How to Calculate Lennard-Jones Parameters from TDDFT?

Question

I am trying to develop parameters for AMBER force fields and to calculate Lennard-Jones parameters for the $\ce{Fe^3+}$ ion. I did a deep search and found that the best way to do this with a high accuracy is to use TDDFT. I found many articles about this, for example (I can put more if there is need):

• Marques, M. A. L.; Castro, A.; Malloci, G.; Mulas, G.; Botti, S. Efficient calculation of van der Waals dispersion coefficients with time-dependent density functional theory in real time: Application to polycyclic aromatic hydrocarbons. The Journal of Chemical Physics 2007, 127 (1), 014107 DOI: 10.1063/1.2746031.

• Banerjee, A.; Autschbach, J.; Chakrabarti, A. Time-dependent density-functional-theory calculation of the van der Waals coefficient $C_6$ of alkali-metal atoms Li, Na, K; alkali-metal dimers $\ce{Li2}$, $\ce{Na2}$, $\ce{K2}$; sodium clusters $\ce{Na_n}$; and fullerene $\ce{C60}$. Phys. Rev. A 2008, 78 (3) DOI: 10.1103/PhysRevA.78.032704.

But, I couldn't understand the articles, because in some there are so many mathematical formulas, in others the method is not clear. So, to get some things clear before starting, I need to answer these questions (let's say I did my TDDFT calculation):

1. After performing TDDFT calculations, are there software or tutorials that I can use to calculate van der Waals parameters?
2. If there is no software or tutorial to calculate this, where can I start?

Answer 1

I'm basing my answer on T. Gould, T. Bucko, J. Chem, Theory Comput. 2016, 12, 3603-3613.

In your TDDFT calculation, what you want to compute is the frequency-dependent dipole polarizability, $\alpha_{X/Y}(i\omega)$, that determines the dispersion coefficient, $C_6$. $$ C_{6,XY} = \frac{3}{\pi}\int d\omega \alpha_X(i\omega)\alpha_Y(i\omega)$$

They also propose some approximate relationships that depend only on the static dipole polarizabilities, $\alpha(0)$. $$ C_{6,XY} \approx \Xi [\alpha_X(0)\alpha_Y(0)]^{0.73\pm 0.01} $$ where $\Xi = (1.5 \pm 0.1)[\text{hartree} \cdot a_0^{1.62} ]$.

I don't know any tutorials for this, but you could start trying to reproduce some of the numbers on these papers.