Research paperEfficient evaluation of poly-electron populations in natural bond orbital analysis
Graphical abstract
Introduction
In an extended series of works, Karafiloglou and co-workers [1] have demonstrated the conceptual usefulness of “poly-electron population analysis” (PEPA) that generalizes conventional orbital populations as provided, e.g., by natural population analysis (NPA) [2]. In Karafiloglou’s formulation, a specific population (or “probability of occupancy”) can be assigned not only to individual localized orbitals (LOs), such as natural atomic orbitals (NAOs), natural hybrid orbitals (NHOs), or natural bond orbitals (NBOs) [3], but also to general electron-hole (e/h) excitation patterns of chemical interest. In the present work we focus on the “natural” NAO/NBO-based (NPEPA) algorithms that are implemented in the forthcoming NBO 7.0 program version [4].
Each e/h pattern of NPEPA analysis specifies a particular combination of occupied (“electron”) and vacant (“hole”) spin-orbitals that may (or may not) occur in Slater-determinantal contributions to the total wavefunction Ψ. As usual, the associated “Born probability” P(e/h) of such contributions is evaluated by summing the squared coefficients of all determinants exhibiting the desired pattern. For example, if the desired e/h patterns are expressed in terms of NBOs, each NBO occupancy pattern can be viewed as a distinct “resonance structure” whose Born probability quantifies the “weighting” of the bonding pattern in Ψ, thereby providing interesting comparisons with corresponding natural resonance theory (NRT) [5] weightings. If instead the analysis of Ψ is posed in terms of NAOs, the specific NAO patterns might correspond to the 2-center/2-electron (2c/2e) “covalent” or “ionic” contributions of classical Heitler-London valence-bond (VB) theory [6], thereby providing interesting comparisons with alternative assessments of VB-character in modern wavefunctions [7]. Many other questions involving deeper details of Pauli compliancy [8], electron correlation [9], or multi-center bonding aspects [10] could be imagined, such as the distinct probabilities for any possible arrangement of six electrons among the six π-type NAOs of benzene. All such questions are in principle addressable with NPEPA.
The original numerical implementation of NPEPA [11] is based on Moffitt’s theorem [12] and fully exploits the powerful constraints of dual e-based vs. h-based determinantal expansions of Ψ [13]. However, the starting Ψ is usually obtained as a single- or multi-configuration expansion in delocalized molecular orbital (MO) determinants. The required transformations from MO to LO form therefore lead to steep factorial increases in the number of LO-based determinants {} from a given MO-based determinant . The goal of the present work is to introduce an algorithmic modification that can successfully deal with the potential factorial explosions of LO-based determinants in NPEPA evaluations.
Section snippets
Poly-electron population analysis
The N-electron wavefunction ψ can be expressed in configuration interaction (CI) form as a linear combination of normalized Slater determinants where each index I represents a unique set of N occupied molecular spin-orbitals {} from which the determinant is constructed. We seek to analyze this wavefunction in a complete, orthonormal basis of localized spin-orbitals {}
The Slater determinants of Eq. (1) can be re-expressed as linear combinations of normalized
Concluding discussion
The foregoing B3LYP/6-31G* calculation for CH3NH2 allows details of a single NPEPA evaluation to be illustrated, but cannot adequately suggest the vastly improved practicality of the frozen-NO method for more complex NPEPA applications to larger systems.
To this end we may return to the question of calculating Born probabilities for every possible pattern of electrons and holes in the six p-type NAOs of the benzene pi system. There are 4096 such probabilities (every possible arrangement of 0–12
Acknowledgment
Computational support of this project was provided in part by National Science Foundation Grant CHE-0840494.
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