Linear response thoery
Linear response (LR) theory relates the response of a molecular property to the excited states of the unperturbed system, or to a time-dependent external perturbation.
Assume we have solved the unperturbed electronic structure problem and obtained the eigenvalues and eigenvectors of unperturbed Hamiltonian
\[\hat{H}\Psi_n = E_n\Psi_n\]Now apply the time dependent perturbation \(\hat{V}(t)\) to the system. The time-evolution of a specific state, e.g., the ground state \(\Psi_0\) can be described by the time-dependent Schrödinger equation
\[i\hbar\frac{d}{dt}\Psi_0(t) = \left(\hat{H}+\hat{V}(t)\right)\Psi_0(t)\]We can expand the time-dependent wavefunction as
\[\Psi_0(t) = \left(\Psi_0 + \delta\Psi_0(t) \right)e^{-iE_0t/\hbar}\]The first-order response of wave function can be obtained by perturbation theory
\[\delta\Psi_0(t) = \sum_{n\neq 0} c_n(t) \Psi_n\] \[i\hbar\frac{d}{dt}c_n(t) = \langle \Psi_n | \hat{V}(t) |\Psi_0\rangle +\left( E_n - E_0 \right)c_n(t)\]or equivalently
\[\begin{aligned} i\hbar c_n(t) &= \int_{-\infty}^t \left[\langle \Psi_n | \hat{V}(t') |\Psi_0\rangle +\left( E_n - E_0 \right)c_n(t')\right] dt'\\ &= \int_{-\infty}^{\infty} \theta(t-t')\left[\langle \Psi_n | \hat{V}(t') |\Psi_0\rangle +\left( E_n - E_0 \right)c_n(t')\right] dt' \end{aligned}\]where \(\theta(t-t')\) is the Heaviside step function. By using the Fourier transformation convolution theorem, we can obtain the response of the wave function in the frequency domain
\[\begin{aligned} i\hbar c_n(\omega) &= \theta(\omega)\left[\langle \Psi_n | \hat{V}(\omega) |\Psi_0\rangle +\hbar\omega_{n0}c_n(\omega)\right] \\ c_n(\omega) &= \frac{\langle \Psi_n | \hat{V}(\omega) |\Psi_0\rangle}{\hbar(\omega-\omega_{n0})} \\ |\delta\Psi_0(\omega)\rangle &= \sum_{n\neq 0} \frac{\langle \Psi_n | \hat{V}(\omega) |\Psi_0\rangle}{\hbar(\omega-\omega_{n0})} |\Psi_n\rangle \end{aligned}\]where \(\omega_{n0} = (E_n - E_0)/\hbar\). In above equations, we have used that the Fourier transformation of the Heaviside step function \(\theta(\omega) = \lim_{\epsilon\rightarrow 0^+} \frac{i}{\omega + i\epsilon}\).
Now consider the frequency-domain response of a molecular property, i.e., the expectation value of an operator \(\hat{a}\), to the perturbation
\[\begin{aligned} \langle \delta\hat{a}(\omega)\rangle & \equiv \langle \Psi_0(\omega) | \hat{a} |\Psi_0(\omega)\rangle - \langle \Psi_0 | \hat{a} |\Psi_0\rangle \\ & = \sum_{n\neq 0} \frac{\langle \Psi_0 | \hat{a} |\Psi_n\rangle\langle \Psi_n | \hat{V}(\omega) |\Psi_0\rangle}{\hbar(\omega-\omega_{n0})} + \text{c.c.} \end{aligned}\]If the perturbation is a time-dependent monochromatic electric field,
\[\hat{V}(t) = \hat{b}\cos(\omega_0 t) = \frac{\hat{b}}{2}(e^{i\omega_0 t} + e^{-i\omega_0 t}) ,\] \[\hat{V}(\omega) = \int_{-\infty}^{\infty} e^{i\omega t} \hat{V}(t) dt = \pi \hat{b} \left[ \delta(\omega+\omega_0) + \delta(\omega-\omega_0) \right]\]we can obtain the linear response of any property \(\hat{a}\) to the electric field
\[\begin{aligned} \langle \delta\hat{a}(t)\rangle &= \frac{1}{2\pi} \int_{-\infty}^{\infty} \sum_{n\neq 0} \frac{\langle \Psi_0 | \hat{a} |\Psi_n\rangle\langle \Psi_n | \hat{V}(\omega) |\Psi_0\rangle}{\hbar(\omega-\omega_{n0})} d\omega + \text{c.c.} \\ &=\frac{1}{2} \sum_{n\neq 0} \frac{\langle \Psi_0 | \hat{a} |\Psi_n\rangle\langle \Psi_n | \hat{b} |\Psi_0\rangle}{\hbar(\omega_0-\omega_{n0})} e^{-i\omega_0 t} \\ &- \frac{1}{2} \sum_{n\neq 0} \frac{\langle \Psi_0 | \hat{a} |\Psi_n\rangle\langle \Psi_n | \hat{b} |\Psi_0\rangle}{\hbar(\omega_0+\omega_{n0})} e^{i\omega_0 t} + \text{c.c.} \\ & = \sum_{n\neq 0} \frac{\langle \Psi_0 | \hat{a} |\Psi_n\rangle\langle \Psi_n | \hat{b} |\Psi_0\rangle}{\hbar(\omega_0^2-\omega_{n0}^2)} \omega_{n0} \cos(\omega_0 t) \\ & - i \sum_{n\neq 0} \frac{\langle \Psi_0 | \hat{a} |\Psi_n\rangle\langle \Psi_n | \hat{b} |\Psi_0\rangle}{\hbar(\omega_0^2-\omega_{n0}^2)} \omega_0 \sin(\omega_0 t) + \text{c.c.} \end{aligned}\]This result is very important, indicating the response of molecular properties is at the same frequency as the external perturbation \(\omega_0\).