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removing conflicts

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Pierre-Francois Loos 2020-11-30 22:47:10 +01:00
commit 4c78b6418a
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Manuscript/EPAWTFT.tex
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@ -1286,11 +1286,11 @@ orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \the
%\begin{equation} %\begin{equation}
% E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon). % E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
%\end{equation} %\end{equation}
With this representation, the parametrised RMP Hamiltonian becomes With this representation, the parametrised \hugh{atomic} RMP Hamiltonian becomes
\begin{widetext} \begin{widetext}
\begin{equation} \begin{equation}
\label{eq:H_RMP} \label{eq:H_RMP}
\Tilde{\bH}_\text{RMP}\qty(\lambda) = \hugh{\bH_\text{atom}\qty(\lambda)} =
\begin{pmatrix} \begin{pmatrix}
2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\ 2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\
-\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\ -\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\
@ -1327,7 +1327,8 @@ In contrast, smaller $\epsilon$ gives a weaker attraction to the atomic site,
representing strong screening of the nuclear attraction by core and valence electrons, representing strong screening of the nuclear attraction by core and valence electrons,
and again a less negative $\lambda$ is required for ionisation to occur. and again a less negative $\lambda$ is required for ionisation to occur.
Both of these factors are common in atoms on the right-hand side of the periodic table, \eg\ \ce{F}, Both of these factors are common in atoms on the right-hand side of the periodic table, \eg\ \ce{F},
\ce{O}, \ce{Ne}, and thus molecules containing these atoms are often class $\beta$ systems with \ce{O}, \ce{Ne}.
Molecules containing these atoms are therefore often class $\beta$ systems with
a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006} a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006}
% EXACT VERSUS APPROXIMATE % EXACT VERSUS APPROXIMATE
@ -1375,13 +1376,12 @@ For $\lambda>1$, the HF potential becomes an attractive component in Stillinger'
Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction
becomes increasingly repulsive. becomes increasingly repulsive.
\titou{Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent \titou{Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent
points where the two-electron repulsion overcomes the attractive HF potential and an electron points where the two-electron repulsion overcomes the attractive HF potential
are successively expelled from the molecule.\cite{Sergeev_2006}} and a single electron dissociates from the molecule.\cite{Sergeev_2006}}
\titou{T2: I'd like to discuss that with you.} \titou{T2: I'd like to discuss that with you.}
Symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons. In contrast, symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
Consider one of the two reference UHF solutions where the spin-up and spin-down electrons are localised on the left and Consider one of the two reference UHF solutions where the spin-up and spin-down electrons are localised on the left and right sites respectively.
right sites respectively.
The spin-up HF potential will then be a repulsive interaction from the spin-down electron The spin-up HF potential will then be a repulsive interaction from the spin-down electron
density that is centred around the right site (and vice-versa). density that is centred around the right site (and vice-versa).
As $\lambda$ becomes greater than 1 and the HF potentials become attractive, there will be a sudden As $\lambda$ becomes greater than 1 and the HF potentials become attractive, there will be a sudden