Fast Emulation of Fermionic Circuits with Matrix Product States.

We describe a matrix product state (MPS) extension for the Fermionic Quantum Emulator (FQE) software library. We discuss the theory behind symmetry-adapted MPSs for approximating many-body wave functions of spin-1/2 Fermions, and we present an open-source, MPS-enabled implementation of the FQE interface (MPS-FQE). The software uses the open-source pyblock3 and block2 libraries for most elementary tensor operations, and it can largely be used as a drop-in replacement for FQE that allows for more efficient but approximate emulation of larger Fermionic circuits. Finally, we show several applications relevant to both near-term and fault-tolerant quantum algorithms where approximate emulation of larger systems is expected to be useful: characterization of state preparation strategies for quantum phase estimation, the testing of different variational quantum eigensolver ansätze, the numerical evaluation of Trotter errors, and the simulation of general quantum dynamics problems. In all these examples, approximate emulation with MPS-FQE allows us to treat systems that are significantly larger than those accessible with a full statevector emulator.

Algorithm for analytic nuclear energy gradients of state averaged DMRG-CASSCF theory with newly derived coupled-perturbed equations.

We present an algorithm for evaluating analytic nuclear energy gradients of the state-averaged density matrix renormalization group complete-active-space self-consistent field (SA-DMRG-CASSCF) theory based on the newly derived coupled-perturbed (CP) DMRG-CASSCF equations. The Lagrangian for the conventional SA-CASSCF analytic gradient theory is extended to the SA-DMRG-CASSCF variant that can fully consider a whole set of constraints on the parameters of multi-root canonical matrix product states formed at all the DMRG block configurations. An efficient algorithm to solve the CP-DMRG-CASSCF equations for determining the multipliers was developed. The complexity of the resultant analytic gradient algorithm is overall the same as that of the unperturbed SA-DMRG-CASSCF algorithm. In addition, a reduced-scaling approach was developed to directly compute the SA reduced density matrices (SA-RDMs) and their perturbed ones without calculating separate state-specific RDMs. As part of our implementation scheme, we neglect the term associated with the constraint on the active orbitals in terms of the active-active rotation in the Lagrangian. Thus, errors from the true analytic gradients may be caused in this scheme. The proposed gradient algorithm was tested with the spin-adapted implementation by checking how accurately the computed analytic energy gradients reproduce numerical gradients of the SA-DMRG-CASSCF energies using a common number of renormalized bases. The illustrative applications show that the errors are sufficiently small when using a typical number of the renormalized bases, which is required to attain adequate accuracy in DMRG's total energies.

Reliably assessing the electronic structure of cytochrome P450 on today's classical computers and tomorrow's quantum computers.

An accurate assessment of how quantum computers can be used for chemical simulation, especially their potential computational advantages, provides important context on how to deploy these future devices. To perform this assessment reliably, quantum resource estimates must be coupled with classical computations attempting to answer relevant chemical questions and to define the classical algorithms simulation frontier. Herein, we explore the quantum computation and classical computation resources required to assess the electronic structure of cytochrome P450 enzymes (CYPs) and thus define a classical-quantum advantage boundary. This is accomplished by analyzing the convergence of density matrix renormalization group plus n-electron valence state perturbation theory (DMRG+NEVPT2) and coupled-cluster singles doubles with noniterative triples [CCSD(T)] calculations for spin gaps in models of the CYP catalytic cycle that indicate multireference character. The quantum resources required to perform phase estimation using qubitized quantum walks are calculated for the same systems. Compilation into the surface code provides runtime estimates to compare directly to DMRG runtimes and to evaluate potential quantum advantage. Both classical and quantum resource estimates suggest that simulation of CYP models at scales large enough to balance dynamic and multiconfigurational electron correlation has the potential to be a quantum advantage problem and emphasizes the important interplay between classical computations and quantum algorithms development for chemical simulation.

Computational Spectroscopy of the Cr-Cr Bond in Coordination Complexes.

We report the accurate computational vibrational analysis of the Cr-Cr bond in dichromium complexes using second-order multireference complete active space methods (CASPT2), allowing direct comparison with experimental spectroscopic data both to facilitate interpreting the low-energy region of the spectra and to provide insights into the nature of the bonds themselves. Recent technological development by the authors has realized such computation for the first time. Accurate simulation of the vibrational structure of these compounds has been hampered by their notorious multiconfigurational electronic structure that yields bond distances that do not correlate with bond order. Some measured Cr-Cr vibrational stretching modes, ν(Cr2), have suggested weaker bonding, even for so-called ultrashort Cr-Cr bonds, while others are in line with the bond distance. Here, we optimize geometries and compute ν(Cr2) with CASPT2 for three well-characterized complexes, Cr2(O2CCH3)4(H2O)2, Cr2(mhp)4, and Cr2(dmp)4. We obtain CASPT2 harmonic ν(Cr2) modes in good agreement with experiment at 282 cm-1 for Cr2(mhp)4 and 353 cm-1 for Cr2(dmp)4, compute 50Cr and 54Cr isotope shifts, and demonstrate that the use of the so-called IPEA shift leads to improved Cr-Cr distances. Additionally, normal mode sampling was used to estimate anharmonicity along ν(Cr2), leading to an anharmonic mode of 272 cm-1 for Cr2(mhp)4 and 333 cm-1 for Cr2(dmp)4.

Multireference Electron Correlation Methods: Journeys along Potential Energy Surfaces.

Multireference electron correlation methods describe static and dynamical electron correlation in a balanced way and, therefore, can yield accurate and predictive results even when single-reference methods or multiconfigurational self-consistent field theory fails. One of their most prominent applications in quantum chemistry is the exploration of potential energy surfaces. This includes the optimization of molecular geometries, such as equilibrium geometries and conical intersections and on-the-fly photodynamics simulations, both of which depend heavily on the ability of the method to properly explore the potential energy surface. Because such applications require nuclear gradients and derivative couplings, the availability of analytical nuclear gradients greatly enhances the scope of quantum chemical methods. This review focuses on the developments and advances made in the past two decades. A detailed account of the analytical nuclear gradient and derivative coupling theories is presented. Emphasis is given to the software infrastructure that allows one to make use of these methods. Notable applications of multireference electron correlation methods to chemistry, including geometry optimizations and on-the-fly dynamics, are summarized at the end followed by a discussion of future prospects.

Efficient and stochastic multireference perturbation theory for large active spaces within a full configuration interaction quantum Monte Carlo framework.

Full Configuration Interaction Quantum Monte Carlo (FCIQMC) has been effectively applied to very large configuration interaction (CI) problems and was recently adapted for use as an active space solver and combined with orbital optimization. In this work, we detail an approach within FCIQMC to allow for efficient sampling of fully internally contracted multireference perturbation theories within the same stochastic framework. Schemes are described to allow for the close control over the resolution of stochastic sampling of the effective higher-body intermediates within the active space. It is found that while complete active space second-order perturbation theory seems less amenable to a stochastic reformulation, strongly contracted N-Electron Valence second-order Perturbation Theory (NEVPT2) is far more stable, requiring a similar number of walkers to converge the sc-NEVPT2 expectation values as to converge the underlying CI problem. We demonstrate the application of the stochastic approach to the computation of sc-NEVPT2 within a (24, 24) active space in a biologically relevant system and show that small numbers of walkers are sufficient for a faithful sampling of the sc-NEVPT2 energy to chemical accuracy, despite the active space already exceeding the limits of practicality for traditional approaches. This raises prospects of an efficient stochastic solver for multireference chemical problems requiring large active spaces, with an accurate treatment of external orbitals.

Imaginary Shift in CASPT2 Nuclear Gradient and Derivative Coupling Theory.

We report the analytical nuclear gradient theory for complete active space second-order perturbation theory (CASPT2) with imaginary shift, which is commonly used to avoid divergence of the perturbation expression. Our formulation is based on the Lagrangian approach and is an extension of the algorithm for CASPT2 nuclear gradients with real shift. The working equations are derived and implemented into an efficient parallel program. Numerical examples are presented for the ground- and excited-state geometries and conical intersections of a green fluorescent protein model chromophore, p-HBDI-. We also report timing benchmarks with adenine, p-HBDI-, and iron porphyrin. It is demonstrated that the energies and geometries obtained with the imaginary shift improve accuracy at a minor additional cost which is mainly associated with evaluating the effective density matrix elements for the imaginary shift term.

Visualizing Complex-Valued Molecular Orbitals.

We report an implementation of a program for visualizing complex-valued molecular orbitals. The orbital phase information is encoded on each of the vertices of triangle meshes using the standard color wheel. Using this program, we visualized the molecular orbitals for systems with spin-orbit couplings, external magnetic fields, and complex absorbing potentials. Our work has not only created visually attractive pictures but also clearly demonstrated that the phases of the complex-valued molecular orbitals carry rich chemical and physical information on the system, which has often been unnoticed or overlooked.

Zero-Field Splitting Parameters from Four-Component Relativistic Methods.

We report an approach for determination of zero-field splitting parameters from four-component relativistic calculations. Our approach involves neither perturbative treatment of spin-orbit interaction nor truncation of the spin-orbit coupled states. We make use of a multi-state implementation of relativistic complete active space perturbation theory (CASPT2), partially contracted N-electron valence perturbation theory (NEVPT2), and multi-reference configuration interaction theory (MRCI), all with the fully internally contracted ansatz. A mapping is performed from the Dirac Hamiltonian to the pseudospin Hamiltonian, using correlated energies and the magnetic moment matrix elements of the reference wave functions. Direct spin-spin coupling is naturally included through the full 2-electron Breit interaction. Benchmark calculations on chalcogen diatomics and pseudotetrahedral cobalt(II) complexes show accuracy comparable to the commonly used state-interaction with spin-orbit (SI-SO) approach, while tests on a uranium(III) single-ion magnet suggest that for actinide complexes the strengths of our approach through the more robust treatment of spin-orbit effects and the avoidance of state truncation are of greater importance.

Large-scale relativistic complete active space self-consistent field with robust convergence.

We report an efficient algorithm using density fitting for the relativistic complete active space self-consistent field (CASSCF) method, which is significantly more stable than the algorithm previously reported by one of the authors [J. E. Bates and T. Shiozaki, J. Chem. Phys. 142, 044112 (2015)]. Our algorithm is based on the second-order orbital update scheme with an iterative augmented Hessian procedure, in which the density-fitted orbital Hessian is directly contracted to the trial vectors. Using this scheme, each microiteration is made less time consuming than one Dirac-Hartree-Fock iteration, and macroiterations converge quadratically. In addition, we show that the CASSCF calculations with the Gaunt and full Breit interactions can be efficiently performed by means of approximate orbital Hessians computed with the Dirac-Coulomb Hamiltonian. It is demonstrated that our algorithm can also be applied to systems under an external magnetic field, for which all of the molecular integrals are computed using gauge-including atomic orbitals.

On the difference between variational and unitary coupled cluster theories.

There have been assertions in the literature that the variational and unitary forms of coupled cluster theory lead to the same energy functional. Numerical evidence from previous authors was inconsistent with this claim, yet the small energy differences found between the two methods and the relatively large number of variational parameters precluded an unequivocal conclusion. Using the Lipkin Hamiltonian, we here present conclusive numerical evidence that the two theories yield different energies. The ambiguities arising from the size of the cluster parameter space are absent in the Lipkin model, particularly when truncating to double excitations. We show that in the symmetry adapted basis under strong correlation, the differences between the variational and unitary models are large, whereas they yield quite similar energies in the weakly correlated regime previously explored. We also provide a qualitative argument rationalizing why these two models cannot be the same. Additionally, we study a generalized non-unitary and non-hermitian variant that contains excitation, de-excitation, and mixed operators with different amplitudes and show that it works best when compared to the traditional, variational, unitary, and extended forms of coupled cluster doubles theories.

Occupied-Orbital Fast Multipole Method for Efficient Exact Exchange Evaluation.

We present an efficient algorithm for computing the exact exchange contributions in the Hartree-Fock and hybrid density functional theory models on the basis of the fast multipole method (FMM). Our algorithm is based on the observation that FMM with hierarchical boxes can be efficiently used in the exchange matrix construction, when at least one of the indices of the exchange matrix is constrained to be an occupied orbital. Timing benchmarks are presented for alkane chains (C400H802 and C150H302), a graphene sheet (C150H30), a water cluster [(H2O)100], and a protein Crambin (C202H317O64N55S6). The computational cost of the far-field exchange evaluation for Crambin is roughly 3% that of a self-consistent field iteration when the multipoles up to rank 2 are used.

Electronic structure of SmO and SmO- via slow photoelectron velocity-map imaging spectroscopy and spin-orbit CASPT2 calculations.

The chemi-ionization reaction of atomic samarium, Sm + O → SmO+ + e-, has been investigated by the Air Force Research Laboratory as a means to modify local electron density in the ionosphere for reduction of scintillation of high-frequency radio waves. Neutral SmO is a likely unwanted byproduct. The spectroscopy of SmO is of great interest to aid in interpretation of optical emission spectra recorded following atmospheric releases of Sm as part of the Metal Oxide Space Cloud (MOSC) observations. Here, we report a joint experimental and theoretical study of SmO using slow photoelectron velocity-map imaging spectroscopy of cryogenically cooled SmO- anions (cryo-SEVI) and high-level spin-orbit complete active space calculations with corrections from second order perturbation theory (CASPT2). With cryo-SEVI, we measure the electron affinity of SmO to be 1.0581(11) eV and report electronic and vibrational structure of low-lying electronic states of SmO in good agreement with theory and prior experimental work. We also obtain spectra of higher-lying excited states of SmO for direct comparison to the MOSC results.

On-the-Fly CASPT2 Surface-Hopping Dynamics.

We report the development of programs for on-the-fly surface-hopping dynamics simulations in the gas and condensed phases on the potential energy surfaces computed by multistate multireference perturbation theory (XMS-CASPT2) with full internal contraction. On-the-fly nonadiabatic dynamics simulations are made possible by improving the algorithm for XMS-CASPT2 nuclear energy gradient and derivative coupling evaluation. The program is interfaced to a surface-hopping dynamics program, Newton-X, and a classical molecular dynamics package, tinker, to realize such simulations. On-the-fly XMS-CASPT2 surface-hopping dynamics simulations of 9H-adenine and an anionic GFP model chromophore (para-hydroxybenzilideneimidazolin-5-one) in water are presented to demonstrate the applicability of our program to sizable systems. Our program is implemented in the bagel package, which is publicly available under the GNU General Public License.

Analytical Derivative Coupling for Multistate CASPT2 Theory.

The probability of nonradiative transitions in photochemical dynamics is determined by the derivative couplings, the couplings between different electronic states through the nuclear degrees of freedom. Efficient and accurate evaluation of the derivative couplings is, therefore, of central importance to realize reliable computer simulations of photochemical reactions. In this work, the derivative couplings for multistate multireference second-order perturbation theory (MS-CASPT2) and its "extended" variant (XMS-CASPT2) are studied, in which we present an algorithm for their analytical evaluation. The computational costs for evaluating the derivative couplings are essentially the same as those for calculating the nuclear energy gradients. The geometries and energies calculated with XMS-CASPT2 for small molecules at minimum energy conical intersections (MECIs) are in good agreement with those computed by multireference configuration interaction. As numerical examples, MECIs are optimized using XMS-CASPT2 for stilbene and a green fluorescent protein model chromophore (the 4-para-hydroxybenzylidene-1,2-dimethyl-imidazolin-5-one anion).

Hyperfine Coupling Constants from Internally Contracted Multireference Perturbation Theory.

We present an accurate method for calculating hyperfine coupling constants (HFCCs) based on the complete active space second-order perturbation theory (CASPT2) with full internal contraction. The HFCCs are computed as a first-order property using the relaxed CASPT2 spin-density matrix that takes into account orbital and configurational relaxation due to dynamical electron correlation. The first-order unrelaxed spin-density matrix is calculated from one- and two-body spin-free counterparts that are readily available in the CASPT2 nuclear gradient program [M. K. MacLeod and T. Shiozaki, J. Chem. Phys. 142, 051103 (2015)], whereas the second-order part is computed directly using the newly extended automatic code generator. The relaxation contribution is then calculated from the so-called Z-vectors that are available in the CASPT2 nuclear gradient program. Numerical results are presented for the CN and AlO radicals, for which the CASPT2 values are comparable (or, even superior in some cases) to the ones computed by the coupled-cluster and density matrix renormalization group methods. The HFCCs for the hexaaqua complexes with V(II), Cr(III), and Mn(II) are also presented to demonstrate the accuracy and efficiency of our code.

Nuclear Energy Gradients for Internally Contracted Complete Active Space Second-Order Perturbation Theory: Multistate Extensions.

We report the development of the theory and computer program for analytical nuclear energy gradients for (extended) multistate complete active space perturbation theory (CASPT2) with full internal contraction. The vertical shifts are also considered in this work. This is an extension of the fully internally contracted CASPT2 nuclear gradient program recently developed for a state-specific variant by us [MacLeod and Shiozaki, J. Chem. Phys. 2015, 142, 051103]; in this extension, the so-called λ equation is solved to account for the variation of the multistate CASPT2 energies with respect to the change in the amplitudes obtained in the preceding state-specific CASPT2 calculations, and the Z vector equations are modified accordingly. The program is parallelized using the MPI3 remote memory access protocol that allows us to perform efficient one-sided communication. The optimized geometries of the ground and excited states of a copper corrole and benzophenone are presented as numerical examples. The code is publicly available under the GNU General Public License.

Relativistic internally contracted multireference electron correlation methods.

We report internally contracted relativistic multireference configuration interaction (ic-MRCI), complete active space second-order perturbation (CASPT2), and strongly contracted n-electron valence state perturbation theory (NEVPT2) on the basis of the four-component Dirac Hamiltonian, enabling accurate simulations of relativistic, quasi-degenerate electronic structure of molecules containing transition-metal and heavy elements. Our derivation and implementation of ic-MRCI and CASPT2 are based on an automatic code generator that translates second-quantized ansätze to tensor-based equations, and to efficient computer code. NEVPT2 is derived and implemented manually. The rovibrational transition energies and absorption spectra of HI and TlH are presented to demonstrate the accuracy of these methods.

Orbital Optimization in the Active Space Decomposition Model.

We report the derivation and implementation of orbital optimization algorithms for the active space decomposition (ASD) model, which are extensions of complete active space self-consistent field (CASSCF) and its occupation-restricted variants in the conventional multiconfiguration electronic-structure theory. Orbital rotations between active subspaces are included in the optimization, which allows us to unambiguously partition the active space into subspaces, enabling application of ASD to electron and exciton dynamics in covalently linked chromophores. One- and two-particle reduced density matrices, which are required for evaluation of orbital gradient and approximate Hessian elements, are computed from the intermediate tensors in the ASD energy evaluation. Numerical results on 4-(2-naphthylmethyl)-benzaldehyde and [36]cyclophane and model Hamiltonian analyses of triplet energy transfer processes in the Closs systems are presented. Furthermore, model Hamiltonians for hole and electron transfer processes in anti-[2.2](1,4)pentacenophane are studied using an occupation-restricted variant.