Adapting algebraic diagrammatic construction schemes for the polarization propagator to problems with multi-reference electronic ground states exploiting the spin-flip ansatz.

For the investigation of molecular systems with electronic ground states exhibiting multi-reference character, a spin-flip (SF) version of the algebraic diagrammatic construction (ADC) scheme for the polarization propagator up to third order perturbation theory (SF-ADC(3)) is derived via the intermediate state representation and implemented into our existing ADC computer program adcman. The accuracy of these new SF-ADC(n) approaches is tested on typical situations, in which the ground state acquires multi-reference character, like bond breaking of H2 and HF, the torsional motion of ethylene, and the excited states of rectangular and square-planar cyclobutadiene. Overall, the results of SF-ADC(n) reveal an accurate description of these systems in comparison with standard multi-reference methods. Thus, the spin-flip versions of ADC are easy-to-use methods for the calculation of "few-reference" systems, which possess a stable single-reference triplet ground state.

Statistical analysis of electronic excitation processes: Spatial location, compactness, charge transfer, and electron-hole correlation.

We report the development of a set of excited-state analysis tools that are based on the construction of an effective exciton wavefunction and its statistical analysis in terms of spatial multipole moments. This construction does not only enable the quantification of the spatial location and compactness of the individual hole and electron densities but also correlation phenomena can be analyzed, which makes this procedure particularly useful when excitonic or charge-resonance effects are of interest. The methods are first applied to bianthryl with a focus on elucidating charge-resonance interactions. It is shown how these derive from anticorrelations between the electron and hole quasiparticles, and it is discussed how the resulting variations in state characters affect the excited-state absorption spectrum. As a second example, cytosine is chosen. It is illustrated how the various descriptors vary for valence, Rydberg, and core-excited states, and the possibility of using this information for an automatic characterization of state characters is discussed.

Analysis and comparison of CVS-ADC approaches up to third order for the calculation of core-excited states.

The extended second order algebraic-diagrammatic construction (ADC(2)-x) scheme for the polarization operator in combination with core-valence separation (CVS) approximation is well known to be a powerful quantum chemical method for the calculation of core-excited states and the description of X-ray absorption spectra. For the first time, the implementation and results of the third order approach CVS-ADC(3) are reported. Therefore, the CVS approximation has been applied to the ADC(3) working equations and the resulting terms have been implemented efficiently in the adcman program. By treating the α and ß spins separately from each other, the unrestricted variant CVS-UADC(3) for the treatment of open-shell systems has been implemented as well. The performance and accuracy of the CVS-ADC(3) method are demonstrated with respect to a set of small and middle-sized organic molecules. Therefore, the results obtained at the CVS-ADC(3) level are compared with CVS-ADC(2)-x values as well as experimental data by calculating complete basis set limits. The influence of basis sets is further investigated by employing a large set of different basis sets. Besides the accuracy of core-excitation energies and oscillator strengths, the importance of cartesian basis functions and the treatment of orbital relaxation effects are analyzed in this work as well as computational timings. It turns out that at the CVS-ADC(3) level, the results are not further improved compared to CVS-ADC(2)-x and experimental data, because the fortuitous error compensation inherent in the CVS-ADC(2)-x approach is broken. While CVS-ADC(3) overestimates the core excitation energies on average by 0.61% ± 0.31%, CVS-ADC(2)-x provides an averaged underestimation of -0.22% ± 0.12%. Eventually, the best agreement with experiments can be achieved using the CVS-ADC(2)-x method in combination with a diffuse cartesian basis set at least at the triple-ζ level.

Experimental benchmark data and systematic evaluation of two a posteriori, polarizable-continuum corrections for vertical excitation energies in solution.

We report the implementation and evaluation of a perturbative, density-based correction scheme for vertical excitation energies calculated in the framework of a polarizable continuum model (PCM). Because the proposed first-order correction terms depend solely on the zeroth-order excited-state density, a transfer of the approach to any configuration interaction-type excited-state method is straightforward. Employing the algebraic-diagrammatic construction (ADC) scheme of up to third order as well as time-dependent density-functional theory (TD-DFT), we demonstrate and evaluate the approach. For this purpose, we assembled a set of experimental benchmark data for solvatochromism in molecules (xBDSM) containing 44 gas-phase to solvent shifts for 17 molecules. These data are compared to solvent shifts calculated at the ADC(1), ADC(2), ADC(3/2), and TD-DFT/LRC-ωPBE levels of theory in combination with state-specific as well as linear-response type PCM-based correction schemes. Some unexpected trends and differences between TD-DFT, the levels of ADC, and variants of the PCM are observed and discussed. The most accurate combinations reproduce experimental solvent shifts resulting from the bulk electrostatic interaction with maximum errors in the order of 50 meV and a mean absolute deviation of 20-30 meV for the xBDSM set.

The third-order algebraic diagrammatic construction method (ADC(3)) for the polarization propagator for closed-shell molecules: efficient implementation and benchmarking.

The implementation of an efficient program of the algebraic diagrammatic construction method for the polarisation propagator in third-order perturbation theory (ADC(3)) for the computation of excited states is reported. The accuracies of ADC(2) and ADC(3) schemes have been investigated with respect to Thiel's recently established benchmark set for excitation energies and oscillator strengths. The calculation of 141 vertical excited singlet and 71 triplet states of 28 small to medium-sized organic molecules has revealed that ADC(3) exhibits mean error and standard deviation of 0.12 ± 0.28 eV for singlet states and -0.18 ± 0.16 eV for triplet states when the provided theoretical best estimates are used as benchmark. Accordingly, the ADC(2)-s and ADC(2)-x calculations revealed accuracies of 0.22 ± 0.38 eV and -0.70 ± 0.37 eV for singlets and 0.12 ± 0.16 eV and -0.55 ± 0.20 eV for triplets, respectively. For a comparison of CC3 and ADC(3), only non-CC3 benchmark values were considered, which comprise 84 singlet states and 19 triplet states. For these singlet states CC3 exhibits an accuracy of 0.23 ± 0.21 eV and ADC(3) an accuracy of 0.08 ± 0.27 eV, and accordingly for the triplet states of 0.12 ± 0.10 eV and -0.10 ± 0.13 eV, respectively. Hence, based on the quality of the existing benchmark set it is practically not possible to judge whether ADC(3) or CC3 is more accurate, however, ADC(3) has a much larger range of applicability due to its more favourable scaling of O(N(6)) with system size.

Calculating core-level excitations and X-ray absorption spectra of medium-sized closed-shell molecules with the algebraic-diagrammatic construction scheme for the polarization propagator.

Core-level excitations are generated by absorption of high-energy radiation such as X-rays. To describe these energetically high-lying excited states theoretically, we have implemented a variant of the algebraic-diagrammatic construction scheme of second-order ADC(2) by applying the core-valence separation (CVS) approximation to the ADC(2) working equations. Besides excitation energies, the CVS-ADC(2) method also provides access to properties of core-excited states, thereby allowing for the calculation of X-ray absorption spectra. To demonstrate the potential of our implementation of CVS-ADC(2), we have chosen medium-sized molecules as examples that have either biological importance or find application in organic electronics. The calculated results of CVS-ADC(2) are compared with standard TD-DFT/B3LYP values and experimental data. In particular, the extended variant, CVS-ADC(2)-x, provides the most accurate results, and the agreement between the calculated values and experiment is remarkable.

Strong enhancement of parity violation effects in chiral uranium compounds.

The effects of parity violation (PV) on the vibrational transitions of chiral uranium compounds of the type N≡UXYZ and N≡UHXY (X, Y, Z = F, Cl, Br, I) are analysed by means of exact two-component relativistic (X2C) Hartree-Fock and density functional calculations using NUFClI and NUHFI as representative examples. The PV contributions to the vibrational transitions were found to be in the Hz range, larger than for any of the earlier proposed chiral molecules. Thus, these systems are very promising candidates for future experimental PV measurements. A detailed comparison of the N≡UHFI and the N≡WHFI homologues reveals that subtle electronic structure effects, rather than exclusively a simple Z(5) scaling law, are the cause of the strong enhancement in PV contributions of the chiral uranium molecules.

New tools for the systematic analysis and visualization of electronic excitations. I. Formalism.

A variety of density matrix based methods for the analysis and visualization of electronic excitations are discussed and their implementation within the framework of the algebraic diagrammatic construction of the polarization propagator is reported. Their mathematical expressions are given and an extensive phenomenological discussion is provided to aid the interpretation of the results. Starting from several standard procedures, e.g., population analysis, natural orbital decomposition, and density plotting, we proceed to more advanced concepts of natural transition orbitals and attachment/detachment densities. In addition, special focus is laid on information coded in the transition density matrix and its phenomenological analysis in terms of an electron-hole picture. Taking advantage of both the orbital and real space representations of the density matrices, the physical information in these analysis methods is outlined, and similarities and differences between the approaches are highlighted. Moreover, new analysis tools for excited states are introduced including state averaged natural transition orbitals, which give a compact description of a number of states simultaneously, and natural difference orbitals (defined as the eigenvectors of the difference density matrix), which reveal details about orbital relaxation effects.

New tools for the systematic analysis and visualization of electronic excitations. II. Applications.

The excited states of a diverse set of molecules are examined using a collection of newly implemented analysis methods. These examples expose the particular power of three of these tools: (i) natural difference orbitals (the eigenvectors of the difference density matrix) for the description of orbital relaxation effects, (ii) analysis of the one-electron transition density matrix in terms of an electron-hole picture to identify charge resonance and excitonic correlation effects, and (iii) state-averaged natural transition orbitals for a compact simultaneous representation of several states. Furthermore, the utility of a wide array of additional analysis methods is highlighted. Five molecules with diverse excited state characteristics are chosen for these tasks: pyridine as a prototypical small heteroaromatic molecule, a model system of six neon atoms to study charge resonance effects, hexatriene in its neutral and radical cation forms to exemplify the cases of double excitations and spin-polarization, respectively, and a model iridium complex as a representative metal organic compound. Using these examples a number of phenomena, which are at first sight unexpected, are highlighted and their physical significance is discussed. Moreover, the generality of the conclusions of this paper is verified by a comparison of single- and multireference ab initio methods.

Calculating X-ray Absorption Spectra of Open-Shell Molecules with the Unrestricted Algebraic-Diagrammatic Construction Scheme for the Polarization Propagator.

X-ray absorption spectroscopy (XAS) is a powerful tool that provides information about the electronic structure of molecules via excitation of electrons from the K-shell core region to the unoccupied molecular levels. These high-lying electronic core-excited states can be accurately calculated using the algebraic-diagrammatic construction scheme of second order ADC(2) by applying the core-valence separation (CVS) approximation to the ADC(2) working equations. For the first time, an efficient implementation of an unrestricted CVS-ADC(2) variant CVS-UADC(2) is presented for the calculation of open-shell molecules by treating α and ß spins separately from each other. The potential of the CVS-UADC(2) method is demonstrated with a set of small organic radicals by comparison with standard TD-DFT/B3LYP values and experimental data. It turns out that the extended variant CVS-UADC(2)-x, in particular, provides the most accurate results with errors of only 0.1% compared to experimental values. This remarkable agreement justifies the prediction of yet nonrecorded experimental XAS spectra like the one of the anthracene cation. The cation exhibits additional peaks due to the half-filled single-occupied molecular orbital, which may help to distinguish cation from the neutral species.

New implementation of high-level correlated methods using a general block tensor library for high-performance electronic structure calculations.

This article presents an open-source object-oriented C++ library of classes and routines to perform tensor algebra.The primary purpose of the library is to enable post-Hartree­Fock electronic structure methods; however, the code is general enough to be applicable in other areas of physical and computational sciences. The library supports tensors of arbitrary order (dimensionality), size, and symmetry. Implemented data structures and algorithms operate on large tensors by splitting them into smaller blocks, storing them both in core memory and in files on disk, and applying divide-and-conquer-type parallel algorithms to perform tensor algebra. The library offers a set of general tensor symmetry algorithms and a full implementation of tensor symmetries typically found in electronic structure theory: permutational, spin, and molecular point group symmetry. The Q-Chem electronic structure software uses this library to drive coupled-cluster, equation-of-motion, and algebraic-diagrammatic construction methods.

Evidence for low-temperature melting of mercury owing to relativity.

An old problem solved: Monte Carlo simulations using the diatomic-in-molecule method derived from accurate ground- and excited-state relativistic calculations for Hg2 show that the melting temperature for bulk mercury is lowered by 105 K, which is due to relativistic effects.

Ultrafast charge separation driven by differential particle and hole mobilities.

The process of a local excitation evolving into an intramolecular charge-separated state is followed and compared for several systems by directly simulating the time propagation of the electronic wavefunction. The wavefunction and Hamiltonian are handled using the extended second-order algebraic diagrammatic construction (ADC(2)-x), which explicitly accounts for electron correlation in the dynamic many-particle state. The details of the charge separation can be manipulated according to the chemical composition of the system; atoms which dope the conjugated system with either particles or holes are shown to effect whether the particle or hole is more mobile. Initially, the charges oscillate between the ends of linear molecules (with different rates), separating periodically, but, at long times, both charges tend to spread over the whole molecule. Charge separation is also shown to occur for asymmetric systems, where it may eventually be experimentally feasible to excite a localized resonance (nonstationary state) on one end of the molecule preferentially and follow the ensuing dynamics.

Tracing molecular electronic excitation dynamics in real time and space.

We present a method for studying the movement of electrons and energy within and between electronically excited molecules. The dynamically changing state is a many-electron wavepacket, for which we numerically integrate the Schrodinger equation using the ADC(2) effective Hamiltonian for the particle-hole propagator. We develop the tools necessary for following the separate motions of the particles and holes. Total particle and hole densities can be used to give an overview of the dynamics, which can be atomically decomposed in a Mulliken fashion, or individual particle and hole states give a more detailed look at the structure of an excitation. We apply our model to a neon chain, as an illustrative example, projecting an excited eigenstate of an isolated atom onto the coupled system as the initial state. In addition to demonstrating our propagation and analysis machinery, the results show a dramatic difference in excitation-energy transfer rates as a consequence of initial polarization. Furthermore, already in a system with three constituents, an important aspect of multiple coupled systems appears, in that one absorbing system essentially shields another, changing the effective sitewise coupling parameters.

Nature of the lowest excited states of neutral polyenyl radicals and polyene radical cations.

Due to the close relation of the polyenyl radicals C(2n+1)H(2n+3) () and polyene radical cations C(2n)H(2n+2) (+) to the neutral linear polyenes, one may suspect their excited states to possess substantial double excitation character, similar to the famous S(1) state of neutral polyenes and thus to be equally problematic for simple excited state theories. Using the recently developed unrestricted algebraic-diagrammatic construction scheme of second order perturbation theory and the equation-of-motion coupled-cluster method, the vertical excitation energies, their corresponding oscillator strengths, and the nature of the wave functions of the lowest excited electronic states of the radicals are calculated and analyzed in detail. For the polyenyl radicals two one-photon allowed states are found as D(1) and D(4) states, with two symmetry-forbidden D(2) and D(3) states in between, while in the polyene radical cations D(1) and D(2) are allowed and D(3) is forbidden. The order of the states is conserved with increasing chain length. It is found that all low-lying excited states exhibit a significant but similar amount of doubly excited configuration in their wave functions of 15%-20%. Using extrapolation, predictions for the excitation energies of the five lowest excited states of the polyene radical cations are made for longer chain lengths.

Excitation energy transfer and carotenoid radical cation formation in light harvesting complexes - a theoretical perspective.

Light harvesting complexes have been identified in all chlorophyll-based photosynthetic organisms. Their major function is the absorption of light and its transport to the reaction centers, however, they are also involved in excess energy quenching, the so-called non-photochemical quenching (NPQ). In particular, electron transfer and the resulting formation of carotenoid radical cations have recently been discovered to play an important role during NPQ in green plants. Here, the results of our theoretical investigations of carotenoid radical cation formation in the major light harvesting complex LHC-II of green plants are reported. The carotenoids violaxanthin, zeaxanthin and lutein are considered as potential quenchers. In agreement with experimental results, it is shown that zeaxanthin cannot quench isolated LHC-II complexes. Furthermore, subtle structural differences in the two lutein binding pockets lead to substantial differences in the excited state properties of the two luteins. In addition, the formation mechanism of carotenoid radical cations in light harvesting complexes LH2 and LH1 of purple bacteria is studied. Here, the energetic position of the S(1) state of the involved carotenoids neurosporene, spheroidene, spheroidenone and spirilloxanthin seems to determine the occurrence of radical cations in these LHCs upon photo-excitation. An elaborate pump-deplete-probe experiment is suggested to challenge the proposed mechanism.

Unrestricted algebraic diagrammatic construction scheme of second order for the calculation of excited states of medium-sized and large molecules.

An unrestricted version of the algebraic diagrammatic construction (ADC) scheme of the polarization propagator in second order perturbation theory [UADC(2)] is derived via the intermediate state representation. The accuracy of the extended UADC(2)-x approach is evaluated by comparison of computed excitation energies of 11 medium-sized radicals with their corresponding experimental literature values and with excitation energies computed at equation-of-motion-CCSD (coupled clusters singles and doubles) level of theory. Overall, our numerical tests show that UADC(2)-x exhibits an averaged mean deviation in the excitation energies of only 0.3-0.4 eV compared to experimental gas phase data. It provides thus an alternative to coupled-cluster based approaches for the calculation of excited states of medium-sized open-shell molecules.

Simple replacement of violaxanthin by zeaxanthin in LHC-II does not cause chlorophyll fluorescence quenching.

Recently, a mechanism for the energy-dependent component (qE) of non-photochemical quenching (NPQ), the fundamental photo-protection mechanism in green plants, has been suggested. Replacement of violaxanthin by zeaxanthin in the binding pocket of the major light harvesting complex LHC-II may be sufficient to invoke efficient chlorophyll fluorescence quenching. Our quantum chemical calculations, however, show that the excited state energies of violaxanthin and zeaxanthin are practically identical when their geometry is constrained to the naturally observed structure of violaxanthin in LHC-II. Therefore, since violaxanthin does not quench LHC-II, zeaxanthin should not either. This theoretical finding is nicely in agreement with experimental results obtained by femtosecond spectroscopy on LHC-II complexes containing violaxanthin or zeaxanthin.

Quantum chemical insights in energy dissipation and carotenoid radical cation formation in light harvesting complexes.

Light harvesting complexes (LHCs) have been identified in all photosynthetic organisms. To understand their function in light harvesting and energy dissipation, detailed knowledge about possible excitation energy transfer (EET) and electron transfer (ET) processes in these pigment proteins is of prime importance. This again requires the study of electronically excited states of the involved pigment molecules, in LHCs of chlorophylls and carotenoids. This paper represents a critical review of recent quantum chemical calculations on EET and ET processes between pigment pairs relevant for the major LHCs of green plants (LHC-II) and of purple bacteria (LH2). The theoretical methodology for a meaningful investigation of such processes is described in detail, and benefits and limitations of standard methods are discussed. The current status of excited state calculations on chlorophylls and carotenoids is outlined. It is focused on the possibility of EET and ET in the context of chlorophyll fluorescence quenching in LHC-II and carotenoid radical cation formation in LH2. In the context of non-photochemical quenching of green plants, it is shown that replacement of the carotenoid violaxanthin by zeaxanthin in its binding pocket of LHC-II can not result in efficient quenching. In LH2, our computational results give strong evidence that the S(1) states of the carotenoids are involved in carotenoid cation formation. By comparison of theoretical findings with recent experimental data, a general mechanism for carotenoid radical cation formation is suggested.

Carotenoid radical cation formation in LH2 of purple bacteria: a quantum chemical study.

In LH2 complexes of Rhodobacter sphaeroides the formation of a carotenoid radical cation has recently been observed upon photoexcitation of the carotenoid S2 state. To shed more light onto the yet unknown molecular mechanism leading to carotenoid radical formation in LH2, the interactions between carotenoid and bacteriochlorophyll in LH2 are investigated by means of quantum chemical calculations for three different carotenoids--neurosporene, spheroidene, and spheroidenone--using time-dependent density functional theory. Crossings of the calculated potential energy curve of the electron transfer state with the bacteriochlorophyll Qx state and the carotenoid S1 and S2 states occur along an intermolecular distance coordinate for neurosporene and spheroidene, but for spheroidenone no crossing of the electron transfer state with the carotenoid S1 state could be found. By comparison with recent experiments where no formation of a spheroidenone radical cation has been observed, a molecular mechanism for carotenoid radical cation formation is proposed in which it is formed via a vibrationally excited carotenoid S1 or S*state. Arguments are given why the formation of the carotenoid radical cation does not proceed via the Qx, S2, or higher excited electron transfer states.