6.9 Structure of Weakly Bound Complexes
157
χ bb =
1
2
χ 0
3 sin
2
θ a − 1
(6.34)
These two equations give θ a = 82.1° and θ a = 82.2°. These two values, although
vibrationally averaged structures, are not too far from the ab initio equilibrium
structure, 94.05° (CCSD(T)/AV5Z + 45 mid-bond functions).
However, the assumption that the electric field gradient is the same in both the
free molecule and the complex, may be rough. Indeed, when the atom is involved
in the intermolecular bond, its electric field gradient may be considerably perturbed
upon complex formation (Ngarï et al. 1999). A typical example is the planar complex
HC≡CH· · · N 2 O where the out-of-plane coupling constant χ cc of the central nitrogen
atom is 21% larger than the unperturbed value (Leung 1997).
When the off-diagonal elements of χ are available, the diagonalization of the
χ-tensor gives the rotational angles from the inertial principal axis system to the
quadrupole principal axis system. If the quadrupole nucleus is at the end of a bond,
this gives with a good precision the angles between the bond and the principal axis.
When the complex has a symmetry plane, it is also possible to derive the angle
between the bond axis and an inertial axis.
Example: 2,5-dihydrofuran· · · HCl (Legon and Thorn 1994).
In this complex, the principal inertial plane (a, c) is a symmetry plane. Hence,
χ ab = χ bc = 0. If z is the direction of the HCl bond, the angle ∠(a, z) = θ can be
obtained from χ. Let χ gg with g = x, y, z, be the elements of the quadrupole coupling
tensor in its principal axis system. If the y-axis is perpendicular to z and coincides
with the b-axis, we have
χ aa = χ zz cos
2
θ + χ xx sin
2
θ
(6.35a)
χ bb =
χ yy
(6.35b)
χ cc = χ zz sin
2
θ + χ xx cos
2
θ
(6.35c)
χ ac =
χ xx − χ zz
cos θ sin θ
(6.35d)
giving
tan 2θ = −2
χ ac
χ aa − χ cc
(6.36)
For C 4 H 6 0· · · H
35 Cl, χ aa = −35.587(5) MHz, χ bb − χ cc = 13.22(1) MHz, and
χ ac = 29.10(4) MHz, give θ = 25.164°. If the assumption that the z-axis coincides
with the bond axis is valid, we should have χ xx =
χ yy
. Indeed, we find 25.3 and
24.9 MHz.
The accuracy of the angles derived from the quadrupole coupling constants is
limited because the bond axis does not always coincide with a principal axis a the
157
χ bb =
1
2
χ 0
3 sin
2
θ a − 1
(6.34)
These two equations give θ a = 82.1° and θ a = 82.2°. These two values, although
vibrationally averaged structures, are not too far from the ab initio equilibrium
structure, 94.05° (CCSD(T)/AV5Z + 45 mid-bond functions).
However, the assumption that the electric field gradient is the same in both the
free molecule and the complex, may be rough. Indeed, when the atom is involved
in the intermolecular bond, its electric field gradient may be considerably perturbed
upon complex formation (Ngarï et al. 1999). A typical example is the planar complex
HC≡CH· · · N 2 O where the out-of-plane coupling constant χ cc of the central nitrogen
atom is 21% larger than the unperturbed value (Leung 1997).
When the off-diagonal elements of χ are available, the diagonalization of the
χ-tensor gives the rotational angles from the inertial principal axis system to the
quadrupole principal axis system. If the quadrupole nucleus is at the end of a bond,
this gives with a good precision the angles between the bond and the principal axis.
When the complex has a symmetry plane, it is also possible to derive the angle
between the bond axis and an inertial axis.
Example: 2,5-dihydrofuran· · · HCl (Legon and Thorn 1994).
In this complex, the principal inertial plane (a, c) is a symmetry plane. Hence,
χ ab = χ bc = 0. If z is the direction of the HCl bond, the angle ∠(a, z) = θ can be
obtained from χ. Let χ gg with g = x, y, z, be the elements of the quadrupole coupling
tensor in its principal axis system. If the y-axis is perpendicular to z and coincides
with the b-axis, we have
χ aa = χ zz cos
2
θ + χ xx sin
2
θ
(6.35a)
χ bb =
χ yy
(6.35b)
χ cc = χ zz sin
2
θ + χ xx cos
2
θ
(6.35c)
χ ac =
χ xx − χ zz
cos θ sin θ
(6.35d)
giving
tan 2θ = −2
χ ac
χ aa − χ cc
(6.36)
For C 4 H 6 0· · · H
35 Cl, χ aa = −35.587(5) MHz, χ bb − χ cc = 13.22(1) MHz, and
χ ac = 29.10(4) MHz, give θ = 25.164°. If the assumption that the z-axis coincides
with the bond axis is valid, we should have χ xx =
χ yy
. Indeed, we find 25.3 and
24.9 MHz.
The accuracy of the angles derived from the quadrupole coupling constants is
limited because the bond axis does not always coincide with a principal axis a the