Atomic Spectral-Product Representations of Molecular Electronic Structure: Metric Matrices and Atomic-Product Composition of Molecular Eigenfunctions

Abstract

Recent progress is reported in development of ab initio computational methods for the electronic structures of molecules employing the many-electron eigenstates of constituent atoms in spectral-product forms. The approach provides a universal atomic-product description of the electronic structure of matter as an alternative to more commonly employed valence-bond- or molecular-orbital-based representations. The Hamiltonian matrix in this representation is seen to comprise a sum over atomic energies and a pairwise sum over Coulombic interaction terms that depend only on the separations of the individual atomic pairs. Overall electron antisymmetry can be enforced by unitary transformation when appropriate, rather than as a possibly encumbering or unnecessary global constraint. The matrix representative of the antisymmetrizer in the spectral-product basis, which is equivalent to the metric matrix of the corresponding explicitly antisymmetric basis, provides the required transformation to antisymmetric or linearly independent states after Hamiltonian evaluation. Particular attention is focused in the present report on properties of the metric matrix and on the atomic-product compositions of molecular eigenstates as described in the spectral-product representations. Illustrative calculations are reported for simple but prototypically important diatomic (H(2), CH) and triatomic (H(3), CH(2)) molecules employing algorithms and computer codes devised recently for this purpose. This particular implementation of the approach combines Slater-orbital-based one- and two-electron integral evaluations, valence-bond constructions of standard tableau functions and matrices, and transformations to atomic eigenstate-product representations. The calculated metric matrices and corresponding potential energy surfaces obtained in this way elucidate a number of aspects of the spectral-product development, including the nature of closure in the representation, the general redundancy or linear dependence of its explicitly antisymmetrized form, the convergence of the apparently disparate atomic-product and explicitly antisymmetrized atomic-product forms to a common invariant subspace, and the nature of a chemical bonding descriptor provided by the atomic-product compositions of molecular eigenstates. Concluding remarks indicate additional studies in progress and the prognosis for performing atomic spectral-product calculations more generally and efficiently.

Quantum-Mechanical Definition of Atoms and Chemical Bonds in Molecules

“…Although elaborate methods have been developed for finite-subspace calculation in atomic spectral-product representations [61,[65][66][67][72][73][74], a factored version of the general development is particularly suited to calculations of the molecular and fragment energies of focus here [66,67,74]. The approach requires for its validity only the linear independence of the antisymmetrized form of the chosen finite subspace [75], providing a Hamiltonian matrix identical in appearance to Eq.…”

“…Molecular (Born-Oppenheimer) Hamiltonian matrices take particularly simple forms in the atomic spectral-product representation as sums over universal atomic and pair-interaction Hamiltonian matrices which can be calculated once and for all and retained for repeated applications [66,67]. Total molecular energies obtained by conventional Hamiltonian matrix diagonalization are seen to take the form of sums over atomic and pairwise-interaction energies, expressed in terms of products of the universal atomic and interaction Hamiltonian matrices and the calculated molecular eigenvectors.…”

“…Atomic energy distributions obtained in this way describe the extent to which individual atoms are excited and their electrons apportioned to atomic bonding partners over the entire molecular volume, whereas the pairwise-atomic interaction energies provide corresponding chemical-bonding energies between constituent atoms. Overall electron antisymmetry of molecular eigenstates is formally achieved in the development by convergence in the limit of closure, or by ex-post-facto enforcement in finite subspaces [66,67].…”

“…(16) in the indicated finite subspace spectral-product basis to accommodate this change of representation [66,67,74]. The total electronic energy obtained from the foregoing expressions for a particular molecular eigenstate (i) is seen to be a sum of atomic energies for the constituent atoms and a sum of atomic-pair bonding energies of the form [cf., Eq.…”

Quantum-mechanical definition of atoms and their mutual interactions in Born-Oppenheimer molecules

“…Quantummechanical operators for atoms in molecules are obtained in this representation with fixed electron-nuclei assignments made in accordance with those assignments employed in the atomic spectral functions. Totally antisymmetric eigenstates supported in this way provide molecular electronic energies which separate naturally into sums of atomic and pairwise-atomic interaction-energy components upon removal of so-called unphysical non-Pauli eigenstates from the development [71][72][73][74][75][76].…”