Entanglement entropy of electronic excitations

Coupled-cluster treatment of complex open-shell systems: the case of single-molecule magnets

We investigate the reliability of two cost-effective coupled-cluster methods for computing spin-state energetics and spin-related properties of a set of open-shell transition-metal complexes. Specifically, we employ the second-order approximate coupled-cluster singles and doubles (CC2) method and projection-based embedding that combines equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) with density functional theory (DFT). The performance of CC2 and EOM-CCSD-in-DFT is assessed against EOM-CCSD. The chosen test set includes two hexaaqua transition-metal complexes containing Fe(II) and Fe(III), and a large Co(II)-based single-molecule magnet with a non-aufbau ground state. We find that CC2 describes the excited states more accurately, reproducing EOM-CCSD excitation energies within 0.05 eV. However, EOM-CCSD-in-DFT excels in describing transition orbital angular momenta and spin-orbit couplings. Moreover, for the Co(II) molecular magnet, using EOM-CCSD-in-DFT eigenstates and spin-orbit couplings, we compute spin-reversal energy barriers, as well as temperature-dependent and field-dependent magnetizations and magnetic susceptibilities that closely match experimental values within spectroscopic accuracy. These results underscore the efficiency of CC2 in computing state energies of multi-configurational, open-shell systems and highlight the utility of the more cost-efficient EOM-CCSD-in-DFT for computing spin-orbit couplings and magnetic properties of complex and large molecular magnets.

Molecular-Orbital Framework of Two-Electron Processes: Application to Auger and Intermolecular Coulomb Decay

States with core- or inner-shell vacancies, which are commonly created by absorption of high-energy photons, can decay by a two-electron process in which one electron fills the core hole and the second one is ejected. These processes accompany many X-ray spectroscopies. Depending on the nature of the initial core- or inner-shell-hole state and the decay valence-hole state, these processes are called Auger decay, intermolecular Coulomb decay, or electron-transfer-mediated decay. To connect many-body wave functions of the initial and final states with the molecular orbital picture of the decay, we introduce the concept of natural Auger orbitals (NAOs). NAOs are obtained by a two-step singular value decomposition of the two-body Dyson orbitals, reduced quantities that enter the expression of the decay rate in the Feshbach-Fano treatment. NAOs afford chemical insight and interpretation of the high-level ab initio calculations of Auger decay and related two-electron relaxation processes.

The inverted singlet-triplet gap: a vanishing myth?

Molecules with an inverted singlet-triplet gap (STG) between the first excited singlet and triplet states, for example, heptazine, have recently been reported and gained substantial attention since they violate the famous Hund's rule. Utilizing state-of-the-art high-level ab initio methods, the singlet-triplet gap vanishes and approaches zero from below whatever is improved in the theoretical description of the molecules: the basis set or the level of electron correlation. Seemingly, the phenomenon of inverted singlet-triplet gaps tends to vanish the closer we observe.

Excited states of ortho-nitrobenzaldehyde as a challenging case for single- and multi-reference electronic structure theory

We present a large set of vertical excitation calculations for the ortho-nitrobenzaldehyde (oNBA) molecule, which exhibits a very challenging excited-state electronic structure like other nitroaromatic compounds. The single-reference methods produce mostly consistent results up to about 5.5 eV. By contrast, the CAS second-order perturbation theory (CASPT2) results depend sensitively on the employed parameters. At the CAS self-consistent field level, the energies of the bright ππ * $$ {\pi \pi}^{\ast } $$ states are strongly overestimated while doubly excited states appear too low and mix with these ππ * $$ {\pi \pi}^{\ast } $$ states. This mixing hampers the CASPT2 step, leading to inconsistent results. Only by increasing the number of states in the state-averaging step to about 40-to cover all bright ππ * $$ {\pi \pi}^{\ast } $$ states embedded in the double excitations-and employing extended multistate CASPT2 could CASPT2 results consistent with experiment be obtained. We assign the four bands in the molecule's spectrum: The weakest band at 3.7 eV arises from the n NO 2 π * $$ {n}_{\mathrm{NO}2}{\pi}^{\ast } $$ states, the second one at 4.4 eV from the ππ * $$ {\pi \pi}^{\ast } $$ ( L b $$ {L}_b $$ ) state, the shoulder at 5.2 eV from the ππ * $$ {\pi \pi}^{\ast } $$ ( L a $$ {L}_a $$ ) state, and the maximum at 5.7 eV from the ππ * $$ {\pi \pi}^{\ast } $$ ( B a / B b $$ {B}_a/{B}_b $$ ) states. We also highlight the importance of modern wave function analysis techniques in elucidating the absorption spectrum of challenging molecules.

Dicarbon defect in hexagonal boron nitride monolayer—a theoretical study

A comprehensive theoretical study of the lowest electronic vertical excitations of the CBCN defect in the monolayer of hexagonal boron nitride has been performed. Both the periodic boundary conditions approach and the finite-cluster simulation of the defect have been utilized at the density-functional theory (DFT) level. Clusters of increasing sizes have been used in order to estimate artefacts resulting from edge effects. The stability of the results with respect to several density functionals and various basis sets has been also examined. High-level ab initio calculations with methods like equation-of-motion coupled cluster method with single and double excitations (EOM-CCSD), algebraic-diagrammatic construction to the second order (ADC(2)), and time-dependent approximate coupled cluster theory to the second order (TD-CC2) were performed for the smallest clusters. It turns out that TD-DFT with the CAM-B3LYP functional gives similar lowest excitation energies as EOM-CCSD, ADC(2), and TD-CC2. The lowest excitation energies resulting from the periodic-boundary calculation utilizing the Bethe–Salpeter equation are in agreement with the results for finite clusters. The analysis of important configurations and transition densities shows that for all studied methods, the lowest excited state is localized on two carbon atoms and their closest neighbours and has a large dipole transition moment. The optimized geometries for the lowest two excited states indicate that in both cases, the carbon–carbon bond becomes a single bond, while for the second excited state, additionally one from boron–nitrogen bonds loses its partially double character. The calculation of the excitation energies at the respective optimal geometry reveals that these two energies become about 0.5 eV lower than vertical excitations from the ground-state geometry.

Photophysics of Anionic Bis(4H-imidazolato)CuI Complexes

In this paper, the photophysical behavior of four panchromatically absorbing, homoleptic bis(4H-imidazolato)Cu^I complexes, with a systematic variation in the electron-withdrawing properties of the imidazolate ligand, were studied by wavelength-dependent time-resolved femtosecond transient absorption spectroscopy. Excitation at 400, 480, and 630 nm populates metal-to-ligand charge transfer, intraligand charge transfer, and mixed-character singlet states. The pump wavelength-dependent transient absorption data were analyzed by a recently established 2D correlation approach. Data analysis revealed that all excitation conditions yield similar excited-state dynamics. Key to the excited-state relaxation is fast, sub-picosecond pseudo-Jahn-Teller distortion, which is accompanied by the relocalization of electron density onto a single ligand from the initially delocalized state at Franck-Condon geometry. Subsequent intersystem crossing to the triplet manifold is followed by a sub-100 ps decay to the ground state. The fast, nonradiative decay is rationalized by the low triplet-state energy as found by DFT calculations, which suggest perspective treatment at the strong coupling limit of the energy gap law.

Atomic Group Decomposition of Charge Transfer Excitation Global Indexes

A model for decomposing the Le Bahers, Adamo, and Ciofini Charge Transfer (CT) Excitations global indexes ( J. Chem. Theory Comput. 2011, 7, 2498-2506) into molecular subdomains contributions is presented and a software, DOCTRINE (atomic group Decomposition Of the Charge TRansfer INdExes) for the implementation of this novel model has been coded. Although our method applies to any fuzzy or to any disjoint exhaustive partitioning of the real space, it is here applied using a definition of chemically relevant molecular subdomains based on the Atoms in Molecules Bader basins. This choice has the relevant advantage of associating intra or inter subdomain contributions to rigorously defined quantum objects, yet bearing a clear chemical meaning. Our method allows for a quantitative evaluation of the subdomain contributions to the charge transfer, the charge transfer excitation length and the dipole moment change upon excitation. All these global indexes may be obtained either from the electron density increment or the electron density depletion upon excitation. However, the subdomain contributions obtained from the two distributions generally differ, therefore allowing to distinguish whether the contribution to a given property of a given subdomain is dominated by one of the two distributions or if both are playing a significant role. As a toy system for the first application of our model, a typical [D-π-A, π = conjugated bridge] compound belonging to the merocyanine dyes family is selected, and the first four excited states of this compound in a strongly polar protic solvent and in a weakly polar solvent are thoroughly investigated.

Providing theoretical insight into the role of symmetry in the photoisomerization mechanism of a non-symmetric dithienylethene photoswitch

Dithienylethene (DTE) molecular photoswitches have shown to be excellent candidates in the design of efficient optoelectronic devices, due to their high photoisomerization quantum yield (QY), for which symmetry is suggested to play a crucial role. Here, we present a theoretical study on the photochemistry of a non-symmetric dithienylethene photoswitch, with a special emphasis on the effect of asymmetric substitution on the photocyclization and photoreversion mechanisms. We used the Spin-Flip Time Dependent Density Functional Theory (SF-TDDFT) method to locate and characterize the main structures (conical intersections and minima) of the ground state and the first two excited states, S₁ and S₂, along the ring-opening/closure reaction coordinate of the photocyclization and photoreversion processes, and to identify the important coordinates governing the radiationless decay pathways. Our results suggest that while the main features that characterize the photoisomerization of symmetric DTEs are also present for the photoisomerization of the non-symmetric DTE, the lower energy barrier on S₁ along the cycloreversion reaction speaks in favor of a more efficient and therefore a higher cycloreversion QY for the non-symmetric DTEs, making them a better candidate for molecular optoelectronic devices than their symmetric counterparts.

Effects of Heterocyclic Ring Fusion and Chain Elongation on Chiroptical Properties of Polyaza[9]helicene: A Computational Study

In the present work, the effect of lateral and helical extensions on the physical and chiroptical properties of azahelicenes is reported. Starting with the experimentally reported polyaza[9]helicene (9Ha), three derivatives, two with laterally fused electron-withdrawing rings and the third with larger helical length, are designed. For the excited-state properties such as UV-vis and CD spectra, performances of different DFT functionals are evaluated by comparing the energies and characters of the excited states against the ADC(2) results. CPL properties are calculated at DFT level. Among the three designed systems, pyrazine-based 9HaP shows an improved g_CPL value compared to that for parent 9Ha. However, quinoxaline-based 9HaQ is found to be the worst CPL emitter with the lowest dissymmetry factor. The helically extended derivative, 11Ha, shows good CPL results, but g_CPL remains smaller than that for the parent system. The CPL results are analyzed in terms of electric dipole transition moment (EDTM) and magnetic dipole transition moment (MDTM) vectors, and angles between these two vectors.

Substituent effects on the UV-visible spectrum and excited electronic states of dithiocarboxylates

The absorption spectra of a series of dithiocarboxylates were investigated in the ultraviolet-visible region. Two questions that this study aimed to address were as follows: (1) What transitions give rise to the features in the electronic spectra? And (2) what are the long- and short-range substituent effects on the absorption spectra? A series of 11 dithiocarboxylates were prepared as organic soluble salts. Time-dependent density functional theory (TDDFT) was used to calculate excited state energies and oscillator strengths of electronic transitions. TDDFT at the CAM-B3LYP/def2-TZVPD level of theory predicts two low-energy n → π* transitions and two π → π* transitions at higher energy, consistent with the experimental spectra. This state ordering and density is in contrast to the better studied thiocarbonyls for which only two transitions within the singlet manifold appear in the UV-visible region. For derivatives of dithiobenzoate, the energy of the three lowest energy states are insensitive to changes to substituents para to the dithiocarboxylate group. In contrast, the energy of the highest ππ* state varies by 0.78 eV. This work shows that the results of TDDFT calculations can be used to predict the electronic absorption spectra of dithiocarboxylates, providing a useful tool for designing dithiocarboxylate light absorbers.

Energy Landscape of State-Specific Electronic Structure Theory

State-specific approximations can provide a more accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere, and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated, and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree-Fock and excited-state mean-field representations of singlet H₂ (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape, and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimization landscape and elucidate the challenges faced by variance optimization algorithms, including the presence of unphysical saddle points or maxima of the variance.

Alkaline earth cations binding mode tailors excited-state charge transfer properties of guanine quadruplex: A TDDFT study

Quadruplexes formed by nucleic acids and their derivates tend to chelate different monovalent and bivalent cations, which simultaneously affect their excited electronic states properties. Cation binding to every and every other cavity of the central ion channel could be exploited for tuning exited-state charge transfer properties. In this work we utilize set of descriptors constructed on the basis of the one-electron transition density matrix obtained using linear-response TDDFT to study excited states properties of four crystallized tetramolecular quadruplexes that chelate alkaline earth cations (Ca²⁺, Sr²⁺ and Ba²⁺). Here, we show that alkaline earth cations situated at adjacent vacancies promote existence of the nucleobase-metal charge separation (CS) states, contrary to the structures with cations that occupy every second available vacancy. We argued that stabilization of these CS states is due to the strong electric field that stabilizes d orbitals of the cations which accept an excited-electron. Moreover, CS content is increased and redshifted below the first bright transition when number of the chelated cations is increased. Hydration effects stabilized CS states and increased their relative content. We also identified electron detachment states in the broad energy range for the Ca²⁺ containing system. These findings are valuable for understanding and development of the novel nanostructures based on the quadruplex scaffold with adjustable optical properties.

Self-assembly of rylene-decorated guanine ribbons on graphene surface for optoelectronic applications: a theoretical study

We are witnessing a change of paradigm from the conventional top-down to the bottom-up fabrication of nanodevices and particularly optoelectronic devices. A promising example of the bottom-up approach is self-assembling of molecules into layers with predictable and reproducible structural, electronic and optical properties. Nucleobases possess extraordinary ability to self-assembly into one-, two-, and three-dimensional structures. Optical properties of nucleotides are not suitable for wider application to optoelectronics and photovoltaics due to their large optical band gap, which is in contrast to rylene-based dyes that have been intensively investigated in organic optoelectronics. However, these lack the self-assembly capability of nucleobases. Combinations of covalently decorated guanine molecules with rylene type chromophores present 'the best of the both worlds'. Due to the large size of such compounds and its flexible nature their self-assemblies have not been fully understood yet. Here, we use a theoretical approach to study the structural, energetic and optical properties of rylene-based dye decorated guanine (GPDI), as self-assembled on a graphene sheet. Particularly we utilize the density-functional based tight-binding method to study atomic structure of these systems including the potential energy surface of GPDI and stability and organization of single- and multilayered GPDIs on graphene sheet. Using density-functional theory (DFT) we employ the energy decomposition analysis to gain a deeper insight into the contributions of different moieties to stability of GPDI films. Using time dependent DFT we analyze optical properties of these systems. We find that atomically thin films consisting of only a few molecular layers with large surface areas are more favorable than isolated thick islands. Our study of excited states indicates existence of charge separated states similar to ones found in the well-studied hydrogen bonded organic frameworks. The self-assembly characterized with a large homogeneous coverage and long-living charge-separated states provide the great potential for optoelectronic applications.

Exploitation of Baird Aromaticity and Clar’s Rule for Tuning the Triplet Energies of Polycyclic Aromatic Hydrocarbons

Polycyclic aromatic hydrocarbons (PAH) are a prominent substance class with a variety of applications in molecular materials science. Their electronic properties crucially depend on the bond topology in ways that are often highly non-intuitive. Here, we study, using density functional theory, the triplet states of four biphenylene-derived PAHs finding dramatically different triplet excitation energies for closely related isomeric structures. These differences are rationalised using a qualitative description of Clar sextets and Baird quartets, quantified in terms of nucleus independent chemical shifts, and represented graphically through a recently developed method for visualising chemical shielding tensors (VIST). The results are further interpreted in terms of a 2D rigid rotor model of aromaticity and through an analysis of the natural transition orbitals involved in the triplet excited states showing good consistency between the different viewpoints. We believe that this work constitutes an important step in consolidating these varying viewpoints of electronically excited states.

Equation-of-motion coupled-cluster method with double electron-attaching operators: Theory, implementation, and benchmarks

We report a production-level implementation of the equation-of-motion (EOM) coupled-cluster (CC) method with double electron-attaching (DEA) EOM operators of 2p and 3p1h types, EOM-DEA-CCSD. This ansatz, suitable for treating electronic structure patterns that can be described as two-electrons-in-many orbitals, represents a useful addition to the EOM-CC family of methods. We analyze the performance of EOM-DEA-CCSD for energy differences and molecular properties. By considering reduced quantities, such as state and transition one-particle density matrices, we compare EOM-DEA-CCSD wave functions with wave functions computed by other EOM-CCSD methods. The benchmarks illustrate that EOM-DEA-CCSD is capable of treating diradicals, bond-breaking, and some types of conical intersections.

Assessing challenging intra- and inter-molecular charge-transfer excitations energies with double-hybrid density functionals

We investigate the performance of a set of recently introduced range-separated double-hybrid functionals, namely ωB2-PLYP, ωB2GP-PLYP, RSX-0DH, and RSX-QIDH models for hard-to-calculate excitation energies. We compare with the parent (B2-PLYP, B2GP-PLYP, PBE0-DH, and PBE-QIDH) and other (DSD-PBEP86) double-hybrid models as well as with some of the most widely employed hybrid functionals (B3LYP, PBE0, M06-2X, and ωB97X). For this purpose, we select a number of medium-sized intra- and inter-molecular charge-transfer excitations, which are known to be challenging to calculate using time-dependent density-functional theory (TD-DFT) and for which accurate reference values are available. We assess whether the high accuracy shown by the newest double-hybrid models is also confirmed for those cases too. We find that asymptotically corrected double-hybrid models yield a superior performance, especially for the inter-molecular charge-transfer excitation energies, as compared to standard double-hybrid models. Overall, the PBE-QIDH and its corresponding range-separated RSX-QIDH functional are recommended for general-purpose TD-DFT applications, depending on whether long-range effects are expected to play a significant role.

A simple molecular orbital picture of RIXS distilled from many-body damped response theory

Ab initio calculations of resonant inelastic x-ray scattering (RIXS) often rely on damped response theory, which prevents the divergence of response solutions in the resonant regime. Within the damped response theory formalism, RIXS moments are expressed as the sum over all electronic states of the system [sum-over-states (SOS) expressions]. By invoking resonance arguments, this expression can be reduced to a few terms, an approximation commonly exploited for the interpretation of computed cross sections. We present an alternative approach: a rigorous formalism for deriving a simple molecular orbital picture of the RIXS process from many-body calculations using the damped response theory. In practical implementations, the SOS expressions of RIXS moments are recast in terms of matrix elements between the zero-order wave functions and first-order frequency-dependent response wave functions of the initial and final states such that the RIXS moments can be evaluated using complex response one-particle transition density matrices (1PTDMs). Visualization of these 1PTDMs connects the RIXS process with the changes in electronic density. We demonstrate that the real and imaginary components of the response 1PTDMs can be interpreted as contributions of the undamped off-resonance and damped near-resonance SOS terms, respectively. By analyzing these 1PTDMs in terms of natural transition orbitals, we derive a rigorous, black-box mapping of the RIXS process into a molecular orbital picture. We illustrate the utility of the new tool by analyzing RIXS transitions in the OH radical, benzene, para-nitroaniline, and 4-amino-4'-nitrostilbene. These examples highlight the significance of both the near-resonance and off-resonance channels.

TheoDORE: A toolbox for a detailed and automated analysis of electronic excited state computations

The advent of ever more powerful excited-state electronic structure methods has led to a tremendous increase in the predictive power of computation, but it has also rendered the analysis of these computations much more challenging and time-consuming. TheoDORE tackles this problem through providing tools for post-processing excited-state computations, which automate repetitive tasks and provide rigorous and reproducible descriptors. Interfaces are available for ten different quantum chemistry codes and a range of excited-state methods implemented therein. This article provides an overview of three popular functionalities within TheoDORE, a fragment-based analysis for assigning state character, the computation of exciton sizes for measuring charge transfer, and the natural transition orbitals used not only for visualization but also for quantifying multiconfigurational character. Using the examples of an organic push-pull chromophore and a transition metal complex, it is shown how these tools can be used for a rigorous and automated assignment of excited-state character. In the case of a conjugated polymer, we venture beyond the limits of the traditional molecular orbital picture to uncover spatial correlation effects using electron-hole correlation plots and conditional densities.

Two Cycling Centers in One Molecule: Communication by Through-Bond Interactions and Entanglement of the Unpaired Electrons

Many applications in quantum information science (QIS) rely on the ability to laser-cool molecules. The scope of applications can be expanded if laser-coolable molecules possess two or more cycling centers, i.e., moieties capable of scattering photons via multiple absorption-emission events. Here we employ the equation-of-motion coupled-cluster method for double electron attachment (EOM-DEA-CCSD) to study the electronic structure of hypermetallic molecules with two alkaline-earth metals connected by an acetylene linker. The electronic structure of the molecules is similar to that of two separated alkali metals; however, the interaction between the two electrons is weak and largely dominated by through-bond interactions. The communication between the two cycling centers is quantified by the extent of the entanglement of the two unpaired electrons associated with the two cycling centers. This contribution highlights the rich electronic structure of hypermetallic molecules that may advance various applications in QIS and beyond.

Toward an understanding of electronic excitation energies beyond the molecular orbital picture

Can we gain an intuitive understanding of excitation energies beyond the molecular picture?

Directional and regioselective hole injection of spiropyran photoswitches intercalated into A/T-duplex DNA

The hole electron transfer of UV excited spiropyran intercalated in dsDNA is directional, asymmetric and regioselective, as shown by quantitative multiscale computations.

Can TDDFT Describe Excited Electronic States of Naphthol Photoacids? A Closer Look with EOM-CCSD

The ¹L_b and ¹L_a excited states of naphthols are characterized by using time-dependent density functional theory (TDDFT), configuration interaction with singles (CIS), and equation-of-motion coupled cluster singles and doubles (EOM-CCSD) methods. TDDFT fails dramatically at predicting the energy and ordering of the ¹L_a and ¹L_b excited states as observed experimentally, while EOM-CCSD accurately predicts the excited states as characterized by natural transition orbital analysis. The limitations of TDDFT are attributed to the absence of correlation from doubly excited configurations as well as the inconsistent description of excited electronic states of naphthol photoacids revealed by excitation analysis based on the one-electron transition density matrix.

Universal Exciton Size in Organic Polymers is Determined by Nonlocal Orbital Exchange in Time-Dependent Density Functional Theory

The exciton size of the lowest singlet excited state in a diverse set of organic π-conjugated polymers is studied and found to be a universal, system-independent quantity of approximately 7 Å in the single-chain picture. With time-dependent density functional theory (TDDFT), its value as well as the overall description of the exciton is almost exclusively governed by the amount of nonlocal orbital exchange. This is traced back to the lack of the Coulomb attraction between the electron and hole quasiparticles in pure TDDFT, which is reintroduced only with the admixture of nonlocal orbital exchange.