Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.6 The variety of interaction potentials 459
LJ
LJ with 2 parameters: E () R = A
AB
12 6
⎛ 1⎞⎛1⎞ ⎜ ⎟ − C ⎜ ⎟
⎝ R⎠
⎝ R⎠
AB AB
[8.71]
LJ expressions are of extensive use in MM to model non-covalent interactions among
molecular sites, often with the addition of local charges (potential 14)
LJ with charge: E ( r ) = A
ms ms ms
12 6
⎛ ⎞
q q
⎜
⎟ Cms
r
⎟
⎝ ms ⎠ rms
rms
−
⎛ ⎞
⎜
⎟
⎝ ⎠
+
1 1
m s
[8.72]
Some potentials add an effective dielectric constant in the denominator of the Coulomb
term to mimic polarization effects.
The Buckhingam potential is similar to LJ with a more physical description of the repulsion
term given in terms of a decaying exponential function:
E
( BU )
AB
r
r
= exp −
− r * r *
⎛ ⎡ ⎞ ⎤
⎢ ⎜ ⎟− ⎥
⎣ ⎝ ⎠ ⎦
−
⎧⎪
6
α ⎛ ⎞
ε⎨ α 1 ⎜ ⎟
⎩⎪α
6
α −6⎝
⎠
−6
⎫⎪
⎬
⎭⎪
[8.73]
where:
ε depth of the attractive well
r RAB r* distance of the minimum
α numerical parameter
The difference in the computational costs of LJ and Buckingham potentials is related
to the difference between computing the square of a value already available (R -12 from R -6 )
and that of computing an exponential. Here again the increment in computational costs of
simulations due to small changes in the potential plays a significant role. Also, the
Buckingham + charge potential is used to describe liquids.
The last potential of Table 8.5 is specialized for water. It has been enclosed in the table
to document the progress in one-site potentials; in this case, the LJ + dipole is supplemented
by a short-range “sticky” tetrahedral interaction.
The listing of one-center potentials is not exhausted by the examples given in Table
8.5. We have, for example, neglected all the potentials in use for rare gases systems in which
much attention is paid to using higher terms to describe the dispersion contribution. Our aim
was just to show with a few examples how it is possible to define a large variety of potentials
remaining within the constraints of using a single center.
The problem of giving a cursory but significant enough view of potentials becomes
harder when one passes to many-site potentials. In Table 8.6 we report some analogues of
HS with a more complex shape.
They consist of fused regular forms (spheres or combination of spheres with other regular
solids). The hard version of these potentials is accompanied by soft modifications and
by versions including dipole, charges, as in Table 8.5.
A step further along this way is given by a potential composed by spheres linked by
‘spacers’ with a constant length but allowing changes of conformation with appropriate torsion
potentials. These potentials are used for polymers or for molecules having long hydrocarbon
chains. This is not, however, the main trend in the evolution of potentials.