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--- howto:tddft [2022/07/18 17:53] – ahehn
+++ howto:tddft [2024/02/24 10:01] (current) – oschuett
@@ Line 1: / Line 1: @@
-====== How to run a LR-TDDFT calculation for absorption and emission spectroscopy ======
+This page has been moved to: https://manual.cp2k.org/trunk/methods/properties/optical/tddft.html
-This is a short tutorial on how to run linear-response TDDFT computations for absorption and emission spectroscopy. The TDDFT module enables a description of excitation energies and excited-state computations within the Tamm-Dancoff approximation (TDA) featuring GGA and hybrid functionals as well as semi-empirical simplified TDA kernels. The details of the implementation can be found in [[https://aip.scitation.org/doi/full/10.1063/1.5078682]] (Ref. [1]) and in [[https://pubs.acs.org/doi/10.1021/acs.jctc.2c00144]] for corresponding excited-state gradients.
-Note that the current module is based on an earlier TDDFT implementation [[https://chimia.ch/chimia/article/view/2005_499]].
-Please cite these papers if you were to use the TDDFT module for the computation of excitation energies or excited-state gradients.
-===== Brief theory recap =====
-The implementation in CP2K is based on the Tamm-Dancoff approximation (TDA), which describes each excited state $p$ with the excitation energy $\Omega^p$ and the corresponding excited-state eigenvectors $\mathbf{X}^p$ as an Hermitian eigenvalue problem
-\begin{equation} \label{tda_equation}
-\begin{aligned}
-      \mathbf{A} \mathbf{X}^p = \Omega^p \mathbf{X}^p \, , \\
-      \sum_{\kappa k} [ F_{\mu \kappa \sigma} \delta_{ik} - F_{ik \sigma} S_{\mu \kappa} ] X^p_{\kappa k \sigma} + \sum_{\lambda} K_{\mu \lambda \sigma} [\mathbf{X}^p] C_{\lambda i \sigma} =  \sum_{\kappa} \Omega^p S_{\mu \kappa} X^p_{\kappa i \sigma} \, .
-    \end{aligned}
-\end{equation}
-The Hermitian matrix $\mathbf{A}$ contains as zeroth-order contributions the difference in the Kohn-Sham (KS) orbital energies $\mathbf{F}$, and to first order kernel contributions $\mathbf{K}$ which comprise --depending on the chosen density functional approximation -- Coulomb $\mathbf{J}$ and exact exchange $\mathbf{K}^{\rm{\tiny{EX}}}$ contributions as well as contributions due to the exchange-correlation (XC) potential $\mathbf{V}^{\rm{\tiny{XC}}}$ and kernel $\mathbf{f}^{\rm{\tiny{XC}}}$,
-\begin{equation} \label{f_and_k_matrix}
-\begin{aligned}
-F_{\mu \nu \sigma} [\mathbf{D}] &= h_{\mu \nu} + J_{\mu \nu \sigma} [\mathbf{D}] - a_{\rm{\tiny{EX}}}K^{\rm{\tiny{EX}}}_{\mu \nu \sigma} [\mathbf{D}] + V_{\mu \nu \sigma}^{\rm{\tiny{XC}}} \, , \\
-K_{\mu \nu \sigma} [\mathbf{D}^{\rm{\tiny{X}}}] &=  J_{\mu \nu \sigma} [\mathbf{D}^{\rm{\tiny{X}}}] - a_{\rm{\tiny{EX}}} K^{\rm{\tiny{EX}}}_{\mu \nu \sigma}[\mathbf{D}^{\rm{\tiny{X}}}] + \sum_{\kappa \lambda \sigma'} f^{\rm{\tiny{XC}}}_{\mu \nu \sigma,\kappa \lambda \sigma'} D_{\kappa \lambda \sigma'}^{\rm{\tiny{X}}} \, .
-\end{aligned}
-\end{equation}
-$\mathbf{S}$ denotes the atomic-orbital overlap matrix, $\mathbf{C}$ the occupied ground-state KS orbitals and $\mathbf{D}$ and $\mathbf{D}^{\rm{\tiny{X}}}$ ground-state and response density matrices,
-\begin{equation}\label{density_matrices}
-\begin{aligned}
-D_{\mu \nu \sigma} = \sum_k C_{\mu k \sigma} C_{\nu k \sigma}^{\rm{T}} \, , \\
-D_{\mu \nu \sigma}^{\rm{\tiny{X}}} = \frac{1}{2} \sum_{k} ( X^p_{\mu k \sigma} C_{\nu k \sigma}^{\rm{T}} + C_{\mu k \sigma} (X^p_{\nu k \sigma})^{\rm{T}} ) \, .
-\end{aligned}
-\end{equation}
-Within the current implementation, symmetrization and orthogonalization of the response density matrix is ensured at each step of the Davidson algorithm.
-The current implementation features to approximate the exact exchange contribution of hybrid functionals using the auxiliary density matrix method (ADMM). Furthermore, the standard kernel can be approximated using the semi-empirical simplified Tamm-Dancoff approximation (sTDA), neglecting in this case XC contributions and approximating both Coulomb and exchange contributions $\mathbf{J}$ and $\mathbf{K}$ using semi-empirical operators $\boldsymbol{\gamma}^{\rm{\tiny{J}}}$ and $\boldsymbol{\gamma}^{\rm{\tiny{K}}}$ depending on the interatomic distance $R_{AB}$ of atoms $A$ and $B$,
-\begin{equation} \label{stda_kernel}
-\begin{aligned}
-\gamma^{\rm{\tiny{J}}}(A,B) &= \left ( \frac{1}{(R_{AB})^{\alpha} + \eta^{-\alpha}}  \right)^{1/\alpha} \, , \\
-\gamma^{\rm{\tiny{K}}}(A,B) &= \left ( \frac{1}{(R_{AB})^{\beta} +( a_{\rm{\tiny{EX}}}\eta)^{- \beta} } \right )^{1/\beta} \, ,
-\end{aligned}
-\end{equation}
-that depend on the chemical hardness $\eta$, the Fock-exchange mixing parameter $a_{\rm{\tiny{EX}}}$ and powers of $\alpha$ and $\beta$ for either Coulomb and exchange interactions.
-Within the current implementation, oscillator strengths can be calculated for molecular systems in the length form and for periodic systems using the velocity form (see Ref.[1]).
-Based on Eq.\ (\ref{tda_equation}), excited-state gradients can be formulated based on a variational Lagrangian for each excited state $p$,
-\begin{equation}
-\begin{aligned}
-L [\mathbf{X}, \mathbf{C}, \Omega, \bar{\mathbf{W}}^{\rm{\tiny{X}}}, \bar{\mathbf{Z}}, \bar{\mathbf{W}}^{\rm{\tiny{C}}} ] &= \Omega -  \sum_{\kappa \lambda k l \sigma} \Omega ( X_{\kappa k \sigma }^{\rm{T}} S_{\kappa \lambda } X_{\lambda l \sigma} - \delta_{kl} ) \\
-&-  \sum_{kl \sigma} ( \bar{W}_{kl \sigma}^{\rm{\tiny{X}}} )^{\rm{T}}  \sum_{\kappa \lambda} \frac{1}{2} ( C_{\kappa k \sigma}^{\rm{T}} S_{\kappa \lambda} X_{\lambda l \sigma}  + X_{\kappa k \sigma}^{\rm{T}}S_{\kappa \lambda} C_{\lambda l \sigma}) \\
-&+ \sum_{\kappa k \sigma}( \bar{Z}_{\kappa k \sigma})^{\rm{T}}  \sum_{\lambda} ( F_{\kappa \lambda \sigma}C_{\lambda k \sigma} - S_{\kappa \lambda } C_{\lambda k \sigma} \varepsilon_{k \sigma}) \\
-&- \sum_{kl\sigma} (\bar{W}^{\rm{\tiny{C}}}_{kl \sigma})^{\rm{T}}  ( S_{kl \sigma} - \delta_{kl})\, .
-\end{aligned}
-\end{equation}
-introducing Lagrange multipliers $\bar{\mathbf{W}}^{\rm{\tiny{X}}}$, $\bar{\mathbf{W}}^{\rm{\tiny{C}}}$, and $\bar{\mathbf{Z}}$ to ensure stationarity of the corresponding ground-state (GS) equations and to account for the geometric dependence of the Gaussian orbitals and thus requiring to solve the Z vector equation iteratively.
-===== The LR-TDDFT input section =====
-To compute absorption spectra, parameters defining the LR-TDDFT computation have to be specified in the ''TDDFPT'' subsection of the section ''PROPERTIES'' of section ''FORCE_EVAL''. Furthermore, ''RUN_TYPE'' has to be set to ENERGY and the underlying KS ground-state reference has to be specified in the ''DFT'' section.
-The most important keywords and subsections of ''TDDFPT'' are:
-  * ''KERNEL'': option for the kernel matrix $\mathbf{K}$ to choose between the full kernel for GGA or hybrid functionals and the simplified TDA kernel
-  * ''NSTATES'': number of excitation energies to be computed
-  * ''CONVERGENCE'': threshold for the convergence of the Davidson algorithm
-  * ''RKS_TRIPLETS'': option to switch from the default computation of singlet excitation energies to triplet excitation energies
-To compute excited-state gradients and thus corresponding fluorescence spectra, the excited state to be optimized furthermore has to be specified by adding the subsection ''EXCITED_STATES'' of the section ''DFT.
-===== Simple examples =====
-==== Acetone molecule====
-===== FAQ =====