Numerical investigation of the validity of Janak path integral and Slater–Janak theorem

Janak's theorem states that the derivative of the total energy with respect to an orbital occupation number equals the corresponding orbital energy. Integration of the Janak relation is referred to as the Janak path integral, and it connects the total energies of the N- and N-1- electron systems, yielding the ΔSCF ionization energy. We numerically assess the validity of the Janak path integral for ionization processes using both Hartree-Fock (HF) and Kohn-Sham (KS) methods. Although convergence of the Janak path integral to the exact value becomes slower as the curvature of the Janak path increases, it nonetheless exactly reproduces the ΔSCF result-even for nonlinear Janak paths and regardless of the approximate functional employed. Furthermore, the total electronic energy can be expressed in terms of highest occupied molecular orbital energies. It is known as the Slater-Janak theorem. We examine this relation using a range of numerical integration schemes. While systematic refinement of the integration yields convergence to the exact result, convergence is rapid for HF but markedly slower for KS. The KS exchange-correlation functional is expressed in terms of the electron density and introduces spurious electron-electron interactions in one-electron systems. Consequently, the Janak path for a one-electron system exhibits anomalous behavior near the cationic limit. This results in slow convergence of the Janak path integral for systems with few electrons and, correspondingly, slow convergence of the KS Slater-Janak relation.