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.

Analytic Construction of One-Electron Reduced Density Matrices from Electron Densities within Finite Basis Sets.

We show how to construct analytically all one-electron reduced density matrices (1-RDMs) compatible with a given electron density within a finite basis set, provided that the density is specified as a symmetric quadratic form in terms of the basis functions. Contrary to the current belief, exact linear dependencies in the basis function products assist, rather than hinder, such constructions. By applying the N-representability conditions to the analytically reconstructed 1-RDMs, one can perform a constrained search over physically acceptable 1-RDMs that yield a given finite-basis-set density. The discussion is illustrated with worked-out examples.

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.

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.

Operationally unsaturated ruthenium complex stabilized by a phosphine 1-azaallyl ligand.

Coordinatively unsaturated transition-metal compounds stabilized by supplemental electron donation from π-basic ligands are described as "operationally unsaturated". Such complexes are useful analogues of active catalyst structures that readily react with substrate molecules. We report that [Ph2P(C6H4)NCHC(CH3)2]- (L1) effectively stabilizes Ru(II) in an operationally unsaturated form. In the absence of Lewis bases, the 1-azaallyl group of L1 dominantly coordinates through a κ1-N mode, but can readily and reversibly isomerize to an η3-NCC coordination mode to stabilize the metal. As an operationally unsaturated complex, Ru(Cp*)(L1) dimerizes at low temperature. At ambient temperature it rapidly reacts with pyridine or PPh3 to form an adduct. These findings with L1 demonstrate that changes in the hapticity of a 1-azaallyl fragment offer an alternative means to stabilize low-coordinate metals.

Using Density Functional Theory for Testing the Robustness of Mobile-Proton Molecular Dynamics Simulations on Electrosprayed Ions: Structural Implications for Gaseous Proteins.

Current experiments only provide low-resolution information on gaseous protein ions generated by electrospray ionization (ESI). Molecular dynamics (MD) simulations can yield complementary insights. Unfortunately, conventional MD does not capture the mobile nature of protons in gaseous proteins. Mobile-proton MD (MPMD) overcomes this limitation. Earlier MPMD data at 300 K indicated that protein ions generated by "native" ESI retain solution-like structures with a hydrophobic core and zwitterionic exterior [Bakhtiari, M.; Konermann, L. J. Phys. Chem. B 2019, 123, 1784-1796]. MPMD redistributes protons using electrostatic and proton affinity calculations. The robustness of this approach has never been scrutinized. Here, we close this gap by benchmarking MPMD against density functional theory (DFT) at the B3LYP/6-31G* level, which is well suited for predicting proton affinities. The computational cost of DFT necessitated the use of small peptides. The MPMD energetic ranking of proton configurations was found to be consistent with DFT single-point energies, implying that MPMD can reliably identify favorable protonation sites. Peptide MPMD runs converged to DFT-optimized structures only when applying 300-500 K temperature cycling, which was necessary to prevent trapping in local minima. Temperature cycling MPMD was then applied to gaseous protein ions. Native ubiquitin converted to slightly expanded structures with a zwitterionic core and a nonpolar exterior. Our data suggest that such inside-out protein structures are intrinsically preferred in the gas phase, and that they form in ESI experiments after moderate collisional excitation. This is in contrast to native ESI (with minimal collisional excitation, simulated by MPMD at 300 K), where kinetic trapping promotes the survival of solution-like structures. In summary, this work validates the MPMD approach for simulations on gaseous peptides and proteins.

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.

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.

Reconstruction of Exchange-Correlation Potentials from Their Matrix Representations.

Within a basis set of one-electron functions that form linearly independent products (LIPs), it is always possible to construct a unique local (multiplicative) real-space potential that is precisely equivalent to an arbitrary given operator. Although standard basis sets of quantum chemistry rarely form LIPs in a numerical sense, occupied and low-lying virtual canonical Kohn-Sham orbitals often do so, at least for small atoms and molecules. Using these principles, we construct atomic and molecular exchange-correlation potentials from their matrix representations in LIP basis sets of occupied canonical Kohn-Sham orbitals. The reconstructions are found to imitate the original potentials in a consistent but exaggerated way. Since the original and reconstructed potentials produce the same ground-state electron density and energy within the associated LIP basis set, the procedure may be regarded as a rigorous solution to the Kohn-Sham inversion problem within the subspace spanned by the occupied Kohn-Sham orbitals.

Noninteracting v-Representable Subspaces of Orbitals in the Kohn-Sham Method.

The notion of noninteracting v-representability is extended from electron densities to finite-dimensional linear subspaces of orbitals. Unlike electron densities, orbital subspaces can be tested for noninteracting v-representability using a transparent necessary condition: the subspace must be invariant under the action of some one-electron Kohn-Sham Hamiltonian. This condition allows one to determine in principle, and sometimes in practice, whether a given one-electron basis set can represent an N-electron density within the Kohn-Sham method and to find the corresponding Kohn-Sham effective potential v if it exists. If the occupied Kohn-Sham orbitals form linearly independent products, then their subspace is determined by the corresponding ground-state electron density. This means that the Kohn-Sham effective potential corresponding to certain finite-basis-set electron densities can be deduced from the basis set itself.

A strongly Lewis-acidic and fluorescent borenium cation supported by a tridentate formazanate ligand.

Lewis acids are highly sought after for their applications in sensing, small-molecule activation, and catalysis. When combined with π-conjugated molecular frameworks, Lewis acids with unique optoelectronic properties can be realized. Here, we use a tridentate formazanate ligand to create a planar, redox-active, fluorescent, and strongly Lewis-acidic borenium cation. We also demonstrate that this compound can act as a colourimetric probe for reactivity.

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.

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.

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.

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.

First Ionization Energy as the Asymptotic Limit of the Average Local Electron Energy.

The first vertical ionization energy of an atom or molecule is encoded in the rate of exponential decay of the exact natural orbitals. For natural orbitals represented in terms of Gaussian basis functions, this property does not hold even approximately. We show that it is nevertheless possible to deduce the first ionization energy from the long-range behavior of Gaussian-basis-set wave functions by evaluating the asymptotic limit of a quantity called the average local electron energy (ALEE), provided that the most diffuse functions of the basis set have a suitable shape and location. The ALEE method exposes subtle qualitative differences between seemingly analogous Gaussian basis sets and complements the extended Koopmans theorem by being robust in situations where the one-electron reduced density matrix is ill-conditioned.

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.

What Is the Accuracy Limit of Adiabatic Linear-Response TDDFT Using Exact Exchange-Correlation Potentials and Approximate Kernels?

Calculation of vertical excitation energies by the adiabatic linear-response time-dependent density-functional theory (TDDFT) requires static Kohn-Sham potentials and exchange-correlation kernels. When these quantities are derived from standard density-functional approximations (DFA), mean absolute errors (MAE) of the method are known to range from 0.2 eV to over 1 eV, depending on the functional and type of excitation. We investigate how the performance of TDDFT varies when increasingly accurate exchange-correlation potentials derived from Hartree-Fock (HF) and post-HF wavefunctions are combined with different approximate kernels. The lowest MAEs obtained in this manner for valence excitations are about 0.15-0.2 eV, which appears to be the practical limit of the accuracy of TDDFT that can be achieved by improving the Kohn-Sham potentials alone. These findings are consistent with previous reports on the benefits of accurate exchange-correlation potentials in TDDFT, but provide new insights and afford more definitive conclusions.

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.