Multipoles

We discussed the interactions between point charges and the use of Coulomb's law to calculate the forces between them. However, many molecules do not have a net charge, but they still exhibit electric fields due to the uneven distribution of charge within the molecule. These electric fields can be described using the concept of multipole interactions.

Types

We categorize different types of multipoles using the order, \(l\), which is related to the angular dependence of the electric potential and field generated by that charge distribution.

Monopole

A monopole (\(l\) = 1) is a single point charge, such as an atomic ion. The electric field generated by a monopole is spherically symmetric and follows Coulomb's law. The electric potential of a monopole varies as 1/r, where r is the distance from the charge.

Dipole

A dipole (\(l\) = 1) is characterized by a separation \(\left( \overrightarrow{r_i} \right)\) of charges (\(q_i\)) within a molecule.

\[ \overrightarrow{\mu} = \sum_{i = 1}^{N} \overrightarrow{r_i} q_i \]

In a polar covalent bond, such as in the SiO molecule, the electronegative element (O) attracts more electron density from the electropositive element (Si). This results in one end of the bond having a net positive charge, while the other end has a net negative charge. The molecule remains neutral overall, but the uneven charge distribution creates a cylindrically symmetric electric field around the bond, known as a dipole field.

The strength of a dipole is measured by its dipole moment, \(\mu\), which is the product of the charge separation and the distance between the charges. For a system of two point charges \(+q\) and \(-q\) we have

\[ \overrightarrow{\mu} = -q \overrightarrow{r_1} + q \overrightarrow{r_2} = q \left(\overrightarrow{r_1} + \overrightarrow{r_2} \right) \]

Note that \(\overrightarrow{r_1} + \overrightarrow{r_2}\) can be defined as the Euclidean distance, or magnitude, \(d\). Thus, the dipole moment is

\[ \mu = q d \]

For a dipole, the electric field is strongest along the axis of the dipole and weakens as you move away from this axis. The field lines point from the positive charge to the negative charge. The electric potential of a dipole varies as \(1/r^2\) and has an angular dependence proportional to \(\cos (\theta)\), where \(\theta\) is the angle between the dipole axis and the position vector.

In some molecules, the dipoles of individual bonds may cancel each other out, resulting in a zero net dipole moment. For example, in CO₂, the dipoles of the two CO bonds cancel each other, leading to a zero dipole moment. However, the uneven distribution of charges still generates an electric field that is cylindrically symmetric about the bond axis and symmetric under inversion. This type of electric field is characteristic of higher-order multipoles.

Quadrupole

The quadrupole (\(l\) = 2) is the next multipole after the dipole; it is a system of four point charges arranged in a square or tetrahedron, with alternating positive and negative charges. Quadrupole interactions are weaker than dipole interactions but can still play a significant role in intermolecular interactions, especially in the absence of dipole moments.

A quadrupole is a system of four point charges arranged in a square or tetrahedron, with alternating positive and negative charges. The electric field generated by a quadrupole has a more complex angular dependence than that of a dipole. The electric potential of a quadrupole varies as \(1/r^3\) and has an angular dependence proportional to \(3 \cos^2 (\theta) - 1\), where \(\theta\) is the angle between the quadrupole axis and the position vector.

Octopole

The octopole (\(l\) = 3) is an even higher-order multipole, generated by a symmetric arrangement of eight point sources or poles of the electric field. Octopole interactions are generally weaker than quadrupole interactions and are rarely measured in intermolecular effects.

Interactions

All interactions can be expressed as a linear combination of multipole contributions

\[ U (r, \theta, \phi) = \sum_{l = 0}^{\infty} \sum_{m = -l}^{l} U_{l, m} (r) Y_l^m \left( \theta, \phi \right) \]

where \(Y_l^m \left( \theta, \phi \right)\) are the spherical harmonics and the radial dependence is captured in \(U_{l, m} \left( r \right)\).

Monopole-dipole

TODO:

\[ u_{1-2} \left( R \right) = \frac{1}{4 \pi \varepsilon_0} \left[ \frac{q_A q_B}{R + (d_A / 2)} - \frac{q_A q_B}{R - (d_A / 2)} \right] \]

\[ u_{1-2} \left( R \right) = - \frac{\mu_A q_B}{\left( 4 \pi \varepsilon_0 \right) R^2} \]

Dipole-dipole

TODO:

\[ u_{2-2} \left( R \right) = \left( \frac{1}{4 \pi \varepsilon_0 } \right) \left[ \frac{\mu_A q_B}{\left[ R + \left(d_B / 2 \right) \right]^2 } - \frac{\mu_A q_B}{\left[ R + \left(d_B / 2 \right) \right]^2 } \right] \]

\[ u_{2-2} \left( R \right) = - \frac{2 \mu_A \mu_B}{\left( 4 \pi \varepsilon_0 \right) R^3} \]

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1. Chapter 2 of Jensen, F. (2017). Introduction to computational chemistry. John wiley & sons. ↩

2. Chapter 10 of Cooksy, A. (2014). Physical Chemistry: Quantum chemistry and molecular interactions. Pearson. ↩