Natural localized orbitals

The natural localized orbitals introduced in Section 1.5 provide a useful alternative to the canonical delocalized MOs (CMOs) that are usually employed to analyze chemical bonding. The NAO and NBO basis sets may be regarded as intermediates in a succession of basis transformations that lead from starting AOs /, to the final canonical MOs , ... 

Each set of natural localized orbitals may be considered to lie along a sequence of transformations T that leads from one-center basis AOs to final delocalized canonical MOs or NOs, viz.. 

The natural localized orbital transformations provide useful tools for analyzing various molecular properties or energy components in localized bonding terms. We briefly summarize the underlying principles of several such analyses that are implemented in the general NBO program, restricting attention primarily to HF-level treatment. 

Fig. 1 Orbital localization measured by the orbital-contribution entropy 5orb for the Hartree-Fock calculation of the N2 molecule. From left to right. 1, canonical orbitals 2, natural localized orbitals 5, Pauli kinetic energy density fy 4, ELF...

In this section, we present a unified picture of the different electronic effects that combine to determine methyl rotor potentials in the S0, Sp and D0 electronic states of different substituted toluenes. Our approach is based on analysis of ab initio wavefunctions using the natural bond orbitals (NBOs)33 of Weinhold and cowork-ers. We will attempt to decompose the methyl torsional potential into two dominant contributions. The first is repulsive steric interactions, which are important only when an ortho substituent is present. The second is attractive donor-acceptor interactions between CH bond pairs and empty antibonding orbitals vicinal to the CH bonds. In the NBO basis, these attractive interactions dominate the barrier in ethane (1025 cm-1) and in 2-methylpropene (1010 cm-1) see Figure 3. By analogy, donor-acceptor attractions are important in toluenes whenever there is a substantial difference in bond order between the two ring CC bonds adjacent to the C-CH3 bond. Viewed the other way around, we can use the measured methyl rotor potential as a sensitive probe of local ring geometry. 

Density matrices, natural localized and delocalized orbitals, and the Lewis-structure picture... 

The N/2 Lewis-type NBOs of highest occupancy ( /AB) 2) can be directly associated with the localized electron pairs of the chemist s Lewis-structure diagram.46 Each Lewis-type NBO f2AB can be decomposed into constituent natural hybrid orbitals (NHOs) hA and hB on atoms A and B,... 

The starting Oa NBO is thereby perturbed to a natural localized molecular orbital (NLMO), distinguished by a tilde from its parent NBO... 

Numerical LMOs of this work are determined by the natural localized-molecular-orbital (NLMO) method A. E. Reed and F. Weinhold, J. Chem. Phys. 83 (1985), 1736. The LMOs determined by other methods (e.g., C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 34 [1963], 457 and J. M. Foster and S. F. Boys, Rev. Mod. Phys. 32 [1960], 300) are rather similar, and could be taken as equivalent for present purposes. 

Electron transfer processes induce variations in the occupancy and/or the nature of orbitals which are essentially localized at the redox centers. However, these centers are embedded in a complex dielectric medium whose geometry and polarization depend on the redox state of the system. In addition, a finite delocalization of the centers orbitals through the medium is essential to-promote long-range electron transfers. The electron transfer process must therefore be viewed as a transition between two states of the whole system. The expression of the probability per unit time of this transition may be calculated by the general formahsm of Quantum Mechanics. 

An important result, found for the SDTQ[N/N] wavefiinctions of all molecules considered, is that the split-localized molecular orbitals yield a considerably faster convergence for truncated expansions than the natural orbitals. For example, for NCCN SDTQ[18/18], millihartree accuracy is achieved by about 50,000 determinants of the ordering based on split-localized orbitals whereas about 150,000 determinants are needed for the natural-orbital-based ordering. This observation calls for the revision of a widely held bias in favor of natural orbitals. 

The final step is the orbital optimization for the truncated SDTQ-CI expansion. We used the Jacobi-rotation-based MCSCF method of Ivanic and Ruedenberg (55) for that purpose. Table 1 contains the results for the FORS 1 and FORS2 wavefiinctions of HNO and NCCN, obtained using cc-pVTZ basis sets (51). In all cases, the configurations were based on split-localized orbitals. For each case, four energies are listed corresponding to (i) whether the full or the truncated SDTQ-CI expansion was used and (ii) whether the split-localized orbitals were those deduced from the SD naturals orbitals or were eventually MCSCF optimized. It is seen that... 

The electron density described by a core orbital space will of course strongly affect the nature of the active orbitals. The form of the inactive orbitals may be influenced by placing symmetry restrictions on them, or by invoking an initial orbital localization [11-15]. The localized orbitals that are not of interest for the VB description may then be placed in the core in the subsequent CASSCF or fully variational VB calculation and, if necessary, some or all of them may be frozen. 

Percent of localization of the methanide lone pair on the carbon atom according to the calculated NLMO (natural localized molecular orbital). 

One way in which the n bonds in these compounds may be examined is by calculating natural bond orbitals (NBO) from the molecular wave function." Here, the MOs are localized, and the nature of the localized orbitals may be examined. The p-n localized orbital is found to have decreased occupation as the carbon-carbon bond is distorted, indicating that the p orbitals are being used in part in forming other localized orbitals (i.e., rehybridized). It is also possible to calculate natural hybrid orbitals, and their deviation from the line of centers between the atoms involved. The deviations are included in Table 15.2, and are seen to increase markedly with increased twisting. 

The most extensive and systematic study of the chemistry of lignin with theoretical methods has been performed by Remko and co-workers. Their work has involved the nature of intramolecular (40-43) and inter-molecular (44-48) hydrogen bonding of lignin model compounds, spectral transitions (49-52), and conformational analysis (53). The methods used have included CNDO/2 and PCILO (Perturbative Configuration Interaction using Localized Orbitals) (54). 

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