Dec., 1962
A SIMPLE MOLECULAR ORBITALSTUDYOF AROMATIC MOLECULES
-4.91 D.(Li-H:+) for excited AIZ+ U H and 3.065 (30) A. M. Karo, 4.' Cham. Phya., 81, 182 (1959). (31) A. M. Karo, ibid., 38, 907 (1960).
2299
D. for the electric transition moment connecting AIZ+ and X12+; we obtain -4.25 and 4.30 D. for these quantities.
A SIMPLE MOLECULAR ORBITAL STUDY OF AROMATIC MOLECULES AND ION3 HAVING ORBITALLY DEGENERATE GROUND STATES BY LAWRENCE C. SNYDER Bell Telephone Laboratories, Inc., Murray Hill, New Jersey Receiued M a y $1, 1966
Modifications of simple Hiickel molecular orbital theory have been employed to describe the static Jahn-Teller geometrical instability of symmetrical aromatic molecule8 having orbitally degenerate a electronic ground states in the Huckel approximation. The parameterization of Hiickel theory by Longuet-Higgins and Salem has been applied in iterative computations to describe cyclobutadiene ( C4H4), C4H4+,tetraphenylcyclobutadiene(C4(ca6)4), C4( C6H6)4+1cyclooctatetraene (CsH8), and CsH8-'. The magnitude of distortion energies and properties of the distorted molecule are estimated. It is noted that the rectangular form of cyclobutadiene is an unstable planar geometry under the arameterization by LennardJones. For the neutral molecules, whose ground states are pseudo-degenerate, the effect o f approximate introduction of electron repulsion is shown to be a reduction of the Jahn-Teller instability found by the simpler theory for the lowest energy singlet state. The results of applications of the modified Hiickel theory t o several other molecules are summarized.
Introduction In 1937 Jahn and Teller' stated that symmetrical molecular systems having an orbitally degenerate ground state may be geometrically unstable with respect to a displacement of atoms which removes that degeneraqy. Orbital degeneracy of the a electrons can occur in a simple Hucke12description of planar aromatic systems having a threefold or greater axis of rotational symmetry. It occurs when the highest occupied pair of orbitals is degenerate and contains l, 2, or 3 electrons. In this case, there is more than one way allowed by the Pauli principle to assign the electrons to these orbitals, and all are degenerate in the Huckel approximation. A list of observed and hypothetical molecules, which if planar would have orhitally degenerate ground states in the Huckel approximation, occurs in Fig. 7 and 8 along with references reporting their experimental properties. One can anticipate for these molecules unusual phenomena associated with the fact that a small perturbation cain remove the apparent degeneracy and cause large fluctuations in properties computed from the wave function. The perturbation may be an interaction with the medium containing the molecule, as has been assumed in an explanation8 of the unusually great electron spin resonance (e.s.r.) line width exhibited by benzene-, coronene-, and tri~henylene-.~ Displacements of C-C bond lengths also may act as the perturbati~n.~-~ A strong interaction can cause the molecule to behave as having a distorted geometry. Weaker interac(1) H. A. Jahn and E. Teller, PTOC.Roy. SOC.(London). A161, 220 (1937). (2) A. Streitwieser, "Molecular Orbital Theory for Organic Chemists," John Wiley and Sons, New York, N. Y., 1961. (3) H. M. McConciell and A. D. McLachlan, J . Chem. Phys., 84, 1 (1961). (4) M . G. Townsend and 8.I. Weissmaa, ibtd., 88, 309 (1960). (5) J. E. Lennard-Jones and J. Turkevioh, PTOC. Roy. SOC.(London), Al68, 297 (1937). (6) A. D. Liehr, 2. phyeik. Cham. (Frankfurt), 9, 338 (1956). (7) L. C. Snyder, J . Chem. Phys., 83, 619 (1960). (8) W. D. Hohey and A. D. McLaohlan, ibid.. 88, 1695 (1960). (9) C. A. Coulson and A. Golebiewaki, Mol. Phgs., 6, 71 (1962).
tions can lead to effects in Raman and infrared spectra. This study is a computational exploration of the importance of bond length displacements as a perturbation removing the T electronic orbital degeneracy. Results of detailed calculations are given for cyclobutadiene (C4H3, C4H4+, tetraphenylcyclobutadiene (Cd(CaH&), C4(CeHs)4+, cyclooctatetraene (CsHs), and CsH8-. A modification of simple Huckel molecular orbital theory is adopted to obtain an estimate of the change in energy when the hypothetical symmetrical molecule of minimum energy is distorted to that set of bond lengths which minimize the total energy. Bond lengths and orders together with charge and spin densities are computed for the distorted molecule. The electronic energy is taken to depend parametrically upon the bond lengths. This is a static Jahn-Teller calculation. The potential energy surfaces obtained for nuclear motion probably can be safely used, with some caution, in classical thinking about the nuclear motions. These calculations are a convenient preliminary to the more complex dynamic description which hears more direct interpretation.*$lo Approximate introduction of electran repulsion is made for the neutral molecules to explore its effect on the estimated magnitude of the energy changes upon distortion in the lowest singlet state. A Simple Molecular Orbital Description.To attain an approximate description of the interaction of the ?r electrons with bond length displacements, we adopt a well known model used earlier by Lennard-Jones" and later by Longuet-Higgins and Salem.lZ The total energy E is taken to be the sum of the ?r electron energy E , computed in a modified Huckel approximation plus an energy E,, taken to be a sum of independent u bond energies V(r,). (10) A. D. Liehr, Ann. R e t Phue. Cham., 13 (1962). (11) J. E. Lennard-Jones, Proc. Rou. SOC.(London), 8168, 280 (1937). (12) H. C. Longuet-Higgins and L. Salem, {bid.. AMl, 172 (1959).
LAWREXCE C. SNYDER
2300
E
=
E,
-+ E ,
(1)
E V(Tk1)
(2)
E, k>l
A electron wave function 1c.* is taken to be a simple product of molecular orbitals pi with occupation numbers pi and which are linear combinations of the a atomic orbitals XI.
+
(7) (8)
pkl = CikCil
The qualitative origin of Jahn-Teller geometrical instability in this approximate description may ,e seen in eq. 7. For the symmetrical molecule, the sum of the partial bond orders of a single chemical type of bond is equal in the degenerate orbitals pm and qn. A perturbation which shortens one bond and thus increases p in that bond causes a mixing of pm and pn, removes the degeneracy, and lowers the energy of the a electrons. The presentation of computational results is conveniently made in terms of the Coulson bond orders pkl and charge densities q1.2 (9) i
Properties of the hypothetical symmetrical molecule are computed with the assumption that its wave function is composed of equal components of functions corresponding to the ways allowed by the Pauli principle for pn vm electrons to be assigned to the degenerate pair of orbitals. The corresponding expression for the a electron energy E T s y m is written
+
+
E,Sym= E,"" where P
(Vm
TWm
+
+P W ~
vn)/2
and EaYm =
In computations of bond orders, and charge densities for the symmetrical molecule, one replaces V, and vn by P in eq. 9 and 10. Parameterization and Computations.-Computations with the expressions of the previous section require the dependence of the resonance integral B and the u bond potential V on bond length. Several pairs of functions have been developed. Two have been used in this study: the recent expressions by Longuet-Higgins and Salem,l 2 and those originated by Lennard-Jones. l1 Longuet-Higgins and Salem fitted the AI, and B,, stretching force constants of benzene and a linear bond order-bond length relation to obtain the constants a and POof the equations
p(Tkl)=
+
E,O" VmWm v~W, (6) The energy of the a electrons is conveniently expressed as the energy of those in lower closed shells ETcs plus the energy of those in the degenerate orbitals pm and p n . The molecular orbital energies Wi are conveniently expressed in this approximation as the sum over all bonds, which join pairs of atoms k and 1, of the ith orbital contribution to the bond order pkli multiplied by the resonance integral B(Tk1). =
Vol. 66
+
ETsYm E,
(11)
V(rkl) =
pOe-('kl'-1.40),'a
2(6.667)
(Tkl
- 1.50
(12)
+
U)B
(13)
a = 0.3106 A.
Po = -25.56 kcal. These equations imply that with a bond order the energy of a bond is minimum when = 6.667(1.500 -
pkl,
(14) Several qualitative features of this parameterization are attractive for this study in which bond lengths may exceed the usual small range of lengths found in unsaturated compounds. First, the resonance integral ,f3 is approximately proportional to the overlap between two ?r atomic orbitals on interacting neighbor atoms. Second, the second derivative of p and of the negative of the overlap have the same sign over the interesting range of bond lengths. Third, the u bond potential has a minimum of about 75 kcal. with respect to the separated atoms; this is approximately the dissociation energy of a carbon-carbon single bond. Finally, the stretching force constant for a bond of zero x bond order is only about 10% greater than the stretching force constant for the C-C bond of ethane. A classical parameterization is due to LennardJones. It was applied by Lennard-Jones and Turkevich,6 by Liehr,6,13and by Snyder7 to study the interaction of ?r electrons with bond length displacements in cyclobutadiene. It is basic in many papers on ?r electron theory. pkl
2p(Tkl) =
'/ZKd(Tkl
- 1.3312 -
7'lm)
'/2Ks(?kl
-
1.54)2 - 54.4 kcal.
V ( T ~ I= )
'/2KS(~ki
K d = 1410.906 kcal./i.2 K ,
- 1.54)' =
(15)
(16)
714.091 ked./&?
The constant -54.4 kcal. is added here to give p the magnitude which Coulson and AltmanI4 found satisfactory in a study of the resonance energy of benzene. These equations imply that a bond of bond order pkl will have minimum energy a t the length ru given by the equation (13) -4. D. Liehr, 2. Natu~forsch.,16a, 641 (1961). (14) C. A. Coulson and 5. L. Altman, Trans. Faraday Soe., 48, 293 (1952).
A S I M P L E hlOLECUL-4R
Dec., 1962 rkl =
+ +
1.54 pk1(1.0878225) 1 pki(0.9758064)
8.
(17)
This parameterization is less satisfactory in several ways. First, the quadratic form of V must be incorrect a t large bond distances. Second, the resonance integral p changes sign a t large bond lengths. Finally, its second derivative is opposite in sign to that of the negative of the overlap of two T atomic orbitals. The author believes that the parameterization by Longuet-Higgins and Salem is superior for this study of geometrical instability. Unless otherwise noted, all com~outationsreported here have used those assumptions. Computations to seek that set of bond lengths which minimizes E s y m for the symmetrical molecule or E for the distorted molecule were performed iteratively on an I.B.M. 7090 computer. The following sequeiice of steps is common to both calculations: (a) The resonance integrals p are given initial values. (b) The eigenvalue problem is solved. (c) The total energy, bond orders, and charge densities are computed from the roots and vectors obtained i n b. (d) Bond lengths are computed from eq. 14 with the bond orders obtained in c. (e) The resonance integrals p are revised with eq. 12 and the bond lengths obtained in d. (f) Control is returned to b until the total energy does not change upon iteration. I n the interations to seek an energy minimum for the symmetrical molecule, all p are initially taken equal. In step e, 7 is substituted for v, and vn in eq. 6,9, and 10 for energy, bond order, and charge density. To seek an energy minimum for the distorted molecule, the initial bond lengths are taken to be those computed to minimize E*ym,plus an arbitrary small displacement of any or all bond lengths. The initial p values are computed from these displaced bond lengths. The electrons are now assigned to orbitals pm and pn computed in b so as to fill the lower energy orbital first. Early in this study, a subroutine was written to generate a complete set of C-C bond length displacements having the character of the symmetry group of the T electron Hamiltonian. The change in E upon adding or subtracting each of this complete set of bond length displacements to the bond lengths found for the 5,ymmetrical molecule was studied. It mas found for molecules with a 3-, 5-, 6-, or Tfold axis of rotation that only a single degenerate pair of bond length displacements gave a stabilizing Jahn-Teller interaction with the T electrons. This was found for d l such molecules in Fig. 7 and 8. Only a single displacement was found to give a stabilizing intemction for molecules with a 4-or 8fold axis of rotation. That displacement alternately lengthens and shortens the bonds around the central ring of the six molecules of this study.16
Results The computed changes in energy upon distortion for C4H4+, C4(C6H&+, and CsHs- are collected in Table I. The stabilization upon distortion is (15) hI S Ch,ld
.l.f07.
Phys
9, 601 (1960).
2301
O R B I T A L STCDY O F ,lROMATIC L I O L E C U L E S
LENGTH
(b)
(a,
(C)
Fig. 1.-Computed Huckel description of distorted (a) C4H4+,(b) C4(CaH&+, and (c) CsH8-. Spin density equals absolute value of charge density.
TABLEI DISTORTION EXERGIES IN MOLECULES HAVING DEGENERATE GROUXD STATES Energy changea
C4H4
+
Ca(CeHrh+
AE -2.851 -1.219 -3.084 AE, -6.397 f1.865 AEC +3.546 LE = Emin- E,,in kcal.
CsHs-
-1.804 -4.503 +a.699
about 2 kcal. for each. Computed properties for the distorted molecules are shown in Fig. 1. It is convenient to discuss the energy of these molecules as a function of a coordinate X , which is a linear combination of displacements of all nearest neighbor bond lengths. We define X = 0 for that set of bond lengt,hs which minimizes Esym, and X = 1for that set of bond length displacements which minimizes E of eq. 1. If t'he bond length displacements in Fig. 1 are taken to correspond to X = 1, then there is a -1. Indeed, second energy minimum a t X according to the analysis of McLachlan and Snyder,16 the ground states of these molecules are doubly degenerate, with the two states corresponding qualitatively to the geometries a t X = 1 and X -1. Also according to that analysis, these three molecules are expected to have smsller spin density fluctuations and e.s.r. line widths than were exhibited by the negative ions of benzene, coronene, and triphenylene. The computed stabilization energies for the neutral molecules having pseudo-degenerate ground states are summarized in Table 11. The gain in TABLEI1 DISTORTION ENERGIESIN MOLECULES HAVINGPSEUDODEGENERATE GROUND STATES Energy change"
a
C4H4
C~(CIHS)~
-11.445 - 6.83 -25.764 -20.092 AE, +14.319 $13.258 AE == Emin- E,,, in kcal. AE LET
CaHa
-
7.213 -17.975 10.762
+
energy upon distortion is computed to be about four times as large as in the ions. The 6 to 11 kcal. of stabilization would suggest that these hypothetical molecules would exist in a rectangular stable geometry. The computed bond lengths and (16) A. D. McLachlan and L. C. Snyder, J . Chem. Phus., 36, 1159 (1962).
LAWRENCEC. SNYDER
2302
1.3501
17.00
1.3771
I .820
I
I
(bl tc) Fig. 2.-Computed Huckel description of distorted (a) C4”, (b) C(CsHo)c, a d (c) CsHs. Y
tions, in which an initial displacement lengthens one single bond and shortens the other, indicate geometrical instability, as shown in Table 111. This novel behavior is not, however, to be interpreted as having much chemical meaning. It is a reflection of the cited basic inadequacies of that parameterization. It is true, however, that under both parameterizations E , decreases under this displacement of the rectangular form. TABLEI11 INSTABILITY OF RECTANGULAR CONFIGURATION OF CYCLOBUTADIENE UNDER LENNARD-JONES PARAMETERIZATION
Y
Iteration
I
0 1 2 3 4 5 6
I
+--
I
I
a
(63 db Fig. 3.-Basic molecular orbitals convenient for study of effect of introduction of electron repulsion.
Vol. 66
AE‘
0.0 - 26.8 - 82.7 - 255.9 - 803.9 -2645.0 -9997.8
rii
TU
TU
141
1.330 1.330 1.330 1.330 1.330 1.331 1.335
1.540 1.547 1.533 1.563 1.582 1.620 1.705
1.330 1.330 1.330 1.330 1.330 1.331 1.335
1.540 1.533 1.528 1.519 1.505 1.482 1.447
In cal.
Approximate Introduction of Electron Repulsion.-If one introduces electron repulsion, then the three singlet and single triplet functions of the bond orders are shown in Fig. 2. This approxima- neutral hydrocarbons, which are degenerate in the tion suggests almost alternating double and single Huckel approximation, are no longer degenerate : bonds in the ground state. The long bonds have a these compounds therefore are said to have pseudomobile ‘II order almost zero in C4H4 and C4(C6Hp)c, degenerate ground states. It is of interest to exsuggesting instability with respect to decomposition plore the effect of the introduction of electron reto acetylene and stilbene. In contrast, the long pulsion on the Jahn-Teller interactions. Elaborate bonds of distorted C4H4+and C~(CEHS)*+ have bond computations have been made to describe the elecorders 0.25 and 0.27. This suggests that these tronic structure of square ~yclobutadiene.~9-22 ions may be more stable and easy to prepare than Complete configuration calculations indicate that the neutral molecules. The bond length alterna- its ground state is a singlet and that the lowest tion shown in Fig. 2 for cyclooctatetraene5 recalls excited state is a triplet. There has been only its actual boat form with alternating double and qualitative discussion of the relation of the introsingle bonds. In this modified Hiiclrel approxima- duction of electron repulsion to the Jahn-Teller tion, the energy required to change distorted CsHs stabilization of the lowest singlet ~ t a t e . ~ ~ , ~ ~ to a planar form with alternating double and It is convenient to adopt basis functions like single bond lengths and orders is 24.39 kcal. This (Pa and (Pb in Fig. 3 for the degenerate pair of molecsuggests that the strain relieved upon buckling to ular orbitals. The function (Pa is antisymmetric non-planarity is of that magnitude. with respect to the line x = 0. With these orbitals The present calculations show that in an expres- one may write the following functions and approxision for the potential energy as a function of the mate expressions for their energy bond lengths in the central cyclic ring, for all six molecules, the bonds will be strongly interacting. ‘*sa = ( v a ( 1 ) (~a(2)} E , = E T 2Wa Jaa Evidence for this in Raman spectra of cyclobutaE, = ETcs 2 w b Jbb diene derivatives will be an unusually low anti- *\Ebb = ( P b ( 1 ) (Pb(2)) symmetric stretching frequency relative to the 1 symmetrical stretching frequency of the central fing.” In the infrared spectra, evidence of this interaction is a tendency of the degenerate E stretching mode to split toward double and single C-C bond stretching frequencies. Under the assumptions of Lennard-Jones, symmetrical cyclobutadiene is unstable by 20.9 kcal. with respect to a rectangular configurati~n.~-~ If the stability of this rectangular form with respect (19) G. W. Wheland, Proc. Roy. SOC.(London), 8 1 6 4 , 397 (1938). (20) D. P. Craig, ibid.. A808, 498 (1950). to a complete set of bond length displacements is (21) R. MoWeeny, ibid., A887, 288 (1955). examined, it is found not to correspond to a mini(22) D.P.Craig, in “Nonbeneenoid Aromatic Compounds,” D. Ginsmum of the energy.I8 Instead iterative computa- burg, Ed., Interscience Publishers, Ino., New York, N. Y., 1959, p. 1.
+ +
(17) T. P. Wilson, J . Cham. Phys.. 11, 369 (1943). (18) L. C. Snyder, “Symposium on Molecular Structure and Spectroscopy,” Ohio State University, Columbua, Ohio, June, 1960, Paper B4, and June, 1961, Paper B7.
+ +
(23) S. Shida, Bull. Cham. SOC.Japan, 87, 243 (1954). (24) C. A. Coulson, Contribution “The Fundamentals of Conjugation in Ring Systems,” to the Chemical Society (London) Symposium, Bristol, 1958, Special Publication No. 12.
Dee., 1962
A SIMPLEMOLECULAR ORBITAL STUDYOF AROMATIC MOLECULES
2303
"
Fig. 5.-Geometrical
stability of C4(CsH6)4states after configuration interaction.
Fig. 6.-Geometrical
stability of CeHs states after configuration interaction.
X.
Fig. 4-Geometril:al
stability of C4H4 states after configuration interaction.
The function '\Pas corresponds to the bond length and order alternation depicted in Fig. 2 , while '!bt,b corresponds to the opposite alternation. Of the listed four functions for the symmetrical molecule, only this pair interacts through electron repulsion. ("pas [ 1/T12 I '\Ebb) Kab (18) For the symmetrical molecule the functions and '\Ebb interact to form the highest and lowest singlet states, separated by 2&b. This important integral contains no contribution of overlap charge distributions. It was computed with the approximations of Parr and P a r i ~ e r , *and ~ with the following integrals over atomic orbitals : ( p p l p p ) = 10.98 e.v., (pplqq) = 7.1 e.v. for nearest neighbors, and a classical electrostatic model which places one half T electron 0.82 A. above and below interacting atoms more distant than nearest neighbors. After configuration interaction, the lowest singlet is J,, - J a b above the lowest triplet function. This separation depends mainly on overlap charge distributions. Although Jaa- J a b has been computed approximately and is listed in Table 111, its magnitude is not critical in this study of the effect of the interaction '\Ea& (and*'Ebb on Jahn-Teller geometrical instability. The qualitative consequence of this interaction is that the minimization of electron repulsion requires an equal mixture of and '\Ebb, yet the maximization of the interaction with (25)
R. G. Parr and R. Pariser, J . Chsm. Phys., 23, 711 (1955).
X.
the changing bonds of the core requires or '*bb alone. Thus, no wave function can achieve maximum Jahn-Teller interaction and minimum electron repulsion simultaneously. To afford an approximate description of this situation, the following approximations have been made.26 It is assumed that the electron interaction integrals are independent of the bond length displacement X and have values appropriate for X = 0. It is assumed that ET('\Eaa:X)
= ET('*aa:O)
- K,X
(19)
(26) Subsequent to this lecture, a paper has been published which introduosa electron repulsion in a similar way t o describe large conjugated polyenes: M. Gouterman and G. Wagniere, J . Chern. Phye., 36,
1188 (1962).
LAWRENCE C.SNYDER
2304
I STRUCTURE
MOLECULE
MINIMAL WAVE
j
ESYM -EM IN (~cAL/MOLE)
FUNCTION
1 -
Vol. 66
I
ESYM- ESAD
CALMO OLE)
2.077(L-5)
2,046 (L-S)
4.463(L-J)
3.079 (L-J)
I
I
I
c
1.459
1.459
2.080
2.053
1.095 1.615
1.085 1.381
@X
1.178
1.178
@Y
1.534
1.534
0.611 -
0.561
,
0 0
I
9
,A,
@X
cp
f
~4Ph4
I
I 0.493
a k
QX -
TRIPHENYLENE
0.503
0.495
a -4-
0.402
co R o NE N E-
0.384 -
SYM -TR I PH E p Y LBENZENE-
Fig. 7.-Summary of modified Huckel description of geometrical instability of molecules having orbitally degenerate ground states. Parameteriaations of Longuet-Higgins and Salem ( L S ) and of Lennard-Jones (L-J) are employed. If the last electron enters a molecular orbital with a vertical nodal plane for the distorted molecule of minimum energy, @y is entered; otherwise ax. Experimental observation of these or closely related molecules has been reported: (a) ref. 4, (b) ref. 27, (c) ref. 28. E,(l\kb),:X)
=
E,("kbb:O)
x) = E,(
E;,('\kab :
+ K,X :0 )
(20) (21)
E,(3\kab:X) = E,(3*ab:0)
where
(22)
A SIMPLE MOLECULAR ORBITALSTUDYOF AROMATIC MOLECULES
Dec., 1962
MOLECULE
MINI MAL WAV E FUNCTION
STRUCTURE
a
c3 : H
(c2v)
C4H4
8
~ 7 H f c8148
b
CeHg
'ol
AI(C2d
DI SSOC
5.246 9.540
A1 P 2 v )
4.202 6.550
4.202 6 400
5.249
'qaa OR'qpbb AI
g (Dah)
A1 ( C 2 d
'p\ C4 Ph 4
,Y'
0:'
d
'$4
OR'qbb
+Q?
A I @2v)
\
F
1
0
0
0
0
2 399
2.058
DISSOC
4.638
6.834 DISSOC
11.207
1.838 4.048
1 .e34 3,761
P
e PENTALENE
HE PTALENE
-
Y
?
Phs
7.213 9.832
20.909
P
C
c
-
7.295 (L S) 15.650( L-J)
11.445 DISSOC
A,
c3 P h i
7.4 29 (L -S) DlSSOC ( L - J )
ESYM- €SAD (kCAL/MOLE)
' 9 4 8 OR I q b b
0 0 0 0
c51-1;
ESYM-EM IN (hCAL/MOLE)
2305
f a
Ag(c2h)
0.090 0,495
A g (C2h)
0.298 0,585
BENZODICYCLOPENTADIENE
0 Ag(D2h)
AI ( C 2 d
1,2 :5,6-DIBENZO PENTALENE
0
0 0 008
-
A'(CS)
Fig. 8.-Summary
of modified Huckel descri tion of geometrical instability of molecules having pseudo-degenerate
Experimental observation of these or closely related molecules has been reported: (a) ref. 27, (b) ref. 28, (c) ref. 29, (d) ref. 30, (e) ref. 31, (f) ref. 32. The lowest four molecules in this table are pseudo-aromatic but not pseudo-degenerate: molecular orbital discussions of their geometric stability have been given in ref. 33 and the second paper of ref. 18.
K,
=
-AE,
(23)
Moreover, the average bond length has been found to be almost independent of X in many computations. Thus, we take
E , = +K,X2 K,
=
AE,
(24) (25)
The values of AEr and U,in eq. 23 and 25 are taken from Table 11. Under these assumptions, the energies of the four states after configuration interaction have been computed a t X = 0, X = 1, and Xminlwhich.minimizes the energy of the lowest singlet state .
Xminz=
("->' -( 5)'(26) 2K,
2306
ROBERT S. MULLIKEN
Tiol. 66
species having orbitally degenerate ground states in a Huckel description of a hypothetical planar ~tate.~’-~O The iterative modified Huckel calculaThe resulting description of the interacted states tions described here have been applied to study the and their energy dependence on the displacement X possible geometrical stability of many molecules. are summarized in Fig. 4 to 6 and Table IV. A partial summary of the computed results is made The important observation to make is that whereas in Fig. 7 and 8. Thk summary should provide a the lowest singlet of the neutral molecules is sta- qualitative picture of the importance of the bilized on distortion by 6 to ll kcal. in the simpler interaction of a electrons with bond length displaceapproximation; with the introduction of electron ments in these molecules. As was noted, in molerepulsion, a shallow minimum of only 1.3 kcal. or cules with a 3-, 5-, 6-, or 7-fold axis, a degenerate less occurs. For these pseudo-degenerate neutral pair of bond length displacements interacts with the molecules, the simple Huckel theory almost n electrons. As has been shown by Liehrl* and there may occur both certainly greatly overestimates the Jahn-Teller by Coulson and Golebie~ski,~ minima and saddle-points in the total energy as a stabilization on distortion. function of these displacements. In Fig. 7 and TABLE IV 8, energy minima are referred to by E m i n , and energy saddle-points are referred to by E m d . More deGEOMETRICAL INSTABILITY WITH CONFIQURATION tailed discussions of the results for some of the INTERACTION aromatic negative ions and for the pseudo-aromatic C4H4 CdCsHdr C8Ha molecules are planned. E m i n - Ea,,” -0.46 -0.46 -1.27 It is the author’s hope that these computations, Xmin f O .53 f O .49 & O . 67 while admittedly approximate, Kill stimulate reCI’ mi: 0.91 0.93 0.95 search in this area. Kab 18.5 11.4 9.0 Acknowledgments.-The author wishes to thank K, 25.8 20 1 17.9 Dr. A. D. Liehr for introducing him to the study of K.3 14.4 13.3 10.8 molecules having orbitally degenerate ground Jaa - Jab 7.85 2.54 5.54 states, and Professor R. Breslow and Dr. H. H. All energies in kcal. * See Fig. 4-6. Freedman for discussions of some of their esperimental findings. Summary (27) R. Breslow and H. W. Chang, J. Am. Chem. S o e , 84, 1484 Molecules having orhitally degenerate ground states usually are chemically labile and difficult to (1962). T.J. Kats and H. L. Strauss, J . Chem. Phus., 32, 1873 (1960); separate and study by classical chemical methods. T. (28) J. Kata, J . Am. Chem. Soc., 83, 3784, 3785 (1960). The advent of electron spin resonance, nuclear (29) H.H. Freedman, ibid., 83, 2195 (1961). (30) R. Breslow and H. W. Chang, ibid., 83, 3727 (1961). magnetic resonance, and other advances in chemical (31) T.J. Kata and M. Rosenberger, ibid., 84, 865 (1962). technology have made these molecules suitable for (32) H.J. Dauben and D. J. Bertelli, ibid., 83, 4658 (1961). scientific study. The vigorous efforts of organic (33) P. C. Don Boer-Veenendaal and D. H. W. Den Boer, Mol. chemists are daily adding to the list of observed Phys., 4, 33 (1961). (1
T-DELOCALIZATION I N BUTADIEXE AND CYANOGES’ BY ROBERT S. MULLIKEN Departments of Physics and Chemistry, Unicersity of Chicago, Chicago S7,Illinois Received June 8 , 1968
An examination of Coulson bond orders and of overlap populations obtained from Clementi and blclean’s all-electron SCF-LCAO-MO calculations on CzNz shows a close parallelism to corresponding quantities obtained from r-electron-only SCF-LCAO-MO calculations on 1,3-butadiene1 thus giving support to the validity of the latter. Comparisons of overlap populations for the hypothetical unconjugated butadiene or cyanogen, for the related molecules CIHa or HCN, and for the actual conjugated molecules, show a consistent picture of the effects of r-electron delocalization. This picture is more instructive than that which is seen in similar comparisons of Coulson bond orders.
Introduction As is well known from r-electron-only LCAOMO calculations on conjugated r-electron systems, conjugation is accompanied by r-electron deloca.1ization which increases the Coulson bond order from zero to R finite value for a single bond located between two double bonds, at the same time slightly decreasing the bond orders of the double bonds. When the self-consistent-field (SCF) form of the (1) This work was seeiated by the O5ce of Naval Researoh under Contraat Nonr-2121(01).
method including overlap is used, the computed effects, which are much smaller than when the simple Huckel method is used, are of reasonable magnitude. However, the calculated results still involve the assumption that the effects of the numerous non-a electrons present can be adequately represented by the usual simplified model for the u-electron core. Recent &electron SCF-LCAO-MO calculations by Clementi and McLean2 on one of the simplest (2) E.Clementi and A. D. McLean, J . Chem. Phus., 36,563 (1962).
