Orbital entropies, pair entropies, and mutual information have become popular tools for analysis of strongly correlated wave functions. They can quantitatively measure how strongly an orbital participates in the electron correlation and reveal the correlation pattern between different orbitals. However, this pattern can become rather complicated and sometimes difficult to interpret for large active spaces and is not invariant with respect to the spin projection (M_s) component of the spin multiplet state. We introduce a modified spin-free orbital entropy, pair entropy, and mutual information, which simplify the correlation analysis and are invariant with respect to M_s. By comparison of these quantities with their "original" spin-including counterparts one can distinguish static correlation due to spin couplings from the "genuine" strong correlation due to a multiconfigurational character of the wave function. Recently, Evangelista introduced mutual correlation computed from the two-body cumulant as an alternative correlation measure, which is easier to compute for some methods. We present here a spin-free analogue of this quantity, which is M_s invariant, and perform a comparison of the entropy-based and cumulant-based correlation measures in both spin-free and original variants. We illustrate the approaches on iron-sulfur bound complexes with one and two iron atoms.

Surface hopping (SH) methods are typically employed to simulate ultrafast nonadiabatic processes, but long timescales often remain beyond their reach. To address this, accelerated SH scheme mitigate this limitation by scaling the driving forces of such process, either nonadiabatic couplings (NACs) in case of internal conversion or spin-orbit couplings (SOCs) for intersystem crossing. However, obtaining the actual time constant requires extrapolation from several ensembles of trajectories with different scaling factors. This introduces a significant computational demand, often restricting the number of trajectories per ensemble and, therefore, reducing the statistical confidence in the resulting time constant. In this work, we investigate the accelerated scheme using silaethylene (CH_2SiH_2) as a case study, evaluating various population fitting methods and extrapolation techniques. We trained machine learning models for potential energy surfaces (PESs) and NACs, and extended our rotate-predict-rotate approach to fit SOCs. These models demonstrate high performance, yielding populations within the confidence interval of the reference MR-CISD/SA-CASSCF(2,2) data; however, the extrapolation itself is highly sensitive to the fitted time constants, leading to discrepancies in the final time constant. Finally, we showcase and discuss how ML models can enhance the reliability of an accelerated SH scheme.

Mixed quantum-classical dynamics (MQCD) methods are effective models for excited-state processes in quasi-classical molecular systems, in which nuclear motion is described by classical trajectories while electronic populations undergo quantum nonadiabatic transitions. This article presents Newton-X 26, a new generation of the Newton-X platform that consolidates two decades of development into a modular ecosystem for generating spectra and initial conditions, propagating dynamics, and analyzing, postprocessing, and archiving data. Newton-X 26 supports multiple MQCD strategies, including surface hopping, decoherence-corrected Ehrenfest dynamics, and ab initio multiple spawning, and connects to a range of electronic-structure engines through dedicated interfaces. The platform emphasizes efficient execution for large trajectory ensembles, enabling systematic convergence analyses and uncertainty estimation. Complementary tools support automated data curation, machine-learning-assisted workflows, and reproducible FAIR-oriented reporting and sharing. Taken together, Newton-X 26 provides an open-source environment for routine MQCD applications and continued method development across multiple electronic-structure levels.

Nonadiabatic molecular dynamics is a key technique for investigating a broad range of photochemical and photophysical processes. Among the established approaches, surface hopping schemes are widely used and can be easily integrated with various quantum chemistry programs or machine learning models. We present a flexible framework in MLatom that includes a newly implemented Tully's fewest-switches surface hopping algorithm and its time-dependent Baeck-An variant. The capabilities of this framework are demonstrated through three representative examples corresponding to typical stages of a surface hopping study. First, we focus on methods providing energy, energy gradients, and nonadiabatic couplings. We show that the flexibility of user-defined custom models can save computational time and that it is useful for benchmarking machine learning models. Next, we compare curvature-driven surface hopping schemes and show that the Landau-Zener approach outperforms the time-dependent Baeck-An scheme. Finally, we showcase easy-to-use analysis tools for both individual trajectories and trajectory ensembles. This framework enables accelerated development of machine learning models and provides deeper insight into nonadiabatic dynamics. It is available as a part of the open-source MLatom package.

The COLUMBUS program system provides the tools for performing high-level multireference (MR) computations, including the multireference configuration interaction (MRCI) method and its multireference averaged quadratic coupled cluster (MR-AQCC) extension, allowing computations on a wide range of fascinating atomic and molecular systems, including the treatment of open-shells and complicated excited state phenomena. The inclusion of spin-orbit coupling (SOC) directly within the MRCI step enables the description of systems containing heavy elements, such as lanthanides and actinides, whose properties are strongly influenced by SOC. Analytic energy gradients and nonadiabatic couplings at the correlated MRCI level provide the foundation for a variety of dynamics studies, giving insight into ultrafast photochemistry. New and ongoing method developments in COLUMBUS include the computation of spin densities, improved descriptions of ionic states, enhancements to the AQCC method, and the porting of COLUMBUS to graphical processing units (GPUs). New external interfaces enable an enhanced description of electronic resonances and molecules in strong laser fields. This work highlights these new developments while providing a detailed account of the diverse applications of COLUMBUS in recent years.

A computational study of I-BODIPY (2-ethyl-4,4-difluoro-6,7-diiodo-1,3-dimethyl-4-bora-3a,4a-diaza-s-indacene) was conducted to investigate its photophysical properties as a potential triplet photosensitizer for singlet oxygen generation. Multireference CASPT2 and CASSCF methods were used to calculate vertical excitation energies and spin-orbit couplings (SOCs) in a model monoiodinated BODIPY molecule to assess the applicability of the single-reference ADC(2) method. Time-dependent density functional theory (TD-DFT) with the Tamm-Dancoff approximation (TDA) was tested against ADC(2) using different exchange-correlation functionals, employing a two-component pseudopotential basis set for iodine. SOC magnitudes between excited states were discussed using the Slater-Condon rules. The geometry dependence of SOCs for the lowest states was also examined. TD-DFT/B3LYP and TD-DFT(TDA)/BHLYP were selected for subsequent absorption spectra and trajectory surface hopping (TSH) molecular dynamics (MD) simulations. Two bright states were identified in I-BODIPY's visible spectrum, showing a red shift due to iodine substitution. Excited-state MD simulations, including nonadiabatic effects and SOCs, were performed to investigate relaxation after photoexcitation to the S1 state. TSH MD simulations revealed that intersystem crossings occur on a similar timescale to internal conversions. After triplet population growth, a "saturation" phase was reached with a triplet-to-singlet ratio of about 4:1. The calculated triplet quantum yield of 0.85 agrees qualitatively with the experimental singlet oxygen generation yield of 0.99.

Orbital entropies, pair entropies, and mutual information have become popular tools for analysis of strongly correlated wave functions. They can quantitatively measure how strongly an orbital (e.g. from the DMRG active space) participates in the strong correlation and reveal the entanglement pattern between different orbitals. However, this pattern can become rather complicated and sometimes difficult to interpret for large active spaces and is not invariant with respect to the spin projection (M_s) component of the spin multiplet state. We introduce a modified spin-free orbital entropy, pair entropy, and mutual information, which simplify the entanglement analysis and are invariant with respect to M_s. By comparison of these quantities with their “original” spin-including counterparts one can distinguish static correlation due to spin couplings from the “genuine” strong correlation due to a multiconfigurational character of the wave function. We illustrate the approach on a model consisting of a non-interacting dimer of triplet diradicals and on a more realistic example of iron-sulfur bound complexes with one and two iron atoms.

Nonadiabatic couplings (NACs) play a crucial role in modeling photochemical and photophysical processes with methods such as the widely used fewest-switches surface hopping (FSSH). There is therefore a strong incentive to machine learn NACs for accelerating simulations. However, this is challenging due to NACs' vectorial, double-valued character and the singularity near a conical intersection seam. For the first time, we design NAC-specific descriptors based on our domain expertise and show that they allow learning NACs with never-before-reported accuracy of R^2 exceeding 0.99. The key to success is also our new ML phase-correction procedure. We demonstrate the efficiency and robustness of our approach on a prototypical example of fully ML-driven FSSH simulations of fulvene targeting the SA-2-CASSCF(6,6) electronic structure level. This ML-FSSH dynamics leads to an accurate description of S_1 decay while reducing error bars by allowing the execution of a large ensemble of trajectories. Our implementations are available in open-source MLatom.

We analyze the structures, stabilities, and thermochemical properties of polyethylene (PE) oligomer chains cross-linked by metal (M) atoms through C-M-C bonds. Representative PEn-Mm-PEn complexes contain between 7 and 15 carbon atoms in each oligomer and one to three Li, Be, Mg, Zn, Ag, or Au cross-linking metal elements. PEn-Mm-PEn complexes are quasiplanar with nearly parallel PE chains. Their stability is determined by covalent C-M-C bonds accompanied by noncovalent dispersion interactions between PEn chains. Using the CAM-B3LYP+D3BJ+ABC functional, the binding energies of PE15-M-PE15 with respect to two PE15 radicals and metal fragments are -225, -230, -322, -551, -289, and -303 kJ/mol for Li, Ag, Au, Be, Mg, and Zn atoms, respectively. Entropy contributions (109 to 121 kJ/mol at 298.15 K) destabilize all complexes significantly. With two cross-linking metal elements in PE15-M2-PE15 complexes, binding energies are about double. Complexes with several open-shell Li, Ag, or Au doublet atoms have spins located on separated C-M-C bonds. High-spin PE15-Mm1-PE15-Mm2-PE15 complexes of three PE oligomers cross-linked by up to five doublet metal atoms create parallel PE tubes, which are suggested as elementary cells for modeling magnetic polymer tubes.

Small molecules possessing a triplet ground state are fundamentally intriguing but also in high demand for applications such as quantum sensing and quantum computing. Such molecules are rare, and most examples involve extended π-systems. Topology and shape of the spin density will be very different for molecules where the triplet state arises from σ-overlap. Drawing inspiration from NV- (anionic nitrogen-vacancy) centres in a diamond crystal, which possess triplet ground states that are robust due to the distortion-preventing crystal lattice, we investigate hetero-atom substituted diamondoids (molecular nanodiamonds) as molecular mimics for NV- centres. It is found that even in these small systems, distortions that stabilize singlet states are energetically costly, and the triplet states are more stable than the singlets. The stabilization of the triplet over the singlet is 13, 16, and 18 kcal mol-1, in anionic C3v-C33H36N- and in the charge-neutral molecules C3v-C33H36O and C3v-C33H36S, respectively, using CAM-B3LYP-D3(BJ)/Def2-QZVPP. Comparable numbers are obtained with other density functional theory (DFT) methods, including double-hybrids. Wavefunction-based approaches on the other hand disagree in their predictions: While the MP2 method applied with the DLPNO approximation predicts a preference for the singlet, density matrix renormalization group (DMRG) calculations qualitatively agree with DFT in their prediction of a triplet ground state, although by a small margin, for C3v-C33H36N- and C3v-C33H36O, but not for C3v-C33H36S. Weighing the evidence, we conclude, with reasonable confidence for C3v-C33H36N- and C3v-C33H36O and lesser confidence for C3v-C33H36S, that the ground state for the molecular nanodiamonds studied is a triplet state.

Heavy atom compounds represent a challenge for computational chemistry due to the need for simultaneous treatment of relativistic and correlation effects. Often such systems also exhibit strong correlation, which hampers the application of perturbation theory or single-reference coupled cluster (CC) methods. As a viable alternative, we have proposed externally correcting the CC method using the density matrix renormalization group (DMRG) wave functions, yielding the DMRG-tailored CC method. In a previous paper [J. Chem. Phys. 2020, 152, 174107], we reported a first implementation of this method in the relativistic context, which was restricted to molecules with real double group symmetry. In this work, we present a fully general implementation of the method, covering complex and quaternion double groups as well. The 4c-TCC method thus becomes applicable to polyatomic molecules, including heavy atoms. For the assessment of the method, we performed calculations of the chiral uranium compound NUHFI, which was previously studied in the context of the enhancement of parity violation effects. In particular, we performed calculations of a cut of the potential energy surface of this molecule along the stretching of the N-U bond, where the system exhibits strong multireference character. Since there are no experimental data for NUHFI, we have performed also an analogous study of the (more symmetric) NUF3 molecule, where the vibrational frequency of the N-U bond can be compared with spectroscopic data.

Unlike with the energy, which is a scalar property, machine learning (ML) prediction of vector or tensor properties poses the additional challenge of achieving proper invariance (covariance) with respect to molecular rotation. For the energy gradients needed in molecular dynamics (MD), this symmetry is automatically fulfilled when taking analytic derivative of the energy, which is a scalar invariant (using properly invariant molecular descriptors). However, if the properties cannot be obtained by differentiation, other appropriate methods should be applied to retain the covariance. Several approaches have been suggested to properly treat this issue. For nonadiabatic couplings and polarizabilities, for example, it was possible to construct virtual quantities from which the above tensorial properties are obtained by differentiation and thus guarantee the covariance. Another possible solution is to build the rotational equivariance into the design of a neural network employed in the model. Here, we propose a simpler alternative technique, which does not require construction of auxiliary properties or application of special equivariant ML techniques. We suggest a three-step approach, using the molecular tensor of inertia. In the first step, the molecule is rotated using the eigenvectors of this tensor to its principal axes. In the second step, the ML procedure predicts the vector property relative to this orientation, based on a training set where all vector properties were in this same coordinate system. As the third step, it remains to transform the ML estimate of the vector property back to the original orientation. This rotate-predict-rotate (RPR) procedure should thus guarantee proper covariance of a vector property and is trivially extensible also to tensors such as polarizability. The RPR procedure has an advantage that the accurate models can be trained very fast for thousands of molecular configurations, which might be beneficial where many training sets are required (e.g., in active learning). We have implemented the RPR technique, using the MLatom and Newton-X programs for ML and MD, and performed its assessment on the dipole moment along MD trajectories of 1,2-dichloroethane.

A swarm of semi-classical quantum mechanics/molecular mechanics molecular-dynamics simulations where OM2/MNDO is combined with the Gromacs program for consideration of explicit water is performed, solving the time-dependent Schrödinger equation in each step of the trajectories together with the Tully's fewest switches algorithm. Within this stochastic treatment, time dependent probabilities of the three lowest electronic states are determined. The fact that nucleobases are quickly deactivated is confirmed in the cytosine case where our best lifetime estimation is τ1=0.82 ps for the model with 100 water molecules with the SPCE force field and a time step of 0.1 fs. Lifetimes of the remaining molecules are visibly longer: 5-azacytosine, 2,4-diamino-1,3,5-triazine (DT), and 2,4,6-triamino-1,3,5-triazine (TT) molecules have an S1 → S0 de-excitation time of slightly above 10 ps. The lifetimes of the triazine family increases with the increasing number of exocyclic amino groups, that is, s-triazine < 2-amino-1,3,5-triazine < DT < TT. This can be explained by a higher mobility of the carbon-bonded hydrogen atoms in comparison with heavier amino groups since their movement is slowed down due to a substantially higher mass than hydrogen atoms, which can easier reach the out-of-plane positions required in the conical intersection structures. Moreover, bulkier NH2 ligands suffer due to greater friction caused by the surrounding water environment. These mechanical aspects caused a change in the explored lifetime dependences in comparison with our previous gas-phase study.

In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants-the state universal one and the Brillouin-Wigner and Mukherjee's state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO-LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations.

Newton-X is an open-source computational platform to perform nonadiabatic molecular dynamics based on surface hopping and spectrum simulations using the nuclear ensemble approach. Both are among the most common methodologies in computational chemistry for photo -physical and photochemical investigations. This paper describes the main features of these methods and how they are implemented in Newton-X. It emphasizes the newest developments, including zero-point-energy leakage correction, dynamics on complex-valued potential energy surfaces, dynamics induced by incoherent light, dynamics based on machine-learning potentials, exciton dynamics of multiple chromophores, and supervised and unsupervised machine learning techniques. Newton-X is interfaced with several third-party quantum-chemistry programs, spanning a broad spectrum of electronic structure methods.

Halogenated and alkylated BODIPY derivatives are reported as suitable candidates for their use as photosensitizers in photodynamic therapy due to their efficient intersystem crossing (ISC) between states of different spin multiplicities. Spin-orbit couplings (SOCs) are evaluated using an effective one-electron spin-orbit Hamiltonian for brominated and alkylated BODIPY derivatives to investigate the quantitative effect of alkyl and bromine substituents on ISC. BODIPY derivatives containing bromine atoms have been found to have significantly stronger SOCs than alkylated BODIPY derivatives outside the Frank-Condon region while they are nearly the same at local minima. Based on calculated time-dependent density functional theory (TD-DFT) vertical excitation energies and SOCs, excited-state dynamics of three BODIPY derivatives were further explored with TD-DFT surface hopping molecular dynamics employing a simple accelerated approach. Derivatives containing bromine atoms have been found to have very similar lifetimes, which are much shorter than those of the derivatives possessing just the alkyl moieties. However, both bromine atoms and alkyl moieties reduce the HOMO/LUMO gap, thus assisting the derivatives to behave as efficient photosensitizers.

In this paper, we discuss the utilization of variational quantum solver (VQE) and recently introduced generalized unitary coupled cluster (GUCC) formalism for the diagonalization of downfolded/effective Hamiltonians in active spaces. In addition to effective Hamiltonians defined by the downfolding of a subset of virtual orbitals we also consider their form defined by freezing core orbitals, which enables us to deal with larger systems. We also consider various solvers to identify solutions of the GUCC equations. We use N2, H2O, and C2H4, as benchmark systems to illustrate the performance of the combined framework.

We present a new implementation of density matrix renormalization group based tailored coupled clusters method (TCCSD), which employs the domain-based local pair natural orbital approach (DLPNO). Compared to the previous local pair natural orbital (LPNO) version of the method, the new implementation is more accurate, offers more favorable scaling, and provides more consistent behavior across the variety of systems. On top of the singles and doubles, we include the perturbative triples correction (T), which is able to retrieve even more dynamic correlation. The methods were tested on three systems: tetramethyleneethane, oxo-Mn(Salen), and iron(II)-porphyrin model. The first two were revisited to assess the performance with respect to LPNO-TCCSD. For oxo-Mn(Salen), we retrieved between 99.8 and 99.9% of the total canonical correlation energy which is an improvement of 0.2% over the LPNO version in less than 63% of the total LPNO runtime. Similar results were obtained for iron(II)-porphyrin. When the perturbative triples correction was employed, irrespective of the active space size or system, the obtained energy differences between two spin states were within the chemical accuracy of 1 kcal/mol using the default DLPNO settings.

There are three essential problems in computational relativistic chemistry: Electrons moving at relativistic speeds, close lying states, and dynamical correlation. Currently available quantum-chemical methods are capable of solving systems with one or two of these issues. However, there is a significant class of molecules in which all the three effects are present. These are the heavier transition metal compounds, lanthanides, and actinides with open d or f shells. For such systems, sufficiently accurate numerical methods are not available, which hinders the application of theoretical chemistry in this field. In this paper, we combine two numerical methods in order to address this challenging class of molecules. These are the relativistic versions of coupled cluster methods and the density matrix renormalization group (DMRG) method. To the best of our knowledge, this is the first relativistic implementation of the coupled cluster method externally corrected by DMRG. The method brings a significant reduction of computational costs as we demonstrate on the system of TlH, AsH, and SbH.

Fe(ii)-porphyrins play an important role in many reactions relevant to material science and biological processes, due to their closely lying spin states. Although the prevalent opinion is that these systems posses the triplet ground state, the recent experiment on Fe(ii)-phthalocyanine under conditions matching those of an isolated molecule points toward the quintet ground state. We present a thorough DFT and DMRG-based tailored CC study of Fe(ii)-porphyrin model, in which we address all previously discussed correlation effects. We examine the importance of geometrical parameters, the Fe-N distances in particular, and conclude that the system possesses the quintet ground state.