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Towards Expanding Conceptual Density Functional Theory to Include New External Perturbations:

Extreme Pressure Polarizable Continuum Model Applied to Atoms

By: J. Eeckhoudt, T. Bettens, P. Geerlings,R. Cammi, B. Chen, M. Alonsoa and F. De Proft

Since its conception, conceptual density functional theory (CDFT) has given rigorous definitions to often vaguely defined chemical concepts which can be numerically evaluated, thus providing a framework to study chemical reactivity in. In this way, concepts such as the electronegativity and Pearson’s hard and soft acids and bases (HSAB) principle have been elucidated and the CDFT response functions have been used in analyzing the selectivity of reactions, a range of spectroscopic properties and even more complex phenomena using QSPR/QSAR and machine learning. It is in this field that the ALGC group at the VUB has long been involved, supported by an international network of collaborators and the HYDRA Tier-2 facility by the VSC.

In recent years, several extensions to the traditional expansion in the number of electrons N and the external potential v have been at stake including electric fields, magnetic fields, temperature, external forces, confinement and now pressure. The response functions to these perturbations expand the framework to describe more specific chemical behavior and rationalize reactivity under a variety of conditions. As an expansion to previous studies on confinement, we now introduce pressure into the mix through the recent extreme pressure polarizable continuum model (XP-PCM) and evaluate its influence and role in conceptual DFT.

It is at this point that the added value of the VSC infrastructure is highlighted. To calculate the CDFT properties of even just atoms, the Kohn-Sham equations need to be solved iteratively in a finite basis, every step involving numerical quadrature on a fine grid and matrix diagonalization resulting in a process that formally scales O(N4) where N represents the number of gaussian functions in the basis set. Combined with XP-PCM, which requires the scanning of a series of cavity constraints, the computing power of the HYDRA cluster constitutes a principal resource.

It is in this field that the ALGC group at the VUB has long been involved, supported by an international network of collaborators and the HYDRA Tier-2 facility by the VSC.

Applying XP-PCM to the atoms of the main group elements from hydrogen to krypton, the first order response of the energy with respect to pressure could be identified as a measure of the system size, i.e. the ‘electronic’ volume. This measure, while generally smaller than other scales, correlated well with known scales of atomic radii. Not surprisingly, the electronegativity and its components, the ionization potential and the electron affinity, all decreased under external pressure while displaying periodic patterns which could be explained on the basis of the aforementioned electronic volume. Similar periodic patterns were found in the chemical hardness, increasing under pressure for isolated atomic systems and its cube inverse, the softness, correlates well with the isotropic polarizability, a known relationship at ambient conditions. These global descriptors were applied to some specific applications, showing promising results for their application to other chemical problems.

Figure 1: Right) Radial Distribution Functions of the electron density at different pressures for the hydrogen atom. Left) Difference in radial distribution function between different pressures and the reference state

Finally, the periodicity in the electron density, a first local response function in CDFT, was studied using the radial distribution function, the quantum similarity index, and the Kullback-Leibler information deficiency. These all agreed on a more sensitive electron density to pressure for elements early on in a period compared to those later on. While this concludes our study on atoms.

Read the full publication in the Chemical Science journal (,

we continue expanding our work on this topic by looking at different pressure models, larger molecular systems, and more complex local response functions

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