Mathematical basis of approximate MO theories: Origin of mulliken's magic formula

Abstract

AbstractsThe mathematical basis for the Ruedenberg's and Mulliken's approximations is given in a general framework.Les bases mathCmatiques des approximations de Ruedenberg et de Mulliken sont prCsenttes dans un cadre g6nCral. Die mathematische Basis de Ruedenberg'schen und Mulliken'schen Naherungen wird in einem allgemeinen Rahmen gegeben.Consider an electronic system defined by a countable set of subsystems (atoms, molecular fragments, molecules, etc.) with associated sets of basis functions Sr = {C#JL ; p E I}. If I and J are two different subsystems, one can write the set Sr in terms of linear combinations of the elements of S J :The {a::] are a set of appropriate coefficients, which can be defined through the equation where are the elements of the system's metric matrix.submatrices, one can construct a hypermatrix With the overlap elements partitioned between intra-and intersubsystems

Theory of time‐dependent nonequilibrium transport through a single molecule in a nonorthogonal basis set

“…Indeed, although each a goes from 0 to a positive value when going from one electrode through the central region to the other electrode, it is at present not known whether they are also, for example, strictly non‐negative, which would be a sufficient condition for their vanishing under the orthogonal case condition, Equation . To estimate the value of these potentials, we use the block‐form of S , Equation , which is decomposed via the identity into its block‐diagonal, , and block‐off‐diagonal, , parts, …”

“…is also gauge-invariant. It consists solely of terms containing characteristic potentials a , which is decomposed via the identity [82] S5S D 1S O into its block-diagonal, S D , and block-off-diagonal, S O , parts,…”

“…Þg aC ðt; t 1 ÞÞ 5 2eTr½ðH aC 1R int aC 2S aC io t Þg Ca ðt; t 1 Þ1r int aC ðt; tÞG Ca ðt2t 1 Þ 2g aC ðt; t 1 ÞðH Ca 1R int Ca 1io t 1 S aC Þ2G aC ðt2t 1 Þr int Ca ðt 1 ; t 1 Þ Þg Ca ðt; t 1 Þ1S Ca ðo t 1o t 1 Þg aC ðt; t 1 ÞÞ;After some algebra involving expressions for the TIH contribution g < from the dot and leads, Equations 78 and 80, and the TIH contribution r from the junction to the dot SE, Equation82, we obtain…”

Discussion on the variable Wolfsberg–Helmholtz parameter, a new simplified Löwdin transformation and the characteristic structure of the transformed EHT Hamiltonian matrices

“…It is a kind of expressing the well-known Mulliken approximation [9,12,13] for constructing the off-diagonal Hamiltonian elements in terms of the diagonal elements.…”

“…Thus, the simplified Lö wdin transformation produces a transformed EHT Hamiltonian matrix, whose main elements are the sum of the inward matrix products (13).…”