Differences

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--- howto:tddft [2022/07/19 08:28] – ahehn
+++ howto:tddft [2022/07/19 11:55] – ahehn
@@ Line 12: / Line 12: @@
 \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} \, .
+      \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{D}^{{\rm{\tiny{X}}}p}] C_{\lambda i \sigma} &=  \sum_{\kappa} \Omega^p S_{\mu \kappa} X^p_{\kappa i \sigma} \, .
     \end{aligned}
 \end{equation}
@@ Line 21: / Line 21: @@
 \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}}} \, .
+K_{\mu \nu \sigma} [\mathbf{D}^{{\rm{\tiny{X}}}p}] &=  J_{\mu \nu \sigma} [\mathbf{D}^{{\rm{\tiny{X}}}p}] - a_{\rm{\tiny{EX}}} K^{\rm{\tiny{EX}}}_{\mu \nu \sigma}[\mathbf{D}^{{\rm{\tiny{X}}}p}] + \sum_{\kappa \lambda \sigma'} f^{\rm{\tiny{XC}}}_{\mu \nu \sigma,\kappa \lambda \sigma'} D_{\kappa \lambda \sigma'}^{{\rm{\tiny{X}}}p} \, .
 \end{aligned}
 \end{equation}
@@ Line 30: / Line 30: @@
 \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}} ) \, .
+D_{\mu \nu \sigma}^{{\rm{\tiny{X}}}p} &= \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}
@@ Line 49: / Line 49: @@
 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 $p$,
+Based on the TDA eigenvalue problem, excited-state gradients can be formulated based on a variational Lagrangian for each excited state $p$,
 \begin{equation}
@@ Line 358: / Line 358: @@
 The most important keywords of the subsection ''sTDA'' are:
   * ''FRACTION'': fraction of exact exchange $a_{\rm{\tiny{EX}}}$
-  * ''MATAGA_NISHIMOTO_CEXP'': keyword to modify the parameter $\alpha$ of $\gamma_{\rm{\tiny{J}}$
+  * ''MATAGA_NISHIMOTO_CEXP'': keyword to modify the parameter $\alpha$ of $\gamma^{\rm{\tiny{J}}}$
-  * ''MATAGA_NISHIMOTO_XEXP'': keyword to modify the parameter $\beta$ of $\gamma_{\rm{\tiny{EX}}}$
+  * ''MATAGA_NISHIMOTO_XEXP'': keyword to modify the parameter $\beta$ of $\gamma^{\rm{\tiny{K}}}$
   * ''DO_EWALD'': keyword to switch on Ewald summation for the Coulomb contributions, required when treating periodic systems. (Exact exchange is treated within the minimum image convention and does not require any adjustment.)