1036 W. NMR SHIELDING AND COUPLING CONSTANTS – DERIVATION
(involving4 the electronic momenta pj and angular momenta LAj with respect to the nucleus A, where j means electron number j)
µmc¶2 A;B j;l γAγB¿ψ(0)¯IA · ih¡rAj ×pj¢R0IB · ih¡rBl ×pl¢ψ(0)Àaver
=µ e ¶2 XXγAγBψ(0)¯IA ·¡rAj ×pj¢R0IB ·¡rBl ×pl¢ψ(0)®aver A;B j;l
=µ e ¶2 XXγAγBψ(0)¯IA ·LAjR0IB ·LBlψ(0)®aver A;B j;l
=µ e ¶2 XXγAγBIA ·IB 1©ψ(0)¯LAj;xR0LBl;xψ(0)® A;B j;l
+ψ(0)¯ˆAj;y ˆ0LBl;yψ(0)®+ψ(0)¯ˆAj;z ˆ0LBl;zψ(0)®ª:
Thus, ﬁnally
EPSO = 3µmc¶2 A;B j;l γAγBIA ·IBψ(0)¯LAjR0LBlψ(0)®:
(0)¯ (0)®
SD 0 6 0 6 0 aver
4Let us have a closer look at the operator ¡∇j × rAj ¢acting on a function (it is necessary to remember
that ∇j in ∇j × rAj is not just acting on the components of rAj alone, but in fact on rAj times a wave Aj Aj Aj
function) f: Let us see: ¶ µ ¶ µ ¶ ∇j × Aj f = i ∇j × Aj f +j ∇j × Aj f +k ∇j × Aj f
Aj Aj x Aj y Aj z
= i ∂ zAj − ∂ yAj f +similarly with y and z j Aj j Aj x
= i −3yAjzAj + zAj ∂yj +3yAjzAj − yAj ∂zj xf +similarly with y and z
µ ¶
= i rAj ∂yj − rAj ∂zj xf +similarly with y and z
µ ¶
= i rAj ∂yj − rAj ∂zj xf +similarly with y and z = −ih¡−rAj ×pj¢f = ih¡rAj ×pj¢f:
2 Coupling constants 1037
= γel X XγAγB¿ψ(0)¯·ˆj ·IA −3(sj ·rAj)(IA ·rAj)¸
j;l=1A;B Aj Aj ×R0·sl ·IB −3(sl ·rBl)(IB ·rBl)¸ψ(0)À
Bl Bl aver
= γel X XγAγBIA ·IB 1½¿ψ(0)¯·sj;x −3(sj ·rAj)xAj ¸ j;l=1A;B Aj Aj
×R0·sl;x −3(sl ·rBl)(xBl)¸ψ(0)À
Bl Bl
(0)¯ sj;y (sj ·rAj)yAj sl;y (sl ·rBl)(yBl) (0)
¿ (0)¯·rAj (sj ·rAj)zAj ¸ 0·rBl (sl ·rBl)(zBl)¸ (0)À¾ 0 r3 j r5 j 0 r3l r5l 0
Therefore,
X X
ESD = γel γAγBIA ·IB
j;l=1A;B
×¿ψ(0)¯· sj −3(sj ·rAj)rAj ¸ ˆ0· sl −3(sl ·rBl)(rBl)¸ψ(0)À:
Aj Aj Bl Bl
• EFC = ψ(0)¯ ˆ7 ˆ0 ˆ7ψ(0)®
= γel γAγBψ(0)¯δ(rAj)sj ·IAR0δ(rBl)sl ·IBψ(0)®aver j;l=1A;B
= γel X XγAγBIA ·IB 1©ψ(0)¯δ(rAj)sj;x ˆ0δ(rBl)sl;xψ(0)® j;l=1A;B
+ψ(0)¯δ(rAj)sj;y ˆ0δ(rBl)sl;yψ(0)® +ψ(0)¯δ(rAj)sj;zR0δ(rBl)sl;zψ(0)®ª:
Hence,
µ ¶
¯FC = γel γAγBIA ·IB ψ(0)¯δ(rAj)sjR0δ(rBl)slψ(0) : j;l=1A;B
The results mean that the coupling constants J are just as reported on p. 671.
X. MULTIPOLE EXPANSION
What is the multipole expansion for?
In the perturbational theory of intermolecular interactions (Chapter 13) the per-turbation operator (V ) plays an important role. The operator contains all the Coulombic charge–charge interactions, where one of the point charges belongs to subsystem A, the second to B. Therefore, according to the assumption behind the perturbational approach (large intermolecular distance) there is a guarantee that both charges are distant in space. For example, for two interacting hydrogen atoms (electron 1 at the nucleus a, electron 2 at nucleus b, a.u. are used)
V =−ra2 + r12 − r11 + R; (X.1)
where R stands for the internuclear distance. A short inspection convinces us that the mean value of the operator −r1 + r1 , with the wave function1 ψA;n (1)ψB;n (2), would give something close to zero, because both distances in the denominators are almost equal to each other, Fig. X.1.a. The same can be said of the two other terms of V . This is why, the situation is similar (see Chapter 13)
to weighing the captain’s hat, which we criticized so harshly in the supermolecular approach to supermolecular forces, see Fig. 13.4.
What could we do to prevent a loss of accuracy? This is precisely the goal of the multipole expansion for each of the operators rij .
Coordinate system
What is the multipole expansion really? We will explain this in a moment. Let us begin quietly with introducing two Cartesian coordinate systems: one on mole-cule A, the second on molecule B (Fig. X.1.b).
This can be done in several ways. Let us begin by choosing the origins of the coordinate systems. How do we choose them? Is it irrelevant? It turns out that the choice is important. Let us stop the problem here and come back to it later on. Just as a signal, let me communicate the conclusion: the origins should be chosen in the neighbourhood of the centres of mass (charges) of the interacting molecules. Let
1ψA;n (1) means an excited state (n1 is the corresponding quantum number) of atom A, ψB;n (2) similarly for atom B. Note that electron 1 is always close to nucleus a, electron 2 close to nucleus b,
while A and B are far distant.
1038
X. MULTIPOLE EXPANSION 1039
Fig. X.1. The coordinate system used in the multipole expansion. (a) Interparticle distances. The large black dots denote the origins of the two Cartesian coordinate systems, labelled aand b, respectively. We assume particle 1 always resides close to a, particle 2 always close to b. The ﬁgure gives a notation re-lated to the distances considered. (b) Two Cartesian coordinate systems (and their polar counterparts): one associated with the centre a, the second one with centre b (the x and y axes are parallel in both systems, the z axes are collinear). Note that the two coordinate systems are not on the same footing: the z axis of a points towards b, while the coordinate system b does not point to a. Sometimes in the
literature we introduce an alternative coordinate system with “equal footing” by changing zb → −zb (then the two coordinate systems point to each other), but this leads to different “handedness” (“right-”
or “left-handed”) of the systems and subsequently to complications for chiral molecules. Let us stick to the “non-equivalent choice”.
us introduce the axes by taking the z axes (za and zb) collinear pointing in the same direction, axes xa and xb as well as ya and yb, pairwise parallel.
The multipole series and the multipole operators of a particle
With such a coordinate system the Coulomb interaction of particles 1 and 2 (with charges q1 and q2) can be expanded using the following approximation2
n n m=+s
1 2 = Akl|m|R−(k+l+1) ˆ (k;m)(1)∗ ˆ (l;m)(2); (X.2) 12 k=0 l=0 m=−s
2It represents an approximation because it is not valid for R < |ra1 −rb2 |, and this may happen in real systems (the electron clouds extend to inﬁnity), also because nk;nl are ﬁnite instead of equal to ∞.
1040 X. MULTIPOLE EXPANSION
where the coefﬁcient
(k+l)!
kl|m| (k+|m|)!(l+|m|)!
whereas
(X.3)
MULTIPOLE MOMENT OPERATORS
M(k;m)(1) and M(l;m)(2) represent, respectively, the m-th components of
the 2k-pole and 2l-pole of particle 1 in the coordinate system on a and of particle 2 in the coordinate system on b:
ˆ (k;m)(1) = q1rk P|m|(cosθa1)exp(imφa1); (X.4) ˆ (l;m)(2) = q2rb2P|m|(cosθb2)exp(imφb2); (X.5)
with r;θ;φ standing for the spherical coordinates of a particle (in coordinate sys-tem a or b, Fig. X.1.b), the associated Legendre polynomials P|m| with |m|6k are deﬁned as (cf. p. 176)
P|m|(x)= 2kk!¡1−x2¢|m|/2 dxk+|m| ¡x2 −1¢k; (X.6)
nk and nl in principle have to be equal to ∞, but in practice take ﬁnite integer values, s is the lower of the summation indices k, l.
Maybe an additional remark would be useful concerning the nomenclature: any multipole may be called a 2k-pole (however strange this name looks), because this “multi” means the number 2k. If we know how to make powers of two, and in ad-dition have some contact with the world of the ancient Greeks and Romans, we will know how to compose the names of the successive multipoles: 20 = 1, hence monopole; 21 = 2, hence dipole, 22 = 4, hence, quadrupole, etc. The names, how-ever, are of no importance. The formulae for the multipoles are important.
Multipole moment operators for many particles
A while ago a deﬁnition of the multipole moments of a single point-like charged particle was introduced. However, the multipole moments will be calculated in future, practically always for a molecule. Then,
THE TOTAL MULTIPOLE MOMENT OPERATOR
The total multipole moment operator represents the sum of the same oper-ators for the individual particles (of course, all them have to be calculated in the same coordinate system): M(k;m)(A)= i∈A M(k;m)(i).
The ﬁrst thing we have to stress about multipole moments is that, in principle, they depend on the choice of the coordinate system (Fig. X.2).
This will soon be seen when inspecting the formulae for multipole moments.
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