Knowledge Bank

Coulomb and exchange integrals are calculated assuming wave-functions of the form $\varphi 1=\zeta (x)\eta (y),\zeta i(z),z$ is the coordinate in the direction of the line connecting adjacent nuclei. x and v are the coordinates perpendicular to z and to one another. $\zeta$ and $\eta$ are for all states taken to be the same normalized optimum trigonometric functions of adequate symmetry calculated by Kuhn and Huber. (Helv. chim. acta \underline{35} 1155 (1952). When the potential may be set {V}{(x, y, z)}={V}{x}+{V}{y}(y)+{V}{z}{(z)}\mbox{ then }\zeta{i}\mbox{ is an Eigenfunction of}(\frac{b^{2}}{2m}+\frac{d^{2}}{dz^{2}}+{V}{z})\zeta={E}\zetaApplication to ethylene gives with {V}{z}=-\int\zeta^{2}\eta^{2}\frac{{e}^{2}{Z}{eff}}{{r}{1{e}}}{dx}{dy}-\int\zeta^{2}\eta^{2}\frac{{e}^{2}{Z}{eff}}{{r}{1{r}}}{dx}{dy} where $r1e$ and $r1r$ are the distances between the volume element and the left and right nucleus respectively, with $Zeff=3.25$ for the lowest excited singlet state an energy V = 11.6eV over the ground state and for the lowest triplet state T = 7.6eV. Further approximation by the Hartree-Fock method does not appreciably alter these values. LCAO MO treatment by Parr and Crawford (J. Chem. Phys. \underline{16},526 (1948)) gives $V=11.5eV$ and $T=3.1eV$. Experimental values are $V=7.6eV$ and $T=3.1-5.6eV$. It is suggested that better agreement with experiment can be achieved taking after Pauling and Sherman (Zs. Krist. \underline{81}.7 (1932)) $Zeff=2.04$. This leads to