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# 10.12.2 Numerical Calculation of Static Polarizabilities

(February 4, 2022)

Where analytic gradients are not available, static polarizabilities (only) can be computed via finite-difference in the applied field, which is known as the finite field (FF) approach. Beginning with Q-Chem 5.1, a sophisticated “Romberg” approach to FF differentiation is available, which includes procedures for assessing the stability of the results with respect to the finite-difference step size. The Romberg approach is described in Section 10.12.3. This section describes Q-Chem’s older approach to FF calculations based on straightforward application of small electric fields along the appropriate Cartesian directions.

Dipole moments can be calculated numerically as the first derivative of the energy with respect to $\vec{F}$ by setting JOBTYPE = DIPOLE and IDERIV = 0. If IDERIV is not specified explicitly, the dipole moment will be calculated analytically, which for post-Hartree–Fock levels of theory invokes a gradient calculation in order to utilize the relaxed wavefunction.

Similarly, set JOBTYPE = POLARIZABILITY for numerical evaluation of the static polarizability tensor $\mathord{\buildrel{\lower 3.0pt\hbox{\scriptscriptstyle\leftrightarrow}}% \over{\alpha}}$. This is performed by either first-order finite difference, taking first-order field derivatives of analytic dipole moments, or by second-order finite difference of the energy. The latter is useful (indeed, required) for methods where analytic gradients are not available, such as CCSD(T) for example. Note, however, that the electron cloud is formally unbound in the presence of static electric fields and therefore a bound solution is a consequence of using a finite basis set. (With analytic derivative techniques the perturbing field is infinitesimal so this is not an issue.) This fact, along with the overall sensitivity of numerical derivatives to the finite-difference step size, means that care must be taken in choosing the strength of the applied finite field.

To control the order for numerical differentiation with respect to the applied electric field, use IDERIV in the same manner as for geometric derivatives, i.e., for polarizabilties use IDERIV = 0 for second-order finite-difference of the energy and IDERIV = 1 for first-order finite difference of gradients. In addition, for numerical polarizabilities at the Hartree-Fock or DFT level set RESPONSE_POLAR = $-1$ in order to disable the analytic polarizability code.

RESPONSE_POLAR

RESPONSE_POLAR
Control the use of analytic or numerical polarizabilities.
TYPE:
INTEGER
DEFAULT:
0 or $-$1 = 0 for HF or DFT, $-$1 for all other methods
OPTIONS:
0 Perform an analytic polarizability calculation. $-$1 Perform a numeric polarizability calculation even when analytic 2nd derivatives are available.
RECOMMENDATION:
None

In finite-difference geometric derivatives the \$rem variable FDIFF_STEPSIZE controls the size of the nuclear displacements (see Section 9.1.2), but here it controls the magnitude of the electric field perturbations:

FDIFF_STEPSIZE

FDIFF_STEPSIZE
Displacement used for calculating derivatives by finite difference.
TYPE:
INTEGER
DEFAULT:
1 Corresponding to $1.88973\times 10^{-5}$ a.u.
OPTIONS:
$n$ Use a step size of $n$ times the default value.
RECOMMENDATION:
Use the default unless problems arise.