3.1. Comparison of polarizabilities of the cumulenes and polyynes chains connecting the same substituents.
Energies of the eight investigated systems (four cumulenes and four polyynes with different R1 and R2 substituents) were plotted as functions of the intensities of the applied field. These functions were represented as practically perfect parabolas, with coefficients of determination in the 0.999–1.000 limits. Two examples of this plot are shown in Fig. 2a for (10)cumulene and in Fig. 3a for (10)polyyne bearing the same substituents, NO2 and CN. The functions of energy (En) against the electric field strength (F) may be expressed by following quadratic Eq. (2), which is correct in the case where the fields that are co-linear with the molecular axis were applied and higher terms in F may be neglected [22]. Figures 2a and 3a display energies of cumulenes and polyynes as functions of the strength of the electric field:
E n = Eo + µo F – ½ α F2 (2)
In Eq. (2), µo is a dipole moment of a neutral molecule without the applied field and α denotes molecular polarizability, a value which is a measure of the energy increment due to the dislocation of electron density within the molecule under the action of the electric field.
Based on Eq. (2), one may determine the dipole moment of the molecule without the field (µo) and in the field (µ(F)) using the following equations:
dEn/dF = µo – αF (3)
µ(F) = µo + αF (4)
In a case where Eq. (3) is valid, the plot of dEn/dF against F should be a straight line with a slope equal to α and an intercept equal to the field-free dipole moment µo. Such plots are shown in Figs. 2b and 3b, for the cumulene and polyyne molecules with NO2 and CN substituents. The same calculations were performed for another cumulenes and polyynes with different substituents R1 and R2 such as NO2, NH2, OH, CN. It should be emphasised that the coefficients of determination of dEn/dF against F for all eight molecules are 1 up to four decimals. Dipole moments of the molecules in the electric field directed along the x-axis, as shown in Fig. 1, were determined according to Eq. (4). As can be seen, dipole moments in the electric field directed along the x-axis of the molecules are larger for substituted cumulenes (Tables 1) than for polyynes (Table 2).
Table 1
Dipole moments of the (10)cumulenes with two substituents, R1 and R2
F (atomic units)
|
R1 = NO2
R2 = CN
|
R1 = NO2
R2 = NH2
|
R1 = NO2
R2 = OH
|
R1 = OH
R2 = NH2
|
-0.015
|
17.5
|
23.9
|
21.1
|
17.0
|
0
|
1.01
|
6.40
|
4.92
|
1.79
|
0.015
|
-15.6
|
-11.1
|
-11.3
|
-13.4
|
Table 2
Dipole moments of the (10)polyynes with two substituents, R1 and R2
F (atomic units)
|
R1 = NO2
R2 = CN
|
R1 = NO2
R2 = NH2
|
R1 = NO2
R2 = OH
|
R1 = OH
R2 = NH2
|
-0.015
|
11.81
|
20.66
|
18.24
|
14.3
|
0
|
0.34
|
6.11
|
4.72
|
1.24
|
0.015
|
-11.13
|
-8.44
|
-8.81
|
-16.74
|
Electric dipole moment atomic unit is e×ao where e – is electron charge and ao – Bohr radius. The electric dipole moment atomic unit equals 8.478×10− 30 C×m or 2.542 D (debye).
Polarizabilities of the substituted cumulenes and polyynes, obtained from the derivative of the electronic energy (En) with respect to F, are given in Table 3. The data show that polarizability values of cumulenes are 7–25% higher than those of polyynes.
Table 3
Polarizability of substituted cumulenes and polyynes in atomic units
|
R1 = NO2
R2 = CN
|
R1 = NO2
R2 = NH2
|
R1 = NO2
R2 = OH
|
R1 = OH
R2 = NH2
|
(10)cumulene-R1R2
|
-1102
|
-1167
|
-1122
|
-1011
|
(10)polyyne-R1R2
|
-975
|
-970
|
-901
|
-949
|
Polarizability atomic unit is e2 ao2 / Eh, where e – is electron charge, ao – Bohr radius, and Eh – hartree. The polarizability atomic unit equals 1.649 × 10− 41 C2m2/J.
3.2.Comparison of two types of the sp-hybridised carbon-carbon double bonds.
Figure 1 shows that the carbon atoms are linked by double bonds in cumulenes, whereas in polyynes there are alternating single and triple bonds. To verify this structure, we compared bond length values for two series of molecules with the same functional groups without the applied electric field and with the field of intensities + 0.01 and − 0.01 a.u. The data is shown in Table 4.
Table 4
Comparison of mean bond lengths of the C-C bonds in 10-cumulene-NO2CN and 10-polyyne-NO2CN and their standard deviations
|
(10)cumulene-R1R2
|
(10)polyyne-R1R2
|
Field, a.u.
|
0
|
0.01
|
-0.01
|
0
|
0.01
|
-0.01
|
Mean value of bond length
|
1.29
|
1.29
|
1.30
|
1.28
|
1.28
|
1.28
|
SD of bond length
|
0.024
|
0.024
|
0.029
|
0.050
|
0.050
|
0.045
|
Data in Table 4 is evidence that the mean bond lengths of the link between two functional groups are nearly the same for the both types of the linker and do not change when the electric field (of the strength of + 0.01 and − 0.01 a.u.) is applied. These values (1.29 and 1.28 Å) can be compared with values of lengths of the single, double, and triple bonds calculated for the CH3-CH3, CH2 = CH2, and CH ≡ CH at the same calculation level (B3LYP/6–31 + G*). The values were 1.53 (1.532), 1.34 (1.335), and 1.21 (1.2075) Å, respectively. The comparison showed that the bonds in both types of molecules can be represented as double bonds with a small addition of triple bond. Cumulenes do not exhibit a substantial bond length alternation as compared with that of polyynes. Standard deviations of bond length for polyyne-NO2CN (0.05Å) are twice as large as that for cumulene-NO2CN (0.02–0.03 Å).