Basis set effects on frontier molecular orbital energies and energy gaps: a comparative study between plane waves and localized basis functions in molecular systems

In order to study the Kohn-Sham frontier molecular orbital energies in the complete basis limit, a comparative study between localized functions and plane waves, obtained with the local density approximation exchange-correlation functional is made. The analyzed systems are ethylene and butadiene, since they are theoretical and experimentally well characterized. The localized basis sets used are those developed by Dunning. For the plane-waves method, the pseudopotential approximation is employed. The results obtained by the localized basis sets suggest that it is possible to get an estimation of the orbital energies in the limit of the complete basis set, when the basis set size is large. It is shown that the frontier molecular orbital energies and the energy gaps obtained with plane waves are similar to those obtained with a large localized basis set, when the size of the supercell and the plane-wave expansion have been appropriately calibrated.     

Second hyperpolarizability (gamma) of singlet diradical system: dependence of gamma on the diradical character

The dependence of the second hyperpolarizability (gamma) on the diradical character (y) for singlet diradical systems is investigated using a model compound, the p-quinodimethane (PQM) molecule with different both-end carbon-carbon (C-C) bond lengths, by several ab initio molecular orbital and density functional theory methods. The diradical character based on UHF calculations indicates that at equilibrium geometry PQM is in a singlet ground state and primarily exhibits a quinoid structure, whereas the diradical character increases when increasing both-end C-C bond lengths. At the highest level of approximation, that is, using the UCCSD(T) method with the 6-31G+diffuse p (zeta = 0.0523) basis set, the longitudinal static gamma of PQM presents a maximum value for intermediate diradical character (y approximately 0.5) while the gamma values are larger for intermediate and large diradical character (y approximately 0.5-0.7) than for small diradical character (y < 0.2). This feature suggests that the gamma values of singlet diradical systems in the intermediate and somewhat strong correlation regimes are significantly enhanced as compared to those in the weak correlation regime. These results are substantiated by a complementary study of the variation in gamma upon twisted ethylene.     

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.     

(Hyper)polarizability density analysis for open-shell molecular systems based on natural orbitals and occupation numbers

We have developed a method for analyzing the (hyper)polarizabilities of open-shell molecular systems. This method employs the (hyper)polarizability densities based on the natural orbitals and occupation numbers, which enables us to analyze the contributions of odd electrons having various open-shell (diradical) characters. Within broken-symmetry, i.e., spin-unrestricted, single-determinant molecular orbital and density functional theory approaches, we can also remove the spin contamination effects on these quantities through spin projection. To do that, an approximate spin projected method has been elaborated and applied to the analysis of the (hyper)polarizability of multi-radical systems. As examples, typical open-shell singlet systems, 1,3-dipoles and rectangular graphene nanoflakes, are examined.     

Multicomponent density functional theory study of the interplay between electron-electron and electron-proton correlation

The interplay between electron-electron and electron-proton correlation is investigated within the framework of the nuclear-electronic orbital density functional theory (NEO-DFT) approach, which treats electrons and select protons quantum mechanically on the same level. Recently two electron-proton correlation functionals were developed from the electron-proton pair densities obtained from explicitly correlated wavefunctions. In these previous derivations, the kinetic energy contribution arising from electron-proton correlation was neglected. In this paper, an electron-proton correlation functional that includes this kinetic energy contribution is derived using the adiabatic connection formula in multicomponent DFT. The performance of the NEO-DFT approach using all three electron-proton correlation functionals in conjunction with three well-established electronic exchange-correlation functionals is assessed. NEO-DFT calculations with these electron-proton correlation functionals capture the increase in the hydrogen vibrational stretching frequencies arising from the inclusion of electron-electron correlation in model systems. Electron-proton and electron-electron correlation are found to be uncoupled and predominantly additive effects to the total energy for the model systems studied. Thus, electron-proton correlation functionals and electronic exchange-correlation functionals can be developed independently and subsequently combined together without re-parameterization.     

Second hyperpolarizabilities of singlet polycyclic diphenalenyl radicals: effects of the nature of the central heterocyclic ring and substitution to diphenalenyl rings

Adopting density functional theory and a hybrid exchange-correlation functional, the relationship between the second hyperpolarizability (gamma) and the diradical character has been investigated for diphenalenyl-based compounds containing different heterocyclic five-membered central rings (C(4)H(4)X, where X = NH, PH, O, S, CH(2), SiH(2), BH, GaH, C=O, C=S, and C=Se) or substituted by donor (NH(2))/acceptor(NO(2)) groups. It turns out that these structural modifications can tune the diradical character from 0.0 to 0.968 and lead to variations of gamma over more than 1 order of magnitude, demonstrating the controllability of gamma in this family of compounds. In particular, when the central ring is strongly aromatic, the diradical character is larger than 0.7, which is associated with pretty large gamma values except for almost the pure diradical case (y approximately 1). On the other hand, when the aromaticity decreases--or the antiaromaticity increases--the diradical character and the second hyperpolarizability get smaller. These relationships are correlated to structural (bond length alternation) and charge distribution (charge transfer between the phenalenyl rings and the central ring) properties, which account for the relative importance of the resonance diradical, zwitterionic, and quinoid forms. Therefore, the diradical character and the second hyperpolarizability can be controlled by the aromaticity of the ring while the paradigm of the enhancement of gamma for intermediate diradical character is globally verified. Then, upon introducing donor groups, the zwitterionic character increases, leading to closed-shell species and small second hyperpolarizabilities. In the case of substitution by acceptor groups, the charge transfer is reduced but the diradical character and the second hyperpolarizability hardly changes.     

Towards understanding of magnetic interactions within a series of tetrathiafulvalene-pi conjugated-verdazyl diradical cation system: a density functional theory study

The intramolecular magnetic exchange coupling constants (J) for a series of tetrathiafulvalene (TTF) and verdazyl diradical cations connected by a range of pi conjugated linkers have been investigated by means of methodology based on unrestricted density functional theory. The magnetic interaction between radicals is transmitted via pi-electron conjugation for all considered compounds. The calculation of J yields strong or medium ferromagnetic coupling interactions (in the range of 56 and 300 K) for diradical cations connected by linkers with an even number of carbon atoms that are able to provide a spin polarization pathway, while antiferromagnetic coupling is predicted when linkers with an odd number of carbon atoms are employed. The topological analysis of spin density distributions have been used to reveal the effects of the spin polarization on both linkers and spin carriers. The absence of heteroatoms that impede the spin polarization pathway, and the existence of a unique spin polarization path instead of several possible competitive routes are factors which contribute to large positive J values favoring ferromagnetic interactions between the two terminal pi-radicals. The magnitude of J depends strongly on the planarity of the molecular structure of the diradical cation since a more effective orbital overlap between the two pi-systems can be achieved. Hence, the dependence of J on the torsion angle (theta) of each spin carrier has been analyzed. In this respect, our findings show that this geometrical distortion reduces largely the calculated J values for ferromagnetic couplings, leading to weak antiferromagnetic interactions for a torsion angle of 90 degrees .     

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.     

双咪唑苯和双三唑苯及其衍生物非线性光学性质的密度泛函研究

采用密度泛函理论(DFT)的UB3LYP(B3LYP)/6-31+G**方法对双咪唑苯和双三咪唑苯双自由基及其衍生物几何结构进行优化,并结合有限场(FF)方法计算这些体系的非线性光学(NLO)系数.结果表明,引入给、受体取代基都能使体系的极化率α和二阶超极化率γ增大.在双自由基体系中,引入给体NH2的α和γ值大于引入受体NO2的值,与闭壳层体系中结果相反.分析自由基成分和电荷对体系的二阶超极化率γ影响的结果表明,处于中间双自由基成分的分子比相似共轭性的闭壳层分子有更大的二阶超极化率γ;带电荷的双自由基体系引入给、受体之后,与中性自由基体系相比具有更大的二阶超极化率γ.     

Koopmans-like approximation in the Kohn-Sham method and the impact of the frozen core approximation on the computation of the reactivity parameters of the density functional theory

A Koopmans-like approximation is introduced in the spin-polarized version of the Kohn-Sham (KS) density functional theory to obtain a relation between KS orbital energies and vertical ionization potential and electron affinity. Expressions for reactivity indexes (like electronegativity, hardness, electrophilicity, and excitation energies) include KS frontier orbital energies and additional contributions associated with the self-interaction correction. Those reactivity parameters were computed with different exchange-correlation functionals to test the approach for a set of small molecules. The results show that the present approximation provides a better way to estimate hardness, electronegativity, and electrophilicity than just the use of frontier orbital energy values. However KS HOMO and LUMO energy gap gives a better agreement with excitation energies.     

Optical absorption and luminescence energies of F centers in CaO from ab initio embedded cluster calculations

We calculated the optical absorption and luminescence energies of electrons trapped at oxygen vacancies in CaO using a consistent embedded cluster method which accounts for the long-range polarization effects and partial covalence of CaO. Optical absorption and luminescence energies of neutral (F center) and positively charged (F+ center) vacancies are calculated by means of time dependent density functional theory using the B3LYP exchange-correlation density functional. Our results demonstrate that using large basis sets to describe a diffuse nature of excited states, and properly accounting for long-range polarization induced by charged and excited defect states, is crucial for accurate predictions of optical excitation and luminescence energies of these defects.     

Communication: Effect of the orbital-overlap dependence in the meta generalized gradient approximation

We study for the first time the effect of the dependence of meta generalized gradient approximation (MGGA) for the exchange-correlation energy on its input, the kinetic energy density, through the dimensionless inhomogeneity parameter, α, that characterizes the extent of orbital overlap. This leads to a simple MGGA exchange functional, which interpolates between the single-orbital regime, where α = 0, and the slowly varying density regime, where α ≈ 1, and then extrapolates to α → ∞. When combined with a variant of the Perdew-Burke-Ernzerhof GGA correlation, the resulting MGGA performs equally well for atoms, molecules, surfaces, and solids.     

Density functional and molecular orbital study of physical process of inversion of nitrogen trifluoride (NF3) molecule

The physical process of the umbrella inversion of the nitrogen trifluoride molecule has been studied invoking the formalisms of the density functional theory, the frontier orbital theory, and the molecular orbital theory. An intuitive structure and dynamics of evolution of the transition state for the event of inversion is suggested. The physical process of dynamic evolution of the molecular conformations between the equilibrium (C_3v) shape and the planar (D_3h) transition state has been followed by a number of molecular orbital and density functional parameters like the total energy, the eigenvalues of the frontier orbitals, the highest occupied molecular orbital and lowest unoccupied molecular orbital, the (HOMO–LUMO) gap, the global hardness and softness, and the chemical potential. The molecular conformations are generated by deforming the ∠FNF angle through steps of 2° from its equilibrium value, and the cycle is continued till the planar transition state is reached, and the geometry of each conformation is optimized with respect to the length of the N? F bond. The geometry optimization demonstrates that the structural evolution entails an associated slow decrease in the length of the N? F bond. The dipole moment at the equilibrium form is small and that at the transition state is zero and shows a strange behavior with the evolution of conformations. As the molecular structure begins to distort from its equilibrium shape by opening of the ∠FNF angle, the dipole moment starts increasing very sharply, and the trend continues very near to the transition state but abruptly vanishes at the transition state. A rationale of the strange variation of dipole moment as a function of evolution of conformations could be obtained in terms of quantum mechanical hybridization of the lone pair on the N atom. The pattern of charge density reorganization as a function of geometry evolution is a continuous depletion of charge from the F center and piling up of charge on the N center. The continuous shortening of bond length and the pattern of variation of net charge densities on atomic sites with evolution of molecular conformations predicts that the bond moment would decrease continuously. The quantum mechanical hybridization of the lone pair of the central N atom shows that the percentage of s character of the lone‐pair hybrid on the N atom decreases at a very accelerated rate, and the lone pair at the transition state is accommodated in a pure p orbital. The result of the continued destruction of asymmetry of charge distribution in the lone pair on the central N atom due to the elimination of contribution of the s orbital with evolution of molecular conformations is the sharp decrease in lone‐pair moment. The decrease in bond moment is overcompensated by the sharp fall of its offsetting component, the lone‐pair moment, resulting in a net gain in dipole moment with the evolution of molecular geometry. Since the offsetting component decreases very sharply, the net effect is a sharp rise of dipole moment with the evolution of molecular conformations just before the transition state. The lone‐pair moment is zero by virtue of the symmetry of the pure p orbital, the lone pair of the central atom in the transition state, and the sum of the bond moments is zero by symmetry of the geometry. The barrier height is quite high at ～65.45 kcal/mol, which is close to values computed through more sophisticated methods. It is argued that an earlier suggestion regarding the development of high barrier value of NF₃ system seems to be misleading and confronting with the conclusions of the density functional theory. An analysis and a comparative study of the physical components of the one‐ and two‐center energy terms reveals that the pattern of the charge density reorganization has the principal role in deciding the origin and the magnitude of barrier of inversion of the molecule and the barrier originates not from a particular energetic effect localized in a particular region of the molecule, rather the barrier originates from a subtle interplay of one‐ and two‐center components of the total energy. The decomposed energy components show that the F?F nonbonded interaction and N? F bonded interaction favor the formation of transition state, while the one‐center energy terms prohibit the formation of the transition state. The barrier principally develops from the one‐center energy components. The profile of the HOMO is isomorphic and that of the LUMO is homomorphic with the potential energy curve for the physical process of the event of umbrella inversion of the molecule. The variation of the HOMO–LUMO gap, ?ε, the global hardness, η, and the softness, S, as a function of the reaction coordinates of angular deformation of NF₃ molecule are quite consistent with the predictions of the molecular orbital and the density functional theories in connection with the deformation of molecular geometry. The profiles of ?ε, η, and S, as a function of reaction coordinates, mimic the potential energy curve of the molecule. The eigenvalues of the frontier orbitals, and the ?ε, η, S parameters are found to be equally effective theoretical parameters, like the total energy, to monitor the physical process of the inversion of pyramidal molecules. The nature of the variation of the global hardness parameter between the equilibrium shape and the transition state form for the inversion is in accordance with the principle of maximum hardness (PMH). © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002     

Effect of double bonds on the conducting properties of ciguatoxin 3C and tetrahydropyrane-based polymers: a theoretical study

The electronic structure of the ciguatoxin 3C is analyzed through the Kohn-Sham model by using two different kinds of basis sets: localized basis set (Gaussian functions) and nonlocalized basis set (plane wave functions). With the localized basis functions, two approximations are used for the exchange-correlation functional: the local density approximation and the generalized gradient approximation. With the nonlocalized basis set, just the local density approximation is used. The energy gap, obtained from the frontier molecular orbitals, for this molecule predicts that this system is a semiconductor, even when the number of double bonds is increased inside the structure. However, as large molecules built with the basic unit--the tetrahydropyrane--of the ciguatoxin 3C are found in nature, it suggests studying the gap in polymeric systems built with the basic unit of this molecule. It is demonstrated that the presence of double bonds reduces considerably the gap, indicating the possibility of forming conducting materials by introducing double bonds in this kind of molecular systems. Thus, molecules strongly linked with biological systems can be used as precursor to build electric conducting systems.     

Induced currents and electron counting in aromatic boron wheels: B8(2-) and B9-)

The newly discovered atom-centered polygonal wheels B8(2-) and B9- are predicted to show ring currents characteristic of aromatic systems. Ipsocentric mapping of induced current density for both molecules attributes a pi diatropic current to the four electrons of the doubly degenerate pi HOMO and a sigma diatropic current to the four electrons of the doubly degenerate sigma HOMO, each orbital pair having an available transition to corresponding LUMO orbitals in which the angular node count increases by one. Thus, on the magnetic criterion, B8(2-) and B9- are each both pi- and sigma-aromatic as a consequence of the nodal properties of the frontier orbitals of the pi- and sigma-stacks.     

Size dependences of the diradical character and the second hyperpolarizabilities in dicyclopenta-fused acenes: relationships with their aromaticity/antiaromaticity

Using long-range corrected density functional theory, the relationships between the electronic, magnetic, and nonlinear optical properties are drawn for two families of organic compounds, the dicyclopenta-fused acenes (DPAs) and the polyacenes (PAs), containing up to N = 12 fused rings. First, the longitudinal second hyperpolarizability (γ) of singlet DPAs is significantly enhanced with increasing system size, in comparison to PAs. This behavior is associated with an increase in the longitudinal spin polarization between the terminal five-membered rings of DPAs and is consistent with previous studies where γ is maximized for intermediate diradical character. The size dependence of the diradical character is also found to cause a hump in the γ/N evolution for singlet DPAs around N = 8. In fact, in the case of singlet PAs, the diradical characters y(0) and y(1), the various magnetic properties and the γ/N values vary monotonically with N, whereas for singlet DPAs, the shielding, the magnetizability, and the γ/N values exhibit extrema near N = 8 due to the appearance of transversal spin polarization in the middle six-membered rings in addition to the longitudinal spin polarization between the terminal five-membered rings. Moreover, it is shown that for singlet DPAs the longitudinal spin polarization (characterized by y(0)) is associated with the antiaromaticity (N ≤ 3) and the slight- or non-aromaticity (N ≥ 4) of the terminal five-membered rings, whereas the appearance of transversal spin-polarization (characterized by y(1)) is associated with the decrease in the aromaticity in the inner six-membered rings as shown for large PAs. Therefore, the exceptional behaviors in singlet DPAs for small N (N < 9) are caused by the increase in diradical character y(0) correlated with the anti-aromaticity or the slight-/non-aromaticity of terminal rings and the corresponding emergence of a global aromatic character. Such a relationship between the aromaticity/antiaromaticity and the diradical character is useful for designing real open-shell NLO molecules through the control of their diradical characters.     

Mechanism of the forbidden [3s,5s]-sigmatropic shift: orbital symmetry influences stepwise mechanisms involving diradical intermediates

The mechanisms of [3s,5s]-sigmatropic shifts of octa-1,3,7-triene and 7-methylenenona-1,3,8-triene have been elaborated using B3LYP and BPW91 density functional theory and CASPT2 methods. These orbital symmetry forbidden rearrangements are stepwise, involving diradical intermediates. A comparison with several [3,3]-sigmatropic shifts of substituted hexadienes and of [5,5]-sigmatropic shifts that are allowed, but nevertheless follow stepwise paths, shows that the activation barrier for the disallowed [3,5] shift is significantly larger than that for the stepwise reactions that are orbital symmetry allowed. Cyclic diradicals that have an aromatic circuit of electrons including the two radical centers and conjugated pi or sigma bonds are stabilized as compared to cyclic diradicals with an antiaromatic circuit of electrons. This applies to the transition states leading to and from the diradicals and influences the activation energies of stepwise sigmatropic shifts. The magnitudes of these effects are small but will have a significant influence on the rates of competing processes. This series of calculations has been used to assess the relative capabilities of the two functionals. We find that BPW91 underestimates the endothermicity of diradical formation and the barrier to diradical formation whereas B3LYP overestimates these quantities.