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@ -281,10 +281,10 @@
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@ -373,35 +379,29 @@
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@ -482,17 +482,16 @@ Combining eqs \ref{E_KS}, \ref{Tn}, and \ref{Ehartree}, the effective potential
|
||||
\begin{align}
|
||||
V_\mathrm{eff}[\rho(\mathbf{r})]&=V_\mathrm{ext}[\rho(\mathbf{r})]+V_\mathrm{H}[\rho(\mathbf{r})]+V_\mathrm{xc}[\rho(\mathbf{r})] \nonumber \\
|
||||
&=V_\mathrm{ext}[\rho(\mathbf{r})]+\int\frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf {r'}|}d\mathbf{r'} +\frac{\delta E_\mathrm{xc}[\rho(\mathbf{r})]}{\delta \rho(\mathbf{r})} \\
|
||||
&=\frac{\delta E_\mathrm{eff}[\rho(\mathbf{r})]}{\delta\rho(\mathbf{r})}
|
||||
&=\frac{\partial E_\mathrm{eff}[\rho(\mathbf{r})]}{\partial\rho(\mathbf{r})}
|
||||
\label{Veff}
|
||||
\end{align}
|
||||
where $V_{H}$[$\rho$(\textbf{r})] refers to the Hartree potential and $V_{xc}$[$\rho$(\textbf{r})] is the exchange-correlation potential.
|
||||
Therefore, for a given $V_\mathrm{eff}$[$\rho$(\textbf{r})], one can obtain $\rho$(\textbf{r}) by making the right-hand side of equation~\ref{Lagrange}
|
||||
independent of \textbf{r} which is done by introducing the molecular orbitals $\psi_{i}$(\textbf{r}) such that:
|
||||
|
||||
\begin{eqnarray}
|
||||
\label{rho}
|
||||
\rho(\mathbf{r})=\sum_{i}^{N} \left |\psi_i(\mathbf{r}) \right |^2
|
||||
\end{eqnarray}
|
||||
%Therefore, for a given $V_\mathrm{eff}$[$\rho$(\textbf{r})], one can obtain $\rho$(\textbf{r}) by making the right-hand side of equation~\ref{Lagrange} independent of \textbf{r} which is done by introducing the molecular orbitals $\psi_{i}$(\textbf{r}) such that:
|
||||
%\begin{eqnarray}
|
||||
%\label{rho}
|
||||
%\rho(\mathbf{r})=\sum_{i}^{N} \left |\psi_i(\mathbf{r}) \right |^2
|
||||
%\end{eqnarray}
|
||||
%<<<<<<< HEAD
|
||||
%where \textbf{x} denotes the four vector-containing space and spin variables, and the integration is implemented over the spin variable $\sigma$. Moreover, the molecular orbitals $\psi_{i}$[$\rho$(\textbf{r})] should satisfy the one-electron KS equation,
|
||||
%\begin{eqnarray}
|
||||
@ -500,19 +499,18 @@ independent of \textbf{r} which is done by introducing the molecular orbitals $\
|
||||
%\underbrace{\left(-\frac{1}{2}\mathbf{\nabla}_i^2+V_\mathrm{eff}[\rho(\mathbf{r})]\right)}_\mathrm{KS ~ operator} \psi_i[\rho(\mathbf{r})]=E_i\psi_i[\rho(\mathbf{r})]
|
||||
%=======
|
||||
|
||||
The molecular orbitals $\psi_{i}$(\textbf{r}) should satisfy the one-electron KS equations:
|
||||
In addition, the molecular orbitals $\phi_{i}$(\textbf{r}) should satisfy the one-electron KS equations:
|
||||
|
||||
\begin{eqnarray}
|
||||
\label{KS}
|
||||
\underbrace{\left(-\frac{1}{2}\mathbf{\nabla}_i^2+V_\mathrm{eff}[\rho(\mathbf{r})]\right)}_\mathrm{KS ~ operator} \psi_{i}(\textbf{r})=E_i \psi_{i}(\textbf{r})
|
||||
\underbrace{\left(-\frac{1}{2}\mathbf{\nabla}_i^2+V_\mathrm{eff}[\rho(\mathbf{r})]\right)}_\mathrm{KS ~ operator} \phi_{i}(\textbf{r})=E_i \phi_{i}(\textbf{r})
|
||||
%>>>>>>> 92023a10c3aa8b7dc4ace43987c1d571fb99a738
|
||||
\end{eqnarray}
|
||||
|
||||
This result can be reobtained within a variational context when looking for those orbitals minimizing the
|
||||
energy functional of equation~\ref{HK}, subject to orthonormality conditions:
|
||||
\begin{eqnarray}
|
||||
\label{orthonormality}
|
||||
\int \psi^*_i(\mathbf{r}) \psi_j(\mathbf{r}) d\mathbf{r}=\delta_{ij}
|
||||
\int \phi^*_i(\mathbf{r})\phi_j(\mathbf{r}) d\mathbf{r}=\delta_{ij}
|
||||
\end{eqnarray}
|
||||
|
||||
%<<<<<<< HEAD
|
||||
@ -570,9 +568,9 @@ Non-self-consistent DFTB scheme is suitable to study systems in which polyatomic
|
||||
This is the case ighly ionic and homonuclear covalent systems. However, uncertainty rises in the non-self-consistent DFTB scheme whem chemical bonds
|
||||
are controlled by a subtler charge balance between atoms especially for polar, semi-conductor and heteronuclear molecules. The self-consistent-charge
|
||||
extension of DFTB, SCC-DFTB, was born as an improvement of standard DFTB to provide a better description of electronic systems in which long-range
|
||||
Coulomb interactions are significant.\cite{Elstner1998, Porezag1995, Seifert1996, Elstner1998, Elstne2007, Oliveira2009} However, this method is not
|
||||
fully suitable for the calculation of molecular aggregates because of the dispersion and charge resonance present in such systems. Further corrections
|
||||
were made to the model in order to address these problems. I know describe in more details how SCC-DFTB can be derived from DFT.
|
||||
Coulomb interactions are significant.\cite{Elstner1998, Porezag1995, Seifert1996, Elstner1998, Elstne2007, Oliveira2009}
|
||||
%However, this method is not fully suitable for the calculation of molecular aggregates because of the dispersion and charge resonance present in such systems. Further corrections were made to the model in order to address these problems.
|
||||
I now describe in more details how SCC-DFTB can be derived from DFT.
|
||||
|
||||
\textbf{General principles of DFTB.} DFTB is derived from DFT based on the following approximations:
|
||||
|
||||
@ -593,7 +591,9 @@ were made to the model in order to address these problems. I know describe in mo
|
||||
\textbf{First-order DFTB} (historically referred to as zeroth-order DFTB) takes into account the first term of Taylor's expansion and which is
|
||||
equivalence with the other tight-binding model. \textbf{Second-order DFTB} (historically the SCC-DFTB) introduces a self-consistent procedure on
|
||||
atomic charges. There is also a more recent \textbf{third-order extension of DFTB} (referred to as DFTB3).\cite{Gaus2011} DFTB3 was not used in
|
||||
this thesis so the third-order expansion term will not be shown in the following equations. According to equation~\ref{E_KS}, $E_\mathrm{DFT}[\rho(\mathbf{r})]$
|
||||
this thesis so the third-order expansion term will not be shown in the following equations.
|
||||
|
||||
According to equation~\ref{E_KS}, $E_\mathrm{DFT}[\rho(\mathbf{r})]$
|
||||
can be written as:
|
||||
%>>>>>>> 92023a10c3aa8b7dc4ace43987c1d571fb99a738
|
||||
\begin{align}
|
||||
@ -620,58 +620,61 @@ E_\mathrm{DFTB}[\rho_{_0}(\mathbf{r})+\delta \rho(\mathbf{r})]=& \overbrace{E_\m
|
||||
\label{Edftb-long}
|
||||
\end{align}
|
||||
|
||||
From the first line of equation~\ref{Edftb-long}, we can define that the reference Hamiltonian $\hat{H}^0$ only depends on the reference
|
||||
electron density $\rho_{_0}$:
|
||||
The right-hand side terms in the first line of equation~\ref{Edftb-long} only depends on with $\rho_0(\mathbf{r})$, and correspond to a repulsive contribution $E_\mathrm{rep}$. The sum of the terms in the second line is the so-called band energy $E_\mathrm{band}$.
|
||||
We can define that a reference Hamiltonian $\hat{H}^0$, which only depends on the reference electron density $\rho_{_0}$:
|
||||
\begin{eqnarray}
|
||||
\label{H0}
|
||||
\hat{H}^0=-\frac{1}{2}\nabla^2 + \underbrace{V_\mathrm{ext}[\rho(\mathbf{r})]+ \int \frac{\rho_{_0}'(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d\mathbf{r'} +V_\mathrm{xc}[\rho_{_0}(\mathbf{r})] }_{V_\mathrm{eff}([\rho_{_0}(\mathbf{r})])}
|
||||
\end{eqnarray}
|
||||
where we combined the last three terms in the operator as $V_\mathrm{eff}([\rho_{_0}(\mathbf{r})])$.
|
||||
|
||||
The right-hand side terms in the first line of equation~\ref{Edftb-long} vary linearly with $\rho_0(\mathbf{r})$, and correspond to a
|
||||
repulsive contribution $E_\mathrm{rep}$. The sum of the terms in the second line is the so-called band energy $E_\mathrm{band}$. The third
|
||||
line is the second-order energy $E_\mathrm{2nd}$. Equation~\ref{Edftb-long} can be rewritten as follows:
|
||||
The third line is the second-order energy $E_\mathrm{2nd}$. Equation~\ref{Edftb-long} can be rewritten as follows:
|
||||
\begin{align}
|
||||
E_\mathrm{DFTB}[\rho_{_0}(\mathbf{r})+\delta \rho(\mathbf{r})]=& E_\mathrm{rep}[\rho_{_0}(\mathbf{r})] + \overbrace{\sum_{i}^\mathrm{occ} n_i \left\langle \psi_i \left\vert \hat{H}^0 \right\vert \psi_i \right\rangle}^{E_\mathrm{band}} ~ + E_\mathrm{2nd}\left[\rho_{_0}(\mathbf{r}), (\delta\rho(\mathbf{r}))^2\right]
|
||||
\label{Eterms}
|
||||
\end{align}
|
||||
|
||||
\textbf{Band energy term.}
|
||||
In DFTB, one relies on the use of LCAO for the description of the Kohn-Sham molecular orbitals $\psi_i$(\textbf{r}). Here the atomic
|
||||
In DFTB, it relies on the use of LCAO for the description of the KS molecular orbitals $\phi_i$(\textbf{r}). Here the atomic
|
||||
orbitals are limited to the valence orbitals of atoms:
|
||||
\begin{align}
|
||||
\psi_i(\mathbf{r}) = \sum_\nu C_{i\nu} \varphi_\nu(\mathbf{r}-\mathbf{R_\alpha})
|
||||
\label{orbitals}
|
||||
\end{align}
|
||||
$E_\mathrm{band}$ is the sum over the energies of all occupied orbitals obtained by diagonalization of the parameterized Hamiltonian matrix.
|
||||
For atoms $\alpha$ and $\beta$, the corresponding atomic orbitals are $\varphi_\mu$ and $\varphi_\nu$.
|
||||
Then $E_\mathrm{band}$ can be rewritten with equations~\ref{Veff} and \ref{orbitals}:
|
||||
|
||||
where $\varphi_\mu$ is the orbital of atom $\alpha$.
|
||||
%$E_\mathrm{band}$ is the sum over the energies of all occupied orbitals obtained by diagonalization of the parameterized Hamiltonian matrix.
|
||||
%For atoms $\alpha$ and $\beta$, the corresponding atomic orbitals are $\varphi_\mu$ and $\varphi_\nu$.
|
||||
$E_\mathrm{band}$ can be rewritten with equations~\ref{Veff} and \ref{orbitals}:
|
||||
\begin{align}
|
||||
\sum_{i}^\mathrm{occ} n_i \left\langle \psi_i \left\vert \hat{H}^0 \right\vert \psi_i \right\rangle=\sum_{i}^\mathrm{occ} n_i \sum_{\mu}^\mathrm{occ} \sum_{\nu}^\mathrm{occ} C_{i\mu}C_{i\nu} \underbrace{\left\langle \varphi_\mu \left\vert \hat{T}_e[\rho(\mathbf{r})] + V_\mathrm{eff}[\rho_{_0}(\mathbf{r})] \right\vert \varphi_\nu \right \rangle}_{H^0_{\mu\nu}}
|
||||
\sum_{i}^\mathrm{occ} n_i \left\langle \psi_i \left\vert \hat{H}^0 \right\vert \psi_i \right\rangle=\sum_{i}^\mathrm{occ} n_i \sum_{\mu}^\mathrm{occ} \sum_{\nu}^\mathrm{occ} C_{i\mu}C_{i\nu} \underbrace{\left\langle \varphi_\mu \left\vert \hat{T}_e[\rho(\mathbf{r})] + V_\mathrm{eff}[\rho_{_0}(\mathbf{r})] \right\vert \varphi_\nu \right \rangle}_{H^0_{\mu\nu}}, \mathrm {with} ~ \mu \in {\alpha}, \nu \in {\beta}
|
||||
\label{Eband}
|
||||
\end{align}
|
||||
$\forall\mu\in {\alpha}, \nu\in{\beta}$. The Hamiltonian matrix element $H^0_{\mu\nu}$ is defined as:
|
||||
%where $\varphi_\nu$ is the orbital of atom $\beta$.
|
||||
%$\forall\mu\in {\alpha}, \nu\in{\beta}$.
|
||||
The Hamiltonian matrix element $H^0_{\mu\nu}$ is defined as:
|
||||
|
||||
\begin{align}
|
||||
H^0_{\mu\nu}&= \left\langle \varphi_\mu \left\vert \hat{H}^0 \right\vert \varphi_\nu \right\rangle, \nonumber \\
|
||||
\label{matrix}
|
||||
\end{align}
|
||||
|
||||
|
||||
The effective potential $V_\mathrm{eff}[\rho_{_0}(\mathbf{r})]$ is defined as the sum of potentials $V_\alpha(\textbf{r})$ centered on the atoms:
|
||||
\begin{align}
|
||||
V_\mathrm{eff}[\rho_{_0}(\mathbf{r})]=\sum_{\alpha}V
|
||||
_\alpha(\mathbf{r}_\alpha)
|
||||
_\alpha(\mathbf{r}-\mathbf{R}_\alpha)
|
||||
\label{Vdftb}
|
||||
\end{align}
|
||||
where $\mathbf{r}_\alpha=\mathbf{r}-\mathbf{R}_\alpha$.
|
||||
%where $\mathbf{r}-\mathbf{R}_\alpha$ equals to $\mathbf{r}_\alpha$.
|
||||
The Hamiltonian matrix elements can be written as follows:
|
||||
%\begin{align}
|
||||
%H^0_{\mu\nu}=\left\langle\varphi_\mu \left\vert -\frac{1}{2}\nabla_\nu^2 + V_\alpha+(1-\delta_{\alpha\beta})V_\beta \right\vert \varphi_\nu \right\rangle, \mathrm {with} ~ \mu \in {\alpha}, \nu \in {\beta}
|
||||
%\label{Hamil}
|
||||
%\end{align}
|
||||
%where $\delta_{\alpha\beta}$ is the Kronecker's delta.
|
||||
|
||||
\begin{align}
|
||||
H^0_{\mu\nu}=\left\langle\varphi_\mu \left\vert -\frac{1}{2}\nabla_\nu^2 + V_\alpha+(1-\delta_{\alpha\beta})V_\beta \right\vert \varphi_\nu \right\rangle, \mathrm {with} ~ \mu \in {\alpha}, \nu \in {\beta}
|
||||
H^0_{\mu\nu}=\left\langle\varphi_\mu \left\vert -\frac{1}{2}\nabla_i^2 + V_\alpha+ V_\beta\right\vert \varphi_\nu \right\rangle
|
||||
\label{Hamil}
|
||||
\end{align}
|
||||
where $\delta_{\alpha\beta}$ is the Kronecker's delta.
|
||||
|
||||
%In practice, $H^0_{\mu\nu}$ are calculated as follows:
|
||||
For diagonal elements, the energy level in the free atom is chosen, which ensures the correct dissociation limits.
|
||||
@ -682,7 +685,7 @@ To summarize, within the electronic density superposition approach, the $H^0_{\m
|
||||
H^0_{\mu\nu}=
|
||||
\begin{cases}
|
||||
\varepsilon_\mu^\mathrm{free~atom}, ~ & \mu=\nu \\
|
||||
\left \langle \varphi_\mu \left\vert -\frac{1}{2}\nabla_\nu^2 + V_\alpha+(1-\delta_{\alpha\beta})V_\beta \right\vert \varphi_\nu \right\rangle, ~ & \mu \in {\alpha}, \nu \in {\beta}, \alpha \neq \beta \\
|
||||
\left \langle \varphi_\mu \left\vert -\frac{1}{2}\nabla_i^2 + V_\alpha+V_\beta \right\vert \varphi_\nu \right\rangle, ~ & \mu \in {\alpha}, \nu \in {\beta}, \alpha \neq \beta \\
|
||||
0, ~ & \mathrm{otherwise}
|
||||
\end{cases}
|
||||
\label{Hmatrix}
|
||||
@ -691,9 +694,6 @@ H^0_{\mu\nu}=
|
||||
It should be noted that the $H^0_{\mu\nu}$ elements only depend on atoms $\alpha$ and $\beta$. Therefore only the two-center matrix elements and
|
||||
the two-center elements of the overlap matrix can be explicitly calculated, in other words, interactions at three or more centers are neglected as stated above.
|
||||
|
||||
\textbf{Repulsive energy term.} $E_\mathrm{rep}$ is a repulsive contribution obtained from the sum of atomic-pair terms, which only depend
|
||||
on the reference electronic density $\rho_{_0}(\textbf{r})$.
|
||||
|
||||
%In order to obtain a good evaluation of $\rho_{_0}(\textbf{r})$, it is approximated as a overlap of atom-like densities centered on the nuclei $\alpha$:
|
||||
%\begin{align}
|
||||
%\rho_{_0}(\mathbf{r})=\sum_{\alpha}^{N}
|
||||
@ -712,29 +712,6 @@ on the reference electronic density $\rho_{_0}(\textbf{r})$.
|
||||
%\end{align}
|
||||
%where $E_\mathrm{long}$ refers to the interactions energy of more than two centers.
|
||||
|
||||
Due to the screening of terms of more than two centers, one can assume that the two-center contributions are short ranged.
|
||||
But $E_\mathrm{rep}$ doesn't decay to zero for long interatomic distances. Instead, it decays to a constant given by the atomic
|
||||
contributions:
|
||||
\begin{align}
|
||||
\displaystyle \lim_{R_{\alpha\beta} \to \infty}E_\mathrm{rep}[\rho_{_0}(\mathbf{r})]=\sum_{\alpha}^{N}
|
||||
E_\mathrm{rep}[\rho_{_0}^\alpha(\mathbf{r}_\alpha)]
|
||||
\label{Erep-infinity}
|
||||
\end{align}
|
||||
|
||||
The right side of equation~\ref{Erep-infinity} is assumed to make $E_\mathrm{rep}$ only rely on the two-center contributions:
|
||||
\begin{align}
|
||||
E_\mathrm{rep}[\rho_{_0}(\mathbf{r})]\approx\frac{1}{2}\sum_{\alpha}^{N}\sum_{\beta}^{N} V(\mathbf{R}_\alpha-\mathbf{R}_\beta)
|
||||
\label{Erep-right}
|
||||
\end{align}
|
||||
|
||||
In practice, it it possible to calculate $E_\mathrm{rep}$ with known values of $\rho_{_0}(\textbf{r})$, but it's more convenient to adjust
|
||||
the expression of $E_\mathrm{rep}$ to $ab ~ initio$ calculations. Therefore, $E_\mathrm{rep}$ is determined by comparing the
|
||||
difference between the DFT energy $E_\mathrm{DFT}$ and $E_\mathrm{band}$+$E_\mathrm{2nd}$ as a function of the interatomic distance $R_{\alpha\beta}$:
|
||||
\begin{align}
|
||||
E_\mathrm{rep}[\rho_{_0}(\mathbf{r})] \equiv E_\mathrm{rep}(R_{\alpha\beta})=E_\mathrm{DFT}(R_{\alpha\beta})-E_\mathrm{band}(R_{\alpha\beta})-E_\mathrm{2nd}(R_{\alpha\beta})
|
||||
\label{Erep-final}
|
||||
\end{align}
|
||||
|
||||
\textbf{Second-order term.}
|
||||
In SCC-DFTB, the electronic density is corrected by including the second-order contribution $E_\mathrm{2nd}$ in equation~\ref{Eterms},
|
||||
which is ignored in first-order DFTB.
|
||||
@ -761,20 +738,21 @@ $E_\mathrm{2nd}$ can be rewritten with equations~\ref{Edftb-long} and \ref{Eterm
|
||||
\begin{align}
|
||||
E_\mathrm{2nd}\approx
|
||||
&\frac{1}{2}\sum_{\alpha}^{N}\sum_{\beta}^{N} \Delta q_{\alpha} \Delta q_{\beta} \overbrace{\iint \left(\frac{1}{|\mathbf{r}-\mathbf {r'}|} + \frac{\delta^2 E_{xc}[\rho_{_0}(\mathbf{r})]}{\delta\rho(\mathbf{r})\delta\rho(\mathbf{r'})} \Big\vert _{\rho_{_0}, ~ \rho_{_0}'} \right) F(\alpha, \beta) d\mathbf{r}d\mathbf{r'}}^{\gamma_{\alpha\beta}} \nonumber \\
|
||||
=& \frac{1}{2}\sum_{\alpha}^{N}\sum_{\beta}^{N} \Delta q_{\alpha} \Delta q_{\beta} \gamma_{\alpha\beta} \\
|
||||
F(\alpha, \beta)= & F_{0}(\mathbf{r}-\mathbf{R}_\alpha)\times F_{0}(\mathbf{r'}-\mathbf{R}_\beta)
|
||||
=& \frac{1}{2}\sum_{\alpha}^{N}\sum_{\beta}^{N} \Delta q_{\alpha} \Delta q_{\beta} \gamma_{\alpha\beta}
|
||||
\label{E2nd}
|
||||
\end{align}
|
||||
where the two-electron integrals $\gamma_{\alpha\beta}$ is introduced for convenience.
|
||||
where $F(\alpha, \beta)= F_{0}(\mathbf{r}-\mathbf{R}_\alpha)\times F_{0}(\mathbf{r'}-\mathbf{R}_\beta) \nonumber$ the two-electron integrals $\gamma_{\alpha\beta}$ is introduced for convenience.
|
||||
|
||||
To calculate equation~\ref{E2nd}, $\gamma_{\alpha\beta}$ must be analyzed. In the limit case, the interatomic distance is very large,
|
||||
$|{\textbf{R}_\alpha - \textbf{R}_\beta}|$ = $|\textbf{r} - \textbf{r}'| \rightarrow \infty$ with GGA-DFT, the exchange-correlation term tends to zero.
|
||||
$\gamma_{\alpha\beta}$ that describes the interaction of two normalized spherical electronic densities reduces to 1/$|\textbf{R}_\alpha-\textbf{R}_\beta|$, so $E_\mathrm{2nd}$ can be expressed as follows:
|
||||
|
||||
\begin{align}
|
||||
E_\mathrm{2nd}=\frac{1}{2}\sum_{\alpha}^{N}\sum_{\beta}^{N} \frac{\Delta q_{\alpha} \Delta q_{\beta}} {|\mathbf{R}_\alpha-\mathbf{R}_\beta|}
|
||||
\label{E2ndsimple}
|
||||
E_\mathrm{2nd}=\frac{1}{2}\sum_{\alpha}^{N}\sum_{\beta}^{N} \frac{\Delta q_{\alpha} \Delta q_{\beta}} {|\mathbf{R}_\alpha-\mathbf{R}_\beta|} + f(U_\alpha, U_\beta, (\mathbf{R}_\alpha-\mathbf{R}_\beta))
|
||||
\label{simpleE2nd}
|
||||
\end{align}
|
||||
where U$_\alpha$, U$_\beta$ are the Hubbard parameters: $\gamma_{\alpha\alpha} \approx I_\alpha-A_\alpha \approx 2\eta\alpha \approx U_\alpha$.
|
||||
$I$ is the ionisation potential and $A$ is the electonic affinity of atom $\alpha$. $\eta_\alpha$ refers to the chemical hardness of atom $\alpha$.\cite{Elstner1998}
|
||||
|
||||
It is worth noting that the electronic density $\rho(\mathbf{r})$ influences explicitly the calculation of the electrostatic energy in DFT.
|
||||
In the context of DFTB, point charges are used and the electronic density around the atom is condensed at a point. In practice, Mulliken's
|
||||
@ -785,7 +763,30 @@ q_\alpha^\mathrm{Mull}=\frac{1}{2}\sum_{\alpha}^\mathrm{occ} n_i
|
||||
\right)
|
||||
\label{Mulliken}
|
||||
\end{align}
|
||||
where $\mu$ denotes the orbitals belonging to the atom $\alpha$. This definition does not allow the bond between two different atoms to polarize.
|
||||
%where $\mu$ denotes the orbitals belonging to the atom $\alpha$. This definition does not allow the bond between two different atoms to polarize.
|
||||
|
||||
\textbf{Repulsive energy term.} $E_\mathrm{rep}$ is a repulsive contribution obtained from the sum of atomic-pair terms, which only depend
|
||||
on the reference electronic density $\rho_{_0}(\textbf{r})$.
|
||||
%Due to the screening of terms of more than two centers, one can assume that the two-center contributions are short ranged. But $E_\mathrm{rep}$ doesn't decay to zero for long interatomic distances. Instead, it decays to a constant given by the atomic contributions:
|
||||
%\begin{align}
|
||||
%\displaystyle \lim_{R_{\alpha\beta} \to \infty}E_\mathrm{rep}[\rho_{_0}(\mathbf{r})]=\sum_{\alpha}^{N}
|
||||
%E_\mathrm{rep}[\rho_{_0}^\alpha(\mathbf{r}_\alpha)]
|
||||
%\label{Erep-infinity}
|
||||
%\end{align}
|
||||
%The right side of equation~\ref{Erep-infinity}
|
||||
It is assumed to make $E_\mathrm{rep}$ only rely on the two-center contributions:
|
||||
\begin{align}
|
||||
E_\mathrm{rep}[\rho_{_0}(\mathbf{r})]\approx\frac{1}{2}\sum_{\alpha}^{N}\sum_{\beta}^{N} V(\mathbf{R}_\alpha-\mathbf{R}_\beta)
|
||||
\label{Erep-right}
|
||||
\end{align}
|
||||
|
||||
In practice, it it possible to calculate $E_\mathrm{rep}$ with known values of $\rho_{_0}(\textbf{r})$, but it's more convenient to adjust
|
||||
the expression of $E_\mathrm{rep}$ to $ab ~ initio$ calculations. Therefore, $E_\mathrm{rep}$ is determined by comparing the
|
||||
difference between the DFT energy $E_\mathrm{DFT}$ and $E_\mathrm{band}$+$E_\mathrm{2nd}$ as a function of the interatomic distance $R_{\alpha\beta}$:
|
||||
\begin{align}
|
||||
E_\mathrm{rep}[\rho_{_0}(\mathbf{r})] \equiv E_\mathrm{rep}(R_{\alpha\beta})=E_\mathrm{DFT}(R_{\alpha\beta})-E_\mathrm{band}(R_{\alpha\beta})-E_\mathrm{2nd}(R_{\alpha\beta})
|
||||
\label{Erep-final}
|
||||
\end{align}
|
||||
|
||||
\textbf{Total energy.}
|
||||
The total energy in SCC-DFTB can be written from the previous different contributions as follows:
|
||||
@ -802,24 +803,26 @@ V(\mathbf{R}_\alpha-\mathbf{R}_\beta) \nonumber \\
|
||||
|
||||
%\textbf{Secular equation}
|
||||
%=======
|
||||
Here the contribution from $H^0$ is exactly the same with the one in standard DFTB scheme and the molecular orbitals $\psi_i$ are expanded in a LCAO model (equation~\ref{orbitals}).
|
||||
|
||||
%Here the contribution from $H^0$ is exactly the same with the one in standard DFTB scheme and the molecular orbitals $\psi_i$ are expanded in a LCAO model (equation~\ref{orbitals}).
|
||||
|
||||
%>>>>>>> 92023a10c3aa8b7dc4ace43987c1d571fb99a738
|
||||
|
||||
\textbf{Secular equations.}
|
||||
From this LCAO model, we can get the secular equations:
|
||||
From this energy expression model, we can get the secular equations:
|
||||
\begin{align}
|
||||
\sum_\nu C_{i\nu} \left({H}^0_{\mu\nu} - \varepsilon_i S_{\mu\nu} \right) =0, ~ \forall \mu, ~ \nu
|
||||
\label{secular-eq}
|
||||
\end{align}
|
||||
where $H^0_{\mu\nu}$ are the Hamiltonian matrix elements and $S_{\mu\nu}$ are the overlap matrix elements.
|
||||
where $H^0_{\mu\nu}$ is the DFTB operator matrix element and $S_{\mu\nu}$ are the overlap matrix elements.
|
||||
|
||||
The secular equations can be rewritten with modified Hamiltonian matrix elements, $H_{\mu\nu}$, defined by:
|
||||
The secular equations can be rewritten using the matrix, $H_{\mu\nu}$, defined by:
|
||||
\begin{align}
|
||||
H_{\mu\nu}=H_{\mu\nu}^0 + \frac{1}{2} S_{\mu\nu} \sum_{\zeta}(\gamma_{\alpha\zeta} + \gamma_{\beta\zeta})\Delta q_\zeta
|
||||
H_{\mu\nu}=H_{\mu\nu}^0 + \underbrace{\frac{1}{2} S_{\mu\nu} \sum_{\zeta}(\gamma_{\alpha\zeta} + \gamma_{\beta\zeta})\Delta q_\zeta}_{H_{\mu\nu}^\prime}
|
||||
\label{Huv}
|
||||
\end{align}
|
||||
|
||||
The $H_{\mu\nu}^0$ and $S_{\mu\nu}$ matrix elements are the same than the ones defined in the standard DFTB method (equation~\ref{matrix}).
|
||||
$H_{\mu\nu}^0$ comes from the band energy and $H_{\mu\nu}^\prime$ is from the second order term.
|
||||
%The $H_{\mu\nu}^0$ and $S_{\mu\nu}$ matrix elements are the same than the ones defined in the standard DFTB method (equation~\ref{matrix}).
|
||||
The $H_{\mu\nu}$ elements rely on the atomic charges explicitly, and the atomic charges depend on the molecular orbitals (see equation~\ref{Mulliken}).
|
||||
Then the resolution can be achieved in a self-consistent way.
|
||||
First, from an initial set of charges
|
||||
@ -843,20 +846,20 @@ charges were proposed by J. Li $et al.$ in 1998 and they have been considered in
|
||||
and the electrostatic potential, partial atomic charges in molecules, and Coulombic intermolecular potential of polycyclic aromatic hydrocarbon clusters.\cite{Li1998, Kalinowski2004, Kelly2005, Rapacioli2009corr}
|
||||
The CM3 charges are defined as:
|
||||
\begin{align}
|
||||
q_{\alpha}^\mathrm{CM3} = q_{\alpha}^\mathrm{Mull} + \sum_{\beta \neq \alpha} (C_{t_\alpha t_\beta} K_{\alpha\beta} + D_{t_\alpha t_\beta} K_{\alpha\beta}^2 )
|
||||
q_{\alpha}^\mathrm{CM3} = q_{\alpha}^\mathrm{Mull} + \sum_{\beta \neq \alpha} (D_{t_\alpha t_\beta} K_{\alpha\beta} + D_{t_\alpha t_\beta} K_{\alpha\beta}^2 )
|
||||
\label{CM3}
|
||||
\end{align}
|
||||
where $K_{\alpha\beta}$ is the Mayer bond order,\cite{Mayer1983, Cances1997, Lu2012} which depends on the density matrix of the orbitals belonging to
|
||||
each of the two atoms $\alpha$ and $\beta$. $C_{t_\alpha t\beta}$ and $D_{t_\alpha t\beta}$ are empirical parameters which are related
|
||||
to the nature of atoms $\alpha$ and $\beta$.
|
||||
where $K_{\alpha\beta}$ is the Mayer bond order,\cite{Mayer1983, Cances1997, Lu2012} between
|
||||
%which depends on the density matrix of the orbitals belonging to each of the two
|
||||
atoms $\alpha$ and $\beta$. $C_{t_\alpha t\beta}$ and $D_{t_\alpha t\beta}$ are empirical parameters which are related to the nature of
|
||||
atoms $\alpha$ and $\beta$.
|
||||
|
||||
In this thesis, in practice for the calculation of electronic energy, the definition of CM3 charges is simplified as:
|
||||
\begin{align}
|
||||
q_{\alpha}^\mathrm{CM3} = q_{\alpha}^\mathrm{Mull} + \sum_{\beta \neq \alpha} C_{t_\alpha t_\beta} K_{\alpha\beta}
|
||||
q_{\alpha}^\mathrm{CM3} = q_{\alpha}^\mathrm{Mull} + \sum_{\beta \neq \alpha} D_{t_\alpha t_\beta} K_{\alpha\beta}
|
||||
\label{CM3-classical}
|
||||
\end{align}
|
||||
|
||||
|
||||
%<<<<<<< HEAD
|
||||
%\textbf{Dispersion energy}
|
||||
%
|
||||
@ -987,15 +990,12 @@ to understand the PES. Stable molecular structures correspond to the minima in t
|
||||
of the valley around a minimum determines the vibrational spectrum.
|
||||
The key points on a PES can be classified according to the first and all second derivatives of the energy with
|
||||
respect nuclear coordinates, which correspond to the gradients and the curvatures, respectively.
|
||||
When points have a zero gradient (stationary points) and their second derivatives are positive, this corresponds
|
||||
to a \textbf{local minimum} (physically stable structure). Among these local minima, the lowest energy minima is called
|
||||
the \textbf{global minimum}. When at least one of the second derivatives is negative, the point is a transition state (saddle point).
|
||||
The points have a zero gradient (stationary points) and their second derivatives are positive, they correspond
|
||||
to \textbf{local minima} (physically stable structure). Among these local minima, the lowest energy minimum is called the \textbf{global minimum}. When at least one of the second derivatives is negative, the point is a transition state (saddle point).
|
||||
|
||||
\figuremacro{PES}{Schematic representation of some key points on a model potential energy surface.}
|
||||
|
||||
Molecular structure, properties, chemical reactivity, dynamics, and vibrational spectra of molecules can be readily
|
||||
understood in terms of PES. Only very simple PES can be obtained from experiment whereas computational chemistry
|
||||
has developed different kinds of methods to efficiently explore PES. To survey the PES, the choice of an exploration
|
||||
Molecular structure, properties, chemical reactivity, dynamics, and vibrational spectra of molecules can be readily understood in terms of PES. Only very simple PES can be obtained from experiment whereas computational chemistry has developed different kinds of methods to efficiently explore PES. To survey the PES, the choice of an exploration
|
||||
method can be guided by the shape of the PES (the statistical set that one wishes to study) and the temporal aspect.
|
||||
|
||||
\textbf{Monte Carlo} and classical \textbf{MD} simulations are widely recognized approaches for the exploration of PES of systems containing a large number of degrees of freedom such as molecular aggregates.
|
||||
@ -1019,60 +1019,13 @@ inherent randomness, and the computing power of the computer can directly simula
|
||||
in nuclear physics, analysis of the transmission process of neutrons in a reactor.
|
||||
The second category applies to problems that can be transformed into randomly distributed characteristic numbers, such as the
|
||||
probability of a random event. Through the random sampling method, the probability of random events is estimated by the
|
||||
frequency of occurrence, and this is used as the solution to the problem. This method is usually used to solve complicated
|
||||
multi-dimensional integration problems.
|
||||
frequency of occurrence, and this is used as the solution to the problem. This method is usually used to solve complicated multi-dimensional integration problems. Monte Carlo methods are mainly used in the field of optimization, numerical integration, and generating draws from a probability distribution.
|
||||
|
||||
Monte Carlo methods are mainly used in the field of optimization, numerical integration, and generating draws from a probability
|
||||
distribution. In physics related problems, Monte Carlo methods are applied to simulat systems diplaying many coupled degrees
|
||||
of freedom, such as fluids, strongly coupled solids, disordered materials, and cellular structures. In statistical physics not
|
||||
related to thermodynamics, Monte Carlo molecular simulation is an alternative to MD simulations. Monte Carlo methods
|
||||
can be used to compute statistical field theories of simple particles and polymer systems.\cite{Rosenbluth1955, Binder1993, Baeurle2009}
|
||||
When solving practical problems using a Monte Carlo methods, it includes two main parts. First, random variables with various
|
||||
probability distributions need to be generated to simulate a certain process. Second, statistical methods need to be used to
|
||||
estimate the numerical characteristics of the model to obtain a numerical solutions to the actual problem. Monte Carlo methods
|
||||
make it possible for the sampling of a PES by performing random shifts in order to correctly reproduce the probability distribution
|
||||
of the configurations. One example for the calculation procedure of a molecular simulation is as follows:
|
||||
|
||||
\begin{enumerate}
|
||||
\item A random molecular configuration is generated using a random number generator.
|
||||
|
||||
\item Random changes are made to the particle coordinates of this molecular configuration, resulting in a new molecular configuration.
|
||||
|
||||
\item Calculate the energy of the new molecular configuration.
|
||||
|
||||
\item Compare the energy change between the new molecular configuration and the former one to determine whether to accept the configuration change or not.
|
||||
%>>>>>>> 92023a10c3aa8b7dc4ace43987c1d571fb99a738
|
||||
|
||||
\begin{itemize}
|
||||
\item[$\bullet$] If the energy of the new molecular configuration is lower than that of the original one, the new configuration change is accepted, and the
|
||||
new configuration is used for the next iteration.
|
||||
|
||||
\item[$\bullet$] If not, the Boltzmann factor is calculated and a random number is generated. If this random number is greater than the calculated
|
||||
Boltzmann factor, then the new configuration is discarded. If not, the new configuration is conserved and is used for the next iteration.
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\item If a new iteration is required, the process is repeated from step 2.
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\textbf{General principles.}
|
||||
Monte Carlo calculations that lead to quantitative results may be regarded as attempts to estimate the value of a multiple integral.
|
||||
This is particularly true for the applications in equilibrium statistical thermodynamics, where one hopes to calculate the thermal
|
||||
average $\langle A \rangle _T$ of an observable $A(\textbf X)$ as an integral over phase space $\Omega$, where $\textbf X$ is a
|
||||
point in $\Omega$:
|
||||
\begin{align}
|
||||
\langle A \rangle _T= \frac{1}{Z} \int_\Omega d\mathbf X A(\mathbf X)e^{-H(\mathbf X)/k_BT}
|
||||
\label{thermalAverage}
|
||||
\end{align}
|
||||
in which $Z$ is the partition function, and $k_B$ is the Boltzmann constant. $T$ refers to the temperature, and $H(\textbf X)$ denotes the Hamiltonian of the system.
|
||||
To illustrate the general application of Monte Carlo techniques, here the standard example of the one-dimensional integral $I$ over integration space $\Omega$ is taken:
|
||||
\begin{align}
|
||||
I=\int_\Omega f(\mathbf x)d\mathbf x
|
||||
\label{SingleIntegral}
|
||||
\end{align}
|
||||
|
||||
Such integral can be evaluated more efficiently using conventional numerical means than using Monte Carlo methods. However, the extension to higher
|
||||
dimensions (number of degrees of freedom greater than 3) is always difficult in practice with conventional numerical integration. It is then possible to
|
||||
use stochastic approaches that one explores the configuration space randomly and the computation of the integral is estimated in the form of an average
|
||||
@ -1132,6 +1085,51 @@ In the case of symmetrical movements, $\eta (\mathbf x_i \rightarrow \mathbf x_j
|
||||
\label{assess}
|
||||
\end{align}
|
||||
|
||||
In physics related problems, Monte Carlo methods are applied to simulat systems diplaying many coupled degrees
|
||||
of freedom, such as fluids, strongly coupled solids, disordered materials, and cellular structures. In statistical physics not
|
||||
related to thermodynamics, Monte Carlo molecular simulation is an alternative to MD simulations. Monte Carlo methods
|
||||
can be used to compute statistical field theories of simple particles and polymer systems.\cite{Rosenbluth1955, Binder1993, Baeurle2009}
|
||||
When solving practical problems using a Monte Carlo methods, it includes two main parts. First, random variables with various probability distributions need to be generated to simulate a certain process. Second, statistical methods need to be used to
|
||||
estimate the numerical characteristics of the model to obtain a numerical solutions to the actual problem. Monte Carlo methods
|
||||
make it possible for the sampling of a PES by performing random shifts in order to correctly reproduce the probability distribution
|
||||
of the configurations.
|
||||
|
||||
\textbf{General principles.}
|
||||
Monte Carlo calculations that lead to quantitative results may be regarded as attempts to estimate the value of a multiple integral.
|
||||
This is particularly true for the applications in equilibrium statistical thermodynamics, where one hopes to calculate the thermal
|
||||
average $\langle A \rangle _T$ of an observable $A(\textbf X)$ as an integral over phase space $\Omega$, where $\textbf X$ is a
|
||||
point in $\Omega$:
|
||||
\begin{align}
|
||||
\langle A \rangle _T= \frac{1}{Z} \int_\Omega d\mathbf X A(\mathbf X)e^{-H(\mathbf X)/k_BT}
|
||||
\label{thermalAverage}
|
||||
\end{align}
|
||||
in which $Z$ is the partition function, and $k_B$ is the Boltzmann constant. $T$ refers to the temperature, and $H(\textbf X)$ denotes the Hamiltonian of the system.
|
||||
|
||||
5
|
||||
One example for the calculation procedure of a molecular simulation is as follows:
|
||||
|
||||
\begin{enumerate}
|
||||
\item A random molecular configuration is generated using a random number generator.
|
||||
|
||||
\item Random changes are made to the particle coordinates of this molecular configuration, resulting in a new molecular configuration.
|
||||
|
||||
\item Calculate the energy of the new molecular configuration.
|
||||
|
||||
\item Compare the energy change between the new molecular configuration and the former one to determine whether to accept the configuration change or not.
|
||||
%>>>>>>> 92023a10c3aa8b7dc4ace43987c1d571fb99a738
|
||||
|
||||
\begin{itemize}
|
||||
\item[$\bullet$] If the energy of the new molecular configuration is lower than that of the original one, the new configuration change is accepted, and the
|
||||
new configuration is used for the next iteration.
|
||||
|
||||
\item[$\bullet$] If not, the Boltzmann factor is calculated and a random number is generated. If this random number is greater than the calculated
|
||||
Boltzmann factor, then the new configuration is discarded. If not, the new configuration is conserved and is used for the next iteration.
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\item If a new iteration is required, the process is repeated from step 2.
|
||||
|
||||
\end{enumerate}
|
||||
\subsection{Classical Molecular Dynamics}
|
||||
|
||||
%<<<<<<< HEAD
|
||||
@ -1167,13 +1165,12 @@ energies were studied using MD in 1960 and MD was also applied to simulate liqui
|
||||
the description of phase equilibria, the relaxation of metastable states and the dynamics of processes
|
||||
at interfaces. In addition, MD can also model chemical reactions in complex environments.
|
||||
|
||||
MD simulations allow to model real time evolution of particles, one can then access time-dependent properties. If the
|
||||
As MD simulations allow to model real time evolution of particles, one can then access time-dependent properties. If the
|
||||
time evolution is obtained by \textbf{integrating Newton's equations of motion} for a system of interacting particles, it is
|
||||
referred to as \textbf{classical MD}.
|
||||
A classical MD simulation needs the definition of a potential describing the interaction between particles in order to calculate
|
||||
the PES. %This is often referred to as a force field in chemistry and biology, and as an interatomic potential in materials physics.
|
||||
Potentials can be of different levels of accuracy as described in the previous section. The most commonly used potentials
|
||||
in chemistry are \textbf{force fields}. In that case, one refers to \textbf{molecular mechanics}, which embodies a classical
|
||||
Potentials can be of different levels of accuracy as described in the previous section. The most commonly used potentials in chemistry are \textbf{force fields}. In that case, one refers to \textbf{molecular mechanics}, which embodies a classical
|
||||
mechanics treatment of the interactions between particles. In classical molecular dynamics, electrons and nuclei are not
|
||||
distinguished and one refers only to particles. As already mentioned, the main drawback of force fields is that
|
||||
they usually can not model chemical reactions. If the potential comes from a quantum chemical treatment of the
|
||||
@ -1208,7 +1205,7 @@ position ($\textbf R_\alpha$) of $\alpha$:
|
||||
\label{Force}
|
||||
\end{align}
|
||||
where $V$ depends on the positions of all atoms or particles. This leads to a system of $f \times N$ second-order differential equations
|
||||
where $f$ is the dimension of space. In our case, the degree of freedom $f$ is equal to 3. According to the known initial positions of
|
||||
where $f$ is the dimension of space. In our case, the number of degrees of freedom $f$ is equal to 3. According to the known initial positions of
|
||||
particles, the potential energy can be obtained.
|
||||
Then, a numerical resolution of the partial derivative equations provided in equation~\ref{Force} can be obtained using a suitable \textbf{integration algorithm}.
|
||||
The integration algorithm gives access to the positions and velocities of atoms or particles and to the forces acting on these atoms
|
||||
@ -1219,11 +1216,12 @@ Many high order integration algorithms have been proposed depending on the desir
|
||||
Euler algorithm,\cite{Batina1991, Butcher2008} Verlet algorithms, predictor-corrector algorithm,\cite{Gear1966, Diethelm2002} and
|
||||
Runge-Kutta algorithm.\cite{Runge1895, Kutta1901, Butcher1996, Butcher2015}
|
||||
The Verlet algorithms include the Simple Verlet (SV),\cite{Verlet1967} the Leapfrog Verlet (LFV),\cite{Fraige2004} and the Velocity Verlet (VV).\cite{Swope1982}
|
||||
The \textbf{Velocity Verlet algorithm} is the most widely used in many MD codes owing to its numerical stability and implementation simplicity.
|
||||
Furthermore, movement constants are very well preserved over time. We briefly describe the basis of the Velocity Verlet algorithm below.
|
||||
The \textbf{VV algorithm} is the most widely used in many MD codes owing to its numerical stability and simplicity implementation.
|
||||
Furthermore, movement constants are very well preserved over time. We briefly describe the basis of the VV algorithm below.
|
||||
|
||||
%The time symmetry in the Verlet algorithm reduces the level of local errors introduced into the integration by the discretization through removing all the odd-degree terms which refers to the three-degree terms in $\delta t$.
|
||||
The local error is quantified by inserting the exact values $\textbf R_\alpha(t_{n-1})$, $\textbf R_\alpha(t_n)$, and $\textbf R_\alpha(t_{n+1})$ into the iteration and calculating the Taylor expansions at time $t=t_n$ of the position vector $\textbf R_\alpha(t\pm\delta t)$ in different time directions:
|
||||
The local error is quantified by inserting the exact values $\textbf R_\alpha(t_{n-1})$, $\textbf R_\alpha(t_n)$, and $\textbf R_\alpha(t_{n+1})$ into the iteration and calculating the Taylor expansions at time $t=t_n$ of the position vector $\textbf R_\alpha(t\pm\delta t)$
|
||||
%in different time directions:
|
||||
|
||||
\begin{align}
|
||||
\mathbf R_\alpha(t+\delta t) = \mathbf R_\alpha(t) + \mathbf v_\alpha(t)\delta t +\frac{\mathbf a_\alpha(t)\delta t^2}{2}+ \frac{\mathbf b_\alpha(t)\delta t^3}{6} + O(\delta t^4)
|
||||
@ -1265,24 +1263,23 @@ The VV algorithm is often applied to solve this problem as it allows velocities
|
||||
\end{align}
|
||||
|
||||
%The error in the VV is of the same order with the one in the basic Verlet. However, compared to the basic Verlet, the VV algorithm requires less memory because two vectors of position are tracked in basic Verlet while one position vector and one velocity vector are tracked in VV.
|
||||
The standard implementation of the VV algorithm is a four steps scheme: firstly to calculate equation~\ref{VV1}, secondly to calculate equation~\ref{VV2}, thirdly to
|
||||
derive $\mathbf a_\alpha(t+ \delta t)$ from the interaction potential with $\mathbf R_\alpha(t+ \delta t)$, finally to compute equation~\ref{VV3}.
|
||||
|
||||
The standard implementation of the VV algorithm is a four steps scheme: firstly to calculate the following equation,
|
||||
\begin{align}
|
||||
\mathbf v_\alpha(t+\frac{1}{2}\delta t) = \mathbf v_\alpha(t) + \frac{1}{2}\mathbf a_\alpha(t)\delta t
|
||||
\label{VV1}
|
||||
\end{align}
|
||||
|
||||
Secondly to calculate equation as follows:
|
||||
\begin{align}
|
||||
\quad \qquad \mathbf R_\alpha(t+ \delta t) =\mathbf R_\alpha(t) + \mathbf v_\alpha(t + \frac{1}{2} \delta t) \delta t
|
||||
\label{VV2}
|
||||
\end{align}
|
||||
|
||||
Thirdly to derive $\mathbf a_\alpha(t+ \delta t)$ from the interaction potential at $\mathbf R_\alpha(t+ \delta t)$, finally to compute the following equation.
|
||||
\begin{align}
|
||||
\qquad \qquad \qquad \mathbf v_\alpha(t+ \delta t) = \mathbf v_\alpha(t + \frac{1}{2} \delta t) + \frac{1}{2} \mathbf a_\alpha(t+ \delta t) \delta t
|
||||
\label{VV3}
|
||||
\end{align}
|
||||
<%<<<<<< HEAD
|
||||
|
||||
%<%<<<<<< HEAD
|
||||
%The VV algorithm needs to ensure two intrinsic properties of classical motion equations.
|
||||
%One is temporal reversibility, the invariance of the trajectories at $t$ and $-t$. This symmetry leads to the independence of the dynamics of the microscopic system from the direction of time. The other one is the conservation of the Hamiltonian function over time. Because of the discretization of the trajectories, this conservation can not be assured. So a stable integration algorithm must impose this conservation for enough long time steps ($\delta t$) to allow the long simulation times. The VV algorithm is able to do this due to its sufficient numerical stability. At the end of each integration step, VV algorithm allows to access to $\mathbf R_\alpha (t+\delta t)$, $\mathbf v_\alpha (t+\delta t)$, $\mathbf F_\alpha (t+\delta t)$ directly.
|
||||
|
||||
@ -1303,9 +1300,11 @@ Temperature is not a state variable in the simulations, an average kinetic tempe
|
||||
|
||||
At the end of each integration step, the VV algorithm gives directly access to $\mathbf R_\alpha (t+\delta t)$, $\mathbf v_\alpha (t+\delta t)$,
|
||||
and $\mathbf F_\alpha (t+\delta t)$.
|
||||
The VV algorithm needs to ensure two intrinsic properties of the classical equations of motion. One is temporal reversibility,
|
||||
The VV algorithm ensures two intrinsic properties of the classical equations of motion. One is temporal reversibility,
|
||||
the invariance of the trajectories at $t$ and $-t$. This symmetry leads to the independence of
|
||||
he dynamics from the direction of time. The other property is the conservation of the Hamiltonian function over time.
|
||||
he dynamics from the direction of time. The other property is the conservation of the
|
||||
%Hamiltonian function
|
||||
total energy over time.
|
||||
Because of the discretization of trajectories, this conservation can not be insured. A stable integration algorithm must
|
||||
impose this conservation for long enough time steps ($\delta t$) to allow for sufficiently long simulation times. The VV algorithm
|
||||
is able to do this due to its sufficient numerical stability.
|
||||
@ -1322,10 +1321,7 @@ forces. \textbf{In this thesis, the SCC-DFTB method has been applied to compute
|
||||
|
||||
\subsection{Parallel-Tempering Molecular Dynamics} \label{sec:PTMD}
|
||||
|
||||
In a number of cases, it is necessary to explore the PES as thoroughly as possible for the study of dynamical,
|
||||
thermodynamical, and structural properties of a given system. For atomic and molecular clusters, PES usually
|
||||
presents a large number of stable configurations linked together by energy barriers. Unfortunately, MD simulations
|
||||
cannot overcome these energy barriers in a reasonable simulation time even within the canonical ensemble ($N, V, T$).
|
||||
In a number of cases, it is necessary to explore the PES as thoroughly as possible for the study of dynamical, thermodynamical, and structural properties of a given system. For atomic and molecular clusters, PES usually presents a large number of stable configurations linked together by energy barriers. Unfortunately, MD simulations cannot overcome these energy barriers in a reasonable simulation time even within the canonical ensemble ($N, V, T$).
|
||||
This leads to non-ergodic simulations that cannot be used to extract meaningful statistical averages. Actually, when
|
||||
a simulation explores the well of a PES, it may often be blocked in this well at low temperature because the energy
|
||||
barriers are too high to be crossed. In this case, if $E_b$ refers to the energy barrier and $T$ is the temperature of
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@ -3,7 +3,7 @@
|
||||
\FN@pp@footnotehinttrue
|
||||
\citation{Brechignac1989,Brechignac1994}
|
||||
\citation{Wong2004,Bush2008}
|
||||
\citation{Holm2010,M. Gatchell2014,Gatchell2017}
|
||||
\citation{Holm2010,Gatchell2014,Gatchell2017}
|
||||
\citation{Boering1992,Wells2005,Zamith2019thermal}
|
||||
\@writefile{toc}{\contentsline {chapter}{\numberline {4}Dynamical Simulation of Collision-Induced Dissociation}{97}{chapter.4}}
|
||||
\@writefile{lof}{\addvspace {10\p@ }}
|
||||
@ -33,6 +33,7 @@
|
||||
\citation{Carl2013,Hofstetter2013,Coates2018}
|
||||
\@writefile{brf}{\backcite{Holm2010}{{98}{4.1}{section.4.1}}}
|
||||
\@writefile{brf}{\backcite{Gatchell2017}{{98}{4.1}{section.4.1}}}
|
||||
\@writefile{brf}{\backcite{Gatchell2014}{{98}{4.1}{section.4.1}}}
|
||||
\@writefile{brf}{\backcite{Zamith2019thermal}{{98}{4.1}{section.4.1}}}
|
||||
\@writefile{brf}{\backcite{Boering1992}{{98}{4.1}{section.4.1}}}
|
||||
\@writefile{brf}{\backcite{Wells2005}{{98}{4.1}{section.4.1}}}
|
||||
|
||||
@ -30,7 +30,7 @@ The stability of cluster can be investigated from dissociation experiments. Clus
|
||||
For instance, the sodium cluster ions and lithium cluster cation were dissociated with a pulsed UV laser source.\cite{Brechignac1989, Brechignac1994}
|
||||
Gaseous hydrated trivalent metal ions were dissociated using blackbody infrared radiative dissociation (BIRD).\cite{Wong2004, Bush2008}
|
||||
The collision between cluster and high or low energetic particles at different pressure also have been investigated.
|
||||
Collisions between the high energetic projectile ions (such as 3 keV Ar$^+$, 22.5 keV He$^{2+}$) and neutral targets were investigated by Gatchell and A. Holm.\cite{Holm2010, M. Gatchell2014, Gatchell2017}
|
||||
Collisions between the high energetic projectile ions (such as 3 keV Ar$^+$, 22.5 keV He$^{2+}$) and neutral targets were investigated by M. Gatchell and A. Holm.\cite{Holm2010, Gatchell2014, Gatchell2017}
|
||||
Collisions between clusters and projectile have been also explored at low collision energy, which allows for the derivation of dissociation energies and the thermal evaporation and stability of clusters. \cite{Boering1992, Wells2005, Zamith2019thermal}
|
||||
|
||||
By colliding a molecule, or a molecular aggregate, with a non-reactive rare gas atom (neon, argon) or a small molecule such as H$_2$O or N$_2$, it is possible to monitor the parent ions and collision products by use, for instance, of tandem mass spectrometry (MS/MS).\cite{Ma1997, Chowdhury2009} The resulting mass spectra provide a wealth of information about the structure of the parent and product ions from which one can infer, for instance, dissociation mechanisms \cite{Nelson1994, Molina2015} or bond and hydration enthalpies \cite{Carl2007}.
|
||||
|
||||
@ -21,7 +21,7 @@
|
||||
\setcounter{ContinuedFloat}{0}
|
||||
\setcounter{pp@next@reset}{1}
|
||||
\setcounter{@fnserial}{0}
|
||||
\setcounter{NAT@ctr}{482}
|
||||
\setcounter{NAT@ctr}{483}
|
||||
\setcounter{Item}{5}
|
||||
\setcounter{Hfootnote}{0}
|
||||
\setcounter{bookmark@seq@number}{65}
|
||||
|
||||
@ -466,68 +466,69 @@
|
||||
\bibcite{Brechignac1994}{{418}{}{{}}{{}}}
|
||||
\bibcite{Wong2004}{{419}{}{{}}{{}}}
|
||||
\bibcite{Bush2008}{{420}{}{{}}{{}}}
|
||||
\bibcite{Boering1992}{{421}{}{{}}{{}}}
|
||||
\bibcite{Wells2005}{{422}{}{{}}{{}}}
|
||||
\bibcite{Ma1997}{{423}{}{{}}{{}}}
|
||||
\bibcite{Chowdhury2009}{{424}{}{{}}{{}}}
|
||||
\bibcite{Sleno2004ion}{{425}{}{{}}{{}}}
|
||||
\bibcite{Cody1982}{{426}{}{{}}{{}}}
|
||||
\bibcite{Olsen2007higher}{{427}{}{{}}{{}}}
|
||||
\bibcite{Hart2011}{{428}{}{{}}{{}}}
|
||||
\bibcite{Gauthier1991}{{429}{}{{}}{{}}}
|
||||
\bibcite{Laskin2005}{{430}{}{{}}{{}}}
|
||||
\bibcite{Mcquinn2009}{{431}{}{{}}{{}}}
|
||||
\bibcite{Carl2013}{{432}{}{{}}{{}}}
|
||||
\bibcite{Hofstetter2013}{{433}{}{{}}{{}}}
|
||||
\bibcite{Coates2017}{{434}{}{{}}{{}}}
|
||||
\bibcite{Haag2009}{{435}{}{{}}{{}}}
|
||||
\bibcite{Li1992}{{436}{}{{}}{{}}}
|
||||
\bibcite{Bobbert2002}{{437}{}{{}}{{}}}
|
||||
\bibcite{Spasov2000}{{438}{}{{}}{{}}}
|
||||
\bibcite{Armentrout2008}{{439}{}{{}}{{}}}
|
||||
\bibcite{Rodgers1998}{{440}{}{{}}{{}}}
|
||||
\bibcite{Armentrout2007}{{441}{}{{}}{{}}}
|
||||
\bibcite{Braud2017}{{442}{}{{}}{{}}}
|
||||
\bibcite{Chirot2006new}{{443}{}{{}}{{}}}
|
||||
\bibcite{Frenzel2004}{{444}{}{{}}{{}}}
|
||||
\bibcite{Spiegelman2020}{{445}{}{{}}{{}}}
|
||||
\bibcite{Simon2017}{{446}{}{{}}{{}}}
|
||||
\bibcite{Rapacioli2018}{{447}{}{{}}{{}}}
|
||||
\bibcite{Simon2018}{{448}{}{{}}{{}}}
|
||||
\bibcite{Warshel1976}{{449}{}{{}}{{}}}
|
||||
\bibcite{Cui2001}{{450}{}{{}}{{}}}
|
||||
\bibcite{Iftner2014}{{451}{}{{}}{{}}}
|
||||
\bibcite{Kukk2015}{{452}{}{{}}{{}}}
|
||||
\bibcite{Rapacioli2018atomic}{{453}{}{{}}{{}}}
|
||||
\bibcite{Dontot2019}{{454}{}{{}}{{}}}
|
||||
\bibcite{Chung2011}{{455}{}{{}}{{}}}
|
||||
\bibcite{Saggese2015}{{456}{}{{}}{{}}}
|
||||
\bibcite{Mao2017}{{457}{}{{}}{{}}}
|
||||
\bibcite{Wang2018}{{458}{}{{}}{{}}}
|
||||
\bibcite{Kyrtopoulos2001}{{459}{}{{}}{{}}}
|
||||
\bibcite{Farmer2003}{{460}{}{{}}{{}}}
|
||||
\bibcite{Aumaitre2019}{{461}{}{{}}{{}}}
|
||||
\bibcite{Rapacioli2005}{{462}{}{{}}{{}}}
|
||||
\bibcite{Berne2008}{{463}{}{{}}{{}}}
|
||||
\bibcite{Eschenbach1998}{{464}{}{{}}{{}}}
|
||||
\bibcite{Goulart2017}{{465}{}{{}}{{}}}
|
||||
\bibcite{Lei2019}{{466}{}{{}}{{}}}
|
||||
\bibcite{Roser2015}{{467}{}{{}}{{}}}
|
||||
\bibcite{Lemmens2019}{{468}{}{{}}{{}}}
|
||||
\bibcite{Delaunay2015}{{469}{}{{}}{{}}}
|
||||
\bibcite{Piacenza2005}{{470}{}{{}}{{}}}
|
||||
\bibcite{Birer2015}{{471}{}{{}}{{}}}
|
||||
\bibcite{Zhao2008truhlar}{{472}{}{{}}{{}}}
|
||||
\bibcite{Bowal2019}{{473}{}{{}}{{}}}
|
||||
\bibcite{Ricca2013}{{474}{}{{}}{{}}}
|
||||
\bibcite{Grafenstein2009}{{475}{}{{}}{{}}}
|
||||
\bibcite{Dontot2016}{{476}{}{{}}{{}}}
|
||||
\bibcite{Dontot2020}{{477}{}{{}}{{}}}
|
||||
\bibcite{Rapacioli2011}{{478}{}{{}}{{}}}
|
||||
\bibcite{Gatchell2016}{{479}{}{{}}{{}}}
|
||||
\bibcite{Gatchell2016knockout}{{480}{}{{}}{{}}}
|
||||
\bibcite{Levine1987}{{481}{}{{}}{{}}}
|
||||
\bibcite{Chen2014}{{482}{}{{}}{{}}}
|
||||
\bibcite{Gatchell2014}{{421}{}{{}}{{}}}
|
||||
\bibcite{Boering1992}{{422}{}{{}}{{}}}
|
||||
\bibcite{Wells2005}{{423}{}{{}}{{}}}
|
||||
\bibcite{Ma1997}{{424}{}{{}}{{}}}
|
||||
\bibcite{Chowdhury2009}{{425}{}{{}}{{}}}
|
||||
\bibcite{Sleno2004ion}{{426}{}{{}}{{}}}
|
||||
\bibcite{Cody1982}{{427}{}{{}}{{}}}
|
||||
\bibcite{Olsen2007higher}{{428}{}{{}}{{}}}
|
||||
\bibcite{Hart2011}{{429}{}{{}}{{}}}
|
||||
\bibcite{Gauthier1991}{{430}{}{{}}{{}}}
|
||||
\bibcite{Laskin2005}{{431}{}{{}}{{}}}
|
||||
\bibcite{Mcquinn2009}{{432}{}{{}}{{}}}
|
||||
\bibcite{Carl2013}{{433}{}{{}}{{}}}
|
||||
\bibcite{Hofstetter2013}{{434}{}{{}}{{}}}
|
||||
\bibcite{Coates2017}{{435}{}{{}}{{}}}
|
||||
\bibcite{Haag2009}{{436}{}{{}}{{}}}
|
||||
\bibcite{Li1992}{{437}{}{{}}{{}}}
|
||||
\bibcite{Bobbert2002}{{438}{}{{}}{{}}}
|
||||
\bibcite{Spasov2000}{{439}{}{{}}{{}}}
|
||||
\bibcite{Armentrout2008}{{440}{}{{}}{{}}}
|
||||
\bibcite{Rodgers1998}{{441}{}{{}}{{}}}
|
||||
\bibcite{Armentrout2007}{{442}{}{{}}{{}}}
|
||||
\bibcite{Braud2017}{{443}{}{{}}{{}}}
|
||||
\bibcite{Chirot2006new}{{444}{}{{}}{{}}}
|
||||
\bibcite{Frenzel2004}{{445}{}{{}}{{}}}
|
||||
\bibcite{Spiegelman2020}{{446}{}{{}}{{}}}
|
||||
\bibcite{Simon2017}{{447}{}{{}}{{}}}
|
||||
\bibcite{Rapacioli2018}{{448}{}{{}}{{}}}
|
||||
\bibcite{Simon2018}{{449}{}{{}}{{}}}
|
||||
\bibcite{Warshel1976}{{450}{}{{}}{{}}}
|
||||
\bibcite{Cui2001}{{451}{}{{}}{{}}}
|
||||
\bibcite{Iftner2014}{{452}{}{{}}{{}}}
|
||||
\bibcite{Kukk2015}{{453}{}{{}}{{}}}
|
||||
\bibcite{Rapacioli2018atomic}{{454}{}{{}}{{}}}
|
||||
\bibcite{Dontot2019}{{455}{}{{}}{{}}}
|
||||
\bibcite{Chung2011}{{456}{}{{}}{{}}}
|
||||
\bibcite{Saggese2015}{{457}{}{{}}{{}}}
|
||||
\bibcite{Mao2017}{{458}{}{{}}{{}}}
|
||||
\bibcite{Wang2018}{{459}{}{{}}{{}}}
|
||||
\bibcite{Kyrtopoulos2001}{{460}{}{{}}{{}}}
|
||||
\bibcite{Farmer2003}{{461}{}{{}}{{}}}
|
||||
\bibcite{Aumaitre2019}{{462}{}{{}}{{}}}
|
||||
\bibcite{Rapacioli2005}{{463}{}{{}}{{}}}
|
||||
\bibcite{Berne2008}{{464}{}{{}}{{}}}
|
||||
\bibcite{Eschenbach1998}{{465}{}{{}}{{}}}
|
||||
\bibcite{Goulart2017}{{466}{}{{}}{{}}}
|
||||
\bibcite{Lei2019}{{467}{}{{}}{{}}}
|
||||
\bibcite{Roser2015}{{468}{}{{}}{{}}}
|
||||
\bibcite{Lemmens2019}{{469}{}{{}}{{}}}
|
||||
\bibcite{Delaunay2015}{{470}{}{{}}{{}}}
|
||||
\bibcite{Piacenza2005}{{471}{}{{}}{{}}}
|
||||
\bibcite{Birer2015}{{472}{}{{}}{{}}}
|
||||
\bibcite{Zhao2008truhlar}{{473}{}{{}}{{}}}
|
||||
\bibcite{Bowal2019}{{474}{}{{}}{{}}}
|
||||
\bibcite{Ricca2013}{{475}{}{{}}{{}}}
|
||||
\bibcite{Grafenstein2009}{{476}{}{{}}{{}}}
|
||||
\bibcite{Dontot2016}{{477}{}{{}}{{}}}
|
||||
\bibcite{Dontot2020}{{478}{}{{}}{{}}}
|
||||
\bibcite{Rapacioli2011}{{479}{}{{}}{{}}}
|
||||
\bibcite{Gatchell2016}{{480}{}{{}}{{}}}
|
||||
\bibcite{Gatchell2016knockout}{{481}{}{{}}{{}}}
|
||||
\bibcite{Levine1987}{{482}{}{{}}{{}}}
|
||||
\bibcite{Chen2014}{{483}{}{{}}{{}}}
|
||||
\FN@pp@footnotehinttrue
|
||||
\@input{6_backmatter/declaration.aux}
|
||||
\providecommand\NAT@force@numbers{}\NAT@force@numbers
|
||||
|
||||
@ -2554,6 +2554,13 @@
|
||||
trivalent metal ions}.
|
||||
\newblock {\em J. Am. Chem. Soc.}, {\bf 130}(28):9122--9128, 2008.
|
||||
|
||||
\bibitem{Gatchell2014}
|
||||
{\sc Michael Gatchell, Patrick Rousseau, A~Domaracka, Mark~H Stockett, Tao
|
||||
Chen, Henning~T Schmidt, JY~Chesnel, A~M{\'e}ry, S~Maclot, L~Adoui, et~al.}
|
||||
\newblock {\bf Ions colliding with mixed clusters of C$_{60}$ and coronene:
|
||||
Fragmentation and bond formation}.
|
||||
\newblock {\em Phys. Rev. A}, {\bf 90}(2):022713, 2014.
|
||||
|
||||
\bibitem{Boering1992}
|
||||
{\sc Kristie~A Boering, Joseph Rolfe, and John~I Brauman}.
|
||||
\newblock {\bf Low energy collision induced dissociation: phase-shifting
|
||||
|
||||
@ -9,14 +9,10 @@ A level-1 auxiliary file: 1_GeneIntro/GeneIntro.aux
|
||||
A level-1 auxiliary file: 2_Introduction/introduction.aux
|
||||
A level-1 auxiliary file: 3/structure_stability.aux
|
||||
A level-1 auxiliary file: 4/collision.aux
|
||||
White space in argument---line 6 of file 4/collision.aux
|
||||
: \citation{Holm2010,M.
|
||||
: Gatchell2014,Gatchell2017}
|
||||
I'm skipping whatever remains of this command
|
||||
A level-1 auxiliary file: 5/general_conclusion.aux
|
||||
The style file: Latex/Classes/PhDbiblio-url2.bst
|
||||
A level-1 auxiliary file: 6_backmatter/declaration.aux
|
||||
Reallocated field_info (elt_size=4) to 18014 items from 5000.
|
||||
Reallocated field_info (elt_size=4) to 18041 items from 5000.
|
||||
Database file #1: 6_backmatter/references.bib
|
||||
I was expecting a `,' or a `}'---line 1252 of file 6_backmatter/references.bib
|
||||
:
|
||||
@ -121,45 +117,45 @@ Warning--empty year in Unger1993
|
||||
Warning--empty title in GaussianCode
|
||||
Warning--empty journal in GaussianCode
|
||||
Warning--empty year in GaussianCode
|
||||
You've used 482 entries,
|
||||
You've used 483 entries,
|
||||
2776 wiz_defined-function locations,
|
||||
3548 strings with 95460 characters,
|
||||
and the built_in function-call counts, 222153 in all, are:
|
||||
= -- 23195
|
||||
> -- 9218
|
||||
3553 strings with 95767 characters,
|
||||
and the built_in function-call counts, 222834 in all, are:
|
||||
= -- 23255
|
||||
> -- 9276
|
||||
< -- 22
|
||||
+ -- 3680
|
||||
- -- 3142
|
||||
* -- 17439
|
||||
:= -- 33780
|
||||
add.period$ -- 1451
|
||||
call.type$ -- 482
|
||||
change.case$ -- 2521
|
||||
+ -- 3703
|
||||
- -- 3164
|
||||
* -- 17505
|
||||
:= -- 33895
|
||||
add.period$ -- 1454
|
||||
call.type$ -- 483
|
||||
change.case$ -- 2533
|
||||
chr.to.int$ -- 0
|
||||
cite$ -- 494
|
||||
duplicate$ -- 6839
|
||||
empty$ -- 18808
|
||||
format.name$ -- 3142
|
||||
if$ -- 50990
|
||||
cite$ -- 495
|
||||
duplicate$ -- 6853
|
||||
empty$ -- 18846
|
||||
format.name$ -- 3164
|
||||
if$ -- 51131
|
||||
int.to.chr$ -- 0
|
||||
int.to.str$ -- 482
|
||||
missing$ -- 504
|
||||
newline$ -- 2409
|
||||
num.names$ -- 964
|
||||
pop$ -- 3749
|
||||
int.to.str$ -- 483
|
||||
missing$ -- 505
|
||||
newline$ -- 2414
|
||||
num.names$ -- 966
|
||||
pop$ -- 3763
|
||||
preamble$ -- 1
|
||||
purify$ -- 2520
|
||||
purify$ -- 2532
|
||||
quote$ -- 0
|
||||
skip$ -- 11290
|
||||
skip$ -- 11314
|
||||
stack$ -- 0
|
||||
substring$ -- 13655
|
||||
swap$ -- 1968
|
||||
substring$ -- 13678
|
||||
swap$ -- 1972
|
||||
text.length$ -- 22
|
||||
text.prefix$ -- 0
|
||||
top$ -- 0
|
||||
type$ -- 1874
|
||||
type$ -- 1878
|
||||
warning$ -- 12
|
||||
while$ -- 1746
|
||||
width$ -- 485
|
||||
write$ -- 5269
|
||||
(There were 22 error messages)
|
||||
while$ -- 1749
|
||||
width$ -- 486
|
||||
write$ -- 5280
|
||||
(There were 21 error messages)
|
||||
|
||||
@ -242,20 +242,20 @@
|
||||
\backcite {Thomas1927}{{22}{2.3.2}{equation.2.3.16}}
|
||||
\backcite {Fermi1928}{{22}{2.3.2}{equation.2.3.16}}
|
||||
\backcite {Kohn1965}{{23}{2.3.2}{equation.2.3.18}}
|
||||
\backcite {Kohn1965}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Kohn1965}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Herman1969}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Perdew1996}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Becke1993}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Becke1993}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Becke1988}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Lee1988}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Stephens1994}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Adamo1999}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Henderson2009}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Krukau2006}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Zhao2008}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Li2018}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Kohn1965}{{25}{2.3.2}{equation.2.3.26}}
|
||||
\backcite {Kohn1965}{{25}{2.3.2}{equation.2.3.26}}
|
||||
\backcite {Herman1969}{{25}{2.3.2}{equation.2.3.26}}
|
||||
\backcite {Perdew1996}{{25}{2.3.2}{equation.2.3.26}}
|
||||
\backcite {Becke1993}{{25}{2.3.2}{equation.2.3.26}}
|
||||
\backcite {Becke1993}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Becke1988}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Lee1988}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Stephens1994}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Adamo1999}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Henderson2009}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Krukau2006}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Zhao2008}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Li2018}{{26}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Foulkes1989}{{26}{2.3.3}{subsection.2.3.3}}
|
||||
\backcite {Porezag1995}{{26}{2.3.3}{subsection.2.3.3}}
|
||||
\backcite {Elstner1998}{{26}{2.3.3}{subsection.2.3.3}}
|
||||
@ -265,29 +265,30 @@
|
||||
\backcite {Elstne2007}{{26}{2.3.3}{subsection.2.3.3}}
|
||||
\backcite {Oliveira2009}{{26}{2.3.3}{subsection.2.3.3}}
|
||||
\backcite {Gaus2011}{{27}{2.3.3}{subsection.2.3.3}}
|
||||
\backcite {Mulliken1955}{{30}{2.3.3}{equation.2.3.44}}
|
||||
\backcite {Mulliken1955}{{31}{2.3.3}{equation.2.3.47}}
|
||||
\backcite {Bader1985}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Bader1990}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Glendening1998}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Glendening2012}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Singh1984}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Besler1990}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Li1998}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Kalinowski2004}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Kelly2005}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Rapacioli2009corr}{{32}{2.3.3}{equation.2.3.51}}
|
||||
\backcite {Mayer1983}{{32}{2.3.3}{equation.2.3.52}}
|
||||
\backcite {Cances1997}{{32}{2.3.3}{equation.2.3.52}}
|
||||
\backcite {Lu2012}{{32}{2.3.3}{equation.2.3.52}}
|
||||
\backcite {Lewis1995}{{32}{2.3.3}{equation.2.3.53}}
|
||||
\backcite {Gianturco1999}{{32}{2.3.3}{equation.2.3.53}}
|
||||
\backcite {Elstner2001}{{32}{2.3.3}{equation.2.3.53}}
|
||||
\backcite {Wu2002}{{32}{2.3.3}{equation.2.3.53}}
|
||||
\backcite {Grimme2004}{{32}{2.3.3}{equation.2.3.53}}
|
||||
\backcite {Zimmerli2004}{{32}{2.3.3}{equation.2.3.53}}
|
||||
\backcite {Neumann2005}{{32}{2.3.3}{equation.2.3.53}}
|
||||
\backcite {Goursot2007}{{32}{2.3.3}{equation.2.3.53}}
|
||||
\backcite {Mulliken1955}{{29}{2.3.3}{equation.2.3.40}}
|
||||
\backcite {Elstner1998}{{30}{2.3.3}{equation.2.3.42}}
|
||||
\backcite {Mulliken1955}{{30}{2.3.3}{equation.2.3.42}}
|
||||
\backcite {Bader1985}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Bader1990}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Glendening1998}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Glendening2012}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Singh1984}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Besler1990}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Li1998}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Kalinowski2004}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Kelly2005}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Rapacioli2009corr}{{32}{2.3.3}{equation.2.3.48}}
|
||||
\backcite {Mayer1983}{{32}{2.3.3}{equation.2.3.49}}
|
||||
\backcite {Cances1997}{{32}{2.3.3}{equation.2.3.49}}
|
||||
\backcite {Lu2012}{{32}{2.3.3}{equation.2.3.49}}
|
||||
\backcite {Lewis1995}{{32}{2.3.3}{equation.2.3.50}}
|
||||
\backcite {Gianturco1999}{{32}{2.3.3}{equation.2.3.50}}
|
||||
\backcite {Elstner2001}{{32}{2.3.3}{equation.2.3.50}}
|
||||
\backcite {Wu2002}{{32}{2.3.3}{equation.2.3.50}}
|
||||
\backcite {Grimme2004}{{32}{2.3.3}{equation.2.3.50}}
|
||||
\backcite {Zimmerli2004}{{32}{2.3.3}{equation.2.3.50}}
|
||||
\backcite {Neumann2005}{{32}{2.3.3}{equation.2.3.50}}
|
||||
\backcite {Goursot2007}{{32}{2.3.3}{equation.2.3.50}}
|
||||
\backcite {Frenkel2001}{{33}{2.3.4}{subsection.2.3.4}}
|
||||
\backcite {Jones1924}{{33}{2.3.4}{subsection.2.3.4}}
|
||||
\backcite {Lennard1924}{{33}{2.3.4}{subsection.2.3.4}}
|
||||
@ -296,18 +297,18 @@
|
||||
\backcite {Stillinger1985}{{33}{2.3.4}{subsection.2.3.4}}
|
||||
\backcite {Kaplan2006}{{33}{2.3.4}{subsection.2.3.4}}
|
||||
\backcite {Bahar1997}{{33}{2.3.4}{subsection.2.3.4}}
|
||||
\backcite {Morse1929}{{34}{2.3.4}{equation.2.3.55}}
|
||||
\backcite {Girifalco1959}{{34}{2.3.4}{equation.2.3.55}}
|
||||
\backcite {Jones1924}{{34}{2.3.4}{equation.2.3.56}}
|
||||
\backcite {Lennard1924}{{34}{2.3.4}{equation.2.3.56}}
|
||||
\backcite {Lennard1931}{{34}{2.3.4}{equation.2.3.56}}
|
||||
\backcite {Monticelli2013}{{34}{2.3.4}{equation.2.3.57}}
|
||||
\backcite {Morse1929}{{33}{2.3.4}{equation.2.3.52}}
|
||||
\backcite {Girifalco1959}{{33}{2.3.4}{equation.2.3.52}}
|
||||
\backcite {Jones1924}{{33}{2.3.4}{equation.2.3.53}}
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