# 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 $j$)

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