refer to temperature more consistently with :math:T instead of *T*
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@ -41,20 +41,21 @@ The pressure is computed by the formula
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P = \frac{N k_B T}{V} + \frac{1}{V d}\sum_{i=1}^{N'} \vec r_i \cdot \vec f_i
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where *N* is the number of atoms in the system (see discussion of DOF
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below), :math:`k_B` is the Boltzmann constant, *T* is the temperature, *d*
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is the dimensionality of the system (2 for 2d, 3 for 3d), and *V* is the
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system volume (or area in 2d). The second term is the virial, equal to
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:math:`-dU/dV`, computed for all pairwise as well as 2-body, 3-body, 4-body,
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many-body, and long-range interactions, where :math:`\vec r_i` and
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:math:`\vec f_i` are the position and force vector of atom *i*, and the
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dot indicates the dot product (scalar product). This is computed in parallel
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for each sub-domain and then summed over all parallel processes. Thus
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:math:`N'` necessarily includes atoms from neighboring sub-domains (so-called
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ghost atoms) and the position and force vectors of ghost atoms are thus
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included in the summation. Only when running in serial and without
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periodic boundary conditions is :math:`N' = N` the number of atoms in the
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system. :doc:`Fixes <fix>` that impose constraints (e.g., the :doc:`fix
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shake <fix_shake>` command) may also contribute to the virial term.
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below), :math:`k_B` is the Boltzmann constant, :math:`T` is the
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temperature, *d* is the dimensionality of the system (2 for 2d, 3 for
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3d), and *V* is the system volume (or area in 2d). The second term is
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the virial, equal to :math:`-dU/dV`, computed for all pairwise as well
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as 2-body, 3-body, 4-body, many-body, and long-range interactions, where
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:math:`\vec r_i` and :math:`\vec f_i` are the position and force vector
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of atom *i*, and the dot indicates the dot product (scalar product).
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This is computed in parallel for each sub-domain and then summed over
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all parallel processes. Thus :math:`N'` necessarily includes atoms from
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neighboring sub-domains (so-called ghost atoms) and the position and
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force vectors of ghost atoms are thus included in the summation. Only
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when running in serial and without periodic boundary conditions is
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:math:`N' = N` the number of atoms in the system. :doc:`Fixes <fix>`
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that impose constraints (e.g., the :doc:`fix shake <fix_shake>` command)
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may also contribute to the virial term.
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A symmetric pressure tensor, stored as a 6-element vector, is also
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calculated by this compute. The six components of the vector are
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@ -52,7 +52,7 @@ concentration of species *j* in particle *i*, :math:`u_j` is the
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internal energy of species j, :math:`\Delta H_{f,j} is the heat of
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formation of species *j*, N is the number of molecules represented
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by the coarse-grained particle, :math:`k_B` is the Boltzmann constant,
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and *T* is the temperature of the system. Additionally, it is
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and :math:`T` is the temperature of the system. Additionally, it is
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possible to modify the concentration-dependent particle internal
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energy relation by adding an energy correction, temperature-dependent
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correction, and/or a molecule-dependent correction. An energy
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@ -261,7 +261,7 @@ pressure of the fictitious gas reservoir by:
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\mu^{id} = & k T \ln{\rho \Lambda^3} \\
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= & k T \ln{\frac{\phi P \Lambda^3}{k_B T}}
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where :math:`k_B` is the Boltzmann constant, *T* is the user-specified
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where :math:`k_B` is the Boltzmann constant, :math:`T` is the user-specified
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temperature, :math:`\rho` is the number density, *P* is the pressure,
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and :math:`\phi` is the fugacity coefficient. The constant
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:math:`\Lambda` is required for dimensional consistency. For all unit
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@ -320,7 +320,7 @@ this will ensure roughly the same behavior whether or not the
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*full_energy* option is used.
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Inserted atoms and molecules are assigned random velocities based on
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the specified temperature *T*. Because the relative velocity of all
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the specified temperature :math:`T`. Because the relative velocity of all
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atoms in the molecule is zero, this may result in inserted molecules
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that are systematically too cold. In addition, the intramolecular
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potential energy of the inserted molecule may cause the kinetic energy
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@ -81,11 +81,11 @@ the particle's velocity. The proportionality constant for each atom is
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computed as :math:`\frac{m}{\mathrm{damp}}`, where *m* is the mass of the
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particle and damp is the damping factor specified by the user.
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:math:`F_r` is a force due to solvent atoms at a temperature *T*
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:math:`F_r` is a force due to solvent atoms at a temperature :math:`T`
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randomly bumping into the particle. As derived from the
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fluctuation/dissipation theorem, its magnitude as shown above is
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proportional to :math:`\sqrt{\frac{k_B T m}{dt~\mathrm{damp}}}`, where
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:math:`k_B` is the Boltzmann constant, *T* is the desired temperature,
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:math:`k_B` is the Boltzmann constant, :math:`T` is the desired temperature,
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*m* is the mass of the particle, *dt* is the timestep size, and damp is
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the damping factor. Random numbers are used to randomize the direction
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and magnitude of this force as described in :ref:`(Dunweg) <Dunweg1>`,
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@ -88,20 +88,23 @@ target temperature Tt obtained from the following equation:
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T_t - T = \frac{\left(\frac{1}{2}\left(P + P_0\right)\left(V_0 - V\right) + E_0 - E\right)}{N_{dof} k_B } = \Delta
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where *T* and :math:`T_t` are the instantaneous and target temperatures,
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*P* and :math:`P_0` are the instantaneous and reference pressures or axial stresses,
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depending on whether hydrostatic or uniaxial compression is being
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performed, *V* and :math:`V_0` are the instantaneous and reference volumes,
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*E* and :math:`E_0` are the instantaneous and reference internal energy (potential
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plus kinetic), :math:`N_{dof}` is the number of degrees of freedom used in the
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definition of temperature, and :math:`k_B` is the Boltzmann constant. :math:`\Delta` is the
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negative deviation of the instantaneous temperature from the target temperature.
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When the system reaches a stable equilibrium, the value of :math:`\Delta` should
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fluctuate about zero.
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where :math:`T` and :math:`T_t` are the instantaneous and target
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temperatures, *P* and :math:`P_0` are the instantaneous and reference
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pressures or axial stresses, depending on whether hydrostatic or
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uniaxial compression is being performed, *V* and :math:`V_0` are the
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instantaneous and reference volumes, *E* and :math:`E_0` are the
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instantaneous and reference internal energy (potential plus kinetic),
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:math:`N_{dof}` is the number of degrees of freedom used in the
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definition of temperature, and :math:`k_B` is the Boltzmann
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constant. :math:`\Delta` is the negative deviation of the instantaneous
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temperature from the target temperature. When the system reaches a
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stable equilibrium, the value of :math:`\Delta` should fluctuate about
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zero.
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The values of :math:`E_0`, :math:`V_0`, and :math:`P_0` are the instantaneous values at the start of
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the simulation. These can be overridden using the fix_modify keywords *e0*,
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*v0*, and *p0* described below.
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The values of :math:`E_0`, :math:`V_0`, and :math:`P_0` are the
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instantaneous values at the start of the simulation. These can be
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overridden using the fix_modify keywords *e0*, *v0*, and *p0* described
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below.
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----------
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@ -73,11 +73,11 @@ the particle's velocity. The proportionality constant for each atom is
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computed as :math:`\frac{m}{\mathrm{damp}}`, where *m* is the mass of
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the particle and damp is the damping factor specified by the user.
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:math:`F_r` is a force due to solvent atoms at a temperature *T*
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:math:`F_r` is a force due to solvent atoms at a temperature :math:`T`
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randomly bumping into the particle. As derived from the
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fluctuation/dissipation theorem, its magnitude as shown above is
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proportional to :math:`\sqrt{\frac{k_B T m}{dt~\mathrm{damp}}}`, where
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:math:`k_B` is the Boltzmann constant, *T* is the desired temperature,
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:math:`k_B` is the Boltzmann constant, :math:`T` is the desired temperature,
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*m* is the mass of the particle, *dt* is the timestep size, and damp is
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the damping factor. Random numbers are used to randomize the direction
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and magnitude of this force as described in :ref:`(Dunweg) <Dunweg5>`,
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@ -57,12 +57,12 @@ multiple times to adjust :math:`\gamma` for several atom types.
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self-consistent.
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In a Brownian dynamics context, :math:`\gamma = \frac{k_B T}{D}`, where
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:math:`k_B =` Boltzmann's constant, *T* = temperature, and *D* = particle
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diffusion coefficient. *D* can be written as :math:`\frac{k_B T}{3 \pi
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\eta d}`, where :math:`\eta =` dynamic viscosity of the frictional fluid
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and d = diameter of particle. This means :math:`\gamma = 3 \pi \eta d`,
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and thus is proportional to the viscosity of the fluid and the particle
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diameter.
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:math:`k_B =` Boltzmann's constant, :math:`T` = temperature, and *D* =
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particle diffusion coefficient. *D* can be written as :math:`\frac{k_B
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T}{3 \pi \eta d}`, where :math:`\eta =` dynamic viscosity of the
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frictional fluid and d = diameter of particle. This means :math:`\gamma
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= 3 \pi \eta d`, and thus is proportional to the viscosity of the fluid
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and the particle diameter.
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In the current implementation, rather than have the user specify a
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viscosity, :math:`\gamma` is specified directly in force/velocity units.
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@ -102,9 +102,9 @@ The excess chemical potential mu_ex is defined as:
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\mu_{ex} = -kT \ln(<\exp(-(U_{N+1}-U_{N})/{k_B T})>)
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where :math:`k_B` is the Boltzmann constant, *T* is the user-specified temperature,
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U_N and U_{N+1} is the potential energy of the system with N and N+1
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particles.
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where :math:`k_B` is the Boltzmann constant, :math:`T` is the
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user-specified temperature, :math:`U_N` and :math:`U_{N+1}` is the
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potential energy of the system with :math:`N` and :math:`N+1` particles.
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The *full_energy* option means that the fix calculates the total
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potential energy of the entire simulated system, instead of just the
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@ -43,7 +43,7 @@ Style *ufm* computes pairwise interactions using the Uhlenbeck-Ford model (UFM)
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where :math:`r_c` is the cutoff, :math:`\sigma` is a distance-scale and
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:math:`\epsilon` is an energy-scale, i.e., a product of Boltzmann constant
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:math:`k_B`, temperature *T* and the Uhlenbeck-Ford p-parameter which
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:math:`k_B`, temperature :math:`T` and the Uhlenbeck-Ford p-parameter which
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is responsible
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to control the softness of the interactions :ref:`(Paula Leite2017) <PL1>`.
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This model is useful as a reference system for fluid-phase free-energy calculations :ref:`(Paula Leite2016) <PL2>`.
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