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Joseph J. BelBruno Dartmouth College Hanover, NH 03755
Calculating Interaction Energies Using First Principle Theories: Consideration of Basis Set Superposition Error and Fragment Relaxation
W
J. Phillip Bowen Center for Drug Discovery, Department of Chemistry and Biochemistry, University of North Carolina at Greensboro, Greensboro, NC 27402 Jennifer B. Sorensen Department of Chemistry, Seattle University, Seattle, WA 98122 Karl N. Kirschner* Department of Chemistry, Center for Molecular Design, Hamilton College, Clinton, NY 13323; *
[email protected]
Over the past decades computational chemistry has grown to become an important research field. Typically, present-day physical chemistry articles have at least one reference to computational data. At the turn of the century the number of peer-reviewed journals that were dedicated to computational and theoretical chemistry was in the teens, which does not include journals published solely on the Internet. In 1998 the Nobel Prize in chemistry was awarded to Walter Kohn and John Pople for their development of computational methodologies. However, many graduate students do not have a basic understanding of this field or the complexities that lie within and even fewer undergraduates know the field exists. In this article we explain basis set superposition error and demonstrate some of the complexities and subtleties when using ab initio theory to calculate molecular interaction energies for weakly bonded gas-phase complexes. For simplicity we will limit our discussion to the interaction between two molecules; however, these ideas can be extended to larger nonbonded clusters. In quantum mechanics, basis sets, which are composed of Gaussian functions, are used to develop the wave function of a system and can be equated to the orbitals about an atom or molecule. Basis sets differ not only in the number of Gaussian functions used, but also in how their exponents and coefficients were determined. A small basis set uses a small number of functions to describe the orbital space about an atom. Using a small number of Gaussian functions to approximate the atomic environment can lead to (i) basis set incompleteness error (BSIE) and (ii) basis set superposition error (BSSE). In both cases, the errors arise from the approximation of using a limited number of mathematical functions to describe atomic and molecular orbitals. BSIE manifests itself in poor geometries and poor representation of potential energy surfaces (and consequently the interaction energy) relative to experimental results or levels of theory using a “complete basis set”. BSSE can be considered as a specific type of error within BSIE. van Mourik and coworkers have explained BSSE as a “balancing error” arising from the difference in the number of orbitals used to describe the dimer www.JCE.DivCHED.org
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complex and its respective monomers (1). Enlarging the basis set by increasing the number of Gaussian functions to describe the orbital space about the atom reduces BSIE and BSSE. In the basis set expansion, one can approach the complete basis set limit where both BSIE and BSSE approach zero or in other words are no longer present in the calculation. However, the drawback of using larger basis sets is that the cost of the calculation, in terms of computer power and in time requirements, increases substantially. The use of extrapolation techniques (2) to obtain geometries and energies at the complete basis set limit is a popular method for obtaining ab initio results that are free of BSIE and consequently of BSSE. So far, this discussion has addressed BSIE and BSSE in terms of weakly bonded complexes, but these errors are also present in covalent and ionic bonds present in the monomers (i.e., N2, HCl, CH3CH3, see ref 1). However, the effects of BSSE are often small in strongly bound systems and, as a result, are often ignored (3). BSSE is often considered in context with interaction energies between nonbonded molecules. An ab initio interaction energy is routinely determined by using the point at the bottom of a potential energy well formed from the forces between two molecules, BSSE also affects the shape of the potential energy surface in general. Therefore, properties such as vibrational frequencies, cross sections, and dynamics are also affected by BSSE. Since BSSE is an error that arises from the approximations made in theories, it is important to correct for this error to allow us to improve upon the underlying theories instead of trying to compensate for the errors (e.g., use of a correction term). Another goal for correcting interaction energies and potential energy surfaces for BSSE is to obtain energies or frequencies that are within chemical accuracy. Currently, chemical accuracy is defined as an energy that is accurate to within 1 kcal mol1 and a vibrational frequency to within 30 cm1. Ultimately, how well one calculates an interaction energy using a theoretical technique can only be gauged when compared to an experimental value. An additional point is that these theoretically determined in-
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Figure 1. (A) The methanol–water complex with the orbital space qualitatively represented by a wire mesh. (B) The first step in the counterpoise procedure where the nuclei of the water are removed while retaining the orbital space of the complex. (C) The second step in the counterpoise procedure where the methanol nuclei are remove while retaining the orbital space the complex.
teraction energies are often uncorrected for zero-point vibrational energy, for thermal corrections, or for entropy, resulting in an energy that represents the electronic portion of the interaction, which is only part of the Gibbs free energy or enthalpy that is measured experimentally. Below we describe four different methods that can be used to obtain an interaction energy using ab initio or density functional methods. These four methods will then be illustrated by calculating the interaction energy of the methanol–water complex as the model system, where the methanol donates a hydrogen to the water (Figure 1A). Additionally, BSSE and fragment relaxation will be examined as a function of a small, medium, and large basis set. Breakdown of the Four Methods The first method is the most intuitive of the four. If a complete basis set could hypothetically be used, then the experimentally accurate method is to subtract the two fully optimized monomers from the fully optimized dimer complex α ∪ β
α
β
∆E (AB ) = E AB (AB ) − E A ( A) − E B (B )
(1)
where the subscripts on the energy terms represent the geometries and the superscripts the basis sets (4, 5): α傼β is the basis set for dimer AB, α is the basis set for monomer A, α傼β and β is the basis set for monomer B. For example, EAB (AB) is the energy of the AB dimer, calculated in the geometry of the AB complex with the dimer basis set α傼β. With the exception of a few specialized theories that correct for BSSE a priori (6–11), corrections to the interaction energy must be made a posteriori to obtain an interaction energy that is BSSE-free. However, one cannot use a complete basis set in the calculations required for eq 1 and as a result the interaction energy computed possesses BSSE. The second method, the Boys–Bernardi functional counterpoise scheme (fCP), is the most common approach used to correct for BSSE (12): α ∪ β
α ∪ β
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α ∪ β
α
β
∆ESP (AB ) = E AB (AB ) − E AB( A ) − E AB(B)
(3)
α ∪ β
∆Ef C P (AB ) = E AB (AB) − E AB (A ) − E AB ( B ) (2) 1226
α傼β In this case EAB (A) is the energy of monomer A, whose geometry is that found in the AB complex and calculated with α傼β the dimer basis set, likewise for EAB (B). Note that the subscript on the ∆E term represents the method. The important concept of the fCP method is that each of these calculations uses the same number of basis functions, specifically the number of functions used in the creation of the dimer basis set. The fCP calculation can be further explained in the following manner: (a) a full geometry optimization is performed on the dimer complex, (b) the nuclei and electrons of one fragment are removed while leaving the orbital space about the dimer intact, (c) a single self-consistent field is obtained (i.e., a single-point calculation is performed) on the remaining monomer with the dimer basis set, (d) steps b and c are repeated for the other fragment. The dimer orbital space with the dimer (fCP step a) and the dimer orbital space with each monomer (fCP step b) is schematically represented in Figure 1.1 Equation 2 gives the BSSE-corrected interaction energy and is considered basis-set consistent since the same number of basis functions are used in each component. However, the fCP method is not universally regarded as the best method for BSSE correction (13, 14). Opponents point out that the method is a posteriori, and thus the system’s potential energy surface will be fundamentally different than determined by a standard gradient-based calculation; the fCP determined intermolecular distances are systematically longer and the intermolecular angles can be significantly different (15, 16). Additionally, the interaction energies are often argued to be overcorrected by this method (17, 18). Nevertheless, this fCP method can be used to obtain useful energetics and an understanding of the how the approximated orbital space introduces errors in the calculation (19). The third method is analogous to the ∆EfCP of eq 2, although here a single point (SP) energy of each monomer is calculated using the monomer basis set rather than the dimer basis set:
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This is the interaction energy of the dimer neglecting BSSE
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and fragment relaxation energy. Fragment relaxation energy is the thermodynamic penalty for distorting the monomers from their isolated gas-phase geometry to their respective geometry found in the dimer. For very weak interactions the penalty is negligible, but as the interaction increases in strength the penalty becomes greater since the monomers are likely to become more perturbed owing to the presence of stronger induction forces. This is particularly true for systems that includes ions or molecules with large dipole moments. Both eqs 2 and 3 neglect fragment relaxation as a result of using monomer geometries as they are found in the dimer complex. The monomer energies in eq 3 are more positive than the energies of the fully optimized monomers in eq 1, corresponding to less energetically favored (strained) monomer structures. This equation was often used when computer resources were minimal during the formative years of computation chemistry and has lost it appeal in the modern age except when dealing with very large systems such as a protein–ligand complex. A certain quantity of energy must be absorbed by the individual monomers as they distort during the formation of the dimer due to induction forces. Subtracting eq 3 from eq 1 yields the overall relaxation energy (Rel) of the monomers α
α
∆E Rel (AB ) = E AB ( A) − E A (A ) β
(4)
β
+ E AB (B) − E B (B)
where the terms inside the square brackets are the individual relaxation energies for the monomers. These energies are a reflection of how perturbed the monomer geometry becomes from being in the presence of the other monomer. The magnitude of the BSSE energy2 can be determined by subtracting eq 3 from eq 2 α
α ∪ β
∆E BSSE (AB ) = E AB ( A) − E AB (A ) β
α ∪ β
+ E AB (B) − E AB
(B )
(5)
where the terms inside the square brackets are the individual
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monomer BSSE energies. The magnitude of ∆EBSSE(AB) is an indication of the ability the given basis set has in describing the orbital space of the dimer and monomers, where a smaller value correlates to a better performance. Finally, a rigorous fourth method (Rig) that incorporates relaxation energy into the counterpoise correction can be determined by adding eq 4 to eq 2: α ∪ β
α ∪ β
α ∪ β
∆E Rig (AB ) = E AB ( AB ) − E AB (A ) − E AB (B ) α
α
β
β
+ E AB(A) − E A (A )
(6)
+ E AB (B) − E AB (B)
The use of this equation has been criticized because it is not basis set consistent and requires a total of 7 calculations: 3 full geometry optimizations, 2 counterpoise single-point calculations, and 2 single-point calculations. Despite these drawbacks eq 6 does, in principle, represent the most theoretically rigorous method for reproducing an experimentally relevant interaction energy using traditional quantum methods. Methanol–Water Dimer The methanol–water dimer is a convenient hetero-molecular model system to use that displays different values of BSSE and fragment relaxation for each constituent. Table 1 displays the number of basis functions and “raw” energies obtained from each of the calculations using HF兾3-21G, HF兾6-31++G**, and HF兾aug-cc-pVTZ levels of theory on the methanol–water dimer. Using the appropriate raw energies with eqs 1–3 and 6 will yield the interaction energy of the dimer complex as determined by the four methods, which are presented in Table 2. In addition to the interaction energies, the sum totals of the fragment relaxation energies and the BSSE energies are also shown. The HF兾3-21G interaction energies have a range from 11.21 to 6.12 kcal mol1. The BSSE energy using this basis set is 4.96 kcal mol1, which is significantly larger than the fragment relaxation energy of 0.13 kcal mol1 (Table 2). In comparison, the HF兾6-31++G**
Table 1. The Number of Basis Functions and the Raw Output Energies from the Individual Calculations Energy/Eh
No. of Basis Functions
Theory
Full Optimizations
Single Point
Counterpoise
HF/3-21G CH3 OH
26
114.3980194
114.3978664
114.3987448
H2 O
13
75.5859598l
75.5859123
75.5929461
39
190.0016497
CH3 OH···H2 O HF/6-31++G** CH3 OH
62
115.0525144
115.0524693
115.0526687
H2 O
31
76.0313091
76.0313062
76.0321103
CH3 OH···H2 O
93
191.0915052
184
115.0930871
115.0929752
115.0930307
92
76.0612031
76.0611520
76.0612192
276
191.1600554
HF/aug-cc-pVTZ CH3 OH H2 O CH3 OH···H2 O
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Table 2. Basis Set Superposition Error, Relaxation Energy, and Interaction Energy of CH3OH···H2O Theory
RO- - - O /Å
∆E(AB)
∆ESP(AB)
∆EfCP(AB)
∆ERig(AB)
∆EBSSE(AB)
∆ERel(AB)
HF/3-21G
2.772
11.088
11.214
6.249
6.123
4.965 (0.551, 4.414)
0.126 (0.096, 0.030)
HF/6-31++G**
2.994
4.820
4.850
4.221
4.191
0.630 (0.125, 0.505)
0.030 (0.028, 0.002)
HF/aug-cc-pVTZ
3.047
3.618
3.720
3.643
3.541
0.077 (0.035, 0.042)
0.102 (0.070, 0.032)
1
NOTE: The data are in units of kcal mol . The data in parenthesis are the individual contribution of methanol and water, respectively.
energies have a much smaller span of interaction energies, ranging from 4.85 to 4.19 kcal mol1. At this level of theory the BSSE energy is 0.63 kcal mol1, with a fragment relaxation energy of 0.03 kcal mol1. Thus, an increase of 54 basis functions in the calculation reduces the BSSE energy by 4.36 kcal mol1. Furthermore, the HF兾aug-cc-pVTZ level of theory has a range of interaction from 3.72 to 3.54 kcal mol1, with a BSSE energy of 0.08 kcal mol1 and a fragment relaxation energy of 0.10 kcal mol1. It is easy to see that as the basis set becomes larger the BSSE energy becomes smaller. With larger basis sets, the individual fragments of the dimer have increasingly adequate orbital space to place their electrons during the calculations. In other words, the calculations are approaching a limit where the addition of a few more basis functions does not alter the energies much. Thus, going from 39 to 93 basis functions has a big impact on the fragments energies, while going from 93 to 276 basis functions has a smaller impact, since a basis set limit is being approached. A slightly different trend is seen for the fragment relaxation energy. Since the methanol–water complex is a charge neutral system, the fragment relaxation energy is very small at all levels of theory studied because the induction forces are small. The fragment relaxation energies obtained from HF兾3-21G and HF兾6-31++G** are much smaller than the corresponding BSSE energy. This signifies that for small basis sets, correcting for BSSE is a higher priority than correcting for fragment relaxation in obtaining a more precise interaction energy. At the HF兾aug-cc-pVTZ level of theory, the fragment relaxation energy is greater than the BSSE energy. Now, when using a large basis set, correcting for fragment relaxation energy has become more important than correcting for BSSE to obtain a more precise energy. Another interesting aspect to note is that the hydrogen donor (methanol) has a larger fragment relaxation value than the hydrogen acceptor (water), indicating its geometry is more distorted. Examining the different configuration of the water– methanol complex, where the water is donating the hydrogen to the methanol, can easily extend this study.3 There are only a few experimental studies reported on the methanol– water and water–methanol complexes (20). Consequently, most of our understanding of this system has come from theoretical studies. The relative stability of the two configurations has been shown to be sensitive to basis set size and the level of theory used in their calculations. At low theory levels with small basis sets (e.g., HF兾6-31G* or HF兾STO3G), the methanol–water is the more stable configuration. Conversely, calculations using larger basis sets (e.g., HF兾augcc-pVDZ) determine that water–methanol is the more stable complex. Extrapolating to the complete basis set limit at the 1228
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MP2 level of theory also results in the water–methanol complex to be more stable by 0.71 kcal mol1 than the methanol–water complex (21). Performing a single-point calculation using the CCSD(T)兾aug-cc-pVDZ theory on an optimized MP2兾aug-cc-pVDZ geometry maintained that the methanol–water is the more stable configuration by 0.57 kcal mol1 (22). Keeping these results in mind, one can explore the sensitivity of the interaction energy and relative stability of the two configurations as a function of basis set size and theory level. In addition to the water and methanol complexes, there are a number of other systems that can be studied to gain a better feel for the impact that BSSE and fragment relaxation have on interaction energy. A proper set of systems would include dispersion forces (e.g., Ar···Ar; N2···N2), dipole–induce dipole (e.g., H2O···N2; H2O···C6H6), dipole–dipole (e.g., H2O···H2O), charge–induced dipole (e.g., OH−···N2), charge–dipole (e.g., H3O+···H2O), and charge–charge interactions (e.g., CH3CO2−···NH4+). These different interactions should display different sensitivities to the level of theory and to the basis set employed, and the fragment relaxation energy contribution should increase as the forces involved in the complex become greater. Conclusion Many scientists and the majority of students who use computational chemistry as a research tool often are not aware that there are four different mathematical and conceptual methods that can be used to determine interaction energies when employing ab initio or density functional theory. In this article we have presented and explained the similarities and differences in these four methods, as well as exemplified them using the methanol–water complex. Additionally, we have also explained basis set superposition error and fragment relaxation error. In choosing which mathematical method to use for calculating interaction energies, one must also consider the size of the basis set employed, the potential magnitude of BSSE, the effects of induction forces on the constituent molecules, and the overall cost of the calculations. There are also several other methods that can be used to eliminate BSSE, including the chemical Hamiltonian approach (23), self consistent field for molecular interactions and modification thereof (24, 25), interacting correlated fragment (26), and by simply increasing the size of the basis set. Computational Methods All calculations were performed using the Gaussian03 program suites (27). Other quantum chemistry programs will differ in the input formats used for running the calculations.
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Note that the counterpoise and single-point calculations have input structures that correspond to the output geometries from the fully optimized complex. Three different basis sets, 3-21G, 6-31++G**, and aug-cc-pVTZ, were used to demonstrate the correlation between BSSE and basis set size. Hartree–Fock (HF) theory was used in all calculations (28). For simplicity, zero-point vibration energy and thermal corrections were not discussed in this article. W
Supplemental Material
The Gaussian input files for each calculation used in this article are available in this issue of JCE Online. Notes 1. The fCP procedure can be used for a dimer complex that is composed either of molecules or of atoms. In this article we will focus on dimers that are composed of molecules but the ideas can be easily transferred to dimers composed of atoms. 2. BSSE energy is due to the different size basis sets and should not be confused with the BSSE corrected interaction energy denoted as ∆EfCP(AB). 3. Note that the first monomer listed in the dimer is the hydrogen donor; thus, water–methanol is different from methanol– water.
Literature Cited 1. van Mourik, T.; Wilson, A. K.; Peterson, K. A.; Woon, D. E.; Dunning, T. H., Jr. Adv. Quant. Chem. 1999, 31, 105–135. 2. Woon, D. E.; Dunning, T. H., Jr. J. Phys. Chem. 1993, 99, 1914–1929. 3. van Mourik, T. University College London, London. Personal communication, 1998. 4. Xantheas, S. S. J. Chem. Phys. 1996, 104, 8821–8824. 5. Rayón, V. M.; Sordo, J. A. Theor. Chem. Acc. 1998, 99, 68– 70. 6. Cullen, J. M. Int. J. Quantum Chem. Symp. 1991, 25, 193– 207. 7. Mayer, I. Int. J. Quantum Chem. 1983, 23, 341–363. 8. Valiron, P.; Vibók, Á.; Mayer, I. J. Comput. Chem. 1993, 14, 401–409. 9. Mayer, I. Int. J. Quantum. Chem. 1998, 70, 41–63. 10. Hamza, A.; Vibók, Á.; Halász, G. J.; Mayer, I. J. Mol. Struct. (Theochem) 2000, 501–502, 427–434. 11. Salvador, P.; Paizs, B.; Duran, M.; Suhai, S. J. Comput. Chem. 2001, 22, 765–786. 12. Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553–566.
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13. Frisch, M. J.; Del Bene, J. E.; Binkley J. S.; Schaefer, H. F., III J. Chem. Phys. 1986, 84, 2279–2289. 14. Liedl, K. R. J. Chem. Phys. 1986, 108, 3199–3204. 15. Hobza P.; Havlas, Z. Theo. Chem. Acc. 1998, 99, 372–377. 16. Hobza, P.; Havlas, Z. Collect. Czech. Chem. Commun. 1998, 63, 1343–1354. 17. Van Duijneveldt, F. B.; van Duijneveldt-van de Rijdt, Jeanne G. C. M.; van Lenthe, J. H. Chem. Rev. 1994, 94, 1873–1885; and references within. 18. Schwenke, D. W.; Truhlar D. G. J. Chem. Phys. 1985, 82, 2418–2426. 19. Cook, D. B.; Sordo, J.; Sordo, T. L. Int. J. Quant. Chem. 1993, 48, 375–384. 20. Stockman, P. A.; Blake, G. A.; Lovas, F. J.; Suenran, R. D. J. Chem. Phys. 1997, 107, 3782–3790. 21. Kirschner, K. N.; Woods, R. J. J. Phys. Chem. A 2001, 105, 4150–4155. 22. Fileti, E. E.; Chaudhuri, P.; Canuto, S. Chem. Phys. Lett. 2004, 400, 494–499. 23. Mayer, I. J. Comput. Chem. 1993, 14, 401–409. 24. Famulari, A.; Gianinette, E.; Raimondi, M.; Sironi, M. Int. J. Quant. Chem. 1998, 69, 151–158. 25. Calderoni, G.; Cargnoni, F.; Famulari, A. J. Chem. Phys. A 2002, 106, 5521–5528. 26. Liu, B.; McLean, A. D. J. Phys. Chem. 1989, 91, 2348–2359. 27. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, Revision C.02; Gaussian, Inc.: Wallingford CT, 2004. 28. For a review of Hartree–Fock theory see Cramer, Christopher J. Essentials of Computational Chemistry Theories and Models, 2nd ed.; John Wiley and Sons, Inc.: Hoboken, NJ, 2004.
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