Multireference Perturbation Theory Research Articles

Corresponding Active Orbital Spaces along Chemical Reaction Paths.

The accuracy of reaction energy profiles calculated with multiconfigurational electronic structure methods and corrected by multireference perturbation theory depends crucially on consistent active orbital spaces selected along the reaction path. However, it has been challenging to choose molecular orbitals that can be considered corresponding in different molecular structures. Here, we demonstrate how active orbital spaces can be selected consistently along reaction coordinates in a fully automatized way. The approach requires no structure interpolation between reactants and products. Instead, it emerges from a synergy of the Direct Orbital Selection orbital mapping ansatz combined with our fully automated active space selection algorithm autoCAS. We demonstrate our algorithm for the potential energy profile of the homolytic carbon-carbon bond dissociation and rotation around the double bond of 1-pentene in the electronic ground state. However, our algorithm also applies to electronically excited Born-Oppenheimer surfaces.

CO Inversion on a NaCl(100) Surface: A Multireference Quantum Embedding Study.

We develop a multireference quantum embedding model to investigate a recent experimental observation of the isomerization of vibrationally excited CO molecules on a NaCl(100) surface [Science 2020, 367, 175-178]. To explore this mechanism, we built a reduced potential energy surface of CO interacting with NaCl(100) using a second-order multireference perturbation theory, modeling the adsorbate-surface interaction with our previously developed active space embedding theory (ASET). We considered an isolated CO molecule on NaCl(100) and a high-coverage CO monolayer (1/1), and for both we generated potential energy surfaces parametrized by the CO stretching, adsorption, and inversion coordinates. These surfaces are used to determine stationary points and adsorption energies and to perform a vibrational analysis of the states relevant to the inversion mechanism. We found that for near-equilibrium bond lengths, CO adsorbed in the C-down configuration is lower in energy than in the O-down configuration. Stretching of the C-O bond reverses the energetic order of these configurations, supporting the accepted isomerization mechanism. The vibrational constants obtained from these potential energy surfaces show a small (< 10 cm-1) blue- and red-shift for the C-down and O-down configurations, respectively, in agreement with experimental assignments and previous theoretical studies. Our vibrational analysis of the monolayer case suggests that the O-down configuration is energetically more stable than the C-down one beyond the 16th vibrational excited state of CO, a value slightly smaller than the one from quasi-classical trajectory simulations (22nd) and consistent with the experiment. Our analysis suggests that CO-CO interactions in the monolayer play an important role in stabilizing highly vibrationally excited states in the O-down configuration and reducing the barrier between the C-down and O-down geometries, therefore playing a crucial role in the inversion mechanism.

Statistical errors in reduced density matrices sampled from quantum circuit simulation and the impact on multireference perturbation theory

In this work, we present a detailed analysis of statistical errors in reduced density matrices (RDMs) of active space wavefunctions sampled from quantum circuit simulation and the impact on results obtained by the multireference theories.

Photodissociation dynamics and UV absorption spectrum of acetone oxide (CH3)2COO.

The photodynamics and B1A' ← X1A' absorption spectrum of acetone oxide, (CH3)2COO, are studied theoretically from first principles. The underlying adiabatic potential energy curves (and surfaces) are computed by a second-order multireference perturbation theory method and diabatized using a diabatization by ansatz scheme. To confirm the results, for selected geometries EOM-CCSD and XMS-RS2C calculations were also performed. The dynamical calculation rests on the multi-configuration time-dependent Hartree wavepacket propagation method. The experimental absorption spectrum is reproduced satisfactorily. This result serves to validate the Hamiltonian model built within the quasi-diabatic representation. Contrary to the smallest Criegee intermediate, CH2OO, it is found that the vibronic coupling between the B and C states of (CH3)2COO plays an essential role in reproducing the experimental absorption spectrum. Time-dependent electronic populations reveal a faster decay than for the smaller system CH2OO. This is interpreted in terms of the stronger coupling between the B and C states in the larger system leading to a shorter lifetime for the B state than in CH2OO.

Block Effective Hamiltonian Theory and Its Application.

Block effective Hamiltonian theory (BEHT) is presented in this work. Configuration interaction functions are divided into P, Q, and R spaces. Effective Hamiltonian is constructed with the partitioning technique within the P space. The eigenvalue problem of the effective Hamiltonian is then solved iteratively. It is demonstrated that the ground-state energies of N2, HF, and F2 calculated with BEHT converge to the multireference configuration interaction energies from below and the iteration number becomes smaller as BEHT energy becomes closer to the exact energy. The accuracy of BEHT is better than that of the second-order multireference perturbation theory, although the matrix elements in both methods are the same. The ionization potentials of the singlet state of HF, the doublet state of F, and the triplet state of NH and the potential energy curves of CH4, C2, and N2 are calculated with BEHT and compared with experiments and results of CASSCF, CCSD, and CCSD(T) and the results of the full configuration interaction if available. The iteration numbers are all less than 10 in this study. These calculations show the good performances of BEHT in comparison with other theoretical approximation methods.

Role of Exact Exchange in Difference Projected Double-Hybrid Density Functional Theory for Treatment of Local, Charge Transfer, and Rydberg Excitations

Difference approaches to the study of excited states have undergone a renaissance in recent years, with the development of a plethora of algorithms for locating self-consistent field approximations to excited states. Density functional theory is likely to offer the best balance of cost and accuracy for difference approaches, and yet there has been little investigation of how the parametrization of density functional approximations affects performance. In this work, we aim to explore the role of the global Hartree-Fock exchange parameter in tuning accuracy of different excitation types within the framework of the recently introduced difference projected double-hybrid density functional theory approach and contrast the performance with conventional time-dependent double-hybrid density functional theory. Difference projected double-hybrid density functional theory was demonstrated to give vertical excitation energies with average error and standard deviation with respect to multireference perturbation theory comparable to more expensive linear-response coupled cluster approaches ( J. Chem. Phys.2020, 153, 074103). However, despite benchmarking of local excitations, there has been no investigation of the methods performance for charge transfer or Rydberg excitations. In this work we report a new benchmark of charge transfer, Rydberg, and local excited state vertical excitation energies and examine how the exact Hartree-Fock exchange affects the benchmark performance to provide a deeper understanding of how projection and nonlocal correlation balance differing sources of error in the ground and excited states.

The d-d transitions and ligand field parameters for Cr3+/Co2+ doped (Mg, Zn)Al2O4: Multi-reference Ab initio investigations

The normal spinels (Mg, Zn)Al2O4 doped with transition metal (TM) ions Cr3+/Co2+ are versatile materials with important electronic, optical and spectral properties. In addition to being used in many applications, they are excellent systems for testing some models and simulation features. The aim of this paper is to present, in the unified frame, the results on d-d transitions and ligand field parameters (LFPs) for the title systems, based on ab initio calculations, combining periodic density functional theory (DFT) supercell approach with ab initio (AI) multi -reference perturbation theory (MRPT) and multi-reference configuration interaction (MRCI) methods. These AI methods, based on complete active space self-consistent-field (CASSCF) reference, allow to calculate and investigate the energy levels of TM ions and the d-d transitions between them. From the AI results the B and C Racah parameters, the spin-orbit coupling (SOC) constant and the LFPs in the frame of the angular overlap model (AOM) were accurately extracted with ab initio ligand field theory (AILFT) protocol, all with subsequent comparison with the experimental data or existing theoretical results in the literature. The calculation technique presented in this paper serves as a predictive formalism for further studies of larger monomer clusters, for which experimental data is unreliable or unavailable.

The many-body expansion for metals. I. The alkaline earth metals Be, Mg, and Ca

We examine the many-body expansion (MBE) for alkaline earth metal clusters, Ben, Mgn, Can (n = 4, 5, 6), at the Møller–Plesset second order perturbation theory, coupled-cluster singles and doubles with perturbative triples, multi-reference perturbation theory, and multi-reference configuration interaction levels of theory. The magnitude of each term in the MBE is evaluated for several geometrical configurations. We find that the behavior of the MBE for these clusters depends strongly on the geometrical arrangement and, to a lesser extent, on the level of theory used. Another factor that affects the MBE is the in situ (ground or excited) electronic state of the individual atoms in the cluster. For most geometries, the three-body term is the largest, followed by a steady decrease in absolute energy for subsequent terms. Though these systems exhibit non-negligible multi-reference effects, there was little qualitative difference in the MBE when employing single vs multi-reference methods. Useful insights into the connectivity and stability of these clusters have been drawn from the respective potential energy surfaces and quasi-atomic orbitals for the various dimers, trimers, and tetramers. Through these analyses, we investigate the similarities and differences in the binding energies of different-sized clusters for these metals.

Perturbative Expansion of Nonorthogonal Product Approach for Charge Transfer States.

Modeling of the excited states of multichromophoric systems is crucial for the understanding of photosynthesis functioning. The excitonic Hamiltonian method is widely used for such calculations. Excited states of the combined system are constructed from the wave functions of individual chromophores while their interactions are described by excitonic couplings. In the current study, we enhance a previously proposed nonorthogonal product approach to incorporate dynamic correlation effects accounted for by the multireference perturbation theory. We discuss the problems of constructing the excitonic Hamiltonian including charge transfer states for the molecular systems where the overlap contribution to the excitonic couplings is non-negligible. The benchmark calculations were performed for a model system. It was shown that the overlap component of the excitonic coupling is of great importance. The enhanced method provides an accurate description of the excited state energies and other properties.

A local pair-natural orbital-based complete-active space perturbation theory using orthogonal localized virtual molecular orbitals.

The multireference second-order perturbation theory (CASPT2) is known to deliver a quantitative description of various complex electronic states. Despite its near-size-consistent nature, the applicability of the CASPT2 method to large, real-life systems is mostly hindered by large computational and storage costs for the two-external tensors, such as two-electron integrals, amplitudes, and residuum. To this end, Menezes and co-workers developed a reduced-scaling CASPT2 scheme by incorporating the local pair-natural orbital (PNO) representation of the many-body wave functions using non-orthonormal projected atomic orbitals (PAOs) into the CASPT theory [F. Menezes et al., J. Chem. Phys. 145, 124115 (2016)]. Alternatively, in this paper, we develop a new PNO-based CASPT2 scheme using the orthonormal localized virtual molecular orbitals (LVMOs) and assess its performance and accuracy in comparison with the conventional PAO-based counterpart. Albeit the compactness, the LVMOs were considered to perform somewhat poorly compared to PAOs in the local correlation framework because they caused enormously large orbital domains. In this work, we show that the size of LVMO domains can be rendered comparable to or even smaller than that of PAOs by the use of the differential overlap integrals for domain construction. Optimality of the MOs from the CASSCF treatment is a key to reducing the LVMO domain size for the multireference case. Due to the augmented Hessian-based localization algorithm, an additional computational cost for obtaining the LVMOs is relatively minor. We demonstrate that the LVMO-based PNO-CASPT2 method is routinely applicable to large, real-life molecules such as Menshutkin SN2 reaction in a single-walled carbon nanotube reaction field.

Automatic derivation of many-body theories based on general Fermi vacua.

This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.

A simple semiempirical model for the static polarizability of electronically excited atoms and molecules

We present a semiempirical analytical model for the static polarizability of electronically excited atoms and molecules, which requires very few readily accessible input data, including the ground-state polarizability, elemental composition, ionization potential, and spin multiplicities of excited and ground states. This very simple model formulated in a semiclassical framework is based on a number of observed trends in polarizability of electronically excited compounds. To adjust the model, both accurate theoretical predictions and reliable measurements previously reported elsewhere for a broad range of multielectron species in the gas phase are utilized. For some representative compounds of general concern that have not yet attracted sufficient research interest, the results of our multireference second-order perturbation theory calculations are additionally engaged. We show that the model we developed has reasonable (given the considerable uncertainties in the reference data) accuracy in predicting the static polarizability of electronically excited species of arbitrary size and excitation energy. These findings can be useful for many applications, where there is a need for inexpensive and quick assessments of the static gas-phase polarizability of excited electronic states, in particular, when building the complex nonequilibrium kinetic models to describe the observed optical refractivity (dielectric permittivity) of nonthermal reacting gas flows.

Reference Energies for Cyclobutadiene: Automerization and Excited States.

Cyclobutadiene is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods. Indeed, due to its high spatial symmetry, especially at the D4h square geometry but also in the D2h rectangular arrangement, the ground and excited states of cyclobutadiene exhibit multiconfigurational characters and single-reference methods, such as standard adiabatic time-dependent density-functional theory (TD-DFT) or standard equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations. In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the automerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies at D2h and D4h equilibrium structures. In particular, selected configuration interaction (SCI), multireference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled-cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed. The spin-flip formalism, which is known to provide a qualitatively correct description of these diradical states, is also tested within TD-DFT (combined with numerous exchange-correlation functionals) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)]. A theoretical best estimate is defined for the automerization barrier and for each vertical transition energy.

Heavy-Atom Quantum Tunnelling in Spin Crossovers of Nitrenes.

We simulate two recent matrix‐isolation experiments at cryogenic temperatures, in which a nitrene undergoes spin crossover from its triplet state to a singlet state via quantum tunnelling. We detail the failure of the commonly applied weak‐coupling method (based on a linear approximation of the potentials) in describing these deep‐tunnelling reactions. The more rigorous approach of semiclassical golden‐rule instanton theory in conjunction with double‐hybrid density‐functional theory and multireference perturbation theory does, however, provide rate constants and kinetic isotope effects in good agreement with experiment. In addition, these calculations locate the optimal tunnelling pathways, which provide a molecular picture of the reaction mechanism. The reactions involve substantial heavy‐atom quantum tunnelling of carbon, nitrogen and oxygen atoms, which unexpectedly even continues to play a role at room temperature.

Heavy‐Atom Quantum Tunnelling in Spin Crossovers of Nitrenes**

We simulate two recent matrix-isolation experiments at cryogenic temperatures, in which a nitrene undergoes spin crossover from its triplet state to a singlet state via quantum tunnelling. We detail the failure of the commonly applied weak-coupling method (based on a linear approximation of the potentials) in describing these deep-tunnelling reactions. The more rigorous approach of semiclassical golden-rule instanton theory in conjunction with double-hybrid density-functional theory and multireference perturbation theory does, however, provide rate constants and kinetic isotope effects in good agreement with experiment. In addition, these calculations locate the optimal tunnelling pathways, which provide a molecular picture of the reaction mechanism. The reactions involve substantial heavy-atom quantum tunnelling of carbon, nitrogen and oxygen atoms, which unexpectedly even continues to play a role at room temperature.

Low-Lying Excited States of Natural Carotenoids Viewed by Ab Initio Methods.

Low-lying excited states of carotenoids (the optically dark 2Ag- and bright 1Bu+) are deeply involved in energy transfer processes in photosynthetic antennas, such as light harvesting and non-photochemical quenching. Though any ab initio modeling of these phenomena has to rely on precise energies of the carotenoid electronic states, the accurate evaluation of these states remains a challenging problem due to their different natures. The paper aims to study the accuracy of the excitation energies of the low-lying excited states of certain open- and closed-chain carotenoids obtained by a state-of-the-art multireference approach for electronic structure calculation. Here, density matrix renormalization group SCF (DMRGSCF) and a perturbative approach based on driven similarity renormalization group second-order multireference perturbation theory (DSRG-MRPT2) were used to treat the static and dynamic correlation, respectively. Nuclear geometries of the electronic states were optimized with DFT-based approaches. It is demonstrated that spin-flip TD-DFT can replace multiconfigurational methods for the geometry optimization of the 2Ag- state but not for the calculation of the excitation energy. Adiabatic excitation energies to the 1Bu+ state were shown to be within a margin of 1000 cm-1 with an appropriate flow parameter value. Adiabatic excitation energies to the 2Ag- state for the open-chain carotenoids lie within a range of experimental values (taking into account the broad range of experimental estimates); for the closed-chain ones, the error does not exceed 2000 cm-1. Ab initio stationary (1Ag- → 1Bu+) and transient (2Ag- → 1Bu+) absorption spectra were modeled for violaxanthin and lycopene, and these spectra showed good agreement with the experimental ones both in terms of the vibronic structure and the transition energies.

Rydberg states of ZnAr complex

Ab initio potential energy curves (PECs) of electronic states of ZnAr complex are calculated up to Rydberg state correlating with the asymptote of Zn atom. Analysis of various sources of errors of presented calculations is performed including finite basis set size, deficiencies in inclusion of the electron correlation and relativistic effects, and size-inconsistency errors of the multireference second-order perturbation theory. In particular, it is emphasised that the inclusion of the midbond bases substantially improves the convergence rate of the binding energies with respect to the basis set size not only for the van der Waals ground state of ZnAr complex, but also it is the case for the excited states including the Rydberg ones. Wherever it is possible, the comparison with the experimental data and other theoretical results is presented. Properties of the double-well PECs of the Σ Rydberg states are interpreted within the simplified theory, in which, the appearance of the energy barrier and, partially, of the outer well originates from the low-energy scattering of the Rydberg electron on the Ar atom.

Multi-reference many-body perturbation theory for nuclei

In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small (\(e_{\mathrm {max}}=4\)) model space demonstrate that exact binding energies of closed- (\({}^{16}\mathrm {O}\)) and open-shell (\({}^{18}\mathrm {O}\), \({}^{20}\mathrm {Ne}\)) nuclei are reproduced within 0.5–\(1.5\%\) at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.

Multi-reference many-body perturbation theory for nuclei

The neon isotopic chain displays a rich phenomenology, ranging from clustering in the ground-state of the self-conjugate doubly open-shell stable $$^{20}$$ Ne isotope to the physics of the island of inversion around the neutron-rich $$^{30}$$ Ne isotope. This second (i.e. Paper II) of the present series proposes an extensive ab initio study of neon isotopes based on two complementary many-body methods, i.e. the quasi-exact in-medium no-core shell model (IM-NCSM) and the projected generator coordinate method (PGCM) that is ideally suited to capturing strong static correlations associated with shape deformation and fluctuations. Calculations employ a state-of-the-art generation of chiral effective field theory Hamiltonians and evaluate the associated systematic uncertainties. In spite of missing so-called dynamical correlations, which can be added via the multi-reference perturbation theory proposed in the first paper (i.e. Paper I) of the present series [1], the PGCM is shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing the physics of the island of inversion constitutes a challenge that seems to require the inclusion of dynamical correlations. This is addressed in the third paper (i.e. Paper III) of the present series [2].

Multi-reference many-body perturbation theory for nuclei

Multi-reference many-body perturbation theory for nuclei