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.     

Hartree-Fock exchange fitting basis sets for H to Rn

For elements H to Rn (except Lanthanides), a series of auxiliary basis sets fitting exchange and also Coulomb potentials in Hartree–Fock treatments (RI-JK-HF) is presented. A large set of small molecules representing nearly each element in all its common oxidation states was used to assess the quality of these auxiliary bases. For orbital basis sets of triple zeta valence and quadruple zeta valence quality, errors in total energies arising from the RI-JK approximation are below ∼1 meV per atom in molecular compounds. Accuracy of RI-JK-approximated HF wave functions is sufficient for being used for post-HF treatments like Møller–Plesset perturbation theory, MP2. Compared to nonapproximated treatments, RI-JK-HF leads to large computational savings for quadruple zeta valence orbital bases and, in case of small to midsize systems, to significant savings for triple zeta valence bases. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2008     

Analytical gradients of the second‐order Møller–Plesset energy using Cholesky decompositions

An algorithm for computing analytical gradients of the second‐order Møller–Plesset (MP2) energy using density fitting (DF) is presented. The algorithm assumes that the underlying canonical Hartree–Fock reference is obtained with the same auxiliary basis set, which we obtain by Cholesky decomposition (CD) of atomic electron repulsion integrals. CD is also used for the negative semidefinite MP2 amplitude matrix. Test calculations on the weakly interacting dimers of the S22 test set (Jure?ka et al., Phys. Chem. Chem. Phys. 2006, 8, 1985) show that the geometry errors due to the auxiliary basis set are negligible. With double‐zeta basis sets, the error due to the DF approximation in intermolecular bond lengths is better than 0.1 pm. The computational time is typically reduced by a factor of 6–7. © 2013 Wiley Periodicals, Inc.     

Hydrogen bonding in the urea dimers and adenine–thymine DNA base pair: anharmonic effects in the intermolecular H-bond and intramolecular H-stretching vibrations

The equilibrium structures, binding energies, vibrational harmonic frequencies, and the anharmonic corrections for two different (cyclic and asymmetric) urea dimers and for the adenine–thymine DNA base pair system have been studied using the second-order Møller–Plesset perturbation theory (MP2) method and different density functional theory (DFT) exchange–correlation (XC) functionals (BLYP, B3LYP, PBE, HCTH407, KMLYP, and BH and HLYP) with the D95V, D95V**, and D95V++** basis sets. The widely used a posteriori Boys–Bernardi or counterpoise correction scheme for basis set superposition error (BSSE) has been included in the calculations to take into account the BSSE effects during geometry optimization (on structure), on binding energies and on the different levels of approximation used for calculating the vibrational frequencies. The results obtained with the ab initio MP2 method are compared with those calculated with different DFT XC functionals; and finally the suitability of these DFT XC functionals to describe intermolecular hydrogen bonds as well as harmonic frequencies and the anharmonic corrections is assessed and discussed.     

An improved generator coordinate Hartree–Fock method applied to the choice of contracted Gaussian basis sets for first‐row diatomic molecules

Accurate Gaussian basis sets (18s for Li and Be and 20s11p for the atoms from B to Ne) for the first‐row atoms, generated with an improved generator coordinate Hartree–Fock method, were contracted and enriched with polarization functions. These basis sets were tested for B₂, C₂, BeO, CN⁻, LiF, N₂, CO, BF, NO⁺, O₂, and F₂. At the Hartree–Fock (HP), second‐order Møller–Plesset (MP2), fourth‐order Møller–Plesset (MP4), and density functional theory (DFT) levels, the dipole moments, bond lengths, and harmonic vibrational frequencies were studied, and at the MP2, MP4, and DFT levels, the dissociation energies were evaluated and compared with the corresponding experimental values and with values obtained using other contracted Gaussian basis sets and numerical HF calculations. For all diatomic molecules studied, the differences between our total energies, obtained with the largest contracted basis set [6s5p3d1f], and those calculated with the numerical HF methods were always less than 3.2 mhartree. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 15–23, 2000     

Theoretical study on intermolecular interaction of epoxyethane dimer

Ab initio calculations at Hartree–Fock and fourth‐order Mø ller–Plesset (MP4) correlation correction levels with 6‐31G* basis set have been performed on the epoxyethane dimer. Dimer binding energies have been corrected for the basis set superposition error (BSSE) and the zero‐point energy. The greatest corrected dimer binding energy is −8.36 kJ/mol at the MP4/6‐31G*//HF/6‐31G* level. The natural bond orbital analysis has been performed to trace the origin of the weak interactions that stabilize dimer. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 94–98, 2000     

Effect of electron correlation on theoretical vibrational frequencies

Theoretical HF /6-31G * (Hartree–Fock, 6-31G * basis set) and MP 2/6-31G * (second-order Møller–Plesset, 6-31G * basis set) vibrational frequencies based on complete quadratic force fields have been obtained for a set of 36 one- and two-heavy-atom molecules comprising first-row elements for which experimental spectroscopic data are available. Frequencies calculated at the HF /6-31G * level are an average of 12.6% higher than experimental values. Partial treatment of electron correlation via the perturbation method of Møller and Plesset, terminated at second order, leads to a significant reduction in this error, although theoretical MP 2/6-31G * frequencies are still larger than the experimental quantities by 7.3%. Part of the difference may be traced to the restriction of quadratic force fields, as comparison with experimental harmonic frequencies shows deviations of only 9.5% and 4.7% for the two levels, respectively. The calculated frequencies are used in conjunction with the corresponding theoretical equilibrium structures to obtain absolute molecular entropies, which may in turn be used to yield entropies of reaction. These latter quantities are generally in good accord with entropies derived using experimental structures and frequencies.     

The GPU‐enabled divide‐expand‐consolidate RI‐MP2 method (DEC‐RI‐MP2)

We report porting of the Divide‐Expand‐Consolidate Resolution of the Identity second‐order Møller–Plesset perturbation (DEC‐RI‐MP2) method to the graphic processing units (GPUs) using OpenACC compiler directives. It is shown that the OpenACC compiler directives implementation efficiently accelerates the rate‐determining step of the DEC‐RI‐MP2 method with minor implementation effort. Moreover, the GPU acceleration results in a better load balance and thus in an overall scaling improvement of the DEC algorithm. The resulting cross‐platform hybrid MPI/OpenMP/OpenACC implementation has scalable and portable performance on heterogeneous HPC architectures. The GPU‐enabled code was benchmarked using a reduced version of the S12L test set of Stefan Grimme (Grimme, Chem. Eur. J. 2012, 18, 9955) consisting of supramolecular complexes up to 158 atoms and 4292 contracted basis functions (cc‐pVTZ). The test set results demonstrate the general applicability of the DEC‐RI‐MP2 method showing results consistent with the DEC‐RI‐MP2 introductory paper (Baudin et al., J. Chem. Phys. 2016, 144, 054102) on molecules of complicated electronic structures. © 2016 Wiley Periodicals, Inc.     

Cyclobutadiene automerization and rotation of ethylene: Energetics of the barriers by using Spin‐polarized wave functions

Spin‐projected spin polarized Møller–Plesset and spin polarized coupled clusters calculations have been made to estimate the cyclobutadiene automerization, the ethylene torsion barriers in their ground state, and the gap between the singlet and triplet states of ethylene. The results have been obtained optimizing the geometries at MP4 and/or CCSD levels, by an extensive Gaussian basis set. A comparative analysis with more complex calculations, up to MP5 and CCSDTQP, together with others from the literature, have also been made, showing the efficacy of using spin‐polarized wave functions as a reference wave function for Møller–Plesset and coupled clusters calculations, in such problems. © 2014 Wiley Periodicals, Inc.     

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     

Perturbation expansion theory corrected from basis set superposition error II. Charge transfer,pair correlationand dispersion terms

The second-order perturbation theory based on the locally projected molecular orbitals is developed. A few test calculations with cc-pVDZ and aug-cc-pVDZ basis sets are carried out for the dimers, (H₂O)₂ and (HF)₂. The charge transfer terms remove the deficiency of the locally projected self-consistent field method for molecular interaction (LP SCF MO MI), and the potential energy curves calculated with aug-cc-pVDZ are very close to the corresponding curves of the counterpoise-corrected SCF energy. Only after adding the spin-exchanged dispersion type to the dispersion and intra-molecular pair correlation terms, the calculated potential energy curves become close to those of the counterpoise-corrected second-order Møller–Plesset (MP2). Pragmatic approaches for reducing the influence of the basis set superposition error are proposed.     

Møller-Plesset perturbation energies and distances for HeC(20) extrapolated to the complete basis set limit

The potential energy surface for the C₂₀–He interaction is extrapolated for three representative cuts to the complete basis set limit using second‐order Møller–Plesset perturbation calculations with correlation consistent basis sets up to the doubly augmented variety. The results both with and without counterpoise correction show consistency with each other, supporting that extrapolation without such a correction provides a reliable scheme to elude the basis‐set‐superposition error. Converged attributes are obtained for the C₂₀– He interaction, which are used to predict the fullerene dimer ones. Time requirements show that the method can be drastically more economical than the counterpoise procedure and even competitive with Kohn‐Sham density functional theory for the title system. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Integral transformation with low-order scaling for large local second-order Møller–Plesset calculations

An algorithm is presented for the four-index transformation of electron repulsion integrals to a localized molecular orbital (MO) basis. Unlike in most programs, the first two indices are transformed in a single step. This and the localization of the orbitals allows the efficient neglect of small contributions at several points in the algorithm, leading to significant time savings. Thresholds are applied to the following quantities: distant orbital pairs, the virtual space before and after the orthogonalizing projection to the occupied space, and small contributions in the transformation. A series of calculations on medium-sized molecules has been used to determine appropriate thresholds that keep the truncation errors small (below 0.01% of the correlation energy in most cases). Benchmarks for local second-order Møller–Plesset perturbation theory (MP2; i.e., MP2 with a localized MO basis in the occupied subspace) are presented for several large molecules with no symmetry, up to 975 contracted basis functions, and 60 atoms. These are among the largest MP2 calculations performed on a single processor. The computational time (with constant basis set) scales with a somewhat lower than cubic power of the molecular size, and the memory demand is moderate even for large molecules, making calculations that require a supercomputer for the traditional MP2 feasible on workstations. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1241–1254, 1998     

Two-level hierarchical parallelization of second-order Møller-Plesset perturbation calculations in divide-and-conquer method

A two‐level hierarchical parallelization scheme including the second‐order Møller–Plesset perturbation (MP2) theory in the divide‐and‐conquer method is presented. The scheme is a combination of coarse‐grain parallelization assigning each subsystem to a group of processors, with fine‐grain parallelization, where the computational tasks for evaluating MP2 correlation energy of the assigned subsystem are distributed among processors in the group. Test calculations demonstrate that the present scheme shows high parallel efficiency and makes MP2 calculations practical for very large molecules. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Proton affinity of five‐membered heterocyclic amines: Assessment of computational procedures

Ab initio quantum chemical calculations, G3B3, second‐order Møller–Plesset (MP2), and the hybrid density functional method B3LYP were employed to compute the proton affinities of 24 heterocyclic amines. A range of basis sets are employed, starting from double‐ζ polarization quality to triple‐ζ quality basis set with augmented diffuse and polarization function. Experimental values were used to calibrate the performance of various theoretical models. The regioselectivity for the protonation has been unambiguously established by performing B3LYP/6‐31G* calculations on the possible putative sites of attack. For the given series of compounds the performance of B3LYP/6‐31++G** and G3B3 levels of theory have been in excellent agreement with the experimental results with the deviations are of the order comparable with the experimental error. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Bond length alternation in cyclic polyenes. V. Local finite-order perturbation theory approach

The finite-order many-body perturbation theory using the localized Wannier orbital basis is applied to the problem of bond length alternation in the Pariser–Parr–Pople model of cyclic polyenes C_N H_N, N = 4v + 2, which may be regarded as a simplified model of polyacetylene. Both the Møller–Plesset and the Epstein–Nesbet-type partitionings of the model Hamiltonian are employed. The localized orbital basis enables an efficient truncation of the perturbation theory summations over the intermediate states as well as an elimination of energetically unimportant diagrams, thus enabling one to obtain the fourth-order Møller–Plesset-perturbation energies with a relatively small computational effort even for large polyenes. The results obtained with the second-, third-, and fourth-order Møller–Plesset and with the third-order Epstein–Nesbet perturbation theories yield very similar bond length distortions (about 0.05 Å) and stabilization energies per site (about 0.04 eV) as obtained earlier with the RHF , one-parameter AMO , and delocalized orbital perturbation theories. The effects of truncation and diagram elimination in the fourth-order Møller–Plesset perturbation theory and the abnormal behavior of the second-order Epstein–Nesbet perturbation theory results in the localized Wannier basis near the instability threshold of the RHF solutions are discussed.     

Ab initio study of the reaction mechanism of water dissociation into the ionic species OH− and H3O+

A reaction mechanism of water dissociation is proposed where solvent effects are accounted for via a minimum stable model that considers the interaction of five water molecules. It is based on the fully self-consistent field (SCF) optimized structures of the reactant, product, and transition state, the calculations being at the Hartree–Fock and configuration interaction level [Møller–Plesset second-order perturbation (MP2) and coupled-cluster single and double excitations (CCSD)]. They were performed with four different basis sets that included polarized and diffuse orbitals. The dissociative mechanism leads to the ionic species OH⁻+H₃O⁺ as stable products and upon analysis of the energy hypersurface, a transition state is found which yields an activation barrier of 21.2 kcal/mol. This value is in good agreement with the experimentally determined enthalpy for the reaction. The contribution of the aggregation energy is emphasized. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 253–259, 1998     

Development and testing of an algorithm for efficient MP2/CCSD(T) energy estimation of molecular clusters with the 2–body approach

This work reports the development and testing of an automated algorithm for estimating the energies of weakly bound molecular clusters employing correlated theory. Firstly, the monomers and dimers of (homo/hetero) clusters are identified, and the sum of one-body and two-body contributions to correlation energy is calculated. The addition of this contribution to the Hartree-Fock full calculation (FC) energies provides a good estimate of the total energies at Møller–Plesset second-order perturbation theory (MP2)/coupled-cluster method with singles and doubles (CCSD) (T)-level theory using augmented Dunning basis sets. The estimated energies for several test clusters show an excellent agreement with their FC counterparts, with a substantial wall-clock time saving employing off-the-shelf hardware. Furthermore, the complete basis set (CBS) limit for MP2 energy computed using the two-body approach also agrees with its CBS energy with its FC counterpart.