Full CI Limit Research Articles

Advances in Parallel Heat Bath Configuration Interaction

Heat-bath configuration interaction (HCI) is a deterministic method that approaches the full CI limit at greatly reduced computational cost. In this work, computational improvements to the HCI algorithm are introduced targeting speed, parallel efficiency, and memory requirements. The new implementation introduces a hash function to distribute determinants and takes advantage of MPI and OpenMP for parallelism allowing for a (22e,168o) active space to be studied, which explicitly includes 2.39 × 107 variational determinants and 8.95 × 1010 perturbative determinants. Benchmarks show up to 86% parallel efficiency of the perturbative step on 32 nodes (4096 cores) and a total efficiency of 74%. The new HCI implementation is benchmarked for accuracy against prior results and applied to study the triplet-quintet gap in the challenging [FeO(NH3)5]2+ complex.

ATOMIC-2 Protocol for Thermochemistry.

ATOMIC is a midlevel thermochemistry protocol that uses Pople's concept of bond separation reactions (BSRs) as a theoretical framework to reduce computational demands in the evaluation of atomization energies and enthalpies of formation. Various composite models are available that approximate bond separation energies at the complete-basis-set limit of all-electron CCSD(T), each balancing computational cost with achievable accuracy. Evaluated energies are then combined with very high-level, precomputed atomization energies of all auxiliary molecules appearing in the BSR to obtain the atomization energy of the molecule under study. ATOMIC-2 is a new version of the protocol that retains the overall concept and all previously defined composite models but improves on ATOMIC-1 in various other ways: Geometry optimization and zero-point-energy evaluation are performed at the density functional level (PBE0-D3/6-311G(d)), which shows significant computational savings and better accuracy than the previously employed RI-MP2/cc-pVTZ. The BSR framework is improved, using more accurate complete-basis-set (CBS) extrapolations toward the Full CI limit for the atomization energies of all auxiliary molecules. Finally, and most importantly, an error and uncertainty model termed ATOMIC-2um is added that estimates average bias and uncertainty for each of the atomization energy contributions that arise from the simplified treatment of some contributions to bond separation energies (CCSD(T)) and the neglect of others (such as higher order, scalar relativistic, or diagonal Born-Oppenheimer corrections) or from residual error in the energies of auxiliary molecules. Large and diverse benchmarks including up to 1179 molecules are used to evaluate necessary reference data and to correlate the observed error for each of the contributions with appropriate proxies that are available without additional quantum-chemical calculations for a particular molecule and represent its size and type. The implementation of ATOMIC-2 considers neutral, closed-shell molecules containing H, C, N, O, and F atoms; compared to ATOMIC-1, the framework has been extended to cover a few challenging but rare bond topologies. In comparison to highly accurate reference data for 184 molecules taken from the ATcT database (V. 1.122r), regular ATOMIC-2 shows noticeable underbinding, but the bias-corrected protocol ATOMIC-2um is found to be more accurate than either ATOMIC-1 or standard Gaussian-4 theory, and the uncertainty model is consistent with statistics of actually observed errors. Problems arising from ambiguous or challenging Lewis-valence structures defining BSRs are discussed, and computational efficiency is demonstrated. Computer code is made available to perform ATOMIC-2um analyses.

TWOE Code: An Efficient Tool for Explicit Partition of Coupled Cluster and Configuration Interaction Energies into Atomic and Diatomic Contributions

The efficient implementation of the TWOE program for evaluating the atomic and interatomic energy components at post-HF level was developed. The systematic convergence of these terms up to a near full-CI limit was performed for the first time for a series of coupled cluster methods: CCSD → CCSDT → CCSDTQ → CCSDTQP. A comparison with corresponding CI approaches (up to fifth excitation level) is additionally discussed. For a set of diatomic systems, it was demonstrated that, along with a full molecular energy convergence, all its components are also converged but with different patterns. It was found that not all components are decreased in their values at increasing computational rank. For instance, atomic energy parts are decreased while interatomic (interaction) energies are increased as the limiting level is approached. Two schemes were employed for atomic partition of molecules: the Baders approach and planes dissection. Influence of dynamical correlation effects on atomic energy components was analyzed in detail. Current TWOE implementation allows one, in principle, to work with any ab initio method providing the two-particle density matrix. It is believed that the developed program will be a useful tool for a real space energy decomposition that helps to reveal the most peculiar points in the structure of the total and correlation energies of a molecule.

Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs.

Quantum chemistry is a discipline which relies heavily on very expensive numerical computations. The scaling of correlated wave function methods lies, in their standard implementation, between and , where N is proportional to the system size. Therefore, performing accurate calculations on chemically meaningful systems requires (i) approximations that can lower the computational scaling and (ii) efficient implementations that take advantage of modern massively parallel architectures. Quantum Package is an open-source programming environment for quantum chemistry specially designed for wave function methods. Its main goal is the development of determinant-driven selected configuration interaction (sCI) methods and multireference second-order perturbation theory (PT2). The determinant-driven framework allows the programmer to include any arbitrary set of determinants in the reference space, hence providing greater methodological freedom. The sCI method implemented in Quantum Package is based on the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm which complements the variational sCI energy with a PT2 correction. Additional external plugins have been recently added to perform calculations with multireference coupled cluster theory and range-separated density-functional theory. All the programs are developed with the IRPF90 code generator, which simplifies collaborative work and the development of new features. Quantum Package strives to allow easy implementation and experimentation of new methods, while making parallel computation as simple and efficient as possible on modern supercomputer architectures. Currently, the code enables, routinely, to realize runs on roughly 2 000 CPU cores, with tens of millions of determinants in the reference space. Moreover, we have been able to push up to 12 288 cores in order to test its parallel efficiency. In the present manuscript, we also introduce some key new developments: (i) a renormalized second-order perturbative correction for efficient extrapolation to the full CI limit and (ii) a stochastic version of the CIPSI selection performed simultaneously to the PT2 calculation at no extra cost.

Deterministic Construction of Nodal Surfaces within Quantum Monte Carlo: The Case of FeS.

In diffusion Monte Carlo (DMC) methods, the nodes (or zeroes) of the trial wave function dictate the magnitude of the fixed-node (FN) error. In standard DMC implementations, the nodes are optimized by stochastically optimizing a short multideterminant expansion in the presence of an explicitly correlated Jastrow factor. Here, following a recent proposal, we pursue a different route and consider the nodes of selected configuration interaction (sCI) expansions built with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm. By increasing the size of the sCI expansion, these nodes can be systematically and deterministically improved. The present methodology is used to investigate the properties of the transition metal sulfide molecule FeS. This apparently simple molecule has been shown to be particularly challenging for electronic structure theory methods due to the proximity of two low-energy quintet electronic states of different spatial symmetry and the difficulty to treat them on equal footing from a one-electron basis set point of view. In particular, we show that, at the triple-ζ basis set level, all sCI results-including those extrapolated at the full CI (FCI) limit-disagree with experiment, yielding an electronic ground state of 5Σ+ symmetry. Performing FN-DMC simulation with sCI nodes, we show that the correct 5Δ ground state is obtained if sufficiently large expansions are used. Moreover, we show that one can systematically get accurate potential energy surfaces and reproduce the experimental dissociation energy as well as other spectroscopic constants.

Cheap and Near Exact CASSCF with Large Active Spaces.

We use the recently developed Heat-bath Configuration Interaction (HCI) algorithm as an efficient active space solver to perform multiconfiguration self-consistent field calculations (HCISCF) with large active spaces. We give a detailed derivation of the theory and show that difficulties associated with non-variationality of the HCI procedure can be overcome by making use of the Lagrangian formulation to calculate the HCI relaxed two-body reduced density matrix. HCISCF is then used to study the electronic structure of butadiene, pentacene, and Fe-porphyrin. One of the most striking results of our work is that the converged active space orbitals obtained from HCISCF are relatively insensitive to the accuracy of the HCI calculation. This allows us to obtain nearly converged CASSCF energies with an estimated error of less than 1 mHa using the orbitals obtained from the HCISCF procedure in which the integral transformation is the dominant cost. For example, an HCISCF calculation on the Fe-porphyrin model complex with an active space of (44e, 44o) took only 412 s per iteration on a single node containing 28 cores, out of which 185 s was spent in the HCI calculation and the remaining 227 s was used mainly for integral transformation. Finally, we also show that active space orbitals can be optimized using HCISCF to substantially speed up the convergence of the HCI energy to the Full CI limit because HCI is not invariant to unitary transformations within the active space.

Excited states using semistochastic heat-bath configuration interaction

We extend our recently developed heat-bath configuration interaction (HCI) algorithm, and our semistochastic algorithm for performing multireference perturbation theory, to calculate excited-state wavefunctions and energies. We employ time-reversal symmetry, which reduces the memory requirements by more than a factor of two. An extrapolation technique is introduced to reliably extrapolate HCI energies to the full CI limit. The resulting algorithm is used to compute fourteen low-lying potential energy surfaces of the carbon dimer using the cc-pV5Z basis set, with an estimated error in energy of 30-50 μHa compared to full CI. The excitation energies obtained using our algorithm have a mean absolute deviation of 0.02 eV compared to experimental values.

Efficient implementation of restricted active space configuration interaction with the hole and particle approximation

The restricted active space configuration interaction (RASCI) formalism with the hole and particle truncation of the wavefunction, that is, RASCI(h,p), holds very nice properties such as balanced treatment of ground and low-lying excited states, spin-completeness, large flexibility of the wavefunction, and moderate computational cost. In this article, I present a new implementation of the RASCI(h,p) method using a general algorithm based on the integral-driven approach. The new implementation allows to choose any electronic configuration as the single reference in combination with an excitation operator with any number of ionization, electron attachment, or spin-flip (SF) excitations. The applicability and good performance of the new computational code is tested in the ground state calculation of water molecule with increasingly large active spaces and up to the full-CI limit, the calculation of all-trans linear polyenes with variable number of SF excitations, and the low-lying states of fluorine molecule with a double-ionization potential operator.

Many-body calculations of molecular electric polarizabilities in asymptotically complete basis sets

The static dipole polarizabilities of Ne, CO, N2, F2, HF, H2O, HCN, and C2H2 (acetylene) have been determined close to the Full-CI limit along with an asymptotically complete basis set (CBS), according to the principles of a Focal Point Analysis. For this purpose the results of Finite Field calculations up to the level of Coupled Cluster theory including Single, Double, Triple, Quadruple and perturbative Pentuple excitations [CCSDTQ(P)] were used, in conjunction with suited extrapolations of energies obtained using augmented and doubly-augmented Dunning's correlation consistent polarized valence basis sets of improving quality. The polarizability characteristics of C2H4 (ethylene) and C2H6 (ethane) have been determined on the same grounds at the CCSDTQ level in the CBS limit. Comparison is made with results obtained using lower levels in electronic correlation, or taking into account the relaxation of the molecular structure due to an adiabatic polarization process. Vibrational corrections to electronic polarizabilities have been empirically estimated according to Born–Oppenheimer Molecular Dynamical simulations employing Density Functional Theory. Confrontation with experiment ultimately indicates relative accuracies of the order of 1 to 2%.

Spin–orbit coupling and electron correlation in relativistic configuration interaction and coupled-cluster methods

We studied convergence characteristics of relativistic effective core potential (RECP) based configuration interaction (CI) and coupled-cluster (CC) schemes in terms of spin–orbit coupling and electron correlation. The relativistic correlated methods can be divided into Kramers restricted (KR) and spin–orbit (SO) methods which differ by the stage of spin–orbit treatment: the KR method employs two-component Kramers restricted Hartree–Fock (HF) spinors as the one-electron basis in which spin–orbit coupling is included, whereas the SO method is based on one-component molecular orbitals generated from scalar relativistic HF and the spin–orbit interaction is then entered in post-HF step. The KR method is usually superior to the SO method for molecules containing heavy elements since spin–orbit coupling is included from the HF step. A performance calibration of the SO method against the KR method is performed by computations of the ground state energies and equilibrium bond lengths of MH (M=Tl, Pb, Bi, Po, and At). Spin–orbit coupling of each molecule was systematically increased by adjusting the spin–orbit operator of RECP to investigate its impact on the SO method. Although KRCI and SOCI converged to the same full-CI limit, for the strong spin–orbit coupling SOCI required higher levels of correlation compared to KRCI to account for the orbital relaxation effect. SOCC, in contrast, was able to recover both spin–orbit interaction and electron correlation in CC steps regardless of the spin–orbit strength, implying that SOCC could be the reliable and efficient relativistic ab initio method for moderate sized molecules containing heavy elements.

An adaptive coupled-cluster theory: @CC approach

A formulation of an adaptive coupled-cluster theory is presented. The method automatically "adjusts" to any state of an electronic system and converges to the full CI limit, thus being capable of describing both single- and multireference phenomena. Adaptivity is accomplished through a guided selection of a compact set of cluster amplitudes as required for a proper description of the electronic system under consideration. The approach suggested is of "black-box" type. A special importance-selection function (discriminatory function) is explicitly introduced for the guided selection of variables involved in the theoretical model. The method is tested on molecules which exhibit strong multireference character in the region of chemical bond elongation. An unambiguous comparison with formally exact full CI solutions shows that the method is capable of providing mHartee accuracy using a rather compact set of cluster amplitudes.

Imaging Momentum Orbital Densities of Conformationally Versatile Molecules: A Benchmark Theoretical Study of the Molecular and Electronic Structures of Dimethoxymethane

The main purpose of the present work is to predict from benchmark many-body quantum mechanical calculations the results of experimental studies of the valence electronic structure of dimethoxymethane employing electron momentum spectroscopy, and to establish once and for all the guidelines that should systematically be followed in order to reliably interpret the results of such experiments on conformationally versatile molecules. In a first step, accurate calculations of the energy differences between stationary points on the potential energy surface of this molecule are performed using Hartree-Fock (HF) theory and post-HF treatments of improving quality (MP2, MP3, CCSD, CCSD(T), along with basis sets of increasing size. This study focuses on the four conformers of this molecule, namely the trans-trans (TT), trans-gauche (TG), gauche-gauche (G+G+), and gauche-gauche (G+G-) structures, belonging to the C2v, C1, C2, and Cs symmetry point groups, respectively. A focal point analysis supplemented by suited extrapolations to the limit of asymptotically complete basis sets is carried out to determine how the conformational energy differences at 0 K approach the full CI limit. In a second step, statistical thermodynamics accounting for hindered rotations is used to calculate Gibbs free energy corrections to the above energy differences, and to evaluate the abundance of each conformer in the gas phase. It is found that, at room temperature, the G+G+ species accounts for 96% of the conformational mixture characterizing dimethoxymethane. In a third step, the valence one-electron and shake-up ionization spectrum of dimethoxymethane is analyzed according to calculations on the G+G+ conformer alone by means of one-particle Green's function [1p-GF] theory along with the benchmark third-order algebraic diagrammatic construction [ADC(3)] scheme. A complete breakdown of the orbital picture of ionization is noted at electron binding energies above 22 eV. A comparison with available (e,2e) ionization spectra enables us to identify specific fingerprints of through-space orbital interactions associated with the anomeric effect. At last, based on our 1p-GF/ADC(3) assignment of spectral bands, accurate and spherically averaged (e,2e) electron momentum distributions at an electron impact energy of 1200 eV are computed from the related Dyson orbitals. Very significant discrepancies are observed with momentum distributions obtained for several outer-valence levels using standard Kohn-Sham orbitals.

Orbital-optimized coupled-cluster theory does not reproduce the full configuration-interaction limit

It is shown that due to the mixing of the usual projection approach of coupled cluster with variational orbital optimization, orbital-optimized coupled cluster (OCC) fails to reproduce the full configuration-interaction (full CI) limit when the cluster operator becomes complete. It is pointed out that the fulfillment of the projected singles equations, which define the orbital gradient in Brueckner coupled cluster (BCC), is mandatory for a correct behavior. As numerical examples we present general OCC and BCC calculations up to the full CI limit on CH(2) and an active-space model of ozone. The observed deviations of OCC from full CI are of the order of the correlation error obtained in calculations with up to quadruples excitations. Thus the failure of OCC may be considered tolerable in more approximate calculations but clearly prohibitive for any benchmark application. For applications to active-space models a hybrid approach for OCC is suggested in which for active particle-hole rotations the Brueckner orbital gradient is employed, whereas for the remaining orbital rotations the variational orbital gradient is retained.

Multireference configuration interaction calculations of some low-lying states of positronium hydride

Multireference configuration interaction (MRCI) calculations of the ground state S2,1 and low-lying excited states P2,1, D2,1, and S2,3 of positronium hydride (PsH) were carried out, and their total energies and two-photon annihilation rates were presented. Our MRCI calculations gave more compact and accurate wave functions than recent large scale full CI ones, and the effectiveness of the MRCI method for positron-atom complexes was shown. Two extrapolation methods derived from an analysis of the convergence pattern of the total energies and two-photon annihilation rates with respect to angular momentum of orbitals were attempted in order to estimate full CI limit of those values. As a result, our reliable extrapolation method gave −0.788 642, −0.732 826, −0.724 642, and −0.739 829 a.u. for the total energies of the S2,1, P2,1, D2,1, and S2,3 states of PsH, respectively. For the two-photon annihilation rates of the respective states, 2.1078, 1.5702, 1.0980, and 1.0784 ns−1 were obtained, respectively.

Passing the one-quadrillion limit in FCl extrapolations on a personal computer

An efficient algorithm for the full configuration interaction (FCI) on a personal computer is presented. Selected configuration interaction and second-order perturbation theory (PT) are combined to approach the full-CI limit. Comparison was made to existing full-CI benchmarks. A test calculation of NH 2 symmetrical dissociation shows that the running time for finding the correlation energy with an accuracy of 0.1 kcal/mol is just 5 min on an IBM 486DX2-66 PC. As another example, the approximate full-CI calculations of the HF dimer in the (4s3 1d/2s 1p) and the (43 3p 1d/2s 1p) basis sets have been carried out. The ground state of (HF) 2 produces 3 × 10 14 and 4 × 10 15 determinants, respectively.

Accurate CI expansion in a spline basis of the helium ground‐state wave function

AbstractCareful study of radial spline basis‐set convergence on the angular limits of the He 1s 2 state has been performed. Improvement of algorithm allows reaching the full ci limit in a large radial (79 spline functions) and angular (Imax=80) expansion. A variational energy of − 2.903724299061 au, with an error of 7.8 × 10−8 au is obtained, improving by over two orders of magnitude the previous available limit. Accurate angular momentum limits (1 × 10−8 au) are obtained up to l = 25. Analysis of the behavior of two‐particle radial wave functions reveals a sharply peaked behavior which allows effective truncation of the two‐electron radial basis for high angular momentum values. A reasonable estimate of computational resources required indicate that an accuracy below 1 × 10−8 au is presently feasible and about 1 × 10−9 au possible in the near future. © 1995 John Wiley & Sons, Inc.

The dipole moment and magnetic hyperfine properties of the excited A 2Σ+(3sσ) Rydberg state of nitric oxide

The dipole moment and magnetic hyperfine properties of the A 2Σ+ Rydberg state of nitric oxide have been evaluated at a variety of levels of theory with extended correlation consistent basis sets. Using the finite field approach to compute the dipole moment, restricted coupled cluster RCCSD(T) and complete active space-configuration interaction CAS-CI+Q methods yield values (1.09–1.12 D) that are essentially identical to experiment. In contrast, dipole moments computed as an expectation value of the dipole moment operator typically differ from experiment by 0.1–0.6 D. The rather unfavorable comparisons with experiment reported in previous theoretical studies may stem, in part, from the method chosen to evaluate the dipole moment. Magnetic hyperfine properties were evaluated using a variety of unrestricted and restricted open-shell Hartree–Fock-based methods. We estimated the full CI limiting properties by exploiting the convergence behavior of a sequence of MRCI wave functions. The isotropic component Aiso(14N) of 39±1 MHz evaluated in this fashion is in excellent accord with the experimental value of 41.4±1.7 MHz. Highly correlated UHF-based methods [e.g., CCSD(T) and QCISD(T)] yield comparable values of 40–41 MHz that are in good agreement with both experiment and the apparent full CI limit. However, for Aiso(17O), the full CI limit (−97±2 MHz) and the UHF-based results (ca.−118 MHz) differ by roughly 20 MHz. It remains unclear how to reconcile this large discrepancy.

On the convergence ofMBPT andCC nuclear magnetic shielding constants ofBH toward the fullCI limit

The performance of many-body perturbation theory (MBPT) and coupled-cluster (CC) methods in the calculation of nuclear magnetic shielding constants for BH is analyzed by comparison with results from full configuration interaction (FCI) calculations. Low-order MBPT computations are shown to be inadequate for BH, while the coupled-cluster ansatz, in particular, in its singles and doubles (CCSD) approximation, provides an efficient treatment of electron correlation, overestimating correlation effects to the 11B shielding by only 3-4 ppm (1.5%). Based on this comparison, CCSD calculations with large atomic natural orbital (ANO) basis sets are used to estimate accurate theoretical values for the isotropic and anisotropic shielding constants of BH. © 1995 John Wiley & Sons, Inc.

Exploiting regularity in systematic sequences of wavefunctions which approach the full CI limit

The ground state total energy and related 1-electron properties are computed for three small molecules (N2, H2O, and H2CN) using several systematic sequences of wavefunctions which approach the full CI. These sequences include multireference CI, averaged coupled pair functional and quasidegenerate variational perturbation theory wavefunctions. It is demonstrated that sufficient regularity exists in the sequence of variationally computed energies to permit extrapolation to the full CI limit using simple analytic expressions. It is furthermore demonstrated that a subset of the original list of configurations employed in the normal singles and doubles CI procedure can be selected using second order perturbation theory without adversely affecting the extrapolation to the full CI limit. This significantly broadens the range of applicability of the method. Along these lines, a scheme is proposed for the extrapolation of the selected CI results to the zero threshold (i.e. unselected) values in cases where the numbers of configurations associated with the latter would render the calculations intractable. Due to the vast reduction in the number of configurations which are handled variationally, the proposed scheme makes it possible to derive estimates of the full CI limit in cases where explicit full CI is either very difficult or currently impossible.

Approximating full configuration interaction with selected configuration interaction and perturbation theory

Selected configuration interaction (CI) calculations and second order perturbation theory are combined to systematically approach the full-CI limit. The resulting algorithm has negligible requirement for memory or disk space, being limited only by available cpu time. Comparison is made to existing full-CI benchmarks (DZ and DZP water, the oxygen atom and its anion, ammonia and the magnesium atom). In all cases the full-CI result is recovered to better than 0.1 kcal/mol.