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fix issue with phi

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This commit is contained in:
Pierre-Francois Loos 2020-07-28 17:58:05 +02:00
parent 0d57b5092a
commit 933ac80e32
1 changed files with 6 additions and 4 deletions
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RapportStage/Rapport.tex
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@ -249,7 +249,7 @@ This evidences that encircling non-Hermitian degeneracies at EPs leads to an int
The eigenvectors associated to the eigenenergies \eqref{eq:E_2x2} are The eigenvectors associated to the eigenenergies \eqref{eq:E_2x2} are
\begin{equation}\label{eq:phi_2x2} \begin{equation}\label{eq:phi_2x2}
\begin{split} \begin{split}
\phi_{\pm}(\lambda) \phi_{\pm}
& = & =
\begin{pmatrix} \begin{pmatrix}
(\epsilon_1 - \epsilon_2 \pm \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2})/2\lambda (\epsilon_1 - \epsilon_2 \pm \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2})/2\lambda
@ -267,9 +267,9 @@ The eigenvectors associated to the eigenenergies \eqref{eq:E_2x2} are
\end{equation} \end{equation}
and, for $\lambda=\lambda_\text{EP}$, they become and, for $\lambda=\lambda_\text{EP}$, they become
\begin{align} \begin{align}
\phi_{\pm}\qty(i\,\frac{\epsilon_1 - \epsilon_2}{2}) & = \begin{pmatrix} -i \\ 1\end{pmatrix}, \phi_{\pm} & = \begin{pmatrix} -i \\ 1\end{pmatrix},
& &
\phi_{\pm}\qty(-i\,\frac{\epsilon_1 - \epsilon_2}{2}) & = \begin{pmatrix} i \\ 1\end{pmatrix}. \phi_{\pm} & = \begin{pmatrix} i \\ 1\end{pmatrix},
\end{align} \end{align}
which are clearly self-orthogonal, i.e., their norm is equal to zero. which are clearly self-orthogonal, i.e., their norm is equal to zero.
%Using Eq.~\eqref{eq:E_EP}, Eq.~\eqref{eq:phi_2x2} can be recast as %Using Eq.~\eqref{eq:E_EP}, Eq.~\eqref{eq:phi_2x2} can be recast as
@ -282,7 +282,7 @@ Then, if the eigenvectors are properly normalized, they behave as $(\lambda - \l
\phi_{\pm}(2\pi) & = \phi_{\mp}(0), \phi_{\pm}(2\pi) & = \phi_{\mp}(0),
& &
\phi_{\pm}(4\pi) & = -\phi_{\pm}(0), \phi_{\pm}(4\pi) & = -\phi_{\pm}(0),
\\ &
\phi_{\pm}(6\pi) & = -\phi_{\mp}(0), \phi_{\pm}(6\pi) & = -\phi_{\mp}(0),
& &
\phi_{\pm}(8\pi) & = \phi_{\pm}(0). \phi_{\pm}(8\pi) & = \phi_{\pm}(0).
@ -349,6 +349,7 @@ is the core Hamiltonian and
v^\text{HF}(\vb{x}) = \sum_i \qty[ J_i(\vb{x}) - K_i(\vb{x}) ] v^\text{HF}(\vb{x}) = \sum_i \qty[ J_i(\vb{x}) - K_i(\vb{x}) ]
\end{equation} \end{equation}
is the HF mean-field potential with is the HF mean-field potential with
\begin{subequations}
\begin{gather} \begin{gather}
\label{eq:CoulOp} \label{eq:CoulOp}
J_i(\vb{x})\phi_p(\vb{x})=\qty[\int\dd\vb{x}'\phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') ] \phi_p(\vb{x}) J_i(\vb{x})\phi_p(\vb{x})=\qty[\int\dd\vb{x}'\phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') ] \phi_p(\vb{x})
@ -356,6 +357,7 @@ is the HF mean-field potential with
\label{eq:ExcOp} \label{eq:ExcOp}
K_i(\vb{x})\phi_p(\vb{x})=\qty[\int\dd\vb{x}'\phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_p(\vb{x}') ] \phi_i(\vb{x}) K_i(\vb{x})\phi_p(\vb{x})=\qty[\int\dd\vb{x}'\phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_p(\vb{x}') ] \phi_i(\vb{x})
\end{gather} \end{gather}
\end{subequations}
being the Coulomb and exchange operators (respectively) in the spin-orbital basis. being the Coulomb and exchange operators (respectively) in the spin-orbital basis.
From hereon, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied (or virtual) orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. From hereon, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied (or virtual) orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational principle to find an approximation of $\Psi$ as a single Slater determinant. Hence a Slater determinant is not an eigenfunction of the exact Hamiltonian $\hH$. However, it is, by definition, an eigenfunction of the so-called (approximated) HF many-electron Hamiltonian defined as the sum of the one-electron Fock operators Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational principle to find an approximation of $\Psi$ as a single Slater determinant. Hence a Slater determinant is not an eigenfunction of the exact Hamiltonian $\hH$. However, it is, by definition, an eigenfunction of the so-called (approximated) HF many-electron Hamiltonian defined as the sum of the one-electron Fock operators