On the spin separation of algebraic two-component relativistic Hamiltonians: Molecular properties

Brightening triplet excitons enable high-performance white-light emission in organic small molecules via integrating n-π*/π-π* transitions

Luminescent materials that simultaneously embody bright singlet and triplet excitons hold great potential in optoelectronics, signage, and information encryption. However, achieving high-performance white-light emission is severely hampered by their inherent unbalanced contribution of fluorescence and phosphorescence. Herein, we address this challenge by pressure treatment engineering via the hydrogen bonding cooperativity effect to realize the mixture of n-π*/π-π* transitions, where the triplet state emission was boosted from 7% to 40% in isophthalic acid (IPA). A superior white-light emission based on hybrid fluorescence and phosphorescence was harvested in pressure-treated IPA, and the photoluminescence quantum yield was increased to 75% from the initial 19% (blue-light emission). In-situ high-pressure IR spectra, X-ray diffraction, and neutron diffraction reveal continuous strengthening of the hydrogen bonds with the increase of pressure. Furthermore, this enhanced hydrogen bond is retained down to the ambient conditions after pressure treatment, awarding the targeted IPA efficient intersystem crossing for balanced singlet/triplet excitons population and resulting in efficient white-light emission. This work not only proposes a route for brightening triplet states in organic small molecules, but also regulates the ratio of singlet and triplet excitons to construct high-performance white-light emission.

Theoretical study on the photophysical properties of thiophene-fused-type BODIPY series molecules in fluorescence imaging and photodynamic therapy

As a class of photosensitizers (PSs) with dual functions of photodynamic therapy (PDT) and fluorescence imaging, the relationship between the structure and dual-function of thiophene-fused-type BODIPY dyes has not been studied in depth before. We found that the thiophene-fused-type BODIPY triplet photosensitizer is produced according to the energy level matching rule and the introduction of the thiophene ring significantly reduces the energy gap ΔEST between singlet and triplet states, as revealed by our investigation of the excited state structures and energies of thieno-fused BODIPY dyes. At the same time, a tiny ΔEST also results in a greatly enhanced intersystem crossing (ISC) rate, kISC. The kISC value of MeO-BODIPY, having the highest singlet oxygen quantum yield (ΦΔ), is the largest. Substitution with a strong electron donor N,N-dimethylaminophenyl (DMA) leads to the vertical configuration in the T1 state. The small ΔE (0.0029 eV) between the HOMO and HOMO-1 triggers the photo induced electron transfer (PET) of inhibiting ISC and fluorescence. When thieno-fused BODIPYs react with pyrrole, the increase of π-conjugation and smaller ΔEHOMO-LUMO explain the redshift in emission wavelength of thieno-pyrrole-fused BODIPY. The more planar configuration of the S1 state and the stronger oscillator intensity reflect a higher fluorescence quantum yield (ΦF). The extension of π-conjugation can cause molecules to transition to higher-level singlet excited states (Sn states, n ≥ 1) after absorbing energy and reduce the energy level of the excited state, resulting in multiple channels and favoring 1O2 production for thieno-pyrrole-fused BODIPYs with electron-withdrawing groups at the para-position of the phenyl groups. Due to ΔES0-T1 < 0.980 eV, the substitution of electron-donating groups cannot produce 1O2. In this work, we have revealed the mechanism of ISC and the fluorescence emission process in the thiophene-fused-type BODIPY dye, which has provided a theoretical foundation and guidance for the future design of BODIPY-based heavy-atom-free PSs for molecular applications in PDT.

Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

Combining Functional Units to Design Organic Materials with Dynamic Room-Temperature Phosphorescence under Continuous Ultraviolet Irradiation

Developing materials with dynamic room-temperature phosphorescence (RTP) properties is crucial for expanding the applications of organic light-emitting materials. In this study, we designed and synthesized two novel RTP molecules by combining functional units, incorporating the folded unit thianthrene into the classic luminescent cores thioxanthone or anthraquinone to construct TASO and TA2O. In this combination, the TA unit contributes to the enhancement of spin-orbit coupling (SOC), while the luminescent core governs the triplet energy level. After the strategic manipulation of SOC using the thianthrene unit, the target molecules exhibited a remarkable enhancement in RTP performance. This strategy led to the successful development of TASO and TA2O molecules with outstanding dynamic RTP properties when exposed to continuous ultraviolet irradiation, a result that can be ascribed to their efficient RTP, improved absorption ability, and oxygen-sensitive RTP properties. Leveraging the oxygen-mediated ultraviolet-radiation-induced RTP enhancement in TASO-doped polymer films, we developed a novel time-resolved detection technique for identifying phase separation in polymers with varying oxygen permeability. This research offers a promising approach for constructing materials with dynamic RTP properties.

Unified construction of relativistic Hamiltonians

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree-Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems.

Block2: A comprehensive open source framework to develop and apply state-of-the-art DMRG algorithms in electronic structure and beyond

block2 is an open source framework to implement and perform density matrix renormalization group and matrix product state algorithms. Out-of-the-box it supports the eigenstate, time-dependent, response, and finite-temperature algorithms. In addition, it carries special optimizations for ab initio electronic structure Hamiltonians and implements many quantum chemistry extensions to the density matrix renormalization group, such as dynamical correlation theories. The code is designed with an emphasis on flexibility, extensibility, and efficiency and to support integration with external numerical packages. Here, we explain the design principles and currently supported features and present numerical examples in a range of applications.

Zero-field splitting parameters within exact two-component theory and modern density functional theory using seminumerical integration

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin-dipole contribution and the response equations of the spin-orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn-Sham equations for the spin-orbit perturbation term. Second, the spin-orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin-orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin-orbit mean field Ansatz.

When do tripdoublet states fluoresce? A theoretical study of copper(II) porphyrin

Open-shell molecules rarely fluoresce, due to their typically faster non-radiative relaxation rates compared to closed-shell ones. Even rarer is the fluorescence from states that have two more unpaired electrons than the open-shell ground state, since they involve excitations from closed-shell orbitals to vacant-shell orbitals, which are typically higher in energy compared to excitations from or out of open-shell orbitals. States that are dominated by the former type of excitations are known as tripdoublet states when they can be described as a triplet excitation antiferromagnetically coupled to a doublet state, and their description by unrestricted single-reference methods (e.g., U-TDDFT) is notoriously inaccurate due to large spin contamination. In this work, we applied our spin-adapted TDDFT method, X-TDDFT, and the efficient and accurate static-dynamic-static second order perturbation theory (SDSPT2), to the study of the excited states as well as their relaxation pathways of copper(II) porphyrin; previous experimental works suggested that the photoluminescence of some substituted copper(II) porphyrins originate from a tripdoublet state, formed by a triplet ligand π → π* excitation antiferromagnetically coupled with the unpaired d electron. Our results demonstrated favorable agreement between the X-TDDFT, SDSPT2 and experimental excitation energies, and revealed noticeable improvements of X-TDDFT compared to U-TDDFT, not only for vertical excitation energies but also for adiabatic energy differences. These suggest that X-TDDFT is a reliable tool for the study of tripdoublet state fluorescence. Intriguingly, we showed that the aforementioned tripdoublet state is only slightly above the lowest doublet excited state and lies only slightly higher than the lowest quartet state, which suggests that the tripdoublet of copper(II) porphyrin is long-lived enough to fluoresce due to a lack of efficient non-radiative relaxation pathways; an explanation for this unusual state ordering is given. Indeed, thermal vibration correlation function (TVCF)-based calculations of internal conversion, intersystem crossing, and radiative transition rates confirm that copper(II) porphyrin emits thermally activated delayed fluorescence (TADF) and a small amount of phosphorescence at low temperature (83 K), in accordance with experiment. The present contribution is concluded by a few possible approaches of designing new molecules that fluoresce from tripdoublet states.

SOiCISCF: Combining SOiCI and iCISCF for Variational Treatment of Spin-Orbit Coupling

It has recently been shown that the SOiCI approach [Zhang, N.; J. Phys.: Condens. Matter 2022, 34, 224007], in conjunction with the spin-separated exact two-component relativistic Hamiltonian, can provide very accurate fine structures of systems containing heavy elements by treating electron correlation and spin-orbit coupling (SOC) on an equal footing. Nonetheless, orbital relaxations/polarizations induced by SOC are not yet fully accounted for due to the use of scalar relativistic orbitals. This issue can be resolved by further optimizing the still real-valued orbitals self-consistently in the presence of SOC, as done in the spin-orbit coupled CASSCF approach [Ganyushin, D.; et al. J. Chem. Phys. 2013, 138, 104113] but with the iCISCF algorithm [Guo, Y.; J. Chem. Theory Comput. 2021, 17, 7545-7561] for large active spaces. The resulting SOiCISCF employs both double group and time reversal symmetries for computational efficiency and the assignment of target states. The fine structures of p-block elements are taken as showcases to reveal the efficacy of SOiCISCF.

Simulating Spin-Orbit Coupling with Quasidegenerate N-Electron Valence Perturbation Theory

We present the first implementation of spin-orbit coupling effects in fully internally contracted second-order quasidegenerate N-electron valence perturbation theory (SO-QDNEVPT2). The SO-QDNEVPT2 approach enables the computations of ground- and excited-state energies and oscillator strengths combining the description of static electron correlation with an efficient treatment of dynamic correlation and spin-orbit coupling. In addition to SO-QDNEVPT2 with the full description of one- and two-body spin-orbit interactions at the level of two-component Breit-Pauli Hamiltonian, our implementation also features a simplified approach that takes advantage of spin-orbit mean-field approximation (SOMF-QDNEVPT2). The accuracy of these methods is tested for the group 14 and 16 hydrides, 3d and 4d transition metal ions, and two actinide dioxides (neptunyl and plutonyl dications). The zero-field splittings of group 14 and 16 molecules computed using SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the available experimental data. For the 3d transition metal ions, the SO-QDNEVPT2 method is significantly more accurate than SOMF-QDNEVPT2, while no substantial difference in the performance of two methods is observed for the 4d ions. Finally, we demonstrate that for the actinide dioxides the results of SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the data from previous theoretical studies of these systems. Overall, our results demonstrate that SO-QDNEVPT2 and SOMF-QDNEVPT2 are promising multireference methods for treating spin-orbit coupling with a relatively low computational cost.

Self-Adaptive Real-Time Time-Dependent Density Functional Theory for X-ray Absorptions

Real-time time-dependent density functional theory (RT-TDDFT) can in principle access the whole absorption spectrum of a many-electron system exposed to a narrow pulse. However, this requires an accurate and efficient propagator for the numerical integration of the time-dependent Kohn-Sham equation. While a low-order time propagator is already sufficient for the low-lying valence absorption spectra, it is no longer the case for the X-ray absorption spectra (XAS) of systems composed even only of light elements, for which the use of a high-order propagator is indispensable. It is then crucial to choose a largest possible time step and a shortest possible simulation time, so as to minimize the computational cost. To this end, we propose here a robust AutoPST approach to determine automatically (Auto) the propagator (P), step (S), and time (T) for relativistic RT-TDDFT simulations of XAS.

Paramagnetic NMR Shielding Tensors Based on Scalar Exact Two-Component and Spin-Orbit Perturbation Theory

The temperature-dependent Fermi-contact and pseudocontact terms are important contributions to the paramagnetic NMR shielding tensor. Herein, we augment the scalar-relativistic (local) exact two-component (X2C) framework with spin-orbit perturbation theory including the screened nuclear spin-orbit correction for the EPR hyperfine coupling and g tensor to compute these temperature-dependent terms. The accuracy of this perturbative ansatz is assessed with the self-consistent spin-orbit two-component and four-component treatments serving as reference. This shows that the Fermi-contact and pseudocontact interaction is sufficiently described for paramagnetic NMR shifts; however, larger deviations are found for the EPR spectra and the principle components of the EPR properties of heavy elements. The impact of the perturbative treatment is further compared to that of the density functional approximation and the basis set. Large-scale calculations are routinely possible with the multipole-accelerated resolution of the identity approximation and the seminumerical exchange approximation, as shown for [CeTi₆O₃(OiPr)₉(salicylate)₆].

The Chemistry of Mercury in the Stratosphere

Mercury, a global contaminant, enters the stratosphere through convective uplift, but its chemical cycling in the stratosphere is unknown. We report the first model of stratospheric mercury chemistry based on a novel photosensitized oxidation mechanism. We find two very distinct Hg chemical regimes in the stratosphere: in the upper stratosphere, above the ozone maximum concentration, Hg⁰ oxidation is initiated by photosensitized reactions, followed by second-step chlorine chemistry. In the lower stratosphere, ground-state Hg⁰ is oxidized by thermal reactions at much slower rates. This dichotomy arises due to the coincidence of the mercury absorption at 253.7 nm with the ozone Hartley band maximum at 254 nm. We also find that stratospheric Hg oxidation, controlled by chlorine and hydroxyl radicals, is much faster than previously assumed, but moderated by efficient photo-reduction of mercury compounds. Mercury lifetime shows a steep increase from hours in the upper-middle stratosphere to years in the lower stratosphere.

SOiCI and iCISO: combining iterative configuration interaction with spin-orbit coupling in two ways

The near-exact iCIPT2 approach for strongly correlated systems of electrons, which stems from the combination of iterative configuration interaction (iCI, an exact solver of full CI) with configuration selection for static correlation and second-order perturbation theory (PT2) for dynamic correlation, is extended to the relativistic domain. In the spirit of spin separation, relativistic effects are treated in two steps: scalar relativity is treated by the infinite-order, spin-free part of the exact two-component (X2C) relativistic Hamiltonian, whereas spin-orbit coupling (SOC) is treated by the first-order, Douglas-Kroll-Hess-like SOC operator derived from the same X2C Hamiltonian. Two possible combinations of iCIPT2 with SOC are considered, i.e., SOiCI and iCISO. The former treats SOC and electron correlation on an equal footing, whereas the latter treats SOC in the spirit of state interaction, by constructing and diagonalizing an effective spin-orbit Hamiltonian matrix in a small number of correlated scalar states. Both double group and time reversal symmetries are incorporated to simplify the computation. Pilot applications reveal that SOiCI is very accurate for the spin-orbit splitting (SOS) of heavy atoms, whereas the computationally very cheap iCISO can safely be applied to the SOS of light atoms and even of systems containing heavy atoms when SOC is largely quenched by ligand fields.

Folding-Induced Spin-Orbit Coupling Enhancement for Efficient Pure Organic Room-Temperature Phosphorescence

For the direct luminescence of triplet excitons, different mechanisms have been proposed for realizing pure organic room-temperature phosphorescence (RTP). To further verify the mechanism of folding-induced spin-orbit coupling (SOC) enhancement, two analogues of thianthrene (TA) were introduced by gradually replacing the sulfur atom with an oxygen atom for a systematical comparison, corresponding to phenoxathiine (PX) and dioxins (DX) molecules with increasing folding dihedral angles (or decreasing degrees of folding). Photophysical measurements show an obviously enhanced RTP efficiency from DX and PX to TA, which is consistent with their greatly enhanced SOC with a decrease in folding dihedral angle. The folding angle-dependent SOC calculations for each molecule reveal that this enhanced RTP is dominated by folding-induced SOC enhancement, in contrast with the negligible heavy-atom effect from oxygen to sulfur. This work further validates the rationality of the folding-induced SOC enhancement mechanism, which provides an innovative molecular design strategy for developing efficient pure organic RTP materials using folding structures.

Efficient Room-Temperature Phosphorescence from Discrete Molecules Based on Thianthrene Derivatives for Oxygen Sensing and Detection

In this work, two thianthrene (TA) derivatives, 1-phenylthianthrene (TA1P) and 2-phenylthianthrene (TA2P), were synthesized with single-phenyl modification for pure organic discrete-molecule room-temperature phosphorescence (RTP). They both show the dual emission of fluorescence and RTP in amorphous polymer matrix after deoxygenation, as a result of a new mechanism of folding-induced spin-orbit coupling (SOC) enhancement. Compared with TA1P, TA2P exhibits a higher RTP efficiency and a larger spectral separation between fluorescence and RTP, which is ascribed to the substituent effect of TA at the 2-position. With decreasing oxygen concentration from 1.61% to 0%, the discrete-molecule TA2P shows an about 18-fold increase in RTP intensity and an almost constant fluorescence intensity, which can make TA2P as a self-reference ratiometric optical oxygen sensing probe at low oxygen concentrations. The oxygen quenching constant (K_SV) of TA2P is estimated as high as 10.22 KPa⁻¹ for polymethyl methacrylate (PMMA)-doped film, and even reach up to 111.86 KPa⁻¹ for Zeonex^®-doped film, which demonstrates a very high sensitivity in oxygen sensing and detection. This work provides a new idea to design pure organic discrete-molecule RTP materials with high efficiency, and TA derivatives show a potential to be applied in quantitative detection of oxygen as a new-generation optical oxygen-sensing material.

The effect of heavy atoms replacement sites on the luminescent ways of D-A-D type diphenyl sulfone molecules: Thermally activated delayed fluorescence and phosphorescence

To obtain efficient pure organic thermally activated delayed fluorescence (TADF) materials, introducing non-metal heavy atoms is the common molecular design strategy, enhancing the intrinsically weak spin-orbit coupling (SOC) between singlet and triplet excited states by heavy-atom effect. However, the effect of heavy atom replacement sites is rarely explored. Herein, two series of molecules are investigated on the basis of different heavy atoms replacement sites to reveal the inherent structure-property relationships. The results show that DMSeC-DPS, which O is replaced with Se in periphery of donor units, could exhibit enhanced TADF performance. Because (i) sufficiently small singlet-triplet states energy gap (ΔE_ST) and enhanced SOC as well as mixed CT/LE character in T₁ state could facilitate reverse intersystem crossing process, and (ii) non-radiative consumption are decreased for S₁→S₀ transition. Additionally, replacement of As at the connection site between donor and acceptor units folds evidently the geometry, leading to much larger ΔE_ST and enhanced exponentially SOC between T₁ and S₀ state due to the great participation of heavy atoms of the frontier molecules orbitals and heavy-atom effect. The pure LE character leads to relative stability and slight non-radiative consumption in T₁ state. The luminescent way of DMOC-As-DPS would be transformed to phosphorescence. This work provides updated theoretical perspective for the effect of heavy atoms replacement sites and proposes a design strategy for the utilization of non-metal heavy atoms in efficiency organic lighting emitting diodes.

The Static-Dynamic-Static Family of Methods for Strongly Correlated Electrons: Methodology and Benchmarking

A series of methods (SDSCI, SDSPT2, iCI, iCIPT2, iCISCF(2), iVI, and iCAS) is introduced to accurately describe strongly correlated systems of electrons. Born from the (restricted) static-dynamic-static (SDS) framework for designing many-electron wave functions, SDSCI is a minimal multireference (MR) configuration interaction (CI) approach that constructs and diagonalizes a [Formula: see text] matrix for [Formula: see text] states, regardless of the numbers of orbitals and electrons to be correlated. If the full molecular Hamiltonian H in the QHQ block (which describes couplings between functions of the first-order interaction space Q) of the SDSCI CI matrix is replaced with a zeroth-order Hamiltonian [Formula: see text] before the diagonalization is taken, we obtain SDSPT2, a CI-like second-order perturbation theory (PT2). Unlike most variants of MRPT2, SDSPT2 treats single and multiple states in the same way and is particularly advantageous in the presence of near degeneracy. On the other hand, if the SDSCI procedure is repeated until convergence, we will have iterative CI (iCI), which can converge quickly from the above to the exact solutions (full CI) even when starting with a poor guess. When further combined with the selection of important configurations followed by a PT2 treatment of dynamic correlation, iCI becomes iCIPT2, which is a near-exact theory for medium-sized systems. The microiterations of iCI for relaxing the coefficients of contracted many-electron functions can be generalized to an iterative vector interaction (iVI) approach for finding exterior or interior roots of a given matrix, in which the dimension of the search subspace is fixed by either the number of target roots or the user-specified energy window. Naturally, iCIPT2 can be employed as the active space solver of the complete active space (CAS) self-consistent field, leading to iCISCF(2), which can further be combined with iCAS for automated selection of active orbitals and assurance of the same CAS for all states and all geometries. The methods are calibrated by taking the Thiel set of benchmark systems as examples. Results for the corresponding cations, a new set of benchmark systems, are also reported.

NAC-TDDFT: Time-Dependent Density Functional Theory for Nonadiabatic Couplings

First-order nonadiabatic coupling (NAC) matrix elements (fo-NACMEs) are the basic quantities in theoretical descriptions of electronically nonadiabatic processes that are ubiquitous in molecular physics and chemistry. Given the large size of systems of chemical interests, time-dependent density functional theory (TDDFT) is usually the first choice of methods. However, the lack of many-electron wave functions in TDDFT renders the formulation of NAC-TDDFT for fo-NACMEs conceptually difficult. Because of this, various variants of NAC-TDDFT have been proposed in the literature from different standing points, including the Hellmann-Feynman-like expression and auxiliary/pseudo-wave function (AWF)-, equation-of-motion (EOM)-, and time-dependent perturbation theory (TDPT)-based formulations. Based on critical analyses, the following conclusions are made here: (1) The Hellmann-Feynman-like expression, which is rooted in exact wave function theory, is hardly useful due to huge demand on basis sets. (2) Although most popular, the AWF variants of NAC-TDDFT are not theoretically founded and become ambiguous particularly for the fo-NACMEs between two excited states, although they do agree with the EOM and TDPT variants under the Tamm-Dancoff approximation. (3) The TDPT variant of NAC-TDDFT is theoretically most rigorous but suffers from numerical instabilities on the one hand and does not differ to a significant extent from the EOM variant on the other hand. (4) As such, the EOM variant of NAC-TDDFT for the fo-NACMEs between the ground and excited states and between two excited states is solely the right choice in practice. These formal analyses are fully supported by numerical experimentations, taking azulene as a showcase. The proper implementation of the EOM variant of NAC-TDDFT is also highlighted, showing that the fo-NACMEs between the ground and excited states and between two excited states are computationally very much the same as the analytic energy gradients of DFT and TDDFT, respectively. Possible future developments of the EOM variant of NAC-TDDFT are also highlighted. Its extensions to spin-adapted open-shell TDDFT and proper treatment of spin-orbit couplings (which are another source of force for electronically nonadiabatic processes) are particularly warranted in the near future.

Dynamic-then-Static Approach for Core Excitations of Open-Shell Molecules

Delta self-consistent-field methods are widely used in studies of electronically excited states. However, the nonaufbau determinants are generally spin-contaminated. Here, we describe a general approach for spin-coupling interactions of open-shell molecules, making use of multistate density functional theory (MSDFT). In particular, the effective exchange integrals that determine spin coupling are obtained by enforcing the multiplet degeneracy of the S+1 state in the M_S = S manifold. Consequently, they are consistent with the energy of the high-spin state that is adequately treated by Kohn-Sham density functional theory (DFT) and, thereby, free of double counting of correlation. The method was applied to core excitations of open-shell molecules and compared with those by spin-adapted time-dependent DFT. An excellent agreement with experiment was found employing the BLYP functional and aug-cc-pCVQZ basis set. Overall, MSDFT provides an effective combination of the strengths of DFT and wave function theory to achieve efficiency and accuracy.

iCAS: Imposed Automatic Selection and Localization of Complete Active Spaces

It is shown that in the spirit of "from fragments to molecule" for localizing molecular orbitals [J. Chem. Theory Comput. 2011, 7, 3643], a prechosen set of occupied/virtual valence/core atomic/fragmental orbitals can be transformed to an equivalent set of localized occupied/virtual pre-localized molecular orbitals (pre-LMO), which can then be taken as probes to select the same number of maximally matching localized occupied/virtual Hartree-Fock (HF) or restricted open-shell HF (ROHF) molecular orbitals as the initial local orbitals spanning the desired complete active space (CAS). In each cycle of the self-consistent field (SCF) calculation, the CASSCF orbitals can be localized by means of the noniterative "top-down least-change" algorithm for localizing ROHF orbitals [J. Chem. Phys. 2017, 146, 104104] such that the maximum matching between the orbitals of two adjacent iterations can readily be monitored, leading finally to converged localized CASSCF orbitals that overlap most the guess orbitals. Such an approach is to be dubbed as "imposed CASSCF" (iCASSCF or simply iCAS in short) for good reasons: (1) it has been assumed that only those electronic states that have largest projections onto the active space defined by the prechosen atomic/fragmental orbitals are to be targeted. This is certainly an imposed constraint but has wide applications in organic and transition metal chemistry where valence (or core) atomic/fragmental orbitals can readily be identified. (2) The selection of both initial and optimized local active orbitals is imposed from the very beginning by the pre-LMOs (which span the same space as the prechosen atomic/fragmental orbitals). Apart from the (imposed) automation and localization, iCAS has two additional merits: (a) the guess orbitals are guaranteed to be the same for all geometries, for the pre-LMOs do not change in character with geometry and (b) the use of localized orbitals facilitates the SCF convergence, particularly for large active spaces. Both organic molecules and transition-metal complexes are taken as showcases to reveal the efficacy of iCAS.

Analytic evaluation of energy first derivatives for spin-orbit coupled-cluster singles and doubles augmented with noniterative triples method: General formulation and an implementation for first-order properties

A formulation of analytic energy first derivatives for the coupled-cluster singles and doubles augmented with noniterative triples [CCSD(T)] method with spin-orbit coupling included at the orbital level and an implementation for evaluation of first-order properties are reported. The standard density-matrix formulation for analytic CC gradient theory adapted to complex algebra has been used. The orbital-relaxation contributions from frozen core, occupied, virtual, and frozen virtual orbitals to analytic spin-orbit CCSD(T) gradients are fully taken into account and treated efficiently, which is of importance to calculations of heavy elements. Benchmark calculations of first-order properties including dipole moments and electric-field gradients using the corresponding exact two-component property integrals are presented for heavy-element containing molecules to demonstrate the applicability and usefulness of the present analytic scheme.

Further Development of iCIPT2 for Strongly Correlated Electrons

The efficiency of the recently proposed iCIPT2 [iterative configuration interaction (iCI) with selection and second-order perturbation theory (PT2); J. Chem. Theory Comput. 2020, 16, 2296] for strongly correlated electrons is further enhanced (by up to 20×) by using (1) a new ranking criterion for configuration selection, (2) a new particle-hole algorithm for Hamiltonian construction over randomly selected configuration state functions (CSF), and (3) a new data structure for the quick sorting of the variational and first-order interaction spaces. Meanwhile, the memory requirement is also significantly reduced. As a result, this improved implementation of iCIPT2 can handle 1 order of magnitude more CSFs than the previous version, as revealed by taking the chromium dimer and an iron-sulfur cluster, [Fe₂S₂(SCH₃)]₄^2-, as examples.

Suppressing triplet decay in quinoidal singlet fission materials: the role of molecular planarity and rigidity

Singlet fission, in which one singlet exciton is split into two triplet excitons, provides the potential to exceed the Shockley-Queisser limit for the power conversion efficiencies of organic solar cells. However, the charge transfer from the triplet state is found to be slow in singlet fission materials, so suppression of the triplet decay is crucial for effective utilization of singlet fission. Here, we first investigated triplet decay for the singlet fission molecular materials of ThBF and TThBF, which are characteristic of twisted and flexible quinoidal backbones. It is found that these compounds show rapid nonradiative decay in the Franck-Condon region and through the T₁/S₀ crossing point. Interestingly, upon locking the backbone twist by methylene, the LThBF and LTThBF compounds exhibit much higher energy barriers from T₁ to the T₁/S₀ crossing point, vanishing spin-orbit couplings, and decreased reorganization energies due to the planar and rigid structures. Consequently, both the triplet decay pathways are effectively suppressed. Our work reveals the importance of molecular planarity and rigidity in suppressing triplet decay and will be very helpful for full utilization of singlet fission in organic photovoltaics.

Investigation of different photochemical reactions of avobenzone derivatives by ultrafast transient absorption spectroscopy

Avobenzone (AB) is one of the most widely used UVA sunscreens, and it is viewed as a model compound for studying the photoisomerization process. In recent years, Miranda and co-workers studied photophysical and photochemical reactions of several AB derivatives. However, there is still a gap in the data of these compounds in the ultrafast time region. To get a better understanding of the photophysical and photochemical reaction mechanisms, selected AB derivatives of AB-Me, AB-Pr, AB-Br and AB-Cl were studied using ultrafast transient absorption spectroscopy and density functional theory calculations in the present study. It is unravelled that alkylated substituted AB compounds of AB-Me and AB-Pr exhibit an efficient intersystem crossing with the generation of the corresponding triplet state species, which further leads to the Norrish type II reaction for AB-Pr. On the other hand, AB-Br and AB-Cl prefer photochemical reactions via the singlet state surface. Based on the DFT calculations, the spin-orbit coupling constant between the singlet and triplet states, the energy difference between the singlet and triplet states and the natural transition orbital separations of the studied AB compounds were found to be the leading reasons accounting for their corresponding photochemical activities via singlet and triplet states.

Density Functional Calculations of Electron Paramagnetic Resonance g- and Hyperfine-Coupling Tensors Using the Exact Two-Component (X2C) Transformation and Efficient Approximations to the Two-Electron Spin-Orbit Terms

A two-component quasirelativistic density functional theory implementation of the computation of hyperfine and g-tensors at exact two-component (X2C) and Douglas-Kroll-Hess method (DKH) levels in the Turbomole code is reported and tested for a series of smaller 3d¹, 4d¹, and 5d¹ complexes, as well as for some larger 5d⁷ Ir and Pt systems in comparison with earlier four-component matrix-Dirac-Kohn-Sham results. A main emphasis is placed on efficient approximations to the two-electron spin-orbit contributions, comparing an existing implementation of two variants of Boettger's "scaled nuclear spin-orbit" (SNSO) approximation in the code with a newly implemented atomic mean-field spin-orbit (AMFSO) approximation. The different variants perform overall comparably well with the four-component data. The AMFSO approximation has the added advantage of being able to include the spin-other-orbit contributions arising from the Gaunt term of relativistic electron-electron interactions. These are of comparably larger importance for the 3d complexes than for their heavier homologues. The excellent agreement between X2C and four-component electron paramagnetic resonance parameter results provides the opportunity to treat large systems efficiently and accurately with the computationally more expedient two-component quasirelativistic methodology.

Relativistic time-dependent density functional theories

The foundations, formalisms, technicalities, and practicalities of relativistic time-dependent density functional theories (R-TD-DFT) for spinor excited states of molecular systems containing heavy elements are critically reviewed.

Combining the spin-separated exact two-component relativistic Hamiltonian with the equation-of-motion coupled-cluster method for the treatment of spin–orbit splittings of light and heavy elements

An accurate and efficient treatment of spin–orbit splittings has been achieved by combining the sf-X2C+soc-DKH1 Hamiltonian with the equation-of-motion coupled-cluster method.

Big picture of relativistic molecular quantum mechanics

Any quantum mechanical calculation on electronic structure ought to choose first an appropriate Hamiltonian H and then an Ansatz for parameterizing the wave function Ψ, from which the desired energy/property E(λ) can finally be calculated. Therefore, the very first question is: what is the most accurate many-electron Hamiltonian H? It is shown that such a Hamiltonian i.e. effective quantum electrodynamics (eQED) Hamiltonian, can be obtained naturally by incorporating properly the charge conjugation symmetry when normal ordering the second quantized fermion operators. Taking this eQED Hamiltonian as the basis, various approximate relativistic many-electron Hamiltonians can be obtained based entirely on physical arguments. All these Hamiltonians together form a complete and continuous ‘Hamiltonian ladder’, from which one can pick up the right one according to the target physics and accuracy. As for the many-electron wave function Ψ, the most intriguing questions are as follows. (i) How to do relativistic explicit correlation? (ii) How to handle strong correlation? Both general principles and practical strategies are outlined here to handle these issues. Among the electronic properties E(λ) that sample the electronic wave function nearby the nuclear region, nuclear magnetic resonance (NMR) shielding and nuclear spin-rotation (NSR) coupling constant are especially challenging: they require body-fixed molecular Hamiltonians that treat both the electrons and nuclei as relativistic quantum particles. Nevertheless, they have been formulated rigorously. In particular, a very robust ‘relativistic mapping’ between the two properties has been established, which can translate experimentally measured NSR coupling constants to very accurate absolute NMR shielding scales that otherwise cannot be obtained experimentally. Since the most general and fundamental issues pertinent to all the three components of the quantum mechanical equation HΨ = EΨ (i.e. Hamiltonian H, wave function Ψ, and energy/property E(λ)) have fully been understood, the big picture of relativistic molecular quantum mechanics can now be regarded as established.