Which equation makes it possible to calculate the forces of attraction between a surface and molecules?

Question

Which equation makes it possible to calculate the forces of attraction between a surface and molecules? Is there any free software that implements such a calculation?

[Clarification 1] Between a surface and molecules there is no chemical reaction, no absorption, adsorption may be but unlikely.

[Clarification 2] Between a surface and molecules of another substance I expect something like adhesion. Such a property is possessed by, for example, honey, clay, glue as well. Undoubtedly some substances may be corrosive in interaction with the surface, but I expect that their being corrosive will be very weakly and appears in a very long time. Thus, I guess, corrosivity can be neglected if to consider not so long time intervals. I suppose that this aspect will allow to simplify a model of this interaction. Another question is how big is a contribution of this aspect into a general force fond? An interesting question is as well how big is a contribution of hydrogen bonds if any?

Answer 1

The long-range (non-chemical) interaction between a neutral molecule and a surface is dominated by van der Waals interactions. The simplest model you could use for this is a Lennard-Jones potential, which captures the approximate long-range $R^{-6}$ dependence of the attractive potential. This is an approximation, of course, that tries to model the effective attraction between fluctuating electrical multipoles.

As summarized by Tao, Perdew, and Ruzsinszky, the van der Waals energy can be written as series expansion

$$E_{vdw} = -\frac{C_6}{R^6} -\frac{C_8}{ R^8} - \frac{C_{10}}{R^{10}} - \dots \tag{1} $$

where “$C_6$ describes the instantaneous dipole-dipole interaction, $C_8$ describes the dipole-quadrupole interaction, and $C_{10}$ describes the quadrupole-quadrupole and dipole-octupole interactions.” For many applications, the leading $C_6$ term is often considered to be sufficient and many models stick with this. The higher order $C_8$ and $C_{10}$ can be important, but they’re much harder to compute and can’t be directly measured so they are somewhat scarce in the literature. Many of the density functional approximations that are designed to accurately reproduce van der Waals interactions incorporate $C_6$ in some way or another. This is especially prominent in Grimme’s DFT-D methods, for example.

Most planewave DFT codes implement VDW functionals, so you could use something like Quantum ESPRESSO, CP2K (this actually uses the GPW method), or VASP (if you have a license) to compute these interactions. Alternatively, if you have validated force field parameters and want to study larger numbers of molecules interacting with a surface, you can use an MD code like LAMMPS or GROMACS to run classical simulations.

Answer 2

The Schrödinger (or relativistic Dirac) equation is what essentially determine the interaction, and there are many free and open source programs that can perform such calculations; a recent listing can be found in our open access WIREs CMS paper. Since you are dealing with gas phase, atomic-orbital approaches will be likely most cost efficient for this as periodicity will only have to be included in the two slab directions.

However, running such calculations can be quite costly and requires much expertise to be able to set up the calculation properly. One needs to not only choose the right quantum chemical model (e.g. density functional), but also converge the many parameters of the calculation (e.g. basis set, slab depth, k point sampling, etc).

Depending on the calculation, force field methods may be a more suitable approach, as they tend to be somewhat easier to use and much faster to run calculations with. For instance, the LAMMPS code might have suitable force fields for such calculations; however, if no suitable force fields exist, one may still have to resort to electronic structure methods.