# How to Calculate Lennard-Jones Parameters from TDDFT?

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?
• I don't have access to the papers mentioned (so I cannot judge their level of detail), but have another reference for you: dx.doi.org/10.1063/1.3382344 – TAR86 Jan 21 at 21:23

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 $$\Xi = (1.5 \pm 0.1)[\text{hartree} \cdot a_0^{1.62} ]$$.