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Reply to comment on “A possible definition of basis set superposition error”
14 July 1995
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CHEMICAL PHYSICS LETTERS Chemical Physics Letters 241 (1995) 146-148
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Reply to Comment on “A possible definition of basis set superposition error’ ’ Ernest R. Davidson, Subhas J. Chakravorty Department
of Chemistry, Indiana Uniuersity, Bloomington,
IN 47405, USA
Received 30 January 1995
Abstract Contrary to the statements of Gutowski et al., the difference between the true dissociation energy and the counterpoisecorrected calculation of the dissociation energy is shown to be entirely a non-additive correction for basis set incompleteness.
Gutowski et al., in the preceding Comment [ll, declare that the NAC, given by their Eq. (7) (equivalent to Eq. (8) of the original Ref. [2]) does not obey the definition of a ‘non-additive correction’. We disagree and claim that not only does NAC, obey such a definition, but so does the quantity DBSI(-D) introduced by Gutowski et al. in their Eq. (10). Only the counterpoise, CPC, itself is of a different type and plays a unique role. Nothing in our preceding paper indicated anything was logically wrong with the CPC, although there is no reason to expect, a priori, that it will improve agreement with the exact energy when applied to conventional basis sets. To see that NAC,, and DBSI(-D) are non-additive corrections, we adopt the commonly understood definition of a non-additive correction. If a numerical quantity f is determined by two independent sets of conditions X and Y (not necessarily continuous variables), then the effect of changing X
and Y away from some reference X,, Y0 may be written as f(X, Y) = f. + A, + A, + NAC,
set of conditions
(1)
where fo =f(Xo,
Yo),
A, =f(X,
Yo) -f(X,,
Y,,),
Y) -f(X,,
Yo),
A, =f(Xo, NAC =f(X,
-f(X,
Y) -f(X,,
Y)
Yo) +f(Xo,
Yo).
(2)
Here, A, and A, give the change in f expected if these conditions acted independently and NAC gives the non-additive correction due to interaction between the conditions. Gutowski et al. recognize that NAC, given by their Eq. (6) is such a non-additive effect where X and Y refer to the basis sets B and C. They fail to recognize that Eq. (7) is also of this form where X
0009-2614/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00612-5
E.R. Dauidson, S.J. Chakrauorty/Chemical
and Y refer to the interfragment distance R and the basis set C. To see this more precisely, consider the energy, E(R (X; Y) calculated for the supermolecule A. . . B as a function of the one geometrical parameter R representing the distance between the monomers (holding all other geometrical parameters constant) with basis set X for fragment A and basis set Y for fragment B. Suppose the energy is computed by a size-consistent method so that for large R, the supermolecule energy is the sum of the fragment energies. Then E(R,)dge;
9’%~)=E,,+A,+Ac+NACAa, (3)
where E,, = E(ml &r!; 9), A,=E(R,I~;~)-E(ccl~;~), A, =E(aI
csz’ge’,;9Fe)
-E(4
NAC.4, = E( R, I s“iFe; LWe) -E(
_w’; 9), - E(ml MC,; s2?‘Ze),
R, I LV’;9’) + E(QJ( a’; .9)
(4)
and %?e is the supplemental basis set for the distance R,. At R,, E(R, I dge’,; 9@e) becomes E(AB I d&5%‘) in the previous notation [1,2]. Clearly this NAC,, satisfies the usual conceptual definition of a non-additive correction for the simultaneous variation in R and the basis set. To see that DBSI(-D) is also of this form, consider a slightly modified reference state, E( R, I @-‘geq; sa/“,9q) =E;+A”+A’,+DBSI(-D), Eb = E(ml dL&;
&&q,
aR = E( R, 1 LZ’S’~; .(s,B’) - E(mI ML&; J;s,9), a;. =E@ DBSI( -0)
dLi7ese;
~e.mze)
- E(4
2dszJe; &!+q,
= E( R, 1edL2$ge; L&GSC,) - E(ml tiqge; - E(R,
I dge;
+ E(ml dsq;
.g$q, ties’) -Qaq.
(5)
Thus DBSI( -0) also represents the non-additive correction for simultaneous variation of R and the
Physics Letters 241 (1995) 146-148
147
basis set. It differs from NAC,, only in the choice of the reference state, E,. A large term in DBSI( -D) for the MP2 energy should clearly be the extra dispersion energy recovered by completing the basis set. This is verified in Table 3 of Ref. [l], On a more substantive note, Gutowskilet al. suggest seeking basis sets for which DBSI( -D) is small so that the CPC-corrected binding energy, - dK, is an accurate estimate of D. This would provide a satisfactory result for van der Waals dimers although it might be difficult to implement for some methods of calculation. The difficulty with this suggestion is illustrated by the MP2 results in Table 3 of Ref. [l]. Unless one relies on accidental cancellation of unlike (‘*) the only way to make effects such as @ST and ?? DBSI(‘- D) small is to rn:ke the basis’ set nearly complete. Because of the form of the MP$ energy, it is unlikely that a basis set will recover a substantially larger fraction of the dispersion energy than it does of the valence shell correlation energy of each fragment. Traditionally, basis sets have been chosen to minimize E. Feller et al. and Del Bene et al. (Refs. [20-231 of Ref. [ll) seem to have found basis sets is more stable than -D,,, to for which -D,,, basis set improvement. This also, however, may rely on cancellation of unlike effects and not be a general result. A more serious unsolved problem is the application of these concepts to reactions and potential energy surfaces of traditional molecules, It is clear that BSSE must have a large effect on calculated potential energy surfaces of unimolecular rearrangements such as the Cope rearrangement or isomerization of difluoroethylene. Since a precise global definition for a CPC-corrected potential energy surface is difficult, a basis set that minimizes the error in AEcrc while leaving a potentially large counterpoise correction would not be useful in this case. Even for van der Waals molecules, AE,,, has a logical problem if a reference configuration other than infinite separation is considered. If X, denotes a set of relative coordinates of rigid monomers, then the CPC definition of the energy difference between X, and X, could be extended to read AE,, =E(X,
I S’S’,; djc%‘) - E(X, 1safe@‘;; d$‘), (6)
E.R. Davidson, S.J. Chakravor&/Chemical
148
where, for example, &gj indicates the normal basis d supplemented by the basis 225’ at the ghost position Xi. When Xj corresponds to infinite separation, this is the usual definition used for the CPC-corrected energy separation. The difficulty with this more general definition is that AE,, # AEi, - AEik. For van der Waals complexes AE,, = AE,
- AEp
(7)
Physics Letters 241 (1995) 146-148
Acknowledgement This work was supported by grant number CHE9007393 from the National Science Foundation.
References
defining (8)
instead of using Eq. (6) seems natural, but a similar approach for unimolecular rearrangements is not.
[l] M. Gutowski, M.M. SzczgSniak and G. Chalasihki, Chem. Phys. Letters 241 (199.5) 140. [2] E.R. Davidson and S.J. Chakravorty, Chem. Phys. Letters 217 (1994) 48.
ELSEVIER
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 241 (1995) 146-148
Reply
Reply to Comment on “A possible definition of basis set superposition error’ ’ Ernest R. Davidson, Subhas J. Chakravorty Department
of Chemistry, Indiana Uniuersity, Bloomington,
IN 47405, USA
Received 30 January 1995
Abstract Contrary to the statements of Gutowski et al., the difference between the true dissociation energy and the counterpoisecorrected calculation of the dissociation energy is shown to be entirely a non-additive correction for basis set incompleteness.
Gutowski et al., in the preceding Comment [ll, declare that the NAC, given by their Eq. (7) (equivalent to Eq. (8) of the original Ref. [2]) does not obey the definition of a ‘non-additive correction’. We disagree and claim that not only does NAC, obey such a definition, but so does the quantity DBSI(-D) introduced by Gutowski et al. in their Eq. (10). Only the counterpoise, CPC, itself is of a different type and plays a unique role. Nothing in our preceding paper indicated anything was logically wrong with the CPC, although there is no reason to expect, a priori, that it will improve agreement with the exact energy when applied to conventional basis sets. To see that NAC,, and DBSI(-D) are non-additive corrections, we adopt the commonly understood definition of a non-additive correction. If a numerical quantity f is determined by two independent sets of conditions X and Y (not necessarily continuous variables), then the effect of changing X
and Y away from some reference X,, Y0 may be written as f(X, Y) = f. + A, + A, + NAC,
set of conditions
(1)
where fo =f(Xo,
Yo),
A, =f(X,
Yo) -f(X,,
Y,,),
Y) -f(X,,
Yo),
A, =f(Xo, NAC =f(X,
-f(X,
Y) -f(X,,
Y)
Yo) +f(Xo,
Yo).
(2)
Here, A, and A, give the change in f expected if these conditions acted independently and NAC gives the non-additive correction due to interaction between the conditions. Gutowski et al. recognize that NAC, given by their Eq. (6) is such a non-additive effect where X and Y refer to the basis sets B and C. They fail to recognize that Eq. (7) is also of this form where X
0009-2614/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00612-5
E.R. Dauidson, S.J. Chakrauorty/Chemical
and Y refer to the interfragment distance R and the basis set C. To see this more precisely, consider the energy, E(R (X; Y) calculated for the supermolecule A. . . B as a function of the one geometrical parameter R representing the distance between the monomers (holding all other geometrical parameters constant) with basis set X for fragment A and basis set Y for fragment B. Suppose the energy is computed by a size-consistent method so that for large R, the supermolecule energy is the sum of the fragment energies. Then E(R,)dge;
9’%~)=E,,+A,+Ac+NACAa, (3)
where E,, = E(ml &r!; 9), A,=E(R,I~;~)-E(ccl~;~), A, =E(aI
csz’ge’,;9Fe)
-E(4
NAC.4, = E( R, I s“iFe; LWe) -E(
_w’; 9), - E(ml MC,; s2?‘Ze),
R, I LV’;9’) + E(QJ( a’; .9)
(4)
and %?e is the supplemental basis set for the distance R,. At R,, E(R, I dge’,; 9@e) becomes E(AB I d&5%‘) in the previous notation [1,2]. Clearly this NAC,, satisfies the usual conceptual definition of a non-additive correction for the simultaneous variation in R and the basis set. To see that DBSI(-D) is also of this form, consider a slightly modified reference state, E( R, I @-‘geq; sa/“,9q) =E;+A”+A’,+DBSI(-D), Eb = E(ml dL&;
&&q,
aR = E( R, 1 LZ’S’~; .(s,B’) - E(mI ML&; J;s,9), a;. =E@ DBSI( -0)
dLi7ese;
~e.mze)
- E(4
2dszJe; &!+q,
= E( R, 1edL2$ge; L&GSC,) - E(ml tiqge; - E(R,
I dge;
+ E(ml dsq;
.g$q, ties’) -Qaq.
(5)
Thus DBSI( -0) also represents the non-additive correction for simultaneous variation of R and the
Physics Letters 241 (1995) 146-148
147
basis set. It differs from NAC,, only in the choice of the reference state, E,. A large term in DBSI( -D) for the MP2 energy should clearly be the extra dispersion energy recovered by completing the basis set. This is verified in Table 3 of Ref. [l], On a more substantive note, Gutowskilet al. suggest seeking basis sets for which DBSI( -D) is small so that the CPC-corrected binding energy, - dK, is an accurate estimate of D. This would provide a satisfactory result for van der Waals dimers although it might be difficult to implement for some methods of calculation. The difficulty with this suggestion is illustrated by the MP2 results in Table 3 of Ref. [l]. Unless one relies on accidental cancellation of unlike (‘*) the only way to make effects such as @ST and ?? DBSI(‘- D) small is to rn:ke the basis’ set nearly complete. Because of the form of the MP$ energy, it is unlikely that a basis set will recover a substantially larger fraction of the dispersion energy than it does of the valence shell correlation energy of each fragment. Traditionally, basis sets have been chosen to minimize E. Feller et al. and Del Bene et al. (Refs. [20-231 of Ref. [ll) seem to have found basis sets is more stable than -D,,, to for which -D,,, basis set improvement. This also, however, may rely on cancellation of unlike effects and not be a general result. A more serious unsolved problem is the application of these concepts to reactions and potential energy surfaces of traditional molecules, It is clear that BSSE must have a large effect on calculated potential energy surfaces of unimolecular rearrangements such as the Cope rearrangement or isomerization of difluoroethylene. Since a precise global definition for a CPC-corrected potential energy surface is difficult, a basis set that minimizes the error in AEcrc while leaving a potentially large counterpoise correction would not be useful in this case. Even for van der Waals molecules, AE,,, has a logical problem if a reference configuration other than infinite separation is considered. If X, denotes a set of relative coordinates of rigid monomers, then the CPC definition of the energy difference between X, and X, could be extended to read AE,, =E(X,
I S’S’,; djc%‘) - E(X, 1safe@‘;; d$‘), (6)
E.R. Davidson, S.J. Chakravor&/Chemical
148
where, for example, &gj indicates the normal basis d supplemented by the basis 225’ at the ghost position Xi. When Xj corresponds to infinite separation, this is the usual definition used for the CPC-corrected energy separation. The difficulty with this more general definition is that AE,, # AEi, - AEik. For van der Waals complexes AE,, = AE,
- AEp
(7)
Physics Letters 241 (1995) 146-148
Acknowledgement This work was supported by grant number CHE9007393 from the National Science Foundation.
References
defining (8)
instead of using Eq. (6) seems natural, but a similar approach for unimolecular rearrangements is not.
[l] M. Gutowski, M.M. SzczgSniak and G. Chalasihki, Chem. Phys. Letters 241 (199.5) 140. [2] E.R. Davidson and S.J. Chakravorty, Chem. Phys. Letters 217 (1994) 48.