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howto:tddft [2022/07/18 18:00] – ahehnhowto:tddft [2024/02/24 10:01] (current) – oschuett
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-====== 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==== +
- +
-<code - acetone_tddft.inp> +
- +
-&GLOBAL +
-  PROJECT S20Acetone +
-  RUN_TYPE ENERGY +
-  PREFERRED_DIAG_LIBRARY SL +
-  PRINT_LEVEL medium +
-&END GLOBAL +
-&FORCE_EVAL +
-  METHOD Quickstep +
-&PROPERTIES +
-  &TDDFPT                          ! input section for TDDFPT +
-   KERNEL FULL                       ! specification of the underlying kernel matrix K +
-                                     ! FULL kernel is for GGA and hybrid functional computations +
-                                     ! sTDA kernel is referring to a semi-empirical sTDA computation +
-   NSTATES 10                      ! specifies the number of excited states to be computed +
-   MAX_ITER   100                  ! number of iterations for the Davidson algorithm +
-   CONVERGENCE [eV] 1.0e-7         ! convergence threshold in eV +
-   RKS_TRIPLETS F                  ! Keyword to choose between singlet and triplet excitations +
- !  &XC                            ! If choosing kernel FULL, the underlying functional can be +
- !   &XC_FUNCTIONAL PBE0             ! specified by adding an XC section +
- !   &END XC_FUNCTIONAL              ! If not choosing ADMM, the functional can be chosen +
- !  &END XC                        ! independently from the chosen GS functional +
-  &END TDDFPT +
-&END PROPERTIES +
-  &DFT +
-    &QS +
-      METHOD GPW +
-      EPS_DEFAULT 1.0E-17 +
-      EPS_PGF_ORB 1.0E-20 +
-    &END QS +
-    &SCF +
-      SCF_GUESS restart +
-      &OT +
-         PRECONDITIONER FULL_ALL +
-         MINIMIZER DIIS +
-      &END OT +
-      &OUTER_SCF +
-         MAX_SCF 900 +
-         EPS_SCF 1.0E-7 +
-      &END OUTER_SCF +
-      MAX_SCF 10 +
-      EPS_SCF 1.0E-7 +
-    &END SCF +
-    POTENTIAL_FILE_NAME POTENTIAL_UZH +
-    BASIS_SET_FILE_NAME BASIS_MOLOPT_UZH +
-    BASIS_SET_FILE_NAME BASIS_ADMM_UZH +
-    &MGRID +
-      CUTOFF 800 +
-      REL_CUTOFF 80 +
-    &END MGRID +
-    &AUXILIARY_DENSITY_MATRIX_METHOD +
-      METHOD BASIS_PROJECTION +
-      EXCH_SCALING_MODEL NONE +
-      EXCH_CORRECTION_FUNC NONE +
-      ADMM_PURIFICATION_METHOD NONE +
-    &END AUXILIARY_DENSITY_MATRIX_METHOD +
-    &POISSON +
-       PERIODIC NONE +
-       POISSON_SOLVER WAVELET +
-    &END +
-    &XC +
-     &XC_FUNCTIONAL PBE0 +
-     &END XC_FUNCTIONAL +
-    &END XC +
-  &END DFT +
-  &SUBSYS +
-    &CELL +
-      ABC [angstrom] 14.0 14.0 14.0 +
-      PERIODIC NONE +
-    &END CELL +
-    &COORD +
-       C 0.000000 1.282877 -0.611721 +
-       C 0.000000 -1.282877 -0.611721 +
-       C 0.000000 0.000000 0.185210 +
-       O 0.000000 0.000000 1.392088 +
-       H 0.000000 2.133711 0.059851 +
-       H -0.876575 1.319344 -1.256757 +
-       H 0.876575 1.319344 -1.256757 +
-       H 0.000000 -2.133711 0.059851 +
-       H 0.876575 -1.319344 -1.256757 +
-       H -0.876575 -1.319344 -1.256757 +
-    &END COORD +
-    &TOPOLOGY +
-     &CENTER_COORDINATES T +
-     &END +
-    &END +
-    &KIND H +
-      BASIS_SET ORB DZVP-MOLOPT-PBE0-GTH-q1 +
-      BASIS_SET AUX_FIT admm-dzp-q1 +
-      POTENTIAL GTH-PBE0-q1 +
-    &END KIND +
-    &KIND O +
-      BASIS_SET ORB DZVP-MOLOPT-PBE0-GTH-q6 +
-      BASIS_SET AUX_FIT admm-dzp-q6 +
-      POTENTIAL GTH-PBE0-q6 +
-    &END KIND +
-    &KIND C +
-      BASIS_SET ORB DZVP-MOLOPT-PBE0-GTH-q4 +
-      BASIS_SET AUX_FIT admm-dzp-q4 +
-      POTENTIAL GTH-PBE0-q4 +
-    &END KIND +
-  &END SUBSYS +
-&END FORCE_EVAL +
- +
-</code> +
- +
-In the resulting output file, there is a ''XAS_TDP'' section reporting the steps of the calculations. Especially important are the parts related to donor core state identification. For each excited atom/core level combination, a small report about Mulliken population analysis and overlap with pure donor level is printed. For the Carbon 1s: +
- +
-===== FAQ ===== +
howto/tddft.1658167238.txt.gz · Last modified: 2022/07/18 18:00 by ahehn