Analytical evaluation of Fukui functions and real-space linear response function

Abstract

Many useful concepts developed within density functional theory provide much insight for the understanding and prediction of chemical reactivity, one of the main aims in the field of conceptual density functional theory. While approximate evaluations of such concepts exist, the analytical and efficient evaluation is, however, challenging, because such concepts are usually expressed in terms of functional derivatives with respect to the electron density, or partial derivatives with respect to the number of electrons, complicating the connection to the computational variables of the Kohn-Sham one-electron orbitals. Only recently, the analytical expressions for the chemical potential, one of the key concepts, have been derived by Cohen, Mori-Sánchez, and Yang, based on the potential functional theory formalism. In the present work, we obtain the analytical expressions for the real-space linear response function using the coupled perturbed Kohn-Sham and generalized Kohn-Sham equations, and the Fukui functions using the previous analytical expressions for chemical potentials of Cohen, Mori-Sánchez, and Yang. The analytical expressions are exact within the given exchange-correlation functional. They are applicable to all commonly used approximate functionals, such as local density approximation (LDA), generalized gradient approximation (GGA), and hybrid functionals. The analytical expressions obtained here for Fukui function and linear response functions, along with that for the chemical potential by Cohen, Mori-Sánchez, and Yang, provide the rigorous and efficient evaluation of the key quantities in conceptual density functional theory within the computational framework of the Kohn-Sham and generalized Kohn-Sham approaches. Furthermore, the obtained analytical expressions for Fukui functions, in conjunction with the linearity condition of the ground state energy as a function of the fractional charges, also lead to new local conditions on the exact functionals, expressed in terms of the second-order functional derivatives. We implemented the expressions and demonstrate the efficacy with some atomic and molecular calculations, highlighting the importance of relaxation effects.

Range-Separation Parameter in Tuned Exchange–Correlation Functionals: Successive Ionizations and the Fukui Function

“…We note that Yang et al 39 recently presented the theory for the analytic evaluation of this derivative. See Refs.…”

First principles view on chemical compound space: Gaining rigorous atomistic control of molecular properties

“…[59]. Analytical expressions for second-order derivatives and linear response functions have very recently been proposed by Yang et al [60] The same authors also derived important constraints for the electronic structure that must be met by the exact exchange-correlation functional. In analogy to using constraints obtained for variable N e , such as piece-wise linear behavior and derivative discontinuities, to design improved density functionals, [61][62][63] A Cohen's current efforts are dedicated to variations in the external potentials that include fractional nuclear charges.…”

“…3, the higher order effects mostly cancel, thereby rendering the first order HF derivative quite predictive. [59,60] To improve the predictive power of the first-order term in Eq. (9), an empirical correction has been introduced that ' 'linearizes' ' the energy through a global yet nonlinear Hamiltonian, [68] If we assume f ðkÞ to be a second-order polynomial in k, two coefficients are determined by the boundary conditions that Hðk ¼ 0Þ ¼ 0, and Hðk ¼ 1Þ ¼ 1, leaving one additional degree of freedom.…”

Bond fukui indices: Comparison of frozen molecular orbital and finite differences through mulliken populations

“…Several authors suggested different approaches towards their calculation beyond finite differences 8,9 . In finite differences, the Fukui function is computed using the electron density of a molecule with N electrons and that with N ± δ electrons and the difference divided by δ is considered to be the Fukui function.…”