Classical Reaction Barriers in DFT: An Adiabatic-Connection Perspective

Classical reaction barriers in density-functional theory are considered from the perspective of the density-fixed adiabatic connection. A 'reaction adiabatic-connection integrand', <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>λ</mml:mi></mml:msub></mml:math>, is introduced, where λ is the electron-electron interaction strength, for which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:msub><mml:mi>R</mml:mi><mml:mi>λ</mml:mi></mml:msub><mml:mi>d</mml:mi><mml:mi>λ</mml:mi></mml:math> equals the barrier, meaning the barrier can be easily visualized as the area under a plot of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>λ</mml:mi></mml:msub></mml:math> vs λ. For five chemical reactions, plots of reference <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>λ</mml:mi></mml:msub></mml:math>, calculated from Lieb maximizations at the coupled-cluster level of theory, are compared with approximate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>λ</mml:mi></mml:msub></mml:math>, calculated from common exchange-correlation functionals using coordinate scaling, for coupled-cluster densities. The comparison provides a simple way to visualize and understand functional-driven errors and trends in barriers from approximate functionals, while allowing a clean separation of the role of exchange and correlation contributions to the barrier. Specifically, the accuracy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is determined entirely by the accuracy of the exchange functional, while the shape of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>λ</mml:mi></mml:msub></mml:math> is determined entirely by the correlation functional. The results clearly illustrate why the optimal amount of exact (orbital) exchange in hybrid functionals differs between reactions, including forward and reverse directions in the same reaction, and hence why simply introducing larger amounts of exact exchange may not be a reliable approach for improving barriers. Instead, the shape of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>λ</mml:mi></mml:msub></mml:math> must be captured more accurately through more accurate correlation functionals, and the numerical data presented may be useful for this purpose. Density-driven errors are then considered, and possible cancellation with functional-driven errors in barriers─noted in prior studies when Hartree-Fock densities are used─is illustrated from the perspective of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>λ</mml:mi></mml:msub></mml:math>.