Configuration Interaction Expansion Research Articles

Strategies to Obtain Reliable Energy Landscapes from Embedded Multireference Correlated Wavefunction Methods for Surface Reactions.

Embedded correlated wavefunction (ECW) theory is a powerful tool for studying ground- and excited-state reaction mechanisms and associated energetics in heterogeneous catalysis. Several factors are important to obtaining reliable ECW energies, critically the construction of consistent active spaces (ASs) along reaction pathways when using a multireference correlated wavefunction (CW) method that relies on a subset of orbital spaces in the configuration interaction expansion to account for static electron correlation, e.g., complete AS self-consistent field theory, in addition to the adequate partitioning of the system into a cluster and environment, as well as the choice of a suitable basis set and number of states included in excited-state simulations. Here, we conducted a series of systematic studies to develop best-practice guidelines for ground- and excited-state ECW theory simulations, utilizing the decomposition of NH3 on Pd(111) as an example. We determine that ECW theory results are relatively insensitive to cluster size, the aug-cc-pVDZ basis set provides an adequate compromise between computational complexity and accuracy, and that a fixed-clean-surface approximation holds well for the derivation of the embedding potential. Additionally, we demonstrate that a merging approach, which involves generating ASs from the molecular fragments at each configuration, is preferable to a creeping approach, which utilizes ASs from adjacent structures as an initial guess, for the generation of consistent potential energy curves involving open-d-shell metal surfaces, and, finally, we show that it is essential to include bands of excited states in their entirety when simulating excited-state reaction pathways.

X marks the spot: Accurate energies from intersecting extrapolations of continuum quantum Monte Carlo data.

We explore the application of an extrapolative method that yields very accurate total and relative energies from variational and diffusion quantum Monte Carlo (VMC and DMC) results. For a trial wave function consisting of a small configuration interaction (CI) wave function obtained from full CI quantum Monte Carlo and reoptimized in the presence of a Jastrow factor and an optional backflow transformation, we find that the VMC and DMC energies are smooth functions of the sum of the squared coefficients of the initial CI wave function and that quadratic extrapolations of the non-backflow VMC and backflow DMC energies intersect within uncertainty of the exact total energy. With adequate statistical treatment of quasi-random fluctuations, the extrapolate and intersect with polynomials of order two method is shown to yield results in agreement with benchmark-quality total and relative energies for the C2, N2, CO2, and H2O molecules, as well as for the C2 molecule in its first electronic singlet excited state, using only small CI expansion sizes.

A nuclear configuration interaction approach to study nuclear spin effects: an application to ortho- and para-3 He2 @C60.

We introduce a non-orthogonal configuration interaction approach to investigate nuclear quantum effects on energies and densities of confined fermionic nuclei. The Hamiltonian employed draws parallels between confined systems and many-electron atoms, where effective non-Coulombic potentials represent the interactions of the trapped particles. One advantage of this method is its generality, as it offers the potential to study the nuclear quantum effects of various confined species affected by effective isotropic or anisotropic potentials. As a first application, we analyze the quantum states of two 3 He atoms encapsulated in C60 . At the Hartree-Fock level, we observe the breaking of spin and spatial symmetries. To ensure wavefunctions with the correct symmetries, we mix the broken-symmetry Hartree-Fock states within the non-orthogonal configuration interaction expansion. Our proposed approach predicts singly and triply degenerate ground states for the singlet (para-3 He2 @C60 ) and triplet (ortho-3 He2 @C60 ) nuclear spin configurations, respectively. The ortho-3 He2 @C60 ground state is 5.69 cm-1 higher in energy than the para-3 He2 @C60 ground state. The nuclear densities obtained for these states exhibit the icosahedral symmetry of the C60 embedding potential. Importantly, our calculated energies for the lowest 85 states are in close agreement with perturbation theory results based on a harmonic oscillator plus rigid rotor model of 3 He2 @C60 .

All-Purpose Measure of Electron Correlation for Multireference Diagnostics.

We present an analytical relationship between two natural orbital occupancy-based indices, and INDmax, and two established electron correlation metrics: the leading term of a configuration interaction expansion, c0, and the D2 diagnostic. Numerical validation revealed that and INDmax can effectively substitute for c0 and D2, respectively. These indices offer three distinct advantages: (i) they are universally applicable across all electronic structure methods, (ii) their interpretation is more intuitive, and (iii) they can be readily incorporated into the development of hybrid electronic structure methods. Additionally, we draw a distinction between correlation measures and correlation diagnostics, establishing MP2 and CCSD numerical thresholds for INDmax, which are to be used as a multireference diagnostic. Our findings further demonstrate that establishing thresholds for other electronic structure methods can be easily accomplished using small data sets.

Time-dependent multiconfiguration self-consistent-field and time-dependent optimized coupled-cluster methods for intense laser-driven multielectron dynamics

We reviewed the time-dependent multiconfiguration self-consistent-field (TD-MCSCF) method and the time-dependent optimized coupled-cluster (TD-OCC) method for first-principles simulations of high-field phenomena such as tunneling ionization and high-order harmonic generation in atoms and molecules irradiated by a strong laser field. These methods provide a flexible and systematically improvable description of the multielectron dynamics by expressing the all-electron wavefunction by configuration interaction expansion or coupled-cluster expansion, using time-dependent one-electron orbital functions. The time-dependent variational principle plays a key role in deriving these methods while satisfying gauge invariance and Ehrenfest theorem. The real-time/real-space implementation with an absorbing boundary condition enables the simulation of high-field processes involving multiple excitation and ionization. We present a detailed, comprehensive discussion of such features of the TD-MCSCF and TD-OCC methods.

Orthogonally Constrained Orbital Optimization: Assessing Changes of Optimal Orbitals for Orthogonal Multireference States.

The choice of molecular orbitals is decisive in configuration interaction calculations. In this letter, a democratic description of the ground and excited states follows an orthogonally constrained orbital optimization to produce state-specific orbitals. The approach faithfully recovers the excitation energy of a four-electron Hubbard trimer, whereas state-average calculations can miss the value by a factor 2.5. The method emphasizes the need for orbital optimization to reduce configuration interaction expansions and to reach spectroscopic accuracy.

Configuration interaction based nonequilibrium steady state impurity solver

We present a solver for correlated impurity problems out of equilibrium based on a combination of the so-called auxiliary master equation approach (AMEA) and the configuration interaction expansion. Within AMEA one maps the original impurity model onto an auxiliary open quantum system with a restricted number of bath sites which can be addressed by numerical many-body approaches such as Lanczos/Arnoldi exact diagonalization (ED) or matrix product states (MPS). While the mapping becomes exponentially more accurate with increasing number of bath sites, ED implementations are severely limited due to the fast increase of the Hilbert space dimension for open systems, and the MPS solver typically requires rather long runtimes. Here, we propose to adopt a configuration interaction approach augmented by active space extension to solve numerically the correlated auxiliary open quantum system. This allows access to a larger number of bath sites at lower computational costs than for plain ED. We benchmark the approach with numerical renormalization group results in equilibrium and with MPS out of equilibrium. In particular, we evaluate the current, the conductance, as well as the Kondo peak and its splitting as a function of increasing bias voltage below the Kondo temperature ${T}_{\text{K}}$. We obtain a rather accurate scaling of the conductance as a function of the bias voltage and temperature rescaled by ${T}_{\text{K}}$ for moderate to strong interactions in a wide range of parameters. The approach combines the fast runtime of ED with an accuracy close to the one achieved by MPS making it an attractive solver for nonequilibrium dynamical mean field theory.

Applying Generalized Variational Principles to Excited-State-Specific Complete Active Space Self-consistent Field Theory.

We employ a generalized variational principle to improve the stability, reliability, and precision of fully excited-state-specific complete active space self-consistent field theory. Compared to previous approaches that similarly seek to tailor this ansatz's orbitals and configuration interaction expansion for an individual excited state, we find the present approach to be more resistant to root flipping and better at achieving tight convergence to an energy stationary point. Unlike state-averaging, this approach allows orbital shapes to be optimal for individual excited states, which is especially important for charge-transfer states and some doubly excited states. We demonstrate the convergence and state-targeting abilities of this method in LiH, ozone, and MgO, showing in the latter that it is capable of finding three excited-state energy stationary points that no previous method has been able to locate.

Optimization of Large Determinant Expansions in Quantum Monte Carlo.

We present a new method for the optimization of large configuration interaction (CI) expansions in the quantum Monte Carlo (QMC) framework. The central idea here is to replace the nonorthogonal variational optimization of CI coefficients performed in usual QMC calculations by an orthogonal non-Hermitian optimization thanks to the so-called transcorrelated (TC) framework, the two methods yielding the same results in the limit of a complete basis set. By rewriting the TC equations as an effective self-consistent Hermitian problem, our approach requires the sampling of a single quantity per Slater determinant, leading to minimal memory requirements in the QMC code. Using analytical quantities obtained from both the TC framework and the usual CI-type calculations, we also propose improved estimators which reduce the statistical fluctuations of the sampled quantities by more than an order of magnitude. We demonstrate the efficiency of this method on wave functions containing 105-106 Slater determinants, using effective core potentials or all-electron calculations. In all the cases, a sub-milli-Hartree convergence is reached within only two or three iterations of optimization.

Atomic structure and electron impact excitation of Al-like ions (Ga–Br)

Dirac atomic R-matrix coupling (DARC) calculations were conducted on Al-like ions (31 ≤ Z ≤ 35), which included a total of 163 fine structure levels in both the configuration interaction and close-coupling expansions. Comparisons with previous theoretical values were made in terms of energy levels, transition wavelengths and transition rates. The results of collision strength (Ω) showed a good consistency between the current DARC and distorted wave methods. The effective collision strengths (γ) were estimated in the temperature range of 105 K–108 K. The results may be useful in the designation and analysis of extreme ultraviolet spectral lines with the exploration of new data from fusion and astrophysical plasma sources.

Selected configuration interaction wave functions in phaseless auxiliary field quantum Monte Carlo

We present efficient algorithms for using selected configuration interaction (sCI) trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC). These advances, geared toward optimizing computational performance for longer configuration interaction expansions, allow us to use up to a million configurations in the trial state for ph-AFQMC. In one example, we found the cost of ph-AFQMC per sample to increase only by a factor of about 3 for a calculation with 104 configurations compared to that with a single one, demonstrating the tiny computational overhead due to a longer expansion. This favorable scaling allows us to study the systematic convergence of the phaseless bias in auxiliary field quantum Monte Carlo calculations with an increasing number of configurations and provides a means to gauge the accuracy of ph-AFQMC with other trial states. We also show how the scalability issues of sCI trial states for large system sizes could be mitigated by restricting them to a moderately sized orbital active space and leveraging the near-cancellation of out of active space phaseless errors.

Implementation of a time-dependent multiconfiguration self-consistent-field method for coupled electron-nuclear dynamics in diatomic molecules driven by intense laser pulses

We present an implementation of a time-dependent multiconfiguration self-consistent-field (TD-MCSCF) method [R. Anzaki et al., Phys. Chem. Chem. Phys. 19, 22008 (2017)] with the full configuration interaction expansion for coupled electron-nuclear dynamics in diatomic molecules subject to a strong laser field. In this method, the total wave function is expressed as a superposition of different configurations constructed from time-dependent electronic Slater determinants and time-dependent orthonormal nuclear basis functions. The primitive basis functions of nuclei and electrons are strictly independent of each other without invoking the Born-Oppenheimer approximation. Our implementation treats the electronic motion in its full dimensionality on curvilinear coordinates, while the nuclear wave function is propagated on a one-dimensional stretching coordinate with rotational nuclear motion neglected. We apply the present implementation to high-harmonic generation and dissociative ionization of a hydrogen molecule and discuss the role of electron-nuclear correlation.

Complete-active-space extended Koopmans theorem method

The complete-active-space (CAS) extended Koopmans theorem (EKT) method is defined as a special case of the EKT in which the reference state is a CAS configuration interaction (CI) expansion and the electron removal operator acts only on the active orbitals. With these restrictions, the EKT is equivalent to the CI procedure involving all hole-state configurations derived from the active space of the reference wavefunction and has properties analogous to those of the original Koopmans theorem. The equivalence is used to demonstrate in a transparent manner that the first ionization energy predicted by the EKT is in general not exact, i.e., not equal to the difference between the full CI energies of the neutral and the ion, but can approach the full CI result with arbitrary precision even within a finite basis set. The findings also reconcile various statements about the EKT found in the literature.

Edge counts for the auxiliary pair graph within the graphical unitary group approach

Closed-form expressions are presented for the numbers of edges in the auxiliary pair graphs (APGs) associated with non spin–orbit and spin–orbit Shavitt graphs for full configuration interaction expansions. A Shavitt graph is a visual representation of a configuration state function expansion space constructed via the graphical unitary group approach (GUGA). An APG is an organisational aid and a programmatic tool generated from a Shavitt graph. The number of edges in an APG determines bounds on the computational scaling as a function of the total numbers of electrons, orbitals, and spin multiplicities. The edge counts extend a suite of Shavitt graph statistics based on these functional parameters. The derivation and the presentation of the formulas for the edge counts has been assisted by the bra–ket interchange symmetry and the particle-hole interchange symmetry in the GUGA formalism. These symmetry operators produce one-to-one correspondences between various sets of edges, and this yields identities among some edge count formulas. There are 208 possible edge types. Of these, some do not contribute to two-electron operators, some are related by bra–ket interchange symmetry, and some are related by particle-hole interchange symmetry. For the remaining unique edge types, explicit expressions are derived for the numbers of edges.

Model protein excited states: MRCI calculations with large active spaces vs CC2 method.

Benchmarking calculations on excited states of models of phenylalanine protein chains are presented to assess the ability of alternative methods to the standard and most commonly used multiconfigurational wave function-based method, the complete active space self-consistent field (CASSCF), in recovering the non-dynamical correlation for systems that become not affordable by the CASSCF. The exploration of larger active spaces beyond the CASSCF limit is benchmarked through three strategies based on the reduction in the number of determinants: the restricted active space self-consistent field, the generalized active space self-consistent field (GASSCF), and the occupation-restricted multiple active space (ORMAS) schemes. The remaining dynamic correlation effects are then added by the complete active space second-order perturbation theory and by the multireference difference dedicated configuration interaction methods. In parallel, the approximate second-order coupled cluster (CC2), already proven to be successful for small building blocks of model proteins in one of our previous works [Ben Amor et al., J. Chem. Phys. 148, 184105 (2018)], is investigated to assess its performances for larger systems. Among the different alternative strategies to CASSCF, our results highlight the greatest efficiency of the GASSCF and ORMAS schemes in the systematic reduction of the configuration interaction expansion without loss of accuracy in both nature and excitation energies of both singlet ππ* and nπ* CO excited states with respect to the equivalent CASSCF calculations. Guidelines for an optimum applicability of this scheme to systems requiring active spaces beyond the complete active space limit are then proposed. Finally, the extension of the CC2 method to such large systems without loss of accuracy is demonstrated, highlighting the great potential of this method to treat accurately excited states, mainly single reference, of very large systems.

Portably parallel construction of a configuration-interaction wave function from a matrix-product state using the Charm++ framework.

The construction of configuration-interaction (CI) expansions from a matrix product state (MPS) involves numerous matrix operations and the skillful sampling of important configurations in a large Hilbert space. In this work, we present an efficient procedure for constructing CI expansions from MPS employing the parallel object-oriented Charm++ programming framework, upon which automatic load-balancing and object migrating facilities can be employed. This procedure was employed in the MPS-to-CI utility (Moritz et al., J. Chem. Phys. 2007, 126, 224109), the sampling-reconstructed complete active-space algorithm (SR-CAS, Boguslawski et al., J. Chem. Phys. 2011, 134, 224101), and the entanglement-driven genetic algorithm (EDGA, Luo et al., J. Chem. Theory Comput. 2017, 13, 4699). It enhances productivity and allows the sampling programs to evolve to their population-expansion versions, for example, EDGA with population expansion (PE-EDGA). Further, examples of 1,2-dioxetanone and firefly dioxetanone anion (FDO- ) molecules demonstrated the following: (a) parallel efficiencies can be persistently improved by simply by increasing the proportions of the asynchronous executions and (b) a sampled CAS-type CI wave function of a bi-radical-state FDO- molecule utilizing the full valence (30e,26o) active space can be constructed within a few hours with using thousands of cores.

Comparative metathermodynamic description of thermal and correlation electron effects

Unified metathermodynamic description of thermal and correlation electron effects is proposed. It is based on the temperature-dependent populations of the common for all temperatures basis states of the (formally accurate) configuration interaction (CI) expansion of the components of the canonical ensemble. These populations are incorporated into the effective ensemble of the metathermodynamic approach through the partial effective tem- peratures. The extended temperature ${T}_{e}$ is introduced as the statistical average of those temperatures. The absolute zero of the ${T}_{e}$ scale represents a ``no-correlation'' Hartree-Fock state, while elevated ${T}_{e}$ describes the increase of the ``correlation motion'' and the conventional thermal motion. The diagonal double configuration interaction (DDCI) approximation is considered, in which the proposed metathermodynamic description of thermal and correlation electron effects is based on the temperature-dependent orbital populations.

Compression of Spin-Adapted Multiconfigurational Wave Functions in Exchange-Coupled Polynuclear Spin Systems.

We present a protocol based on unitary transformations of molecular orbitals to reduce the number of nonvanishing coefficients of spin-adapted configuration interaction expansions. Methods that exploit the sparsity of the Hamiltonian matrix and compactness of its eigensolutions, such as the full configuration interaction quantum Monte Carlo (FCIQMC) algorithm in its spin-adapted implementation, are well suited to this protocol. The wave function compression resulting from this approach is particularly attractive for antiferromagnetically coupled polynuclear spin systems, such as transition-metal cubanes in biocatalysis, and Mott and charge-transfer insulators in solid-state physics. Active space configuration interaction calculations on N2 and CN– at various bond lengths, the stretched square N4 compounds, the chromium dimer, and a [Fe2S2]2– model system are presented as a proof-of-concept. For the Cr2 case, large and intermediate bond distances are discussed, showing that the approach is effective in cases where static and dynamic correlations are equally important. The [Fe2S2]2– case shows the general applicability of the method.

Energy of fermionic ground states with low-entanglement single-reference expansions, and tensor-based strong-coupling extensions of the coupled-cluster method

We consider a fermionic system for which there exist a single-reference configuration-interaction (CI) expansion of the ground state wave function that converges, albeit not necessarily rapidly, with respect to excitation number. We show that, if the coefficients of Slater determinants (SD) with $l\leq k$ excitations can be defined with a number of free parameters $N_{\leq k}$ bounded polynomially in $k$, the ground state energy $E$ only depends on a small fraction of all the wave function parameters, and is the solution of equations of the coupled-cluster (CC) form. This generalizes the standard CC method, for which $N_{\leq k}$ is bounded by a constant. Based on that result and low-rank tensor decompositions (LRTD), we discuss two possible extensions of the CC approach for wave functions with general polynomial bound for $N_{\leq k}$. First, one can use LRTD to represent the amplitudes of the CC cluster operator $T$ which, unlike in the CC case, is not truncated with respect to excitation number, and the energy and tensor parameters are given by a LRTD-adapted version of standard CC equations. Second, the LRTD can also be used to directly parametrize the CI coefficients, which involves different equations of the CC form. We derive those equations for up to quadruple-excitation coefficients, using a different type of excitation operator in the CC wave function ansatz, and a Hamiltonian representation in terms of excited particle and hole operators. We complete the proposed CC extensions by constructing compact tensor representations of coefficients, or $T$-amplitudes, using superpositions of tree tensor networks which take into account different possible types of entanglement between excited particles and holes. Finally, we discuss why the proposed extensions are theoretically applicable at larger coupling strengths than those treatable by the standard CC method.

On the time evolution of fermionic occupation numbers.

We derive an approximate equation for the time evolution of the natural occupation numbers for fermionic systems. The evolution of such numbers is connected with the symmetry-adapted generalized Pauli exclusion principle, as well as with the evolution of the natural orbitals and a set of many-body relative phases. We then relate the evolution of these phases to a geometrical and a dynamical term attached to some of the Slater determinants appearing in the configuration-interaction expansion of the wave function. Our approach becomes exact for highly symmetric systems whenever the wave function possesses as many Slater determinants as independent occupation numbers.