Valency interaction formula

A singlet carbene (CH₂) molecule superimposed on its VIF structure

The Valency Interaction Formula, or VIF is a method based in quantum mechanics that describes numerous qualities of molecules through mathematics. More specifically, it utilizes the wave functions that describe the vibrations of the molecule and applies a one-electron Hamiltonian operator to solve. The VIF method is a pictoril representation of this mathematical process, and makes solving with the pictoral rules much easier than solving the equation itself. The result is an accurate description of electron density in atomic orbitals. Because VSEPR theory accurately predicts stable molecular geometries^[1], there is an assumed link between energy and electron density. This implies that when solving for electron density using the VIF method, electron repulsion energy can also be predicted. Such information can be used to predict whether or not a molecular reaction will occur, as well as give relative rates of reaction using the Arrhenius equation.

While rigorous testing of this method is ongoing, it is accepted by the scientific community as a legitimate way of calculating these molecular properties.^[2]^[3]

Mathematical Basis

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 atomic 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.

Quantum Mechanical Functions

The VIF method is rooted in quantum mechanics. Wave functions used to describe molecular properties are projected into a Hilbert space to get discernable results via Bra-ket notation. Bras and Kets are the building blocks of the unity operators that are multiplied by the wave function which is projected into the Hilbert space so the results can be reasonably interpreted.

Interpreting results

^[4]

References

^ Parr, R. G.; Yang, W. "Density-Functional Theory of Atoms and Molecules" Oxford University Press: Oxford, U.K., 1989.
^ 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. Am. Chem. Soc. 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
^ Alia, J. "Graph Representation of Quantum Mechanical Operators as Molecular Structural Formulas" 2008.

[1] Parr, R. G.; Yang, W. "Density-Functional Theory of Atoms and Molecules" Oxford University Press: Oxford, U.K., 1989.

[2] 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. Am. Chem. Soc. 1994, 98, 5867-5877.

[3] Alia, J.; Vlaisavljevich, B. "Prediction of Molecular Properties Including Symmetry from Quantum-Based Molecular Structural Formulas, VIF" jp-2007-120214.R1

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

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