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ztGaA8{Y#gG%|8F?P=~?bz}fT^w?kdmVXB{pJ}FsbWu7>Ahq%hZp6f8;Prbe6J$c(P zF$=_4QntlRJH)&GGCDp^y(e|QPbR;jErLC}Od~DlldN2520niwdK5@bgi{t`Z8X&> zT7t6xVqjTaP*|VM%oJRFo_5KFnOz-yeil^lG^mn+t27O1bR@Gk^*%m_VlWI(&TFCB-Byzw6` z8v?#iSn;=g4HY<16D{5~Ue=_2_{sW3zKuj1&{rL59Q^Fc^S)T)$s)=95S6@S)~QxP z1pSnZ`?sDZv|u!_WufhNJ$XuCH2wP;D`T!Ff~(X|_TR@LjVsZqOe!LRb3P(1W$x#8 z(XWX34gF-OAHP~SIXiwd{qm1xXJ&)S1%iP7*+C!>4-bU%e=x^ioS;fJ0>{f$M?zkT zLIvUE<>3DNivG=GRkE{?G-+BInad7^34GwO;fAe_$ lSD%0N{y+RbjORa#utB&uc@k_Hl~B3a*->d|q*SC){~z|PXy*U` diff --git a/RapportStage/Rapport.tex b/RapportStage/Rapport.tex index d227a92..09c1ff8 100644 --- a/RapportStage/Rapport.tex +++ b/RapportStage/Rapport.tex @@ -243,39 +243,29 @@ and we have \end{align} This evidences that encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. -The eigenvectors associated to the energies \eqref{eq:E_2x2} are -\begin{equation}\label{ev2x2} -\phi_{\pm}=\begin{pmatrix} -\frac{1}{2\lambda}(\epsilon_1 - \epsilon_2 \pm \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2}) \\ 1 -\end{pmatrix}, -\end{equation} The eigenvectors associated to the eigenenergies \eqref{eq:E_2x2} are -\begin{equation} \label{eq:phi_2x2} -\phi_{\pm}= - \begin{pmatrix} - (E_{\pm}-2\epsilon_2)/\lambda - \\ - 1 - \end{pmatrix}, +\begin{equation}\label{eq:phi_2x2} +\phi_{\pm}=\begin{pmatrix} +(\epsilon_1 - \epsilon_2 \pm \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2})/2\lambda \\ 1 +\end{pmatrix}, \end{equation} and, for $\lambda=\lambda_\text{EP}$, they become -\begin{equation} -\phi_{\pm}=\begin{pmatrix} -\mp i \\ 1 -\end{pmatrix}, -\end{equation} -which are clearly self-orthogonal. -\titou{what do you mean by self-orthogonal?} +\begin{align} + \phi_{\pm}(i\,\frac{\epsilon_1 - \epsilon_2}{2}) & = \begin{pmatrix} -i \\ 1\end{pmatrix}, + & + \phi_{\pm}(-i\,\frac{\epsilon_1 - \epsilon_2}{2}) & = \begin{pmatrix} i \\ 1\end{pmatrix}. +\end{align} +which are 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 -\begin{equation} +\begin{equation}\label{eq:phi_EP} \phi_{\pm}= \begin{pmatrix} - (\epsilon_1-\epsilon_2)/\lambda \pm \sqrt{2\lambda_\text{EP}/\lambda} \sqrt{1 - \lambda_\text{EP}/\lambda} + (\epsilon_1-\epsilon_2)/2\lambda \pm \sqrt{2\lambda_\text{EP}} \sqrt{\lambda - \lambda_\text{EP}}/\lambda \\ 1 - \end{pmatrix}, + \end{pmatrix}. \end{equation} -where one clearly see that, if normalised, they behave as $(\lambda - \lambda_\text{EP})^{-1/4}$ resulting in the following pattern when looping around one EP: +We have seen that the EP inherits its topology from the double valued function $\sqrt{\lambda - \lambda_\text{EP}}$. Then if the eigenvectors are normalised they behave as $(\lambda - \lambda_\text{EP})^{-1/4}$ which gives the following pattern when looping around one EP: \begin{align} \phi_{\pm}(2\pi) & = \phi_{\mp}(0), & @@ -286,9 +276,7 @@ where one clearly see that, if normalised, they behave as $(\lambda - \lambda_\t \end{align} In plain words, four loops around the EP are necessary to recover the initial state. We can also see that looping the other way around leads to a different pattern. -\titou{I don't think it is too obvious that the eigenvectors behave as $(\lambda - \lambda_\text{EP})^{-1/4}$.} - - +\antoine{Is this a bit better like this ?} %============================================================% \section{Perturbation theory} @@ -375,18 +363,18 @@ But as mentioned before \textit{a priori} there are no reasons that $E_{\text{MP The MP partitioning is not the only one possible in electronic structure theory. An other possibility, even more natural than the MP one, is to take the diagonal elements of $\bH$ as the zeroth-order Hamiltonian. Hence, the off-diagonal elements of $\bH(\lambda)$ are the perturbation operator. This partitioning leads to the Epstein-Nesbet (EN) perturbation theory. The zeroth-order eigenstates for this partitioning are Slater determinants as for the MP partitioning. The expression of the second order correction to the energy is given for both MP and EN. The energies at the MP denominator are the orbitals energies whereas in the EN case it is the excitation energies. The i,j indices represent the occupied orbitals and r,s the virtual orbitals of the basis sets. \begin{equation}\label{eq:EMP2} -E_{\text{MP2}}=\sum\limits_{\substack{iWl&sA)UF-e-C@w+!9#F&Cpf`1xVu|ma19WGySuv&kl^kN4DRmok>|YcsrstU zpS^4L%v9H2-M!YmuDfZzN=PxYvwT6OX*{WVLgu7kqi`^>MivmDU{x@;w{W$j`11Zp z6`6v9f>p}O*45nk{cdaQYA#`J>R@J$EF^^N;_7T}Y=`Wbaojmpb=-zd%XOzV+Y(g4 zqMH(sl@s(KtpF$rAA{V+O>BXYNVa{Ge+ItUs%M^H?%?JG%6Lx!ZZW7dAZ0 zzB_u;S)Q0Td|R>cd((TmS#G-8vwb=`v(XQT?Zg1x)OR&62v@t2U~6*R)(5e-HAO|+ zAauTUthAwmUg|5$c^2Fgrda>%3lGhYwA_(ILo7C$7UtK5W6;K@8k2-hDQz69vhO1C zk0#pd|Gv0%@DnYNuyaED#{C-}rR^!Yf_OTSoX`z^6x4SaAXuo<40w*9>wuTd%oG#c0jqD51gdilL-D~Zw7a4e}%pWoCJP( z<7@~H?zQN zjT)_>Uxi~$gTY08rl4Rj#M#nb8ZQ$EU;&3;DDRD6zPVaLrT1`X6}?$zB0 zime6R!JFNcec|G1X+)KpLFPj@#JG#alK`y+9t$-Xg|)H)dwFV&@SSgCmsV5lIYOK+ zl-xFS#D>P^q~WzDjjq%0`tNk*EgeB)Vchk1fByQC9OXD&#QGd5^E)^@Lg0$6a$^c> zFPoszwU)uhca);-r&=L}D>IK|1iMgEBn@)oxw?xHcs_|P)`GE;Tvb3xz zVj+g$jqmhzuRXz~!#K^H6eCax-Zg@j0RIfEr;fwZ{C>Cm65^t7>GFhM8!O!xluoIw 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