New approach for the electronic energies of the hydrogen molecular ion

How delocalized are the polyacenes?

In an attempt to quantify electron delocalization in polyacenes with up to 50 carbon atoms, we have performed self-consistent field calculations in which the π electrons are constrained to occupy highly localized molecular orbitals (HILOs) centered on a maximum of two, six or ten adjacent carbon atoms. We have also performed similar calculations on simple polyacene analogs consisting only of hydrogen atoms and exhibiting electron delocalization in the σ framework. We find that the energetic cost of localizing the π electrons in the polyacenes is roughly 60, 5 or 0.1 kJ/mol per ring atom for the two-, six- and ten-atom HILOs, respectively, and the use of these localized models overestimates the predicted hydrogenation energies of the acenes by roughly 50%, 4% and 0.1%, respectively. We conclude that the chemistry of polyacenes can be modeled well using highly localized descriptions of the π electrons.

Nested gausslet basis sets

We introduce nested gausslet bases, an improvement on previous gausslet bases that can treat systems containing atoms with much larger atomic numbers. We also introduce pure Gaussian distorted gausslet bases, which allow the Hamiltonian integrals to be performed analytically, as well as hybrid bases in which the gausslets are combined with standard Gaussian-type bases. All these bases feature the diagonal approximation for the electron-electron interactions so that the Hamiltonian is completely defined by two Nb × Nb matrices, where Nb ≈ 104 is small enough to permit fast calculations at the Hartree-Fock level. In constructing these bases, we have gained new mathematical insight into the construction of one-dimensional diagonal bases. In particular, we have proved an important theorem relating four key basis set properties: completeness, orthogonality, zero-moment conditions, and diagonalization of the coordinate operator matrix. We test our basis sets on small systems with a focus on high accuracy, obtaining, for example, an accuracy of 2 × 10-5 Ha for the total Hartree-Fock energy of the neon atom in the complete basis set limit.

Exact Analytical Form of Diatomic Molecular Orbitals

We provide the exact analytical form of diatomic molecular orbitals, as given by the solutions of a single-electron diatomic molecule with arbitrary nuclear charges, using our recently developed method for solving Schrödinger equations. We claim that the best representation of the wave function is a factorized form including a power prefactor, an exponentially decaying term, a modulator function on the exponential, and additional factors accounting for nodal surfaces and the magnetic quantum number. Applying our method, we have identified unexpected extreme points along the potential energy curves, hence revealing the limitations of the well-known concepts of bonding and antibonding.

Semiclassical Treatment of High-Lying Electronic States of H2. J Phys Chem A 2018;122:8565-8575. [PMID: 30299945 DOI: 10.1021/acs.jpca.8b07878]

We report quantum-mechanical and semiclassical WKB calculations for energies and wave functions of high-lying ²Σ states of H₂⁺ in atomic units. The high-lying states we present lie in an unexplored regime, corresponding asymptotically to H ( n ≤ 146) plus a proton, with R ≤ 120 000 a₀. We compare quantum-mechanical energies, spectroscopic constants, dipole matrix elements, and phases with semiclassical results and demonstrate a high level of agreement. Our quantum-mechanical phases were determined by using Milne's phase-amplitude procedure. We compare our semiclassical energies for low-lying states with those of other researchers.

Bonding in phase change materials: concepts and misconceptions

Bonding concepts originating in chemistry are surveyed from a condensed matter perspective, beginning around 1850 with 'valence' and the word 'bond' itself. The analysis of chemical data in the 19th century resulted in astonishing progress in understanding the connectivity and stereochemistry of molecules, almost without input from physicists until the development of quantum mechanics in 1925 and afterwards. The valence bond method popularized by Pauling and the molecular orbital methods of Hund, Mulliken, Bloch, and Hückel play major roles in the subsequent development, as does the central part played by the kinetic energy in covalent bonding (Ruedenberg and others). 'Metallic' (free electron) and related approaches, including pseudopotential and density functional theories, have been remarkably successful in understanding structures and bonding in molecules and solids. We discuss these concepts in the context of phase change materials, which involve the rapid and reversible transition between amorphous and crystalline states, and note the confusion that some have caused, in particular 'resonance' and 'resonant bonding'.

Chemistry in one dimension

Orbital basis functions in a one-dimensional triatomic molecule ABC.

Covalent bonds are created by the drive of electron waves to lower their kinetic energy through expansion

An analysis based on the variation principle shows that in the molecules H2 (+), H2, B2, C2, N2, O2, F2, covalent bonding is driven by the attenuation of the kinetic energy that results from the delocalization of the electronic wave function. For molecular geometries around the equilibrium distance, two features of the wave function contribute to this delocalization: (i) Superposition of atomic orbitals extends the electronic wave function from one atom to two or more atoms; (ii) intra-atomic contraction of the atomic orbitals further increases the inter-atomic delocalization. The inter-atomic kinetic energy lowering that (perhaps counter-intuitively) is a consequence of the intra-atomic contractions drives these contractions (which per se would increase the energy). Since the contractions necessarily encompass both, the intra-atomic kinetic and potential energy changes (which add to a positive total), the fact that the intra-atomic potential energy change renders the total potential binding energy negative does not alter the fact that it is the kinetic delocalization energy that drives the bond formation.

Physical understanding through variational reasoning: electron sharing and covalent bonding

Energy changes of stationary states resulting from geometric parameter changes in the Hamiltonian can be understood by variational reasoning in terms of the physical attributes of the kinetic and the potential energy functionals. In atoms as well as molecules, the energy minimization determines the ground state as the optimal compromise between the potential pull of the nuclear attractions and the localization-resisting kinetic pressure of the electron cloud. This variational competition is analyzed for the exact ab initio ground-state wave function of the hydrogen molecule ion to elucidate the formation of the bond. Its electronic wave function is shown to differ from the ground-state wave function of the hydrogen atom by polarization, sharing, and contraction, and the corresponding contributions to the binding energy are examined in detail. All told, the critical feature is that a molecular orbital, contracting (in the variational context) toward two nuclei simultaneously, can lower its potential energy while maintaining a certain degree of delocalization. As a consequence, its kinetic energy functional has a lower value than that of an orbital contracting toward a single nucleus equally closely. By contrast, the potential energy functional is lowered equally effectively whether the orbital contracts toward one nucleus or simultaneously toward two nuclei. Because of this weaker kinetic energy pressure, the electrostatic potential pull of the nuclei in the molecule is able to attach the orbital more tightly to each of the nuclei than the pull of the single nucleus in the atom is able to do. The role of the virial theorem is clarified. Generalizations to other molecules are discussed.

Why does electron sharing lead to covalent bonding? A variational analysis

Ground state energy differences between related systems can be elucidated by a comparative variational analysis of the energy functional, in which the concepts of variational kinetic pressure and variational electrostatic potential pull are found useful. This approach is applied to the formation of the bond in the hydrogen molecule ion. A highly accurate wavefunction is shown to be the superposition of two quasiatomic orbitals, each of which consists to 94% of the respective atomic 1s orbital, the remaining 6% deformation being 73% spherical and 27% nonspherical in character. The spherical deformation can be recovered to 99.9% by scaling the 1s orbital. These results quantify the conceptual metamorphosis of the free-atom wavefunction into the molecular wavefunction by orbital sharing, orbital contraction, and orbital polarization. Starting with the 1s orbital on one atom as the initial trial function, the value of the energy functional of the molecule at the equilibrium distance is stepwise lowered along several sequences of wavefunction modifications, whose energies monotonically decrease to the ground state energy of H2+. The contributions of sharing, contraction and polarization to the overall lowering of the energy functional and their kinetic and potential components exhibit a consistent pattern that can be related to the wavefunction changes on the basis of physical reasoning, including the virial theorem. It is found that orbital sharing lowers the variational kinetic energy pressure and that this is the essential cause of covalent bonding in this molecule.