Electronic States of the Benzene Dimer: A Simple Case of Complexity

Electronic structure calculations of the excited states of the benzene dimer using equation-of-motion coupled-cluster method are reported. The calculations reveal large density of electronic states, including multiple valence, Rydberg, and mixed Rydberg-valence states. The calculations of the oscillator strengths for the transitions between the excimer state (i.e., the lowest excited state of the dimer, 1¹<i>B</i>_1<i>g</i>) and other excited states allowed us to identify the target state responsible for the excimer absorption as the E_1<i>u</i> state of a mixed Rydberg-valence character at 3.04 eV above the excimer (1¹<i>B</i>_1<i>g</i>). Although at <i>D</i>_6<i>h</i> the 1¹<i>B</i>_1<i>g</i> → <i>E</i>_1<i>u</i> transition is symmetry-forbidden, small geometric displacements (to <i>D</i>_2<i>h</i>) that have a negligible effect on the excitation energy split this degenerate state into the dark (4<i>B</i>_3<i>u</i>) and bright (4<i>B</i>_2<i>u</i>) components (oscillator strength of 0.3 au). The excitation energy for this transition depends strongly on the dimer structure, which explains the broad character of the experimentally observed excimer absorption spectrum

Non-Condon Effects in the One- and Two-Photon Absorption Spectra of the Green Fluorescent Protein

We report calculations of one and two photon absorption (OPA and TPA, respectively) spectra of the green fluorescent protein (GFP) chromophore using the double-harmonic parallel-mode approximation and including explicit dependence of the electronic transition moments on nuclear geometry. The non-Condon effects are found to be more significant for TPA resulting in a different shape of the spectra and a blue shift of 500 cm⁻¹ of the TPA peak absorption relative to OPA. The computed shift is in excellent agreement with the experimentally observed 700 cm⁻¹

Visualizing the Contributions of Virtual States to Two-Photon Absorption Cross Sections by Natural Transition Orbitals of Response Transition Density Matrices

Observables such as two-photon absorption cross sections cannot be computed from the wave functions of initial and final states alone because of their nonlinear nature. Rather, they depend on the entire manifold of the excited states, which follows from the familiar sum-over-states expressions of second- and higher-order properties. Consequently, the interpretation of the computed nonlinear optical properties in terms of molecular orbitals is not straightforward and usually relies on approximate few-states models. Here, we show that the two-photon absorption (2PA) transitions can be visualized using response one-particle transition density matrices, which are defined as transition density matrices between the zero-order and first-order perturbed states. We also extend the concept of natural transition orbitals to 2PA transitions. We illustrate the utility of this new tool, which provides a rigorous black box alternative to traditional qualitative few-states analysis, by considering 2PA transitions in ethylene, <i>trans</i>-stilbene, and <i>para</i>-nitroaniline

On the Nature of an Extended Stokes Shift in the mPlum Fluorescent Protein

Far-red fluorescent proteins (FPs) enable deep-tissue in vivo imaging. Combining FPs with large and small Stokes shifts enables single-excitation/dual-emission multicolor applications. Using a quantum mechanics/molecular mechanics (QM/MM) scheme, we carried out a series of simulations to identify the origin of an extended Stokes shift (0.2 eV) observed in mPlum, one of the most far-red-shifted FPs. We demonstrated that the red shift of emission is largely due to the excited-state relaxation of the chromophore itself. Rigid protein environment suppresses the relaxation; however, if the hydrogen-bond network around the chromophore is sufficiently flexible, it can rearrange upon electronic excitation, allowing the chromophore to relax. The reorganization of the hydrogen-bond network is driven by changes in bonding and charge distributions of the chromophore in the excited state. The ILE65 and GLU16 residues play the most important role. The MD simulations reveal two ground-state populations with the direct (Chro-ILE65···GLU16) and water-mediated (Chro-ILE65···Wat321···GLU16) hydrogen-bond patterns. In the excited state, both populations relax to a single emitting state with the water-mediated (Chro-ILE65···Wat321···GLU16) hydrogen-bond pattern, which provides a better match for the excited-state charge distribution (the acylimine’s oxygen has a larger negative charge in S₁ than in S₀). The extended Stokes shift arises due to the conversion of the direct hydrogen-bond pattern to the water-mediated one accompanied by large structural relaxation of the electronically excited chromophore. This conclusion is supported by calculations for the GLU16LEU mutant, which has only one hydrogen-bond pattern. Consequently, no interconversion is possible, and the computed Stokes shift is small, in agreement with the experiment. Our theoretical findings provide support to a recent study of the Stokes shifts in mPlum and its mutants

Real and Imaginary Excitons: Making Sense of Resonance Wave Functions by Using Reduced State and Transition Density Matrices

Within non-Hermitian quantum mechanics, metastable electronic states can be represented by isolated <i>L</i>²-integrable complex-valued wave functions with complex energies. An analysis scheme of the real and imaginary parts of resonance wave functions by using reduced transition density matrices and natural transition orbitals is presented. While the real parts of excitons describe changes in the electron density corresponding to the bound part of the resonance, the imaginary excitons can be interpreted as virtual states facilitating one-electron decay into the continuum. The different nature of real and imaginary excitons is revealed by exciton descriptors, in particular hole-particle separation and their correlation. Singular values and respective participation ratios quantify the extent of collectivity of the excitation and a number of distinct decay channels. The utility of the new tool is illustrated by the analysis of bound and metastable excited states of cyanopolyyne anions

Modeling Photoelectron Spectra of CuO, Cu₂O, and CuO₂ Anions with Equation-of-Motion Coupled-Cluster Methods: An Adventure in Fock Space

The experimental photoelectron spectra of di- and triatomic copper oxide anions have been reported previously. We present an analysis of the experimental spectra of the CuO^–, Cu₂O^–, and CuO₂^– anions using equation-of-motion coupled-cluster (EOM-CC) methods. The open-shell electronic structure of each molecule demands a unique combination of EOM-CC methods to achieve an accurate and balanced representation of the multiconfigurational anionic- and neutral-state manifolds. Analysis of the Dyson orbitals associated with photodetachment from CuO^– reveals the strong non-Koopmans character of the CuO states. For the lowest detachment energy, a good agreement between theoretical and experimental values is obtained with CCSD­(T) (coupled-cluster with single and double excitations and perturbative account of triple excitations). The (T) correction is particularly important for Cu₂O^–. Use of a relativistic pseudopotential and matching basis set improves the quality of results in most cases. EOM-DIP-CCSD analysis of the low-lying states of CuO₂^– reveals multiple singlet and triplet anionic states near the triplet ground state, adding an extra layer of complexity to the interpretation of the experimental CuO₂^– photoelectron spectrum

Complex Absorbing Potential Equation-of-Motion Coupled-Cluster Method Yields Smooth and Internally Consistent Potential Energy Surfaces and Lifetimes for Molecular Resonances

The recently developed equation-of-motion electron-attachment coupled-cluster singles and doubles (EOM-EA-CCSD) method augmented by a complex absorbing potential (CAP) is applied to the ²Π_g resonance of N₂^– and the ²Σ_u⁺ resonance of H₂^– at various internuclear distances. The results illustrate the advantages of EOM-CC for treating resonance states over state-specific approaches. CAP-EOM-EA-CCSD produces smoothly varying potential energy curves and lifetimes for both Σ and Π resonances. The computed lifetimes and energy differences between the neutral and electron-attached states are internally consistent, that is, the resonance width becomes zero at the same internuclear distance where the energy of the electron-attached state drops below that of the neutral state. Such smooth and internally consistent behavior is only achieved when the perturbation due to the CAP is removed using the first-order deperturbative correction that we introduced earlier; the evaluation of resonance positions and widths from raw (uncorrected) energies leads to unphysical discontinuities and fails to correctly describe the conversion of a resonance to a bound state at large internuclear distances

Probing Electronic Wave Functions of Sodium-Doped Clusters: Dyson Orbitals, Anisotropy Parameters, and Ionization Cross-Sections

We apply high-level ab initio methods to describe the electronic structure of small clusters of ammonia and dimethyl ether (DME) doped with sodium, which provide a model for solvated electrons. We investigate the effect of the solvent and cluster size on the electronic states. We consider both energies and properties, with a focus on the shape of the electronic wave function and the related experimental observables such as photoelectron angular distributions. The central quantity in modeling photoionization experiments is the Dyson orbital, which describes the difference between the initial <i>N</i>-electron and final (<i>N</i>–1)-electron states of a system. Dyson orbitals enter the expression of the photoelectron matrix element, which determines total and partial photoionization cross-sections. We compute Dyson orbitals for the Na­(NH₃)_<i>n</i> and Na­(DME)_<i>m</i> clusters using correlated wave functions (obtained with equation-of-motion coupled-cluster model for electron attachment with single and double substitutions) and compare them with more approximate Hartree-Fock and Kohn-Sham orbitals. We also analyze the effect of correlation and basis sets on the shapes of Dyson orbitals and the experimental observables

On the Photodetachment from the Green Fluorescent Protein Chromophore

Motivated by the discrepancies in recent experimental and theoretical studies of photodetachment from isolated model chromophores of the green fluorescent protein (GFP), this study reports calculations of the electron detachment energies and photoelectron spectra of the phenolate and deprotonated <i>p</i>-hydroxybenzylidene-2,3-dimethylimidazolinone (HBDI) anions. The spectra were computed using double-harmonic parallel normal mode approximation. High-level coupled-cluster methods as well as density functional theory were used to compute vertical and adiabatic detachment energies of the phenolate anion serving as a model system representing anionic GFP-like chromophores (HBDI). The benchmark calculations reveal that the basis set has significant effect on the computed detachment energies, whereas the results are less sensitive to the level of electron correlation treatment. At least aug-cc-pVTZ basis set is required. The best ωB97X-D and CCSD­(T) estimates of phenolate’s adiabatic detachment energy are 2.12 and 2.19 eV; these values are very close to the experimental value, 2.253 eV [Gunion et al. <i>Int. J. Mass Spectrom. Ion Proc.</i> <b>1992</b>, <i>117</i>, 601]. The best estimate of the vertical detachment energy of deprotonated HBDI is 2.76 eV, which supports bound character of the bright excited state in the Franck–Condon region. The most intense transition in the computed photoelectron spectra of both phenolate and deprotonated HBDI is the 0–0 S₀–D₀ transition, which is 0.11 eV below vertical detachment energy. Therefore, the position of the maximum of the photoelectron spectrum does not represent vertical detachment energy, and the direct comparison between theory and experiment must involve spectrum modeling

Supplementary Information Files for: libwfa: Wavefunction analysis tools for excited and open‐shell electronic states

Supplementary Information Files for: libwfa: Wavefunction analysis tools for excited and open‐shell electronic statesAn open-source software library for wavefunction analysis, libwfa, provides a comprehensive and flexible toolbox for post-processing excited-state calculations, featuring a hierarchy of interconnected visual and quantitative analysis methods. These tools afford compact graphical representations of various excited-state processes, provide detailed insight into electronic structure, and are suitable for automated processing of large data sets. The analysis is based on reduced quantities, such as state and transition density matrices (DMs), and allows one to distill simple molecular orbital pictures of physical phenomena from intricate correlated wavefunctions. The implemented descriptors provide a rigorous link between many-body wavefunctions and intuitive physical and chemical models, for example, exciton binding, double excitations, orbital relaxation, and polyradical character. A broad range of quantum-chemical methods is interfaced with libwfa via a uniform interface layer in the form of DMs. This contribution reviews the structure of libwfa and highlights its capabilities by several representative use cases.<br