S. ÖBERG
Department of Mathematics, University of Luleå, Luleå, S95187,
Sweden
P. R. BRIDDON
Department of Physics, The University of Newcastle upon Tyne,
Newcastle upon Tyne, NE1 7RU, UK
B. R. EGGEN
School of Chemistry, Physics and Environmental Science,
The University of Sussex,
Falmer, Brighton BN1 9QJ, UK
The
molecular structure of ferrocene is analyzed using a first principles
self-consistent local density pseudopotential method. The
cyclopentadienyl rings are found to be almost planar and the vertical
metal-ligand distance of 1.55 Å in fair agreement with the
experimental value of 1.66 Å. The reorientational barrier
between the D 
and D 
symmetric structures is found
to be 1.15 kcal/mole.
Although the basic molecular structure of ferrocene has been
established to consist of an Fe atom sandwiched between two parallel
cyclopentadienyl rings with a hapitacy of five, i.e.,
the number of C atoms in a ligand ring, there has been uncertainly on
the relative orientation of the rings. Early crystal structure
measurements [1] were interpreted in terms of a staggered
ring arrangement. Later, neutron diffraction experiments on
the crystal suggests a mixture of staggered and eclipsed, 
molecules [2]. However, further neutron diffraction studies
[9] gave a structure close to the eclipsed form and this is
supported by X-ray analysis [4] and gas phase
electron-diffraction studies [5]. The latter gave a barrier of
only 1.1 kcal/mole between these structures. This low barrier is
probably the source of the difficulty in determining the exact
molecular symmetry.
There is also controversy over the planarity of the rings. The
neutron diffraction studies, [9] found the H atoms displaced
towards the Fe atom leading to an angle of 1.6 
between
the C-H bonds and the plane of the six carbon atoms. However, the
gas-phase electron diffraction studies [5] gave an angle of
3.7 .
Theoretical studies have had mixed successes in obtaining the experimental structure. Several Hartree-Fock investigations have been reported [3]. These give a (vertical) metal-ligand distance of 1.88 Å in serious disagreement with experimental value of 1.66 Å [5]. This is reduced by the inclusion of electron correlation and MP2 theory gives 1.468 Å [10]. Nevertheless, there is still a significant error which is eliminated by methods treating correlation at higher levels of accuracy [10]. All calculations have resulted in planar rings.
We determine here the structure of ferrocene using self-consistent
local density functional pseudopotential method (AIMPRO
[8]). Ceperley-Alder exchange correlation is used
[6] and the norm-conserving pseudopotentials for Fe and C taken
from ref. [7]. The scalar relativistic pseudopotential of Fe
was generated from the Dirac equation [7]. The full Coulomb
potential was used for H. The Kohn-Sham orbitals were expanded in
Cartesian Gaussian orbitals of the form 
. Six values of a and ten polynomial functions ( 
) for each were placed on Fe; four values of a and
polynomials with 
were sited on the ten C
atoms; two similar Cartesian Gaussian orbitals were centered on each H
atom as well as at the center of the 10 Fe-C bonds. The exponents of
the atom sited Gaussian orbitals were selected by minimizing the
energy of the neutral atoms [8].
The charge density found from the occupied Kohn-Sham orbitals of the molecule was fitted to a set of simple Gaussian functions sited at nuclei and the bond centers. Twelve functions with different exponents were placed at the Fe atom; four on each C atom; three on each H atom and one at each bond Fe-C bond center. Again these exponents were found from atomic fits. The use of pseudopotentials eliminated the need to consider the rapidly varying core orbitals. The self-consistent energy was found together with the analytic forces on each atom. The structure was then relaxed using a conjugate gradient algorithm. Further details have been given earlier [8].
The perfect eclipsed and staggered
structure were relaxed. We found the arrangement has lower
energy by 0.05 eV or 1.15 kcal/mole. The energy of structures
intermediate between these were found to lie between these
energies. The barrier to reorientation is then 1.15 kcal/mole.
We found the vertical metal-ring distance of 1.55 Å in fair
agreement with the experimental value of 1.66 Å. The Fe-C, C-C
and C-H bonds lengths were found to be 1.951 ( 
),
1.413 ( 
) and 1.083 ( 
) Å,
respectively for the arrangement and the H atoms moved out of
the plane of the ring, but away from Fe, by 3.0
( 
). These are within 5% of the experimental values
given in parenthesis [5] although the direction of movement of
the H atoms is opposite to that found experimentally.
It is clear that the bonding is described by the splitting of the
d-orbitals of Fe by the rings. The two highest filled levels have
and 
symmetries respectively. The first unoccupied
level has 
symmetry and is an anti-bonding orbital. The
corresponding bonding 
molecular orbital is the main
contribution to the stability of this molecule.
In conclusion, we have found that the density functional pseudopotential method is able to reproduce the short Fe-C bonds in ferrocene as well as a low rotational barrier to the cyclopentadienyl rings.