GIAO methods

The GIAO method has roots in the old idea of London who introduced field-dependent gauge-invariant atomic orbitals. These are constructed from real atomic orbitals multiplied by a field-dependent phase factor 

The general aspect of gauge invariance is that the addition of the gradient of an arbitrary function A(r) to the vector potential A leaves the magnetic field B = V x A unaffected. Hence the vector potential transforms as 

For example, the translation of a position vector r by a constant vector d 

The molecular orbitals satisfy the Brillouin condition which states that the matrix elements of the Fock operator between occupied and unoccupied molecular orbitals vanish 

The Fock matrix for a closed-shell system is defined as 

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GIAO method (Gauge Independent Atomic Orbitals), 126... 

There are several alternative computational methods for obtaining NMR data such as the GIAO method (Wolinski et al., 1990), the LORG method (Bouman and Hansen, 1989), and IGAIM (Keith and Bader, 1992). 

To compare with the experimental NMR data, the 13C isotropic chemical shifts of all of the acetylide and vinylidene complexes were calculated. These systems were too large for us to be able to do chemical shift calculations at the MP2 level. We thus used the GIAO method at the RHF level. Fortunately, the RHF level of theory appears to be adequate for these complexes. We used a mixed basis set for the GIAO, with tzplarge+ on C, tzplarge on H, tzp on the Mg and O associated with the adsorbed species, and dzp on the remaining Mg and O (26). 

We refer to the literature for a more detailed discussion of this point (3,7), as well as for the actual implementation within the framework of the DFT-GIAO method for NMR shieldings (6,7,9). 

Computational Details. Restricted Hartree-Fock (RHF) calculations were carried out using Gaussian 94 (45) and ACES II (46) on an IBM RISC/6000 computer. The gauge independent atomic orbitals (GIAO) method was used for the shielding calculations (47). All second-order many-body perturbation theory (MBPT2, also referred to as MP2) calculations were performed with ACES II (46). 

Table 5. Be O aiMj ljj NMR shieldings (in ppm) calculated for Be aquo and hydroxo complexes using the 6-31G basis set, the GIAO method and 6-31G optimized geometries.

NMR shielding tensors were calculated using the Gauge-Independent Atomic Orbital (GIAO) method as implemented in Gaussian 98 (44-46). The basis sets and level of theory are the same as used in the geometry optimizations and frequency calculations mentioned above. 

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