Stationarity of electron distribution in ground-state molecular systems
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Stationarity of electron distribution in ground-state molecular systems Dariusz Szczepanik · Janusz Mrozek
Received: 10 December 2012 / Accepted: 15 February 2013 / Published online: 27 February 2013 © The Author(s) 2013. This article is published with open access at Springerlink.com
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 orbital 1 Introduction In the last half-century 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, EPA-procedures 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 EPA-procedures among which the most commonly used are those proposed by: Mulliken [1,2], Löwdin [3,4], Weinhold [5], Bader [6], Hirshfeld [7] and Merz-Kollman [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 closed-shell ground states at two levels of theory,
D. Szczepanik (B) · J. Mrozek Department of Computational Methods in Chemistry, Faculty of Chemistry, Jagiellonian University, R. Ingardena St. 3, 30-060 Cracow, Poland e-mail: [email protected]
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J Math Chem (2013) 51:1388–1396
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Hartree-Fock (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 [10–12] and the theory of Markov Processes [13]. 2 Stationarity from idempotency For simplicity and transparency of definitions we assume without loss of generality the wavefunctions to be real. First, let us consider the one-determinant wavefunction of the ground-state molecular system, | H F , with N electrons doubly occupying the n o lowest of MOs, |Φ o , from the Hilbert space of the all n orthonormal (spinless) molecular orbitals, |Φ = |Φ1o , . . . , |Φnoo , |Φnvo +1 , . . . , |Φnv ,
(1)
generated as the linear combinations of basis functions |χ, representing orthogonal atomic orbitals, |Φ = |χC,
(2)
where square matrix C groups the relevant LCAO MO expansion coefficients, C
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