J. Phys. Chem. 1992,96,1943-1952 (23) Smith, C. M. J . Mol. Struct. (THEOCHEM) 1989, 184, 103, 343. (24) Smith, C. M. J . Mol. Struct. (THEOCHEM) 1990, 209, 273. (25) Frisch, M. J.; Head-Gordon, M.; Schlegel, H. B.; Raghavachari,K.; Binkley, J. S.;Gonzalez, C.; Defrees, D.J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R.; Kahn, L. R.; Stewart, J. J. P.; Fluder, E. M.; Topiol, S.;Pople, J. A. GAUSSIAN 88; Gaussian Inc.: Pittsburgh, PA, 1988. (26) Politzer, P.; Abrahmsen, L.; Sjoberg, P. J. Am. Chem. Soc. 1%4, 106, 855. (27) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (28) Allen, F. H.; Kennard, 0.;Watson, D.G.; Brammer, L.; Orpen, A. G.; Taylor, R. J. Chem. Soc., Perkin Trans. 2 1987, S1. (29) Bader, R. F. W.; Carroll, M. T.; Checseman, J. R.; Chang, C. J . Am. Chem. Soc. 1987. 109, 7968.
7943
(30) Peck, D.G.; Mehta, A. J.; Johnston, K. P. J . Phys. Chem. 1989, 93, 4297. (31) Yamasaki, K.; Kajimoto, 0. Chem. Phys. Left. 1990, 172, 271. (32) OShea, K. E.; Kirmse, K. M.; Fox, M.-A.;Johnston, K. P. J . Phys. Chem. 1991,95, 7863. (33) Murray, J. S.; Lane, P.; Politzer, P. J . Mol. Strucr. (THEOCHEM) 1990, 209, 163. (34) March, J. Advanced Organic Chemistry, 3rd ed.;Wiley-Interscience: New York, 1985. (35) Murray, J. S . ; Politzer, P. Chem. Phys. Lett. 1987, 136, 283. (36) Hamilton, W. C. Statisrics in Physical Science; Ronald Press: New York, 1964. (37) Hermann, R. B. J . Phys. Chem. 1972, 76, 2754. (38) McAuliffe, C. J . Phys. Chem. 1966, 70, 1267.
Spin-Coupled Description of Cyciobutadiene and 2,4-Dimethyienecyciobutane-l,3-diyi: Antipairs Stuart C. Wright, David L. Cooper, Department of Chemistry, University of Liverpool, P.O. Box 147, Liverpool L69 3BX. U.K. Joseph Cerratt,* Department of Theoretical Chemistry, University of Bristol. Cantock's Close, Bristol BS8 1 TS, U.K. and Mario Raimondi Dipartimento di Chimica Fisica ed Elettrochimica, Universitb di Milano, Via Golgi 19, 20133 Milano, Italy (Received: March 3, 1992)
Spin-coupled theory is applied to the description of the ?r electrons in cyclobutadiene, C4H4,and its dimethylene derivatives. This approach is greatly clarified by first considering the simple case of square-planar H4. For C4H4,the theory predicts that the ground state is IBlg,with a 3A2gstate lying at 39.6 kJ mol-' (9.46 kcal mol-') higher, which is in good agreement with experiment. The theory also shows that in the ground state the square-planar geometry is unstable and distorts to a rectangle, while in the triplet state the square-planar geometry is stable. This is also in accordance with observation. A central feature of the present description is that the shapes of the spin-coupled orbitals and the pairing of the spins in these states are highly unusual. Both orbitals and spin pairings are quite different from those of MO theory and bear little relation to those of classical valence bond theory. However by considering the related molecule 2,4-dimethylenecyclobutane-1,2-diyl (DMCBD), we show that our results are consistent and make excellent physical and chemical sense. In particular we see the emergence of what we believe to be a distinctive feature of antiaromatic systems: pairs of electrons in semilocalized orbitals whose spins are coupled to form a triplet. We call these antipairs. C4H4possesses two such antipairs, whereas DMCBD has one. Preliminary results for BBB (bismethylenebiscyclobutylidene,CloHs,a non-Kekul6 isomer of naphthalene) and for the ground IE2/state of C5H5+show that these each have two antipairs.
1. Introduction One of the mast useful results of simple molecular orbital theory is the establishment of the "4n 2" and "4n"rules for conjugated systems. According to these, a planar molecule with a total of 4n 2 conjugated a electrons (n = 1, 2, ...) is aromatic in character, whereas a molecule with 4n such electrons is antiaromatic. By contrast such rules can be derived in valence bond theory only as a result of rather intricate arguments.' In recent years, we have applied the spin-coupled valence bond theory to a wide variety of molecules in both ground and excited states and have shown that this approach incorporates a considerable degree of electron correlation in a compact and highly visual manner.2 In particular, we have used the method to describe the r electrons in a series of planar aromatic molecules including benzene,' naphthalene? and five- and six-membered heterocyclic system^.^-^ The results for benzene, for example, show that the correlation energy included is ca. 92% of that given by a fullvalence (* only) CASSCF calculation. In the spin-coupled description, the a electrons are accommodated in six highly localized
+
+
orbitals which, except for small distortions, closely resemble isolated C ( 2 p ) atomic orbitals. In addition, the special stability of the ring system appears to arise from a symmetric coupling of the electron spins around the ring. In other words, if we include practically all of the nondynamical correlation among the a electrons, the picture which transpires is astonishingly close to that originally due to KekulE. However the crucial difference is that this now emerges as the result of an a b initio calculation, without the imposition of any constraints or preconceptions as to the form of the T orbitals or of the type of spin pairing. An important aspect of these results which is relevant to the present paper is as follows: The "classical" VB picture describes the a electrons of benzene in terms of a single covalent configuration formed from undeformed C(2px) atomic orbitals with five possible spin pairings (two KekulE plus three para-bond or "Dewar" structures). However the resulting energy is always considerably higher than that of a closed-shell M O wave function using the selflsame A 0 basis. To improve matters, it is necessary to include
QQ22-3654/92/2Q96-1943$Q3.0Q/Q 0 1992 American Chemical Society
7944 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
Wright et al. Generally speaking, the correct ordering of the states in MO theory is attained only by including cofliguration interaction. In C4H4this can be done by means of a full CI calculation involving the four valence r molecular orbitals (and the four T electrons). This gives rise to just 20 configurationsfor singlet states and 15 for triplet states. In almost all recent works, such a full CI calculation has been carried O U ~ . ~ ~ - ~ ~ In contrast, the simplest VB wave function for the moleculels leads to the correct sequence of the states. This description considers just four 2p, orbitals, one located at each of the carbon atoms of the molecule, and from them we form two structures which may be represented as '0, =
In
Q
The linear combination 1@l-1+2 gives a wave function of 'BI symmetry, while the 3A2sstate is formed from a combination of structures of the type
Figure 1. Potential surface of singlet and triplet states of C4Hl. The coordinate Q measures the displacement from square planar geometry.
ionic structures. For benzene there are 170 of this type and it is possible to incorporate all of them together with the five covalent functions. The resulting occupation numbers apparently show that the importance of the ionic structures in the final wave function ourweighs that ofthe couulent structures? In actual fact, as everyone knows, benzene is one of the most covalent solvents known. However, as shown by the spin-coupled wave function, the true significanceof the ionic structures in classical VB theory is simply to provide the necessary deformation of the atomic orbitals-which in this case is relatively minor. (The deformation of the orbitals from isolated atomic form is considerably larger in some heterocyclic systems.s,6) The purpose of this paper is to apply the spin-coupled approach to antiaromatic systems. The simplest of these is the cyclobutadiene molecule, C4H4,which is usually considered to possess four ?r electrons. We show that spin-coupled theory describes the electronic states of C4H4to good accuracy for all geometries of the carbon atom framework. At least as important as this is the fact that the form of the orbitals tums out to be highly unusual. The same is true of the type of spin pairing: Both the orbitals and spin pairings bear little or no relation to those of classical VB theory. However, by considering the related molecule 2,4-dimethylenecyclobutane-1,3-diyl, we show that our results are consistent and make excellent physical and chemical sense. In particular, we see the emergence of a distinctive feature of antiaromatic systems: pairs of electrons in characteristic semilocalized orbitals whose spins are coupled to a triplet. We call these "antipairs", This description of antiaromatic systems leads to further consequences, such as the prediction of novel organic materials with unusual kinds of electron spin alignments and consequently with highly unusual magnetic properties. It is now well-established that the ground electronic state of cyclobutadienein the square-planar nuclear configuration (D4,, symmetry) is lB!, with a 3A2gexcited state lying at ca. 0.43 eV (41.5 kJ mol-') in energy above it. In molecular orbital terms, both of these states arise from the same configuration, (a2?e 2, and according to Hund's first rule, the 3A28state should have tbe lower energy. The observed ordering of levels clearly contravenes this rule. In actual fact, the equilibrium nuclear configuration of the molecule in the ground state is known to be rectang~lar,~ the distortion from the square-planar form usually being discussed in terms of a %econd-order Jahn-Teller effect". (A true or "first-order" Jahn-Teller effect requires the electronic state in a nonlinear polyatomic molecule to be degenerate and the energy to vary linearly with the nuclear displacement.) Furthermore it is now reasonably certain that the singlet state lies below the triplet9*I0for all nuclear configurations in the neighborhood of the square-planar geometry, as shown in Figure 1. There are a number of other states which arise from four electrons in the a2" and eg orbitals such as 'Alg and IBZg,but these all occur at much higher energies.
'02=
%,=
ij
in which the dots represent the two electrons which are triplet coupled. Since the structures 1.1 are of much lower energy than those of 1.2, it is obvious that the energy of the lBle state will be lower than that of 3A2,. The ground-state wave function thus incorporates resonance similar to that in benzene, and consequently it is not obvious why this should lead to increased stability in the one case but not in the other. This is a conspicuous shortcoming of the simple VB model. The elementary VB description is greatly improved by the 'generalized VB" (GVB) wave function, in which the orbitals appearing in (1.1) and (1-2) are optimized.I6 As a result of this, pairs of orbitals polarize toward one another so as to increase the overlap between them. Otherwise the orbitals remain highly localized and are generally similar to those of classical VB. This approach affords a good description of the low-lying stat= of C4H4 and in particular gives a value of ca. 0.56 eV for the splitting between the lBls and 3A2gstates. The spinaupled description of the ?r electrons of cyclobutadiene is greatly clarified if we tum fmt to an even simpler system. This is square-planar H4, which we consider in some detail in the following section. We use this to explain the essential features of the spinaupled model and how this differs both from the MO approach and from that of classical VB. We return to C4H4in section 3. Related molecules such as 2,4-dimethylenecyclobutane-l,3-diyl (DMCBD) and 3,4-dimethylenecyclobuteneare briefly discussed in section 4 and the wider implications of this model for antiaromatic systems are discussed in section 5 . 2. Spin-Coupled Wave Functions for Square Planar H,
The wave functions for this system are of the form1' *SM
= CCSk34(919293948~,M;kl k
(2.1)
The four electrons each occupy one of four distinct but nonorthogonal orbitals which are denoted by 41-44. These are not assumed to have the form of localized ls(H) functions but instead are fully optimized as described briefly below. Since each orbital is singly occupied, there are a number of different ways of pairing the spins of the individual electrons to form the required resultant spin S. This is shown by the occurrence in wave function (2.1) of the linear combination
Each term in the sum is a four-electron spin function of net spin S and z component M. The different spin functions are distinguished by the index k. This denotes the particular mode of coupling the individualspins. The coefficients of the spin functions, csk, are determined variationally (seebelow). For the sinslet state (S = 0) of H4, there are just two terms in the sum, and for the triplet state (S = 1) there are three. In general, for a system with
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 1945
Spin-Coupled Descriptions of Antipairs
512 2 312
/ k=2
,,k = l
1
N b
0
1
2 3 4 5 Figure 2. Branching diagram for N = 4, S = 0.
0
5 Figure 3. Branching diagram for N = 4, S = 1.
6
N electrons and resultant spin S, the number of possible spin functionsE, is given by ( 2 s 1)N! = (Y2N S l)! (Y2N - S)!
’
+
+ +
2
3
4
6
upon pairs of electrons. The spins of electrons 1 and 2 , 3 and 4, .,.,N - 1 and N, are first coupled either to a singlet or to a triplet, and these are subquently coupled in a given sequence to produce the required resultant. The individual spin functions in this basis are denoted by indexes of the type
Each term in wave function (2.1) ask = 34(414243~4e$,~;kl
1
k = ((...(~1z~jq)S4;~56S~-N-Z;S”)S )S6;... (2.2)
corresponds to an orbital codiguration in which the electron spins are coupled in a specific manner. We refer to askas a structure. A very important ingredient of this whole approach is the fact that the spin eigenfunctions can be constructed in a number of different ways, each of which highlights a particular aspect of the electronic interactions. We now consider briefly the methods which are most relevant to this work. More details can be found in refs 18 and 19. One of the simplest procedures is the well-known method of Rumer which is used extensively in classical VB theory: The singly occupied orbitals are arranged in a ring and the different spin functions (for a singlet state) are determined by joining up all possible pairs of orbitals in such a way that no two of the joining lines cross one another. This is illustrated for the case of four electrons with zero spin by equation 1.1 : We see that there are two linearly independent ways of pairing the electrons. Another very useful procedure, which is much used in s incoupled work, is that due to Kotani.l8J9 In this method, the are formed by successively coupling the individual spin functions, ai or Bi, according to the rules for angular momentum coupling in quantum theory. We refer to the spin functions formed in this way as the ‘standard basis”. The index k for this case can be represented as a series of partial spins, k (S1S2..SFSw1),where S, is the resultant spin after coupling i electrons. (It is unnecessary to indicate the spin SN since this is just the same as the total spin S.) Thus for the singlet state of four electrons, the two spin functions are represented by
(2.3)
where si+ is 1 or 0 depending upon whether electron spins ui-I and ui are coupled to a triplet or singlet. In the case of N = 4, S = 1, we have in this basis
k = 1 (1,l)l; k = 2 =(l,O)l; k = 3 = (0,l)l (2.4) The index for the first spin function shows that electrons 1 and 2, as well as 3 and 4, are coupled to a net spin of 1, indicated by (l,l), and the two triplet pairs to the required net triplet: (1,l)l. In spin function 2, electrons 1 and 2 form a spin of 1, but 3 and 4 give a spin of zero. The net spin is then naxssarily unity. F i i y in spin function 3, it is electrons 1 and 2 which have a zero resultant spin, 3 and 4 a spin of 1. For N = 4, S = 0, we have
e!,&.: k = 1 (1,l)O; k = 2
(0,O)O
(2.5)
In this last instance, provided that one takes care to choose the correct phases, the standard and Serber bases are identical. A very general program has been written which transforms the spin-coupling coefficients CSk between the Rumer, Kotani, and Serber basesaZ2 The spin-coupled orbitals 41-44 are expanded in a set of basis functions xp, m in number, drawn from all the centers in the molecule, much as in MO-based approaches:
qMk
k = 1 = (hlY2)
k = 2 = (&O’/,)
This shows that in the first spin function, electrons 1 and 2 form a net spin of 1, and similarly electrons 3 and 4. These two triplet subsystems combine to give an overall spin of zero, Le., spin function e&,.l provides an ‘antiferromagnetic” type of coupling. In the second spin function, the spins of electrons 1 and 2 are coupled to give a zero resultant, as are the spins of electrons 3 and 4. Spin function e&,.. thus simply represents two singlet electron pairs. Similar considerations go through for the three triplet spin functions. These are represented as k 1 (f/zlYz) k = 2 E (y21yz) k = 3 I (f/20j/2) In general, the functions in the standard basis ae orthonormal. They can be represented very conveniently by means of the ‘branching diagram” in which the number of electrons Ni is plotted against the partial resultant spin St.This is shown in Figures 2 and 3 for the singlet and triplet cases, respectively. Another basis of spin functions which is particularly convenient in the present application is the ‘Serber basis”.*OJ1 This focuses
m
4fi = cc,x,
(P
P’ 1
= 1-41
(2.6)
The coefficients cw are determined simultaneously with the spinaupling coeffiaentscsk by a variational procedure. In actual practice, we frequently carry out an initial SCF calculation and then take the complete set of MOs, both occupied and virtual, as the basis functions xp. If for a moment we represent each spin-coupled orbital +,, by just a single atomic orbital centered on one of the atoms, xpsay, then wave function 2.1 reduces to a classical VB wave function, e.g., for the singlet state 9 = C l q c*@2 (2.7)
+
where
4 k = ~ ~ X l X * X 3 X 4 ~ ) k(kJ = 192) in which each x, is a hydrogen-like 1s function centered on the
relevant nucleus. If the spin functions el, are taken to be those of Rumer, then the coefficients c1 and c2 are +1 and -1, respectively (wave function 2.7 however still remains to be normalized), and wave function 2.7 is actually the same as that depicted in eq 1.1 for C4He From this perspective, it can be seen that by simultaneously optimizing the orbitals 4,-44 and the
Wright et al.
7946 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 TABLE I: Wave Functions and Energies of H, Singlet States labeling of centers: a--b I
d-c
geometry D4*, 1 A side D4h,1 A side D4h, 5 A side D4*,5 A side rectangled
energy (au) -1.915 105598 -1.915071935 -1.866 327 842 -1.866327842 -1.965 184948
I
spin-coupling CoefP 0.999889 1.0 1.0 1.0 0.238362
-0.01487Ib 0.W 0.0' 0.W 0.971 177'
"Spin-couplingcoefficients are in the Kotani basis. In the present case of N = 4, S = 0, this is also identical to the Serber basis. 'Orbitals are in the "standard ordering", eq 2.8: 4I H a + c, 42 H a - c, @3 H 6 d, 44 H 6 - d. 'Localized orbitals, @ I 3: a; b2 H c; b3 H b; b4 N d; see text. dSides (a$), (c,d): 1.100 A; sides ( 6 4 , (a&: 0.985 774 A. eLocalized orbitals: 4I H a + Ad; 42 d + Xa; 43 H b Ac; b4 H c + Ab; see text.
+
+
TABLE II: Wave FUDCHOMand Energies of H, Triplet States (Labeling of Centers as for Table I) spin-coupling CoefP geometry energy (au) D4h, 1 A side -1.900778498 0.968311 -0.176597 0.176597' rectangleC -1.91 1277 353 0.971 526 -0.167 536 0.167 536 "Spin-coupling coefficients are in the Serber basis. See text. bOrbitalsare in the "standard ordering", eq 2.8: = a + c, 42 H a - c, #3 H 6 + d, 44 N b - d . 'Sides (a$), (c,d): 1.100A; sides (6,c), (a&: 0.985 774 A.
coefficients c1 and c2, the spin-coupled approach provides a very flexible generalization of the VB approach. The significance of this will become clear in the following. Total energies, orbitals, and spin-coupling coefficients for H4 were determined using the standard basis of spin functions and an STO-3G basis set for the orbitals. This last was used primarily to facilitate comparison with another calculation,u see below. The results are shown in Table I for the singlet state and in Table I1 for the triplet state. Let a-d denote localized distorted s-like orbitals on successive hydrogen atoms as we move clockwise around H4. The converged spin-coupled orbitals 41-44 for the square-planar ground-state geometry can be characterized as follows:
dl H a + c;
42 44
N-
s
a - c; b-d
43- b + d ;
(2.8)
Orbital 41stretches diagonally across the H4square and represents an in-phase combination of the functions a and c. Orbital 42does so as well but is the out-of-phase combination of a and c. As a result orbital (b2 possesses an additional node in a plane which is perpendicular to the molecular plane and passes through the &d axis. Similar considerations hold for 43and 44,except that these involve the other diagonal of the H4 square. From this it is easy to see that only orbitals 4, and 43overlap; 42and 44are orthogonal to them and to each other. It is worth emphasizing that this property of the orbitals emerges from the calculation and is not imposed beforehand and indeed occurs only as long as the D4h symmetry remains. We take the ordering of the spin-coupled orbitals as given in (2.8) as our "standard" ordering. Thus the symmetry of the orbitals of H4 is such that they fall and ($1~,4~). Each set is invariant into two sets of two: under one of the Dlh subgroups of the molecular point group D4h, the two sets being interchanged by a rotation by u/2 about the C4axis. (For a general discussion of the symmetry properties of spin-coupled orbitals, see ref 17.) As a result of this (see Appendix), each structure @Ok
= 34(41424344%.0;k1
in wave function 2.1 possesses the correct IBI, symmetry of the ground state. From Table I we see that the spin-coupling coefficients show a great preponderance of spin function 0&,.,(cl = 0.999 889,c2
= -0.014871). This corresponds to the "antiferromagnetic" type of coupling mentioned above. However it is important to note that, at least for the square-planar geometry with sides of length 1 A, the spin-coupling coefficient c2, though small, is not zero. Since the form of the orbitals is so different from that expected on the basis of classical VB theory, it is useful to show the relationship between the two approaches. For this purpose we use the explicit decomposition of the spin-coupled orbitals 41-44 in terms of atom-based functions, 2.8. Substituting in eq 2.1 and using the standard spin functions, we obtain after a little algebra
% = cI [ (1/2).4abcded,o,ll + (fi/2).4abcdB:,,,,I] + c2A((a2- c?)(b2- &)ed,o,21 (2.9) The first term, cl[ ...I, has the form of the classical VB wave function for H4 expressed in the standard basis of spin functions rather than in terms of the more familiar Rumer functions. It incorporates the resonance of classical VB theory. The parts involving coefficient c2 consist of a series of doubly ionic structures. However if the ionic structures are neglected, i.e., if coefficient c2 becomes zero-as indeed occurs as the size of the H4 square is increased (see below)-the spin-coupled wave function reduces instead to the single structure:
(2.10) This function is unchanged by any transformation amongst the orbital pairs (41,42)and (43,44),and as can be appreciated from eq 2.8, one possible form for the orbitals is the completely localizad representation: $1
S!
a;
42
b; 43
N-
C;
44
N
d
(2.11)
It should be noted that this freedom is absent in the full wave function 2.1 or 2.9. From this it is clear that the presence of the ionic structures, even though their contribution is small, profoundly alters the form of the spin-coupled wave function. The comparison with the case of benzene is instructive. In both instances, the role of ionic structures in the spin-coupled wave function is small but crucial. In benzene, their presence provides a minor degree of deformation of the orbitals from isolated atomic form but more importantly confers on the wave function a quality and a consequent conviction which is absent from the classical VB description with covalent structures only. In the present case, the contribution of the ionic structures is equally small but nevertheless leads to quite a different form for the orbitals which, as we attempt to show here, gives rise in turn to a new and useful view of antiaromatic systems. As can be seen from Table I, it is possible to converge upon another root of 'B1, symmetry whose wave function is precisely of the form (2.10)and which lies at only 9.16 X l@ eV in energy (0.088 kJ mol-') above the ground state. The invariance of this wave function to transformations among the pairs (~$],4~) and ( 4 1 ~ ~causes 4 ~ ) singularities in the second-derivative matrix which we use in our "stabilized Newton-Raphson" procedurez4 for minimizing the energy. These singularities may be removed by sufficiently constrainingone or more of the coefficients c,, in eq 2.6 so that the orbitals adopt a definite representation, (i.8)or (2.1l), for example. Thus the orbitals in this very low-lying solution for H4 possess an unusual degree of freedom. However this is at the expense of a more important degree of freedom: the linear combination of different spin functions. If the form of the orbitals is constrained so that of symmetrically distorted atomic functions (as indeed one might expect on the basis of the spin-coupled result for benzene3),by imposing the form 61 3 u + h(b + d) + IC; 42 b + X(U + C) + pd; 43 I? c + X(b + d) pa; 44 -d + X(a + c) + p b (2.12)
+
then there is only where A and p are parameters to be a single spin function which provides the correct 'B, symmetry. In the standard basis, this is just the combination &own in the first part of expression 2.9. Consequently we believe that an excited solution of the form 2.10does not represent anything more
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 1941
Spin-Coupled Descriptions of Antipairs than a stationary point on the surface of parameter space. Certainly it does not have the "feel" of a true electronic state, and it has no analogue in the case of cyclobutadiene (following section). As the length of the side of the H4 square RH is increased, the role of the ionic structures in the ground state diminishes and coefficient c2tends to zero. Indeed at RH = 5 A, only one solution for the singlet state is found, that corresponding to wave function 2.10. The orbitals assume either the form 2.8 or the localized form 2.11 depending on the initial guess. As can be seen in Table I, the energy in the two cases is identical. When the bond lengths of H4 in the ground state are distorted to a rectangular geometry, the energy decreases very sharply and the orbitals localize. The spin-coupling coefficients, both of which remain nonzero, assume values characteristic of two H2 units. This change occurs with astonishing rapidity. We define as 100% distortion from the square if the a,b and c,d distances are increased from 1 to 1.100 A, and the a,d and b,c distances decreased from 1 to 0.985 774 A, so that the ratio of the lengths of the sides is the same as for rectangular C4H4(as given in Table IV). As a result of such a distortion, the orbitals assume the form
41 N (a + Ad);
42
Y
(d + XU);
43
( b + Ac); 44 Y (C + Ab) (2.13)
E
From Table I we see that the corresponding spin-coupling coefficients are ~2 = 0.971 177 CI = 0.238 362 which clearly indicates the formation of (a,d) and (b,c)pairs. That is, the system very rapidly shows the formation of two H-H bonds, each of which is similar to the bond in H2. The value of the coefficient c2 is worth noting. If the corresponding spin function were neglected-as one might be tempted to do for DO,+ symmetry-it would not be possible to describe the potential energy surface of H4 correctly. Even if the a,b and c,d distances are increased from 1 A by just O.OO0 25 A, and the a,d and b,c distances decreased by 3.557 X A, we observe significant localization of the orbitals. As defined above, this corresponds to a 0.2595 deviation from D4h symmetry. The lability of the orbitals in the square-planar geometry shows that in a real sense, this conformation should be regarded as a transition state. In the lowest triplet state of H4, the orbitals possess the same unusual form 2.8 as those in the square-planar singlet case. The symmetry of this state in the square-planar geometry is 3A2 and is most clearly displayed by employing the Serber basis ofspin functions 2.4. In this the wave function has the form (see Appendix) W3A2J = c l @ l+~ 4% - @ I 3 1 in which the structures @lk
(2.14)
alkare given by 34{41424344e:.M;k)
(cf. expression 2.2). Total energies and spin-coupling coefficients for both the square-planar and rectangular geometries are shown in Table 11. Note that the spin coupling coefficients are given there as cl, c2, and -c2. The predominance of structure all(Le. of spin coupling 1 in q 2.4), cI = 0.968 3 11, shows not only that are the orbitals of the 3A28state very similar to those of lBls but also the mode of coupling of the spins in the two states is also very similar. It is also worth noting that on distortion to a rectangle, the orbitals do not localize. Instead the form 2.8 is stable. A similar behavior is found in C4H4, to which we now turn.
3. Cyclobutadiene: Results and DLscussion In the present work, the wave functions for the system are of the form
Here the 24 electrons which constitute the core of the molecule are explicitly shown as occupying 12 doubly occupied orbitals fii
TABLE III: Wave Functions and Energies of C4H4Singlet States labeling 01 centers: a-b I
d-c
I ~
geometry' of square:
origin is at center of mass one C is at ( x y ) = (0.700,0.700) one H atom is at (x,y) = (1.464,1.464)
geometry' of rectangle:
origin is at center of mass one C atom is at ( x y ) = (0.7700,0.6900) one H atom is at ( x y ) = (1.5478,1.4678)
geometry D.,
rectangle rectangle
energy (au) -153.738435968' -153.678 529 207d -153.750979 118
spin-coupling cocffo 0.999865
0.016468'
0.153851
0.988094e
"Spin-coupling coefficients are in the Kotani basis. In the present case of N = 4, S = 0, this is identical to the Serber basis. Core taken from SCF calculation on state. See note b of Table V. 'Orbitals are in the "standard ordering", eq 2.8: 4I y a + c, 42 a - c, 63 b d, N b - d. dSCF wave function for singlet state. cLocalized orbitals: dl N a + Ad; 42 Y d + Xa; 43 N b Xc; &, c + Ab; see text. /All distances in angstroms.
'
+
+
TABLE IW Wave Function and Energies of C,H4 Triplet States (Geometries and Labeling of Centers PI)for Table 111) geometry energy (au) spin-coupling coeffo -153.689 615 410' -153.723 384385 0.978645 -0.145 354 0.145 354' rectangle -1 53.667 792 749d rectangle -153.709 742030 0.982694 -0.130981 0.130981'
Dir O4h
1Blg-3A28Splitting (Spin-Coupled, D4hSymmetry): 0.410 eV (39.6 kJ mol-')
"Spin-coupling coefficients are in the Serber basis. See text. bSCF wave function for lAZpstate. 'Orbitals are in the "standard ordering", b + d, 44 N b - d. dSCF wave eq 2.8: N a + c, 42 Y a - c, @3 function for triplet state.
( i = 1,2, ..., 12) of u symmetry. The corresponding spin function represents 12 singlet pairs:
e&f= fl(a182
- bla2)***fl(a23b24
- bZ3a24)
(3.2)
The last four electrons each occupy one of four nonorthogonal orbitals which we denote by &+4, similar to the case of H,, These orbitals are now of 7r symmetry, and we focus our attention on these. The associated spin functions are exactly as desrribed above. The basis set for determining 41-44 consists of a complete set of occupied and virtual molecular orbitals of u symmetry, obtained from an initial SCF calculation on the cyclobutadiene system. This is strictly equivalent to expanding the orbitals in terms of all the atom-centered basis functions of 7r symmetry. This same SCF calculation also furnishes the doubly occupied m e orbitals fi,-+12. These last remain "frozen" in the spin-coupled calculations. The energies and spin-coupling coefficients for C4H4 were obtained by using a tripletvalence plus polarization (TZVP) basii. For C/H, this consists of (lOS6p/Ss) Gaussian functions contracted to [5s3p/6s] and augmented by polarization functions with exponents d(C) = 0.72 and p(H) = 1.0. The exponents used for H are 33.64, 5.058, 1.147,0.3211, and 0.1013.25The results are shown in Table I11 for the singlet state and in Table IV for the triplet state. The orbitals for the singlet ground state in the square-planar and rectangular geometries are shown in Figures 4 and 5, respectively. The corresponding orbitals for the triplet state are shown in Figures 6 and 7. As can be seen from the tables, the gain in energy on passing from the SCF to the spin-coupled description is considerable, ranging from 0.034 au for the square-planar triplet state to as much as 0.072 au for the rectangular singlet state. (It should be noted in Table I11 that the spin-coupled calculation for the square-planar geometry uses RHF orbitals from the triplet state.) The magnitude of this last energy difference highlights the in-
Wright et al.
'948 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
41 #2 Figure 4. Orbitals of the singlet ground state of C4H4;square-planar geometry; 4, N a c; 42 N a - c; 43 N b d; 44 N b - d. Contours of l4,,(r)l2 are plotted in a plane la, above the molecular plane.
+
+
Figure 41 6. Orbitals for the triplet state42of C4H4square-planar geometry; - C; ,p3 N b d; 44 N b - d. The contours are 4I a + c; 42 plotted as in Figure 4.
+
Ii X
x
QJ X
@/
-_
& 04 Figure 5. Orbitals for the singlet ground state of C4H4, rectangular geometry: 4, N a Ad; 42 N d + Xa; 4 N b + Xc; d4 N c + Ab. The contours are plotted as in Figure 4.
Figure 7. Orbitals for the triplet state of C4H4,rectangular geometry: 4, N a + c; 42 N a - c; 43 N b d; 44 N b - d. The contours are plotted as in Figure 4.
consistency in the MO description of the distortion of the molecule from the square-planar to rectangular geometry. We were unable to locate a second very low lying root of lB1, symmetry, in contrast to the case of H4.This solution may well be a saddle point in the present case. The separation between the singlet and triplet states in the square planar configuration is 0.410 eV (39.6 kJ mol-', 9.45 kcal mol-'). It is clear from Figures 4-7 that the general form of the s orbitals for cyclobutadiene is very similar to those of H4, i.e. &'et a + c ; & a - c ; tp3 b + d ; $4 6 - d (3.3)
but where a-d now change sign under reflection in the plane of symmetry. This leads to a slight modification in their behavior under the operations of D4h,as compared to H4 (see Appendix). However, each individual structure in wave function 3.1 continues to possess the correct lB1,symmetry in the square-planar configuration of the ground state just as in the case of H4. As can be seen from Table 111, spin function plays the dominant role, with a coefficient of 0,999 865. In the 3A2,state, the three structures all,a12, and 9 1 3 behave in the Serber basis as described by eq 2.14; i.e., allpossesses the full 3A2,symmetry, whereas only the combination @12-@13 does
+
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The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 1949
Spin-Coupled Descriptions of Antipairs
a
TABLE V: Energies md Wave F~mctiolrsfor 2&Dimetbykmeyclobutme-1,3-diyl 3B2uState total energy (au)
RHF -230.61153
b
so. This is reflected in the values of the spin-coupling coefficients given in Table IV. It can be seen from Tables 111 and IV that the electronic structure of the IB,, and ’A, states of squareplanar C4H4is very similar. In particular, both states show a remarkable preponderance of an antiferromagnetic coupling, in which the electrons in orbitals and 43,44are coupled to form a triplet, the two triplets then coupled to give the required overall spin. The substitution of orbitals 3.3 into the total wave function 3.1 leads to a d m m p i t i o n of the spin-coupled wave function in terms of YclassicalnVB structures similar to that discussed in section 2 for H,. It is necessary to bear in mind only the u symmetry of the functions involved here. When we move to a rectangular geometry, we see that the four spin-coupled orbitals of the ground state immediately become highly localized: For the geometry changes indicated in Table 111, in which the shorter C-C bonds are (a,d) and (b,c),the orbitals assume the forms 41 Y u + M; 42
H
d
+ XU;
Spin Functions in the Serber Basis‘ S4 $56 spin-couplingcoeff 2 1 8.56534 X lo-’ (1‘1) 4.90744 X (190) 1 0 (190) 1 1 4.323467 1 0 2.63941 X lo-* (191) (191) 1 1 1.99718 X 10-* (091) 1 1 4.323 467 (091) 1 0 4.90744X 1W2 0 1 -6.93279 X (191) (0s)) 0 1 0.883766
k
Figure 8. 2,4-Dimethylenecyclobutane-1,3-diyl(DMCBD). The p i tions of the carbon atoms marked 1-4 and a, b are a s d a t e d with highly localized functions of r symmetry.
4 3 b~+ XC; 44 N
c
+ Ab
very much as in eq 2.13. The values of the associated spin-coupling coefficients, 0.153 851 and 0.988094, clearly show that essentially we see the formation of two C-C double bonds, similar to those in ethene. Just as in H4,the singlet square-planar geometry is to be regarded in a real sense as a transition state. However on distortion to a rectangular geometry, the orbitals of the )Aze s t a t e i n sharp contrast to those of the ground s t a t e r e t a m their form, and the total energy is found to increase: The squareplanar conformation,with two electron pairs, is stable.
4. Rehted Systems: 54- and 3,CMmethylene Derivative8 of Cyclobutaaiene The overall consistency of the spin-coupled results for cyclobutadiene is better appreciated when seen in a wider context. For this purpose we have examined the 2,4- and 3,4-dimethylene derivatives of C4H4,which possess six r electrons. A preliminary account of our fmdings for these systems has already appeared.26 The 2,4diiethylenecyclobutane-1,3-diyl molecule (DMCBD) is a nonaromatic isomer of benzene. As shown in Figure 8, it may be imagined as being derived from cyclobutadiene by substitution of CH, group for the hydrogen atoms at opposite comers of the ring. DMCBD is produced by photolysis of an appropriate diazine pre~ursor.~’It is yellow-orange and displays highly structured absorption (A, = 506 nm) and fluorescence (Lx =: 510 nm) spectra. The triplet spin and planar geometry have been confirmed In particular, proton hyperfine recently using EPR splitting has been observed, and this shows that the spin densities at the ring and methylene carbons are almost equal. We return to the significance of this below. All competent theoretical models are in agreement that the ground state of DMCBD should be 3B2uin DShsymmetry. In particular Davidson et al.29 have carried out complete CI calculations in the r space using a minimal basis set. Besides the 3B2u ground state, they predicted the lowest singlet state to be ‘&-for which a two-configuration SCF description was found to be essential-rather than the open-shell lBzustate, as might be expected on the basis of HUckel theory.
spin-coupled -230.679 051
(812j34)
1
2 3 4 5 6 7 8
9
IA, State
total energy (au)
RHFb
spin-coupled -230.643 347
SDin Functions in the Serber Basis’ (SI~J;~) s4 s56 spin-coupling coeff 1 1 -3.69170 X (190) 1 1.52951 X lo4 (1J) 1 (091) 1 1 -3.69069 X 10” 6.32680X (191) 0 0 (090) 0 0 0.997997
/c
1
2 3 4 5
‘Spin functions are labeled by k = (($I2jj4)S4;(56)S, as described in separate RHF calculation on the ‘A state was carried section 2. out. The doubly occupied orbitals are those of the fBzustate. The spin-coupled wave functions for this system are similar to those for C4H4: *S,M
C C s k ~ ( + ~ . : . . . ~ 8 4 1 ~ 2 . . . 4 ~ e ~ ~ ~(4.1) ~,M;k~ k
but where there are now 18 doubly occupied orbitals of u symmetry and six singly occupied nonorthogonal orbitals of r symmetry. For six electrons there are nine spin functions for S = 1 and five for S = 0. The Serber basis was again used (see equation for& preceding eq 2.2), since thii displays particularly clearly both the significant spin couplings, and the effects of the symmetry of the system. The spin functions for the triplet and singlet states in this basis are listed in Table V). (Calculations were actually carried out using the standard basis of spin functions, and the resulting spin-coupling coefficients were subequently transformed as described in ref 22. See also the Appendix). Calculations were carried out on the lowest triplet and singlet states of DMCBD using the same kind of TZVP basis as used for cyclobutadiene. Using the partially optimized nuclear geometry determined by Davidson et ~ 1 . : ~a standard restricted HartreeFock (RHF) calculation was first performed for the 3Bh state. Since those authors found that molecular orbitals optimized for this state also gave the best CI description for the ‘Agstate, we utilized these MOs for the orbitals +1+12 in (4.1). Spincoupled calculations were then carried out for the electrons of the mystem, yielding the orbitals t#q-& and the spin-coupling coefficients cSk. The results are shown in Table V,and plots of are displayed in Figure 9. From Figure 9 it can be seen that orbitals and 43,44are two pairs of well-localized functions associated with the carbon atoms shown in Figure 8. If we associate two localized functions of ?r symmetry, a and b, with the two remaining C atoms of the ring in Figure 8, then orbitals 45 and 46 may be characterized as
45-a+b
46-0-b
(4.2)
Wright et al.
7950 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 45 X
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Figure 9. Orbitals 41-66 of the 3B2ustate of DMCBD: 45 Na-b.
= a + b; 46
In other words, orbitals 45and $6 form precisely the same kind of triplet pair as found in C4H4. The two orbitals are orthogonal. It may be seen from the spin-coupling coefficients given in Table V,that the major contribution to the 3B2ustate of DCMBD arises from spin function 09 = ((0,0)0;1) (4.3) in which the methylene C atoms are connected to the ring by two normal (Le., singlet-coupled) 7r bonds, and the triplet character arises entirely from the $5946 pair. However the coefficient for this coupling is only -0.884, and an important contribution to the total wave function arises from the combination of spin functions 0 3 + 0 6 = ((1,O)l;l) + ((0,l)l;l) (4.4) for which the coeffcient is -4.323. This shows that in addition to $5,$6, the exocyclic ?r bonds also possess triplet character. The triplet nature of the 3B2ustate is thus significantly spread throughout the three electron pairs. This accords with the observed proton hyperfine couplings in the EPR spectrum. The spin polarization at carbon atoms 1-4, which this indicates, can arise onZy from a triplet coupling of the corresponding pair of 7~ orbitals, or $3,$4. [To obtain spin polarization at a particular carbon atom, we would need to replace at least one of the doubly occupied MOs of symmetry by two appropriate singly occupied orbitals of the same symmetry, ($'&",.). A partial triplet character of the resulting configuration, combined with a triplet coupling of the correspondingpair of ?r orbitals or $3,$4, would then produce the required nonzero spin density at the relevant carbon atoms. However this extension takes us beyond the scope of the present work.] If the coefficient of spin function (4.3) were -0.999 say-as one might surely expect on simple chemical grounds-then it would be difficult to explain this phenomenon. It is worth noting that the coefficients of spin functions e5= ((i,i)i;i) e4= ((1,i)i;o) are zero to all intents and purposes, since these do not yield a state of symmetry Bzu. Spin-coupled calculations on the lA, state of DMCBD were carried out at the same geometry as for the 3B2u state. However it should be noted that the planar geometry is not a true minimum
-
Figure 10. Orbitals 41-46 of the 'A, state of DMCBD. The contours are plotted as in Figure 4.
1
5
6
3 Figure 11. 3,4-Dimethylenecyclobutene.The positions of carbon atoms marked 1-6 are associated with highly localized functions of ?r symmetry.
for this state but is instead a transition state for the formation of a nonplanar structure, 2,4-dimethylenebicyclo[ 1.1.O]butane, via the development of a transannular bond.30 Results of the calculations are given in Table V, and the orbitals are displayed in Figure 10. From this last it can be seen that in contrast to the triplet ground state, they are now all highly localized. Thus instead of eq 4.2 we now have $5-a $6-b (4.5) The overlap between these two orbitals is -0.309, indicating the incipient formation of a fairly strong bond across the ring. This suggests that a nonplanar structure which allows this bond to shorten would be more stable. The other orbitals, $1-$4, are similar to their counterparts in the 3Bh,state. As can be seen from Table V, the coefficient of spin function e2 = ((1,1)1;1) is zero, since this does not give rise to a state of symmetry lA,. Otherwise, the spin-coupling coefficients for this state indicate an overwhelming preponderance of three normal electron-pair bonds. The value given by the spin-coupled calculations for the singlet-triplet splitting in this molecule, assuming an identical planar nuclear configuration in the two states, is 0.972 eV (22.4 kcal mol-I). However this value is likely to be reduced substantially
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 7951
Spin-Coupled Descriptions of Antipairs -~
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