Molecular Models

3 Interaction Potentials

All models in the database ”Molecular Models of the Boltzmann-Zuse Society” consist of Lennard-Jones 12-6 interaction sites and a variation of point charges, point-dipoles and point quadrupoles. The last two are computationally much cheaper compared to the corresponding configuration of point charges.

Lennard-Jones 12-6

Repulsion and dispersion interaction between two particles $i,j$ of the same kind at a distance $r$ is modelled throughout the database by the standard Lennard-Jones 12-6 potential:

\begin{equation} u_{ij}^\mathrm{LJ}(r)=4\varepsilon\left[ \left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right]. \end{equation}

The potential model itself consists of two parts – the first part with the positive sing represents the repulsion and the negative the attraction. The potential has two parameters: The size parameter $\sigma$ with a dimension of length defines the distance where the potential energy is zero and the energy-parameter $\varepsilon$, which defines the depth of the potential and thereby sets the dispersion energy.

Figure 2: Lennard-Jones potential between two particles.

For unlike interactions – the interaction of two sites with different $\varepsilon$ and/ or $\sigma$ – mixing rules can be applied. The parameters $\eta_{kl}$ and $\xi_{kl}$ in eq. (2) and (3) are used to correct the binary interaction parameters of the components $k$ and $l$ (if necessary). $\eta_{kl}$ and $\xi_{kl}$ are mostly a constant for a certain mixture. The extensive study of the influence of different mixing rules by Schnabel et al. [Schnabel, 2007 C] showed, that mixture bubble densities are accurately obtained using the arithmetic mean of the two size parameters $\sigma_k,~\sigma_l$ as proposed by the Lorentz combining rule (3). That results in $\eta_{kl}=1$ being a very accurate description of the unlike size parameter. The vapor pressure turns out to be dependent on both unlike Lennard-Jones parameters. It was therefore recommended by Schnabel et al. to adjust the unlike Lennard-Jones energy parameter to the vapour pressure.

\begin{equation} \sigma_{kl}=\eta_{kl}\frac{\sigma_k+\sigma_l}{2}\label{eq:sigma_combination}\\ \end{equation}
\begin{equation} \varepsilon_{kl}=\xi_{kl}\sqrt{\varepsilon_k\varepsilon_l} \end{equation}

Point Charge

Point charges are first order electrostatic interaction sites. These sites are indicated in the database with an ’e’. The electrostatic interaction between two point charges $q_i$ and $q_j$ is given by Coulomb’s law:

\begin{equation} u^\mathrm{ee}_{ij}(r_{ij})=\frac{1}{4\epsilon_0\pi}\frac{q_iq_j}{r_{ij}} \end{equation}

with $q$ the magnitude of the charge and $r_{ij}$ the distance between two charges.

Figure 3: Coulomb potential between two point charges.

Dipole

A point dipole describes the electrostatic field of two point charges with equal magnitude, but opposite sign at a mutual distance $a\rightarrow 0$. Point dipole interaction sites are labeled throughout the database with a '$d$'. The magnitude of a dipole moment is defined by $\mu=qa$, where $q$ is the magnitude of the two point charges. The electrostatic interaction between two point dipoles with the moments $\mu_i$ and $\mu_j$ at a distance $r_{ji}$ is given by:

\begin{equation} u_{ij}^\mathrm{dd}(r_{ij},\theta_i,\theta_j,\phi_{ij},\mu_i,\mu_j)=\frac{1}{4\pi\epsilon_0}\frac{\mu_i\mu_j}{r^3_{ij}}\left[(\sin\theta_i \sin\theta_j \cos\phi_{ij} -2\cos\theta_i \cos\theta_j\right], \end{equation}

[Gray, 1984] where the angles $\theta_i$, $\theta_j$ and $\phi_{ij}$ indicate the relative angular orientation of the two point dipoles with $\theta$ being the angle between the dipole direction and the distance vector of the two interacting dipoles and $\phi_{ij}$ being the azimuthal angle of the two dipole directions, cf. Fig. 4.

Figure 4: Scheme of the angles $\theta_i$, $\theta_j$ and $\phi_{ij}$ indicating the relative angular orientation of the two point dipole $i$ and $j$, which are situated at a distance $r_{ij}$.The orientation of the different dipoles are indicated by arrows.

Since some simulation programs cannot handle point dipoles explicitly, point dipoles are on the fly converted into dipoles, which are assembled by two point charges; equivalently, a point dipole can be replaced by two point charges. Note that the dipole moment is defined by $\mu=qa$. This results in two parameters that need to be determined: The distance between the two charges $a$ and the magnitude of the charges $q$. This problem is addressed in the work of Engin et al. ([Engin, 2011 B]) for the molecular model class of 2CLJD and 2CLJQ, which is illustrated in Fig. 5. Engin et al. are proposing to set the distance $a$ between the two point charges to ${\sigma}/{20}$ in [Engin, 2011 B], where $\sigma$ is the size parameter of the Lennard-Jones site. The magnitude of the two point charges $q$ is than straightforwardly computed as:

\begin{equation} q=\frac{\mu}{a}=\frac{20\mu}{\sigma} \end{equation}

Figure 5: Model class of 2CLJD and 2CLJQ, which was investigated by Engin et al. in [Engin, 2011 B]: The models consist of two equal Lennard-Jones sites located at a specified distance from each other and a massless dipole or quadrupole in the center of mass whose orientation is parallel to the distance vector between the two Lennard Jones sites.

The method proposed by Engin et al. in [Engin, 2011 B] was extended in the database to arbitrary molecular structures by calculating the parameter $a$ by means of the Lennard-Jones interaction site closest to the point dipole according to the Euclidean norm.

Quadrupole

Figure 6: Charge distribution of a linear quadrupole.

A linear point quadrupole describes the electrostatic field (cf. Fig. 6) induced either by two collinear point dipoles with the same moment, but opposite orientation at a distance $d\rightarrow 0$ or three point charges. Point quadrupole interaction sites are labeled as '$q$' in the database. The magnitude of a point quadrupole $Q$ is defined as $Q=2qd^2$, where $q$ is the magnitude of the three similar charges and $d$ their distance (cf. Fig. 6 or 7). Note that the central charge has twice the magnitude as the edge charges in a linear quadrupole, cf. Fig. 6 or 7. The interaction potential is given by:

$$\begin{eqnarray} u_{ij}^\mathrm{qq}(r_{ij},\theta_i,\theta_j,\phi_{ij},Q_i,Q_j) = \frac{1}{4\pi\epsilon_0}\frac{3}{4}\frac{Q_iQ_j}{r^5_{ij}}\left[1-5((\cos\theta_i)^2+\cos(\theta_j)^2) \\[5pt] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-15(\cos\theta_i)^2(\cos\theta_j)^2+2(\sin\theta_i \sin\theta_j \cos\phi_{ij} -4\cos\theta_i \cos\theta_j)^2\right], \end{eqnarray}$$

where the angles $\Theta_i$, $\Theta_j$ and $\Phi_{ij}$ indicate the relative angular orientation of the two point quadrupoles (cf. Fig. 4).

Figure 7: Arrangement of the point charges in a linear quadrupole (for a positive quadrupole moment), where the arrow describes the orientation of the quadrupole. In the case of a negative quadrupole moment, the signs of the point charges are inverted.

Similar to the point dipoles, not all simulation programs can handle explicitly point quadrupoles. Therefore point quadrupoles are converted into quadrupoles constituted of point charges. This results in a problem similar to the conversion of dipoles, which was also addressed byEngin et al. in [Engin, 2011 B]. The quadrupole conversion is analogous to the dipole conversion by means of the value $a=\sigma/20$ proposed by Engin et al. in [Engin, 2011 B].