Handbook of Solvents - George Wypych - ChemTech - Ventech!

446 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
Other polarizabilities that may have practical importance are the quadrupole
polarizability tensors A, B, etc. whose elements can be defined in terms of the expansion of
a molecular quadrupole θ subjected to a uniform electric field:
θ
αβ
∂E
( 0)
1
=− 3 = θ + A αβ γαβ , Fγ+ Bγδ, αβFγFδ+ � [8.54]
∂F
2
αβ
The A, B, ... terms are also present in the expansion series of the dipole when the molecule
is subjected to higher derivatives of F. For example, the tensor A determines both the
quadrupole induced by a uniform field and the dipole induced by a field gradient.
The electrical influence of a molecule on a second molecule cannot be reduced to a
constant field F or to the combination of it with a gradient∇F. So an appropriate handling of
the formal expansion [8.52] may turn out to be a delicate task. In addition, the formal analysis
of convergence of the series gives negative answers: there are only demonstrations that
in special simple cases there is divergence. It is advisable not to push the higher limit of
truncated expansions much with the hope of making the result better.
The use of several expansion centers surely improves the situation. The averaged
value of dipole first polarizability α can be satisfactorily expressed as a sum of transferable
group contributions, but the calculations of IND with distributed polarizabilities are rather
unstable, probably divergent, and it is convenient to limit the calculation to the first term
alone, as is actually done in almost all the practical implementations. Instability and lack of
convergence are factors suggesting an accurate examination of the function to fit the opportune
values for these distributed α polarizabilities.
The dispersion term
The dispersion term DIS can be formally treated as IND. Asymptotically (at large R), DIS
can be expressed in terms of the dynamic multipole polarizabilities of the monomers.
∞ ∞
l<
k<
∞
1
− ( l+ l′+ k+ k′+
2)
m m ′ A
B
DIS ≈− ∑∑R ∑ ∑ Cll′ Ckk ′ × ∫πlk(
m, m′ ; iω)
πlk ′ ′ ( −m, − m′ ; iω) dω
[8.55]
2π
ll , ′= 1kk
, ′= 1
m=−l< m′=− k<
The dynamic polarizabilities are similar to the static ones, but they are dependent on the frequency
of the applied field, and they have to be computed at the imaginary frequency iω.
Here again the standard PT techniques are not accurate enough, and one has to employ
others techniques. It is worth remarking that at short or intermediate distances the dynamic
multipole polarizabilities give an appreciation of DIS of poor quality, unless computed on
the dimer basis set. This is a consequence of the BSS errors: the PT theory has thus to abandon
the objective of computing everything solely relying on monomers properties.
At the variational level there are no formal problems to use the dynamic
polarizabilities, for which there are efficient variational procedures.
The expansion is often truncated to the first term (dipole contributions only), giving
origin to a single term with distance dependence equal to R -6 . From eq. [8.49] it turns out
that all the members of this expansion have an even negative dependence on R. For accurate
studies, especially for simple systems in the gas phase, the next terms, C 8R -8 ,C 10R -10 , ... are
often considered. To have an odd term, one has to consider third order elements in the PT
theory, which generally are rarely used.
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