diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 94d74c6..83b41dc 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -283,7 +283,7 @@ The latters are determined by solving the ensemble KS equation where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and \begin{equation} \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})} - = \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} + = \fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} \end{equation} is the Hxc potential, with \begin{subequations} @@ -670,117 +670,79 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{ & & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\ & & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\ \\ - B & LYP & aug-mcc-pV8Z\fnm[1] & & & & 28.42\fnm[2] \\ - B3 & LYP & aug-mcc-pV8Z\fnm[1] & & & & 27.77\fnm[2] \\ - HF & LYP & aug-mcc-pV8Z\fnm[1] & & & & 29.18\fnm[2] \\ - HF & & aug-mcc-pV8Z\fnm[1] & & & & 28.65\fnm[2] \\ + B & LYP & aug-mcc-pV8Z & & & & 28.42 \\ + B3 & LYP & aug-mcc-pV8Z & & & & 27.77 \\ + HF & LYP & aug-mcc-pV8Z & & & & 29.18 \\ + HF & & aug-mcc-pV8Z & & & & 28.65 \\ \\ - HF & FCI & aug-mcc-pV8Z\fnm[1] & & & & 28.75\fnm[2] \\ + \hline + \mc{5}{l}{Accurate (FCI/aug-mcc-pV8Z)\fnm[1]} & 28.75 \\ \end{tabular} \end{ruledtabular} -\fnt[1]{Reference \onlinecite{Mielke_2005}.} -\fnt[2]{Reference \onlinecite{Barca_2018a}.} +\fnt[1]{Reference \onlinecite{Barca_2018a}.} \end{table*} %%% %%% %%% %%% %%% TABLE I %%% -\begin{table*} +\begin{table} \caption{ -Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 3.7$ bohr for various methods, combinations of xc functionals, and basis sets. +Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 3.7$ bohr obtained with the aug-cc-pVTZ basis set for various methods and combinations of xc functionals. \label{tab:BigTab_H2st} } \begin{ruledtabular} -\begin{tabular}{llccccc} - \mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\ - \cline{1-2} \cline{4-5} - exchange & correlation & Basis & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\ +\begin{tabular}{llcccc} + \mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\ + \cline{1-2} \cline{3-4} + exchange & correlation & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\ \hline - HF & & aug-cc-pVDZ & 19.08 & 6.58 & 12.92 & 6.52 \\ - & & aug-cc-pVTZ & 19.09 & 6.59 & 12.92 & 6.52 \\ - \\ - S & & aug-cc-pVDZ & 5.31 & 5.60 & 5.46 & 5.56 \\ - & & aug-cc-pVTZ & 5.31 & 5.60 & 5.46 & 5.56 \\ - \\ - S & VWN5 & aug-cc-pVDZ & 5.34 & 5.57 & 5.46 & 5.53 \\ - & & aug-cc-pVTZ & 5.34 & 5.57 & 5.46 & 5.52 \\ - \\ - S & eVWN5 & aug-cc-pVDZ & 5.53 & 5.76 & 5.56 & 5.72 \\ - & & aug-cc-pVTZ & 5.53 & 5.76 & 5.56 & 5.72 \\ - & & aug-cc-pVQZ & & & & \\ - \\ - GIC-S & & aug-cc-pVDZ & 5.56 & 5.56 & 5.56 & 5.56 \\ - & & aug-cc-pVTZ & 5.55 & 5.56 & 5.56 & 5.56 \\ - \\ - GIC-S & VWN5 & aug-cc-pVDZ & 5.59 & 5.53 & 5.57 & 5.53 \\ - & & aug-cc-pVTZ & 5.58 & 5.53 & 5.57 & 5.52 \\ - \\ - GIC-S & eVWN5 & aug-cc-pVDZ & 5.78 & 5.72 & 5.66 & 5.72 \\ - & & aug-cc-pVTZ & 5.77 & 5.72 & 5.66 & 5.72 \\ - \\ - B & LYP & aug-cc-pVTZ & & & & 5.28 \\ - B3 & LYP & aug-cc-pVTZ & & & & 5.55 \\ - HF & LYP & aug-cc-pVTZ & & & & 6.68 \\ - \\ -% HF & FCI & aug-cc-pVDZ & & & & 8.78 \\ -% HF & FCI & aug-cc-pVTZ & & & & 8.71 \\ -% HF & FCI & aug-cc-pVQZ & & & & 8.70 \\ - HF & FCI & aug-cc-pV5Z & & & & 8.69 \\ + HF & & 19.09 & 6.59 & 12.92 & 6.52 \\ + S & & 5.31 & 5.60 & 5.46 & 5.56 \\ + S & VWN5 & 5.34 & 5.57 & 5.46 & 5.52 \\ + S & eVWN5 & 5.53 & 5.76 & 5.56 & 5.72 \\ + GIC-S & & 5.55 & 5.56 & 5.56 & 5.56 \\ + GIC-S & VWN5 & 5.58 & 5.53 & 5.57 & 5.52 \\ + GIC-S & eVWN5 & 5.77 & 5.72 & 5.66 & 5.72 \\ + B & LYP & & & & 5.28 \\ + B3 & LYP & & & & 5.55 \\ + HF & LYP & & & & 6.68 \\ + \hline + \mc{5}{l}{Accurate (FCI/aug-cc-pV5Z)\fnm[1]} & 8.69 \\ \end{tabular} \end{ruledtabular} -\end{table*} +\fnt[1]{FCI calculations performed with QUANTUM PACKAGE. \cite{QP2}} +\end{table} %%% %%% %%% %%% %%% TABLE I %%% -\begin{table*} +\begin{table} \caption{ -Excitation energies (in hartree) associated with the lowest double excitation of \ce{He} for various methods, combinations of xc functionals, and basis sets. +Excitation energies (in hartree) associated with the lowest double excitation of \ce{He} obtained with the d-aug-cc-pVQZ basis set for various methods and combinations of xc functionals. \label{tab:BigTab_He} } \begin{ruledtabular} -\begin{tabular}{llccccc} - \mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\ - \cline{1-2} \cline{4-5} - exchange & correlation & Basis & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\ +\begin{tabular}{llcccc} + \mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\ + \cline{1-2} \cline{3-4} + exchange & correlation & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\ \hline - HF & & d-aug-cc-pVDZ & 1.897 & 2.209 & 2.078 & 2.143 \\ - & & d-aug-cc-pVTZ & 1.874 & 2.213 & 2.080 & 2.143 \\ - & & d-aug-cc-pVQZ & 1.874 & 2.212 & 2.080 & 2.142 \\ - \\ - S & & d-aug-cc-pVDZ & 1.075 & 2.055 & 1.546 & 2.030 \\ - & & d-aug-cc-pVTZ & 1.062 & 2.057 & 1.547 & 2.031 \\ - & & d-aug-cc-pVQZ & 1.062 & 2.056 & 1.547 & 2.030 \\ - \\ - S & VWN5 & d-aug-cc-pVDZ & 1.172 & 2.102 & 1.612 & \\ - & & d-aug-cc-pVTZ & 1.163 & & & \\ - & & d-aug-cc-pVQZ & & & & \\ - \\ - S & eVWN5 & d-aug-cc-pVDZ & & & & \\ - & & d-aug-cc-pVTZ & & & & \\ - & & d-aug-cc-pVQZ & & & & \\ - \\ - GIC-S & & d-aug-cc-pVDZ & & & & \\ - & & d-aug-cc-pVTZ & & & & \\ - & & d-aug-cc-pVQZ & & & & \\ - \\ - GIC-S & VWN5 & d-aug-cc-pVDZ & & & & \\ - & & d-aug-cc-pVTZ & & & & \\ - & & d-aug-cc-pVQZ & & & & \\ - \\ - GIC-S & eVWN5 & d-aug-cc-pVDZ & & & & \\ - & & d-aug-cc-pVTZ & & & & \\ - & & d-aug-cc-pVQZ & & & & \\ - \\ - B & LYP & d-aug-cc-pVQZ & & & & \\ - B3 & LYP & d-aug-cc-pVQZ & & & & \\ - HF & LYP & d-aug-cc-pVQZ & & & & \\ - \\ - \mc{2}{l}{Exact} & & & & & 2.126\fnm[1] \\ + HF & & 1.874 & 2.212 & 2.080 & 2.142 \\ + S & & 1.062 & 2.056 & 1.547 & 2.030 \\ + S & VWN5 & 1.163 & 2.104 & 1.612 & 2.079 \\ + S & eVWN5 & 1.174 & 2.108 & 1.615 & 2.083 \\ + GIC-S & & 1.996 & 2.044 & 1.988 & 2.030 \\ + GIC-S & VWN5 & 2.107 & 2.097 & 2.060 & 2.079 \\ + GIC-S & eVWN5 & 2.118 & 2.100 & 2.063 & 2.083 \\ + B & LYP & & & & 2.147 \\ + B3 & LYP & & & & 2.150 \\ + HF & LYP & & & & 2.171 \\ + \hline + \mc{5}{l}{Exact (explicitly-correlated method)\fnm[1] } & 2.126 \\ \end{tabular} \end{ruledtabular} -\fnt[1]{Reference \onlinecite{Burges_1995}} -\end{table*} +\fnt[1]{Reference \onlinecite{Burges_1995}.} +\end{table} %%%%%%%%%%%%%%%%%% %%% CONCLUSION %%%