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The magnitude of intramolecular basis set superposition error
1 November
1996
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical
Physics
Letters 261 (1996)
633-636
The magnitude of intramolecular basis set superposition error Frank Jensen ’ Research
School
of Chemistry,
Australian
Received14 June
National
University.
1996; in final form
Canberra,
1 August
ACT
0200,
Australia
1996
Abstract
It is shownthat part of what normally is considereda basisset effect on relative energies,more properly may be interpretedasintramolecularbasissetsuperposition error.
1. Introduction The generation of approximate solutions to the Schrodinger equation for molecular systemsis done almost exclusively by expanding the molecular orbitals in a Gaussianbasis set located on the nuclei. The representation of the MOs is improved as the basis set is enlarged, however, in practical calculations the basis is usually far from complete. When calculating weak intermolecular interactions, such as van der Waals or hydrogen bonding, it is widely recognized that basis set incompletenessintroduces errors. The interaction calculated by subtracting the energy of the isolated components from the energy of the molecular complex is overestimated. This is easily understood since the wavefunction for one of the components in the complex can benefit from basisfunctions located on the other (and vice versa), an effect known as basis set superposition error (BSSE). The effect is relatively small in absolute value, normally less than a few kcal/mol, but may
’ Permanent address: Department of Chemistry, versity, DK-5230 Odense M., Denmark.
Odense
0009-2614/96/$12.00 Copyright PII SOOOS-2614(96)01033-O
by Elsevier
0 1996 Published
Uni-
Science
be significant as the total non-bonded interaction is of the samemagnitude. Although the origin of BSSE is well understood, there is at present no easy way of correction for it. The most commonly usedmethod is the counterpoise (CP) correction [l], where the BSSE is estimated as the energy difference between calculations in the individual basis and combined basis,for each of the components. The literature regarding BSSE and CP corrections is large, for some recent work see Ref. [2-61. It should be noted that the CP correction does not provide either upper or lower bounds for the BSSE. The only method for eliminating BSSE is to increase the basis set until the interaction energy is stable to the desired accuracy, however, such a brute force approach is slowly convergent and only possible for small systems. What is perhapsnot as widely appreciated, is that BSSE is always present when comparing energiesof two different systems. As the functions follow the nuclei, the basis set is different for each geometrical configuration. It is usually assumedthat this effect is small, and it is essentially always ignored (for an exception, see Ref. [7]). What is realized, however, is that relative energies of different configurations are sensitive to the size of the basis set, often B.V. All rights reserved.
634
F. Jensen/Chemical
Physics Letters
referred to as the basis set effect. Part of the basis set effect is rooted in imbalances created by the nuclear centred functions, and could more properly be considered as intramolecular BSSE.
2. Computational
details
The basis sets used in this work are the correlation consistent basis sets developed by Dunning, denoted as cc-pVXZ, where X = D, T, Q, or 5 for double-, triple-, quadruple- and quintuple-zeta quality [8]. All calculations have been performed with the Gaussian-94 program packages [9], and geometries have been optimized at the MP2/cc-pVDZ level. For H,O and NH, the heavy atom is fixed at the origin, the C,/C, rotation axis is conserved, and the CP basis functions are located at the hydrogen positions in the bend and linear/planar optimized geometries, respectively. For CH,CH, the C, axis is conserved and the eclipsed conformation is formed by rotating one of the methyl group 60” from the staggered conformation. The CP functions are located at 60” between the hydrogens at each end.
3. Results and discussion
Three examples of energy differences have been chosen for illustration: the barrier to linearity in H,O, the inversion barrier in NH, and the rotational barrier in CH,CH,. A conventional way of estimating the infinite basis set limit of a given property is to perform a series of calculations with increasingly larger basis set and extrapolate the results. The ccpVXZ basis sets are well suited in this respect as the results for X = D, T, Q, or 5 usually can be fitted quite accurately by an exponential function, A(n) = A(m) + B exp( - Cn>, with n = 2, 3, 4, or 5. Given geometries for two different configurations, a normal procedure is to assign a set of basis functions to each nucleus and perform two calculations to get the relative energy. The two energies are thus calculated with diflerent basis sets, since the position of the functions change. In the spirit of the CP method, we will compare this value with the result obtained in a basis set consisting of the combined set of functions for the two geometries. The latter will
261 (1996)
633-636
Table 1 Barrier towards
linearity
Basis
for H,O
(kcal/mol)
SCF
cc-pVDZ cc-pW-2 cc-pVQZ cc-pvsz extrapolated Experimental
MP2
normal
CP
normal
CP
34.62 32.50 31.99 31.69 31.66
33.80 32.12 31.84 31.69 31.71
36.02 33.18 31.99 31.31 30.84
34.21 32.09 31.47 31.18 31.10
value is 32.16 kcal/mol
[IO].
be denoted the CP basis, and it is identical for the two calculations, having the same number of functions at the same positions for both calculations. The difference between the regular and CP basis will be denoted the intramolecular BSSE. We will especially concentrate on the magnitude of the intramolecular BSSE with the cc-pVDZ basis, as this represents a typical size of basis which can be used for a variety of systems. The MP2/cc-pVDZ optimi?ed geometry of H,O has a tend length of 0.965 A which decreases to 0.935 A for the linear geometry. The calculated barrier to linearity as a function of basis set at the SCF and MP2 levels is shown in Table 1. The convergence of the MP2 barrier with and without the CP basis functions are shown in Fig. 1. The extrapolated barriers are slightly different, but the CP value appears to be better converged, and the result of 31.10 kcal/mol is probably the better number. Of the 4.92 kcal/mol basis set error (relative to
.
cc-pvxz+cP
32
31 1
Fig. I. Basis set convergence the MP2 level.
for the H,O
barrier
to linearity
at
F. Jensen/Chemical Table 2 Inversion Basis
cc-pVDZ cc-pvTZ cc-pVQZ cc-pvsz extrapolated Experimental
barrier
for NH,
Physics Letters
261 (1996)
633-636
635
(kcal/mol) MP2
SCF normal
CP
normal
CP
6.96 4.78 4.39 4.10 4.12
6.04 4.31 4.12 4.04 4.06
8.24 5.82 5.15 4.68 4.58
7.18 5.12 4.72 4.54 4.54
value is 5.24 kcal/mol
[l I]. 2
the extrapolated value) with the normal cc-pVDZ basis, 1.81 kcal/mol is recovered by addition of the CP functions. Even with the cc-pV5Z basis there is a 0.13 kcal/mol difference between the normal and CP basis. The picture is the same at the SCF level, although the convergence is somewhat faster than at the MP2 level. The value at the CCSD(T)/cc-pVTZ level is 33.87 kcal/mol, i.e. correlation beyond MP2 increases the barrier by 0.69 kcal/mol. The experimental value is 32.16 kcal/mol [ 101, which may be compared to the value of 3 1.79 obtained by addition of the CCSD(T) correction to the MP2 extrapolated value. The difference is most likely due to geometry errors and coupling between higher order electron correlation and basis set effects. As for the H,O case, the NH bond length in NH, decreases (from 1.024 to 1.003 A> as the geometry becomes planar. The results for the inversion barrier are shown in Table 2, and the MP2 convergence in Fig. 2. The extrapolated values for the regular and CP basis are in this case similar. The basis set error with the cc-pVDZ basis is 3.70 kcal/mol, of which 1.06 kcal/mol can be attributed to intramolecular BSSE. The estimate of correlation effects beyond MP2 by the CCSD(T)/cc-pVTZ method is +OSO kcal/mol. Addition of this to the MP2 extrapolated value gives a barrier of 5.04 kcal/mol, which can be compared to the experimental value of 5.24 kcal/mol
t111. For the internal rotation in ethane, there are three geometry parameters which change when going from the staggered to the eclipsed form. The C-C bond length increases (from 1.530 to 1.544 A), the C-H bond length decreases (from 1.103 to 1.102 ft> and the HCC angle increases (from 111.3 to 111.8”).
3
Fig. 2. Basis set convergence MP2 level.
4 ccpvxz basis
6
for the NH 3 inversion
6
barrier
at the
Since the C-C bond length changes, a rigorous application of the CP principle would mean that two sets of basis functioas for carbon should be positioned only 0.007 A apart. Clearly such closely spaced functions will effectively span the same space, and in practice lead to numerical problems due to linear dependence. We have therefore only added the CP basis functions for the hydrogens, and the results are shown in Table 3. The rotational barrier is an example of an ‘easy’ computational problem, already with a regular cc-pVQZ basis is the energy difference converged to 0.01 kcal/mol. Nevertheless, of the 0.40 kcal/mol error with the cc-pVDZ basis, 0.37 kcal/mol is due to intramolecular BSSE. The effect of higher order correlation is small ( - 0.13 kcal/mol from CCSD(T)/cc-pVDZ calculations) and addition of this to the limiting MP2 result gives a value slightly lower than the experimental value of 2.88 kcal/mol [12], analogous to the H,O and NH, cases.
Table 3 Rotational Basis
cc-pVDZ cc-pVTZ cc-pVQZ cc-pvsz Experimental
barrier
for CH .CH,
(kcal /mol)
SCF
MP2
normal
CP
normal
CP
3.30 3.06 3.06 3.06
3.12 3.06
3.25 2.92 2.85 2.86
2.88 2.84
value is 2.88 kcal/mol
[ 121.
636
F. Jensen/Chemical
Physics
4. Conclusion The present results show that changes which normally are considered a basis set effect, in part are due to intramolecular BSSE. The magnitude of the intramolecular BSSE is similar to that observed in intermolecular complexes, i.e. a few kcal/mol. Nevertheless, this may be a significant fraction of the total basis set effect. It should be noted that addition of CP basis functions is normally not a cost-efficient method for improving the quality of the results. For comparing two geometries, a CP-corrected basis contains significantly more functions. In the present cases, the cc-pVDZ basis with CP functions is intermediate in size between cc-pVDZ and cc-pVTZ, and gives also results which are of intermediate quality. Furthermore, if more than two geometries are compared, addition of CP basis functions for all conformations becomes infeasible, and a two-by-two comparison becomes cumbersome.
Acknowledgements This work was supported by grants from the Danish Natural Science Research Council. The author thanks RSC, ANU for a visiting fellowship.
Letters 261 (1996)
633-636
References [II SF. I21 M.
Boys and F. Bemardi, Mol. Phys. 19 (1970) 553. Gutowski, F.B. van Duijneveldt, G. Chalasinski and L. Piela, Mol. Phys. 61 (1987). Chem. Phys. Lett. 217 [31 E.R. Davidson and S.J. Chakravarty, (1994) 48. [41 M. Gutowski, M.M. Szczesniak and G. Chalasinski, Chem. Phys. Lett. 241 (1995) 140. Chem. Phys. I.&t. 241 151 E.R. Davidson and S.J. Chakravarty, (1995) 146. PI A.J. Ahkowicz, Z. Latajka, S. Scheiner and G. Chalasinski, J. Mol. Stmct. (Theochem) 342 (1995) 153. I71 S. Reiling, J. Brickmann, M. Schlenkrich and P.A. Bopp, J. Comput. Chem. 17 (1996) 133. @I T.H. Dunning, Jr., J. Chem. Phys. 90 (1989) 1007. [91 M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stephanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, ES. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. DeFrees, J. Baker, J.J.P. Stewart, M. Head-Gordon, C. Gonzales and J.A. Pople, GAUSSIAN 94 (Gaussian, Inc., Pittsburgh, PA, 1995). 1101P. Jensen, J. Mol. Spectrosc. 133 (1989) 438. [ill V. Spirko, J. Mol. Spectrosc. 101 (1983) 30. 1121 E. Hirota, Y. Endo, S. Saito and J.L. Duncan, J. Mol. Spectrosc. 89 (1981) 285.
1996
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical
Physics
Letters 261 (1996)
633-636
The magnitude of intramolecular basis set superposition error Frank Jensen ’ Research
School
of Chemistry,
Australian
Received14 June
National
University.
1996; in final form
Canberra,
1 August
ACT
0200,
Australia
1996
Abstract
It is shownthat part of what normally is considereda basisset effect on relative energies,more properly may be interpretedasintramolecularbasissetsuperposition error.
1. Introduction The generation of approximate solutions to the Schrodinger equation for molecular systemsis done almost exclusively by expanding the molecular orbitals in a Gaussianbasis set located on the nuclei. The representation of the MOs is improved as the basis set is enlarged, however, in practical calculations the basis is usually far from complete. When calculating weak intermolecular interactions, such as van der Waals or hydrogen bonding, it is widely recognized that basis set incompletenessintroduces errors. The interaction calculated by subtracting the energy of the isolated components from the energy of the molecular complex is overestimated. This is easily understood since the wavefunction for one of the components in the complex can benefit from basisfunctions located on the other (and vice versa), an effect known as basis set superposition error (BSSE). The effect is relatively small in absolute value, normally less than a few kcal/mol, but may
’ Permanent address: Department of Chemistry, versity, DK-5230 Odense M., Denmark.
Odense
0009-2614/96/$12.00 Copyright PII SOOOS-2614(96)01033-O
by Elsevier
0 1996 Published
Uni-
Science
be significant as the total non-bonded interaction is of the samemagnitude. Although the origin of BSSE is well understood, there is at present no easy way of correction for it. The most commonly usedmethod is the counterpoise (CP) correction [l], where the BSSE is estimated as the energy difference between calculations in the individual basis and combined basis,for each of the components. The literature regarding BSSE and CP corrections is large, for some recent work see Ref. [2-61. It should be noted that the CP correction does not provide either upper or lower bounds for the BSSE. The only method for eliminating BSSE is to increase the basis set until the interaction energy is stable to the desired accuracy, however, such a brute force approach is slowly convergent and only possible for small systems. What is perhapsnot as widely appreciated, is that BSSE is always present when comparing energiesof two different systems. As the functions follow the nuclei, the basis set is different for each geometrical configuration. It is usually assumedthat this effect is small, and it is essentially always ignored (for an exception, see Ref. [7]). What is realized, however, is that relative energies of different configurations are sensitive to the size of the basis set, often B.V. All rights reserved.
634
F. Jensen/Chemical
Physics Letters
referred to as the basis set effect. Part of the basis set effect is rooted in imbalances created by the nuclear centred functions, and could more properly be considered as intramolecular BSSE.
2. Computational
details
The basis sets used in this work are the correlation consistent basis sets developed by Dunning, denoted as cc-pVXZ, where X = D, T, Q, or 5 for double-, triple-, quadruple- and quintuple-zeta quality [8]. All calculations have been performed with the Gaussian-94 program packages [9], and geometries have been optimized at the MP2/cc-pVDZ level. For H,O and NH, the heavy atom is fixed at the origin, the C,/C, rotation axis is conserved, and the CP basis functions are located at the hydrogen positions in the bend and linear/planar optimized geometries, respectively. For CH,CH, the C, axis is conserved and the eclipsed conformation is formed by rotating one of the methyl group 60” from the staggered conformation. The CP functions are located at 60” between the hydrogens at each end.
3. Results and discussion
Three examples of energy differences have been chosen for illustration: the barrier to linearity in H,O, the inversion barrier in NH, and the rotational barrier in CH,CH,. A conventional way of estimating the infinite basis set limit of a given property is to perform a series of calculations with increasingly larger basis set and extrapolate the results. The ccpVXZ basis sets are well suited in this respect as the results for X = D, T, Q, or 5 usually can be fitted quite accurately by an exponential function, A(n) = A(m) + B exp( - Cn>, with n = 2, 3, 4, or 5. Given geometries for two different configurations, a normal procedure is to assign a set of basis functions to each nucleus and perform two calculations to get the relative energy. The two energies are thus calculated with diflerent basis sets, since the position of the functions change. In the spirit of the CP method, we will compare this value with the result obtained in a basis set consisting of the combined set of functions for the two geometries. The latter will
261 (1996)
633-636
Table 1 Barrier towards
linearity
Basis
for H,O
(kcal/mol)
SCF
cc-pVDZ cc-pW-2 cc-pVQZ cc-pvsz extrapolated Experimental
MP2
normal
CP
normal
CP
34.62 32.50 31.99 31.69 31.66
33.80 32.12 31.84 31.69 31.71
36.02 33.18 31.99 31.31 30.84
34.21 32.09 31.47 31.18 31.10
value is 32.16 kcal/mol
[IO].
be denoted the CP basis, and it is identical for the two calculations, having the same number of functions at the same positions for both calculations. The difference between the regular and CP basis will be denoted the intramolecular BSSE. We will especially concentrate on the magnitude of the intramolecular BSSE with the cc-pVDZ basis, as this represents a typical size of basis which can be used for a variety of systems. The MP2/cc-pVDZ optimi?ed geometry of H,O has a tend length of 0.965 A which decreases to 0.935 A for the linear geometry. The calculated barrier to linearity as a function of basis set at the SCF and MP2 levels is shown in Table 1. The convergence of the MP2 barrier with and without the CP basis functions are shown in Fig. 1. The extrapolated barriers are slightly different, but the CP value appears to be better converged, and the result of 31.10 kcal/mol is probably the better number. Of the 4.92 kcal/mol basis set error (relative to
.
cc-pvxz+cP
32
31 1
Fig. I. Basis set convergence the MP2 level.
for the H,O
barrier
to linearity
at
F. Jensen/Chemical Table 2 Inversion Basis
cc-pVDZ cc-pvTZ cc-pVQZ cc-pvsz extrapolated Experimental
barrier
for NH,
Physics Letters
261 (1996)
633-636
635
(kcal/mol) MP2
SCF normal
CP
normal
CP
6.96 4.78 4.39 4.10 4.12
6.04 4.31 4.12 4.04 4.06
8.24 5.82 5.15 4.68 4.58
7.18 5.12 4.72 4.54 4.54
value is 5.24 kcal/mol
[l I]. 2
the extrapolated value) with the normal cc-pVDZ basis, 1.81 kcal/mol is recovered by addition of the CP functions. Even with the cc-pV5Z basis there is a 0.13 kcal/mol difference between the normal and CP basis. The picture is the same at the SCF level, although the convergence is somewhat faster than at the MP2 level. The value at the CCSD(T)/cc-pVTZ level is 33.87 kcal/mol, i.e. correlation beyond MP2 increases the barrier by 0.69 kcal/mol. The experimental value is 32.16 kcal/mol [ 101, which may be compared to the value of 3 1.79 obtained by addition of the CCSD(T) correction to the MP2 extrapolated value. The difference is most likely due to geometry errors and coupling between higher order electron correlation and basis set effects. As for the H,O case, the NH bond length in NH, decreases (from 1.024 to 1.003 A> as the geometry becomes planar. The results for the inversion barrier are shown in Table 2, and the MP2 convergence in Fig. 2. The extrapolated values for the regular and CP basis are in this case similar. The basis set error with the cc-pVDZ basis is 3.70 kcal/mol, of which 1.06 kcal/mol can be attributed to intramolecular BSSE. The estimate of correlation effects beyond MP2 by the CCSD(T)/cc-pVTZ method is +OSO kcal/mol. Addition of this to the MP2 extrapolated value gives a barrier of 5.04 kcal/mol, which can be compared to the experimental value of 5.24 kcal/mol
t111. For the internal rotation in ethane, there are three geometry parameters which change when going from the staggered to the eclipsed form. The C-C bond length increases (from 1.530 to 1.544 A), the C-H bond length decreases (from 1.103 to 1.102 ft> and the HCC angle increases (from 111.3 to 111.8”).
3
Fig. 2. Basis set convergence MP2 level.
4 ccpvxz basis
6
for the NH 3 inversion
6
barrier
at the
Since the C-C bond length changes, a rigorous application of the CP principle would mean that two sets of basis functioas for carbon should be positioned only 0.007 A apart. Clearly such closely spaced functions will effectively span the same space, and in practice lead to numerical problems due to linear dependence. We have therefore only added the CP basis functions for the hydrogens, and the results are shown in Table 3. The rotational barrier is an example of an ‘easy’ computational problem, already with a regular cc-pVQZ basis is the energy difference converged to 0.01 kcal/mol. Nevertheless, of the 0.40 kcal/mol error with the cc-pVDZ basis, 0.37 kcal/mol is due to intramolecular BSSE. The effect of higher order correlation is small ( - 0.13 kcal/mol from CCSD(T)/cc-pVDZ calculations) and addition of this to the limiting MP2 result gives a value slightly lower than the experimental value of 2.88 kcal/mol [12], analogous to the H,O and NH, cases.
Table 3 Rotational Basis
cc-pVDZ cc-pVTZ cc-pVQZ cc-pvsz Experimental
barrier
for CH .CH,
(kcal /mol)
SCF
MP2
normal
CP
normal
CP
3.30 3.06 3.06 3.06
3.12 3.06
3.25 2.92 2.85 2.86
2.88 2.84
value is 2.88 kcal/mol
[ 121.
636
F. Jensen/Chemical
Physics
4. Conclusion The present results show that changes which normally are considered a basis set effect, in part are due to intramolecular BSSE. The magnitude of the intramolecular BSSE is similar to that observed in intermolecular complexes, i.e. a few kcal/mol. Nevertheless, this may be a significant fraction of the total basis set effect. It should be noted that addition of CP basis functions is normally not a cost-efficient method for improving the quality of the results. For comparing two geometries, a CP-corrected basis contains significantly more functions. In the present cases, the cc-pVDZ basis with CP functions is intermediate in size between cc-pVDZ and cc-pVTZ, and gives also results which are of intermediate quality. Furthermore, if more than two geometries are compared, addition of CP basis functions for all conformations becomes infeasible, and a two-by-two comparison becomes cumbersome.
Acknowledgements This work was supported by grants from the Danish Natural Science Research Council. The author thanks RSC, ANU for a visiting fellowship.
Letters 261 (1996)
633-636
References [II SF. I21 M.
Boys and F. Bemardi, Mol. Phys. 19 (1970) 553. Gutowski, F.B. van Duijneveldt, G. Chalasinski and L. Piela, Mol. Phys. 61 (1987). Chem. Phys. Lett. 217 [31 E.R. Davidson and S.J. Chakravarty, (1994) 48. [41 M. Gutowski, M.M. Szczesniak and G. Chalasinski, Chem. Phys. Lett. 241 (1995) 140. Chem. Phys. I.&t. 241 151 E.R. Davidson and S.J. Chakravarty, (1995) 146. PI A.J. Ahkowicz, Z. Latajka, S. Scheiner and G. Chalasinski, J. Mol. Stmct. (Theochem) 342 (1995) 153. I71 S. Reiling, J. Brickmann, M. Schlenkrich and P.A. Bopp, J. Comput. Chem. 17 (1996) 133. @I T.H. Dunning, Jr., J. Chem. Phys. 90 (1989) 1007. [91 M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stephanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, ES. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. DeFrees, J. Baker, J.J.P. Stewart, M. Head-Gordon, C. Gonzales and J.A. Pople, GAUSSIAN 94 (Gaussian, Inc., Pittsburgh, PA, 1995). 1101P. Jensen, J. Mol. Spectrosc. 133 (1989) 438. [ill V. Spirko, J. Mol. Spectrosc. 101 (1983) 30. 1121 E. Hirota, Y. Endo, S. Saito and J.L. Duncan, J. Mol. Spectrosc. 89 (1981) 285.