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The atom-atom polarizability and the transmission probability at the Fermi level, as obtained through the source-and-sink-potential method for every possible configuration of contacts simultaneously, are compared for polycyclic aromatic compounds. This comparison leads to the conjecture that a positive atom-atom polarizability is a necessary condition for transmission to take place in alternant hydrocarbons without non-bonding orbitals and that the relative transmission probability for different configurations of the contacts can be predicted by analyzing the corresponding atom-atom polarizability. A theoretical link between the two considered properties is derived, leading to a mathematical explanation for the observed trends for transmission based on the atom-atom polarizability. [ABSTRACT FROM AUTHOR]

We present the finalized Buckingham-Hirshfeld method (BHD-DFT) for the evaluation of interaction energies of non-bonded dimers with Density Functional Theory (DFT). In the method, dispersion energies are evaluated from static multipole polarizabilities, obtained on-the-fly from Coupled Perturbed Kohn-Sham calculations and partitioned into diatomic contributions using the iterative Hirshfeld partitioning method. The dispersion energy expression is distributed over four atoms and has therefore a higher delocalized character compared to the standard pairwise expressions. Additionally, full multipolar polarizability tensors are used as opposed to effective polarizabilities, allowing to retain the anisotropic character at no additional computational cost. A density dependent damping function for the BLYP, PBE, BP86, B3LYP, and PBE0 functionals has been implemented, containing two global parameters which were fitted to interaction energies and geometries of a selected number of dimers using a bi-variate RMS fit. The method is benchmarked against the S22 and S66 data sets for equilibrium geometries and the S22x5 and S66x8 data sets for interaction energies around the equilibrium geometry. Best results are achieved using the B3LYP functional with mean average deviation values of 0.30 and 0.24 kcal/mol for the S22 and S66 data sets, respectively. This situates the BHD-DFT method among the best performing dispersion inclusive DFT methods. Effect of counterpoise correction on DFT energies is discussed. [ABSTRACT FROM AUTHOR]

On the basis of the zero-temperature grand canonical ensemble generalization of the energy E[N,Ns,v,B] for fractional particle N and spin Ns numbers, the energy surface over the (N,Ns) plane is displayed and analyzed in the case of homogeneous external magnetic fields B(r[accent:_right_hook_over]). The (negative of the) left-/right-side derivatives of the energy with respect to N, N↑, and N↓ give the fixed-Ns, spin-up, and spin-down ionization potentials/electron affinities, respectively, while the derivative of E[N,Ns,v,B] with respect to Ns gives the (signed) half excitation energy to the lowest-lying state with Ns increased (or decreased) by 2. The highest occupied and lowest unoccupied Kohn-Sham spin-orbital energies are identified as the corresponding spin-up and spin-down ionization potentials and electron affinities. The excitation energies to the lowest-lying states with Ns±2 can be obtained as the differences between the lowest unoccupied and the opposite-spin highest occupied spin-orbital energies, if the (N,Ns) representation of the Kohn-Sham spin-potentials is used. The cases where the convexity condition on the energy does not hold are also discussed. Finally, the discontinuities of the energy derivatives and the Kohn-Sham potential are analyzed and related. [ABSTRACT FROM AUTHOR]

A benchmark theoretical study of the electronic ground state and of the vertical and adiabatic singlet-triplet (ST) excitation energies of benzene (n=1) and n-acenes (C4n+2H2n+4) ranging from naphthalene (n=2) to heptacene (n=7) is presented, on the ground of single- and multireference calculations based on restricted or unrestricted zero-order wave functions. High-level and large scale treatments of electronic correlation in the ground state are found to be necessary for compensating giant but unphysical symmetry-breaking effects in unrestricted single-reference treatments. The composition of multiconfigurational wave functions, the topologies of natural orbitals in symmetry-unrestricted CASSCF calculations, the T1 diagnostics of coupled cluster theory, and further energy-based criteria demonstrate that all investigated systems exhibit a 1Ag singlet closed-shell electronic ground state. Singlet-triplet (S0-T1) energy gaps can therefore be very accurately determined by applying the principles of a focal point analysis onto the results of a series of single-point and symmetry-restricted calculations employing correlation consistent cc-pVXZ basis sets (X=D, T, Q, 5) and single-reference methods [HF, MP2, MP3, MP4SDQ, CCSD, CCSD(T)] of improving quality. According to our best estimates, which amount to a dual extrapolation of energy differences to the level of coupled cluster theory including single, double, and perturbative estimates of connected triple excitations [CCSD(T)] in the limit of an asymptotically complete basis set (cc-pV∞Z), the S0-T1 vertical excitation energies of benzene (n=1) and n-acenes (n=2–7) amount to 100.79, 76.28, 56.97, 40.69, 31.51, 22.96, and 18.16 kcal/mol, respectively. Values of 87.02, 62.87, 46.22, 32.23, 24.19, 16.79, and 12.56 kcal/mol are correspondingly obtained at the CCSD(T)/cc-pV∞Z level for the S0-T1 adiabatic excitation energies, upon including B3LYP/cc-PVTZ corrections for zero-point vibrational energies. In line with the absence of Peierls distortions, extrapolations of results indicate a vanishingly small S0-T1 energy gap of 0 to ∼4 kcal/mol (∼0.17 eV) in the limit of an infinitely large polyacene. [ABSTRACT FROM AUTHOR]

The effect of the recently uncovered nonuniqueness of the external magnetic field B(r[accent:_right_hook_over]) corresponding to a given pair of density n(r[accent:_right_hook_over]) and spin density ns(r[accent:_right_hook_over]) on the derivative of the energy functional of spin-polarized density functional theory, and its implications for the definition of chemical reactivity descriptors, is examined. For ground states, the nonuniqueness of B(r[accent:_right_hook_over]) implies the nondifferentiability of the energy functional Ev,B[n,ns] with respect to ns(r[accent:_right_hook_over]). It is shown, on the other hand, that this nonuniqueness allows the existence of the one-sided derivatives of Ev,B[n,ns] with respect to ns(r[accent:_right_hook_over]). Although the N-electron ground state can always be obtained from the minimization of Ev,B[n,ns] without any constraint on the spin number Ns=∫ns(r[accent:_right_hook_over])dr[accent:_right_hook_over], the Lagrange multiplier μs associated with the fixation of Ns does not vanish even for ground states. μs is identified as the left- or right-side derivative of the total energy with respect to Ns, which justifies the interpretation of μs as a (spin) chemical potential. This is relevant not only for the spin-polarized generalization of conceptual density functional theory, the spin chemical potential being one of the elementary reactivity descriptors, but also for the extension of the thermodynamical analogy of density functional theory for the spin-polarized case. For higher-order reactivity indices, B(r[accent:_right_hook_over])’s nonuniqueness has similar implications as for μs, leading to a split of the indices with respect to Ns into one-sided reactivity descriptors. [ABSTRACT FROM AUTHOR]

Spin-polarized density functional theory (SP-DFT) allows both the analysis of charge-transfer (e.g., electrophilic and nucleophilic reactivity) and of spin-polarization processes (e.g., photophysical changes arising from electron transitions). In analogy with the dual descriptor introduced by Morell et al. [J. Phys. Chem. A 109, 205 (2005)], we introduce new dual descriptors intended to simultaneously give information of the molecular regions where the spin-polarization process linking states of different multiplicity will drive electron density and spin density changes. The electronic charge and spin rearrangement in the spin forbidden radiative transitions S0→T(n,π*) and S0→T(π,π*) in formaldehyde and ethylene, respectively, have been used as benchmark examples illustrating the usefulness of the new spin-polarization dual descriptors. These quantities indicate those regions where spin-orbit coupling effects are at work in such processes. Additionally, the qualitative relationship between the topology of the spin-polarization dual descriptors and the vertical singlet triplet energy gap in simple substituted carbene series has been also discussed. It is shown that the electron density and spin density rearrangements arise in agreement with spectroscopic experimental evidence and other theoretical results on the selected target systems. [ABSTRACT FROM AUTHOR]

A hardness based similarity index for studying the quantum similarity for atoms is analyzed. The investigation of hardness and Fukui functions of atoms leads to the construction of a quantum similarity measure, which can be interpreted as a quantified comparison of chemical reactivity of atoms. Evaluation of the new measure reveals periodic tendencies throughout Mendeleev’s table. Moreover on the diagonal the global hardness was recovered. Considering a corresponding quantum similarity index reveals that renormalization of the measure can mask periodic patterns. The hardness was calculated for atoms with nuclear charge 3≤=Z≤=103, using the best single configuration electron density functions available. Different hardness kernels were used and the importance of the different contributions to the kernel was investigated. The atomic self-similarities constructed in this way show a fair correlation with experimental atomic polarizability [ABSTRACT FROM AUTHOR]

A novel quantum similarity measure (QSM) is constructed based on concepts from information theory. In an application of QSM to atoms, the new QSM and its corresponding quantum similarity index (QSI) are evaluated throughout the periodic table, using the atomic electron densities and shape functions calculated in the Hartree-Fock approximation. The periodicity of Mendeleev’s table is regained for the first time through the evaluation of a QSM. Evaluation of the information theory based QSI demonstrates, however, that the patterns of periodicity are lost due to the renormalization of the QSM, yielding chemically less appealing results for the QSI. A comparison of the information content of a given atom on top of a group with the information content of the elements in the subsequent rows reveals another periodicity pattern. Relativistic effects on the electronic density functions of atoms are investigated. Their importance is quantified in a QSI study by comparing for each atom, the density functions evaluated in the Hartree-Fock and Dirac-Fock approximations. The smooth decreasing of the relevant QSI along the periodic table illustrates in a quantitative way the increase of relativistic corrections with the nuclear charge. [ABSTRACT FROM AUTHOR]

The electrophilicity index is analyzed within the framework of spin-polarized density-functional theory. In this context, constrained philicities, ωN≡(μN)2/(2ηNN), are introduced in order to define the capability of a system to acquire or donate electrons in a process at constant spin number. The spin-philicity/spin-donicity indices, ωS±≡(μS±)2/(2ηSS), are examined and rationalized here as the philicity of a given system to change its spin-polarization state, as being defined through the spin potential μS and spin hardness ηSS for a process at constant number of electrons. The local extension of these indices has been also outlined and numerical results have been discussed on the analysis of the electrophilic nature of some simple carbene systems both in the singlet and triplet states. [ABSTRACT FROM AUTHOR]

An extension of Cohen’s nuclear Fukui function is presented in the spin-polarized framework of density-functional theory (SP-DFT). The resulting new nuclear Fukui function indices [uppercase_phi_synonym]Nα and [uppercase_phi_synonym]Sα are intended to be the natural descriptors for the responses of the nuclei to changes involving charge transfer at constant multiplicity and also the spin polarization at constant number of electrons. These generalized quantities allow us to gain new insights within a perturbative scheme based on DFT. Calculations of the electronic and nuclear SP-DFT quantities are presented within a Kohn-Sham framework of chemical reactivity for a sample of molecules, including H2O, H2CO, and some simple nitrenes (NX) and phosphinidenes (PX), with X=H, Li, F, Cl, OH, SH, NH2, and PH2. Results have been interpreted in terms of chemical bonding in the context of Berlin’s theorem, which provides a separation of the molecular space into binding and antibinding regions. [ABSTRACT FROM AUTHOR]

The central role of the shape function σ(r_) from the density functional theory (DFT), the ratio of the electron density ρ(r_) and the number of electrons N of the system (density per particle), is investigated. Moreover, its relationship with DFT based reactivity indices is established. In the first part, it is shown that an estimate for the chemical hardness can be obtained from the long range behavior of the shape function and its derivative with respect to the number of electrons at a fixed external potential. Next, the energy of the system is minimized with the constraint that the shape function should integrate to unity; the associated Lagrange multiplier is shown to be related to the electronic chemical potential μ of the system. Finally, the importance of the shape function for both molecular structure, reactivity, and similarity is outlined. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An overview of mathematical properties of the non-local second order derivatives of the canonical, grand canonical, isomorphic, and grand isomorphic ensembles is given. The significance of their positive or negative semidefiniteness and the implications of these properties for atoms and molecules are discussed. Based on this property, many other interesting properties can be derived, such as the expansion in eigenfunctions, bounds on the diagonal and off-diagonal elements, and the eigenvalues of these kernels. We also prove Kato's theorem for the softness kernel and linear response and the dissociation limit of the linear responses as the sum of the linear responses of the individual fragments when dissociating a system into two non-interacting molecular fragments. Finally, strategies for the practical calculation of these kernels, their eigenfunctions, and their eigenvalues are discussed.

Within the context of spin polarized conceptual density functional theory, the spin polarized linear response functions are introduced both in the [N, N(s)] and [N(α), N(β)] representations. The mathematical relations between the spin polarized linear response functions in both representations are examined and an analytical expression for the spin polarized linear response functions in the [N(α), N(β)] representation is derived. The spin polarized linear response functions were calculated for all atoms up to and including argon. To simplify the plotting of our results, we integrated χ(r, r') to a quantity χ(r, r'), circumventing the θ and ϕ dependence. This allows us to plot and to investigate the periodicity throughout the first three rows in the periodic table within the two different representations. For the first time, χ(αβ)(r, r'), χ(βα)(r, r'), and χ(SS)(r, r') plots have been calculated and discussed. By integration of the spin polarized linear response functions, different components to the polarisability, α(αα), α(αβ), α(βα), and α(ββ) have been calculated.

Many useful concepts developed within density functional theory provide much insight for the understanding and prediction of chemical reactivity, one of the main aims in the field of conceptual density functional theory. While approximate evaluations of such concepts exist, the analytical and efficient evaluation is, however, challenging, because such concepts are usually expressed in terms of functional derivatives with respect to the electron density, or partial derivatives with respect to the number of electrons, complicating the connection to the computational variables of the Kohn-Sham one-electron orbitals. Only recently, the analytical expressions for the chemical potential, one of the key concepts, have been derived by Cohen, Mori-Sánchez, and Yang, based on the potential functional theory formalism. In the present work, we obtain the analytical expressions for the real-space linear response function using the coupled perturbed Kohn-Sham and generalized Kohn-Sham equations, and the Fukui functions using the previous analytical expressions for chemical potentials of Cohen, Mori-Sánchez, and Yang. The analytical expressions are exact within the given exchange-correlation functional. They are applicable to all commonly used approximate functionals, such as local density approximation (LDA), generalized gradient approximation (GGA), and hybrid functionals. The analytical expressions obtained here for Fukui function and linear response functions, along with that for the chemical potential by Cohen, Mori-Sánchez, and Yang, provide the rigorous and efficient evaluation of the key quantities in conceptual density functional theory within the computational framework of the Kohn-Sham and generalized Kohn-Sham approaches. Furthermore, the obtained analytical expressions for Fukui functions, in conjunction with the linearity condition of the ground state energy as a function of the fractional charges, also lead to new local conditions on the exact functionals, expressed in terms of the second-order functional derivatives. We implemented the expressions and demonstrate the efficacy with some atomic and molecular calculations, highlighting the importance of relaxation effects.

This paper presents an approach for obtaining accurate interaction energies at the density functional theory level for systems where dispersion interactions are important. This approach combines Becke and Johnson's [J. Chem. Phys. 127, 154108 (2007)] method for the evaluation of dispersion energy corrections and a Hirshfeld method for partitioning of molecular polarizability tensors into atomic contributions. Due to the availability of atomic polarizability tensors, the method is extended to incorporate anisotropic contributions, which prove to be important for complexes of lower symmetry. The method is validated for a set of 18 complexes, for which interaction energies were obtained with the B3LYP, PBE, and TPSS functionals combined with the aug-cc-pVTZ basis set and compared with the values obtained at the CCSD(T) level extrapolated to a complete basis set limit. It is shown that very good quality interaction energies can be obtained by the proposed method for each of the examined functionals, the overall performance of the TPSS functional being the best, which with a slope of 1.00 in the linear regression equation and a constant term of only 0.1 kcal/mol allows to obtain accurate interaction energies without any need of a damping function for complexes close to their exact equilibrium geometry.

The local softness of MgO, CaO, SrO, and BaO (100) surfaces has been studied using a model based on the local density of states. In all the species, the local softness (chemical reactivity) of oxygen atoms at the surface is enhanced as compared to the bulk. The results for the local and the global softness are in agreement with the ionic pattern of the metal-oxygen bond of the series.

The recent approach presented by Becke and Johnson [J. Chem. Phys. 122, 154104 (2005); 123, 024101 (2005); 123, 154101 (2005); 124, 174104 (2006); 124, 014104 (2006)] for the evaluation of dispersion interactions based on the properties of the exchange-hole dipole moment is combined with a Hirshfeld-type partitioning for the molecular polarizabilities into atomic contributions, recently presented by some of the present authors [A. Krishtal et al., J. Chem. Phys. 125, 034312 (2006)]. The results on a series of nine dimers, involving neon, methane, ethene, acetylene, benzene, and CO(2), taken at their equilibrium geometry, indicate that when the C(6), C(8), and C(10) terms are taken into account, the resulting dispersion energies can be obtained deviating 3% or 8% from high level literature data [E. R. Johnson and A. D. Becke, J. Chem. Phys. 124, 174104 (2006)], without the use of a damping function, the only outlier being the parallel face-to-face benzene dimer.

The Fukui function is a frequently used DFT concept in the description of a system's regioselective preferences to undergo electrophilic, nucleophilic, or radical attacks. Until now, this function has usually been evaluated using finite difference approximations. The first paper in this series proposed a method for obtaining the Fukui function by a direct calculation of the functional derivative of the chemical potential with respect to the external potential. This paper extends the method to condensed Fukui functions and applies it to an extensive testing set of molecules. Results are promising, which demonstrates the usefulness of the new formalism.

By using perturbations in the molecular external potential, the authors deduce the Fukui function from the change in Kohn-Sham orbital energies, avoiding the troublesome differentiation of the density with respect to electron number. Though this paper focuses on the Fukui function, the same general technique can be used to compute the functional derivative of any observable with respect to the external potential. In this paper, the method is used to compute the Fukui function for the beryllium atom and the formaldehyde molecule. The follow-up paper (part II) addresses the problem of computing condensed reactivity indicators.

The weight of the energetic components (electronic kinetic, electron-nucleus and electron-electron Coulombic, and correlation energies) of the ionization potential, electron affinity, chemical potential, and global hardness is evaluated and contrasted with the energetic components of the hardness kernel and the experimental values of these properties for 40 systems. The contrast of the hardness terms obtained from finite difference and hardness kernel gives some insight on the possible implications to differentiate the electronic energy with respect to the number electrons or the electron density.

The Woodward-Hoffmann rules for pericyclic reactions, a fundamental set of reactivity rules in organic chemistry, are formulated in the language of conceptual density functional theory (DFT). DFT provides an elegant framework to introduce chemical concepts and principles in a quantitative manner, partly because it is formulated without explicit reference to a wave function, on whose symmetry properties the Woodward-Hoffmann [J. Am. Chem. Soc. 87, 395 (1965)] rules are based. We have studied the initial chemical hardness response using a model reaction profile for two prototypical pericyclic reactions, the Diels-Alder cycloaddition of 1,3-butadiene to ethylene and the addition of ethylene to ethylene, both in the singlet ground state and in the first triplet excited state. For the reaction that is thermally allowed but photochemically forbidden, the initial hardness response is positive along the singlet reaction profile. (By contrast, for the triplet reaction profile, a negative hardness response is observed.) For the photochemically allowed, thermally forbidden reaction, the behavior of the chemical hardness along the initial stages of the singlet and triplet reaction profiles is reversed. This constitutes a first step in showing that chemical concepts from DFT can be invoked to explain results that would otherwise require invoking the phase of the wave function.

Generalized hardness and softness reactivity kernels are defined within a spin-polarized density-functional theory (SP-DFT) conceptual framework. These quantities constitute the basis for the global, local (i.e., r-position dependent), and nonlocal (i.e., r and r'-position dependents) indices devoted to the treatment of both charge-transfer and spin-polarization processes in such a reactivity framework. The exact relationships between these descriptors within a SP-DFT framework are derived and the implications for chemical reactivity in such context are outlined.

A derivation of the density-functional-theory- (DFT) based reactivity indices in the ensemble unrestricted Hartree-Fock (eUHF) method is presented. The comparison between the properties of the reactivity indices evaluated in one and two sets of spin-orbital approach of the eUHF and hyper-unrestricted Hartree-Fock (UHF) methods are shown. All approaches give similar Fukui function irrespective of methodology used, but significantly differ for the global indices, containing important chemical information, and so their interpretation in terms of DFT- based indices can be questionable. The calculation scheme for the indices using the first- and second-order coupled perturbed eHF equations is proposed. A method for the identification of the spinorbitals involved in the change of the total number of electrons is included. The illustrative examples (water and hydrogen cyanide) show that the ground-state (GS) properties of the (Z +/- 1)-electron systems can be predicted from the GS properties of the Z-electron systems with an accuracy comparable with the UHF calculations. The relaxation effect, important for the HCN system in which a change in the symmetry of the highest-occupied spin-orbital occurs, is effectively predicted.

Correct evaluation of the reactivity indices, such as chemical potential, hardness, and Fukui function demands for the extension of the formalism beyond the integer particle picture. An ensemble approach is used as an extension of the unrestricted Hartree-Fock (UHF) method for noninteger electron number systems. A prescription is given for the construction of an ensemble Fock operator for a system with partially filled spin-orbitals. The comparison between the ensemble HF method and the hyper-HF method in terms of density matrices and spin-orbitals is presented. The equivalence of the equiensemble case and the ensemble UHF case with unequal weight factors is shown.