Through-space and through-bridge components of chemical bonds

Roman F. Nalewajski The direct (through-space) and indirect (through-bridge) components of chemical interactions between atomic orbitals are identified in both the Wiberg bondorder formalism and the Orbital Communication Theory of the chemical bond. The illustrative examples using the Hckel description of the conjugated -bonds in benzene and butadiene are given and the existence of the through-bridge bond between bridgehead carbons in small propellanes is conjectured. 1 Introduction In Molecular Orbital (MO) theory the chemical interaction between, say, two (valence) Atomic-Orbitals (AO) or general basis functions originating from different atoms is strongly influenced by their direct overlap/interaction, which conditions the bonding effect experienced by electrons occupying their bonding combination in the molecule, compared to the non-bonding reference of electrons on separated AO. This throughspace bonding mechanism is then associated with typical accumulation of the valence electrons in the region between the two nuclei, due to the constructive interference Throughout the paper A denotes a scalar quantity, A stands for a row-vector, and A represents a square or rectangular matrix. between the two AO, which exhibits some polarization reflecting the initial electronegativity difference of the two atoms involved. In other words, such shared bond charge is synonymous with the presence of the bond-covalency in direct (throughspace) interaction between the two AO, which is also reflected by the associated covalent Valence-Bond (VB) structure. Similar effect of the bonding accumulation of the information densities relative to the promolecular distribution is detected in maps of alternative densities of the entropy deficiency and displacement in Shannons entropy [15]. Accordingly, the complementary bond-ionicity aspect is manifested in MO theory by the MO polarization, or alternativelyby the participation of the orthogonal part of the ionic VB structure in the ground-state wave function. Let us recall, that on the elementary CID-level of the Configuration-Interaction (CI) in the minimum basis set both MO and VB descriptions of the chemical bond in H2 are exactly equivalent, differing only in specific routes of arriving at the same two-electron ground-state wave function describing the singlet-paired electrons. The inter-orbital bonding interaction lacking such an accumulation of the bond charge (information), e.g., in the smallest propellanes, can be also realized indirectly, through the neighboring AO intermediaries forming a bridge for an effective interaction between distant terminal AO, e.g., those located on bridgehead carbons in propellanes, or the meta- and para-carbons in the benzene ring [1,2,6,7]. This indirect (through-bridges) mechanism reflects the implicit dependence between AO resulting from the overall participation of the AO-intermediaries in the system chemical bonds determined by the subspace of the occupied MO. The associated through-bridge bondorder of the central bond in [1.1.1] propellane has been estimated using the generalized quadratic Wiberg-type indices [811] from the two-electron difference approach [1217] to be of the order of 0.8 bond, with the full, single-bond in [2.2.2] propellane including the additional 0.2 bond originating from the direct, through-space component [1,2,6], which is clearly detected in both the density-difference, and local information-theoretic (IT) probes, e.g., the entropy-deficiency, entropy-difference diagrams [13], Electron Localization Function (ELF) [1821], and Contra-Gradience (CG) plots [2225]. Thus, such a generalized outlook on the bond-order concept [1,2], emerging from both the Wiberg [8] or quadratic-difference approaches in MO theory [1,1217] and the IT bond-multiplicity in the Orbital Communication Theory (OCT) of the chemical bond [2,7,2629], identifies the chemical bond multiplicity as a measure of the statistical dependence (non-additivity) between orbitals on different atomic centers. On one hand, this dependence between basis functions of different atoms can be partly realized directly (through space), by the constructive interference of orbitals (probability amplitudes) on two atoms, which increases the electron density between them. On the other hand, it can also have an indirect origin, through the dependence on orbitals of the remaining Atoms-in-Molecules (AIM) used to construct the system occupied MO. The latter component is due to the orthonormality relations of the occupied MO, which determine the framework of chemical bonds in the molecule. Therefore, each pair of AO or AIM exhibits the partial through-space and through-bridge components: the bond-order of the former quickly vanishes with an increase of the inter-atomic separation or when the interacting AO are heavily engaged in forming other chemical bonds, while the latter can still assume appreciable values, when the remaining atoms form an effective bridge of the neighboring, chemically bonded atoms, which links the specified AO/AIM in question. In the present analysis, which was prompted by the discussion at the international meeting Twenty Years ELF (Paris, June 2124, 2010), we shall identify both these components of chemical interactions using the Wiberg [8] measure of bond multiplicities. The corresponding IT-covalencies will also be examined within the recently proposed OCT of the chemical bond [2,7,2629]. 2 Bond projections and density matrix In standard SCF MO theory the network of chemical bonds is determined by the occupied MO in the system ground-state. Let us assume the closed-shell (cs) configuration of N = 2n electrons in the standard spin-restricted Hartree-Fock (RHF) description, which involves n lowest, doubly-occupied (orthonormal) MO. In the familiar LCAO MO approach they are generated as linear combinations of the (Lwdin-orthogonalized) AO (basis functions) = (1, 2, . . . , m ) = {i } contributed by the system constituent atoms, | = {i, j } I, = [(1, 2, . . . , n ), (n+1, . . .m )] o, v where the diagonal matrix d groups the MO occupations, d = {s,s (2, s n; 0, s > n)}, and the basis set projections onto the occupied (bond) subspace o, = Po |i = i b where the rectangular matrices Co = |o and Cv = |v group the relevant expansion coefficients of the n (doubly-occupied) and m n virtual (empty) MO, respectively, to be determined using the iterative self-consistent-field (SCF) procedure. The full LCAO MO matrix C is unitary, C = C1, since it rotates orthonormal AO into the orthonormal MO, and hence the inverse transformation reads: = C. The molecular electron density, (r) = 2o(r)o(r) = (r) 2CoCo (r) (r) (r) = N p(r), and hence also the one-electron probability distribution p(r) = (r)/N , the shapefactor of , are determined by the 1-density matrix , also called the Charge-andBond-Order (CBO) matrix, o = 2CoCo = 2 | Po | = 2 = CdC S = b b = /2 = CoCo = Po = o o |s s | It thus satisfies the f (...truncated)