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 = ∫ r²ρ(r) d³r

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_ir_jρ(r) d³r

where r₁=x, r₂=y, r₃=z.

So the trace of this tensor, which is generally written Q_xx+Q_yy+Q_zz rather than Q₁₁+Q₂₂+Q₃₃, is

tr(Q_ij) = ∫ (x²+y²+z²)ρ(r) d³r
tr(Q_ij) = Q = ∫ r²ρ(r) d³r

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 CO₂, the methyl group (so ethane and chloromethane), benzene, and many other molecules.

Units

Units might be eŲ, eBohr², 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 eBohr² is about 0.28002852 eŲ, so only about a third bigger than the Buckingham.