Wannier functions have become a powerful tool in the
electronic
structure calculations of extended systems. The generalized Pipek-Mezey
Wannier functions exhibit appealing characteristics (e.g., reaching
an optimal localization and the separation of the σ–π
orbitals) compared with other schemes. However, when applied to giant
nanoscale systems, the orbital localization suffers from a large computational
cost overhead if one is interested in localized states in a small
fragment of the system. Herein, we present a swift, efficient, and
robust approach for obtaining regionally localized orbitals of a subsystem
within the generalized Pipek-Mezey scheme. The proposed algorithm
introduces a reduced work space and sequentially exhausts the entire
orbital space until the convergence of the localization functional.
It tackles systems with ∼10000 electrons within 0.5 h with
no loss in localization quality compared to the traditional approach.
Regionally localized orbitals with a higher extent of localization
are obtained via judiciously extending the subsystem’s size.
Exemplifying on large bulk and a 4 nm wide slab of diamond with an
NV
–
center, we demonstrate the methodology and discuss
how the choice of the localization region affects the excitation energy
of the defect. Furthermore, we show how the sequential algorithm is
easily extended to stochastic methodologies that do not provide individual
single-particle eigenstates. It is thus a promising tool to obtain
regionally localized states for solving the electronic structure problems
of a subsystem embedded in giant condensed systems.
In this paper, symmetric structure of induction motor system in stationary aflO coordinates is studied by the geometric approach. The results show that the system possesses symmetry (G, 0, ~) and infinitesimal symmetry. Under certain conditions, the system can be transformed into a form possessing state-space symmetry ( G, • ) and infinitesimal state-space symmetry by means of state feedback and input coordinate base transform. The results can be extended to the fifth order induction motor system fed by hysteresis-band current-controlled PWM inverter.
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