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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]

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]

A benchmark theoretical determination of the electron affinities of benzene and linear oligoacenes ranging from naphthalene to hexacene is presented, using the principles of a focal point analysis. These energy differences have been obtained from a series of single-point calculations at the Hartree–Fock, second-, third-, and partial fourth-order Mo\ller–Plesset (MP2, MP3, and MP4SDQ) levels and from coupled cluster calculations including single and double excitations (CCSD) as well as perturbative estimates of connected triple excitations [CCSD(T)], using basis sets of improving quality, containing up to 1386, 1350, 1824, 1992, 1630, and 1910 basis functions in the computations, respectively. Studies of the convergence properties of these energy differences as a function of the size of the basis set and order attained in electronic correlation enable a determination of the vertical electron affinities of the four larger terms of the oligoacene (C2+4nH2+2n) series within chemical accuracy (0.04 eV). According to our best estimates, these amount to +0.28, +0.82, +1.21, and +1.47 eV when n=3, 4, 5, and 6. Adiabatic electron affinities have been further calculated by incorporating corrections for zero-point vibrational energies and for geometrical relaxations. The same procedure was applied to determine the vertical electron affinities of benzene and naphthalene, which are found to be markedly negative (∼-1.53 and ∼-0.48 eV, respectively). Highly quantitative insights into experiments employing electron transmission spectroscopy on these compounds were also amenable from such an approach, provided diffuse atomic functions are deliberately removed from the basis set, in order to enforce confinement in the molecular region and enable a determination of pseudoadiabatic electron affinities (with respect to the timescale of nuclear motions). Comparison was made with calculations employing density functional theory and especially designed models that exploit the integer discontinuity in the potential or incorporate a potential wall in the unrestricted Kohn–Sham orbital equation for the anion. [ABSTRACT FROM AUTHOR]

The performance of several density-functional theory (DFT) methods for the calculation of current densities induced by a uniform magnetic field is examined. Calculations are performed using the BLYP and KT3 generalized-gradient approximations, together with the B3LYP hybrid functional. For the latter, both conventional and optimized effective potential (OEP) approaches are used. Results are also determined from coupled-cluster singles-and-doubles (CCSD) electron densities by a DFT constrained search procedure using the approach of Wu and Yang (WY). The current densities are calculated within the CTOCD-DZ2 distributed origin approach. Comparisons are made with results from Hartree-Fock (HF) theory. Several small molecules for which correlation is known to be especially important in the calculation of magnetic response properties are considered—namely, O3, CO, PN, and H2CO. As examples of aromatic and antiaromatic systems, benzene and planarized cyclooctatetraene molecules are considered, with specific attention paid to the ring current phenomenon and its Kohn-Sham orbital origin. Finally, the o-benzyne molecule is considered as a computationally challenging case. The HF and DFT induced current maps show qualitative differences, while among the DFT methods the maps show a similar qualitative structure. To assess quantitative differences in the calculated current densities with different methods, the maximal moduli of the induced current densities are compared and integration of the current densities to yield shielding constants is performed. In general, the maximal modulus is reduced in moving from HF to B3LYP and BLYP, and further reduced in moving to KT3, OEP(B3LYP), and WY(CCSD). The latter three methods offer the most accurate shielding constants in comparison with both experimental and ab initio data and hence the more reliable route to DFT calculation of induced current density in molecules. [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]

Computational study of electronic resonances is still a very challenging topic, with the phenomenon of dissociative electron attachment (DEA) being one of the multiple features worth investigating. Recently, we extended the charge stabilization method from energies to properties of conceptual density functional theory and applied this to metastable anionic states of ethene and chlorinated ethene derivatives to study the DEA mechanism present in these compounds. We now present an extension to spatial functions, namely, the electronic Fukui function and the electron localization function. The results of our analysis show that extrapolated spatial functions are relevant and useful for more precise localization of the unbound electron. Furthermore, we report for the first time the combination of the electron localization function with Berlin's binding function for these challenging electronic states. This promising methodology allows for accurate predictions of when and where DEA will happen in the molecules studied and provides more insight into the process., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

The charge stabilization method has often been used before for obtaining energies of temporary anions. Herein, we combine this method for the first time with conceptual density functional theory (DFT) and quantum theory of atoms in molecules by extending it to the study of nuclear Fukui functions, atom-condensed electronic Fukui functions, and bond critical points. This is applied to temporary anions of ethene and chlorinated ethene compounds, which are known to undergo dissociative electron attachment (DEA). It appears that the method is able to detect multiple valence resonance states in the same molecule, namely, a Π and a Σ state. The obtained nuclear and atom-condensed electronic Fukui functions are interpreted as nuclear forces and electron distributions, respectively, and show clear differences between the Π and Σ states. This enables a more profound characterization and understanding of how the DEA process proceeds. The conclusions are in line with findings from earlier publications, proving that the combination of conceptual DFT with the charge stabilization method yields reasonable results at rather low computational cost.

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.

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.

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.

Two approaches are investigated for modeling electron densities of temporary anions in density functional theory (DFT). Both rely on an artificial binding of the excess electron, in one case by a compact basis set and in the other by a potential wall. The key feature of the calculations is that the degree of binding is controlled in both cases by knowledge of the negative electron affinity of the corresponding neutral, approximated in terms of DFT local functional frontier orbital eigenvalues and vertical ionization potential, A=-(epsilon(LUMO)+epsilon(HOMO))-I. To illustrate the two approaches, Fukui functions for nucleophilic attack are determined in four molecules with increasingly negative electron affinities. They yield very similar results, which are notably different to those determined without artificial electron binding. The use of a potential wall has the attractive feature that large, diffuse basis sets can be used, avoiding the need for a compact basis, tailored to a particular molecule.

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 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.