Exploring Intrinsic Bond Properties with the Fukui Matrix from Conceptual Density Matrix Functional Theory

We extend the traditional conceptual density functional theory (CDFT) to conceptual density matrix functional theory (CDMFT) by replacing the external potential v(r) by the one-electron integral h_rs in the energy functional. This approach provides a new path for investigating intrinsic bond properties such as bond reactivity. The derivation of the Fukui matrix, i.e., derivative of the density matrix P with respect to the number of electrons N, is elucidated, and the result is illustrated in a case study on H₂O. The matrix is shown to play a crucial role in quantifying changes of bond strength for electron removal or addition processes via the bond order derivative ${\left(\frac{\partial B}{\partial N}\right)}^{-}$. Using the Mayer bond order and different atoms-in-molecules partitioning methods, we show that as a first-order response quantity, the bond order derivative agrees well with the finite difference bond order changes. The bond order derivative (bond Fukui function) is a bond reactivity descriptor. We demonstrate this by predicting the regioselectivity of a classical electrophilic addition reaction (the bromination of alkenes) and predicting the initial electron-driven bond cleavage in mass spectrometry. Specifically, the bond order derivative captures all of the major signals from the experimental mass spectra for a series of small molecules with a variety of functional groups.