The energy of the hyperfine electrical interaction between the charge distribution of the core and the extranuclear static electric field can be extended to multipoles. The monopole term only causes an energy shift and the dipole term disappears, so that the first relevant expansion term is the quadrupole term:
:     ij=1;2;3
This can be written as a product of the quadrupole moment  and the electric field gradient . Both [tensor]s are of second order. Higher orders have too small effect to be measured with PAC.
The electric field gradient is the second derivative of the electric potential  at the core:
 becomes diagonalized, that:
The matrix is free of traces in the main axis system (Laplace equation)
Typically, the electric field gradient is defined with the largest proportion  and :
The energy difference between two substates,  and , is given by:
The quadrupole frequency  is introduced.
The formulas in the colored frames are important for the evaluation:
The publications mostly list .  as elementary charge and  as Planck constant are well known or well defined. 
The nuclear quadrupole moment  is often determined only very inaccurately (often only with 2-3 digits).
Because  can be determined much more accurately than , it is not useful to specify only  because of the error propagation.
In addition,  is independent of spin! This means that measurements of two different isotopes of the same element can be compared, such as  Hg(5/2−), Hg(5/2−) and Hg(9/2−). Further,  can be used as finger print method.
For the energy difference then follows:
If , then:
with:
For integer spins applies:
:         und          
For half integer spins applies:
:          und          
The perturbation factor is given by:
With the factor for the probabilities of the observed frequencies:
As far as the magnetic dipole interaction is concerned, the electrical quadrupole interaction also induces a precision of the angular correlation in time and this modulates the quadrupole interaction frequency. This frequency is an overlap of the different transition frequencies . The relative amplitudes of the various components depend on the orientation of the electric field gradient relative to the detectors (symmetry axis) and the asymmetry parameter . For a probe with different probe nuclei, one needs a parameter that allows a direct comparison: Therefore, the quadrupole coupling constant  independent of the nuclear spin  is introduced.