Handbook of Solvents - George Wypych - ChemTech - Ventech!

442 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
V B(r 1) is the electrostatic potential of molecule B (often called MEP or MESP, according
to the authors 3,18,19 ). This a true molecular quantity, not depending on interactions, and it
is often used to look at local details of the electrostatic interactions between molecules as required,
for example, in chemical reactivity and molecular docking problems.
To model simplified expressions of the intermolecular potential the direct use of eq.
[8.45] does not introduce significant improvements. The MEP, however, may be used in
two ways. We consider here the first, consisting of defining, and using, a multipolar expansion
of it. The theory of multipolar expansion is reported in all the textbooks on
electrostatics and on molecular interactions, as well as in many papers, using widely different
formalisms. It can be used to separately expandρ A,ρ B, and 1/r 12 of eq. [8.44] or V B andρ A
of eq. [8.45]. We adopt here the second choice. The multipolar expansion of V B(r 1) may be
so expressed:
∞
B − ( l+
1)
m
() 1 = ∑ ∑ ,
l ( , )
V r M R Y
B l m
l=
0 m=−l l
θϕ [8.47]
It is a Taylor expansion in powers of the distance R from the expansion center. There
are negative powers of R only, because this expansion is conceived for points lying outside
a sphere containing all the elements of the charge distribution.
m
The other elements of [8.47] are the harmonic spherical functions Yl and the
B
multipole elements M l,m which have values specific for the molecule:
B ⎛ 4π
⎞ l m
3
m
Mlm
, = ⎜ ⎟ rYl ( , ) B() rd r= B| Ml
B
⎝2l
+ 1 ∫ θϕρ Ψ Ψ [8.48]
⎠
The spherical harmonics are quite appropriate to express the explicit orientational dependence
of the interaction, but in the chemical practice it is customary to introduce a linear
m
transformation of the complex spherical functions Yl into real functions expressed over
Cartesian coordinates, which are easier to visualize. In Table 8.3 we report the expressions
of the multipole moments.
Table 8.3. Multipole moments expressed with the aid of real Cartesian harmonics
M 0 1 element Charge Q
m
M1 m
M2 m
M3 3 elements Dipole µ x,µ y,µ z
5 elements Quadrupole θxy, θxz, θyz, θ 2
x − y 2, θ 2
z
7 elements Octupole ω xxy , etc.
The leading parameter is l, which defines the 2 l poles. They are, in order, the monopole
(l=0, a single element corresponding to the net molecular charge), the dipole (l=1, three elements,
corresponding to the 3 components of this vector), the quadrupole (l=2, five components,
corresponding to the 5 distinct elements of this first rank tensor), the octupole, etc.
We have replaced the potential given by the diffuse and detailed charge distribution ρ B with
that of a point charge, plus a point dipole, plus a point quadrupole, etc., placed all at the expansion
center. Note that the potential of the 2 l pole is proportional to r -(l+1) . This means that