Phenomenological rules play a central role in the design of chemical reactions and materials with targeted properties. Typically, these are formulated heuristically in terms of non-interacting orbitals and bands, yet show remarkable accuracy in predicting the complex behavior of intrinsically interacting many-body systems. While their non-interacting formulation makes them easy to interpret, it potentially hinders the development of new rules for systems governed by strong correlation, such as transition metal-based materials. In this work, we present a rigorous framework that allows bridging between fully interacting, even strongly correlated, systems and an effective one-body picture in terms of quasiparticles. Further, we present a computational strategy to efficiently and accurately access the main components of such a description: the embedding approximation of the ghost Gutzwiller Ansatz. We illustrate the capabilities of this quasiparticle formulation on the Woodward-Hoffmann rules, and apply their reformulated version to toy ``reactions'' which exemplify the main scenarios covered by them.
Chemistry and materials science tackle extraordinarily complex many-body systems and, consequently, phenomenological rules often lie at the center of the successful design of new synthesis pathways and devices of tailored opto-electronic properties. These are often so instrumental that they become a regular part of undergraduate curricula, as is the case with Hückel [1][2][3], Goodenough-Kanamori [4,5] or Woodward-Hoffmann rules [6][7][8][9][10][11]. These last ones, for example, concern predicting whether certain types of chemical reactions are likely to be thermally activated or not. While they can be formulated in different ways, commonly the Woodward-Hoffmann rules are stated in terms of molecular orbital symmetry: In essence, along a symmetry-preserving reaction pathway, if the frontier orbitals involved in the transformation belong to different irreducible representations and cross in the HOMO-LUMO gap, the reaction is not expected to proceed thermally. One calls such reactions Woodward-Hoffmann forbidden, while reactions without such a crossing are Woodward-Hoffmann allowed. This nicely illustrates the nature of such phenomenological rules: they connect the complex chemical reality to a simple, intuitive model in terms of non-interacting molecular orbitals, somehow without compromising their accuracy. This last point is particularly remarkable when one considers that electron correlation is prevalent in molecules and materials, and crucially becomes dominant along reactions breaking chemical bonds. This begs the question: Is it possible to derive, or at least convincingly justify, these rules from a fully interacting formalism? This would not only cement the theoretical underpinnings of already existing rules, but could potentially open a systematic path to discovering new ones tailored to strongly correlated systems, such as transition metal catalysts or quantum materials.
One strategy towards this goal involves justifying these rules using explicitly many-body concepts. An insightful example concerns recent work by Xie et al. [12], where they observe that the crossing of non-interacting orbitals, central to the original formulation of the Woodward-Hoffmann rules, may directly translate into the crossing of zeros of the one-body Green’s function. While this direction is highly interesting, fully foregoing the noninteracting picture poses two inconvenient challenges: (i) the interpretation, or distilling, of phenomenological rules from the frequency dependent Green’s function is more difficult than from an orbital theory, and (ii) accessing the witnesses for the fulfillment/violation of these rules is computationally much more expensive in a fully interacting framework. Ideally, one would wish for a formulation firmly rooted in the interacting limit, yet somehow leveraging the language of non-interacting orbitals. This is precisely the philosophy we adopt in this work. Starting from a fully interacting perspective, we derive an interpretable, one-body framework which is amenable to formulating phenomenological rules governing correlated electrons and compatible with an efficient computational implementation. We refer to this as the quasiparticle picture [13,14]. In essence, the idea is representing the features of strongly correlated electrons using auxiliary noninteracting systems, as one does in Kohn-Sham density functional theory for weakly-correlated materials [15]. Within such a quasiparticle language, it is possible to justify and formulate phenomenological rules in terms of molecular orbitals, yet remaining by construction in a fully interacting framework. We present the theoretical ingredients of this quasiparticle formalism, together with a computational strategy to efficiently and accurately access its main witnesses: the embedding approximation of the ghost Gutzwiller Ansatz [16][17][18][19][20][21][22]. We will exemplify this programme on the Woodward-Hoffmann rules, using two toy “reactions” to illustrate how they can be reformu-lated in the quasiparticle language. Looking ahead, this strategy has the potential to unveil new phenomenological rules for strongly correlated molecules and materials.
We start by reexamining the arguments of [12] from a different perspective, partly following Refs. [14,23,24]. We assume a generic basis of single-particle molecular orbitals ψ α (r), where α = 1, . . . , 2N includes the spin. Although the basis is complete only for N → ∞, we will work with a finite N henceforth. The zerotemperature Green’s function of the imaginary frequency iϵ, ϵ ∈ [-∞, ∞], is generally a matrix G(iϵ) = G(-iϵ) † in this basis, with elements
where n ≥ 0 runs over all many-body eigenstates |n⟩ with eigenvalues E n . The ground state is identified by n = 0 and assumed to be non-degenerate. We emphasize that zero imaginary frequency, ϵ = 0, separates the processes of adding an electron to the ground state (ϵ > 0) from those of removing one (ϵ < 0). We write G(iϵ) in (1) as
with
This content is AI-processed based on open access ArXiv data.