Electron localization function for non-collinear spins
Abstract: Understanding of bonding is key to modelling materials and predicting properties thereof. A widely adopted indicator of bonds and atomic shells is the electron localization function (ELF). The building blocks of the ELF are also used in the construction of modern density functional approximations. Here we demonstrate that the ELF breaks down when applied beyond regular non-relativistic quantum states. We show that for tackling general non-collinear open-shell solutions, it is essential to address both the U(1) gauge invariance -- i.e., invariance under a multiplication by a position dependent phase factor -- and SU(2) gauge invariance -- i.e. invariance under local spin rotations -- {\em conjointly}. Remarkably, we find that the extended ELF also improves the description of paradigmatic collinear states.
- L. Pauling, Oregon State University Libraries Special Collections (1946).
- Here and elsewhere we use Einstein summation over repeated indices, unless explicitly specified.
- In the many-electron case, the value of the ELF~~ELF\widetilde{\mathrm{ELF}}over~ start_ARG roman_ELF end_ARG at large 𝐫𝐫{\bf r}bold_r depends on the asymptotic behaviour of Dunifsuperscript𝐷unifD^{\mathrm{unif}}italic_D start_POSTSUPERSCRIPT roman_unif end_POSTSUPERSCRIPT vs. D~ncsubscript~𝐷nc\widetilde{D}_{\mathrm{nc}}over~ start_ARG italic_D end_ARG start_POSTSUBSCRIPT roman_nc end_POSTSUBSCRIPT.
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