Chemistry - Multipole expansion

Contents

1 Multipole expansion for electric potentials

2 Multipole expansion for electric fields

3 Multipole expansion for magnetic vector potentials

4 Multipole expansion for magnetic fields

5 Applications of the multipole expansion

Multipole expansion for electric potentials

The (scalar) potential at the point x for an arbitrary charge distribution ρ(x) is given by

$V(\mathbf{x}) = {1\over4\pi\epsilon_0}\int{\rho(\mathbf{y})d\mathbf{y}\over|\mathbf{x}-\mathbf{y}|}.$

This can be expanded in (negative) powers of $|\mathbf{x}|$ , obtaining (after some work) the multipole expansion

$V(\mathbf{x}) = {1\over4\pi\epsilon_0}\sum_{n=0}^{\infty}\left[|\mathbf{x}|^{-(n+1)}\int |\mathbf{y}|^n P_n(\cos\theta)\rho(\mathbf{y})d\mathbf{y}\right]$

where this integral, like the previous one, is over all of space, P_n is the degree-n Legendre polynomial, and θ is the angle between the vectors x and y.

The first couple of terms in the expansion are familiar:

$V(\mathbf{x}) = {1\over4\pi\epsilon_0}\left( {1\over|\mathbf{x}|} \int\rho(\mathbf{y})d\mathbf{y} + {1\over|\mathbf{x}|^2} \hat{\mathbf{x}} \cdot \int\mathbf{y}\rho(\mathbf{y})d\mathbf{y} + \cdots \right)$

where $\hat{\mathbf{x}}$ is the unit vector parallel to x. The first term here is the field of a point charge equal to the total charge, located at the origin. The second is the field of an electric dipole; the integral is the dipole moment of the configuration of charges.

Higher terms in the expansion include higher powers of 1/|x|, and therefore become less and less important at large distances. Hence the multipole expansion is a practical tool for the approximation of fields; far away from a given configuration of charges, the first few terms are typically dominant.

We may term the charge a "monopole moment"; it is a scalar. The dipole moment is a vector. In general, the order-n term in the sum is 1/|x|ⁿ⁺¹ times the contraction of a certain nth-rank tensor with n copies of $\hat{\mathbf{x}}$ ; the tensor is the 2ⁿ-pole moment of the configuration of charges.

The gravitational field is formally identical to the electrical field, so there is also a multipole expansion for gravitational potentials.

Multipole expansion for electric fields

We may take gradients of the expansion above to yield an expansion of the electric or gravitational field.

Multipole expansion for magnetic vector potentials

Suppose we have a current loop with a current I flowing in it. Then the vector potential of the induced magnetic field is

$A(\mathbf{x}) = {\mu_0I\over4\pi} \oint{d\mathbf{y}\over|\mathbf{x}-\mathbf{y}|}$

and as before we can expand in negative powers of $|\mathbf{x}|$ , obtaining another multipole expansion:

$A(\mathbf{x}) = {\mu_0I\over4\pi}\sum_{n=0}^{\infty}\left[|\mathbf{x}|^{-(n+1)}\oint |\mathbf{y}|^n P_n(\cos\theta)d\mathbf{y}\right]$

The n=0 term is always zero, since it equals the integral of a constant function around a closed loop. (This term, if present, would describe magnetic monopoles; if those existed, there would be no such thing as a magnetic field's vector potential.)

The n=1 term is the dipole term; applying Stokes' theorem we recover its usual form in terms of the area of the loop.

Multipole expansion for magnetic fields

We may take gradients of the expansion above to yield an expansion of the magnetic field.

Applications of the multipole expansion

The fast multipole algorithm of Greengard and Rokhlin is a general technique for accelerating computer simulations of particle dynamics and electrostatics. The idea is to decompose the force on a particle, or the potential at a given point, into two terms: one comes from nearby particles and can be computed quickly because there aren't too many of them, and the other comes from distant particles and can be computed quickly (with known bounds on the error) by aggregating many distant particles and using only the first few terms in a multipole expansion.