According to Koopmans theorem, the derivative of a canonical molecular orbitals (MO) energies with respect to a bond length (dynamic orbital force [DOF]) can be used as a tool to estimate the bonding/antibonding character of valence MOs along this bond, with a focus on lone pair MOs, in a series of small molecules. The HOMO DOF agrees with the calculated variation of bond length and force constant in the corresponding ground state cation, and of bond length variation by protonation. These results also agree with available experimental data. It is worthy to note that the p-type HOMOs in AH and AH2 are found bonding. The lone pair MO is bonding in NH3, while it is antibonding in PH3, AsH3, and AF3.
DOF of 2a1 MO associated to the lone pair is bonding for NH3 and antibonding for PH3 and AsH3
In the [1.1.1]propellane 1 inverted bond case, the DOF study leads to an essentially non-bonding or slightly repulsive character of the so-called ‘inverted’ central CC bond of propellane. MO analysis using dynamic orbital forces (DOF) shows that the -type interactions in the central CC bond of propellane are slightly repulsive. This bonding is thus only due to -type (or ‘banana’) bonds. The correlation between the bond energy and the sum of DOFs over occupied MOs allows to expect an in-situ bond energy of ca. 60 kcal/mol for this bond. On the same grounds, a weak bond (15-20 kcal/mol) should exist between the corresponding carbon atoms of bicyclopentane 2 despite a strong -type repulsion and the absence of a formal bond.
HOMO of propellane and bicyclopentane with DOFs (a.u.) of the central (purple; left value) and wing (black; right value) CC bonds.
The CC bonding is analyzed using Dynamic Orbital Forces in the series cyclopropane-ethane- benzene-ethylene-acetylene. First, the sum of the DOF is found linearly correlated to bond energies and thus can be used as a tool to for determination of CC bond strength. A partition of bonding into σ and π components indicates a weakening of the σ bonding along the series, mainly due to the decrease of the bonding character of the highest σ MO.
Relative
σ-π contribution in bonding, from the sum of the corresponding DOFs: Σt, total sum;
Σσ sum of DOFs of σ-type MOs; Σπ sum of DOFs of π-type MOs.
The study of the DOF along a RC allows a deeper insight on reaction processes. It highlights the nature of the main MO reorganizations, and at what stage of the RC they occur. For instance, in the Diels-Alder reaction, it is possible to identify a part of the reaction dominated by repulsive 4-electron interaction and a part dominated by attractive 2-electron interactions. Also, the shape of the DOFs as a function of the RC reveals the existence of avoided MO crossings and their precise location. Even in spontaneous reactions with a monotonous potential energy variation, like the addition of hydride ion to carbonyl, extremums of MO energy and sudden electron rearrangements can be put in evidence.
Diels-Alder reaction. MO correlation diagram and variation of the DOF (a.u) of the three highest MOs: A” (a), 2A’ (b) and 1A’(c) along the reaction coordinate R(Å). For each MO, its energy derivative with respect to R, r1 and r2 is displayed.