Insights on spin polarization through the spin density source function

Abstract

Understanding how spin information is transmitted from paramagnetic to non-magnetic centers is crucial in advanced materials research and calls for novel interpretive tools. Herein, we show that the spin density at a point may be seen as determined by a local source function for such density, operating at all other points of space. Integration of the local source over Bader's quantum atoms measures their contribution in determining the spin polarization at any system's location. Each contribution may be then conveniently decomposed in a magnetic term due to the magnetic natural orbital(s) density and in a reaction or relaxation term due to the remaining orbitals density. A simple test case, ³B₁ water, is chosen to exemplify whether an atom or group of atoms concur or oppose the paramagnetic center in determining a given local spin polarization. Discriminating magnetic from reaction or relaxation contributions to such behaviour strongly enhances chemical insight, though care needs to be paid to the large sensitivity of the latter contributions to the level of the computational approach and to the difficulty of singling out the magnetic orbitals in the case of highly correlated systems. Comparison of source function atomic contributions to the spin density with those reconstructing the electron density at a system's position, enlightens how the mechanisms which determine the two densities may in general differ and how diverse may be the role played by each system's atom in determining each of the two densities. These mechanisms reflect the quite diverse portraits of the electron density and electron spin density Laplacians, hence the different local concentration/dilution of the total and (α-β) electron densities throughout the system. Being defined in terms of an observable, the source function for the spin density is also potentially amenable to experimental determination, as customarily performed for its electron density analogue.

Fig. 1. Electron density Laplacian, electron spin density s(r) and its Laplacian in the (y,z) plane for ³B₁ H₂O, at (top) CASSCF(8,8) and (bottom) UHF/UHF spin-contamination annihilated computational levels. Atomic units (a.u.) are used throughout. Contour maps are drawn at intervals of ±(2,4,8) × 10ⁿ, –4 ≤ n ≤ 0 (s, ∇²s) and –3 ≤ n ≤ 0 (∇²ρ). Dotted blue (full red) lines indicate negative (positive) values and full black lines mark boundaries of atomic basins. The O–H bond critical point (bcp, 1) and the bonded charge concentration point (bcc, 2) are shown as black and green dots.

Fig. 2. Electron density Laplacian, spin density and its Laplacian in the (x,z) plane, at (top) CASSCF(8,8) and (bottom) UHF/UHF spin contamination annihilated computational levels. Contour levels as in Fig. 1. The non-bonded charge concentration (nbcc, 3) and the (3,+1) L(r) rcps (4) are shown as green and red dots.

Fig. 3. SF and SF_S percentage contributions at some reference points (rps) for ³B₁ H₂O at the CASSCF(8,8) level. The separate contributions to SF_S due to the magnetic (SF_S mag) and the remaining (SF_S – SF_S mag) natural orbitals are also shown (for SF only those due to magnetic orbitals, denoted as SF mag). Each atom is displayed as a sphere, whose volume is proportional to the source percentage contribution to ρ(r) or s(r) values at the rp (first column). Colour codes: blue (yellow) atoms act as positive (negative) sources for ρ at rps; green (red) atoms act as positive (negative) sources for s at rp, hence yielding an α(β) effect at rp (the sign of percentage source is instead positive or negative whether the atomic source concurs with or opposes s at rp).