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.     

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.     

An explicitly correlated local coupled cluster method for calculations of large molecules close to the basis set limit

A new explicitly correlated local coupled-cluster method with single and double excitations and a perturbative treatment of triple excitations [DF-LCCSD(T0)-F12x (x = a,b)] is presented. By means of truncating the virtual orbital space to pair-specific local domains (domain approximation) and a simplified treatment of close, weak and distant pairs using LMP2-F12 (pair approximation) the scaling of the computational cost with molecular size is strongly reduced. The basis set incompleteness errors as well as the errors due to the domain approximation are largely eliminated by the explicitly correlated terms. All integrals are computed using efficient density fitting (DF) approximations. The accuracy of the method is investigated for 52 reactions involving medium size molecules. A comparison of DF-LCCSD(T0)-F12x reaction energies with canonical CCSD(T)-F12x calculations shows that the errors introduced by the domain approximation are indeed very small. Care must be taken to keep the errors due to the additional pair approximation equally small, and appropriate distance criteria are recommended. Using these parameters, the root mean square (RMS) deviations of DF-LCCSD(T0)-F12a calculations with triple-ζ basis sets from estimated CCSD(T) complete basis set (CBS) limits and experimental data amount to only 1.5 kJ mol(-1) and 2.9 kJ mol(-1), respectively. For comparison, the RMS deviation of the CCSD(T)/CBS values from the experimental values amounts to 3.0 kJ mol(-1). The potential of the method is demonstrated for five reactions of biochemical or pharmacological interest which include molecules with up to 61 atoms. These calculations show that molecules of this size can now be treated routinely and yield results that are close to the CCSD(T) complete basis set limits.     

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.     

On the accuracy of explicitly correlated coupled-cluster interaction energies--have orbital results been beaten yet?

The basis set convergence of weak interaction energies for dimers of noble gases helium through krypton is studied for six variants of the explicitly correlated, frozen geminal coupled-cluster singles, doubles, and noniterative triples [CCSD(T)-F12] approach: the CCSD(T)-F12a, CCSD(T)-F12b, and CCSD(T)(F12*) methods with scaled and unscaled triples. These dimers were chosen because CCSD(T) complete-basis-set (CBS) limit benchmarks are available for them to a particularly high precision. The dependence of interaction energies on the auxiliary basis sets has been investigated and it was found that the default resolution-of-identity sets cc-pVXZ/JKFIT are far from adequate in this case. Overall, employing the explicitly correlated approach clearly speeds up the basis set convergence of CCSD(T) interaction energies, however, quite surprisingly, the improvement is not as large as the one achieved by a simple addition of bond functions to the orbital basis set. Bond functions substantially improve the CCSD(T)-F12 interaction energies as well. For small and moderate bases with bond functions, the accuracy delivered by the CCSD(T)-F12 approach cannot be matched by conventional CCSD(T). However, the latter method in the largest available bases still delivers the CBS limit to a better precision than CCSD(T)-F12 in the largest bases available for that approach. Our calculations suggest that the primary reason for the limited accuracy of the large-basis CCSD(T)-F12 treatment are the approximations made at the CCSD-F12 level and the non-explicitly correlated treatment of triples. In contrast, the explicitly correlated second-order Mo?ller-Plesset perturbation theory (MP2-F12) approach is able to pinpoint the complete-basis-set limit MP2 interaction energies of rare gas dimers to a better precision than conventional MP2. Finally, we report and analyze an unexpected failure of the CCSD(T)-F12 method to deliver the core-core and core-valence correlation corrections to interaction energies consistently and accurately.     

Explicitly correlated second-order Møller-Plesset perturbation theory employing pseudospectral numerical quadratures

We implemented explicitly correlated second-order M?ller-Plesset perturbation theory with numerical quadratures using pseudospectral construction of grids. Introduction of pseudospectral approach for the calculation of many-electron integrals gives a possibility to use coarse grids without significant loss of precision in correlation energies, while the number of points in the grid is reduced about nine times. The use of complementary auxiliary basis sets as the sets of dealiasing functions is justified at both theoretical and computational levels. Benchmark calculations for a set of 16 molecules have shown the possibility to keep an error of second-order correlation energies within 1 milihartree (mH) with respect to MP2-F12 method with dense grids. Numerical tests for a set of 13 isogyric reactions are also performed.     

Implementation of the CCSD(T)-F12 method using cusp conditions

The explicitly-correlated coupled-cluster singles and doubles with perturbative triples method (CCSD(T)-F12) is implemented using the cusp conditions. Numerical tests for a set of 16 molecules have shown agreement of correlation energies within 1 mE(h) between the cusp-condition and fully-optimized CCSD(T)-F12 methods. Benchmark calculations on 13 chemical reactions with the cusp-condition CCSD(T)-F12 method reproduce experimental enthalpies within 2 kJ mol(-1). It is also shown that regular unitary-invariant ansatz cannot exactly satisfy singlet and triplet cusp conditions in open-shell situations. We present an extended ansatz which can handle both conditions exactly.     

Analytical energy gradients for local second-order Møller-Plesset perturbation theory using density fitting approximations

An efficient method to compute analytical energy derivatives for local second-order M?ller-Plesset perturbation energy is presented. Density fitting approximations are employed for all 4-index integrals and their derivatives. Using local fitting approximations, quadratic scaling with molecular size and cubic scaling with basis set size for a given molecule is achieved. The density fitting approximations have a negligible effect on the accuracy of optimized equilibrium structures or computed energy differences. The method can be applied to much larger molecules and basis sets than any previous second-order M?ller-Plesset gradient program. The efficiency and accuracy of the method is demonstrated for a number of organic molecules as well as for molecular clusters. Examples of geometry optimizations for molecules with 100 atoms and over 2000 basis functions without symmetry are presented.     

Combining explicitly correlated R12 and Gaussian geminal electronic structure theories

Explicitly correlated R12 methods using a single short-range correlation factor (also known as F12 methods) have dramatically smaller basis set errors compared to the standard wave function counterparts, even when used with small basis sets. Correlations on several length scales, however, may not be described efficiently with one correlation factor. Here the authors explore a more general MP2-R12 method in which each electron pair uses a set of (contracted) Gaussian-type geminals (GTGs) with fixed exponents, whose coefficients are optimized linearly. The following features distinguish the current method from related explicitly correlated approaches published in the literature: (1) only two-electron integrals are needed, (2) the only approximations are the resolution of the identity and the generalized Brillouin condition, (3) only linear parameters are optimized, and (4) an arbitrary number of (non-)contracted GTGs can appear. The present method using only three GTGs and a double-zeta quality basis computed valence correlation energies for a set of 20 small molecules only 2.2% removed from the basis set limit. The average basis set error reduces to 1.2% using a near-complete set of seven GTGs with the double-zeta basis set. The conventional MP2 energies computed with much larger quadruple, quintuple, and sextuple basis sets all had larger average errors: 4.6%, 2.4%, and 1.5%, respectively. The new method compares well to the published MP2-R12 method using a single Slater-type geminal (STG) correlation factor. For example, the average basis set error in the absolute MP2-R12 energy obtained with the exp(-r12) correlation factor is 1.7%. Correlation contribution to atomization energies evaluated with the present method and with the STG-based method only required a double-zeta basis set to exceed the precision of the conventional sextuple-zeta result. The new method is shown to always be numerically stable if linear dependencies are removed from the two-particle basis and the zeroth-order Hamiltonian matrix is made positive definite.     

Some aspects of the model potential method

The theoretical justification of the model potential method is studied in some detail. The correct equations within the framework of Roothaan's open-shell scheme are derived and the approximations necessary to get a workable method are discussed. Analysis of the local part of the model potential suggests a new analytical form for it. The new expression is theoretically more consistent than the original one, and it can be determined in a more straightforward way. A basis set approximation, which is particularly suitable for approximate evaluation of two-electron integrals when only valence orbitals are involved, is discussed and tested with encouraging results. The ideas are tested on the Fe and I atoms.     

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.     

Basis set limits of the second order Moller-Plesset correlation energies of water, methane, acetylene, ethylene, and benzene

We report second order Moller-Plesset (MP2) and MP2-F12 total energies on He, Ne, Ar, H(2)O, CH(4), C(2)H(2), C(2)H(4), and C(6)H(6), using the correlation consistent basis sets, aug-cc-pVXZ (X=D-7). Basis set extrapolation techniques are applied to the MP2 and MP2-F12/B methods. The performance of the methods is tested in the calculations of the atoms, He, Ne, and Ar. It is indicated that the two-point extrapolation of MP2-F12/B with the basis sets (X=5,6) is the most reliable. Similar accuracy is obtained using two-point extrapolated conventional MP2 with the basis sets (X=6,7). For the molecules investigated the valence MP2 correlation energy is estimated within 1 mE(h).     

Explicit correlation and intermolecular interactions: investigating carbon dioxide complexes with the CCSD(T)-F12 method

We have optimized the lowest energy structures and calculated interaction energies for the CO(2)-Ar, CO(2)-N(2), CO(2)-CO, CO(2)-H(2)O, and CO(2)-NH(3) dimers with the recently developed explicitly correlated coupled cluster singles doubles and perturbative triples [CCSD(T)]-F12 methods and the associated VXZ-F12 (where X = D,T,Q) basis sets. For a given cardinal number, we find that results obtained with the CCSD(T)-F12 methods are much closer to the CCSD(T) complete basis set limit than the conventional CCSD(T) results. The relatively modest increase in the computational cost between explicit and conventional CCSD(T) is more than compensated for by the impressive accuracy of the CCSD(T)-F12 method. We recommend use of the CCSD(T)-F12 methods in combination with the VXZ-F12 basis sets for the accurate determination of equilibrium geometries and interaction energies of weakly bound electron donor acceptor complexes.     

A hybrid scheme for the resolution-of-the-identity approximation in second-order Møller-Plesset linear-r(12) perturbation theory

In the framework of second-order M?ller-Plesset linear-r(12) (MP2-R12) perturbation theory, a method is developed and implemented that uses an auxiliary basis set for the resolution-of-the-identity (RI) approximation for the three- and four-electron integrals. In contrast to previous work, the two-electron integrals that must be evaluated never involve more than one auxiliary basis function. The new method therefore scales linearly with the number of auxiliary basis functions and is much more efficient than the previous one, which scaled quadratically. A general formulation of MP2-R12 theory is presented for various ansatze, approximations, and orbitals (canonical or localized). The new method is assessed by computations of the valence-shell second-order M?ller-Plesset correlation energy of a few small closed-shell systems. The preliminary calculations indicate that the difference between the new and previous methods is about one order of magnitude smaller than the errors that occur due to basis-set truncations and RI approximations and under the assumptions of generalized and extended Brillouin conditions.     

Application of Gaussian-type geminals in local second-order Møller-Plesset perturbation theory

In this work Gaussian-type Geminals (GTGs) are applied in local second-order Moller-Plesset perturbation theory to improve the basis set convergence. Our implementation is based on the weak orthogonality functional of Szalewicz et al., [Chem. Phys. Lett. 91, 169 (1982); J. Chem. Phys. 78, 1420 (1983)] and a newly developed program for calculating the necessary many-electron integrals. The local approximations together with GTGs in the treatment of the correlation energy are introduced and tested. First results for correlation energies of H(2)O, CH(4), CO, C(2)H(2), C(2)H(4), H(2)CO, and N(2)H(4) as well as some reaction and activation energies are presented. More than 97% of the valence-shell correlation energy is recovered using aug-cc-pVDZ basis sets and six GTGs per electron pair. The results are compared with conventional calculations using correlation-consistent basis sets as well as with MP2-R12 results.     

On the convergence of Z-averaged perturbation theory

Very high order open-shell Z-averaged perturbation theory (ZAPT) energies, equilibrium bond lengths, and harmonic vibrational frequencies have been computed for a suite of small molecules using a determinantal algorithm. The convergence of ZAPTn energies is compared to alternative Moller-Plesset (MP) perturbation theories built on restricted open-shell Hartree-Fock (ROMP, RMP, OPT1, and OPT2) and unrestricted Hartree-Fock (UMP) reference wave functions for NH(2) at three N-H bond lengths and for CN. The ZAPTn energy series closely parallel those of RMPn and ROMPn theories for these systems. Further, we examine the convergence of ZAPTn energies, equilibrium bond lengths (r(e)), and harmonic vibrational frequencies (omega(e)) for X (2)Sigma(g)(+) CN, X (4)Sigma(g) (-) C(2)(+), and b (2)Delta(g) C(2)(+), tracking oscillations in the energy series for the challenging latter system to order 1000. Finally, we obtain r(e) and omega(e) values from explicit ZAPT2 and ZAPT4 computations with a triple-zeta plus double polarization basis set. The ensuing results are very close to those from second- and fourth-order RMP and ROMP for the NO and CN molecules but are significantly closer to experiment in the case of (3)Sigma(g)(-) O(2). The ZAPTn series exhibit all the fascinating diversity of behavior previously observed for closed-shell MPn theory. Particularly encouraging is the ability of Feenberg transformations to remove erratic, strongly oscillatory, and divergent behavior that may occur in ZAPTn series and provide systematic improvements toward the full configuration interaction limit. In light of the appealing mathematical properties of ZAPT and similarity of results to those from the oft-applied RMP theory, coupled with the reductions in computational cost inherent in the ZAPT method relative to theories requiring different orbitals for different spins, we recommend low-order ZAPT for general applications to open-shell systems, particularly in cases where spin contamination is of concern.     

When does the non-variational nature of second-order Møller-Plesset energies manifest itself? All-electron correlation energies for open-shell atoms from K to Br

All-electron correlation energies E(c) are not very well known for open-shell atoms with more than 18 electrons. The complete basis-set (CBS) limits of second-order M?ller-Plesset (MP2) perturbation theory energies are obtained for open-shell atoms by computations in large basis sets combined with a knowledge of the MP2/CBS limit for the next larger closed-shell atom with the same valence shell structure. Then higher-order correlation corrections are found by coupled-cluster calculations using basis sets that are not quite as large. The method is validated for the open-shell atoms from Al to Cl for which E(c) is reasonably well established. Then, the method is used to obtain non-relativistic E(c) values, probably accurate to 3%, for the open-shell atoms of the fourth period: K, Sc-Cu, and Ga-Br. These energies are compared with the predictions of 19 density functionals and may be useful for the parameterization of new ones. The results show that MP2 overestimates |E(c)| for atoms heavier than Fe.     

Explicitly correlated local second-order perturbation theory with a frozen geminal correlation factor

The recently introduced MP2-R122*A(loc) and LMP2-R122*A(loc) methods are modified to use a short-range correlation factor expanded as a fixed linear combination of Gaussian geminals. Density fitting is used to reduce the effort for integral evaluation, and local approximations are introduced to improve the scaling of the computational resources with molecular size. The MP2-F122*A(loc) correlation energies converge very rapidly with respect to the atomic orbital basis set size. Already with the aug-cc-pVTZ basis the correlation energies computed for a set of 21 small molecules are found to be within 0.5% of the MP2 basis set limit. Furthermore the short-range correlation factor leads to an improved convergence of the resolution of the identity, and eliminates problems with long-range errors in density fitting caused by the linear r12 factor. The DF-LMP2-F122*A(loc) method is applied to compute second-order correlation energies for molecules with up to 49 atoms and more than 1600 basis functions.     

A simple and efficient CCSD(T)-F12 approximation

A new explicitly correlated CCSD(T)-F12 approximation is presented and tested for 23 molecules and 15 chemical reactions. The F12 correction strongly improves the basis set convergence of correlation and reaction energies. Errors of the Hartree-Fock contributions are effectively removed by including MP2 single excitations into the auxiliary basis set. Using aug-cc-pVTZ basis sets the CCSD(T)-F12 calculations are more accurate and two orders of magnitude faster than standard CCSD(T)/aug-cc-pV5Z calculations.     

Basis set and electron correlation effects on the polarizability and second hyperpolarizability of model open-shell pi-conjugated systems

The basis set and electron correlation effects on the static polarizability (alpha) and second hyperpolarizability (gamma) are investigated ab initio for two model open-shell pi-conjugated systems, the C(5)H(7) radical and the C(6)H(8) radical cation in their doublet state. Basis set investigations evidence that the linear and nonlinear responses of the radical cation necessitate the use of a less extended basis set than its neutral analog. Indeed, double-zeta-type basis sets supplemented by a set of d polarization functions but no diffuse functions already provide accurate (hyper)polarizabilities for C(6)H(8) whereas diffuse functions are compulsory for C(5)H(7), in particular, p diffuse functions. In addition to the 6-31G(*)+pd basis set, basis sets resulting from removing not necessary diffuse functions from the augmented correlation consistent polarized valence double zeta basis set have been shown to provide (hyper)polarizability values of similar quality as more extended basis sets such as augmented correlation consistent polarized valence triple zeta and doubly augmented correlation consistent polarized valence double zeta. Using the selected atomic basis sets, the (hyper)polarizabilities of these two model compounds are calculated at different levels of approximation in order to assess the impact of including electron correlation. As a function of the method of calculation antiparallel and parallel variations have been demonstrated for alpha and gamma of the two model compounds, respectively. For the polarizability, the unrestricted Hartree-Fock and unrestricted second-order M?ller-Plesset methods bracket the reference value obtained at the unrestricted coupled cluster singles and doubles with a perturbative inclusion of the triples level whereas the projected unrestricted second-order M?ller-Plesset results are in much closer agreement with the unrestricted coupled cluster singles and doubles with a perturbative inclusion of the triples values than the projected unrestricted Hartree-Fock results. Moreover, the differences between the restricted open-shell Hartree-Fock and restricted open-shell second-order M?ller-Plesset methods are small. In what concerns the second hyperpolarizability, the unrestricted Hartree-Fock and unrestricted second-order M?ller-Plesset values remain of similar quality while using spin-projected schemes fails for the charged system but performs nicely for the neutral one. The restricted open-shell schemes, and especially the restricted open-shell second-order M?ller-Plesset method, provide for both compounds gamma values close to the results obtained at the unrestricted coupled cluster level including singles and doubles with a perturbative inclusion of the triples. Thus, to obtain well-converged alpha and gamma values at low-order electron correlation levels, the removal of spin contamination is a necessary but not a sufficient condition. Density-functional theory calculations of alpha and gamma have also been carried out using several exchange-correlation functionals. Those employing hybrid exchange-correlation functionals have been shown to reproduce fairly well the reference coupled cluster polarizability and second hyperpolarizability values. In addition, inclusion of Hartree-Fock exchange is of major importance for determining accurate polarizability whereas for the second hyperpolarizability the gradient corrections are large.