# Quadrupole Moments

Firstly one must state that the quadrupole moment is only independent of origin if the lower order moments (monopole = charge, and dipole) are zero. However, once periodic boundary conditions are imposed, it can be hard to tell what the dipole moment is, for the dipole moment with PBCs is not well-defined.

Once those issues are resolved, there is then the problem that there are several competing definitions of the quadrupole moment, and it is important to know which one is being referred to.

## A scalar

Q = ∫ r^{2}ρ(r) d^{3}r

This definition is known as the second moment of the charge distribution, the scalar quadrupole moment, or the trace of the quadrupole tensor. It is a useful quantity, appearing in the Makov-Payne correction for charged systems in PBCs.

## A tensor

Q_{ij} = ∫ r_{i}r_{j}ρ(r) d^{3}r

where r_{1}=x, r_{2}=y, r_{3}=z.

So the trace of this tensor, which is generally written
Q_{xx}+Q_{yy}+Q_{zz} rather than
Q_{11}+Q_{22}+Q_{33}, is

tr(Q_{ij}) = ∫
(x^{2}+y^{2}+z^{2})ρ(r)
d^{3}r

tr(Q_{ij}) = Q = ∫ r^{2}ρ(r) d^{3}r

which is the first "scalar" definition.

## A traceless tensor

Partly because many experiments cannot detect the spherically-symmetric part of the quadrupole tensor, and this spherically-symmetric part is that which gives rise to the trace, it is common to see quadrupole moments written in the form of traceless tensors.

Q'_{ij} = Q_{ij} - Q δ_{ij}/3

But that definition is not unique. Some people appear to prefer

Q'_{ij} = 3 Q_{ij} - Q δ_{ij}

and others use

Q'_{ij} = ( 3 Q_{ij} - Q δ_{ij} )/2

This final form is probably the most common, and is the form endorsed by IUPAC in its Quantities, Units and Symbols in Physical Chemistry. However, Wikipedia's article on quadrupole moments lacks the factor of a half. Note that most of these definitions cause even the off-diagonal elements to change by a multiplicative factor in the conversion between the traced and traceless forms.

Fortunately traceless tensors are easy to spot, for the trace of the quadrupole moment is very unlikely to be exactly zero for any physical molecular charge distribution. However, working out which of the three common definitions of a traceless tensor is being used can be harder, particularly as they differ by relatively small factors.

## Symmetry

The quadrupole tensor is symmetric, Q_{ij}=Q_{ji},
so has no more than six independent components.

If the system has tetrahedral symmetry (e.g. methane), then the
tensor is diagonal with
Q_{xx}=Q_{yy}=Q_{zz}. The traceless tensor
is the zero tensor. So symmetry can leave between one and six
independent components for the full tensor, or zero to five for the
traceless version.

Unfortunately if the system has cylindrical symmetry, with the z
axis being the cylinder's axis, then quadrupole tensor is diagonal
with Q_{xx}=Q_{yy}. This is unfortunate because in
the traceless version one then has
Q'_{zz}=-2Q'_{xx}=-2Q'_{yy} and Q'_{zz}
then tends to be referred to as *the* quadrupole moment, and is
a scalar, but is certainly not the scalar quadrupole moment as
defined at the top of this page. All diatomic molecules have
cylindrical symmetry, as does CO_{2}, the methyl group (so
ethane and chloromethane), benzene, and many other molecules.

## Units

Units might be eÅ^{2}, eBohr^{2}, or
DÅ, where D is the Debye and is about
0.2081943 eÅ. The DÅ is also called the
Buckingham, and is abbreviated B. Again the differences between
these units are not large, especially as the eBohr^{2} is
about 0.28002852 eÅ^{2}, so only about a third
bigger than the Buckingham.