Chirgwin–Coulson weights
A method of computing the weight of a constituent orbital,
[4] A method of creating a linearly independent, complete set of valence bond structures for a molecule was proposed by Yuri Rumer.
[5] Covalent, or uncharged, structures can be created by connecting all of the orbitals with one another.
The resulting VB structures can be represented by a linear combination of determinants
spin, while a letter with over-line indicates an electron with
The VB structure for 1, for example would be a linear combination of the determinants
For a monoanionic species, the VB structure for 11 would be a linear combination of
is a complete, linearly independent set of VB structures and
, for a molecule, consider the determinant of a given orbital population, represented by
can be written as a linear combination of atomic orbitals
can be further decomposed into the half determinants for an ordering of atomic orbitals
can be represented as a combination of the half determinants of the atomic orbitals,
[6][7][8] The hydrogen molecule can be considered to be a linear combination of two
:[9] Where the negative sign arises from the antisymmetry of electron exchange.
The overlap matrix between the atomic orbitals between the three valence bond configurations
A sample output is given below:[6] Finding the eigenvectors of the matrix
is energy due to orbital overlap, yields the VB-vector
using density functional theory yields the coefficients
Thus, the Coulson-Chrigwin weights can be computed:[6] To check for consistency, the inverse weights can be computed by first determining the inverse of the overlap matrix: Next, the normalization constant
Informally, the computed weights indicate that the wave function for the
molecule has a minor contribution from an ionic species not predicted from a strictly MO model for bonding.
Consider, the VB contributions for the ground state of
is: Which implies that the ground state has the following coefficients: Given the following overlap matrix for the half determinants:[6] The overlap between two VB structures represented by the product of two VB determinants
As such:[3] This compares well with reported Chirgwin–Coulson weights of 0.226 for the standard Lewis structure of ozone in the ground state.
) is a cyclic, planar compound that is isoelectronic with benzene.
Given the lone pair in the nitrogen p orbital out of the plane and the empty p orbital of boron, the following resonance structure is possible:[citation needed] However, VB calculations using a double-zeta D95 basis set indicate that the predominant resonance structures are the structure with all three lone pairs on the nitrogen (labeled 1 below) and the six resonance structures with one double bond between boron and nitrogen (labeled 2 below).
The data, together, indicate that, despite the similarity in appearance and structure, the electrons on borazine are less delocalized than those on benzene.
[11] Disulfur dinitride is a square planar compound that contains a 6 electron conjugated
The primary diradical resonance structures (1 and 2) and a secondary zwitterionic structure (3) are shown below:[citation needed] Valence bond calculations using the Dunning's D95 full double-zeta basis set indicate that the dominant resonance structure is the singlet diradical with a long nitrogen-nitrogen bond (structure 1), with Chirgwin-Coulson weight 0.47.
[13] This result corresponds nicely with the general rules regarding Lewis structures, namely that formal charges ought to be minimized, and contrasts with earlier computational results indicating that 1 is the dominant structure.
