corrections, which are free from BSSE, amounts to the CP-corrected interaction energy [7-161. Unfortunately, these arguments remain largely unnoticed. A breakthrough in this impasse is welcome and we believe that the recent series of papers from Davidson’s group [31-331 offers a convenient start-
141
M. Gutowski et al. /Chemical Physics Letters 241 (1995) 140-145
ing point to revisit the issue. The recent contribution by Davidson and Chakravorty (D&Ch) [33] is of particular relevance as the authors discussed the efficacy of the counterpoise method. For this purpose, they studied the HF and H,O dimers at the SCF and MP2 levels of theory. In their work, the dimer basis set JZ!L%’of the 6-31G * * (HF) and 6-31G * (H,O) quality was augmented with an orthogonal complement E’. For convenience, it is assumed that the resulting &LZ?Z??basis set is complete. D&Ch invoked some non-additive terms of the monomer and dimer type to interpret the disrepancy between the CP-corrected interaction energy obtained in the tiL&’ basis set and the complete basis set (CBS) interaction energy [33]. The presence of the non-additive terms, especially of the monomer type, might mislead readers that something is fundamentally wrong with CP. Moreover, the authors did not offer any practical guide as to how to minimize the discrepancy between the CBS and CP-corrected interaction energies. In this Comment we discuss the nature of the CP-corrected interaction energy within the framework proposed by D & Ch. We demonstrate that their approach leads, after minor modifications, to essentially the same conclusions as those reached in earlier studies, in which the supermolecular interaction energy was analyzed in terms of the perturbation theory of intermolecular forces [7-141. It should be mentioned that D&Ch also addressed other important issues which will not be discussed in this Comment. In particular, we note their conclusion that the occupied orbitals of the partner should be included in the monomer energy calculation, a point strongly advocated by us for a long time [7-91.
In the D& Ch analysis, the CBS interaction ergy between A and B, denoted -D,
-D=E(AB(dLZ?~) -E(Al&‘L&Z’) -E(B\
JZ’L%?‘),
was split into the following -D
+ NAC,
+ NAC,
First, the nominal
We will follow the notation from Ref. [33] where E(X 1$V) stands for the energy of the system X calculated in basis set 9. The analysis presented below applies to any size-consistent method such as SCF, Moller-Plesset perturbation theory, coupled cluster theory etc.
terms [33]:
.
+ NAC,,
(counterpoise
(2)
uncorrected)
energy
-Q,,, -D,,,
=E(ABl&~)
-E(AId)
-E(BIL%‘), (3)
has to be corrected by the counterpoise and CPC,, CPC,
= +(A(
ML&‘) +E(AI
terms CPC,
JZ’)
(4)
(and similarly for CPC,), to deliver the counterpoise corrected interaction energy - Dcpc,
-D cpc= -On,,+ CPC, + CPC a.
(5)
Obtaining the CBS interaction energy -D requires further inclusion of some NAC,, NAC,, and NAC,, corrections. The monomer non-additive correction NAC,, NAC,
= -E(A +E(AI
1_w’c%W’)+ E(A 1~~‘58’) ti’Z’> -E(AI
.@‘)
(6)
(and similarly for NAC,), describes the non-additive effect resulting from extension of the monomercentered basis set ti by the orthogonal complement 55’ and the partner basis set L@. The correction NAC,, has the form =E(ABI~&?%)-E(AB(ti.%J) -E(A).&Z)
Discrepancy between the counterpoise-corrected and complete basis set interaction energies
(1)
= -On,,,, + CPC, + CPC,
NAC,,
2.
en-
+E(Aj
-E(B~L%‘%Y) +E(B(s)
&‘) (7)
and was dubbed the dimer non-additive correction [33]. We should point out that the expression for NAC, does not fit the definition of non-additivity applied for the monomer NAC, and NAC, terms. The decomposition of -D given by Eq. (2) resulted from a specific sequence in which +@, L&‘,and E’ were combined to form LZ?L&“Z?? [33]. First, monomer basis sets & and L@ were supplemented
hf. Gutowski et al. / Chemical Physics Letters 241 (1995) 140-145
142
with the orthogonal complement ‘&’(&-t&V’, L8’ +L%‘%). In the next step, &L%? and L&Z’ were supplemented with z&’and &‘, respectively, to form the complete basis set &&5”Z. The partitioning given by Eqs. (2)--(7) shows that the counterpoise-corrected interaction energy - DCpc would be equal to -D if all NAC terms vanished or cancelled each other. D & Ch demonstrated, however, that separate NAC terms are significant for the 631G* * and 6-31G * basis sets and that a partial cancellation of the dimer and monomer NAC terms occurs at the MP2 level only. What is the meaning of the NAC terms present in Eq. (2)? Do their nonzero values disqualify the counterpoise procedure? An alternative splitting of -D will be presented below to provide a physically straightforward interpretation of the sum of non-additive terms. Let us change the order in which the complete basis set &L%%? is approached from a monomer perspective. For monomer A, for instance, we first supplement & with .%’ to allow for CPC,. In the second step, we supplement &L%’with the orthogonal complement $? which allows us to reproduce the dimer basis set incompleteness (DBSI) terms. We define the DBSI term for a quantity E(X) (which may be a monomer or dimer energy as well as a component of the interaction energy) as DBSI( E(X))
= E(X 1&L&Z’) - E(X 1ML%‘). (8)
Then, the CBS energy of monomer
E(A ) s’B!F)
A is expressed as
= E(A 1s’) - CPC,
+ DBSI( E(A))
(9)
and similarly for E(B I ~.GZW). For the dimer, CBS energy may be written as E(AB 1d.B?)
the
energy,
defined by Eq.
+ DBSI( -D),
(11)
where the term denoted DBSI( - D) has this form: DBSI( -D)
which is close to
= E(AEl 1MB’) + DBSI( E(AE3)).
Finally, the CBS interaction (l), takes the form = -Dcgc
3. Guidelines to obtain -D,,,
the complete basis set result (10)
-D
Eq. (11) offers an alternative to Eq. (2). The sum of the non-additive NAC,, NAC,, and NAC,, terms simplifies to the component of the interaction energy, DBSI(-D). Due to the reduction of the monomer E(X 1~) and E(X I%5?) terms (X= A, B) present in Eqs. (6) and (71, DBSI( -D) can no longer be interpreted as non-additive with respect to different basis set extensions. Eq. (11) follows the decomposition of the complete basis set .@‘L%Z’into ~Y.58’and the orthogonal complement %Yand offers a natural separation of the and CBS interaction energy -D into -Dcpc DBSI( - 0). - DCPC is the interaction energy which can be reproduced within the finite dimer basis set _&L%‘. Since the monomer energy artifacts were eradicated by the CPC corrections, -DC,, is a meaningful interaction energy for a finite d&5’ basis set. However, it is not equal to the CBS energy -D due to the missing DBSI( -D) part. DBSI( -D) is the part of the interaction energy which cannot be reproduced within the finite dimer basis &LlY, as it requires the orthogonal complement 5?. The nonzero values of DBSI( -D) are often used as an argument against the counterpoise method [19-281. The counterpoise method, however, cannot make a finite basis set any larger (not to mention complete!) and thus is not intended to compensate for this kind of deficiency. As concluded elsewhere interaction energy is as close to [6-141, the -DC,, the CBS limit as the finite dimer basis set allows it to be.
= DBSI( E(AB)) - DBSI( E(B))
- DBSI( E(A))
.
(12)
Eqs. (11) and (12) indicate that -DC,, would be equal to -D if the monomer and dimer DBSI terms vanished or cancelled each other. The first possibility is impractical as it would require the d.58’ basis set to be complete. The second possibility is much more attractive. In fact, this is the way the supermolecular approach works in the case of both weak intermolecular interactions and strong chemical bonds. The monomer and dimer DBSI terms would cancel each other providing the interaction energy terms
M. Gutowski et al. /Chemical
Physics Letters 241 (1995) 140-145
were already saturated within the .@A%’basis set. Then, inclusion of the orthogonal complement %? affects monomer energies only; the DBSI( -D) term becomes zero (see Eq. (12)), and -D = -DC,,. This ideal is hard to achieve in practice, but recent studies indicate that the goal is feasible [15]. A remarkable agreement with experimental data was achieved in recent ab initio studies in which the counterpoise method was employed [4-6,15,34,35]. The theoretical results were proved to be saturated with respect to further extensions of one-electron basis sets and the level of electron correlation treatment. A common feature of these studies is that much effort has been spent to design basis sets which accurately reproduce dominant components of the interaction energy and hence minimize IDBSI(-D) ( rather than the monomer energies. Interaction energies thus obtained are quite accurate even though the monomer DBSI terms may remain large. It should be mentioned that the basis sets optimized for atomic energies (or energies of atomic anions) were also used in a number of recent studies [21-231. By their design, these basis sets are suitable to make CPCs and the monomer DBSI terms relatively small. Unfortunately, their small values do not guarantee that the interaction energy is accurately reproduced within ti.9. In fact, the quality of separate interaction energy components has never been tested for these basis sets. This seems to be an urgent
Table 1 The HF dimer interaction energies CD,,,,,, Dc,,, and Dl, counterpoise corrections (CPC) and the dimer basis set incompleteness terms (DBSI) for the basis sets and geometry from Ref. [33]. FC stands for the ‘frozen core’ results. All entries in kcal/mol SCF - D,,, CPC, CPC, - DCPC DBSI(E@B)l - DBSI(E(A)) - DBSI( E(B)) DBSI( - D) -D
AMP2(FC)
AMP2
- 6.04
- 0.94
- 1.02
1.30 0.13
0.77 0.13
0.82 0.16
- 4.61
- 0.04
- 0.04
- 72.14 35.88 37.04 0.78
- 124.55 61.91 62.45 - 0.20
- 153.47 76.35 76.90 - 0.22
- 3.83
- 0.24
- 0.26
143
Table 2 The Ha0 dimer interaction energies CD,,,,,, D,,c, and D), counterpoise corrections (CPC) and the dimer basis set incompleteness terms (DBSI) for the basis sets and geometry from Ref. [33]. FC stands for the ‘frozen core’ results. All entries in kcal/mol SCF - D,,, CPC, CPC, - DCPC DBSI(E(AB)l - DBSI(E(Al) - DBSI( E(B)) DBSI( - D) -D
-5.65 0.23 0.67
A MP2(FCl -1.37 0.16 0.75
-4.74
-0.45
- 69.33 35.45 35.01 1.13
- 111.88 56.00 55.41 - 0.47
-3.61
-0.92
Ak4P2 - 1.45 0.20 0.79 - 0.46 - 163.77 81.93 81.34 - 0.50 - 0.96
task since the convergence of the - Dcpc interaction energies was reported to be far from smooth 121-231.
4. Numerical
results
Numerical aspects of the splitting of -D, given by Eqs. (ll)-(121, are presented in Tables 1 and 2 for the HF and H,O dimers, respectively. Geometries, basis sets and other details are d&cussed in Ref. [33]. The counterpoise corrections CPC, and CPC, are significant at both the SCF and MP2’ level. The CP-corrected interaction energy, - DcRc, is the meaningful interaction energy for the LX?& basis set and the theoretical model used. The separate monomer and dimer DBSI terms are huge, some two orders of magnitude larger than the CPC terms. These numbers reflect the fact that the 6-31G * * type basis sets leave a lot of room for the improvement of monomer energies. The huge DBSI terms cancel each other to a large extent but the residue DBSI ( -D) is still significant. This is expected, since the 6-31G * * basis set is not suitable to deliver accurate interaction energies [2,12]. It is worth noting that for both dimers the DBSI(-D) terms are positive at the SCF level and negative at the MP2 level, see Tabled 1 and 2, showing that even the sign of the discmpancy between -Dcpc and the CBS result cannot1 be a priori
M. Gutowski et al. /Chemical
144
Table 3 Perturbation components (for definitions see Ref. [15]) of the interaction energy in the HF dimer. All entries in kcal/mol
SCF decomposition (10) %s %xh
*& MP2 decomposition (20)
?? ;$ Edis
A&h
6-31G * ’
WTdf(b-ext)
DBSJ
-7.51 4.46 - 1.60
- 6.22 4.13 - 1.85
1.29 - 0.33 - 0.25
- 0.91 -0.19 0.99
- 1.55 0.27 0.90
-0.64 0.46 - 0.09
taken for granted. Such a behavior, however, can easily be rationalized using arguments based on the perturbation theory of intermolecular forces. Let us briefly discuss the HF dimer, for which the basis set dependence of the interaction energy components was recently analyzed in Refs. [12,15] and the results are presented in Table 3. For convenience, the results obtained with the WTdRb-ext) basis set from Ref. [15] are considered as the CBS limits. Due to the differences in dimer geometries and basis sets employed in Refs. [15,33], the perturbation interpretation of the DBSI(-D) terms from Table 1 is semiquantitative. The SCF/6-31G * * counterpoise-corrected interaction energy is too attractive primarily due to overestimation of the first-order electrostatic interaction (lo), by 1.29 kcal/mol, see Table 3. Two es energy, ?? other components of the SCF interaction energy, ,tki, and AEdS,r, are also affected by the dimer basis set incompleteness, and the corresponding DBSI terms are - 0.33 and - 0.25 kcal/mol, respectively. The sum of the above DBSI terms of 0.71 kcal/mol reproduces well the SCF DBSI(-D) term of 0.78 kcal/mol reported in Table 1. The MP2/6-31G * * counterpoise-corrected interaction energy is too repulsive primarily due to underestimation of the second-order dispersion energy, , by 0.64 kcal/mol. Another component of the MP2 interaction energy which is inaccurate in the 6-31G * * basis set is the second-order intramonomer correlation correction to the electrostatic term, earn). It differs from the WTdf(b-ext) result by 0.46 kcal/mol. The sum of the DBSI terms of -0.27 kcal/mol reproduces satisfactorily the MP2
??
??
Physics Letters 241 (1995) 140-145
DBSI( -D) term of - 0.22 kcal/mol, reported in Table 1. To conclude, the NAC terms (Eq. (2)) are shown to sum up to DBSI(-D), a part of the interaction energy. The perturbation theory results demonstrate that the DBSI(-D) term can be made to vanish by selecting the d.58’ basis set so as to saturate the leading components of the interaction energy.
5. Conclusions The current analysis of the counterpoise issue within the framework proposed by Davidson and Chakravorty [33] and the previous discussions based on the perturbation theory of intermolecular forces [7-141 lead to essentially the same conclusion that the counterpoise-corrected interaction energy is the meaningful interaction energy for a given finite basis set. It differs from the complete basis set result by the dimer basis set incompleteness term, DBSI( -D), defined by Eq. (12). Our analysis demonstrates that DBSI( -0) should be interpreted as a genuine component of the interaction energy. It may be reduced to a small magnitude if the dimer basis set accurately reproduces the interaction energy terms. Conversely, the large values of DBSI( -D) are caused by a poor choice of the dimer basis set and not by a failure of the counterpoise procedure.
Acknowledgement This work was supported by the National Science Foundation (Grant No. CHE-9215082) and by the Polish Committee for Scientific Research KBN (Grant No. 2 0556 91 01).
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