Configuration Interaction Algorithm Research Articles

Compactification of determinant expansions via transcorrelation.

Although selected configuration interaction (SCI) algorithms can tackle much larger Hilbert spaces than the conventional full CI method, the scaling of their computational cost with respect to the system size remains inherently exponential. In addition, inaccuracies in describing the correlation hole at small interelectronic distances lead to the slow convergence of the electronic energy relative to the size of the one-electron basis set. To alleviate these effects, we show that the non-Hermitian, transcorrelated (TC) version of SCI significantly compactifies the determinant space, allowing us to reach a given accuracy with a much smaller number of determinants. Furthermore, we note a significant acceleration in the convergence of the TC-SCI energy as the basis set size increases. The extent of this compression and the energy convergence rate are closely linked to the accuracy of the correlation factor used for the similarity transformation of the Coulombic Hamiltonian. Our systematic investigation of small molecular systems in increasingly large basis sets illustrates the magnitude of these effects.

Approximate Excited-State Geometry Optimization with the State-Averaged Adaptive Sampling Configuration Interaction Algorithm with Large Active Spaces.

The selected configuration interaction (SCI) wave function is a useful approximation to the full configuration interaction (FCI) one. The adaptive sampling CI (ASCI) method is a deterministic SCI method. By combining ASCI and orbital optimization, the ASCI self-consistent field (ASCI-SCF) method, which is an approximation of the complete active space self-consistent field (CASSCF) method, can be formulated as well. However, their applicability has been tested mainly on the systems in their electronically ground states. In this work, we implement the state-average (SA) ansatz in ASCI-SCF calculations to calculate excited states. We also derive expressions for the approximate analytical gradient and implement them as a computer program. We demonstrate the applicability of the current method for calculating vertical and adiabatic excitation energies and optimizing the molecular geometries of thermally activated delayed fluorescence (TADF) molecules.

Transcorrelated selected configuration interaction in a bi-orthonormal basis and with a cheap three-body correlation factor.

In this work, we develop a mathematical framework for a selected configuration interaction (SCI) algorithm within a bi-orthogonal basis for transcorrelated (TC) calculations. The bi-orthogonal basis used here serves as the equivalent of the standard Hartree-Fock (HF) orbitals. However, within the context of TC, it leads to distinct orbitals for the left and right vectors. Our findings indicate that the use of such a bi-orthogonal basis allows for a proper definition of the frozen core approximation. In contrast, the use of HF orbitals results in bad error cancellations for ionization potentials and atomization energies (AE). Compared to HF orbitals, the optimized bi-orthogonal basis significantly reduces the positive part of the second-order energy (PT2), thereby facilitating the use of standard extrapolation techniques of hermitian SCI. While we did not observe a significant improvement in the convergence of the SCI algorithm, this is largely due to the use in this work of a simple three-body correlation factor introduced in a recent study. This correlation factor, which depends only on atomic parameters, eliminates the need for re-optimization of the correlation factor for molecular systems, making its use straightforward and user-friendly. Despite the simplicity of this correlation factor, we were able to achieve accurate results on the AE of a series of 14 molecules on a triple-zeta basis. We also successfully broke a double bond until the full dissociation limit while maintaining the size consistency property. This work thus demonstrates the potential of the BiO-TC-SCI approach in handling complex molecular systems.

A configuration-based heatbath-CI for spin-adapted multireference electronic structure calculations with large active spaces.

This work reports on a spin-pure configuration-based implementation of the heatbath configuration interaction (HCI) algorithm for selective configuration interaction. Besides the obvious advantage of being spin-pure, the presented method combines the compactness of the configurational ansatz with the known efficiency of the HCI algorithm and a variety of algorithmic and conceptual ideas to achieve a high level of performance. In particular, through pruning of the selected configurational space after HCI selection by means of a more strict criterion, a more compact wavefunction representation is obtained. Moreover, the underlying logic of the method allows us to minimize the number of redundant matrix-matrix multiplications while making use of just-in-time compilation to achieve fast diagonalization of the Hamiltonian. The critical search for 2-electron connections within the configurational space is facilitated by a tree-based representation thereof as suggested previously by Gopal et al. Usage of a prefix-based parallelization and batching during the calculation of the PT2-correction leads to a good load balancing and significantly reduced memory requirements for these critical steps of the calculation. In this way, the need for a semistochastic approach to the PT2 correction is avoided even for large configurational spaces. Finally, several test-cases are discussed to demonstrate the strengths and weaknesses of the presented method.

Extension of selected configuration interaction for transcorrelated methods.

In this work, we present an extension of popular selected configuration interaction (SCI) algorithms to the Transcorrelated (TC) framework. Although we used in this work the recently introduced one-parameter correlation factor [E. Giner, J. Chem. Phys. 154, 084119 (2021)], the theory presented here is valid for any correlation factor. Thanks to the formalization of the non-Hermitian TC eigenvalue problem as a search of stationary points for a specific functional depending on both left- and right-functions, we obtain a general framework, allowing for different choices for both the selection criterion in SCI and the second order perturbative correction to the energy. After numerical investigations on different second-row atomic and molecular systems in increasingly large basis sets, we found that taking into account the non-Hermitian character of the TC Hamiltonian in the selection criterion is mandatory to obtain a fast convergence of the TC energy. In addition, selection criteria based on either the first order coefficient or the second order energy lead to significantly different convergence rates, which is typically not the case in the usual Hermitian SCI. Regarding the convergence of the total second order perturbation energy, we find that the quality of the left-function used in the equations strongly affects the quality of the results. Within the near-optimal algorithm proposed here, we find that the SCI expansion in the TC framework converges faster than the usual SCI in terms of both the basis set and the number of Slater determinants.

Floquet Time-Dependent Configuration Interaction for Modeling Ultrafast Electron Dynamics

Time-dependent electronic structure methods are a valuable tool for simulating spectroscopic experiments. Recent advances in time-dependent configuration interaction (TDCI) algorithms have made them an attractive means of modeling many-electron dynamics, particularly for cases where multireference effects are essential. Here we present an extension to TDCI, Floquet TDCI, where the electronic wave function is expanded in a basis of light-dressed determinants. Our approach is based on our high-performance graphics processing unit (GPU) accelerated implementation of complete active space configuration interaction (CASCI). Simulations of two-photon absorption demonstrate that Floquet TDCI is well-suited for modeling dynamics in intense, ultrashort laser pulses. Accurate results are obtained for pulse energies up to ∼4 × 10-4 J/cm2 per pulse in the most difficult case explored here. By simulation of a set of molecules under continuous wave coupling, we demonstrate the ability of Floquet to describe the entanglement of light and multiple molecules in a cavity (i.e., a cavity polariton). Excellent computational performance is observed: a 320 fs propagation of a large dye (C30N2H22) with a 2 as timestep and a large active space (10 electrons in 11 orbitals), including a monochromatic pulse with three photon states, was performed in 3 h 6 min on a single Tesla V100 GPU. Our Floquet TDCI algorithm scales linearly with the number of photon states and exponentially with the number of photon colors included in the calculation. We argue that its energy-conserving nature makes Floquet TDCI well-suited to drive nonadiabatic molecular dynamics simulations.

Modern Approaches to Exact Diagonalization and Selected Configuration Interaction with the Adaptive Sampling CI Method.

Recent advances in selected configuration interaction methods have made them competitive with the most accurate techniques available and, hence, creating an increasingly powerful tool for solving quantum Hamiltonians. In this work, we build on recent advances from the adaptive sampling configuration interaction (ASCI) algorithm. We show that a useful paradigm for generating efficient selected CI/exact diagonalization algorithms is driven by fast sorting algorithms, much in the same way iterative diagonalization is based on the paradigm of matrix vector multiplication. We present several new algorithms for all parts of performing a selected CI, which includes new ASCI search, dynamic bit masking, fast orbital rotations, fast diagonal matrix elements, and residue arrays. The ASCI search algorithm can be used in several different modes, which includes an integral driven search and a coefficient driven search. The algorithms presented here are fast and scalable, and we find that because they are built on fast sorting algorithms they are more efficient than all other approaches we considered. After introducing these techniques, we present ASCI results applied to a large range of systems and basis sets to demonstrate the types of simulations that can be practically treated at the full-CI level with modern methods and hardware, presenting double- and triple-ζ benchmark data for the G1 data set. The largest of these calculations is Si2H6 which is a simulation of 34 electrons in 152 orbitals. We also present some preliminary results for fast deterministic perturbation theory simulations that use hash functions to maintain high efficiency for treating large basis sets.

Using computed infrared intensities for the reduction of vibrational configuration interaction bases.

The Adaptive Vibrational Configuration Interaction (A-VCI) algorithm is an iterative process that computes the anharmonic spectrum of a molecule using nested bases to discretize the Hamiltonian operator. For large molecular systems, the size of the discretization space and the computation time quickly become prohibitive. It is therefore necessary to develop new methods to further limit the number of basis functions. Most of the time, the interpretation of an experimental infrared spectrum does not require the calculation of all eigenvalues but only those corresponding to vibrational states with significant intensity. In this paper, a technique that uses infrared intensities is introduced to select a subset of eigenvalues to be precisely calculated. Thus, we build smaller nested bases and reduce both the memory footprint and the computational time. We validate the advantages of this new approach on a well-studied 7-atom molecular system (C2H4O), and we apply it on a larger 10-atom molecule (C4H4N2).

Efficient calculations of a large number of highly excited states for multiconfigurational wavefunctions.

Electronically excited states play important roles in many chemical reactions and spectroscopic techniques. In quantum chemistry, a common technique to solve excited states is the multiroot Davidson algorithm, but it is not designed for processes like X-ray spectroscopy that involves hundreds of highly excited states. We show how the use of a restricted active space wavefunction together with a projection operator to remove low-lying electronic states offers an efficient way to reach single and double-core-hole states. Additionally, several improvements to the stability and efficiency of the configuration interaction (CI) algorithm for a large number of states are suggested. When applied to a series of transition metal complexes the new CI algorithm does not only resolve divergence issues but also leads to typical reduction in computational time by 70%, with the largest savings for small molecules and large active spaces. Together, the projection operator and the improved CI algorithm now make it possible to simulate a wide range of single- and two-photon spectroscopies. © 2019 Wiley Periodicals, Inc.

Influence of pseudopotentials on excitation energies from selected configuration interaction and diffusion Monte Carlo

Due to their diverse nature, the faithful description of excited states within electronic structure theory methods remains one of the grand challenges of modern theoretical chemistry. Quantum Monte Carlo (QMC) methods have been applied very successfully to ground state properties but still remain generally less effective than other non-stochastic methods for electronically excited states. Nonetheless, we have recently reported accurate excitation energies for small organic molecules at the fixed-node diffusion Monte Carlo (FN-DMC) within a Jastrow-free QMC protocol relying on a deterministic and systematic construction of nodal surfaces using the selected configuration interaction (sCI) algorithm known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). Albeit highly accurate, these all-electron calculations are computationally expensive due to the presence of core electrons. One very popular approach to remove these chemically-inert electrons from the QMC simulation is to introduce pseudopotentials (also known as effective core potentials). Taking the water molecule as an example, we investigate the influence of Burkatzki-Filippi-Dolg (BFD) pseudopotentials and their associated basis sets on vertical excitation energies obtained with sCI and FN-DMC methods. Although these pseudopotentials are known to be relatively safe for ground state properties, we evidence that special care may be required if one strives for highly accurate vertical transition energies. Indeed, comparing all-electron and valence-only calculations, we show that using pseudopotentials with the associated basis sets can induce differences of the order of 0.05 eV on the excitation energies. Fortunately, a reasonable estimate of this shift can be estimated at the sCI level.

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.

A-VCI: A flexible method to efficiently compute vibrational spectra.

The adaptive vibrational configuration interaction algorithm has been introduced as a new method to efficiently reduce the dimension of the set of basis functions used in a vibrational configuration interaction process. It is based on the construction of nested bases for the discretization of the Hamiltonian operator according to a theoretical criterion that ensures the convergence of the method. In the present work, the Hamiltonian is written as a sum of products of operators. The purpose of this paper is to study the properties and outline the performance details of the main steps of the algorithm. New parameters have been incorporated to increase flexibility, and their influence has been thoroughly investigated. The robustness and reliability of the method are demonstrated for the computation of the vibrational spectrum up to 3000 cm-1 of a widely studied 6-atom molecule (acetonitrile). Our results are compared to the most accurate up to date computation; we also give a new reference calculation for future work on this system. The algorithm has also been applied to a more challenging 7-atom molecule (ethylene oxide). The computed spectrum up to 3200 cm-1 is the most accurate computation that exists today on such systems.

Benchmarking Post-Hartree-Fock Methods To Describe the Nonlinear Optical Properties of Polymethines: An Investigation of the Accuracy of Algebraic Diagrammatic Construction (ADC) Approaches.

Third-order nonlinear optical (NLO) properties of polymethine dyes have been widely studied for applications such as all-optical switching. However, the limited accuracy of the current computational methodologies has prevented a comprehensive understanding of the nature of the lowest excited states and their influence on the molecular optical and NLO properties. Here, attention is paid to the lowest excited-state energies and their energetic ratio, as these characteristics impact the figure-of-merit for all-optical switching. For a series of model polymethines, we compare several algebraic diagrammatic construction (ADC) schemes for the polarization propagator with approximate second-order coupled cluster (CC2) theory, the widely used INDO/MRDCI approach and the symmetry-adapted cluster configuration interaction (SAC-CI) algorithm incorporating singles and doubles linked excitation operators (SAC-CI SD-R). We focus in particular on the ground-to-excited state transition dipole moments and the corresponding state dipole moments, since these quantities are found to be of utmost importance for an effective description of the third-order polarizability γ and two-photon absorption spectra. A sum-overstates expression has been used, which is found to quickly converge. While ADC(3/2) has been found to be the most appropriate method to calculate these properties, CC2 performs poorly.

Balancing accuracy and efficiency in selecting vibrational configuration interaction basis states using vibrational perturbation theory

This work describes the benchmarking of a vibrational configuration interaction (VCI) algorithm that combines the favourable computational scaling of VPT2 with the algorithmic robustness of VCI, in which VCI basis states are selected according to the magnitude of their contribution to the VPT2 energy, for the ground state and fundamental excited states. Particularly novel aspects of this work include: expanding the potential to 6th order in normal mode coordinates, using a double-iterative procedure in which configuration selection and VCI wavefunction updates are performed iteratively (micro-iterations) over a range of screening threshold values (macro-iterations), and characterisation of computational resource requirements as a function of molecular size. Computational costs may be further reduced by a priori truncation of the VCI wavefunction according to maximum extent of mode coupling, along with discarding negligible force constants and VCI matrix elements, and formulating the wavefunction in a harmonic oscillator product basis to enable efficient evaluation of VCI matrix elements. Combining these strategies, we define a series of screening procedures that scale as O(Nmode6)−O(Nmode9) in run time and O(Nmode6)−O(Nmode7) in memory, depending on the desired level of accuracy. Our open-source code is freely available for download from http://www.sourceforge.net/projects/pyvci-vpt2.

Heat-Bath Configuration Interaction: An Efficient Selected Configuration Interaction Algorithm Inspired by Heat-Bath Sampling

We introduce a new selected configuration interaction plus perturbation theory algorithm that is based on a deterministic analog of our recent efficient heat-bath sampling algorithm. This Heat-bath Configuration Interaction (HCI) algorithm makes use of two parameters that control the trade-off between speed and accuracy, one which controls the selection of determinants to add to a variational wave function and one which controls the selection of determinants used to compute the perturbative correction to the variational energy. We show that HCI provides an accurate treatment of both static and dynamic correlation by computing the potential energy curve of the multireference carbon dimer in the cc-pVDZ basis. We then demonstrate the speed and accuracy of HCI by recovering the full configuration interaction energy of both the carbon dimer in the cc-pVTZ basis and the strongly correlated chromium dimer in the Ahlrichs VDZ basis, correlating all electrons, to an accuracy of better than 1 mHa, in just a few minutes on a single core. These systems have full variational spaces of 3 × 10(14) and 2 × 10(22) determinants, respectively.

PyVCI: A flexible open-source code for calculating accurate molecular infrared spectra

The PyVCI program package is a general purpose open-source code for simulating accurate molecular spectra, based upon force field expansions of the potential energy surface in normal mode coordinates. It includes harmonic normal coordinate analysis and vibrational configuration interaction (VCI) algorithms, implemented primarily in Python for accessibility but with time-consuming routines written in C. Coriolis coupling terms may be optionally included in the vibrational Hamiltonian. Non-negligible VCI matrix elements are stored in sparse matrix format to alleviate the diagonalization problem. CPU and memory requirements may be further controlled by algorithmic choices and/or numerical screening procedures, and recommended values are established by benchmarking using a test set of 44 molecules for which accurate analytical potential energy surfaces are available.Force fields in normal mode coordinates are obtained from the PyPES library of high quality analytical potential energy surfaces (to 6th order) or by numerical differentiation of analytic second derivatives generated using the GAMESS quantum chemical program package (to 4th order). Program summaryProgram title: PyVCICatalogue identifier: AFAC_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAC_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: MIT LicenseNo. of lines in distributed program, including test data, etc.: 401703No. of bytes in distributed program, including test data, etc.: 7091446Distribution format: tar.gzProgramming language: Python, C.Computer: PC.Operating system: Linux, MacOSX, Windows.RAM: Varies widelyClassification: 16.3.External routines: Numpy, Scipy, CythonNature of problem: The simulation of accurate molecular vibrational spectra is a significant and long-standing problem in computational chemistry. There are two major challenges: constructing an accurate ab initio potential energy surface and solving the nuclear vibrational problem. Both scale poorly with respect to molecular size, requiring large amounts of CPU time and memory.Solution method: We have implemented a straightforward numerical differentiation algorithm to construct quartic force fields in normal mode coordinates using second derivatives of the energy with respect to nuclear displacement obtained from ab initio quantum chemical calculations, for nuclear vibrational structure algorithm development and testing purposes. We have also provided an interface to the PyPES library of high quality semi-global potential energy surfaces, which enable quantitative prediction of molecular vibrational spectra. To solve the nuclear vibrational problem, we use a vibrational configuration interaction algorithm in a harmonic oscillator basis.Unusual features: One of the unusual features of our code is its flexibility, with multiple ways of generating or supplying force field data, dynamic memory allocation, adjustable screening thresholds, and explicit user control over terms in the VCI wave-function (maximum excitation level and extent of mode-coupling). We employ sparse matrix linear algebra libraries to reduce the memory required for VCI matrix storage and diagonalization, and provide for parallel VCI matrix construction to reduce required wall times.Additional comments: User Manual and examples (tutes) includedRunning time: Varies widely

Using full configuration interaction quantum Monte Carlo in a seniority zero space to investigate the correlation energy equivalence of pair coupled cluster doubles and doubly occupied configuration interaction.

Over the past few years, pair coupled cluster doubles (pCCD) has shown promise for the description of strong correlation. This promise is related to its apparent ability to match results from doubly occupied configuration interaction (DOCI), even though the latter method has exponential computational cost. Here, by modifying the full configuration interaction quantum Monte Carlo algorithm to sample only the seniority zero sector of Hilbert space, we show that the DOCI and pCCD energies are in agreement for a variety of 2D Hubbard models, including for systems well out of reach for conventional configuration interaction algorithms. Our calculations are aided by the sign problem being much reduced in the seniority zero space compared with the full space. We present evidence for this and then discuss the sign problem in terms of the wave function of the system which appears to have a simplified sign structure.

Graphical unitary group approach based on hole-particle symmetry using in multi-reference configuration interaction

Graphical unitary group approach (GUGA) based on the hole-particle symmetry is used to develop an efficient multi-reference configuration interaction (MRCI) algorithm. By dividing the orbital into the doubly occupied, active and external orbitals, the coupling coefficient can be calculated as the product of three segmental factors in each orbital space. Only segmental factors in the active space needs to be stored that reduces the storage requirements significantly. Therefore, MRCI calculation with more than 32 internal orbitals can be performed by our program easily. The application of hole-particle symmetry also allows us to divide the configuration space into a serial of sub-CI spaces according to the number of holes in the doubly occupied space and the number of electrons in the external space. The internal contraction of configuration space can be performed within the individual sub-CI space. Consequently, a new internally contracted MRCI (ic-MRCI) algorithm based on GUGA is proposed and developed. Benchmark calculations illustrate that the large scale MRCI calculation with several billions of un-contracted configuration state functions (CSFs) has already become a routine task.

Energy error bars in direct configuration interaction iteration sequence.

A computational scheme for approximate lower bound to eigenvalues of linear operators is elaborated, based on Löwdin's bracketing function. Implementation in direct full configuration interaction algorithm is presented, generating essentially just input-output increase. While strict lower bound property is lost due to approximations, test calculations result lower bounds of the same order of magnitude, as the usual upper bound, provided by the expectation value. Difference of upper and lower bounds gives an error bar, characterizing the wavefunction at the given iteration step.