Tim Joachim Zuehlsdorff

Linear-scaling time-dependent density-functional theory in the linear response formalism

We present an implementation of time-dependent density-functional theory (TDDFT) in the linear response formalism enabling the calculation of low energy optical absorption spectra for large molecules and nanostructures. The method avoids any explicit reference to canonical representations of either occupied or virtual Kohn-Sham states and thus achieves linear-scaling computational effort with system size. In contrast to conventional localised orbital formulations, where a single set of localised functions is used to span the occupied and unoccupied state manifold, we make use of two sets of in situ optimised localised orbitals, one for the occupied and one for the unoccupied space. This double representation approach avoids known problems of spanning the space of unoccupied Kohn-Sham states with a minimal set of localised orbitals optimised for the occupied space, while the in situ optimisation procedure allows for efficient calculations with a minimal number of functions. The method is applied to a number of medium sized organic molecules and a good agreement with traditional TDDFT methods is observed. Furthermore, linear scaling of computational cost with system size is demonstrated on (10,0) carbon nanotubes of different lengths.

Solvent effects on electronic excitations of an organic chromophore

In this work we study the solvatochromic shift of a selected low-energy excited state of alizarin in water by using a linear-scaling implementation of large-scale time-dependent density functional theory (TDDFT). While alizarin, a small organic dye, is chosen as a simple example of solute-solvent interactions, the findings presented here have wider ramifications for the realistic modeling of dyes, paints, and pigment-protein complexes. We find that about 380 molecules of explicit water need to be considered in order to yield an accurate representation of the solute-solvent interaction and a reliable solvatochromic shift. By using a novel method of constraining the TDDFT excitation vector, we confirm that the origin of the slow convergence of the solvatochromic shift with system size is due to two different effects. The first factor is a strong redshift of the excitation due to an explicit delocalization of a small fraction of the electron and the hole from the alizarin onto the water, which is mainly confined to within a distance of 7 Å from the alizarin molecule. The second factor can be identified as long-range electrostatic influences of water molecules beyond the 7 Å region on the ground-state properties of alizarin. We also show that these electrostatic influences are not well reproduced by a QM/MM model, suggesting that full QM studies of relatively large systems may be necessary in order to obtain reliable results.

Linear-scaling time-dependent density-functional theory beyond the Tamm-Dancoff approximation: Obtaining efficiency and accuracy with in situ optimised local orbitals

We present a solution of the full time-dependent density-functional theory (TDDFT) eigenvalue equation in the linear response formalism exhibiting a linear-scaling computational complexity with system size, without relying on the simplifying Tamm-Dancoff approximation (TDA). The implementation relies on representing the occupied and unoccupied subspaces with two different sets of in situ optimised localised functions, yielding a very compact and efficient representation of the transition density matrix of the excitation with the accuracy associated with a systematic basis set. The TDDFT eigenvalue equation is solved using a preconditioned conjugate gradient algorithm that is very memory-efficient. The algorithm is validated on a small test molecule and a good agreement with results obtained from standard quantum chemistry packages is found, with the preconditioner yielding a significant improvement in convergence rates. The method developed in this work is then used to reproduce experimental results of the absorption spectrum of bacteriochlorophyll in an organic solvent, where it is demonstrated that the TDA fails to reproduce the main features of the low energy spectrum, while the full TDDFT equation yields results in good qualitative agreement with experimental data. Furthermore, the need for explicitly including parts of the solvent into the TDDFT calculations is highlighted, making the treatment of large system sizes necessary that are well within reach of the capabilities of the algorithm introduced here. Finally, the linear-scaling properties of the algorithm are demonstrated by computing the lowest excitation energy of bacteriochlorophyll in solution. The largest systems considered in this work are of the same order of magnitude as a variety of widely studied pigment-protein complexes, opening up the possibility of studying their properties without having to resort to any semiclassical approximations to parts of the protein environment.

Combining the ensemble and Franck-Condon approaches for calculating spectral shapes of molecules in solution

The correct treatment of vibronic effects is vital for the modeling of absorption spectra of many solvated dyes. Vibronic spectra for small dyes in solution can be easily computed within the Franck-Condon approximation using an implicit solvent model. However, implicit solvent models neglect specific solute-solvent interactions on the electronic excited state. On the other hand, a straightforward way to account for solute-solvent interactions and temperature-dependent broadening is by computing vertical excitation energies obtained from an ensemble of solute-solvent conformations. Ensemble approaches usually do not account for vibronic transitions and thus often produce spectral shapes in poor agreement with experiment. We address these shortcomings by combining zero-temperature vibronic fine structure with vertical excitations computed for a room-temperature ensemble of solute-solvent configurations. In this combined approach, all temperature-dependent broadening is treated classically through the sampling of configurations and quantum mechanical vibronic contributions are included as a zero-temperature correction to each vertical transition. In our calculation of the vertical excitations, significant regions of the solvent environment are treated fully quantum mechanically to account for solute-solvent polarization and charge-transfer. For the Franck-Condon calculations, a small amount of frozen explicit solvent is considered in order to capture solvent effects on the vibronic shape function. We test the proposed method by comparing calculated and experimental absorption spectra of Nile red and the green fluorescent protein chromophore in polar and non-polar solvents. For systems with strong solute-solvent interactions, the combined approach yields significant improvements over the ensemble approach. For systems with weak to moderate solute-solvent interactions, both the high-energy vibronic tail and the width of the spectra are in excellent agreement with experiments.

Predicting solvatochromic shifts and colours of a solvated organic dye: The example of nile red

The solvatochromic shift, as well as the change in colour of the simple organic dye nile red, is studied in two polar and two non-polar solvents in the context of large-scale time-dependent density-functional theory (TDDFT) calculations treating large parts of the solvent environment from first principles. We show that an explicit solvent representation is vital to resolve absorption peak shifts between nile red in n-hexane and toluene, as well as acetone and ethanol. The origin of the failure of implicit solvent models for these solvents is identified as being due to the strong solute-solvent interactions in form of π-stacking and hydrogen bonding in the case of toluene and ethanol. We furthermore demonstrate that the failures of the computationally inexpensive Perdew-Burke-Ernzerhof (PBE) functional in describing some features of the excited state potential energy surface of the S1 state of nile red can be corrected for in a straightforward fashion, relying only on a small number of calculations making use of more sophisticated range-separated hybrid functionals. The resulting solvatochromic shifts and predicted colours are in excellent agreement with experiment, showing the computational approach outlined in this work to yield very robust predictions of optical properties of dyes in solution.

Identifying and tracing potential energy surfaces of electronic excitations with specific character via their transition origins: application to oxirane

We show that the transition origins of electronic excitations identified by quantified natural transition orbital (QNTO) analysis can be employed to connect potential energy surfaces (PESs) according to their character across a wide range of molecular geometries. This is achieved by locating the switching of transition origins of adiabatic potential surfaces as the geometry changes. The transition vectors for analysing transition origins are provided by linear response time-dependent density functional theory (TDDFT) calculations under the Tamm-Dancoff approximation. We study the photochemical CO ring opening of oxirane as an example and show that the results corroborate the traditional Gomer-Noyes mechanism derived experimentally. The knowledge of specific states for the reaction also agrees well with that given by previous theoretical work using TDDFT surface-hopping dynamics that was validated by high-quality quantum Monte Carlo calculations. We also show that QNTO can be useful for considerably larger and more complex systems: by projecting the excitations to those of a reference oxirane molecule, the approach is able to identify and analyse specific excitations of a trans-2,3-diphenyloxirane molecule.

Large-Scale Applications

To test the methods developed in the last couple of chapters, a number of realistic large scale applications are focused on. Two different problems are considered in detail, the first one of which, the study of environmental effects on the excitons localised on bacteriochlorophyll (Bchl) sites in the Fenna-Matthews-Olson complex, has already been briefly introduced in the last chapter.

Unraveling electronic absorption spectra using nuclear quantum effects: Photoactive yellow protein and green fluorescent protein chromophores in water

Many physical phenomena must be accounted for to accurately model solution-phase optical spectral line shapes, from the sampling of chromophore-solvent configurations to the electronic-vibrational transitions leading to vibronic fine structure. Here we thoroughly explore the role of nuclear quantum effects, direct and indirect solvent effects, and vibronic effects in the computation of the optical spectrum of the aqueously solvated anionic chromophores of green fluorescent protein and photoactive yellow protein. By analyzing the chromophore and solvent configurations, the distributions of vertical excitation energies, the absorption spectra computed within the ensemble approach, and the absorption spectra computed within the ensemble plus zero-temperature Franck-Condon approach, we show how solvent, nuclear quantum effects, and vibronic transitions alter the optical absorption spectra. We find that including nuclear quantum effects in the sampling of chromophore-solvent configurations using ab initio path integral molecular dynamics simulations leads to improved spectral shapes through three mechanisms. The three mechanisms that lead to line shape broadening and a better description of the high-energy tail are softening of heavy atom bonds in the chromophore that couple to the optically bright state, widening the distribution of vertical excitation energies from more diverse solvation environments, and redistributing spectral weight from the 0-0 vibronic transition to higher energy vibronic transitions when computing the Franck-Condon spectrum in a frozen solvent pocket. The absorption spectra computed using the combined ensemble plus zero-temperature Franck-Condon approach yield significant improvements in spectral shape and width compared to the spectra computed with the ensemble approach. Using the combined approach with configurations sampled from path integral molecular dynamics trajectories presents a significant step forward in accurately modeling the absorption spectra of aqueously solvated chromophores.

Linear-Scaling TDDFT in ONETEP

In this section, an approach to solve for the lowest eigenstates of the TDDFT linear-response equation in linear scaling effort is outlined and its implementation in the ONETEP code is discussed in some detail.

Implicit and explicit host effects on excitons in pentacene derivatives

An ab initio study of the effects of implicit and explicit hosts on the excited state properties of pentacene and its nitrogen-based derivatives has been performed using ground state density functional theory (DFT), time-dependent DFT, and ΔSCF. We observe a significant solvatochromic redshift in the excitation energy of the lowest singlet state (S1) of pentacene from inclusion in a p-terphenyl host compared to vacuum; for an explicit host consisting of six nearest neighbour p-terphenyls, we obtain a redshift of 65 meV while a conductor-like polarisable continuum model (CPCM) yields a 78 meV redshift. Comparison is made between the excitonic properties of pentacene and four of its nitrogen-based analogs, 1,8-, 2,9-, 5,12-, and 6,13-diazapentacene with the latter found to be the most distinct due to local distortions in the ground state electronic structure. We observe that a CPCM is insufficient to fully understand the impact of the host due to the presence of a mild charge-transfer (CT) coupling between the chromophore and neighbouring p-terphenyls, a phenomenon which can only be captured using an explicit model. The strength of this CT interaction increases as the nitrogens are brought closer to the central acene ring of pentacene.