Electric Charges and their Properties - Molecular Modelling - page 18

Molecule

p e

Electric field

Figure 1.14 Induced molecular dipole

Electric field

Figure 1.15 More general case

The dipole polarizability is a special type of physical property called a tensor, just like
the electric second moment. We can represent asa3 × 3 real symmetric matrix

⎛ ⎞

α xx α xy α xz

⎝ ⎠

= α yx α yy α yz (1.27)

α zx α zy α zz

and I will write the more general expression as

p e (induced) = p e (permanent) + α. E + higher order terms (1.28)

The higher order terms are the hyperpolarizabilities ; they feature in advanced texts of
this kind. We are not going to meet them again. We interpret Equation (1.28) as a matrix
equation; the p 's are column vectors.

1.12.1 Properties of Polarizabilities

The matrix can always be written in diagonal form by a suitable rotation of the Cartesian
axes (see Appendix A) to give

⎛ ⎞

α aa 00
0 α bb 0
00α cc

⎝ ⎠

=