RT-TDDFT
We refer the interested reader to the following paper detailing the RT-TDDFT implementation within FHI-aims:
Basic Theory Overview
Starting with time-dependent Hartree-Fock, numerous theoretical methods have been developed for studying non-equilbrium response properties of systems with the real-time propagation approach. Real-time TDDFT (RT-TDDFT) has become one of the most popular methods over the past few decades due to its balance of accuracy and efficiency, and its rigorus justification from the Runge-Gross theorem.
Not limited to the linear response regiem, RT-TDDFT can be used to study both linear and nonlinear responses to external purturbation of various types and strengths. Importantly both nonperiodic and periodic boundary conditions are supported and simulations of molecules and exteneded systems can be be performed on the same footing.
While the list of applications of RT-TDDFT is long, we focus only on one in this tutorial: optical absorption. Additional applications can be found in the paper listed above. Finally we present only the very basic equations at the heart of RT-TDDFT to conclude the theoretical overview. Again, please refer to the above paper for additonal implementation details.
TDDFT is based on the 1-1 correspondence between the time-dependent one-particle probability denisty and the time-dependent external potenital. This is the correspondence established by the Runge-Gross theorem, and the Hohenberg-Kohn theorem extends this to the time-dependent regiem. Unlike many time independent electronic structure problems that can be case as some form of eigenvalue problem, RT-TDDFT amounts to solving a set of coupled non-linear parital differential equation. The central time-dependent Kohn-Sham equation is:
\(i \frac{\partial}{\partial t} \psi_n^{\mathrm{KS}}(\mathbf{r}, t)=\hat{H}^{\mathrm{KS}}[\rho(t), t] \psi_n^{\mathrm{KS}}(\mathbf{r}, t)\)
where \(\psi_n^{\mathrm{KS}}(\mathbf{r}, t)\) are the single-particle KS wavefunctions propagated in time by the KS Hamiltonian \(\hat{H}^{\mathrm{KS}}[\rho(t), t]\):
\(\hat{H}^{\mathrm{KS}}[\rho(t), t]=\hat{T}_{\mathrm{el}}+\hat{V}_{\mathrm{ext}}(t)+\hat{V}_H[\rho(t)]+\hat{V}_{\mathrm{XC}}[\rho(t)]\)
where \(\hat{T}_{\mathrm{el}}\) is the electronic kinetic operator, \(\hat{V}_H[\rho(t)]\) is the Hartree potenital, and \(\hat{V}_{\mathrm{XC}}[\rho(t)]\) is the exchange-correlation potential. Multiple avenues exist for applying \(\hat{V}_{\mathrm{ext}}(t)\), the external potenitial, and this point is further discused within the individual tutorial examples.
While the TD-KS equation shares its formal structure with the time-dependent Schrodinger equation, additional complexity aries because the electron density depends on the wavefunctions themselves, and the dependence of the Hartree and XC potential on the KS wavefunctions makes the TD-KS equation nonlinear. Further discussion on time propagation and implementation with an NAO basis is disucssed in the above paper.