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Update pair_lj_pirani.rst

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Mateo Rodríguez
2025-04-23 19:34:06 +02:00
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@ -30,7 +30,12 @@ Description
.. versionadded:: TBD .. versionadded:: TBD
Pair style *lj/pirani* computes pairwise interactions from an Improved Pair style *lj/pirani* computes pairwise interactions from an Improved
Lennard-Jones (ILJ) potential according to :ref:`(Pirani) <Pirani>`: Lennard-Jones (ILJ) potential according to :ref:`(Pirani) <Pirani>`.
The ILJ force field is adequate to model both equilibrium and
non-equilibrium properties of matter, in gaseous and condensed phases,
and at gas-surface interfaces. In particular, its use improves the
description of elementary process dynamics where the traditional
Lennard-Jones (LJ) formulation is usually applied.
.. math:: .. math::
@ -46,37 +51,40 @@ Lennard-Jones (ILJ) potential according to :ref:`(Pirani) <Pirani>`:
An additional parameter, :math:`\alpha`, has been introduced in order An additional parameter, :math:`\alpha`, has been introduced in order
to be able to recover the traditional Lennard-Jones (LJ) 12-6 with a specific to be able to recover the traditional Lennard-Jones 12-6 with a specific
choice of parameters. With :math:`R_m \equiv r_0 = \sigma \cdot 2^{1 / 6}`, 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` :math:`\alpha = 0`, :math:`\beta = 12` and :math:`\gamma = 6`
it is straightforward to prove that LJ 12-6 is obtained. it is straightforward to prove that LJ 12-6 is obtained. Also, it can be
verified that using :math:`\alpha= 4`, :math:`\beta= 8` and
:math:`\gamma = 6`, at the equilibrium distance, the first and second
derivatives of ILJ match those of LJ 12-6. The parameter :math:`R_m`
corresponds to the equilibrium distance and :math:`\epsilon` to the
well depth.
This potential provides some advantages with respect to the standard LJ This potential provides some advantages with respect to the standard LJ
potential, as explained in :ref:`(Pirani) <Pirani>`. potential, as explained in :ref:`(Pirani) <Pirani>`: it provides a more
It can be used for neutral-neutral (:math:`\gamma = 6`), realistic description of the long range behaviour and an attenuation of
the hardness of the repulsive wall.
This force field can be used for neutral-neutral (:math:`\gamma = 6`),
ion-neutral (:math:`\gamma = 4`) or ion-ion systems (:math:`\gamma = 1`). ion-neutral (:math:`\gamma = 4`) or ion-ion systems (:math:`\gamma = 1`).
These settings remove issues at short- and long-range for these systems when Notice that this implementation does not include possible electrostatic
a standard LJ model is used. interactions which should be eventually added by means of a hybrid style
(e.g. :doc:`pair_style hybrid/overlay <pair_hybrid_overlay>` or variants).
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.
As discussed in :ref:`(Pirani) <Pirani>`, analyses of a As discussed in :ref:`(Pirani) <Pirani>`, analyses of a
variety of systems showed that :math:`\alpha= 4` generally works very well. 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) 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` this factor may take a slightly different value. The parameter
codifies the hardness (polarizability) of the interacting partners, and for :math:`\beta` codifies the hardness (polarizability) of the interacting
neutral-neutral systems it ranges from 6 to 11. Moreover, the modulation of partners, and for neutral-neutral systems it usually ranges from 6 to 11.
:math:`\beta` can model additional interaction effects, such as charge Moreover, the modulation of :math:`\beta` can model additional interaction
transfer in the perturbative limit, and can mitigate the effect of some effects, such as charge transfer in the perturbative limit, and can
uncertainty in the data used to build up the potential function. 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 The following coefficients must be defined for each pair of atoms