Natural Bond Orbitals (NBO)

Some of the most important information about chemistry can be obtained from solutions of the Schrödinger equation. It is through the use of quantum mechanics that we are able to arrive to solutions that can predict properties of molecules with a high level of accuracy. While there are many excellent teachers, books, and lectures describing the fundamental theory of quantum mechanics (QM), it is the interpretation and practicality of many existing QM schemes that chemists find difficult to translate into their everyday observations and understanding. By solving the Schrodinger equation one obtains "electronic energy" for a specific spatial and spin state of electron, called molecular orbital. As an exact solution of Schrödinger equation has only been achieved for one-electron system (hydrogen),  there are many computational techniques developed that approximate the solution for multiple-electron system.

To understand properties and reactivity of organic compounds, chemists have to rely on concepts such as, chemical bond, lone electron pair, conjugation, charges, aromaticity, and other properties, which are often difficult to predict or quantify. By successfully applying those understandings to molecular structures, more complex properties or concepts including anomeric effect, rotational barriers, hyperconjugation, resonance, and basicity can be described and predicted.

Molecular Orbitals:
The most commonly used technique to approximately solve the Schrödinger equation is called a Hartree-Fock method (HF), which  decomposes many-electron wavefunction into a combination of molecular orbitals that are normalized and orthogonal. Molecular orbitals are then expanded in terms of atomic orbitals as linear combination of atomic orbitals (LCAO). Conceptually different, but similar in mathematical formalism is Density functional theory (DFT), which has evolved rather recently from the Hohenberg and Kohn theorem, determines the energy of a molecule from the electron density instead of a wavefunction. In the HF analogy, a determinant of Kohn-Scham (K-S) orbitals is formed and the electron density is used to calculate energy. While there has been some debate over the interpretation of K-S orbitals (being a pure mathematical construct used to re-construct the density), their shapes tend to be remarkably similar to canonical HF MOs. Today, DFT formalism including the K-S orbitals are frequently used in qualitative analysis of chemical properties.

Although "classical" Hartree-Fock and Kohn-Sham orbitals are, by definition, the best possible for a single-configuration wavefunction or electron density, respectively, there is even a "better" set of orbitals, called "natural" orbitals, to describe the correlated electron density. And this is the point, where the concept of Natural Bond Orbitals comes in. NBO localization algorithm permits the form of the BOs to be fully optimized with respect to a maximum-occupancy criterion based solely on the first-order reduced density matrix. In NBO analysis, the input basis set is transformed successively into various localized basis sets, first to natural atomic orbitals and then to hybrid orbitals which are used to form bond orbitals (NBOs). Finally the NBOs are transformed into localized MOs. The NBO analysis can illuminate interesting chemical aspects of the bonding and allow explanations of the various chemistry phenomena, such as, reactivity, stereoselectivity, basicity, and intramolecular and intermolecular energy barriers, all based on orbital interaction concepts. Most importantly, from the chemist's perspective, NBO analysis provides an orbital picture that is as close as possible to a classical "textbook" Lewis structure for a molecule.

NBO methodology:

The NBO analysis involves sequential transformation of nonorthogonal atomic orbitals (AOs) to the complete and orthonormal sets of “natural” atomic orbitals (NAOs), hybrid orbitals (NHOs), and bond orbital (NBOs). These localized basis sets describe electron density and other properties by the smallest number of filled orbitals in the most rapidly convergent fashion. As mentioned earlier, these orbital are closely related to the localized orbitals (bonds and lone pairs) used by organic chemists. The NBO method was developed by Weinhold and co-authors and it is becoming a powerful and popular method for study of bonding concepts. 

On a more technical level, the NBO localization protocol divides NBOs into core, bonding, anti-bonding and ‘‘Rydberg’’(remaining) orbitals. The core and bonding orbitals describe the strictly localized Lewis structure of a molecule. The sequence goes through various localized basis sets in the following order:

AOs-> NAOs -> NHOs -> NBOs -> NLMOs -> CMOs 

Examples of properties that can be calculated in NBO basis include:

• Dipole, polarizability, atomic charges,
• Bonding-anti-bonding orbital interactions, 
• Resonance structures (second order perturbation theory),
• Bond orders,
• Energy decomposition,
• Chemical shift, J-couplings,
• Steric analysis,
• Canonical MO analysis.
• 
  

Differences between  canonical and NBO MOs:

• Canonical MOs (CMOs) are delocalized over the whole molecule,
• They usually bear no resemblance to localized s and p bonds, or to lone pairs, so they cannot be used to support familiar chemical reasoning 
• On the path to NBOs, an ordinary MO calculation is done at a high enough level to reproduce measured geometry or energies (delocalized MOs are obtained), 
• Then the atomic basis set is transformed into an equal number of natural atomic orbitals (NAOs), and the MOs into an equal number of natural bond orbitals (NBOs) 
• These additional transformations are cheap in computer time
• More intuitive NBOs are result of those mathematical transformations 

While analysis of the NBO results is instructive and informative, one usually wishes to visualize newly generated NBOs. In the next post, I will describe the use of Jmol for visualization of NBOs.

References:

• F. Weinhold, "Natural Bond Orbital Methods" in, Encyclopedia of Computational Chemistry, P. v.R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger., P. A. Kollman, H. F. Schaefer III, P. R. Schreiner (Eds.), (John Wiley & Sons, Chichester, UK, 1998), Vol. 3, pp. 1792-1811. 

• F. Weinhold and C. R. Landis, Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective (Cambridge University Press, Sept 2005). 

• A. E. Reed, L. A. Curtiss, and F. Weinhold, Chem. Rev. 88, 899-926 (1988).

• L. Pauling, "The Nature of the Chemical Bond", (Cornell University Press, 3rd ed., 1960)