General considerations

The presence of an external electric field can be modeled by adding a term to our ordinary field-free, non-relativistic Hamiltonian corresponding to the interaction between the dipole moment operator and the external electric field:

$$\mathcal{H} = \mathcal{H}^{0} - \mathbf{Ed}_{e}$$ (14.1)

where $\mathbf{d}_{e}$ is the electric dipole moment operator defined as

$$\mathbf{d}_{e} = \sum_{i}\mathbf{r}_{i}$$ (14.2)

and $\mathcal{H}^{0}$ is our ordinary field-free, non-relativistic Hamiltonian operator. It is noteworthy that we do not include the nuclear dipole moment operator, and the total electronic energy will thus depend on the position of the molecule in the Cartesian coordinate frame.

The electric field dependence of different molecular properties are obtainable by adding fields in different directions and with different signs and then extract the information by numerical differentiation. Note that care has to be taken to choose a field that is weak enough for the numeric differentiation to be valid, yet large enough to give numerically significant changes in the molecular properties (see for instance Ref. [91]). Note also that it may be necessary to increase the convergence threshold for the solution of the response equations if molecular properties are being evaluated.

Whereas the finite field approach may be combined with any property that can be calculated with the RESPONSE program, more care need to be taken if the finite field method is used with the ABACUS program. Properties that involve perturbation-dependent basis sets, like nuclear shieldings and molecular Hessians, will often introduce extra reorthonormalization terms due to the finite field operator, and care has to be taken to ensure that these terms indeed have been included in DALTON.

NOTE: In the current release, the only properties calculated with perturbation dependent basis sets that may be numerically differentiated using finite field, are the nuclear shieldings and magnetizabilities using the implementation described in Ref. [92], and molecular gradients.