Low-lying Excitation Energies Research Articles

Designing an Efficient Singlet Fission Material with B-N Substitution in Pyrene: A Model Exact Study.

The electronic structure of boron (B)-nitrogen (N)-substituted pyrene molecules is the center of attraction in designing an efficient intermolecular singlet fission (x-SF) material. Thermodynamic energy criteria required for x-SF are obtained by captodative substitution with B and N in pristine pyrene to increase the lowest singlet-triplet energy gap. We computed low-lying excited states of BN-embedded pyrene molecules by exactly solving the Pariser-Parr-Pople (PPP) model Hamiltonian and compared these results with the TDDFT and EOM-CCSD values. Exact diagonalization of the PPP model Hamiltonian suggests that pristine pyrene, which is endothermic for x-SF, becomes isoergic with certain (BN)2 substitution. The low-lying excited state energies calculated using the model Hamiltonian match very well with experimental values over EOM-CCSD and TDDFT. Moreover, the low value of the spin-orbit coupling constant calculated for BN-substituted pyrene strengthens its applicability as an SF material.

Linear Scaling Calculations of Excitation Energies with Active-Space Particle-Particle Random-Phase Approximation.

We developed an efficient active-space particle-particle random-phase approximation (ppRPA) approach to calculate accurate charge-neutral excitation energies of molecular systems. The active-space ppRPA approach constrains both indexes in particle and hole pairs in the ppRPA matrix, which only selects frontier orbitals with dominant contributions to low-lying excitation energies. It employs the truncation in both orbital indexes in the particle-particle and the hole-hole spaces. The resulting matrix, whose eigenvalues are excitation energies, has a dimension that is independent of the size of the systems. The computational effort for the excitation energy calculation, therefore, scales linearly with system size and is negligible compared with the ground-state calculation of the (N - 2)-electron system, where N is the electron number of the molecule. With the active space consisting of 30 occupied and 30 virtual orbitals, the active-space ppRPA approach predicts the excitation energies of valence, charge-transfer, Rydberg, double, and diradical excitations with the mean absolute errors (MAEs) smaller than 0.03 eV compared with the full-space ppRPA results. As a side product, we also applied the active-space ppRPA approach in the renormalized singles (RS) T-matrix approach. Combining the non-interacting pair approximation that approximates the contribution to the self-energy outside the active space, the active-space GRSTRS@PBE approach predicts accurate absolute and relative core-level binding energies with the MAEs around 1.58 and 0.3 eV, respectively. The developed linear scaling calculation of excitation energies is promising for applications to large and complex systems.

Systematic shell-model study of 98−130Cd isotopes and 8+ isomeric states

We present systematic shell-model studies of even-even 98−130Cd isotopes using a realistic effective shell-model interaction derived from the G-matrix approach with an inert core 88Sr. Our calculated low-lying excited energy spectra and electromagnetic properties are compared with the experimental data. On the basis of recently available experimental data, we predict spins and parities corresponding to unconfirmed states. We also discuss the properties of 8+ isomeric states in 98−104,130Cd isotopes.

Setup for the Ionic Lifetime Measurement of the 229mTh3+ Nuclear Clock Isomer

For the realization of an optical nuclear clock, the first isomeric excited state of thorium-229 (229mTh) is currently the only candidate due to its exceptionally low-lying excitation energy (8.338±0.024 eV). Such a nuclear clock holds promise not only to be a very precise metrological device but also to extend the knowledge of fundamental physics studies, such as dark matter research or variations in fundamental constants. Considerable progress was achieved in recent years in characterizing 229mTh from its first direct identification in 2016 to the only recent observation of the long-sought-after radiative decay channel. So far, nuclear resonance as the crucial parameter of a nuclear frequency standard has not yet been determined with laser-spectroscopic precision. To determine another yet unknown basic property of the thorium isomer and to further specify the linewidth of its ground-state transition, a measurement of the ionic lifetime of the isomer is in preparation. Theory and experimental investigations predict the lifetime to be 103–104 s. To precisely target this property using hyperfine structure spectroscopy, an experimental setup is currently being commissioned at LMU Munich. It is based on a cryogenic Paul trap providing long-enough storage times for 229mTh ions, that will be sympathetically cooled with 88Sr+. This article presents a concept for an ionic lifetime measurement and discusses the laser-optical part of a setup specifically developed for this purpose.

Electronic, redox, and optical property prediction of organic π-conjugated molecules through a hierarchy of machine learning approaches.

Accelerating the development of π-conjugated molecules for applications such as energy generation and storage, catalysis, sensing, pharmaceuticals, and (semi)conducting technologies requires rapid and accurate evaluation of the electronic, redox, or optical properties. While high-throughput computational screening has proven to be a tremendous aid in this regard, machine learning (ML) and other data-driven methods can further enable orders of magnitude reduction in time while at the same time providing dramatic increases in the chemical space that is explored. However, the lack of benchmark datasets containing the electronic, redox, and optical properties that characterize the diverse, known chemical space of organic π-conjugated molecules limits ML model development. Here, we present a curated dataset containing 25k molecules with density functional theory (DFT) and time-dependent DFT (TDDFT) evaluated properties that include frontier molecular orbitals, ionization energies, relaxation energies, and low-lying optical excitation energies. Using the dataset, we train a hierarchy of ML models, ranging from classical models such as ridge regression to sophisticated graph neural networks, with molecular SMILES representation as input. We observe that graph neural networks augmented with contextual information allow for significantly better predictions across a wide array of properties. Our best-performing models also provide an uncertainty quantification for the predictions. To democratize access to the data and trained models, an interactive web platform has been developed and deployed.

An exact one-particle theory of bosonic excitations: from a generalized Hohenberg–Kohn theorem to convexified N-representability

Motivated by the Penrose–Onsager criterion for Bose–Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh–Ritz variational principle to ensemble states with spectrum w and prove a corresponding generalization of the Hohenberg–Kohn theorem: the underlying one-particle reduced density matrix determines all properties of systems of N identical particles in their w -ensemble states. Then, to circumvent the v-representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy–Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body w -ensemble N-representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli’s exclusion principle for fermions and recently discovered generalizations thereof.

Low-lying excited state energy trap induced by cross-relaxation – The main origin of concentration quenching in lanthanide upconversion nanoparticles

In lanthanide-doped upconversion nanoparticles (UCNPs), the concentration of emitter ions, also known as activator ions, is usually limited to 1 − 5 mol% due to concentration quenching effects. This circumstance limits the luminescent efficiency of UCNPs’ and their use in a variety of application areas. Earlier studies have attributed the activator concentration quenching to migration of energy to the nanoparticle surface, while indicating that cross-relaxation between activator ions had a minor role therein. In this work, we carried out comparative studies on Er3+-doped and Yb3+-Er3+ codoped UCNPs and could, in contrast to this notion, prove a general adverse effect of cross-relaxation between activator ions, here Er3+ ions, on upconversion luminescence (UCL). The direct result of the cross-relaxation is that the energy of the excitation light is accumulated into a low-lying excited state of Er3+ in the infrared region, so forming a "low-lying excited state energy trap". As a result, the excitation energy is used for generating down-conversion luminescence or for indirectly facilitating UCL channels that are directly related to the low-lying excited state energy trap. The identified effect can be used to regulate UCL channels to achieve a concentrated UCL band that is more favorable for certain applications, e.g., biological imaging.

Study of nuclear low-lying excitation spectra with the Bayesian neural network approach

Nuclear low-lying excitation spectra are studied with the Bayesian neural network (BNN) approach by taking 02+, 21+, and 41+ states as examples. The BNN approach can well describe the low-lying excitation energies in a large energy scale from about 0.1 MeV to about several MeV, by including an input related to nuclear collectivity besides proton and neutron numbers and employing the logarithm of excitation energy as the output. Comparing with the sophisticated microscopic collective Hamiltonian model, the BNN approach significantly improves the description of nuclear low-lying excitation energies, which can generally reproduce the experimental data within about 1.12 times including those of transitional nuclei and magic nuclei. Taking Mg, Ca, Kr, Sm, and Pb isotopes as examples, it is found that the BNN approach well describes the isotopic trend of low-lying excitation energies, including those abrupt increases at magic numbers due to the shell effect, very low excitation energies of 02+ states due to the shape coexistence, and complex nuclear shape evolution due to the shape phase transition.

Multi-reference many-body perturbation theory for nuclei

In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small (\(e_{\mathrm {max}}=4\)) model space demonstrate that exact binding energies of closed- (\({}^{16}\mathrm {O}\)) and open-shell (\({}^{18}\mathrm {O}\), \({}^{20}\mathrm {Ne}\)) nuclei are reproduced within 0.5–\(1.5\%\) at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.

Excited States Computation of Models of Phenylalanine Protein Chains: TD-DFT and Composite CC2/TD-DFT Protocols.

The present benchmark calculations testify to the validity of time-dependent density functional theory (TD-DFT) when exploring the low-lying excited states potential energy surfaces of models of phenylalanine protein chains. Among three functionals suitable for systems exhibiting charge-transfer excited states, LC-ωPBE, CAM-B3LYP, and ωB97X-D, which were tested on a reference peptide system, we selected the ωB97X-D functional, which gave the best results compared to the approximate coupled-cluster singles and doubles (CC2) method. A quantitative agreement for both the geometrical parameters and the vibrational frequencies was obtained for the lowest singlet excited state (a ππ* state) of the series of capped peptides. In contrast, only a qualitative agreement was met for the corresponding adiabatic zero-point vibrational energy (ZPVE)-corrected excitation energies. Two composite protocols combining CC2 and DFT/TD-DFT methods were then developed to improve these calculations. Both protocols substantially reduced the error compared to CC2 and experiment, and the best of both even led to results of CC2 quality at a lower cost, thus providing a reliable alternative to this method for very large systems.

For an ab initio calculation of the magnetic excitations: RelaxSE!

In this paper, we present a novel efficient and parallel implementation, RelaxSE, for the calculation of the low-lying excited states and energies of strongly correlated systems. RelaxSE is based on the fully uncontracted multi-reference method of Selected Active Space + Single excitations. This method has been specifically designed to be able to tackle systems with numerous open shells per atoms. It is, however, computationally challenging due to the rapid scaling of the number of determinants and their non-trivial ordering induced by the selection process. We propose a combined determinant-driven and integral-driven approach designed for hybrid OpenMP/MPI parallelization. The performances of RelaxSE are evaluated on a controlled test set and show linear scaling with respect to the number of determinants and a small overhead due to the parallelization. Systems with up to 1 × 109 determinants are successfully computed.

Assessment of the Ab Initio Bethe-Salpeter Equation Approach for the Low-Lying Excitation Energies of Bacteriochlorophylls and Chlorophylls.

Bacteriochlorophyll and chlorophyll molecules are crucial building blocks of the photosynthetic apparatus in bacteria, algae, and plants. Embedded in transmembrane protein complexes, they are responsible for the primary processes of photosynthesis: excitation energy and charge transfer. Here, we use ab initio many-body perturbation theory within the GW approximation and Bethe–Salpeter equation (BSE) approach to calculate the electronic structure and optical excitations of bacteriochlorophylls a, b, c, d, and e and chlorophylls a and b. We systematically study the effects of the structure, basis set size, partial self-consistency in GW, and the underlying exchange–correlation approximation and compare our calculations with results from time-dependent density functional theory, multireference RASPT2, and experimental literature results. We find that optical excitations calculated with GW+BSE are in excellent agreement with experimental data, with an average deviation of less than 100 meV for the first three bright excitations of the entire family of (bacterio)chlorophylls. Contrary to state-of-the-art time-dependent density functional theory (TDDFT) with an optimally tuned range-separated hybrid functional, this accuracy is achieved in a parameter-free approach. Moreover, GW+BSE predicts the energy differences between the low-energy excitations correctly and eliminates spurious charge transfer states that TDDFT with (semi)local approximations is known to produce. Our study provides accurate reference results and highlights the potential of the GW+BSE approach for the simulation of larger pigment complexes.

Role of chiral symmetry in the nucleon excitation spectrum

The origin of the low-lying nature of the $N$*(1440), or Roper resonance, has been the subject of significant interest for many years, including several investigations using lattice QCD. The majority of lattice studies have not observed a low-lying excited state energy level in the region of the Roper resonance. However, it has been claimed that chiral symmetry could play an important role in our understanding of this resonance. The purpose of this study is to systematically examine the role of chiral symmetry in the low-lying nucleon spectrum by directly comparing the clover and overlap fermion actions. To ensure any differences in results are attributable to the choice of fermion action, simulations are performed on the same set of gauge field configurations at matched pion masses. Correlation matrix techniques are employed to determine the excitation energy of the first positive-parity excited state for each action. The clover and overlap actions show a remarkable level of agreement. We do not find any evidence that fermion action chiral symmetry plays a significant role in understanding the Roper resonance on the lattice.

All-electron ab initio Bethe-Salpeter equation approach to neutral excitations in molecules with numeric atom-centered orbitals.

The Bethe-Salpeter equation (BSE) based on GW quasiparticle levels is a successful approach for calculating the optical gaps and spectra of solids and also for predicting the neutral excitations of small molecules. We here present an all-electron implementation of the GW+BSE formalism for molecules, using numeric atom-centered orbital (NAO) basis sets. We present benchmarks for low-lying excitation energies for a set of small organic molecules, denoted in the literature as "Thiel's set." Literature reference data based on Gaussian-type orbitals are reproduced to about one millielectron-volt precision for the molecular benchmark set, when using the same GW quasiparticle energies and basis sets as the input to the BSE calculations. For valence correlation consistent NAO basis sets, as well as for standard NAO basis sets for ground state density-functional theory with extended augmentation functions, we demonstrate excellent convergence of the predicted low-lying excitations to the complete basis set limit. A simple and affordable augmented NAO basis set denoted "tier2+aug2" is recommended as a particularly efficient formulation for production calculations. We finally demonstrate that the same convergence properties also apply to linear-response time-dependent density functional theory within the NAO formalism.

Absolutely Localized Projection-Based Embedding for Excited States.

We present a quantum embedding method that allows for calculation of local excited states embedded in a Kohn-Sham density functional theory (DFT) environment. Projection-based quantum embedding methodologies provide a rigorous framework for performing DFT-in-DFT and wave function in DFT (WF-in-DFT) calculations. The use of absolute localization, where the density of each subsystem is expanded in only the basis functions associated with the atoms of that subsystem, provide improved computationally efficiency for WF-in-DFT calculations by reducing the number of orbitals in the WF calculation. In this work, we extend absolutely localized projection-based quantum embedding to study localized excited states using EOM-CCSD-in-DFT and TDDFT-in-DFT. The embedding results are highly accurate compared to the corresponding canonical EOM-CCSD and TDDFT results on the full system, with TDDFT-in-DFT frequently more accurate than canonical TDDFT. The absolute localization method is shown to eliminate the spurious low-lying excitation energies for charge-transfer states and prevent overdelocalization of excited states. Additionally, we attempt to recover the environment response caused by the electronic excitations in the high-level subsystem using different schemes and compare their accuracy. Finally, we apply this method to the calculation of the excited-state energy of green fluorescent protein and show that we systematically converge to the full system results. Here we demonstrate how this method can be useful in understanding excited states, specifically which chemical moieties polarize to the excitation. This work shows absolutely localized projection-based quantum embedding can treat local electronic excitations accurately and make computationally expensive WF methods applicable to systems beyond current computational limits.

Four-component relativistic time-dependent density-functional theory using a stable noncollinear DFT ansatz applicable to both closed- and open-shell systems.

We present a formulation of relativistic linear response time-dependent density functional theory for the calculation of electronic excitation energies in the framework of the four-component Dirac-Coulomb Hamiltonian. This approach is based on the noncollinear ansatz originally developed by Scalmani and Frisch [J. Chem. Theory Comput. 8, 2193 (2012)] and improves upon the past treatment of the limit cases in which the spin density approaches zero. As a result of these improvements, the presented approach is capable of treating both closed- and open-shell reference states. Robust convergence of the Davidson-Olsen eigenproblem algorithm for open-shell reference states was achieved through the use of a solver which considers both left and right eigenvectors. The applicability of the present methodology on both closed- and open-shell reference states is demonstrated on calculations of low-lying excitation energies for Group 3 atomic systems (Sc3+-Ac3+) with nondegenerate ground states, as well as for Group 11 atomic systems (Cu-Rg) and octahedral actinide complexes (PaCl6 2-, UCl6 -, and NpF6) with effective doublet ground states.

Charge transfer excitation energies from ground state density functional theory calculations.

Calculating charge transfer (CT) excitation energies with high accuracy and low computational cost is a challenging task. Kohn-Sham density functional theory (KS-DFT), due to its efficiency and accuracy, has achieved great success in describing ground state problems. To extend to excited state problems, our group recently demonstrated an approach with good numerical results to calculate low-lying and Rydberg excitation energies of an N-electron system from a ground state KS or generalized KS calculations of an (N - 1)-electron system via its orbital energies. In the present work, we explore further the same methodology to describe CT excitations. Numerical results from this work show that performance of conventional density functional approximations (DFAs) is not as good for CT excitations as for other excitations due to the delocalization error. Applying localized orbital scaling correction (LOSC) to conventional DFAs, a recently developed method in our group to effectively reduce the delocalization error, can improve the results. Overall, the performance of this methodology is better than time dependent DFT (TDDFT) with conventional DFAs. In addition, it shows that results from LOSC-DFAs in this method reach similar accuracy to other methods, such as ΔSCF, G0W0 with Bethe-Salpeter equations, particle-particle random phase approximation, and even high-level wavefunction methods like CC2. Our analysis shows that the correct 1/R trend for CT excitation can be captured from LOSC-DFA calculations, stressing that the application of DFAs with the minimal delocalization error is essential within this methodology. This work provides an efficient way to calculate CT excitation energies from ground state DFT.

Combining symmetry breaking and restoration with configuration interaction: Extension to z -signature symmetry in the case of the Lipkin model

Background: Ab initio many-body methods whose numerical cost scales polynomially with the number of particles have been developed over the past fifteen years to tackle closed-shell mid-mass nuclei. Open-shell nuclei have been further addressed by implementing variants based on the concept of spontaneous symmetry breaking (and restoration). Purpose: In order to access the spectroscopy of open-shell nuclei more systematically while controlling the numerical cost, we design a novel many-body method that combines the merit of breaking and restoring symmetries with those brought about by low-rank individual excitations. Methods: The recently proposed truncated configuration-interaction method based on optimized symmetry-broken and -restored states is extended to the z-signature symmetry associated with a discrete subgroup of SU(2). The highly-truncated N-body Hilbert subspace within which the Hamiltonian is diagonalized is spanned by a z-signature broken and restored Slater determinant vacuum and associated low-rank excitations. Results: The proposed method provides an excellent reproduction of the ground-state energy and of low-lying excitation energies of various z-signatures and total angular momenta. In doing so, the successive benefits of (i) breaking the symmetry, (ii) restoring the symmetry, (iii) including low-rank particle-hole excitations and (iv) optimizing the amount by which the underlying vacuum breaks the symmetry are illustrated. Conclusions: The numerical cost of the newly designed variational method is polynomial with respect to the system size. The present study confirms the results obtained previously for the attractive pairing Hamiltonian in connection with the breaking and restoration of U(1) global gauge symmetry. These two studies constitute a strong motivation to apply this method to realistic nuclear Hamiltonians.

Equation of motion approach for describing allowed transitions in Ne and Al3+ under classical and quantum plasmas

The equation of motion coupled cluster methodology within relativistic framework has been applied to analyze the electron correlation effects on the low lying dipole allowed excited states of Ne and Al3+ under classical and quantum plasma environments. The effect of confinement due to classical plasma has been incorporated through screened Coulomb potential, while that of quantum plasma has been treated by exponential cosine screened Coulomb potential. The confined structural properties investigated are the depression of ionization potential, low lying excitation energies (dipole allowed), oscillator strengths, transition probabilities, and frequency dependent polarizabilities under systematic variation of the plasma-atom coupling strength determined through the screening parameter. Specific atomic systems are chosen for their astrophysical importance and availability of experimental data related to laboratory plasma with special reference to Al3+ ion. Here, we consider 1s22s22p6(1S0)→1s22s22p5 ns/nd (1P1) (n=3,4) dipole allowed transitions of Ne and Al3+. Results for the free (isolated) atomic systems agree well with those available in the literature. Spectroscopic properties under confinement show systematic and interesting pattern with respect to plasma screening parameter.

Speed-Up of the Excited-State Benchmarking: Double-Hybrid Density Functionals as Test Cases.

The EX6-0, EX7-0, and EX7-1 representative benchmark sets are developed for the fast evaluation of the performance of a density functional, or more generally of a computational protocol, in modeling low-lying valence singlet-singlet excitation energies of organic dyes within the range of 1.5 to 4.5 eV. All sets share the advantage of being small (a maximum of 7 molecules) but providing statistical errors representative of larger and extended databases. To that extent, the EX7-1 benchmark set goes a step further and is composed of systems as small as possible in order to alleviate the associated computational cost. The reliability of all the sets is assessed through the benchmarking of 15 modern double-hybrid density functionals. The investigation shows not only that the 3 benchmark sets provide close error metrics for each density functional but also that when taking advantage of the Resolution-of-the-Identity and a balanced triple-ζ basis set (e.g., def2-TZVP), double hybrids overperform the "popular" hybrids in modeling vertical absorption, emission, and adiabatic energies.