# Atom-condensed softness matrix

In the ACKS2 polarizable force field paper, I found a thing called the atom-condensed softness matrix. In another paper, I found this expression for it:

$$\chi_{kl} = 2 \sum_{i}^{\text{occ MOs}} \sum_{j}^{\text{unocc MOs}} \frac{\langle\psi_{i}|g_{k}|\psi_{j}\rangle\langle\psi_{j}|g_{l}|\psi_{i}\rangle}{\epsilon_{i} - \epsilon_{j}} \delta_{\sigma_{i}\sigma_{j}},$$ where

• $\epsilon_{i}$: orbital energy of the $i$th KS orbital
• $\chi$: (non-interacting) response matrix
• $\psi_{i}$: spatial orbital
• $g_{i}$: potential basis function
• What is the physical meaning of it, or at least what information we can get from this matrix?

• If in KS-DFT we consider the system as non-interacting, why do we consider interaction between two species in this equation? (if I understand it right - between molecules $i$ and $j$)

## 1 Answer

Your first misconception is what i and j mean. These are indices of molecular orbitals.

A response matrix is something used in respone theory. It stems from Perturbation Theory.

Response Theory is a mathematical formalism to compute time-dependent molecular properties (in theoretical chemistry, that is).

This pdf (no guarantee for anything) describes it pretty well and concise. http://www.lct.jussieu.fr/pagesperso/toulouse/enseignement/molecular_properties.pdf

• Thank you very much. One more question then - in calculations I get matrix size of 10x10 for every atom in molecule (i.e. for ethylene it would be 60x60) - does this depend on basis function used and does every matrix element represent interaction between different orbitals (for s, px, py, pz, d etc.)? – cinnamon Nov 8 '16 at 13:31
• @BrigitaP. How many MO do you have? How big is your basis set for every atom? Over what does the matrix iterate? What integrals are being calculated there? The answers to these questions should answer yours. I think you would help yourself the most if you get a book about Computer Chemistry (Jensen maybe) and read a little bit in there. – Fl.pf. Nov 8 '16 at 15:15