Added .tex files

appendix_CFT.tex

@@ -0,0 +1,64 @@
+
+\chapter{Crystal Field Splitting Energy}
+\label{ap: cft}
+
+Table~\ref{tab: cft-full} lists representative values of
+the crystal field splitting energy~$\Delta_\text{oct}$
+for octahedral complexes of the form $\mathrm{[ML_6]}^{n+}$
+for monodentate ligands L and $\mathrm{[ML_3]}^{n+}$ for bidentate ones.
+The ligands are sorted according to the spectrochemical series,
+in increasing order of ligand field strength~\cite{Ryutaro1938a, Ryutaro1938b, FiggisBook}.
+%
+\begin{table}[ht!]
+  \centering
+  {\renewcommand*{\arraystretch}{1.5}
+    \begin{tabular}{ c c c c c c c c c c }
+      \toprule
+      \multirow{2}{*}{
+        \begin{minipage}[c]{1.75cm} \centering Electron System \end{minipage}
+      } & \multirow{2}{*}{Ion}
+      & \multicolumn{8}{c }{$\Delta_\text{oct}$ ($\times  10^3$~cm$^{-1}$)} \\
+      \cline{3-10}
+       &  & Br$^-$ & Cl$^-$ & F$^-$ & ox$^{2-}$ & OH$_2$ & NH$_3$ & en & CN$^-$ \\
+      \midrule
+      d$^1$ & Ti$^{3+}$ &  & 13 & 18.9 &  & 20.1 & 17 &  & 22 \\
+       & V$^{4+}$ &  & 15 & 20.1 &  &  &  &  &  \\
+      d$^2$ & V$^{3+}$ &  & 12 & 16.1 & 18 & 20 & 18 &  & 24 \\
+      d$^3$ & V$^{2+}$ &  & 8 &  &  & 12.4 &  &  &  \\
+       & Cr$^{3+}$ & 13 & 13 & 14.5 & 17.4 & 17 & 21.5 & 22 & 26 \\
+       & Mo$^{3+}$ & 14.5 & 19.2 &  &  &  &  &  &  \\
+      d$^4$ & Cr$^{2+}$ &  & 10 & 14 &  & 13 &  & 18 &  \\
+       & Mn$^{3+}$ &  & 17.5 & 22 & 20 & 20 &  &  & 31 \\
+      d$^5$ & Mn$^{2+}$ & 7 & 7.5 & 7.8 &  & 8.5 &  &  & 33 \\
+       & Fe$^{3+}$ &  &  & 14 & 14 & 14 &  &  & 35 \\
+      d$^6$ & Fe$^{2+}$ &  &  & 10 &  & 10 &  &  & 32 \\
+       & Co$^{3+}$ &  &  & 13 & 18 & 20.8 & 22.9 & 23.2 &  \\
+       & Ru$^{2+}$ &  &  &  &  & 19.8 & 28.1 &  &  \\
+       & Rh$^{2+}$ & 19 & 20.4 &  & 26 & 27 & 34 & 35 &  \\
+       & Ir$^{3+}$ & 23 & 25 &  &  &  & 41 & 41 &  \\
+       & Pt$^{4+}$ & 25 & 29 & 33 &  &  &  &  &  \\
+      d$^7$ & Co$^{2+}$ & 6.5 & 7.65 & 8.3 & 11 & 9.3 & 10.2 & 11 &  \\
+      d$^8$ & Ni$^{2+}$ & 6.8 & 7.2 & 7.25 &  & 8.5 & 11.5 & 11.5 &  \\
+      d$^9$ & Cu$^{2+}$ &  &  &  &  & 12 & 16 & 16 & \\
+      \bottomrule
+    \end{tabular}
+  }
+  \caption{Octahedral CFT splitting energy~$\Delta_\text{oct}$
+    of some transition-metal complexes of the form [ML\textsubscript{6}]\textsuperscript{n+}
+    for all monodentate ligands L. Ligands $\mathrm{L = ox^{2-}}$ and en are bidentate
+    and form complexes of the form $\mathrm{[ML_3]}^{n+}$.
+    Note that $\mathrm{ox^{2-} = (COO)_2^{2-}}$ is the oxalate dianion
+    and $\mathrm{en = C_2 H_4 (NH_2)_2}$ is ethylenediamine.
+    Data herein is taken from Ref.~\cite{FiggisBook}.
+  }
+  \label{tab: cft-full}
+\end{table}
+
+Note that values of $\Delta_\text{oct}$ also follow an increasing trend as a function
+of the oxidation state and period number of the metal cation,
+e.g. $\Delta_\text{oct}(\mathrm{Fe^{3+}}) < \Delta_\mathrm{o}(\mathrm{Fe^{2+}})$
+and $\Delta_\mathrm{o}(\mathrm{Fe^{2+}}) < \Delta_\mathrm{o}(\mathrm{Ru^{2+}})$.
+Both effects are consequences of $\Delta_\text{oct} \propto \frac{1}{r^5}$,
+where $r$ is the metal--ligand interatomic distance;
+a more highly charged metal cation pulls its ligands closer while
+a larger one allows them to approach by lessening steric hindrance~\cite{FiggisBook}.

appendix_MottBethe.tex

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+\chapter{Derivation of the Mott-Bethe Formula}
+\label{ap: MottBethe}
+
+The Mott-Bethe formula is often used to convert in between
+the X-ray scattering factor
+%
+\begin{equation}
+  f_\text{X}(\boldsymbol{q}) = \int n_\text{e}(\boldsymbol{r}) \mathrm{e}^{ - \mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r
+\end{equation}
+%
+and the electron scattering factor
+%
+\begin{equation}
+  f_\text{e}(\boldsymbol{q}) = - \frac{m_\text{e}}{2 \unslant[-.2]\pi \hbar^2} \int U(\boldsymbol{r}) \mathrm{e}^{ - \mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r
+\end{equation}
+%
+for the purpose of calculations.
+However, the source of this handy relationship is rarely stated.
+Thus, for the completeness of this thesis,
+the derivation for this equation is described below,
+as laid out in Ref.~\cite{LandauLifshitzBook}.
+
+During a scattering event with an atom, X-rays interact mainly with
+the electron number density $n_\text{e}(\boldsymbol{r})$
+while electrons are deflected by the electrostatic potential $\phi(\boldsymbol{r})$
+caused by the bound electrons and the nucleus.
+%
+A starting point to relate these two quantities is the Poisson Equation
+and the total atomic charge density~$\rho(\boldsymbol{r})$,
+%
+\begin{equation}
+  \nabla^2 \phi(\boldsymbol{r}) = -\frac{\rho(\boldsymbol{r})}{\epsilon_0}
+\end{equation}
+%
+in which their Fourier transforms are inserted,
+%
+\begin{equation}
+  \begin{aligned}
+    \nabla^2 \left( \int \phi_{\boldsymbol{q}} \mathrm{e}^{ \mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r} } \mathrm{d}^3 q \right)
+      & = -\frac{1}{\epsilon_0} \left( \int \rho_{\boldsymbol{q}} \mathrm{e}^{\mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 q \right) \\
+    \int \phi_{\boldsymbol{q}} \left( -q^2 \mathrm{e}^{\mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \right) \mathrm{d}^3 q
+      & = -\frac{1}{\epsilon_0} \int \rho_{\boldsymbol{q}} \mathrm{e}^{\mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 q \\
+    \phi_{\boldsymbol{q}} & = \frac{1}{\epsilon_0 q^2} \rho_{\boldsymbol{q}}
+  \end{aligned}
+\end{equation}
+%
+Therefore,
+%
+\begin{equation}
+  \begin{aligned}
+    \int \phi(\boldsymbol{r}) \mathrm{e}^{ - \mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r
+      & = \frac{1}{\epsilon_0 q^2} \int \rho(\boldsymbol{r}) \mathrm{e}^{ - \mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r
+  \end{aligned}
+\end{equation}
+%
+On the right side, $\rho(\boldsymbol{r})$ can be expanded into a point-like nuclear charge density
+$\rho_\text{N}(\boldsymbol{r}) = Z e \delta^3(\boldsymbol{r})$ and an extended electronic charge density $\rho_\text{e}(\boldsymbol{r}) = e n_\text{e}(\boldsymbol{r})$,
+%
+\begin{equation}
+  \begin{aligned}
+    \frac{1}{\epsilon_0 q^2} \int \rho(\boldsymbol{r}) \mathrm{e}^{\mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r
+      & = \frac{1}{\epsilon_0 q^2} \int \left( Z e \delta^3(\boldsymbol{r}) - e n_\text{e}(\boldsymbol{r}) \right) \mathrm{e}^{\mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r \\
+      & = \frac{e}{\epsilon_0 q^2} \left( Z - \int n_\text{e}(\boldsymbol{r}) \mathrm{e}^{\mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r \right) \\
+      & = \frac{e}{\epsilon_0 q^2} \left( Z - f_\text{X}(\boldsymbol{q}) \right)
+  \end{aligned}
+\end{equation}
+%
+where $Z$ is the atomic number of the atom and $n_\text{e}(\boldsymbol{r})$ is the number density of the bound electrons.
+%
+On the left side, recall that $U(\boldsymbol{r}) = -e \phi(\boldsymbol{r})$,
+%
+\begin{equation}
+  \begin{aligned}
+    \int \phi(\boldsymbol{r}) \mathrm{e}^{ - \mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r
+      & = -\frac{1}{e} \int U(\boldsymbol{r}) \mathrm{e}^{ - \mathrm{i} \boldsymbol{q} \cdot \boldsymbol{r}} \mathrm{d}^3 r \\
+      & = \frac{2 \unslant[-.2]\pi \hbar^2}{m_\text{e} e} f_\text{e}(\boldsymbol{q})
+  \end{aligned}
+\end{equation}
+%
+By recombining the result from the left and right sides, the Mott-Bethe formula is thus recovered:
+%
+\begin{equation}
+  \begin{aligned}
+    f_\text{e}(\boldsymbol{q}) = \frac{m_\text{e} e^2}{2 \unslant[-.2]\pi \hbar^2 \epsilon_0 q^2} \left( Z - f_\text{X}(\boldsymbol{q}) \right)
+  \end{aligned}
+\end{equation}

appendix_SCO.tex

@@ -0,0 +1,165 @@
+
+\chapter{Theory of Spin Crossover}
+\label{ap: sco}
+
+In Section~\ref{sec: SCO-theory}, the theory of thermal and photoinduced
+spin crossover is discussed.
+%
+In particular, the non-adiabatic multiphonon relaxation model
+by Buhks et al~\cite{Buhks1980} is mentioned in the context of
+the LS$\leftarrow$HS relaxation following LIESST.
+%
+Here, this model is described in more detail for reference.
+
+Following the lead of Hauser in Ref.~\cite{SCO-II},
+consider Fermi's Golden Rule~\cite{Dirac1927, Fermi1950},
+%
+\begin{equation}
+  \begin{aligned}
+    w_{i \rightarrow f} & = \frac{2 \unslant[-.2]\pi}{\hbar} |V_{i f}|^2 \rho_f
+  \end{aligned}
+  \label{eq: fermi-golden}
+\end{equation}
+%
+where $w_{i \rightarrow f}$ is the probability of transition from an initial state~$i$
+(in the $m$-th vibrational level of the HS state)
+to a final state~$f$ (in the $m^\prime$-th vibrational level of the LS state),
+$|V_{i f}|^2$ is the matrix element of the perturbing potential
+that couples states $i$ and $f$, and $\rho_f$ is the density of states at $f$.
+
+Given that the perturbing potential is
+the spin--orbit interaction~$\hat{H}_\text{SO} = \zeta \hat{\boldsymbol{L}} \cdot \hat{\boldsymbol{S}}$,
+the coupling matrix element can be expanded as
+%
+\begin{equation}
+  \begin{aligned}
+    |V_{i f}|^2 & = |\langle \Psi_f | \hat{H}_\text{SO} | \Psi_i \rangle|^2 \\
+      & = |\langle \psi_\text{HS} | \hat{H}_\text{SO} | \psi_\text{LS} \rangle|^2
+      | \langle \chi_{m^\prime} | \chi_m \rangle |^2
+  \end{aligned}
+\end{equation}
+%
+where $| \psi \rangle, | \chi \rangle $ are the electronic and vibrational parts
+of the total wavefunction~$| \Psi \rangle$ and,
+according to the Condon~approximation, $| \Psi_i \rangle = | \psi_\text{HS} \rangle | \chi_m \rangle$
+and $| \Psi_f \rangle = | \psi_\text{LS} \rangle | \chi_{m^\prime} \rangle$.
+%
+To evaluate this quantity, consider that
+the electronic states are mixed to some degree
+due to the presence of the spin--orbit interaction~$\hat{H}_\text{SO}$
+and thus need to be expressed using first-order perturbation theory,
+%
+\begin{equation}
+  \begin{aligned}
+    | \psi_\text{LS} \rangle
+      & \approx | \psi_\text{LS}^{(0)} \rangle
+        + \sum_{j}
+          \frac{\langle \psi_j^{(0)} | \hat{H}_\text{SO} | \psi_\text{LS}^{(0)} \rangle}{E_\text{LS}^{(0)} - E_j^{(0)}} | \psi_j^{(0)} \rangle \\
+    \langle \psi_\text{HS} |
+      & \approx \langle \psi_\text{HS}^{(0)} |
+        + \sum_{j}
+          \frac{\langle \psi_\text{HS}^{(0)} | \hat{H}_\text{SO} | \psi_j^{(0)} \rangle}{E_\text{HS}^{(0)} - E_j^{(0)}} \langle \psi_j^{(0)} |
+  \end{aligned}
+\end{equation}
+%
+where the energy denominators are evaluated in
+the equilibrium nuclear configuration of the respective states.
+Then,
+%
+\begin{equation}
+  \begin{aligned}
+    \langle \psi_\text{HS} | \hat{H}_\text{SO} | \psi_\text{LS} \rangle
+      & = \langle \psi_\text{HS}^{(0)} | \hat{H}_\text{SO} | \psi_\text{LS}^{(0)} \rangle
+        + \sum_{j} \langle \psi_\text{HS}^{(0)} | \hat{H}_\text{SO} | \psi_j^{(0)} \rangle
+        \langle \psi_j^{(0)} | \hat{H}_\text{SO} | \psi_\text{LS}^{(0)} \rangle \\
+      & \quad \left( \frac{1}{E_\text{LS}^{(0)} - E_j^{(0)}} + \frac{1}{E_\text{HS}^{(0)} - E_j^{(0)}} \right) \\
+      & = 0 + \langle \mathrm{^5T_{2g}} | \hat{H}_\text{SO} | \mathrm{^3T_{1g}} \rangle
+        \langle \mathrm{^3T_{1g}} | \hat{H}_\text{SO} | \mathrm{^1A_{1g}} \rangle
+        \left( \frac{1}{\Delta E_\text{LI}^{(0)}} + \frac{1}{\Delta E_\text{HI}^{(0)}} \right)
+  \end{aligned}
+\end{equation}
+%
+where $\Delta E_\text{LI}^{(0)}, \Delta E_\text{HI}^{(0)}$ are the energy difference
+between the triplet intermediate state $\mathrm{^3T_{1g}}$ and the other states.
+From Ref.~\cite{Griffith1964}, $\mathrm{^3T_{1g}}$,
+with electronic configuration $\mathrm{(t_{2g})^5 (e_g^*)^1}$,
+is the only term which has non-vanishing spin--orbit matrix elements
+with both $\mathrm{^1A_{1g}}$ and $\mathrm{^5T_{2g}}$,
+%
+\begin{equation}
+  \begin{aligned}
+    \langle \mathrm{^5T_{2g}} | \hat{H}_\text{SO} | \mathrm{^1A_{1g}} \rangle
+      & = 0 \\
+    \langle \mathrm{^5T_{2g}} | \hat{H}_\text{SO} | \mathrm{^3T_{1g}} \rangle
+      & = - \sqrt{6} \zeta \\
+    \langle \mathrm{^3T_{1g}} | \hat{H}_\text{SO} | \mathrm{^1A_{1g}} \rangle
+      & = \sqrt{3} \zeta
+  \end{aligned}
+\end{equation}
+
+Assume that the LS and HS potentials are identical and harmonic of
+the same frequency~$\omega$, with the latter displaced energetically by $\Delta E_\text{HL}^{(0)}$ and
+configurationally by $\Delta Q_\text{HL} = \sqrt{6} \Delta r_\text{HL}$
+along a single internal vibrational coordinate~$Q$, namely the totally symmetric metal--ligand stretch mode.
+Then, energy conservation requires simply $m^\prime = m + n$,
+where $n = \frac{\Delta E_\text{HL}^{(0)}}{\hbar \omega}$ is the reduced energy gap between the LS and HS states,
+and the density of states~$\rho_f$ becomes $\frac{g_f}{\hbar \omega}$,
+where $g_f = 1$ is the degeneracy of the final electronic state.
+
+To obtain the LS$\leftarrow$HS relaxation rate constant~$k_\text{HL}(T)$,
+Eq.~\eqref{eq: fermi-golden} is combined with those above and
+ensemble-averaged over all $m$,
+%
+\begin{equation}
+  \begin{aligned}
+    k_\text{HL}(T)
+      & = \frac{2 \unslant[-.2]\pi}{\hbar^2 \omega}
+        |\langle \psi_\text{HS} | \hat{H}_\text{SO} | \psi_\text{LS} \rangle|^2 \bar{F}_n(T)
+  \end{aligned}
+\end{equation}
+%
+where $\bar{F}_n$ is the ensemble average of the Franck-Condon factor~$F_{m n}$,
+%
+\begin{equation}
+  \begin{aligned}
+    F_{m n}(T) & = | \langle \chi_{m + n} | \chi_m \rangle |^2 \\
+    \bar{F}_n(T)
+      & = \frac{\sum \limits_m F_{m n}(T) \mathrm{e}^{-m \hbar \omega / k_\text{B} T}}{\sum \limits_m \mathrm{e}^{-m \hbar \omega / k_\text{B} T}}
+  \end{aligned}
+\end{equation}
+
+% Sousa, de Graaf calculations
+In a similar procedure, Sousa et~al~\cite{Sousa2013} have evaluate
+more spin--orbit coupling matrix elements using high-level quantum-chemical methods.
+%
+Table~\ref{tab: sco-so} shows some of these computational results.
+%
+\begin{table}[ht!]
+  \centering
+  {\renewcommand*{\arraystretch}{1.5}
+    \begin{tabular}{| c | c | c c c c c c c c |}
+      \cline{3-10}
+      \multicolumn{2}{c|}{} & \multicolumn{8}{c |}{$\psi_f$} \\
+      \cline{3-10}
+      \multicolumn{2}{c|}{} & $\mathrm{^1 A_{1g}}$
+        & $\mathrm{^1 T_{1g}}$ & $\mathrm{^1 MLCT}$ & $\mathrm{^3 T_{1g}}$
+        & $\mathrm{^3 T_{2g}}$ & $\mathrm{^3 MLCT}$ & $\mathrm{^5 T_{2g}}$ & $\mathrm{^5 MLCT}$ \\
+      \hline
+      \multirow{8}{*}{$\psi_i$} & $\mathrm{^1 A_{1g}}$ &  &  &  & 527.7 & 83.7 & 81.6 & 0 & 0 \\
+       & $\mathrm{^1 T_{1g}}$ &  &  &  & 75.5 & 131.4 & 164.7 & 0 & 0 \\
+       & $\mathrm{^1 MLCT}$ &  &  &  & 96.0 & 214.3 & 199.9 & 0 & 0 \\
+       & $\mathrm{^3 T_{1g}}$ & 527.7 & 75.5 & 96.0 &  &  &  & 417.7 &  \\
+       & $\mathrm{^3 T_{2}}$ & 83.7 & 131.4 & 214.3 &  &  &  & 219.9 &  \\
+       & $\mathrm{^3 MLCT}$ & 81.6 & 164.7 & 199.9 &  &  &  & 6.2 & 344.3 \\
+       & $\mathrm{^5 T_{2g}}$ & 0 & 0 & 0 & 417.7 & 219.9 & 6.2 &  &  \\
+       & $\mathrm{^5 MLCT}$ & 0 & 0 & 0 &  &  & 344.3 &  & \\
+      \hline
+    \end{tabular}
+  }
+  \caption{Select spin--orbit coupling matrix elements
+    $\langle \psi_f | \hat{H}_\text{SO} | \psi_i \rangle$
+    of $\mathrm{[Fe^{II}(bpy)_3]^{2+}}$ in the equilibrium nuclear configuration
+    of the $\mathrm{^1 A_{1g}}$ LS state, calculated at the CASSCF/CASPT2 level~\cite{Sousa2013}.
+  }
+  \label{tab: sco-so}
+\end{table}

appendix_SCO_BPY.tex

@@ -0,0 +1,33 @@
+% \chapter{SVD Analysis of $\mathrm{[Fe^{II}(bpy)_3](PF_6)_2}$ TA Data}
+\chapter{SVD Analysis of [Fe\textsuperscript{II}(bpy)\textsubscript{3}](PF\textsubscript{6})\textsubscript{2} TA Data}
+\label{ap: sco-bpy}
+
+% Figure: aqueous SVD analysis
+\begin{figure}[hp]
+  \centering
+  \includegraphics[width = \textwidth]{Figures/fig_BPY_data_aqueous_svd.pdf}
+  \caption[SVD analysis of solvated BPY TA data.]{
+    SVD analysis of solvated BPY TA data:
+    (a) UV short-time, (b) UV long-time, (c) Vis short-time, and (d) Vis long-time.
+    From left to right, the panels show
+    the principal wavelenght-dependent singular vectors~$u_i(\lambda)$,
+    the first 100 singular values~$s_i$,
+    and the principal time-dependent singular vectors~$v_i(t)$.
+  }
+  \label{fig: BPY-data-aqueous-svd}
+\end{figure}
+
+% Figure: aqueous SVD analysis
+\begin{figure}[p]
+  \centering
+  \includegraphics[width = \textwidth]{Figures/fig_BPY_data_crystal_svd.pdf}
+  \caption[SVD analysis of single-crystal BPY TA data.]{
+    SVD analysis of single-crystal BPY TA data:
+    (a) UV short-time, (b) UV long-time, (c) Vis short-time, and (d) Vis long-time.
+    From left to right, the panels show
+    the principal wavelenght-dependent singular vectors~$u_i(\lambda)$,
+    the first 100 singular values~$s_i$,
+    and the principal time-dependent singular vectors~$v_i(t)$.
+  }
+  \label{fig: BPY-data-crystal-svd}
+\end{figure}