Multipole Moment - Molecular Electrostatic Multipole Moments

Molecular Electrostatic Multipole Moments

All atoms and molecules (except S-state atoms) have one or more non-vanishing permanent multipole moments. Different definitions can be found in the literature, but the following definition in spherical form has the advantage that it is contained in one general equation. Because it is in complex form it has as the further advantage that it is easier to manipulate in calculations than its real counterpart.

We consider a molecule consisting of N particles (electrons and nuclei) with charges eZ_i. (Electrons have the Z-value unity, for nuclei it is the atomic number). Particle i has spherical polar coordinates r_i, θ_i, and φ_i and Cartesian coordinates x_i, y_i, and z_i. The (complex) electrostatic multipole operator is

$Q^m_\ell \equiv \sum_{i=1}^N e Z_i \; R^m_{\ell}(\mathbf{r}_i),$

where is a regular solid harmonic function in Racah's normalization (also known as Schmidt's semi-normalization). If the molecule has total normalized wave function Ψ (depending on the coordinates of electrons and nuclei), then the multipole moment of order of the molecule is given by the expectation (expected) value:

$M^m_\ell \equiv \langle \Psi | Q^m_\ell | \Psi \rangle.$

If the molecule has certain point group symmetry, then this is reflected in the wave function: Ψ transforms according to a certain irreducible representation λ of the group ("Ψ has symmetry type λ"). This has the consequence that selection rules hold for the expectation value of the multipole operator, or in other words, that the expectation value may vanish because of symmetry. A well-known example of this is the fact that molecules with an inversion center do not carry a dipole (the expectation values of vanish for m = −1, 0, 1). For a molecule without symmetry no selection rules are operative and such a molecule will have non-vanishing multipoles of any order (it will carry a dipole and simultaneously a quadrupole, octupole, hexadecapole, etc.).

The lowest explicit forms of the regular solid harmonics (with the Condon-Shortley phase) give:

(the total charge of the molecule). The (complex) dipole components are:

$M^1_1 = - \sqrt{\tfrac{1}{2}} \sum_{i=1}^N e Z_i \langle \Psi | x_i+iy_i | \Psi \rangle\quad \hbox{and} \quad M^{-1}_{1} = \sqrt{\tfrac{1}{2}} \sum_{i=1}^N e Z_i \langle \Psi | x_i - iy_i | \Psi \rangle.$

$M^0_1 = \sum_{i=1}^N e Z_i \langle \Psi | z_i | \Psi \rangle.$

Note that by a simple linear combination one can transform the complex multipole operators to real ones. The real multipole operators are of cosine type or sine type . A few of the lowest ones are:

$\begin{align} C^0_1 &= \sum_{i=1}^N eZ_i \; z_i \\ C^1_1 &= \sum_{i=1}^N eZ_i \;x_i \\ S^1_1 &= \sum_{i=1}^N eZ_i \;y_i \\ C^0_2 &= \frac{1}{2}\sum_{i=1}^N eZ_i\; (3z_i^2-r_i^2)\\ C^1_2 &= \sqrt{3}\sum_{i=1}^N eZ_i\; z_i x_i \\ C^2_2 &= \frac{1}{3}\sqrt{3}\sum_{i=1}^N eZ_i\; (x_i^2-y_i^2) \\ S^1_2 &= \sqrt{3}\sum_{i=1}^N eZ_i\; z_i y_i \\ S^2_2 &= \frac{2}{3}\sqrt{3}\sum_{i=1}^N eZ_i\; x_iy_i \\ \end{align}$

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