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 motion induced by molecular 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 conceptionally divided into the external valence electrons responsible for the bond and the remaining ion, the electrons of which are unaffected by the molecule formation.
- 2.
The interaction energy E of of two neutral system at a distance d′ much larger than their size is proportional to −E 1(d′)·p 2, where E 1(d′) ~ p 1/d′3 is the electric field created by the dipole moment p 1 at a distance d′; p 2 = α 2·E 1(d′) is the dipole moment of the system 2 induced by the field E 1(d) and α 2 is the polarizability of the system 2. For dimensional reasons, α 2 is proportional to the volume r 3a2 (see Sect. 5.3); the latter can be written with the help of (8.3) and p 22 ∼ e 2·r 2a2 as r 3a2 = r 2a2 r a2 = r 2a2 c p2 e 2/I P2 ∝ p 22 /I P2. Substituting E 1(d′), p 2, and α 2 in E = −E 1(d′), p 2 we obtain E = −A/d′6, where A ∼ p 21 p 22 /I P2. If we have started with the equivalent relation −E 2(d′)·p 1 for E instead of −E 1(d′)·p 2, the result for A would have been A ~ p 21 p 22 /I p1; it is not unreasonable to assume, for symmetry reasons, that the correct expression for A is the average of the two: \( A = c_{W} p_{1}^{2} \;p_{2}^{2} \left( {{\tfrac{1}{{I_{P1} }}} + {\tfrac{1}{{I_{P2} }}}} \right) \). For the hydrogen–hydrogen case, where p 21 = p 22 = 3e 2 a 2 B , the numerical factor c W is equal to 0.18.
- 3.
The electrons will not be squeezed between the two nuclei as to screen their repulsion; on the contrary, 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 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′ 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 |E o | is much smaller than usually.
- 5.
If the molecule is linear, there are only two rotational degrees of freedom and 3N−5 vibrational ones.
- 6.
If a waveparticle is in a state ϕ, the average value of a physical quantity A is obtained by the following formula: 〈\(\langle A\rangle= \left\langle \phi \right|A\left| \phi \right\rangle /\left\langle \phi \right|\left. \phi \right\rangle \)〉 where by definition we have \( \left\langle \phi \right|A\left| \phi \right\rangle \equiv \int \phi A\phi d^{3} r\;{\text{and}}\;\left\langle \phi \right|\left. \phi \right\rangle \equiv \int \phi \phi d^{3} r; \) ϕ, c 1, c 2 are assumed to be real.
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© 2011 Eleftherios N. Economou
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Economou, E.N. (2011). From Atoms to Molecules. In: A Short Journey from Quarks to the Universe. SpringerBriefs in Physics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20089-2_9
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DOI: https://doi.org/10.1007/978-3-642-20089-2_9
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