Induced Electric Moments and Polarizabilities

Traditionally, the dependence of the cartesian components of the electric moments, fj,a , ) and ap , ), on powers of the strength of an external field and field gradient a is expressed in the following way (Buckingham, 1967) 

These expansions serve mainly as definitions of the polarizabilities and hyperpolarizabilities as proportionality constants in the correction terms to the permanent moments. The dipole polarizability a is a second-rank tensor with nine cartesian components the dipole-quadrupole polarizability and first dipole 

Based on the expansions, one can express the polarizabilities and hyperpolarizabilities as first and higher derivatives of the field-dependent moments ft , ) and , ) with respect to the components of the electric field and field gradient  

The expansion of the electric moments in Eqs. (4.44) and (4.45) also explains why the polarizabilities are sometimes called the linear response of the moments to an electric field or field gradient, the first hyperpolarizabilities the quadratic response 

The isotropic or mean polarizabilities normally measured for molecules in the liquid or gas phase are defined as (Buckingham, 1967) 

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