Theor Chem Acc (2015) 134:147
1 3
the latter, which gives more fl exibility to the molecule to
change its conformation and form the overlap. Under CT,
2b and 3 undergo a larger change in radius than 1b . Also,
the large standard deviations of the radius for 2b
2+ and
for 3
2− ( > 0.15 Å) indicate that the use of Eq. ( 2 ) is more
approximate in these cases. This statement is confi rmed by
the larger average deviations from helicity for those two
cases ( ¯
D > 0.15 Å).
In Table 3 , d CC and ¯
p provide a comparable metric, as
expected, but d CC is usually larger than ¯
p . The exception
is for 2b
2+ for the reasons mentioned above. It is important
to note that in 1b and 3 , the average pitch is smaller than
the carbon–carbon vdW distance (3.4 Å [ 34 ]), even for the
neutral case. This small through space contact distance is
partially responsible for the large actuation effect on the
pitch due to CT.
Table 4 lists the strain data for all three systems and
both directions of the CT based on Eq. ( 1 ) using a subscript to identify which of the three measures of elongation are used. Large strain is obtained for 1b
2−
, 3
2+ , and
2b
2−
. Most of the s(q) values are negative, indicating that
the effect is mostly quantum mechanical, and, when negative, in opposition to the Coulombic repulsion due to the
extra charge on the molecule. The numbers in bold identify
Fig. 9 HOMO of a 3 , b 3
2+ and c 3
2−
. Isosurface was generated with a contour value of 0.030 a.u. for the neutral and oxidized molecule ( a , b ),
and 0.025 a.u. for the reduced molecule ( c )
Table 2 Average radius, ¯
r (Å), and its standard deviation, σ ¯
r (Å), average deviation from helicity, ¯
D (Å), and its standard deviation, σ ¯
D , for 1b ,
2b , and 3 in different charge states
For the charged molecules, the percentages of change in the radius from the neutral ones are given in parentheses
Neutral
q = +2
q = −2
¯
r
σ ¯
r
¯
D
σ ¯
D
¯
r
σ ¯
r
¯
D
σ ¯
D
¯
r
σ ¯
r
¯
D
σ ¯
D
1b
1.347
0.040
0.033
0.023
1.345 (−0.2)
0.046
0.042
0.025
1.384 (2.7)
0.038
0.030
0.023
2b
2.903
0.040
0.054
0.027
2.784 (−4.1)
0.174
0.156
0.082
3.120 (7.5)
0.058
0.109
0.062
3
2.189
0.100
0.092
0.049
2.129 (−2.8)
0.100
0.082
0.062
2.011 (−8.1)
0.168
0.165
0.051
Table 3 Changes upon charge
transfer of various measures of
the molecular length: average
pitch, ¯
p (Å), z (Å), and
carbon–carbon intramolecular
distances, d CC (Å), for 1b , 2b
and 3
a Underlined numbers refer to cases where the helicity is partially lost due to CT (see text)
Neutral
q = +2
q = −2
¯
p
z
d CC
¯
p
z
d CC
¯
p
z
d CC
1b
3.299
4.352
3.378
3.258
4.254
3.334
3.130
4.046
3.182
2b
3.561
4.431
3.688
3.402
4.415
a
3.884
3.437
3.982
3.500
3
3.308
4.030
3.422
3.148
3.939
3.273
3.246
4.283
3.373
Table 4 Computed strain due
to CT, s ( q ) (in %), using the
three different metrics for 1b ,
2b , and 3
a Numbers in bold indicate large actuation (see text)
b Underlined numbers indicate partial loss of helicity upon CT (see text)
q = +2
q = −2
s ¯
p
s z
s d CC
s ¯
p
s z
s d CC
1b
−1.24
−2.25
−1.30
−5.11
−7.02
−5.81
2b
−4.45
b
−0.35
5.31
−3.49
−10.13
−5.09
3
−4.82
a
−2.26
−4.34
−1.86
6.28
−1.43
52
Reprinted from the journal
1 3
the latter, which gives more fl exibility to the molecule to
change its conformation and form the overlap. Under CT,
2b and 3 undergo a larger change in radius than 1b . Also,
the large standard deviations of the radius for 2b
2+ and
for 3
2− ( > 0.15 Å) indicate that the use of Eq. ( 2 ) is more
approximate in these cases. This statement is confi rmed by
the larger average deviations from helicity for those two
cases ( ¯
D > 0.15 Å).
In Table 3 , d CC and ¯
p provide a comparable metric, as
expected, but d CC is usually larger than ¯
p . The exception
is for 2b
2+ for the reasons mentioned above. It is important
to note that in 1b and 3 , the average pitch is smaller than
the carbon–carbon vdW distance (3.4 Å [ 34 ]), even for the
neutral case. This small through space contact distance is
partially responsible for the large actuation effect on the
pitch due to CT.
Table 4 lists the strain data for all three systems and
both directions of the CT based on Eq. ( 1 ) using a subscript to identify which of the three measures of elongation are used. Large strain is obtained for 1b
2−
, 3
2+ , and
2b
2−
. Most of the s(q) values are negative, indicating that
the effect is mostly quantum mechanical, and, when negative, in opposition to the Coulombic repulsion due to the
extra charge on the molecule. The numbers in bold identify
Fig. 9 HOMO of a 3 , b 3
2+ and c 3
2−
. Isosurface was generated with a contour value of 0.030 a.u. for the neutral and oxidized molecule ( a , b ),
and 0.025 a.u. for the reduced molecule ( c )
Table 2 Average radius, ¯
r (Å), and its standard deviation, σ ¯
r (Å), average deviation from helicity, ¯
D (Å), and its standard deviation, σ ¯
D , for 1b ,
2b , and 3 in different charge states
For the charged molecules, the percentages of change in the radius from the neutral ones are given in parentheses
Neutral
q = +2
q = −2
¯
r
σ ¯
r
¯
D
σ ¯
D
¯
r
σ ¯
r
¯
D
σ ¯
D
¯
r
σ ¯
r
¯
D
σ ¯
D
1b
1.347
0.040
0.033
0.023
1.345 (−0.2)
0.046
0.042
0.025
1.384 (2.7)
0.038
0.030
0.023
2b
2.903
0.040
0.054
0.027
2.784 (−4.1)
0.174
0.156
0.082
3.120 (7.5)
0.058
0.109
0.062
3
2.189
0.100
0.092
0.049
2.129 (−2.8)
0.100
0.082
0.062
2.011 (−8.1)
0.168
0.165
0.051
Table 3 Changes upon charge
transfer of various measures of
the molecular length: average
pitch, ¯
p (Å), z (Å), and
carbon–carbon intramolecular
distances, d CC (Å), for 1b , 2b
and 3
a Underlined numbers refer to cases where the helicity is partially lost due to CT (see text)
Neutral
q = +2
q = −2
¯
p
z
d CC
¯
p
z
d CC
¯
p
z
d CC
1b
3.299
4.352
3.378
3.258
4.254
3.334
3.130
4.046
3.182
2b
3.561
4.431
3.688
3.402
4.415
a
3.884
3.437
3.982
3.500
3
3.308
4.030
3.422
3.148
3.939
3.273
3.246
4.283
3.373
Table 4 Computed strain due
to CT, s ( q ) (in %), using the
three different metrics for 1b ,
2b , and 3
a Numbers in bold indicate large actuation (see text)
b Underlined numbers indicate partial loss of helicity upon CT (see text)
q = +2
q = −2
s ¯
p
s z
s d CC
s ¯
p
s z
s d CC
1b
−1.24
−2.25
−1.30
−5.11
−7.02
−5.81
2b
−4.45
b
−0.35
5.31
−3.49
−10.13
−5.09
3
−4.82
a
−2.26
−4.34
−1.86
6.28
−1.43
52
Reprinted from the journal