Accurate TDDFT absorption spectra over full spectral range

含时密度泛函理论(TDDFT)已被证明是获得吸收光谱和其他相关激发态量的可靠工具。然而，整个吸收光谱的计算可能非常苛刻，而且它们的评估在计算上被禁止用于大型系统。TDDFT大致可分为线性响应或频域含时密度泛函理论(LR-TDDFT)和实时含时密度泛函理论(RT-TDDFT)。在后者中，含时Kohn-Sham态在含时外势的影响下显式地随时间传播。然而，通常需要大量的时间步长来获得所需的精度，并且对于大型系统的计算很快变得不可行。LR-TDDFT的目标是解决一个非厄米本征值问题，以获得激发态。一般来说，这种激发态的数量随着系统规模的增大而增加，对于大系统来说，求解线性代数问题也变得禁止计算。在该项目中，我们将开发一种新的方法来计算完整的TDDFT谱，其中计算成本不应显著超过基础基态计算的成本。特别地，我们将结合混合泛函DFT得到可靠的近似的整个TDDFT谱。因此，我们的目标是在大光谱范围和大系统规模上提供准确的吸收光谱。为此，我们提出了一种结合了LR-TDDFT和RT-TDDFT的优点的方法，以克服目前系统规模的限制。为了将计算成本降到最低，我们的目标是在RT-TDDFT中使用很短的时间传播。由于光谱的分辨率在很大程度上取决于模拟时间，因此可以预期，对于以状态连续或准连续为特征的光谱，这种方法已经很好地工作了。然而，如果频谱还包括离散激励，则该方法将失败，因为模拟时间可能太短，无法达到所需的分辨率。使用LR-TDDFT可以包括关于离散激励的遗漏信息。求解与线性响应形式相关的特征值问题，给出了特定问题的特征值形式的不同激发能量。为了将计算成本保持在最小，我们将使用所谓的小矩阵近似(SMA)来近似求解LR方程。SMA将作为更复杂的近似的基础，这将允许在描述某些激发或某些光谱区域时具有高精度。通过这种方法，我们设想了一种既能保证快速评估整个TDDFT谱又能保证高精度的方法。

Time dependent density functional theory (TDDFT) has proven to be a reliable work horse for obtaining absorption spectra and other related excited state quantities. Yet, calculation of entire absorption spectra can be very demanding and their evaluation is computationally prohibited for large system sizes. Roughly speaking TDDFT can be divided in two regimes, the linear-response or frequency domain time dependent density functional theory (LR-TDDFT) and in real-time time dependent density functional theory (RT-TDDFT). In the latter the time dependent Kohn-Sham states are explicitly propagated in time under the influence of a time dependent external potential. However, a large number of time steps is usually required to obtain the required accuracy and calculations for large systems becomes quickly unfeasible. LR-TDDFT aims to solve a non-hermitian eigenvalue problem in order to obtain the excited states. Generally speaking, the number of such excited states grows with system size and solving the linear algebra problem becomes also computational prohibited for large systems. Within the project we will develop a new approach for calculating full TDDFT spectra where the computational cost should not dramatically exceed the cost of the underlying ground state calculation. In particular we will obtain reliable approximated entire TDDFT spectra in combination with hybrid functional DFT. Thus our target is to provide accurate absorption spectra over a large spectral range and for large system sizes. For this purpose we suggest a method which combines the strengths of LR-TDDFT and RT-TDDFT to overcome the current limitations in system size. In order to reduce the computational cost to a minimum we aim at using very short time propagation in RT-TDDFT. As the resolution of the spectra significantly depends on the simulation time one can expect that this works already quite well for a spectrum which is characterized by a continuum or quasi continuum of states. However, it will fail if the spectrum also includes discrete excitations as the simulation time might be too short to achieve the desired resolution. Missing information about discrete excitations can be included using LR-TDDFT. Solving the eigenvalue problem associated with the linear-response formalism gives distinct excitation energies in form of eigenvalues of the particular problem. To keep the computational cost at a minimum we will approximately solve the LR equations using the so called small matrix approximation (SMA). The SMA will serve as a basis for more sophisticated approximations which will allow high accuracy in describing certain excitations or certain spectral regions. By this we envision an approach which guarantees the fast evaluation of entire TDDFT spectra in combination with high accuracy.