Francesco Aquilante

The frozen natural orbital (FNO) method is a well-known approximation that significantly reduces the cost of many-body perturbation theory treatments for obtaining correlation energies in a systematic and controlled manner, enabling the simulation of larger and more realistic molecular systems. Here, we implement FNO on top of restricted active-space self-consistent field (RASSCF) wave functions within second-order perturbation theory, leading to the FNO-RASPT2 method. To account for the additional approximations incurred when using RASSCF wave functions, as well as to apply it to states of different molecular charge and spin multiplicity, we introduce a regularization approach via a level-shift in the orbital deletion procedure, which helps avoid singularities resulting from vanishing denominators due to the reduced occupation of restricted (active) virtual orbitals and that we show does not significantly influence the final energies obtained. Numerical results on vertical excitation energies showcase the FNO to be up to ∼4.0 times faster than the parent computation while retaining ∼95% of the dynamic correlation energy and being within 0.1 eV of the reference calculation, with the CASPT2 equations – the main bottleneck in large molecular systems – registering over an order of magnitude speed-up. An in-depth analysis shows the regularized FNO approach to retain electron correlation appropriately for internal and semi-internal contributions, while displaying larger deviations for the external term, which originates from the initial FNO truncation based on an approximated one-electron reduced density matrix. Ongoing efforts are directed toward improving and extending our current implementation to the calculation of analytic gradients and nonadiabatic couplings to enable faster nonadiabatic molecular dynamics simulations in larger and more realistic molecular systems.

The frozen natural orbital (FNO) method is a well-known approximation that significantly reduces the cost of many-body perturbation theory treatments for obtaining correlation energies in a systematic and controlled manner, enabling the simulation of larger and more realistic molecular systems. Here, we implement FNO on top of restricted active-space self-consistent field (RASSCF) wave function within second-order perturbation theory, leading to the FNO-RASPT2 method. To account for the additional approximations incurred when using restricted RASSCF wave functions, as well as to apply it to states of different molecular charge and spin multiplicity, we introduce a regularisation approach via a level-shift in the orbital deletion procedure, which helps avoid singularities resulting from vanishing denominators due to the reduced occupation of restricted (active) virtual orbitals and that we show does not significantly influence the final energies obtained. Numerical results on vertical excitation energies showcase the FNO to be up to ∼3.5 faster than the parent computation while retaining ∼95% of the dynamic correlation energy and being within 0.1 eV of the reference calculation, with the CASPT2 equations-the main bottleneck in large molecular systems-registering over an order of magnitude speed-up. An in-depth analysis shows the regularized FNO approach to retain electron correlation appropriately for internal and semi-internal contributions, while displaying larger deviations for the external term, which originates from the initial FNO truncation based on an approximated one-electron reduced density matrix. Ongoing efforts are directed towards improving and extending our current implementation to the calculation of analytic gradients and non-adiabatic couplings to enable faster non-adiabatic molecular dynamics simulations in larger and more realistic molecular systems.

The frozen natural orbital (FNO) method is a well-known approximation that significantly reduces the cost of many-body perturbation theory treatments for obtaining correlation energies in a systematic and controlled manner, enabling the simulation of larger and more realistic molecular systems. Here, we implement FNO on top of restricted active-space self-consistent field (RASSCF) wave function within second-order perturbation theory, leading to the FNO-RASPT2 method. To account for the additional approximations incurred when using restricted RASSCF wave functions, as well as to apply it to states of different molecular charge and spin multiplicity, we introduce a regularisation approach via a level-shift in the orbital deletion procedure, which helps avoid singularities resulting from vanishing denominators due to the reduced occupation of restricted (active) virtual orbitals and that we show does not significantly influence the final energies obtained. Numerical results on vertical excitation energies showcase the FNO to be up to ∼3.5 faster than the parent computation while retaining ∼95% of the dynamic correlation energy and being within 0.1 eV of the reference calculation, with the CASPT2 equations-the main bottleneck in large molecular systems-registering over an order of magnitude speed-up. An in-depth analysis shows the regularized FNO approach to retain electron correlation appropriately for internal and semi-internal contributions, while displaying larger deviations for the external term, which originates from the initial FNO truncation based on an approximated one-electron reduced density matrix. Ongoing efforts are directed towards improving and extending our current implementation to the calculation of analytic gradients and non-adiabatic couplings to enable faster non-adiabatic molecular dynamics simulations in larger and more realistic molecular systems.

It is shown that if a triple of distinct positive integers ( a,b,c ) were to exist such that a n + b n = c n for some odd integer n ≥3, then it must be Pythagorean, i.e. a 2 + b 2 = c 2 must hold too, from which a contradiction arises since this is possible only if either a or b are zero. We arrive at this conclusion by investigating the (partial) trace of a model hamiltonian operator whose energy levels correspond to those of the so-called Hückel hamiltonian applied to rings containing an odd number of atoms lying on a Möbius strip rather than a planar topology. Furthermore, the contradictory nature of our result implies the correctness of the associated statement contained in the famous Fermat’s Last Theorem. Given the use of concepts from quantum mechanics and matrix algebra unknown at his time, and the fact that the essence of the present proof may not fit within a margin of a typical book, mystery still remains over Pierre de Fermat’s demonstrationem mirabilem .

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 ¹n_O/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.

The developments of the open-source OpenMolcas chemistry software environment since spring 2020 are described, with a focus on novel functionalities accessible in the stable branch of the package or via interfaces with other packages. These developments span a wide range of topics in computational chemistry and are presented in thematic sections: electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report offers an overview of the chemical phenomena and processes OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

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.

In this article the recent developments of the open-source OpenMolcas chemistry software environment, since spring 2020, are described, with the main focus on novel functionalities that are accessible in the stable branch of the package and/or via interfaces with other packages. These community developments span a wide range of topics in computational chemistry, and are presented in thematic sections associated with electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report represents a useful summary of these developments, and it offers a solid overview of the chemical phenomena and processes that OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

We present a derivation of the exact expression for Pulay forces in density-functional theory calculations augmented with extended Hubbard functionals and arising from the use of orthogonalized atomic orbitals as projectors for the Hubbard manifold. The derivative of the inverse square root of the orbital overlap matrix is obtained as a closed-form solution of the associated Lyapunov (Sylvester) equation. The expression for the resulting contribution to the forces is presented in the framework of ultrasoft pseudopotentials and the projector-augmented-wave method and using a plane-wave basis set. We have benchmarked the present implementation with respect to finite differences of total energies for the case of NiO, finding excellent agreement. Owing to the accuracy of Hubbard-corrected density-functional theory calculations—provided the Hubbard parameters are computed for the manifold under consideration—the present work paves the way for systematic studies of solid-state and molecular transition-metal and rare-earth compounds.