Large Orbital Basis Research Articles

Role of the pair potential for the saturation of generalized Pauli constraints

The dependence of the (quasi-)saturation of the generalized Pauli constraints on the pair potential is studied for ground states of few-fermion systems. For this, we consider spinless fermions in one dimension which are harmonically confined and interact by pair potentials of the form $|{x}_{i}\ensuremath{-}{x}_{j}{|}^{s}$ with $\ensuremath{-}1\ensuremath{\le}s\ensuremath{\le}5$. We use the density matrix renormalization group approach and large orbital basis to achieve the convergence on more than ten digits of both the variational energy and the natural occupation numbers. Our results confirm that the conflict between energy minimization and fermionic exchange symmetry results in a universal and nontrivial quasisaturation of the generalized Pauli constraints (quasipinning), implying tremendous structural simplifications of the fermionic ground state for all $s$. Those numerically exact results are complemented by an analytical study based on a self-consistent perturbation theory which we develop for this purpose. The respective results for the weak-coupling regime eventually elucidate the singular behavior found for the specific values $s=2,4,...,$ resulting in an extremely strong quasipinning.

A general second order complete active space self-consistent-field solver for large-scale systems

We present a new second order complete active space self-consistent field implementation to converge wavefunctions for both large active spaces and large atomic orbital (AO) bases. Our algorithm decouples the active space wavefunction solver from the orbital optimization in the microiterations, and thus may be easily combined with various modern active space solvers. We also introduce efficient approximate orbital gradient and Hessian updates, and step size determination. We demonstrate its capabilities by calculating the low-lying states of the Fe(II)-porphine complex with modest resources using a density matrix renormalization group solver in a CAS(22,27) active space and a 3000 AO basis.

Highly correlated configuration interaction calculations on water with large orbital bases

A priori selected configuration interaction (SCI) with truncation energy error [C. F. Bunge, J. Chem. Phys. 125, 014107 (2006)] and CI by parts [C. F. Bunge and R. Carbó-Dorca, J. Chem. Phys. 125, 014108 (2006)] are used to approximate the total nonrelativistic electronic ground state energy of water at fixed experimental geometry with CI up to sextuple excitations. Correlation-consistent polarized core-valence basis sets (cc-pCVnZ) up to sextuple zeta and augmented correlation-consistent polarized core-valence basis sets (aug-cc-pCVnZ) up to quintuple zeta quality are employed. Truncation energy errors range between less than 1 μhartree, and 100 μhartree for the largest orbital set. Coupled cluster CCSD and CCSD(T) calculations are also obtained for comparison. Our best upper bound, -76.4343 hartree, obtained by SCI with up to sextuple excitations with a cc-pCV6Z basis recovers more than 98.8% of the correlation energy of the system, and it is only about 3 kcal/mol above the "experimental" value. Despite that the present energy upper bounds are far below all previous ones, comparatively large dispersion errors in the determination of the extrapolated energies to the complete basis set do not allow to determine a reliable estimation of the full CI energy with an accuracy better than 0.6 mhartree (0.4 kcal/mol).

Reply to “Comment on ‘Nonanalyticity of the optimized effective potential with finite basis sets’ ”

The Comment by Friedrich et al. does not dispute the central result of our paper [Phys. Rev. A 85, 052508 (2012)] that nonanalytic behavior is present in long-established mathematical pathologies arising in the solution of finite basis optimized effective potential (OEP) equations. In the Comment, the terms ``balancing of basis sets'' and ``basis-set convergence'' imply a particular order towards the limit of a large orbital basis sets where the large-orbital-base limit is always taken first, before the large-auxiliary-base limit, until overall convergence is achieved, at a high computational cost. The authors claim that, on physical grounds, this order of limits is not only sufficient, but also necessary in order to avoid the mathematical pathologies. In response to the Comment, we remark that it is already written in our paper that the nonanalyticity trivially disappears with large orbital basis sets. We point out that the authors of the Comment give an incorrect proof of this statement. We also show that the order of limits towards convergence of the potential is immaterial. A recent paper by the authors of the Comment proposes a partial correction for the incomplete orbital basis error in the full-potential linearized augmented-plane-wave method. Similar to the correction developed in our paper, this correction also benefits from an effectively complete orbital basis, even though only a finite orbital basis is employed in the calculation. This shows that it is unnecessary to take, in practice, the limit of an infinite orbital basis in order to avoid mathematical pathologies in the OEP. Our paper is a significant contribution in that direction with general applicability to any choice of basis sets. Finally, contrary to an allusion in the abstract and assertions in the main text of the Comment that unphysical oscillations of the OEP are supposedly attributed to the common energy denominator approximation, in fact, such anomalies arise with the full treatment of the small eigenvalues of the density response function. This characteristic of the finite basis OEP is well known in the literature but also is clearly demonstrated in our paper.

A partitioned correlation function interaction approach for describing electron correlation in atoms

The traditional multiconfiguration Hartree–Fock (MCHF) and configuration interaction (CI) methods are based on a single orthonormal orbital basis. For atoms with many closed core shells, or complicated shell structures, a large orbital basis is needed to saturate the different electron correlation effects such as valence, core–valence and correlation within the core shells. The large orbital basis leads to massive configuration state function (CSF) expansions that are difficult to handle, even on large computer systems. We show that it is possible to relax the orthonormality restriction on the orbital basis and break down the originally very large calculations into a series of smaller calculations that can be run in parallel. Each calculation determines a partitioned correlation function (PCF) that accounts for a specific correlation effect. The PCFs are built on optimally localized orbital sets and are added to a zero-order multireference (MR) function to form a total wave function. The expansion coefficients of the PCFs are determined from a low dimensional generalized eigenvalue problem. The interaction and overlap matrices are computed using a biorthonormal transformation technique (Verdebout et al 2010 J. Phys. B: At. Mol. Phys. 43 074017). The new method, called partitioned correlation function interaction (PCFI), converges rapidly with respect to the orbital basis and gives total energies that are lower than the ones from ordinary MCHF and CI calculations. The PCFI method is also very flexible when it comes to targeting different electron correlation effects. Focusing our attention on neutral lithium, we show that by dedicating a PCF to the single excitations from the core, spin- and orbital-polarization effects can be captured very efficiently, leading to highly improved convergence patterns for hyperfine parameters compared with MCHF calculations based on a single orthogonal radial orbital basis. By collecting separately optimized PCFs to correct the MR function, the variational degrees of freedom in the relative mixing coefficients of the CSFs building the PCFs are inhibited. The constraints on the mixing coefficients lead to small off-sets in computed properties such as hyperfine structure, isotope shift and transition rates, with respect to the correct values. By (partially) deconstraining the mixing coefficients one converges to the correct limits and keeps the tremendous advantage of improved convergence rates that comes from the use of several orbital sets. Reducing ultimately each PCF to a single CSF with its own orbital basis leads to a non-orthogonal CI approach. Various perspectives of the new method are given.

Quasi-Classical Trajectory Study of the N(2D) + H2(X1ΣG+) → NH(X3Σ–) + H(2S) Reaction Based on an Analytical Potential Energy Surface

Using the multireference configuration interaction method with the Davidson correction and a large orbital basis set (aug-cc-pV5Z), we obtain an energy grid that includes 17,500 points for the construction of a new analytical potential energy surface (APES) for the N((2)D) + H(2)(X(1)Σ(g)(+)) → NH(X(3)Σ(-)) + H((2)S) reaction. The APES, which contains 145 parameters and is represented with a many-body expansion and a new switch function, is fitted from the ab initio energies using an adaptive nonlinear least-squares algorithm. The geometric characteristics of the reported APES in the literature and those of our APES are also compared. On the basis of the APES that we obtained, reaction cross sections are computed by means of quasi-classical trajectory calculations and compared with the experimental and theoretical values available in the literature.

Quantum wave packet calculation of the O(3P) + H2 reaction on the new potential energy surfaces for the two lowest states

New accurate potential energy surfaces (PESs) for the lowest states (3A″ and 3A′) of the O(3P)+H2 reaction are proposed using an ab initio multireference configuration interaction method (MRCI) with Davidson correction and a large orbital basis set (aug-cc-pv5z). The many-body expansion procedure is employed to describe the analytical PES function. The topographical features of the new global PESs are presented and compared with previous surfaces. The quantum reaction scattering dynamics calculations are carried out over the collision energies range of 0.3–1.0eV on the new PESs. The integral cross-sections and rate coefficients for the title reaction were calculated. The calculated coefficients are lower than the experimental ones at the low temperature.

An ab initio potential energy surface and dynamics of the [formula omitted] → ArH + + H reaction

An ab initio potential energy surface (PES) for the ground state (1 2 A ′ ) of the chemical reaction Ar + H 2 + → ArH + + H has been constructed from a set of accurate ab initio data, which we have computed using the coupled-cluster theory including all single and double excitations plus perturbative corrections for the triples UCCSD(T) with a large orbital basis set of aug-cc-pV5Z. The new PES has a root-mean-square (rms) error of 0.5341 kcal/mol. The total integral reaction cross-sections have been calculated at three collision energies by means of the quasi-classical trajectory (QCT) calculation based on the new PES and compared with previous TSH results.

Quasi-classical Trajectory Study of the Ne + H2+ → NeH+ + H Reaction Based on Global Potential Energy Surface

Using the multireference configuration interaction method with a Davidson correction and a large orbital basis set (aug-cc-pVQZ), we obtain an energy grid that includes 32 038 points for the construction of a new analytical potential energy surface (APES) for the Ne + H(2)(+) → NeH(+) + H reaction. The APES is represented as a many-body expansion containing 142 parameters, which are fitted from 31 000 ab initio energies using an adaptive nonlinear least-squares algorithm. The geometric characteristics of the reported APES and the one presented here are also compared. On the basis of the APES we obtained, reaction cross sections are computed by means of quasi-classical trajectory (QCT) calculations and compared with the experimental and theoretical data in the literature.

Obtaining stable solutions of the optimized-effective-potential method in the basis set representation

Equations of the optimized-effective-potential method in a basis set representation are solved with the use of the incomplete Cholesky decomposition technique. The resulting local potential is expanded in terms of the products of occupied and virtual Kohn-Sham orbitals thus avoiding the use of auxiliary basis sets. It is demonstrated that, for a sufficiently large orbital basis set satisfying the condition of linear dependence of these products, stable and numerically accurate solutions of the OEP method can be obtained with the use of the suggested computational approach.

Exploring biorthonormal transformations of pair-correlation functions in atomic structure variational calculations

Multiconfiguration expansions frequently target valence correlation and correlation between valence electrons and the outermost core electrons. Correlation within the core is often neglected. A large orbital basis is needed to saturate both the valence and core–valence correlation effects. This in turn leads to huge numbers of configuration state functions (CSFs), many of which are unimportant. To avoid the problems inherent to the use of a single common orthonormal orbital basis for all correlation effects in the multiconfiguration Hartree–Fock (MCHF) method, we propose to optimize independent MCHF pair-correlation functions (PCFs), bringing their own orthonormal one-electron basis. Each PCF is generated by allowing single- and double-excitations from a multireference (MR) function. This computational scheme has the advantage of using targeted and optimally localized orbital sets for each PCF. These pair-correlation functions are coupled together and with each component of the MR space through a low dimension generalized eigenvalue problem. Nonorthogonal orbital sets being involved, the interaction and overlap matrices are built using biorthonormal transformation of the coupled basis sets followed by a counter-transformation of the PCF expansions. Applied to the ground state of beryllium, the new method gives total energies that are lower than the ones from traditional complete active space (CAS)-MCHF calculations using large orbital active sets. It is fair to say that we now have the possibility to account for, in a balanced way, correlation deep down in the atomic core in variational calculations.

An ab initio potential energy surface of the [formula omitted] reaction

A highly accurate potential energy surface (PES) for the ground state ( 1 2 A ′ ) of the title reaction has been computed using an ab initio multireference configuration interaction method (MRCI) with Davidson correction and a large orbital basis set (aug-cc-pv5z). The many-body expansion and neural network method have been used to fit a set of 10 190 ab initio points. The new potential energy surface has a root-mean-square (rms) error of 0.095 kcal/mol. The total integral reaction cross-sections have been calculated as functions of the collision energy in the range of 0–2.0 eV by means of the quasi-classical trajectory calculation (QCT).

Potential energy surface for interactions between two hydrogen molecules

Nonrelativistic clamped-nuclei energies of interaction between two ground-state hydrogen molecules with intramolecular distances fixed at their average value in the lowest rovibrational state have been computed. The calculations applied the supermolecular coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)] and very large orbital basis sets-up to augmented quintuple zeta size supplemented with bond functions. The same basis sets were used in symmetry-adapted perturbation theory calculations performed mainly for larger separations to provide an independent check of the supermolecular approach. The contributions beyond CCSD(T) were computed using the full configuration interaction method and basis sets up to augmented triple zeta plus midbond size. All the calculations were followed by extrapolations to complete basis set limits. For two representative points, calculations were also performed using basis sets with the cardinal number increased by one or two. For the same two points, we have also solved the Schrodinger equation directly using four-electron explicitly correlated Gaussian (ECG) functions. These additional calculations allowed us to estimate the uncertainty in the interaction energies used to fit the potential to be about 0.15 K or 0.3% at the minimum of the potential well. This accuracy is about an order of magnitude better than that achieved by earlier potentials for this system. For a near-minimum T-shaped configuration with the center-of-mass distance R=6.4 bohrs, the ECG calculations give the interaction energy of -56.91+/-0.06 K, whereas the orbital calculations in the basis set used for all the points give -56.96+/-0.16 K. The computed points were fitted by an analytic four-dimensional potential function. The uncertainties in the fit relative to the ab initio energies are almost always smaller than the estimated uncertainty in the latter energies. The global minimum of the fit is -57.12 K for the T-shaped configuration at R=6.34 bohrs. The fit was applied to compute the second virial coefficient using a path-integral Monte Carlo approach. The achieved agreement with experiment is substantially better than in any previous work.

Pair potential for helium from symmetry-adapted perturbation theory calculations and from supermolecular data

Symmetry-adapted perturbation theory (SAPT) was applied to the helium dimer for interatomic separations R from 3 to 12 bohrs. The first-order interaction energy and the bulk of the second-order contribution were obtained using Gaussian geminal basis sets and are converged to about 0.1 mK near the minimum and for larger R. The remaining second-order contributions available in the SAPT suite of codes were computed using very large orbital basis sets, up to septuple-zeta quality, augmented by diffuse and midbond functions. The accuracy reached at this level was better than 1 mK in the same region. All the remaining components of the interaction energy were computed using the full configuration interaction method in bases up to sextuple-zeta quality. The latter components, although contributing only 1% near the minimum, have the largest uncertainty of about 10 mK in this region. The total interaction energy at R=5.6 bohrs is -11.000+/-0.011 K. For R< or =6.5 bohrs, the supermolecular (SM) interaction energies computed by us recently turned out to be slightly more accurate. Therefore, we have combined the SM results for R< or =6.5 bohrs with the SAPT results from 7.0 to 12 bohrs to fit analytic functions for the potential and for its error bars. The potential fit uses the best available van der Waals constants C(6) through C(16), including C(11), C(13), and C(15), and is believed to be the best current representation of the Born-Oppenheimer (BO) potential for helium. Using these fits, we found that the BO potential for the helium dimer exhibits the well depth D(e)=11.006+/-0.004 K, the equilibrium distance R(e)=5.608+/-0.012 bohrs, and supports one bound state for (4)He(2) with the dissociation energy D(0)=1.73+/-0.04 mK, and the average interatomic separation R=45.6+/-0.5 A.

Ab initio design of picosecond infrared laser pulses for controlling vibrational-rotational excitation of CO molecules

Optimal control of rovibrational excitations of the CO molecule using picosecond infrared laser pulses is described in the framework of the electric-nuclear Born-Oppenheimer approximation [G. G. Balint-Kurti et al., J. Chem. Phys. 122, 084110 (2005)]. The potential energy surface of the CO molecule in the presence of an electric field is calculated using coupled cluster theory with a large orbital basis set. The quantum dynamics of the process is treated using a full three dimensional treatment of the molecule in the laser field. The detailed mechanisms leading to efficient control of the selected excitation processes are discussed.

Quantum control of molecular motion including electronic polarization effects with a two-stage toolkit

A method for incorporating strong electric field polarization effects into optimal control calculations is presented. A Born-Oppenheimer-type separation, referred to as the electric-nuclear Born-Oppenheimer (ENBO) approximation, is introduced in which variations of both the nuclear geometry and the external electric field are assumed to be slow compared with the speed at which the electronic degrees of freedom respond to these changes. This assumption permits the generation of a potential energy surface that depends not only on the relative geometry of the nuclei but also on the electric field strength and on the orientation of the molecule with respect to the electric field. The range of validity of the ENBO approximation is discussed in the paper. A two-stage toolkit implementation is presented to incorporate the polarization effects and reduce the cost of the optimal control dynamics calculations. As an illustration of the method, it is applied to optimal control of vibrational excitation in a hydrogen molecule aligned along the field direction. Ab initio configuration interaction calculations with a large orbital basis set are used to compute the H-H interaction potential in the presence of the electric field. The significant computational cost reduction afforded by the toolkit implementation is demonstrated.

Ab initio potential energy surfaces, total absorption cross sections, and product quantum state distributions for the low-lying electronic states of N2O

Adiabatic potential energy surfaces for the six lowest singlet electronic states of N(2)O (X (1)A('), 2 (1)A('), 3 (1)A('), 1 (1)A("), 2 (1)A(") and 3 (1)A(")) have been computed using an ab initio multireference configuration interaction (MRCI) method and a large orbital basis set (aug-cc-pVQZ). The potential energy surfaces display several symmetry related and some nonsymmetry related conical intersections. Total photodissociation cross sections and product rotational state distributions have been calculated for the first ultraviolet absorption band of the system using the adiabatic ab initio potential energy and transition dipole moment surfaces corresponding to the lowest three excited electronic states. In the Franck-Condon region the potential energy curves corresponding to these three states lie very close in energy and they all contribute to the absorption cross section in the first ultraviolet band. The total angular momentum is treated correctly in both the initial and final states. The total photodissociation spectra and product rotational distributions are determined for N(2)O initially in its ground vibrational state (0,0,0) and in the vibrationally excited (0,1,0) (bending) state. The resulting total absorption spectra are in good quantitative agreement with the experimental results over the region of the first ultraviolet absorption band, from 150 to 220 nm. All of the lowest three electronically excited states [(1)Sigma(-)(1 (1)A(")), (1)Delta(2 (1)A(')), and (1)Delta(2 (1)A("))] have zero transition dipole moments from the ground state [(1)Sigma(+)(1 (1)A('))] in its equilibrium linear configuration. The absorption becomes possible only through the bending motion of the molecule. The (1)Delta(2 (1)A('))<--X (1)Sigma(+)((1)A(')) absorption dominates the absorption cross section with absorption to the other two electronic states contributing to the shape and diffuse structure of the band. It is suggested that absorption to the bound (1)Delta(2 (1)A(")) state makes an important contribution to the experimentally observed diffuse structure in the first ultraviolet absorption band. The predicted product rotational quantum state distribution at 203 nm agrees well with experimental observations.

Cracking Electron Correlation

After reviewing current perceptions about the electron correlation problem, a general selected configuration interaction (CI) method to calculate highly accurate energies for atoms and medium-sized molecules is described. For a given orbital basis, selected CI may be implemented to yield the best energy and corresponding wave function for given selection thresholds together with an estimate of the full CI energy. The linked cluster expansion is used as an intermediate device to reveal relationships between variational coefficients in the CI expansion. These relationships allow to predict CI coefficients that are used in an a priori and fast-to-evaluate variational estimate of individual configuration energy contributions via Brown's formula both for selection purposes, and to estimate the truncation energy error caused by unselected configurations. Concepts based on natural orbitals are used likewise for configurations not amenable to the above analysis. An application to the ground state of the Ne atom achieves an accuracy of two µhartrees using very large orbital bases, increasing accuracy more than a hundredfold and at fair computational cost.

Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Core–valence and quintuple-ζ basis sets for H to Ar and QZVPP basis sets for Li to Kr

An implementation of analytic basis set gradients is reported for the optimization of auxiliary basis sets in resolution-of-the-identity second-order Moller–Plesset perturbation theory (RI-MP2) and approximate coupled-cluster singles-and-doubles (RI-CC2) calculations. The analytic basis set gradients are applied in the optimization of auxiliary basis sets for a number of large one-electron orbital basis sets which provide correlation energies close to the basis set limit: the core–valence basis sets cc-pwCVXZ (B–Ne, Al–Ar) with X = D, T, Q, 5, the quintuple-ζ basis sets cc-pV5Z (H–Ar) and cc-pV(5 + d)Z (Al–Ar) and the doubly-polarized valence quadruple-ζ basis sets QZVPP for Li–Kr. The quality of the optimized auxiliary basis sets is evaluated for several test sets with small and medium sized molecules.

The rotational spectrum and theoretical study of a dinuclear complex, MnRe(CO)(10).

The first rotational spectrum of a dinuclear complex, MnRe(CO)(10), has been obtained using a high-resolution pulsed beam microwave spectrometer. Sixty-four hyperfine components of the J=11-->J(')=12 and J=12-->J(')=13 rotational transitions were measured for two rhenium isotopomers. The B values obtained from the experiment are B=200.36871(18) MHz for the (187)Re isotopomer and B=200.5561(10) MHz for the (185)Re isotopomer. The measured rotational constants are in reasonably good agreement with the B values calculated from the x-ray diffraction structural data, and from theoretical calculations. The gas-phase Mn-Re bond distance is approximately 2.99 A, and the calculated value is only slightly longer. The experimental quadrupole coupling constant for the manganese atom is eQq(aa) ((55)Mn)=-16.52(5) MHz, and the corresponding quadrupole coupling constants for the two rhenium isotopomers are eQq(aa) ((187)Re)=370.4(4) MHz and eQq(aa) ((185)Re)=390.9(6) MHz. The quadrupole coupling constants were also determined from a variety of theoretical calculations, with very large Gaussian orbital bases. The best estimates, at a nonrelativistic level, are eQq(aa) ((55)Mn)=0.68 MHz and eQq(aa) ((187)Re)=327.6 MHz with a 874 GTO basis set, but the results are very basis set dependent, especially the sign of the Mn quadrupole coupling. Very slight bending of angles MnC(eq)O(eq) and ReC(eq)O(eq) angles is found in the calculations.