Slope of the Delocalization Function Is Proportional to Analytical Hardness

Area of Science:

• Computational Chemistry
• Quantum Chemistry
• Materials Science

Background:

• Density Functional Theory (DFT) is crucial for chemical reactivity but suffers from limitations like delocalization error.
• Delocalization error causes energy dependence on electron number (N) to deviate from exact linear behavior in common DFT approximations (DFAs).
• Previous work established analytical hardness (η±) for pure functionals.

Purpose of the Study:

• Extend the application of analytical hardness (η±) to hybrid and range-separated functionals.
• Investigate the relationship between analytical hardness and the delocalization function slope.
• Develop an approximate scheme to construct energy vs. N curves without fractional electron calculations.

Main Methods:

• Applied analytical hardness calculations to hybrid and range-separated functionals.
• Compared analytical hardness with the slope of the delocalization function (Hait and Head-Gordon).
• Presented an approximate scheme for energy vs. N curve construction.

Main Results:

• Demonstrated a linear relationship between the slope of the delocalization function and analytical hardness.
• Successfully extended analytical hardness calculations beyond pure functionals.
• Proposed a method to approximate energy vs. N curves without fractional occupations.

Conclusions:

• The analytical hardness provides a valuable tool for understanding and mitigating delocalization errors in DFAs.
• The linear relationship offers a new perspective on the delocalization error and functional development.
• The approximate scheme offers a computationally efficient alternative for constructing energy vs. N curves.