Koopmans’-Type Theorem in Kohn–Sham Theory with Optimally Tuned Long-Range-Corrected (LC) Functionals

Abstract

In the present study, we have investigated the applicability of long-range-corrected (LC) functionals to a Kohn–Sham (KS) Koopmans’-type theorem. Specifically, we have examined the performance of optimally tuned LCgau-core functionals (in combination with BOP and PW86-PW91 exchange-correlation functionals) by calculating the ionization potential (IP) within the context of Koopmans’ prediction. In the LC scheme, the electron repulsion operator, 1/r 12, is divided into short-range and long-range components using a standard error function, with a range separation parameter μ determining the weight of the two ranges. For each system that we have examined (H2O, CO, benzene, N2, HF, H2CO, C2H4, and five-membered ring compounds cyclo-C4H4X, with X = CH2, NH, O, and S, and pyridine), the value of μ is optimized to minimize the deviation of the negative HOMO energy from the experimental IP. Our results demonstrate the utility of optimally tuned LC functionals in predicting the IP of outer valence levels. The accuracy is comparable to that of highly accurate ab initio theory. However, our Koopmans’ method is less accurate for the inner valence and core levels. Overall, our results support the notion that orbitals in KS-DFT, when obtained with the LC functional, provide an accurate one-electron energy spectrum. This method represents a one-electron orbital theory that is attractive in its simple formulation and effective in its practical application.

Taking Advantage of a Systematic Energy Non-linearity Error in Density Functional Theory for the Calculation of Electronic Energy Levels

“…26−28 As the energy non-linearity error is, perhaps not surprisingly, system-and orbital-dependent, the accuracy of the orbitalenergy estimated IP and EA (and by extension, excitation energies) depends on the mixing proportions of HF and DFT in an LC-DFT. 13 Let us now revisit Figure 1 and again note that the errors for estimated IP by M + and M are opposite in signs: IP(by M + ) − IP(exact) is positive, and IP(by M) − IP(exact) is negative. Thus, averaging the two would benefit from error cancellation.…”

“…11 In comparison, by mixing Hartree−Fock (HF) into DFT, in particular by incorporating long-range HF in LC-DFT, one largely remedies this shortcoming. 12 The optimal mixing of DFT and HF in these "hybrid" methods, however, varies from system to system, 13 which somewhat hinders the straightforward application of this approach. In this work, we present a different orbital-based scheme to address this issue by turning the fundamental source of the problem into a tool for improving the accuracy in estimated orbital energies.…”

Meta-Hybrid Density Functional Theory Prediction of the Reactivity, Stability, and IGM of Azepane, Oxepane, Thiepane, and Halogenated Cycloheptane

“…Quantum chemical descriptors such as hardness (η), electronegativity (χ), chemical potential (μ), and softness, which provide substantial insight into the stability and reactivity of chemical systems, were estimated through Koopmans' approximation and conceptual density functional theory (CDFT) including the ground-state ionization potential (IP), second vertical ionization potential, and electron affinity value of chemical systems. 36,37 The electronic properties such as ionization potential (IP), EA, and X as evaluated by Koopmans' approximation are presented in Table 4. Quantities such as these give insight into the reactivity of the substituted cycloheptane molecules and other systems.…”

Excited-State Properties of Some Thermally Activated Delayed Fluorescence Emitters: Quest for an Accurate and Reliable Computational Method

“…36−39 Further, it has been reported that the optimally tuned RS functionals can accurately reproduce the IPs of outer valence levels compared to highly accurate methods. 40 Very recently, Chan et al 41 have proposed an approximate approach for predicting reasonable IPs and electron affinities with a low level of theory such as BLYP/6-31G (d), which can be further used to calculate the excitation energies.…”

“…In particular, range-separated (RS) density functionals have gained much attention because of reproducing properties such as orbital energies, polarizability, hyperpolarizability, CT excitation energy, Rydberg excitation, and so forth. − The range separation parameter (μ) in RS density functionals is a critical parameter in predicting ground- and excited-state properties. For instance, non-empirical tuning of μ in the RS functionals results in more accurate predictions of various ground- and excited-state properties such as electron density, fundamental gap, CT excitation energies, and so forth. − Recently, Hirao and co-workers have applied the ionization potential (IP) tuning method along with their proposed scheme in calculating the excitation energies of polyenes, DNA bases, in prediction of UV/vis spectra, CT excitation energies, and core-level excitation energies accurately from orbital energies. − Further, it has been reported that the optimally tuned RS functionals can accurately reproduce the IPs of outer valence levels compared to highly accurate methods . Very recently, Chan et al have proposed an approximate approach for predicting reasonable IPs and electron affinities with a low level of theory such as BLYP/6-31G (d), which can be further used to calculate the excitation energies.…”