Knowledge Bank

“The penetration energy or Coulomb energy (in atomic units) for an N-electron diatomic molecule, for an internucler distance R, can be defined as {C}\equiv -\int \Psi_{a}^{(1, 2,......, {m})}, \Psi_{b}^{({m}+1, ......, {N})}{H}^{\prime}\Psi_{a}^{(1, 2,......, {m})}\Psi_{b}^{({m}+1,...... {N})}{dv}{1}......{dv}{N}, where $H\prime =Zb\Sigma i=1m(1/rbi)-Za\Sigma i=m+1N(1/rai)+1+rij+ZaZb/R$ is the interaction Hamiltonian. This yields -{C}\Sigma_{i}\Sigma_{j}(\chi_{{ai}}^{(1)}\chi_{{ai}}^{(1)}|\chi_{{bj}}^{(2)}\chi_{{bj}}^{(2)})+{Z}{a}{Z}{b}/{R}-{Z}{b}\Sigma{i}(\chi_{ai}|{r}{b}|\chi{ai})-{Z}{a}\Sigma{j}(\chi_{bi}|{r}{a}|\chi{bi}) C represents the classical electrostatic energy for the mutual penetration of two atoms if their charge clouds remained unchanged as the atoms come together. The Coulomb energy for $H2$ has been computed over a broud range of internuclear distances for various configurations in which the two electrons (one on each atom) are assigned to a variety of combinations of the atomic orbitals $1s,2s,2p\sigma$, and $2p\pi$, and to hybrid orbitals of the form $1-\beta 22s+\beta 2p\sigma$. The dependence of C on $\beta$ when hybrid orbitals were used was studied. The computed Coulomb energies are in some cases very large; the largest C (4.1 e.v. at $R=1.96\backslash AA$) is found for the configuration $2dia$ $2dib$, where 2di indicates a diagonal hybrid AO ($\beta =1/2$). These results suggest (in agreement with one or two usually-forgotten early computations by others) that Coulomb energies make major contributions to bonding energies when 2p$\sigma$ or, especially, 2s - 2p$\sigma$ hybrid AOs are involved. The results of Conlomb energy computations for $N2$ and CH will also be presented.”