# Theoretical Studies on Structures and Electronic State of Alkyl-Substituted Ethyl Cations

Authors: Toshiaki Shimasaki; Ken Nakayasu; Mitsuhiro Shibata; Tatsuaki Yamaguchi
Journal: Journal of Chemistry
 (2013)

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## Abstract

This paper describes the gas phase structural details for a series of alkyl-substituted ethyl cations1–11 by mainly natural bond orbital (NBO) analysis. For the simplest ethyl cation, two kinds of optimized structures (nonclassical 1 and classical 2) were found to have minima on the potential surface, and the nonclassical 1 was more stable than the corresponding classical 2 by 1.19 kcal/moL at Møller-Plesset (MP) level calculations in the gas phase. The calculated values of Wiberg’s bond index revealed that the central ethylic C1–C2 bonds of the nonclassical cations 1, 6, 7, and 10 have 1.10–1.46 bond multiplicities. The compliance constant (Cii) values indicated that the cation center C1 of nonclassical 2-butyl cation 6 (0.209 Å/mdyn) acquires stronger hyperconjugative stabilization from C2–R3 (C–H) bond than that of the nonclassical ethyl cation 1 (0.388 Å/mdyn). Moreover, the gas phase stabilities for the alkyl-substituted ethyl cations 1–11 were also estimated by the hydride affinities based on isodesmic equations.


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## Body

## 1. Introduction

The carbocations are not only important chemical species as intermediates in various organic reactions but also have been gathering much attention on their structure and electronic state [1]. Since the 1960s, Von Schleyer and coworkers have examined alkyl cations (carbonium ions), in part alkyl dications, by magic acid techniques [2]. Yannoni et al. have also studied the alkyl cation and especially revealed the structural details of tert-butyl cation (1,1-dimethylethyl cation) by NMR methods in 1989 [3].Fǎrcaşiu et al. have theoretically investigated the conformations of carbocations and the reactions of their precursors with borane at high-levelab initiocalculations [4, 5]. In 2007, Ricks and coworkers revealed that extraordinary stability of C4H9+ species is brought about the charge distribution and hyperconjugation by means of IR spectral analysis [6]. Feng et al. reported the structural properties and energetics at high-level MO calculations on tert-butyl cation [7]. Structural details of ethyl cation, which is the most basic and popular alkyl cation, were reported by Andrei and coworkers in 2008 [8]. They found that the ethyl cation takes the nonclassical structure from both the experimental results of Ar-tagging IRPD spectra and ab initio calculations. Furthermore, Reed and coworkers have recently isolated alkyl and any other cations as salts with carborane anions and characterized by X-ray crystallography [9].However, the detailed information on the structure and electronic state of carbocation such as relative isodesmic energy, bond multiplicity, and the relative energy difference has not been fully elucidated in the past theoretical and experimental studies. Especially, systematical consideration on the detailed electronic state of alkyl cation is still needed to understand the mechanisms for several basic organic reactions and structural aspects of fundamental carbon species. The present study describes the results of theoretical investigations on the detailed structure and electronic state of various carbocations (ethyl cation and its alkylsubstitutes) by using Møller-Plesset method.


## 2. Computational Methods

Ab initio theoretical calculations at second order Møller-Plesset perturbation theory (MP2) level were performed by using Gaussian 03 program for all the alkyl-substituted ethyl cations 1–11 (Table 1). First geometry optimization was performed at the B3LYP/6-31G(d) level and the optimized structure was used as an input structure for the calculation at the MP2/6-311+G(d,p) level of theory. The SCF energy, Wiberg’s bond index, and compliance constant calculation for each optimized ethyl cation were verified using the same basis set, and their results were summarized in Tables 3 and 4. The isodesmic equation and the relative reaction energy (ΔEiso) were summarized in Table 5.Table 1
Alkyl-substituted ethyl cations1–11.
Cation
R1
R2
R3
R4
R5
1
a
H
H
H
H
H
2
H
H
H
H
H
3
Me
H
H
H
H
4
Me
Me
H
H
H
5
H
H
Me
H
Me
6
a
H
Me
H
Me
H
7
a
Me
H
H
Me
H
8
Me
H
H
Me
H
9
Me
Me
Me
Me
H
10
a
Me
Me
H
Me
Me
11
Me
Me
Me
Me
Me
aNonclassical cation.


## 3. Results and Discussions

### 3.1. Calculated Molecular Structures and Molecular Orbitals of Alkyl-Substituted Ethyl Cations1–11

The optimized structures of cations1–11 are shown in Figure 1, and their selected bond length and angle parameters are summarized in Table 2.Table 2
Calculated structural parameters for multisubstituted ethyl cations1–11. Selected bond lengths are in angstrom (Å) and selected torsion angles in degrees (°).
Cation
Bond lengths (Å)
Torsion angles (°)
C1–C2
R1–C1
C2–R3
C2–R5
C1–C2–R3
R1–C1–C2
R1–C1–R2
C1–C2–R5
1
1.39
1.09
1.31
1.09
58.1
120.5
119.1
120.5
2
1.42
1.09
1.11
1.11
106.3
120.1
116.8
106.3
3
1.44
1.44
1.12
1.09
97.9
125.3
117.4
114.1
4
1.46
1.46
1.11
1.09
102.7
119.5
120.7
113.8
5
1.41
1.09
1.58
1.58
109.4
118.1
120.0
109.4
6
1.40
1.09
1.31
1.49
58.0
116.5
118.5
124.8
7
1.40
1.49
1.32
1.09
57.4
126.9
117.4
115.5
8
1.41
1.48
1.66
1.08
76.2
124.1
118.2
118.8
9
1.45
1.47
1.60
1.09
96.8
121.2
118.7
110.5
10
1.42
1.50
1.32
1.50
57.2
122.4
115.1
122.5
11
1.45
1.48
1.62
1.53
92.6
121.2
116.5
116.8Table 3
Bond length, Wiberg’s bond index (WBI), and compliance constants (Cii) for multisubstituted ethyl cations 1, 3, 6, and 9 and corresponding alkanes.
Bond
Bond length (Å)
WBI
C
i
i (Å/mdyn)
C1–C2
(1)
1.39
1.46
0.388
(1–H)
1.53
1.04
0.510
(3)
1.44
1.17
0.377
(3–H)
1.53
1.02
0.617
(6)
1.40
1.38
0.209
(6–H)
1.53
1.02
0.578
(9)
1.45
1.10
0.436
(9–H)
1.54
0.99
0.644
C2–R3
(1)
1.31
0.45
2.236
(1–H)
1.09
0.96
0.755
(3)
1.12
0.83
2.718
(3–H)
1.10
0.95
0.702
(6)
1.31
0.45
3.609
(6–H)
1.10
0.95
0.702
(9)
1.60
0.90
1.126
(9–H)
1.53
1.01
0.612Table 4
Bond orders, bond indexes, hybridizations, and occupancies parameters for multisubstituted ethyl cations1–11.
Cation
Wiberg’s bond indexes (NAO bond orders)
Hybridizations
Occupancies
C1–C2
R1–C1
C2–R3
C2–R5
C1
C2
C1–C2
C2–R5
1
1.46 (1.20)
0.92 (0.78)
0.45 (0.46)
0.92 (0.78)
sp2.69
sp2.69
1.99493
0.69157
2
1.23 (1.06)
0.95 (0.77)
0.84 (0.75)
0.84 (0.75)
sp1.00
sp3.16
1.99705
1.91297
3
1.17 (1.02)
1.17 (1.02)
0.83 (0.75)
0.91 (0.77)
sp2.32
sp3.17
1.99051
1.88072
4
1.12 (0.99)
1.12 (0.99)
0.86 (0.76)
0.92 (0.78)
sp2.37
sp3.21
1.98454
1.91306
5
1.16 (1.01)
0.95 (0.78)
0.93 (0.79)
0.93 (0.79)
sp2.12
sp3.27
1.99111
1.91993
6
1.38 (1.16)
0.91 (0.77)
0.45 (0.47)
1.05 (0.91)
sp2.47
sp2.47
1.98604
0.69074
7
1.38 (1.16)
1.05 (0.91)
0.44 (0.46)
0.91 (0.77)
sp2.90
sp2.90
1.98649
0.70556
8
1.22 (1.07)
1.07 (0.93)
0.77 (0.63)
1.05 (0.92)
sp2.67
sp2.13
1.98615
1.73687
9
1.10 (0.97)
1.10 (0.97)
0.90 (0.75)
0.90 (0.76)
sp2.40
sp3.39
1.97682
1.88461
10
1.32 (1.13)
1.03 (0.90)
0.43 (0.46)
1.04 (0.90)
sp3.05
sp3.05
1.97706
0.71843
11
1.10 (0.96)
1.09 (0.96)
0.87 (0.72)
0.99 (0.83)
sp3.05
sp3.06
1.97307
1.85722Table 5
Total energiesEtotal (Hartree) of alkyl-substituted ethyl cations 1–11and reaction energies (kcal/mol) of corresponding isodesmic reactionsa.
Cation
R1
R2
R3
R4
R5
E
total
Alkane
E
total
Δ
E
iso
1
H
H
H
H
H
−78.2804
1–H
−79.1823
31.75
2
H
H
H
H
H
−78.2785
1–H
−79.1823
30.56
3
Me
H
H
H
H
−117.339
3–H
−118.200
57.11
4
Me
Me
H
H
H
−156.380
4–H
−157.220
70.59
5
H
H
Me
H
Me
−156.323
5–H
−157.220
34.83
6
H
Me
H
Me
H
−156.353
6–H
−157.219
54.28
7
Me
H
H
Me
H
−156.351
6–H
−157.219
53.02
8
Me
H
H
Me
H
−156.347
6–H
−157.219
50.51
9
Me
Me
Me
Me
H
−234.418
9–H
−235.251
74.99
10
Me
Me
H
Me
Me
−234.407
9–H
−235.251
68.08
11
Me
Me
Me
Me
Me
−273.433
11–H
−274.265
75.61
aThe structural optimizations and total energies Etotal (Hartree) of methane and methyl cation were separately carried out at the same basis sets, respectively.Figure 1
Optimized structures of alkyl-substituted ethyl cations at MP2/6-311+G(d,p) level of theory.Our calculations for the simplest ethyl cation by MP2 level of theory demonstrated that the cation has two energies-minimized structures (1 and 2) in agreement with their previous studies [7, 8]. The cation center of optimized 2 is almost localized on one carbon atom, whereas the cation centers of 1 are splitting to two carbons (Figure 1). Therefore, the calculated structure of 1 is categorized into a nonclassical cation, while that of 2 is considered as a classical ethyl cation. The central C1–C2 bond length (1.39 Å) of nonclassical 1 is similar to that of a double bond (1.34 Å). On the other hand, the C1–C2 bond length of classical 2 is to be 1.42 Å. The C2–R3 bond lengths of 1 and 2 are envisaged to be 1.31 and 1.11 Å, and the bond angles of each C1–C2–R3 are 58.1 and 106.3°, respectively. The single point energies of each structure involving ZPE showed that nonclassical 1 is more stable than classical 2 by 1.19 kcal/moL.The gas phase structural feature of isopropyl cation3 was also investigated by the same level of theory as 1 and 2. The C1–C2 bond length of the calculated 3 is 1.44 Å, which became shorter than that of the corresponding alkane 3-H (1.53 Å). In case of 3, the cation center is almost localized on the C1 carbon. Actually, the C1–C2–R3 bond angle of the calculated 3 is 97.9°, which is much smaller than an ideal sp3 bond angle (109.3°) due to the hyperconjugative effect from C–H bond to the cation center. Whereas LUMO of the 2-isopropyl cation 3 is mainly formed in C1 cation center, HOMO is bound on the C–H bonds of two methyl groups (Figure 2).Figure 2
HOMO orbitals of ethyl cations1–11.With respect to optimized structures oftert-butyl cation 4 and isobutyl cation 5, the C1–C2 bond lengths are 1.46 and 1.41 Å at MP2 level in the gas phase, respectively. The calculation predicted that 4 has a Cs symmetric structure where the four carbons are coplanar (R1–C1–C2 angle is 119.5°), and its cation center (C1) takes almost sp2 hybridization. In fact, one C1–C2–H (R3) bond angle of the optimized 4 becomes smaller (102.7°) than that of 4-H (110.2°). The isobutyl cation 5 possesses a classicalform, in which the cationic charge is localized on the C1 (Figure 3). Two methyl groups connected to the C2 are located on up and down sides of the cation plane (R1R2–C1–C2), because efficient hyperconjugation from the two C–C bonds to the cation center contributes for this conformation. The tert-butyl cation 4 is much more stable (by >35 kcal/moL) than 5 in gas phase, due to a strong hyperconjugative stabilization effect from three methyl groups as discussed later (Section 3.3).Figure 3
LUMO orbitals of ethyl cations1–11.A similar trend in1 and 2 was also observed in cases of 2-butyl cations 6, 7, and 8. Both 6 and 7 are considered as nonclassical cations because the optimized structure has almost planar framework whose cation center is splitting between C1 and C2. In contrast, another “kinked” form (classical cation) 8 was inferred by MP2 optimizations. LUMOs of the nonclassical cations 6 and 7 mainly exist on C1 and C2 carbons, while they spread to the hydrogen of the “kinked” methyl group for the classical 8. Both the central C1–C2 bond lengths of 6 (nonclassical: trans) and 7 (nonclassical: cis) are estimated to be 1.40, whose value is almost the same as that of the classical 8 (1.41 Å). The C2–R3 (C–C) bond length of the classical 8 is expected to be 1.66 Å, which is elongated from a typical C(sp3)–C(sp3) bond length (1.54 Å) due to the hyperconjugative effect. The C1–C2–R3 bond angles of the nonclassical 6 and 7 are 57.8 and 58.1°, respectively, whereas the classical 8 has a larger angle (76.2°). The nonclassical 6 and 7 are estimated to be more stable than the classical 8 by 3.77 and 2.51 kcal/moL, respectively, suggesting that the hyperconjugations of the methyl groups of 6 and 7 are more effective for the stabilization than that of “kinked” methyl group of 8. Moreover, the trans-nonclassical 6 is a little more stable than the cis-nonclassical 7 by 1.26 kcal/moL.The 2,3-dimethylbutyl cations9 (classical: 1.44 Å for C1–C2 bond length, 1.60 Å for C2–R3 bond length, and 96.8° for C1–C2–R3 torsion angle) and 10 (nonclassical: 1.42 Å for C1–C2 bond length, 1.32 Å for C2–R3 bond length, and 57.2° for C1–C2–R3 torsion angle) possess the similar structural features with the ethyl and 2-butyl cations. However, our calculations revealed that classical 9 is more stable than nonclassical 10 by 8.79 kcal/moL in the gas phase. Although the reason is not clear, there is a possibility that four methyl groups of nonclassical 10 are more sterically hindered. LUMO orbital of classical 9 exists at the cation center (C1) and methyl groups connecting to the C1 and C2, whereas the LUMO of nonclassical 10 is almost localized on C1 and C2 carbons in a similar manner to ethyl and 2-butyl cations (Figure 3).As a result of MP2 calculation, trimethylbutyl cation11 is clearly categorized into a classical cation, because the cation center is localized on the C1 (Figure 3). The central ethylic C1–C2 bond length (1.45 Å) of 11 is significantly shorter than that of corresponding alkane 11-H (1.55 Å) and slightly longer than those of classical ethyl and 2-butyl cations 2 and 4. The C2–R3 bond length of 11 (1.62 Å) is longer than typical sp3–sp3 single bond length (1.54 Å). The C1–C2–R3 bond angle of 11 (92.6°) is smaller than that of corresponding alkane 1-H (111.9°).


### 3.2. The Comparison of Structural and Bond Strength Parameters between Selected Cations and Their Alkanes

Structural and bond strength parameters at MP2 level calculations between cations1–11 and corresponding alkanes 1-H–11-H were compared in order to estimate the hyperconjugative effect (Figure 4 and Table 3).Figure 4
Structural comparisons between alkyl-substituted ethyl cations1, 3, 5, and 9 and their alkanes calculated by MP2/6-311+G(d,p) level of theory.The C–C bond of ethane1-H is 1.53 Å in length, which should be an almost ideal sp3 C–C bond, whereas that of 1 generated by hydride elimination from 1-H becomes shorter by 0.15 Å (Figure 4). As shown in Table 3, the compliance constant Cii of C1–C2 bond is changed from 0.510 (1-H) to 0.388 Å/mdyn (1), indicating that the central C1–C2 bond of simple ethyl cation 1 is stronger than that of 1-H. Such a change is attributed to the hyperconjugative effect caused by the electron transfer from C–H covalent bond to the cation center. In contrast, the C2–R3 (C–H) bond strength of cation 1 becomes much weaker than that of 1-H (from 0.755 to 2.236 Å/mdyn), indicating that the bonding electrons of C2–R3 are moved to the unfilled p orbital (cation center). In fact, the value of Wiberg’s bond index (WBI) is changed from 1.04 (1) to 1.46 (1-H) (Figure 3), meaning that the C1–C2 bond becomes stronger by the hydride elimination. Details of the WBIs with bond orders of the 1–11 are discussed in the following section.The C1–C2 bond length of 2-propylcation 3 (1.44 Å) became shorter than that of 3-H (1.53 Å) by the hydride elimination (Figure 4).  A similar tendency is observed for the bond strengths and WBI values of 3 and 3-H. Thus, the compliance constant Cii of C1–C2 bond changed from 0.617 (1.04) to 0.377 Å/mdyn (1.17). On the other hand, the C2–R3 bond strength of 3 became weaker (from 0.702 to 2.718 Å/mdyn) from that of 3-H. Moreover, the bond angle of C1–C2–R3 changed from 110.7 (3-H) to 97.94° (3) according to our calculations. These structural changes can also be explained by the concept of hyperconjugative effect.Similarly, gas phase calculations predict that the ethylic C1–C2 bond of trans-2-butyl cation 5 is shorter and stronger (1.40 Å and 0.578 Å/mdyn) than that of corresponding alkane 5-H(1.53 Å and 0.209 Å/mdyn), while the C2–R3 bond strength of 5 (0.702 Å/mdyn) becomes much weaker than that of 5-H (3.609 Å/mdyn) by hydride elimination.On the other hand, MP2 level calculation in gas phase demonstrated that both 2,3-dimethylbutyl cations8 and its hydride adduct 8-H have the same central C1–C2 bond length (1.54 Å). Also, accompanied by the structural change from 8 to 8-H, a decrease of Cii (from 0.644 to 0.436 Å/mdyn) and a slight increase of bond order (from 0.99 to 1.10) of the C1–C2 bond were estimated by the calculation. Moreover, the C2–R3 bond length and strength of 8 (1.53 Å and 0.612 Å/mdyn) were shorter and stronger than those of 8-H (1.60 Å and 1.126 Å/mdyn), respectively. Although the hyperconjugative effect from C2–R3 bond to cation center of 8 is expected from the MP2 calculation, steric repulsion between the methyl groups would be inhibiting the shrinking of C1–C2 bond. The pronounced deformation of C1–C2–R3 bond angle of the cation 8 (96.77°) relative to alkane 8-H (111.3°) supports the occurrence of the steric repulsion.


### 3.3. NBO Analysis of the Alkyl-Substituted Ethyl Cations

Table3 summarizes Wiberg’s bond indexes (WBIs) and bond orders of ethyl cations 1–11 evaluated by the NBO analysis at MP2/6-311++G(3df,3pd) level of theory. The character of chemical bond can be intuitively understood by the use of WBI bond indexes. For example, a bond index of 1 means that the bond takes single bond.The WBIs of nonclassical1 and classical 2 are evaluated to be 1.46 and 1.23, respectively, meaning that the more stable nonclassical ethyl cation 1 is essentially the same as a protonated ethylene. Actually, the NBO calculation by MP2/6-311++G(3df,3dp) level suggests that both the C1 and C2 of 1 possess sp1.99 hybridization which is nearly equal to sp2. On the other hand, the C1 and C2 of 2 are sp1.55 and sp2.45 carbons, respectively, indicating that 2 is a classical cation.A similar trend was also observed in 2-butyl cations6 (trans-nonclassical), 7(cis-nonclassical), and 8 (classical) in the gas phase. The WBIs for 6, 7, and 8 are 1.38, 1.38, and 1.22, respectively. The hybridizations of C1 and C2 of both 6 and 7 are sp2.09 and sp2.08, while those of 8 are sp2.00 and sp2.26, respectively.According to our calculations, the hybridizations of C1 and C2 for classical 2,3-dimethylbutyl cation 9 are provided as sp1.86 and sp2.51, respectively. On the other hand, both of them were sp1.86 for nonclassical 2,3-dimethylbutyl cation 10 by NBO calculation in the gas phase. The bond order and bond index of central ethylic C1–C2 bond for 10 are 0.96 and 1.10, respectively.Our calculations revealed that the hybridization values of central C1 and C2 (sp1.77 and sp2.66) of the trimethylbutyl cation 11 are also close to those of 9 (Table 4).


### 3.4. Total Energies and Hydrogen Affinities of Each Cation

The zero point corrected total energies [Etotal’s] of alkyl-substituted ethyl cations and their isodesmic reaction energy [ΔEiso] by MP2/6-311++G(3df,3dp) level are summarized in Table 5.EachΔEiso is derived from the difference of total energies between alkane plus methyl cation and alkyl cation plus methane according to the equation shown in Table 5. The difference of ΔEiso (ΔΔEiso) between nonclassical ethyl cation 1 and classical cation 2 is 1.19 kcal/moL. The gas phase MP2 calculation revealed that the hydride affinity of the t-butyl cation 3 is considerably lower than that of simple ethyl cation by 38–40 kcal/moL. This fact means that the tert-butyl cation 3 is much more stable than ethyl, propyl, and other butyl cations.2-Methylpropan-1-yl cation5 with a similar stability to ethyl cation is less stable than tert-butyl cation 4 (35.8 kcal/moL). In case of 2-butyl cations, both the nonclassical cations 5 and 6 possess almost the same isodesmic reaction energy (ΔEiso=54.28 kcal/moL for 5 and 53.02 kcal/moL for 6) in the gas phase. On the other hand, MP2 calculations revealed that classical 7 is less stable than nonclassical 5 and 6 by 3 kcal/moL.Although nonclassical ethyl and butyl cations are more stable than the corresponding classical cations, classical and nonclassical 2,3-dimethylbutyl cations9 and 10 showed an opposite trend (ΔΔEiso=6.91 kcal/moL). That classical 9 is more stable than nonclassical 10 is attributed to a higher steric repulsion between four methyl groups in 10 with a shorter C1–C2 bond length than the corresponding alkane.Tetramethylbutyl cation11 had the highest gas phase isodesmic reaction energy (ΔEiso=75.61 kcal/moL), indicating the most stable cation among a series of alkyl-substituted ethyl cations in this report.


## 3.1. Calculated Molecular Structures and Molecular Orbitals of Alkyl-Substituted Ethyl Cations1–11

The optimized structures of cations1–11 are shown in Figure 1, and their selected bond length and angle parameters are summarized in Table 2.Table 2
Calculated structural parameters for multisubstituted ethyl cations1–11. Selected bond lengths are in angstrom (Å) and selected torsion angles in degrees (°).
Cation
Bond lengths (Å)
Torsion angles (°)
C1–C2
R1–C1
C2–R3
C2–R5
C1–C2–R3
R1–C1–C2
R1–C1–R2
C1–C2–R5
1
1.39
1.09
1.31
1.09
58.1
120.5
119.1
120.5
2
1.42
1.09
1.11
1.11
106.3
120.1
116.8
106.3
3
1.44
1.44
1.12
1.09
97.9
125.3
117.4
114.1
4
1.46
1.46
1.11
1.09
102.7
119.5
120.7
113.8
5
1.41
1.09
1.58
1.58
109.4
118.1
120.0
109.4
6
1.40
1.09
1.31
1.49
58.0
116.5
118.5
124.8
7
1.40
1.49
1.32
1.09
57.4
126.9
117.4
115.5
8
1.41
1.48
1.66
1.08
76.2
124.1
118.2
118.8
9
1.45
1.47
1.60
1.09
96.8
121.2
118.7
110.5
10
1.42
1.50
1.32
1.50
57.2
122.4
115.1
122.5
11
1.45
1.48
1.62
1.53
92.6
121.2
116.5
116.8Table 3
Bond length, Wiberg’s bond index (WBI), and compliance constants (Cii) for multisubstituted ethyl cations 1, 3, 6, and 9 and corresponding alkanes.
Bond
Bond length (Å)
WBI
C
i
i (Å/mdyn)
C1–C2
(1)
1.39
1.46
0.388
(1–H)
1.53
1.04
0.510
(3)
1.44
1.17
0.377
(3–H)
1.53
1.02
0.617
(6)
1.40
1.38
0.209
(6–H)
1.53
1.02
0.578
(9)
1.45
1.10
0.436
(9–H)
1.54
0.99
0.644
C2–R3
(1)
1.31
0.45
2.236
(1–H)
1.09
0.96
0.755
(3)
1.12
0.83
2.718
(3–H)
1.10
0.95
0.702
(6)
1.31
0.45
3.609
(6–H)
1.10
0.95
0.702
(9)
1.60
0.90
1.126
(9–H)
1.53
1.01
0.612Table 4
Bond orders, bond indexes, hybridizations, and occupancies parameters for multisubstituted ethyl cations1–11.
Cation
Wiberg’s bond indexes (NAO bond orders)
Hybridizations
Occupancies
C1–C2
R1–C1
C2–R3
C2–R5
C1
C2
C1–C2
C2–R5
1
1.46 (1.20)
0.92 (0.78)
0.45 (0.46)
0.92 (0.78)
sp2.69
sp2.69
1.99493
0.69157
2
1.23 (1.06)
0.95 (0.77)
0.84 (0.75)
0.84 (0.75)
sp1.00
sp3.16
1.99705
1.91297
3
1.17 (1.02)
1.17 (1.02)
0.83 (0.75)
0.91 (0.77)
sp2.32
sp3.17
1.99051
1.88072
4
1.12 (0.99)
1.12 (0.99)
0.86 (0.76)
0.92 (0.78)
sp2.37
sp3.21
1.98454
1.91306
5
1.16 (1.01)
0.95 (0.78)
0.93 (0.79)
0.93 (0.79)
sp2.12
sp3.27
1.99111
1.91993
6
1.38 (1.16)
0.91 (0.77)
0.45 (0.47)
1.05 (0.91)
sp2.47
sp2.47
1.98604
0.69074
7
1.38 (1.16)
1.05 (0.91)
0.44 (0.46)
0.91 (0.77)
sp2.90
sp2.90
1.98649
0.70556
8
1.22 (1.07)
1.07 (0.93)
0.77 (0.63)
1.05 (0.92)
sp2.67
sp2.13
1.98615
1.73687
9
1.10 (0.97)
1.10 (0.97)
0.90 (0.75)
0.90 (0.76)
sp2.40
sp3.39
1.97682
1.88461
10
1.32 (1.13)
1.03 (0.90)
0.43 (0.46)
1.04 (0.90)
sp3.05
sp3.05
1.97706
0.71843
11
1.10 (0.96)
1.09 (0.96)
0.87 (0.72)
0.99 (0.83)
sp3.05
sp3.06
1.97307
1.85722Table 5
Total energiesEtotal (Hartree) of alkyl-substituted ethyl cations 1–11and reaction energies (kcal/mol) of corresponding isodesmic reactionsa.
Cation
R1
R2
R3
R4
R5
E
total
Alkane
E
total
Δ
E
iso
1
H
H
H
H
H
−78.2804
1–H
−79.1823
31.75
2
H
H
H
H
H
−78.2785
1–H
−79.1823
30.56
3
Me
H
H
H
H
−117.339
3–H
−118.200
57.11
4
Me
Me
H
H
H
−156.380
4–H
−157.220
70.59
5
H
H
Me
H
Me
−156.323
5–H
−157.220
34.83
6
H
Me
H
Me
H
−156.353
6–H
−157.219
54.28
7
Me
H
H
Me
H
−156.351
6–H
−157.219
53.02
8
Me
H
H
Me
H
−156.347
6–H
−157.219
50.51
9
Me
Me
Me
Me
H
−234.418
9–H
−235.251
74.99
10
Me
Me
H
Me
Me
−234.407
9–H
−235.251
68.08
11
Me
Me
Me
Me
Me
−273.433
11–H
−274.265
75.61
aThe structural optimizations and total energies Etotal (Hartree) of methane and methyl cation were separately carried out at the same basis sets, respectively.Figure 1
Optimized structures of alkyl-substituted ethyl cations at MP2/6-311+G(d,p) level of theory.Our calculations for the simplest ethyl cation by MP2 level of theory demonstrated that the cation has two energies-minimized structures (1 and 2) in agreement with their previous studies [7, 8]. The cation center of optimized 2 is almost localized on one carbon atom, whereas the cation centers of 1 are splitting to two carbons (Figure 1). Therefore, the calculated structure of 1 is categorized into a nonclassical cation, while that of 2 is considered as a classical ethyl cation. The central C1–C2 bond length (1.39 Å) of nonclassical 1 is similar to that of a double bond (1.34 Å). On the other hand, the C1–C2 bond length of classical 2 is to be 1.42 Å. The C2–R3 bond lengths of 1 and 2 are envisaged to be 1.31 and 1.11 Å, and the bond angles of each C1–C2–R3 are 58.1 and 106.3°, respectively. The single point energies of each structure involving ZPE showed that nonclassical 1 is more stable than classical 2 by 1.19 kcal/moL.The gas phase structural feature of isopropyl cation3 was also investigated by the same level of theory as 1 and 2. The C1–C2 bond length of the calculated 3 is 1.44 Å, which became shorter than that of the corresponding alkane 3-H (1.53 Å). In case of 3, the cation center is almost localized on the C1 carbon. Actually, the C1–C2–R3 bond angle of the calculated 3 is 97.9°, which is much smaller than an ideal sp3 bond angle (109.3°) due to the hyperconjugative effect from C–H bond to the cation center. Whereas LUMO of the 2-isopropyl cation 3 is mainly formed in C1 cation center, HOMO is bound on the C–H bonds of two methyl groups (Figure 2).Figure 2
HOMO orbitals of ethyl cations1–11.With respect to optimized structures oftert-butyl cation 4 and isobutyl cation 5, the C1–C2 bond lengths are 1.46 and 1.41 Å at MP2 level in the gas phase, respectively. The calculation predicted that 4 has a Cs symmetric structure where the four carbons are coplanar (R1–C1–C2 angle is 119.5°), and its cation center (C1) takes almost sp2 hybridization. In fact, one C1–C2–H (R3) bond angle of the optimized 4 becomes smaller (102.7°) than that of 4-H (110.2°). The isobutyl cation 5 possesses a classicalform, in which the cationic charge is localized on the C1 (Figure 3). Two methyl groups connected to the C2 are located on up and down sides of the cation plane (R1R2–C1–C2), because efficient hyperconjugation from the two C–C bonds to the cation center contributes for this conformation. The tert-butyl cation 4 is much more stable (by >35 kcal/moL) than 5 in gas phase, due to a strong hyperconjugative stabilization effect from three methyl groups as discussed later (Section 3.3).Figure 3
LUMO orbitals of ethyl cations1–11.A similar trend in1 and 2 was also observed in cases of 2-butyl cations 6, 7, and 8. Both 6 and 7 are considered as nonclassical cations because the optimized structure has almost planar framework whose cation center is splitting between C1 and C2. In contrast, another “kinked” form (classical cation) 8 was inferred by MP2 optimizations. LUMOs of the nonclassical cations 6 and 7 mainly exist on C1 and C2 carbons, while they spread to the hydrogen of the “kinked” methyl group for the classical 8. Both the central C1–C2 bond lengths of 6 (nonclassical: trans) and 7 (nonclassical: cis) are estimated to be 1.40, whose value is almost the same as that of the classical 8 (1.41 Å). The C2–R3 (C–C) bond length of the classical 8 is expected to be 1.66 Å, which is elongated from a typical C(sp3)–C(sp3) bond length (1.54 Å) due to the hyperconjugative effect. The C1–C2–R3 bond angles of the nonclassical 6 and 7 are 57.8 and 58.1°, respectively, whereas the classical 8 has a larger angle (76.2°). The nonclassical 6 and 7 are estimated to be more stable than the classical 8 by 3.77 and 2.51 kcal/moL, respectively, suggesting that the hyperconjugations of the methyl groups of 6 and 7 are more effective for the stabilization than that of “kinked” methyl group of 8. Moreover, the trans-nonclassical 6 is a little more stable than the cis-nonclassical 7 by 1.26 kcal/moL.The 2,3-dimethylbutyl cations9 (classical: 1.44 Å for C1–C2 bond length, 1.60 Å for C2–R3 bond length, and 96.8° for C1–C2–R3 torsion angle) and 10 (nonclassical: 1.42 Å for C1–C2 bond length, 1.32 Å for C2–R3 bond length, and 57.2° for C1–C2–R3 torsion angle) possess the similar structural features with the ethyl and 2-butyl cations. However, our calculations revealed that classical 9 is more stable than nonclassical 10 by 8.79 kcal/moL in the gas phase. Although the reason is not clear, there is a possibility that four methyl groups of nonclassical 10 are more sterically hindered. LUMO orbital of classical 9 exists at the cation center (C1) and methyl groups connecting to the C1 and C2, whereas the LUMO of nonclassical 10 is almost localized on C1 and C2 carbons in a similar manner to ethyl and 2-butyl cations (Figure 3).As a result of MP2 calculation, trimethylbutyl cation11 is clearly categorized into a classical cation, because the cation center is localized on the C1 (Figure 3). The central ethylic C1–C2 bond length (1.45 Å) of 11 is significantly shorter than that of corresponding alkane 11-H (1.55 Å) and slightly longer than those of classical ethyl and 2-butyl cations 2 and 4. The C2–R3 bond length of 11 (1.62 Å) is longer than typical sp3–sp3 single bond length (1.54 Å). The C1–C2–R3 bond angle of 11 (92.6°) is smaller than that of corresponding alkane 1-H (111.9°).


## 3.2. The Comparison of Structural and Bond Strength Parameters between Selected Cations and Their Alkanes

Structural and bond strength parameters at MP2 level calculations between cations1–11 and corresponding alkanes 1-H–11-H were compared in order to estimate the hyperconjugative effect (Figure 4 and Table 3).Figure 4
Structural comparisons between alkyl-substituted ethyl cations1, 3, 5, and 9 and their alkanes calculated by MP2/6-311+G(d,p) level of theory.The C–C bond of ethane1-H is 1.53 Å in length, which should be an almost ideal sp3 C–C bond, whereas that of 1 generated by hydride elimination from 1-H becomes shorter by 0.15 Å (Figure 4). As shown in Table 3, the compliance constant Cii of C1–C2 bond is changed from 0.510 (1-H) to 0.388 Å/mdyn (1), indicating that the central C1–C2 bond of simple ethyl cation 1 is stronger than that of 1-H. Such a change is attributed to the hyperconjugative effect caused by the electron transfer from C–H covalent bond to the cation center. In contrast, the C2–R3 (C–H) bond strength of cation 1 becomes much weaker than that of 1-H (from 0.755 to 2.236 Å/mdyn), indicating that the bonding electrons of C2–R3 are moved to the unfilled p orbital (cation center). In fact, the value of Wiberg’s bond index (WBI) is changed from 1.04 (1) to 1.46 (1-H) (Figure 3), meaning that the C1–C2 bond becomes stronger by the hydride elimination. Details of the WBIs with bond orders of the 1–11 are discussed in the following section.The C1–C2 bond length of 2-propylcation 3 (1.44 Å) became shorter than that of 3-H (1.53 Å) by the hydride elimination (Figure 4).  A similar tendency is observed for the bond strengths and WBI values of 3 and 3-H. Thus, the compliance constant Cii of C1–C2 bond changed from 0.617 (1.04) to 0.377 Å/mdyn (1.17). On the other hand, the C2–R3 bond strength of 3 became weaker (from 0.702 to 2.718 Å/mdyn) from that of 3-H. Moreover, the bond angle of C1–C2–R3 changed from 110.7 (3-H) to 97.94° (3) according to our calculations. These structural changes can also be explained by the concept of hyperconjugative effect.Similarly, gas phase calculations predict that the ethylic C1–C2 bond of trans-2-butyl cation 5 is shorter and stronger (1.40 Å and 0.578 Å/mdyn) than that of corresponding alkane 5-H(1.53 Å and 0.209 Å/mdyn), while the C2–R3 bond strength of 5 (0.702 Å/mdyn) becomes much weaker than that of 5-H (3.609 Å/mdyn) by hydride elimination.On the other hand, MP2 level calculation in gas phase demonstrated that both 2,3-dimethylbutyl cations8 and its hydride adduct 8-H have the same central C1–C2 bond length (1.54 Å). Also, accompanied by the structural change from 8 to 8-H, a decrease of Cii (from 0.644 to 0.436 Å/mdyn) and a slight increase of bond order (from 0.99 to 1.10) of the C1–C2 bond were estimated by the calculation. Moreover, the C2–R3 bond length and strength of 8 (1.53 Å and 0.612 Å/mdyn) were shorter and stronger than those of 8-H (1.60 Å and 1.126 Å/mdyn), respectively. Although the hyperconjugative effect from C2–R3 bond to cation center of 8 is expected from the MP2 calculation, steric repulsion between the methyl groups would be inhibiting the shrinking of C1–C2 bond. The pronounced deformation of C1–C2–R3 bond angle of the cation 8 (96.77°) relative to alkane 8-H (111.3°) supports the occurrence of the steric repulsion.


## 3.3. NBO Analysis of the Alkyl-Substituted Ethyl Cations

Table3 summarizes Wiberg’s bond indexes (WBIs) and bond orders of ethyl cations 1–11 evaluated by the NBO analysis at MP2/6-311++G(3df,3pd) level of theory. The character of chemical bond can be intuitively understood by the use of WBI bond indexes. For example, a bond index of 1 means that the bond takes single bond.The WBIs of nonclassical1 and classical 2 are evaluated to be 1.46 and 1.23, respectively, meaning that the more stable nonclassical ethyl cation 1 is essentially the same as a protonated ethylene. Actually, the NBO calculation by MP2/6-311++G(3df,3dp) level suggests that both the C1 and C2 of 1 possess sp1.99 hybridization which is nearly equal to sp2. On the other hand, the C1 and C2 of 2 are sp1.55 and sp2.45 carbons, respectively, indicating that 2 is a classical cation.A similar trend was also observed in 2-butyl cations6 (trans-nonclassical), 7(cis-nonclassical), and 8 (classical) in the gas phase. The WBIs for 6, 7, and 8 are 1.38, 1.38, and 1.22, respectively. The hybridizations of C1 and C2 of both 6 and 7 are sp2.09 and sp2.08, while those of 8 are sp2.00 and sp2.26, respectively.According to our calculations, the hybridizations of C1 and C2 for classical 2,3-dimethylbutyl cation 9 are provided as sp1.86 and sp2.51, respectively. On the other hand, both of them were sp1.86 for nonclassical 2,3-dimethylbutyl cation 10 by NBO calculation in the gas phase. The bond order and bond index of central ethylic C1–C2 bond for 10 are 0.96 and 1.10, respectively.Our calculations revealed that the hybridization values of central C1 and C2 (sp1.77 and sp2.66) of the trimethylbutyl cation 11 are also close to those of 9 (Table 4).


## 3.4. Total Energies and Hydrogen Affinities of Each Cation

The zero point corrected total energies [Etotal’s] of alkyl-substituted ethyl cations and their isodesmic reaction energy [ΔEiso] by MP2/6-311++G(3df,3dp) level are summarized in Table 5.EachΔEiso is derived from the difference of total energies between alkane plus methyl cation and alkyl cation plus methane according to the equation shown in Table 5. The difference of ΔEiso (ΔΔEiso) between nonclassical ethyl cation 1 and classical cation 2 is 1.19 kcal/moL. The gas phase MP2 calculation revealed that the hydride affinity of the t-butyl cation 3 is considerably lower than that of simple ethyl cation by 38–40 kcal/moL. This fact means that the tert-butyl cation 3 is much more stable than ethyl, propyl, and other butyl cations.2-Methylpropan-1-yl cation5 with a similar stability to ethyl cation is less stable than tert-butyl cation 4 (35.8 kcal/moL). In case of 2-butyl cations, both the nonclassical cations 5 and 6 possess almost the same isodesmic reaction energy (ΔEiso=54.28 kcal/moL for 5 and 53.02 kcal/moL for 6) in the gas phase. On the other hand, MP2 calculations revealed that classical 7 is less stable than nonclassical 5 and 6 by 3 kcal/moL.Although nonclassical ethyl and butyl cations are more stable than the corresponding classical cations, classical and nonclassical 2,3-dimethylbutyl cations9 and 10 showed an opposite trend (ΔΔEiso=6.91 kcal/moL). That classical 9 is more stable than nonclassical 10 is attributed to a higher steric repulsion between four methyl groups in 10 with a shorter C1–C2 bond length than the corresponding alkane.Tetramethylbutyl cation11 had the highest gas phase isodesmic reaction energy (ΔEiso=75.61 kcal/moL), indicating the most stable cation among a series of alkyl-substituted ethyl cations in this report.


## 4. Conclusions

Gas phase structural and electronic properties of alkyl-substituted ethyl cations1–11 were investigated at MP2 level. It was revealed that there are nonclassical and classical forms for ethyl, 2-butyl, and 2,3-dimethylbutyl cations, whose nonclassical cationic state is more stable than the classical one (1.19 and 3.48 kcal/moL). On the contrary, the calculation for 2,3-dimethylbutyl cation demonstrated that the classical cation is more stable by 8.79 kcal/moL than the nonclassical one. The WBI parameters of the central C1–C2 bonds for the nonclassical and classical cations were 1.32–1.46 and 1.10–1.23 multiplicities in the gas phase, respectively. It was estimated from compliance constant Cii that the C1–C2 bonds for the nonclassical cations are stronger than those of the classical cations, while the former C2–R3 bonds are weaker than the latter ones due to the hyperconjugative effects. The gas phase hydride affinities based on isodesmic equations demonstrated that the nonclassical cations are strongly stabilized by electron donations from C2–R3 bond. In case of 2,3-dimethylbutyl cations, however, the classical cation was predicted to be more stable than nonclassical one by 8.79 kcal/moL due to the steric repulsions between four methyl groups substituted on the C1 and C2 carbons.



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