Multiconfigurational Wave Functions Research Articles

Vibrational Entanglement through the Lens of Quantum Information Measures.

We introduce a quantum information analysis of vibrational wave functions to understand complex vibrational spectra of molecules with strong anharmonic couplings and vibrational resonances. For this purpose, we define one- and two-modal entropies to guide the identification of strongly coupled vibrational modes and to characterize correlations within modal basis sets. We evaluate these descriptors for multiconfigurational vibrational wave functions which we calculate with the n-mode vibrational density matrix renormalization group algorithm. Based on the quantum information measures, we present a vibrational entanglement analysis of the vibrational ground and excited states of CO2, which display strong anharmonic effects due to the symmetry-induced and accidental (near-) degeneracies. We investigate the entanglement signature of the Fermi resonance and discuss the maximally entangled state arising from the two degenerate bending modes.

Relativistic Quantum Chemical Investigation of Actinide Covalency Measured by Electron Paramagnetic Resonance Spectroscopy.

We investigate actinide covalency effects in two [AnCptt3] (An = Th, U) complexes recently studied with pulsed electron paramagnetic resonance spectroscopy, using the Hyperion package to obtain relativistic hyperfine coupling constants from relativistic multiconfigurational wave functions. 1H and 13C HYSCORE simulations using the computed parameters show excellent agreement with the experimental data, highlighting the accuracy of modern relativistic ab initio methods. The extent of covalency indicated from the calculations on [ThCptt3] is in agreement with the original report based on traditional spectral fitting methods, while the covalency in [UCptt3] is found to be previously overestimated. The latter is due to the paramagnetic spin-orbit effect that arises naturally in a relativistic theory of hyperfine coupling and yet was not accounted for in the original study, thus highlighting the necessity of relativistic approaches for the interpretation of magnetic resonance data pertaining to actinides.

Quantum Algorithmic Approach to Multiconfigurational Valence Bond Theory: Insights from Interpretable Circuit Design.

Efficient ways to prepare Fermionic ground states on quantum computers are in high demand, and different techniques have been developed over the past few years. Despite having a vast set of methods, it is still unclear which method performs well for which system. In this work, we combine interpretable circuit designs with an effective basis approach in order to optimize a multiconfigurational valence bond wave function. Based on selected model systems, we show how this leads to an explainable performance. We demonstrate that the developed methodology outperforms related methods in terms of the size of the effective basis, as well as individual quantum resources for the involved circuits.

Prediction Challenge: Simulating Rydberg photoexcited cyclobutanone with surface hopping dynamics based on different electronic structure methods.

This research examines the nonadiabatic dynamics of cyclobutanone after excitation into the n → 3s Rydberg S2 state. It stems from our contribution to the Special Topic of the Journal of Chemical Physics to test the predictive capability of computational chemistry against unseen experimental data. Decoherence-corrected fewest-switches surface hopping was used to simulate nonadiabatic dynamics with full and approximated nonadiabatic couplings. Several simulation sets were computed with different electronic structure methods, including a multiconfigurational wavefunction [multiconfigurational self-consistent field (MCSCF)] specially built to describe dissociative channels, multireference semiempirical approach, time-dependent density functional theory, algebraic diagrammatic construction, and coupled cluster. MCSCF dynamics predicts a slow deactivation of the S2 state (10ps), followed by an ultrafast population transfer from S1 to S0 (<100fs). CO elimination (C3 channel) dominates over C2H4 formation (C2 channel). These findings radically differ from the other methods, which predicted S2 lifetimes 10-250 times shorter and C2 channel predominance. These results suggest that routine electronic structure methods may hold low predictive power for the outcome of nonadiabatic dynamics.

Zero-Field Splitting Tensor of the Triplet Excited States of Aromatic Molecules: A Valence Full-π Complete Active Space Self-Consistent Field Study.

A method to predict the D tensor in the molecular frame with multiconfigurational wave functions in large active space was proposed, and the spin properties of the lowest triplets of aromatic molecules were examined with full-π active space; such calculations were challenging because the size of active space grows exponentially with the number of π electrons. In this method, the exponential growth of complexity is resolved by the density matrix renormalization group (DMRG) algorithm. From the D tensor, we can directly determine the direction of the magnetic axes and the ZFS parameters, D- and E-values, of the phenomenological spin Hamiltonian with their signs, which are not usually obtained in ESR experiments. The method using the DMRG-CASSCF wave function can give correct results even when the sign of D- and E-values is sensitive to the accuracy of the prediction of the D tensor and existing methods fail to predict the correct magnetic axes.

Photoionization of the Propargyl Radical Using the R-Matrix Method.

The absolute photoionization cross section of the propargyl radical measured by Savee et al. [J. Chem. Phys. 2012, 136, 134307] is about 2-3 times larger than the experimental result from Robinson et al. [J. Chem. Phys. 2003, 119, 5311]. Meanwhile, there is a significant discrepancy in the shapes of the propargyl photoionization spectra between these two measurements and the result measured by Zhang et al. [J. Chem. Phys. 2006, 124, 074302]. To estimate the accuracy of the various results, we carried out comprehensive calculations with the multichannel R-matrix method, which uses multiconfigurational wave functions to generate accurate initial and final states. Three scattering models including static-exchange, correlated 1 state, and 80 state close-coupling (CC) approximations are employed to reveal the dynamic interaction. The autoionization peaks near the ionization threshold are predicted for the first time. Two resonances are detected with the photon energy around 41.4 and 49.6 eV in the 80 state CC model. We checked the sensitivity of the results to change descriptions of the active spaces, the different partial waves, and the R-matrix radii in the theoretical model. Our best results are compared with the experimental measurements, and discussions are provided. The cross sections and asymmetry parameters for the valence orbitals are reported for photon energies from ionization thresholds of 70 eV.

Fully relativistic distorted wave calculations of electron impact excitation of gallium atom: Cross sections relevant for plasma kinetic modeling

Electron impact excitation cross sections of neutral Ga are calculated using the relativistic distorted wave approximation with the aid of multiconfigurational Dirac-Fock wavefunctions. The cross sections are calculated from the fine structure resolved levels of the 4s24p manifold, i.e., from the ground 2P1/2 and the first excited state 2P3/2, to the fine structure resolved levels of the manifolds 4s24pdf, 4s4p2(4P1/2,4P3/2, and 4P5/2), 4s25spdf, 4s26spdf, 4s27spdf, and 4s28spdf for incident electron energies ranging from threshold to 500 eV. Where available, a comparison of the calculated oscillator strengths, transition probabilities, and cross sections with the previously reported results has been performed. The cross sections for the excitation from 4s24p2P1/2 to 4s24f, 4s4p2(4P1/2,4P3/2, and 4P5/2), 4s25f, 4s26f, 4s27df, and 4s28spdf and all the excitation cross sections from 4s24p2P3/2 are calculated and reported for the first time for a wide range of electron energies. Furthermore, the rate coefficients are also calculated using the present cross sections for an electron temperature varying from 0.5 to 5 eV. An analytic fitting of the calculated rate coefficients is provided to facilitate their inclusion in various plasma kinetic models.

Variational Pair-Density Functional Theory: Dealing with Strong Correlation at the Protein Scale.

Multiconfigurational pair-density functional theory (MC-PDFT) offers a promising solution to the challenges faced by traditional density functional theory (DFT) in addressing molecular systems containing transition metals, open-shells, or strong correlations in general. By utilizing both the density and on-top pair-density, MC-PDFT can make use of a more flexible multiconfigurational wave function to capture the necessary static correlation, while the pair-density functional also includes the effect of dynamic correlation. So far, MC-PDFT has been used after a multiconfigurational self-consistent field (MCSCF) step, using the orbitals and configuration interaction coefficients from the converged MCSCF wave function to compute PDFT energies and properties. Here, instead, we propose to perform a direct optimization of the wave function using the pair-density functionals, resulting in a variational formulation of MC-PDFT. We derive the expressions for the wave function gradient and illustrate their similarity to standard MCSCF equations. Furthermore, we illustrate the accuracy on a set of singlet-triplet gaps as well as dissociation curves. Our findings highlight one of MC-PDFT's standout features: a reduced dependency on the active space size compared to conventional multiconfigurational wave function methodologies. Additionally, we show that the computational cost of MC-PDFT is potentially lower than MCSCF and often on-par with standard Kohn-Sham DFT, which is demonstrated by performing a MC-PDFT calculation of the entire ferredoxin protein with 1447 atoms and nearly 12000 basis functions.

Separation of metric in Wick's theorem.

In quantum chemistry, Wick's theorem is an important tool to reduce products of fermionic creation and annihilation operators. It is especially useful in computations employing reference states. The original theorem has been generalized to tackle multiconfigurational wave functions or nonorthogonal orbitals. One particular issue of the resulting structure is that the metric and density matrices are intertwined despite their different origin. Here, an alternative, rather general tensorial formulation of Wick's theorem is proposed. The main difference is the separation of the metric-the coefficients at normal-ordered operators become products of an n-electron density matrix element and the Pfaffian of a matrix formed by orbital overlaps. Different properties of the formalism are discussed, including the use of density cumulants, the particle-hole symmetry, and applications to transition density matrices, i.e., the case of different bra and ket reference states. The metric-separated version of Wick's theorem provides a platform for the derivation of various quantum chemical methods, especially those complicated by non-trivial reference states and nonorthogonality issues.

Characterizing Conical Intersections in DNA/RNA Nucleobases with Multiconfigurational Wave Functions of Varying Active Space Size.

We characterize the photochemically relevant conical intersections between the lowest-lying accessible electronic excited states of the different DNA/RNA nucleobases using Cholesky decomposition-based complete active space self-consistent field (CASSCF) algorithms. We benchmark two different basis set contractions and several active spaces for each nucleobase and conical intersection type, measuring for the first time how active space size affects conical intersection topographies in these systems and the potential implications these may have toward their description of photoinduced phenomena. Our results show that conical intersection topographies are highly sensitive to the electron correlation included in the model: by changing the amount (and type) of correlated orbitals, conical intersection topographies vastly change, and the changes observed do not follow any converging pattern toward the topographies obtained with the largest and most correlated active spaces. Comparison across systems shows analogous topographies for almost all intersections mediating population transfer to the dark 1nO/Nπ* states, while no similarities are observed for the "ethylene-like" conical intersection ascribed to mediate the ultrafast decay component to the ground state in all DNA/RNA nucleobases. Basis set size seems to have a minor effect, appearing to be relevant only for purine-based derivatives. We rule out structural changes as a key factor in classifying the different conical intersections, which display almost identical geometries across active space and basis set change, and we highlight instead the importance of correctly describing the electronic states involved at these crossing points. Our work shows that careful active space selection is essential to accurately describe conical intersection topographies and therefore to adequately account for their active role in molecular photochemistry.

Quantification of the Ionic Character of Multiconfigurational Wave Functions: The Qat Diagnostic.

The complete active space self-consistent field (CASSCF) method is a cornerstone in modern excited-state quantum chemistry providing the starting point for most common multireference computations. However, CASSCF, when used with a minimal active space, can produce significant errors (>2 eV) even for the excitation energies of simple hydrocarbons if the states of interest possess ionic character. After illustrating this problem in some detail, we present a diagnostic for ionic character, denoted as Q at, that is readily computed from the transition density. A set of 11 molecules is considered to study errors in vertical excitation energies. State-averaged CASSCF obtains a mean absolute error (MAE) of 0.87 eV for the 34 singlet states considered. We highlight a strong correlation between the obtained errors and the Q at diagnostic, illustrating its power to predict problematic cases. Conversely, using multireference configuration interaction with single and double excitations and Pople's size extensivity correction (MR-CISD+P), excellent results are obtained with an MAE of 0.11 eV. Furthermore, correlations with the Q at diagnostic disappear. In summary, we hope that the presented diagnostic will facilitate reliable and user-friendly multireference computations on conjugated organic molecules.

Simulations of the Aqueous "Brown-Ring" Complex Reveal Fluctuations in Electronic Character.

Ab initio molecular dynamics (AIMD) simulations of the aqueous [Fe(H2O)5(NO)]2+ "brown-ring" complex in different spin states, in combination with multiconfigurational quantum chemical calculations, show a structural dependence on the electronic character of the complex. Sampling in the quartet and sextet ground states show that the multiplicity is correlated with the Fe-N distance. This provides a motivation for a rigid Fe-N scan in the isolated "brown-ring" complex to investigate how the multiconfigurational wave function and the electron density change around the FeNO moiety. Our results show that subtle changes in the Fe-N distance produce a large response in the electronic configurations underlying the quartet wave function. However, while changes in spin density and potential energy are pronounced, variations in charge are negligible. These trends within the FeNO moiety are preserved in structural sampling of the AIMD simulations, despite distortions present in other degrees of freedom in the bulk solution.

Deep-Learning Approach for the Atomic Configuration Interaction Problem on Large Basis Sets.

High-precision atomic structure calculations require accurate modeling of electronic correlations typically addressed via the configuration interaction (CI) problem on a multiconfiguration wave function expansion. The latter can easily become challenging or infeasibly large even for advanced supercomputers. Here, we develop a deep-learning approach which allows us to preselect the most relevant configurations out of large CI basis sets until the targeted energy precision is achieved. The large CI computation is thereby replaced by a series of smaller ones performed on an iteratively expanding basis subset managed by a neural network. While dense architectures as used in quantum chemistry fail, we show that a convolutional neural network naturally accounts for the physical structure of the basis set and allows for robust and accurate CI calculations. The method was benchmarked on basis sets of moderate size allowing for the direct CI calculation, and further demonstrated on prohibitively large sets where the direct computation is not possible.

Toward Benchmark-Quality Ab Initio Predictions for 3d Transition Metal Electrocatalysts: A Comparison of CCSD(T) and ph-AFQMC.

Generating accurate ab initio ionization energies for transition metal complexes is an important step toward the accurate computational description of their electrocatalytic reactions. Benchmark-quality data is required for testing existing theoretical methods and developing new ones but is complicated to obtain for many transition metal compounds due to the potential presence of both strong dynamical and static electron correlation. In this regime, it is questionable whether the so-called gold standard, coupled cluster with singles, doubles, and perturbative triples (CCSD(T)), provides the desired level of accuracy─roughly 1-3 kcal/mol. In this work, we compiled a test set of 28 3d metal-containing molecules relevant to homogeneous electrocatalysis (termed 3dTMV) and computed their vertical ionization energies (ionization potentials) with CCSD(T) and phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) in the def2-SVP basis set. A substantial effort has been made to converge away the phaseless bias in the ph-AFQMC reference values. We assess a wide variety of multireference diagnostics and find that spin-symmetry breaking of the CCSD wave function and the PBE0 density functional correlate well with our analysis of multiconfigurational wave functions. We propose quantitative criteria based on symmetry breaking to delineate correlation regimes inside of which appropriately performed CCSD(T) can produce mean absolute deviations from the ph-AFQMC reference values of roughly 2 kcal/mol or less and outside of which CCSD(T) is expected to fail. We also present a preliminary assessment of density functional theory (DFT) functionals on the 3dTMV set.

Complex Linear Response Functions for a Multiconfigurational Self-Consistent Field Wave Function in a High Performance Computing Environment.

We present novel developments for the highly efficient evaluation of complex linear response functions of a multiconfigurational self-consistent field (MCSCF) wave function as implemented in MultiPsi. Specifically, expressions for the direct evaluation of linear response properties at given frequencies using the complex polarization propagator (CPP) approach have been implemented, within both the Tamm-Dancoff approximation (TDA) and the random phase approximation (RPA). Purely real algebra with symmetric and antisymmetric trial vectors in a shared subspace is used wherein the linear response equations are solved. Two bottlenecks of large scale MC-CPP calculations, namely, the memory footprint and computational time, are addressed. The former is addressed by limiting the size of the subspace of trial vectors by using singular value decomposition (SVD) on either orbital or CI subspaces. The latter is addressed using an efficient parallel implementation as well as the strategy of dynamically adding linear response equations at near-convergence to neighboring roots. Furthermore, a novel methodology for decomposing MC-CPP spectra in terms of intuitive orbital excitations in an approximate fashion is presented. The performance of the code is illustrated with several numerical examples, including the X-ray spectrum of a molecule with nearly one hundred atoms. Additionally, for X-ray spectroscopy, the effect of including or excluding the core orbital in the active space on small covalent metal complexes is discussed.

Electronic Structure of the Complete Series of Gas-Phase Manganese Acetylacetonates by X-ray Absorption Spectroscopy.

Metal centers in transition metal-ligand complexes occur in a variety of oxidation states causing their redox activity and therefore making them relevant for applications in physics and chemistry. The electronic state of these complexes can be studied by X-ray absorption spectroscopy, which is, however, due to the complex spectral signature not always straightforward. Here, we study the electronic structure of gas-phase cationic manganese acetylacetonate complexes Mn(acac)1-3+ using X-ray absorption spectroscopy at the metal center and ligand constituents. The spectra are well reproduced by multiconfigurational wave function theory, time-dependent density functional theory as well as parameterized crystal field and charge transfer multiplet simulations. This enables us to get detailed insights into the electronic structure of ground-state Mn(acac)1-3+ and extract empirical parameters such as crystal field strength and exchange coupling from X-ray excitation at both the metal and ligand sites. By comparison to X-ray absorption spectra of neutral, solvated Mn(acac)2,3 complexes, we also show that the effect of coordination on the L3 excitation energy, routinely used to identify oxidation states, can contribute about 40-50% to the observed shift, which for the current study is 1.9 eV per oxidation state.

Relativistic calculation of electron impact excitation of tin: cross sections of importance in plasma modeling

The present work includes the calculation of electron impact excitation cross sections for the fine structure resolved transitions in Sn I and Sn II, using the relativistic distorted wave approximation with the aid of Dirac-Fock multiconfigurational wavefunctions. For Sn I, the cross sections are calculated for the excitations from the fine structure resolved energy levels of the manifolds, , , , and including the ground state, to the fine structure resolved levels of , , , , and manifolds for a wide range of projectile electron energy from 0 to 500 eV. Further, the cross sections are also calculated for the excitations of Sn II from the fine structure resolved energy levels of the manifolds, , , , and including the ground state, to the fine structure resolved levels of , , , , and manifolds. All the cross sections for Sn I except for the excitation from the ground state to the fine-structure resolved excited states of , and manifolds and all the cross sections for Sn II are reported for the first time. In order to ensure the consistency of the present calculations, a comparison of the oscillator strengths and transition probabilities is carried out with the available calculations and measurements. The present cross sections are also compared with previous calculations and observed to be in reasonable agreement. The calculated cross sections are of great interest in the modeling and diagnostic of various plasmas such as laser induced plasma, fusion plasma, etc. Further, to ease the use of calculated cross sections, the analytic fitting of the cross sections is also provided.

A hybrid stochastic configuration interaction-coupled cluster approach for multireference systems.

The development of multireference coupled cluster (MRCC) techniques has remained an open area of study in electronic structure theory for decades due to the inherent complexity of expressing a multiconfigurational wavefunction in the fundamentally single-reference coupled cluster framework. The recently developed multireference-coupled cluster Monte Carlo (mrCCMC) technique uses the formal simplicity of the Monte Carlo approach to Hilbert space quantum chemistry to avoid some of the complexities of conventional MRCC, but there is room for improvement in terms of accuracy and, particularly, computational cost. In this paper, we explore the potential of incorporating ideas from conventional MRCC-namely, the treatment of the strongly correlated space in a configuration interaction formalism-to the mrCCMC framework, leading to a series of methods with increasing relaxation of the reference space in the presence of external amplitudes. These techniques offer new balances of stability and cost against accuracy, as well as a means to better explore and better understand the structure of solutions to the mrCCMC equations.

Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory.

State-specific electronic structure theory provides a route toward balanced excited-state wave functions by exploiting higher-energy stationary points of the electronic energy. Multiconfigurational wave function approximations can describe both closed- and open-shell excited states and avoid the issues associated with state-averaged approaches. We investigate the existence of higher-energy solutions in complete active space self-consistent field (CASSCF) theory and characterize their topological properties. We demonstrate that state-specific approximations can provide accurate higher-energy excited states in H2 (6-31G) with more compact active spaces than would be required in a state-averaged formalism. We then elucidate the unphysical stationary points, demonstrating that they arise from redundant orbitals when the active space is too large or symmetry breaking when the active space is too small. Furthermore, we investigate the singlet-triplet crossing in CH2 (6-31G) and the avoided crossing in LiF (6-31G), revealing the severity of root flipping and demonstrating that state-specific solutions can behave quasi-diabatically or adiabatically. These results elucidate the complexity of the CASSCF energy landscape, highlighting the advantages and challenges of practical state-specific calculations.

A Multiconfigurational Wave Function Implementation of the Frenkel Exciton Model for Molecular Aggregates.

We present an implementation of the Frenkel exciton model into the OpenMolcas program package enabling calculations of collective electronic excited states of molecular aggregates based on a multiconfigurational wave function description of the individual monomers. The computational protocol avoids using diabatization schemes and, thus, supermolecule calculations. Additionally, the use of the Cholesky decomposition of the two-electron integrals entering pair interactions enhances the efficiency of the computational scheme. The application of the method is exemplified for two test systems, that is, a formaldehyde oxime and a bacteriochlorophyll-like dimer. For the sake of comparison with the dipole approximation, we restrict our considerations to situations where intermonomer exchange can be neglected. The protocol is expected to be beneficial for aggregates composed of molecules with extended π systems, unpaired electrons such as radicals or transition metal centers, where it should outperform widely used methods based on time-dependent density functional theory.