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Polarizabilities

The (static) electric dipole polarizability tensor \(\boldsymbol{\alpha}\) measures the first-order response of the molecular dipole to a uniform external electric field \(\mathbf{F}\). Expanding the molecular energy in the field strength,

\[ E(\mathbf{F}) \;=\; E_0 \;-\; \boldsymbol{\mu}_0\!\cdot\!\mathbf{F} \;-\; \tfrac{1}{2}\sum_{ij} \alpha_{ij}\, F_i F_j \;+\;\mathcal{O}(F^3), \]

the polarizability appears as the second derivative

\[ \alpha_{ij} \;=\; \left.\frac{\partial \mu_i}{\partial F_j}\right|_{\mathbf{F}=0} \;=\; -\left.\frac{\partial^2 E}{\partial F_i\,\partial F_j}\right|_{\mathbf{F}=0}. \]

For a single-determinant wavefunction the analytic evaluation requires the first-order orbital response \(\mathbf{U}^j_{ai}\) to a dipole perturbation \(\hat\mu_j\), which is obtained by solving the coupled-perturbed Hartree-Fock (CPHF) or coupled-perturbed Kohn-Sham (CPKS) equations. In closed-shell HF/RKS,

\[ \alpha_{ij} \;=\; -\,4\sum_{a,i'}\, \mu^{(i)}_{a i'}\,U^{(j)}_{a i'}, \]

where \(\mu^{(i)}_{ai'} = \langle\psi_a|\hat\mu_i|\psi_{i'}\rangle\) is the dipole integral between occupied orbital \(i'\) and virtual orbital \(a\), and \(U^{(j)}\) is the orbital-response coefficient for perturbation \(\partial/\partial F_j\). Analogous expressions, with the appropriate spin sums and ROHF mixed blocks, are implemented for UHF and ROHF open-shell reference wavefunctions; polarizabilities for open-shell DFT are not available (TeraChem prints a warning and skips the calculation).

Set polarizability yes to invoke this evaluation. The full \(3\times 3\) tensor is reported in \(\mathring{A}^3\). Two derived quantities are commonly needed:

  • the rotationally-invariant isotropic average \(\bar\alpha = \tfrac{1}{3}\mathrm{Tr}\,\boldsymbol{\alpha} = \tfrac{1}{3}(\alpha_{xx}+\alpha_{yy}+\alpha_{zz})\), and
  • the anisotropy \(\Delta\alpha = \left\{\frac{1}{2}\left[(\alpha_{xx}-\alpha_{yy})^2 + (\alpha_{yy}-\alpha_{zz})^2 + (\alpha_{zz}-\alpha_{xx})^2 + 6(\alpha_{xy}^2+\alpha_{xz}^2+\alpha_{yz}^2)\right]\right\}^{1/2}.\)

The same CPHF/CPKS machinery yields the dipole moment derivatives \(\partial\mu_i/\partial R_{A\beta}\) — the quantities that determine infrared absorption intensities — at essentially no extra cost. Request them with dipolederivative yes; this implicitly enables polarizability yes.

A purely numerical alternative is available through finite_field with field components efx, efy, efz (see the full parameter list). Running several single-point energies at small displacements \(\pm F_i\) and combining them by finite difference gives \(\alpha_{ij}\) without CPSCF; this is the route for methods (or environments) for which the analytic response is not yet implemented.

The polarizability derivatives with respect to nuclear displacement, \(\partial\alpha_{ij}/\partial R_{A\beta}\), are needed to compute Raman intensities and are evaluated during numerical-frequency runs (run frequencies or run initcond); see Frequencies & Thermochemistry.