In modern valence bond (VB) theory calculations, Chirgwin–Coulson weights (also called Mulliken weights) are the relative weights of a set of possible VB structures of a molecule. Related methods of finding the relative weights of valence bond structures are the Löwdin and the inverse weights. The weights are named after B. H. Chirgwin and Charles Coulson who developed it in 1950.
Background For a wave function Ψ = ∑ i C i Φ i {\displaystyle \Psi =\sum \limits _{i}C_{i}\Phi _{i}} where Φ 1 , Φ 2 , … , Φ n {\displaystyle \Phi _{1},\Phi _{2},\dots ,\Phi _{n}} are a linearly independent, orthogonal set of basis orbitals, the weight of a constituent orbital Ψ i {\displaystyle \Psi _{i}} would be C i 2 {\displaystyle C_{i}^{2}} since the overlap integral, S i j {\displaystyle S_{ij}} , between two wave functions Ψ i , Ψ j {\displaystyle \Psi _{i},\Psi _{j}} would be 1 for i = j {\displaystyle i=j} and 0 for i ≠ j {\displaystyle i\neq j} . In valence bond theory, however, the generated structures are not necessarily orthogonal with each other, and oftentimes have substantial overlap between the two structures. As such, when considering non-orthogonal constituent orbitals (i.e. orbitals with non-zero overlap) the non-diagonal terms in the overlap matrix would be non-zero, and must be included in determining the weight of a constituent orbital. A method of computing the weight of a constituent orbital, Φ i {\displaystyle \Phi _{i}} , proposed by Chirgwin and Coulson would be:
Application of the Chirgwin–Coulson formula to a molecular orbital yields the Mulliken population of the molecular orbital.
Rigorous formulation
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 b ¯ c d ¯ | {\displaystyle |a{\overline {b}}c{\overline {d}}|} , where a letter without an over-line indicates an electron with α {\displaystyle \alpha } spin, while a letter with over-line indicates an electron with β {\displaystyle \beta } spin. The VB structure for 1, for example would be a linear combination of the determinants | 1 2 ¯ 3 4 ¯ | {\displaystyle |1{\overline {2}}3{\overline {4}}|} , | 2 1 ¯ 3 4 ¯ | {\displaystyle |2{\overline {1}}3{\overline {4}}|} , | 1 2 ¯ 4 3 ¯ | {\displaystyle |1{\overline {2}}4{\overline {3}}|} , and | 2 1 ¯ 4 3 ¯ | {\displaystyle |2{\overline {1}}4{\overline {3}}|} . For a monoanionic species, the VB structure for 11 would be a linear combination of | 1 2 ¯ 4 4 ¯ | {\displaystyle |1{\overline {2}}4{\overline {4}}|} and | 2 1 ¯ 4 4 ¯ | {\displaystyle |2{\overline {1}}4{\overline {4}}|} , namely:
ϕ 11 = 1 2 ( | 1 2 ¯ 4 4 ¯ | + | 2 1 ¯ 4 4 ¯ | ) {\displaystyle \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 | φ 1 φ 2 ¯ φ 3 φ 4 ¯ … | {\displaystyle |\varphi _{1}{\overline {\varphi _{2}}}\varphi _{3}{\overline {\varphi _{4}}}\dots |} containing n {\displaystyle n} electrons, represented by the electron indices 1 , 2 , … , n {\displaystyle 1,2,\dots ,n} , and n {\displaystyle n} orbitals, represented by φ 1 , φ 2 , … , φ n {\displaystyle \varphi _{1},\varphi _{2},\dots ,\varphi _{n}} , can be represented by the following Slater determinant:
| φ 1 φ 2 ¯ φ 3 φ 4 ¯ … | = 1 n ! | φ 1 ( 1 ) α ( 1 ) φ 1 ( 2 ) α ( 2 ) … φ 1 ( n ) α ( n ) φ 2 ( 1 ) β ( 1 ) φ 2 ( 2 ) β ( 2 ) … φ 2 ( n ) β ( n ) ⋮ ⋮ ⋱ ⋮ | {\displaystyle |\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 α ( k ) {\displaystyle \alpha (k)} and β ( k ) {\displaystyle \beta (k)} represent an α {\displaystyle \alpha } or β {\displaystyle \beta } spin on the k th {\displaystyle k^{\text{th}}} electron, respectively. For the case of a two electron system with orbitals a {\displaystyle a} and b {\displaystyle b} , the VB structure, | a b ¯ | {\displaystyle |a{\overline {b}}|} , can be represented: | a b ¯ | = 1 2 | a ( 1 ) α ( 1 ) a ( 2 ) α ( 2 ) b ( 1 ) β ( 1 ) b ( 2 ) β ( 2 ) | {\displaystyle |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 b ¯ | = 1 2 ( a ( 1 ) b ( 2 ) [ α ( 1 ) β ( 2 ) ] − a ( 2 ) b ( 1 ) [ α ( 2 ) β ( 1 ) ] ) {\displaystyle |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 Ψ = ∑ i C i Φ i {\displaystyle \Psi =\sum \limits _{i}C_{i}\Phi _{i}} where Φ 1 , Φ 2 , … , Φ N {\displaystyle \Phi _{1},\Phi _{2},\dots ,\Phi _{N}} is a complete, linearly independent set of VB structures and C k {\displaystyle C_{k}} is the coefficient of each structure, the Chirgwin-Coulson weight W K {\displaystyle W_{K}} of a VB structure Φ K {\displaystyle \Phi _{K}} can be computed in the following manner:
W i = ∑ j C i C j ⟨ Φ i | Φ j ⟩ = ∑ j C i C j S i j {\displaystyle W_{i}=\sum \limits _{j}C_{i}C_{j}\langle \Phi _{i}|\Phi _{j}\rangle =\sum \limits _{j}C_{i}C_{j}S_{ij}}
Where S {\displaystyle S} is the overlap matrix satisfying ⟨ Φ i | Φ j ⟩ = S i j {\displaystyle \langle \Phi _{i}|\Phi _{j}\rangle =S_{ij}} . Other methods of computing weights of VB structure include Löwdin weights, where W i Lowdin = ∑ j , k S i j 1 / 2 C j S i k 1 / 2 C k {\displaystyle W_{i}^{\text{Lowdin}}=\sum \limits _{j,k}S_{ij}^{1/2}C_{j}S_{ik}^{1/2}C_{k}} , and inverse weights, where W i inverse = 1 N ( C i 2 ( S − 1 ) i i ) {\displaystyle W_{i}^{\text{inverse}}={\frac {1}{N}}{\bigg (}{\frac {C_{i}^{2}}{(S^{-1})_{ii}}}{\bigg )}} with N {\displaystyle N} being a normalization factor defined by N = ∑ i C i 2 ( S − 1 ) i i {\displaystyle N=\sum \limits _{i}{\frac {C_{i}^{2}}{(S^{-1})_{ii}}}} . The use of Löwdin and inverse weights is appropriate when the Chirgwin–Coulson weights either exceed 1 or are negative.
Half determinant decomposition of molecular orbitals Given a set of molecular orbitals, Ψ 1 , Ψ 2 , … , Ψ m {\displaystyle \Psi _{1},\Psi _{2},\dots ,\Psi _{m}} , for a molecule, consider the determinant of a given orbital population, represented by D MO {\displaystyle D_{\text{MO}}} . The determinant can be written as the following Slater determinant:
D MO = | Ψ 1 Ψ ¯ 1 Ψ 2 Ψ ¯ 2 … | {\displaystyle D_{\text{MO}}=|\Psi _{1}{\overline {\Psi }}_{1}\Psi _{2}{\overline {\Psi }}_{2}\dots |}
Computing the determinant explicitly by multiplying this expression can be a computationally difficult task, given that each molecular orbital is composed of a combination of atomic orbitals. On the other hand, because the determinant of a product of matrices is equal to the product of determinants, the determinant can be regrouped to half-determinants, one of which contains only electrons with α {\displaystyle \alpha } spin and the only with electrons of β {\displaystyle \beta } spin, that is: D MO = h MO α h MO β {\displaystyle D_{\text{MO}}=h_{\text{MO}}^{\alpha }h_{\text{MO}}^{\beta }} where h MO α = | ϕ 1 ϕ 2 … | {\displaystyle h_{\text{MO}}^{\alpha }=|\phi _{1}\phi _{2}\dots |} and h MO β = | ϕ ¯ 1 ϕ ¯ 2 … | {\displaystyle h_{\text{MO}}^{\beta }=|{\overline {\phi }}_{1}{\overline {\phi }}_{2}\dots |} . Note that any given molecular orbital Ψ MO {\displaystyle \Psi _{\text{MO}}} can be written as a linear combination of atomic orbitals ϕ 1 , ϕ 2 , … , ϕ n {\displaystyle \phi _{1},\phi _{2},\dots ,\phi _{n}} , that is for each Ψ i {\displaystyle \Psi _{i}} , there exist C i j {\displaystyle C_{ij}} such that Ψ i = ∑ j C i j ϕ j {\displaystyle \Psi _{i}=\sum \limits _{j}C_{ij}\phi _{j}} . As such, the half determinant h MO α {\displaystyle h_{\text{MO}}^{\alpha }} can be further decomposed into the half determinants for an ordering of atomic orbitals h r α = | ϕ 1 , ϕ 2 , … , ϕ n | {\displaystyle h_{r}^{\alpha }=|\phi _{1},\phi _{2},\dots ,\phi _{n}|} corresponding to a VB structure r {\displaystyle r} . As such, the molecular orbital Ψ i {\displaystyle \Psi _{i}} can be represented as a combination of the half determinants of the atomic orbitals, h MO α = ∑ r C r α h r α {\displaystyle h_{\text{MO}}^{\alpha }=\sum \limits _{r}C_{r}^{\alpha }h_{r}^{\alpha }} . The coefficient C r α {\displaystyle C_{r}^{\alpha }} can be determined by evaluating the following matrix:
C r α = | C 11 C 21 … C n 1 C 12 C 22 … C n 2 ⋮ ⋮ ⋱ C 1 n C 2 n … C n n | {\displaystyle 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 β {\displaystyle \beta } electrons, h MO β {\displaystyle h_{\text{MO}}^{\beta }} . As such, the determinant D MO {\displaystyle D_{\text{MO}}} can be expressed as D MO = ∑ r , s C r α C r β h r α h s β {\displaystyle D_{\text{MO}}=\sum \limits _{r,s}C_{r}^{\alpha }C_{r}^{\beta }h_{r}^{\alpha }h_{s}^{\beta }} , where r , s {\displaystyle r,s} index across all possible VB structures.
Sample computations for simple molecules
Computations for the hydrogen molecule The hydrogen molecule can be considered to be a linear combination of two H {\displaystyle {\ce {H}}} 1 s {\displaystyle 1s} orbitals, indicated as φ 1 {\displaystyle \varphi _{1}} and φ 2 {\displaystyle \varphi _{2}} . The possible VB structures for H 2 {\displaystyle {\ce {H_2}}} are the two covalent structures, | φ 1 φ 2 ¯ | {\displaystyle |\varphi _{1}{\overline {\varphi _{2}}}|} and | φ 2 φ 1 ¯ | {\displaystyle |\varphi _{2}{\overline {\varphi _{1}}}|} indicated as 1 and 2 respectively, as well as the ionic structures | φ 1 φ 1 ¯ | {\displaystyle |\varphi _{1}{\overline {\varphi _{1}}}|} and | φ 2 φ 2 ¯ | {\displaystyle |\varphi _{2}{\overline {\varphi _{2}}}|} indicated as 3 and 4 respectively, shown below.
Because structures 1 and 2 both represent covalent bonding in the hydrogen molecule and exchanging the electrons of structure 1 yields structure 2, the two covalent structures can be combined into one wave function. As such, the Heitler-London model for bonding in H 2 {\displaystyle {\ce {H_2}}} , Φ H L {\displaystyle \Phi _{HL}} , can be used in place of the VB structures | φ 1 φ 2 ¯ | {\displaystyle |\varphi _{1}{\overline {\varphi _{2}}}|} and | φ 1 ¯ φ 2 | {\displaystyle |{\overline {\varphi _{1}}}\varphi _{2}|} :
Φ H L = | φ 1 φ 2 ¯ | − | φ 1 ¯ φ 2 | {\displaystyle \Phi _{HL}=|\varphi _{1}{\overline {\varphi _{2}}}|-|{\overline {\varphi _{1}}}\varphi _{2}|}
Where the negative sign arises from the antisymmetry of electron exchange. As such, the wave function for the H 2 {\displaystyle {\ce {H_2}}} molecule, Ψ H 2 {\displaystyle \Psi _{{\text{H}}_{2}}} , can be considered to be a linear combination of the Heitler-London structure and the two ionic valence bond structures.
Ψ H 2 = C 1 Φ H L + C 2 | φ 1 φ 1 ¯ | + C 3 | φ 2 φ 2 ¯ | {\displaystyle \Psi _{{\text{H}}_{2}}=C_{1}\Phi _{HL}+C_{2}|\varphi _{1}{\overline {\varphi _{1}}}|+C_{3}|\varphi _{2}{\overline {\varphi _{2}}}|}
The overlap matrix between the atomic orbitals between the three valence bond configurations Φ H L {\displaystyle \Phi _{HL}} , | φ 1 φ 1 ¯ | {\displaystyle |\varphi _{1}{\overline {\varphi _{1}}}|} , and | φ 2 φ 2 ¯ | {\displaystyle |\varphi _{2}{\overline {\varphi _{2}}}|} is given in the output for valence bond calculations. A sample output is given below:
S = | S 11 S 21 S 22 S 31 S 32 S 33 | = | 1 0.77890423 1 0.77890423 0.43543258 1 | {\displaystyle S={\begin{vmatrix}S_{11}\\S_{21}&S_{22}\\S_{31}&S_{32}&S_{33}\\\end{vmatrix}}={\b