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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).\n",
"\n",
"## Parameterization\n",
"\n",
"SCF states are parameterized as:\n",
"\n",
"\\begin{equation*}\n",
"\\label{psi-kappa-param}\n",
" | \\bar{\\psi}(t) \\rangle =\n",
" e^{-i\\hat{\\kappa}(t)} |0\\rangle ; \\quad\n",
" \\hat{\\kappa}(t) =\n",
" \\sum_a^\\mathrm{unocc} \\sum_i^\\mathrm{occ} \\left[\n",
" \\kappa_{ai}(t) \\hat{a}^\\dagger_a \\hat{a}_i + \\kappa_{ai}^*(t) \\hat{a}^\\dagger_i \\hat{a}_a\n",
" \\right] ,\n",
"\\end{equation*}\n",
"\n",
"where the creation, $\\hat{a}^\\dagger$, and annihilation, $\\hat{a}$, operators act on unoccupied, secondary, $a$ and occupied, inactive, $i$ molecular orbitals.\n",
"\n",
"## Equation of motion\n",
"The time evolution of the SCF state is determined from the Ehrenfest theorem:\n",
"\n",
"\\begin{equation*}\n",
"\\label{ehren-equiv}\n",
" \\frac{\\partial}{\\partial t} \n",
" \\langle \\bar{\\psi}(t) | \\hat{\\Omega}_{pq}^\\dagger | \\bar{\\psi}(t) \\rangle \n",
" -\n",
" \\langle \\bar{\\psi}(t) | \n",
" \\frac{\\partial \\hat{\\Omega}_{pq}^\\dagger}{\\partial t} \n",
" | \\bar{\\psi}(t) \\rangle = \n",
" \\frac{1}{i\\hbar}\n",
" \\langle \\bar{\\psi}(t) | [\\hat{\\Omega}_{pq}^\\dagger, \\hat{H}] \n",
" | \\bar{\\psi}(t) \\rangle \n",
"\\end{equation*}\n",
"\n",
"We require that this equation is fulfilled for the set of time-transformed electron-transfer operators:\n",
"\n",
"\\begin{equation*}\n",
" \\hat{\\Omega}_{ai}^\\dagger = \n",
" e^{-i\\hat{\\kappa}} \\, \\hat{a}^\\dagger_a \\hat{a}_i \\, e^{i\\hat{\\kappa}}\n",
"\\end{equation*}\n",
"\n",
"as well as the corresponding Hermitian conjugate. There will be two equations associated with each single-electron excited determinant, so there are twice as many equations as there are unknown *complex* parameters, $\\kappa_{ai}(t)$, with independent real and imaginary parts in the time-dependent phase-isolated wave function, $| \\bar{\\psi}(t) \\rangle$.\n",
"\n",
"At this point, we adopt perturbation theory:\n",
"\n",
"\\begin{equation*}\n",
" \\kappa_{ai}(t) = \\kappa_{ai}^{(1)} + \\kappa_{ai}^{(2)} + \\kappa_{ai}^{(3)} + \\cdots ,\n",
"\\end{equation*}\n",
"\n",
"which, after finding a first-order solution in the frequency domain, will eventually results in\n",
"\n",
"\\begin{equation*}\n",
" \\boldsymbol{\\kappa}^{(1)}(\\omega) = - i\n",
" \\left(\\mathbf{E}^{[2]} - \\hbar \\omega \\mathbf{S}^{[2]}\\right)^{-1} \n",
" \\mathbf{V}^{\\omega, [1]} .\n",
"\\end{equation*}\n",
"\n",
"The vector $\\boldsymbol{\\kappa}^{(1)}$ collects the set of parameters and their complex conjugate and matrices $\\mathbf{E}^{[2]}$ and $\\mathbf{S}^{[2]}$ are known as the electronic Hessian and overlap matrices, respectively, and $\\mathbf{V}^{\\omega, [1]}$ is known as the property gradient of the perturbation operator.\n",
"\n",
"## Response functions\n",
" The response functions of an observable $\\hat{\\Omega}$ are defined by:\n",
"\n",
"\\begin{equation*}\n",
" \\langle \\bar{\\psi}(t) | \\hat{\\Omega} | \\bar{\\psi}(t) \\rangle =\n",
" \\langle 0 | \\hat{\\Omega} | 0 \\rangle + \n",
" \\sum_{\\omega}\n",
" \\langle \\! \\langle \\hat{\\Omega}; \\hat{V}^{\\omega} \\rangle \\! \\rangle \n",
" F^{\\omega}\n",
" e^{-i\\omega t} + \\cdots \n",
"\\end{equation*}\n",
"\n",
"and we are able to identify the following expression for the linear response function\n",
"\n",
"\\begin{equation*}\n",
" \\langle \\! \\langle \\hat{\\Omega}; \\hat{V}^{\\omega} \\rangle \\!\n",
" \\rangle = -\n",
"\\left[\\boldsymbol{\\Omega}^{[1]}\\right]^\\dagger\n",
" \\left(\\mathbf{E}^{[2]} - \\hbar \\omega \\mathbf{S}^{[2]}\\right)^{-1} \\mathbf{V}^{\\omega, [1]} ,\n",
"\\end{equation*}\n",
"\n",
"where $\\boldsymbol{\\Omega}^{[1]}$ (analogously to $\\mathbf{V}^{\\omega, [1]}$) is known as the property gradient associated with the observable. This expression for the linear response functions is also known as the random phase approximation (RPA) equation in the case a Hartree--Fock reference state. It is the principal equation to be solved in time-dependent Hartree--Fock (TDHF) and time-dependent density functional theory (TDDFT) theories."
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