Rethinking first-principles electron transport theories with projection operators: The problems caused by partitioning the basis set

Abstract

We revisit the derivation of electron transport theories with a focus on the projection operators chosen to partition the system. The prevailing choice of assigning each computational basis function to a region causes two problems. First, this choice generally results in oblique projection operators, which are non-Hermitian and violate implicit assumptions in the derivation. Second, these operators are defined with the physically insignificant basis set and, as such, preclude a well-defined basis set limit. We thus advocate for the selection of physically motivated, orthogonal projection operators (which are Hermitian) and present an operator-based derivation of electron transport theories. Unlike the conventional, matrix-based approaches, this derivation requires no knowledge of the computational basis set. In this process, we also find that common transport formalisms for nonorthogonal basis sets improperly decouple the exterior regions, leading to a short circuit through the system. We finally discuss the implications of these results for first-principles calculations of electron transport.

Invariance of molecular charge transport upon changes of extended molecule size and several related issues

“…This is a minimal mandatory requirement for any valid theoretical approach. However, although conceptually arbitrary, for practical purposes, it is convenient that the partitioning fulfills certain conditions [ 36 ]. Unnecessary, more demanding computational effort can be avoided if the junction is partitioned such that: (i) the properties of the electrodes are homogeneous and do not differ from those of the bulk materials (metals), (ii) there is no direct interaction between the left and right electrodes, and (iii) interactions between the extended molecule and the two electrodes are merely confined to the extended molecule–electrode interfaces.…”

Theoretische Chemie 2013

“…Für die oben beschriebene Standardmethode können aus großen Basissätzen unphysikalisch hohe Leitwerte resultieren 310,311. Reuter et al schlugen als Ursache vor,312 dass der Strom zwar durch die Grenzfläche zwischen (erweitertem) Molekül und Elektrode berechnet werden soll, diese Grenzfläche aber in der Berechnung nicht im dreidimensionalen kartesischen Raum definiert wird, sondern durch überlappende atomzentrierte Basisfunktionen (ähnlich wie in der Populationsanalyse313). Stattdessen sollte den Autoren zufolge die Aufteilung in Molekül und Elektrode schon auf der Stufe der Operatoren erfolgen, also bevor diese als Matrizen in einer atomzentrierten Basis dargestellt werden.…”

Theory of projections with nonorthogonal basis sets: Partitioning techniques and effective Hamiltonians

“…Some of the mathematical and conceptual issues discussed here have been previously addressed in the literature [27,[32][33][34][35][36][37][38][39], but we believe that our approach to the problem may shed a new light on some of the unresolved or controversial points. We begin by revisiting some basic operator operations in a nonorthogonal metric in Sec.…”