The extrapolation of one-electron basis sets in electronic structure calculations: how it should work and how it can be made to work

We consider the extrapolation of the one-electron basis to the basis set limit in the context of coupled cluster calculations. We produce extrapolation coefficients that produce much more accurate results than previous extrapolation forms. These are determined by fitting to accurate benchmark results. For coupled cluster singles doubles energies, we take our benchmark results from the work of Klopper that explicitly includes the interelectronic distance. For the perturbative triples energies, our benchmark results are obtained from large even-tempered basis set calculations.     

Can extrapolation to the basis set limit be an alternative to the counterpoise correction? A study on the helium dimer

Configuration interaction and coupled cluster calculations are reported for He₂ using various orbital basis sets of the d-aug-AVXZ type, with the results being extrapolated to the one electron basis set limit both with counterpoise and without counterpoise correction. A generalized uniform singlet- and triplet-pair extrapolation scheme has been utilized for such a purpose. Using appropriate corrections to mimic full configuration interaction, the energies were predicted in excellent agreement with the best available estimates. The results also suggest that extrapolation to the complete basis set limit may be a general alternative to the counterpoise correction that yields a more accurate potential energy while being more economical.     

Near-Dirac–Hartree–Fock results for first-row atoms calculated with GTO basis sets

Relativistic basis sets for first-row atoms have been constructed by using the near-Hartree–Fock (nonrelativistic) eigenvectors calculated by Partridge. These bases generate results of near-Dirac–Hartree–Fock quality. Relativistic total and orbital energies, relativistic corrections to the total energy, and magnetic interaction energies for the first-row atoms have been presented. The smallest Gaussian expansions (13s8 p expansions) yield Dirac–Hartree–Fock total energies accurate through six significant digits, while the largest expansions (18s13p expansions) give these energies accurate through seven significant digits. These results are more accurate than some of the results reported earlier, particularly for the open-shell atoms, indicating that the basis employed is reasonably economical for relativistic calculations. © 1995 John Wiley & Sons, Inc.     

Accurate ab initio binding energies of alkaline earth metal clusters

The effects of basis set superposition error (BSSE) and core-correlation on the electronic binding energies of alkaline earth metal clusters Y(n) (Y = Be, Mg, Ca; n = 2-4) at the Moller-Plesset second-order perturbation theory (MP2) and the single and double coupled cluster method with perturbative triples correction (CCSD(T)) levels are examined using the correlation consistent basis sets cc-pVXZ and cc-pCVXZ (X = D, T, Q, 5). It is found that, while BSSE has a negligible effect for valence-electron-only-correlated calculations for most basis sets, its magnitude becomes more pronounced for all-electron-correlated calculations, including core electrons. By utilizing the negligible effect of BSSE on the binding energies for valence-electron-only-correlated calculations, in combination with the negligible core-correlation effect at the CCSD(T) level, accurate binding energies of these clusters up to pentamers (octamers in the case of the Be clusters) are estimated via the basis set extrapolation of ab initio CCSD(T) correlation energies of the monomer and cluster with only the cc-pVDZ and cc-pVTZ sets, using the basis set and correlation-dependent extrapolation formula recently devised. A comparison between the CCSD(T) and density functional theory (DFT) binding energies is made to identify the most appropriate DFT method for the study of these clusters.     

Equation-of-motion coupled-cluster study on exciton states of polyethylene with periodic boundary condition

Equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) method has been applied to exciton states of polyethylene using ab initio crystal Hartree-Fock method with one-dimensional periodic boundary condition. Full transformation of two-electron integrals from atomic-orbital basis to crystal-orbital basis has been performed for EOM-CCSD calculations. In order to make transformed integrals to have correct properties of translational symmetry, a lattice summation scheme has been proposed. The EOM-CCSD excitation energies have been obtained for the lowest singlet and triplet exciton states of polyethylene. The excitation energies converge with system size much faster than oligomer calculations using n-alkanes. Quasiparticle energy-level calculations by second-order many-body perturbation theory and by solving the inverse Dyson equation have also been performed to obtain exciton binding energies. Basis set dependencies on excitation energy, quasiparticle band gap, and exciton binding energy have been investigated. At the 6-31+G level, the excitation energy of the lowest singlet-exciton state and its binding energy are calculated to be 8.1 and 3.2 eV, respectively. The calculated excitation energy is well comparable with the corresponding experimental value, 7.6 eV.     

On the effectiveness of CCSD(T) complete basis set extrapolations for atomization energies

The leading cause of error in standard coupled cluster theory calculations of thermodynamic properties such as atomization energies and heats of formation originates with the truncation of the one-particle basis set expansion. Unfortunately, the use of finite basis sets is currently a computational necessity. Even with basis sets of quadruple zeta quality, errors can easily exceed 8 kcal/mol in small molecules, rendering the results of little practical use. Attempts to address this serious problem have led to a wide variety of proposals for simple complete basis set extrapolation formulas that exploit the regularity in the correlation consistent sequence of basis sets. This study explores the effectiveness of six formulas for reproducing the complete basis set limit. The W4 approach was also examined, although in lesser detail. Reference atomization energies were obtained from standard coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) calculations involving basis sets of 6ζ or better quality for a collection of 141 molecules. In addition, a subset of 51 atomization energies was treated with explicitly correlated CCSD(T)-F12b calculations and very large basis sets. Of the formulas considered, all proved reliable at reducing the one-particle expansion error. Even the least effective formulas cut the error in the raw values by more than half, a feat requiring a much larger basis set without the aid of extrapolation. The most effective formulas cut the mean absolute deviation by a further factor of two. Careful examination of the complete body of statistics failed to reveal a single choice that out performed the others for all basis set combinations and all classes of molecules.     

Accurate ab intio determination of binding energies for rare-gas dimers by basis set extrapolation

To investigate the electron correlation effect on the binding energies of very weakly bound complexes at highly correlated levels, an extrapolation scheme exploiting the convergent behavior of the binding energy differences between two correlation levels with the correlation-consistent basis set aug-cc-pVXZ was explored. The scheme is based on extrapolating the binding energy differences between the lower and higher correlation levels (such as second-order Møller–Plesset perturbation theory and the single and double coupled-cluster method with perturbative triple correction level), CCSD(T), by X^–3 for relatively small basis set calculations to estimate the corresponding basis set limit, which is then added to the complete basis set(CBS) limit binding energy at the lower correlation level to derive the CBS limit binding energy at the higher level. Test results on rare-gas dimers Rg₂ (Rg is He, Ne, Ar) show that the CCSD(T) CBS limit binding energies estimated by this scheme with aug-cc-pVXZ and aug-cc-pV(X+1)Z basis sets are more accurate than the CBS limit estimated by direct extrapolation of correlation energies by X^–3 with aug-cc-pV(X+1)Z and aug-cc-pV(X+2)Z basis sets in most cases, which signifies the utility of the proposed extrapolation scheme as the level of electron correlation treatment increases. The nonnegligible discrepancy in the well depth near equilibrium between the experimental and the full connected single, double, and triple coupled-cluster method CBS limit estimate obtained by this procedure in the case of Ar₂ suggests that the previous semiempirical potential may be too attractive near equilibrium compared with the actual one.Acknowledgement The major portion of this work was carried out while the author was visiting the Quantum Theory Project (QTP) at the University of Florida. The author is thankful to Rodney Bartlett for hospitality and support during the visit. The author is also thankful to Ajith Perera for assistance in using the ACESII program package. Computational support from the QTP at the University of Florida and the Institute for Basic Science at Ajou University is gratefully acknowledged.     

Improved relativistic energy-consistent pseudopotentials for 3d-transition metals

Energy-consistent relativistic pseudopotentials for 3d-transition metals Sc to Ni based on modified valence energies are proposed. The pseudopotentials are adjusted at the finite difference level within the intermediate coupling scheme with respect to multi-configuration Dirac–Hartree–Fock data based on the Dirac–Coulomb Hamiltonian with an estimate of the Breit contributions in quasidegenerate perturbation theory. Typically a few hundred to thousand J levels arising from about 35 to 40 configurations ranging from the anion down to the highly charged cation are considered as references. It is shown that introducing a small common energetic shift of all valence energies reduces the errors in the parameter adjustment considerably. Results of highly correlated atomic and molecular test calculations using large basis sets and basis set extrapolation techniques are presented. To be submitted to Theoretical Chemistry Accounts (special volume on the occasion of Prof. Dr. H. Stoll's 60th birthday)     

An overview of coupled cluster theory and its applications in physics

Summary What has since become known as the normal coupled cluster method (NCCM) was invented about thirty years ago to calculate ground-state energies of closed-shell atomic nuclei. Coupled cluster (CC) techniques have since been developed to calculate excited states, energies of open-shell systems, density matrices and hence other properties, sum rules, and the sub-sum-rules that follow from imbedding linear response theory within the NCCM. Further extensions deal both with systems at nonzero temperature and with general dynamical behaviour. More recently, a new version of CC theory, the so-called extended coupled cluster method (ECCM) has been introduced. It has the potential to describe such global phenomena as phase transitions, spontaneous symmetry breaking, states of topological excitation, and nonequilibrium behaviour. CC techniques are now widely recognized as providing one of the most universally applicable, most powerful, and most accurate of all microscopicab initio methods in quantum many-body theory. The number of successful applications within physics is now impressively large. In most such cases the numerical results are either the best or among the best available. A typical case is the electron gas, where the CC results for the correlation energy agree over the entire metallic density range to within less than 1 millihartree (or <1%) with the essentially exact Green's function Monte Carlo results. The role of CC theory within modern quantum many-body theory is first surveyed, by a comparison with other techniques. Its full range of applications in physics is then reviewed. These include problems in nuclear physics, both for finite nuclei and infinite nuclear matter; the electron gas; various integrable and nonintegrable models; various relativistic quantum field theories; and quantum spin chain and lattice models. Particular applications of the ECCM include the quantum hydrodynamics of a zero-temperature, strongly-interacting condensed Bose fluid; a charged impurity in a polarizable medium (e.g., positron annihilation in metals); and various anharmonic oscillator and spin systems.     

Encoding and Processing of Alphanumeric Information by Chemical Mixtures

A novel infochemical device that is based on ¹H NMR readout of chemical information is presented. This chemical encoding system utilizes two measurable parameters of homogeneous mixtures, chemical shift and peak integration, for three different applications: 1) a text‐encoding device that is based on spectral representation of a sequence of symbols, 2) encoding of 21‐digit binary numbers, each represented by an NMR spectrum, and their algebraic manipulations, such as addition and subtraction, and 3) encoding of 21‐digit decimal numbers. The first application enables molecular information storage and encryption. The relative concentration of each component, as measured by the relevant peak integration, can represent a symbol. The second application of this system, in addition to its obvious memory capability, enables mathematical operations. The NMR spectrum of a given mixture represents a 21‐digit binary number where each of the peaks encodes for a specific digit. In any of the input mixtures (numbers) each compound is either present or absent, representing either 1 or 0, respectively. We used the various binary numbers to carry out addition operations by combining two or more solutions (numbers). Subtraction operations were also preformed by digital processing of the information. The third application is the representation of decimal numbers. As before, each of the peaks encodes for a specific digit. In any of the input mixtures each compound is present in one of 10 different relative concentrations, representing the 10 digits of a decimal number.     

Improved correlation energy extrapolation schemes based on local pair natural orbital methods

It is well-known that the basis set limit is difficult to reach in correlated post Hartree-Fock ab initio calculations. One possible route forward is to employ basis set extrapolation schemes. In order to avoid prohibitively expensive calculations, the highest level calculation (typically based on the "gold standard" coupled cluster theory with single, double, and perturbative triple excitations, CCSD(T)) is only performed with the smallest basis set, and the remaining basis set incompleteness is estimated at a lower level of theory, typically second-order M?ller-Plesset perturbation theory (MP2). In this work, we provide a comprehensive investigation of alternative schemes where the MP2 extrapolation is replaced by the coupled-electron pair approximation, version 1 (CEPA/1) or the local pair natural orbital version of this method (LPNO-CEPA/1). It is shown that the MP2 method achieves apparent accuracy only due to error cancellation. Systematically more accurate results at small additional computational cost are obtained if the MP2 step is replaced by LPNO-CEPA/1. The errors of LPNO-CEPA/1 relative to canonical CEPA/1 are negligible. Owing to the highly systematic nature of the deviations between canonical and LPNO methods, basis set extrapolation reduces the LPNO errors in the total energies by 1 order of magnitude (~0.2 kcal/mol) and errors in energy differences to essentially zero. Using the CCSD(T)/LPNO-CEPA/1-based extrapolation scheme, new reference values are proposed for the recently published S66 set of interaction energies. The deviations between the new values and the original interactions energies are mostly very small but reach values up to 0.3 kcal/mol.     

Interaction energy of a water molecule with a single-layer graphitic surface modeled by hydrogen- and fluorine-terminated clusters

In ab initio calculations a finite graphitic cluster model is often used to approximate the interaction energy of a water molecule with an infinite single-layer graphitic surface (graphene). In previous studies, the graphitic cluster model is a collection of fused benzene rings terminated by hydrogen atoms. In this study, the effect of using fluorine instead of hydrogen atoms for terminating the cluster model is examined to clarify the role of the boundary. The interaction energy of a water molecule with the graphitic cluster was computed using ab initio methods at the MP2 level of theory and with the 6-31G(d = 0.25) basis set. The interaction energy of a water molecule with graphene is estimated by extrapolation of two series of increasing size graphitic cluster models (C(6n2)H(6n) and C(6n2)F(6n), n = 1-3). Two fixed orientations of water molecule are considered: (a) both hydrogen atoms of water pointing toward the cluster (mode A) and (b) both hydrogen atoms of water pointing away from the cluster (mode B). The interaction energies for water mode A are found to be -2.39 and -2.49 kcal/mol for C(6n2)H(6n) and C(6n2)F(6n) cluster models, respectively. For water mode B, the interaction energies are -2.32 and -2.44 kcal/mol for C(6n2)H(6n) and C(6n2)F(6n) cluster models, respectively.     

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.     

Ab initio vibrational predissociation dynamics of He-I2(B) complex

Three-dimensional quantum mechanical calculations on the vibrational predissociation dynamics of HeI2 B state complex are performed using a potential energy surface accurately fitted to unrestricted open-shell coupled cluster ab initio data, further enabling extrapolation for large I2 bond lengths. A Lanczos iterative method with an optimized complex absorbing potential is used to determine energies and lifetimes of the vibrationally predissociating He,I2(B,v') complex for v'相似文献   

Higher-order equation-of-motion coupled-cluster methods for electron attachment

High-order equation-of-motion coupled-cluster methods for electron attachment (EA-EOM-CC) have been implemented with the aid of the symbolic algebra program TCE into parallel computer programs. Two types of size-extensive truncation have been applied to the electron-attachment and cluster excitation operators: (1) the electron-attachment operator truncated after the 2p-1h, 3p-2h, or 4p-3h level in combination with the cluster excitation operator after doubles, triples, or quadruples, respectively, defining EA-EOM-CCSD, EA-EOM-CCSDT, or EA-EOM-CCSDTQ; (2) the combination of up to the 3p-2h electron-attachment operator and up to the double cluster excitation operator [EA-EOM-CCSD(3p-2h)] or up to 4p-3h and triples [EA-EOM-CCSDT(4p-3h)]. These methods, capable of handling electron attachment to open-shell molecules, have been applied to the electron affinities of NH and C2, the excitation energies of CH, and the spectroscopic constants of all these molecules with the errors due to basis sets of finite sizes removed by extrapolation. The differences in the electron affinities or excitation energies between EA-EOM-CCSD and experiment are frequently in excess of 2 eV for these molecules, which have severe multideterminant wave functions. Including higher-order operators, the EA-EOM-CC methods predict these quantities accurate to within 0.01 eV of experimental values. In particular, the 3p-2h electron-attachment and triple cluster excitation operators are significant for achieving this accuracy.     

Theoretical study of the lowest pi-->pi* excitation energies for neutral and doped polyenes

In earlier theoretical studies, it has been widely noticed that the electron correlation effect played an important role in determining the excitation energies of low-lying pi-->pi(*) excited states for neutral polyenes and their radical cations and dications. In this paper, neutral and doped polyene oligomers of medium to large sizes are investigated with the Pariser-Parr-Pople model, and the pi-electron correlation effect is fully taken into consideration by virtue of the density-matrix renormalization group method. The excitation properties in the polymer limit are also obtained by exponential extrapolation from the finite oligomers. The reasonable agreement of our results with the available experimental observations and advanced ab initio calculations is witnessed. It is also observed that while charge doping can significantly lower the exciting energy, the odd-charged oligomers show lower excitation energies than the even-charged ones.     

Infinite Basis set extrapolation for Double Hybrid Density Functional Theory 1: effect of applying various extrapolation functions

We applied the Infinite Basis (IB) set extrapolation and Double Hybrid Density Functional Theory (DHDF) to calculate the databases of atomization energies, ionization energies, electron affinities, reaction barrier heights, proton affinities, alkyl bond dissociation energies, and noncovalent interactions. The Complete Basis Set (CBS) limit is estimated by extrapolating the hybrid density functional theory and PT2 energies using extrapolation functions including exponential, inverse power, modified exponential, and the combination of the these functions. We found that the combination of B2KPLYP/cc-pV[D|T]Z (which is the extrapolation based on the energies calculated in cc-pVDZ and cc-pVTZ) gives results in quadruple-ζ quality. However, if we want to reach the ～2 kcal/mol chemical accuracy limit, the cc-pV[T|Q]Z is required. Similar results with various extrapolation functions obtained, because the IB parameters were determined by minimizing the averaged mean unsigned error of the calculated databases. We generalized the IB set extrapolation to include more than two basis sets, but we found that extrapolation with two basis sets is satisfactory to give reasonable results. The largest error occurred in the databases of the electron affinities and the weak interactions between the noble gas and the nonpolar molecules. We expect that performing the DHDF-IB scheme with the basis sets augmented by diffuse basis functions will further improve the results.     

Multireference configuration interaction calculations for complexes of positronium and B, C, N, and O atoms

Positronium (Ps) binding energies for complexes of Ps and atoms with open shell electrons, PsX (X=B, C, N, and O), are calculated using the multireference singly and doubly excited configuration interaction (MRSDCI) method. The effectiveness of this method for the complexes is verified. The MRSDCI calculations are carried out with a frozen-core approximation so as to incorporate only the most important valence correlation effects. Many-body correlation effects and contributions from higher angular momentum orbitals are estimated by extrapolation techniques. The resulting Ps binding energies agree well with the results of diffusion Monte Carlo simulations by Bressanini et al. (Phys Rev A 57:1678,1998) and by Jiang and Schrader (J Chem Phys 109:9430,1998). For PsO the Ps binding energy obtained by Jiang and Schrader is about 1.8 times larger than that of Bressanini et al.; the present calculated value is close to that of Jiang and Schrader.     

Bulk properties from finite cluster calculations. VIII. Benchmark calculations of the efficiency of extrapolation methods for the HF and MP2 energies of polyacenes

The total energies per unit cell of both the undistorted and the Peierls-distorted polyacene polymers are computed from the respective HF/6-311G** and MP2/6-311G** finite-cluster data using 40 different extrapolation schemes. The benchmark calculations, which aim at assessing the efficiency of extrapolation methods, clearly show that the best procedure for obtaining rapidly converging bulk properties should involve computation of the energy differences, followed by rational extrapolation techniques such Wynn's p algorithm or its iteration, both with the interpolation points x_n = (n + 1)², and closely related extrapolation methods, or Wynn's ε algorithm and its close relative, Aitken's iterated Δ² algorithm. © John Wiley & Sons, Inc.