Theoretical and technical aspects

Within the non-relativistic theory of nuclear spin-spin coupling, there are four terms contributing to the NSSCC between two nuclei A and B: the paramagnetic and diamagnetic orbital terms (OP and OD, respectively), and the electron-spin dependent Fermi-contact (FC) and spin-dipole term (SD). In the literature, the OP and OD terms are often named PSO and DSO (for paramagnetic and diamagnetic spin-orbital). In the more general ZORA formulation, very similar operators are responsible for the NSSCC, therefore we use the same terminology for the individual contributions. In general, the interpretation of the results for a heavy atom system is basically equivalent to a non-relativistic situation.

In most cases, the FC term yields the most important contribution to the NSSCC. However, many exceptions are known for which one or each of the other terms can be non-negligible or even dominant. We therefore suggest that you always check, at least for a smaller but similar model system, or by using a smaller basis set, which of the four terms are negligible and which are dominant.

By default, the CPL program computes the FC coupling between the first and all other nuclei of the molecule, respectively. Other couplings or the computation of the OP, OD and SD terms can be requested by input switches (see the 'Running CPL' section of this document for details).

All contributions to the NSSCC are evaluated with the help of the numerical integration scheme implemented into ADF. In general, the computation of the OD term is computationally very cheap, since only integrals involving the electron density have to be evaluated. The next expensive term is the OP term. For this contribution, the first-order perturbed MOs have to be computed. With the available density functionals in ADF, the OP term does not cause a change in the Kohn-Sham potential, and the first-order MOs can be computed directly (i.e. without an iterative procedure). This is equivalent to the approach that has been implemented in the NMR code for ADF.

Both the FC and the SD terms induce electron spin-density to first-order as a perturbation. Equivalent to the iterative solution of the unperturbed Kohn-Sham equations, the first-order MOs depend on that first-order spin-density, which in turn depends on the first-order MOs. Therefore, in order to evaluate the FC and SD NSSCC contributions, the CPL program carries out a SCF cycle. In the scalar or non-relativistic case, the computational cost for the FC term is comparable to an ADF single point calculation with a local density functional. The evaluation of the SD term is more expensive. The current implementation utilizes the CPL spin-orbit code to compute the combined FC+SD contribution and therefore leaves some room for future speed-ups. In most cases, the SD term yields a negligible NSSCC and the much faster code for the scalar- or non-relativistic FC term can be used. However, it is very important to include the SD term in the computation if coupling anisotropies are to be evaluated.

In the case where the NSSCC computation is based on spin-orbit coupled relativistic two component ZORA MOs, the SD term causes only a marginal increase in computational time as compared to the FC term alone. Generally, in this case the computational cost for the FC term is already approximately one order of magnitude higher than in the scalar or non-relativistic case, since the 3 (x, y, z) components of the spin-density with respect to 3 components of the perturbation, respectively, have to be determined self-consistently. The additional presence of the SD term only shows up in a somewhat more costly evaluation of the matrix elements of the perturbation operator. However, CPL spends most of its computational time in the SCF cycle. Therefore, in spin-orbit computations the computation of the FC+SD terms is the default. The OP term has to be evaluated self-consistently, too, in this case and is added as a perturbation in the SCF cycle upon request.

We use the terminology 'perturbing' and 'responding nucleus' within the CPL output. The 'perturbing' nucleus is the one, for which the first-order MOs have to be computed (self-consistently), while the NSSCC is then determined by these first-order MOs and the FC, SD, and OP matrix elements of the second, 'responding' nucleus. For the OD term, this distinction makes no sense but is used in the output for reasons of consistency.

Experimental NSSCCs between two nuclei A and B are usually reported as J(A,B) in Hertz. From a computational point of view, the so-called reduced NSSCCs K(A,B) are more convenient for comparisons. CPL outputs both. The J's are set to zero in case the nuclear magneto-gyric ratio of one of the nuclei A or B is not available at run time.