These dopant atoms can be arranged into arrays [@Pok07], or geometric patterns for wires [@Ruess07] and associated tunnel junctions [@Ruess07a], gates, and quantum dots [@Ruess07c; @Fuhrer09] – all of which are necessary components of a functioning device [@Fuechsle10]. The patterns themselves define atomically abrupt regions of doped and undoped silicon. While silicon, bulk-doped silicon, and the physics of the phosphorus incorporation [@Wilson04] are well-understood, models of this quasi-two-dimensional phosphorus sheet are still in their initial stages. In particular, it is critical in many applications to understand the effect of this confinement on the conduction band valley degeneracy, inherent in the band structure of silicon. For example, the degeneracy of the valleys has the potential to cause decoherence in a spin-based quantum computer [@Koiller01; @Boykin04], and the degree of valley degeneracy lifting (valley splitting) defines the conduction properties of highly confined planar quantum dots[@Fuechsle10].

The importance of understanding valley splitting in monolayer $\delta$-doped Si:P structures has led to a number of theoretical works in recent years spanning several techniques, from pseudopotential theories via planar Wannier orbital (PWO) bases [@Qian05], density functional theory (DFT) via linear combination of atomic orbital (LCAO) bases [@Carter09; @Carter11], to tight-binding (TB) models [@Cartoixa05; @Lee09; @Ryu10; @Ryu11; 