We want to compare here the weights of each Lewis structure given by the two
different Hückel-Lewis schemes (HL-CI and HL-P) discussed in Sect. 13.2 with
those obtained by the NRT method [9, 13, 14] (see Table 13.2).
In the two-structure mesomerism, the weights for W L and W R are equal to 50 %
for the allyl series, as expected. The trust parameters given by the two schemes are
equal and agree well with the NRT method (from about 92–99 %).
3
There is however a notable exception for the allyl radical: the trust parameter,
s ¼ 0 %, indicates an incorrect HL-CI wave function. This surprising result can be
traced back in the assumption B\0 (13.19) and inspection of the symmetry of the
allyl radical state provides an explanation. As mentioned earlier in Sect. 13.2.1, this
assumption implies an in-phase interaction between the Lewis structures (positive c i
coefficients in (13.16)). As pointed out by Goddard [25], this actually corresponds
to an excited state. In the allyl radical case, the ground state space symmetry is A 2
(C 2v point group), as can be seen from Fig. 13.4. The Hückel wave function is
antisymmetric with respect to the r v plane perpendicular to the molecular plane.
Thus, only an out-of-phase combination of the resonant structures W L and W R can
account for the correct symmetry of the ground state, since each structure W L and
Table 13.2 Weights w i of resonant Lewis structures for the cationic, radical and anionic allyl
molecules obtained with the two schemes HL-CI and HL-P and compared with the standard NRT
(B3LYP/6-31+G(d)) weights
Molecule
HL-CI
HL-P
NRT
w i
s
w i
s
w i
s
þ Allyl
50/50/
92
50/50/
92
50/50/
97
46/46/8
98
44/44/12
99
45/45/10
97.5
Allyl
50/50/
0
50/50
92
50/50/
96
39/39/22
0
50/50/0
92
50/50/0
–
À Allyl
50/50/
92
50/50/
92
50/50
94
46/46/8
98
44/44/12
99
44/44/12
96.5
The weights w i (in %) are given in the order W L =W R =W C and gathered along with the trust
parameters s (in %) for each method
Fig. 13.5 Mesomerism scheme for the allyl radical: the delocalized Hückel wave function W ref is
represented as a linear combination of resonating Lewis structures W L , W R and W C
3
Note here that the s value for the NRT results is defined as the sum of the weights of the p/
Lewis resonant structures built from the NBO analysis of the molecular density. The NRT
weights given here are renormalized so that the sum of the weights of all considered contributors
is equal to 100 %. The same definition will apply in Sect. 13.3.2.
13 Localized Structures at the Hückel Level …
349
different Hückel-Lewis schemes (HL-CI and HL-P) discussed in Sect. 13.2 with
those obtained by the NRT method [9, 13, 14] (see Table 13.2).
In the two-structure mesomerism, the weights for W L and W R are equal to 50 %
for the allyl series, as expected. The trust parameters given by the two schemes are
equal and agree well with the NRT method (from about 92–99 %).
3
There is however a notable exception for the allyl radical: the trust parameter,
s ¼ 0 %, indicates an incorrect HL-CI wave function. This surprising result can be
traced back in the assumption B\0 (13.19) and inspection of the symmetry of the
allyl radical state provides an explanation. As mentioned earlier in Sect. 13.2.1, this
assumption implies an in-phase interaction between the Lewis structures (positive c i
coefficients in (13.16)). As pointed out by Goddard [25], this actually corresponds
to an excited state. In the allyl radical case, the ground state space symmetry is A 2
(C 2v point group), as can be seen from Fig. 13.4. The Hückel wave function is
antisymmetric with respect to the r v plane perpendicular to the molecular plane.
Thus, only an out-of-phase combination of the resonant structures W L and W R can
account for the correct symmetry of the ground state, since each structure W L and
Table 13.2 Weights w i of resonant Lewis structures for the cationic, radical and anionic allyl
molecules obtained with the two schemes HL-CI and HL-P and compared with the standard NRT
(B3LYP/6-31+G(d)) weights
Molecule
HL-CI
HL-P
NRT
w i
s
w i
s
w i
s
þ Allyl
50/50/
92
50/50/
92
50/50/
97
46/46/8
98
44/44/12
99
45/45/10
97.5
Allyl
50/50/
0
50/50
92
50/50/
96
39/39/22
0
50/50/0
92
50/50/0
–
À Allyl
50/50/
92
50/50/
92
50/50
94
46/46/8
98
44/44/12
99
44/44/12
96.5
The weights w i (in %) are given in the order W L =W R =W C and gathered along with the trust
parameters s (in %) for each method
Fig. 13.5 Mesomerism scheme for the allyl radical: the delocalized Hückel wave function W ref is
represented as a linear combination of resonating Lewis structures W L , W R and W C
3
Note here that the s value for the NRT results is defined as the sum of the weights of the p/
Lewis resonant structures built from the NBO analysis of the molecular density. The NRT
weights given here are renormalized so that the sum of the weights of all considered contributors
is equal to 100 %. The same definition will apply in Sect. 13.3.2.
13 Localized Structures at the Hückel Level …
349