One may first discuss the amplitude of the impact of the spin polarization on the
energies. As seen from Tables 14.1 and 14.5 this effect is not extremely sensitive to
the extent of the spin delocalization. It only increases slightly when going from 1a
to 3a. One may notice that this energy is almost constant in the series 1a, 2c, 3c,
which is in line with the localized character of the SOMO in the series, as predicted
by Hückel calculation.
Tables 14.2, 14.3 and 14.4 compare the Hückel spin densities of the monoradicals of Fig. 14.1 with those obtained by the RODFT calculations [55, 56].
A general comment concerns the delocalization of the unpaired electron from the
external CH 2 group to the benzene rings. It is larger in the Hückel calculation, by a
factor close to 1.2. A good fit with the RODFT values would be obtained by fixing
the value of the extra-cyclic CC hopping integral t′ to 0.8t, which would only
change the ratios between the spin density of the extra cyclic carbon and those of
the rings. Actually the relative amplitudes in the rings given by the RODFT calculations are in good agreement with those of the topological approach, in the series
2b, 4b, they are divided by a factor 4 when going from a ring to the next one. The
Table 14.1 Spin polarization energy (in a.u) (difference between the RDFT and UDFT solutions)
and S
2
of the UDFT solution of the monoradicals
Compound
1a,b
2a
3a
2b
3b
2c
3c
ΔE
−0.077
−0.084
−0.104
−0.081
−0.081
−0.079
−0.079
S
2
0.7785
0.7896
0.8229
0.7859
0.7880
0.7829
0.7836
Table 14.2 Mulliken spin densities on the various atoms of the series a monoradicals for the
RDFT and UDFT solutions
Atoms
number
1a,b
2a
3a
Hückel RDFT UDFT Hückel RDFT UDFT Hückel RDFT UDFT
1
0.571
0.652 0.813
0.529
0.616 0.783
0.471
0.549
0.729
2
–
0.009 −0.185 –
0.010 −0.192 –
0.010 −0.203
3
0.143
0.104 0.237
0.253
0.181 0.363
0.264
0.228
0.422
4
–
0.003 −0.128 –
0.004 −0.104 –
0.005 −0.127
5
O.143 0.111 0.259
0.059
0.047 0.130
0.029
0.027
0.107
6
–
–
–
–
0.001 −0.116 –
0.001 −0.123
7
–
–
–
0.059
0.041 0.139
–
0.019
0.125
8
–
–
–
–
0.039 0.134
0.117
0.078
0.247
9
–
–
–
–
0.001 −0.082 –
0.002 −0.081
10
–
–
–
0.056
0.046 0.142
0.029
0.025
0.096
11
–
–
–
–
0.001 −0.094 –
0.001 −0.136
12
–
–
–
–
–
–
0.029
0.017
0.099
13
–
–
–
–
–
–
–
0.001 −0.073
14
–
–
–
–
–
–
0.029
0.023
0.102
15
–
–
–
–
–
–
–
0.001 −0.082
386
J.-P. Malrieu et al.
energies. As seen from Tables 14.1 and 14.5 this effect is not extremely sensitive to
the extent of the spin delocalization. It only increases slightly when going from 1a
to 3a. One may notice that this energy is almost constant in the series 1a, 2c, 3c,
which is in line with the localized character of the SOMO in the series, as predicted
by Hückel calculation.
Tables 14.2, 14.3 and 14.4 compare the Hückel spin densities of the monoradicals of Fig. 14.1 with those obtained by the RODFT calculations [55, 56].
A general comment concerns the delocalization of the unpaired electron from the
external CH 2 group to the benzene rings. It is larger in the Hückel calculation, by a
factor close to 1.2. A good fit with the RODFT values would be obtained by fixing
the value of the extra-cyclic CC hopping integral t′ to 0.8t, which would only
change the ratios between the spin density of the extra cyclic carbon and those of
the rings. Actually the relative amplitudes in the rings given by the RODFT calculations are in good agreement with those of the topological approach, in the series
2b, 4b, they are divided by a factor 4 when going from a ring to the next one. The
Table 14.1 Spin polarization energy (in a.u) (difference between the RDFT and UDFT solutions)
and S
2
of the UDFT solution of the monoradicals
Compound
1a,b
2a
3a
2b
3b
2c
3c
ΔE
−0.077
−0.084
−0.104
−0.081
−0.081
−0.079
−0.079
S
2
0.7785
0.7896
0.8229
0.7859
0.7880
0.7829
0.7836
Table 14.2 Mulliken spin densities on the various atoms of the series a monoradicals for the
RDFT and UDFT solutions
Atoms
number
1a,b
2a
3a
Hückel RDFT UDFT Hückel RDFT UDFT Hückel RDFT UDFT
1
0.571
0.652 0.813
0.529
0.616 0.783
0.471
0.549
0.729
2
–
0.009 −0.185 –
0.010 −0.192 –
0.010 −0.203
3
0.143
0.104 0.237
0.253
0.181 0.363
0.264
0.228
0.422
4
–
0.003 −0.128 –
0.004 −0.104 –
0.005 −0.127
5
O.143 0.111 0.259
0.059
0.047 0.130
0.029
0.027
0.107
6
–
–
–
–
0.001 −0.116 –
0.001 −0.123
7
–
–
–
0.059
0.041 0.139
–
0.019
0.125
8
–
–
–
–
0.039 0.134
0.117
0.078
0.247
9
–
–
–
–
0.001 −0.082 –
0.002 −0.081
10
–
–
–
0.056
0.046 0.142
0.029
0.025
0.096
11
–
–
–
–
0.001 −0.094 –
0.001 −0.136
12
–
–
–
–
–
–
0.029
0.017
0.099
13
–
–
–
–
–
–
–
0.001 −0.073
14
–
–
–
–
–
–
0.029
0.023
0.102
15
–
–
–
–
–
–
–
0.001 −0.082
386
J.-P. Malrieu et al.