The electronic density and energy of the C_{2}H_{2}, C_{2}H_{4}, and C_{2}H_{6} molecules are computed in the Hartree‐Fock approximation and the results compared. Each molecule is computed at three different carbon‐carbon internuclear distances, namely, those experimentally known for the equilibrium geometry in C_{2}H_{2}, C_{2}H_{4}, and C_{2}H_{6}. The comparison is extended by reporting computations on the lowest singlet state of the C_{2} molecule. The analysis is performed by making use of Mulliken's electron population analysis and our technique for partitioning the energy (bond energy analysis). For C_{2}H_{4} we have also considered the twisted configuration (where one CH_{2} group is perpendicular to the other) and for C_{2}H_{6} we have considered both staggered and eclipsed configurations (and therefore, we are in a position to add a few comments on the barrier to internal rotation). The basis set is sufficiently large and extended so as to be near to the Hartree‐Fock limit for all the computations reported. The binding energy computed in the Hartree‐Fock approximation has been corrected for the correlation error making use of Wigner's formula relating the electronic density to the correlation energy. It is shown that the one‐center energies, the molecular orbital valence state (MOVS) energies, are regular functions of the 2*s* and 2*p* population at the C atom, complementing previous work where it was shown that the MOVS energies are functions of the gross atomic charges. Also, it has been shown that the two‐center energy terms are proportional to bond strengths traditionally used in the chemical literature. The difficulty of analyzing small energy differences in the barrier to the internal rotation of C_{2}H_{6} has been removed by analyzing the barrier at smaller $R(C\u2013C)$ distances where the barrier is considerably larger. It has been proposed that the observed (or computed) small barrier is the result of internal charge transfer and conjugation between the pseudo‐2*p* orbitals on H_{3} and the 2*p* orbitals on C.

## REFERENCES

*basis set dependent*and this dependency brings about an uncertainty with regard to the physical accuracy of the analysis (not to the

*numerical*accuracy, which in general, exceeds the

*physical*accuracy of the method). The use of a large basis set, obtained from atomic computations, and incremented by polarization functions ($2p$ for hydrogen, $3d$ for carbon or better, $2p$ and $3d$ for hydrogen, $3d$ and $4f$ for carbon) tends to decrease the dependency on the basis set; the use of the same basis set for different molecules, tends to increase confidence in the

*comparison*of data between molecules. However, some basis set dependence still remains in the E.P.A. and B.E.A. (What one would like to have is a numerical Hartree‐Fock scheme for molecular computations, parallel to what is done for example in atomic computations. However, the rapid and numerous density gradient variations in a molecule make such a scheme a numerically difficult one if one wishes high accuracy.)

*Quantum Theory of Atoms, Molecules, and Solid State*edited by P. O. Lowdin (Academic, New York, 1966), p. 263;

*Quantum Chemistry*(Prentice‐Hall, New York, 1954), Chap. 8.

*Dissociation Energies and Spectra of Diatomic Molecules*(Chapman & Hall, London, 1947).

*The Strengths of Chemical Bonds*(Butterworths, London, 1954).

*The Nature of the Chemical Bond*(Cornell U.P., Ithaca, New York, 1939).

*Thermochemistry of Chemical Substances*(Reinhold, New York, 1936).

*The Chemistry of the Cyano Group*, edited by Zvi Rappoport (Interscience, New York, 1970), Chap. 1.

*Pseudopotential*(Springer, New York, 1967). (b) See Ref. 13(b), pp. 27–35.

*The Electronic Structure of Molecules*(Wiley, New York, 1964).

*Note*: The two‐center terms include the nuclear‐nuclear repulsion. If we remove this quantity nearly

*all*terms become negative in sign. In $C2H2,$ e.g., at $R1,$ $R2,$ $R3,$ the $C(1)\u2212H(l)$ terms are $\u22124.057,$ $\u22124.047,$ $\u22124.047,$ the $C(l)\u2212H(2)$ terms are $\u22121.100,$ $\u22121.057,$ $\u22121.012,$ the $C(l)\u2212C(2)$ terms are $\u22127.871,$ $\u22127.593,$ $\u22127.208,$ and the $H(l)\u2212H(2)$ terms are $\u22120.128,$ $\u22120.120,$ $\u22120.112,$ respectively (all data in atomic units).

*Hartree‐Fock limit*and differs sufficiently from the limit (0.005 to 0.010 a.u.) as to make any discussion on the exact value due to correlation somewhat meaningless. In an extreme sense one could even raise the question whether one can conclusively state anything at all concerning the barrier in $C2H6$ since we estimate that the energy difference from the Hartree‐Fock limit is

*larger*than the quantity we discuss (the experimental value of the varrier is 0.00466 a.u.). The numerous computations previously reported on the $C2H6$ barrier using either small or intermediate or larger basis sets, all give a barrier about 90% in agreement with experimental data. Thus, we are confident in discussing the barrier of rotation in the Hartree‐Fock model, even if the Hartree‐Fock limit has not been reached. In addition, by analyzing $C2H6$ at small $R(C\u2010C)$ distance, where the barrier is a

*large quantity*and by simultaneously analyzing the barrier of $C2H4$ (where the barrier is large at equilibrium distances), we have drastically decreased the probability of attributing physical significance to variation (either in the density or in the energy) which might be largely basis set dependent. Concerning the binding energies of $C2H6$ we should note that the optimization of the geometry was done with reference to the Hartree‐Fock energy: The computed barrier is in fair agreement with the experimental barrier, but the optimization of the geometry has not been carried out in full, since the C‐H distances have been kept constant.