Discontinuities of Kinetic Energy Densities within Finite and Complete Basis Sets.

Although electron densities are always continuous, other ingredients of density-functional approximations can be sharply discontinuous at isolated points. In particular, the positive-definite, Weizsäcker, and Pauli kinetic energy densities expressed in terms of Slater-type orbitals all have discontinuities at the positions of the atomic nuclei in molecules. The first two of those quantities are similarly discontinuous even in the basis-set limit. These striking features are not as widely recognized as they deserve to be. We show in detail how discontinuities of kinetic energy densities arise from asymmetric electron-nucleus cusps of molecular wave functions and point out instances of their significance in electronic structure theory.

Maximization of linear independence of basis function products.

Basis sets consisting of functions that form linearly independent products (LIPs) have remarkable applications in quantum chemistry but are scarce because of mathematical limitations. We show how to linearly transform a given set of basis functions to maximize the linear independence of their products by maximizing the determinant of the appropriate Gram matrix. The proposed method enhances the utility of the LIP basis set technology and clarifies why canonical molecular orbitals form LIPs more readily than atomic orbitals. The same approach can also be used to orthogonalize basis functions themselves, which means that various orthogonalization techniques may be viewed as special cases of a certain nonlinear optimization problem.

Local Potentials Reconstructed within Linearly Independent Product Basis Sets of Increasing Size.

Given a matrix representation of a local potential v(r) within a one-electron basis set of functions that form linearly independent products (LIP), it is possible to construct a well-defined local potential v~(r) that is equivalent to v(r) within that basis set and has the form of an expansion in basis function products. Recently, we showed that for exchange-correlation potentials vXC(r) defined on the infinite-dimensional Hilbert space, the potentials v~XC(r) reconstructed from matrices of vXC(r) within minimal LIP basis sets of occupied Kohn-Sham orbitals bear only qualitative resemblance to the originals. Here, we show that if the LIP basis set is enlarged by including low-lying virtual Kohn-Sham orbitals, the agreement between v~XC(r) and vXC(r) improves to the extent that the basis function products are appropriate as a basis for vXC(r). These findings validate the LIP technology as a rigorous potential reconstruction method.

Derivation and reinterpretation of the Fermi-Amaldi functional.

The Fermi-Amaldi correction to the electrostatic self-repulsion of the particle density is usually regarded as a semi-classical exchange functional that happens to be exact only for one- and closed-shell two-electron systems. We show that this functional can be derived quantum-mechanically and is exact for any number of fermions or bosons of arbitrary spin as long as the particles occupy the same spatial orbital. The Fermi-Amaldi functional is also size-consistent for such systems, provided that the factor N in its expression is understood as an orbital occupation number rather than the total number of particles. These properties of the Fermi-Amaldi functional are ultimately related to the fact that it is a special case of the self-exchange energy formula. Implications of our findings are discussed.

Multiplicative potentials for kinetic energy and exact exchange.

Harriman showed that within finite basis sets of one-electron functions that form linearly independent products (LIP), differential and integral operators can be represented exactly and unambiguously by multiplicative (local) potentials. Although almost no standard basis sets of quantum chemistry form LIPs in a numerical sense, occupied self-consistent field (SCF) orbitals routinely do so. Using minimal LIP basis sets of occupied SCF orbitals, we construct multiplicative potentials for electronic kinetic energy and exact exchange that reproduce the Hartree-Fock and Kohn-Sham Hamiltonian matrices and electron densities for atoms and molecules. The results highlight fundamental differences between local and nonlocal operators and suggest a practical possibility of developing exact kinetic energy functionals within finite basis sets by using effective local potentials.

Near-Infrared Boron Difluoride Formazanate Dyes.

Near-infrared (NIR) dyes are sought after for their utility in light harvesting, bioimaging, and light-mediated therapies. Since long-wavelength photoluminescence typically involves extensive π-conjugated systems of double bonds and aromatic rings, it is often assumed that NIR dyes have to be large molecules that require complex syntheses. We challenge this assumption by demonstrating that facile incorporation of tertiary amine groups into readily available 3-cyanoformazans affords efficient production of relatively simple NIR-active BF2 formazanate dyes (λabs =691-760 nm, λPL =834-904 nm in toluene). Cyclic voltammetry experiments on these compounds reveal multiple reversible redox waves linked to the interplay between the tertiary amine and BF2 formazanate moieties. Density-functional calculations indicate that the NIR electronic transitions in BF2 formazanates are of π→π*-type, but do not always involve strong charge transfer.

Complete-active-space extended Koopmans theorem method.

The complete-active-space (CAS) extended Koopmans theorem (EKT) method is defined as a special case of the EKT in which the reference state is a CAS configuration interaction (CI) expansion and the electron removal operator acts only on the active orbitals. With these restrictions, the EKT is equivalent to the CI procedure involving all hole-state configurations derived from the active space of the reference wavefunction and has properties analogous to those of the original Koopmans theorem. The equivalence is used to demonstrate in a transparent manner that the first ionization energy predicted by the EKT is in general not exact, i.e., not equal to the difference between the full CI energies of the neutral and the ion, but can approach the full CI result with arbitrary precision even within a finite basis set. The findings also reconcile various statements about the EKT found in the literature.

Visualizing atomic sizes and molecular shapes with the classical turning surface of the Kohn-Sham potential.

The Kohn-Sham potential [Formula: see text] is the effective multiplicative operator in a noninteracting Schrödinger equation that reproduces the ground-state density of a real (interacting) system. The sizes and shapes of atoms, molecules, and solids can be defined in terms of Kohn-Sham potentials in a nonarbitrary way that accords with chemical intuition and can be implemented efficiently, permitting a natural pictorial representation for chemistry and condensed-matter physics. Let [Formula: see text] be the maximum occupied orbital energy of the noninteracting electrons. Then the equation [Formula: see text] defines the surface at which classical electrons with energy [Formula: see text] would be turned back and thus determines the surface of any electronic object. Atomic and ionic radii defined in this manner agree well with empirical estimates, show regular chemical trends, and allow one to identify the type of chemical bonding between two given atoms by comparing the actual internuclear distance to the sum of atomic radii. The molecular surfaces can be fused (for a covalent bond), seamed (ionic bond), necked (hydrogen bond), or divided (van der Waals bond). This contribution extends the pioneering work of Z.-Z. Yang et al. [Yang ZZ, Davidson ER (1997) Int J Quantum Chem 62:47-53; Zhao DX, et al. (2018) Mol Phys 116:969-977] by our consideration of the Kohn-Sham potential, protomolecules, doubly negative atomic ions, a bond-type parameter, seamed and necked molecular surfaces, and a more extensive table of atomic and ionic radii that are fully consistent with expected periodic trends.

Cationic Boron Formazanate Dyes*.

Incorporation of cationic boron atoms into molecular frameworks is an established strategy for creating chemical species with unusual bonding and reactivity but is rarely thought of as a way of enhancing molecular optoelectronic properties. Using boron formazanate dyes as examples, we demonstrate that the wavelengths, intensities, and type of the first electronic transitions in BN heterocycles can be modulated by varying the charge, coordination number, and supporting ligands at the cationic boron atom. UV-vis absorption spectroscopy measurements and density-functional (DFT) calculations show that these modulations are caused by changes in the geometry and extent of π-conjugation of the boron formazanate ring. These findings suggest a new strategy for designing optoelectronic materials based on π-conjugated heterocycles containing boron and other main-group elements.

Optoelectronic Properties of Carbon-Bound Boron Difluoride Hydrazone Dimers.

The creation of dimeric boron difluoride complexes of chelating N-donor ligands is a proven strategy for the enhancement of the optoelectronic properties of fluorescent dyes. We report dimers based on the boron difluoride hydrazone (BODIHY) framework, which offer unique and sometimes unexpected substituent-dependent absorption, emission, and electrochemical properties. BODIHY dimers have low-energy absorption bands (λmax =421 to 479 nm, ϵ=17 200 to 39 900 m-1 cm-1 ) that are red-shifted relative to monomeric analogues. THF solutions of these dimers exhibit aggregation-induced emission upon addition of water, with emission enhancement factors ranging from 5 to 18. Thin films of BODIHY dimers are weakly emissive as a result of the inner-filter effect, attributed to intermolecular π-type interactions. BODIHY dimers are redox-active and display two one-electron oxidation and two one-electron reduction waves that strongly depend on the N-aryl substituents. These properties are rationalized using density-functional theory calculations and X-ray crystallography experiments.

Asymptotic behavior of the average local ionization energy in finite basis sets.

The average local ionization energy (ALIE) has important applications in several areas of electronic structure theory. Theoretically, the ALIE should asymptotically approach the first vertical ionization energy (IE) of the system, as implied by the rate of exponential decay of the electron density; for one-determinantal wavefunctions, this IE is the negative of the highest-occupied orbital energy. In practice, finite-basis-set representations of the ALIE exhibit seemingly irregular and sometimes dramatic deviations from the expected asymptotic behavior. We analyze the long-range behavior of the ALIE in finite basis sets and explain the puzzling observations. The findings have implications for practical calculations of the ALIE, the construction of Kohn-Sham potentials from wavefunctions and electron densities, and basis-set development.

Optoelectronic, Aggregation, and Redox Properties of Double-Rotor Boron Difluoride Hydrazone Dyes.

We develop the chemistry of boron difluoride hydrazone dyes (BODIHYs) bearing two aryl substituents and explore their properties. The low-energy absorption bands (λmax =427-464 nm) of these dyes depend on the nature of the N-aryl groups appended to the BODIHY framework. Electron-donating and extended π-conjugated groups cause a redshift, whereas electron-withdrawing groups result in a blueshift. The title compounds were weakly photoluminescent in solution and strongly photoluminescent as thin films (λPL =525-578 nm) with quantum yields of up to 18 % and lifetimes of 1.1-1.7 ns, consistent with the dominant radiative decay through fluorescence. Addition of water to THF solutions of the BODIHYs studied causes molecular aggregation which restricts intramolecular motion and thereby enhances photoluminescence. The observed photoluminescence of BODIHY thin films is likely facilitated by a similar molecular packing effect. Finally, cyclic voltammetry studies confirmed that BODIHY derivatives bearing para-substituted N-aryl groups could be reversibly oxidized (Eox1 =0.62-1.02 V vs. Fc/Fc+ ) to their radical cation forms. Chemical oxidation studies confirmed that para-substituents at the N-aryl groups are required to circumvent radical decomposition pathways. Our findings provide new opportunities and guiding principles for the design of sought-after multifunctional boron difluoride complexes that are photoluminescent in the solid state.

Oxoborane Formation Turns on Formazanate-Based Photoluminescence.

The synthesis of compounds containing multiple bonds to boron has challenged main-group chemists for decades. Despite significant progress, the possibility that the formation of such bonds can turn on photoluminescence has received minimal attention. We report an oxoborane (B=O) complex that is electronically stabilized by a formazanate ligand in the absence of significant steric bulk and, unlike the common BX2 (X=F, Cl) formazanate adducts, exhibits intense photoluminescence. The latter property was rationalized through density-functional calculations which indicated that the B=O bond enhances photoluminescence by drastically reducing differences between the ligand's geometries in the ground and excited states. The title oxoborane compound was synthesized from an air- and moisture-stable BCl2 formazanate complex and subsequently converted to a redox-active boroxine. Each of these species may also serve as a precursor to functional materials.

X-ray Absorption Near-Edge Structure Spectroscopy of a Stable 6-Oxoverdazyl Radical and Its Diamagnetic Precursor.

The electronic structure of 1,3,5-triphenyl-6-oxoverdazyl, a heteroatom-rich stable organic radical, and its diamagnetic 1,3,5-triphenyl-6-oxotetrazane precursor are probed using X-ray absorption near-edge structure (XANES) spectroscopy. The N K-edge XANES spectra of the 6-oxoverdazyl radical contain strong N 1s → π* resonances for each set of equivalent nitrogen atoms. The fact that these resonances are absent from the analogous spectra of the 6-oxotetrazane, whereas the O K-edge and C K-edge XANES spectra of both species are very similar, demonstrates that the unpaired electron of the radical is localized primarily on the N atoms of the 6-oxoverdazyl heterocycle. The O K-edge XANES spectra of both species contain strong O 1s → π* (C═O) peaks, but the peak of the radical is red-shifted by 0.5 eV relative to that of the 6-oxotetrazane, which indicates that the C═O bond in the radical is part of a larger π-conjugated system. The proposed interpretations of the XANES spectra are aided by density-functional calculations.

Formazanate Complexes of Hypervalent Group 14 Elements as Precursors to Electronically Stabilized Radicals.

The stability of molecular radicals containing main-group elements usually hinges on the presence of bulky substituents that shield the reactive radical center. We describe a family of Group 14 formazanate complexes whose chemical reduction allows access to radicals that are stabilized instead by geometric and electron-delocalization effects, specifically by the square-pyramidal coordination geometry adopted by the Group 14 atom (Si, Ge, Sn) within the framework of the heteroatom-rich formazanate ligands. The reduction potentials of the Si, Ge, and Sn complexes as determined by cyclic voltammetry become more negative in that order. Examination of the solid-state structures of these complexes suggested that their electron-accepting ability decreases with increasing size of the Group 14 atom because a larger central atom increases the nonplanarity of the ligand-based conjugated π-electron system of the complex. The experimental findings were supported by density-functional calculations on the parent complexes and the corresponding radical anions.

Aluminum Complexes of N2O23- Formazanate Ligands Supported by Phosphine Oxide Donors.

The synthesis and characterization of a new family of phosphine oxide supported aluminum formazanate complexes (7a,b, 8a, 9a) are reported. X-ray diffraction studies showed that the aluminum atoms in the complexes adopt an octahedral geometry in the solid state. The equatorial positions are occupied by an N2O23- formazanate ligand, and the axial positions are occupied by L-type phosphine oxide donors. UV-vis absorption spectroscopy revealed that the complexes were strongly absorbing (ε ≈ 30000 M-1 cm-1) between 500 and 700 nm. The absorption maxima in this region were simulated using time-dependent density functional theory. With the exception of 3-cyano-substituted complex 7b, which showed maximum luminescence intensity in the presence of excess phosphine oxide, the title complexes are nonemissive in solution and the solid state. The electrochemical properties of the complexes were probed using cyclic voltammetry. Each complex underwent sequential one-electron oxidations in potential ranges of -0.12 to 0.29 V and 0.62 to 0.97 V, relative to the ferrocene/ferrocenium redox couple. Electrochemical reduction events were observed at potentials between -1.34 and -1.75 V. In combination with tri-n-propylamine as a coreactant, complex 7b acted as an electrochemiluminescence emitter with a maximum electrochemiluminescence intensity at a wavelength of 735 nm, red-shifted relative to the photoluminescence maximum of the same compound.

Exact exchange-correlation potentials of singlet two-electron systems.

We suggest a non-iterative analytic method for constructing the exchange-correlation potential, vXC(r), of any singlet ground-state two-electron system. The method is based on a convenient formula for vXC(r) in terms of quantities determined only by the system's electronic wave function, exact or approximate, and is essentially different from the Kohn-Sham inversion technique. When applied to Gaussian-basis-set wave functions, the method yields finite-basis-set approximations to the corresponding basis-set-limit vXC(r), whereas the Kohn-Sham inversion produces physically inappropriate (oscillatory and divergent) potentials. The effectiveness of the procedure is demonstrated by computing accurate exchange-correlation potentials of several two-electron systems (helium isoelectronic series, H2, H3+) using common ab initio methods and Gaussian basis sets.

Improved method for generating exchange-correlation potentials from electronic wave functions.

Ryabinkin, Kohut, and Staroverov (RKS) [Phys. Rev. Lett. 115, 083001 (2015)] devised an iterative method for reducing many-electron wave functions to Kohn-Sham exchange-correlation potentials, vXC(𝐫). For a given type of wave function, the RKS method is exact (Kohn-Sham-compliant) in the basis-set limit; in a finite basis set, it produces an approximation to the corresponding basis-set-limit vXC(𝐫). The original RKS procedure works very well for large basis sets but sometimes fails for commonly used (small and medium) sets. We derive a modification of the method's working equation that makes the RKS procedure robust for all Gaussian basis sets and increases the accuracy of the resulting exchange-correlation potentials with respect to the basis-set limit.

Structurally Diverse Boron-Nitrogen Heterocycles from an N2 O23- Formazanate Ligand.

Five new compounds comprised of unprecedented boron-nitrogen heterocycles have been isolated from a single reaction of a potentially tetradentate N2 O23- formazanate ligand with BF3 ⋅OEt2 and NEt3 . Optimized yields for each product were obtained through variation of experimental conditions and rationalized in terms of relative Gibbs free energies of the products as determined by electronic structure calculations. Chemical reduction of two of these compounds resulted in the formation of a stable anion, radical anion, and diradical dianion. Structural and electronic properties of this new family of redox-active heterocycles were characterized using UV/vis absorption spectroscopy, cyclic voltammetry and X-ray crystallography.

Origin of the step structure of molecular exchange-correlation potentials.

The exact exchange-correlation potential of a stretched heteronuclear diatomic molecule exhibits a localized upshift in the region around the more electronegative atom; by this device the Kohn-Sham scheme ensures that the molecule dissociates into neutral atoms. Baerends and co-workers showed earlier that the upshift originates in the response part of the exchange-correlation potential. We describe a reliable numerical method for constructing the response potential of a given many-electron system and report accurate plots of this quantity. We also demonstrate that the step feature itself can be obtained directly from the interacting wavefunction of the system by computing the so-called average local electron energy. These findings illustrate in previously unavailable detail the mechanism of the formation of the upshift and the role played by static correlation in this process.