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Update pair_lj_pirani.rst

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Mateo Rodríguez
2025-04-22 21:13:22 +02:00
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@ -46,37 +46,37 @@ Lennard-Jones (ILJ) potential according to :ref:`(Pirani) <Pirani>`:
An additional parameter, :math:`\alpha`, has been introduced in order
to be able to recover the traditional Lennard-Jones (LJ) 12-6 with an adequate
to be able to recover the traditional Lennard-Jones (LJ) 12-6 with a specific
choice of parameters. With :math:`R_m \equiv r_0 = \sigma \cdot 2^{1 / 6}`,
:math:`\alpha = 0`, :math:`\beta = 12` and :math:`\gamma = 6`
it is straightforward to prove that LJ 12-6 is obtained.
This potential provides some advantages with respect to the LJ
potential and can be really useful for molecular dynamics simulations,
as one can see from :ref:`(Pirani) <Pirani>`.
This potential provides some advantages with respect to the standard LJ
potential, as explained in :ref:`(Pirani) <Pirani>`.
It can be used for neutral-neutral (:math:`\gamma = 6`),
ion-neutral (:math:`\gamma = 4`) or ion-ion systems (:math:`\gamma = 1`).
It removes most of the issues at short- and long-range of the LJ model.
These settings remove issues at short- and long-range for these systems when
a standard LJ model is used.
It is possible to verify that using (:math:`\alpha= 4`), (:math:`\beta= 6`)
and (:math:`\gamma = 6`), at the equilibrium distance,
the first and second derivatives of ILJ coincide with those of LJ 12-6
( and the reduced force constant amounts to the typical 72).
In this case, LJ provides a long-range coefficient with a factor of 2 compared
with the ILJ. Also, the short-range interaction is overestimated by LJ.
It is possible to verify that using :math:`\alpha= 4`, :math:`\beta= 6`
and :math:`\gamma = 6`, at the equilibrium distance, the first and second
derivatives of ILJ match those of LJ 12-6. In this case, the standard LJ
energy is two times stronger than ILJ at long distances. Also, strength
of the short-range interaction is overestimated by LJ.
The ILJ potential solves both problems.
The analysis of a diverse amount of systems verified that (:math:`\alpha= 4`)
works very well. In some special cases (e.g. those involving very small
multiple charged ions) this factor may take a slightly different value.
The parameter (:math:`\beta`) codifies the hardness (polarizability) of the
interacting partners, and for neutral-neutral systems it ranges from 6 to 11.
Moreover, the modulation of (:math:`\beta`) permits to indirectly consider the
role of further interaction components (such as the charge transfer in the
perturbative limit) and mitigates the effect of some uncertainty in the data.
As discussed in :ref:`(Pirani) <Pirani>`, analyses of a
variety of systems showed that :math:`\alpha= 4` generally works very well.
In some special cases (e.g. those involving very small multiple charged ions)
this factor may take a slightly different value. The parameter :math:`\beta`
codifies the hardness (polarizability) of the interacting partners, and for
neutral-neutral systems it ranges from 6 to 11. Moreover, the modulation of
:math:`\beta` can model additional interaction effects, such as charge
transfer in the perturbative limit, and can mitigate the effect of some
uncertainty in the data used to build up the potential function.
The following coefficients must be defined for each pair of atoms