ISSN 00360244, Russian Journal of Physical Chemistry A, 2014, Vol. 88, No. 4, pp. 629–633. © Pleiades Publishing, Ltd., 2014. Original Russian Text © V.V. Belikov, D.A. Bokhan, D.N. Trubnikov, 2014, published in Zhurnal Fizicheskoi Khimii, 2014, Vol. 88, No. 4, pp. 615–619.
STRUCTURE OF MATTER AND QUANTUM CHEMISTRY
Estimating the Basis Set Superposition Error in the CCSD(T)(F12) Explicitly Correlated Method Using the Example of a Water Dimer V. V. Belikova,b, D. A. Bokhana, and D. N. Trubnikova a
Department of Chemistry, Moscow State University, Moscow, 119991 Russia b Russian State Financial University, Moscow, 125933 Russia email:
[email protected] Received March 5, 2013
Abstract—Changes in the basis set superposition errors upon transitioning from conventional CCSD(T) to the CCSD(T)(F12) explicitly correlated method is studied using the example of a water dimer. A comparison of the compensation errors for CCSD(T) and CCSD(T)(F12) reveals a substantial reduction in the superpo sition error upon use of the latter. Numerical experiments with water dimers show it is possible theoretically predict an equilibrium distance between oxygen atoms that is similar to the experimental data (2.946 Å), as is the predicted energy of dissociation of a dimer (5.4 ± 0.7 kcal/mol). It is found that the structural and energy parameters of hydrogen bonds in water dimers can be calculated precisely even with twoexponential corre lationconsistent basis sets if we use the explicitly correlated approach and subsequently correct the basis set superposition error. Keywords: CCSD(T)(F12) explicitly correlated method, superposition of basis sets, errors, water dimer. DOI: 10.1134/S0036024414040037
INTRODUCTION Systematic studies of the equilibrium geometry of water dimers and their energy of dissociation began with Feller [1] at the MP2 level. In the works of Fey ereisen [2] and Xantheas [3, 4], MP2 calculations with five and sixexponential correlationconsistent Dun ning basis sets were performed [5–7]. Extrapolating the results to a complete basis set (CBS) yielded an estimate of the energy of dissociation (4.94 kcal/mol). The first study to use the MP2R12 explicitly corre lated method was conducted by Klopper [8]; the pre dicted energy of dissociation for a dimer was 4.94 kcal/mol. Both results are in good agreement with the experimental value of 5.4 ± 0.7 kcal/mol [9, 10]. Methods of a higher order (MP4, QCISD(T), and CCSD(T)) correct the energies of dissociation by only ~0.1 kcal/mol [1, 11, 12]. Detailed studies of the prop erties of water dimers by means of CCSD(T) were conducted by Halkier [13] and Klopper [8]; their value for the energy of dissociation was 5.0 kcal/mol. The improvement in the precision of the results was due to the use of extended molecular basis sets with subse quent extrapolation to the CBS. Theoretical studies of the molecular geometry of a water dimer have been described in detail in the litera ture. The predicted equilibrium distance between the oxygen atoms varies from 2.905 to 2.955 Å, while the complex has Cs symmetry [14–17]. The oxygen–oxy gen distance obtained experimentally is 2.976 Å; after allowing for anharmonic corrections, however, this value falls to 2.946 Å [18, 19]. Optimizing the geome
try of a water dimer is a difficult problem, due to the planar shape of the potential curve around the equilib rium point, and the correlation energy must be esti mated with a high degree of precision in order to per form it. Energies of correlation obtained with the use of the CCSD(T)(F12) explicitly correlated method [20–22] display rapid convergence along the angular moment of the basis set due to the inclusion of Slatertype hem inals. The CCSD(T)(F12) method could thus be a good tool for studying weakly bonded molecular com plexes. The use of small basis sets leads to the emer gence of basis set superposition error, due to the basis set of one fragment of the system being supplemented by functions centered on other fragments of the same system upon a reduction in the distance between frag ments. Detailed studies of the basis set superposition error for water dimers have been conducted by Simon [25] at the MP2 level and Halkier [13] by means of CCSD(T). Despite the relatively large distance between oxygen atoms, the effect of basis set superpo sition on the equilibrium geometry and energy of dis sociation was found to be remarkably strong. The aim of this work was to analyze the basis set superposition errors for the CCSD(T)(F12) explicitly correlated method in order to study the possibility of using small basis sets to describe weakly bound systems with hydrogen bonds.
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sequent subtraction of the obtained energies from the energies of monomers in their intrinsic basis sets:
Δ 1 2 3 4 5
0.003
CP
A
AB
Δ BSSE ( R A…B ) = ( E A – E A ( R A…B ) ) B
(3)
AB
+ ( E B – E B ( R A…B ) ). CP
Here, Δ BSSE (RA…B) is the compensation correction for
0.002
A
0.001
0
2.5
3.0
3.5
4.0
4.5 5.0 R(O–O), a.u.
Fig. 1. Compensation correction (Δ, a.u.) for a water dimer using different methods and basis sets: (1) CCSD(T)/augccpCVDZ, (2) CCSD(T)(F12)/augcc pCVDZ, (3) CCSD(T)/augccpCVTZ, (4) CCSD(T)(F12)/augccpCVTZ, and (5) SCF.
CP
Calculations were performed with the ACES II program [26]; the CCSD(T)(F12) version was used with the addition of cusp conditions, and was imple mented in B approximation [23]. Multielectron inte grals were calculated using pseudospectral numerical quadratures [27] and radial and angular arrays with 30 and 50 points. The Slater exponent was γ = 1.5. Expanded correlationconsistent Dunning basis sets augccPCVDZ and augccPCVTZ were also used [5–7]. The corresponding molecular geometry of each monomer in a water dimer was taken from [22] and fixed in all of our calculations. Angular parameters related to the mutual positions of monomers were also taken from [22], and only molecular distance was con sidered a parameter in this work. As was mentioned in [23], when there are twoexponential basis sets in CCSD(T)(F12), errors at the Hartrey–Fock level make the main contribution to the errors in measuring enthalpies of chemical reactions. To prevent such errors, we used Hartrey–Fock energies calculated in a threeexponential basis, while correlation energies (Ecorr) were calculated in two and threeexponential basis sets: + E corr ( augccpCVDZ ), E total ( tripleζ ) = E SCF ( augccpCVTZ ) + E corr ( augccpCVTZ ).
CP
E ( R A…B ) = E ( R A…B ) + Δ BSSE ( R A…B ). (4) With a water dimer, we have two nonequivalent frag ments, since proton donor and acceptor molecules have somewhat different geometries.
METHOD OF CALCULATION
E total ( doubleζ ) = E SCF ( augccpCVTZ )
B
a particular molecular distance RA…B; E A and E B are the energies of A and B fragments, calculated in their AB AB intrinsic basis sets; and E A (RA…B) and E B (RA…B) are the corresponding energies calculated in the basis set CP of the entire system. The Δ BSSE (RA…B) correction should thus be added to the energy of entire system in order to obtain the corrected potential curve:
(1)
(2)
Boys–Bernardi compensation corrections were used to prevent basis set superposition error [28]. To calculate the value of this correction for system AB, we must calculate the energies of individual fragments A and B using the basis set of the entire system with sub
RESULTS AND DISCUSSION Compensation Correction The compensation correction for potential ener gies calculated according to CCSD(T) and CCSD(T)(F12) is written as the function of distance between oxygen atoms. The corresponding results are given in Fig. 1. Owing to the experimental conver gence of the energy according to the Hartrey–Fock method relative to maximum angular moment of the basis set, the corresponding values of corrections are very small and drop drastically with an increase in molecular distance. The highest value of correction is observed for CCSD(T) in the twoexponential basis set. The corresponding explicitly correlated results show substantially lower values for the compensation correction, an additional consequence of directly incorporating electron distance into the structure of test wave functions. A similar tendency is observed for threeexponential basis sets with an overall drop in the magnitude of the error. The rate at which this error decreases with an increase in the oxygen–oxygen distance depends on the degree of diffusion (a value inversely proportional to the degree of decay) of the basis functions centered on each fragment. Since this diffusion grows with the maximum angular moment of the basis set that we use, the corresponding curves obtained by means of CCSD(T)(F12) and presented in Fig. 1 always decline more slowly than those of normal CCSD(T). The explicitly correlated approach simulates functions with high angular moments in the molecular basis set, reducing both the value of the superposition error and the rate at which it diminishes. The drop in the abso lute value of the error is always greater than the rate at which it diminishes and we always obtain better results
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ESTIMATING THE BASIS SET SUPERPOSITION ERROR IN THE CCSD(T)(F12)
when using the explicitly correlated approach. The values of the oxygen–oxygen equilibrium distance and the energy of dissociation of a water dimer with the necessary allowance for the above error are refined below.
152.64 (a)
152.65 152.66
1
152.67 152.68 ~ ~
Equilibrium Distance
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152.80 152.81 −Et, arb. units
Potential curves of a water dimer were calculated by means of bonded clusters, CCSD(T), and CCSD(T)(F12). The results for twoexponential basis sets, augccpCVDZ, and threeexponential augcc pCVTZ are given in Fig. 2. It is evident that for both methods and basis sets, the corresponding curves of potential energies have a planar bottom; the equilib rium geometry should therefore be very sensitive to the basis set we use and the necessary allowance for the basis set superposition error. Additional information on the values of the potential energy near equilibrium is given in Tables 1 and 2. The results confirm a very slow change in the potential curve in the range of 2.90 to 3.00 Å for all basis sets and methods. In the augcc pCVDZ basis set, CCSD(T) predicts very different molecular distances: we obtain 2.92 and 2.98 Å with and without corrections, respectively. Note that the uncorrected calculated values are closer to the experimental value (2.946 Å) than the corresponding corrected result. This is because the twoexponential basis set is insufficient for describing the correlation energy at CCSD(T) level, and the pres ence of basis functions centered on neighboring frag ments is equivalent to supplementing the basis set of a considered fragment with functions that have high angular moments and provide a better description of a supersystem at certain geometric parameters. In our CCSD(T)(F12) calculations, the predicted uncor rected distance between oxygen atoms was 2.91 Å, while subsequent compensation correction yielded a result of 2.93 Å. Introducing Slatertype heminals into the structure of our test wave functions allows simula tion of CCSD(T) calculations with basis functions that have high angular moments, reducing the basis set superposition error and weakening its effect. The above is of particular imporatnce because the Boys–Bernardi correction initially proposed for cor recting the energies of the Hartrey–Fock method, assumes low values of the superposition errors. The use of Boys–Bernardi corrections thus makes more sense when employing the CCSD(T)(F12) method. When there is a more complete threeexponential basis set, the difference between the uncorrected and corrected methods is smaller (2.91 as opposed to 2.93 Å for CCSD(T), and 2.91 as opposed to 2.92 Å for CCSD(T)(F12)). Correcting the superposition error thus yields more precise results for molecular distance in a water dimer, while the explicitly correlated method yields quite precise distances between frag ments even in a twoexponential basis set.
631
2
152.82 3 152.78
4
5
(b)
152.79 1
152.80 152.83 ~ ~ 152.84 152.85
2
152.86 3
4
5 R(O–O), Å
Fig. 2. Potential curve for a water dimer as a function of oxygen–oxygen distance, calculated in the (a) augcc pCVDZ and (b) augccpCVTZ basis sets: (1) CCSD(T), (2) CCSD(T)(F12). Et is the total energy.
Energies of Dissociation Values of the energy of dissociation obtained by dif ferent means and in different basis sets are given in Table 3. The corresponding experimental value is esti mated at 5.4 ± 0.7 kcal/mol. The strongest effect of the basis set superposition error is observed for the results of CCSD(T) in a twoexponential basis set in which the value of the energy of dissociation was intention ally raised by 0.78 kcal/mol. For comparison, the con tribution from the superposition error in the CCSD(T)(F12) results is ~0.25 kcal/mol, due to sub stantially lower dependence of the explicitly correlated results from the basis set. The use of rapidly diminish ing Slater correlation multipliers allows us to ignore the heminal superposition error [29]. When there is a threeexponential basis set, the contribution from the superposition error is 0.4 kcal/mol for CCSD(T) and 0.25 kcal/mol for CCSD(T)(F12). Note that CCSD(T)(F12) yields an estimate of the energy of dissociation within the boundaries of the measurement error for the experimental energy of dis Vol. 88
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Table 1. Energy of a water dimer (–E, a.u.), calculated in augccpCVDZ (I) without correction and (II) with cor rection R(O…O), Å
I
II CCSD(T)
I
II
CCSD(T)(F12)
2.90
152.67654 152.67523 152.82484 152.82443
2.91
152.67654 152.67525 152.82484 152.82444
2.92
152.67655 152.67527 152.82484 152.82444
2.93
152.67654 152.67529 152.82484 152.82444
2.94
152.67654 152.67530 152.82483 152.82444
2.95
152.67653 152.67531 152.82482 152.82444
3.00
152.67645 152.67531 152.82472 152.82437
ligibly small) contribution to the considered fragment. For relatively short covalent bonds, however, the effect of basis set superposition can be so great that correctly allowing for this error becomes critically important. The estimates of the energy of dissociation at 5 kcal/mol in [8] are more precise because of the high number of diffusion functions in the basis sets that were used. In order to improve the precision of calcu lations at the level of twoexponential basis sets in CCSD(T)(F12), we must introduce these additional sets of diffusion functions. ACKNOWLEDGMENTS The authors are grateful to Moscow State Univer sity Supercomputing Center for providing access to “SKIF” supercomputer. REFERENCES
Table 2. Energy of a water dimer (–E, a.u.), calculated in aug ccpCVTZ (I) without correction and (II) with correction R(O…O), Å
I
II CCSD(T)
I
II
CCSD(T)(F12)
2.90
152.80754 152.80687 152.86150 152.86108
2.91
152.80754 152.80689 152.86150 152.86109
2.92
152.80754 152.80689 152.86150 152.86109
2.93
152.80754 152.80690 152.86149 152.86109
2.94
152.80753 152.80689 152.86148 152.86108
2.95
152.80752 152.80689 152.86147 152.86107
3.00
152.80742 152.80683 152.86137 152.86098
Table 3. Energies of dissociation of a water dimer (in kcal/mol), calculated using different methods and basis sets augcc pCVDZ
augcc pCVTZ
CCSD(T), without correction
5.07
5.10
CCSD(T), with correction
4.29
4.69
CCSD(T)(F12), without correction
5.01
5.04
CCSD(T)(F12), with correction
4.76
4.78
Method
sociation even if we use twoexponential basis sets. The obtained results for a water dimer allow us to sug gest that similar situations could be characteristic of the most weakly bound clusters with hydrogen bonds, where the distance between fragments can be as great as several angstroms and the basis functions of neigh boring fragments make a very limited (though not neg
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