Valency interaction formula

The Valency Interaction Formula, or VIF is a method for drawing molecular structural formulas based in quantum mechanics. The mathematical basis for VIF was formulated by Oktay Sinanoglu in a series of five papers published in 1984. [ Sinanoğlu, O, “On the algebraic construction of chemistry from quantum mechanics. A fundamental valency vector field defined on the Euclidean 3-space and its relation to Hilbert space”,Theoret. Chim. Acta (Berl.), 1984, 65, 243-248.] [Sinanoğlu, O., “A principle of linear covariance for quantum mechanics and electronic structure theory of molecules and other atom clusters”, Theoret. Chim. Acta (Berl.) 1984, 6., 233-242] [Sinanoğlu, O., Non-unitary classification of molecular electronic structures and other atom clusters”, Theoret. Chim. Acta (Berl.) 1984, 6., 249-254] [Sinanoğlu, O., “Structural Covariance of Graphs”, Theoret. Chim. Acta (Berl.) 1984, 6., 255-265] [Sinanoğlu, O., “Deformational Covariance of Graphs”, Theoret. Chim. Acta (Berl.) 1984, 6., 267-270] In the first implementations of the VIF method, the VIF pictures represented one-electron Hamiltonian operators. [Sinanoğlu, O. Chem. Phys. Lett., 1984, 103(4), 315-322] [Sinanoğlu, O., Alia, J., Hastings, M. J. Phys. Chem. 1994, 98, 5867-5877] Alia has shown how the method is easily extended so that VIF pictures represent one-electron density operators and the practical value of doing this. [Alia, J., “Molecular Structural Formulas as One-Elelctron Density and Hamiltonian Operators: The VIF Method Extended” J. Phys. Chem. A, 2007, 111(12), 2307-2318. doi|10.1021/jp065855y]

Chemical deductions are made from a VIF picture with the application of two pictorial rules. These are linear transformations applied to the quantum operator and preserve invariants crucial to the characterization of the molecules electronic properties, the numbers of bonding, non-bonding, and anti-bonding orbitals and/or the number of doubly, singly, and unoccupied valence orbitals.

Frequently VIF molecular structural formulas as one-electron Hamiltonian and one-electron density operators are the same picture. As a consequence VSEPR theory applied to VIF pictures can be used to interpret the energies of natural bond orbitals, NBO, calculated using Hatree-Fock and Density Functional levels of theory. [Alia, Vlaisavljevich, Abbot, Warneke, Mastin, "Prediction of Molecular Properties Including Symmetry from Quantum-Based Molecular Structural Formulas, VIF" "J. Phys. Chem. A", in press]

The link between electron density and energy is intrinsic to a great deal of chemical reasoning and increasingly important in computational methods. [Parr, R. G.; Yang, W. "Density-Functional Theory of Atoms and Molecules" Oxford University Press: Oxford, U.K., 1989.]

As quantum-based molecular structural formulas, VIF pictures allow a great deal of flexibility in describing a variety of chemical bonding types. See Table 1 below. Alia and Vlaisavljevich have applied the VIF method toward understanding the unconventional bonding in a number of boranes.

Current research is being done utilizing the VIF method to understand the selectivity of concerted reactions. Anderson is conducting an in depth study of VIF applied to carbene reactions. The method has also been found to work well in describing the selectivity of H-shift reactions and concerted ring-closures.

While rigorous testing of this method is ongoing, it is accepted by the scientific community as a legitimate way of calculating these molecular properties. [Sinanoglu, O.; Alia, J.; Hastings, M. "Valency Interactions in AH_m⁰ and MO Energy Level Patterns Directly from the Pictorial "VIF" Method Compared with Computer Calculations" J. Phys. Chem. 1994, 98, 5867-5877.] [Alia, J.; Vlaisavljevich, B. "Prediction of Molecular Properties Including Symmetry from Quantum-Based Molecular Structural Formulas, VIF" jp-2007-120214.R1]

Two Pictorial Rules

The Multiplication Rule

The multiplication rule is based on matrix calculations, specifically the row multiplications that are allowed on such matrices. In matrix form, one can multiply an entire row by any non-zero number. Pictorially, this allows for any valency point in the VIF picture to be multiplied by any non-zero value. This allows for diagonalization of the matrix using the addition rule.

The Addition Rule

The addition rule is used in conjunction with the multiplication rule. Pictorally, line segments can be rotated and superimposed on other line segments, canceling the latter while retaining the former. This is used to simplify the picture and ultimately gives an indication of the number of electrons associated with the molecular orbitals (represented by the dots on either side of a given line segment). Mathematically, this results in adding to and canceling rows on a matrix. This matrix is the foundation of the operator that is used on the wave function.

Results Found Through VIF

OOI and LPI

The structural formulas underlying the VIF method represent one electron hamiltonian operators, which are in turn used to find the Orbital Occupancy Index (OOI) and the Level Pattern Index (LPI). These are important when considering the properties of the molecule, since they provide information on what molecular orbitals are doubly, singly, or unoccupied as well as their relative energy in accordance with the reference energy (-13.6 eV).

Eigenvalues

The eigenvalues of the matrix associated with the wave function are best found by using the Mathematica program. While the values of the eigenvalues are unimportant, the signs of the eigenvalues - positive or negative - indicate bonding or anti-bonding molecular orbitals respectively.

Interpreting Results

[Alia, J. "Graph Representation of Quantum Mechanical Operators as Molecular Structural Formulas" 2008.]

ee also

* VSEPR Theory
* Molecular Geometry
* Structural Formula
* Lewis Structure

References