Assessment of the ΔSCF density functional theory approach for electronic excitations in organic dyes

DFTB+, a software package for efficient approximate density functional theory based atomistic simulations

DFTB+ is a versatile community developed open source software package offering fast and efficient methods for carrying out atomistic quantum mechanical simulations. By implementing various methods approximating density functional theory (DFT), such as the density functional based tight binding (DFTB) and the extended tight binding method, it enables simulations of large systems and long timescales with reasonable accuracy while being considerably faster for typical simulations than the respective ab initio methods. Based on the DFTB framework, it additionally offers approximated versions of various DFT extensions including hybrid functionals, time dependent formalism for treating excited systems, electron transport using non-equilibrium Green's functions, and many more. DFTB+ can be used as a user-friendly standalone application in addition to being embedded into other software packages as a library or acting as a calculation-server accessed by socket communication. We give an overview of the recently developed capabilities of the DFTB+ code, demonstrating with a few use case examples, discuss the strengths and weaknesses of the various features, and also discuss on-going developments and possible future perspectives.

Excited State Orbital Optimization via Minimizing the Square of the Gradient: General Approach and Application to Singly and Doubly Excited States via Density Functional Theory

We present a general approach to converge excited state solutions to any quantum chemistry orbital optimization process, without the risk of variational collapse. The resulting square gradient minimization (SGM) approach only requires analytic energy/Lagrangian orbital gradients and merely costs 3 times as much as ground state orbital optimization (per iteration), when implemented via a finite difference approach. SGM is applied to both single determinant ΔSCF and spin-purified restricted open-shell Kohn-Sham (ROKS) approaches to study the accuracy of orbital optimized DFT excited states. It is found that SGM can converge challenging states where the maximum overlap method (MOM) or analogues either collapse to the ground state or fail to converge. We also report that ΔSCF/ROKS predict highly accurate excitation energies for doubly excited states (which are inaccessible via TDDFT). Singly excited states obtained via ROKS are also found to be quite accurate, especially for Rydberg states that frustrate (semi)local TDDFT. Our results suggest that orbital optimized excited state DFT methods can be used to push past the limitations of TDDFT to doubly excited, charge-transfer, or Rydberg states, making them a useful tool for the practical quantum chemist's toolbox for studying excited states in large systems.

Highly Accurate Prediction of Core Spectra of Molecules at Density Functional Theory Cost: Attaining Sub-electronvolt Error from a Restricted Open-Shell Kohn-Sham Approach

We present the use of the recently developed square gradient minimization (SGM) algorithm for excited-state orbital optimization to obtain spin-pure restricted open-shell Kohn-Sham (ROKS) energies for core excited states of molecules. The SGM algorithm is robust against variational collapse and offers a reliable route to converging orbitals for target excited states at only 2-3 times the cost of ground-state orbital optimization (per iteration). ROKS/SGM with the modern SCAN/ωB97X-V functionals is found to predict the K-edge of C, N, O, and F to a root mean squared error of ∼0.3 eV. ROKS/SGM is equally effective at predicting L-edge spectra of third period elements, provided a perturbative spin-orbit correction is employed. This high accuracy can be contrasted with traditional time-dependent density functional theory (TDDFT), which typically has greater than 10 eV error and requires translation of computed spectra to align with experiment. ROKS is computationally affordable (having the same scaling as ground-state DFT and a slightly larger prefactor) and can be applied to geometry optimizations/ab initio molecular dynamics of core excited states, as well as condensed phase simulations. ROKS can also model doubly excited/ionized states with one broken electron pair, which are beyond the ability of linear response based methods.

Variational calculations of excited states via direct optimization of the orbitals in DFT

A direct optimization method for obtaining excited electronic states using density functionals is presented.

A correlation–relaxation-balanced direct method at the second order perturbation theory for accurate ionization potential predictions

With almost no extra computational cost after a normal MP2 procedure, the CRB-MP2 method proposed here yields high quality valence and core IPs for a wide range of species.

Imaginary-Time Time-Dependent Density Functional Theory and Its Application for Robust Convergence of Electronic States

Reliable and robust convergence to the electronic ground state within density functional theory (DFT) Kohn-Sham (KS) calculations remains a thorny issue in many systems of interest. In such cases, charge sloshing can delay or completely hinder the convergence. Here, we use an approach based on transforming the time-dependent DFT equations to imaginary time, followed by imaginary-time evolution, as a reliable alternative to the self-consistent field (SCF) procedure for determining the KS ground state. We discuss the theoretical and technical aspects of this approach and show that the KS ground state should be expected to be the long-imaginary-time output of the evolution, independent of the exchange-correlation functional or the level of theory used to simulate the system. By maintaining self-consistency between the single-particle wave functions (orbitals) and the electronic density throughout the determination of the stationary state, our method avoids the typical difficulties encountered in SCF. To demonstrate dependability of our approach, we apply it to selected systems which struggle to converge with SCF schemes. In addition, through the van Leeuwen theorem, we affirm the physical meaningfulness of imaginary-time TDDFT, justifying its use in certain topics of statistical mechanics such as in computing imaginary-time path integrals.

Accurate Band Gap Predictions of Semiconductors in the Framework of the Similarity Transformed Equation of Motion Coupled Cluster Theory

In this work, we present a detailed comparison between wave-function-based and particle/hole techniques for the prediction of band gap energies of semiconductors. We focus on the comparison of the back-transformed Pair Natural Orbital Similarity Transformed Equation of Motion Coupled-Cluster (bt-PNO-STEOM-CCSD) method with Time Dependent Density Functional Theory (TD-DFT) and Delta Self Consistent Field/DFT (Δ-SCF/DFT) that are employed to calculate the band gap energies in a test set of organic and inorganic semiconductors. Throughout, we have used cluster models for the calculations that were calibrated by comparing the results of the cluster calculations to periodic DFT calculations with the same functional. These calibrations were run with cluster models of increasing size until the results agreed closely with the periodic calculation. It is demonstrated that bt-PNO-STEOM-CC yields accurate results that are in better than 0.2 eV agreement with the experiment. This holds for both organic and inorganic semiconductors. The efficiency of the employed computational protocols is thoroughly discussed. Overall, we believe that this study is an important contribution that can aid future developments and applications of excited state coupled cluster methods in the field of solid-state chemistry and heterogeneous catalysis.

In this work, it is shown that a combination of the embedded cluster approach with wave-function-based ab initio methods in the framework of the Similarity Transformed Equation of Motion Coupled Cluster (bt-PNO STEOM-CC) provides an accurate protocol for band gap energy predictions in classes of organic and inorganic semiconductors.

Insights into Direct Methods for Predictions of Ionization Potential and Electron Affinity in Density Functional Theory

Vertical ionization potential (IP) and electron affinity (EA) are fundamental molecular properties, while the Δ method and the direct method are the widely used approaches to compute these properties. The Δ method is calculated by taking the total energy difference of the initial and final states, whose reliability is seriously affected by the issue associated with the imbalanced treatment of these two states. The direct method based on the derivatives involving only one single-state calculation can yield a quasi-particle spectrum whose accuracy, on the other hand, is mostly affected by the levels of approximate molecular structure theories. Because of the aforementioned issues, EA prediction can be particularly problematic. Here we present, for the first time, an analytic theory on the derivation and realization of generalized Kohn-Sham (KS) eigenvalues of doubly hybrid (DH) functionals that depend on both occupied and unoccupied orbitals. The method based on the KS eigenvalues of neutral systems, termed the NKS method, is found to suffer little from the imbalance issue, while it is only the NKS method that can offer accurate EA prediction from a good functional approximation, such as the XYG3 type of DH functionals. Being less sensitive to the size of basis sets, the NKS method is of great significance for its application to large systems. The insights gained in this work are useful for the calculation of properties associated with small energy differences while emphasizing the importance of the development of generalized functionals that rely on both occupied and unoccupied orbitals.

Excited-State Potential Energy Surfaces, Conical Intersections, and Analytical Gradients from Ground-State Density Functional Theory

Kohn-Sham density functional theory (KS-DFT) has been a well-established theoretical foundation for ground-state electronic structure and has achieved great success in practical calculations. Recently, utilizing the eigenvalues from KS or generalized KS (GKS) calculations as an approximation to the quasiparticle energies, our group demonstrated a method to calculate the excitation energies from (G)KS calculation on the ground-state ( N - 1)-electron system. This method is now called QE-DFT (quasiparticle energies from DFT). In this work, we extend this QE-DFT method to describe excited-state potential energy surfaces (PESs), conical intersections, and the analytical gradients of excited-state PESs. The analytical gradients were applied to perform geometry optimization for excited states. In conjunction with several commonly used density functional approximations, QE-DFT can yield PESs in the vicinity of the equilibrium structure with accuracy similar to that from time-dependent DFT (TD-DFT). Furthermore, it describes conical intersection well, in contrast to TD-DFT. Good results for geometry optimization, especially bond length, of low-lying excitations for 14 small molecules are presented. The capability of describing excited-state PESs, conical intersections, and analytical gradients from QE-DFT and its efficiency based on just ground-state DFT calculations should be of great interest for describing photochemical and photophysical processes in complex systems.

Approximating Quasiparticle and Excitation Energies from Ground State Generalized Kohn-Sham Calculations

Quasiparticle energies and fundamental band gaps in particular are critical properties of molecules and materials. It was rigorously established that the generalized Kohn-Sham HOMO and LUMO orbital energies are the chemical potentials of electron removal and addition and thus good approximations to band edges and fundamental gaps from a density functional approximation (DFA) with minimal delocalization error. For other quasiparticle energies, their connection to the generalized Kohn-Sham orbital energies has not been established but remains highly interesting. We provide the comparison of experimental quasiparticle energies for many finite systems with calculations from the GW Green function and localized orbitals scaling correction (LOSC), a recently developed correction to semilocal DFAs, which has minimal delocalization error. Extensive results with over 40 systems clearly show that LOSC orbital energies achieve slightly better accuracy than the GW calculations with little dependence on the semilocal DFA, supporting the use of LOSC DFA orbital energies to predict quasiparticle energies. This also leads to the calculations of excitation energies of the N-electron systems from the ground state DFA calculations of the ( N - 1)-electron systems. Results show good performance with accuracy similar to TDDFT and the delta SCF approach for valence excitations with commonly used DFAs with or without LOSC. For Rydberg states, good accuracy was obtained only with the use of LOSC DFA. This work highlights the pathway to quasiparticle and excitation energies from ground density functional calculations.

Non-adiabatic molecular dynamics with ΔSCF excited states

Accurate modelling of nonadiabatic transitions and electron-phonon interactions in extended systems is essential for understanding the charge and energy transfer in photovoltaic and photocatalytic materials. The extensive computational costs of the advanced excited state methods have stimulated the development of many approximations to study the nonadiabatic molecular dynamics (NA-MD) in solid-state and molecular materials. In this work, we present a novel ▵SCF-NA-MD methodology that aims to account for electron-hole interactions and electron-phonon back-reaction critical in modelling photoinduced nuclear dynamics. The excited states dynamics is described using the delta self-consistent field (▵SCF) technique within the density functional formalism and the trajectory surface hopping. The technique is implemented in the open-source Libra-X package freely available on the Internet (https://github.com/Quantum-Dynamics-Hub/Libra-X). This work illustrates the general utility of the developed ▵SCF-NA-MD methodology by characterizing the excited state energies and lifetimes, reorganization energies, photoisomerization quantum yields, and by providing the mechanistic details of reactive processes in a number of organic molecules.

Ultrafast non-radiative dynamics of atomically thin MoSe2

Photo-induced non-radiative energy dissipation is a potential pathway to induce structural-phase transitions in two-dimensional materials. For advancing this field, a quantitative understanding of real-time atomic motion and lattice temperature is required. However, this understanding has been incomplete due to a lack of suitable experimental techniques. Here, we use ultrafast electron diffraction to directly probe the subpicosecond conversion of photoenergy to lattice vibrations in a model bilayered semiconductor, molybdenum diselenide. We find that when creating a high charge carrier density, the energy is efficiently transferred to the lattice within one picosecond. First-principles nonadiabatic quantum molecular dynamics simulations reproduce the observed ultrafast increase in lattice temperature and the corresponding conversion of photoenergy to lattice vibrations. Nonadiabatic quantum simulations further suggest that a softening of vibrational modes in the excited state is involved in efficient and rapid energy transfer between the electronic system and the lattice.

Electronic structure of BN-aromatics: Choice of reliable computational tools

The importance of having reliable calculation tools to interpret and predict the electronic properties of BN-aromatics is directly linked to the growing interest for these very promising new systems in the field of materials science, biomedical research, or energy sustainability. Ionization energy (IE) is one of the most important parameters to approach the electronic structure of molecules. It can be theoretically estimated, but in order to evaluate their persistence and propose the most reliable tools for the evaluation of different electronic properties of existent or only imagined BN-containing compounds, we took as reference experimental values of ionization energies provided by ultra-violet photoelectron spectroscopy (UV-PES) in gas phase-the only technique giving access to the energy levels of filled molecular orbitals. Thus, a set of 21 aromatic molecules containing B-N bonds and B-N-B patterns has been merged for a comparison between experimental IEs obtained by UV-PES and various theoretical approaches for their estimation. Time-Dependent Density Functional Theory (TD-DFT) methods using B3LYP and long-range corrected CAM-B3LYP functionals are used, combined with the ΔSCF approach, and compared with electron propagator theory such as outer valence Green's function (OVGF, P3) and symmetry adapted cluster-configuration interaction ab initio methods. Direct Kohn-Sham estimation and "corrected" Kohn-Sham estimation are also given. The deviation between experimental and theoretical values is computed for each molecule, and a statistical study is performed over the average and the root mean square for the whole set and sub-sets of molecules. It is shown that (i) ΔSCF+TDDFT(CAM-B3LYP), OVGF, and P3 are the most efficient way for a good agreement with UV-PES values, (ii) a CAM-B3LYP range-separated hybrid functional is significantly better than B3LYP for the purpose, especially for extended conjugated systems, and (iii) the "corrected" Kohn-Sham result is a fast and simple way to predict IEs.

An Ab Initio Exciton Model Including Charge-Transfer Excited States

The Frenkel exciton model is a useful tool for theoretical studies of multichromophore systems. We recently showed that the exciton model could be used to coarse-grain electronic structure in multichromophoric systems, focusing on singly excited exciton states [ Acc. Chem. Res. 2014 , 47 , 2857 - 2866 ]. However, our previous implementation excluded charge-transfer excited states, which can play an important role in light-harvesting systems and near-infrared optoelectronic materials. Recent studies have also emphasized the significance of charge-transfer in singlet fission, which mediates the coupling between the locally excited states and the multiexcitonic states. In this work, we report on an ab initio exciton model that incorporates charge-transfer excited states and demonstrate that the model provides correct charge-transfer excitation energies and asymptotic behavior. Comparison with TDDFT and EOM-CC2 calculations shows that our exciton model is robust with respect to system size, screening parameter, and different density functionals. Inclusion of charge-transfer excited states makes the exciton model more useful for studies of singly excited states and provides a starting point for future construction of a model that also includes double-exciton states.

The role of sulfur oxidation in controlling the electronic properties of sulfur-containing host molecules for phosphorescent organic light-emitting diodes

In this study a series of dibenzothiophene (DBT) derivatives having different valence states of sulfur atoms have been reported as host materials for blue phosphorescent organic light-emitting diodes. Their electronic properties have also been thoroughly investigated to develop structure-property relationships which include the consideration of the effect of various oxidation states of the sulfur atom in the core moiety and linking (C-N linkage) of subunits with the core at different positions. The results obtained from the electronic structure calculations highlight that the triplet energy (E_T), singlet-triplet energy difference (ΔE_ST), reorganization energy for the hole and the injection barrier for the electron decrease with an increase in the oxidation state of the sulfur atom in DBT. On the other hand, the injection barrier for the hole and the reorganization energy for the electron increase upon increasing the oxidation state of the sulfur atom present in the DBT.

Theoretical rationalization for reduced charge recombination in bulky carbazole-based sensitizers in solar cells

The search for greater efficiency in organic dye-sensitized solar cells (DSCs) and in their perovskite cousins is greatly aided by a more complete understanding of the spectral and morphological properties of the photoactive layer. This investigation resolves a discrepancy in the observed photoconversion efficiency (PCE) of two closely related DSCs based on carbazole-containing D-π-A organic sensitizers. Detailed theoretical characterization of the absorption spectra, dye adsorption on TiO₂ , and electronic couplings for charge separation and recombination permit a systematic determination of the origin of the difference in PCE. Although the two dyes produce similar spectral features, ground- and excited-state density functional theory (DFT) simulations reveal that the dye with the bulkier donor group adsorbs more strongly to TiO₂ , experiences limited π-π aggregation, and is more resistant to loss of excitation energy via charge recombination on the dye. The effects of conformational flexibility on absorption spectra and on the electronic coupling between the bright exciton and charge-transfer states are revealed to be substantial and are characterized through density-functional tight-binding (DFTB) molecular dynamics sampling. These simulations offer a mechanistic explanation for the superior open-circuit voltage and short-circuit current of the bulky-donor dye sensitizer and provide theoretical justification of an important design feature for the pursuit of greater photocurrent efficiency in DSCs. © 2017 Wiley Periodicals, Inc.

Rational design of carbazolyl and aryl phosphine oxide (APO) based ambipolar host materials for blue electrophosphorescence: a density functional theory study

Carbazolyl and Aryl Phosphine Oxide (APO) based ambipolar host materials.

Exploring excited state properties of 7-hydroxy and 7-methoxy 4-methycoumarin: a combined time-dependent density functional theory/effective fragment potential study

Hydrogen bond dynamics, C–OH bond contracting, O–H bond stretching and O–H⋯O HB strengthening reveal the ESHT in 4MU at the S₁state.

An extended DFTB-CI model for charge-transfer excited states in cationic molecular clusters: model studies versus ab initio calculations in small PAH clusters

We present an extension of the constrained density functional tight binding scheme combined with configuration interaction (DFTB-CI) to efficiently compute excited states of molecular cluster cations.

Quantum chemical design of carbazole- and pyridoindole-based ambipolar host materials for blue phosphorescent OLEDs

Structure–property relationship of carbazole- and pyridoindole-type host materials for blue PhOLEDs using quantum chemical calculations.

Derivation of the RPA (Random Phase Approximation) Equation of ATDDFT (Adiabatic Time Dependent Density Functional Ground State Response Theory) from an Excited State Variational Approach Based on the Ground State Functional

The random phase approximation (RPA) equation of adiabatic time dependent density functional ground state response theory (ATDDFT) has been used extensively in studies of excited states. It extracts information about excited states from frequency dependent ground state response properties and avoids, thus, in an elegant way, direct Kohn-Sham calculations on excited states in accordance with the status of DFT as a ground state theory. Thus, excitation energies can be found as resonance poles of frequency dependent ground state polarizability from the eigenvalues of the RPA equation. ATDDFT is approximate in that it makes use of a frequency independent energy kernel derived from the ground state functional. It is shown in this study that one can derive the RPA equation of ATDDFT from a purely variational approach in which stationary states above the ground state are located using our constricted variational DFT (CV-DFT) method and the ground state functional. Thus, locating stationary states above the ground state due to one-electron excitations with a ground state functional is completely equivalent to solving the RPA equation of TDDFT employing the same functional. The present study is an extension of a previous work in which we demonstrated the equivalence between ATDDFT and CV-DFT within the Tamm-Dancoff approximation.

Gentlest ascent dynamics for calculating first excited state and exploring energy landscape of Kohn-Sham density functionals

We develop the gentlest ascent dynamics for Kohn-Sham density functional theory to search for the index-1 saddle points on the energy landscape of the Kohn-Sham density functionals. These stationary solutions correspond to excited states in the ground state functionals. As shown by various examples, the first excited states of many chemical systems are given by these index-1 saddle points. Our novel approach provides an alternative, more robust way to obtain these excited states, compared with the widely used ΔSCF approach. The method can be easily generalized to target higher index saddle points. Our results also reveal the physical interest and relevance of studying the Kohn-Sham energy landscape.

Simulation of X-ray absorption spectra with orthogonality constrained density functional theory

Orthogonality constrained density functional theory (OCDFT) [F. A. Evangelista, P. Shushkov and J. C. Tully, J. Phys. Chem. A, 2013, 117, 7378] is a variational time-independent approach for the computation of electronic excited states. In this work we extend OCDFT to compute core-excited states and generalize the original formalism to determine multiple excited states. Benchmark computations on a set of 13 small molecules and 40 excited states show that unshifted OCDFT/B3LYP excitation energies have a mean absolute error of 1.0 eV. Contrary to time-dependent DFT, OCDFT excitation energies for first- and second-row elements are computed with near-uniform accuracy. OCDFT core excitation energies are insensitive to the choice of the functional and the amount of Hartree-Fock exchange. We show that OCDFT is a powerful tool for the assignment of X-ray absorption spectra of large molecules by simulating the gas-phase near-edge spectrum of adenine and thymine.

Ensemble DFT Approach to Excited States of Strongly Correlated Molecular Systems

Ensemble density functional theory (DFT) is a novel time-independent formalism for obtaining excitation energies of many-body fermionic systems. A considerable advantage of ensemble DFT over the more common Kohn-Sham (KS) DFT and time-dependent DFT formalisms is that it enables one to account for strong non-dynamic electron correlation in the ground and excited states of molecular systems in a transparent and accurate fashion. Despite its positive aspects, ensemble DFT has not so far found its way into the repertoire of methods of modern computational chemistry, probably because of the perceived lack of practically affordable implementations of the theory. The spin-restricted ensemble-referenced KS (REKS) method is perhaps the first computationally feasible implementation of the ideas behind ensemble DFT which enables one to describe accurately electronic transitions in a wide class of molecular systems, including strongly correlated molecules (biradicals, molecules undergoing bond breaking/formation), extended π-conjugated systems, donor-acceptor charge transfer adducts, etc.