Are dispersion constant calculated differently for elements that are heavier than Xe?

Question

How does one obtain the value of dispersion coefficient $C_6$ for elements that come after Xe. I am going through the original paper [1]. I can see that: $$\tag{1}C^a_6 = 0.05N(I^a \alpha^a),$$ where $\alpha^a$ is the dipole polarization and $I^a$ is ionization potential. This does not seem that complicated but in the subsequent paragraphs they tend to complicate things. Where they are talking about if the element in the compounds vary a lot from free elements then we need to derive coefficients from molecular properties and there is quite a lot of other things too.

Is there any simpler way to go about doing this?

Answer 1

There's many ways to calculate a $C_6$ value, for which it doesn't matter whether the element is heavier or lighter than Xe.

One of the simplest equations for $C_6$ is in the abstract of a 1931 paper by Slater and Kirkwood (often called the Slater-Kirkwood approximation):

Therefore we have (in the notation of Brandt in 1955, and probably others between 1931-1955):

$$\tag{1} C_6 = -\frac{3}{4}\left(a_0N\alpha^3\right)^{1/2}e^2, $$

with $a_0$ being the Bohr radius, $N$ being the number of electrons in the molecular valence shell, $\alpha$ being the average dipole polarizability, and $e$ being the charge of an electron.

For a heteronuclear system another simple formula is given here:

$$\tag{2} C_6^{AB} = -\frac{3}{2}\frac{I_AI_B}{I_A+I_B}\alpha_A\alpha_B, $$

with $I_A$ and $I_B$ being the first ionization energies for the systems $A$ and $B$, and $\alpha_A$ and $\alpha_B$ being again their average dipole polarizabilities.

You are correct that other formulas exist, for example, I found at least two in this paper:

I tried to turn the question What are the different ways of calculating dispersion constants? into a list of all the known methods to approximate $C_6$, but I wanted it to be in the form of one-topic-per-answer with each user explaining just one method (otherwise you might get 10 answers from me, which I think would look weird).

Beyond the element Xe, relativistic effects will become quite important when attempting to calculate the dipole polarizability $\alpha$ or the ionization energy $I$ using ab initio methods. Another famous relativistic effect was described in this 1948 paper by Casimir and Polder, in which they showed that for very large distances the $C_6$ actually becomes a $C_7$.

Answer 2

You already have a potentially good answer from Nike Dattani, however I might frame this as an entirely different question. Unless you have a good reason to determine you own dispersion constants / corrections, can you just use an existing dispersion correction such as DFT-D4 (rather than D2). Dispersion should work with any element on the periodic table, as of this commit.

You can interface this onto any calculator using the ASE-D4 calculator if you need to. This might solve your issue from a practical standpoint, even if this isn't your original question.