RPA from linearized TDHF

To be concise, below we present a less general derivation than the standard method [12,13] to derive the RPA equations from linearized time-dependent Hartree-Fock (TDHF) equations. For density independent forces or functionals with terms quadratic in density, the density matrix and mean field of a time-dependent nuclear state are expressed as

$$\rho(t) = \rho_0 + {\tilde\rho}^{}_\omega e^{i\omega t} + {\tilde\rho}^\dagger _\omega e^{-i\omega t}\,,$$ (1)

$$h(t) = h_0 + {\tilde h}^{}_\omega e^{i\omega t} + {\tilde h}^\dagger _\omega e^{-i\omega t}\,,$$ (2)

where $\rho_0$ and $h_0$ are the Hartree-Fock (HF) ground-state density matrix and mean field, respectively. Inserting $\rho(t)$ and $h(t)$ of Eqs. (1) and (2) into the TDHF equation, and keeping only terms linear in the fluctuating quantities $\tilde\rho$ and $\tilde h$, we get a linearized TDHF equation, or the RPA equations:

$$\hbar\omega {\tilde\rho}_{\omega,mi} = \left(\epsilon_m-\epsilon_i\right) {\tilde\rho}_{\omega,mi} + {\tilde h}_{\omega,mi}\,,$$ (3)

$$\hbar\omega {\tilde\rho}_{\omega,im} = \left(\epsilon_i-\epsilon_m\right) {\tilde\rho}_{\omega,im} - {\tilde h}_{\omega,im}\,,$$ (4)

where we use the letter $m$ for particle states and $i$ for hole states, and where $\epsilon_{m,i}$ are the HF single-particle energies. The fields $\tilde h$ are the first functional derivatives of the used EDF, evaluated using the density amplitudes $\tilde\rho$ of Eq. (1). Density dependence of the used EDF beyond quadratic gives rise to rearrangement fields in $\tilde h$. These rearrangement parts of $\tilde h$ must be linearized around $\rho_0$ to make our RPA equations explicitly first order in $\tilde\rho$.

One way to achieve this is by calculating functional derivatives of the rearrangement parts of $\tilde h$ with respect to density, which technically makes our mean-field routine differ from the standard HF routines. Since in our implementation we use the standard Skyrme forces that have simple density dependencies of the coupling constants, the explicit functional differentiation does not cause any mathematical or performance problems. Had a EDF with more complex density dependence been used it would have been an advantage to instead use the FAM method [6] for linearization.

If the matrix elements of $\tilde h$ in Eqs. (3) and (4) are expanded in terms of the particle-hole (p-h) and hole-particle (h-p) matrix elements of $\tilde\rho$, we obtain the traditional RPA equations. In this work, we do not construct the RPA matrix, but directly solve Eqs. (3) and (4) by calculating the matrix elements of fields $\tilde h$ using a HF mean-field routine that uses the time-reversal-invariance breaking density matrix $\tilde\rho$. Since the same routine is used to evaluate the HF and RPA mean fields, the method is always fully self-consistent [14,15]. In the following equations, we use the standard abbreviations $X_{mi}={\tilde\rho}_{\omega,mi}$ and $Y_{mi}={\tilde\rho}_{\omega,im}$. The density vector that contains the p-h matrix elements of ${\tilde \rho}^\omega$ is defined as ${\mathcal X}^\omega=\left(X_{m_1,i_1}, X_{m_2,i_2}, \ldots,X_{m_D,i_D}\right)$, and similarly for the vector ${\mathcal Y}$ of h-p elements, where $D$ is the number of allowed p-h configurations. Overlaps of RPA vectors are defined as

$$\langle X,Y\vert X',Y' \rangle = \left( \begin{array}{cc... ...{\mathcal X}'^T \\ -{\mathcal Y}'^T \\ \end{array} \right)$$ (5)

and the minus sign results from the RPA norm matrix.