Abstract
Atoms attract each other to form molecules. For each molecule we would like to know the relative positions of its atoms, their vibration properties, and the changes in the electronic states induced by molecule formation. Dimensional analysis and various computational methods such as the so-called LCAO are useful tools in obtaining this information.
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Notes
- 1.
Each atom in a molecule is divided into the external valence electrons responsible for the bonding and the remaining ion, the electronic states of which are considered unaffected by the molecule formation.
- 2.
In electrostatics the interaction energy E of two neutral, initially unpolarized, systems at a distance \(d^{\prime}\) much larger than their size is proportional to \(- E_{1} (d^{\prime}) \cdot p_{2}\), where \(E_{1} (d^{\prime}) \propto p_{1} /d^{\prime 3}\) is the electric field created by the dipole moment \(p_{1}\) at a distance \(d^{\prime}\); \(p_{2} = \alpha_{2} \cdot \,E_{1} (d^{\prime})\) is the dipole moment of the system 2 induced by the field \(E_{1} (d^{\prime})\) and \(\alpha_{2}\) is the polarizability of the system 2. (See (6.12)). For dimensional reasons, \(\alpha_{2}\) is proportional to the volume \(r_{a2}^{3}\), where \(r_{a2}\) is the radius of system 2; this volume can be written with the help of (10.3) and \(p_{2}^{2} \propto e^{2} \cdot r_{a2}^{2}\) as \(r_{a2}^{3} = r_{a2}^{2} \,r_{a2} = r_{a2}^{2} \,c_{p2} e^{2} /I_{P2} \propto p_{2}^{2} /I_{P2}\). Substituting \(E_{1} (d^{\prime})\), \(p_{2}\), and \(\alpha_{2}\) in \(E = - E_{1} (d^{\prime}) \cdot p_{2}\) we obtain \(E = - A/d^{\prime 6}\), where \(A \propto p_{1}^{2} \,p_{2}^{2} /I_{P2}\). If we had started with the equivalent relation \(- E_{2} (d^{\prime}) \cdot p_{1}\) for E instead of \(- E_{1} (d^{\prime}) \cdot p_{2} ,\) the result for A would have been \(A \propto p_{1}^{2} \,p_{2}^{2} /I_{P1}\); it is not unreasonable to assume, for symmetry reasons, that the correct expression for A is the average of the two: \(A = c_{W} p_{1}^{2} \,p_{2}^{2} (\tfrac{1}{{I_{P1} }} + \tfrac{1}{{I_{P2} }}).\) For the hydrogen-hydrogen case, where \(p_{1}^{2} = p_{2}^{2} = 3\,e^{2} a_{B}^{2}\), the numerical factor \(c_{W}\) is equal to 0.72.
- 3.
In the case \(d^{\prime} \ll d,\) the electrons will not be squeezed between the two nuclei as to screen their repulsion; on the contrary, as \(d^{\prime} \to 0\), they will approach the ground state configuration of an atom of atomic number \(Z_{1} + Z_{2} ,\) where \(Z_{1} ,\,\,Z_{2}\) are the atomic numbers of the two atoms under consideration.
- 4.
Atoms of the noble gases are an exception: Because of their fully completed outer shells and the large energy separation of the next empty level, no overlap of atomic orbitals belonging to different atoms is tolerated, and the curve E versus \(d^{\prime}\) Â (see Fig. 11.1) Â starts moving upwards before any overlap occurs and before any molecular orbital is formed. As a result, the equilibrium distance d is considerably larger than the sum \(r_{a1} + r_{a2}\) and the energy gain \(\left| {E_{o} } \right|\) is much smaller than usually.
- 5.
If the molecule is linear, there are only two rotational degrees of freedom and \(3N_{a} - 5\) vibrational ones.
- 6.
If a waveparticle is in a state \(\phi ,\) the average value of a physical quantity A is obtained by the following formula: \(\left\langle A \right\rangle = \left\langle \phi \right|\hat{A}\left| \phi \right\rangle /\left\langle \phi \right|\left. \phi \right\rangle\) where \(\hat{A}\) is the operator corresponding to A and by definition we have \(\left\langle \phi \right|\hat{A}\left| \phi \right\rangle \equiv \int {\phi \,} (\hat{A}\,\phi \,)d^{3} r\,\,{\text{and}}\,\,\left\langle \phi \right|\left. \phi \right\rangle \equiv \int {\phi \,} \phi \,d^{3} r\); \(\phi ,\,\,c_{1} ,\,\,c_{2}\) are assumed to be real.
References
J. McMurry, Organic Chemistry (Brooks/Cole, KY, 1996)
W. Harrison, Electronic Structure and the Properties of Solids (Dover Publications, New York, 1989)
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Economou, E.N. (2016). From Atoms to Molecules. In: From Quarks to the Universe. Springer, Cham. https://doi.org/10.1007/978-3-319-20654-7_11
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DOI: https://doi.org/10.1007/978-3-319-20654-7_11
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