# 11.14 Finite-Field Calculation of (Hyper)Polarizabilities

The dipole moment vector ($\vec{\mu}$), polarizability tensor ($\mathord{\buildrel{\lower 3.0pt\hbox{\scriptscriptstyle\leftrightarrow}}% \over{\alpha}}$), first hyperpolarizability ($\vec{\mathord{\buildrel{\lower 3.0pt\hbox{\scriptscriptstyle\leftrightarrow}% }\over{\beta}}}$), and higher-order hyperpolarizabilities determine the response of the system to an applied electric field:

 $E(\vec{F})=E(0)-\vec{\mu}(0)\cdot\vec{F}-\frac{1}{2!}\mathord{\buildrel{\lower 3% .0pt\hbox{\scriptscriptstyle\leftrightarrow}}\over{\alpha}}:\vec{F}\vec{F}-% \frac{1}{3!}\vec{\mathord{\buildrel{\lower 3.0pt\hbox{\scriptscriptstyle% \leftrightarrow}}\over{\beta}}}\vdots\vec{F}\vec{F}\vec{F}-\cdots\;.$ (11.77)

The various polarizability tensor elements are therefore derivatives of the energy with respect to one or more electric fields, which might be frequency-dependent (dynamic polarizabilities) or not (static polarizabilities). The most efficient way to compute these properties is by analytic gradient techniques, assuming that the required derivatives have been implemented at the desired level of theory. For DFT calculations using LDA, GGAs, or global hybrid functionals the requisite analytic gradients have been implemented and their use to compute static and dynamic (hyper)polarizabilities is described in Section 11.12.