Supplemental data
Combining symmetry-separated and bent-bond spin-coupled models of
cylindrically-symmetric multiple bonding
Fabio E. Penotti†,* and David L. Cooper‡
†
Consiglio Nazionale delle Ricerche, Istituto di Scienze e Tecnologie Molecolari, Via Golgi 19,
I-20133 Milano MI, Italy
‡
Department of Chemistry, University of Liverpool, Liverpool L69 7ZD, United Kingdom
* Corresponding Author: E-mail: f.penotti@istm.cnr.it
Contents
S-A: Results obtained using additional constraints on the spin-coupling coefficients
S2
S-B: Spin and symmetry considerations for symmetry-separated SC(σ,π) configurations
S2
S-C: Spin and symmetry considerations for bent-bond SC(ω) configurations
S6
Additional references
S10
S1
S-A: Results obtained using additional constraints on the spin-coupling coefficients
Instead of mixing VB wavefunctions with rotated versions of themselves, as is described in the
paper, rigorous 1+g symmetry can also be achieved by imposing appropriate additional constraints
on the spin-coupling coefficients. The theoretical background to this for the symmetry-separated
(σ,π) and bent-bond (ω) configurations is explained in detail in Sections S-B and S-C, respectively.
The results of such calculations are summarised in Table S1. For both molecules, SC(σ,π) lies lower
than SC(ω) and relatively little is achieved by combining these two descriptions, unless the valence
orbitals are reoptimised. Mixing in also an RHF-like configuration leads to a further energy
improvement. Nonetheless, in all cases, the energies are inferior to the analogous ones that were
obtained with our preferred strategy in the paper (see Tables 1 and 2).
Table S1. Energies (in hartree) for C2H2 and N2 with rigorous 1g+ symmetry imposed in the VB
calculations via additional constraints on the spin-coupling coefficients.a
Wavefunction
a
C2H2
N2
SC(ω)
–76.94194
–109.07828
SC(σ,π)
–76.94499
–109.08420
SC(σ,π)  SC(ω)
–76.94555
–109.08543
SC-like(σ,π)  SC-like(ω)
–76.96215
–109.10472
SC-like(σ,π)  SC-like(ω)  RHF-like –76.98192
–109.12408
All of the active orbitals were (re)optimised in all cases except SC(σ,π)  SC(ω). Inactive (core)
electrons were accommodated in doubly-occupied MOs taken directly from RHF.
S-B: Spin and symmetry considerations for symmetry-separated SC(σ,π) configurations
When examining the symmetry properties of the string of four singly-occupied π orbitals it is
useful to substitute electron variables x and y: these have the same functional dependence on the
angle of rotation about the internuclear axis (z) as the corresponding orbitals. With this in mind we
consider the following product
x y x y
(S1)
in which the first factor refers to electron 1, the second to electron 2, and so on. This product is
obviously invariant to reflections in either the x = 0 or y = 0 planes, but for it also to be invariant
under reflection in the x=y plane, this overall space-spin configuration must be invariant under the
S2
simultaneous transpositions of x with y and of x with y. [These simultaneous swaps are equivalent
to a rotation around z of 90°.] A necessary and sufficient condition for invariance under this double
swap is that the vector of spin-coupling coefficients is restricted to the invariant eigenspace of the
corresponding permutation [1]. This then ensures that this overall space-spin configuration
transforms as the totally symmetric irreducible representation of the C4v molecular point group.
In the present study, the symmetry relations amongst the singly-occupied π orbitals in the
converged SC(σ,π) wavefunctions also give us the conditions x = σ̂h x and y = σ̂h y. These are of
course in addition to the symmetry relations for σ̂h-related σ orbitals: the vector of spin-coupling
coefficients must be an eigenvector of the permutation that corresponds to performing the σ̂h
operation on the full orbital string. Restriction of the vector of spin-coupling coefficients to the
intersection of the two eigenspaces that we have identified then ensures full symmetry in the D4h
group. In practice, such spin symmetry conditions typically appear ‘spontaneously’ in the
converged spin-coupling coefficients, although imposing appropriate constraints can sometimes aid
convergence from a poor starting guess.
The situation is rather less straightforward when we consider instead a rotation by an arbitrary
angle around the z axis. In general, such a rotation cannot simply be equivalent to a permutation of
the factors in Equation S1 amongst themselves. Nonetheless, such a single product can be made
invariant under such a rotation by simultaneously enforcing triplet coupling of the spins associated
with x and y, and similarly those for x and y. Triplet coupling implies that an electron pair is spin
symmetric (and therefore space-antisymmetric) so that the x y product has the symmetry properties
of (xy−yx), i.e. 3u. This is analogous to noting that the RHF wavefunction for the z-component of
the 3P ground state of carbon can be written 1s22s22px2py, in which the spins associated with the
singly-occupied orbitals are necessarily coupled to a triplet. Returning to the molecular case, we
note that the x y product has the same symmetry as does x y and, of course, that the direct product
of two 3u terms is 1+g .
This rather strict requirement for triplet coupling for the x y and x y pairs, when combined with
the conditions for σ̂h invariance, leads to a total of 29 linear constraints on the coefficients of
Yamanouchi-Kotani (YK) spin functions, as listed in Table S2 (for the specific ordering of the
orbitals stated in the Table caption, which is meant to maximise the weight of perfect pairing).
These strict conditions on the spin-coupling coefficients, which subsume the weaker ones described
above, do not in general appear ‘spontaneously’ in the converged wavefunction. We plot in Figure
S1(a) the overlap between the D4h-invariant SC(σ,π) solution for C2H2 and the wavefunctions that
are obtained from it by successive rotations of 10° about z. Figure S1(b) is the corresponding plot
for N2. The symmetry contamination is obviously small, but it is undeniably present. We observe a
cos(4φ) modulation in both cases.
S3
Table S2. The 29 linear constraints imposed on the coefficients of Yamanouchi-Kotani (YK) spin
functions for the ten-electron valence part of the symmetry-separated configuration with orbital
product σ1 σ2 σ̂hσ1 σ̂hσ2 σ3 σ̂hσ3 πx σ̂hπx πy σ̂hπy so as to ensure rigorous 1+g symmetry.a
c2 = − √(⅔) c1
c5 = − √(⅔) c3 + √(⅓) c4
c6 = √(⅔) c29
c7 = √(⅔) c30
c8 = √(⅔) c31
c9 = √⅔ c32
c10 = √(½) (−√(⅓) c31 + c32)
c11 = − √(1∕6) c30 − ½√(⅓) c31 − ½ c32
c12 = √(½) c30 − ½ c31 + ½√(⅓) c32
c13 = √(⅔) c36
c14 = √(⅔) c37
c15 = − √(⅓) c29
c16 = − √(⅓) c30
c17 = − √(⅓) c31
c18 = − √(⅓) c32
c19 = ½ (√(⅓) c31 – c32)
1
c20 = ½ [√(⅓) c30 + √( ∕6) c31 + √(½) c32]
c21 = ½ [−c30 + √(½) c31 − √(1∕6) c32]
c22 = − √(⅓) c36
c23 = − √⅓ c37
c24 = − √(⅓) c38
c25 = − √⅓ c39
c26 = ⅓ [√2 c38 − c39]
c27 = − √⅓ c41
c28 = − √⅓ c42
c33 = ½ (−c31 + √3 c32)
3
c34 = −½ [c30 + √(½) c31 + √( ∕2) c32]
c35 = ½ [√3 c30 − √(3∕2) c31 + √(½) c32]
c40 = − √(⅔) c38 + √(⅓) c39
a
The YK functions are in standard, genealogical order.
In order to understand better the nature of the contaminant, it proves useful to consider the
consequences of singlet coupling for the x y pair. The resultant space-symmetric x y product then
has the symmetry properties of (xy+yx), i.e. one component of a 1Δg term, the other being of course
(x2−y2). Given that the direct product of two Δ terms corresponds to the direct sum of Σ+, [Σ−] and Γ
terms, it seems that we could reasonably expect an unwanted contribution of 1Γg symmetry, which
would then be consistent with the observed cos(4φ) behaviour. In order to explore this further, we
notice that each 1Δxy term transforms under a rotation about z just like
sin(φ1) cos(φ2) + cos(φ1) sin(φ2) = sin(φ1+φ2)
(S2)
in which subscripts have been used to keep track of the order of the factors. It follows that the
symmetric combination of two 1Δxy terms transforms like
sin(φ1+φ2) sin(φ3+φ4) + sin(φ3+φ4) sin(φ1+φ2)
= cos[(φ1+φ2)−(φ3+φ4)] − cos[(φ3+φ4)+(φ1+φ2)]
(S3)
For an arbitrary rotation angle φ about the z axis, the first cosine term on the right hand side of
Equation S3 is invariant. Specifically, it corresponds to Σ+, given that it is also invariant under the
transformation φi → −φi for i = 1 to 4. [This is equivalent to reflection in any plane that contains the
S4
z axis.] On the other hand, the second cosine term on the right hand side of Equation S3 transforms
like cos(4φ), i.e. like one of the two real components of a Γ term. This is the symmetry contaminant
that showed up in Figure S1. More specifically, the unwanted component is 1Γg. Excluding it by
means of the spin constraints listed in Table S2 also means that a 1+g term is rejected, leading to the
energy penalties that become clear when comparing entries in Table S1 with those in Tables 1 and
2.
Figure S1. Overlaps (•) between a D4h-invariant SC(σ,π) solution and wavefunctions obtained from
it by successive 10° rotations (a) for C2H2 and (b) for N2. The solid curves represent fits (with very
low residuals) to the functional form p + q cos(4φ).
Both of the 1g+ and Γg irreducible representations of D∞h correspond to the Ag irreducible
representation for the highest Abelian subgroup (D2h) that is commonly used in quantum chemistry
codes. This helps to mask the (small) presence of the symmetry contaminant. If instead we perform
a symmetry decomposition of the total wavefunction using the D12h group, then 1g+ and Γg do now
correspond to different irreducible representations, specifically A1g and E4g, respectively. In the case
of the D4h-invariant SC(σ,π) solution for C2H2, we find contributions of 0.994 for A1g and of 0.0065
for E4g. The corresponding values for N2 are 0.993 and 0.0075, respectively.
S5
A further consequence of some of our analysis is that it becomes clear that if only the perfect
pairing spin function is included (the last YK function in standard order), no ordering of the orbitals
can ever yield rigorous Σ+ symmetry unless πx and πx are identical, aside from a possible phase
factor, and similarly πy and πy (as is the case for the SCF wavefunction). In general, at least one
other YK spin function must be included.
S-C: Spin and symmetry considerations for bent-bond SC(ω) configurations
The bent-bond configurations considered in the present work feature two σ̂h-related triads of ω
orbitals. The ω orbitals within each triad are of course related to one another by successive rotations
by 120° around the internuclear axis (z). Just as in the symmetry-separated (σ,π) case, various spin
constraints can reasonably be expected to be obeyed spontaneously by the variationally-optimised
spin-coupling coefficients for the SC(ω) wavefunction. The vector of spin-coupling coefficients
must in the present case be a simultaneous eigenvector of three permutations, one corresponding to
σ̂h reflection of the full orbital string, one for reflection in the xz plane, and one for rotation by 120°
around the z axis. If the permutation matrices are taken to be in the ‘space’ representation the
corresponding eigenvalue should be +1 for all three permutations [1,2]. These conditions are
sufficient for the overall space-spin configuration to be invariant under all symmetry operations of
the D3h point group, but they do not ensure invariance under arbitrary rotations about z. We plot in
Figure S2(a) the overlap between the D3h-invariant SC(ω) solution for C2H2 and the wavefunctions
that are obtained from it by successive rotations of 10° about z. Figure S2(b) is the corresponding
plot for N2. This time, cos(3φ) modulations are apparent, as one might have easily guessed. This
identifies the symmetry species of the contaminant as Φ. Further analysis shows it to be 1Φu.
It seems reasonable to suppose, by analogy to the symmetry-separated case that was analysed in
some detail in Section S-B, that rigorous Σ+ symmetry is likely to require the three ω orbitals within
each triad to be coupled with parallel spins, i.e. to a resultant spin S=3∕2. The requirement for quartet
spin multiplicity for a given triad is equivalent to restricting the vector of spin-coupling coefficients
to the invariant eigenspace of a cyclic permutation of the spins of the three orbitals, i.e. of the
(even) permutation equivalent to a rigid rotation of those three orbitals by 120° about z. Strict spin
constraints that ensure rigorous Σ+ symmetry then require the vector of spin-coupling coefficients to
be a simultaneous invariant eigenvector of the two permutations representing separate rotation of
either triad, rather than an eigenvector of the one permutation representing simultaneous rotation of
both triads.
The Φu irreducible representation of D∞h corresponds to the B3u and B2u irreducible
representations of D2h, so that the (small) presence of the symmetry contaminant is already apparent
when working within this subgroup. In the case of D12h, the relevant irreducible representation is
S6
E3u. A symmetry decomposition of the D3h-invariant SC(ω) solution for C2H2 using the D12h group
reveals contributions of 0.994 for A1g and of 0.0064 for E3u. The corresponding values for N2 are
0.993 and 0.0067, respectively.
Figure S2. Overlaps (•) between a D3h-invariant SC(ω) solution and wavefunctions obtained from it
by successive 10° rotations (a) for C2H2 and (b) for N2. The solid curves represent fits (with very
low residuals) to the functional form p + q cos(3φ).
The vector of spin-coupling coefficients should of course also be required to be an eigenvector
of two further permutations of the full orbital string, namely those equivalent to reflection in the σh
(xy) or in the xz planes (albeit these correspond to conditions that are already satisfied in the
standard D3h-invariant SC(ω) solution). The full set of 37 linear constraints on the coefficients of
Yamanouchi-Kotani (YK) spin functions are listed in Table S3 (for the specific ordering of the
orbitals stated in the Table caption, which is meant to maximise the weight of perfect pairing).
Given that the full spin space is spanned by just 42 YK functions, the final total number of spin
constraints is clearly rather high. Note that if only the perfect pairing spin function is included (the
last YK function in standard order), no ordering of the orbitals can ever yield rigorous Σ+ symmetry
S7
unless each ω orbital is identical to its σ̂h-related counterpart. In general, at least three other YK spin
functions must be included.
Table S3. The 37 linear constraints imposed on the coefficients of Yamanouchi-Kotani spin
functions for the ten-electron valence part of the bent-bonds configuration with orbital product
σ1 σ2 σ̂hσ1 σ̂hσ2 ω σ̂hω C3+ω S3+ω C3−ω S3−ω so as to ensure rigorous 1+g symmetry.a
c1 = √(3∕5) c29
c2 = − √(2∕5) c29
c3 = − 2 c38 / √15
c4 = − 2 c39 / √15
c5 = ⅔ (√2 c38 – c39) / √5
c6 = √(⅔) c29
c7 = ⅔ c38
c8 = ⅔ c39
c9 = ⅔√(⅓) (−√2 c38 + c39)
c10 = − ⅓√2 c38
c11 = − ⅓√2 c39
c12 = ⅔√(⅓) c38 − ⅓√(⅔) c39
c13 = − ⅓√2 c41
c14 = − ⅓√2 c42
c15 = − √(⅓) c29
c16 = − ⅓√2 c38
c17 = − ⅓√2 c39
c18 = ⅔√(⅓) c38 − ⅓√(⅔) c39
c19 = ⅓ c38
c20 = ⅓ c39
c21 = ⅓ (−√(⅔) c38 + √(⅓) c39)
c22 = ⅓ c41
c23 = ⅓ c42
c24 = − √(⅓) c38
c25 = − √(⅓) c39
c26 = ⅓ (√2 c38 − c39)
c27 = − √(⅓) c41
c28 = − √(⅓) c48
c30 = √(⅔) c38
c31 = √(⅔) c39
c32 = ⅓ (−2 c38 + √2 c39)
c33 = − √(⅓) c38
c34 = − √(⅓) c39
c35 = ⅓ (√2 c38 − c39)
c36 = − √(⅓) c41
c37 = − √(⅓) c42
c40 = √(⅓) (−√2 c38 + c39)
a
The YK functions are in standard, genealogical order.
In order to understand the inversion-symmetry properties of the unwanted Γ and Φ components
for the D4h-invariant SC(σ,π) and D3h-invariant SC(ω) solutions, respectively, it proves informative
to consider two Cartesian coordinate systems for a centrosymmetric linear molecule, with one
centred on fragment A and the other on the symmetry-equivalent fragment B, such that σ̂h xA = xB
and σ̂h yA = yB. We now consider the effects of the σ̂h reflection and of the î inversion operation on a
generic product (xA)p(yA)q. Clearly,
σ̂h (xA)p(yA)q = (xB)p(yB)q
(S4)
whereas
î (xA)p(yA)q = (−1)p+q (xB)p(yB)q
(S5)
A suitable prototype basis for the Γ representation, restricted to being σ̂h invariant as it must be,
could consist of a sum of spherical harmonics with l=4, m=+4, centred on A and B, and a similar
sum of their counterparts for l=4, m=−4. Such spherical harmonics are themselves just linear
combinations of (xA)p(yA)q products, with p+q=4. Given that p+q is even, it follows from Equation
S5 that the contaminant in the D4h-invariant SC(σ,π) solution must therefore be gerade, i.e. it
S8
transforms as Γg. An entirely analogous argument holds for the Φ representation, except that a
suitable σ̂h-invariant basis would instead consist of a sum of spherical harmonics with l=3, m=+3,
centred on A and B, and a corresponding sum of their counterparts for l=3, m=−3. This time, such
spherical harmonics are linear combinations of (xA)p(yA)q products with p+q=3. Given that p+q is
now odd, the contaminant in the D3h-invariant SC(ω) solution must be ungerade, i.e. it transforms
as Φu.
Our conjecture about the requirement for parallel spins within each triad can be lent support
perhaps most easily by considering an analogy with the 5S excited state of carbon. The spins
associated with the four valence orbitals in the reference configuration 1s22s2px2py2pz are
necessarily coupled to a quintet (S=5∕2), and so the spins for the three singly-occupied 2p orbitals are
exclusively coupled to a quartet. Indeed, if one considers the C∞v subgroup that has the C∞ axis
aligned along the body diagonal of the first octant, it becomes clear that a 120° rotation around that
axis then simply permutes the three 2p orbitals amongst themselves, just like the three ω orbitals
within each triad in the linear molecules. Actually, the constraint of quartet coupling within such a
triad is a special case of a more general requirement that applies when some of the orbitals in a
given configuration are not simply permuted amongst themselves under a continuous rotation, but
are instead turned into linear combinations of one another. Then, if the overall space-spin
configuration is at most to change sign under that rotation, those orbitals must have their spins
mutually parallel.
Consider the effect on the ω orbitals in a triad of a rotation Ȓ around z by a given (but
completely arbitrary) angle:
3
Ȓ ωl(rl) =  ali ωi(rl)
(for l=1, 2, 3)
(S6)
i=1
so that
3
Ȓ [ω1(r1) ω2(r2) ω3(r3)] = 
3
3
 
a1i a2j a3k [ωi(r1) ωj(r2) ωk(r3)]
(S7)
i=1 j=1 k=1
For this configuration to at most change sign under Ȓ it is important that no terms survive on the
right hand side of Equation S7 in which there are multiply occupied orbitals. Given that there is
only a finite number of these ali, which are continuous functions of the infinitely-variable rotation
angle, any fortuitous cancellation of unwanted products could not possibly hold for all rotations.
Only antisymmetrisation can come to the rescue, and only then if all three electrons have parallel
spins, i.e. they are coupled to a quartet. Otherwise, contributions that are not just permutations could
survive in the antisymmetrised space-spin configuration. No other mode of spin coupling could lead
to their cancellation for any arbitrary choice of rotation angle.
S9
All this is of course predicated on Equation S6, in which each triad is closed under an arbitrary
rotation around z. This in turn means that if the ω orbitals include contributions from σ and π basis
functions, then they cannot include any contribution from other types of basis functions – otherwise,
the three orbitals could not possibly constitute a set that is closed under arbitrary rotations and so
the space-spin configuration would be of mixed symmetry. Given that each ω orbital is a linear
combination of only σ and π basis functions, the reducible representation of the C∞v group that is
carried by each triad is the direct sum of Σ+ and Π.
Much of the argument just presented obviously generalizes to n orbitals connected by
successive rotations through 2π/n around a C∞ axis (z). For example, five such symmetry-related
orbitals could be obtained from σ, π and δ basis functions. A space-spin configuration that features
two such pentads could be made invariant under arbitrary rotations around z by requiring the spins
associated with the five orbitals within each pentad to be mutually parallel, i.e. coupled to a sextet
(S=5∕2). This sort of reasoning can also be applied to the case of two πx πy orbital pairs, leading to the
same conclusions as were reached in Section S-B by a more detailed analysis that also established
the symmetry species of the contaminant.
Additional references
[1]
F.E. Penotti, Int. J. Quantum Chem. 78 (1), 24−31 (2000).
[2]
M. Kotani, A. Amemiya, E. Ishiguro, and T. Kimura, Tables of Molecular Integrals
(Maruzen, Tokyo, 1955); Chapter 1, pp. 2−28.
S10