Systematically convergent basis sets for explicitly correlated wavefunctions: the atoms H, He, B-Ne, and Al-Ar

Correlation consistent basis sets have been optimized for use with explicitly correlated F12 methods. The new sets, denoted cc-pVnZ-F12 (n=D,T,Q), are similar in size and construction to the standard aug-cc-pVnZ and aug-cc-pV(n+d)Z basis sets, but the new sets are shown in the present work to yield much improved convergence toward the complete basis set limit in MP2-F12/3C calculations on several small molecules involving elements of both the first and second row. For molecules containing only first row atoms, the smallest cc-pVDZ-F12 basis set consistently recovers nearly 99% of the MP2 valence correlation energy when combined with the MP2-F12/3C method. The convergence with basis set for molecules containing second row atoms is slower, but the new DZ basis set still recovers 97%-99% of the frozen core MP2 correlation energy. The accuracy of the new basis sets for relative energetics is demonstrated in benchmark calculations on a set of 15 chemical reactions.     

Extrapolating to the one-electron basis-set limit in electronic structure calculations

A simple, yet reliable, scheme based on treating uniformly singlet-pair and triplet-pair interactions is suggested to extrapolate atomic and molecular electron correlation energies calculated at two basis-set levels of ab initio theory to the infinite one-electron basis-set limit. The novel dual-level method is first tested on extrapolating the full correlation in single-reference coupled-cluster singles and doubles energies for the closed-shell systems CH2((1)A1), H2O, HF, N2, CO, Ne, and F2 with correlation-consistent basis sets of the type cc-pVXZ (X=D,T,Q,5,6) reported by Klopper [Mol. Phys. 6, 481 (2001)] against his own benchmark calculations with large uncontracted basis sets obtained from explicit correlated singles and doubles coupled-cluster theory. Comparisons are also reported for the same data set but using both single-reference Moller-Plesset and coupled-cluster doubles methods. The results show a similar, often better, accordance with the target results than Klopper's extrapolations where singlet-pair and triplet-pair energies are extrapolated separately using the popular X(-3) and X(-5) dual-level laws, respectively. Applications to the extrapolation of the dynamical correlation in multireference configuration interaction calculations carried out anew for He, H2, HeH+, He2 ++, H3+(1 (1)A'), H3+(1 (3)A'), BH, CH, NH, OH, FH, B2, C2, N2, O2, F2, BO, CO, NO, BN, CN, SH, H2O, and NH3 with standard augmented correlation-consistent basis sets of the type aug-cc-pVXZ (X=D,T,Q,5,6) are also reported. Despite lacking accurate theoretical or experimental data for comparison in the case of most diatomic systems, the new method also shows in this case a good performance when judged from the results obtained with the traditional schemes which extrapolate using the two largest affordable basis sets. For the Hartree-Fock and complete-active space self-consistent field energies, a simple pragmatic extrapolation rule is examined whose results are shown to compare well with the ones obtained from the best reported schemes.     

Accurate extrapolation of electron correlation energies from small basis sets

A new two-point scheme is proposed for the extrapolation of electron correlation energies obtained with small basis sets. Using the series of correlation-consistent polarized valence basis sets, cc-pVXZ, the basis set truncation error is expressed as deltaE(X) proportional, variant(X + xi(i))(-gamma). The angular momentum offset xi(i) captures differences in effective rates of convergence previously observed for first-row molecules. It is based on simple electron counts and tends to values close to 0 for hydrogen-rich compounds and values closer to 1 for pure first-row compounds containing several electronegative atoms. The formula is motivated theoretically by the structure of correlation-consistent basis sets which include basis functions up to angular momentum L = X-1 for hydrogen and helium and up to L = X for first-row atoms. It contains three parameters which are calibrated against a large set of 105 reference molecules (H, C, N, O, F) for extrapolations of MP2 and CCSD valence-shell correlation energies from double- and triple-zeta (DT) and triple- and quadruple-zeta (TQ) basis sets. The new model is shown to be three to five times more accurate than previous two-point schemes using a single parameter, and (TQ) extrapolations are found to reproduce a small set of available R12 reference data better than even (56) extrapolations using the conventional asymptotic limit formula deltaE(X) proportional, variantX(-3). Applications to a small selection of boron compounds and to neon show very satisfactory results as well. Limitations of the model are discussed.     

On the accuracy of DFT-SAPT, MP2, SCS-MP2, MP2C, and DFT+Disp methods for the interaction energies of endohedral complexes of the C(60) fullerene with a rare gas atom

Selected points on the potential energy surface for the complexes Rg@C(60) (Rg = He, Ne, Ar, Kr) are calculated with various theoretical methods, like symmetry-adapted perturbation theory with monomers described by density functional theory (DFT-SAPT), supermolecular M?ller-Plesset theory truncated on the second order (MP2), spin-component-scaled MP2 (SCS-MP2), supermolecular density functional theory with empirical dispersion correction (DFT+Disp), and the recently developed MP2C method that improves the MP2 method for long-range electron correlation effects. A stabilization of the endohedral complex is predicted by all methods, but the depth of the potential energy well is overestimated by the DFT+Disp and MP2 approaches. On the other hand, the MP2C model agrees well with DFT-SAPT, which serves as the reference. The performance of SCS-MP2 is mixed: it produces too low interaction energies for the two heavier guests, while its accuracy for He@C(60) and Ne@C(60) is similar to that of MP2C. Fitting formulas for the main interaction energy components, i.e. the dispersion and first-order repulsion energies are proposed, which are applicable for both endo- and exohedral cases. For all examined methods density fitting is used to evaluate two-electron repulsion integrals, which is indispensable to allow studies of noncovalent complexes of this size. It has been found that density-fitting auxiliary basis sets cannot be used in a black-box fashion for the calculation of the first-order SAPT electrostatic energy, and that the quality of these basis sets should be always carefully examined in order to avoid an unphysical long-range behavior.     

Effects of counterpoise correction and basis set extrapolation on the MP2 geometries of hydrogen bonded dimers of ammonia, water, and hydrogen fluoride

We study the combined effects of counterpoise correction and basis set extrapolation on the second-order M?ller-Plesset (MP2) geometries of three hydrogen bonded dimers, namely (NH(3))(2), (H(2)O)(2) and (HF)(2). For (NH(3))(2), we study three characteristic structures on its potential energy surface. In addition, we look at the basis set convergence when diffuse functions on the hydrogen atoms are left out, as well as the errors introduced by including core correlation with valence-only correlation-consistent basis sets. Overall, the counterpoise-corrected and extrapolated geometries appear to be very reliable and are in convincing agreement with the geometries from explicitly correlated MP2-F12 calculations. Obtaining geometries with errors of less than 0.001 ?ngstrom and 0.5 degrees compared to the basis set limit is, however, even with these advanced methods a difficult task.     

Polarization consistent basis sets. 4: the elements He, Li, Be, B, Ne, Na, Mg, Al, and Ar

Polarization consistent basis sets, optimized for density functional calculations, are proposed for the elements He, Li, Be, B, Ne, Na, Mg, Al, and Ar. The basis sets for He, B, Ne, Al, and Ar are assigned based on the previously proposed basis sets for H, C-F, and Si-Ar. The basis sets for Li, Be, Na, and Mg are defined based on energetic analysis along the lines used in previous work and the performance for molecular systems. The performance for atomization energies is comparable to those for systems composed of the elements H, C-F, and Si-Ar.     

Extrapolation of electron correlation energies to finite and complete basis set targets

The electron correlation energy of two-electron atoms is known to converge asymptotically as approximately (L+1)(-3) to the complete basis set limit, where L is the maximum angular momentum quantum number included in the basis set. Numerical evidence has established a similar asymptotic convergence approximately X(-3) with the cardinal number X of correlation-consistent basis sets cc-pVXZ for coupled cluster singles and doubles (CCSD) and second order perturbation theory (MP2) calculations of molecules. The main focus of this article is to probe for deviations from asymptotic convergence behavior for practical values of X by defining a trial function X(-beta) that for an effective exponent beta=beta(eff)(X,X+1,X+N) provides the correct energy E(X+N), when extrapolating from results for two smaller basis sets, E(X) and E(X+1). This analysis is first applied to "model" expansions available from analytical theory, and then to a large body of finite basis set results (X=D,T,Q,5,6) for 105 molecules containing H, C, N, O, and F, complemented by a smaller set of 14 molecules for which accurate complete basis set limits are available from MP2-R12 and CCSD-R12 calculations. beta(eff) is generally found to vary monotonically with the target of extrapolation, X+N, making results for large but finite basis sets a useful addition to the limited number of cases where complete basis set limits are available. Significant differences in effective convergence behavior are observed between MP2 and CCSD (valence) correlation energies, between hydrogen-rich and hydrogen-free molecules, and, for He, between partial-wave expansions and correlation-consistent basis sets. Deviations from asymptotic convergence behavior tend to get smaller as X increases, but not always monotonically, and are still quite noticeable even for X=5. Finally, correlation contributions to atomization energies (rather than total energies) exhibit a much larger variation of effective convergence behavior, and extrapolations from small basis sets are found to be particularly erratic for molecules containing several electronegative atoms. Observed effects are discussed in the light of results known from analytical theory. A carefully calibrated protocol for extrapolations to the complete basis set limit is presented, based on a single "optimal" exponent beta(opt)(X,X+1,infinity) for the entire set of molecules, and compared to similar approaches reported in the literature.     

Infinite-basis calculations of binding energies for the hydrogen bonded and stacked tetramers of formic acid and formamide and their use for validation of hybrid DFT and ab initio methods

Benchmark stabilization energies for planar H-bonded and stacked structures of formic acid tetramers and formamide tetramers were determined as the sum of the infinite basis set limit of MP2 energies and a CCSD(T) correction term evaluated with the 6-31G*(0.25) basis set. The infinite basis (IB) set limit of MP2 energies was determined by two-point extrapolation using the aug-cc-pVXZ basis sets for X = D and T and separate extrapolation of the Hartree-Fock and correlation energies with new IB parameters for augmented basis sets determined here. Final stabilization energies (kcal/mol) for the tetramer studied are in the range of 4.6 to approximately 6.7 kcal/mol and they were used as reference data to test 14 density functionals. Among the tested DFT methods, PWB6K gives the best performance with an average error equal to only 30% of the average binding energy. In contrast, the popular B3LYP functional has an average error of 85%. We recommend the PWB6K method for exploring the potential energy surfaces of organic complexes and clusters and supramolecular assemblies.     

Accurate calculations of intermolecular interaction energies using explicitly correlated wave functions

Explicitly correlated second-order M?ller-Plesset (MP2-F12) calculations of intermolecular interaction energies for the S22 benchmark set of Jurecka, Sponer, Cerny, and Hobza (Chem. Phys. Phys. Chem. 2006, 8, 1985) are presented and compared with standard MP2 results. The MP2 complete basis set limits are estimated using basis set extrapolation and augmented quadruple-zeta and quintuple-zeta basis sets. Already with augmented double-zeta basis sets the MP2-F12 interaction energies are found to be closer to the complete basis set limits than standard MP2 calculations with augmented quintuple-zeta basis sets. Various possible approximations in the MP2-F12 method are systematically tested. Best results are obtained with localized orbitals and the diagonal MP2-F12/C(D) ansatz. Hybrid approximations, in which some contributions of the auxiliary basis set are neglected and which considerably reduce the computational cost, have a negligible effect on the interaction energies. Also the orbital-invariant fixed-amplitude approximation of Ten-no leads to only slightly less accurate results. Preliminary results for the neon and benzene dimers, obtained with the recently proposed CCSD(T)-F12a approximation, indicate that the CCSD(T) basis set limits can also be very closely approached using augmented triple-zeta basis sets.     

Theoretical Study of Decomposition Pathways for Rare-gas-containing Compounds HRgX （Rg = He, Ne, Ar, Kr; X = Cl, Br）

1 INTRODUCTION transfer species with common formula H-Rgδ -Xδ, where X represents a strongly electrone-gative atom Since xenon hexafluoraplatinate, XePtF6 , the [1] or fragment, and Rg is a rare-gas atom. These species first rare gas-containing compound was discovered have linear equi- librium geometries and are mainly by Bartlett in 1962, rare gases are getting more and bound up by strong columbic attraction between (H- more attention and have been found to be possible to Rg) and…     

Theoretical studies of host-guest interaction in gas hydrates

Ab initio calculations and atoms-in-molecules (AIM) analysis have been used to investigate the host-guest interaction in dodecahedral water cages using a variety of guest species that include monatomic (He, Ne, Ar, Kr, and Xe), diatomic (CO, H(2), N(2), O(2), and NO), triatomic (CO(2), NO(2), and O(3)), and polyatomic (CH(4) and NH(3)) molecules. Geometry optimization for the guest species, host cage, and their complexes was carried out using the second order M?ller-Plesset perturbation method with the 6-31G** basis set. Single point energy calculations using the same method but different basis sets (6-31++G**, 6-311++G**, aug-cc-pVDZ, and aug-cc-pVTZ) were carried out for the MP2/6-31G** optimized geometries. The interaction energy between the guest species and the host cage has been obtained in the complete basis set limit by basis set extrapolation.     

Franck-Condon simulation, including anharmonicity, of the photodetachment spectrum of P2H(-): restricted-spin coupled-cluster single-double plus perturbative triple and unrestricted-spin coupled-cluster single-double plus perturbative triple -F12x potential energy functions of P2H and P2H(-)

Geometry optimization and harmonic vibrational frequency calculations have been carried out on the X?(2)A(') state of P(2)H and the X?(1)A(') state of P(2)H(-) using the restricted-spin coupled-cluster single-double plus perturbative triple excitation [RCCSD(T)] and explicitly correlated unrestricted-spin coupled-cluster single-double plus perturbative triple excitation [UCCSD(T)-F12x] methods. For RCCSD(T) calculations, basis sets of up to the augmented correlation-consistent polarized valence quintuple-zeta (aug-cc-pV5Z) quality were employed, and contributions from extrapolation to the complete basis set limit and from core correlation of the P 2s(2)2p(6) electrons were also included. For UCCSD(T)-F12x calculations, different atomic orbital basis sets of triple-zeta quality with different associated complementary auxiliary basis sets and different geminal Slater exponents were used. When the P 2s(2)2p(6) core electrons were correlated in these F12x calculations, appropriate core-valence basis sets were employed. In addition, potential energy functions (PEFs) of the X?(2)A(') state of P(2)H and the X?(1)A(') state of P(2)H(-) were computed at different RCCSD(T) and UCCSD(T)-F12x levels, and were used in variational calculations of anharmonic vibrational wavefunctions, which were then utilized to calculate Franck-Condon factors (FCFs) between these two states, employing a method which includes allowance for anharmonicity and Duschinsky rotation. The photodetachment spectrum of P(2)H(-) was then simulated using the computed FCFs. Simulated spectra obtained using the RCCSD(T)/aug-cc-pV5Z and UCCSD(T)-F12x(x = a or b)/aug-cc-pCVTZ PEFs are compared and found to be essentially identical. Based on the computed FCFs, a more detailed assignment of the observed vibrational structure than previously reported, which includes "hot bands," has been proposed. Comparison between simulated and available experimental spectra has been made, and the currently most reliable sets of equilibrium geometrical parameters for P(2)H and its anion have been derived. The photodetachment spectrum of P(2)D, yet to be recorded, has also been simulated.     

True stabilization energies for the optimal planar hydrogen-bonded and stacked structures of guanine...cytosine, adenine...thymine, and their 9- and 1-methyl derivatives: complete basis set calculations at the MP2 and CCSD(T) levels and comparison with experiment

Planar H-bonded and stacked structures of guanine...cytosine (G.C), adenine...thymine (A...T), 9-methylguanine...1-methylcytosine (mG...mC), and 9-methyladenine...1-methylthymine (mA...mT) were optimized at the RI-MP2 level using the TZVPP ([5s3p2d1f/3s2p1d]) basis set. Planar H-bonded structures of G...C, mG...mC, and A...T correspond to the Watson-Crick (WC) arrangement, in contrast to mA...mT for which the Hoogsteen (H) structure is found. Stabilization energies for all structures were determined as the sum of the complete basis set limit of MP2 energies and a (DeltaE(CCSD(T)) - DeltaE(MP2)) correction term evaluated with the cc-pVDZ(0.25,0.15) basis set. The complete basis set limit of MP2 energies was determined by two-point extrapolation using the aug-cc-pVXZ basis sets for X = D and T and X = T and Q. This procedure is required since the convergency of the MP2 interaction energy for the present complexes is rather slow, and it is thus important to include the extrapolation to the complete basis set limit. For the MP2/aug-cc-pVQZ level of theory, stabilization energies for all complexes studied are already very close to the complete basis set limit. The much cheaper D-->T extrapolation provided a complete basis set limit close (by less than 0.7 kcal/mol) to the more accurate T-->Q term, and the D-->T extrapolation can be recommended for evaluation of complete basis set limits of more extended complexes (e.g. larger motifs of DNA). The convergency of the (DeltaE(CCSD(T)) - DeltaE(MP2)) term is known to be faster than that of the MP2 or CCSD(T) correlation energy itself, and the cc-pVDZ(0.25,0.15) basis set provides reasonable values for planar H-bonded as well as stacked structures. Inclusion of the CCSD(T) correction is essential for obtaining reliable relative values for planar H-bonding and stacking interactions; neglecting the CCSD(T) correction results in very considerable errors between 2.5 and 3.4 kcal/mol. Final stabilization energies (kcal/mol) for the base pairs studied are very substantial (A...T WC, 15.4; mA...mT H, 16.3; A...T stacked, 11.6; mA...mT stacked, 13.1; G...C WC, 28.8; mG...mC WC, 28.5; G...C stacked, 16.9; mG...mC stacked, 18.0), much larger than published previously. On the basis of comparison with experimental data, we conclude that our values represent the lower boundary of the true stabilization energies. On the basis of error analysis, we expect the present H-bonding energies to be fairly close to the true values, while stacked energies are still expected to be about 10% too low. The stacking energy for the mG...mC pair is considerably lower than the respective H-bonding energy, but it is larger than the mA...mT H-bonding energy. This conclusion could significantly change the present view on the importance of specific H-bonding interactions and nonspecific stacking interactions in nature, for instance, in DNA. Present stabilization energies for H-bonding and stacking energies represent the most accurate and reliable values and can be considered as new reference data.     

Benchmark database of accurate (MP2 and CCSD(T) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs

MP2 and CCSD(T) complete basis set (CBS) limit interaction energies and geometries for more than 100 DNA base pairs, amino acid pairs and model complexes are for the first time presented together. Extrapolation to the CBS limit is done by using two-point extrapolation methods and different basis sets (aug-cc-pVDZ - aug-cc-pVTZ, aug-cc-pVTZ - aug-cc-pVQZ, cc-pVTZ - cc-pVQZ) are utilized. The CCSD(T) correction term, determined as a difference between CCSD(T) and MP2 interaction energies, is evaluated with smaller basis sets (6-31G** and cc-pVDZ). Two sets of complex geometries were used, optimized or experimental ones. The JSCH-2005 benchmark set, which is now available to the chemical community, can be used for testing lower-level computational methods. For the first screening the smaller training set (S22) containing 22 model complexes can be recommended. In this case larger basis sets were used for extrapolation to the CBS limit and also CCSD(T) and counterpoise-corrected MP2 optimized geometries were sometimes adopted.     

General orbital invariant MP2-F12 theory

A general form of orbital invariant explicitly correlated second-order closed-shell Moller-Plesset perturbation theory (MP2-F12) is derived, and compact working equations are presented. Many-electron integrals are avoided by resolution of the identity (RI) approximations using the complementary auxiliary basis set approach. A hierarchy of well defined levels of approximation is introduced, differing from the exact theory by the neglect of terms involving matrix elements over the Fock operator. The most accurate method is denoted as MP2-F12/3B. This assumes only that Fock matrix elements between occupied orbitals and orbitals outside the auxiliary basis set are negligible. For the chosen ansatz for the first-order wave function this is exact if the auxiliary basis is complete. In the next lower approximation it is assumed that the occupied orbital space is closed under action of the Fock operator [generalized Brillouin condition (GBC)]; this is equivalent to approximation 2B of Klopper and Samson [J. Chem. Phys. 116, 6397 (2002)]. Further approximations can be introduced by assuming the extended Brillouin condition (EBC) or by neglecting certain terms involving the exchange operator. A new approximation MP2-F12/3C, which is closely related to the MP2-R12/C method recently proposed by Kedzuch et al. [Int. J. Quantum Chem. 105, 929 (2005)] is described. In the limit of a complete RI basis this method is equivalent to MP2-F12/3B. The effect of the various approximations (GBC, EBC, and exchange) is tested by studying the convergence of the correlation energies with respect to the atomic orbital and auxiliary basis sets for 21 molecules. The accuracy of relative energies is demonstrated for 16 chemical reactions. Approximation 3C is found to perform equally well as the computationally more demanding approximation 3B. The reaction energies obtained with smaller basis sets are found to be most accurate if the orbital-variant diagonal Ansatz combined with localized orbitals is used for the first-order wave function. This unexpected result is attributed to geminal basis set superposition errors present in the formally more rigorous orbital invariant methods.     

Basis set convergence of explicitly correlated double-hybrid density functional theory calculations

The basis set convergence of explicitly correlated double-hybrid density functional theory (DFT) is investigated using the B2GP-PLYP functional. As reference values, we use basis set limit B2GP-PLYP-F12 reaction energies extrapolated from the aug(')-cc-pV(Q+d)Z and aug(')-cc-pV(5+d)Z basis sets. Explicitly correlated double-hybrid DFT calculations converge significantly faster to the basis set limit than conventional calculations done with basis sets saturated up to the same angular momentum (typically, one "gains" one angular momentum in the explicitly correlated calculations). In explicitly correlated F12 calculations the VnZ-F12 basis sets converge faster than the orbital A(')VnZ basis sets. Furthermore, basis set convergence of the MP2-F12 component is apparently faster than that of the underlying Kohn-Sham calculation. Therefore, the most cost-effective approach consists of combining the MP2-F12 correlation energy from a comparatively small basis set such as VDZ-F12 with a DFT energy from a larger basis set such as aug(')-cc-pV(T+d)Z.     

Effect of fluorination: gas-phase structures of N,N-dimethylvinylamine and perfluoro-N,N-dimethylvinylamine.

The gas-phase structures of N,N-dimethylvinylamine, (CH(3))(2)NC(H)=CH(2) (1), and perfluoro-N,N-dimethylvinylamine, (CF(3))(2)NC(F)=CF(2) (2), were determined by gas electron diffraction and quantum chemical methods (B3LYP and MP2 with 6-31G basis sets). The configuration around nitrogen is slightly pyramidal in both compounds, with the sum of the nitrogen bond angles 351.2(12) degrees and 354.8(6) degrees in 1 and 2, respectively. In the parent compound 1, the (CH(3))(2)N group lies nearly in the plane of the vinyl group, and the nitrogen lone pair (lp) is almost perpendicular to this plane (Phi(C=C-N-lp) = 98(6) degrees). In the perfluorinated species 2, however, the (CF(3))(2)N group is oriented perpendicular to the vinyl plane, and the lone pair is parallel to the C=C bond (Phi(C=C-N-lp) = 2(5) degrees). A natural bond orbital analysis provides a qualitative explanation for this conformational change upon fluorination. The sterically unfavorable in-plane orientation of the dimethylamino group in 1 is stabilized by conjugation between the nitrogen lone pair and the C=C pi-bond. The anomeric effect between the lone pair and the C(alpha)-F sigma-bond in addition to steric effects favors the perpendicular orientation of the (CF(3))(2)N group in 2. Both quantum chemical methods reproduce the experimental structures satisfactorily.     

Noble‐Gas Complexes: Theoretical Investigation of Multicenter Polynuclear Species

Ab initio calculations at the MP2 level of theory disclose the conceivable existence of neutral complexes containing four or five distinct noble gases (Ng) each bound to a distinct Be‐atom. These multicenter polynuclear Ng molecules are formally obtained by replacing the H‐atoms of CH₄ and but‐2‐yne with ? NBeNg moieties, which behave as independent monovalent ‘functional groups’. Our investigated complexes include the five homotetranuclear [C(NBeNg)₄] complexes 1 – 5 (Ng=He? Xe), the five heterotetranuclear complexes [CN₄Be₄(He)(Ne)(Ar)(Kr)] ( 6 ), [CN₄Be₄(He)(Ne)(Ar)(Xe)] ( 7 ), [CN₄Be₄(He)(Ne)(Kr)(Xe)] ( 8 ), [CN₄Be₄(He)(Ar)(Kr)(Xe)] ( 9 ), and [CN₄Be₄(Ne)(Ar)(Kr)(Xe)] ( 10 ), and the heteropentanuclear complex [HC₄N₅Be₅(He)(Ne)(Ar)(Kr)(Xe)] ( 11 ). We also investigated the five model complexes [H₃CNBeNg] (Ng=He? Xe) containing a single ? NBeNg moiety. The geometries and vibrational frequencies of all these species, invariably characterized as minimum‐energy structures, were computed at the MP2(full)/6‐31G(d,p)/SDD level of theory, and their stability with respect to the loss of the various Ng‐atoms was evaluated by single‐point calculations at the MP2(full)/6‐311G(d)/SDD level of theory. The beryllium‐Ng binding energies range from ca. 17 (Ng=He) to ca. 63 (Ng=Xe) kJ/mol, and the results of natural‐bond‐orbital (NBO) and atoms‐in‐molecules (AIM) analysis reveal that the Be? Ng interaction is essentially electrostatic for helium, neon, argon, and krypton, and has probably a small covalent contribution for xenon.     

Basis set convergence of the coupled-cluster correction, δ(MP2)(CCSD(T)): best practices for benchmarking non-covalent interactions and the attendant revision of the S22, NBC10, HBC6, and HSG databases

In benchmark-quality studies of non-covalent interactions, it is common to estimate interaction energies at the complete basis set (CBS) coupled-cluster through perturbative triples [CCSD(T)] level of theory by adding to CBS second-order perturbation theory (MP2) a "coupled-cluster correction," δ(MP2)(CCSD(T)), evaluated in a modest basis set. This work illustrates that commonly used basis sets such as 6-31G*(0.25) can yield large, even wrongly signed, errors for δ(MP2)(CCSD(T)) that vary significantly by binding motif. Double-ζ basis sets show more reliable results when used with explicitly correlated methods to form a δ(MP2-F12)(CCSD(T(*))-F12) correction, yielding a mean absolute deviation of 0.11 kcal mol(-1) for the S22 test set. Examining the coupled-cluster correction for basis sets up to sextuple-ζ in quality reveals that δ(MP2)(CCSD(T)) converges monotonically only beyond a turning point at triple-ζ or quadruple-ζ quality. In consequence, CBS extrapolation of δ(MP2)(CCSD(T)) corrections before the turning point, generally CBS (aug-cc-pVDZ,aug-cc-pVTZ), are found to be unreliable and often inferior to aug-cc-pVTZ alone, especially for hydrogen-bonding systems. Using the findings of this paper, we revise some recent benchmarks for non-covalent interactions, namely the S22, NBC10, HBC6, and HSG test sets. The maximum differences in the revised benchmarks are 0.080, 0.060, 0.257, and 0.102 kcal mol(-1), respectively.     

Complexes of type C6H7(+)·L (L = N2 and CO2) studied by explicitly correlated coupled cluster theory

Complexes of the benzenium ion (C(6)H(7)(+)) with N(2) or CO(2) have been studied by explicitly correlated coupled cluster theory at the CCSD(T)-F12x (x = a, b) level [T. B. Adler et al., J. Chem. Phys. 127, 221106 (2007)] and the double-hybrid density functional B2PLYP-D [T. Schwabe and S. Grimme, Phys. Chem. Chem. Phys. 9, 3397 (2007)]. Improved harmonic vibrational wavenumbers for C(6)H(7)(+) have been obtained by CCSD(T?)-F12a calculations with the VTZ-F12 basis set. Combining them with previous B2PLYP-D anharmonic contributions we arrive at anharmonic wavenumbers which are in excellent agreement with recent experimental data from p-H(2) matrix isolation IR spectroscopy [M. Bahou et al., J. Chem. Phys. 136, 154304 (2012)]. The energetically most favourable conformer of C(6)H(7)(+)·N(2) shows a π-bonded structure similar to C(6)H(7)(+)·Rg (Rg = Ne, Ar) [P. Botschwina and R. Oswald, J. Phys. Chem. A 115, 13664 (2011)] with D(e) ≈ 870 cm(-1). For C(6)H(7)(+)·CO(2), a slightly lower energy is calculated for a conformer with the CO(2) ligand lying in the ring-plane of the C(6)H(7)(+) moiety (D(e) ≈ 1508 cm(-1)). It may be discriminated from other conformers through a strong band predicted at 1218 cm(-1), red-shifted by 21 cm(-1) from the corresponding band of free C(6)H(7)(+).