almost done with HF part
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@ -189,6 +189,7 @@ More importantly here, although EPs usually lie off the real axis, these singula
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\caption{%
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Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}).
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Only the interacting closed-shell singlets are plotted in the complex plane, becoming degenerate at the EP (black dot).
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The contour followed around the EP in order to interchange states is also represented.
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\label{fig:FCI}}
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\end{figure*}
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@ -205,21 +206,21 @@ The exact Hamiltonian is then
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\label{eq:H_FCI}
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\bH =
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\begin{pmatrix}
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U & - t & + t & 0 \\
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U & - t & - t & 0 \\
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- t & 0 & 0 & - t \\
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+ t & 0 & 0 & + t \\
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0 & - t & + t & U \\
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- t & 0 & 0 & - t \\
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0 & - t & - t & U \\
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\end{pmatrix},
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\end{equation}
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where $t$ is the hopping parameter and $U$ is the on-site Coulomb repulsion.
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We refer the interested reader to Refs.~\onlinecite{Carrascal_2015,Carrascal_2018} for more details about this system.
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The parameter $U$ dictates the correlation regime.
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In the weak correlation regime (\ie, small $U$), the kinetic energy dominates and the electrons are delocalised over both sites.
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For large $U$ (or strong correlation regime), the electron repulsion term drives the physics and the electrons localise on opposite sites to minimise their Coulomb repulsion.
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In the large-$U$ (or strong correlation) regime, the electron repulsion term drives the physics and the electrons localise on opposite sites to minimise their Coulomb repulsion.
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This phenomenon is sometimes referred to as a Wigner crystallisation. \cite{Wigner_1934}
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To illustrate the formation of an exceptional points, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$.
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When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) eigenvalues
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To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$.
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When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies
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\begin{subequations}
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\begin{align}
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E_{\mp} &= \frac{1}{2} \qty(U \mp \sqrt{ (4 \lambda t)^2 + U^2 } ),
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@ -240,12 +241,12 @@ with energy
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\label{eq:E_EP}
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E_\text{EP} = \frac{U}{2}.
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\end{equation}
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These $\lambda$ values correspond to so-called EPs and connect the ground and excited state in the complex plane.
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These $\lambda$ values correspond to so-called EPs and connect the ground and excited states in the complex plane.
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Crucially, the energy surface becomes non-analytic at $\lambda_{\text{EP}}$ and a square-root singularity forms with two branch cuts running along the imaginary axis from $\lambda_{\text{EP}}$ to $\pm \i \infty$ (see Fig.~\ref{subfig:FCI_cplx}).
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On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda = \Re(\lambda_{\text{EP}})$.
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The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction.
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Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states.
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Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states (see Fig.~\ref{subfig:FCI_cplx}).
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This behaviour can be seen by expanding the radicand in Eq.~\eqref{eq:singletE} as a Taylor series around $\lambda_{\text{EP}}$ to give
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\begin{equation}
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E_{\pm} \approx E_{\text{EP}} \pm \sqrt{32t^2 \lambda_{\text{EP}}} \sqrt{\lambda - \lambda_{\text{EP}}}.
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@ -328,25 +329,25 @@ is the HF mean-field potential with
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\begin{subequations}
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\begin{gather}
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\label{eq:CoulOp}
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\Hat{J}_i(\vb{x})\phi_j(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') \dd\vb{x}' ] \phi_j(\vb{x})
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\Hat{J}_i(\vb{x})\phi_j(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') \dd\vb{x}' ] \phi_j(\vb{x}),
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\\
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\label{eq:ExcOp}
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\Hat{K}_i(\vb{x})\phi_j(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_j(\vb{x}') \dd\vb{x}'] \phi_i(\vb{x})
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\Hat{K}_i(\vb{x})\phi_j(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_j(\vb{x}') \dd\vb{x}'] \phi_i(\vb{x}),
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\end{gather}
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\end{subequations}
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being the Coulomb and exchange operators (respectively) in the spin-orbital basis. \cite{SzaboBook}
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The HF energy is then defined as
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\begin{equation}
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\label{eq:E_HF}
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E_\text{HF} = \sum_i^{N} h_i + \frac{1}{2} \sum_{ij}^{N} \qty( J_{ij} - K_{ij} )
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E_\text{HF} = \sum_i^{N} h_i + \frac{1}{2} \sum_{ij}^{N} \qty( J_{ij} - K_{ij} ),
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\end{equation}
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with
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\begin{align}
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h_i & = \mel{\phi_i}{h}{\phi_i}
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h_i & = \mel{\phi_i}{\Hat{h}}{\phi_i},
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&
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J_{ij} & = \mel{\phi_i}{\Hat{J}_j}{\phi_i}
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J_{ij} & = \mel{\phi_i}{\Hat{J}_j}{\phi_i},
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&
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K_{ij} & = \mel{\phi_i}{\Hat{K}_j}{\phi_i}
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K_{ij} & = \mel{\phi_i}{\Hat{K}_j}{\phi_i}.
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\end{align}
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If the spatial part of the spin-orbitals are restricted to be the same for spin-up and spin-down electrons, one talks about restricted HF (RHF) theory, whereas if one does not enforce this constrain it leads to the so-called unrestricted HF (UHF) theory.
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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.
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@ -369,38 +370,44 @@ Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational princi
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Coming back to the Hubbard dimer, the HF energy is [see Eq.~\eqref{eq:E_HF}]
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\begin{equation}
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E_\text{HF} = -t \qty[ \sin \theta_\alpha + \sin \theta_\beta ] + \frac{U}{2} \qty[ 1 + \cos \theta_\alpha \cos \theta_\beta ]
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E_\text{HF} = -t \qty[ \sin \theta_\alpha + \sin \theta_\beta ] + \frac{U}{2} \qty[ 1 + \cos \theta_\alpha \cos \theta_\beta ],
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\end{equation}
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where
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\begin{align}
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\mathcal{B}^{\sigma} & = \cos(\frac{\theta_\sigma}{2}) \Lsi - \sin(\frac{\theta_\sigma}{2}) \Rsi
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\mathcal{B}^{\sigma} & = \cos(\frac{\theta_\sigma}{2}) \Lsi + \sin(\frac{\theta_\sigma}{2}) \Rsi,
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\\
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\mathcal{A}^{\sigma} & = \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi
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\mathcal{A}^{\sigma} & = - \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi
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\end{align}
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are the bonding $\mathcal{B}^{\sigma}$ and anti-bonding $\mathcal{A}^{\sigma}$ molecular orbitals for the spin-$\sigma$ electrons and the angles which minimises the HF energy, \ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$, are
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are the bonding $\mathcal{B}^{\sigma}$ and anti-bonding $\mathcal{A}^{\sigma}$ molecular orbitals for the spin-$\sigma$ electrons.
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The angles which minimises the HF energy, \ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$, are
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\begin{equation}
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\theta_\text{RHF}^\alpha = \theta_\text{RHF}^\beta = \pi/4
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\theta_\text{RHF}^\alpha = \theta_\text{RHF}^\beta = \pi/4,
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\end{equation}
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for $0 \le U \le 4t$, and
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for $0 \le U \le 4t$, yielding
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\begin{align}
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\theta_\text{UHF}^\alpha & = \arctan (-\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U})
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\\
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\theta_\text{UHF}^\beta & = \arctan (+\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U})
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\mathcal{B}_\text{RHF}^{\sigma} & = \frac{\Lsi + \Rsi}{\sqrt{2}},
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&
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\mathcal{A}_\text{RHF}^{\sigma} & = \frac{\Lsi - \Rsi}{\sqrt{2}},
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\end{align}
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otherwise.
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In the RHF formalism, the two electrons are restricted to ``live'' in the same spatial orbital.
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The RHF ground-state energy is then
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and the following RHF ground-state energy:
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\begin{equation}
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E_\text{RHF} = -2t + \frac{U}{2}
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\end{equation}
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In the RHF formalism, the two electrons are restricted to ``live'' in the same spatial orbital.
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The RHF wave function cannot model properly the physics of the system at large $U$ because the spatial orbitals are restricted to be the same, and, \textit{a fortiori}, it cannot represent two electrons on opposite sites.
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Within the HF approximation, at the critical value of $U = 4t$, famously known as the Coulson-Fischer point, \cite{Coulson_1949} a symmetry-broken UHF solution appears with lower in energy than the RHF one.
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The UHF ground-state energy is
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Within the HF approximation, at the critical value of $U = 4t$, famously known as the Coulson-Fischer point, \cite{Coulson_1949} a symmetry-broken UHF solution appears with a lower energy than the RHF one.
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Indeed, for $U \ge 4t$, we have
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\begin{align}
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\theta_\text{UHF}^\alpha & = \titou{\arctan (-\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U})},
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\\
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\theta_\text{UHF}^\beta & = \titou{\arctan (+\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U})},
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\end{align}
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and the corresponding UHF ground-state energy is
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\begin{equation}
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E_\text{UHF} = - \frac{2t^2}{U}
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E_\text{UHF} = - \frac{2t^2}{U}.
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\end{equation}
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for $U \ge 4t$.
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Note that, for $U \ge 4t$, the RHF wave function remains a genuine solution of the HF equations but corresponds to a saddle point, not a minimum.
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%=====================================================%
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\subsection{M{\o}ller-Plesset perturbation theory}
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@ -419,14 +426,14 @@ This yields
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+ \lambda\sum_{i<j}^{N}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}}
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\Bigg].
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\end{multline}
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If one considers a RHF or UHF reference wave functions, it leads to the RMP or UMP series, respectively.
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The nature of the HF wave function is up for grabs, and if one considers a RHF or UHF reference wave functions, it leads to the RMP or UMP series, respectively.
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As mentioned earlier, in perturbation theory, the energy is a power series of $\lambda$ and the physical energy is obtained at $\lambda = 1$.
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The MP$n$ energy is defined as
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\begin{equation}
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E_{\text{MP}n}= \sum_{k=0}^n E^{(k)},
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E_{\text{MP}n}= \sum_{k=0}^n E_{\text{MP}}^{(k)},
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\end{equation}
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where $E^{(k)}$ is the $k$th-order MP correction, and it is well known that $E_{\text{MP1}} = E^{(0)} + E^{(1)} = E_\text{HF}$. \cite{SzaboBook}
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where $E_{\text{MP}}^{(k)}$ is the $k$th-order MP correction, and it is well known that $E_{\text{MP1}} = E_{\text{MP}}^{(0)} + E_{\text{MP}}^{(1)} = E_\text{HF}$. \cite{SzaboBook}
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The MP2 energy reads
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\begin{equation}\label{eq:EMP2}
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E_{\text{MP2}} = \frac{1}{4} \sum_{ij} \sum_{ab} \frac{\abs{\mel{ij}{}{ab}}^2}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b},
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@ -437,7 +444,7 @@ with $\mel{pq}{}{rs} = \braket{pq}{rs} - \braket{pq}{sr}$, and where
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\end{equation}
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are two-electron integrals in the spin-orbital basis. \cite{Gill_1994}
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As mentioned earlier, there is, \textit{a priori}, no guarantee that the MP$m$ series converges to the exact energy when $m \to \infty$.
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As mentioned earlier, there is, \textit{a priori}, no guarantee that the MP$n$ series converges to the exact energy as $n \to \infty$.
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In fact, it is known that when the HF wave function is a poor approximation to the exact wave function, for example in multi-reference systems, the MP method yields deceptive results. \cite{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988,Leininger_2000}
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A convenient way to investigate the convergence properties of the MP series is to analytically continue the coupling parameter $\lambda$ into the complex variable.
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By doing so, the Hamiltonian and the energy become complex-valued functions of $\lambda$, and the energy becomes a multivalued function on $K$ Riemann sheets (where $K$ is the number of basis functions).
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