changes in index.md and adapt title

feos-org · prehner · Jan 23, 2023 · Jan 19, 2023 · Jan 19, 2023 · Jan 19, 2023
commit a339c6207a10835556f18a69918432905fc82820
diff --git a/docs/theory/dft/index.md b/docs/theory/dft/index.md
@@ -8,6 +8,7 @@ This section explains the implementation of the core expressions from classical
    euler_lagrange_equation
    functional_derivatives
    solver
+   pdgt
 ```
 
 It is currently still under construction. You can help by [contributing](https://github.com/feos-org/feos/issues/70).
diff --git a/docs/theory/dft/pdgt.md b/docs/theory/dft/pdgt.md
@@ -1,21 +1,21 @@
-# Predictive Density Gradient Theory (pDGT)
+# Predictive density gradient theory
 
 Predictive density gradient theory (pDGT)  is an efficient approach for the prediction of surface tensions, which is derived from non-local DFT, see [Rehner, 2018](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.063312). A gradient expansion is applied to the weighted densities of the Helmholtz energy functional to second order as well as to the Helmholtz energy density to first order. 
 
 Weighted densities (in non-local DFT) are determined from
 $$
-	n_\alpha(\mathbf{r})=\sum_in_\alpha^i(\mathbf{r})=\sum_i\int\rho_i(\mathbf{r}- \mathbf{r}')\omega_\alpha^i(\mathbf{r}')\diff\mathbf{r}'.
+	n_\alpha(\mathbf{r})=\sum_in_\alpha^i(\mathbf{r})=\sum_i\int\rho_i(\mathbf{r}- \mathbf{r}')\omega_\alpha^i(\mathbf{r}')\mathrm{d}\mathbf{r}'.
 $$
-These convolutions are time-consuming calculations. Therefore, these equations are simplified by using a Taylor expansion around $\rb$ for the density of each component $\rho_i$ as
+These convolutions are time-consuming calculations. Therefore, these equations are simplified by using a Taylor expansion around $\mathbf{r}$ for the density of each component $\rho_i$ as
 $$
 	\rho_i(\mathbf{r}-\mathbf{r}')=\rho_i(\mathbf{r})-\nabla\rho_i(\mathbf{r})\cdot \mathbf{r}'+\frac{1}{2}\nabla\nabla\rho(\mathbf{r}):\mathbf{r}'\mathbf{r}'+\ldots  
 $$
 
 In the convolution integrals, the integration over angles can now be performed analytically for the spherically symmetric weight functions $\omega_\alpha^i(\mathbf{r})=\omega_\alpha^i(r)$
 which provides
 $$
-	n_\alpha^i(\mathbf{r})=\rho_i(\mathbf{r})\underbrace{4\pi\int_0^\infty \omega_\alpha^i(r)r^2\diff r}_{\omega_\alpha^{i0}}
-	+\nabla^2\rho_i(\mathbf{r})\underbrace{\frac{2}{3}\pi\int_0^\infty\omega_\alpha^i(r)r^4\diff r}_{\omega_\alpha^{i2}}+\ldots
+	n_\alpha^i(\mathbf{r})=\rho_i(\mathbf{r})\underbrace{4\pi\int_0^\infty \omega_\alpha^i(r)r^2\mathrm{d} r}_{\omega_\alpha^{i0}}
+	+\nabla^2\rho_i(\mathbf{r})\underbrace{\frac{2}{3}\pi\int_0^\infty\omega_\alpha^i(r)r^4\mathrm{d} r}_{\omega_\alpha^{i2}}+\ldots
 
 $$
 with the weight constants $\omega_\alpha^{i0}$ and $\omega_\alpha^{i2}$. 
@@ -29,14 +29,14 @@ $$
 The second simplification is the expansion of the reduced residual
 Helmholtz energy density $\phi(\{ n_\alpha\})$ around the local density approximation truncated after the second term
 $$
-	\Phi(\lbrace n_\alpha\rbrace)
-	=\Phi(\lbrace n_\alpha^0\rbrace)
+	\Phi(\lbrace n_\alpha\mathbf{r}race)
+	=\Phi(\lbrace n_\alpha^0\mathbf{r}race)
 	+\sum_i\sum_\alpha\frac{\partial\Phi}{\partial n_\alpha}\omega_\alpha^{i2}\nabla^2\rho_i + \ldots
 
 $$
 The Helmholtz energy functional (which was introduced in the section about the \href{www.euler-lagrange-equation.de}{Euler-Lagrange equation}) then reads
 $$
-	F[\bm{\rho}(\mathbf{r})]=\int\left(f(\bm{\rho})+\sum_{ij}\frac{c_{ij}}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\diff\mathbf{r}
+	F[\bm{\rho}(\mathbf{r})]=\int\left(f(\bm{\rho})+\sum_{ij}\frac{c_{ij}}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\mathrm{d}\mathbf{r}
 $$
 with the density dependent influence parameter
 $$