Transition matrices and orbitals from reduced density matrix theory

Many-Body Excited States with a Contracted Quantum Eigensolver

Calculating ground and excited states is an exciting prospect for near-term quantum computing applications, and accurate and efficient algorithms are needed to assess viable directions. We develop an excited-state approach based on the contracted quantum eigensolver (ES-CQE), which iteratively attempts to find a solution to a contraction of the Schrödinger equation projected onto a subspace and does not require a priori information on the system. We focus on the anti-Hermitian portion of the equation, leading to a two-body unitary ansatz. We investigate the role of symmetries, initial states, constraints, and overall performance within the context of the model strongly correlated rectangular H4 system. We show that the ES-CQE achieves near-exact accuracy across the majority of states, covering regions of strong and weak electron correlation, while also elucidating challenging instances for two-body unitary ansatz.

Visualizing and characterizing excited states from time-dependent density functional theory

Time-dependent density functional theory (TD-DFT) is the most widely-used electronic structure method for excited states, due to a favorable combination of low cost and semi-quantitative accuracy in many contexts, even if there are well recognized limitations. This Perspective describes various ways in which excited states from TD-DFT calculations can be visualized and analyzed, both qualitatively and quantitatively. This includes not just orbitals and densities but also well-defined statistical measures of electron-hole separation and of Frenkel-type exciton delocalization. Emphasis is placed on mathematical connections between methods that have often been discussed separately. Particular attention is paid to charge-transfer diagnostics, which provide indicators of when TD-DFT may not be trustworthy due to its categorical failure to describe long-range electron transfer. Measures of exciton size and charge separation that are directly connected to the underlying transition density are recommended over more ad hoc metrics for quantifying charge-transfer character.

Computational study of the photophysical properties and electronic structure of iridium(III) photosensitizer complexes with electron-withdrawing groups

A series of novel [Ir(tpy)(btp)Cl]+ complexes (Ir1-Ir4) have been reported to show excellent performance as photosensitizers. The introduction of electron-withdrawing groups increases visible light absorption and the lifetime of triplet states. To improve the photophysical properties, we theoretically design Ir5-Ir9 with electron-withdrawing groups (Cl, F, COOH, CN and NO2). Surprisingly, our findings indicate that the photosensitizer performance does not strictly increase with the electron-withdrawing ability of the substituents. In this work, the geometric and electronic structures, transition features, and photophysical properties of Ir1-Ir9 are investigated. The natural transition orbital (NTO) analysis indicates that the T1 and T2 states play a role in the photochemical pathways. Ultraviolet-visible (UV-vis) absorption spectra and charge-transfer spectra (CTS) have been investigated to show that the introduction of electron-withdrawing groups not only improves the visible light absorbing ability, but also changes the nature of electron excitation, providing a future molecular design strategy for similar series of photosensitizers. The rates of (reverse) intersystem crossing and the Huang-Rhys factors are evaluated to interpret the experimental results within the framework of Marcus theory. For complexes Ir1-Ir7, the introduction of electron-withdrawing groups leads to a lower efficiency of reverse intersystem crossing and a strong non-radiative process T2 → T1, resulting in a long triplet lifetime and excellent performance as a photosensitizer. Furthermore, some newly designed complexes (Ir7-Ir9) show great potential as thermally activated delayed fluorescence emitters, contrary to our initial expectations.

Fundamental Variable and Density Representation in Multistate DFT for Excited States

Complementary to the theorems of Hohenberg and Kohn for the ground state, Theophilou's subspace theory establishes a one-to-one relationship between the total eigenstate energy and density ρ_V(r) of the subspace spanned by the lowest N eigenstates. However, the individual eigenstate energies are not directly available from such a subspace density functional theory. Lu and Gao (J. Phys. Chem. Lett. 2022, 13, 7762) recently proved that the Hamiltonian projected on to this subspace is a matrix functional H[D] of the multistate matrix density D(r) and that variational optimization of the trace of the Hamiltonian matrix functional yields exactly the individual eigenstates and densities. This study shows that the matrix density D(r) is the necessary fundamental variable in order to determine the exact energies and densities of the individual eigenstates. Furthermore, two ways of representing the matrix density are introduced, making use of nonorthogonal and orthogonal orbitals. In both representations, a multistate active space of auxiliary states can be constructed to exactly represent D(r) with which an explicit formulation of the Hamiltonian matrix functional H[D] is presented. Importantly, the use of a common set of orthonormal orbitals makes it possible to carry out multistate self-consistent-field optimization of the auxiliary states with singly and doubly excited configurations (MS-SDSCF).

Impact of packing arrangement on the optical properties of C60 cluster aggregates

The packing arrangement of organic π-conjugated molecules in a nanoscale material can have a strong impact on their optical properties. Here, using real-time-propagation time dependent density functional theory (rt-TDDFT) calculations with the support of transition contribution maps, we study how modifications in the packing arrangement (cubic-like and chain-like aggregates composed of eight C60 molecules) and packing density (assembled at close distances with center-to-center inter-fullerene distances (d) varying from 9 Å to 11 Å) of C60 molecules affect the optical properties of cluster aggregates. The important conclusions drawn from this work are summarized as follows. For d = 9 Å, the charge transfer excitons produced by cubic and chain-like C60 cluster aggregates have highly different optical characteristics, as evidenced by the transition contribution maps. On the other hand, for d = 10 Å and 11 Å, both kinds of aggregates produce qualitatively similar optical features with the emergence of Wannier-like delocalized excitons having distinct degrees of localization and spatial distribution. The theoretical findings in this study elucidate the optical excitations in C60 cluster aggregates and could help in the design of more efficient organic devices.

Computational Study of Photochemical Relaxation Pathways of Platinum(II) Complexes

A series of functional platinum(II) complexes (Pt1-Pt3), which present high activity in four-photon absorption, in vivo imaging, and precise cancer therapy, as previously reported by the experimental work of Zhang et al. (Inorg. Chem. 2021, 60, 2362-2371), are computationally investigated in the article. We find that after the complex goes through four-photon absorption to the S₁ state, it undergoes intersystem crossing to the T₂ state and eventually reaches the T₁ state through internal conversion. On the T₁ state, both radiative and nonradiative decay to S₀ exit. The radiative decay forms the basis for the phosphorescence imaging in tissues as reported in the original paper. In addition, the nonradiative decay can simultaneously generate cytotoxic singlet oxygen by the excited energy transfer process, also known as triplet oxygen's quenching of triplet states. We conclude that the phosphorescence property as well as the photosensitizer character jointly bring high activity of in vivo imaging and photodynamic therapy to these complexes.

Exploring Dyson's Orbitals and Their Electron Binding Energies for Conceptualizing Excited States from Response Methodology

The molecular orbital (MO) concept is a useful tool, which relates the molecular ground-state energy with the energies (and occupations) of the individual orbitals. However, analysis of the excited states from linear response computations is performed in terms of the initial state MOs or some other forms of orbitals, e.g., natural or natural transition orbitals. Because these orbitals lack the respective energies, they do not allow developing a consistent orbital picture of the excited states. Herein, we argue that Dyson's orbitals enable description of the response states compatible with the concepts of molecular orbital theory. The Dyson orbitals and their energies obtained by mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) for the response ground state are remarkably similar to the canonical MOs obtained by the usual DFT calculation. For excited states, the Dyson orbitals provide a chemically sensible picture of the electronic transitions, thus bridging the chasm between orbital theory and response computations.

Chemically Modified Quinoidal Oligothiophenes for Enhanced Linear and Third-Order Nonlinear Optical Properties

In the present investigation, quantum chemical calculations have been performed in a systematic way to explore the optoelectronic, charge transfer, and nonlinear optical (NLO) properties of different bis(dicyanomethylene) end-functionalized quinoidal oligothiophenes. The effect of different conformations (linking modes of thiophene rings) on conformational, optoelectronic, and NLO properties are studied from the best-performed dimer to octamer. The optical and NLO properties of all the selected systems (1-7) are calculated by means of density functional theory (DFT) methods. Among all the designed compounds, the largest linear isotropic (αiso) polarizability value of 603.1 × 10-24 esu is shown by compound 7 which is ∼12, ∼16, ∼9, ∼11, ∼10, and ∼4 times larger as compared to compounds 1-6, respectively. A relative investigation is performed considering the expansion in third-order NLO polarizability as a function of size and conformational modes. Among all the investigated systems, system 7 shows the highest value of static second hyperpolarizability ⟨γ⟩ with an amplitude of 7607 × 10-36 esu at the M06/6-311G** level of theory, which is ∼521, ∼505, ∼38, ∼884, ∼185, and ∼15 times more than that of compounds 1-6, respectively. The extensively larger ⟨γ⟩ amplitude of compound 7 with higher oscillator strength and lower transition energy indicates that NLO properties are remarkably dependent upon linking modes of thiophene rings and its chain length. Furthermore, to trace the origin of higher nonlinearities, TD-DFT calculations are also performed at the same TD-M06/6-311G** level of theory. Additionally, a comprehensive understanding of the effect of structure/property relationship on the NLO polarizabilities of these investigated quinoidal oligothiophenes is obtained through the inspection of Frontier molecular orbitals, the density of states (TDOS and PDOS), and molecular electrostatic potential diagrams including the transition density matrix. Hence, the current examination will not just feature the NLO capability of entitled compounds yet additionally incite the interest of experimentalists to adequately modify the structure of these oligothiophenes for efficient optical and NLO applications.

On the Low-Lying Electronically Excited States of Azobenzene Dimers: Transition Density Matrix Analysis

Azobenzene-containing molecules may associate with each other in systems such as self-assembled monolayers or micelles. The interaction between azobenzene units leads to a formation of exciton states in these molecular assemblies. Apart from local excitations of monomers, the electronic transitions to the exciton states may involve charge transfer excitations. Here, we perform quantum chemical calculations and apply transition density matrix analysis to quantify local and charge transfer contributions to the lowest electronic transitions in azobenzene dimers of various arrangements. We find that the transitions to the lowest exciton states of the considered dimers are dominated by local excitations, but charge transfer contributions become sizable for some of the lowest ππ* electronic transitions in stacked and slip-stacked dimers at short intermolecular distances. In addition, we assess different ways to partition the transition density matrix between fragments. In particular, we find that the inclusion of the atomic orbital overlap has a pronounced effect on quantifying charge transfer contributions if a large basis set is used.

Designing and theoretical study of fluorinated small molecule donor materials for organic solar cells

A recently synthesized photoactive donor named fluorinated thienyl-substituted benzodithiophene (DRTB-FT), modified with four novel end capped acceptor molecules, has been investigated through different electrical, quantum, and spectrochemical techniques for its enhanced electro-optical and photovoltaic properties. DRTB-FT was connected to 2-methylenemalononitrile (D-1), 2-methylene-3-oxobutanenitrile (D-2), 2-(2-methylene-3-oxo-2,3-dihydro-1H-inden-1-ylidene) malononitrile (D-3), and 3-methyl-5methylene-2-thioxothiazolidin-4-one (D-4) as terminal acceptor moieties. The architectural D-1 and D-3 molecules owe reduced optical band gap of 2.45 and 2.28 eV benefited from A-D-A configuration and have broaden maximum absorption band (λ_max) at 617 and 602 nm in polar organic solvent (chloroform). Reduced optical band gap sets the ease for enhanced absorption. Reorganization energy of electron (λ_e) of D-3 molecule (0.00397 eV) was smaller among all which disclosed its greater mobility of conducting electrons (ICT). Larger values of dipole moment (μ) of D-1 (5.939 Debye) and D-3 (3.661 Debye) molecules in comparison to R indicated greater solubilities of the targeted molecules. Among the tailored molecules, D-3 showed the lowest binding energy of 0.25 eV in solvent phase and 0.08 eV in gaseous phase. The voltaic strength of the designed molecules was examined with respect to fullerene derivative (PC₆₁BM) which exposed that D-1 is the best choice for achieving higher PCE. TDM (transition density matrix), DOS (density of states) analysis, and binding energies all were estimated at MPW1PW91/6-31G (d, p) level of DFT (density functional theory). All the architecture molecules show reduced band gap and high electron transfer rate due to the lowest reorganization energy (RE) of electron. The results show that there is greater contribution of acceptor and conjugated donor core towards the total absorption into the visible region of the spectrum. When tailored molecules D-1, D-2, D-3, and D-4 were blended with fullerene derivative polymer (PC₆₁BM), they give high values of voltage at zero current level (V_oc) compared to R.

Dynamical transition orbitals: A particle-hole description in real-time TDDFT dynamics

We expand the concept of natural transition orbitals in the context of real-time time-dependent density functional theory (RT-TDDFT) and show its application in practical calculations. Kohn-Sham single-particle wavefunctions are propagated in RT-TDDFT simulation, and physical properties remain invariant under their unitary transformation. In this work, we exploit this gauge freedom and expand the concept of natural transition orbitals, which is widely used in linear-response TDDFT, for obtaining a particle-hole description in RT-TDDFT simulation. While linear-response TDDFT is widely used to study electronic excitation, RT-TDDFT can be employed more generally to simulate non-equilibrium electron dynamics. Studying electron dynamics in terms of dynamic transitions of particle-hole pairs is, however, not straightforward in the RT-TDDFT simulation. By constructing natural transition orbitals through projecting time-dependent Kohn-Sham wave functions onto occupied/unoccupied eigenstate subspaces, we show that linear combinations of a pair of the resulting hole/particle orbitals form a new gauge, which we refer to as dynamical transition orbitals. We demonstrate the utility of this framework to analyze RT-TDDFT simulations of optical excitation and electronic stopping dynamics in the particle-hole description.

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?

Exciton localization in excited-state dynamics of a tetracene trimer: a surface hopping LC-TDDFTB study

Excitons in the molecular aggregates of chromophores are key participants in important processes such as photosynthesis or the functioning of organic photovoltaic devices. Therefore, the exploration of exciton dynamics is crucial. Here we report on exciton localization during excited-state dynamics of the recently synthesized tetracene trimer [Liu et al., Org. Lett., 2017, 19, 580]. We employ the surface hopping approach to nonadiabatic molecular dynamics in conjunction with the long-range corrected time-dependent density functional tight binding (LC-TDDFTB) method [Humeniuk and Mitrić, Comput. Phys. Commun., 2017, 221, 174]. Utilizing a set of descriptors based on the transition density matrix, we perform comprehensive analysis of exciton dynamics. The obtained results reveal an ultrafast exciton localization to a single tetracene unit of the trimer during excited-state dynamics, along with exciton transfer between units.

CC2 oscillator strengths within the local framework for calculating excitation energies (LoFEx)

In a recent work [P. Baudin and K. Kristensen, J. Chem. Phys. 144, 224106 (2016)], we introduced a local framework for calculating excitation energies (LoFEx), based on second-order approximated coupled cluster (CC2) linear-response theory. LoFEx is a black-box method in which a reduced excitation orbital space (XOS) is optimized to provide coupled cluster (CC) excitation energies at a reduced computational cost. In this article, we present an extension of the LoFEx algorithm to the calculation of CC2 oscillator strengths. Two different strategies are suggested, in which the size of the XOS is determined based on the excitation energy or the oscillator strength of the targeted transitions. The two strategies are applied to a set of medium-sized organic molecules in order to assess both the accuracy and the computational cost of the methods. The results show that CC2 excitation energies and oscillator strengths can be calculated at a reduced computational cost, provided that the targeted transitions are local compared to the size of the molecule. To illustrate the potential of LoFEx for large molecules, both strategies have been successfully applied to the lowest transition of the bivalirudin molecule (4255 basis functions) and compared with time-dependent density functional theory.

Solvent effects on the excited state characteristics of adenine–thymine base pairs

A systematic analysis of the excited state characteristics of the DNA base pair adenine–thymine in stacked and Watson–Crick hydrogen bonded configurations has been carried out in this study.

Truncated Transition Densities for Analysis of (Nonlinear) Optical Properties of carbo-Chromophores

The optical properties of several quadrupolar carbo-benzene derivatives are investigated at various levels of calculation (TDDFT and CASPT2) and analyzed using a new theoretical tool here disclosed: The "visualization" of the transition dipole moment from the transition density truncated to the main monoexcitations involved in the electronic transition (TTD). The experimental or calculated one-photon UV-visible absorption spectra of the carbo-benzene derivatives fit with the Gouterman model originally proposed for porphyrins, where the first four excited states involve linear combinations of monoexcitations of the same four frontier molecular orbitals. The relative intensities of the absorption bands are analyzed from the transition dipole moments calculated from the TTDs and an analogy between porphyrins and carbo-benzenes is argued. The two-photon absorption (TPA) cross section related to the third-order nonlinear optical response is calculated for each two-photon-allowed excited state |f⟩ from the contribution of all possible intermediate excited states |i⟩ using the "sum-over-state" (SOS) scheme. The quadrupolar carbo-benzene derivatives fit into the three-level model, as their TPA cross section exhibits a dominant contribution of one of the intermediate excited states. The origin of TPA efficiency (enhancement) upon carbo-merisation of the C-C link to the para-substituents is discussed from the excitation energies of the intermediate and final excited states and from the two corresponding transition dipole moments (μ0i and μif). The latter may be calculated from the TTDs.