Quantum Crystallography:N‐Representability Big and Small**

Abstract

This paper is to honor the memory of Linus Pauling on the occasion of his 120 th birthday (2021).

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

“…Clearly, every 1-RDM in may be written as (5) where b ij = (2 − δ ij )Γ i,j . Electron densities corresponding to 1-RDMs in are regarded as elements of a vector space of the "collapsed" functions…”

“…9 A more ambitious goal is to associate a given density with a nonidempotent N-representable 1-RDM. 7,8 In a parallel but independent development, Harriman showed 10−12 that, within finite basis sets of functions that form linearly independent products (LIPs), the problem of reconstructing 1-RDMs from densities has a straightforward and unique solution independent of any energy functional. This unique 1-RDM is obtained by solving a certain system of linear equations involving the overlap matrix of the basis function products.…”

“…This problem does not have a unique solution even when restricted to ground states because the search for 1-RDMs requires assuming a specific energy functional . A more ambitious goal is to associate a given density with a nonidempotent N -representable 1-RDM. , …”

“…The problem of constructing reduced density matrices compatible with a specified electron density has attracted considerable attention for many years − and gave rise to an entire subfield of computational chemistry known as quantum crystallography. , In the simplest version of this problem, the objective is to find an idempotent one-electron reduced density matrix (1-RDM) γ­( r , r ′) corresponding, as closely as possible, to a given electron density ρ­( r ). This problem does not have a unique solution even when restricted to ground states because the search for 1-RDMs requires assuming a specific energy functional .…”

Spin Energy Contributions of the Kinetic Energy Density in the Stabilization of the Metal–Ligand Interactions

“…16−20 This latter has been recently supported by experimental and theoretical evidence comparing the free Be atom with that in a crystal environment. 21 Classically, the local kinetic energy of a system can be defined without ambiguity; however, in quantum mechanics, there exist an infinite number of expressions that are integrated throughout the space and recover the total kinetic energy of the system. Most of these expressions are valid and mathematically justified; however, their conceptual usefulness is limited.…”