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.     

Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory

The conventional analysis of Perdew and Levy, and Sham and Schlu?ter shows that the functional derivative discontinuity of the exchange-correlation density functional plays a critical role in the correct prediction of bandgaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for bandgap prediction with all common types of exchange-correlation functionals have been derived without invoking the concept of exchange-correlation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital (HOMO) in describing electron removal. (2) Every term in the total energy functional contributes to the energy gap because of the discontinuity of the derivative of the density (or density matrix) with respect to the number of electrons, ((?ρ(s)(r('),r))/?N)(v(s) ), at integers. (3) Consistent with the Perdew-Levy-Sham-Schlu?ter conclusion that the exact Kohn-Sham energy gap differs from the fundamental bandgap by a finite correction due to the functional derivative discontinuity of the exchange-correlation energy, we show that the exchange-correlation functional cannot be an explicit and differentiable functional of the electron density, either local or nonlocal. The last result is further strengthened when we consider Mott insulators. There, the exact exchange-correlation functional needs to have an explicitly discontinuous (nondifferentiable) dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spin-up and spin-down number of electrons.     

SCF?CI and SCF open-shell studies of the base components of the nucleic acids

The π-electron structure of adenine, guanine, cytosine, thymine and uracil in their ground, ionized, singlet and triplet excited states are investigated by means of the SCF ? CI and SCF open-shell methods. The calculations for singlets fit the maxima of the absorption bands well. The energy difference between the first and the second singlet states of adenine is found to be very small. The open-shell method leads to the same relative ionization potential as does the SCF (with the integrals empirically corrected). The calculated energies of the triplet states almost coincide in the SCF open-shell and the SCF ? CI approximation. The calculated transition energies to the first triplet state of the pyrimidines are higher than in the case of the purines. The value of the singlet–triplet separation energy of purines is in agreement with experimental data.     

Computation of the hardness and the problem of negative electron affinities in density functional theory

The absolute hardness in density functional theory (DFT) is discussed, emphasizing the charge-transfer excitation interpretation. Direct evaluation from the computed ionization potential and electron affinity is intrinsically problematic when the affinity is negative; the calculated affinity exhibits a strong basis set dependence, becoming near zero as diffuse functions are added. An alternative Koopmans-based approximation using local functional eigenvalues uniformly and significantly underestimates the hardness. A simple correction to the Koopmans expression is highlighted on the basis of a consideration of the integer discontinuity. The resulting hardness expression does not require the explicit computation of the affinity and has a straightforward interpretation in terms of the electronegativity. The correction eliminates the underestimation and gives hardness values that do not degrade as the electron affinity becomes more negative. For systems with large negative affinities, the values are an improvement over those from the other approaches. The success can be traced to an implicit, unconventional approximation for the electron affinity, which outperforms the standard approach when the affinity is significantly negative and which does not break down as the basis set becomes more diffuse.     

Difficulties in the computation of physical observables due to a truncated CI

It is proved that the values of various types of physical observables issued from a truncated CI are not size consistent. These observables include electronic transitions, ionization potential, and all the quantities depending on the one-electron density matrix γ. Numerical examples are given. It is found that the error due to this size inconsistency can be important for systems of moderate size (e.g., 16% on a matrix element of γ for a 20-electron system).     

Density functionals from many-body perturbation theory: the band gap for semiconductors and insulators

Theoretically the Kohn-Sham band gap differs from the exact quasiparticle energy gap by the derivative discontinuity of the exchange-correlation functional. In practice for semiconductors and insulators the band gap calculated within any local or semilocal density approximations underestimates severely the experimental energy gap. On the other hand, calculations with an "exact" exchange potential derived from many-body perturbation theory via the optimized effective potential suggest that improving the exchange-correlation potential approximation can yield a reasonable agreement between the Kohn-Sham band gap and the experimental gap. The results in this work show that this is not the case. In fact, we add to the exact exchange the correlation that corresponds to the dynamical (random phase approximation) screening in the GW approximation. This accurate exchange-correlation potential provides band structures similar to the local density approximation with the corresponding derivative discontinuity that contributes 30%-50% to the energy gap. Our self-consistent results confirm substantially the results for Si and other semiconductors obtained perturbatively [R. W. Godby et al., Phys. Rev. B 36, 6497 (1987)] and extend the conclusion to LiF and Ar, a wide-gap insulator and a noble-gas solid.     

Atomic and molecular impurities in4He clusters

Recoil-ion charge distributions produced in single collisions of 8 MeV/u Kr¹³⁺ and Kr³²⁺ projectiles with Xe atoms have been measured using time-of-flight spectroscopy. The post-collision charge states of the projectile ions were determined by magnetic dispersion onto a position sensitive microchannel plate detector. The recoil-ion distributions for ionization accompanied by electron loss from Kr¹³⁺ projectiles were bell-shaped with averages that ranged from 7.4 for 1-electron loss to 13.9 for 5-electron loss. The recoil-ion distributions for ionization accompanied by electron capture to Kr³²⁺ projectiles were also bell-shaped, but had much higher average charges that ranged from 20.4 for 1-electron capture to 28.5 for 4-electron capture. The large difference in the average charges produced in the two types of collisions is mainly attributable to charge magnification by Auger decay. A simple model quantitatively explains the variation of the capture-ionization charge distribution width and average charge as a function of the number of captured electrons.     

Solvation effect on the vertical ionization energy of adenine-thymine base pair: From microhydration to bulk

In the present work, we investigate the effect of aqueous environment on the vertical ionization potential (VIP) of adenine-thymine (AT) base pair using a multilayer equation of the motion-coupled cluster method. The microsolvation can cause both blue-shift and red-shit of the IP values. However, the bulk water environment always results in the red-shift of the vertical ionization potential. Our study shows that the correct treatment of the short-range interaction plays an essential role in determining the magnitude of the red-shift. We have developed a biased sampling scheme based on Koopmans' energy, which can significantly speed up the convergence with respect to the number of solvent-solute configurations.     

Longitudinal polarizability of long polymeric chains: quasi-one-dimensional electrostatics as the origin of slow convergence

The longitudinal linear polarizability alpha(N) of a stereoregular oligomer of size N is proportional to N in the large-N limit, provided the system is nonconducting in that limit. It has long been known that the convergence of alpha(N)/N to the asymptotic alpha(infinity) value is slow. We show that the leading term in the difference between alpha(N)/N and alpha(infinity) is of the order of 1/N. The difference [alpha(N)-alpha(N-1)], as well as alpha(center)(N) (when computationally accessible), also converge to alpha(infinity), but faster, the leading term being of the order of 1/N(2). We also present evidence that in these cases the power law convergence behavior is due to quasi-one-dimensional electrostatics, with one exception. Specifically, in molecular systems the difference between alpha(N)/N and alpha(infinity) has not just one but two sources of the O(1/N) term, with one being due to the aforementioned Coulomb interactions, and the second due to the short ranged exponentially decaying perturbations on chain ends. The major role of electrostatics in the convergence of the remainders is demonstrated by means of a Clausius-Mossotti-type classical model. The conclusions derived from the model are also shown to be applicable in molecular systems, by means of test-case ab initio calculations on linear stacks of H(2) molecules, and on polyacetylene chains. The implications of the modern theory of polarization for extended systems are also discussed.     

Femtosecond laser ionization of organic amines with very low ionization potentials: relatively small suppressed ionization features

We examined the femtosecond nonresonant ionization of organic amines with vertical ionization potentials as low as 5.95 eV. The quantitative evaluation of suppressed ionization relative to the single active electron approximation model was done by comparing the saturation intensity, I(sat), in experiments and theory. ADK theory was found to be useful in predicting the ionization yield in the I(sat) scale within a factor of 2, even for molecules with very low ionization potentials. The degree of suppression was, however, smaller than that of benzene. The localization of electrons on the nitrogen atom was found to affect the ionization behavior under the strong laser field. The delocalized pi electrons in benzene could not follow the laser field adiabatically, while those in localized molecular orbitals could. In addition, the growth of a tunneling barrier due to the screening effect in amines may be relatively smaller than that in benzene.     

Correlation problems in atomic and molecular systems. VII. Application of the open-shell coupled-cluster approach to simple π-electron model systems

The coupled-cluster variational-like direct approach to the calculation of ionization and electron attachment energies and of excitation energies is applied to several π-electron model systems, using the PPP Hamiltonian with various parametrizations. A simple approximation, which represents the triexcited clusters in terms of disconnected W₁T₂ terms, is employed. All the necessary diagrams for both excitation energy and ionization potential (electron affinity) calculations are given in the compact Hugenholtz nonoriented form. The results of the calculations for benzene, trans-butadiene, all-trans-hexatriene, and fulvene are compared with the corresponding full CI results, and the conclusions about the validity and efficiency of this approach are drawn.     

Bound state properties of H2 by the many-body green's function method

The Dyson equation for the one-electron Green's function is solved through approximating the optical potential by means of the functional derivative method. The results for the H₂ molecule, including ground state energy and the first six vertical ionization energies, are discussed and compared with experimental values.     

BrCl紫外光电子能谱实验及理论研究

采用紫外光电子能谱研究了影响大气臭氧浓度的重要卤素互化物一氯化溴的精细电离能谱．实验得到BrCl的第一绝热电离能和垂直电离能分别为10．95eV和11．00eV．BrCl的最高占据轨道6π电离产生了明显的旋轨分裂谱带．这对旋轨分裂谱带分别清晰地显示出4个振动精细结构峰．频率分析显示BrCl分子最高占据轨道为弱反键性质．比较了HF方法和外壳层格林函数方法(OVGF)对电离能的计算结果,并对实验值进行了分析比较及指认．采用实验构型OVGF方法给出的电离能结果无论在低电离能区还是在高电离能区都和实验值一致,特别是第一垂直电离能10．988eV与实验值11．00eV非常好地吻合．     

氯代苯阳离子的密度泛函理论研究

采用B3LYP方法及6-311G(d,p)和6-311+G(d,p)基组,对12种氯代苯阳离子进行了理论研究,优化其电子基态的结构,计算了对应分子的垂直电离势(VIP)和绝热电离势(AIP).依据Jahn-Teller理论,确定了1,3,5-C6H3Cl3+和C6Cl6+离子分别具有C2v(2B1)和D2h(2B2g)结构(对应分子分别为D3h和D6h结构).其余10个离子的构型的对称点群与对应分子相同,但构型参数值有明显差别.用B3LYP方法计算的各氯代苯分子的垂直电离势和绝热电离势与实验值符合得很好.     

Electrodonating and electroaccepting powers

By introducing an electron bath that represents the chemical environment in which a chemical species is immersed, and by making use of the second-order Taylor series expansions of the energy as a function of the number of electrons in the intervals between N - 1 and N, and N and N + 1, we show that the electrodonating (omega-) and the electroaccepting (omega+) powers may be defined as omega-/+ = (mu-/+)2/2eta-/+, where mu-/+ are the chemical potentials and eta-/+ are the chemical hardnesses, in their corresponding intervals. Approximate expressions for omega- and omega+ in terms of the ionization potential I and the electron affinity A are established by assuming that eta- = eta+ = eta = mu+ - mu-. The functions omega-/+(r) = omega-/+f -/+(r), where f -/+(r) are the directional Fukui functions, derived from a functional Taylor series for the energy functional truncated at second order, represent the local electrodonating and electroaccepting powers.     

The structural and electronic properties of In(n)N(n = 1-13) clusters

The structural and electronic properties of In(n)N(n=1-13) clusters have been investigated by density-functional theory with the generalized gradient approximation. The results indicate that the equilibrium structures of In(n)N are linear for n=1,2, planar for n=3-5, and three dimensional for n=6-13. Maximum peaks were observed for In(n)N clusters at n=3,7,9 on the size dependence for second-order energy difference. These imply that these clusters possess relatively higher stability, which is consistent with the case of binding energy per atom. Moreover, the results show that the bonding in small In(n)N clusters has a little ionic character by Mulliken population analysis. The energy gap between the highest occupied and lowest unoccupied molecular orbitals, the vertical ionization potential and electron vertical affinity (VIP and VEA) form an even-odd alternating pattern with increasing cluster size. In general, the VIP tends to lower as the cluster size increases, while the VEA tends to increase as the cluster size increases.     

Intermediate state representation approach to physical properties of dicationic states

The second-order algebraic construction (ADC(2)) approach to the two-particle (pp) propagator, devised to compute double ionization energies and associated spectroscopic amplitudes, is reformulated and extended using the concept of intermediate state representations (ISR). The ISR formulation allows one to go beyond the general limitations inherent to the propagator approach, as here (N-2)-electron wave functions and properties become directly accessible. The (N-2)-electron ISR(2) equations for a general one-particle operator have been derived and implemented in a recent version of the double ionization ADC(2) program. As a first test of the method, the dipole moments of a series of 2h states of LiH, HF, and H(2)O were computed and compared to the results of a full configuration interaction (FCI) treatment. The dipole moments obtained at the ADC(2)/ISR(2) computational level are in good agreement with the FCI results.     

内掺三金属氮化物富勒烯Er3N@C80的几何结构与电子性质理论研究

采用密度泛函理论中的广义梯度近似对Ih-Er3N@C80的几何结构和电子性质进行理论研究. 结构优化发现当Er3N的三个Er原子偏离Ih-C80中五边形和六边形公共边中点时形成的结构最稳定. Er—C和Er—N键是离子键. 能级分析可知: 掺入Er3N后, 碳笼的稳定性提高. 局部态密度图和自旋聚居数分析表明: Er3N的掺入使Ih-C80磁性变大, 而Er3N仍然具有一定的磁性. 垂直电离能和垂直亲和能分析表明: 掺入Er3N后, 碳笼的得失电子能力都有所降低.