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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - High level CDFT (Conquering New Frontiers in Conceptual Density Functional Theory: Going Beyond the Single Slater Determinant.)

Teaser

Summary

Although quantum chemistry is now accepted as a useful partner to experimental chemistry, the link between the abstract mathematics of quantum mechanics and the practical chemical concepts that characterize the structure and reactivity of atoms, molecules and solids remains problematic. The powerful computational methods needed to obtain quantitative agreement with experiment diverge from the simplified interpretations familiar to most chemists. Refining old concepts - and finding new ones - that are also applicable to state-of-the-art quantum chemistry methods is the overarching goal of this project.

The approach I follow is based on a framework for chemical reactivity indices called Conceptual Density Functional Theory (CDFT) (sometimes also called DFT reactivity theory, chemical DFT or Chemical Reactivity Theory). In CDFT, chemical concepts and reactivity indices are identified with the response of the energy of the chemical system, E, to perturbations in its number of electrons, N, and its external potential, v(r). (For an isolated system, v(r) is just the potential due to the atomic nuclei.) Many concepts that are commonly used by chemists, but often vaguely defined, have been reformulated using these response functions.

The motivation for this work is the significant progress made in accurately calculating the wave function and energy. Modern computer technology makes traditional high-level ab initio methods like Complete Active Space Self Consistent Field (CASSCF), Full Configuration Interaction (Full-CI) and multi-reference Configuration-Interaction (MRCI) calculations more feasible.
Accurate computation of the energy does not imply an understanding of the reaction considered. This is especially true for many of these high-level wavefunction methods, since they give accurate results but are not readily interpretable with classical orbital-based chemical concepts. The CDFT concepts are, however, defined in a universal manner, and are thus applicable at all levels of modern electronic structure theory.

To obtain qualitative insight into the chemical reactions of molecules using the above mentioned high-level methods, I combined the best of these two worlds by analytically calculating the Fukui function and linear response, two important reactivity indices known from CDFT, using accurate wave function methods, more specifically Full-CI and MRCI. In the frame of this project, I have derived the equations of these response functions, together with Prof. Paul W. Ayers (McMaster University, Canada), an expert in CDFT, high-level ab initio methods. These equations have been implemented using the open-source HORTON (Helpful Open-source Research TOol for N-fermion systems) software and CHEMTOOLS (https://chemtools.org) packages. As such we provide conceptual tools and the software to interpret accurate multideterminantal wavefunctions within the framework of CDFT.
The equations ant the implementation have been tested and benchmarked on small molecules and test-systems and this method and the code can now be applied to systems which cannot be accurately described with single Slated determinant methods, like certain transition states, biradicals and carbenes. Extending the CDFT framework to these high-level ab initio methods opens new collaboration opportunities, with both experimentalists (e.g. groups doing catalysis with transition and main group metals with carbene ligands) and theoreticians (e.g. groups specializing in accurate wave function methods).

The complexity of the calculation of these CDFT concepts for multi-reference wavefunctions, however, increases significantly, making the routine calculations of them hard and only reserved for special cases where the single slater determinant is known of suspected to be incorrect.

Work performed

After becoming more familiar with the so-called second quantization formalism, I derived the analytical formulaâ€™s for the linear response in the most general way possible. This was done in collaboration with Prof. Ayers. The derived equations are valid for high-level ab initio methods, but also for lower level (Single Slater-Determinant) methods, just by using the desired wavefunction method in the equations. The equations were tested against the known expression for the linear response, at the Hartree-Fock and the Full-CI level.

The equations have been implemented into the open-source HORTON package and have been tested on small molecules. We are in the process of writing down the method and results before publicly releasing the source-code. The implementation was done in close collaboration with the lead developers of HORTON in Prof. Ayersâ€™ group. We have also made significant progress in generalizing the expression for the Fukui function. At this point we are searching for a way to make these expressions practically implementable.

As part of the theoretical investigation of these CDFT indices, I have also looked into the mathematical properties of the CDFT response kernels, among which the linear response function (â€œProperties of the Density Functional Response Kernels and its Implications on Chemistryâ€ Fias, S.; Ayers, P.W.; De Proft, F. and Geerlings, P., to be submitted soon) and the analogies between these and the thermodynamic state functions Geerlings, P.; De Proft, F.; Fias, S. Acta Phys. -Chim. Sin. 2018, 34, p. 699).

In collaboration with Prof. Joubert of the UniversitÃ© de Rouen (France) we looked at the behaviour of the linear response kernel when changing the amount of exact (or Hartree-Fock) exchange in the DFT functionals. The results suggest that the calculation of the linear response might be used to test the sanity of DFT functionals.

I have also been involved in a project with Prof. O Anatole von Lilienfeld from the university of Basel (Switzerland) where we introduced so-called alchemical normal modes (ANMs) (for which the CDFT reactivity indices are needed) to navigate and better understand the chemical compound space, the space of all stable chemical compounds. We hope this research will lead to more efficient inverse design of new compounds with pre-determined properties. (Fias, S.; Samuel Chang, K.Y.; and von Lilienfeld, A., J. Phys. Chem. Lett., 2019, 10 (1), p. 30).

Final results

The general, analytical formula for the linear response function that was derived is applicable on any multi-reference wavefunction. As such, this important CDFT reactivity index, which is used to understand the aromaticity, bonding, reactivity and even conductivity of molecules, is extended to a wide range of very accurate methods in quantum chemistry. The recent application of the linear response function to the problem of inverse design, where one tries to find the ideal compound based on certain pre-determined desired properties, opens a wide range of applications, now based on this very accurate wavefunction methods.

The equations were implemented and benchmarked for different wavefunction methods. The equations can now be applied to the research currently conducted at the ALGC group in Brussels, to transition states, biradicals and carbenes, systems which cannot be accurately described with single Slater determinant methods.

By releasing the implementation in the open-source HORTON package, this method will be freely available to the whole scientific community. This would provide an important tool to help in the understanding of the reactivity of molecules and in the inverse design of new compounds.