diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 441593f..b43d39b 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -331,14 +331,14 @@ Expanding the wave function and energy as power series in $\lambda$ as \end{align} \end{subequations} and solving the corresponding perturbation equations up to a given order $k$, then -yields approximate solutions to Eq.~\eqref{eq:SchrEq} \titou{by setting $\lambda = 1$}. +yields approximate solutions to Eq.~\eqref{eq:SchrEq} by setting $\lambda = 1$. % MATHEMATICAL REPRESENTATION Mathematically, Eq.~\eqref{eq:E_expansion} corresponds to a Taylor series expansion of the exact energy around the reference system $\lambda = 0$. The energy of the target ``physical'' system is then recovered at the point $\lambda = 1$. However, like all series expansions, the Eq.~\eqref{eq:E_expansion} has a radius of convergence $\rc$. -When $\rc \titou{\le} 1$, the Rayleigh--Sch\"{r}odinger expansion will diverge +When $\rc \le 1$, the Rayleigh--Sch\"{r}odinger expansion will diverge for the physical system. The value of $\rc$ can vary significantly between different systems and strongly depends on the particular decomposition of the reference and perturbation Hamiltonians in Eq.~\eqref{eq:SchrEq-PT}.\cite{Mihalka_2017b} @@ -377,7 +377,7 @@ We will demonstrate how the choice of reference Hamiltonian controls the positio ultimately determines the convergence properties of the perturbation series. -\titou{Practically, to locate EPs in a more complicated systems, one must solve simultaneously the following equations:\cite{Cejnar_2007} +Practically, to locate EPs in a more complicated systems, one must simultaneously solve\cite{Cejnar_2007} \begin{subequations} \begin{align} \label{eq:PolChar} @@ -388,9 +388,11 @@ ultimately determines the convergence properties of the perturbation series. \end{align} \end{subequations} where $\hI$ is the identity operator. -Equation \eqref{eq:PolChar} is the well-known secular equation providing us with the (eigen)energies of the system. -If an energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy value is, at least, two-fold degenerate. -These degeneracies can be conical intersections between two states with different symmetries for real values of $\lambda$ \cite{Yarkony_1996} or EPs between two states with the same symmetry for complex values of $\lambda$.} +Equation \eqref{eq:PolChar} is the well-known secular equation providing the (eigen)energies of the system. +If the energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy value is at least two-fold degenerate. +These degeneracies can be conical intersections between two states with different symmetries +for real values of $\lambda$\cite{Yarkony_1996} or EPs between two states with the +same symmetry for complex values of $\lambda$. %============================================================%