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arXiv:1701.01278

Time-dependent density-functional calculation of nuclear response functions

Takashi Nakatsukasa

Published Jan 5, 2017• 0 citations
arXivHTMLPDF

Abstract

Basic issues of the time-dependent density-functional theory are discussed, especially on the real-time calculation of the linear response functions. Some remarks on the derivation of the time-dependent Kohn-Sham equations and on the numerical methods are given.
DOI
10.48550/arxiv.1701.01278

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A time-dependent Kohn-Sham theory is presented for obtaining the time-dependent density which has a periodic dependence on time. A set of coupled single-particle equations $\ensuremath{-}\frac{1}{2}{\ensuremath{\nabla}}^{2}{\ensuremath{\chi}}_{i}+{v}_{\mathrm{eff}}{\ensuremath{\chi}}_{i}={\ensuremath{\epsilon}}_{i}{\ensuremath{\chi}}_{i}$ and $\frac{\ensuremath{\partial}{\ensuremath{\chi}}_{i}^{2}}{\ensuremath{\partial}t}+\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\chi}}_{i}^{2}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}{S}_{i})=0$ are obtained. The ${\ensuremath{\chi}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ and ${S}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ are the phase and amplitude, respectively, of the time-dependent Kohn-Sham orbitals, ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ is the time-dependent Kohn-Sham effective potential, and ${\ensuremath{\epsilon}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)=\ensuremath{-}\frac{\ensuremath{\partial}{S}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)}{\ensuremath{\partial}t}$. The density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ is equal to the sum of the squares of the ${\ensuremath{\chi}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$.
A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ leading to a given vvv-representable density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v\ensuremath{\rightarrow}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.
Time-Independent and Periodic Time-Dependent Potentials ...................... 73 6.3 Formal Foundations of Density Functional Theory for Time-Dependent Electric and Magnetic Fields ........................................... 74 6.4 Practical Schemes for the Calculation of Density and Current Density Variables in TDDFT ......................................................... 77 6.5 Linear Response within TDDFT ................................................................. 79 6.6 Concluding Remarks.................................................................................... 80 Acknowledgments................................................................................................... 80 References ............................................................................................................... 81 The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external field to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser field, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach.