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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. +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 a,b the virtual orbitals of the basis sets. \begin{equation}\label{eq:EMP2} E_{\text{MP2}}=\sum\limits_{\substack{i3/2 the matrix elements become real. This interaction corresponds to the spin contamination of the wave function. For $R<3/2$ the matrix elements are complex, this corresponds to the holomorphic solution of \autoref{fig:SpheriumNrj}, the singularities in this case will be treated in \autoref{sec:uhfSing}. The matrix elements become real again for $R<-75/62$, this corresponds to the sb-UHF solution for negative value of $R$ observed in \autoref{sec:spherium}. We will refer to the domain where the matrix element are complex as the holomorphic domain. \begin{equation}\label{eq:MatrixElem} H_{24}=H_{34}=H_{42}=H_{43}=\sqrt{-3+2R}\sqrt{75+62R}\frac{25+2R}{280R^3} \end{equation} -The singularity structure in this case is more complex because of the spin contamination of the wave function. We can not use configuration state function in this case. So when we compute all the degeneracies using \eqref{eq:PolChar} and \eqref{eq:DPolChar} some correspond to EPs and some correspond to conical intersections. The numerical distinction of those singularities is very difficult so we will first look at the energies $E(\lambda)$ obtained with this basis set. The \autoref{fig:UHFMiniBas} is the analog of \autoref{fig:RHFMiniBas} in the UHF formalism. We see that in this case the sp\textsubscript{z} triplet interacts with the s\textsuperscript{2} and the p\textsubscript{z}\textsuperscript{2} singlets. Those avoided crossings are due to the spin contamination of the wave function. The exceptional points resulting from those avoided crossings will be discussed in \autoref{sec:uhfSing}. \\ +The singularity structure in this case is more complex because of the spin contamination of the wave function. We can not use configuration state function in this case. So when we compute all the degeneracies using \eqref{eq:PolChar} and \eqref{eq:DPolChar} some correspond to EPs and some correspond to conical intersections. The numerical distinction of those singularities is very difficult so we will first look at the energies $E(\lambda)$ obtained with this basis set. +\autoref{fig:UHFMiniBas} is the analog of \autoref{fig:RHFMiniBas} in the UHF formalism. We see that in this case the sp\textsubscript{z} triplet interacts with the s\textsuperscript{2} and the p\textsubscript{z}\textsuperscript{2} singlets. Those avoided crossings are due to the spin contamination of the wave function. The exceptional points resulting from those avoided crossings will be discussed in \autoref{sec:uhfSing}. \\ \begin{figure}[h!] \centering @@ -725,7 +727,7 @@ In the RHF case there are only $\alpha$ singularities and large avoided crossing As shown before, some matrix elements of the Hamiltonian become complex in the holomorphic domain. Therefore the Hamiltonian becomes non-Hermitian for those value of $R$. In \cite{Burton_2019a} Burton et al. proved that for the \ce{H_2} molecule the unrestricted Hamiltonian is not \pt -symmetric in the holomorphic domain. An analog reasoning can be done with the spherium model to prove the same result. The \pt -symmetry (invariance with respect to combined space reflection $\mathcal{P}$ and time reversal $\mathcal{T}$) is a property which ensures that a non-Hermitian Hamiltonian has a real energy spectrum. Thus \pt -symmetric Hamiltonian can be seen as an intermediate between Hermitian and non-Hermitian. - In our case we can see in \autoref{fig:UHFPT} that a part of the energy spectrum becomes complex when R is in the holomorphic domain. This part of the spectrum where the energy becomes complex is called the broken \pt symmetry region. This is consistent with the fact that in the holomorphic domain the Hamiltonian break the \pt -symmetry. +\autoref{fig:UHFPT} shows that for the spherium model a part of the energy spectrum becomes complex when R is in the holomorphic domain. The parameter domain of value where the energy becomes complex is called the broken \pt symmetry region. This is consistent with the fact that in the holomorphic domain the Hamiltonian break the \pt -symmetry. \begin{figure}[h!] \centering @@ -735,13 +737,12 @@ As shown before, some matrix elements of the Hamiltonian become complex in the h \label{fig:UHFPT} \end{figure} -For a non-Hermitian Hamiltonian the exceptional points can lie on the real axis. In particular, at the point of PT transition (the point where the energies become complex) the two energies are degenerate so it is an exceptional. We can see this phenomenon on Figure x, the points of PT transition are indicate by . +For a non-Hermitian Hamiltonian the exceptional points can lie on the real axis. In particular, at the point of PT transition (the point where the energies become complex) the two energies are degenerate resulting in such an exceptional point on the real axis. This degeneracy can be seen in \autoref{fig:UHFPT}. \section{Conclusion} \newpage - \printbibliography \newpage diff --git a/RapportStage/WCElargebasis.pdf b/RapportStage/WCElargebasis.pdf index cb48095b149f891832c51cf0d5715e2899481485..ee59b5d0f177e66e1d3fbdca9aee6faea3ecf1b8 100644 GIT binary patch delta 10485 zcmZXZRZtwjwyqN-xVu~MA;18^AxN;nB}j0W1a});gS!nrxD(vnbs!KN1_iW9=)vMP3($%ZF$4x`5m+LJa6KOV~T($p-6iWZ4HgO4$QK!l09 z7L5eBf0EAj6sSMUg0zxY!qzq-orPY;b_Cpuc*H+EzB&M3DyAMi{Zw8v{_b7mmNm;x zKFzxL!1t{mE{t9;@1KUjMMujtx6>PccXyg!T}A)ibZhuOtF%#rKfgZvJc7f)MKm?T z;w6uOmy2I<>kzjIS&m-~u;Dvdnk&IauKwoXjb*F@L)Y$ozF&ogJF8CLAB_t2k6NnP zU2Z#PjlL?xs|D@f_gmymXxr$A$+#WHt>*qzMMFIWl2(h5-dTN(6M-BZVh%UVzg&4w zWsq`??Wv8q{E%^+QDV}H9|OyVKHaA8X8iKz9dmU#6&8P5piUZ76}1L3bs3*teqi#N*=|kD9gIKA z!AukhNH#0U9$$dOP_)Bdva>KX+n4Y6il_JWGZK|;f4#_=XTsjVD8R+BR@Bi1Cqowg z$&0Lb8xZuX%J%u)Q1a+`SZ0DnKE3jx?pWzrOWIe4`Fd%Y@;GIdrGxv0zJ z2(TvdN9j*bO4WQwtBXu@{kMj`J{$yHTd&(plVwutXz$Ent4ZEuexP13TdPTeL}zKq 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