On Koopmans' theorem in density functional theory

This paper clarifies why long-range corrected (LC) density functional theory gives orbital energies quantitatively. First, the highest occupied molecular orbital and the lowest unoccupied molecular orbital energies of typical molecules are compared with the minus vertical ionization potentials (IPs) and electron affinities (EAs), respectively. Consequently, only LC exchange functionals are found to give the orbital energies close to the minus IPs and EAs, while other functionals considerably underestimate them. The reproducibility of orbital energies is hardly affected by the difference in the short-range part of LC functionals. Fractional occupation calculations are then carried out to clarify the reason for the accurate orbital energies of LC functionals. As a result, only LC functionals are found to keep the orbital energies almost constant for fractional occupied orbitals. The direct orbital energy dependence on the fractional occupation is expressed by the exchange self-interaction (SI) energy through the potential derivative of the exchange functional plus the Coulomb SI energy. On the basis of this, the exchange SI energies through the potential derivatives are compared with the minus Coulomb SI energy. Consequently, these are revealed to be cancelled out only by LC functionals except for H, He, and Ne atoms.     

Excitation energies expressed as orbital energies of Kohn–Sham density functional theory with long-range corrected functionals

A new simple and conceptual theoretical scheme is proposed for estimating one-electron excitation energies using Kohn–Sham (KS) solutions. One-electron transitions that are dominated by the promotion from one initially occupied orbital to one unoccupied orbital of a molecular system can be expressed in a two-step process, ionization, and electron attachment. KS with long-range corrected (LC) functionals satisfies Janak's theorem and LC total energy varies almost linearly as a function of its fractional occupation number between the integer electron points. Thus, LC reproduces ionization energies (IPs) and electron affinities (EAs) with high accuracy and one-electron excitation energies are expressed as the difference between the occupied orbital energy of a neutral molecule and the corresponding unoccupied orbital energy of its cation. Two such expressions can be used, with one employing the orbital energies for the neutral and cationic systems, while the other utilizes orbital energies of just the cation. Because the EA of a molecule is the IP of its anion, if we utilize this identity, the two expressions coincide and give the same excitation energies. Reasonable results are obtained for valence and core excitations using only orbital energies.     

Spin-orbit relativistic long-range corrected time-dependent density functional theory for investigating spin-forbidden transitions in photochemical reactions

A long-range corrected (LC) time-dependent density functional theory (TDDFT) incorporating relativistic effects with spin-orbit couplings is presented. The relativistic effects are based on the two-component zeroth-order regular approximation Hamiltonian. Before calculating the electronic excitations, we calculated the ionization potentials (IPs) of alkaline metal, alkaline-earth metal, group 12 transition metal, and rare gas atoms as the minus orbital (spinor) energies on the basis of Koopmans' theorem. We found that both long-range exchange and spin-orbit coupling effects are required to obtain Koopmans' IPs, i.e., the orbital (spinor) energies, quantitatively in DFT calculations even for first-row transition metals and systems containing large short-range exchange effects. We then calculated the valence excitations of group 12 transition metal atoms and the Rydberg excitations of rare gas atoms using spin-orbit relativistic LC-TDDFT. We found that the long-range exchange and spin-orbit coupling effects significantly contribute to the electronic spectra of even light atoms if the atoms have low-lying excitations between orbital spinors of quite different electron distributions.     

Atomic ionization potentials and electron affinities with relativistic and mass corrections

Relativistic corrections to ionization potentials (IPs) and electron affinities (EAs) of atoms with an atomic number Z≤54 are examined based on the first-order perturbation theory with an approximate Schr?dinger form of the Dirac-Coulomb-Breit Hamiltonian. Using a Hartree-Fock (HF) wave function from the numerical HF method as the unperturbed function, both the LS-non-splitting and fine-structure corrections are evaluated together with the normal and specific mass corrections. The LS-non-splitting corrections are found to be important for IPs and EAs of transition metal atoms. The fine-structure corrections are generally larger in magnitude than the LS-non-splitting corrections for the atoms of groups 13–18 with Z≥31, and can never be neglected. Comparison of the IPs and EAs presented here and experimental IPs and EAs gives an estimation of the electron correlation correction for these properties. For some light atoms, the estimated values agree with the results directly obtained from correlated calculations. Received: 28 January 1997 / Accepted: 4 March 1997     

Assessing the role of Hartree‐Fock exchange,correlation energy and long range corrections in evaluating ionization potential,and electron affinity in density functional theory

Accurate determination of ionization potentials (IPs), electron affinities (EAs), fundamental gaps (FGs), and HOMO, LUMO energy levels of organic molecules play an important role in modeling and predicting the efficiencies of organic photovoltaics, OLEDs etc. In this work, we investigate the effects of Hartree Fock (HF) Exchange, correlation energy, and long range corrections in predicting IP and EA in Hybrid Functionals. We observe increase in percentage of HF exchange results in increase of IPs and decrease in EAs. Contrary to the general expectations inclusion of both HF exchange and correlation energy (from the second order perturbation theory MP2) leads to poor prediction. Range separated Hybrid Functionals are found to be more reliable among various DFT Functionals investigated. DFT Functionals predict accurate IPs whereas post HF methods predict accurate EAs. © 2017 Wiley Periodicals, Inc.     

Design of low band gap polymers employing density functional theory—hybrid functionals ameliorate band gap problem

Band gaps in solids and excitation energies in finite systems are underestimated significantly if estimated from differences between eigenvalues obtained within the local spin density approximation (LSDA). In this article we present results on 20 small- and medium-sized π-systems which show that HOMO–LUMO energy differences obtained with the B3LYP, B3P86, and B3PW91 functionals are in good agreement with vertical excitation energies from UV-absorption spectra. The improvement is a result of the use of the exact Hartree–Fock exchange with hybrid methods. Negative HOMO energies and negative LUMO energies do not provide good estimates for IPs and EAs. In contrast to Hartree–Fock theory, where IPs are approximated well and EAs are given poorly, DFT hybrid methods underestimate IPs and EAs by about the same amount. LSDA yields reasonable EAs but poor IPs. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1943–1953, 1997     

Halogenated 6,13-bis(triisopropylsilylethynyl)-5,7,12,14-tetraazapentacene: applications for ambipolar air-stable organic field-effect transistors

Density functional theory calculations were performed to explore the influence of halogenation on the reorganization energies (λ), adiabatic ionization potentials (IPs), adiabatic electron affinities (EAs), and air stabilities of a series of pentacene (PENT) and tetraceno[2,3-b]thiophene (TbTH) derivatives. According to calculated IP and EA values, all well-known PENT and TbTH derivatives in this paper are air-stable p-channel but not air-stable n-channel organic field-effect transistors (OFETs) due to insufficient EAs, consistent with experimental observations. The calculated results show that attaching two or more halogen atoms onto air-unstable 6,13-bis(triisopropylsilylethynyl)-5,7,12,14-tetraazapentacene (TIPS-N4PENT) is sufficient for promoting ambipolar air-stable properties. The electronic coupling and band structure calculations indicate that halogenated TIPS-N4PENT derivatives have potential applications in high-performance ambipolar air-stable OFETs. They also provide rational guidelines for the design of ambipolar air-stable organic semiconductors (OSCs).     

Addition by subtraction in coupled cluster theory. II. Equation-of-motion coupled cluster method for excited, ionized, and electron-attached states based on the nCC ground state wave function

New iterative double and triple excitation corrections to the equation-of-motion coupled cluster (EOM-CC) based upon the recently developed nCC methods [Bartlett and Musia?, J. Chem. Phys. 125, 204105-1 (2006)] are applied to excitation energies (EEs), ionization potentials (IPs), and electron affinities (EAs). The methods have been tested by the evaluation of the vertical EEs, IPs, and EAs for Ne, BH, CH(2), H(2)O, N(2), C(2), CH(+), CO, and C(2)H(4) compared to full configuration interaction, EOM-CCSD, EOM-CCSDT, and experimental data.     

有机半导体的电子电离能、亲和势和极化能的密度泛函理论研究

准确预测有机半导体的能级(如电子电离能和亲和势等)对设计新型有机半导体材料和理解相关机理至关重要。从理论计算的角度看,主要挑战来自于缺少一种不仅能够在定性上合理而且在定量上精确预测,同时并不显著增加计算成本的理论方法。本文中,我们证明了通过结合极化连续介质模型(PCM)和"最优调控"区间分离密度泛函方法能够准确预测一系列有机半导体的电子电离能(IP)、亲和势(EA)和极化能,其预测结果与实验数据吻合得很好。重要的是,经过调控后分子的前线分子轨道能量(即-~εHOMO和-~εLUMO)与对应的IP和EA计算值很接近。调控方法的成功可以进一步归因于其能够根据不同分子体系或同种分子所处的不同状态(气态和固态)"最优"地平衡泛函中分别用于描述电子局域化和离域化的作用。相比而言,其它常见的密度泛函方法由于包含的HF%比例过低(如PBE)或过高(如M06HF和未调控的区间分离泛函),均不能给予合理的预测。因此,我们相信这种PCM-调控的方法能够为研究其它更加复杂的有机体系的能级问题提供一种更加可靠和便捷的理论工具。     

DFT study of the monomers and dimers of 2-pyrrolidone: equilibrium structures, vibrational, orbital, topological, and NBO analysis of hydrogen-bonded interactions

A computational study of the monomers and hydrogen-bonded dimers of 2-pyrrolidone was executed at different DFT levels and basis sets. The above dimeric complexes were treated theoretically to elucidate the nature of the intermolecular hydrogen bonds, geometry, thermodynamic parameters, interaction energies, and charge transfer. The processes of dimer formation from monomers and concerted reactions of double proton transfer were considered. The evolution of geometry, vibrational frequencies, charge distribution, and AIM properties in going from monomers to dimers was systematically followed. The solvent effects upon dimer formation were investigated in terms of the self-consistent reaction field (SCRF Onsager model). For the monomers and three dimers, vibrational frequencies were calculated and the changes in frequencies of the vibrations most sensitive to complexation were discussed. The orbital interactions were shown to lengthen the X-H (X = N, O) bond and lower its vibrational frequency (a red shift). To better understand the nature of the corresponding intermolecular interactions, we performed natural bond orbital (NBO) analysis. Topological analysis of electron density at bond critical points (BCP) was executed for complex molecules using the Bader's atoms in molecules (AIM) theory. The interaction energies were calculated, and the basis set superposition errors (BSSE) were estimated systematically. Satisfactory correlations between the structural parameters, interaction energies, and electron density characteristics at BCP were found.     

Long‐range corrected functionals satisfy Koopmans' theorem: Calculation of correlation and relaxation energies

In this article, we show that the long‐range‐corrected (LC) density functionals LC‐BOP and LCgau‐BOP reproduce frontier orbital energies and highest‐occupied molecular orbital (HOMO)—lowest‐unoccupied molecular orbital (LUMO) gaps better than other density functionals. The negative of HOMO and LUMO energies are compared with the vertical ionization potentials (IPs) and electron affinities, respectively, using CCSD(T)/6‐311++G(3df,3pd) for 113 molecules, and we found LC functionals to satisfy Koopmans' theorem. We also report that the frontier orbital energies and the HOMO‐LUMO gaps of LC‐BOP and LCgau‐BOP are better than those of recently proposed ωM05‐D (Lin et al., J. Chem. Phys. 2012, 136 , 154109). We express the exact IP in terms of orbital relaxation, and correlation energies and hence calculate the relaxation and correlation energies for the same set of molecules. It is found that the LC functionals, in general, includes more relaxation effect than Hartree–Fock and more correlation effect than the other density functionals without LC scheme. Finally, we scan μ parameter in LC scheme from 0.1 to 0.6 bohr^?1 for the above test set molecules with LC‐BOP functional and found our parameter value, 0.47 bohr^?1, is usefully applicable to our tested systems. © 2013 Wiley Periodicals, Inc.     

Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies

A systematic evaluation of the ionization energy within the GW approximation is carried out for the first row atoms, from H to Ar. We describe a Gaussian basis implementation of the GW approximation, which does not resort to any further technical approximation, besides the choice of the basis set for the electronic wavefunctions. Different approaches to the GW approximation have been implemented and tested, for example, the standard perturbative approach based on a prior mean-field calculation (Hartree-Fock GW@HF or density-functional theory GW@DFT) or the recently developed quasiparticle self-consistent method (QSGW). The highest occupied molecular orbital energies of atoms obtained from both GW@HF and QSGW are in excellent agreement with the experimental ionization energy. The lowest unoccupied molecular orbital energies of the singly charged cation yield a noticeably worse estimate of the ionization energy. The best agreement with respect to experiment is obtained from the total energy differences within the random phase approximation functional, which is the total energy corresponding to the GW self-energy. We conclude with a discussion about the slight concave behavior upon number electron change of the GW approximation and its consequences upon the quality of the orbital energies.     

Structural,electronic, and optical properties of phenol‐pyridyl boron complexes for light‐emitting diodes

The coordination chemistry of polydentate chelating ligands that contain mixed pyridinephenol donor sets has been a sought‐after target of study and is a possible extension to the chemistry of polypyridines. In this article, seven compounds, which are the four‐coordinate boron complexes containing the mixed phenol‐pyridyl group, have been studied by theoretical calculation. They can function as charge transport materials and emitters, with high efficiency and stability. To reveal the relationship between the structures and properties of these bifunctional or multifunctional electroluminescent materials, the ground and excited state geometries were optimized at the B3LYP/6‐31G(d), HF/6‐31G(d), and CIS/6‐31G(d) levels, respectively. The ionization potentials (IPs) and electron affinities (EAs) were computed. The mobilities of hole and electron in these compounds were studied computationally based on the Marcus electron transfer theory. The lowest excitation energies, and the maximum absorption and emission wavelengths of these compounds were calculated by time‐dependent density functional theory method. As a result of these calculations, the values of HOMO, LUMO, energy gaps, IPs, EAs, and the balance between the hole‐ and electron‐transfer are greatly improved with the substitution of carbazole in compound 6 . The calculated emission spectra of the seven studied molecules can almost cover the full UV‐vis range (from 447.4 to 649.3 nm). Also, the Stokes shifts are unexpectedly large, ranging from 139.4 to 335.1 nm. This will result in the relatively long fluorescence lifetimes. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

The correlation consistent composite approach (ccCA): an alternative to the Gaussian-n methods

An alternative to the Gaussian-n (G1, G2, and G3) composite methods of computing molecular energies is proposed and is named the "correlation consistent composite approach" (ccCA, ccCA-CBS-1, ccCA-CBS-2). This approach uses the correlation consistent polarized valence (cc-pVXZ) basis sets. The G2-1 test set of 48 enthalpies of formation (DeltaHf), 38 adiabatic ionization potentials (IPs), 25 adiabatic electron affinities (EAs), and 8 adiabatic proton affinities (PAs) are computed using this approach, as well as the DeltaHf values of 30 more systems. Equilibrium molecular geometries and vibrational frequencies are obtained using B3LYP density functional theory. When applying the ccCA-CBS method with the cc-pVXZ series of basis sets augmented with diffuse functions, mean absolute deviations within the G2-1 test set compared to experiment are 1.33 kcal mol(-1) for DeltaHf,0.81 kcal mol(-1) for IPs, 1.02 kcal mol(-1) for EAs, and 1.51 kcal mol(-1) for PAs, without including the "high-level correction" (HLC) contained in the original Gn methods. Whereas the HLC originated in the Gaussian-1 method as an isogyric correction, it evolved into a fitted parameter that minimized the error of the composite methods, eliminating its physical meaning. Recomputing the G1 and G3 enthalpies of formation without the HLC reveals a systematic trend where most DeltaHf values are significantly higher than experimental values. By extrapolating electronic energies to the complete basis set (CBS) limit and adding G3-like corrections for the core-valence and infinite-order electron correlation effects, ccCA-CBS-2 often underestimates the experimental DeltaHf, especially for larger systems. This is desired as inclusion of relativistic and atomic spin-orbit effects subsequently improves theoretical DeltaHf values to give a 0.81 kcal mol(-1) mean absolute deviation with ccCA-CBS-2. The ccCA-CBS method is a viable "black box" method that can be used on systems with at least 10-15 heavy atoms.     

Calculations of ionization energies and electron affinities for atoms and molecules: A comparative study with different methods

In the present work, we examined the performance of 36 density functionals, including the newly developed doubly hybrid density functional XYG3 (Y. Zhang, X. Xu, and W. A. Goddard III, Proc. Natl. Acad. Sci, USA, 2009, 106, 4963), to calculate ionization energies (IEs) and electron affinities (EAs). We used the well-established G2-1 set as reference, which contains 14 atoms and 24 molecules for IE, along with 7 atoms and 18 molecules for EA. XYG3 leads to mean absolute deviations (MADs) of 0.057 and 0.080 eV for IEs and EAs, respectively, using the basis set of 6-311 + G (3df,2p). In comparison with some other functionals, MADs for IEs are 0.109 (B2PLYP), 0.119 (M06-2X), 0.159 (X3LYP), 0.161 (PBE), 0.162 (B3LYP), 0.165 (PBE0), 0.173 (TPSS), 0.200 (BLYP), and 0.215 eV (LC-BLYP). MADs for EAs are 0.090 (X3LYP), 0.090 (B2PLYP), 0.102 (PBE), 0.103 (M06-2X), 0.104 (TPSS), 0.105 (BLYP), 0.106 (B3LYP), 0.126 (LC-BLYP), and 0.128 eV (PBE0).     

DFT study of ionization potentials and electron affinities of formamide and its methylation derivatives

The ionization potentials (IPs) and electron affinities (EAs) of formamide in the gas phase have been calculated using density functional theory (DFT), ab initio HF and Møller-Plesset perturbational theory (MP) at 6-311++G** basis set. The results indicate that the IPs of formamide obtained with DFT and MP are in agreement with the results obtained from experiment. And B3LYP has been confirmed to be the most accurate method in calculating the AIPs and VIPs of formamide through our work. IPs and EAs of formamide in solution are not known experimentally, therefore IPs and EAs of formamide in chloroform, acetone, and dimethylsulfoxide have been calculated using polarized continuum model (PCM) with B3LYP/6-311++G** level and have been compared with the values in the gas phase. The AIPs and VIPs of formamide have been compared with those of its methylation derivatives. All EAs of methylation derivatives of formamide are bigger than those of formamide conformers in the gas phase with BLYP, B3LYP, and B3P86 methods at 6-311++G** basis set. All these indicate that all anions of methylation derivatives of formamide are more stable than anions of formamide with respect to electron detachment adiabatically and vertically in the gas phase.     

Dependence of relativistic effects on electronic configuration in the neutral atoms of d- and f-block elements

Although most neutral d- and f-block atoms have nd(g-2)(n + 1)s(2) and (n - 1)f(g-2)(n + 1)s(2) ground configurations, respectively, where g is the group number (i.e., number of valence electrons), one-third of these 63 atoms prefer a higher d-population, namely via (n + 1)s-->nd "outer" to "inner" electron shift (particularly atoms from the second d-row), or via (n - 1)f-->nd "inner" to "outer" electron shift (particularly atoms from the second f-row). Although the response to the modified self-consistent field is orbital destabilization and expansion for (n + 1)s-->nd, and stabilization and contraction for (n - 1)f-->nd, the relativistic modification of the valence orbital responses is stabilization in both cases. This is explained by double perturbation theory. Accordingly, electron configuration and relativity trigger the orbital energies, the orbital populations and the chemical shell effects in different ways. The particularly pronounced relativistic effects in groups 10 and 11, the so-called gold maximum, occur because of particularly efficient cooperative nonrelativistic shell effects and relativistic stabilization effects (inverse indirect effect) at the end of the d-block.     

Electronic structures and electron detachment energies of halogen substituted acetate anions, XCH2COO- (X=F,Cl,Br)

The electronic structures and the halogen inductive effects on the acetate anion were investigated in XCH2COO- (X=F,Cl,Br) by photoelectron spectroscopy (PES) and ab initio calculations. The PES spectra indicated that the electron binding energies increased in the order of FCl>Br. These systematic changes of detachment energy and IPs were explained by examining the charge redistributions upon detaching electrons.     

Ionization potentials, electron affinities, resonance excitation energies, oscillator strengths, and ionic radii of element Uus (Z = 117) and astatine

Multiconfiguration Dirac-Fock (MCDF) method was employed to calculate the first five ionization potentials, electron affinities, resonance excitation energies, oscillator strengths, and radii for the element Uus and its homologue At. Main valence correlation effects were taken into account. The Breit interaction and QED effects were also estimated. The uncertainties of calculated IPs, EAs, and IR for Uus and At were reduced through an extrapolation procedure. The good consistency with available experimental and other theoretical values demonstrates the validity of the present results. These theoretical data therefore can be used to predict some unknown physicochemical properties of element Uus, Astatine, and their compounds.     

Core ionization potentials from self-interaction corrected Kohn-Sham orbital energies

We propose a simple self-interaction correction to Kohn-Sham orbital energies in order to apply ground state Kohn-Sham density functional theory to accurate predictions of core electron binding energies and chemical shifts. The proposition is explored through a series of calculations of organic compounds of different sizes and types. Comparison is made versus experiment and the "DeltaKohn-Sham" method employing separate state optimizations of the ground and core hole states, with the use of the B3LYP functional and different basis sets. A parameter alpha is introduced for a best fitting of computed and experimental ionization potentials. It is found that internal parametrizations in terms of basis set expansions can be well controlled. With a unique alpha=0.72 and basis set larger than 6-31G, the core ionization energies (IPs) of the self-interaction corrected Kohn-Sham calculations fit quite well to the experimental values. Hence, self-interaction corrected Kohn-Sham calculations seem to provide a promising tool for core IPs that combines accuracy and efficiency.