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.
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. The interactions of the interaction sites are divided into two categories in the following. The interaction potentials are divided in bonded and non-bonded Interactions.

The MolMod database contains rigid and flexible molecular models. The vast majority of force fields is currently of the 'rigid' type. For rigid molecular models, the evidently only non-bonded interactions are applicable (dispersive and repulsive interactions and electrostatic interactions). For flexible molecular models, in addition, bonded interactions are given (stretching bonds, angles, and dihederals). Furthermore, for flexible molecular models, the given Z-matrix is only given as an exemplary equilibrium condition of the molecule that can be used for the initialization in simulations.

# 3.1 Non-Bonded Interactions

The molecular models contained in the MolMod database are built from four types of non-bonded intermolecular interaction sites: Lennard-Jones 12-6, point Charge, point charge and quadrupole.

# 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:

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

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.

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

# 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:

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

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:

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

[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:

$$q=\frac{\mu}{a}=\frac{20\mu}{\sigma}$$

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.

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].

# 3.2 Bonded Interactions

The intramolecular interactions of the molecular models in the MolMod database are described by the bond length, bond angle and the dihedral angle of two bonds.

# Bond

The intramolecular potential of the bond length between two consecutive interaction sites in a flexible molecule is described by the following potential:
$$\begin{eqnarray} u_{\mathrm{bond}}=\frac{k_{\mathrm{bond}}}{2}(l-l_0)^2 \end{eqnarray}$$
where $l$ is the bond length between two interaction sites, $l_0$ is the equilibrium bond length and $k_{\mathrm{bond}}$ is a force constant characterising the effect of a change of the bond length.

# Angle

The intramolecular potential of the bond angle between three consecutive interaction sites in a flexible molecule is described by the following potential:
$$\begin{eqnarray} u_{\mathrm{angle}}=\frac{k_{\mathrm{angle}}}{2}(\alpha-\alpha_0)^2 \end{eqnarray}$$
where $\alpha$ is the bond angle between three interaction sites, $\alpha_0$ is the equilibrium bond angle and $k_{\mathrm{angle}}$ is a force constant characterising the effect of a change of the bond angle.

# Torsion

The intramolecular potential of the dihedral angle between two bonds of four consecutive interaction sites in a flexible molecule is described by the following potential:
$$\begin{eqnarray} u_{\mathrm{tor}}=\sum_{i=0}^5c_i(1+cos(i\phi-\phi_0)) \end{eqnarray}$$
where $\phi$ is the dihedral angle between two bonds of four consecutive interaction sites, $\phi_0$ is the equilibrium dihedral angle and $c_i$ are force constants characterising the effect of a change of the dihedral angle.