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"# Foundations\n",
"\n",
"In Hartree--Fock and Kohn--Sham density functional theory, the reference state is described by a single Slater determinant, also referred to as the self-consistent field (SCF) approximation. The detailed derivation of response functions in this approximation can be found in {cite}`Norman2018`, but the key steps will be indicated below."
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"## Runge-Gross theorem\n",
"\n",
"The basics of TDDFT comes from the work of Work of [Runge and Gross](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.52.997), who presented a theory which underlies TDDFT:\n",
"\n",
"> For every single-particle potential $V(r,t)$, which can be expanded in a Taylor series around $t_0$, there exists a one-to-one mapping $G:V(r,t) \\rightarrow \\rho (r,t)$. This mapping is defined by solving the time-dependent SchrÃ¶dinger equation with a fixed initial state $\\phi$ and calculating the corresponding densities $\\rho (r,t)$.\n",
"\n",
"They also presented schemes for practical schemes, the most important being:\n",
"\n",
"> The exact time-dependent density can be expressed as a sum over states: $\\rho (r,t) = \\sum | \\phi (r,t)|^2$\n",
"\n",
"We calculate time-dependent orbitals from:\n",
"\n",
"$$\n",
"i \\frac{\\partial}{\\partial t} \\phi_i (r,t) = \\left( -\\frac{1}{2} \\nabla^2 + V_{\\textrm{eff}} (r,t) \\right) \\phi_i (r,t)\n",
"$$\n",
"\n",
"This requires knowledge of exact time-dependent xc-functional, which is even less known than the ground-state correspondance.\n",
"\n",
"\n",
"## Adiabatic approximation\n",
"\n",
"In the adiabatic approximation one siplifies the xc-functional as one which do not change over time, such that standard functionals can be used. This is a reasonable approximations for systems changing slowly in time, for which the time-dependence of the energy then changes through the time-dependence of the density.\n",
"\n",
"It can be note that the search for non-adiabatic functional may be somewhat futile, as it would technically require the solution of the full time-dependent SchrÃ¶dinger equation, as [has](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.75.022513) [been](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.78.056501) [discussed](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.78.056502)."
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