Second-order quasi-degenerate perturbation theory with quasi-complete active space self-consistent field reference functions

Quantum chemistry simulation of ground- and excited-state properties of the sulfonium cation on a superconducting quantum processor

The computational description of correlated electronic structure, and particularly of excited states of many-electron systems, is an anticipated application for quantum devices. An important ramification is to determine the dominant molecular fragmentation pathways in photo-dissociation experiments of light-sensitive compounds, like sulfonium-based photo-acid generators used in photolithography. Here we simulate the static and dynamical electronic structure of the H3S+ molecule, taken as a minimal model of a triply-bonded sulfur cation, on a superconducting quantum processor of the IBM Falcon architecture. To this end, we generalize a qubit reduction technique termed entanglement forging or EF [A. Eddins et al., Phys. Rev. X Quantum, 2022, 3, 010309], currently restricted to the evaluation of ground-state energies, to the treatment of molecular properties. While in a conventional quantum simulation a qubit represents a spin-orbital, within EF a qubit represents a spatial orbital, reducing the number of required qubits by half. We combine the generalized EF with quantum subspace expansion [W. Colless et al., Phys. Rev. X, 2018, 8, 011021], a technique used to project the time-independent Schrodinger equation for ground- and excited-states in a subspace. To enable experimental demonstration of this algorithmic workflow, we deploy a sequence of error-mitigation techniques. We compute dipole structure factors and partial atomic charges along ground- and excited-state potential energy curves, revealing the occurrence of homo- and heterolytic fragmentation. This study is an important step towards the computational description of photo-dissociation on near-term quantum devices, as it can be generalized to other photodissociation processes and naturally extended in different ways to achieve more realistic simulations.

Analytic gradients for restricted active space second-order perturbation theory (RASPT2)

The computational cost of analytic derivatives in multireference perturbation theory is strongly affected by the size of the active space employed in the reference self-consistent field calculation. To overcome previous limits on the active space size, the analytic gradients of single-state restricted active space second-order perturbation theory (RASPT2) and its complete active space second-order perturbation theory (CASPT2) have been developed and implemented in a local version of OpenMolcas. Similar to previous implementations of CASPT2, the RASPT2 implementation employs the Lagrangian or Z-vector method. The numerical results show that restricted active spaces with up to 20 electrons in 20 orbitals can now be employed for geometry optimizations.

Single-Root Multireference Brillouin-Wigner Perturbative Approach to Excitation Energies

The state-specific Brillouin-Wigner multireference perturbation theory [which employs Jeziorski-Monkhorst parametrization of the wave function] using improved virtual orbitals, denoted as IVO-BWMRPT, is applied to calculate excitation energies (EEs) for methylene, ethylene, trimethylenemethane, and benzyne systems exhibiting various degrees of diradical character. In IVO-BWMRPT, all of the parameters appearing in the wave function ansatz are optimized for a specific electronic state. For these systems, the IVO-BWMRPT method provides EEs that are in close agreement with the benchmark results and experiments, where available, indicating that the method does not introduce imbalance in the target-specific treatment of closed- and open-shell states involved. The good performance of the present methodology is primarily related to structural compactness of the formalism. Overall, present findings are encouraging for both further development of the approach and chemical applications on the energy differences of strongly correlated systems.

Assessment of the Density-Fitted Second-Order Quasidegenerate Perturbation Theory for Transition Energies: Accurate Computations of Singlet-Triplet Gaps for Charge-Transfer Compounds

Organic light-emitting diodes (OLEDs) have been of significant interest because of their superior performance and low cost of production. Thermally activated delayed fluorescence (TADF) has attracted significant interest in the OLED technology because it improves the efficiency of OLEDs by harvesting triplet excitons. Therefore, the accurate computation of singlet-triplet transition energies (ΔE_S1-T1) of charge-transfer molecules is very important. However, the accurate computation of the ΔE_S1-T1 values is a challenging problem for single-reference methods because of the multireference character of excited states. In this research, an assessment of density-fitted second-order quasidegenerate perturbation theory (DF-QDPT2) [Bozkaya, U.; J. Chem. Theory Comput. 2019, 15, 4415-4429] for singlet-triplet transition energies (ΔE_S1-T1) of charge-transfer compounds is presented. The performance of the DF-QDPT2 method has been compared with those of several density-functional theory functionals, such as B3LYP, PBE0, M06-2X, ωB97X-D, and MN15; density-fitted state-averaged CASSCF (DF-SA-CASSCF); and single-state single-reference second-order perturbation theory (SS-SR-CASPT2) methods. For the TADF molecules considered, the DF-QDPT2 method provides a mean absolute error (MAE) of 0.13 eV, while the MAE values of DF-SA-CASSCF and SS-SR-CASPT2 are 0.65 and 0.74 eV, respectively. The performances of B3LYP and PBE0 are slightly better than that of DF-QDPT2, while M06-2X and ωB97X-D provide noticeably higher errors compared with DF-QDPT2. Furthermore, the standard CASSCF without state-averaging yields dramatic errors with an MAE value of 3.0 eV. Our results demonstrate that eigenvalues of the DF-QDPT2-effective Hamiltonian can be reliably used for the prediction of singlet-triplet transition energies, while eigenvalues of DF-CASSCF/DF-SA-CASSCF fail to provide accurate predictions. Overall, we conclude that the DF-QDPT2 method emerges as a very useful tool for the computation of excited-state properties.

A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment. I. Principles, second-order versions, and tests on ground state potential energy curves

The present paper introduces a new multi-reference perturbation approach developed at second order, based on a Jeziorski-Mokhorst expansion using individual Slater determinants as perturbers. Thanks to this choice of perturbers, an effective Hamiltonian may be built, allowing for the dressing of the Hamiltonian matrix within the reference space, assumed here to be a CAS-CI. Such a formulation accounts then for the coupling between the static and dynamic correlation effects. With our new definition of zeroth-order energies, these two approaches are strictly size-extensive provided that local orbitals are used, as numerically illustrated here and formally demonstrated in the Appendix. Also, the present formalism allows for the factorization of all double excitation operators, just as in internally contracted approaches, strongly reducing the computational cost of these two approaches with respect to other determinant-based perturbation theories. The accuracy of these methods has been investigated on ground-state potential curves up to full dissociation limits for a set of six molecules involving single, double, and triple bond breaking together with an excited state calculation. The spectroscopic constants obtained with the present methods are found to be in very good agreement with the full configuration interaction results. As the present formalism does not use any parameter or numerically unstable operation, the curves obtained with the two methods are smooth all along the dissociation path.

Molcas 8: New capabilities for multiconfigurational quantum chemical calculations across the periodic table

In this report, we summarize and describe the recent unique updates and additions to the Molcas quantum chemistry program suite as contained in release version 8. These updates include natural and spin orbitals for studies of magnetic properties, local and linear scaling methods for the Douglas-Kroll-Hess transformation, the generalized active space concept in MCSCF methods, a combination of multiconfigurational wave functions with density functional theory in the MC-PDFT method, additional methods for computation of magnetic properties, methods for diabatization, analytical gradients of state average complete active space SCF in association with density fitting, methods for constrained fragment optimization, large-scale parallel multireference configuration interaction including analytic gradients via the interface to the Columbus package, and approximations of the CASPT2 method to be used for computations of large systems. In addition, the report includes the description of a computational machinery for nonlinear optical spectroscopy through an interface to the QM/MM package Cobramm. Further, a module to run molecular dynamics simulations is added, two surface hopping algorithms are included to enable nonadiabatic calculations, and the DQ method for diabatization is added. Finally, we report on the subject of improvements with respects to alternative file options and parallelization.

Aspects of Size-Consistency of Orbitally Noninvariant Size-Extensive Multireference Perturbation Theories: A Case Study Using UGA-SSMRPT2 as a Prototype

Profiling a potential energy surface (PES), all the way to dissociate a molecular state into particular fragments and to display real or avoided crossings, requires a multireference description and the maintenance of size-consistency. The many body methods, which suit this purpose, should thus be size-extensive. Size-extensive theories, which are invariant with respect to transformation among active orbitals are, in principle, size-consistent. Relatively cheaper size-extensive theories, which do not possess this invariance, can still be size-consistent if the active orbitals are localized on the asymptotic fragments. Such methods, if perturbative in nature, require the use of an unperturbed Hamiltonian, which has orbital invariance with respect to the transformation within active, core, and virtual orbitals. The principal focus of this paper is to numerically realize size-consistency with localized active orbitals using our recently developed orbitally noninvariant Unitary Group Adapted State Specific Multireference second order Perturbation Theory (UGA-SSMRPT2) as a prototype method. Our findings expose certain generic potential pitfalls of size-extensive but orbitally noninvariant MRPT theories, which are mostly related to the inability of reaching proper localized active orbitals in the fragments due to the artifacts of the orbital generation procedure. Despite the invariance of the zeroth order CAS function, lack of invariance of the MRPT itself then leads to size-inconsistency. In particular, reaching symmetry broken fragment active orbitals is an issue of concern where suitable state-averaging might ameliorate the problem, but then one has to abandon full orbital optimization. Additionally, there can be situations where the orbitals of the fragment reached as an asymptote of the supermolecule are not the same as those obtained from the optimization of the fragments individually and will require additional transformation. Moreover, for a certain PES, one may either abandon the use of optimized orbitals for that state to preserve proper symmetry and degeneracy in the fragment orbitals or be satisfied with the use of optimized orbitals, which generate broken symmetric orbitals in the fragmentation limit. All these pathologies are illustrated using the PES of various electronic states of multiply bonded systems like N2, C2H2, HCN, C2, and O2. Subject to such proviso, the UGA-SSMRPT2 turns out to be an excellent theory for studying the PES leading to fragmentation of strongly correlated systems satisfying the requirements of size-consistency with localized active orbitals. An unexpected spin-off of our studies is the realization that the size-inextensive MRMP2, which bears a close structural similarity with our theory, might under certain situations display size-consistency. We analyze this feature concretely in our paper. Our studies may serve as a benchmark for monitoring numerically the size-consistency of any state specific multireference theory which is size-extensive but not invariant with respect to transformation of active orbitals.

RECENT ADVANCES IN ELECTRONIC STRUCTURE THEORY

Accurate quantum computational chemistry has evolved dramatically. The size of molecular systems, which can be studied accurately using molecular theory is increasing very rapidly. Theoretical chemistry has opened up a world of new possibilities. It can treat real systems with predictable accuracy. Computational chemistry is becoming an integral part of chemistry research. Theory can now make very significant contribution to chemistry.

This review will focus on our recent developments in the theoretical and computational methodology for the study of molecular structure and molecular interactions. We are aiming at developing accurate molecular theory on systems containing hundreds of atoms. We continue our research in the following three directions: (i) development of new ab initio theory, particularly multireference-based perturbation theory, (ii) development of exchange and correlation functionals in density functional theory, and (iii) development of molecular theory including relativistic effects.

We have enjoyed good progress in each of the above areas. We are very excited about our discoveries of new theory and new algorithms and would like to share this enthusiasm with readers.

Relativistic quasidegenerate perturbation theory with four-component general multiconfiguration reference functions

Relativistic quasidegenerate perturbation theory (QDPT) using general multiconfiguration (GMC) reference functions is developed and implemented. It is the relativistic counterpart of the nonrelativistic QDPT with GMC reference and thus retains all the advantages of the nonrelativistic GMC reference QDPT, such as applicability to any configuration space and small computational cost compared to the complete configuration-space case. The method is applied to the potential-energy curves of the ground states of I(2) and Sb(2) molecules, the excitation energies of CH(3)I, and the energies of the lowest terms of C, Si, and Ge atoms, and is shown to provide a balanced description of potential-energy curves and accurate transition energies for systems containing heavy elements and to provide much better results compared to the reference function (i.e., active space configuration interaction) level.

A Case Study of State-Specific and State-Averaged Brueckner Equation-of-Motion Coupled-Cluster Theory: The Ionic-Covalent Avoided Crossing in Lithium Fluoride

State-specific Brueckner equation-of-motion coupled-cluster theory (SS-B-EOMCC) is summarized, which can be considered an internally contracted version of a state-selective multireference coupled-cluster theory, which, however, is not entirely size-consistent. The method is applicable to general multireference problems, adheres to the space and spin symmetries of the molecular system, is straightforwardly extended to a state-averaged version, and has an associated perturbative variant which yields results close to the full coupled-cluster treatment. A key strength is that Brueckner orbitals are used, such that orbitals are optimized in the presence of dynamic correlation. A number of variations on the theme of SS-EOMCC is applied to study the ionic-covalent avoided crossing in LiF in a 6-311++G(3df,3pd) basis set. While reasonable results are obtained at the state-averaged level, the iterative solution process does not consistently converge for SS-EOMCC, due to the non-Hermiticity of the transformed Hamiltonian which may yield complex eigenvalues upon truncated diagonalization. This leads to an irrevocable breakdown of the state-specific EOMCC approach. We indicate some future directions that can resolve some of the problems with the SS-EOMCC methodology, as revealed by the demanding test case of the LiF potential energy curves.

Accurate calculation of core-electron binding energies: Multireference perturbation treatment

Multireference perturbation theory (MRPT) with multiconfigurational self-consistent field (MCSCF) reference functions is applied to the calculations of core-electron binding energies (CEBEs) of atoms and molecules. Orbital relaxations in a core-ionized state and electron correlation are both taken into account in a conventional MCSCF-MRPT procedure. In the MCSCF calculation, the target core ionized state is directly optimized as an excited state and this treatment can completely prevent a variational collapse. Multireference Moller-Plesset perturbation theory and multiconfigurational self-consistent field reference quasidegenerated perturbation theory were used to treat electron correlation. The present method quite accurately reproduced the 1s CEBEs of CH4, NH3, H2O, and FH; the average deviation from the experimental data is 0.11 eV using Ahlrichs' VTZ basis set. The C 1s and O 1s CEBEs of formic acid and acetic acid were calculated and the results are consistent with the bonding characters of the atoms in these molecules. The present procedure can also be applied to CEBEs of higher angular momentum orbitals by including spin-orbit coupling. The calculated CEBEs of Ar 2p, HCl 2p, Kr 3d, and HBr 3d are in reasonable agreement with the available experimental values. In the calculation of the 3d CEBEs, a relativistic correction significantly improves the agreements. The effect of polarization functions is also discussed.

Quasi-degenerate perturbation theory with general multiconfiguration self-consistent field reference functions

The quasi-degenerate perturbation theory (QDPT) with complete active space (CAS) self-consistent field (SCF) reference functions is extended to the general multiconfiguration (MC) SCF references functions case. A computational scheme that utilizes both diagrammatic and sum-over-states approaches is presented. The second-order effective Hamiltonian is computed for the external intermediate configurations (including virtual or/and core orbitals) by the diagrammatic approach and for internal intermediate configurations (including only active orbitals) by the configuration interaction matrix-based sum-over-states approach. The method is tested on the calculations of excitation energies of H(2)O, potential energy curves of LiF, and valence excitation energies of H(2)CO. The results show that the present method yields very close results to the corresponding CAS-SCF reference QDPT results and the available experimental values. The deviations from CAS-SCF reference QDPT values are less than 0.1 eV on the average for the excitation energies of H(2)O and less than 1 kcal/mol for the potential energy curves of LiF. In the calculation of the valence excited energies of H(2)CO, the maximum deviation from available experimental values is 0.28 eV.