Quadratically convergent algorithm for orbital optimization in the orbital-optimized coupled-cluster doubles method and in orbital-optimized second-order Møller-Plesset perturbation theory

Using a Lagrangian-based approach, we present a more elegant derivation of the equations necessary for the variational optimization of the molecular orbitals (MOs) for the coupled-cluster doubles (CCD) method and second-order M?ller-Plesset perturbation theory (MP2). These orbital-optimized theories are referred to as OO-CCD and OO-MP2 (or simply "OD" and "OMP2" for short), respectively. We also present an improved algorithm for orbital optimization in these methods. Explicit equations for response density matrices, the MO gradient, and the MO Hessian are reported both in spin-orbital and closed-shell spin-adapted forms. The Newton-Raphson algorithm is used for the optimization procedure using the MO gradient and Hessian. Further, orbital stability analyses are also carried out at correlated levels. The OD and OMP2 approaches are compared with the standard MP2, CCD, CCSD, and CCSD(T) methods. All these methods are applied to H(2)O, three diatomics, and the O(4)(+) molecule. Results demonstrate that the CCSD and OD methods give nearly identical results for H(2)O and diatomics; however, in symmetry-breaking problems as exemplified by O(4)(+), the OD method provides better results for vibrational frequencies. The OD method has further advantages over CCSD: its analytic gradients are easier to compute since there is no need to solve the coupled-perturbed equations for the orbital response, the computation of one-electron properties are easier because there is no response contribution to the particle density matrices, the variational optimized orbitals can be readily extended to allow inactive orbitals, it avoids spurious second-order poles in its response function, and its transition dipole moments are gauge invariant. The OMP2 has these same advantages over canonical MP2, making it promising for excited state properties via linear response theory. The quadratically convergent orbital-optimization procedure converges quickly for OMP2, and provides molecular properties that are somewhat different than those of MP2 for most of the test cases considered (although they are similar for H(2)O). Bond lengths are somewhat longer, and vibrational frequencies somewhat smaller, for OMP2 compared to MP2. In the difficult case of O(4)(+), results for several vibrational frequencies are significantly improved in going from MP2 to OMP2.     

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.     

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.     

An improved algorithm for analytical gradient evaluation in resolution-of-the-identity second-order Møller-Plesset perturbation theory: application to alanine tetrapeptide conformational analysis

We present a new algorithm for analytical gradient evaluation in resolution‐of‐the‐identity second‐order Møller‐Plesset perturbation theory (RI‐MP2) and thoroughly assess its computational performance and chemical accuracy. This algorithm addresses the potential I/O bottlenecks associated with disk‐based storage and access of the RI‐MP2 t‐amplitudes by utilizing a semi‐direct batching approach and yields computational speed‐ups of approximately 2–3 over the best conventional MP2 analytical gradient algorithms. In addition, we attempt to provide a straightforward guide to performing reliable and cost‐efficient geometry optimizations at the RI‐MP2 level of theory. By computing relative atomization energies for the G3/99 set and optimizing a test set of 136 equilibrium molecular structures, we demonstrate that satisfactory relative accuracy and significant computational savings can be obtained using Pople‐style atomic orbital basis sets with the existing auxiliary basis expansions for RI‐MP2 computations. We also show that RI‐MP2 geometry optimizations reproduce molecular equilibrium structures with no significant deviations (>0.1 pm) from the predictions of conventional MP2 theory. As a chemical application, we computed the extended‐globular conformational energy gap in alanine tetrapeptide at the extrapolated RI‐MP2/cc‐pV(TQ)Z level as 2.884, 4.414, and 4.994 kcal/mol for structures optimized using the HF, DFT (B3LYP), and RI‐MP2 methodologies and the cc‐pVTZ basis set, respectively. These marked energetic discrepancies originate from differential intramolecular hydrogen bonding present in the globular conformation optimized at these levels of theory and clearly demonstrate the importance of long‐range correlation effects in polypeptide conformational analysis. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2007     

Local explicitly correlated second-order Møller-Plesset perturbation theory with pair natural orbitals

We explore using a pair natural orbital analysis of approximate first-order pair functions as means to truncate the space of both virtual and complementary auxiliary orbitals in the context of explicitly correlated F12 methods using localised occupied orbitals. We demonstrate that this offers an attractive procedure and that only 10-40 virtual orbitals per significant pair are required to obtain second-order valence correlation energies to within 1-2% of the basis set limit. Moreover, for this level of virtual truncation, only 10-40 complementary auxiliary orbitals per pair are required for an accurate resolution of the identity in the computation of the three- and four-electron integrals that arise in explicitly correlated methods.     

General biorthogonal projected bases as applied to second-order Møller-Plesset perturbation theory

With low-order scaling correlated wave function theories in mind, we present second quantization formalism as well as biorthonormalization procedures for general--singular or nonsingular--bases. Of particular interest are the so-called projected atomic orbital bases, which are obtained from a set of atom-centered functions and feature a separation of occupied and virtual spaces. We demonstrate the formalism by deriving and implementing second-order M?ller-Plesset perturbation theory in it, and discuss the convergence and preconditioning of the iterative amplitude equations in detail.     

Quasirelativistic theory for the magnetic shielding constant. III. Quasirelativistic second-order Møller-Plesset perturbation theory and its application to tellurium compounds

The quasirelativistic (QR) generalized unrestricted Hartree-Fock method for the magnetic shielding constant [R. Fukuda, M. Hada, and H. Nakatsuji, J. Chem. Phys. 118, 1015 (2003); R. Fukuda, M. Hada, and H. Nakatsuji, J. Chem. Phys.118, 1027 (2003)] has been extended to include the electron correlation effect in the level of the second-order M?ller-Plesset perturbation theory (MP2). We have implemented the energy gradient and finite-perturbation methods to calculate the magnetic shielding constant at the QR MP2 level and applied to the magnetic shielding constants and the NMR chemical shifts of 125Te nucleus in various tellurium compounds. The calculated magnetic shielding constants and NMR chemical shifts well reproduced the experimental values. The relations of the chemical shifts with the natures of ligands, and the tellurium oxidation states were investigated. The chemical shifts in different valence states were explained by the paramagnetic shielding and spin-orbit terms. The tellurium 5p electrons are the dominant origin of the chemical shifts in the Te I and Te II compounds and the chemical shifts were explained by the p-hole mechanism. The tellurium d electrons also play an important role in the chemical shifts of the hypervalent compounds.     

Intermolecular potentials of the methane dimer calculated with Møller-Plesset perturbation theory and density functional theory

We have calculated the intermolecular interaction potentials of the methane dimer at the minimum-energy D(3d) conformation using the Hartree-Fock (HF) self-consistent theory, the correlation-corrected second-order M?ller-Plesset (MP2) perturbation theory, and the density functional theory (DFT) with the Perdew-Wang (PW91) functional as the exchange or the correlation part. The HF calculations yield unbound potentials largely due to the exchange-repulsion interaction. In the MP2 calculations, the basis set effects on the repulsion exponent, the equilibrium bond length, the binding energy, and the asymptotic behavior of the calculated intermolecular potentials have been thoroughly studied. We have employed basis sets from the Slater-type orbitals fitted with Gaussian functions (STO-nG) (n=3-6) [Quantum Theory of Molecular and Solids: The Self-Consistent Field for Molecular and Solids (McGraw-Hill, New York, 1974), Vol. 4], Pople's medium size basis sets of Krishnan et al. [J. Chem. Phys. 72, 650 (1980)] [up to 6-311++G(3df,3pd)] to Dunning's correlation consistent basis sets [J. Chem. Phys. 90, 1007 (1989)] (cc-pVXZ and aug-cc-pVXZ) (X=D, T, and Q). With increasing basis size, the repulsion exponent and the equilibrium bond length converge at the 6-31G** basis set and the 6-311++G(2d,2p) basis set, respectively, while a large basis set (aug-cc-pVTZ) is required to converge the binding energy at a chemical accuracy (approximately 0.01 kcal/mol). Up to the largest basis set used, the asymptotic dispersion coefficient has not converged to the destined C6 value from molecular polarizability calculations. The slow convergence could indicate the inefficacy of using the MP2 calculations with Gaussian-type functions to model the asymptotic behavior. Both the basis set superposition error (BSSE) corrected and uncorrected results are presented to emphasize the importance of including such corrections. Only the BSSE corrected results systematically converge to the destined potential curve with increasing basis size. The DFT calculations generate a wide range of interaction patterns, from purely unbound to strongly bound, underestimating or overestimating the binding energy. The binding energy calculated using the PW91PW91 functional and the equilibrium bond length calculated using the PW91VP86 functional are close to the MP2 results at the basis set limit.     

New implementation of second-order Møller-Plesset perturbation theory with an analytic Slater-type geminal

The author introduces a new method for the exchange commutator integrals in explicitly correlated M?ller-Plesset second order perturbation theory. The method is well suited with an analytic Slater-type geminal correlation factor. He also explains the scheme for auxiliary integrals needed for the correlation factor. Based on different Ans?tze, he analyzes the performance of the method on correlation energies and reaction enthalpies in detail.     

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.     

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.     

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.     

Auxiliary basis sets for density fitting second-order Møller-Plesset perturbation theory: correlation consistent basis sets for the 5d elements Hf-Pt

Auxiliary basis sets specifically matched to the correlation consistent cc-pVnZ-PP, cc-pwCVnZ-PP, aug-cc-pVnZ-PP, and aug-cc-pwCVnZ-PP orbital basis sets (used in conjunction with pseudopotentials) for the 5d transition metal elements Hf-Pt have been optimized for use in density fitting second-order M?ller-Plesset perturbation theory and other correlated ab initio methods. Calculations of the second-order M?ller-Plesset perturbation theory correlation energy, for a test set of small to medium sized molecules, indicate that the density fitting error when utilizing these sets is negligible at three to four orders of magnitude smaller than the orbital basis set incompleteness error.     

Intermolecular potentials of the silane dimer calculated with Hartree-Fock theory, Møller-Plesset perturbation theory, and density functional theory

We have calculated the intermolecular interaction potentials of the silane dimer at the D3d conformation using the Hartree-Fock (HF) self-consistent theory, the correlation-corrected second-order M?ller-Plesset (MP2) perturbation theory, and the density functional theory (DFT) with 108 functionals chosen from the combinations of 9 exchange and 12 correlation functionals. Single-point coupled cluster [CCSD(T)] calculations have also been carried out to calibrate the correlation effect. The HF calculations yield unbound potentials largely because of the exchange-repulsion interaction. In the MP2 calculations, the basis set effects on the repulsion exponent, the equilibrium bond length, the binding energy, and the asymptotic behavior of the calculated intermolecular potentials have been thoroughly studied. We have employed basis sets from the Slater type orbitals fitted with Gaussian functions (STO-nG, n = 3 approximately 6), Pople's medium size basis sets [up to 6-311++G(3df,3pd)], to Dunning's correlation consistent basis sets (cc-pVXZ and aug-cc-pVXZ, X = D, T, Q). With increasing basis size, the repulsion exponent and the equilibrium bond length converge at the 6-31G** basis set and the 6-311++G(3d,3p) basis set, respectively, while a large basis set (aug-cc-pVTZ) is required to converge the binding energy at a chemical accuracy ( approximately 0.05 kcal/mol). Up to the largest basis set used, the asymptotic dispersion coefficient has not converged to the expected C6 value from molecular polarizability calculations. We attribute the slow convergence partly to the inefficacy of using the MP2 calculations with Gaussian type functions to model the asymptotic behavior. Both the basis set superposition error (BSSE) corrected and uncorrected results are presented to emphasize the importance of including such corrections. Only the BSSE corrected results systematically converge to the expected potential curve with increasing basis size. The DFT calculations generate a wide range of interaction patterns, from purely unbound to strongly bound, underestimating or overestimating the binding energy. The binding energies calculated using the OPTXHCTH147, PBEVP86, PBEP86, PW91TPSS, PW91PBE, and PW91PW91 functionals and the equilibrium bond lengths calculated using the MPWHCTH93, TPSSHCTH, PBEVP86, PBEP86, PW91TPSS, PW91PBE, and PW91PW91 functionals are close to the MP2 results using the 6-311++G(3df,3pd) basis set. A correlation between the calculated DFT potentials and the exchange and correlation enhancement factors at the low-density region has been elucidated. The asymptotic behaviors of the DFT potentials are also analyzed.     

A kinetic energy fitting metric for resolution of the identity second-order Møller-Plesset perturbation theory

A kinetic-energy-based fitting metric for application in the context of resolution of the identity second-order M?ller-Plesset perturbation theory is presented, which is derived from the Poisson equation. Preliminary tests of the applicability include the evaluation of the error in the correlation energy, compared to standard M?ller-Plesset perturbation theory, with respect to the auxiliary basis set employed. We comment on the potential merits of this fitting metric, compared to standard resolution of the identity second-order M?ller-Plesset perturbation theory, and discuss its scaling behavior in the limit of large molecules.     

Analytic energy gradient in combined second-order Møller-Plesset perturbation theory and polarizable force field calculation

Second-order M?ller-Plesset perturbation theory (MP2) is used to describe electronic correlation on the basis of Hartree-Fock (HF) variational calculations that incorporate induced dipole polarizable force fields (i.e., QM/MMpol style HF and MP2). The Z-vector equations for regular closed shell and open shell MP2 methods (RMP2, ZAPT2, and UMP2) are extended to include induced dipole contributions to determine the MP2 response density so that nuclear gradient and other properties can be efficiently evaluated. A better estimation of the induced dipole polarization energy can be obtained using the MP2 relaxed density. QM/MMpol style MP2 molecular dynamics simulations are performed for the ground state and first triplet state of acetone solvated by 1024 polarizable water molecules. A switching function is used to ensure energy conservation in QM/MM simulation under periodic boundary condition.     

An atomic orbital-based reformulation of energy gradients in second-order Møller-Plesset perturbation theory

A fully atomic orbital (AO)-based reformulation of second-order M?ller-Plesset perturbation theory (MP2) energy gradients is introduced, which provides the basis for reducing the computational scaling with the molecular size from the fifth power to linear. Our formulation avoids any transformation between the AO and the molecular orbital (MO) basis and employs pseudodensity matrices similar to the AO-MP2 energy expressions within the Laplace scheme for energies. The explicit computation of perturbed one-particle density matrices emerging in the new AO-based gradient expression is avoided by reformulating the Z-vector method of Handy and Schaefer [J. Chem. Phys. 81, 5031 (1984)] within a density matrix-based scheme.     

Potential energy curve for isomerization of N2H2 and C2H4 using the improved virtual orbital multireference Møller-Plesset perturbation theory

Multireference M?ller-Plesset (MRMP) perturbation theory [K. Hirao, Chem. Phys. Lett. 190, 374 (1992)] is modified to use improved virtual orbitals (IVOs) and is applied to study ground state potential energy curves for isomerization and dissociation of the N2H2 and C2H4 molecules. In contrast to traditional MRMP or multistate multiconfiguration quasidegenerate perturbation theory where the reference functions are obtained from (often difficult to converge) state averaged multiconfiguration self-consistent field methods, our reference functions are represented in terms of computationally efficient IVOs. For convenience in comparisons with other methods, a first order complete active space configuration interaction (CASCI) calculation with the IVOs is followed by the use of the IVOs in MRMP to incorporate residual electron correlation effects. The potential energy curves calculated from the IVO-MRMP method are compared with computations using state-of-the-art coupled cluster singles and doubles (CCSD) methods and variants thereof to assess the efficacy of the IVO-MRMP scheme. The present study clearly demonstrates that unlike the CCSD and its variants, the IVO-MRMP approach provides smooth and reliable ground state potential energy curves for isomerization of these systems. Although the rigorously size-extensive completely renormalized CC theory with noniterative triples corrections (CR-CC(2,3)) likewise provides relatively smooth curves, the CR-CC(2,3) calculations overestimate the cis-trans barrier height for N2H2. The ground state spectroscopic constants predicted by the IVO-CASCI method agree well with experiment and with other highly correlated ab initio methods.     

Orbital-optimized third-order Møller-Plesset perturbation theory and its spin-component and spin-opposite scaled variants: application to symmetry breaking problems

In this research, orbital-optimized third-order M?ller-Plesset perturbation theory (OMP3) and its spin-component and spin-opposite scaled variants (SCS-OMP3 and SOS-OMP3) are introduced. Using a Lagrangian-based approach, an efficient, quadratically convergent algorithm for variational optimization of the molecular orbitals (MOs) for third-order M?ller-Plesset perturbation theory (MP3) is presented. Explicit equations for response density matrices, the MO gradient, and Hessian are reported in spin-orbital form. The OMP3, SCS-OMP3, and SOS-OMP3 approaches are compared with the second-order M?ller-Plesset perturbation theory (MP2), MP3, coupled-cluster doubles (CCD), optimized-doubles (OD), and coupled-cluster singles and doubles (CCSD) methods. All these methods are applied to the O(4)(+), O(3), and seven diatomic molecules. Results demonstrate that the OMP3 and its variants provide significantly better vibrational frequencies than MP3, CCSD, and OD for the molecules where the symmetry-breaking problems are observed. For O(4)(+), the OMP3 prediction, 1343 cm(-1), for ω(6) (b(3u)) mode, where symmetry-breaking appears, is even better than presumably more reliable methods such as Brueckner doubles (BD), 1194 cm(-1), and OD, 1193 cm(-1), methods (the experimental value is 1320 cm(-1)). For O(3), the predictions of SCS-OMP3 (1143 cm(-1)) and SOS-OMP3 (1165 cm(-1)) are remarkably better than the more robust OD method (1282 cm(-1)); the experimental value is 1089 cm(-1). For the seven diatomics, again the SCS-OMP3 and SOS-OMP3 methods provide the lowest average errors, ∣Δω(e)∣ = 44 and ∣Δω(e)∣ = 35 cm(-1), respectively, while for OD, ∣Δω(e)∣ = 161 cm(-1)and CCSD ∣Δω(e)∣ = 106 cm(-1). Hence, the OMP3 and especially its spin-scaled variants perform much better than the MP3, CCSD, and more robust OD approaches for considered test cases. Therefore, considering both the computational cost and the reliability, SCS-OMP3 and SOS-OMP3 appear to be the best methods for the symmetry-breaking cases, based on present application results. The OMP3 method offers certain advantages: it provides reliable vibrational frequencies in case of symmetry-breaking problems, especially with spin-scaling tricks, its analytic gradients are easier to compute since there is no need to solve the coupled-perturbed equations for the orbital response, and the computation of one-electron properties are easier because there is no response contribution to the particle density matrices. The OMP3 has further advantages over standard MP3, making it promising for excited state properties via linear response theory.     

Analytic energy gradient for second-order Møller-Plesset perturbation theory based on the fragment molecular orbital method

The first derivative of the total energy with respect to nuclear coordinates (the energy gradient) in the fragment molecular orbital (FMO) method is applied to second order M?ller-Plesset perturbation theory (MP2), resulting in the analytic derivative of the correlation energy in the external self-consistent electrostatic field. The completely analytic energy gradient equations are formulated at the FMO-MP2 level. Both for molecular clusters (H(2)O)(64) and a system with fragmentation across covalent bonds, a capped alanine decamer, the analytic FMO-MP2 energy gradients with the electrostatic dimer approximation are shown to be complete and accurate by comparing them with the corresponding numeric gradients. The developed gradient is parallelized with the parallel efficiency of about 97% on 32 Pentium4 nodes connected by Gigabit Ethernet.