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.     

An efficient local coupled cluster method for accurate thermochemistry of large systems

An efficient local coupled cluster method with single and double excitation operators and perturbative treatment of triple excitations [DF-LCCSD(T)] is described. All required two-electron integrals are evaluated using density fitting approximations. These have a negligible effect on the accuracy but reduce the computational effort by 1-2 orders of magnitude, as compared to standard integral-direct methods. Excitations are restricted to local subsets of non-orthogonal virtual orbitals (domain approximation). Depending on distance criteria, the correlated electron pairs are classified into strong, close, weak, and very distant pairs. Only strong pairs, which typically account for more than 90% of the correlation energy, are optimized in the LCCSD treatment. The remaining close and weak pairs are approximated by LMP2 (local second-order Mo?ller-Plesset perturbation theory); very distant pairs are neglected. It is demonstrated that the accuracy of this scheme can be significantly improved by including the close pair LMP2 amplitudes in the LCCSD equations, as well as in the perturbative treatment of the triples excitations. Using this ansatz for the wavefunction, the evaluation and transformation of the two-electron integrals scale cubically with molecular size. If local density fitting approximations are activated, this is reduced to linear scaling. The LCCSD iterations scale quadratically, but linear scaling can be achieved by neglecting some terms involving contractions of single excitations. The accuracy and efficiency of the method is systematically tested using various approximations, and calculations for molecules with up to 90 atoms and 2636 basis functions are presented.     

Application of second-order Møller-Plesset perturbation theory with resolution-of-identity approximation to periodic systems

Efficient periodic boundary condition (PBC) calculations by the second-order M?ller-Plesset perturbation (MP2) method based on crystal orbital formalism are developed by introducing the resolution-of-identity (RI) approximation of four-center two-electron repulsion integrals (ERIs). The formulation and implementation of the PBC RI-MP2 method are presented. In this method, the mixed auxiliary basis functions of the combination of Poisson and Gaussian type functions are used to circumvent the slow convergence of the lattice sum of the long-range ERIs. Test calculations of one-dimensional periodic trans-polyacetylene show that the PBC RI-MP2 method greatly reduces the computational times as well as memory and disk sizes, without the loss of accuracy, compared to the conventional PBC MP2 method.     

Coupled-cluster theory with simplified linear-r(12) corrections: the CCSD(R12) model

A simplified singles-and-doubles linear-r(12) corrected coupled-cluster model, denoted CCSD(R12), is proposed and compared with the complete singles-and-doubles linear-r(12) coupled-cluster method CCSD-R12. An orthonormal auxiliary basis set is used for the resolution-of-the-identity approximation to calculate three-electron integrals needed in the linear-r(12) Ansatz. Basis-set convergence is investigated for a selected set of atoms and small molecules. In a large basis, the CCSD(R12) model provides an excellent approximation to the full linear-r(12) energy contribution, whereas the magnitude of this contribution is significantly overestimated at the level of second-order perturbation theory.     

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.     

The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development

The computation of the two-electron four-center integrals over gaussian basis functions is a significant component of the overall work of many ab initio methods used today. Improvements in the computational efficiency of the base algorithms have provided significant impact. Somewhat overlooked are methods that provide approximations to these integrals and their implementation in application software. A partial review of approximate integral techniques focused on the resolution of the identity (RI) four-center, two-electron integral approximation is given. The past and current uses of the RI algorithms are presented along with possibilities for further exploitation of the technology. Received: 14 January 1997 / Accepted: 11 March 1997     

Explicit correlation and basis set superposition error: the structure and energy of carbon dioxide dimer

We have investigated the slipped parallel and t-shaped structures of carbon dioxide dimer [(CO(2))(2)] using both conventional and explicitly correlated coupled cluster methods, inclusive and exclusive of counterpoise (CP) correction. We have determined the geometry of both structures with conventional coupled cluster singles doubles and perturbative triples theory [CCSD(T)] and explicitly correlated cluster singles doubles and perturbative triples theory [CCSD(T)-F12b] at the complete basis set (CBS) limits using custom optimization routines. Consistent with previous investigations, we find that the slipped parallel structure corresponds to the global minimum and is 1.09 kJ mol(-1) lower in energy. For a given cardinal number, the optimized geometries and interaction energies of (CO(2))(2) obtained with the explicitly correlated CCSD(T)-F12b method are closer to the CBS limit than the corresponding conventional CCSD(T) results. Furthermore, the magnitude of basis set superposition error (BSSE) in the CCSD(T)-F12b optimized geometries and interaction energies is appreciably smaller than the magnitude of BSSE in the conventional CCSD(T) results. We decompose the CCSD(T) and CCSD(T)-F12b interaction energies into the constituent HF or HF CABS, CCSD or CCSD-F12b, and (T) contributions. We find that the complementary auxiliary basis set (CABS) singles correction and the F12b approximation significantly reduce the magnitude of BSSE at the HF and CCSD levels of theory, respectively. For a given cardinal number, we find that non-CP corrected, unscaled triples CCSD(T)-F12b/VXZ-F12 interaction energies are in overall best agreement with the CBS limit.     

Distributed memory parallel implementation of energies and gradients for second-order Møller-Plesset perturbation theory with the resolution-of-the-identity approximation

We present a parallel implementation of second-order M?ller-Plesset perturbation theory with the resolution-of-the-identity approximation (RI-MP2). The implementation is based on a recent improved sequential implementation of RI-MP2 within the Turbomole program package and employs the message passing interface (MPI) standard for communication between distributed memory nodes. The parallel implementation extends the applicability of canonical MP2 to considerably larger systems. Examples are presented for full geometry optimizations with up to 60 atoms and 3300 basis functions and MP2 energy calculations with more than 200 atoms and 7000 basis functions.     

On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions

The correlation energy in the direct random phase approximation (dRPA) can be written, among other possibilities, either in terms of the interaction strength averaged correlation density matrix, or in terms of the coupled cluster doubles amplitudes obtained in the direct ring approximation (drCCD). Although the corresponding dRPA correlation density matrix on the one hand, and the drCCD amplitude matrix on the other hand, differ significantly, they yield identical energies. Similarly, the analogous RPA and rCCD correlation energies calculated from antisymmetrized two-electron integrals are identical to each other despite very different underlying working equations. In the present communication, a direct correspondence between amplitudes and densities is established and investigated with perturbation theory arguments. Our analysis also sheds some light on the properties of recently proposed RPA/rCCD variants which use antisymmetrized integrals in part of the equations and nonantisymmetrized integrals in others.     

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.     

Coupled-cluster singles and doubles for extended systems

Coupled-cluster theory with connected single and double excitation operators (CCSD) and related approximations, such as linearized CCSD, quadratic configuration interaction with single and double excitation operators, coupled-cluster with connected double excitation operator (CCD), linearized CCD, approximate CCD, and second- and third-order many-body perturbation theories, are formulated and implemented for infinitely extended one-dimensional systems (polymers), on the basis of the periodic boundary conditions and distance-based screening of integrals, density matrix elements, and excitation amplitudes. The variation of correlation energies with the truncation radii of short- and long-range lattice sums and with the number of wave vector sampling points in the first Brillouin zone is examined for polyethylene, polyacetylene, and polyyne, and is shown to be a function of the degree of pi-electron conjugation or the fundamental band gaps. The t2 and t1 amplitudes in the atomic orbital (AO) basis are obtained by first computing the t amplitudes in the Bloch-orbital basis and subsequently back-transforming them into the AO basis. The plot of these AO-based t amplitudes as a function of unit cells also indicates that the t2 amplitudes of polyacetylene and polyyne exhibit appreciably slower decay than those of polyethylene, although the asymptotic decay behavior is invariably 1/r3. The AO-based t1 amplitudes appear to correlate strongly with the electronic structure, and they decay seemingly exponentially for polyethylene whereas they stay at a constant magnitude across the seventh nearest neighbors of polyacetylene and polyyne, which attests to far reaching effects of nondynamical electron correlation mediated by orbital rotation. Nonetheless, the unit cell contributions to the correlation energies taper below 10(-6) hartree after 15 A for all three polymers. The basis set dependence of the decay behavior of t2 amplitudes is also examined for linear hydrogen fluoride polymer (HF)infinity and linear beryllium polymer (Be)infinity employing the STO-3G, 6-31G, and 6-31G* basis sets, and proves to be rather small.     

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.     

Parallelization of four-component calculations. II. Symmetry-driven parallelization of the 4-Spinor CCSD algorithm

Given the importance of the Coupled-cluster (CC) method as an efficient and accurate way to take electron correlation into account, we extend the parallelization technique in the second part of this series also to the 4-Spinor CCSD algorithm implemented in the Dirac-Fock packages DIRAC and MOLFDIR. The present implementation is based on the availability of the transformed molecular two-electron integrals on an external storage medium. The linearity of the CC equations in these two-electron integrals is used in a parallelization strategy that is based on distribution of the two largest integral classes that carry three or four virtual spinor indices. The corresponding partial contributions to the T(1) and T(2) amplitudes are calculated on each node and added using Message Passing Interface (MPI) library calls. Although we did not employ a master/slave principle, one specific node was assigned to also perform the remaining serial parts of the algorithm. In the critical sections considerable savings in storage requirements and computer time could be achieved, and this allows for computations on larger systems in the framework of four-component theory.     

Resolution of the identity atomic orbital Laplace transformed second order Møller-Plesset theory for nonconducting periodic systems

An improvement in performance of the atomic orbital Laplace transformed second-order M?ller-Plesset (AO-LT-MP2) method for periodic systems is reported using the resolution of identity (RI) technique. Transformation of the two-electron integrals constitutes the main computational bottleneck of the AO-LT-MP2 method. A substitution of regular four-center integrals by their three center counterparts in the RI approximation naturally reduces the computational cost of the integral transformation step. The RI divergence problem in the presence of periodic boundary conditions is solved in our implementation by restricting the fitting domain. Accuracy and computational efficiency of the RI-AO-LT-MP2 approach are assessed on a set of one-dimensional test systems: trans-polyacetylene and anti-transoid polymethineimine.     

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.     

Low-order tensor approximations for electronic wave functions: Hartree-Fock method with guaranteed precision

Here we report a formulation of the Hartree-Fock method in an adaptive multiresolution basis set of spectral element type. A key feature of our approach is the use of low-order tensor approximations for operators and wave functions to reduce the steep rise of storage and computational costs with the number of degrees of freedom that plague finite element computations. As a proof of principle we implemented Hartree-Fock method without explicit storage of the full-dimensional wave function and with guaranteed precision (microhartree precision for up to 14 electron systems is demonstrated). Even for the one-electron method the use of low-order tensor approximation reduces storage relative to the full representation, albeit with modest increase in cost. Preliminary tests for explicitly-correlated two-electron (six-dimensional) wave function suggest a factor of 50 savings in storage. At least correlated two-electron methods should be feasible with our approach on modern workstations with guaranteed precision.     

Coupled-cluster methods with perturbative inclusion of explicitly correlated terms: a preliminary investigation

We propose to account for the large basis-set error of a conventional coupled-cluster energy and wave function by a simple perturbative correction. The perturbation expansion is constructed by L?wdin partitioning of the similarity-transformed Hamiltonian in a space that includes explicitly correlated basis functions. To test this idea, we investigate the second-order explicitly correlated correction to the coupled-cluster singles and doubles (CCSD) energy, denoted here as the CCSD(2)(R12) method. The proposed perturbation expansion presents a systematic and easy-to-interpret picture of the "interference" between the basis-set and correlation hierarchies in the many-body electronic-structure theory. The leading-order term in the energy correction is the amplitude-independent R12 correction from the standard second-order M?ller-Plesset R12 method. The cluster amplitudes appear in the higher-order terms and their effect is to decrease the basis-set correction, in accordance with the usual experience. In addition to the use of the standard R12 technology which simplifies all matrix elements to at most two-electron integrals, we propose several optional approximations to select only the most important terms in the energy correction. For a limited test set, the valence CCSD energies computed with the approximate method, termed , are on average precise to (1.9, 1.4, 0.5 and 0.1%) when computed with Dunning's aug-cc-pVXZ basis sets [X = (D, T, Q, 5)] accompanied by a single Slater-type correlation factor. This precision is a roughly an order of magnitude improvement over the standard CCSD method, whose respective average basis-set errors are (28.2, 10.6, 4.4 and 2.1%). Performance of the method is almost identical to that of the more complex iterative counterpart, CCSD(R12). The proposed approach to explicitly correlated coupled-cluster methods is technically appealing since no modification of the coupled-cluster equations is necessary and the standard M?ller-Plesset R12 machinery can be reused.     

New coupled cluster approaches based on the unrestricted Hartree-Fock reference for treating molecules with multireference character

Here we review the basic formalism, implementation details, and performance of two newly developed coupled cluster (CC) methods based on the unrestricted Hartree-Fock (UHF) reference for treating molecules with multireference character. These two approaches can be considered to be approximations to the CC singles, doubles, and triples (CCSDT) method. The key concept of these two approaches is the corresponding orbitals, which are unitary transformations of canonical UHF molecular orbitals so that all spin orbitals are grouped into unique orbital pairs. In one approach called CCSDT(5P), a subset of triple excitations involving up to five-pair indices is included. In another approach called CCSD(T)-h, the contribution of connected triple excitations is treated in a hybrid way. With the concept of active corresponding orbitals, triple excitations can be automatically partitioned into two subsets, and the amplitudes of these two subsets are determined via solving different equations. Both CCSD(T)-h and CCSDT(5P) computationally scale as the seventh power of the system size. A survey of a number of applications demonstrates that CCSD(T)-h is an excellent approximation to the full CCSDT method, and CCSDT(5P) provides a good approximation to CCSDT for single-bond breaking processes. The overall performance of CCSDT(5P) is less accurate than that of CCSD(T)-h, but significantly better than that of the widely used CCSD(T).     

Analytic derivatives for the Cholesky representation of the two-electron integrals

We propose a formalism for calculating analytic derivatives of the electronic energy with respect to nuclear coordinates using Cholesky decomposition of the two-electron integrals. The formalism is derived by exploiting the equivalence of Cholesky decomposition and density fitting when a suitable auxiliary basis set is used for expanding atomic orbital product densities in the latter. An implementation of gradients at the nonhybrid density functional theory level is presented, and sample calculations demonstrate that the errors in equilibrium geometries due to the Cholesky representation of the integrals can be controlled by adjusting the decomposition threshold.     

Density-functional theory-symmetry-adapted intermolecular perturbation theory with density fitting: a new efficient method to study intermolecular interaction energies

The previously developed DFT-SAPT approach, which combines symmetry-adapted intermolecular perturbation theory (SAPT) with a density-functional theory (DFT) representation of the monomers, has been implemented by using density fitting of two-electron objects. This approach, termed DF-DFT-SAPT, scales with the fifth power of the molecular size and with the third power upon increase of the basis set size for a given dimer, thus drastically reducing the cost of the conventional DFT-SAPT method. The accuracy of the density fitting approximation has been tested for the ethyne dimer. It has been found that the errors in the interaction energies due to density fitting are below 10(-3) kcal/mol with suitable auxiliary basis sets and thus one or two orders of magnitude smaller than the errors due to the use of a limited atomic orbital basis set. An investigation of three prominent structures of the benzene dimer, namely, the T shaped, parallel displaced, and sandwich geometries, employing basis sets of up to augmented quadruple-zeta quality shows that DF-DFT-SAPT outperforms second-order Moller-Plesset theory (MP2) and gives total interaction energies which are close to the best estimates inferred from combining the results of MP2 and coupled-cluster theory with single, double, and perturbative triple excitations.