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Hugh Burton 2020-11-19 12:55:38 +00:00
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@ -331,14 +331,14 @@ Expanding the wave function and energy as power series in $\lambda$ as
\end{align} \end{align}
\end{subequations} \end{subequations}
and solving the corresponding perturbation equations up to a given order $k$, then 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 % MATHEMATICAL REPRESENTATION
Mathematically, Eq.~\eqref{eq:E_expansion} corresponds to a Taylor series expansion of the exact energy Mathematically, Eq.~\eqref{eq:E_expansion} corresponds to a Taylor series expansion of the exact energy
around the reference system $\lambda = 0$. around the reference system $\lambda = 0$.
The energy of the target ``physical'' system is then recovered at the point $\lambda = 1$. 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 $\rc$. 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. for the physical system.
The value of $\rc$ can vary significantly between different systems and strongly depends on the particular decomposition 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} 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. 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{subequations}
\begin{align} \begin{align}
\label{eq:PolChar} \label{eq:PolChar}
@ -388,9 +388,11 @@ ultimately determines the convergence properties of the perturbation series.
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\hI$ is the identity operator. 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. Equation \eqref{eq:PolChar} is the well-known secular equation providing 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. 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 $\lambda$.} 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 $\lambda$.
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