PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics
Reliable structures and energetics for two new delocalized p p prototypes: cyanogen dimer and diacetylene dimerwz
Brian W. Hopkins, Adel M. ElSohly and Gregory S. Tschumper*
Received 22nd November 2006, Accepted 3rd January 2007 First published as an Advance Article on the web 7th February 2007 DOI: 10.1039/b616878g
Two new prototype delocalized p p complexes are introduced: the dimers of cyanogen,
(NRC–CRN)2, and diacetylene, (HCRC–CRCH)2. These dimers have properties similar to larger delocalized p p systems such as benzene dimer but are small enough that they can be probed in far greater detail with high accuracy electronic structure methods. Parallel-slipped and T-shaped structures of both cyanogen dimer and diacetylene dimer have been optimized with 15 different procedures. The effects of basis set size, theoretical method, counterpoise correction, and the rigid monomer approximation on the structure and energetics of each dimer have been examined. MP2 and CCSD(T) optimized geometries for all four dimer structures are reported, as well as estimates of the CCSD(T) complete basis set (CBS) interaction energy for every optimized geometry. The data reported here suggest that future optimizations of delocalized p p clusters should be carried out with basis sets of triple-z quality. Larger basis sets and the expensive counterpoise correction to the molecular geometry are not necessary. The rigid monomer approximation has very little effect on structure and energetics of these dimers and may be used without consequence. Due to a consistent cancellation of errors, optimization with the MP2 method leads to CCSD(T)/CBS interaction energies that are within 0.2 kcal mol 1 of those for structures optimized with the CCSD(T) method. Future studies that aim to resolve structures separated by a few tenths of a kcal mol 1 should consider the effects of optimization with the CCSD(T) method.
1. Introduction dimer, is much too large to study at this level of detail. Hence, much work has been carried out to develop less demanding Complexes of molecules with delocalized (conjugated) p elec- electronic structure techniques that can accurately describe tron clouds are exceptionally difficult to study even in com- this type of non-covalent interaction.12–14 If we are to gain an 1–10 parison with other van der Waals complexes. In fact, a understanding of the detailed physics of conjugated p p recent invited article in this journal presented a benchmark systems, we will need a newer, smaller prototype system that database of weakly bound complexes and highlighted the need can be examined in more detail than is possible for benzene to move beyond the MP2 level of theory when the dispersion dimer. 9 contribution to binding becomes significant. Consider, for Cyanogen (NRC–CRN) is one of the smallest known example, the dimers listed in Table 1. MP2 and CCSD(T) closed-shell, neutral, conjugated molecules. As such, its dimer interaction energies are quite similar for (N2)2 and (C2H2)2 is an ideal prototype for conjugated p p interactions. Un- while MP2 tends to substantially overestimate the attractive fortunately, little is known about the (NRC–CRN)2 system. forces of p p stacking in complexes with delocalized p The first theoretical work on cyanogen dimer dates to 1984, electron clouds. Recent work has suggested that highly accu- when Hasanein and Evans studied nine dimer structures at the rate studies of these complexes require correlated methods that HF level of theory and with an empirical atom–atom potential 11 include the effects of quadruple excitations. For instance, the method.15 Unfortunately, the HF level of theory was demon- inclusion of perturbative approximations to quadruple excita- strably inadequate for describing the system; five of the nine tions increases the binding energy in the cyanogen dimer HF potential curves plotted by Hasanein and Evans were parallel-slipped and T-shaped configurations by 0.10 and purely repulsive. Because dispersion forces are extremely im- 1 11 0.07 kcal mol , respectively. This represents a considerable portant in the binding of p-type dimers, MP2 theory may be obstacle since the classic prototype p p system, benzene considered the lowest level of theory that can reasonably be expected to describe their structure and energetics. For this Department of Chemistry and Biochemistry, University of Mississippi, reason, neither Hartree Fock nor density functional methods University, MS 38677-1848, USA. E-mail: [email protected] have been employed in this study. w The HTML version of this article has been enhanced with colour In 1991, molecular beam electric resonance spectroscopy images. 16 z Electronic supplementary information (ESI) available: Cartesian performed by Klemperer and coworkers indicated a coordinates for all optimized structures. See DOI: 10.1039/b616878g T-shaped (NRC–CRN)2 structure in C2v symmetry. No
1550 | Phys.Chem.Chem.Phys., 2007, 9, 1550–1558 This journal is c the Owner Societies 2007 Table 1 Results of previous studies that demonstrate the difference metry; a T-shaped transition state in C2v symmetry; and a 27 between MP2 and CCSD(T) p p stacking interaction energies in stacked second-order saddle point in D2h symmetry. In the parallel-slipped systems with and without delocalized p electron same work, Karpfen examined the acetylene dimer in con- clouds. All energies are in kcal mol 1 siderably more detail; this included performing optimizations Interaction energies with larger basis sets and computing single-point interaction energies at the MP4, CCSD, and CCSD(T) levels of theory. System MP2 CCSD(T) These calculations on acetylene dimer showed virtually no a Nitrogen dimer 0.58 0.51 change in the dimer structure with increasing basis set size, and Acetylene dimera 1.99 1.72 b only minute changes in the interaction energy of (C H ) were Benzene dimer 4.95 2.78 2 2 2 Pyrrole dimerc 0.63 þ0.45 observed as more sophisticated treatments of electron correla- Pyrimidine dimerc 3.87 2.64 tion were applied. Karpfen therefore concluded that the more c Triazine dimer 3.77 2.79 rigorous theoretical treatment was not necessary for the study a Ref. 11. b Ref. 4. c Ref. 3. of the larger diacetylene complex. These results, however, are not really applicable to the dimer of diacetylene. As seen in Table 1, interactions between molecules with delocalized or conjugated p electron clouds are fundamentally different from other structures were observed in the experiment. Subse- those in small p p prototypes like acetylene dimer or the quently the group of de Almeida examined T-shaped, linear, dimer of molecular nitrogen. MP2 calculations are known to and parallel structures at the HF level of theory.17 Important give accurate results for simple p p systems.3,11 For con- nonplanar structures (including some of those studied by jugated systems, however, the results of MP2 calculations are Hasanein and Evans) were neglected entirely by de Almeida erroneous, often overstating the binding energy of conjugated et al., and the work went on to conclude that the T-shaped p p complexes by as much as 100%.1,3,9,28,29 structure was the global minimum on the dimer potential Despite its well known tendency to overestimate the stability energy surface. of stacked, delocalized p p complexes, the MP2 method Interestingly, despite the similarity of their methods, the HF continues to be used for geometry optimizations of dimers results of the de Almeida and Evans studies disagree with bound by delocalized p p interactions. The effect of using respect to the nature of the end-to-end linear structure. Using MP2 optimized geometries on the energetics of these types of the 6-31G basis, Evans’s HF calculations show a purely dimers is not well known, as most delocalized p p complexes repulsive potential curve as the molecules approach end-to- are much too large to optimize with more accurate theoretical end. By contrast, de Almeida’s HF/4-31G calculations do methods. Only recently has this effect been examined when identify an end-to-end linear stationary point on the dimer Sherill and co-workers estimated CCSD(T) optimized struc- surface. This disagreement may be due to the rigid monomer tures of the benzene dimer by using MP2 potential energy approximation, which was employed by Evans but not by de curves that had been corrected for the difference between MP2 Almeida. Such a result is notable in that it highlights the and CCSD(T) interaction energies and found that changes to sensitivity of the dimer surface to subtle geometrical effects; their best estimates of the interaction energies were on the because the minima are very shallow, even slight changes in order of 0.2 kcal mol 1.7 The fact remains, however, that it is theoretical methods can qualitatively change the nature of the currently unknown exactly which theoretical methods are potential energy surface (PES). adequate to study the shape of such a surface. The thrust of The dimer of diacetylene (HCRC–CRCH) has been this work aims to determine how the energetics of delocalized studied somewhat more thoroughly.18 Diacetylene is the smal- p p systems are affected by geometrical perturbations intro- lest of the polyynes thought to be of value as molecular ‘‘rods’’ duced when popular approximations are implemented during in the construction of nanoscale molecular machines.19 The optimization procedures (e.g., smaller basis set, lower theore- interaction between diacetylene units has already been used in tical method, counterpoise correction, rigid monomer approx- the construction of a ‘‘molecular zipper’’ by Shu et al.20 In imation). Having identified the most reliable procedures for addition to their importance in the construction of nano- the characterization of p p interactions, detailed character- materials, both polyacetylenes and cyanopolyacetylenes are izations of the PESs of these new prototypes will soon be known to be present in extraterrestrial space.21–25 These presented. This work will greatly assist the development of less molecules have been spectroscopically observed in the atmo- demanding computational methods for p p interactions sphere of Saturn’s largest moon, Titan,21–24 where they are (e.g., force fields, density functional methods, semi-empirical thought to be formed in the upper polar stratosphere and to methods). play an important role in various photochemical processes.24 They are also thought to exist in the atmospheres of the giant 2. Theoretical methods planets that comprise most of our solar system,25 and have been observed in the envelope of gases that surrounds the solar The stationary points selected for study here are the very low system beyond the outermost planetary orbits.25,26 lying and high symmetry parallel-slipped and T-shaped struc- Karpfen examined the surface of the diacetylene dimer at tures (see Fig. 1). These structures are analogous to the the MP2 level and identified six stationary points: a Y-shaped parallel-slipped and T-shaped structures known to be impor- minimum in Cs symmetry; two distinct parallel-slipped mini- tant on the potential energy surfaces of benzene dimer and 7 ma in C2h symmetry; a cross-shaped minimum in D2d sym- other aromatic complexes. These two points on each dimer
This journal is c the Owner Societies 2007 Phys. Chem. Chem. Phys., 2007, 9, 1550–1558 | 1551 Fig. 1 The intermolecular geometrical parameters reported for the parallel-slipped (left) and T-shaped (right) structures of (HCRC–CRCH)2 (top) and (NRC–CRN)2 (bottom). See text for precise definitions of R1 and R2.
surface have been subjected to substantial scrutiny in order to of basis sets (cc-pVXZ for H and aug-cc-pVXZ for N and C). determine the precise theoretical treatment necessary to accu- The a5Z basis set is also used in this work but only for single rately describe the structure of each dimer. Broadly, we have point energy computations. In addition to the aXZ basis sets, considered the effects of theoretical method, basis set, counter- we have employed both the DZPþþ and TZ2P(f,d)þþ basis poise correction, and the rigid monomer approximation on the sets. These are the standard Huzinaga–Dunning DZ and TZ overall structure and energetics of both the parallel-slipped basis sets35,36 augmented with polarization functions as well as and T-shaped structures of the cyanogen and diacetylene with one set of even-tempered diffuse functions37 on all atoms. dimers. In order to quantify the effects of all of these factors For consistency, we use the terms aXZ, DZPþþ, and on the structure of each dimer, we report the intermolecular TZ2P(f,d)þþ to describe the basis sets used for both cyanogen geometrical parameters of the dimers. For T-shaped structures and diacetylene dimer. It is important to note, however, that
(see Fig. 1), there is only one such parameter, R1, that indicates cyanogen contains no hydrogen atoms, and in such cases the the distance between the center of mass of the cross of the T aXZ, DZPþþ and TZ2P(f,d)þþ basis sets are simply the and the proximal atom on the leg of the T. For the parallel- aug-cc-pVXZ, the DZPþ, and the TZ2P(f)þ basis sets, re- slipped structures, there are two relevant parameters: the spectively. 38 distance R1 between the two lines containing the C–C single Basis set superposition error is well known to be a concern bonds in the monomers and the parallel distance R2 of slip in weakly bound clusters such as those studied here. The between the midpoints of the C–C single bonds of the two counterpoise correction39 of Boys and Bernardi40 is a popular monomers. method for correcting this error and has been incorporated The parallel-slipped and T-shaped structures of both into energy derivatives.41 It is therefore possible to compute a
(NRC–CRN)2 and (HCRC–CRCH)2 have been opti- CP-corrected optimized geometry in addition to a CP-cor- mized at the MP2 and CCSD(T)30–34 levels of theory in rected energy. The differences between the CP-corrected and conjunction with a series of double-, triple-, and quadruple-x uncorrected geometries can be used to quantify the importance basis sets. Three of the basis sets, denoted aXZ where X ¼ D, of BSSE in the geometry of the dimer. For this reason, we have T, Q, are taken from Dunning’s correlation consistent family computed CP-corrected geometries at the MP2 level with the
1552 | Phys.Chem.Chem.Phys., 2007, 9, 1550–1558 This journal is c the Owner Societies 2007 aDz, aTz, DZPþþ and TZ2P(f,d)þþ basis sets for both the Table 2 Important intermolecular geometrical parameters for all parallel-slipped and T-shaped structures. optimized structures. Definitions of R1 and R2 for the parallel-slipped and T-shaped dimer structures are shown in Fig. 1. All distances are Due to the small magnitude of p p interactions, it is in A˚ commonly assumed that the these intermolecular forces have negligible effects on the intramolecular geometries of mono- Optimization level mers in p p complexes, and many theoretical studies of such CP CP RM RM Method Basis R1 R2 R1 R2 R1 R2 systems very frequently employ the rigid monomer approx- Parallel-slipped (NRC–CRN) imation. In order to assess the quality of this approximation, 2 MP2 aDZ 3.183 2.697 3.268 2.848 we have optimized the T-shaped and parallel-slipped cyanogen MP2 aTZ 3.157 2.770 3.210 2.758 and diacetylene dimer structures within the rigid monomer MP2 aQZ 3.158 2.766 (RM) approximation in addition to performing full geometry MP2 DZPþþ 3.233 2.745 3.375 3.074 3.229 2.751 MP2 TZ2P(f,d)þþ 3.215 2.818 3.253 2.899 3.208 2.833 optimizations. CCSD(T) DZPþþ 3.332 2.855 3.327 2.828 For each optimized structure, an estimate of the CCSD(T) CCSD(T) TZ2P(f,d)þþ 3.354 2.836 3.334 2.984 complete basis set interaction energy (EBestEst.) was determined INT R R to gauge the impact of the various optimization procedures on T-shaped (N C–C N)2 MP2 aDZ 3.000 3.103 p p stacking energies. This best estimate was constructed by MP2 aTZ 2.999 3.051 MP2 combining the MP2 interaction energy at the CBS limit (EINT ) MP2 aQZ 3.001 with the difference between MP2 and CCSD(T) interaction MP2 DZPþþ 3.043 3.222 3.052 CCSD(T) MP2 TZ2P(f,d)þþ 3.054 3.097 3.061 energies computed with the aTZ basis set (dMP2 ). The MP2 CCSD(T) DZPþþ 3.118 3.116 CBS limit was generated via the two point (aQZ/a5Z) extra- CCSD(T) TZ2P(f,d)þþ 3.143 3.152 42 BestEst. polation of Helgaker et al. EINT is used to judge the R R reliability of each optimization procedure and to generate Parallel-slipped (HC C–C CH)2 MP2 aDZ 3.359 1.694 3.525 1.780 guidelines for the optimization methods needed to obtain MP2 aTZ 3.361 1.787 3.399 1.762 quantitatively accurate structures and energetics for similar MP2 aQZ 3.360 1.786 complexes in the future. Note, however, that CP corrections MP2 DZPþþ 3.394 1.882 3.703 2.016 3.388 1.879 MP2 TZ2P(f,d)þþ 3.428 1.792 3.506 1.848 3.419 1.796 have not been applied to the interaction energies because the CCSD(T) DZP 3.516 2.004 3.512 1.989 CCSD(T) þþ dMP2 term converges very rapidly with respect to the AO CCSD(T) TZ2P(f,d)þþ 3.603 1.900 3.597 1.900 basis set and BSSE tends to be negligible for the expansive R R aQZ and a5Z basis sets. T-shaped (HC C–C CH)2 MP2 aDZ 2.570 2.707 Calculations for this work were carried out with a variety of MP2 aTZ 2.581 2.644 quantum chemistry software packages. MP2 optimizations MP2 aQZ 2.586 were performed using the analytic gradients available in the MP2 DZPþþ 2.623 2.845 2.593 Gaussian 03 suite of programs.43 CCSD(T) optimizations were MP2 TZ2P(f,d)þþ 2.611 2.685 2.615 CCSD(T) DZPþþ 2.648 2.647 carried out with the analytic gradients in the ACES II pack- CCSD(T) TZ2P(f,d)þþ 2.694 2.700 age.44 Geometries were considered converged when Cartesian 5 1 45 46 gradients fell below 1 10 Eh a0 . MPQC and PSI3 were used for most MP2 and CCSD(T) single point energy 7 calculations, and all energies were converged to 1 10 Eh. T-shaped dimers appear to be somewhat less sensitive to the All calculations (optimizations and single point energies) optimization procedures than the parallel-slipped structures. incorporated the frozen-core approximation. Interestingly, most of the monomers in the fully optimized dimers shown in Fig. 1 are distorted ever so slighlty from linearity (by E11). Nevertheless, all of the T-shaped structures
3. Results and discussion belong to the C2v point group. The monomer forming the top of the T bows slightly away from the monomer forming the leg 3.1 Dimer structures of the T in the cyanogen dimer, while it bows toward the leg of The T-shaped and parallel-slipped dimer structures have been the T in the diacetylene dimer. Likewise, all of the parallel- optimized at the MP2 level of theory with five different basis slipped structures identified still belong to the C2h point group sets (aDZ, aTZ, aQZ, DZPþþ, and TZ2P(f,d)þþ). Despite despite the fact that the ‘‘free’’ terminal atoms of the mono- the fact that the dimers of cyanogen and diacetylene are mers bow slightly away from each other while the other substantially smaller than benzene dimer, optimizations with terminal atoms of the monomers bow toward each other. the CCSD(T) method were not feasible with the large aTZ and Cartesian coordinates for all optimized structures can be aQZ basis sets. Additionally, CCSD(T) optimizations were not found in the ESI.z performed with the aDZ basis set due to peculiar discrepancies Optimizations of the diacetylene monomer warrant special with the diacetylene monomer (vide infra). attention. Both the aDZ basis used here and the parent fully Although the intermonomer geometrical parameters de- augmented aug-cc-pVDZ basis set have difficulty describing picted in Fig. 1 and listed in Table 2 can change by a few the nature of the diacetylene monomer. MP2 and CCSD(T) tenths of an A˚ with the various optimization procedures, the calculations using these basis sets erroneously indicate that the optimized structures are energetically quite similar given the linear structure of the diacetylene molecule has a doubly extremely flat nature of the potential energy surfaces. The degenerate imaginary vibrational frequency of 105i cm 1
This journal is c the Owner Societies 2007 Phys. Chem. Chem. Phys., 2007, 9, 1550–1558 | 1553 and 186i cm 1, respectively, that corresponds to the two consistent basis sets has been observed elsewhere for other degenerate bending modes that move the H atoms into a trans weakly bound systems.48–51 configuration. This result is reminiscent of the spurious ima- ginary frequencies recently reported for MP2 optimized struc- tures of benzene and other arenes.47 The diacetylene situation 3.1.4 Effect of the rigid monomer approximation on struc- is unique, however, insofar as the spurious frequencies arise ture. Despite the fact that the monomers are not perfectly linear in the fully optimized dimers, the rigid monomer here with a correlation consistent basis set. Previous examples approximation has almost no effect on the intermonomer of this phenomenon were explained by ‘‘an insidious basis set geometrical parameters. Comparing the rigid monomer para- incompleteness error that varies strongly with [out-of-plane] meters (RRM and RRM) in Table 2 to the fully optimized values geometric distortions’’ specific to the split valence basis sets 1 2 (R and R ), one sees that the parameters typically change by (e.g., 6-311G). The correlation-consistent basis sets were de- 1 2 less than 0.01 A˚ . scribed by Moran et al. as a remedy to this problem; that the problem seems to be occurring in these basis sets is thus a matter demanding further investigation. 3.2 Dimer energetics Having discussed the effects of the various optimization 3.1.1 Effect of basis set on structure. It is clear from the procedures on the structures of the dimers, we can now MP2/aXZ data in Table 2 that the intermolecular parameters examine how the p p interaction energies are affected by R1 and R2 converge very quickly with respect to the cardinal these geometrical changes. Such effects must be carefully number of the basis set (X ¼ D, T, Q). While the aDZ and aTZ dissected because the typical procedure for computing high- ˚ BestEst. geometrical parameters may differ by as much as 0.093 A, accuracy p p interaction energies (EINT ) involves combin- MP2 results obtained with aTZ basis set are nearly identical to those ing large basis set MP2 computations (EINT ) with corrections CCSD(T) 9,52 from the large aQZ basis set. The aTZ and aQZ results differ from a small basis set CCSD(T) computation (dMP2 ). by only a few thousandths of an A˚ which is quite remarkable Consequently, the three energetic parameters associated with given the extraordinarily flat nature of the PES along R1 and R2. The data in Table 2 also show slightly larger MP2 inter- Table 3 Effects of basis set and correlation treatment used for monomer parameters for the fairly compact DZPþþ and optimization on the MP2 and CCSD(T) interaction energies of the 1 TZ2P(f,d)þþ basis sets relative to results obtained with the dimers of cyanogen and diacetylene. All energies are in kcal mol much larger aTZ and aQZ correlation consistent basis sets. Optimization level For example, R1 for the MP2/TZ2P(f,d)þþ optimized struc- MP2 CCSD(T) BestEst. tures is, on average, only 0.053 A˚ longer than in the MP2/aTZ Method Basis EINT dMP2 EINT optimized structures. Parallel-slipped (NRC–CRN)2 MP2 aDZ 2.55 þ1.02 1.53 3.1.2 Effect of electron correlation on structure. Electron MP2 aTZ 2.51 þ0.99 1.52 MP2 aQZ 2.51 þ0.96 1.55 correlation generally has a larger effect on the structure of MP2 DZPþþ 2.52 þ0.92 1.60 these p p dimers than does the basis set. CCSD(T) inter- MP2 TZ2P(f,d)þþ 2.50 þ0.87 1.63 CCSD(T) DZPþþ 2.41 þ0.68 1.73 monomer parameters (R1 and R2 in Table 2) obtained with the CCSD(T) TZ2P(f,d)þþ 2.33 þ0.62 1.72 DZPþþ and TZ2P(f,d)þþ basis sets are on the order of 0.1 A˚ longer than the corresponding MP2 values. Interestingly, the T-shaped (NRC–CRN)2 differences between the MP2 and CCSD(T) intermonomer MP2 aDZ 2.47 þ0.66 1.81 parameters tend to be more significant with the TZ2P(f,d)þþ MP2 aTZ 2.44 þ0.62 1.81 MP2 aQZ 2.44 þ0.62 1.81 basis set than with the DZPþþ basis set. MP2 DZPþþ 2.47 þ0.62 1.85 MP2 TZ2P(f,d)þþ 2.43 þ0.58 1.86 3.1.3 Effect of counterpoise correction on structure. Exam- CCSD(T) DZPþþ 2.42 þ0.50 1.92 CP CCSD(T) TZ2P(f,d)þþ 2.36 þ0.46 1.91 ining the CP corrected intermonomer parameters (R1 and CP R2 ) in Table 2, one sees rather large changes when a small Parallel-slipped (HCRC–CRCH)2 double-z quality basis set is employed. With the DZPþþ basis MP2 aDZ 2.41 þ1.45 0.97 set, the CP correction increases the monomer separation and MP2 aTZ 2.49 þ1.28 1.21 ˚ MP2 aQZ 2.48 þ1.26 1.22 the slip by well over 0.1 A in every case and by as much as MP2 DZPþþ 2.52 þ1.25 1.27 0.3 A˚ in two cases. However, the effect of the CP correction MP2 TZ2P(f,d)þþ 2.46 þ1.14 1.32 quickly decreases when a larger basis set is used. CP correc- CCSD(T) DZPþþ 2.39 þ0.96 1.43 CCSD(T) TZ2P(f,d) 2.21 0.82 1.39 tions with the TZ2P(f,d)þþ and aTZ basis sets increase the þþ þ ˚ intermonomer parameters by only 0.062 and 0.041 A on T-shaped (HCRC–CRCH)2 average, respectively. Interestingly, the MP2/TZ2P(f,d)þþ MP2 aDZ 1.98 þ0.36 1.61 optimized parameters are quite similar to the MP2/aTZ data. MP2 aTZ 1.94 þ0.34 1.59 ˚ MP2 aQZ 1.94 þ0.34 1.60 For example, the R1 values differ by only 0.018 A on average. MP2 DZPþþ 1.97 þ0.33 1.65 This similarity between optimized geometrical paramerters MP2 TZ2P(f,d)þþ 1.94 þ0.32 1.61 obtained with the TZ2P(f,d)þþ basis set and those from CCSD(T) DZPþþ 2.01 þ0.30 1.71 counterpoise corrected optimizations with large correlation CCSD(T) TZ2P(f,d)þþ 1.91 þ0.27 1.64
1554 | Phys.Chem.Chem.Phys., 2007, 9, 1550–1558 This journal is c the Owner Societies 2007 Table 4 Effects of counterpoise corrected optimization on the MP2 and CCSD(T) interaction energies of the dimers of cyanogen and diacetylene. All energies are in kcal mol 1
Optimization level Non-CP corrected CP corrected
MP2 CCSD(T) BestEst. MP2 CCSD(T) BestEst. CP Method Basis EINT dMP2 EINT EINT dMP2 EINT DEINT
Parallel-slipped (NRC–CRN)2 MP2 aDZ 2.55 þ1.02 1.53 2.47 þ0.85 1.62 0.09 MP2 aTZ 2.51 þ0.99 1.52 2.51 þ0.90 1.61 0.09 MP2 DZPþþ 2.52 þ0.92 1.60 2.31 þ0.66 1.65 0.05 MP2 TZ2P(f,d)þþ 2.50 þ0.87 1.63 2.46 þ0.79 1.68 0.04
T-shaped (NRC–CRN)2 MP2 aDZ 2.47 þ0.66 1.81 2.42 þ0.57 1.85 0.04 MP2 aTZ 2.44 þ0.62 1.81 2.44 þ0.58 1.85 0.04 MP2 DZPþþ 2.47 þ0.62 1.85 2.31 þ0.47 1.83 þ0.01 MP2 TZ2P(f,d)þþ 2.43 þ0.58 1.86 2.42 þ0.53 1.88 0.02
Parallel-slipped (HCRC–CRCH)2 MP2 aDZ 2.41 þ1.45 0.97 2.31 þ1.12 1.18 0.21 MP2 aTZ 2.49 þ1.28 1.21 2.47 þ1.19 1.28 0.07 MP2 DZPþþ 2.52 þ1.25 1.27 2.08 þ0.77 1.31 0.04 MP2 TZ2P(f,d)þþ 2.46 þ1.14 1.32 2.38 þ1.00 1.38 0.06
T-shaped (HCRC–CRCH)2 MP2 aDZ 1.98 þ0.36 1.61 1.91 þ0.29 1.63 0.02 MP2 aTZ 1.94 þ0.34 1.59 1.93 þ0.31 1.62 0.03 MP2 DZPþþ 1.97 þ0.33 1.65 1.82 þ0.22 1.60 þ0.05 MP2 TZ2P(f,d)þþ 1.94 þ0.32 1.61 1.92 þ0.28 1.64 0.02
MP2 CCSD(T) BestEst. this additive scheme (EINT , dMP2 , and EINT ) are re- observations are remarkably consistent with the predictions of ported for the various optimized structures. Sinnokrot and Sherrill for the parallel-slipped benzene dimer The interaction energies in Table 3 that correspond to where estimated CCSD(T)/aug-cc-pVQZ* potential energy MP2/aXZ optimized structures also exhibit rapid convergence curves suggest that the interaction energy will decrease by with respect to the cardinal number of the basis set (X ¼ D, T, 0.15 kcal mol 1 (from 2.78 to 2.63 kcal mol 1) when the Q). While the energetic parameters for the aDZ and aTZ CCSD(T) optimized structure is used rather than the optimized structures differ by more than 0.1 kcal mol 1, the MP2 one.7 values never deviate by more than 0.03 kcal mol 1 for the MP2/aTZ and MP2/aQZ optimized structures (which, as noted earlier, are virtually identical). Even the energetic data 3.2.2 Effect of counterpoise correction on energetics. Simi- larly, the CP correction does not appear to have any effect on associated with the MP2/DZPþþ and MP2/TZ2P(f,d)þþ BestEst. 1 the energetics when looking at EINT in Table 4. However, optimized structures are still within 0.12 kcal mol of the MP2 CCSD(T) MP2 CCSD(T) BestEst. with the DZPþþ basis set the EINT and dMP2 change by EINT , dMP2 , and EINT values for the MP2/aQZ 1 structures. 0.44 and þ0.48 kcal mol , respectively, for parallel-slipped diacetylene dimer when the CP procedure is applied during the geometry optimization. Again, error cancellation produces 3.2.1 Effect of electron correlation on energetics. By simply BestEst. best estimates of the interaction energy that differ by only E 1 comparing the INT values in Table 3 for CCSD(T) opti- 0.04 kcal mol . mized structures to those optimized at the MP2 level with the When one progresses to the larger TZ2P(f,d)þþ basis set, same basis set, it would appear that there is no need to the magnitude of these deviations are substantially reduced. optimize structures with the ‘‘Gold Standard’’ of quantum With this basis set, the energetic parameters change by no i.e. 1 chemistry ( , the CCSD(T) method). For example, the best more than 0.12 kcal mol when structures are reoptimized estimates of the interaction energies for with CP-corrected gradients. CCSD(T)/TZ2P(f,d)þþ fully optimized structures differ by no more than 0.09 kcal mol 1 from the values for the MP2 MP2/TZ2P(f,d)þþ structures. However, changes to EINT 3.2.3 Effect of rigid monomer approximation on energetics. CCSD(T) and dMP2 can be much larger, especially for the parallel- Interaction energies computed at the CCSD(T) and MP2 levels slipped structures. In the case of the parallel-slipped diacety- for fully optimized and rigid monomer structures are reported lene dimer, the MP2 CBS limit of the interaction energy in Table 5. The two sets of values are virtually indistinguish- changes by þ0.25 kcal mol 1 when the MP2/TZ2P(f,d)þþ able from each other; both the MP2 and CCSD(T) interaction structure is reoptimized with the CCSD(T) method. At the energies for the dimers change by less than 0.08 kcal mol 1 CCSD(T) same time, the dMP2 correction changes by 0.32 kcal when the rigid monomer approximation is applied. The devia- mol 1. Due to error cancellation, the best estimate of the tions tend to be slighly smaller for the TZ2P(f,d)þþ basis set interaction energy changes by only 0.07 kcal mol 1. These than for the DZPþþ basis set.
This journal is c the Owner Societies 2007 Phys. Chem. Chem. Phys., 2007, 9, 1550–1558 | 1555 Table 5 Effects of the rigid monomer approximation on the MP2 and CCSD(T) interaction energies of the dimers of cyanogen and diacetylene. All energies are in kcal mol 1
Optimization level Fully optimized Rigid monomer
MP2 CCSD(T) BestEst. MP2 CCSD(T) BestEst. RM Method Basis EINT dMP2 EINT EINT dMP2 EINT DEINT
Parallel-slipped (NRC–CRN)2 MP2 DZPþþ 2.52 þ0.92 1.60 2.55 þ0.88 1.66 0.06 CCSD(T) DZPþþ 2.41 þ0.68 1.73 2.40 þ0.70 1.70 þ0.04 MP2 TZ2P(f,d)þþ 2.50 þ0.87 1.63 2.50 þ0.83 1.67 0.04 CCSD(T) TZ2P(f,d)þþ 2.33 þ0.62 1.72 2.30 þ0.59 1.70 þ0.01
T-shaped (NRC–CRN)2 MP2 DZPþþ 2.47 þ0.62 1.85 2.41 þ0.63 1.78 þ0.07 CCSD(T) DZPþþ 2.42 þ0.50 1.92 2.38 þ0.54 1.84 þ0.08 MP2 TZ2P(f,d)þþ 2.43 þ0.58 1.86 2.42 þ0.57 1.84 þ0.02 CCSD(T) TZ2P(f,d)þþ 2.36 þ0.46 1.91 2.36 þ0.48 1.88 þ0.02
Parallel-slipped (HCRC–CRCH)2 MP2 DZPþþ 2.52 þ1.25 1.27 2.54 þ1.20 1.34 0.07 CCSD(T) DZPþþ 2.39 þ0.96 1.43 2.36 þ0.95 1.40 þ0.03 MP2 TZ2P(f,d)þþ 2.46 þ1.14 1.32 2.46 þ1.09 1.36 0.04 CCSD(T) TZ2P(f,d)þþ 2.21 þ0.82 1.39 2.18 þ0.80 1.38 þ0.01
T-shaped (HCRC–CRCH)2 MP2 DZPþþ 1.97 þ0.33 1.65 1.99 þ0.33 1.66 0.02 CCSD(T) DZPþþ 2.01 þ0.30 1.71 1.97 þ0.29 1.67 þ0.04 MP2 TZ2P(f,d)þþ 1.94 þ0.32 1.61 1.92 þ0.30 1.62 0.00 CCSD(T) TZ2P(f,d)þþ 1.91 þ0.27 1.64 1.88 þ0.26 1.63 þ0.01
MP2 BestEst. 4. Conclusions ergies (EINT and EINT ) along with the maximum deviation from the mean provide some insight into how widely the 4.1 Structures interaction energies vary with the optimization procedures. BestEst. Fifteen different optimization procedures have been applied to As can be seen from Table 6, the 15 EINT values tend to lie 1 the parallel-slipped and T-shaped structures of the cyanogen within 0.12 kcal mol of the average ( 1.64 0.12 1 and diacetylene dimers. Use of the correlation consistent aDZ kcal mol for parallel-slipped (NRC–CRN)2, 1.85 basis set (cc-pVDZ for H and aug-cc-pVDZ for N and C) is not recommended for these systems because of anomalous imaginary frequencies associated with the diacetylene mono- Table 6 Largest, smallest, and average interaction energies of the mer. In contrast, the fairly compact TZ2P(f,d) basis set dimers of cyanogen and diacetylene as well as the maximum deviation þþ from the average. All data are in kcal mol 1 gives optimized intermonomer geometrical parameters that MP2 BestEst. are very similar to those obtained from counterpoise corrected EINT EINT optimizations with the much larger aTZ basis set. As long as Parallel-slipped (NRC–CRN)2 appropriate polarization and diffuse functions are included, Largest 2.55 1.73 the choice of basis has only a modest effect on the intermo- Smallest 2.30 1.52 nomer geometrical parameters (on the order 0.05 A˚ ). The Average 2.45 1.64 Max. dev. 0.15 0.12 structures of the dimers are more sensitive to electron correla- tion than basis set. CCSD(T) intermonomer parameters are T-shaped (NRC–CRN)2 usually E0.1 A˚ longer than the MP2 values. With double-x Largest 2.47 1.92 Smallest 2.31 1.78 basis sets, counterpoise corrected optimized structures are Average 2.41 1.85 significantly different (by 0.1–0.3 A˚ ) from their uncorrected Max. dev. 0.10 0.07 counterparts. However, the effect of the CP correction rapidly R R diminishes when a larger basis set is used. Differences are on Parallel-slipped (HC C–C CH)2 ˚ Largest 2.54 1.43 the order of 0.05 A when the TZ2P(f,d)þþ basis set is used. Smallest 2.08 0.97a The rigid monomer approximation has a negligible effect on Average 2.38 1.31b c the dimer structures. Max. dev. 0.30 0.34 R R 4.2 Energetics T-shaped (HC C–C CH)2 Largest 2.01 1.71 Overall, the interaction energies of (NRC–CRN)2 and Smallest 1.82 1.59 (HCRC–CRCH) are relatively insensitive to the optimiza- Average 1.94 1.63 2 Max. dev. 0.12 0.08 tion procedure. Some statistics associated with the 15 different a b optimization procedures are reported in Table 6 for each Changes to 1.21 when aDZ data omitted. Changes to 1.33 when aDZ data omitted. c Changes to 0.12 when aDZ data omitted. structure. The largest, smallest, and average interaction en-
1556 | Phys.Chem.Chem.Phys., 2007, 9, 1550–1558 This journal is c the Owner Societies 2007 1 0.07 kcal mol for T-shaped (NRC–CRN)2, and 1.63 11 B. W. Hopkins and G. S. Tschumper, J. Phys. Chem. A, 2004, 108, 1 2941–2948. 0.08 kcal mol for T-shaped (HCRC–CRCH)2). The R R 12 M. Elstner, P. Hobza, T. Frauenheim, S. Suhai and E. Kaxiras, J. parallel-slipped structure of (HC C–C CH)2 is the only Chem. Phys., 2001, 114(12), 5149–5155. exception where the maximum deviation from the mean 13 S. Grimme, J. Chem. Phys., 2003, 118(20), 9095–9102. BestEst. 1 EINT value jumps to more than 0.3 kcal mol ( 1.31 14 X. Xu and W. A. Goddard, Proc. Natl. Acad. Sci. U. S. A., 2004, 0.34 kcal mol 1). However, if data from the problematic aDZ 101, 2673–2677. 15 A. A. Hasanein and M. Evans, J. Mol. Liq., 1984, 29, 45–59. basis (vide supra) is omitted, this distribution is narrowed to 16 I. Suni, S. Lee and W. Klemperer, J. Phys. Chem., 1991, 95, 1.33 0.12 kcal mol 1. 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