Finite hydrocarbon chains incorporating cumulenic structures: prediction by ab initio calculations of their equilibrium geometry and electric polarizability
V. P. Bodart, J. Delhalle, M. Dory, J. G. Fripiat, and J.-M. André
V. P. Bodart, J. Delhalle, M. Dory, J. G. Fripiat, and J.-M. André, "Finite hydrocarbon chains incorporating cumulenic structures: prediction by ab initio calculations of their equilibrium geometry and electric polarizability," J. Opt. Soc. Am. B 4, 1047-1054 (1987)
Using ab initio STO-3G calculations, we compute and analyze the equilibrium geometry and the electric polarizability of five series of polycumulenes H2—[—Ck—]x—H2, k = 2–6, and x = 1–3. The importance of structural relaxations depends on the parity of k, and the average polarizabilities, calculated with the finite-field method, are significantly influenced by these structural changes. The trends in polarizability are briefly discussed in terms of the optical gap and the dipole transition moment.
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Theoretical and Experimental (in Parentheses)a Carbon–Carbon Bond Distances (in Å) for the Five Series of Polycumulenes Shown in Fig. 2b
H—[—C1H=C2H—]x—H
H—[—C1H=C2=C3H—]x—H
H—[—C1H=C2=C3=C4H—]x—H
x
RC1=C2
RC—C
RC1=C2
RC2=C3
RC—C
x
RC1=C2
RC2=C3
RC3=C4
RC—C
1
1.3060
1.2879
1.2879
1
1.2958
1.2564
1.2958
(1.339)
(1.305)
(1.305)
(1.328)
(1.256)
(1.328)
2
1.3134
1.4892
1.2869
1.2951
1.4956
2
1.2984
1.2525
1.3072
1.4833
3
1.3220
1.4849
1.2939
1.2939
1.4967
3
1.3113
1.2473
1.3113
1.4794
H—[—C1H=C2=C3=C4=C5H—]x—H
H—[—C1H=C2=C3=C4=C5=C6H—]x—H
RC1=C2
RC2=C3
RC3=C4
RC5=C6
RC—C
x
RC1=C2
RC2=C3
RC3=C4
RC4=C5
RC5=C6
RC—C
1.2946
1.2634
1.2634
1.2946
1
1.2972
1.2591
1.2729
1.2591
1.2972
(1.315)
(1.276)
(1.276)
(1.315)
(1.329)
(1.259)
(1.299)
(1.259)
(1.329)
1.2939
1.2660
1.2594
1.3056
1.4844
2
1.2987
1.2570
1.2772
1.2528
1.3115
1.4767
1.3049
1.2619
1.2619
1.3049
1.4848
3
1.3146
1.2493
1.2832
1.2493
1.3146
1.4736
Ref. 6.
For the trimers, the data refer to the central unit.
Table 2
Alternation Degree Δa (in Å) in the Series of Polycumulenes Represented in Fig. 2
k
3
5
4
6
1
0.000
0.032
0.040
0.038
x
2
0.008
0.046
0.055
0.059
3
0.000
0.042
0.064
0.066
Defined as the largest difference between two adjacent bonds in a monomer unit.
Table 3
Average Polarizability (〈α〉) and Its Ratios with the Number of Electrons (ne), π Electrons (nπ), Carbon Atoms (nC), and the Vector Length of the Carbon Framework (LC) for the Five Series of Polycumulenesa
x
〈α〉 (au)
ne
〈α〉/ne
nπ
〈α〉/nπ
nC
〈α〉/nC
LC (Å)
〈α〉/LC
H—[—C1H=C2H—]x—H
1
10.74
16
0.67
2
5.37
2
5.37
1.31
8.20
1
1
1
1
1
1
1
1
1
2
25.15
30
0.84
4
6.29
4
6.29
3.67
6.85
2.34
1.88
1.25
2
1.17
2
1.17
2.80
0.84
3
43.93
44
1.00
6
7.32
6
7.32
6.13
7.17
4.09
2.75
1.49
3
1.36
3
1.36
4.68
1.36
H—[—C1H=C2=C3H—]x—H
1
18.17
22
0.83
4
4.54
3
6.06
2.58
7.04
1
1
1
1
1
1
1
1
1
2
46.08
42
1.10
8
5.76
6
7.68
6.12
7.53
2.54
1.91
1.32
2
1.27
2
1.27
2.37
1.07
3
79.88
62
1.29
12
6.66
9
8.88
9.58
8.34
4.40
2.82
1.55
3
1.47
3
1.47
3.71
1.18
H—[—C1H=C2=C3=C4H—]x—H
1
30.19
28
1.08
6
5.03
4
7.55
3.85
7.84
1
1
1
1
1
1
1
1
1
2
91.11
54
1.69
12
7.59
8
11.39
8.62
10.57
3.02
1.93
1.56
2
1.51
2
1.51
2.24
1.35
3
184.85
80
2.31
18
10.27
12
15.40
13.45
13.74
6.12
2.86
2.14
3
2.04
3
2.04
3.49
1.75
H—[—C1H=C2=C3=C4=C5H—]x—H
1
44.83
34
1.32
8
5.60
5
8.97
5.12
8.75
1
1
1
1
1
1
1
1
1
2
138.61
66
2.10
16
8.66
10
13.86
11.14
12.44
3.09
1.94
1.56
2
1.55
2
1.55
2.18
1.42
3
266.61
98
2.72
24
11.11
15
17.77
17.12
15.57
5.95
2.88
2.13
3
1.98
3
1.98
3.34
1.78
H—[—C1H=C2=C3=C4=C5=C6H—]x—H
1
65.57
40
1.64
10
6.56
6
10.93
6.39
10.26
1
1
1
1
1
1
1
1
2
229.47
78
2.94
20
11.47
12
19.12
13.67
16.79
3.50
1.95
1.79
2
1.75
2
1.75
2.14
1.64
3
508.57
116
43.8
30
16.95
18
28.25
20.98
24.24
7.76
2.90
2.67
3
2.58
3
2.58
3.28
2.36
Figures in italic type correspond to the values of 〈α〉 and its ratios divided by the first entry in each column.
Table 4
Highest Occupied Molecular Orbital Energy (∊HOMO), Lowest Unoccupied Molecular Orbital Energy (∊LUMO), and Optical Gap ΔE (= ∊HOMO − ∊LUMO) for the Five Series of Polycumulenes H2—[—Ck—]x—H2 (k = 2–6) with Respect to xa
x
k
2
3
4
5
6
1
∊HOMO
−9.129
−8.799
−7.408
−7.239
−6.495
∊LUMO
8.934
8.665
5.695
5.793
4.232
ΔE
18.063
17.464
13.103
13.032
10.727
2
∊HOMO
−7.516
−7.119
−5.999
−5.858
−5.293
∊LUMO
6.965
6.891
4.121
4.270
2.893
ΔE
14.481
14.010
10.120
10.128
8.126
3
∊HOMO
−6.680
−7.118
−5.348
−5.900
−4.761
∊LUMO
6.002
6.785
3.393
4.222
2.286
ΔE
12.682
13.903
8.741
10.122
7.047
All values are in electron volts.
Table 5
Average Polarizability 〈α〉, Optical Gap ΔE, and Inverse of ΔE (ΔE−1) for the Three C6 Compounds: Hexatriene H—[—CH=CH—]3—H (A), Diallene H—[—CH=C=CH—]2—H (B) and Hexapentaene CH2=C=C=C=C=CH2 (C)
《α〉 (au)
ΔE (eV)
ΔE−1
A
43.93
12.68
0.07886
1
1
B
46.08
14.01
0.07138
1.05
0.91
C
65.57
10.73
0.09320
1.49
1.18
Tables (5)
Table 1
Theoretical and Experimental (in Parentheses)a Carbon–Carbon Bond Distances (in Å) for the Five Series of Polycumulenes Shown in Fig. 2b
H—[—C1H=C2H—]x—H
H—[—C1H=C2=C3H—]x—H
H—[—C1H=C2=C3=C4H—]x—H
x
RC1=C2
RC—C
RC1=C2
RC2=C3
RC—C
x
RC1=C2
RC2=C3
RC3=C4
RC—C
1
1.3060
1.2879
1.2879
1
1.2958
1.2564
1.2958
(1.339)
(1.305)
(1.305)
(1.328)
(1.256)
(1.328)
2
1.3134
1.4892
1.2869
1.2951
1.4956
2
1.2984
1.2525
1.3072
1.4833
3
1.3220
1.4849
1.2939
1.2939
1.4967
3
1.3113
1.2473
1.3113
1.4794
H—[—C1H=C2=C3=C4=C5H—]x—H
H—[—C1H=C2=C3=C4=C5=C6H—]x—H
RC1=C2
RC2=C3
RC3=C4
RC5=C6
RC—C
x
RC1=C2
RC2=C3
RC3=C4
RC4=C5
RC5=C6
RC—C
1.2946
1.2634
1.2634
1.2946
1
1.2972
1.2591
1.2729
1.2591
1.2972
(1.315)
(1.276)
(1.276)
(1.315)
(1.329)
(1.259)
(1.299)
(1.259)
(1.329)
1.2939
1.2660
1.2594
1.3056
1.4844
2
1.2987
1.2570
1.2772
1.2528
1.3115
1.4767
1.3049
1.2619
1.2619
1.3049
1.4848
3
1.3146
1.2493
1.2832
1.2493
1.3146
1.4736
Ref. 6.
For the trimers, the data refer to the central unit.
Table 2
Alternation Degree Δa (in Å) in the Series of Polycumulenes Represented in Fig. 2
k
3
5
4
6
1
0.000
0.032
0.040
0.038
x
2
0.008
0.046
0.055
0.059
3
0.000
0.042
0.064
0.066
Defined as the largest difference between two adjacent bonds in a monomer unit.
Table 3
Average Polarizability (〈α〉) and Its Ratios with the Number of Electrons (ne), π Electrons (nπ), Carbon Atoms (nC), and the Vector Length of the Carbon Framework (LC) for the Five Series of Polycumulenesa
x
〈α〉 (au)
ne
〈α〉/ne
nπ
〈α〉/nπ
nC
〈α〉/nC
LC (Å)
〈α〉/LC
H—[—C1H=C2H—]x—H
1
10.74
16
0.67
2
5.37
2
5.37
1.31
8.20
1
1
1
1
1
1
1
1
1
2
25.15
30
0.84
4
6.29
4
6.29
3.67
6.85
2.34
1.88
1.25
2
1.17
2
1.17
2.80
0.84
3
43.93
44
1.00
6
7.32
6
7.32
6.13
7.17
4.09
2.75
1.49
3
1.36
3
1.36
4.68
1.36
H—[—C1H=C2=C3H—]x—H
1
18.17
22
0.83
4
4.54
3
6.06
2.58
7.04
1
1
1
1
1
1
1
1
1
2
46.08
42
1.10
8
5.76
6
7.68
6.12
7.53
2.54
1.91
1.32
2
1.27
2
1.27
2.37
1.07
3
79.88
62
1.29
12
6.66
9
8.88
9.58
8.34
4.40
2.82
1.55
3
1.47
3
1.47
3.71
1.18
H—[—C1H=C2=C3=C4H—]x—H
1
30.19
28
1.08
6
5.03
4
7.55
3.85
7.84
1
1
1
1
1
1
1
1
1
2
91.11
54
1.69
12
7.59
8
11.39
8.62
10.57
3.02
1.93
1.56
2
1.51
2
1.51
2.24
1.35
3
184.85
80
2.31
18
10.27
12
15.40
13.45
13.74
6.12
2.86
2.14
3
2.04
3
2.04
3.49
1.75
H—[—C1H=C2=C3=C4=C5H—]x—H
1
44.83
34
1.32
8
5.60
5
8.97
5.12
8.75
1
1
1
1
1
1
1
1
1
2
138.61
66
2.10
16
8.66
10
13.86
11.14
12.44
3.09
1.94
1.56
2
1.55
2
1.55
2.18
1.42
3
266.61
98
2.72
24
11.11
15
17.77
17.12
15.57
5.95
2.88
2.13
3
1.98
3
1.98
3.34
1.78
H—[—C1H=C2=C3=C4=C5=C6H—]x—H
1
65.57
40
1.64
10
6.56
6
10.93
6.39
10.26
1
1
1
1
1
1
1
1
2
229.47
78
2.94
20
11.47
12
19.12
13.67
16.79
3.50
1.95
1.79
2
1.75
2
1.75
2.14
1.64
3
508.57
116
43.8
30
16.95
18
28.25
20.98
24.24
7.76
2.90
2.67
3
2.58
3
2.58
3.28
2.36
Figures in italic type correspond to the values of 〈α〉 and its ratios divided by the first entry in each column.
Table 4
Highest Occupied Molecular Orbital Energy (∊HOMO), Lowest Unoccupied Molecular Orbital Energy (∊LUMO), and Optical Gap ΔE (= ∊HOMO − ∊LUMO) for the Five Series of Polycumulenes H2—[—Ck—]x—H2 (k = 2–6) with Respect to xa
x
k
2
3
4
5
6
1
∊HOMO
−9.129
−8.799
−7.408
−7.239
−6.495
∊LUMO
8.934
8.665
5.695
5.793
4.232
ΔE
18.063
17.464
13.103
13.032
10.727
2
∊HOMO
−7.516
−7.119
−5.999
−5.858
−5.293
∊LUMO
6.965
6.891
4.121
4.270
2.893
ΔE
14.481
14.010
10.120
10.128
8.126
3
∊HOMO
−6.680
−7.118
−5.348
−5.900
−4.761
∊LUMO
6.002
6.785
3.393
4.222
2.286
ΔE
12.682
13.903
8.741
10.122
7.047
All values are in electron volts.
Table 5
Average Polarizability 〈α〉, Optical Gap ΔE, and Inverse of ΔE (ΔE−1) for the Three C6 Compounds: Hexatriene H—[—CH=CH—]3—H (A), Diallene H—[—CH=C=CH—]2—H (B) and Hexapentaene CH2=C=C=C=C=CH2 (C)
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