SUPPLEMENTAL DATA
Bonding in benzodicylobutadiene isomers:
insights from modern valence bond theory
David L. Cooper†,* and Peter B. Karadakov≠
†
Department of Chemistry, University of Liverpool, Liverpool L69 7ZD, United Kingdom
‡
Department of Chemistry, University of York, Heslington, York, YO10 5DD, United Kingdom
* dlc@liv.ac.uk
Contents
(A)
Full coordinates for benzodicyclobutadiene
S1
(B)
The ‘alternative’ SC(10) solutions for benzo[1,2:4,5]dicyclobutadiene
S1
(C)
Weights for the complete set of Rumer spin functions
S3
(D)
Additional ‘standard’ SC(10) calculations for benzodicyclobutadiene
S3
(E)
SC(6) calculations for benzene
S4
(A)
Full coordinates for benzodicyclobutadiene
The full Cartesian coordinates for both geometries of benzo[1,2:4,5]dicyclobutadiene
and for benzo[1,2:3,4]dicyclobutadiene are listed in Table S1.
Using the same basis set, we found that lower-level calculations, including -space
CASSCF(10,10), MP2 and RB3LYP3, can also provide reasonable geometries for
benzo[1,2:4,5]dicyclobutadiene, as has been noted in previous work [1-5]. We did,
however, also find that the RB3LYP3 description of II is not singlet stable. Furthermore, a
geometry optimization of II with UB3LYP3 converged to a different geometry, in which the
lengths of the annulated bonds were reduced from 1.564 Å to 1.521 Å.
(B)
The ‘alternative’ SC(10) solutions for benzo[1,2:4,5]dicyclobutadiene
Fully-variational SC(10) calculations for benzo[1,2:4,5]dicyclobutadiene at geometry
II converge without symmetry constraints to a ‘standard’ description in which each SC
S1
orbital is mostly associated with a specific carbon atom. On the other hand, again without
symmetry constraints, fully-variational SC(10) calculations for geometry I converge to an
‘alternative’ solution that features four symmetry–unique orbitals: - 1, - 2, - 3 and - 4 (see
Figure S1a). Whereas orbital - 1 (symmetry counterpart - 8) and orbital - 4 (symmetry
counterparts - 5, - 9, - 10) are fairly localized on specific carbon atoms, orbital - 2 (symmetry
counterpart - 6) and orbital - 3 (symmetry counterpart - 7) mostly resemble in- and
out-of-phase combinations, respectively, of functions associated with two symmetryrelated carbon atoms.
The symmetry relations amongst the SC(10) active orbitals in the ‘standard’ solution
for geometry II and in the ‘alternative’ solution for geometry I arose spontaneously from the
fully–variational optimizations, but we may choose to apply appropriate symmetry
constraints on the orbitals so as to generate a ‘standard’ solution at geometry I and an
‘alternative’ solution at geometry II. For both geometries, orbitals - 1 and - 4 in the
‘alternative’ solution (see Figure S1) closely resemble 1 and 3, respectively, in the
corresponding ‘standard’ solution (Figure 7). Similarly, orbitals - 2 and - 3 in the ‘alternative’
solution are much as one should expect for in- and out-of-phase combinations,
respectively, of orbitals 2 and 10 in the corresponding ‘standard’ solution. As such, the
interpretation of the differences between the ‘alternative’ solutions at the two geometries is
much the same as for the ‘standard’ solutions. Furthermore, the differences in energy
between the ‘standard’ and ‘alternative’ solutions are somewhat trivial in the case of
geometry I – the two types of solution are almost degenerate. Which of the ‘standard’ or
‘alternative’ descriptions gives the lowest energy for a particular geometry depends on the
balance between the effects of the different numbers of orbital degrees of freedom and
spin degrees of freedom for the two solutions.
For completeness, orbital overlaps and spin correlation matrix elements for the
‘alternative’ SC(10) solutions are reported in Table S2. According to the value of Q2 3,
which exceeds 0.24 for both geometries, the spins associated with (orthogonal) orbitals - 2
and - 3 are essentially triplet coupled. The occurrence of so-called ‘antipair’ orbitals of this
type, with predominantly triplet coupling, was initially associated in spin-coupled theory
with antiaromatic or diradical character [6, 7], but it has since been shown that they can
even be observed within an alternative -space SC(6) wavefunction for the archetypal
aromatic molecule benzene [8]. McWeeny has shown for the H4 model system how the
existence of such ‘alternative’ SC wavefunctions can be linked to invariances in the total
spin function against one or more subgroups of spin permutations, and has emphasized
the conceptual advantages of using instead the ‘standard’ SC solutions [9].
S2
(C)
Weights for the complete set of Rumer spin functions
Starting with the atom (and orbital) numbering scheme shown in Figure 6, we
superimpose the pattern of singlet-coupled  pairs onto the -bonded framework, using full
lines for singlet-coupled pairs involving spins on adjacent centres and broken lines for
nonadjacent centres. All 42 spin-coupling patterns for benzo[1,2:4,5]dicyclobutadiene are
presented in their standard order [10-12] in Figure S2, along with their inverse-overlap
weights for geometries I and II.
Analogous information is presented for benzo[1,2:3,4]dicyclobutadiene in Figure S3
(using the orbital numbering scheme implied by Figure 10). Starting with Ŕ23 (56.1%), Ŕ32
(5.9%), Ŕ16 (3.8%), Ŕ18 (3.2%) and Ŕ26 (2.2%), we can notionally remove the left-hand
four-membered ring, putting back the H atoms of the benzene ring. As can be seen in
Figure S4, the corresponding bonding patterns of benzocyclobutadiene have weights of
66.0%, 4.0%, 9.4%, 5.0%, and 4.7%, respectively [13].
(D)
Additional ‘standard’ SC(10) calculations for benzodicyclobutadiene
We carried out additional ‘standard’ SC(10) calculations for various subsets of the
different spin eigenfunctions. For some of these, we fixed all of the orbitals ( and ) to be
the same as for the fully-variational calculations based on the full spin space, so that we
optimized only the spin-coupling coefficients ck. In other calculations, we also reoptimized
all of the orbitals. The various energies are reported in Table S3 and the corresponding
inverse-overlap (and Chirgwin-Coulson) weights from selected calculations are listed in
Table S4.
We find for benzo[1,2:4,5]dicyclobutadiene at geometry I that restricting the spin
space to just the five Rumer spin functions shown in Figure 8 has a relatively modest
effect on the SC(10) energy, and relatively little is gained by reoptimizing the orbitals (see
Table S3a). Bonding pattern R19, with double bonds in both cyclobutadiene moieties, does
however have a somewhat reduced weight (see Table S4a). Using fixed orbitals from the
full calculation, it is instructive to compare the calculation that used only R15 and R24 with
the one that used only one of them. The relatively large energy difference of 19.1 kcal/mol
indicates the importance of resonance between the two dominant Kekulé-like modes. For
comparison, the corresponding value for benzene is 19.8 kcal/mol.
Restricting the full spin space of benzo[1,2:4,5]dicyclobutadiene at geometry II to the
eight Rumer spin functions shown in Figure 9 has a relatively small effect on the SC(10)
energy, and little is gained by reoptimizing the orbitals (see Table S3b). A further
S3
restriction of the spin space, so as to include only R8, R35, R1 and R23 gives an energy that
is within 4 millihartree of the full calculation, whether or not the orbitals are reoptimized. As
in the full calculation, the inverse-overlap weights of R8 and R35 are larger than those of R1
and R23, but the Chirgwin-Coulson weights of the four structures are more similar to one
another (see Table S4b). The calculations based only on structures R1 and R23 have a
lower energy than the corresponding ones that are based only on R 8 and R35, but all of
these descriptions are energetically rather poor. At a minimum, we need to include all four
of these bonding patterns, whether or not the orbitals are reoptimized.
For benzo[1,2:3,4]dicyclobutadiene, restricting the spin space to the nine Rumer spin
functions shown in Figure 12 has a very small effect on the energy, regardless of whether
the orbitals are reoptimized (see Table S3c). There is an increase in the relative weight of
structure Ŕ26 (see Table S4c). Even though Kekulé-like mode Ŕ23 never accounts in any of
our calculations for more than ca. 56% of the total, the -electron delocalization that is
introduced by the other eight Rumer spin functions shown in Figure 12 stabilizes the
system by only ca. 3 kcal/mol.
(E)
SC(6) calculations for benzene
For ease of comparison, we report in Table S5 various results for benzene using the
same basis set as was used for benzodicyclobutadiene. The inverse-overlap (and
Chirgwin-Coulson) weights of each Kekulé-like and each Dewar-like bonding pattern in the
SC(6) total spin function are 46.70% (40.46%) and 2.20% (6.36%), respectively.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
R. Boese, J. Benet-Buchholz, A. Stanger, K. Tanaka and F. Toda, Chem. Commun.
(Cambridge, U. K.) (4), 319-320 (1999).
I. Despotović, M. Eckert-Maksić, Z. B. Maksić and D. M. Smith, J. Phys. Chem. A
107 (48), 10396-10405 (2003).
I. Antol, M. Eckert-Maksić, H. Lischka and Z. B. Maksić, ChemPhysChem 5 (7),
975-981 (2004).
S. Sakai and Y. Kita, J. Phys. Org. Chem. 25 (10), 840-849 (2012).
S. Sakai and Y. Kita, Chem. Phys. Lett. 578, 49-53 (2013).
S. C. Wright, D. L. Cooper, J. Gerratt and M. Raimondi, J. Phys. Chem. 96 (20),
7943-7952 (1992).
P. B. Karadakov, J. Gerratt, G. Raos, D. L. Cooper and M. Raimondi, J. Am. Chem.
Soc. 116 (5), 2075-2084 (1994).
P. B. Karadakov, J. G. Hill and D. L. Cooper, Faraday Discuss. 135, 285-297
(2007).
R. McWeeny, Theor. Chim. Acta 73 (2-3), 115-122 (1988).
G. Rumer, Göttinger Nachr. 3, 337 (1932).
S4
11.
12.
13.
M. Simonetta, E. Gianinetti and I. Vandoni, J. Chem. Phys. 48 (4), 1579-1594
(1968).
R. Pauncz, The Symmetric Group in Quantum Chemistry. (CRC Press, Boca Raton,
1995).
P. B. Karadakov, J. Gerratt, D. L. Cooper, M. Raimondi and M. Sironi, Int. J.
Quantum Chem. 60 (1), 545-552 (1996).
S5
Table S1
Cartesian coordinates (in atomic units) from CCSD(T)/6-31G(d) optimizations.
(a) Benzo[1,2:4,5]dicyclobutadiene.
Geometry I
C
C
C
C
C
C
C
C
C
C
H
H
H
H
H
H
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
–2.823054293
–1.335710926
–1.285643776
1.285643776
1.335710926
2.823054293
1.335710926
1.285643776
–1.285643776
–1.335710926
–4.884048799
4.884048799
–2.724827107
2.724827107
2.724827107
–2.724827107
Geometry II
0.000000000
2.186075533
5.100903359
5.100903359
2.186075533
0.000000000
–2.186075533
–5.100903359
–5.100903359
–2.186075533
0.000000000
0.000000000
6.567974755
6.567974755
–6.567974755
–6.567974755
(b) Benzo[1,2:3,4]dicyclobutadiene.
C
C
C
C
C
C
C
C
C
C
H
H
H
H
H
H
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
–1.271283380
–3.816852260
–5.026493335
–2.579463511
–1.397692345
1.397692345
2.579463511
5.026493335
3.816852260
1.271283380
–4.464214397
–7.006157551
–2.413201571
2.413201571
7.006157551
4.464214397
–1.426284452
–2.661964489
–0.360924158
1.056452363
3.333085339
3.333085339
1.056452363
–0.360924158
–2.661964489
–1.426284452
–4.613276501
0.196665983
5.127248804
5.127248804
0.196665983
–4.613276501
S6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
–2.956966876
–1.467569676
–1.383493464
1.383493464
1.467569676
2.956966876
1.467569676
1.383493464
–1.383493464
–1.467569676
–5.019604271
5.019604271
–2.802719187
2.802719187
2.802719187
–2.802719187
0.000000000
2.177790573
4.832714856
4.832714856
2.177790573
0.000000000
–2.177790573
–4.832714856
–4.832714856
–2.177790573
0.000000000
0.000000000
6.321275244
6.321275244
–6.321275244
–6.321275244
Table S2
Orbital overlaps - |-  (upper triangle) and spin correlation matrix elements
Q  (lower triangle) for the ‘alternative’ SC(10) solutions (a) at geometry I and
(b) at geometry II.
(a) Geometry I.
(1)
–0.560
–0.424
0.073
–0.077
0.157
0.197
–0.113
–0.077
0.073
0.744
(2)
0.243
–0.089
0.082
–0.248
–0.328
0.157
0.082
–0.089
0.000 0.049 0.028 0.129 0.000 0.100 0.028 0.049
0.000 0.191 0.107 0.298 0.000 0.129 0.107 0.191
(3)
0.268 0.166 0.000 0.620 0.000 –0.166 –0.268
–0.101
(4)
0.627 0.107 0.166 0.028 0.021 0.010
0.091 –0.729
(5)
0.191 0.268 0.049 0.010 0.021
–0.328 0.082 –0.089
(6)
0.000 0.744 0.191 0.107
–0.420 0.091 –0.101 0.243
(7)
0.000 –0.268 –0.166
0.197 –0.077 0.073 –0.560 –0.424
(8)
0.049 0.028
0.091 –0.032 0.032 –0.089 –0.101 0.073
(9)
0.627
–0.101 0.032 –0.032 0.082 0.091 –0.077 –0.729
(10)
(b) Geometry II.
(1)
–0.555
–0.418
0.197
–0.154
0.090
0.114
–0.066
–0.154
0.197
0.747
(2)
0.243
–0.298
0.166
–0.116
–0.148
0.090
0.166
–0.298
0.000 0.071 0.090 0.089 0.000 0.107
0.000 0.246 0.145 0.130 0.000 0.089
(3)
0.434 0.160 0.000 0.187 0.000
–0.371
(4)
0.610 0.145 0.160 0.090
0.199 –0.455
(5)
0.246 0.434 0.071
–0.148 0.166 –0.298
(6)
0.000 0.747
–0.197 0.199 –0.371 0.243
(7)
0.000
0.114 –0.154 0.197 –0.555 –0.418
(8)
0.199 –0.257 0.223 –0.298 –0.371 0.197
–0.371 0.223 –0.257 0.166 0.199 –0.154
S7
0.090 0.071
0.145 0.246
–0.160 –0.434
0.031 –0.023
–0.023 0.031
0.246 0.145
–0.434 –0.160
0.071 0.090
(9)
0.610
–0.455
(10)
Table S3
Energies from additional SC(10) calculations (in hartree). ‘Fixed’ signifies that
all orbitals ( and ) were taken from the corresponding fully-variational
calculations based on the full spin space.
(a) Benzo[1,2:4,5]dicyclobutadiene at geometry I.
Structures included
All 42
All of those in Figure 8
All of those in Figure 8
R15 and R24 (Kekulé–like)
R15 and R24 (Kekulé–like)
R15 or R24 (Kekulé–like)
Orbitals
Energy
fully–optimized
reoptimized
fixed
reoptimized
fixed
fixed
–382.08781
–382.08554
–382.08531
–382.08457
–382.08397
–382.05358
Label
a1
a2
(b) Benzo[1,2:4,5]dicyclobutadiene at geometry II.
Structures included
All 42
All of those in Figure 9
All of those in Figure 9
R8, R35, R1 and R23
R8, R35, R1 and R23
R1 and R23 (Kekulé–like)
R1 and R23 (Kekulé–like)
R8 and R35 (Dewar–like)
R8 and R35 (Dewar–like)
Orbitals
Energy
fully–optimized
reoptimized
fixed
reoptimized
fixed
reoptimized
fixed
reoptimized
fixed
–382.07430
–382.07281
–382.07216
–382.07091
–382.07039
–382.05982
–382.05756
–382.05701
–382.04530
Label
b1
b2
b3
b4
(c) Benzo[1,2:3,4]dicyclobutadiene.
Structures included
All 42
All of those in Figure 12
All of those in Figure 12
Ŕ23 (Kekulé–like)
Ŕ23 (Kekulé–like)
Orbitals
Energy
fully–optimized
reoptimized
fixed
optimized
fixed
–382.11527
–382.11485
–382.11482
–382.11012
–382.10925
S8
Label
c1
c2
Table S4
Inverse-overlap (and Chirgwin-Coulson) weights from SC(10) calculations
labelled in Table S3.
(a) Benzo[1,2:4,5]dicyclobutadiene at geometry I.
a1
R15
R24
R20
R27
R19
42.62%
42.62%
7.25%
7.25%
0.03%
a2
(37.27%)
(37.27%)
(12.15%)
(12.15%)
(1.18%)
42.34%
42.34%
7.45%
7.45%
0.04%
(37.00%)
(37.00%)
(12.27%)
(12.27%)
(1.45%)
(b) Benzo[1,2:4,5]dicyclobutadiene at geometry II.
R8
R35
R1
R23
R7
R12
R31
R37
b1
25.03% (17.72%)
25.03% (17.72%)
10.62% (20.05%)
10.62% (20.05%)
7.18% (6.11%)
7.18% (6.11%)
7.18% (6.11%)
7.18% (6.11%)
b2
23.93% (16.33%)
23.93% (16.33%)
14.58% (23.37%)
14.58% (23.37%)
5.75% (5.15%)
5.75% (5.15%)
5.75% (5.15%)
5.75% (5.15%)
b3
35.35% (23.56%)
35.35% (23.56%)
14.65% (26.44%)
14.65% (26.44%)
(c) Benzo[1,2:3,4]dicyclobutadiene.
Ŕ23
Ŕ24
Ŕ32
Ŕ9
Ŕ22
Ŕ16
Ŕ18
Ŕ21
Ŕ26
c1
54.61% (52.41%)
8.77% (8.68%)
8.77% (8.68%)
7.87% (8.14%)
7.87% (8.14%)
2.25% (1.94%)
1.31% (2.00%)
1.31% (2.00%)
7.25% (8.02%)
S9
c2
55.65% (53.01%)
8.50% (8.54%)
8.50% (8.54%)
7.63% (8.01%)
7.63% (8.01%)
2.43% (2.03%)
1.17% (1.89%)
1.17% (1.89%)
7.30% (8.08%)
b4
32.54% (21.39%)
32.54% (21.39%)
17.46% (28.61%)
17.46% (28.61%)
Table S5
Results for benzene (D6h) using rCC = 1.39516976 Å and rCH = 1.07556701 Å.
(a) RHF, SC(6) and CASSCF(6,6) energies (in hartree), together with the percentage of
CASSCF correlation energy recovered.
RHF
–230.70264
(0%)
SC(6)
–230.76887 (89.7%)
CASSCF(6,6) –230.77645 (100%)
(b) Energies (in hartree) from additional SC(6) calculations. ‘Fixed’ signifies that all
orbitals ( and ) were taken from the corresponding fully-variational calculation based
on the full spin space.
Structures included
All 5
Both Kekulé–like
Both Kekulé–like
One Kekulé–like
Orbitals
Energy
fully–optimized
reoptimized
fixed
fixed
–230.76887
–230.76856
–230.76853
–230.73696
(c) Orbital overlaps | (upper triangle) and spin correlation matrix elements Q 
(lower triangle) from SC(6) calculations for benzene. The SC orbitals, each of which is
mostly associated with a specific carbon centre, are ordered sequentially around the
ring.
(1)
0.524 0.032 –0.153 0.032 0.524
–0.463
(2)
0.524 0.032 –0.153 0.032
0.184 –0.463
(3)
0.524 0.032 –0.153
–0.191 0.184 –0.463
(4)
0.524 0.032
0.184 –0.191 0.184 –0.463
(5)
0.524
–0.463 0.184 –0.191 0.184 –0.463
(6)
S10
Figure captions
Figure S1.
Symmetry–unique orbitals - 1, - 2, - 3 and - 4 for the ‘alternative’ SC(10)
solutions for benzo[1,2:4,5]dicyclobutadiene (a) at geometry I (left-hand column) and (b) at
geometry II (right-hand column).
Figure S2.
Inverse-overlap
weights
of
all
42
Rumer
spin
functions
for
benzo[1,2:4,5]dicyclobutadiene. The first value is for geometry I and the second one is for
geometry II.
Figure S3.
Inverse-overlap
weights
of
all
42
Rumer
spin
functions
for
weights
of
all
14
Rumer
spin
functions
for
benzo[1,2:3,4]dicyclobutadiene.
Figure S4.
Inverse-overlap
benzocyclobutadiene (taken from Ref. 13).
S11