Systematic Gaussian basis-set limit using completeness-optimized primitive sets. A case for magnetic properties

We discuss the connection between the completeness of a basis set, measured by the completeness profiles introduced by Chong (Can J Chem 1995, 73, 79) at a certain exponent interval, and the possibility of reproducing molecular properties that arise either in the region close to the atomic nuclei or in the valence region. We present a scheme for generating completeness-optimized Gaussian basis sets, in which a preselected range of exponents is covered to an arbitrary accuracy. This is done by requiring Gaussian functions, the exponents of which are selected without reference to the atomic structure, to span the range with completeness profile as close to unity as wanted with as few functions as possible. The initial exponent range can be chosen suitable for calculations of molecular energetics or other valence-like properties. By extending the exponent range, properties requiring augmentation of the basis at a given angular momentum value and/or in a given distance range from the nucleus may be straightforwardly and systematically treated. In this scheme a universal, element-independent exponent set is generated in an automated way. The relation of basis-set completeness and performance in the calculation of magnetizability, nuclear magnetic shielding, and spin-spin coupling is tested with the completeness-optimized primitive sets and literature basis sets.     

Explicitly correlated second-order perturbation theory calculations on molecules containing heavy main-group elements

Slater-type geminals (STGs) have been used as explicitly correlated two-electron basis functions for calculations on the hydrides of N–As and Sb (as well as on the hydrides of O–Se and F–Br with similar, not reported results) in various one-electron basis sets of Gaussian atomic orbitals. The performance of the explicitly correlated theory has been assessed with respect to the exponent of the STG, for example, by using different exponents for individual pair correlation functions and pair energies. It is shown that a correlation factor with an exponent of ${\gamma = 1.4 a_{0}^{-1}}$ can give reliable results within 1% from the basis-set limit for all investigated molecules in an aug-cc-pVQZ basis set for the valence shells, using fixed amplitudes for the STGs in a diagonal orbital-invariant formulation of the theory. The use of relativistic effective core potentials (RECPs) in explicitly correlated second-order perturbation theory has been investigated.     

Polarization-consistent versus correlation-consistent basis sets in predicting molecular and spectroscopic properties

Compared to the correlation-consistent basis sets, it is not known if polarization-consistent pc-n basis sets, which were initially developed for HF and DFT calculations, can provide a monotonic and faster convergence toward the basis-set limit for results at correlated levels as well as better accuracy for a similar number of basis functions. It is also not known whether the pc-n basis sets can compute second derivatives of energy, such as nuclear magnetic shielding tensors, efficiently. To address these questions, the pc-n (n = 1-4), cc-pVxZ, and/or aug-cc-pVxZ (x = D, T, Q, 5, and 6) basis sets were used to compute the molecular and/or spectroscopic parameters of H2, H2O, and NH3 at the RHF, B3-LYP, MP2, and/or CCSD(T) levels of theory. The results show that compared to the cc-pVxZ and/or aug-cc-pVxZ basis sets the pc-n basis sets yield faster convergence toward the basis-set limit but equivalent molecular and/or spectroscopic parameters in the basis-set limit at the RHF, DFT, MP2, and CCSD(T) levels. Because the pc-n basis sets show faster convergence, fewer basis-set functions are needed to reach the accuracy obtained with the aug-cc-pVxZ basis sets, enabling faster calculations and less computer storage space. The results also show that the pc-n basis sets, in conjunction with the "locally dense" basis-set approach, could be applied to predict accurate parameters; thus, they could be used to estimate accurate molecular or spectroscopic properties (e.g., NMR parameters) for larger systems such as the active site of enzymes.     

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.     

Extrapolating to the one-electron basis-set limit in electronic structure calculations

A simple, yet reliable, scheme based on treating uniformly singlet-pair and triplet-pair interactions is suggested to extrapolate atomic and molecular electron correlation energies calculated at two basis-set levels of ab initio theory to the infinite one-electron basis-set limit. The novel dual-level method is first tested on extrapolating the full correlation in single-reference coupled-cluster singles and doubles energies for the closed-shell systems CH2((1)A1), H2O, HF, N2, CO, Ne, and F2 with correlation-consistent basis sets of the type cc-pVXZ (X=D,T,Q,5,6) reported by Klopper [Mol. Phys. 6, 481 (2001)] against his own benchmark calculations with large uncontracted basis sets obtained from explicit correlated singles and doubles coupled-cluster theory. Comparisons are also reported for the same data set but using both single-reference Moller-Plesset and coupled-cluster doubles methods. The results show a similar, often better, accordance with the target results than Klopper's extrapolations where singlet-pair and triplet-pair energies are extrapolated separately using the popular X(-3) and X(-5) dual-level laws, respectively. Applications to the extrapolation of the dynamical correlation in multireference configuration interaction calculations carried out anew for He, H2, HeH+, He2 ++, H3+(1 (1)A'), H3+(1 (3)A'), BH, CH, NH, OH, FH, B2, C2, N2, O2, F2, BO, CO, NO, BN, CN, SH, H2O, and NH3 with standard augmented correlation-consistent basis sets of the type aug-cc-pVXZ (X=D,T,Q,5,6) are also reported. Despite lacking accurate theoretical or experimental data for comparison in the case of most diatomic systems, the new method also shows in this case a good performance when judged from the results obtained with the traditional schemes which extrapolate using the two largest affordable basis sets. For the Hartree-Fock and complete-active space self-consistent field energies, a simple pragmatic extrapolation rule is examined whose results are shown to compare well with the ones obtained from the best reported schemes.     

Dialane anion: three-center two-electron or two-center one-electron bonded

Dialane anions can be formed via a single three-center two-electron (3c-2e) or two-center one-electron (2c-1e) bond. The 2c-1e bonded anion Al(2)H(6)(-)(D(3)(d)) and the 3c-2e bonded anion Al(2)H(6)(-)(C(s)) have significant thermodynamic stabilities with respect to the neutral Al(2)H(6)(D(2)(h)) and correspond to 0.22 and 0.32 eV of the adiabatic electron affinities, respectively. In particular, the 2c-1e bond plays an essential role in stabilizing the Al(2)H(6)(-)(D(3)(d)) anion.     

Analytic second derivatives for the spin-free exact two-component theory

The formulation and implementation of the spin-free (SF) exact two-component (X2c) theory at the one-electron level (SFX2c-1e) is extended in the present work to the analytic evaluation of second derivatives of the energy. In the X2c-1e scheme, the four-component one-electron Dirac Hamiltonian is block diagonalized in its matrix representation and the resulting "electrons-only" two-component Hamiltonian is then used together with untransformed two-electron interactions. The derivatives of the two-component Hamiltonian can thus be obtained by means of simple manipulations of the parent four-component Hamiltonian integrals and derivative integrals. The SF version of X2c-1e can furthermore exploit available nonrelativistic quantum-chemical codes in a straightforward manner. As a first application of analytic SFX2c-1e second derivatives, we report a systematic study of the equilibrium geometry and vibrational frequencies for the bent ground state of the copper hydroxide (CuOH) molecule. Scalar-relativistic, electron-correlation, and basis-set effects on these properties are carefully assessed.     

Oxidation of catechin and rutin by pentaammineruthenium(III) complexes

The reaction of catechin and rutin with Ru(NH(3))(5)L(3+) (L = N-methylpyrazinium (pzCH(3)(+)), pyrazine (pz), and isonicotinamide (isn)) complexes underwent a two-electron oxidation on the catechol ring (B ring) with the formation of quinone products. The kinetics of the oxidation, carried out at [H(+)] = 0.01-1.0 M and pH = 4.0-7.6, suggested that the reaction process involves the rate determining one-electron oxidation of the flavonoids in the form of H(2)X (k(0)), HX(-) (k(1)), and X(2-) (k(2)) by Ru(NH(3))(5)L(3+) complexes to form the corresponding semiquinone radicals, followed by the rapid scavenge of the radicals by the Ru(III) complexes. The specific rate constants (k(0), k(1), and k(2)) were measured and the results together with the application of the Marcus theory were used to estimate the self-exchange parameters for the one-electron couples of the flavonoids, H(2)X/H(2)X(+*), HX(-)/HX(*), and X(2-)/X(-*).     

Probing the effects of heterogeneity on delocalized pi...pi interaction energies

Dimers composed of benzene (Bz), 1,3,5-triazine (Tz), cyanogen (Cy) and diacetylene (Di) are used to examine the effects of heterogeneity at the molecular level and at the cluster level on pi...pi stacking energies. The MP2 complete basis set (CBS) limits for the interaction energies (E(int)) of these model systems were determined with extrapolation techniques designed for correlation consistent basis sets. CCSD(T) calculations were used to correct for higher-order correlation effects (deltaE(CCSD)(T)(MP2)) which were as large as +2.81 kcal mol(-1). The introduction of nitrogen atoms into the parallel-slipped dimers of the aforementioned molecules causes significant changes to E(int). The CCSD(T)/CBS E(int) for Di-Cy is -2.47 kcal mol(-1) which is substantially larger than either Cy-Cy (-1.69 kcal mol(-1)) or Di-Di (-1.42 kcal mol(-1)). Similarly, the heteroaromatic Bz-Tz dimer has an E(int) of -3.75 kcal mol(-1) which is much larger than either Tz-Tz (-3.03 kcal mol(-1)) or Bz-Bz (-2.78 kcal mol(-1)). Symmetry-adapted perturbation theory calculations reveal a correlation between the electrostatic component of E(int) and the large increase in the interaction energy for the mixed dimers. However, all components (exchange, induction, dispersion) must be considered to rationalize the observed trend. Another significant conclusion of this work is that basis-set superposition error has a negligible impact on the popular deltaE(CCSD)(T)(MP2) correction, which indicates that counterpoise corrections are not necessary when computing higher-order correlation effects on E(int). Spin-component-scaled MP2 (SCS-MP2 and SCSN-MP2) calculations with a correlation-consistent triple-zeta basis set reproduce the trends in the interaction energies despite overestimating the CCSD(T)/CBS E(int) of Bz-Tz by 20-30%.     

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.     

Testing the parametric two-electron reduced-density-matrix method with improved functionals: application to the conversion of hydrogen peroxide to oxywater

Parametrization of the two-electron reduced density matrix (2-RDM) has recently enabled the direct calculation of electronic energies and 2-RDMs at the computational cost of configuration interaction with single and double excitations. While the original Kollmar energy functional yields energies slightly better than those from coupled cluster with single-double excitations, a general family of energy functionals has recently been developed whose energies approach those from coupled cluster with triple excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. In this paper we test the parametric 2-RDM method with one of these improved functionals through its application to the conversion of hydrogen peroxide to oxywater. Previous work has predicted the barrier from oxywater to hydrogen peroxide with zero-point energy correction to be 3.3-to-3.9 kcal/mol from coupled cluster with perturbative triple excitations [CCSD(T)] and -2.3 kcal/mol from complete active-space second-order perturbation theory (CASPT2) in augmented polarized triple-zeta basis sets. Using a larger basis set than previously employed for this reaction-an augmented polarized quadruple-zeta basis set (aug-cc-pVQZ)-with extrapolation to the complete basis-set limit, we examined the barrier with two parametric 2-RDM methods and three coupled cluster methods. In the basis-set limit the M parametric 2-RDM method predicts an activation energy of 2.1 kcal/mol while the CCSD(T) barrier becomes 4.2 kcal/mol. The dissociation energy of hydrogen peroxide to hydroxyl radicals is also compared to the activation energy for oxywater formation. We report energies, optimal geometries, dipole moments, and natural occupation numbers. Computed 2-RDMs nearly satisfy necessary N-representability conditions.     

Ab initio molecular orbital study of structures and energetics of Si(3)H(2), Si(3)H(2) (+), and Si(3)H(2) (-)

The geometric structures, isomeric stabilities, and potential energy profiles of various isomers and transition states in Si(3)H(2) neutral, cation and anion are investigated at the coupled-cluster singles, doubles (triples) level of theory. For the geometrical survey, the basis sets used are of the Dunning's correlation consistent basis sets of triple-zeta quality (cc-pVTZ) for the neutral and cation and the Dunning's correlation consistent basis sets of double-zeta quality with diffuse functions (aug-cc-pVDZ) for the anion. For the final energy calculations, the aug-cc-pVTZ: Dunning's correlation consistent basis sets of triple-zeta quality with diffuse functions and cc-pVQZ: Dunning's correlation consistent basis sets of quadruple-zeta quality basis sets are used for the neutral and the aug-cc-pVTZ ones for the cation and anion. The global minimum neutral (I-1: (1)A(1)) has the same framework as that (cyclopropenylidene) of the C(3)H(2) molecule. Other low-lying three isomers (I-2, I-3, and I-4) are also predicted to be within 20 kJ/mol. Five transition states are optimized and their energy relationships with the isomers are clarified. The geometric structure of the global minimum cation (C-1: (2)A(1)) has the same framework as that of the neutral, but that of the anion (A-1: (2)A(')) differs very much from those of the neutral and cation. The calculated vertical and adiabatic ionization potentials from the global minimum neutral (I-1) are 7.85 and 7.77 eV, respectively. The adiabatic electron affinity of the neutral I-1 and the electron detachment energy of the global minimum anion (A-1) are predicted to be 1.21 and 1.92 eV, respectively. The two-electron three-centered bond is widely observed in the present Si(3)H(2) neutral, cation, and anion. The contour plots of their localized molecular orbitals clearly show the existence of such nonclassical chemical bonds.     

The limitations of Slater's element-dependent exchange functional from analytic density-functional theory

Our recent formulation of the analytic and variational Slater-Roothaan (SR) method, which uses Gaussian basis sets to variationally express the molecular orbitals, electron density, and the one-body effective potential of density-functional theory, is reviewed. Variational fitting can be extended to the resolution of identity method, where variationality then refers to the error in each two-electron integral and not to the total energy. However, a Taylor-series analysis shows that all analytic ab initio energies calculated with variational fits to two-electron integrals are stationary. It is proposed that the appropriate fitting functions be charge neutral and that all ab initio energies be evaluated using two-center fits of the two-electron integrals. The SR method has its root in Slater's Xalpha method and permits an arbitrary scaling of the Slater-Gàspàr-Kohn-Sham exchange-correlation potential around each atom in the system. The scaling factors are Slater's exchange parameters alpha. Of several ways of choosing these parameters, two most obvious are the Hartree-Fock (HF) alpha(HF) values and the exact atomic alpha(EA) values. The former are obtained by equating the self-consistent Xalpha energy and the HF energies, while the latter set reproduces exact atomic energies. In this work, we examine the performance of the SR method for predicting atomization energies, bond distances, and ionization potentials using the two sets of alpha parameters. The atomization energies are calculated for the extended G2 set of 148 molecules for different basis-set combinations. The mean error (ME) and mean absolute error (MAE) in atomization energies are about 25 and 33 kcal/mol, respectively, for the exact atomic alpha(EA) values. The HF values of exchange parameters alpha(HF) give somewhat better performance for the atomization energies with ME and MAE being about 15 and 26 kcal/mol, respectively. While both sets give performance better than the local-density approximation or the HF theory, the errors in atomization energy are larger than the target chemical accuracy. To further improve the performance of the SR method for atomization energies, a new set of alpha values is determined by minimizing the MAE in atomization energies of 148 molecules. This new set gives atomization energies half as large (MAE approximately 14.5 kcal/mol) and that are slightly better than those obtained by one of the most widely used generalized-gradient approximations. Further improvements in atomization energies require going beyond Slater's functional form for exchange employed in this work to allow exchange-correlation interactions between electrons of different spins. The MAE in ionization potentials of 49 atoms and molecules is about 0.5 eV and that in bond distances of 27 molecules is about 0.02 A. The overall good performance of the computationally efficient SR method using any reasonable set of alpha values makes it a promising method for study of large systems.     

Introduction

Silicon is taken as a test system for assessing present-day feasibility of calculations for crystalline solids of near-Hartree–Fock quality. The calculations have been performed using CRYSTAL, an ab initio Hartree–Fock crystalline-orbital LCAO program for periodic systems. The influence of the computational parameters that control the truncation of infinite sums on the results has been investigated; it is shown that a reasonable accuracy (numerical errors on total energy per atom below 10^?3 a.u.) can be obtained while keeping the computational burden within manageable limits. The effect on the results of basis-set size and quality is discussed. A number of basis sets have been tested, from minimal to relatively extended sets (28 atomic orbitals per atom). The quality of the wave function has been checked using variational criteria and also through a comparison with experimental data, such as equilibrium geometry, bulk modulus, electron charge density, and electron momentum distribution. For the latter quantities, which are a measure of the accuracy of the one-electron density matrix, the best basis sets provide agreement with experiment that is almost within the experimental error. The correlation energy has been estimated using nonlocal density functionals, based on the one-electron density matrix: After this correction, the atomization energy agrees with experiment to within 2%. The generalization of the above analysis to other crystals is briefly discussed.     

Calculation of transition density matrices by nonunitary orbital transformations

An efficient scheme for calculating one- and two-electron transition density matrices for two wave functions is described. The method applies to CAS (complete active space) wave functions and certain multireference CI expansions. The orbital sets of the two wave functions are not assumed to be equal. They are transformed to a biorthonormal basis, and the corresponding transformation of the CI coefficients is carried out directly, using the one-electron coupling coefficients.     

A new direct CI method for large CI expansions in a small orbital space

A new direct CI method is presented, which is particularly suited for large CI expansions in a small orbital space. These are the type of expansions which are common in the CAS SCF method. Only one-electron coupling coefficients are stored, which leads to reduced elapsed times and storage requirements compared to earlier approaches. The two-electron coupling coefficients are implicitly created in the diagonalization step. The algorithm for updating the CI vector is formulated as the trace of a product of three matrices, Tr(A · D · I). By ordering the one-electron coupling coefficients (A) in a certain way the matrix D is easilly created as a sparse scalar product between these coefficients and the trial CI vector. The main computational step is then a simple matrix multiplication between the matrix D and the symmetry blocked integral matrix (1). This operation vectorizes very well on most vector processors. Another sparse scalar product between the resultant matrix and the coupling coefficients leads to the update of the CI coefficients. In a calculation on CRAY-1 with 30700 configurations, the two-electron part in a CI iteration required 10 s of which half went into the handling of the one-electron formula tape.     

Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian

We report the implementation of analytic energy gradients for the evaluation of first-order electrical properties and nuclear forces within the framework of the spin-free (SF) exact two-component (X2c) theory. In the scheme presented here, referred to in the following as SFX2c-1e, the decoupling of electronic and positronic solutions is performed for the one-electron Dirac Hamiltonian in its matrix representation using a single unitary transformation. The resulting two-component one-electron matrix Hamiltonian is combined with untransformed two-electron interactions for subsequent self-consistent-field and electron-correlated calculations. The "picture-change" effect in the calculation of properties is taken into account by considering the full derivative of the two-component Hamiltonian matrix with respect to the external perturbation. The applicability of the analytic-gradient scheme presented here is demonstrated in benchmark calculations. SFX2c-1e results for the dipole moments and electric-field gradients of the hydrogen halides are compared with those obtained from nonrelativistic, SF high-order Douglas-Kroll-Hess, and SF Dirac-Coulomb calculations. It is shown that the use of untransformed two-electron interactions introduces rather small errors for these properties. As a first application of the analytic geometrical gradient, we report the equilibrium geometry of methylcopper (CuCH(3)) determined at various levels of theory.     

Accurately solving the electronic Schrodinger equation of small atoms and molecules using explicitly correlated (r12-)MR-CI. VIII. Valence excited states of methylene (CH2)

We compute the adiabatic transition energies of methylene (CH(2)) from the ground state to the lowest electronically excited valence states using the r(12)-MR-ACPF-2 method with a large basis set and an extended reference space. We recall that this method aims at reaching the basis-set and full configuration interaction (CI) limits simultaneously. Our best excitation energies, T(e) (T(0)), are 9.22 (8.87) (a (1)A(1), corrected for relativistic and adiabatic effects), 31.98 (31.86) (b (1)B(1)), and 57.62 (57.18) kcal mol(-1) (c (1)A(1)) (both uncorrected). We are able to reach the respective basis-set limits that closely that the remaining errors of our (uncorrected) calculations are clearly due to the MR-ACPF-2 method. While we are unable to assess the error of the latter method in a systematic way, we still believe that it is rather unlikely that the errors of our excitation energies exceed +/-0.10 kcal mol(-1). We finally observe that our (corrected) a state values deviate by only -0.10 (-0.10) kcal mol(-1) from the results of Csaszar et al. [J. Chem. Phys. 118, 10631 (2003)]--who did careful extrapolations to the valence full-CI and basis-set limits and added a correction for the core correlation--and that the deviation from experiment is only -0.13 (-0.13) kcal mol(-1). From these excellent agreements we conclude that our excitation energies to the b and c states are similarly accurate.     

Basis-set quality and basis-set bias in molecular property calculations

Quality measures for Gaussian basis sets are proposed that are based on principal angles between the basis set and reference molecular orbitals. The principal angles are obtained from the cosine-sine (CS) decomposition of orthogonal matrices and yield detailed information about basis-set convergence with respect to different regions of space. Principal angles for occupied orbitals show excellent correlation with basis-set errors in ground-state energies. Furthermore, ground-state bias in finite basis sets can be estimated from the relation between principal angles for occupied and Rydberg orbitals. Ground-state bias is observed in basis sets including extensive diffuse augmentation and affects the quality of computed molecular response properties. Principal angles and ground-state bias are investigated for the H-Ne atoms and a series of diatomics using numerical Hartree-Fock calculations as a reference. Convergence of ground-state energies and static polarizabilities is studied for the hierarchies of correlation-consistent and Karlsruhe segmented def2 basis sets including different levels of diffuse augmentation.