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Concerning the theory of chirality functions

VII. The concept of qualitative supercompleteness

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Abstract

Within the scope of the theory of chirality functions, qualitatively complete chirality functions are subject to restrictions concerning both generality and applicability. In contrast thereto, the concept of qualitative supercompleteness results in less restrictive requirements for chirality functions. Consequently, the applicability of qualitatively supercomplete chirality functions is unlimited with respect to the number of ligand kinds. Given this concept, a group theoretical treatment is performed supplying the formal conditions of qualitative supercompleteness. Subsequently a construction rule for qualitiatively supercomplete chirality functions is presented, which is elaborated in detail in the appendix. On combining physical considerations with the requirement of qualitative supercompleteness the resulting chirality functions appear to include all the possible interactions within and/or between ligands and skeleton. From both a mathematical and a physical point of view these chirality functions should be adequate for describing the chiroptical properties of molecules belonging to a given skeletal class. Nevertheless, all the other critical objections to the theory of chirality functions remain.

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Abbreviations

f :

ensemble operator which is an element of the group algebra of the\(\mathfrak{S}_N \)

\(\mathfrak{G}\) :

group of the symmetry operations of the skeleton

\(\mathfrak{G}_{\text{f}} \) :

group of the covering operations of the fictively extended skeleton

g :

number of enantiomeric pairs of molecules withn different ligands and a skeleton ofn ligand sites

h :

number of enantiomeric pairs of molecules withN different ligands and a fictively extended skeleton ofN ligand sites

\(\mathfrak{H}_{nN}^{(r)} \) :

TU-half order of the partition diagramsγ N (w) witht wr 1 = 1

\(\mathfrak{h}_n^{(r)} \) :

half order of the partition diagramsδ v (w)

L :

vector of the symbolsl i of theN ligands attached to the proper or to the fictive ligand sites

L e :

vector of the symbolsl i of then ligands attached to the proper ligand sites

L f :

vector of the symbolsl i of them ligands attached to the fictive ligand sites

L v :

vector of the symbolsl i ofv ligands attached tov proper ligand sites,v ≤n

l i :

symbol of the ligand attached to theith ligand site

M i (w) :

see Eq. (B21)

m :

number of fictive ligand sites or additional ligand sorts

m w :

number of times av-ligands function induces Δ v (w)

N :

number of ligand sites of the fictively extended skeleton or total number of ligand sorts

n :

number of proper ligand sites or number of ligand sorts of a molecule with pairwise different ligands

n r,n w :

dimension of Λ (r) n or Γ (w) N resp.

\(\mathcal{O}_{(s)} \) :

permutation operator induced by the permutations in the function space, element of\(\mathfrak{S}(\mathcal{O}),\mathfrak{S}_\mathfrak{m} (\mathcal{O}),\mathfrak{S}_\mathfrak{n} (\mathcal{O}),\mathfrak{S}_N (\mathcal{O})\) or\(\mathfrak{S}_\nu (\mathcal{O})\) resp.

o 1 (r),o 1 (w) :

length of the first row ofγ (r)n orγ (w)N resp.

o i (w) :

sum of the lengths of the firsti rows ofγ N (w)

o max :

o max=maxr o (r)1

p (r) ij ,p (w) ij :

projection and shift operators of Γ (r) n or Γ (w) N resp.

p (v) m ,P (v) m :

character projectors onto Γ (v) m of\(\mathfrak{S}_m \) or\(\mathfrak{S}_m (\mathcal{O})\) resp.

p (r) n ,p (r) n :

character projectors onto Γ (r) n of\(\mathfrak{S}_n \) or\(\mathfrak{S}_n (\mathcal{O})\) resp.

p (w) N ,P (w) N :

character projectors onto Γ (w) N of\(\mathfrak{S}_N \) or\(\mathfrak{S}_N (\mathcal{O})\) resp.

P (w) v :

character projector onto Δ (w) v of\(\mathfrak{S}_\nu (\mathcal{O})\)

p σ,P σ :

projection operators onto Γgs of\(\mathfrak{U} = \mathfrak{S} \times \mathfrak{S}_m \) or\(\mathfrak{S}{\text{(}}\mathcal{O}) \times \mathfrak{S}_m {\text{(}}\mathcal{O})\) resp.

p χ,P χ :

projection operators onto Γ x of\(\mathfrak{S}\) or\(\mathfrak{S}{\text{(}}\mathcal{O})\) resp.

q w :

dimension of Δ (w) v

r :

index of an irreducible representation Γ (r) n of the\(\mathfrak{S}_n \)

r :

index of an irreducible representation Λ (r) n of the\(\mathfrak{S}_n \) containing Γ X

\(\mathfrak{S}\) :

group of the ligand permutations which correspond to symmetry operations of the skeleton

\(\mathfrak{S}_o \) :

subgroup of\(\mathfrak{S}\) the elements of which correspond to proper rotations of the skeleton

\(\mathfrak{S}^ * \) :

coset of\(\mathfrak{S}_o \) in\(\mathfrak{S}\) the elements of which correspond to improper rotations of the skeleton

\(\mathfrak{S}_m \) :

symmetric group of the permutations of them ligands attached to the fictive ligand sites

\(\mathfrak{S}_n \) :

symmetric group of the permutations of then ligands attached to the proper ligand sites

\(\mathfrak{S}_N \) :

symmetric group of the permutations of allN ligands

\(\mathfrak{S}_\nu \) :

symmetric group of the permutations ofv ligands attached to proper ligand sites,v ≤ n

\(\begin{gathered} \mathfrak{S}(\mathcal{O}),\mathfrak{S}_m (\mathcal{O}), \hfill \\ \mathfrak{S}_n (\mathcal{O}),\mathfrak{S}_N (\mathcal{O}), \hfill \\ \mathfrak{S}_\nu (\mathcal{O}) \hfill \\ \end{gathered} \) :

symmetric groups the elements of which are permutation operators\(\mathcal{O}(s)\)

s (w)u :

see Eq. (B13)

s w :

see Eq. (B4)

t wrv :

multiplicity of Γ (w) N in Λ (r) n ⊗ Λ (v) m , equal to the multiplicity of Λ (r) n × Λ (v) m in Λ (w) N

\(\mathfrak{U}\) :

group of the ligand permutations which correspond to covering operations of the fictively extended skeleton

\(\mathfrak{U}_o \) :

subgroup of\(\mathfrak{U}\) the elements of which correspond to proper rotations of the fictively extended skeleton

\(\mathfrak{U}^* \) :

coset of\(\mathfrak{U}_o \) in\(\mathfrak{U}\) the elements of which correspond to improper rotations of the fictively extended skeleton

v :

index of an irreducible representation Λ (v) m of the\(\mathfrak{S}_m \)

w :

index of an irreducible representation Λ (w) N of the\(\mathfrak{S}_N \)

w :

index of an irreducible representation Λ (w)N of the\(\mathfrak{S}_N \) containing Γσ

w 0r :

index of the largest diagram\(\gamma _N^{(w)} \in \mathfrak{H}_{nN}^{(r)} \) or\(\delta _\nu ^{(w)} \in \mathfrak{h}_n^{(r)} \) resp.

w *r :

index of the smallest diagram\(\gamma _N^{(w)} \in \mathfrak{H}_{nN}^{(r)} \) or\(\delta _\nu ^{(w)} \in \mathfrak{h}_n^{(r)} \) resp.

x w :

multiplicity of Λσ in Λ (w) N

y (w) ij :

Young operator (ifi = j) or Young unit of Δ (w) v of\(\mathfrak{S}_\nu (\mathcal{O})\)

N y (w) ij :

Young operator (ifi = j) or Young unit of Λ (w) N of\(\mathfrak{S}_N (\mathcal{O})\)

z r :

multiplicity of Λ X in Λ (r n

α (w)ρu :

linear combination coefficient, see Eq. (48), (B41)

Λ (v) m , Λ (r) n , Λ (w) N :

irreducible representation of\(\mathfrak{S}_m \) or\(\mathfrak{S}_m (\mathcal{O})\) resp.,\(\mathfrak{S}_n \) or\(\mathfrak{S}_n (\mathcal{O})\) resp., or\(\mathfrak{S}_N \) or\(\mathfrak{S}_N (\mathcal{O})\), resp.

Λσ :

σ-chirality representation

Λ X :

chirality representation

γ (v) m , γ (r) n , γ (w) N :

partition diagrams corresponding to Λ (v) m , Λ (r) n or Λ (w) N resp.

γ (s) N :

partition diagram corresponding to an assortment ofN ligands

Δ (w) v :

irreducible representation of\(\mathfrak{S}_\nu \) or\(\mathfrak{S}_\nu (\mathcal{O})\) resp.

δ (w)v :

partition diagram corresponding to Δ (w) v , got from γ (w) N by removing the first row

λρ/(w) (li):

ρth parameter of the ligandl i in a set up to the component\(\tilde x^{(w)} (L_e )\) of\(\tilde x(L_e )\)

v :

number of ligands on which the set up\(\tilde \omega ^{(w)} (L_e )\) depends (=order of interaction), equal to the number of boxes ofδ (w)v ,v ≤ n

v (w)i :

length of theith row of γ (w) N

ρ:

subscript which distinguishes between different parameter sets for a givenw, ρ = 1,..., s w

σ ij :

permutation transforming a Young tableauj into a Young tableaui

\(\tilde \upsilon _{\rho k}^{(w)} (L_e ),\) :

product of the Vandermonde determinants of the Young tableauk multipled by

\(\tilde \tilde \upsilon _{\rho k}^{(w)} (L_\nu )\) :

the product of all parameters ofL v

φ(L), φ(L e):

function describing physical properties of a molecule

\(\chi (L),\tilde \chi (L{}_e)\) :

chirality function of a molecule derived by a “Näherungsverfahren”

\(\begin{gathered} \chi ^{(w)} (L), \hfill \\ \tilde \chi ^{(w)} (L{}_e) \hfill \\ \end{gathered} \) :

wth component ofX (L) or\(\tilde \chi (L{}_e)\) resp.

ψρ(γ (w) N ;L):

see Eq.(B19)

ω(L), ω(L e):

set up to the chirality functionX(L) or\(\tilde x(L_e )\) resp.

ω (w) (L) :

v-ligands function, set up toχ (w)(L) or\(\tilde x^{(w)} (L_e )\)

\(\begin{gathered} \tilde \omega ^{(w)} (L_e ), \hfill \\ \tilde \tilde \omega ^{(w)} (L_\nu ) \hfill \\ \hat \omega ^{(w)} (L), \hfill \\ \hat \tilde \omega ^{(w)} (L_e ), \hfill \\ \hat \tilde \tilde \omega ^{(w)} (L_\nu ) \hfill \\ \end{gathered} \) :

Δ (w)v -component of\(\omega ^{(w)} (L),\tilde \omega ^{(w)} (L_e )\) or\(\tilde \tilde \omega ^{(w)} (L_\nu )\) resp.

\(\begin{gathered} \tilde \omega _u^{(w)} (L_e ), \hfill \\ \tilde \tilde \omega _u^{(w)} (L_v ) \hfill \\ \end{gathered} \) :

set up of theuth component,u = 1, ..., x w, to a\(\chi ^{(w)} (L)\) or\(\tilde \tilde \omega ^{(w)} (L_\nu )\) resp., see Eq.

\(\begin{gathered} \tilde \omega _u^{(w)} (L_e ), \hfill \\ \tilde \tilde \omega _u^{(w)} (L_v ) \hfill \\ \end{gathered} \) :

(B16)

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Author notes

  1. Hildegard Keller

    Present address: Department of Organic Chemistry, University of Groningen, Zernikelaan, Groningen, Netherlands

Authors and Affiliations

  1. Institut für Statistik der Wirtschaftsuniversität Wien, Franz Klein-Gasse 1, A-1190, Wien, Österreich

    Gerhard Derflinger & Hildegard Keller

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  1. Gerhard Derflinger
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Dedicated to Prof. O. E. Polansky on the occasion of his 60th birthday.

Part VI: Langer, E., Lehner, H.: Monatsh. Chem. 110, 1003 (1979).

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Derflinger, G., Keller, H. Concerning the theory of chirality functions. Theoret. Chim. Acta 56, 1–43 (1980). https://doi.org/10.1007/BF00716678

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