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"# Theory\n",
"\n",
"## Response theory\n",
"\n",
"Compared to atomic fields, the externally applied electromagnetic fields are several orders of magnitude weaker and perturbation theory is called for. Such an approach provide response functions a distinct separation of one-, two-, three-photon, *etc.*, optical processes, and virtually all spectroscopic properties are encompassed by the theory as possible perturbations include\n",
"\n",
"- Time-independent or time-dependent \n",
"- Electric or magnetic\n",
"- Internal or external\n",
"- Geometric distortions\n",
"\n",
"When an external field is applied to a molecular system the system interacts with this field and responds to the applied perturbation. If such field is of sufficiently small field strength, *i.e.* sufficiently small intensity, we can address the response using perturbation theory with the perturbed Hamiltonian\n",
"\n",
"$$\n",
"\\hat{H} = \\hat{H}_0 + \\hat{V}(t) \n",
"$$\n",
"\n",
"where $\\hat{H}_0$ is the Hamiltonian of the isolated system and $\\hat{V}(t)$ described the quantum mechanical field-coupling. The requirement of a weak perturbation is met for most externally applied fields --- for example, a conventional laser with an intensity of 0.3 GW/cm$^3$ corresponds to an electric field amplitude of $\\sim$$5 \\times 10^{-5}$ a.u., which indeed has a small influence on a molecular system.\n",
"\n",
"If the perturbing field is taken as a static field, the problem is reduced to the time-independent case. The energy can then be considered with a Taylor expansion at zero perturbing field strength\n",
"\n",
"$$\n",
"E = E_0 + \\left. \\frac{\\partial E }{\\partial F_{\\alpha}} \\right| _{F_{\\alpha}=0} F_{\\alpha} \n",
"+\n",
"\\left. \\frac{1}{2}\\frac{\\partial^2 E}{\\partial F_{\\alpha}\\partial F_{\\beta}} \\right| _{F_{\\alpha}=F_{\\beta}=0} F_{\\alpha}F_{\\beta}\n",
"+ ...\n",
"$$\n",
"\n",
"with implied summation over indices. Molecular properties are identified as field derivatives of different orders and different perturbing fields, as the observables associated with the properties changes according to these derivatives. For example, the polarizability may be obtained through the second derivative of the energy with respect to a perturbing electric field, while the first derivative corresponds to the permanent dipole moment, both at zero field strength.\n",
"\n",
"For time-dependent perturbing fields, the interaction between the field and the molecular system yields ill-defined energies due to exchange of energy between the field and the molecule. Here we move instead to a slightly different approach, where the time-dependent molecular properties can be defined through the time-dependent expectation value of an observable, associated with an operator $\\hat{\\Omega}$. We first expand the wave function in orders of the perturbation\n",
"\n",
"$$\n",
"\\left| \\psi(t) \\right> = | \\psi^{(0)} \\rangle +| \\psi^{(1)} \\rangle + | \\psi^{(2)} \\rangle + \\dots\n",
"$$\n",
"\n",
"noting that this wave function is to be normalized at all times. The expectation value of the operator is expanded with terms collecting corrections to the different orders\n",
"\n",
"$$\n",
"\\langle \\psi (t) | \\hat{\\Omega} | \\psi (t) \\rangle = \\langle \\hat{\\Omega} ^{(0)} \\rangle + \\langle \\hat{\\Omega} ^{(1)} \\rangle + \\langle \\hat{\\Omega} ^{(2)} \\rangle + \\dots,\n",
"$$\n",
"\n",
"In the case of a molecule irradiated by light, we are interested in computing the polarizability $\\alpha_{\\alpha\\beta}(-\\omega; \\omega)$, and in the electric-dipole approximation the coupling between the molecular system and external electric field is provided by minus the electric dipole moment operator\n",
"\n",
"$$\n",
"\\hat{V}(t) = - \\hat{\\mu} F(t)\n",
"$$\n",
"\n",
"where $F(t)$ is the electric field amplitude. $\\alpha_{\\alpha\\beta}(-\\omega; \\omega)$ then corresponds to the response with $\\hat{\\Omega} = \\hat{\\mu}_\\alpha$ and $\\hat{V}^\\omega = - \\hat{\\mu}_\\beta$, where $\\alpha, \\beta \\in \\{x,y,z\\}$. For simplicity, we assume the external electric field to be monochromatic and adiabatically switched on at time zero according to\n",
"\n",
"$$\n",
"F(t) = F^\\omega \\sin \\omega t \\times \\mathrm{erf}(a t)\n",
"$$\n",
"\n",
"The first terms in the expectation value of a general operator is\n",
"\n",
"$$\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",
"$$\n",
"\n",
"And we now seek to calculate $\\langle \\! \\langle \\hat{\\Omega}; \\hat{V}^{\\omega} \\rangle$.\n",
"\n",
"### In SCF theory\n",
"\n",
"The derivation of expressions for response functions can appear very different from one source to another and the vast number of technicalities can at first appear overwhelming. The principle, however, is quite straightforward and clear: Form a well-defined quantity of interest, e.g., the electric dipole moment, $\\mu(t)$, and identify response functions in the order expansions of these quantities. In Hartree--Fock and Kohn--Sham density functional theory, the reference state is described by a single Slater determinant, and we parameterize the\n",
"time-dependent wave function as a single phase-isolated Slater determinant\n",
"\n",
"\n",
"$$\n",
"\\left| \\bar{\\Psi} (t) \\right> = e^{-i \\bar{\\kappa} (t)} \\left| 0 \\right> \n",
"$$\n",
"\n",
"with the time-dependent Hermitian operator\n",
"\n",
"$$\n",
"\\bar{\\kappa} (t) = \\sum_s^{\\textrm{unocc}} \\sum_i^{\\textrm{occ}} \\left[ \\kappa_{si} (t) \\hat{a}_s^{\\dagger} \\hat{a}_i + \\kappa_{si}^{\\ast} (t) \\hat{a}_i^{\\dagger} \\hat{a}_s \\right]\n",
"$$\n",
"\n",
"where the creation, $\\hat{a}^\\dagger$, and annihilation, $\\hat{a}$, operators act on unoccupied, secondary, $a$ and occupied, inactive, $i$ molecular orbitals.\n",
"\n",
"The time evolution of the SCF state is determined from the Ehrenfest theorem\n",
"\n",
"$$\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",
"$$\n",
"\n",
"Adopting perturbation theory\n",
"\n",
"$$\n",
" \\kappa_{ai}(t) = \\kappa_{ai}^{(1)} + \\kappa_{ai}^{(2)} + \\kappa_{ai}^{(3)} + \\cdots ,\n",
"$$\n",
"\n",
"we eventually obtain the following expression\n",
"\n",
"$$\n",
" \\boldsymbol{\\kappa}^{(1)}(\\omega) = - i\n",
" \\left(\\mathbf{E}^{[2]} - \\hbar \\omega \\mathbf{S}^{[2]}\\right)^{-1} \n",
" \\mathbf{V}^{\\omega, [1]} .\n",
"$$\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. We are then 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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