Spin Density Topology

Abstract

Despite its role in spin density functional theory and it being the basic observable for describing and understanding magnetic phenomena, few studies have appeared on the electron spin density subtleties thus far. A systematic full topological analysis of this function is lacking, seemingly in contrast to the blossoming in the last 20 years of many studies on the topological features of other scalar fields of chemical interest. We aim to fill this gap by unveiling the kind of information hidden in the spin density distribution that only its topology can disclose. The significance of the spin density critical points, the 18 different ways in which they can be realized and the peculiar topological constraints on their number and kind, arising from the presence of positive and negative spin density regions, is addressed. The notion of molecular spin graphs, spin maxima (minima) joining paths, spin basins and of their valence is introduced. We show that two kinds of structures are associated with a spin-polarized molecule: the usual one, defined through the electron density gradient, and the magnetic structure, defined through the spin density gradient and composed in general by at least two independent spin graphs, related to spin density maxima and minima. Several descriptors, such as the spin polarization index, are introduced to characterize the properties of spin density critical points and basins. The study on the general features of the spin density topology is followed by the specific example of the water molecule in the ³B₁ triplet state, using spin density distributions of increasing accuracy.

Keywords: molecular spin graph; quantum chemical topology; spin density; spin density basins; spin density critical points; spin density source function; spin maxima and minima joining paths; topology; water 3B1 triplet.

Figure 1

CAS contour plots in the σ_v plane of ³B₁ H₂O containing the three nuclei: (a) electron spin density s and (b) electron spin density Laplacian ∇²s. Contour maps are drawn in intervals of ± (2, 4, 8) × 10⁻ⁿ, 0 ≤ n ≤ 6 au (e⁻(bohr)⁻³ and e⁻(bohr)⁻⁵ for electron spin density and electron spin density Laplacian, respectively). Spin density CPs are shown on both maps, the unique ones labelled also by their identification number N (Table 2 and Table 3). CPs’ 6 locations are projected on this map (see text for more details). Solid red contour lines corresponds to s or ∇²s positive values, while the dashed blue contour lines correspond to negative s or ∇²s values. The solid squares mark (3, −3) CPs, the solid triangles (3, −1) CPs, gray triangles (3, +1) CPs and gray squares (3, +3) CPs. Note that, in this figure, in Figure 2 and in all figures of the Supplementary Materials, the atomic nuclei are located in the close neighbourhood of the atomic labels.

Figure 2

CAS contour plots in the σ_v’ plane of ³B₁ H₂O containing the O nucleus: (a) electron spin density s and (b) electron spin density Laplacian ∇²s. For all other details, see the caption of Figure 1.

Figure 3

CAS spin density maxima joining paths: α–α, positive spin density regions (s₊, left); β–β, negative spin density regions (s₋, right). In both panels, the ∇ρ bond paths are also shown (from H nucleus to the bond critical point (bcp) in white and from bcp to O nucleus in red). Critical points of s(r) are labelled as in Table 2 and Table 3 and portrayed as colored balls ((3, −3) s maxima, orange; (3, −1) s saddles, violet; (3, +3) s minima or –s maxima, blue; (3, +1) s ring or –s (3, −1) saddles, yellow). The α–α and β–β spin maxima joining paths are represented as thick green wires. The β–β spin density maxima joining paths all lie in the σ_v plane of ³B₁ H₂O, i.e., that defined by CPs 2, 2’ and 3 in the left panel. The pictures were obtained through the code Diamond v3.21 [39].

Figure 4

Unrestricted Hartree Fock (UHF) spin density maxima joining paths: α–α, positive spin density regions (s₊, left); β–β, negative spin density regions (s_–, right). In both panels, the ∇ρ bond paths are also shown (from H nucleus to the bcp in white and from bcp to O nucleus in red). For all other details, see the caption of Figure 3.

Figure 5

³B₁ H₂O CAS spin density basins. The basins are labelled by their enclosed (3, −3) spin maximum or, for the basin labelled 15, by its enclosed (3, +3) spin minimum (see Table 2 and Table 3). Each basin retains the same color in all illustrations. Panels a and b refer to basins bounded by local Zero–Flux Surfaces (ZFS) of ∇s, while panels c and d to basins bounded by s = 0 isosurfaces (see text). Panel b (d) differs from panel a (c) for the removal of basins 1 and 1’, so disclosing the small and embedded basin 3 around the O nucleus. Basins are represented through their spin density isovalue surfaces ((bohr)⁻³): 0.0003 (basins 1,1’; ∇s ZFS); 0.00009 (basins 1,1’; s = 0); 0.00031 (basins 2,2’); 0.0751 (basin 3); −0.000017 (basins 15, 15’). The drawings in the four panels were performed with UCSF Chimera, developed by the Resource for Biocomputing, Visualization, and Informatics at the University of California, San Francisco, with support from NIH P41–GM103311 [40].