Koopmans' theorem in the ROHF method: canonical form for the Hartree-Fock Hamiltonian

Since the classic work of Roothaan [Rev. Mod. Phys. 32, 179 (1960)], the one-electron energies of a ROHF method are known as ambiguous quantities having no physical meaning. Together with this, it is often assumed in present-day computational studies that Koopmans' theorem is valid in a ROHF method. In this work we analyze the specific dependence of orbital energies on the choice of the basic equations in a ROHF method which are the Euler equations and different forms of the generalized Hartree-Fock equation. We first prove that the one-electron open-shell energies epsilon(m) derived by the Euler equations can be related to the respective ionization potentials I(m) via the modified Koopmans' formula I(m)= -epsilon(m)f(m) where f(m) is an occupation number. As compared to this, neither the closed-shell orbital energies nor the virtual ones derived by the Euler equations can be related to the respective ionization potentials and electron affinities via Koopmans' theorem. Based on this analysis, we derive the new (canonical) form for the Hamiltonian of the Hartree-Fock equation, the eigenvalues of which obey Koopmans' theorem for the whole energy spectrum. A discussion of new orbital energies is presented on the examples of a free N atom and an endohedral N@C(60) (I(h)). The vertical ionization potentials and electron affinities estimated via Koopmans' theorem are compared with the respective observed data and, for completeness, with the respective estimates derived via a DeltaSCF method. The agreement between observed data and their estimates via Koopmans' theorem is qualitative and, in general, appears to possess the same accuracy level as in the closed-shell SCF.     

Orbital energies and negative electron affinities from density functional theory: Insight from the integer discontinuity

Orbital energies in Kohn-Sham density functional theory (DFT) are investigated, paying attention to the role of the integer discontinuity in the exact exchange-correlation potential. A series of closed-shell molecules are considered, comprising some that vertically bind an excess electron and others that do not. High-level ab initio electron densities are used to calculate accurate orbital energy differences, Deltavarepsilon, between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO), using the same potential for both. They are combined with accurate vertical ionization potentials, I(0), and electron affinities, A(0), to determine accurate "average" orbital energies. These are the orbital energies associated with an exchange-correlation potential that averages over a constant jump in the accurate potential, of magnitude Delta(XC)=(I(0)-A(0))-Deltavarepsilon, as given by the discontinuity analysis. Local functional HOMO energies are shown to be almost an order of magnitude closer to these average values than to -I(0), with typical discrepancies of just 0.02 a.u. For systems that do not bind an excess electron, this level of agreement is only achieved when A(0) is set equal to the negative experimental affinity from electron transmission spectroscopy (ETS); it degrades notably when the zero ground state affinity is instead used. Analogous observations are made for the local functional LUMO energies, although the need to use the ETS affinities is less pronounced for systems where the ETS values are very negative. The application of an asymptotic correction recovers the preference, leading to positive LUMO energies (but bound orbitals) for these systems, consistent with the behavior of the average energies. The asymptotically corrected LUMO energies typically agree with the average values to within 0.02 a.u., comparable to that observed with the HOMOs. The study provides numerical support for the view that local functionals exhibit a near-average behavior based on a constant jump of magnitude Delta(XC). It illustrates why a recently proposed DFT expression involving local functional frontier orbital energies and ionization potential yields reasonable estimates of negative ETS affinities and is consistent with earlier work on the failure of DFT for charge-transfer excited states. The near-average behavior of the exchange-correlation potential is explicitly illustrated for selected systems. The nature of hybrid functional orbital energies is also mentioned, and the results of the study are discussed in terms of the variation in electronic energy as a function of electron number. The nature of DFT orbital energies is of great importance in chemistry; this study contributes to the understanding of these quantities.     

Calculations of the optical spectra of hydrocarbon radical cations based on Koopmans' theorem

The first few bands in the optical spectra of radical cations can often be interpreted in terms of A-type transitions that involve electron promotions from doubly occupied to the singly occupied molecular orbital (SOMO) and/or B-type transition which involve electron promotion from the SOMO to virtual molecular orbitals. We had previously demonstrated that, by making use of Koopmans' theorem, the energies of A-type transitions can be related to orbital energy differences between lower occupied MOs and the highest occupied MO (HOMO) in the neutral molecule, calculated at the geometry of the radical cation. We now propose that the energies of B-type transitions can be related similarly to energy differences between the lowest unoccupied MO (LUMO) and higher virtual MOs in the dication, also calculated at the geometry of the radical cation, by way of an extension of Koopmans' theorem to virtual MOs similar to that used sometimes to model resonances in electron scattering experiments. The optical spectra of the radical cations of several polyenes and aromatic compounds, the matrix spectra of which are known (or presented here for the first time), and for which CASSCF/CASPT2 calculations are available, are discussed in terms of these Koopmans-based models. Then the spectra of five poly(bicycloalkyl)-protected systems and that of hexabenzocoronene, compounds not amenable to higher level calculations, are examined and it is found that the Koopmans-type calculations allow a satisfactory interpretation of most of the features in these spectra. These simple calculations therefore provide a computationally inexpensive yet effective way to assign optical transitions in radical ions. Limitations of the model are discussed.     

The virial theorem as an entail for Koopmans' theorem in atomic calculations

The frozen approximation in the Koopmans' theorem suggests that the virial theorem is not obeyed. By imposing the virial theorem to Koopmans' theorem, we observe that the ionization potential of an atomic orbital is directly related to the respective kinetic energy and that the virial theorem introduces some reorganization effect on the electronic cloud. The quantity of reorganization introduced is not hazard, depending on the type of atom as well as the atomic orbital.     

Comparison of DFT methods for molecular orbital eigenvalue calculations

We report how closely the Kohn-Sham highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) eigenvalues of 11 density functional theory (DFT) functionals, respectively, correspond to the negative ionization potentials (-IPs) and electron affinities (EAs) of a test set of molecules. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin density approximation (LSDA), five generalized gradient approximation (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH&HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with average absolute errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, respectively. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (approximately 0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calculated HOMO eigenvalue and the experimental -IP and calculated HOMO-LUMO gap and experimental lowest excitation energy enables us to derive a simple correction formula.     

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.     

A modification of Koopmans' theorem by imposition of the virial theorem in molecules

Imposition of the virial theorem on Koopmans' theorem permits the introduction of some relaxation effect in the electronic cloud of atomic (less than 5%) or molecular (less than 1.3% for the systems studied) systems and a partitioning of the ionization energy. The method is applied in some diatomic hydrides. It is observed that the imposition of the virial theorem improves the ionization of the innermost molecular orbitals significantly, while the improvement is negligible for the outermost orbitals. The ionization energy is divided among three different terms that elucidate some aspects of the nature of the ionization process.     

Low energy (e,2e) studies from CH4: results from symmetric coplanar experiments and molecular three-body distorted wave theory

Low energy experimental and theoretical triply differential cross sections are presented for electron impact ionization of methane (CH(4)) for both the highest occupied molecular orbital (HOMO) and next highest occupied molecular orbital (NHOMO). The HOMO is a predominantly p-type orbital which is labeled 1t(2) and the NHOMO is predominantly s-type labeled 2a(1). Coplanar symmetric (symmetric both in final state electron energies and observation angles) are presented for final state electron energies ranging from 2.5 to 20 eV. The theoretical M3DW (molecular three-body distorted wave) results are in surprisingly good agreement with experiment for the HOMO state and less satisfactory agreement for the NHOMO state. The molecular NHOMO results are also compared with the ionization of the 2s shell of neon which is the isoelectronic atom.     

Calculation of vertical ionization potentials with the Piris natural orbital functional

The Piris natural orbital functional (PNOF) based on a new approach for the two-electron cummulant has been used to predict vertical ionization potentials of 15 molecules. The ionization energies have been calculated using the extended Koopmans' theorem. The calculated PNOF values are in good agreement with the corresponding experimental data.     

Calculation of vertical ionization potentials of ketene by perturbation corrections to Koopmans' theorem

The vertical ionization potentials of ketene are calculated by perturbation corrections to Koopmans' theorem. The present results are compared with those from the pseudonatural orbital coupled electron pair approach and the experimental values.     

Modulation of the electronic structure of semiconducting nanotubes resulting from different metal contacts

Examined in this paper is the role of the metal electrode influencing the structure and electronic properties of semiconducting carbon nanotubes near the interface at low bias. Specifically, we present quantum-chemical calculations of finite sections of a (8,0) semiconducting single wall nanotube contacted with gold and palladium clusters. The calculations at the density functional level of theory, which included full geometry optimizations, indicate the formation of bonds between the metal atoms of the electrode and the carbon atoms of the nanotube. The local work function of the metal electrode can be expected to exhibit significant variations as a result of this bond formation. Compared to the gold-contacted nanotubes, the palladium-contacted nanotubes have a small but interesting increase in both length and diameter. The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of the gold-contacted nanotube are shown localized at the edges. In contrast, the HOMO and LUMO of the palladium-contacted nanotube are extended over the entire nanotube and the metal cluster contacted to it, providing thereby a better conduction path in the contact region of the electrode and the nanotube. The involvement of the highly directional d orbitals in the interactions involving the palladium cluster leads to an enhanced pi electron density in the nanotube. This enhanced pi electron density is synonymous with an improved electron transmission.     

Ab initio investigation of 2,2′‐bis(4‐trifluoromethylphenyl)‐5,5′‐bithiazole for the design of efficient organic field‐effect transistors

The initial molecular structure of 2,2′‐bis(4‐trifluoromethylphenyl)‐ 5,5′‐bithiazole has been optimized in the ground state using density functional theory (DFT). The distribution patterns of highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) have also been evaluated. To shed light on the charge transfer properties, we have calculated the reorganization energy of electron λ_e, the reorganization energy of hole λ_h, adiabatic electron affinity (EA_a), vertical electron affinity (EA_v), adiabatic ionization potential (IP_a), and vertical ionization potential (IP_v) using DFT. Based on the evaluation of hole reorganization energy, λ_h, and electron reorganization energy, λ_e, it has been predicted that 2,2′‐bis(4‐trifluoromethylphenyl)‐5,5′‐bithiazole would be a better electron transport material. Finally, the effect of electric field on the HOMO, LUMO, and HOMO–LUMO gap were observed to check its suitability for the use as a conducting channel in organic field‐effect transistors. © 2015 Wiley Periodicals, Inc.     

Accurate Kohn–Sham ionization potentials from scaled‐opposite‐spin second‐order optimized effective potential methods

One important property of Kohn–Sham (KS) density functional theory is the exact equality of the energy of the highest occupied KS orbital (HOMO) with the negative ionization potential of the system. This exact feature is out of reach for standard density‐dependent semilocal functionals. Conversely, accurate results can be obtained using orbital‐dependent functionals in the optimized effective potential (OEP) approach. In this article, we investigate the performance, in this context, of some advanced OEP methods, with special emphasis on the recently proposed scaled‐opposite‐spin OEP functional. Moreover, we analyze the impact of the so‐called HOMO condition on the final quality of the HOMO energy. Results are compared to reference data obtained at the CCSD(T) level of theory. © 2016 Wiley Periodicals, Inc.     

Au12M (M=Na,Mg, Al,Si, P,S, Cl)团簇的结构、稳定性和电子性质

采用基于密度泛函理论的第一性原理方法系统地研究了Au12M(M=Na,Mg,Al,Si,P,S,Cl)团簇的结构、稳定性和电子性质.对团簇的平均结合能、镶嵌能、垂直离化势、最高占据分子轨道(HOMO)和最低未占据分子轨道(LUMO)的能级差、电荷布居分析、自然键轨道(NBO)进行了计算和讨论.对于Au12M(M=Na,Mg,Al)团簇,它们形成了内含M原子的最稳定的笼状结构.然而对于Au12M(M=Si,P,S,Cl)团簇,它们却形成了以M元素为顶点的稳定锥形结构.在这些团簇中发现Au12S团簇相对是最稳定的,这是由于Au12S团簇形成了稳定的满壳层的电子结构.自然电荷布居分析表明:对于所有的Au12M(M=Na,Mg,Al,Si,P,S,Cl)团簇电荷总是从Au原子转向M原子.自然键轨道和HOMO分析表明Au12M团簇中发生了Au原子的s-d轨道和M原子的p轨道间的杂化现象.     

Electron smearing in DFT calculations: A case study of doxorubicin interaction with single‐walled carbon nanotubes

To address the choice of an appropriate value of electron smearing to facilitate self‐consistent field (SCF) convergence, we studied the interaction of doxorubicin with short armchair and zigzag single‐walled carbon nanotube models with closed caps, at the PWC/DNP level of density functional theory. By gradually reducing the electron smearing value from a large and most commonly used one of 0.005 Ha to zero (Fermi occupation), we monitored the changes in close contacts between the interacting species, total energy of the molecular system, highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energy and isosurfaces, HOMO‐LUMO gap energy, and plots of electrostatic potential. It became evident that the commonly used smearing values of ≥0.001 Ha can alter the results significantly (for example, by one order of magnitude for HOMO–LUMO gap energy). We suggest the setting of electron smearing value at 0.0001 Ha, which does not imply too high computation cost and can guarantee the results close to the ones obtained with Fermi occupation. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Theoretical and experimental investigation on the electronic properties of the shuttlecock shaped and the double-decker structured metal phthalocyanines, MPc and M(Pc)2 (M = Sn and Pb)

The molecular geometries, electronic structures, and excitation energies of tin and lead phthalocyanine compounds, SnPc, PbPc, Sn(Pc)(2), and Pb(Pc)(2), were investigated using the B3LYP method within a framework of density functional theory (DFT). The geometries of SnPc, PbPc, Sn(Pc)(2), and Pb(Pc)(2) were optimized under C(4v), C(4v), D(4d), and D(4d) molecular symmetries, respectively. The excitation energies of these molecules were computed by the time-dependent DFT (TD-DFT) method. The calculated results for the excited states of three compounds other than the unknown Pb(Pc)(2) corresponded well with the experimental results of electronic absorption spectroscopy. The non-planar C(4v) molecular structure of SnPc and PbPc influences especially on the orbital energy of the HOMO-1 through mixing of the s-type atomic orbital of the central metal atom to the π system of the Pc ring in an anti-bonding way; however, the HOMO and the LUMO have little effect of the deviation from the planar structure because they have no contribution from the atomic orbital of the central metal. This orbital mixing pushes up the orbital energy of the HOMO-1, and reduces the energy of the metal-to-ligand charge transfer band of SnPc and PbPc. The calculated results also reproduced well the excitation profile of Sn(Pc)(2), which was quite different from that of SnPc. The strong interactions between the π-type orbitals of two Pc moieties altered the electronic structure resulting in the characteristic excitation profile of Sn(Pc)(2). In addition, this caused a reduction of about 0.8 eV in the ionization potential as compared to usual MPcs including SnPc, which was consistent with the experimental results.     

电场对(4, 0)Zigzag模型单壁碳纳米管的影响

The structural and electronic properties of a (4, 0) zigzag single-walled carbon nanotube (SWCNT) under parallel and transverse electric fields with strengths of 0-1.4×10~(-2) a.u. Were studied using the density functional theory (DFT) B3LYP/6-31G~* method. Results show that the properties of the SWCNT are dependent on the external electric field. The applied external electric field strongly affects the molecular dipole moments. The induced dipole moments increase linearly with increase in the electrical field intensities. This study shows that the application of parallel and transverse electric fields results in changes in the occupied and virtual molecular orbitals (Mos) but the energy gap between the highest occupied MO (HOMO) and the lowest unoccupied MO (LUMO) of this SWCNT is less sensitive to the electric field strength. The electronic spatial extent (ESE) and length of the SWCNT show small changes over the entire range of the applied electric field strengths. The natural bond orbital (NBO) electric charges on the atoms of the SWCNT show that increase in the external electric field strength increases the separation of the center of the positive and negative electric charges of the carbon nanotube.     

Structural,spectroscopic aspects,and electronic properties of (TiO2)n clusters: A study based on the use of natural algorithms in association with quantum chemical methods

In this article, we propose a stochastic search‐based method, namely genetic algorithm (GA) and simulated annealing (SA) in conjunction with density functional theory (DFT) to evaluate global and local minimum structures of (TiO₂)_n clusters with n = 1–12. Once the structures are established, we evaluate the infrared spectroscopic modes, cluster formation energy, vertical excitation energy, vertical ionization potential, vertical electron affinity, highest occupied molecular orbital (HOMO)‐lowest unoccupied molecular orbital (LUMO) gaps, and so forth. We show that an initial determination of structure using stochastic techniques (GA/SA), also popularly known as natural algorithms as their working principle mimics certain natural processes, and following it up with density functional calculations lead to high‐quality structures for these systems. We have shown that the clusters tend to form three‐dimensional networks. We compare our results with the available experimental and theoretical results. The results obtained from SA/GA‐DFT technique agree well with available theoretical and experimental data of literature. © 2013 Wiley Periodicals, Inc.     

DFT study of the geometrical and electronic structure of substituted cumulenes in netral and cationic forms

The geometrical and basic energy parameters of monosubstituted cumulenes and their singly and doubly charged cations were calculated by the Hartree-Fock and density functional (DFT) methods at a B3LYP level of theory using the 6-31G(d) basis set. The substituent was fluorine, cyan, amino group, phenyl, cyanophenyl, aminophenyl, or dimethylaminophenyl. In extended linear carbon systems based on cumulene, rotation of a terminal fragment depends on the character of the highest occupied molecular orbital (HOMO) from which electrons are removed. The terminal group rotates through 90 only when the contribution of electron density from the π molecular orbital (MO) of unsubstituted cumulene to the HOMO of substituted cumulene is over 70%. Otherwise, the terminal group rotates through a smaller angle; with a contribution of less than 30%, the dication is planar in any substituted cumulene. Thus quantitative criteria have been determined to evaluate the specific structural effect due to ionization of substituted cumulenes.