Stationarity of electron distribution in groundstate molecular systems
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Abstract
Stationarity of electron probability distribution within the resolution of atomic orbitals is considered involving some concepts from Orbital Communication Theory and the theory of Markov Processes. A new method of evaluating electron conditional probabilities based on natural orbitals is proposed and briefly discussed.
Keywords
Stationary distribution Markov chain Density matrix Conditional probability Natural orbital1 Introduction
In the last halfcentury the electron population analyses (EPA) turned out to be very useful and commonly used tools in probing the electronic structure and chemical reactivity of molecules. In brief, EPAprocedures give rise to partition the electron density of the whole molecule between atoms, chemical bonds, molecular fragments, etc. Such partition can be performed within either physical or Hilbert space of molecular orbitals (MO) and there is a multitude of diverse EPAprocedures among which the most commonly used are those proposed by: Mulliken [1, 2], Löwdin [3, 4], Weinhold [5], Bader [6], Hirshfeld [7] and MerzKollman [8].
Although this is a field of research that seems to be regarded by some scientists as exhausted, in this short paper we would like to address some of the general issues respecting stationarity and uniqueness of electron probability distributions within the framework of MO theory and resolution of atomic orbitals (AO). Our considerations will be confined only to closedshell ground states at two levels of theory, HartreeFock (HF) and configuration interaction (CI) [9]. The main purpose of this article is to briefly introduce a new method of determining stationary electron probability distributions, based on natural orbitals (NO) and involving some concepts from the Orbital Communication Theory (OCT) of the chemical bond [10, 11, 12] and the theory of Markov Processes [13].
2 Stationarity from idempotency

matrix P represents an aperiodic Markov chain which virtually means that for all atomic orbitals \(\{\chi _\mu \rangle \}\) the direct selfcommunication \(\mu \rightarrow \mu \) is always possible, \(P(\mu \mu )>0\). In practice, aperiodicity can be always attained by appropriate reduction of the AOspace size, i.e. by removing atomic orbitals for which \(\gamma _{\mu ,\mu }<\epsilon \), where \(\epsilon \) represents the fixed threshold value;

matrix P represents an irreducible Markov chain. i.e. if all atomic orbitals are directly and/or indirectly communicated with each other. Also this requirement is straightforward to achieve by eliminating (if present) those AOs for which \(\gamma _{\mu \mu }>2\epsilon \) (ussually core orbitals). The subspace od mutually communicated atomic orbitals defines a single communicating class.
3 Stationarity without idempotency
4 Summary
It was the main goal of this short paper to demonstrate that idempotency property of the density operator is not an indispensable condition for the stationarity of electron probability distribution in molecular systems. We have introduced a simple method of evaluation of conditional probabilities for the multideterminant wavefunction. Moreover, the newly proposed definition based on natural orbitals successfully displaces hitherto prevailing one for the onedeterminant wavefunctions. Since presented methodology introduces some arbitrariness due to orbital order, it has been argued to use the criterion of maximal separation of nearlydegenerated orbital subspaces in groundstate calculations.
The proposed method can be also extended to cover openshell molecular systems as well as the excited states. However, this requires a more insightful investigation and will be the subject of the seperate article.
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