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An accurate computational determination of the relative stabilities and structures of small carbonium ions
;-__^ ‘->&emfA
_
Physics 89 (1984)
North-_HoUand;&sterd~-_ _ -- _ _. -_ _ ,_- r: _ - ^ L
237-244
:
.
-
_-
__
-.
AN ACCU&4TE COMPUTAtiONAL OF THE RELATIVE STAB--
_ ‘-.
_ -
- _
--_
DETERMINATION AND STRUCTURES
_. -
_ _
OF -SMALL
_-
__
..
_^
237 -
-
.
_-
‘; _.
CARBONIUM
IONS
- -
K. HIRAO Depa-tment
of
Chenustry, Cdege
of General Educatron. Nagoya Unrversrty. Nagoya, Japan
and S. YAMABE Department of Chenustry, Nara Universrty of Educatron, Nara. Japan Received
12 March
1984
A highly accurate computational
determination
of the relative
stab&ties
between
classical
open
and non-classical
bridged
&Hz. C-H:. and C2Ht ) is made The cluster expansron of the wavefunction theory IS used with a double-zeta plus polarization basislsset. The results compare well with available experiments and confirm the qualitattve conclusion derived previously from ab imtio calculations. The inclusion of electron correlation has a greater stabthung effect on the non-classical forms of Cz Hz and C,Hf . For C, HT. the C-C protonated bridged form is more stable by 4 kcal/mol than the classical form of the weak intermolecular complex of CzHf and H,. structures
of small carbonium
Ions (CH:.
1.Introduction Carbonium ions are known to play an important role as highly reactive intermediates in many organic reactions. They can have two possible structures, a classical open structure and a non-classical bridged structure. A knowledge of the relative stabilities of both forms is needed for a better understanding of their chemistry. Many ions are detectable by experimental techniques such as mass spectroscopy and some information on their heats of formation may be available. But, experimental determination of structure is difficult because of low concentrations and the amount of good data is relatively small. In the lack of experimental data, these properties have to be deduced from theoretical calculations and the prediction of the equilibrium geometries and stabilities of these ions has been continued as a major challenge to quantum chemistry. Indeed, small carbonium ions have been investigated extensively by a number of authors [l-9]. The main conclusions drawn from
these previous calculations are that the energies of structures involving three-center bonds (nonclassical bridged form) are strongly lowered by inclusion of higher-order angular functions (polarization functions) and that electron correlation effects play an important role in the relative stabilities of these ions. In view of the present situation that the absolute determination of stable geometries of small carbonium ions is desirable, we study systematically a series of structures of these species. In this paper, ab initio calculations for CHZ, C2HT, G-H,+, and G-H; are reported with a good basis set and with a highly accurate wavefunction including correlation effects. The method employed here has been intensively scrutinized with respect to reproducing geometrical structures and relative energies of small organic molecules. Hence, *here is a good reason to believe that, if carefully applied, it will lead -to proper predictions of these quantities where they have not as yet been experimentally determined.
0301-0104/84/$03.00 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K Hirao, S- Yam&f/
238
R&t&e _ - srabiiks
In section 2, the computational method will be described_ The results and a discussion will be given in section 3.
2. Method of calculation The basis set for carbon atoms in the calculations is the Huzinaga (9s/5p) gaussian set contracted to Dunning’s scheme [LO] to [4s/2p]. A (4s)/[2s] set is used on hydrogens. Polarization functions consisting of a set of p functions (a = 1.0) on each hydrogen atom and a set of d functions (a = 0.75) on each carbon atom are added. To optimize the geometry of carbonium ions. the energy gradient method in GAUSSIAN 80 [ll] is used. All the internal freedoms are fully optimized on the SCF level. To obtain the vibrational frequency for the evaluation of the zero-point energy (ZPE). the force constant matrix is made by calculating the numerical second derivatives of the electronic energy. The electron correlation effects are estimated by the single and double CI (SD Cl) calculations at the optimized geometry on the SCF level. In addition, the symmetry-adapted-cluster (SAC) calculation [12] is performed_ In the SAC calculation, we include all single and double excitations from the reference function as a linked cluster_ Quadruply excited configurations are represented by the disjoint clusters of the product of
Table 1 Theoretical SCF equdtbrium geometries and total merges ~fo!eeuics
Geometric
smdcarbonim
&JNS
=’ 11 :: __
el&tron pair cl&ers. This approximation that the one- and -two-particle~*chisters and their interactions are considered has been generally accepted to account for most of the electron correlation effects in the closed-shell ground state [13]. The core orbitals are treated as frozen in the present calculations.
3. Remits and discussion First, geometries and total energies for the neutral molecules obtained after minimization with respect to geometric parameters are given in table 1. The SCF method with a double-zeta plus polarization basis set gives a reasonable accuracy to the equilibrium bond distances and angles. The computational error is believed to be less than 0.1 A for bond lengths and 0.4” for angles. 3.1. CH,f Protonated methane CHC was among the first ion-molecule reaction products observed with mass spectrometers [14]. It has been investigated extensively by a number of authors. Dyczmons et al. [2] have made SCF and IEPA calculations and found that the C,(I) configuration (corresponding to a three-center complex between CHT and H,) is the most stable. The CzV(II), C,,(III) and D,,(W)
(m hnrtree) for neutral molecules
SCF
vahtes =’
andst~ctur~of
SD CI
SAC
parameters HZ CH,
r(HH) r(CH)
C2H2
rKC)
C2H,
rW3
r(CH)
r(CH) -=
CzH,
HCH
r(CC) r(CH) -= HCH
0 733
(0 74130) b,
LOS5 1.191 1060
(1 085) (1 203) (1061)
- 40 207594 - 76 S32544
- 40 382263 ( - 0.174669) c) -77 081363 (-0.248819)
-40.389889 (-0.182295) -77.100735 (-0 268191)
1 326 (1.330) 1.077 (1.076) 116 2 (116 6) 1.531 (1531) 1 OS7 (1096) 107.9 (107-S)
- 78 050546
- 78 326016 ( -0 277470)
-78.351413 (-0
- 79.249260
- 79 556556 (-0.307296)
- 79 584303 (-0.335043)
*’ Distances in A. angles in degree. ” Experimental data c’ Correlation energy.
- 1.131241
300867)
_.__
-
_- K. Himo, S_ Yamabe / Refatme stabdttiesand &uctures of small carbomam eons -
_L
inbdel~~showed i&%easing energy, all being _uqsta-ble with &spect to deformation to C$(I). In another &curate SCF calctilati& using _doul$z-zeta plus poIar&tio& Hariharan et al. [Sj have confirmed the Dyczmons’ finding and they reported that the C,(I) model is 16 kcal/mol more stable than the D&IV) model. Geometries for all of these structures are optirnized and given in fig. 1. The total energies and the relative energies are given in tables 2 and 3, respectively. The C,(I) structure is the most stable by any accuracy adopted. When correIation cor-
239-
rection is made, the _ener& difference between C,(I) and D&V) is compressed :into a smaller range by = 3-4 kcal/niol: In particular, &e energy of C,(II) falls to only 0.7 kcal/mol above C,(I). This close proximity has been noted previously and implies a low activation barrier for interchange or “scrambling” of the hydrogen nuclei. We have z&o made a vibrational analysis and confirmed that the C&I), C,,(III) and D&V) structures are all in the transition states (saddle point) and unstable wfh respect to deformation into the C,(I) structure.
(0.736)
(0.756)
C,(I) c*vw)
Dsh(IV> Fig. 1. The optunized geometries of CHZ.
1015icm
-1
1030icm-1
Bond length m A. Vah~es in parentheses are SCF M&liken atom populations_ In the models
C&D. C&II) and J&W), the dispiacement vector representmg the motton toward the most stable geometry, C,, is sketched together with the corresponding imaginary frequency.
240
K. Hiruo, S Yamabe / Relative stabdrties and structures of small carbomum iok
Table 2 Calculated total energy (hartree) for cations*) SCF
SD CI
SAC
- 40.414425 -40.409405 - 40.401820 -40 387039
-40 593579 (-0.179154) -40.592250 (-0.182845) - 40.586845 ( - 0.185025) - 40572642 ( - 0.185603)
-4040.601308(-0.186883) - 40.600197 ( - 0.190792) - 40.594960 ( y 0.193140) -40.580921 (- 0.193882)
open (I) bridged (II)
- 77.100153 - 77 091775
-m-340371 ( - 0.240218) - 77 340024 (- 0 248249)
- 77_357592( - 0.257439) - 77.357948 (-0 266173)
C,HJ open, echpsed (1) open. brsected (II) bndged (III)
- ?8_331395 - 78.330074 -78 331188
- 78.598738 (-0 267343) - 78 596476 ( - 0 266402) - 78 604407 ( - 0.273219)
- 78 618959 ( -0 287564) - 78 616562 (-0 286488) - 78.625479 ( - 0 294291)
- 79 469946 -79480664 - 79.477293
- 79.779994 ( - 0 310048) - 79.786800 ( - 0 306136)
-79 807911 (-0 -79 814775 (-0
337965) 334111)
-79_782606(-0305314)
- 79.810925 ( -0
333632)
Cations CH; C,(I) Cz,(II) C_,(III) Djh(W C,H;
C,H; CJI) C,,(II) D,,(III)
a) Values m parentheses are recovered correlation energies.
The C,(I) structure is formed by the attachment of H, to CH,I. The geometrical change of Hz and CHJ can be schematically explained in terms of a charge-transfer interaction.
the non-classical bridged (II) structures may be compared. McLean [7] has made a CI calculation on these structures and found that both structures correspond to minima in the potential energy surface with the same energy to within l-2 kcal/mol_ Table 3 and fig. 2 report results for C,Hc. The SCF theory gave lower energies for the classical form (I) by = 5 kcal/mol. However, electron cor-
The vinyl cation is of interest as the simplest carbonium ion for which the classical open (I) and
Table 3 Reiatlre encrges (m kcal/mol) of four protonated molrcules Protonated species
Structure or symmetry
CH;
C,(I) G.(II) C,,(III) D,,(IV)
C2H; C,H;
C2H;
open (I) bridged (II) open. bisected(I) open, eclipsed (II) bndged(II1) C,(I) C&W) D&III)
” Zero-point vibrational energy correction_
SCF
SD CI
SAC
SAC + ZPE u)
Prewous calculations
0.00
000
000
0.0
3.15 7.91 17.19
083
0.0
422 13.14
000 5 26 000 0 83 0.13
000 022 0.00 1.42 - 3.56
0.70 3 98 12.79 000 -022 0.00 1.50 -409 0.00 -4 31 - 1.89
OO@ - 6.73 - 4.61
000 -427 -164
b, Ref. [2]_ =) Ref. [S].
‘) Ref. [6].
6.3 10.7.16 0 0 OC’. 0 0 d,s0.0 =) 5.7. -7, l-2 0.0 =), 0.0 d’ 05 -0-9. -9 0 0 n, 0 0 *’ -99.8. -6.3 -5.2
0.00
- 1.30 000 -4.11 000 -403
=’ Ref. [7].
b,, 0 -0 =I
n Ref. [I]
g) Ref. [9].
K. Hirao,S_ Yayabe / ReiatiueAhhties andstructuresof smallcarbhkm ions
-
(0.706)
-1.432
ts.ao7r
Open (1)
Open, bisected
(I)
91
Bridged (I I) Fig 2. The optimized geometries of C,H:.
relations lead to greater energy lowerings for the non-classical bridged form (II) and this becomes more stable than the classical one. The zero-point vibrational energy correction produces a further lowering of the bridged structure relative to the classical one. At the highest level of theory considered here, the bridged structure is more stable by 1.3 kcal/mol than the classical one.
The ethyl cation also has possible classical open and non-classical bridged structures. In its classical form, it may have one of the two C, rotameric forms, eclipsed (I) or bisected (II), while the bridged form (III) has C,, symmetry. Lischka and Kohler [8] have made PNO CI and CEPA calculausing a rather good basis set tions on C;Hf including carbon d polarization. They found the non-classical bridged structure to be more stable by 6.4 kcal/moi than the classical structure. The optimized geometries and relative energies - of C,Hz are given in fig. 3 and table 3, respectively. The SCF theory predicts the classical form
6 (0.761)
v
Open, eclipsed
(0.760
(II)
to.801)
.---90. 571
Bridged (III) Fig. 3. The optimized geometries of C,H;.
242
K. Hwao. S
Yam&e / Relatme rtabd&
and structures of small carhnim_ ions
_
(I) of C,Hz to be slightly more stabe than the non-classical form (III). Addition of correlation favors the bridged form in turn. The ethyl cation exhibits preference neither for an eclipsed (I) nor for a bisected (II) arrangement_ The present theory predicts that the non-classical bridged structure (III) is more stable by 4.1 kcal/mol classical one (I).
__ _ _ :
to-8271
than the
Protonated ethane has also been observed by mass spectrometric investigations on ion-molecule reactions. Two different structures may be derived; in one case H, is attached to CzHc (I) whereas in the other case the proton is attached to the C-C bond of C,H, in a bridged structure (II). The present results are summarized in table 3 and fig. 4. The &H,‘(I) has a marked resemblance with the structure of CHc(C,) which is a weak intcraction complex between CH_T and H,. Hiraoka and Kebarle [17.1S] have studied the kinetics and equilibria of the following reaction: C,Hf + H, = CzHT_ They reported :he existence of two isomeric species of C, H,’ . one present at low temperatures (- I30 to - 160 “C) and the other one at high temperatures (40-200 “C). The low-temperature species was attributed to the classical structure whereas the high-temperature species was attributed to the bridged structure_ Ab initio SCF calculations (STO-3G, 4-31G and 6-31G) [I] have been performed for both isomers and the structure (II) was found to be more stable by lo-11 kcal/mol_ More recently_ Kohler and Lischka [9] calculated the energies for several conformations using a polarized basis set in a SCF of C,H;. calculation. follo\\ed by valence-shell correlation and found that the most stable conformation is structure (II). In the present calculation, the non-classical bridged form (II) is more stable than the classical form (I). The most stable structure is the C-C protonated form (II) with C,, symmetry_ Inclusion of electron correlation reduced the gap between the energy of the non-classical structure and that of the classical form. At the highest level of theory used here, the C,,(II) is more stable by 4.03
115.6
(0.9OG)
I .a73
C#I) Fig_ 4. The optmuzed
geometries
of C,H:.
kcal/mol than the C,(I). We also calculated another bridged form (III) with DXd symmetry but it is found to be less stable than that of C, symmetry (II) by 2.4 kcal/mol. In the classical form of the ethyl cation and a hydrogen molecule, the hydrogen molecule plays the role of electron donor and its bond is axially oriented toward the electron acceptor C,Hl. In the C-C protonated bridged form the proton is the electron acceptor and it is inserted directly into the center of the two methyl groups.
- _
_-
K--‘hhm, S- Yamabe/Xelnriw~staikztiesandstructz&sof sma!! carbozzzwit ions
_
aTable4 _‘- . __ F Stabilization energies for carbonium ionsa!
_
R&&on
SDCI
-
.. i
.
-_ -
_ .-_..
_=
25.7
39.8
39.8
420
62.8
620
602
502
CzH; (II)= cZH3‘(I)+H,
62.6
65.5
64.3
24 3
C,H;(I)=C,H;(I)+H, C,Hf(II)= C,Hf(I)+H, &H;(I) = CH; +CH, C,H;(II) = CH; +CH,
46 113 12.6 19.3
13.7 18.0 27.5 318
14.7 19.0 295 33.8
7.3 11.4 25.8 29.8
- 243
_
^
Table 4 lists the energies of several significant ion-molecule reactions_ The dissociation process CH; = CHC + H, is exothermic and may be compared with experimental data. The SCF stabilization energy is 25.7 kcal/mol. An inclusion of correlation effects led to a greater energy lowering for CHf . When such correction was included, A E becomes 39.8 kcal/mol which can be compared with an experimental value of 40 kcal/mol (this value is not corrected for zero-point vibrationai energies). For C,Hc = C,Hz + H,, the effect of electron correlation on AE turns out to be small but the zero-point vibrational correction is large (= 10 kcal/mol). The present theory predicts that the dissociation energy of the reaction is 54.3 kcal/ mol.
SD CI
129 8 168 0 162.7 176 3 176.1 138.5 145.5
n’ Values given in khcalorie b’ From refs. [15,16]. ‘) From ref. [17].
132.6 162.6 162 3 169 9 173.5 1402 1445 per moIe.
--
4 0 -=’ 118”
The calculated energy of the dissociation process C,HT = C2Hc + H, is also compared with experimental thermochemical data obtained by Hiraoka and Kebarle [17,18]. An inclusion of correlation effects and zero-point vibrational energy corrections certainly improves the agreement. It is of interest to note that the correlation effect is the significant component of the binding energy of the C,H,f and H, fragments_ 3.6. Proton affznities Table 5 compares theoretical and experimental proton affinities. The experimental numbers are not necessarily the best ones available. The present theory gives a satisfactory agreement with experiment. The SCF proton affinities are systematically too large. The energy shifts calculated by the SD CI are considerably larger but the values are still
SAC
SAC plus zerc+point
I+P-
energy correction CH, C,H,(I) C,H,(II) &H,(I) C,H,(III) C,H,(I) C&(II)
-
40bl
Table 5 Calculated proton affinities of hydrocarbons”) SCF
-_-1
based on the Lterature values of refs. [15.16]_
3.5. Dissociatzon energy
Molecule
-
Exp.
-SAC PIUS zero-point ; enirgycorrection I _
SAC
CH,C(I)=CH;+H,
*) Values given m khcalorie per mole. b’ Thermochemical data for CH, and CHf =’ From ref. [17].
-
C,H;(I)
= C,H;(I)+H,
__
_: -
- SCF
-.
132.7 1612 161.4 167.9 172.0 140.3 1446
127.7 155.6 156.9 159.7 163 8 135 6 139.7
127 b, 151 159 131.8 =’ 139 6 =’
too large compared to the experimental~ones. An inclusion of disjoint quadruply excited clusters (SAC) does not affect the SD CI values The overestimated values are corrected by the inclusion of zero-point vibrational energy corrections.
_ the Institute -for~h$olihlar Sciegc& Th&:sCk@is~ _ supported in part -by- a Grant&-Aid !Sci$$.ific Research from the:-Jab&&e‘ Ministry OF @uca-. tion, Science and Culture.
References 4. Concluding
remarks
PI WA_ Lathan, WJ. Hehre and JA
The present rigorous and reliable calculations on small carbonium ions confirm the qualitative conclusions derived from previous theoretical calculations_ For CHf . the C,(I) geometry is the best by all levels of wavefunctions. Correlation correction reduces the instability of the other models identified as transition states. For C,H,’ and C,H;. the correlation correction switches the best structure from the classical model to the nonclassical bridged one. The best C-H; structure is also of a bridge type. The bridged structure of -ains 4 kcal/mol stability relaC,Hf and C,HT D tive to the classical structure. Even if this value is revised by a more sophisticated wavefunction in the future, the superiority or the inferiority w111be invariant_ Judging from the small energetic difference, such geometric distinction may be difficult under thermal conditions. However, in view of the development of the experimental techniques with which the stabilization and reactivities of these ions in their dilute gas-phase may be directly studied, many of the predictions of the present theory will shortly be brought to test.
Pople. J. Am. Chem. Sot. 98 (1973) 875. 121V. Dyczmons, V_ Staemmler and W. Kutzelnigg. Chem. Phys Letters 5 (1970) 361; V. Dy-ons and W. Kutzelnigg, Theoret. Chim. Acta 33 (1974) 239. [31 J E Wtlliams, J.V. Buss, L.C. Allen, P. van R. Schleyer. WJ. Hehre and J.A. Pople. J. Am Chem. Sot. 92 (1970) 2141. [41 R Hoffmann. L. Radom. JA. Pople. P. van R Schleyer. W J. Hehre and L Salem, 1. Am. Chem. Sot. 94 (1972) 6221 PI PC. Hanharan, W.A. Lathan and J.A. Pople, Chem. Phys Letters 14 (1972) 385. P51 B. Zura~skt, R. Ahhichs and W_ Kuzehtigg. Chem. Phys. Letters 21 (1973) 309. [71 J. Weber and A D McLean, J. Am. Chem. Sot. 98 (1976) PI A WI 1111
Wl I131 1141
Acknowledgement
IW
The authors thank Ms. Yuko Kato for assistance in performing the calculations. Calculations were carried out on M-200 computers at the Nagoya University Computational Center and at
WI I171 PSI
875. H. Ltschka and H-J. Kohler. J. Am Chem Sot 100 (1978) 5297. H-J. Kohler and K Ltschka. Chem. Phys Letters 58 (1978) 175. T H. Dunning, J. Chem. Phys. 53 (1970) 2823. J S Binkley. R A_ Whiteside. R. Krishman. R. Seeger. D J. DeFrees, H B. Schlegel. S Topiol, L R. Kahn and J-A. Pople. GAUSSIAN 80, QCPE No. 406 (1980). K. Htrao. J. Chem. Phys 79 (1983) 5000. K. Htrao and Y. Hatano, Chem. Phys. Letters 100 (1983) 519. V.L. Talroze and A L. Lyubimova, DokI. Akad. Nauk. SSSR 86 (1952) 509 WA_ Chupka and J. Berkowitz, J. Chem. Phys. 54 (1971) 4256. D K. Bohme. m. Interacttons between tons and molecules, rd. P. Ausloos (Plenum Press, New York. 1975) K. Hiraoka and P. Kebarle, J. Am Chem. Sot 98 (1976) 6119 K Hiraoka and P Kebarle. Radiat. Phys Chem. 20 (1982) 41.
_
Physics 89 (1984)
North-_HoUand;&sterd~-_ _ -- _ _. -_ _ ,_- r: _ - ^ L
237-244
:
.
-
_-
__
-.
AN ACCU&4TE COMPUTAtiONAL OF THE RELATIVE STAB--
_ ‘-.
_ -
- _
--_
DETERMINATION AND STRUCTURES
_. -
_ _
OF -SMALL
_-
__
..
_^
237 -
-
.
_-
‘; _.
CARBONIUM
IONS
- -
K. HIRAO Depa-tment
of
Chenustry, Cdege
of General Educatron. Nagoya Unrversrty. Nagoya, Japan
and S. YAMABE Department of Chenustry, Nara Universrty of Educatron, Nara. Japan Received
12 March
1984
A highly accurate computational
determination
of the relative
stab&ties
between
classical
open
and non-classical
bridged
&Hz. C-H:. and C2Ht ) is made The cluster expansron of the wavefunction theory IS used with a double-zeta plus polarization basislsset. The results compare well with available experiments and confirm the qualitattve conclusion derived previously from ab imtio calculations. The inclusion of electron correlation has a greater stabthung effect on the non-classical forms of Cz Hz and C,Hf . For C, HT. the C-C protonated bridged form is more stable by 4 kcal/mol than the classical form of the weak intermolecular complex of CzHf and H,. structures
of small carbonium
Ions (CH:.
1.Introduction Carbonium ions are known to play an important role as highly reactive intermediates in many organic reactions. They can have two possible structures, a classical open structure and a non-classical bridged structure. A knowledge of the relative stabilities of both forms is needed for a better understanding of their chemistry. Many ions are detectable by experimental techniques such as mass spectroscopy and some information on their heats of formation may be available. But, experimental determination of structure is difficult because of low concentrations and the amount of good data is relatively small. In the lack of experimental data, these properties have to be deduced from theoretical calculations and the prediction of the equilibrium geometries and stabilities of these ions has been continued as a major challenge to quantum chemistry. Indeed, small carbonium ions have been investigated extensively by a number of authors [l-9]. The main conclusions drawn from
these previous calculations are that the energies of structures involving three-center bonds (nonclassical bridged form) are strongly lowered by inclusion of higher-order angular functions (polarization functions) and that electron correlation effects play an important role in the relative stabilities of these ions. In view of the present situation that the absolute determination of stable geometries of small carbonium ions is desirable, we study systematically a series of structures of these species. In this paper, ab initio calculations for CHZ, C2HT, G-H,+, and G-H; are reported with a good basis set and with a highly accurate wavefunction including correlation effects. The method employed here has been intensively scrutinized with respect to reproducing geometrical structures and relative energies of small organic molecules. Hence, *here is a good reason to believe that, if carefully applied, it will lead -to proper predictions of these quantities where they have not as yet been experimentally determined.
0301-0104/84/$03.00 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K Hirao, S- Yam&f/
238
R&t&e _ - srabiiks
In section 2, the computational method will be described_ The results and a discussion will be given in section 3.
2. Method of calculation The basis set for carbon atoms in the calculations is the Huzinaga (9s/5p) gaussian set contracted to Dunning’s scheme [LO] to [4s/2p]. A (4s)/[2s] set is used on hydrogens. Polarization functions consisting of a set of p functions (a = 1.0) on each hydrogen atom and a set of d functions (a = 0.75) on each carbon atom are added. To optimize the geometry of carbonium ions. the energy gradient method in GAUSSIAN 80 [ll] is used. All the internal freedoms are fully optimized on the SCF level. To obtain the vibrational frequency for the evaluation of the zero-point energy (ZPE). the force constant matrix is made by calculating the numerical second derivatives of the electronic energy. The electron correlation effects are estimated by the single and double CI (SD Cl) calculations at the optimized geometry on the SCF level. In addition, the symmetry-adapted-cluster (SAC) calculation [12] is performed_ In the SAC calculation, we include all single and double excitations from the reference function as a linked cluster_ Quadruply excited configurations are represented by the disjoint clusters of the product of
Table 1 Theoretical SCF equdtbrium geometries and total merges ~fo!eeuics
Geometric
smdcarbonim
&JNS
=’ 11 :: __
el&tron pair cl&ers. This approximation that the one- and -two-particle~*chisters and their interactions are considered has been generally accepted to account for most of the electron correlation effects in the closed-shell ground state [13]. The core orbitals are treated as frozen in the present calculations.
3. Remits and discussion First, geometries and total energies for the neutral molecules obtained after minimization with respect to geometric parameters are given in table 1. The SCF method with a double-zeta plus polarization basis set gives a reasonable accuracy to the equilibrium bond distances and angles. The computational error is believed to be less than 0.1 A for bond lengths and 0.4” for angles. 3.1. CH,f Protonated methane CHC was among the first ion-molecule reaction products observed with mass spectrometers [14]. It has been investigated extensively by a number of authors. Dyczmons et al. [2] have made SCF and IEPA calculations and found that the C,(I) configuration (corresponding to a three-center complex between CHT and H,) is the most stable. The CzV(II), C,,(III) and D,,(W)
(m hnrtree) for neutral molecules
SCF
vahtes =’
andst~ctur~of
SD CI
SAC
parameters HZ CH,
r(HH) r(CH)
C2H2
rKC)
C2H,
rW3
r(CH)
r(CH) -=
CzH,
HCH
r(CC) r(CH) -= HCH
0 733
(0 74130) b,
LOS5 1.191 1060
(1 085) (1 203) (1061)
- 40 207594 - 76 S32544
- 40 382263 ( - 0.174669) c) -77 081363 (-0.248819)
-40.389889 (-0.182295) -77.100735 (-0 268191)
1 326 (1.330) 1.077 (1.076) 116 2 (116 6) 1.531 (1531) 1 OS7 (1096) 107.9 (107-S)
- 78 050546
- 78 326016 ( -0 277470)
-78.351413 (-0
- 79.249260
- 79 556556 (-0.307296)
- 79 584303 (-0.335043)
*’ Distances in A. angles in degree. ” Experimental data c’ Correlation energy.
- 1.131241
300867)
_.__
-
_- K. Himo, S_ Yamabe / Refatme stabdttiesand &uctures of small carbomam eons -
_L
inbdel~~showed i&%easing energy, all being _uqsta-ble with &spect to deformation to C$(I). In another &curate SCF calctilati& using _doul$z-zeta plus poIar&tio& Hariharan et al. [Sj have confirmed the Dyczmons’ finding and they reported that the C,(I) model is 16 kcal/mol more stable than the D&IV) model. Geometries for all of these structures are optirnized and given in fig. 1. The total energies and the relative energies are given in tables 2 and 3, respectively. The C,(I) structure is the most stable by any accuracy adopted. When correIation cor-
239-
rection is made, the _ener& difference between C,(I) and D&V) is compressed :into a smaller range by = 3-4 kcal/niol: In particular, &e energy of C,(II) falls to only 0.7 kcal/mol above C,(I). This close proximity has been noted previously and implies a low activation barrier for interchange or “scrambling” of the hydrogen nuclei. We have z&o made a vibrational analysis and confirmed that the C&I), C,,(III) and D&V) structures are all in the transition states (saddle point) and unstable wfh respect to deformation into the C,(I) structure.
(0.736)
(0.756)
C,(I) c*vw)
Dsh(IV> Fig. 1. The optunized geometries of CHZ.
1015icm
-1
1030icm-1
Bond length m A. Vah~es in parentheses are SCF M&liken atom populations_ In the models
C&D. C&II) and J&W), the dispiacement vector representmg the motton toward the most stable geometry, C,, is sketched together with the corresponding imaginary frequency.
240
K. Hiruo, S Yamabe / Relative stabdrties and structures of small carbomum iok
Table 2 Calculated total energy (hartree) for cations*) SCF
SD CI
SAC
- 40.414425 -40.409405 - 40.401820 -40 387039
-40 593579 (-0.179154) -40.592250 (-0.182845) - 40.586845 ( - 0.185025) - 40572642 ( - 0.185603)
-4040.601308(-0.186883) - 40.600197 ( - 0.190792) - 40.594960 ( y 0.193140) -40.580921 (- 0.193882)
open (I) bridged (II)
- 77.100153 - 77 091775
-m-340371 ( - 0.240218) - 77 340024 (- 0 248249)
- 77_357592( - 0.257439) - 77.357948 (-0 266173)
C,HJ open, echpsed (1) open. brsected (II) bndged (III)
- ?8_331395 - 78.330074 -78 331188
- 78.598738 (-0 267343) - 78 596476 ( - 0 266402) - 78 604407 ( - 0.273219)
- 78 618959 ( -0 287564) - 78 616562 (-0 286488) - 78.625479 ( - 0 294291)
- 79 469946 -79480664 - 79.477293
- 79.779994 ( - 0 310048) - 79.786800 ( - 0 306136)
-79 807911 (-0 -79 814775 (-0
337965) 334111)
-79_782606(-0305314)
- 79.810925 ( -0
333632)
Cations CH; C,(I) Cz,(II) C_,(III) Djh(W C,H;
C,H; CJI) C,,(II) D,,(III)
a) Values m parentheses are recovered correlation energies.
The C,(I) structure is formed by the attachment of H, to CH,I. The geometrical change of Hz and CHJ can be schematically explained in terms of a charge-transfer interaction.
the non-classical bridged (II) structures may be compared. McLean [7] has made a CI calculation on these structures and found that both structures correspond to minima in the potential energy surface with the same energy to within l-2 kcal/mol_ Table 3 and fig. 2 report results for C,Hc. The SCF theory gave lower energies for the classical form (I) by = 5 kcal/mol. However, electron cor-
The vinyl cation is of interest as the simplest carbonium ion for which the classical open (I) and
Table 3 Reiatlre encrges (m kcal/mol) of four protonated molrcules Protonated species
Structure or symmetry
CH;
C,(I) G.(II) C,,(III) D,,(IV)
C2H; C,H;
C2H;
open (I) bridged (II) open. bisected(I) open, eclipsed (II) bndged(II1) C,(I) C&W) D&III)
” Zero-point vibrational energy correction_
SCF
SD CI
SAC
SAC + ZPE u)
Prewous calculations
0.00
000
000
0.0
3.15 7.91 17.19
083
0.0
422 13.14
000 5 26 000 0 83 0.13
000 022 0.00 1.42 - 3.56
0.70 3 98 12.79 000 -022 0.00 1.50 -409 0.00 -4 31 - 1.89
OO@ - 6.73 - 4.61
000 -427 -164
b, Ref. [2]_ =) Ref. [S].
‘) Ref. [6].
6.3 10.7.16 0 0 OC’. 0 0 d,s0.0 =) 5.7. -7, l-2 0.0 =), 0.0 d’ 05 -0-9. -9 0 0 n, 0 0 *’ -99.8. -6.3 -5.2
0.00
- 1.30 000 -4.11 000 -403
=’ Ref. [7].
b,, 0 -0 =I
n Ref. [I]
g) Ref. [9].
K. Hirao,S_ Yayabe / ReiatiueAhhties andstructuresof smallcarbhkm ions
-
(0.706)
-1.432
ts.ao7r
Open (1)
Open, bisected
(I)
91
Bridged (I I) Fig 2. The optimized geometries of C,H:.
relations lead to greater energy lowerings for the non-classical bridged form (II) and this becomes more stable than the classical one. The zero-point vibrational energy correction produces a further lowering of the bridged structure relative to the classical one. At the highest level of theory considered here, the bridged structure is more stable by 1.3 kcal/mol than the classical one.
The ethyl cation also has possible classical open and non-classical bridged structures. In its classical form, it may have one of the two C, rotameric forms, eclipsed (I) or bisected (II), while the bridged form (III) has C,, symmetry. Lischka and Kohler [8] have made PNO CI and CEPA calculausing a rather good basis set tions on C;Hf including carbon d polarization. They found the non-classical bridged structure to be more stable by 6.4 kcal/moi than the classical structure. The optimized geometries and relative energies - of C,Hz are given in fig. 3 and table 3, respectively. The SCF theory predicts the classical form
6 (0.761)
v
Open, eclipsed
(0.760
(II)
to.801)
.---90. 571
Bridged (III) Fig. 3. The optimized geometries of C,H;.
242
K. Hwao. S
Yam&e / Relatme rtabd&
and structures of small carhnim_ ions
_
(I) of C,Hz to be slightly more stabe than the non-classical form (III). Addition of correlation favors the bridged form in turn. The ethyl cation exhibits preference neither for an eclipsed (I) nor for a bisected (II) arrangement_ The present theory predicts that the non-classical bridged structure (III) is more stable by 4.1 kcal/mol classical one (I).
__ _ _ :
to-8271
than the
Protonated ethane has also been observed by mass spectrometric investigations on ion-molecule reactions. Two different structures may be derived; in one case H, is attached to CzHc (I) whereas in the other case the proton is attached to the C-C bond of C,H, in a bridged structure (II). The present results are summarized in table 3 and fig. 4. The &H,‘(I) has a marked resemblance with the structure of CHc(C,) which is a weak intcraction complex between CH_T and H,. Hiraoka and Kebarle [17.1S] have studied the kinetics and equilibria of the following reaction: C,Hf + H, = CzHT_ They reported :he existence of two isomeric species of C, H,’ . one present at low temperatures (- I30 to - 160 “C) and the other one at high temperatures (40-200 “C). The low-temperature species was attributed to the classical structure whereas the high-temperature species was attributed to the bridged structure_ Ab initio SCF calculations (STO-3G, 4-31G and 6-31G) [I] have been performed for both isomers and the structure (II) was found to be more stable by lo-11 kcal/mol_ More recently_ Kohler and Lischka [9] calculated the energies for several conformations using a polarized basis set in a SCF of C,H;. calculation. follo\\ed by valence-shell correlation and found that the most stable conformation is structure (II). In the present calculation, the non-classical bridged form (II) is more stable than the classical form (I). The most stable structure is the C-C protonated form (II) with C,, symmetry_ Inclusion of electron correlation reduced the gap between the energy of the non-classical structure and that of the classical form. At the highest level of theory used here, the C,,(II) is more stable by 4.03
115.6
(0.9OG)
I .a73
C#I) Fig_ 4. The optmuzed
geometries
of C,H:.
kcal/mol than the C,(I). We also calculated another bridged form (III) with DXd symmetry but it is found to be less stable than that of C, symmetry (II) by 2.4 kcal/mol. In the classical form of the ethyl cation and a hydrogen molecule, the hydrogen molecule plays the role of electron donor and its bond is axially oriented toward the electron acceptor C,Hl. In the C-C protonated bridged form the proton is the electron acceptor and it is inserted directly into the center of the two methyl groups.
- _
_-
K--‘hhm, S- Yamabe/Xelnriw~staikztiesandstructz&sof sma!! carbozzzwit ions
_
aTable4 _‘- . __ F Stabilization energies for carbonium ionsa!
_
R&&on
SDCI
-
.. i
.
-_ -
_ .-_..
_=
25.7
39.8
39.8
420
62.8
620
602
502
CzH; (II)= cZH3‘(I)+H,
62.6
65.5
64.3
24 3
C,H;(I)=C,H;(I)+H, C,Hf(II)= C,Hf(I)+H, &H;(I) = CH; +CH, C,H;(II) = CH; +CH,
46 113 12.6 19.3
13.7 18.0 27.5 318
14.7 19.0 295 33.8
7.3 11.4 25.8 29.8
- 243
_
^
Table 4 lists the energies of several significant ion-molecule reactions_ The dissociation process CH; = CHC + H, is exothermic and may be compared with experimental data. The SCF stabilization energy is 25.7 kcal/mol. An inclusion of correlation effects led to a greater energy lowering for CHf . When such correction was included, A E becomes 39.8 kcal/mol which can be compared with an experimental value of 40 kcal/mol (this value is not corrected for zero-point vibrationai energies). For C,Hc = C,Hz + H,, the effect of electron correlation on AE turns out to be small but the zero-point vibrational correction is large (= 10 kcal/mol). The present theory predicts that the dissociation energy of the reaction is 54.3 kcal/ mol.
SD CI
129 8 168 0 162.7 176 3 176.1 138.5 145.5
n’ Values given in khcalorie b’ From refs. [15,16]. ‘) From ref. [17].
132.6 162.6 162 3 169 9 173.5 1402 1445 per moIe.
--
4 0 -=’ 118”
The calculated energy of the dissociation process C,HT = C2Hc + H, is also compared with experimental thermochemical data obtained by Hiraoka and Kebarle [17,18]. An inclusion of correlation effects and zero-point vibrational energy corrections certainly improves the agreement. It is of interest to note that the correlation effect is the significant component of the binding energy of the C,H,f and H, fragments_ 3.6. Proton affznities Table 5 compares theoretical and experimental proton affinities. The experimental numbers are not necessarily the best ones available. The present theory gives a satisfactory agreement with experiment. The SCF proton affinities are systematically too large. The energy shifts calculated by the SD CI are considerably larger but the values are still
SAC
SAC plus zerc+point
I+P-
energy correction CH, C,H,(I) C,H,(II) &H,(I) C,H,(III) C,H,(I) C&(II)
-
40bl
Table 5 Calculated proton affinities of hydrocarbons”) SCF
-_-1
based on the Lterature values of refs. [15.16]_
3.5. Dissociatzon energy
Molecule
-
Exp.
-SAC PIUS zero-point ; enirgycorrection I _
SAC
CH,C(I)=CH;+H,
*) Values given m khcalorie per mole. b’ Thermochemical data for CH, and CHf =’ From ref. [17].
-
C,H;(I)
= C,H;(I)+H,
__
_: -
- SCF
-.
132.7 1612 161.4 167.9 172.0 140.3 1446
127.7 155.6 156.9 159.7 163 8 135 6 139.7
127 b, 151 159 131.8 =’ 139 6 =’
too large compared to the experimental~ones. An inclusion of disjoint quadruply excited clusters (SAC) does not affect the SD CI values The overestimated values are corrected by the inclusion of zero-point vibrational energy corrections.
_ the Institute -for~h$olihlar Sciegc& Th&:sCk@is~ _ supported in part -by- a Grant&-Aid !Sci$$.ific Research from the:-Jab&&e‘ Ministry OF @uca-. tion, Science and Culture.
References 4. Concluding
remarks
PI WA_ Lathan, WJ. Hehre and JA
The present rigorous and reliable calculations on small carbonium ions confirm the qualitative conclusions derived from previous theoretical calculations_ For CHf . the C,(I) geometry is the best by all levels of wavefunctions. Correlation correction reduces the instability of the other models identified as transition states. For C,H,’ and C,H;. the correlation correction switches the best structure from the classical model to the nonclassical bridged one. The best C-H; structure is also of a bridge type. The bridged structure of -ains 4 kcal/mol stability relaC,Hf and C,HT D tive to the classical structure. Even if this value is revised by a more sophisticated wavefunction in the future, the superiority or the inferiority w111be invariant_ Judging from the small energetic difference, such geometric distinction may be difficult under thermal conditions. However, in view of the development of the experimental techniques with which the stabilization and reactivities of these ions in their dilute gas-phase may be directly studied, many of the predictions of the present theory will shortly be brought to test.
Pople. J. Am. Chem. Sot. 98 (1973) 875. 121V. Dyczmons, V_ Staemmler and W. Kutzelnigg. Chem. Phys Letters 5 (1970) 361; V. Dy-ons and W. Kutzelnigg, Theoret. Chim. Acta 33 (1974) 239. [31 J E Wtlliams, J.V. Buss, L.C. Allen, P. van R. Schleyer. WJ. Hehre and J.A. Pople. J. Am Chem. Sot. 92 (1970) 2141. [41 R Hoffmann. L. Radom. JA. Pople. P. van R Schleyer. W J. Hehre and L Salem, 1. Am. Chem. Sot. 94 (1972) 6221 PI PC. Hanharan, W.A. Lathan and J.A. Pople, Chem. Phys Letters 14 (1972) 385. P51 B. Zura~skt, R. Ahhichs and W_ Kuzehtigg. Chem. Phys. Letters 21 (1973) 309. [71 J. Weber and A D McLean, J. Am. Chem. Sot. 98 (1976) PI A WI 1111
Wl I131 1141
Acknowledgement
IW
The authors thank Ms. Yuko Kato for assistance in performing the calculations. Calculations were carried out on M-200 computers at the Nagoya University Computational Center and at
WI I171 PSI
875. H. Ltschka and H-J. Kohler. J. Am Chem Sot 100 (1978) 5297. H-J. Kohler and K Ltschka. Chem. Phys Letters 58 (1978) 175. T H. Dunning, J. Chem. Phys. 53 (1970) 2823. J S Binkley. R A_ Whiteside. R. Krishman. R. Seeger. D J. DeFrees, H B. Schlegel. S Topiol, L R. Kahn and J-A. Pople. GAUSSIAN 80, QCPE No. 406 (1980). K. Htrao. J. Chem. Phys 79 (1983) 5000. K. Htrao and Y. Hatano, Chem. Phys. Letters 100 (1983) 519. V.L. Talroze and A L. Lyubimova, DokI. Akad. Nauk. SSSR 86 (1952) 509 WA_ Chupka and J. Berkowitz, J. Chem. Phys. 54 (1971) 4256. D K. Bohme. m. Interacttons between tons and molecules, rd. P. Ausloos (Plenum Press, New York. 1975) K. Hiraoka and P. Kebarle, J. Am Chem. Sot 98 (1976) 6119 K Hiraoka and P Kebarle. Radiat. Phys Chem. 20 (1982) 41.