1.1 Hierarchy of Materials
5
monograph is devoted to showing how analysis and understanding of materials are
possible based on molecules.
Before concluding this introductory section, it is reminded that the separation
in the energy scale is not perfect between intermolecular interaction and chemical
phenomena. There exist some examples that reflect this incompleteness. There exist
rare instances of phase transitions, in which molecular structures change with rearrangement of chemical bonds, include the melting of PCl 4 ·PCl 6 [1, 2] to molecular
liquid and that in liquid sulfur (changing from S 8 to linear polymer) [3].
1.2 Intermolecular Interaction
1.2.1 Charge Distribution in a Molecule
Among the four fundamental interactions between particles in nature, i.e., strong
and weak interactions, electrostatic interaction, and gravity, those workable between
atoms and molecules are the two latter of them, which have no characteristic length
scale. The interaction energy of both is inversely proportional to the distance between
two interacting subjects. The gravity is always attractive, whereas the electricity can
have not only attractive but also repulsive character depending on the signs of the electric charge. Because of the large difference in their magnitudes, however, the consideration only on the electric term is sufficient in molecular physics. Indeed, the gravitational and electrostatic interactions between two electrons (with the mass m ≈ 9.1 ·
10
−31 kg and the charge q ≈ −1.6 · 10
−19 C) 1 Å apart are −5.5 · 10
−61 J and 2.3 ·
10
−18 J, respectively. The electrostatic interaction between charges of molecules is
dominant if they carry net charges, i.e., if they are ions.
Even if molecules are neutral, electrostatic interaction governs the interaction
between them. We can easily imagine that the antiparallel arrangement is more stable
than the parallel arrangement for a pair of dipolar but neutral molecules because the
head with the positive partial charge of each molecule is closer to the negatively
charged tail of the other molecule. Thus, the consideration of charge distribution in
a molecule, ρ(r), is necessary. Even for ions, a similar consideration is necessary
to validate the naïve expectation that the interaction between them resembles that
between point charges when the molecular size is non-negligible compared to their
separation.
The following identity is known as the multipole expansion for |R| > |r| in the
three-dimensional space:
1
|R − r|
=
∞
n=0
|r|
n
|R| n+1 P n (cos θ Rr ),
(1.2)
where P n (·) is n-th order Legendre polynomial, and θ Rr the angle between vectors R
and r (cos θ Rr = R · r/|R||r|). Suppose that the center of gravity of a molecule with
5
monograph is devoted to showing how analysis and understanding of materials are
possible based on molecules.
Before concluding this introductory section, it is reminded that the separation
in the energy scale is not perfect between intermolecular interaction and chemical
phenomena. There exist some examples that reflect this incompleteness. There exist
rare instances of phase transitions, in which molecular structures change with rearrangement of chemical bonds, include the melting of PCl 4 ·PCl 6 [1, 2] to molecular
liquid and that in liquid sulfur (changing from S 8 to linear polymer) [3].
1.2 Intermolecular Interaction
1.2.1 Charge Distribution in a Molecule
Among the four fundamental interactions between particles in nature, i.e., strong
and weak interactions, electrostatic interaction, and gravity, those workable between
atoms and molecules are the two latter of them, which have no characteristic length
scale. The interaction energy of both is inversely proportional to the distance between
two interacting subjects. The gravity is always attractive, whereas the electricity can
have not only attractive but also repulsive character depending on the signs of the electric charge. Because of the large difference in their magnitudes, however, the consideration only on the electric term is sufficient in molecular physics. Indeed, the gravitational and electrostatic interactions between two electrons (with the mass m ≈ 9.1 ·
10
−31 kg and the charge q ≈ −1.6 · 10
−19 C) 1 Å apart are −5.5 · 10
−61 J and 2.3 ·
10
−18 J, respectively. The electrostatic interaction between charges of molecules is
dominant if they carry net charges, i.e., if they are ions.
Even if molecules are neutral, electrostatic interaction governs the interaction
between them. We can easily imagine that the antiparallel arrangement is more stable
than the parallel arrangement for a pair of dipolar but neutral molecules because the
head with the positive partial charge of each molecule is closer to the negatively
charged tail of the other molecule. Thus, the consideration of charge distribution in
a molecule, ρ(r), is necessary. Even for ions, a similar consideration is necessary
to validate the naïve expectation that the interaction between them resembles that
between point charges when the molecular size is non-negligible compared to their
separation.
The following identity is known as the multipole expansion for |R| > |r| in the
three-dimensional space:
1
|R − r|
=
∞
n=0
|r|
n
|R| n+1 P n (cos θ Rr ),
(1.2)
where P n (·) is n-th order Legendre polynomial, and θ Rr the angle between vectors R
and r (cos θ Rr = R · r/|R||r|). Suppose that the center of gravity of a molecule with