Determination of VB Structures Rumer's method A method of creating a linearly independent, complete set of valence bond structures for a molecule was proposed by
Yuri Rumer. For a system with n
electrons and n orbitals, Rumer's method involves arranging the orbitals in a circle and connecting the orbitals together with lines that do not intersect one another. Covalent, or uncharged, structures can be created by connecting all of the orbitals with one another. Ionic, or charged, structures for a given atom can be determined by assigning a charge to a molecule, and then following Rumer's method. For the case of
butadiene, the 20 possible Rumer structures are shown, where 1 and 2 are the covalent structures, 3-14 are the monoionic structures, and 15-20 are the diionic structures. The resulting VB structures can be represented by a linear combination of determinants |a\overline{b}c\overline{d}|, where a letter without an over-line indicates an electron with \alpha spin, while a letter with over-line indicates an electron with
\beta spin. The VB structure for 1, for example would be a linear combination of the determinants |1\overline{2}3\overline{4}|, |2\overline{1}3\overline{4}|,|1\overline{2}4\overline{3}|, and |2\overline{1}4\overline{3}|. For a monoanionic species, the VB structure for 11 would be a linear combination of |1\overline{2}4\overline{4}| and |2\overline{1}4\overline{4}|, namely: :\phi_{11}=\frac{1}{\sqrt{2}}(|1\overline{2}4\overline{4}|+|2\overline{1}4\overline{4}|)
Matrix representation of VB structures An arbitrary VB structure |\varphi_1\overline{\varphi_2}\varphi_3\overline{\varphi_4}\dots| containing n electrons, represented by the electron indices 1,2,\dots,n, and n orbitals, represented by \varphi_1,\varphi_2,\dots, \varphi_n, can be represented by the following
Slater determinant: :|\varphi_1\overline{\varphi_2}\varphi_3\overline{\varphi_4}\dots|=\frac{1}{\sqrt{n!}} \begin{vmatrix} \varphi_1(1)\alpha(1) & \varphi_1(2)\alpha(2) & \dots & \varphi_1(n)\alpha(n)\\ \varphi_2(1)\beta(1) & \varphi_2(2)\beta(2) & \dots & \varphi_2(n)\beta(n)\\ \vdots & \vdots & \ddots & \vdots \end{vmatrix} Where \alpha(k) and \beta(k) represent an \alpha or \beta spin on the k^\text{th} electron, respectively. For the case of a two electron system with orbitals a and b, the VB structure, |a\overline{b}|, can be represented:|a\overline{b}|=\frac{1}{\sqrt{2}} \begin{vmatrix} a(1)\alpha(1) & a(2)\alpha(2)\\ b(1)\beta(1) & b(2)\beta(2) \end{vmatrix} Evaluating the determinant yields: :|a\overline{b}|=\frac{1}{\sqrt{2}}(a(1)b(2)[\alpha(1)\beta(2)]-a(2)b(1)[\alpha(2)\beta(1)])
Definition of Chirgwin–Coulson weights Given a
wave function \Psi=\sum\limits_{i}C_i\Phi_i where \Phi_1,\Phi_2,\dots,\Phi_N is a complete, linearly independent set of VB structures and C_k is the coefficient of each structure, the Chirgwin-Coulson weight W_K of a VB structure \Phi_K can be computed in the following manner: The use of Löwdin and inverse weights is appropriate when the Chirgwin–Coulson weights either exceed 1 or are negative. Note that any given molecular orbital \Psi_{\text{MO}} can be written as a
linear combination of atomic orbitals \phi_1,\phi_2,\dots,\phi_n, that is for each \Psi_i, there exist C_{ij} such that \Psi_i=\sum\limits_{j}C_{ij}\phi_j. As such, the half determinant h^\alpha_\text{MO} can be further decomposed into the half determinants for an ordering of atomic orbitals h^\alpha_r=|\phi_1,\phi_2,\dots,\phi_n| corresponding to a VB structure r. As such, the molecular orbital \Psi_i can be represented as a combination of the half determinants of the atomic orbitals, h^\alpha_\text{MO}=\sum\limits_rC^\alpha_rh^\alpha_r. The coefficient C_r^\alpha can be determined by evaluating the following matrix: :C_r^\alpha= \begin{vmatrix} C_{11} & C_{21} & \dots C_{n1}\\ C_{12} & C_{22} & \dots C_{n2}\\ \vdots & \vdots & \ddots\\ C_{1n} & C_{2n} & \dots C_{nn}\\ \end{vmatrix} The same method can be used to evaluate the half determinant for the \beta electrons, h^\beta_\text{MO}. As such, the determinant D_\text{MO} can be expressed as D_\text{MO}=\sum\limits_{r,s}C^\alpha_rC^\beta_rh^\alpha_rh^\beta_s, where r, s index across all possible VB structures. == Sample computations for simple molecules ==