Accuracy of basis-set extrapolation schemes for DFT-RPA correlation energies in molecular calculations

We construct a reference benchmark set for atomic and molecular random phase approximation (RPA) correlation energies in a density functional theory framework at the complete basis-set limit. This set is used to evaluate the accuracy of some popular extrapolation schemes for RPA all-electron molecular calculations. The results indicate that for absolute energies, accurate results, clearly outperforming raw data, are achievable with two-point extrapolation schemes based on quintuple- and sextuple-zeta basis sets. Moreover, we show that results in good agreement with the benchmark can also be obtained by using a semiempirical extrapolation procedure based on quadruple- and quintuple-zeta basis sets. Finally, we analyze the performance of different extrapolation schemes for atomization energies.     

Importance of electron correlation effects and basis set superposition error in calculations of interaction energies and interaction-induced electric properties in hydrogen-bonded complexes: a model study

A detailed study of the interaction energies and interaction-induced electric dipole properties in model linear hydrogen cyanide complexes (HCN) _m (m?=?2?C4) is carried out within the finite field HF SCF, MP2, CCSD and CCSD(T) approximations using the recently developed LPol-n (n?=?ds, fs, dl, fl) basis sets. The importance of high-order correlation effects and the basis set superposition error is evaluated. To correct for the latter is crucial for obtaining accurate interaction energy values, but the error can safely be neglected in the estimation of induced electric properties when the LPol-n (n?=?ds, fs, dl, fl) basis sets are used. Correlation effects are important in the evaluation of both the interaction energies and the induced electric properties of the systems.     

Minimal basis sets in calculations of intermolecular interaction energies

Several minimal (7, 3/3) Gaussian basis sets have been used to calculate the energies and some other properties of CH₄ and H₂O. Improved basis sets developed for these molecules have been extended to NH₃ and HF and employed to H₂CO and CH₃OH. Interaction energies between XH_n molecules have been calculated using the old and the new minimal basis sets. The results obtained with the new basis sets are comparable in accuracy to those calculated with significantly more extended basis sets involving polarization functions. Binding energies calculated using the counterpoise method are not much different for the new and the old minimal basis sets, and are likely to be more accurate than the results of much more extended calculations.     

Linear dependence in basis-set calculations for extended systems

The problem of linear dependence in basis-set calculations for extended systems is discussed. We show that the problem is intrinsic in three-dimensionally extended systems, but is not as serious in systems with extension in fewer dimensions. The possibility of choosing suitable basis sets that avoid linear dependence is discussed. It is shown that for systems extended in three dimensions in which the orbitals near the Fermi-level are well described by plane waves a mixed atomicorbital/plane-wave (AO –PW ) basis set with tight orbitals to describe the cores avoids the problem in the most efficient way. Numerical experiments with 1s Slater-type orbitals and plane waves on a simple cubic lattice are presented for illustration. © 1992 John Wiley & Sons, Inc.     

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.     

On the calculations of interaction energies and induced electric properties within the polarizable continuum model

In this work we investigate the influence of a polarizable environment on the interaction energies and the interaction-induced (excess) static electric dipole properties for the selected model hydrogen-bonded complexes. The excess properties were estimated for water and hydrogen fluoride dimers using the supermolecular approach and assuming the polarizable continuum model (PCM) as a representation of the polarizable environment. We analyze in this context the performance of the counterpoise correction and the consequences of various possible monomer cavity choices. The polarizable environment reduces the absolute magnitudes of interaction energies and interaction-induced dipole moments, whereas an increase is observed for the absolute magnitudes of induced polarizabilities and first hyperpolarizabilities. Our results indicate that the use of either monomeric (MC) or dimeric (DC) cavities in calculations of monomer properties does not change qualitatively the resultant excess properties. We conclude that the DC scheme is more consistent with the definition of the interaction energy and consequently also the interaction-induced property, whereas the MC scheme corresponds to the definition of stabilization energy. Our results indicate also a good performance of the counterpoise correction scheme for the self-consistent methods in the case of all studied properties.     

Potential-Derived point-charge model study of electrostatic interaction energies in some hydrogen-bonded systems

Mulliken's atomic charges (MC ) and potential derived (PD ) point charges obtained from STO -3G wave functions are used to study the electrostatic interaction energies for a series of representative hydrogenbonded complexes. The results of the above-mentioned models are compared with the more accurate results of segmental multipole moment (SMM ) expansion, and it is shown that the PD model is superior to the Mc model. The results of PD model are shown to be well correlated with the results of SMM expansion technique. Results of our calculations using 6-31G and 6-31G** PD charges are also reported here. Electrostatic interaction energies obtained using 6-31G** PD charges are compared with the 6-31G** SCF interaction energies available for the nine hydrogen-bonded dimers of ammonia, water, and hydrogen fluoride and a good con-elation between the two is shown. The interrelationship between the results of different basis sets are also examined for the PD point-charge model. The electrostatic interaction energies obtained using STO -3G PD model are shown to be well correlated to the results of 6-31G and 6-31G** PD models.     

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.     

Perturbation calculations of molecular interaction energies: an example,HF....HF

An expression for the interaction energy of two molecules, obtained by expanding the wavefunction for the whole system in terms of antisymmetrized products of free-molecule functions and using a matrix perturbation scheme, is used in ab initio calculations on the HF dimer. The computed interaction energy is fitted with considerable accuracy by a simple analytical formula. The equilibrium geometry and hydrogen-bond energy are in satisfactory agreement with available experimental results.     

Accurate calculations of intermolecular interaction energies using explicitly correlated wave functions

Explicitly correlated second-order M?ller-Plesset (MP2-F12) calculations of intermolecular interaction energies for the S22 benchmark set of Jurecka, Sponer, Cerny, and Hobza (Chem. Phys. Phys. Chem. 2006, 8, 1985) are presented and compared with standard MP2 results. The MP2 complete basis set limits are estimated using basis set extrapolation and augmented quadruple-zeta and quintuple-zeta basis sets. Already with augmented double-zeta basis sets the MP2-F12 interaction energies are found to be closer to the complete basis set limits than standard MP2 calculations with augmented quintuple-zeta basis sets. Various possible approximations in the MP2-F12 method are systematically tested. Best results are obtained with localized orbitals and the diagonal MP2-F12/C(D) ansatz. Hybrid approximations, in which some contributions of the auxiliary basis set are neglected and which considerably reduce the computational cost, have a negligible effect on the interaction energies. Also the orbital-invariant fixed-amplitude approximation of Ten-no leads to only slightly less accurate results. Preliminary results for the neon and benzene dimers, obtained with the recently proposed CCSD(T)-F12a approximation, indicate that the CCSD(T) basis set limits can also be very closely approached using augmented triple-zeta basis sets.     

Differential correlation effects in chemisorption cluster model calculations: an FCI study

The interaction of atomic hydrogen with Cu₅ and Ag₅ cluster models simulating the Cu(100) and Ag(100) surfaces has been studied at the full configuration interaction (FCI) level in order to establish the transferability of differential correlation arising from the valence shell. It is shown that pseudopotentials that deal explicitly with one electron or eleven electrons lead to differential correlation effects which agree to within 1–2 mhartree when a localization procedure is used to separate d-shell MOs from the valence ones.     

Accurate pair interaction energies for helium from supermolecular Gaussian geminal calculations

Nonrelativistic clamped-nuclei pair interaction energy for ground-state helium atoms has been computed for 12 interatomic separations ranging from 3.0 to 9.0 bohr. The calculations applied the supermolecular approach. The major part of the interaction energy was obtained using the Gaussian geminal implementation of the coupled-cluster theory with double excitations (CCD). Relatively small contributions from single, triple, and quadruple excitations were subsequently included employing the conventional orbital coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)] and the full configuration interaction (FCI) method. For three distances, the single-excitation contribution was taken from literature Gaussian-geminal calculations at the CCSD level. The orbital CCSD(T) and FCI calculations used very large basis sets, up to doubly augmented septuple- and sextuple-zeta size, respectively, and were followed by extrapolations to the complete basis set limits. The accuracy of the total interaction energies has been estimated to be about 3 mK or 0.03% at the minimum of the potential well. For the attractive part of the well, the relative errors remain consistently smaller than 0.03%. In the repulsive part, the accuracy is even better, except, of course, for the region where the potential goes through zero. For interatomic separations smaller than 4.0 bohr, the relative errors do not exceed 0.01%. Such uncertainties are significantly smaller than the expected values of the relativistic and diagonal Born-Oppenheimer contributions to the potential.     

Multiple substitution effects in configuration interaction calculations

The common way of approximating multiple substitution effects by the unlinked cluster contribution based on the linked cluster theorem is investigated. Formally the analysis is done in a configuration interaction language without reference to diagrams. Using the linked cluster theorem a series expansion is then developed for the correction of a CI calculation, including double replacements only, for the erratic treatment of unlinked clusters. The first term in this series is closely related to what is generally referred to as Davidson's correction. The next term is proportional to the square of the correlation energy and is also very easy to calculate. Finally numerical examples are given illustrating the accuracy of the different approximations.     

Configuration interaction calculations of the valence ionization energies of the water dimer

The eight vertical valence ionization energies of the water dimer are calculated by the ΔCI method. Excellent agreement with measurements of the first and second ionization energies is found. Calculations of the remaining six ionization energies is found. Calculations of the remaining six ionization energies are sufficiently accurate to be of value in the identification and assignment of the dimer photoelectron spectrum.     

A comparative basis-set study of NeH+ using coupled-cluster techniques

Unrestricted Hartree-Fock, coupled-cluster calculations are reported for the ground state of NeH⁺ using atomic basis sets of increasing size and accuracy for both Ne and H. The goal is to determine the basis set and coupled-cluster level of calculation needed to obtain a NeH⁺ potential energy curve of known accuracy. Here, it is shown that calculations using a quintuple zeta basis at the coupled-cluster singles and doubles level with noniterative triples, CCSD(T) , predict a Ne—H bond dissociation energy that is within about 0.01 eV of the exact Born–Oppenheimer molecular electronic structure result. Spectroscopic constants determined using the Simons–Parr–Finlan procedure are found to be in very good agreement with the experimental results. Calculations at the augmented quadruple zeta level for the two lowest triplet excited states of the NeH⁺ species are presented. Both of these states separate into ground-state Ne⁺ and H(1s). The resulting potential curves predict stable minima at the SCF, CCSD, and CCSD(T) levels with dissociation energies of about 0.07 eV. Spectroscopic constants from the potential curves and dissociation constants are reported. © 1994 John Wiley & Sons, Inc.     

Quantitative calculations of antibody--antigen binding: steroid--DB3 binding energies by the linear interaction energy method

Linear interaction energy/molecular dynamics calculations have been used to compute steroid/antibody binding energies. The absolute binding affinities of 10 steroids to antibody DB3 and of a hapten to catalytic antibody 1E9 are computed and compared to experiment. A detailed analysis of the molecular origins of the observed binding patterns is provided. The binding energy of an untested steroid is predicted.     

Electrostatic energy calculations by a Finite-difference method: Rapid calculation of charge–solvent interaction energies

Finite-difference Poisson–Boltzmann (FDPB) methods allow a fast and accurate calculations of the reaction field (charge–solvent) energies for molecular systems. Unfortunately, the energy in the FDPB calculations includes the self-energies and the finite-difference approximation to the Coulombic energies as well as the reaction field energy. A second finite-difference calculation, in a uniform dielectric, is therefore necesssary to eliminate these contributions. In this article we describe a rapid and accurate method to calculate the self energy and finite-difference Coulombic energies in a uniform dielectric thus eliminating the need for a second finite-difference calculation. The computational savings for this method range from a factor of 4 for a typical protein to a factor of 10³ for small molecules. © 1992 by John Wiley & Sons, Inc.     

Binding energies in benzene dimers: Nonlocal density functional calculations

The interaction energy and minimum energy structure for different geometries of the benzene dimer have been calculated using the recently developed nonlocal correlation energy functional for calculating dispersion interactions. The comparison of this straightforward and relatively quick density functional based method with recent calculations provides a promising first step to elucidate how the former, quicker method might be exploited in larger more complicated biological, organic, aromatic, and even infinite systems such as molecules physisorbed on surfaces and van der Waals crystals.