Article pubs.acs.org/JPCA
Application of Symmetry-Adapted Perturbation Theory to Small Ionic Systems Piotr Matczak* Department of Theoretical and Structural Chemistry, University of Łódź, Pomorska 163/165, 90-236 Łódź, Poland S Supporting Information *
ABSTRACT: The application of symmetry-adapted perturbation theory (SAPT) to small ionic systems was investigated in the context of the accuracy of calculated interaction energies for alkali halides. Two forms of alkali halides were considered: ion pairs M+X− (M = Li, Na, K, Rb, and X = F, Cl, Br, I) and dimers (MX)2. The influence of the order of energy correction terms included in SAPT and the effect of the so-called hybrid approach to SAPT on the accuracy of the calculated energies (such as the interaction energies in the ion pairs and the binding energies in the dimers with respect to two free monomers) were studied. The effects of the size of basis sets, combined with SAPT, on the accuracy were also established.
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INTRODUCTION Accurate determination of intermolecular (noncovalent) interactions is of great importance in physics, chemistry, and molecular biology because of the essential role of such interactions in the formation of a broad range of chemical systems, from the smallest possible aggregates of rare gases to supramolecules and crystals.1−5 From the computational point of view, modeling of intermolecular interactions may, however, be a serious challenge because many conventional and computationally inexpensive methods (e.g., density functional theory (DFT) methods) fail to predict an important part of intermolecular interactions, namely the dispersion forces.3 Unlike those conventional methods, symmetry-adapted perturbation theory (SAPT)6,7 proved to be successful in determining intermolecular interactions. SAPT represents intermolecular interaction energy as a series of energy correction terms that include various, and often very subtle, energetic effects required to reproduce intermolecular interactions accurately. In addition, the energy correction terms present in SAPT provide a means of the intuitive analysis of intermolecular interaction energy in terms of physically meaningful components. On the other hand, one must be aware of high computational cost of the regular wave function-based SAPT method (formally the scaling is O(N7)), as well as of the fact that the series of energy correction terms in SAPT was expected to be essentially divergent for many-electron systems.3,8,9 Fortunately, it does not prevent the SAPT method from reproducing interaction energies accurately for many systems with intermolecular interactions, but a careful monitoring of SAPT results is sometimes needed.8 More recent studies reviewed in ref 7 show that the use of the regularization technique makes the SAPT series of energy correction terms convergent. Significant work has also been performed on SAPT extensions, lowering the computational cost.10−12 The density-fitting (DF) approximation has been applied to the regular SAPT method, and the resulting DF-SAPT version12 exhibits an improved computa© 2012 American Chemical Society
tional efficiency: the DF approximation reduces the scaling of certain energy correction terms. For the dispersion energy correction term involving triple electron excitations, the nominal scaling is not changed but in this case the DF approximation removes excessive disk I/O operations, and therefore it improves the SAPT efficiency. It should be noted that, apart from the regular wave function-based SAPT method, there is also a formulation in which SAPT is combined with DFT. For the SAPT(DFT) method13,14 the computational cost scales as O(N6) and the application of the DF approximation to SAPT(DFT) reduces the scaling of the resulting DF-SAPT(DFT) version15,16 to O(N5). Recently, SAPT calculations have been carried out for systems in which intermolecular interactions had a strongly ionic character, and the corresponding interaction energies were equal or greater than dozens of kcal/mol. Zahn et al.17 investigated an ionic liquid in the form of the ion pair of 1,3dimethylimidazolium chloride in various conformations, and they calculated the SAPT interaction energy between the cation and the anion at the equilibrium and larger distances. SAPT was also used by Bankiewicz et al.18 to gain an insight into double charge-assisted hydrogen bonds in NH3+−H···Cl−, PH3+−H···Cl−, and NH3+−H···Br−. The SAPT interaction energies in these systems amounted to 116−123 kcal/mol at the equilibrium distances. These values are not too distant from those corresponding to typical ionic bonds, e.g., in alkali halide molecules. It is known that, in general, perturbation theory methods are unsatisfactory for describing strong interactions as the perturbation series converges slowly or even diverges and does not yield chemical accuracy. Thus, a question arises whether SAPT can be applied successfully to interactions between ions separated by the equilibrium distance and where Received: March 16, 2012 Revised: July 21, 2012 Published: August 6, 2012 8731
dx.doi.org/10.1021/jp302548u | J. Phys. Chem. A 2012, 116, 8731−8736
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Table 1. Basis Sets Used for Atoms in Alkali Halides description basis set a
BS I BS IIa BS IIIa BS IVb BS Vf
Li
Na
K
Rb
F
Cl
Br
I
def2-SVP def2-TZVPP def2-QZVPP def2-QZVPP+c def2-QZVPPD
def2-SVP def2-TZVPP def2-QZVPP def2-QZVPP+c def2-QZVPPD
def2-SVP def2-TZVPP def2-QZVPP def2-QZVPP+c def2-QZVPPD
SVPall TZVPPall TZVPPall TZVPPall+d TZVPPDall
def2-SVP def2-TZVPP def2-QZVPP def2-QZVPP+c def2-QZVPPD
def2-SVP def2-TZVPP def2-QZVPP def2-QZVPP+c def2-QZVPPD
def2-SVP def2-TZVPP def2-QZVPP def2-QZVPP+c def2-QZVPPD
SVPall TZVPPall TZVPPall TZVPPall+e TZVPPDall
a
Reference 21. bReferences 21−24. cdef2-QZVPP was supplemented by an additional sp diffuse function taken from the 6-311+G* basis set. TZVPPall was supplemented by an additional sp diffuse function with the exponent equal to 0.008. eTZVPPall was supplemented by an additional sp diffuse function with the exponent equal to 0.035. fReference 25. d
the interaction energy reaches a large value. The computational point of view usually reformulates such a question to the more technical one: “How accurate is SAPT for these interactions?” The present work focuses on the application of SAPT to small ionic systems in the context of the accuracy of the calculated interaction energies. Alkali halides seem to be the simplest yet most representative example of ionic systems, and alkali halides in two forms are investigated here: ion pairs M+X− (M = Li, Na, K, Rb, and X = F, Cl, Br, I) and dimers (MX)2. For the ion pairs, their SAPT interaction energies are compared with the experimental and CCSD(T) binding energies of monomers MX with respect to ions M+ and X−. In the case of the dimers, their binding energies with respect to two free monomers are obtained from the SAPT interaction energies of the dimers and the ion pairs. These binding energies are then juxtaposed with the available experimental values. The influence of the order of energy correction terms included in SAPT, the effect of the so-called hybrid approach to SAPT, and basis-set effects on the accuracy of the calculated interaction energies are discussed. The comparison of the SAPT results with the reference data allows us to establish the accuracy of SAPT for alkali halides quantitatively and, on this basis, to draw a more general conclusion about the potential application of SAPT to ionic systems.
[3] [3] 3 HF ESAPT + HF = ESAPT + δ E int ,resp
(4)
HF HF (10) (20) (20) (10) δE int ,resp = E int − Eelst − Eexch − E ind,resp − Eexch − ind,resp
(5) HF HF (10) (20) (20) (10) δ 3E int ,resp = E int − Eelst − Eexch − E ind,resp − Eexch − ind,resp (30) (30) − E ind − Eexch − ind
(6)
HF HF The δEint,resp and δ3Eint,resp terms are calculated as the subtraction of the correction energy terms of the zeroth order in the intramonomer correlation operator from the supermolecular interaction energy at the Hartree−Fock level of theory. These terms are viewed as approximately estimating higher-order induction effects and their exchange counterparts. Their inclusion in the resulting interaction energy was recommended for some systems20 because it improved the convergence of the SAPT series considerably.3 In this work, all SAPT interaction energies were calculated at [2] [3] the above-mentioned four levels, that is, ESAPT , ESAPT , [2] [3] ESAPT + HF, and ESAPT + HF, in the dimer-centered basis set scheme. The calculations were carried out using five mixtures of Ahlrichs’s all-electron basis sets. For brevity, these mixtures are marked from BS I to BS V, and Table 1 lists them. The SAPT2008.2 program19 was employed for all SAPT calculations. In order to establish the accuracy of SAPT, the interaction energies (E) between ion pairs M+ and X− were calculated for 16 alkali halides and then compared with the corresponding experimental results, that is, the binding energies in the MX monomers with respect to the corresponding ions M+ and X− (these reference energies are listed in Table S1, Supporting Information). Each pair of M+ and X− ions was separated by the distance equal to the experimental bond length of free monomer MX (the experimental bond lengths of 16 MX are reported in Table S1). The comparison of the calculated E values with the experimental data was made in a statistical manner, that is, the average absolute error (AAE) and the maximum absolute error (MAE) were evaluated for E in a series of 16 alkali halide ion pairs. In addition, the SAPT interaction energies were determined for ions separated by the distances obtained from CCSD(T)26 optimizations of MX. In this case, both the SAPT values of E and the CCSD(T) binding energies of MX relative to ions were compared with experiment. The implementation of CCSD(T) provided with GAMESS27,28 was employed. All details of the CCSD(T)
METHODOLOGY In SAPT, the interaction energy between two closed-shell atoms/molecules is expanded in a perturbative series of energy correction terms of increasing order. In practical calculations this series is truncated, and in this work the energy correction terms up to the second order and up to the third order with respect to intermolecular interaction operator were included, resulting in the SAPT interaction energies at the second[3] (E[2] SAPT) and third-order (ESAPT) levels, respectively. [2] (10) (1) (10) (12) (13) (CCSD) ESAPT = Eelst + Eelst,resp + Eelst,resp + Eexch + εexch (20) (20) t (22) t (22) + E ind,resp + Eexch − ind,resp + E ind + Eexch − ind
(1)
[3] [2] (30) (30) (30) (30) ESAPT = ESAPT + E ind + Eexch − ind + Edisp + Eexch − disp (30) (30) + E ind − disp + Eexch − ind − disp
(3)
where
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(20) (20) (2) + Edisp + Eexch − disp + εdisp(2)
[2] [2] HF ESAPT + HF = ESAPT + δE int ,resp
(2)
A detailed explanation of the energy correction terms on the rhs’s of eqs 1 and 2 is given in refs 6 and 19. One often uses the so-called hybrid (i.e., SAPT+HF) approach which can be defined for the second- and third-order SAPT interaction energies in the following way 8732
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differences among these BSs are observed for MAE(E[2] SAPT). In is influenced by the size of basis general, the accuracy of E[2] SAPT sets: the more the extended basis sets are used, the lower the values of the errors obtained. In the case of E[3] SAPT the BS I mixture of basis sets exhibits the AAE value close to those of BS II to BS V. Larger differences among BSs occur for MAE(E[3] SAPT) and then, surprisingly, the use of BS II and BS III leads to the largest MAE value. It is particularly important to establish the influence of the order of energy correction terms included in SAPT on the accuracy of interaction energy. When the errors for E[2] SAPT and E[3] SAPT are compared with one another, there are dramatic increases in the values of the errors for E[3] SAPT. These increases are observed irrespective of the BS applied. The large values of [3] [3] ) and MAE(ESAPT ) suggest that some energy AAE(ESAPT correction terms of the third order are quenched insufficiently by the corresponding exchange terms,29 and such a behavior may be a consequence of the divergent nature of the SAPT series of energy correction terms for the interaction in the investigated ion pairs. The hybrid approach to the SAPT interaction energies improves the accuracy of both second- and third-order energies. An exceptional behavior is presented only by BS I for E[2] SAPT + HF. The decreases in the values of the errors are particularly considerable for E[3] SAPT + HF. On this basis, the conclusion can be drawn that the hybrid approach is required for the reliable determination of the E values in the alkali halide ion pairs. However, this approach does not ensure that chemical accuracy is reached (AAE and MAE are still too large). The necessity of using the hybrid approach to SAPT seems not to be restricted to ion pairs. This approach performed well for charge-transfer complexes having a nearly dative bond30 (it is worth noting that the application of SAPT to strong charge-transfer complexes is undoubtedly nonstandard and so it is for ionic systems). Lao and Herbert29 have very recently reported that for induction-dominated systems, the hybrid approach is needed not only to obtain quantitative interaction energies but also to determine qualitatively correct potential energy surfaces. The necessity of including the δ3EHF int,resp term in third-order SAPT interaction energies has also been discussed in detail in the review paper by Hohenstein and Sherrill.31 These authors considered interaction energies for the S22 test set32 (22 small weakly bound complexes whose intermolecular binding exhibits electrostatic-, dispersion-dominated, or mixed character), and they came to the conclusion that the δ3EHF int,resp term should be included for the entire S22 test set when a small basis set is applied. With HF should only be included for larger basis sets, δ3Eint,resp electrostatic-dominated complexes. Our observations for the ion pairs are in line with this conclusion. The application of the hybrid approach is accompanied by a basis-set effect in the values of AAE: the change from the splitvalence (BS I) to the quadruple-ζ valence (BS III) basis sets results in an increase in the accuracy of the calculated E values. In the case of AAE(E[3] SAPT + HF), this effect also covers the sets of diffuse functions in BS IV and BS V. Some basis-set effect is evident when the accuracy of E[2] SAPT + HF is compared with that of E[3] SAPT + HF: for the standard BSs, that is, BS I to BS III, the E[2] SAPT + HF level of SAPT turns out to be superior, whereas the use of BSs augmented with diffuse functions is accompanied by a slight preference for the E[3] SAPT + HF level. For the completeness of the investigation presented in this work, it is necessary to collate the accuracy of SAPT with that
calculations can be found in section S2, Supporting Information. For the (MX)2 dimers, their binding energies (ΔE) relative to two free monomers were determined, making use of SAPT interaction energies (E) via ΔE = 2[E(M+X−) − E(M+X− in (MX)2 )] − E((MX)(MX))
(7)
where the terms in the square bracket denote the interaction energies between two ions at the experimental distances characteristic of the free monomer and the dimer, respectively, and the last term on the rhs is the interaction energy between two MX monomers in the (MX)2 dimer. The square bracket represents a change in the interaction energy in the MX monomer, resulting from its bond length relaxation (the deformation energy). Hence, ΔE indicates the binding energy in a dimer with respect to two free monomers. The dimers were considered in their experimental equilibrium geometries, that is, in the form of planar rings with the D2h symmetry (see Table S2, Supporting Information for the experimental values of bond lengths and angles). Like the calculated E values for the ion pairs, the SAPT energies ΔE were compared with the available experimental results, and the accuracy of SAPT was assessed by means of AAE(ΔE) and MAE(ΔE).
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RESULTS AND DISCUSSION Interaction Energies in the Ion Pairs. For the pairs of the M+ and X− ions, separated by the distances equal to the experimental bond lengths in the corresponding free MX monomers, their interaction energies were calculated at the four levels of SAPT and using BS I to BS V. Subsequently, the E values were compared with the experimental binding energies of the MX monomers. The resulting AAE(E) and MAE(E) values are shown in Table 2. Table 2. Average Absolute Errors (AAE) and Maximum Absolute Errors (MAE) in the SAPT Interaction Energies (E) between Ions in 16 Pairs of Alkali Halides with Respect to the Experimental Binding Energies of the Alkali Halide Monomersa E[2] SAPT basis set BS BS BS BS BS
I II III IV V
E[3] SAPT
E[2] SAPT + HF
E[3] SAPT + HF
AAE
MAE
AAE
MAE
AAE
MAE
AAE
MAE
3.5 2.1 2.1 2.0 1.7
12.0 7.5 6.4 6.2 6.0
16.3 16.8 17.1 15.9 16.0
37.5 44.1 44.1 37.8 38.2
5.0 2.0 1.6 1.9 1.7
19.9 5.0 4.5 5.5 5.2
5.3 2.3 2.0 1.8 1.6
21.7 5.8 4.9 5.5 5.1
a
The distances between ions in the ion pairs were equal to the experimental bond lengths of the alkali halide monomers. All values in kcal/mol.
Let us first focus only on the results obtained by the secondorder level of SAPT combined with various BSs. An examination of the errors in the E[2] SAPT values in Table 2 indicates that this level of SAPT combined with BS I yields results with the poorest accuracy: both errors are much larger than those for the remaining BSs. It is evident that the most significant increase in the errors occurs when one goes from BS V to BS I. The latter gives the errors twice as large as the former. BS II to BS V exhibit similar values of AAE(E[2] SAPT) that are close or equal to 2 kcal/mol. Slightly more significant 8733
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Table 3. Average Absolute Errors (AAE) and Maximum Absolute Errors (MAE) in the CCSD(T) Binding Energies and in the SAPT Interaction Energies (E) between Ions in 16 Pairs of Alkali Halides with Respect to the Experimental Binding Energies of the Alkali Halide Monomersa E[2] SAPT
CCSD(T) basis set BS BS BS BS BS a
I II III IV V
E[3] SAPT
E[3] SAPT + HF
AAE
MAE
AAE
MAE
AAE
MAE
AAE
MAE
AAE
MAE
6.8 2.8 2.2 2.7 2.6
25.7 7.3 7.1 8.3 6.6
4.1 2.2 2.2 2.2 1.9
14.6 7.3 6.1 6.0 5.8
10.5 12.8 14.3 12.9 13.2
18.0 30.2 30.8 28.6 30.0
4.7 2.0 1.6 1.9 1.9
19.0 5.1 4.6 5.4 5.0
4.9 2.3 2.0 1.8 1.7
20.1 5.9 5.0 5.6 4.9
The distances between ions in the ion pairs were equal to the CCSD(T) bond lengths of the alkali halide monomers. All values in kcal/mol.
greater accuracy than ΔE[3] SAPT + HF, but the differences between the values of AAE and MAE obtained from ΔE[2] SAPT + HF and do not exceed 0.7 kcal/mol. ΔE[3] SAPT + HF The comparison of the accuracy of the ΔE values with that of the E values reveals that the accuracy of SAPT is strongly dependent on the BS applied. For the BS I mixture of basis sets, the accuracies of the second-order and two hybrid levels of SAPT are improved when the ΔE values of the alkali halide dimers are calculated. The smaller values of the errors may be explained in terms of some fortuitous cancelation of errors in the subtraction occurring in eq 7. However, such a cancelation of errors does not take place for more extended basis sets, that is, BS II to BS V. Then, the AAE(ΔE) errors increase by 0.3− 3.0 kcal/mol compared to the corresponding AAE(E) values. Some fortuitous cancelations of errors were previously observed10,34 when SAPT (especially at very simplified levels not considered here) was combined with relatively poor Dunning’s basis sets. The hybrid approach, both in the [2] [3] ΔESAPT + HF and ΔESAPT + HF forms, exhibits the smallest change in accuracy when the size of alkali halide systems grows. Correlation with Trends in Alkali Halide Series. It is sometimes useful to group SAPT energy correction terms into a few fundamental components that can be interpreted in terms of such physically meaningful contributions to interaction energy as electrostatics, induction, dispersion, and exchangerepulsion.7 It is not unreasonable to expect that some of these contributions or certain combinations of them may correlate with parameters describing some trends in the alkali halide series. It is obvious that the electrostatic contribution dominates for the ion pairs of alkali halides. Within a crude approximation, this contribution can be associated with the difference of electronegativities of atoms in MX because, in general, a large electronegativity difference is a very simple indicator of ionic character of a bond. In Figure 1 the ratio Eelst/ E[2] SAPT + HF is plotted against electronegativity differences (Δχ) according to Pauling’s scale. The ion pairs that contain the same alkali or halogen ion are connected in this figure. The electrostatic component (Eelst) collects three energy correction (12) (13) terms E(10) elst , Eelst,resp, and Eelst,resp. It should be noted that Eelst [2] and ESAPT + HF obtained from BS III were used. For all series of M+X− pairs sharing the same ion, the relationship between Eelst/E[2] SAPT + HF and Δχ is far from being linear, but the existence of a quite symmetric pattern in the figure suggests that these two quantities are related to one another.
of another advanced and computationally expensive method, namely CCSD(T). The resulting AAE and MAE are given in Table 3. For E[2] SAPT and both hybrid levels of SAPT, their AAE(E) and MAE(E) values are equal or, more often, smaller than the corresponding values for CCSD(T). This tendency is valid for all BSs. For BS II to BS V, the differences in AAE between CCSD(T) and the three levels of SAPT are very small (
