Page 269 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

252 V. BARONE ET AL.
correlation energy being then introduced by second order many-body perturbation (UMP2)
theory [20]. All electrons were always correlated, for we have shown [21] that core
electrons play an important role in the calculation of hyperfine coupling constants. The most
serious criticism to this approach would be that the wave function consisting of UHF
orbitals does not represent a correct spin state of the molecular system under consideration.
Since, however, all the computations reported in this study give a very low spin
contamination we can expect quite accurate values of spin dependent
properties.
Basis set effects were not in the ground of this study, so that the 6-311G** [22] basis set
has been chosen as a compromise between reliability and computation times.

Isotropic Hyperfine coupling constants are related to the spin densities at the
corresponding nuclei by

where ' is the ratio of the isotropic g value for the radical to that of the free electron,
and are the nuclear magnetogyric ratio and nuclear magneton, respectively. In turn, the
spin density at nucleus N can be calculated as the expectation value of the spin density
operator over the electronic wave function

where the index v runs on all electrons, and Sz is the quantum number of the total electron
spin (1/2 for radicals).
In the framework of the Born-Oppenheimer approximation, we can speak of a potential
energy surface (PES) and of a "property surface", which can be obtained from electronic
wave functions at different nuclear configurations. In this scheme, expectation values of
observables (e.g. hyperfine coupling constants) are obtained by averaging the "property
surface" on the nuclear wave functions. To proceed further, let us introduce a curvilinear
path continuously describing the large amplitude motion (LAM) joining two (possibly
equivalent) energy minima through a first order saddle point (SP). Next, the path is
parametrized in terms of the signed arc length s in mass weighted (MW) cartesian
coordinates. The only necessary condition on the path is that it must not contain any
translational or rotational component. For the remaining f-1 internal degrees of freedom,
{Qi} (which will be referred to as the small amplitude, SA,coordinates) the potential energy
contributions are approximated to second order terms along the LA path. These local
vibrational coordinates must be orthogonal to the path tangent, to translations and to
infinitesimal rotations. In the adiabatic approach [23], the components of the SA
coordinates in the space of the MW cartesian coordinates are the eigenvectors of the
Hessian matrix from which translations, rotations and path tangent are projected out. The
{Qi} are further assumed to adjust adiabatically to the motion along s, thus giving rise to
the following effective potential: