Interaction energies for the water dimer by supermolecular methods and symmetry-adapted perturbation theory: the role of bond functions and convergence of basis subsets

 Using a systematic series of basis sets in supermolecular and symmetry-adapted intermolecular perturbation theory calculations it is examined how interaction energies of various water dimer structures change upon addition and shifting of bond functions. Their addition to augmented double- and triple-zeta basis sets brings the sum of the electron correlation contributions to the second-order interaction energy nearly to convergence, while accurate first-order electrostatic and exchange contributions require better than augmented quadruple-zeta quality. A scheme which combines the different perturbation energy contributions as computed in different basis subsets performs uniformly well for the various dimer structures. It yields a symmetry-adapted perturbation theory value of −21.08 kJ/mol for the energy of interaction of two vibrationally averaged water molecules compared to −21.29 kJ/mol when the full augmented triple-zeta basis set is used throughout. Received: 4 November 1999 / Accepted: 8 February 2000 / Published online: 12 May 2000     

Finite-field many-body perturbation theory. X. Electric field gradients and other properties of N2

Finite-field MBPT calculations have been carried out for the electric field gradient and other electric properties of the nitrogen molecule. On the basis of correlation corrections computed through the fourth order in the electron correlation perturbation the infinite order MBPT result for the electric field gradient at the nitrogen nucleus has been estimated. The corresponding result combined with the NQR coupling constant for N₂ leads to the ¹⁴N nuclear quadrupole moment of 0.0205 ± 0.0010 barn in agreement with the experimental atomic measurement and other molecular calculations. The MBPT estimate of the quadrupole moment of N₂ gives −1.107 ± 0.038 au in agreement with the most recent experimental value.     

Relativistic Gaussian basis sets for radon through plutonium

Summary Relativistic Gaussian basis sets of neutral atoms Rn-Pu and ions Th⁺⁴, U⁺³ and Pu⁺³ in the configurations of average energies are presented. The exponent parameters of the basis sets are determined by least-squares fitting to the numerical Dirac-Fock wave functions. The total energies obtained are within 0.155 a.u. of the Dirac-Fock limits and the qualities of the basis sets are between double-zeta and triple-zeta in the valence parts. Using the exponent parameters the Breit interaction energies have been calculated by perturbation theory and the self-consistent field treatment.     

Perturbational approach to the electron correlation cusp applied to helium‐like atoms

A recently proposed perturbational approach to the electron correlation cusp problem 1 is tested in the context of three spherically symmetrical two‐electron systems: helium atom, hydride anion, and a solvable model system. The interelectronic interaction is partitioned into long‐ and short‐range components. The long‐range interaction, lacking the singularities responsible for the electron correlation cusp, is included in the reference Hamiltonian. Accelerated convergence of orbital‐based methods for this smooth reference Hamiltonian is shown by a detailed partial wave analysis. Contracted orbital basis sets constructed from atomic natural orbitals are shown to be significantly better for the new Hamiltonian than standard basis sets of the same size. The short‐range component becomes the perturbation. The low‐order perturbation equations are solved variationally using basis sets of correlated Gaussian geminals. Variational energies and low‐order perturbation wave functions for the model system are shown to be in excellent agreement with highly accurate numerical solutions for that system. Approximations of the reference wave functions, described by fewer basis functions, are tested for use in the perturbation equations and shown to provide significant computational advantages with tolerable loss of accuracy. Lower bounds for the radius of convergence of the resulting perturbation expansions are estimated. The proposed method is capable of achieving sub‐μHartree accuracy for all systems considered here. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Perturbation calculation of the interaction energy using orthogonalized orbitals

The diagrammatic Rayleigh-Schr?dinger perturbation theory for the interaction of two closed-shell systems is developed up to the third order of pertur-bation using orthogonalized orbitals. The interaction energy is expressed by the Rayleigh-Schr?dinger perturbation expansion. A simple approach for the estimation of basis set superposition error is introduced. The preliminary calculations of the intermolecular interactions for the He dimer within the augmented cc-pVTZ basis set are compared with the supermolecular approach, perturbation calculation in biorthogonal basis sets and symmetry adapted perturbation theory results. Received: 17 December 1996 / Accepted: 5 November 1997     

Auxiliary basis sets for density fitting-MP2 calculations: Nonrelativistic triple-zeta all-electron correlation consistent basis sets for the 3d elements Sc-Zn

Auxiliary basis sets for density fitting second-order Moller-Plesset perturbation theory (DF-MP2) have been optimized for use with the triple-zeta nonrelativistic all-electron correlation consistent orbital basis sets, cc-pVTZ-NR and aug-cc-pVTZ-NR, for the 3d elements Sc-Zn. The relative error in using these auxiliary basis sets is found to be around four orders of magnitude smaller than that from utilizing triple-zeta orbital basis sets rather than corresponding quadruple-zeta basis sets, in calculation of the correlation energy for a test set of 54 small to medium sized transition metal complexes.     

Near-Hartree—Fock calculation of the electric field gradient tensor at the hydrogen nucleus in water

The electric field gradient tensor at the hydrogen nucleus in water has been calculated in an SCF calculation with local near-Hartree—Fock quality using asymmetric basis sets. The results are in excellent agreement with results obtained by Davidson and Feller with a basis set of uniform near-Hartree—Fock quality and deviate only a few percent from experimental results. Calculated and experimental data are combined to improve the transformed experimental values.     

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.     

Highly accurate equilibrium structure of the C2h symmetric N1‐to‐O2 hydrogen‐bonded uracil‐dimer

The highly accurate ab initio equilibrium geometry of the hydrogen‐bonded uracil dimer is derived using a composite geometry extrapolation scheme based on all‐electron, complete basis set extrapolated Møller–Plesset perturbation theory using the jun‐pwCV[T,Q]Z basis sets combined with a valence CCSD(T)/cc‐pVTZ high‐level correction. Geometrical changes on dimerization are discussed and the performance of the several density functional approximations (among others SCAN, ωB97M‐V, DSD‐PBEP86‐D3(BJ), and DSD‐PBEP86‐NL) is evaluated. Orbital‐optimized MP2.5 is discussed as a reduced‐cost alternative to the CCSD(T) gradient in the composite scheme. A new reference interaction energy is calculated with explicitly correlated F12‐CCSD(T).     

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.     

On the Importance of CP-corrected Gradient Optimization in the Study of Hydrogen Bonded Systems

Geometries, harmonic vibrational frequencies and interaction energies of the water-hydrogen sulfide dimer, hydrogen fluoride dimer and glycine zwitterion-water dimer were determined by the counterpoise-corrected (CP-corrected) gradient optimization that explicitly corrects for the basis set superpusition error (BSSE) and CP-uncorrected (normal) gradient opfimization respectively at the B3LYP and MP2 levels of theory, employing the popular Pople‘s standard 6-31G(d), 6-31G(d,p) and 6-311 G(d,p) basis sets in order to assess the importance of CP-corrected gradient optimiTation in the study of hydrogen bonded systems. The normal optimization of these three H-bonded systems obtained using these popular basis sets all yielded erratic results, whereas use of CP-corrected gradient optimization led to consistent results with those from larger basis sets. So this CP receipt becomes useful and necessary to correctly describe large systems, where the use of small basis sets may be necessary.     

Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn

The introduction of the resolution-of-the-identity (RI) approximation for electron repulsion integrals in quantum chemical calculations requires in addition to the orbital basis so-called auxiliary or fitting basis sets. We report here such auxiliary basis sets optimized for second-order Møller–Plesset perturbation theory for the recently published (Weigend and Ahlrichs Phys Chem Chem Phys, 2005, 7, 3297–3305) segmented contracted Gaussian basis sets of split, triple-ζ and quadruple-ζ valence quality for the atoms Rb–Rn (except lanthanides). These basis sets are designed for use in connection with small-core effective core potentials including scalar relativistic corrections. Hereby accurate resolution-of-the-identity calculations with second-order Møller–Plesset perturbation theory (MP2) and related methods can now be performed for molecules containing elements from H to Rn. The error of the RI approximation has been evaluated for a test set of 385 small and medium sized molecules, which represent the common oxidation states of each element, and is compared with the one-electron basis set error, estimated based on highly accurate explicitly correlated MP2–R12 calculations. With the reported auxiliary basis sets the RI error for MP2 correlation energies is typically two orders of magnitude smaller than the one-electron basis set error, independent on the position of the atoms in the periodic table.     

New relativistic atomic natural orbital basis sets for lanthanide atoms with applications to the Ce diatom and LuF3

New basis sets of the atomic natural orbital (ANO) type have been developed for the lanthanide atoms La-Lu. The ANOs have been obtained from the average density matrix of the ground and lowest excited states of the atom, the positive ions, and the atom in an electric field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second-order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calculations of ionization energies and some excitation energies. Computed ionization energies have an accuracy better than 0.1 eV in most cases. Two molecular applications are included as illustration: the cerium diatom and the LuF3 molecule. In both cases it is shown that 4f orbitals are not involved in the chemical bond in contrast to an earlier claim for the latter molecule.     

Calculation of the fourth-rank molecular hypermagnetizability of some small molecules

A computational scheme has been developed within the framework of Rayleigh-Schr?dinger perturbation theory to evaluate nonlinear interaction energy contributions for a molecule in the presence of an external spatially uniform, time-independent magnetic field. Terms connected with the fourth power of the perturbing field, representing the fourth-rank hypermagnetizabilities of five small molecules, have been evaluated at the coupled Hartree-Fock level of accuracy within the conventional common-origin approach. Gaugeless basis sets of increasing size and flexibility have been employed in a numerical test, adopting two different coordinate systems to estimate the degree of convergence of theoretical tensor components.     

Approaching the theoretical limit in periodic local MP2 calculations with atomic-orbital basis sets: the case of LiH

The atomic orbital basis set limit is approached in periodic correlated calculations for solid LiH. The valence correlation energy is evaluated at the level of the local periodic second order M?ller-Plesset perturbation theory (MP2), using basis sets of progressively increasing size, and also employing "bond"-centered basis functions in addition to the standard atom-centered ones. Extended basis sets, which contain linear dependencies, are processed only at the MP2 stage via a dual basis set scheme. The local approximation (domain) error has been consistently eliminated by expanding the orbital excitation domains. As a final result, it is demonstrated that the complete basis set limit can be reached for both HF and local MP2 periodic calculations, and a general scheme is outlined for the definition of high-quality atomic-orbital basis sets for solids.     

Self-consistent, constrained linear-combination-of-atomic-potentials approach to quantum mechanics

Variational fitting gives a stationary linear-combination of atomic potentials (LCAP) approximation to the Kohn-Sham (KS) potential, V. That potential is central to density-functional theory because it generates all orbitals, occupied as well as virtual. Perturbation theory links two self-consistent field (SCF) calculations that differ by the perturbation. Using the same variational LCAP methods and basis sets in the two SCF calculations gives precise KS potentials for each order. Variational V perturbation theory, developed herein through second order, gives stationary potentials at each order and stationary even-order perturbed energies that precisely link the two SCF calculations. Iterative methods are unnecessary because the dimension of the matrix that must be inverted is the KS basis size, not the number of occupied times virtual orbitals of coupled-perturbed methods. With variational perturbation theory, the precision of derivatives and the fidelity of the LCAP KS potential are not related. Finite differences of SCF calculations allow the precision of analytic derivatives from double-precision code to be verified to roughly seven significant digits. For a simple functional, the fourth derivatives of the energy and the first and second derivative of the KS potentials with respect to orbital occupation are computed for a standard set of molecules and basis sets, with and without constraints on the fit to the KS potential. There is no significant difference between the constrained and unconstrained calculations.     

Convergence of electric field and electric field gradient versus atomic basis sets in all-siliceous and Mg substituted phillipsites

Electrostatic potential (EP), electric field (EF), and electric field gradient (EFG) values are calculated in periodic models of magnesium substituted phillipsite (MgPHI) zeolite forms using periodic DFT (PDFT) hybrid B3LYP level with fourteen different basis sets. Relative root mean square differences between the EP, EF, or EFG values estimated between different basis sets are evaluated in several spatial domains available for adsorbate molecules in the zeolite. In these areas, the EF increase in MgPHI is evaluated relative to all-siliceous PHI types. The EP is interpreted in terms of framework ionicity for MgPHI and all-siliceous PHI models. Angular Si-O-Si dependence of the EFG asymmetry at (17)O atoms in all-siliceous zeolites is discussed.     

Scaling in second-order electron correlation calculations using systematic sequences of even-tempered basis sets

Improved results can often be obtained from second-order Rayleigh-Schrödinger perturbation calculations of electron correlation energies using large basis sets by introducing a scaling factor in the zero-order Hamiltonian. The scaling parameter may be determined from full third-order calculations using a smaller basis set. This scaling procedure can be applied in a systematic fashion by employing a sequence of even-tempered basis sets. Calculations illustrating this approach for the beryllium atom and the neon atom are presented. The scaling procedure is also employed in conjunction with a universal systematic sequence of basis functions. Calculations illustrating this Correlation energy — Mang-body perturbation theory.Work supported in part by S.R.C. Research Grant GR/B/4738.6.S.R.C. Advanced Fellow.