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.     

Second-order Møller-Plesset theory with linear R12 terms (MP2-R12) revisited: auxiliary basis set method and massively parallel implementation

Ab initio electronic structure approaches in which electron correlation explicitly appears have been the subject of much recent interest. Because these methods accelerate the rate of convergence of the energy and properties with respect to the size of the one-particle basis set, they promise to make accuracies of better than 1 kcal/mol computationally feasible for larger chemical systems than can be treated at present with such accuracy. The linear R12 methods of Kutzelnigg and co-workers are currently the most practical means to include explicit electron correlation. However, the application of such methods to systems of chemical interest faces severe challenges, most importantly, the still steep computational cost of such methods. Here we describe an implementation of the second-order M?ller-Plesset method with terms linear in the interelectronic distances (MP2-R12) which has a reduced computational cost due to the use of two basis sets. The use of two basis sets in MP2-R12 theory was first investigated recently by Klopper and Samson and is known as the auxiliary basis set (ABS) approach. One of the basis sets is used to describe the orbitals and another, the auxiliary basis set, is used for approximating matrix elements occurring in the exact MP2-R12 theory. We further extend the applicability of the approach by parallelizing all steps of the integral-direct MP2-R12 energy algorithm. We discuss several variants of the MP2-R12 method in the context of parallel execution and demonstrate that our implementation runs efficiently on a variety of distributed memory machines. Results of preliminary applications indicate that the two-basis (ABS) MP2-R12 approach cannot be used safely when small basis sets (such as augmented double- and triple-zeta correlation consistent basis sets) are utilized in the orbital expansion. Our results suggest that basis set reoptimization or further modifications of the explicitly correlated ansatz and/or standard approximations for matrix elements are necessary in order to make the MP2-R12 method sufficiently accurate when small orbital basis sets are used. The computer code is a part of the latest public release of Sandia's Massively Parallel Quantum Chemistry program available under GNU General Public License.     

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.     

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 quantum-chemical calculations using Gaussian-type geminal and Gaussian-type orbital basis sets: applications to atoms and diatomics

We have implemented the use of mixed basis sets of Gaussian one- and two-electron (geminal) functions for the calculation of second-order M?ller-Plesset (MP2) correlation energies. In this paper, we describe some aspects of this implementation, including different forms chosen for the pair functions. Computational results are presented for some closed-shell atoms and diatomics. Our calculations indicate that the method presented is capable of yielding highly accurate second-order correlation energies with rather modest Gaussian orbital basis sets, providing an alternative route to highly accurate wave functions. For the neon atom, the hydrogen molecule, and the hydrogen fluoride molecule, our calculations yield the most accurate MP2 energies published so far. A critical comparison is made with established MP2-R12 methods, revealing an erratic behaviour of some of these methods, even in large basis sets.     

Auxiliary basis sets for density-fitted correlated wavefunction calculations: weighted core-valence and ECP basis sets for post-d elements

We report optimised auxiliary basis sets for the resolution-of-the-identity (or density-fitting) approximation of two-electron integrals in second-order M?ller-Plesset perturbation theory (MP2) and similar electronic structure calculations with correlation-consistent basis sets for the post-d elements Ga-Kr, In-Xe, and Tl-Rn. The auxiliary basis sets are optimised such that the density-fitting error is negligible compared to the one-electron basis set error. To check to which extent this criterion is fulfilled we estimated for a test set of 80 molecules the basis set limit of the correlation energy at the MP2 level and evaluated the remaining density-fitting and the one-electron basis set errors. The resulting auxiliary basis sets are only 2-6 times larger than the corresponding one-electron basis sets and lead in MP2 calculations to speed-ups of the integral evaluation by one to three orders of magnitude. The density-fitting errors in the correlation energy are at least hundred times smaller than the one-electron basis set error, i.e. in the order of only 1-100 μH per atom.     

SF-[2]R12: a spin-adapted explicitly correlated method applicable to arbitrary electronic states

The [2](R12) method [M. Torheyden and E. F. Valeev, J. Chem. Phys. 131, 171103 (2009)] is an explicitly correlated perturbative correction that can greatly reduce the basis set error of an arbitrary electronic structure method for which the two-electron density matrix is available. Here we present a spin-adapted variant (denoted as SF-[2](R12)) that is formulated completely in terms of spin-free quantities. A spin-free cumulant decomposition and multi-reference generalized Brillouin condition are used to avoid three-particle reduced density matrix completely. The computational complexity of SF-[2](R12) is proportional to the sixth power of the system size and is comparable to the cost of the single-reference MP2-R12 method. The SF-[2](R12) method is shown to decrease greatly the basis set error of multi-configurational wave functions.     

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).     

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.     

Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn

The introduction of the resolution-of-the-identity (RI) approximation for electron repulsion integrals in quantum chemical calculations requires in addition to the orbital basis so-called auxiliary or fitting basis sets. We report here such auxiliary basis sets optimized for second-order Møller–Plesset perturbation theory for the recently published (Weigend and Ahlrichs Phys Chem Chem Phys, 2005, 7, 3297–3305) segmented contracted Gaussian basis sets of split, triple-ζ and quadruple-ζ valence quality for the atoms Rb–Rn (except lanthanides). These basis sets are designed for use in connection with small-core effective core potentials including scalar relativistic corrections. Hereby accurate resolution-of-the-identity calculations with second-order Møller–Plesset perturbation theory (MP2) and related methods can now be performed for molecules containing elements from H to Rn. The error of the RI approximation has been evaluated for a test set of 385 small and medium sized molecules, which represent the common oxidation states of each element, and is compared with the one-electron basis set error, estimated based on highly accurate explicitly correlated MP2–R12 calculations. With the reported auxiliary basis sets the RI error for MP2 correlation energies is typically two orders of magnitude smaller than the one-electron basis set error, independent on the position of the atoms in the periodic table.     

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.     

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.     

Approaching the theoretical limit in periodic local MP2 calculations with atomic-orbital basis sets: the case of LiH

The atomic orbital basis set limit is approached in periodic correlated calculations for solid LiH. The valence correlation energy is evaluated at the level of the local periodic second order M?ller-Plesset perturbation theory (MP2), using basis sets of progressively increasing size, and also employing "bond"-centered basis functions in addition to the standard atom-centered ones. Extended basis sets, which contain linear dependencies, are processed only at the MP2 stage via a dual basis set scheme. The local approximation (domain) error has been consistently eliminated by expanding the orbital excitation domains. As a final result, it is demonstrated that the complete basis set limit can be reached for both HF and local MP2 periodic calculations, and a general scheme is outlined for the definition of high-quality atomic-orbital basis sets for solids.     

Some investigations of the MP2-R12 method

Summary The MP2-R12 method was introduced by Kutzelnigg and Klopper to overcome the problem caused by truncation of the one electron basis set in correlation energy calculations at the Møller-Plesset second order level of approximation. Here, we have evaluated the integrals required by their simplest scheme using the Rys-quadrature procedure. Results are presented for Ne, H₂O, and HF using largespdf gaussian basis sets.     

Analytic calculation of first-order molecular properties at the explicitly correlated second-order Moller-Plesset level: basis-set limits for the molecular quadrupole moments of BH and HF

The analytic calculation of first-order properties has been implemented in the DALTON program at the level of explicitly correlated second-order Moller-Plesset perturbation theory (MP2-R12). The implementation has been accomplished for MP2-R12 theory based on standard approximations A, A', and B, using an auxiliary basis for the resolution-of-the-identity approximation, with and without a frozen core. MP2-R12 first-order molecular properties have been calculated analytically for a few small test molecules. For BH and HF, the MP2-R12 results were supplemented with explicitly correlated coupled-cluster calculations (but at this level from numerical derivatives) including vibrational and relativistic corrections.     

Theoretical reference values for the AE6 and BH6 test sets from explicitly correlated coupled-cluster theory

We propose a new computational protocol to obtain highly accurate theoretical reference data. This protocol employs the explicitly correlated coupled-cluster method with iterative single and double excitations as well as perturbative triple excitations, CCSD(T)(F12), using quadruple-z\zeta basis sets. Higher excitations are accounted for by conventional CCSDT(Q) calculations using double-z\zeta basis sets, while core/core-valence correlation effects are estimated by conventional CCSD(T) calculations using quadruple-z\zeta basis sets. Finally, scalar-relativistic effects are accounted for by conventional CCSD(T) calculations using triple-z\zeta basis sets. In the present article, this protocol is applied to the popular test sets AE6 and BH6. An error analysis shows that the new reference values obtained by our computational protocol have an uncertainty of less than 1 kcal/mol (chemical accuracy). Furthermore, concerning the atomization energies, a cancellation of the basis set incompleteness error in the CCSD(T)(F12) perturbative triples contribution with the corresponding error in the contribution from higher excitations is observed. This error cancellation is diminished by the CCSD(T*)(F12) method. Thus, we recommend the use of the CCSD(T*)(F12) method only for small- and medium-sized basis sets, while the CCSD(T)(F12) approach is preferred for high-accuracy calculations in large basis sets.     

Second order explicitly correlated R12 theory revisited: a second quantization framework for treatment of the operators' partitionings

Second order R12 theory is presented and derived alternatively using the second quantized hole-particle formalism. We have shown that in order to ensure the strong orthogonality between the R12 and the conventional part of the wave function, the explicit use of projection operators can be easily avoided by an appropriate partitioning of the involved operators to parts which are fully describable within the computational orbital basis and complementary parts that involve imaginary orbitals from the complete orbital basis. Various Hamiltonian splittings are discussed and computationally investigated for a set of nine molecules and their atomization energies. If no generalized Brillouin condition is assumed, with all relevant partitionings the one-particle contribution arising in the explicitly correlated part of the first order wave function has to be considered and has a significant role when smaller atomic orbital basis sets are used. The most appropriate Hamiltonian splitting results if one follows the conventional perturbation theory for a general non-Hartree-Fock reference. Then, no couplings between the R12 part and the conventional part arise within the first order wave function. The computationally most favorable splitting when the whole complementary part of the Hamiltonian is treated as a perturbation fails badly. These conclusions also apply to MP2-F12 approaches with different correlation factors.     

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.     

Estimates of the ab initio limit for pi-pi interactions: the benzene dimer

State-of-the-art electronic structure methods have been applied to the simplest prototype of aromatic pi-pi interactions, the benzene dimer. By comparison to results with a large aug-cc-pVTZ basis set, we demonstrate that more modest basis sets such as aug-cc-pVDZ are sufficient for geometry optimizations of intermolecular parameters at the second-order M?ller-Plesset perturbation theory (MP2) level. However, basis sets even larger than aug-cc-pVTZ are important for accurate binding energies. The complete basis set MP2 binding energies, estimated by explicitly correlated MP2-R12/A techniques, are significantly larger in magnitude than previous estimates. When corrected for higher-order correlation effects via coupled cluster with singles, doubles, and perturbative triples [CCSD(T)], the binding energies D(e) (D(0)) for the sandwich, T-shaped, and parallel-displaced configurations are found to be 1.8 (2.0), 2.7 (2.4), and 2.8 (2.7) kcal mol(-1), respectively.