modernize boldface font selection in LaTeX sections
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@ -49,7 +49,7 @@ For each atom, :math:`Q_\ell` is a real number defined as follows:
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.. math::
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\bar{Y}_{\ell m} = & \frac{1}{nnn}\sum_{j = 1}^{nnn} Y_{\ell m}\bigl( \theta( {\bf r}_{ij} ), \phi( {\bf r}_{ij} ) \bigr) \\
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\bar{Y}_{\ell m} = & \frac{1}{nnn}\sum_{j = 1}^{nnn} Y_{\ell m}\bigl( \theta( \mathbf{r}_{ij} ), \phi( \mathbf{r}_{ij} ) \bigr) \\
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Q_\ell = & \sqrt{\frac{4 \pi}{2 \ell + 1} \sum_{m = -\ell }^{m = \ell } \bar{Y}_{\ell m} \bar{Y}^*_{\ell m}}
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The first equation defines the local order parameters as averages
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@ -204,7 +204,7 @@ components summed separately for each LAMMPS atom type:
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.. math::
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-\sum_{i' \in I} \frac{\partial {B^{i'}_{j_1,j_2,j} }}{\partial {\bf r}_i}
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-\sum_{i' \in I} \frac{\partial {B^{i'}_{j_1,j_2,j} }}{\partial \mathbf{r}_i}
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The sum is over all atoms *i'* of atom type *I*\ . For each atom *i*,
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this compute evaluates the above expression for each direction, each
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@ -216,7 +216,7 @@ derivatives:
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.. math::
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-{\bf r}_i \otimes \sum_{i' \in I} \frac{\partial {B^{i'}_{j_1,j_2,j}}}{\partial {\bf r}_i}
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-\mathbf{r}_i \otimes \sum_{i' \in I} \frac{\partial {B^{i'}_{j_1,j_2,j}}}{\partial \mathbf{r}_i}
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Again, the sum is over all atoms *i'* of atom type *I*\ . For each atom
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*i*, this compute evaluates the above expression for each of the six
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@ -60,9 +60,9 @@ With this fix active, the force on the *j*\ th atom is given as
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.. math::
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{\bf F}_{j}(t) = & {\bf F}^C_j(t)-\int \limits_{0}^{t} \Gamma_j(t-s) {\bf v}_j(s)~\text{d}s + {\bf F}^R_j(t) \\
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\mathbf{F}_{j}(t) = & \mathbf{F}^C_j(t)-\int \limits_{0}^{t} \Gamma_j(t-s) \mathbf{v}_j(s)~\text{d}s + \mathbf{F}^R_j(t) \\
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\Gamma_j(t-s) = & \sum \limits_{k=1}^{N_k} \frac{c_k}{\tau_k} e^{-(t-s)/\tau_k} \\
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\langle{\bf F}^R_j(t),{\bf F}^R_j(s)\rangle = & \text{k$_\text{B}$T} ~\Gamma_j(t-s)
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\langle\mathbf{F}^R_j(t),\mathbf{F}^R_j(s)\rangle = & \text{k$_\text{B}$T} ~\Gamma_j(t-s)
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Here, the first term is representative of all conservative (pairwise,
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bonded, etc) forces external to this fix, the second is the temporally
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@ -130,7 +130,7 @@ calculated as:
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.. math::
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{\bf F}_{j \alpha} = \gamma \left({\bf v}_n - {\bf u}_f \right) \zeta_{j\alpha}
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\mathbf{F}_{j \alpha} = \gamma \left(\mathbf{v}_n - \mathbf{u}_f \right) \zeta_{j\alpha}
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where :math:`\mathbf{v}_n` is the velocity of the MD particle,
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:math:`\mathbf{u}_f` is the fluid velocity interpolated to the particle
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@ -101,7 +101,7 @@ by the following equations:
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.. math::
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Z = & \int d{\bf q} d{\bf p} \cdot \textrm{exp} [ -\beta H_{eff} ] \\
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Z = & \int d\mathbf{q} d\mathbf{p} \cdot \textrm{exp} [ -\beta H_{eff} ] \\
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H_{eff} = & \bigg(\sum_{i=1}^P \frac{p_i^2}{2M_i}\bigg) + V_{eff} \\
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V_{eff} = & \sum_{i=1}^P \bigg[ \frac{mP}{2\beta^2 \hbar^2} (q_i - q_{i+1})^2 + \frac{1}{P} V(q_i)\bigg]
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@ -97,7 +97,7 @@ inverse temperature :math:`\beta` is given by :ref:`(Tuckerman)
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.. math::
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Z \propto \int d{\bf q} \cdot \frac{1}{N!} \sum_\sigma \textrm{exp} [ -\beta \left( E^\sigma + V \right) ].
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Z \propto \int d\mathbf{q} \cdot \frac{1}{N!} \sum_\sigma \textrm{exp} [ -\beta \left( E^\sigma + V \right) ].
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Here, :math:`V` is the potential between different particles at the same
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imaginary time slice, which is the same for bosons and distinguishable
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@ -57,8 +57,8 @@ materials as described in :ref:`(Feng1) <Feng1>` and :ref:`(Feng2) <Feng2>`.
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\left [ \epsilon + f(\rho_{ij}) + f(\rho_{ji})\right ] -
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\frac{1}{1+e^{-d\left [ \left ( r_{ij}/\left (s_R \cdot r^{eff} \right ) \right )-1 \right ]}}
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\cdot \frac{C_6}{r^6_{ij}} \right \}\\
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\rho_{ij}^2 = & r_{ij}^2 - ({\bf r}_{ij} \cdot {\bf n}_i)^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - ({\bf r}_{ij} \cdot {\bf n}_j)^2 \\
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\rho_{ij}^2 = & r_{ij}^2 - (\mathbf{r}_{ij} \cdot \mathbf{n}_i)^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - (\mathbf{r}_{ij} \cdot \mathbf{n}_j)^2 \\
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f(\rho) = & C e^{ -( \rho / \delta )^2 } \\
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\mathrm{Tap}(r_{ij}) = & 20\left ( \frac{r_{ij}}{R_{cut}} \right )^7 -
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70\left ( \frac{r_{ij}}{R_{cut}} \right )^6 +
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@ -48,8 +48,8 @@ in :ref:`(Kolmogorov) <Kolmogorov2>`.
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\left [ \epsilon + f(\rho_{ij}) + f(\rho_{ji})\right ] -
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\frac{1}{1+e^{-d\left [ \left ( r_{ij}/\left (s_R \cdot r^{eff} \right ) \right )-1 \right ]}}
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\cdot \frac{C_6}{r^6_{ij}} \right \}\\
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\rho_{ij}^2 = & r_{ij}^2 - ({\bf r}_{ij} \cdot {\bf n}_i)^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - ({\bf r}_{ij} \cdot {\bf n}_j)^2 \\
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\rho_{ij}^2 = & r_{ij}^2 - (\mathbf{r}_{ij} \cdot \mathbf{n}_i)^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - (\mathbf{r}_{ij} \cdot \mathbf{n}_j)^2 \\
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f(\rho) = & C e^{ -( \rho / \delta )^2 } \\
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\mathrm{Tap}(r_{ij}) = & 20\left ( \frac{r_{ij}}{R_{cut}} \right )^7 -
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70\left ( \frac{r_{ij}}{R_{cut}} \right )^6 +
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@ -45,8 +45,8 @@ as described in :ref:`(Ouyang7) <Ouyang7>` and :ref:`(Jiang) <Jiang>`.
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\left [ \epsilon + f(\rho_{ij}) + f(\rho_{ji})\right ] -
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\frac{1}{1+e^{-d\left [ \left ( r_{ij}/\left (s_R \cdot r^{eff} \right ) \right )-1 \right ]}}
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\cdot \frac{C_6}{r^6_{ij}} \right \}\\
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\rho_{ij}^2 = & r_{ij}^2 - ({\bf r}_{ij} \cdot {\bf n}_i)^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - ({\bf r}_{ij} \cdot {\bf n}_j)^2 \\
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\rho_{ij}^2 = & r_{ij}^2 - (\mathbf{r}_{ij} \cdot \mathbf{n}_i)^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - (\mathbf{r}_{ij} \cdot \mathbf{n}_j)^2 \\
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f(\rho) = & C e^{ -( \rho / \delta )^2 } \\
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\mathrm{Tap}(r_{ij}) = & 20\left ( \frac{r_{ij}}{R_{cut}} \right )^7 -
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70\left ( \frac{r_{ij}}{R_{cut}} \right )^6 +
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@ -67,7 +67,7 @@ calculating the normals.
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normal vectors used for graphene and h-BN is no longer valid for TMDs.
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In :ref:`(Ouyang7) <Ouyang7>`, a new definition is proposed, where for
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each atom `i`, its six nearest neighboring atoms belonging to the same
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sub-layer are chosen to define the normal vector `{\bf n}_i`.
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sub-layer are chosen to define the normal vector `\mathbf{n}_i`.
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The parameter file (e.g. TMD.ILP), is intended for use with *metal*
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:doc:`units <units>`, with energies in meV. Two additional parameters,
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@ -37,8 +37,8 @@ No simplification is made,
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E = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
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V_{ij} = & e^{-\lambda (r_{ij} -z_0)} \left [ C + f(\rho_{ij}) + f(\rho_{ji}) \right ] - A \left ( \frac{r_{ij}}{z_0}\right )^{-6} \\
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\rho_{ij}^2 = & r_{ij}^2 - ({\bf r}_{ij}\cdot {\bf n}_{i})^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - ({\bf r}_{ij}\cdot {\bf n}_{j})^2 \\
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\rho_{ij}^2 = & r_{ij}^2 - (\mathbf{r}_{ij}\cdot \mathbf{n}_{i})^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - (\mathbf{r}_{ij}\cdot \mathbf{n}_{j})^2 \\
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f(\rho) & = e^{-(\rho/\delta)^2} \sum_{n=0}^2 C_{2n} { (\rho/\delta) }^{2n}
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It is important to have a sufficiently large cutoff to ensure smooth
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@ -33,7 +33,7 @@ elemental bulk material in the form
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.. math::
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E_\mathrm{tot}({\bf R}_1 \ldots {\bf R}_N) = NE_\mathrm{vol}(\Omega )
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E_\mathrm{tot}(\mathbf{R}_1 \ldots \mathbf{R}_N) = NE_\mathrm{vol}(\Omega )
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+ \frac{1}{2} \sum _{i,j} \mbox{}^\prime \ v_2(ij;\Omega )
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+ \frac{1}{6} \sum _{i,j,k} \mbox{}^\prime \ v_3(ijk;\Omega )
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+ \frac{1}{24} \sum _{i,j,k,l} \mbox{}^\prime \ v_4(ijkl;\Omega )
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@ -45,8 +45,8 @@ potential (ILP) potential for hetero-junctions formed with hexagonal
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\left [ \epsilon + f(\rho_{ij}) + f(\rho_{ji})\right ] -
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\frac{1}{1+e^{-d\left [ \left ( r_{ij}/\left (s_R \cdot r^{eff} \right ) \right )-1 \right ]}}
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\cdot \frac{C_6}{r^6_{ij}} \right \}\\
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\rho_{ij}^2 = & r_{ij}^2 - ({\bf r}_{ij} \cdot {\bf n}_i)^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - ({\bf r}_{ij} \cdot {\bf n}_j)^2 \\
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\rho_{ij}^2 = & r_{ij}^2 - (\mathbf{r}_{ij} \cdot \mathbf{n}_i)^2 \\
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\rho_{ji}^2 = & r_{ij}^2 - (\mathbf{r}_{ij} \cdot \mathbf{n}_j)^2 \\
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f(\rho) = & C e^{ -( \rho / \delta )^2 } \\
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\mathrm{Tap}(r_{ij}) = & 20\left ( \frac{r_{ij}}{R_{cut}} \right )^7 -
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70\left ( \frac{r_{ij}}{R_{cut}} \right )^6 +
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@ -63,8 +63,8 @@ calculating the normals.
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.. note::
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To account for the isotropic nature of the isolated gold atom
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electron cloud, their corresponding normal vectors (`{\bf n}_i`) are
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assumed to lie along the interatomic vector `{\bf r}_ij`. Notably, this
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electron cloud, their corresponding normal vectors (`\mathbf{n}_i`) are
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assumed to lie along the interatomic vector `\mathbf{r}_ij`. Notably, this
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assumption is suitable for many bulk material surfaces, for
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example, for systems possessing s-type valence orbitals or
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metallic surfaces, whose valence electrons are mostly
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