*

# Fast, Accurate and Spin-Pure Calculation of the Low-Lying Electronically
Excited States of Organic Hydrocarbon Radicals

James D. Green  Department of Chemistry, Christopher Ingold Building,
University College London, WC1H 0AJ, UK  Timothy J. H. Hele<EMAIL_ADDRESS>Department of Chemistry, Christopher Ingold Building, University College
London, WC1H 0AJ, UK

###### Abstract

Recent years have seen an explosion of interest in organic radicals due to
their promise for highly efficient organic light-emitting diodes (OLEDs) and
molecular qubits. However, predicting their electronic structure has
historically been challenging, especially for excited states, due to spin-
contamination. Computationally inexpensive methods commonly use an
unrestricted basis, leading to highly spin-contaminated excited states,
whereas high-level methods such as Multiconfigurational SCF and Full-CI can
give accurate, spin-pure results but are very computationally expensive. In
this article we task ourselves with finding the simplest possible method to
compute spin-pure excited states of radicals accurately. To do this, we note
previous suggestions that adding a small number of double excitations into a
radical CIS calculation could resolve the spin-contamination problem and
combine this idea with the semi-empirical approach of Pariser-Parr-Pople (PPP)
theory. To our surprise, this method, which we call ExROPPP (Extended
Restricted Open-shell PPP theory), produces excited state energies and
oscillator strengths with similar accuracy to experimental results as high-
level GMC-QDPT for a series of hydrocarbon radicals and at minimal
computational cost. Furthermore, by deriving Hamiltonian and dipole moment
matrix elements we obtain a series of rules for the excited states of all
alternant hydrocarbon radicals which should aid molecular design. These
results pave the way for the highly accurate and efficient computation and
prediction of the excited states of organic radicals.

##  I Introduction

It was long thought that the search for stable emissive organic radicals was
hopeless, as none were known experimentally.[1] However, since the discovery
of stable, highly emissive organic radicals and the development of highly
emissive radical OLEDs, there has been an explosion of interest in the
field.[2, 3, 4, 5, 6, 7, 8] External quantum efficiencies in excess of 25% and
internal quantum efficiencies of near unity[3] for some radical-based devices
have been reported, such figures being very hard to achieve in conventional
OLEDs based on closed-shell molecules and conventional solid-state LEDs due to
spin-statistics (although phosphorescent OLEDs based on molecules containing
heavy elements and those based on organic closed-shell molecules which exhibit
thermally activated delayed fluorescence also have high efficiencies).[9, 10,
11, 12, 13, 14, 15] Such high-performing OLEDs have potential applications in
the next generation of lightweight and flexible displays and lighting.[3]
Moreover, radical-based devices can achieve this intense emission in the near
infrared (NIR) region, a characteristic which is unusual for small molecules
and highly desirable for the development of fluorescent probes and NIR sources
for medical diagnostic applications.[16, 17, 18, 19] Emissive organic radicals
have since been incorporated into emissive conjugated polymers,[7] which opens
up the possibility of having the efficiency and spectral benefits of a radical
emitter with the processability advantages of conjugated polymers and also the
possibility of semiconductivity.[11, 20] In addition, emissive organic
radicals have potential applications as spin-optical interfaces in quantum
information processing.[21]

Extensive work over the years has yielded some elegant and quite general rules
for the electronic structure of radicals, diradicals and polyradicals, however
they are inherently complicated to study theoretically, largely due to spin-
contamination.[22, 23] The first problem is the dilemma of whether to use
unrestricted or restricted orbitals, as the most obvious choice which give the
most accurate energies and the best description of the ground state are the
unrestricted orbitals however, using them results in spin-contamination. The
second is the choice of determinants for the excited states, as including all
single-excitations in the most simple CI method – configuration interaction
singles (CIS) results in spin-contaminated excited states.[24, 25, 26, 27]

In this article, focusing on organic radicals with a single unpaired electron
(monoradicals), we give a detailed overview of some of the existing theories
for radical electronic structure calculation, the challenges that arise when
it comes to excited states of radicals and how or if they can be overcome. We
then expand on this previous work seeking to provide new insight into the
properties of the excited electronic states of radicals.

When studying radicals we have a choice between two classes of open-shell
determinants, neither of which captures all of the essential features of the
systems which we are trying to describe. These are the unrestricted Hartree-
Fock (UHF) method of Pople and Nesbet[28] and the restricted open-shell
Hartree-Fock (ROHF) method which as been developed separately by Roothaan,
Huzinaga, Davidson and Binkley among others in various guises.[29, 30, 31,
32]. UHF is often preferred as its equations are simpler and more general than
ROHF, and it gives lower, more accurate energies.[33] However, due to having
different spatial orbitals for each electron, any UHF determinant or any
finite linear combination of UHF determinants is generally not an
eigenfunction of the total spin operator $\hat{S}^{2}$.[33] While UHF (and
Unrestricted Density Functional Theory, DFT) is a useful tool for calculating
accurate ground state properties and in many such cases the spin-contamination
can be ignored[34, 35, 36, 37, 38, 39, 22, 40, 41], it can be problematic for
excited states where spin contamination is unavoidable.[42, 43] When it comes
to predicting electronic state spectra, it is highly desirable to be able to
assign states to a single spin multiplicity (since optical transitions only
occur between states of the same multiplicity) and therefore it is clear that
UHF or any other unrestricted reference will be less than ideal and could lead
to unphysical or uninterpretable results.[24, 27, 44, 42, 43, 22]

The alternative method, ROHF, uses a wavefunction containing both closed-shell
and open-shell parts. As the closed-shell part imposes the constraint that
pairs of $\alpha$ and $\beta$ electrons share the same spacial function (as in
RHF), spin-contamination is eliminated by construction for states where all
unpaired electrons are of the same spin, including the ground state.
Furthermore, it is possible to form finite linear combinations of spin-impure
ROHF determinants (for all the other states) which are themselves spin-pure.
Therefore, in general, one can construct a ROHF wavefunction which is an
eigenfunction of $\hat{S}^{2}$ for a given state using a finite number of
determinants, making it practically possible to satisfy spin-purity. This is
the great advantage of ROHF compared with UHF.[24, 29, 30, 31, 32, 33, 40, 27]
Another useful feature of using a ROHF reference is that the molecular
orbitals are easier to interpret than for UHF, as there is one set of spacial
orbitals for pairs of $\alpha$ and $\beta$ spin functions. However, as there
are several ways of doing ROHF, all of which having seemingly equal merit and
none of which being exactly equivalent, this gives rise to issues such as
orbital energies not being uniquely defined.[45, 46]

Although both UHF and ROHF clearly have their own limitations, for the
calculation of the photophysical properties of radicals we believe that spin-
purity must take precedence over a slight improvement in accuracy. However,
using a restricted reference is not on its own a sufficient criterion for
satisfying spin-purity of the excited states, as one needs to ensure that the
correct linear combinations of determinants are chosen for states with more
than one open-shell.

This brings us to the second major challenge of preserving spin-purity of the
excited states. In the case of closed-shell molecules, from the set of all
single-excitations of $\alpha$ and $\beta$ electrons,
$|\Phi_{i}^{j^{\prime}}\rangle$ and
$|\Phi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$ respectively, one can form a
complete set of spin-adapted singlet $|^{1}\Psi_{i}^{j^{\prime}}\rangle$ and
triplet $|^{3}\Psi_{i}^{j^{\prime}}\rangle$ configurations or configuration
state functions (CSFs) from a restricted determinant. This is the
configuration interaction singles (CIS) method — the simplest configuration
interaction method and the formal starting point for many other more accurate
methods.[33, 40, 47] Usually we are only interested in the singlet states with
regard to absorption/emission as triplet states are formally dark. Caution
must be taken, however, when choosing a basis for the excited states of
radicals. Firstly, if one were to use a basis of all single excitations, as in
the closed-shell case, and form a complete set of states, this would in
general result in spin-contamination. This is demonstrated pictorially in Fig.
1 below, and also numerically in Tables VIII and IX in the supplementary
information where we have presented the results of this approach for the
benzyl radical. The general spin properties of closed-shell molecules and
radicals are summarized in Table 1.

One solution to the spin-contamination problem in radical CIS is to restrict
the basis of single excitations to only include those which are pure doublets,
which are those which have only one open-shell, in which case the resulting
states would also be pure doublets. Alternatively, one could go a step further
and also add in some spin-adapted doublet states for those configurations
containing three open-shells,[48] however, neither method captures quartet
states. Furthermore, some excited doublet states with three open-shells are
also not captured by these methods, which, despite being dark, could affect
the spectrum. This problem is solved if one also includes a certain type of
double excitations in the basis, as shown in previous literature, where in
this case a complete set of spin-pure doublet and quartet states can be
constructed on the same theoretical footing, in a similar manner to closed-
shell CIS.[24, 25] This formulation of open-shell CI, which is discussed
further in Section II is sometimes referred to as extended CIS or XCIS[25] for
short. This is different to CISD where _all_ double excitations are included
and which is not spin-pure. Also XCIS has the same scaling as CIS, where the
number of determinants scales with
$\mathcal{O}(N_{\mathrm{occ}}N_{\mathrm{vir}})$ whereas CISD scales with
$\mathcal{O}(N_{\mathrm{occ}}^{2}N_{\mathrm{vir}}^{2})$, where
$N_{\mathrm{occ}}$ is the number of occupied orbitals and $N_{\mathrm{vir}}$
the number of virtual (unoccupied) orbitals in the ground state.

Figure 1: Illustration of the problem of using only single excitations to describe the excited states of a doublet radical which consist of three open-shells. Four excited configurations are pictured: a) the spin-pure singlet-coupled doublet (2S) state (‘+’ combination) formed of only single excitations, b) the ‘$-$’ combination of two single excitations which is spin-contaminated, c) the spin-pure quartet state composed of single and double excitations and d) the spin-pure triplet-coupled doublet (2T) state composed of of single and double excitations. Table 1: Comparison of the general spin properties of the singly-excited electronic states for closed-shell molecules and radicals. Property | Closed-shell | Radical
---|---|---
Ground state multiplicity | Singlet | Doublet
Determinant | RHF | ROHF or UHF
Spin-multiplicity of single-excitations |  $|\Psi_{i}^{j^{\prime}}\rangle+|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$ is singlet |  $|\Psi_{\bar{i}}^{\bar{0}}\rangle$, $|\Psi_{0}^{j^{\prime}}\rangle$, $|\Psi_{i}^{j^{\prime}}\rangle+|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$
|  $|\Psi_{i}^{j^{\prime}}\rangle-|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$ is triplet | are doublets
|  |  $|\Psi_{i}^{j^{\prime}}\rangle-|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$ contaminated
Usual multiplicity of | Triplet | Doublet
lowest excited state |  |
Spin restriction alone sufficient | Yes | No
for spin-purity? |  |

We must stress here that there are many other existing methods to obtain
accurate, spin-pure excited states of radicals. There are the
multiconfigurational SCF (MCSCF) method, multiconfigurational second-order
perturbation theory such as CASPT2 and GMC-QDPT mentioned later, full-CI
(FCI), coupled-cluster (CC) theory and so on.[49, 50, 51, 33, 52] These
methods achieve spin-pure results for radicals by using a restricted open-
shell reference and by either including all possible excitations (all within a
given active space in MCSCF and CASPT2/GMCQDPT) or including fewer than all
possible excitations and employing spin-adaption techniques similar to but
more general than that of XCIS.[53, 54, 55] When used correctly these methods
can give accurate results,[56] although for very accurate excitation energies
one must also include the effect of nuclear quantum motion.[57] However, these
calculations are computationally expensive, their computational cost
increasing by many orders of magnitude with increasing molecular size, and
also are by no means black-box methods, requiring chemical intuition and
experience to accurately utilize.[33] One potential avenue is to decrease the
computational cost of such methods by, for example, combining MCSCF or FCI
with quantum Monte Carlo, however, discussions on this topic are beyond the
scope of this article.[58, 59] On the other hand, time-dependent density
functional theory (TD-DFT) allows for the rapid calculation of electronic
excited state spectra which does not require any non-trivial input from the
user and has become the standard method for the simulation of UV/Visible
spectra.[60, 61] However, TD-DFT has several drawbacks, namely that: a) its
results can vary significantly depending on the chosen functional [60, 61], b)
it is inaccurate at describing long-range phenomena such as charge-transfer
without the use of specifically tailored long-range functionals (essentially
point a)),[62, 63, 64, 65] and c) it does not typically give spin-pure states,
as an unrestricted version of the Kohn-Sham equations is often preferred in
favour of more accurate energies, and even if a restricted open-shell
reference is used, conventional TD-DFT does not form spin-adapted excited
states.[42, 43] A spin-adapted version of TD-DFT for radicals has recently
been developed, called X-TDDFT, which is clearly one faster alternative to the
more expensive MCSCF/CASPT2/GMC-QDPT/FCI/CC methods.[43] However, X-TDDFT is
by no means theoretically straight-forward, for example the required double
excitations are not compatible with conventional TD-DFT so a work-around using
spin-flip single excitations and reference states with different values of
$S_{z}$ needs to be implemented, and furthermore this method is bound to
suffer from the first two aforementioned challenges of DFT. Finally, although
TD-DFT is faster than the post-Hartree-Fock methods mentioned above, even it
becomes unfeasibly expensive for large numbers of calculations, and there is a
growing need for ever faster property determination methods for high-
throughput virtual screening of molecular candidates and AI algorithm search
methods, and for property determination of large molecules. [66, 67, 68]

This brings us to two key aims which we will address in this paper:

##### An inexpensive method for screening radicals:

What is the simplest and least computationally expensive method which can
calculate spin-pure doublet and quartet states of radicals with reasonable
accuracy?

##### Design rules for emissive radicals based on alternant hydrocarbons:

Many separate principles have been discovered for designing emissive radicals
which stem from theory, experiment or a combination of the two.[69, 70, 71, 8]
The idea is that, before extensive calculations or synthesis are performed,
one could initially filter out the most promising molecular candidates using a
set of simple design principles, saving time and resources. While this search
for design rules in previous work has been very successful, we believe a
general framework for the interaction of excited states of radicals is needed.
Building on this previous work and considering that many emissive radicals are
derivatives of alternant hydrocarbons such as the triphenylmethyl radical,[2,
3, 4, 5, 6, 7] can we deduce general design rules for such radicals by
applying the ideas of XCIS, with a small number of approximations, to the
special case of alternant hydrocarbon radicals?

The article is structured as follows. Firstly, in Section II we define the key
theoretical concepts which will inform the discussion of our methodology. Here
we discuss the XCIS method for doublet radicals,[24, 25] a restricted open-
shell method for the molecular orbitals,[24] Pariser-Parr-Pople theory and its
approximations[72, 73, 74, 75, 76, 77] and also give a brief overview of the
theories relating to alternant hydrocarbons.[78, 75, 77] In Section III we
then explore the underlying features of the XCIS method by considering various
rotations of the basis, namely, excitations, spin-adpated states and spin-and-
psuedoparity-adapted states for the XCIS equations for the Hamiltonian and
dipole moment operator, presenting a set of matrix elements for XCIS which we
believe have not be published before in their entirety. We then directly apply
the approximations of PPP theory to the XCIS equations in the spin-and-
pseudoparity-adapted basis in order to derive rules for the interaction of
excited states of alternant hydrocarbon radicals, similar to those shown for
closed-shell molecules in previous literature.[77] We also show how XCIS and
PPP theory with a restricted open-shell determinant can be combined and
implemented in a method we call ExROPPP, for the rapid calculation of spin-
pure exicted states of radicals which is not limited to just alternant
hydrocarbons. Finally, the alternacy rules for radicals and the numerical
results of computations with ExROPPP are presented in IV.

##  II Background Theory

Here we discuss the XCIS method which was first eluded to by Longuet-Higgins
and Pople and subsequently elaborated in a more general formalism by Maurice
and Head-Gordon, for the excited states of radicals and which is central to
our work,[24, 25] the restricted open-shell method of Loguet-Higgins and
Pople[24] which allows us to obtain self-consistent molecular orbitals for a
system with one unpaired electron in its ground state, the various
approximations of Pariser-Parr-Pople (PPP) theory,[72, 73, 74, 75, 76, 77] and
finally the theories of Coulson, Rushbrooke, Pople and Pariser which apply
alternant hydrocarbons which we will call upon to derive alternacy rules for
radicals.[78, 75, 77]

###  II.1 From CIS to XCIS

We start by defining the ground state electronic structure of the radical in
question, which is described by restricted orbitals with $k$ doubly-occupied
closed-shells and one singly-occupied open-shell such that the system is an
overall doublet with $M_{S}=1/2$. There are therefore $2k+1$ electrons in
total. We label the orbitals choosing $0$ to be the singly-occupied molecular
orbital (SOMO), and successively number the doubly occupied orbitals in order
of descending energy starting from $1$ (the HOMO), and the vacant orbitals in
order of ascending energy with primes starting from $1^{\prime}$ (the LUMO) as
shown in Fig. 2. We choose this notation as it is the same as that used by
Pariser[77] and Ref. 27. We now discuss the excited configurations. Single
excitations from the ground state $|^{2}\Psi_{0}\rangle$ give rise to the
following configurations: the configuration
$|^{2}\Psi_{\bar{i}}^{\bar{0}}\rangle$ is created from the ground state by
exciting an electron from the occupied $\beta$ spinorbital $\bar{i}$ to the
vacant $\beta$ spinorbital of the SOMO, the configuration
$|^{2}\Psi_{0}^{j^{\prime}}\rangle$ corresponds to excitation of an $\alpha$
electron from the occupied SOMO spinorbital to the vacant spinorbital
$j^{\prime}$, the configuration $|\Psi_{i}^{j^{\prime}}\rangle$ corresponds to
excitation of an $\alpha$ electron from the occupied spinorbital $i$ to
$j^{\prime}$, and $|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$ is the
corresponding excitation for the $\beta$ spinorbitals ${\bar{i}}$ and
${\bar{j}^{\prime}}$, where the bar indicates a down ($\beta$) spin. These
configurations are labelled a) – e) respectively in Fig. 2.

As we are concerned with finding those states which are eigenstates of the
total spin operator $\hat{S}^{2}$, we shall ascertain the effect of
$\hat{S}^{2}$ on the above configurations. The states a) – c) are eigenstates
of $\hat{S}^{2}$ where $S=1/2$ — these configurations are pure doublet states.
This is to be expected as a) – c) have only one open-shell. On the other hand,
those configurations which have three open-shells, d) and e), are not
eigenstates of $\hat{S}^{2}$ and are in fact mixtures of doublet and quartet
states. Naturally we can make two orthonormal linear combinations of
$|\Psi_{i}^{j^{\prime}}\rangle$ and
$|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$,

$\displaystyle|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|\Psi_{i}^{j^{\prime}}\rangle+|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle)$
(1a) $\displaystyle|^{\mathrm{DQ}}\Psi_{i}^{j^{\prime}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|\Psi_{i}^{j^{\prime}}\rangle-|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle)$
(1b)

to see if we can construct two spin-adapted states. We find that the ‘$+$’
combination is a pure doublet with $S=1/2$, however, the ‘$-$’ combination is
not an eigenstate of $\hat{S}^{2}$. It is clearly impossible then to construct
a spin-pure basis for doublet and quartet excited states of a radical from
just single excitations from the ground state.

As previous literature has shown, one must therefore go to higher-order
excitations to attempt to satisfy spin-purity whilst calculating doublet and
quartet states on the same theoretical footing.[24, 25] They considered double
excitations of the type $|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$ as,
along with the single-excitations $|\Psi_{i}^{j^{\prime}}\rangle$ and
$|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$, they complete all permutations of
three open-shells in orbitals $i$, $0$ and $j^{\prime}$ with $M_{S}=1/2$.

Figure 2: Depictions of the electronic structure of a radical using the
orbital numbering scheme used throughout this paper of: a) the restricted
open-shell ground state, b) – e) four types of singly-excited configurations
and f) one type of doubly-excited configuration.

One can construct the matrix representation of $\hat{S}^{2}$ in the basis of
three excitations: $|\Psi_{i}^{j^{\prime}}\rangle$,
$|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle$ and
$|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$

$\displaystyle\mathrm{\mathbf{S^{2}}}=\begin{pmatrix}7/4&-1&1\\\ -1&7/4&-1\\\
1&-1&7/4\\\ \end{pmatrix}$ (2)

which has eigenvalues of 3/4 twice and 15/4, corresponding to two doublet
eigenstates and one quartet eigenstate

$\displaystyle|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|\Psi_{i}^{j^{\prime}}\rangle+|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle)$
(3a) $\displaystyle|^{2\mathrm{T}}\Psi_{i}^{j^{\prime}}\rangle$
$\displaystyle=\frac{1}{\sqrt{6}}(-|\Psi_{i}^{j^{\prime}}\rangle+|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle+2|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle)$
(3b) $\displaystyle|^{4}\Psi_{i}^{j^{\prime}}\rangle$
$\displaystyle=\frac{1}{\sqrt{3}}(|\Psi_{i}^{j^{\prime}}\rangle-|\Psi_{\bar{i}}^{\bar{j}^{\prime}}\rangle+|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle).$
(3c)

These three states then, along with $|^{2}\Phi_{0}\rangle$,
$|^{2}\Psi_{\bar{i}}^{\bar{0}}\rangle$ and
$|^{2}\Psi_{0}^{j^{\prime}}\rangle$, constitute a basis for the spin-pure
excited doublet and quartet states of a radical and will be the starting point
for our methodology.[24, 25, 27] We note here that while the quartet
eigenstate $|^{4}\Psi_{i}^{j^{\prime}}\rangle$ is uniquely defined, the two
doublet eigenstates are not since they are degenerate (they share the same
eigenvalue of $\hat{S}^{2}$) and can be mixed. We have chosen the above
convention for the two doublet states [Eqs. (3c)–(3b)] as it simplifies things
in the following way: the doublet state
$|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$ usually has a non-vanishing
dipole moment from the ground state as noted in Ref. 24 and Ref. 27 and does
not mix directly with the ground state, whereas
$|^{2\mathrm{T}}\Psi_{i}^{j^{\prime}}\rangle$ is always completely dark and
contains contributions from double-excitations, in general having non-zero
ground state mixing. We have used the notation of 2S and 2T to differentiate
between these two states as they are sometimes referred to as the ‘singlet-
coupled’ and ‘triplet-coupled’ doublet states due to the fact that their
dipole moments are analogous to those of singlet and triplet states in closed-
shell molecules.[79] A short comparison of the differences between CIS and
XCIS are summarised in Table 2.

Table 2: Comparison of CIS for closed-shell molecules and XCIS for radicals. Property | CIS | XCIS
---|---|---
Brillouin’s theorem satisfied? | Yes | Not AlwaysaaaBrillouin’s theorem is usually satisfied in ROHF as the orbitals are solved for variationally, minimising interaction of the ground state with singly-excited states. However, in some forms of ROHF, like the one which is used to produce the results in this paper and discussed later, this is done in an approximate way such that the mixing between certain single-excitations are close to but not exactly zero.[29, 30, 31, 32]
Koopmans’ theorem satisfied? | Yes | To a lesser extentbbbIPs and EAs depend on spin of the electron being ionized / attached. IPs and EAs cannot in general be correlated with the eigenvalues of the Fock matrix in ROHF as is the case for closed-shell systems in RHF. However, an analog of Koopmans’ theorem exists in the open-shell case whereby IPs and EAs for different electrons can be correlated to diagonal elements of the respective Fock matrices under a specific unitary transformation (canonical form) of the orbitals and only for the special case of system with a half-filled open-shell with all spins parallel.[80, 81]
Doubly excited configurations required for | No | Yes
calculating high-spin (triplet/quartet) states? |  |

###  II.2 The restricted open-shell method for the orbitals

While there are many variations of restricted open-shell methods to choose
from, we elect to use the method of Longuet-Higgins and Pople.[24] In this
method, the ground state orbitals are solved for variationally as per usual
HF, but in this case minimising the ground state energy with respect to mixing
of small amounts of single excitations $|\Psi_{\bar{i}}^{\bar{0}}\rangle$,
$|^{2}\Psi_{0}^{j^{\prime}}\rangle$ and
$|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$ into the ground state. The
change in the energy of the ground state upon mixing of the above excitations
is

$\displaystyle\delta
E=\sum_{i=1}^{k}\lambda_{i0}\langle\Psi_{0}|\hat{H}|\Psi_{\bar{i}}^{\bar{0}}\rangle+\sum_{j=1}^{\infty}\lambda_{0j}\langle\Psi_{0}|\hat{H}|\Psi_{0}^{j^{\prime}}\rangle+\sum_{i=1}^{k}\sum_{j=1}^{\infty}\lambda_{ij}\sqrt{2}\langle\Psi_{0}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle.$
(4)

The condition for $|\Psi_{0}\rangle$ to be stationary is that $\delta E$ must
be zero for all values of $\\{\lambda_{ij}\\}$, which can only be achieved if
all of the matrix elements in Eq. (4) are equal to zero. These matrix elements
are:

$\displaystyle\langle\Psi_{0}|\hat{H}|\Psi_{\bar{i}}^{\bar{0}}\rangle$
$\displaystyle=h_{i0}+\sum_{l=1}^{k}[2(i0|ll)-(il|l0)]+(i0|00),$ (5a)
$\displaystyle\langle\Psi_{0}|\hat{H}|\Psi_{0}^{j^{\prime}}\rangle$
$\displaystyle=h_{0j^{\prime}}+\sum_{l=1}^{k}[2(0j^{\prime}|ll)-(0l|lj^{\prime})],$
(5b)
$\displaystyle\langle\Psi_{0}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$
$\displaystyle=\sqrt{2}\bigl{\\{}h_{ij^{\prime}}+\sum_{l=1}^{k}[2(ij^{\prime}|ll)-(il|lj^{\prime})]+\frac{1}{2}[2(ij^{\prime}|00)-(i0|0j^{\prime})]\bigr{\\}}.$
(5c)

They propose a single Fock operator for both closed and open-shell
spinorbitals

$\displaystyle\langle\psi_{p}|\hat{H}|\psi_{q}\rangle=F_{pq}$
$\displaystyle=h_{pq}+\sum_{l=1}^{k}[2(pq|ll)-(pl|lq)]+\frac{1}{2}[2(pq|00)-(p0|0q)],$
(6)

which gives

$\displaystyle F_{i0}+\frac{1}{2}(i0|00)=0$ (7a) $\displaystyle
F_{0j^{\prime}}-\frac{1}{2}(0j^{\prime}|00)=0$ (7b) $\displaystyle
F_{ij^{\prime}}=0,$ (7c)

meaning that the ground state mixing of the configuration
$|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$ is exactly zero, and the ground
state mixing of the configurations $|\Psi_{\bar{i}}^{\bar{0}}\rangle$ and
$|^{2}\Psi_{0}^{j^{\prime}}\rangle$ is _almost_ but not exactly zero as
$(i0|00)$ and $(0j^{\prime}|00)$ will likely be small (and in some cases equal
to zero due to symmetry). Therefore, Brillouin’s theorem is _almost_ but not
completely satisfied. As a note here, there are other restricted open-shell
approaches for the orbitals which are more general in that they can be applied
to systems with more than one open shell in the ground state, and which also
satisfy Brillouin’s theorem exactly.[29, 30, 31, 32] However, these more
general methods involve defining separate sets of orbitals for closed-shells
and open-shells. In this case one solves two sets of coupled Roothaan
equations involving two coupled Fock operators for closed-shell and open-shell
spinorbitals (in a similar fashion to unrestricted Hartree-Fock) whilst
imposing the constraint of restricting the spatial parts of the $\alpha$ and
$\beta$ closed-shell spinorbitals to be the same using a method of Lagrange
multipliers.[29, 30, 31, 32] Whilst this ensures Brillouin’s theorem is
satisfied exactly, these methods are complicated to implement and we therefore
elect to use the approximate method of Longuet-Higgins and Pople for systems
with a single open-shell doublet ground state, despite it only approximately
satisfying Brillouin’s theorem, as it is theoretically simpler and only
involves diagonalizing a single Fock matrix for all orbitals.

In the linear combination of atomic orbitals (LCAO) basis, the Fock operator
of Ref.24 takes the form[27]

$\displaystyle
F_{\mu\nu}=h_{\mu\nu}+\sum_{\rho}\sum_{\sigma}\bigl{[}P_{\rho\sigma}(\mu\nu|\rho\sigma)-\frac{1}{2}(\mu\sigma|\rho\nu)\bigr{]}$
(8)

where the corresponding density matrix $P$ is

$\displaystyle P_{\rho\sigma}=\biggl{(}\sum_{l=1}^{k}2C^{*}_{\rho l}C_{\sigma
l}\biggr{)}+C^{*}_{\rho 0}C_{\sigma 0}.$ (9)

Notice that both the Fock operator and density matrix contain two two-electron
parts, the first term being for doubly-occupied orbitals and the second term
for singly-occupied orbitals and whose leading coefficient is half that of the
first term. This procedure for calculating the converged restricted open-shell
orbitals is discussed in further detail in the SI.

###  II.3 Pariser-Parr-Pople theory

Thus far, in our discussion of the method to obtain the molecular orbitals, we
have assumed an arbitrary basis of atomic orbitals and have not detailed the
method to be used to obtain the one-electron (core) integrals $h_{\mu\nu}$ and
two-electron integrals $(\mu\nu|\rho\sigma)$ needed to form the Fock matrix.
We could elect to use Gaussian orbitals and compute the integrals for the set
of all atomic orbitals exactly, however, this comes at a computational expense
which we aim to minimise. We therefore call upon Pariser-Parr-Pople (PPP)
theory as it is arguably the simplest and least computationally expensive
method which is known to accurately reproduce the excited state spectra of
$\pi$-conjugated closed-shell molecules, and we will see that it performs
similarly well for $\pi$-conjugated organic radicals when using restricted
orbitals and coupled with XCIS in a method we call ExROPPP.[72, 73, 74, 75,
76, 77] Moreover, due to the approximations within PPP theory, many integrals
vanish and many different integrals become equivalent, simplifying the form of
the Hamiltonian and other electron operators. Therefore, applying the
approximations of Pariser-Parr-Pople (PPP) theory to the Hamiltonian and
dipole moment operators within XCIS allows us to derive alternacy rules.

PPP theory is a semiempirical method for the electronic structure calculation
of conjugated $\pi$ systems. The $\pi$ system is treated using an implicit
basis of $p$ orbitals centred on the atomic nuclei, where neighbouring $p$
orbitals interact via a Hückel $\beta$ (resonance integral) term. The $\sigma$
system is treated as core-like and only interacts with the $\pi$ system
through a Hückel $\alpha$ (core integral) term which also accounts for nuclear
charge.

In PPP theory, the neglect of differential overlap (NDO) approximation is
applied, where the overlap integral between any two atomic orbitals is equal
to the Kronecker delta function of the two orbitals:
$\langle\phi_{\mu}|\phi_{\nu}\rangle=\delta_{\mu\nu}$. Firstly, this means
that the overlap matrix $\mathrm{\mathbf{S}}$ simply equal to the identity
matrix $\mathrm{\mathbf{I}}$. Therefore, the Roothaan equations simplify to
$\mathrm{\mathbf{FC}}=\mathrm{\mathbf{CE}}$, where $\mathbf{E}$ is a diagonal
matrix of orbital energies $\\{\epsilon_{p}\\}$. The second implication of NDO
is that the expressions for the two-electron integrals are greatly simplified,
as those integrals which depend on the overlap of two $p$ orbitals on
different atoms are neglected. Therefore, the two-electron integrals take the
form:

$\displaystyle(\mu\nu|\rho\sigma)=(\mu\mu|\rho\rho)\delta_{\mu\nu}\delta_{\rho\sigma}=\gamma_{\mu\rho}\delta_{\mu\nu}\delta_{\rho\sigma},$
(10)

where $\gamma_{\mu\rho}$ is a function of the distance between the two atomic
centres.[27]

###  II.4 Alternant hydrocarbon radicals

We now discuss the theoretical concepts that relate to alternant hydrocarbon
radicals which will be key to our discussions later on. Alternant hydrocarbons
are a special class of molecules whose atoms can be divided into two sets,
often referred to as _starred_ and _un-starred_ , where no two atoms of the
same set may be directly bonded. This structural symmetry results in
underlying symmetries in the energies and coefficients of pairs of orbitals,
and degeneracy of pairs of single-excitations. This pairing theorem was first
proven by Coulson and Rushbrooke, where it was shown to hold under the
approximations of neglect of differential overlap (NDO) and resonance
interaction only between nearest neighbouring atoms, such approximations being
present in Pariser-Parr-Pople theory (see Section II.3).[78] Subsequently,
Pople[75] and then Pariser[77] demonstrated that this pairing theorem allowed
the excited states of alternant hydrocarbons to be grouped into in-phase
_plus_ and out-of-phase _minus_ combinations which do not interact with each
other via the Hamiltonian, a property sometimes referred to as pseudoparity.

####  II.4.1 Orbital energy symmetry

According to the theorem, for every bonding orbital there exists an
antibonding orbital with an equal and opposite energy relative to the energy
of a non-interacting atomic orbital. In the case of radicals, the SOMO has
non-bonding character and its energy is the same as that of a non-interacting
orbital, and all _other_ orbitals can be grouped into bonding and antibonding
pairs which are equally spaced in energy about the SOMO. This is shown by the
following relation:

$\displaystyle\frac{\epsilon_{i}+\epsilon_{i^{\prime}}}{2}=\frac{F_{i,i}+F_{i^{\prime},i^{\prime}}}{2}=\epsilon_{0}=F_{0,0}.$
(11)

####  II.4.2 Orbital coefficient symmetry

In alternant hydrocarbon radicals the SOMO has amplitude on only the _starred_
atoms, and that for every Coulson-Rushbrooke orbital pair, the orbital
coefficients in bonding and antibonding pairs: i) are equal in magnitude, and
ii) have equal signs on the _starred_ atoms and opposite signs on the _un-
starred_ atoms. In the convention followed in this paper, for odd alternant
hydrocarbon radicals there are $\frac{N+1}{2}$ _starred_ atoms and
$\frac{N-1}{2}$ _un-starred_ atoms, where $N$ is the total number of atoms.
The following is then true for two electron integrals for alternant
hydrocarbon radicals:

$\displaystyle(ij|kl)$
$\displaystyle=(ij|k^{\prime}l^{\prime})=(i^{\prime}j^{\prime}|kl)=(i^{\prime}j^{\prime}|k^{\prime}l^{\prime}),$
(12a) $\displaystyle(i^{\prime}j|kl)$ $\displaystyle=(ij^{\prime}|kl),$ (12b)
$\displaystyle(i0|jk)$ $\displaystyle=(i^{\prime}0|jk).$ (12c)

Eq. (12c) holds as a special case of Eqs. (12a) and (12b) because the SOMO is
unchanged by changing the signs of the _un-starred_ atoms, i.e.
$\psi_{0}=\psi_{0^{\prime}}$. Similar identities also exist for dipole
moments:

$\displaystyle\mu_{ij}$ $\displaystyle=\mu_{i^{\prime}j^{\prime}},$ (13a)
$\displaystyle\mu_{ij^{\prime}}$ $\displaystyle=\mu_{i^{\prime}j},$ (13b)
$\displaystyle\mu_{i0}$ $\displaystyle=\mu_{i^{\prime}0},$ (13c)

where we use the standard definition of the dipole moment in atomic units

$\displaystyle\mu_{ij}=\langle i|\hat{\mu}|j\rangle=-\langle
i|\mathbf{r}|j\rangle.$ (14)

####  II.4.3 Pseudoparity of the excited states with a single open-shell

These symmetry relations result in degeneracies in pairs of excitations, even
when two electron terms are taken into account within the regime of the above
approximations. The excitations $|\Psi_{\bar{i}}^{\bar{0}}\rangle$ and
$|^{2}\Psi_{0}^{i^{\prime}}\rangle$ are degenerate for corresponding pairs of
bonding and antibonding orbitals $\psi_{i}$ and $\psi_{i^{\prime}}$. They can
be factored into two-determinant linear combinations with a ‘$+$’ or ‘$-$’
sign, where _plus_ states do not interact with _minus_ states, shown below in
Eq. (15).[24, 27]

$\displaystyle|^{2}\Psi_{0i}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|^{2}\Psi_{\bar{i}}^{\bar{0}}\rangle\pm|^{2}\Psi_{0}^{i^{\prime}}\rangle)$
(15)

This is exactly analogous to the formation of singlet and triplet _plus_ and
_minus_ states in closed-shell molecules.[77]

##  III Methodology

This is the point where we expand on previous work, offering new theoretical
insights into the fundamental properties of the excited states of radicals.
Firstly, we present the matrix elements for the Hamiltonian and dipole moment
operators for XCIS in the basis of excitations and of spin-adapted states
which we believe have not yet been published in their entirety. We then show
how rotation of the XCIS states into a basis of spin-adapted _plus_ and
_minus_ states and subsequently applying the approximations of PPP theory
allows us to define a set of general rules for the interaction of the excited
states of alternant hydrocarbons, similar but not identical to those
discovered by Pariser for closed-shell molecules.[77] Finally, we will show
how the methods of XCIS, PPP theory and restricted open-shell electronic
structure theory can be combined into a method called ExROPPP for the rapid
calculation of the spin-pure excited electronic states of radicals.

###  III.1 Functional form of the XCIS Hamiltonian and dipole moment operators

Having defined two sets of basis states for the excited states of a radical in
Section II.1, both ensuring that the final states after CI will be spin-pure,
we now examine the functional form of the XCIS Hamiltonian and dipole moment
operator in these two bases. These two bases are i) excitations and ii) spin-
adapted states.

####  III.1.1 Hamiltonian

##### Excitations

The XCIS Hamiltonian $\mathrm{\mathbf{H_{\textrm{XCIS}}^{x}}}$ in the basis of
excitations (configurations a) – f) in Fig. 2) is presented in Table 8 in
Appendix A. Some of its elements have already been worked out for us in Refs.
24 and 27, which are those on the leading diagonal and some off-diagonal
elements of excitations involving the same orbitals, however many of its
elements were not given in previous literature[24, 25, 27] and thus we derived
them using standard electronic structure theory[33]. Even though some of the
excitations themselves are not spin-pure, we know that these states can be
rotated such that they diagonalize $\mathrm{\mathbf{S^{2}}}$ and these rotated
states are spin-pure. As $\hat{S}^{2}$ and the Hamiltonian $\hat{H}$ commute,
on diagonalizing $\mathrm{\mathbf{H_{\textrm{XCIS}}^{x}}}$ one will obtain a
set of spin-pure eigenstates which also diagonalize $\mathrm{\mathbf{S^{2}}}$
and which will be linear combinations of the spin-adapted states, where
doublets only mix with other doublets and quartets only mix with other
quartets.

##### Spin-adapted states

Alternatively, one can construct the Hamiltonian
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{s}}}$ in the basis of the spin-adapted
states, otherwise known as configuration state functions (CSFs). These matrix
elements are presented in Appendix B in Table 9 and we believe this is the
first time they have been fully written out. This rotation of the basis only
mixes those configurations with three open-shells, so the matrix elements
which contain only configurations with one open-shell will be unchanged and
are hence not duplicated in Table 9. Clearly, diagonalization of
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{s}}}$ will result in a set of spin-pure
eigenstates as the off-diagonal elements between doublet states and quartet
states (matrix elements 1, 4, 7, 11 and 12 in Table 9) are all equal to zero.
As an added note, the Hamiltonian matrix element between two 2S states
$\langle^{2\mathrm{S}}\Psi_{k}^{l^{\prime}}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$
has the same functional form as for between two excited singlet states in a
closed-shell molecule (see Table 9 in Appendix B).[27, 33]

####  III.1.2 Dipole moment

The matrix elements for the dipole moment operator in the basis of excitations
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{x}}}$ and of spin-adapted states
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{s}}}$ can also be found in Appendices E
and F in Table 12 and Table 13 respectively. The transition dipole moment
elements from the ground state were given in Ref. 27, however all the
remaining elements were derived for this article and we believe this is the
first time they have been presented.

###  III.2 XCIS for alternant hydrocarbons

We now consider the special case of alternant hydrocarbon radicals in relation
to the XCIS method. The excited states of closed-shell alternant hydrocarbons
can be factored into _plus_ and _minus_ combinations where there is no
interaction between the two sets and we aim to test whether the same is true
for radicals. Therefore, it would be very useful to us if we could express the
XCIS Hamiltonian and dipole moment operators in terms of these states.

Figure 3: An overview of our methodology for deriving excited-state rules for
alterant hydrocarbons. Firstly we derive the XCIS Hamiltonian and dipole
moment operators in the basis of excitations, then spin-adapted CSFs, and then
pseudoparity (+/-) and spin-adapted states. We then apply the approximations
of PPP/NDO to finally arrive at a set of matrix elements for the XCIS
Hamiltonian and dipole moment operators for alternant hydrcarbons from which
we infer a set of rules.

The _plus_ and _minus_ states with one open-shell have already been worked out
for us by Longuet-Higgins and Pople and are given in Section II.[24] However,
as far as we are aware, Longuet-Higgins and Pople did not consider
pseudoparity adapted states of three open-shells and we must now define them.
The doublet and quartet states with three open-shells clearly also come in
degenerate pairs for the same reasons as mentioned in Section II.4, and we can
make the following linear combinations of these pairs of states,

$\displaystyle|^{2\mathrm{S}}\Psi_{ij}^{\pm}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle\pm|^{2\mathrm{S}}\Psi_{j}^{i^{\prime}}\rangle),$
(16a) $\displaystyle|^{2\mathrm{T}}\Psi_{ij}^{\pm}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle\mp|^{2\mathrm{S}}\Psi_{j}^{i^{\prime}}\rangle),$
(16b) $\displaystyle|^{4}\Psi_{ij}^{\pm}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|^{4}\Psi_{i}^{j^{\prime}}\rangle\pm|^{4}\Psi_{j}^{i^{\prime}}\rangle)$
(16c)

where $i\neq j$. When $i=j$, Eq. (16a)–Eq. (16c) are no longer valid and only
one state $|^{X}\Psi_{i}^{i^{\prime}}\rangle$ exists (where $X=2S$, 2T or 4):

$\displaystyle|^{2\mathrm{S}}\Psi_{i}^{i^{\prime}}\rangle$
$\displaystyle\in\\{+\textrm{ states}\\}$ (17a)
$\displaystyle|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$
$\displaystyle\in\\{-\textrm{ states}\\}$ (17b)
$\displaystyle|^{4}\Psi_{i}^{i^{\prime}}\rangle$ $\displaystyle\in\\{+\textrm{
states}\\}.$ (17c)

Note that we have defined the $|^{2\mathrm{T}}\Psi_{ij}^{\pm}\rangle$ linear
combinations with the opposite signs to the other two pairs of states: i.e.
for $|^{2\mathrm{S}}\Psi_{ij}^{\pm}\rangle$ and $|^{4}\Psi_{ij}^{\pm}\rangle$,
the _plus_ state has a ‘$+$’ sign and the _minus_ state has a ‘$-$’ sign in
their expansions in Eq. (16a) and Eq. (16c) respectively. However, for
$|^{2\mathrm{T}}\Psi_{ij}^{\pm}\rangle$, the _plus_ state has a ‘$-$’ sign and
_minus_ state a ‘$+$’ sign. The reason for this seemingly inconsistent
definition will become apparent when we make the approximations relating to
alternant hydrocarbons.

Now that we have defined a complete basis of pseudoparity and spin adapted
states, we can derive the exact matrix elements for the XCIS Hamiltonian
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{p}}}$ in this basis, using the equations
in Table 8 and Table 9 discussed above. These are given in Appendix C in Table
10. We also derive the exact dipole moments
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{p}}}$ in the basis of _plus_ and _minus_
states which are presented in Table 14 in Appendix G.

Considering the ground state and single-excitations, Pariser proposed a set of
rules for the interaction via the Hamiltonian and via the dipole moment
operator of _plus_ and _minus_ singlet and triplet states of closed-shell
molecules under the approximations of NDO and resonance integrals being non-
zero only between nearest neighbouring atoms. While the idea of pseudoparity
has been known for a long time to also apply to radicals,[24] and some of
Pariser’s rules have been explored in special cases, to the best of our
knowledge there has yet to be an attempt to derive a general set of rules for
alternant hydrocarbon radicals as Pariser did for closed-shell alternant
hydrocarbons. In this section we test Pariser’s rules to see if they also
apply to radicals. It seems natural to apply the relevant equations for the
XCIS method for radicals mentioned above to derive the Hamiltonian and dipole
moment interaction elements of _plus_ and _minus_ states, as it is analogous
to CIS for closed-shell systems.

Now we consider the effect of the same approximations as Pariser did on the
XCIS Hamiltonian and dipole moments, in addition to the assumption that the
Fock matrix is diagonalized, i.e. $F_{i,j}=\delta_{ij}\epsilon_{i}$ (in other
words, this requires that the orbitals used in forming the Hamiltonian are the
converged molecular orbitals). After making this series of approximations the
matrix elements simplify greatly and many of which vanish. The XCIS
Hamiltonian $\mathrm{\mathbf{H_{\textrm{XCIS}}^{a}}}$ and dipole moment
operator $\mathrm{\mathbf{M_{\textrm{XCIS}}^{a}}}$ in the basis of
pseudoparity-and-spin-adapted states under the aforementioned approximations
are given in Table 11 and Table 15 respectively in Appendices D and H. Clear
rules for the excited state spectra of alternant radical hydrocarbons emerge
which are similar but not identical to those for closed-shell molecules and
which will be presented in Section IV.[77] Fig. 3 summarises the methodology
which we have used to derive these rules, and Table 3 gives a summary of all
the matrix elements derived in this article.

Matrix | Operator | Basis | PPP/NDO | Where to find
---|---|---|---|---
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{x}}}$ | Hamiltonian | XCIS excitations | No | Appendix A Table 8
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{s}}}$ | Hamiltonian | Spin-adapted XCIS states | No | Appendix B Table 9
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{p}}}$ | Hamiltonian | Spin-adapted _plus_ and _minus_ | No | Appendix C Table 10
|  | XCIS states |  |
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{a}}}$ | Hamiltonian | Spin-adapted _plus_ and _minus_ | Yes | Appendix D Table 11
|  | XCIS states |  |
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{x}}}$ | Dipole moment | XCIS excitations | No | Appendix E Table 12
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{s}}}$ | Dipole moment | Spin-adapted XCIS states | No | Appendix F Table 13
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{p}}}$ | Dipole moment | Spin-adapted _plus_ and | No | Appendix G Table 14
|  |  _minus_ XCIS states |  |
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{a}}}$ | Dipole moment | Spin-adapted _plus_ and | Yes | Appendix H Table 15
|  |  _minus_ XCIS states |  |
Table 3: A summary of the Hamiltonian and dipole moment matrices referred to
in this paper for the XCIS method derived in various rotations of the XCIS
states showing where to find them in the appendices.

###  III.3 ExROPPP

We now introduce a novel method, which we call Extended Restricted Open-shell
Pariser-Parr-Pople theory or ExROPPP for short, for the rapid and accurate
simulation of spin-pure excited states of organic $\pi$-conjugated radicals.
This method marries the idea of XCIS[24, 25] with PPP theory[72, 73, 74, 75,
76, 77] using the restricted open-shell method of Ref. 24. We implement the
equations in Sections II.2 and II.3 in an in-house code, using the same
parameters as those for closed-shell molecules, to form the restricted open-
shell Fock matrix in PPP theory and to subsequently obtain the optimized
molecular orbital coefficients and orbital energies by solving the Roothaan
equations iteratively until convergence of the total energy. The essential
equations of XCIS discussed in Section II.1 (see
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{x}}}$,
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{s}}}$,
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{x}}}$ and
$\mathrm{\mathbf{M_{\textrm{XCIS}}^{s}}}$ in Tables 8–13) are then implemented
to form and diagonalize the XCIS Hamiltonian to give the energies and
expansion coefficients of the XCIS excited states, and to compute their dipole
moments and oscillator strengths for plotting of linear absorption spectra.
This method was benchmarked on a series of alternant hydrocarbon radicals and
the results of these computations can be found in Section IV. However, we must
stress that this method is not limited to just alternant hydrocarbon radicals.

##  IV Results

In this section we now present the central results of this paper. Firstly, we
inspect matrix elements of the XCIS Hamiltonian and dipole moment operators in
the basis of pseudoparity-and-spin-adapted states derived in Section III.2 to
reveal simple and elegant rules for the interaction of the excited states of
alternant hydrocarbon radicals, similar to those found by Pariser for closed-
shell molecules[77] and which we believe could aid the design of radicals.
Secondly, we present the results of numerical calculations on a range of
alternant hydrocarbon radicals with our new method: ExROPPP, where ExROPPP is
shown to be accurate in comparison to experimental data and higher-level
methods, spin-pure and computationally inexpensive.

###  IV.1 Alternacy rules

After much theoretical groundwork, we now present a set of rules for the
interaction of the excited states of alternant hydrocarbon radicals. These
rules for radicals are compared with those found previously for closed-shell
alternant hydrocarbons.[77]

It has been previously shown in the 1950s that there is zero interaction
between _plus_ and _minus_ states of closed-shell alternant hydrocarbons.[77]
We show here in Table 11 that this is also true for radicals, with the
aforementioned caveat that we have to define the
$|^{2\mathrm{T}}\Psi_{ij}^{\pm}\rangle$ states with the opposite signs in
their expansion in terms of $|^{2\mathrm{T}}\Psi_{i}^{j^{\prime}}\rangle$ and
$|^{2\mathrm{T}}\Psi_{j}^{i^{\prime}}\rangle$, and additionally we notice that
$|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$ states behave as _minus_ states
whereas $|^{2\mathrm{S}}\Psi_{i}^{i^{\prime}}\rangle$ and
$|^{4}\Psi_{i}^{i^{\prime}}\rangle$ behave as _plus_ states. The interaction
of the ground state $|\Psi_{0}\rangle$ with _plus_ and _minus_ states is the
same in radicals as was found by Pariser for closed-shell alternant
hydrocarbons where it behaves like a _minus_ state in both cases.[77] Fig. 4
summarises the excited state Hamiltonian mixing rules for alternant
hydrocarbon radicals.

Figure 4: Interaction of the electronic excited states of an alternant
hydrocarbon radical via the Hamiltonian $\hat{H}$. Doublet states are shown on
the top row and quartet states on the bottom, and _minus_ states are shown on
the left and _plus_ states on the right. Lines drawn between states represent
an interaction via $\hat{H}$. States may interact if and only if they are in
the same quadrant. $|^{2}\Psi_{0}\rangle$ and
$|^{2\mathrm{S}}\Psi_{ij}^{-}\rangle$ do not mix directly despite being in the
same quadrant, however they may mix indirectly at second order via other
states in the same quadrant. This is a weak version of Brillouin’s theorem for
radicals.

Furthermore, Pariser showed that for closed-shell molecules, the Hamiltonian
in the basis of singlet _minus_ states $|^{1}\Psi_{ij}^{-}\rangle$ is exactly
the same as the Hamiltonian in the basis of triplet _minus_ states
$|^{3}\Psi_{ij}^{-}\rangle$. While this is an interesting observation, we find
no corresponding rule for radicals as no two non-zero Hamiltonian matrix
elements are in general the same.

Dipole moments between states of different multiplicity are formally exactly
zero — this is of course the well known spin selection rule which applies to
all molecules.[82, 33] However, it was previously shown by Pariser for
alternant hydrocarbon closed-shell molecules that a further selection rule
applies — the dipole moments between two _plus_ states or between two _minus_
states are exactly zero (under the approximations of NDO and a vanishing
resonance integral between non-directly bonded atoms). Therefore — the
transition dipole moment between two states may be non-zero if and only if the
states are of opposite pseudoparity and equal multiplicity.[77] This has the
further implication that the ground-state and any singly-excited states have
zero permanent dipole moment. As far as we are aware, it was not known whether
this rule would extent to radicals and although it may have been assumed to,
it was never proven. Here we show that this dipole moment selection rule is in
fact also true for alternant hydrocarbon radicals, as illustrated below in
Fig. 5, using the above definitions of _plus_ and _minus_ states (see Table 15
in Appendix H). As a result of this rule, the ground state and all other
states comprised of single-excitations, and double excitations of the form
$|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$, have zero permanent dipole
moment.

All of the above rules hold even after configuration interaction. That is,
there exist four sets of states grouped by pseduoparity (_plus_ or _minus_)
and multiplicity (singlet/doublet or triplet/quartet) where any member of one
set does not interact via the Hamiltonian with any member of any other set,
before and after configuration interaction. Furthermore, states of the same
pseudoparity and/or opposite multiplicity have zero dipole moment between one
another before and after configuration interaction.

Specifically concerning their dipole moment from the ground state
$|\Psi_{0}\rangle$ (which is a _minus_ state of the lowest multiplicity),
_plus_ states of the same multiplicity may be bright and have a non-zero
transition dipole moment where as _minus_ states are always dark, even after
configuration interaction. There are however cases where _plus_ states of the
same multiplicity as the ground state may be completely dark (such as
$|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$) or may have vanishingly small
transition dipole moments, however, this is not in violation of any of the
above rules. As a result of the dipole pseudoparity selection rule for
alternant hydrocarbons and it holding after configuration interaction, it is
usually (but _not_ always) the case that the lowest energy electronic excited
state of an alternant hydrocarbon radical is completely dark at this level of
theory[83, 84] and expected to have a vanishingly small dipole moment in
experiment and in calculations at higher levels of theory.[85, 86, 87, 88, 89,
90, 91, 92, 93, 69] The reason for this is that the lowest state is usually a
doublet _minus_ state of the form $|\Psi_{i0}^{-}\rangle$.[24, 71] The lowest
excited state cannot be $|\Psi_{i0}^{+}\rangle$ as there will always be a
corresponding _minus_ state of lower energy.[24, 71] If the first excited
state is not $|\Psi_{i0}^{-}\rangle$ then it must contain at least some
contribution from configurations of three open-shells. Inspecting the relevant
matrix elements in Table 11, we find that $|^{4}\Psi_{ij}^{+}\rangle$ is
always the lowest energy configuration out of all of those with open shells in
$i$, 0 and $j^{\prime}$ and $|^{4}\Psi_{i}^{i^{\prime}}\rangle$ is always the
lowest energy configuration out of all of those with open shells in $i$, 0 and
$i^{\prime}$, so it is likely but not certain that in the case that
$|\Psi_{i0}^{-}\rangle$ is not lowest in energy, the lowest energy state will
be a quartet state which will also be dark.

For practical purposes, the alternacy rules presented here mean that the UV-
Vis spectrum of an alternant hydrocarbon radical will show a very weak
absorption at low energies (corresponding to $D_{1}$ that will usually be
$|\Psi_{10}^{-}\rangle$) and an intense absorption at higher energies,
(usually corresponding to $|\Psi_{10}^{+}\rangle)$. Since emission is usually
from the lowest excited state by Kasha’s rule, emission from alternant
radicals is likely to be slow and outcompeted by nonradiative decay.
Consequently, organic radicals for OLEDs should not be alternant hydrocarbons.
These rules are consistent with, but stronger than, previous design rules that
emissive radicals should be non-alternant[69, 27]. It was previously known
that $D_{1}$ would be $|\Psi_{10}^{-}\rangle$ and dark, but this did not
exclude the possibility of $|\Psi_{10}^{-}\rangle$ mixing with, and borrowing
intensity[94] from, higher-lying bright excitations. However, the alternacy
rules derived here show that $|\Psi_{10}^{-}\rangle$ can mix only with other
minus excitations which also have zero dipole moment with the ground state,
such that after configuration interaction $|\Psi_{10}^{-}\rangle$ will remain
dark.

In addition, although the alternacy rules derived here strictly speaking only
hold for purely hydrocarbon radicals, they are likely to hold qualitatively
for chlorinated radicals such as the tris-(2,4,6-trichlorophenyl)-methyl (TTM)
and perchlorotriphenylmethyl (PTM) radicals where the high electronegativity
of chlorine compared to carbon and the mismatch between its 3p orbitals and
the 2p orbitals of carbon mean that it is unlikely to participate
significantly in the electronic structure in the conjugated system.[69, 71]

Table 4: A summary of Pariser’s alternacy rules[77] for closed-shell molecules
compared to the rules derived in this paper for alternant hydrocarbon radicals
using the same set of approximations.

| Closed-shell | Radical
---|---|---
Rule 1 | All states belong to one of two | All states belong to one of two
| pseudoparity groups, _plus_ or _minus_. | pseudoparity groups, _plus_ or _minus_.
Rule 2 | Degeneracy: | Degeneracy:
| $E(^{1}\Psi_{i}^{j^{\prime}})=E(^{1}\Psi_{j}^{i^{\prime}})$ | $E(^{2}\Psi_{\bar{i}}^{\bar{0}})=E(^{2}\Psi_{0}^{i^{\prime}})$
| $E(^{3}\Psi_{i}^{j^{\prime}})=E(^{3}\Psi_{j}^{i^{\prime}})$ | $E(^{2\mathrm{S}}\Psi_{i}^{j^{\prime}})=E(^{2\mathrm{S}}\Psi_{j}^{i^{\prime}})$
|  | $E(^{2\mathrm{T}}\Psi_{i}^{j^{\prime}})=E(^{2\mathrm{T}}\Psi_{j}^{i^{\prime}})$
|  | $E(^{4}\Psi_{i}^{j^{\prime}})=E(^{4}\Psi_{j}^{i^{\prime}})$
Rule 3 |  _plus_ and _minus_ states which are linear |  _plus_ and _minus_ states which are linear
| combinations of degenerate states: | combinations of degenerate states:
| $|^{1}\Psi_{i}^{j^{\prime},\pm}\rangle=\frac{1}{\sqrt{2}}(|^{1}\Psi_{i}^{j^{\prime}}\rangle\pm|^{1}\Psi_{j}^{i^{\prime}}\rangle)$ | $|^{2}\Psi_{0i}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|^{2}\Psi_{\bar{i}}^{\bar{0}}\rangle\pm|^{2}\Psi_{0}^{i^{\prime}}\rangle)$
| $|^{3}\Psi_{i}^{j^{\prime},\pm}\rangle=\frac{1}{\sqrt{2}}(|^{3}\Psi_{i}^{j^{\prime}}\rangle\pm|^{1}\Psi_{j}^{i^{\prime}}\rangle)$ | $|^{2\mathrm{S}}\Psi_{ij}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle\pm|^{2\mathrm{S}}\Psi_{j}^{i^{\prime}}\rangle)$
|  | $|^{2\mathrm{T}}\Psi_{ij}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle\mp|^{2\mathrm{S}}\Psi_{j}^{i^{\prime}}\rangle)$
|  | $|^{4}\Psi_{ij}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|^{4}\Psi_{i}^{j^{\prime}}\rangle\pm|^{4}\Psi_{j}^{i^{\prime}}\rangle)$
Rule 4 |  _plus_ and _minus_ states which are not |  _plus_ and _minus_ states which are not
| combinations of degenerate states: | combinations of degenerate states:
|  $|\Psi_{0}\rangle$ is a _minus_ state. |  $|\Psi_{0}\rangle$ is a _minus_ state.
|  $|^{1}\Psi_{i}^{i^{\prime}}\rangle$ and $|^{3}\Psi_{i}^{i^{\prime}}\rangle$ are _plus_ states. |  $|^{2\mathrm{S}}\Psi_{i}^{i^{\prime}}\rangle$ and $|^{4}\Psi_{i}^{i^{\prime}}\rangle$ are _plus_ states but
|  |  $|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$ is a _minus_ state.
Rule 5 | There is zero Hamiltonian interaction | There is zero Hamiltonian interaction
| between _plus_ and _minus_ states. | between _plus_ and _minus_ states.
Rule 6 | $\langle^{1}\Psi_{ij}^{-}|\hat{H}|^{1}\Psi_{kl}^{-}\rangle=\langle^{3}\Psi_{ij}^{-}|\hat{H}|^{3}\Psi_{kl}^{-}\rangle$ | No corresponding rule
Rule 7 | For any two arbitrary states $|\Psi_{u}\rangle$ and $|\Psi_{v}\rangle$, | For any two arbitrary states $|\Psi_{u}\rangle$ and $|\Psi_{v}\rangle$,
|  $\langle\Psi_{u}|\hat{\mu}|\Psi_{v}\rangle$ may be non-zero if and only if |  $\langle\Psi_{u}|\hat{\mu}|\Psi_{v}\rangle$ may be non-zero if and only if
|  $|\Psi_{u}\rangle$ and $|\Psi_{v}\rangle$ have equal multiplicity and |  $|\Psi_{u}\rangle$ and $|\Psi_{v}\rangle$ have equal multiplicity and
| opposite pseudoparity. | opposite pseudoparity.

TABLE 4 continued on next page …
---

TABLE 4 continued from previous page …
---

| Closed-shell | Radical
---|---|---
Rule 8 |  $S_{1}$ state usually bright. |  $D_{1}$ state usually dark.
Rule 9 | Rules 1–8 hold after configuration | Rules 1–5 and 7–9 hold after configuration
| interaction. | interaction, there is no rule 6 for radicals.

Figure 5: Interaction of the electronic excited states of an alternant
hydrocarbon radical via the dipole moment operator $\hat{\mu}$. Doublet states
are shown on the top row and quartet states on the bottom, and _minus_ states
are shown on the left and _plus_ states on the right. Lines drawn between
states represent an interaction via $\hat{\mu}$. Only states with the same
multiplicity _and_ opposite pseudoparity may interact through the dipole
moment operator. In addition, there is zero direct transition dipole moment
between $|^{2}\Psi_{0}\rangle$ and $|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$ and
between any 2S state and any 2T state.

###  IV.2 Numerical results from ExROPPP on alternant hydrocarbon radicals

Here we present the results of our new method: ExROPPP, benchmarked against
experimental data and the results of calculations with General Multi-
Configurational Quasi-Degenerate Perturbation Theory (GMC-QDPT), an accurate
post-Hatree-Fock method for calculating excited states. The UV/Visible
vertical absorption spectra of a series of odd alternant hydrocarbon radicals
pictured in Fig. 6 were simulated using the ExROPPP method described in
Section III, and these results were also compared to the results of
calculations using GMC-QDPT (for all radicals except TXM) and experimental
spectroscopic data in the literature where available. We must stress here that
we choose to use alternant hydrocarbons to demonstrate the utility of ExROPPP,
despite the fact ExROPPP is not limited to just alternant hydrocarbons, as
they are generally stable and thus there is a large amount of experimental
spectroscopic data available for them in the literature. Also, since many
emissive radicals are non-alternant derivatives of alternant hydrocarbons, the
results of these computations are still of relevance to the design of emissive
radicals. The ExROPPP and GMC-QDPT simulated spectra of the radicals are
presented together in figures 7 – 9 and their lowest lying states computed by
the two theoretical methods are compared to the experimental spectroscopic
data and are presented in Tables I – VII in the supplementary information. The
results of the two theoretical methods are generally consistent with each
other and closely reproduce the experimental excitation energies.

Figure 6: The odd alternant hydrocarbons considered in this paper. The
abbreviations were used for the following radicals: DPM – diphenylmethyl, DPXM
– diphenyl-p-xenylmethyl, PDXM – phenyl-di-p-xenylmethyl and TXM – tri-p-
xenylmethyl.

####  IV.2.1 Accuracy

##### The allyl, benzyl and diphenylmethyl radicals.

The allyl and benzyl radicals have two strong absorption bands in the UV and
the diphenylmethyl (DPM) radical has three, in the range of 200–400 nm. They
have no bright bands in the visible, above 400 nm.[85, 86, 87, 88, 89, 90, 91,
92] The ExROPPP and GMC-QDPT simulated spectra for these three radicals are
presented in Fig. 7 and Fig. 8 and their spectroscopic data are presented in
Tables I – VII in the supplementary information. The number of bands and their
energies and relative intensities are consistent between the two theoretical
methods and with the experimental data.

Figure 7: UV-Visible vertical absorption spectra of the allyl radical
simulated by ExROPPP (blue) and GMC-QDPT (red). Figure 8: UV-Visible vertical
absorption spectra of benzyl and diphenylmethyl radicals simulated by ExROPPP
and GMC-QDPT.

For the allyl radical, in the $C_{2v}$ point group, the $1^{2}B_{2}$ state
($D_{1}$) is $|\Psi_{01}^{-}\rangle$ and appears at 404 nm in the
experiment[85] and at 360 and 407 nm in GMC-QDPT and ExROPPP calculations
respectively, showing a good agreement especially for ExROPPP. The intensity
of the $1^{2}B_{2}$ state is too weak to be recorded experimentally which is
in agreement with the low oscillator strengths computed by GMC-QDPT and
ExROPPP of 0.005 and 0 respectively (at the ExROPPP level of theory this state
is completely dark). The $2^{2}B_{2}$ state, $|\Psi_{01}^{+}\rangle$, appears
at 248 nm in the experiment[85] which is close to the wavelengths predicted by
GMC-QDPT and ExROPPP of 192 and 220 nm respectively, and is bright with an
extinction coefficient of 7340
$\textrm{dm}^{3}\textrm{mol}^{-1}\textrm{cm}^{-1}$ as predicted by GMC-QDPT
and ExROPPP with oscillator strengths of 0.567 and 0.747 respectively.

The weak $1^{2}A_{2}$ state ($D_{1}$) of the benzyl radical is seen around 459
nm, identified from the $0-0$ band in the fluorescence spectrum[88].
$1^{2}A_{2}$ is predicted to be at 409 nm in GMC-QDPT and 430 nm in ExROPPP
and similar to the allyl radical is predicted to be completely dark by
ExROPPP, with a very small oscillator strength predicted by GMC-QDPT and
composed of $|\Psi_{01}^{-}\rangle$. The $3^{2}B_{2}$ state
($|\Psi_{02}^{-}\rangle$), which appears at 275 nm with $f=0$ in ExROPPP and
249 nm with $f\approx 0$ in GMC-QDPT, is not seen in experiment likely due to
it having a vanishingly small intensity. The two bright states of the benzyl
radical, $2^{2}A_{2}$ and $4^{2}B_{2}$, which are composed of
$|\Psi_{01}^{+}\rangle$ and $|\Psi_{02}^{+}\rangle$ respectively are seen in
the experimental spectra at 318 and 258 nm respectively.[87] These states are
also well captured by GMC-QDPT ($2^{2}A_{2}$ at 295 nm and $4^{2}B_{2}$ at 236
nm) and ExROPPP ($2^{2}A_{2}$ at 311 nm and $4^{2}B_{2}$ at 252 nm) with
particularly good agreement in ExROPPP. The relative intensities of these
transitions are also in agreement between the two theoretical methods and the
experiment with the $4^{2}B_{2}$ state predicted to be brighter than the
$2^{2}A_{2}$ by GMC-QDPT and ExROPPP, which is indeed the case in the
experimental spectrum. While the agreement between ExROPPP and experiment is
slightly better for the bright states than dark states, the accuracy of
ExROPPP for the benzyl radical is overall as good as if not better than GMC-
QDPT.

The results for the benzyl radical are also supported by early theoretical
results.[95, 96] In particular, Ref. 95 shows the same energy ordering and
relative intensities of the excited states for the benzyl radical:
$1^{2}A_{2}$ (weak absorption), $2^{2}B_{2}$ (weak absorption), $2^{2}A_{2}$
(strong absorption), $3^{2}B_{2}$ (strong absorption, corresponds to
$4^{2}B_{2}$ in ExROPPP). Ref. 95 only considered the first two orbitals above
and below the SOMO in the benzyl radical, derived from the splitting of the
$1^{2}E$ and $2^{2}E$ levels in benzene, so did not account for the extra dark
state $3^{2}B_{2}$ in ExROPPP which involves orbitals 3 and 3′ in the benzyl
radical, so the energetic ordering in both methods is essentially the same for
the states captured by both.

For DPM, which belongs to the $C_{2}$ point group, the $1^{2}A$ state
($D_{1}$) appears at 515 nm in the experiment[91], and at 485 and 470 nm
respectively in GMC-QDPT and ExROPPP. This state has a weak absorption (300
$\textrm{dm}^{3}\textrm{mol}^{-1}\textrm{cm}^{-1}$) in the experiment and is
dark in GMC-QDPT and ExROPPP. The bright $4^{2}A$ state is seen at 336 nm in
experiment[91] and its energy is well reproduced by GMC-QDPT and ExROPPP with
wavelengths of 400 and 417 nm respectively. It has a high measured intensity
of 31000 $\textrm{dm}^{3}\textrm{mol}^{-1}\textrm{cm}^{-1}$ which is in
agreement with the large oscillator strengths calculated by GMC-QDPT and
ExROPPP of 0.714 and 1.362 respectively. Similarly as for the benzyl radical,
the ExROPPP calculated energy of the bright state is in closer agreement with
experiment than for the “dark” ($D_{1}$) state. The literature also describes
a weaker higher energy absorption at 315 nm which appears as a shoulder to the
left of the 336 nm peak with half its intensity.[92] This absorption is
captured by ExROPPP and appears blue-shifted from the experimental value at
246 nm as seen in Fig. 8, however, due to the size of the active space used it
is not captured by GMC-QDPT. The energetic ordering of the doublet states in
DPM is not the same in ExROPPP and GMC-QDPT due to near degeneracies and a
high density of states at low energy.

##### The triphenylmethyl radical and its p-xenyl- derivatives.

The trityl (triphenylmethyl) radical and its p-xenyl derivatives: di-phenyl-p-
xenylmethyl (DPXM), phenyl-di-p-xenylmethyl (PDXM) and tri-p-xenylmethyl (TXM)
have one or two intense absorption bands, depending on the molecule’s point
symmetry, in the near-UV and the start of the visible region around 300–500
nm. They also have one or two very weak bands in the region of 500–700 nm in
the visible.[93] Trityl and TXM, in the $D_{3}$ point group, both have a
single intense near-UV band owing to the $2^{2}E$ state which contains
$|\Psi_{01}^{+}\rangle$ and $|\Psi_{02}^{+}\rangle$ states (excitations
between the SOMO and the doubly-degenerate HOMO/LUMO). DPXM and PDXM, which
both belong to the $C_{2}$ group, have two intense bands owing to the
$|\Psi_{01}^{+}\rangle$ and $|\Psi_{02}^{+}\rangle$ states due to the removal
of their degeneracy in descending from $D_{3}$ to $C_{2}$ symmetry, one of
which having the irreducible representation $A$ and the other $B$. The
relative ordering of these two states differs between the two radicals in the
computed spectra (see Tables V and VI in the SI). The ExROPPP and GMC-QDPT
simulated spectra of these four radicals are presented in Fig. 9 and their
spectroscopic data are presented in Tables IV – VII in the SI.

There is a good agreement between ExROPPP and the experimental data for these
four radicals, which is better for the bright UV ‘$+$’ states as opposed to
the “dark” lowest energy excited ‘$-$’ states in the visible which are
completely dark in ExROPPP (due to the radicals all being alternant) and which
only have a very small extinction coefficient in the experimental spectra. For
the bright transitions, the ExROPPP values are within 25 nm of the
experimental value for trityl, 24 nm for DPXM, 47 nm for PDXM and 56 nm for
TXM. Chu and Weissman[93] suggest that the lowest excited state of the trityl
and TXM radicals is of $A_{1}$ symmetry from analysis of fluorescence
polarisation data, however we believe this not to be that case as this lowest
excited state of the trityl radical has been shown to be a doubly-degenerate
$E$ state in a later publication[84] and in our results. The results of GMC-
QDPT calculations for the trityl and DPXM radicals are also in good agreement
with ExROPPP and the relative energy ordering of the doublet states is the
same in both methods for both radicals.

The excitation energies and relative intensities of the lowest excited states
for the triphenylmethyl radical are also largely supported theoretical studies
based loosely on Ref. 24 where the first excited state in their results is a
${}^{2}E$ state predicted to have a weak absorption at 515 nm (2.41 eV) and
their fifth excited state is a ${}^{2}E$ state predicted to have an intense
absorption at 322 nm (3.84 eV) corresponding to $2^{2}E$ in our results.[84]
The number of states is different in both methods however, the energetic
ordering of the states captured by both ExROPPP and the method of Ref. 84 is
the same. The energies in Ref. 84 were shifted such that the energy of the
lowest excited state matched experimental data, whereas our results are not
manipulated in any way to match experiment.

Figure 9: UV-Visible vertical absorption spectra of the trityl radical and its
derivatives: DPXM, PDXM and TXM simulated by ExROPPP (bottom) and of trityl,
DPXM and PDXM simulated by GMC-QDPT (top). Due to computational expense, TXM
was not simulated by GMC-QDPT.

##### Quartet states

The agreement between the experimental, GMC-QDPT and ExROPPP calculated
energies for the quartet state of the allyl radical, and between the GMC-QDPT
and ExROPPP calculated energies of the two quartet states of the benzyl
radical is not so good as for the doublet states. The experimental and GMC-
QDPT energies of 6.33 eV and 6.07 eV respectively are in agreement for the
quartet energy of the allyl radical, but the ExROPPP energy of 4.40 eV is
significantly lower. Similarly, for the benzyl radical, GMC-QDPT predicts
energies of 4.42 eV and 5.42 eV for the $1^{4}B_{2}$ and $1^{4}A_{2}$ states
respectively, but the ExROPPP values are 3.62 eV and 4.80 eV respectively,
again seeming to underestimate the quartet energy. ExROPPP also seems to
largely underestimate the quartet energies for the trityl radical for the
$1^{4}A_{2}$ and $1^{4}E$ states when compared to the GMC-QDPT values. In
addition, the relative energetic ordering of doublet and quartet states is
different in the GMC-QDPT and ExROPPP results. One possible reason for this
discrepancy is that the set of PPP parameters used in this paper (and PPP
theory in general) have mainly been applied to calculating the energies of
singlet-singlet transitions in closed-shell molecules and in particular,
simulating their spectra, in which only those transitions which are bright are
of importance.[97] Therefore, by analogy, one would expect these parameters to
be more accurate for calculating the energies of excited doublet states and
less accurate for (dark) excited quartet states in radicals, which is what we
observe from our results.

Moreover it is known to be very difficult to experimentally measure quartet
state energies as quartet states are formally dark. Also in the case of odd
alternant hydrocarbon radicals, the first quartet state is always higher in
energy than the lowest excited doublet state so it is not normally observed
through intersystem crossing from the doublet state in the same way that a
triplet state is observed via intersystem crossing from the singlet state in a
closed-shell molecule.[98] As a side note, this is a useful property of such
radicals which makes them desirable for making OLEDs, but also means that the
quartet state is rarely observed.[24, 99, 100] The experimental quartet
energies for the odd alternant hydrocarbon radicals tested, other than allyl,
may be absent from the literature so it is impossible to comment on the
accuracy of ExROPPP for quartet states with any certainty.

Finally, on analysing the CI expansion of the quartet states calculated by
ExROPPP, we notice that the quartet states for the trityl, DPXM, PDXM and TXM
radicals have significant contribution from higher excitations ($i>2$,
$j^{\prime}>2^{\prime}$) and therefore it is expected that their computed
energies will be dependent on inclusion of such higher excitations.

##### Overall accuracy

Fig. 10 shows a comparison of the experimental excitation energies and those
computed by the two theoretical methods. The ExROPPP calculated energies are
all close to the experimental values except for the outlier at 6.33 eV in the
experiment[85] and 4.40 eV calculated by ExROPPP which is for the quartet
state of the allyl radical. As mentioned earlier, it is difficult to comment
on the accuracy of ExROPPP for the calculation of quartet state energies. The
root mean squared (RMS) errors of the ExROPPP and GMC-QDPT results
$E_{i}^{\textrm{calc.}}$ compared to the experimental excitation energies
$E_{i}^{\textrm{exp.}}$ have been calculated using the formula

$\displaystyle\epsilon_{\textrm{RMS}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(E_{i}^{\textrm{calc.}}-E_{i}^{\textrm{exp.}})^{2}}.$
(18)

The $\epsilon_{\textrm{RMS}}$ values for the ExROPPP and GMC-QDPT results are
presented in Table 5. It can be seen that ExROPPP calculates excitation
energies with a similar accuracy to GMC-QDPT for the alternant hydrocarbon
radicals chosen. There is only one quartet state included in the former set of
results for which we have gathered literature experimental data for, which is
that of the allyl radical and is an outlier in ExROPPP. When this quartet
state is omitted the ExROPPP results are in closer agreement to experiment
than GMC-QDPT. We must stress that the results of ExROPPP computations
presented here are the raw results which are in no way shifted to agree with
experiment (or GMC-QDPT) and the same parameters are used as in previous
studies on closed-shell molecules.[101, 68] Computations using the bases of
excitations and of spin-adapted states yield identical results.

Comparing the numerical results from the ExROPPP calculations on alternant
hydrocarbons with the alternacy rules derived in Section IV.1, we see that the
results of ExROPPP are entirely consistent with these rules. This can be seen
in Tables I–VII in Section I of the SI, where the energies, oscillator
strengths and CI expansion coefficients of the ExROPPP states are tabulated.

Table 5: RMS errors of i) ExROPPP and ii) GMC-QDPT calculated excited state energies compared to the experimental values and iii) ExROPPP calculated energies compared to GMC-QDPT for the odd alternant hydrocarbon radicals presented in this paper. Numbers without brackets are for all states, whereas numbers inside brackets are calculated for doublet states only. There is no GMC-QDPT data for TXM so these results are omitted, however ExROPPP results for TXM are included. |  $\epsilon_{\textrm{RMS}}$ / eV (doublets only)
---|---
GMC-QDPT vs Exp. | 0.51 (0.52)
ExROPPP vs Exp. | 0.56 (0.33)
ExROPPP vs GMC-QDPT | 0.66 (0.52)
Figure 10: Comparison of the excitation energies computed by ExROPPP and GMC-
QDPT calculations and experimental data from the literature. The outlier
around 6.3 eV in the experiment, 6 eV in GMC-QDPT but only 4.4 eV in ExROPPP
is the quartet state of the allyl radical.

In general, the key features of the low lying electronic states of radicals
are well captured by the simple and computationally inexpensive ExROPPP
method. For the alternant hydrocarbon radicals studied, the lowest lying
electronic state above the ground state is always a doublet $D_{1}$ and always
has a very weak absorption in the experiment. This state is always
$|^{2}\Psi_{0i}^{-}\rangle$ at the ExROPPP level of theory and always dark[24,
99], but at the higher level of theory in GMC-QDPT it may have a very small
oscillator strength.[24, 99] The doublet states $|^{2}\Psi_{0i}^{+}\rangle$
are always responsible for the low energy intense bands. There are higher
lying doublet states of the type $|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$
which are also bright, however for the molecules tested they lie far into the
UV and are of less relevance to the optical properties. Furthermore, it is
shown by ExROPPP calculations that only two excited states,
$|\Psi_{01}^{+}\rangle$ and $|\Psi_{02}^{+}\rangle$, are responsible for the
significant features of the UV/Visible spectrum for these molecules, involving
transitions between the SOMO and the orbitals $1$, $2$, $1^{\prime}$ and
$2^{\prime}$.

####  IV.2.2 Spin-purity

We calculated $\langle S^{2}\rangle_{i}$ for each of the ExROPPP states $i$,
and then calculated a quantity $\Delta\langle S^{2}\rangle_{i}$ which we
define, using the standard definition of uncertainty[82], as

$\displaystyle\Delta\langle S^{2}\rangle_{i}$ $\displaystyle=\sqrt{\langle
S^{4}\rangle_{i}-\langle S^{2}\rangle_{i}^{2}}$ (19)

and in matrix form

$\displaystyle\mathrm{\mathbf{\Delta}}\langle S^{2}\rangle$
$\displaystyle=\left[\mathrm{\mathbf{S^{2}}}\mathrm{\mathbf{S^{2}}}-(\mathrm{\mathbf{S^{2}}})^{2}\right]^{\frac{1}{2}}$
(20)
$\displaystyle=\left[\mathrm{\mathbf{C}}^{\mathrm{T}}\mathrm{\mathbf{S^{\prime
2}}}\mathrm{\mathbf{S^{\prime
2}}}\mathrm{\mathbf{C}}-(\mathrm{\mathbf{C}}^{\mathrm{T}}\mathrm{\mathbf{S^{\prime
2}}}\mathrm{\mathbf{C}})^{2}\right]^{\frac{1}{2}}$

where $\mathrm{\mathbf{S^{2}}}$ and $\mathrm{\mathbf{S^{\prime 2}}}$ are the
matrix representations of $S^{2}$ in the excited state basis and in the basis
of excitations respectively and $\mathrm{\mathbf{C}}$ are the CI coefficients
of the excited states in the basis of excitations. While there exist other
definitions of spin-contamination $\Delta\langle S^{2}\rangle$,[102, 42] some
of which give non-zero $\Delta\langle S^{2}\rangle$ for spin-pure quartet
states[42] and others require evaluation of inner products of orbitals[102]
and therefore, we use the above definition as it is theoretically simpler and
allows us to define spin-purity for both doublet and quartet states. We find
that $\Delta\langle S^{2}\rangle_{i}=0$ for all ExROPPP states. However, in
the basis of all single excitations only (ROPPP-CIS), $\Delta\langle
S^{2}\rangle_{i}\neq 0$ for all states (apart from for the allyl radical where
only some states are contaminated, but this is a special case and not the
trend), and $\Delta\langle S^{2}\rangle_{i}$ is generally significantly large,
as shown in Tables VIII and IX in the SI.

####  IV.2.3 Speed

The speed of the ExROPPP method was compared to that of the accurate post-
Hartree-Fock method, GMC-QDPT for the radicals studied. These results are
shown in Table 6. GMC-QDPT calculations on allyl, benzyl, trityl and DPXM
radicals were performed using GAMESS-US using 2 processors at a remote
computational facility, and for PDXM using two processors on a desktop
computer. All ExROPPP calculations were performed using Python 3 on dual core
laptop PC. The timings from ExROPPP presented here are for proof of concept
purposes only since the ExROPPP code has not been optimised and was coded in
Python; however, it clearly offers a substantial reduction in computation time
compared to GMC-QDPT. We leave optimisation of the ExROPPP code and rigorous
benchmarking for future work.

Table 6: Comparison of the speed of the ExROPPP method, described in this paper, with GMC-QDPT, an existing accurate CI method, for the calculation of the excited states of radicals. As separate GMC-QDPT calculations were performed for doublet and quartet states, the timings presented here are for doublet calculations only. dddnc stands for not calculated. Molecule |  $t_{\textrm{GMC-QDPT}}$ / s |  $t_{\textrm{ExROPPP}}$ / s
---|---|---
Allyl | 5 | 0.03
Benzyl | 1025 | 0.04
DPM | 12435 | 0.13
Trityl | 5016 | 0.42
DPXM | 17468 | 0.97
PDXM | 219666 | 2.10
TXM | nc | 5.00

##  V Computational details

In this section we outline technical details of the calculations performed by
ExROPPP and GMC-QDPT whose results were presented in Section IV, such details
being key to the interpretation and understanding of these results. Any
additional, less essential details are not included in this section for the
sake of length and are instead included in the supplementary information.

###  V.1 Raw results

Details of the GMC-QDPT calculations including active spaces and number of CI
states are presented below in Table 7.

Table 7: Comparison of the active spaces used and number of CI states calculated in GMC-QDPT calculations for the odd alternant hydrocarbon radicals presented in this paper (except for TXM for which no GMC-QDPT calculations were performed). | Active Space | State Selection
---|---|---
Molecule | $N_{\pi}$ | $N_{\textrm{elec.}}$ | $N_{\textrm{orbs}}$ | $N_{\textrm{Doublets}}$ | $N_{\textrm{Quartets}}$
Allyl | 3 | 3 | 3 | Lowest 5 | Lowest 1
Benzyl | 7 | 7 | 7 | Lowest 6 | Lowest 2
DPM | 13 | 7 | 7 | Lowest 7 | Lowest 3
Trityl | 19 | 5 | 5 | Lowest 5 | Lowest 3
DPXM | 25 | 5 | 5 | Lowest 5 | None
PDXM | 31 | 5 | 5 | Lowest 5 | None

Details such as PPP parameters, optimised geometries and orbitals, and
parameters used for plotting of spectra are given in the supplementary
information.

###  V.2 Simulation of spectra

The lowest 25 XCIS states (in energy, including the ground state) were
calculated for the simulation of vertical absorption spectra by ExROPPP. All
doublet states calculated by GMC-QDPT were included in the simulation of
vertical absorption spectra. Different numbers of states and different active
spaces were chosen for each molecule which can be found in Table 7. Lorentzian
line broadening was applied to the peaks in the simulated spectra for both
ExROPPP and GMC-QDPT with a full width at half maximum (FWHM) value of 20 nm
at 300 nm.

##  VI Conclusions

In this article we have combined previous ideas that adding certain double
excitations to a radical CIS calculation can lead to spin purity[24, 25, 27]
with Pariser-Parr-Pople theory [74, 75, 72, 73, 103] to obtain a computational
method for the low-lying electronically excited states of hydrocarbon
radicals. This method, which we call ExROPPP, reproduces experimental doublet
excitation energies and intensities for a series of odd alternant hydrocarbon
radicals accurately and with minimal computational effort, with a similar
degree of accuracy to General Multiconfigurational Quasi-Degenerate
Perturbation Theory (GMC-QDPT), an accurate multiconfigurational SCF method.
In addition, using the approximations of PPP theory we have derived widely-
applicable rules for alternant hydrocarbons which should aid in the
prediction, assignment and interpretation of their spectra.

There are many avenues for future research. The proof-of-concept calculations
in this paper are for hydrocarbon radicals (for which there already exists
tested parameterization) and we plan to extend this method to radicals
containing nitrogen and chlorine such as the breakthrough fluorescent radical
TTM-3NCz[3], sulfur containing radicals such as those based on thiophene[7]
and non-alternant and ionic radicals. There are possibilities of using machine
learning[104] to optimize ExROPPP parameterization, especially for heteroatoms
and for quartet states, given that our initial results suggest that ExROPPP is
less accurate for quartet state energies than doublet state energies. The
ExROPPP method could be extended to radicals with more than one unpaired
electron in their ground state[22], and the alternacy rules we have derived
for monoradicals may be extendable to these systems. It is our hope that
ExROPPP can be extended to accurately and quickly simulate a wide range of
conjugated organic radicals, becoming a useful tool for rapid spectral
determination. This can then be used to design highly efficient and emissive
radicals[27] for applications such as OLEDs[3] and quantum computing.[21]

## Supplementary Material

The supplementary material consists of tables containing the detailed
energies, oscillator strengths/intensities and composition of excited states
calculated by ExROPPP and GMC-QDPT and obtained from experimental data. Also
deposited here are molecular geometries used for the calculations, and any
other details of the computational methods used in this paper.

## Acknowledgements

The authors thank Emrys W. Evans for helpful suggestions on the research. TJHH
acknowledges a Royal Society University Research Fellowship URF\R1\201502.

## Data Availability Statement

The data that supports the findings of this study are available within the
article and its supplementary material.

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## Appendix A - XCIS Hamiltonian matrix elements in the basis of excitations

Table 8: Matrix elements for the XCIS Hamiltonian $\mathrm{\mathbf{H_{\textrm{XCIS}}^{x}}}$ in the basis of all singly-excited configurations, and doubly-excited configurations of the type $|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$. Some of the excitations themselves are non spin-pure, however, spin-pure linear combinations of excitations will always be formed on diagonalization of the Hamiltonian. # | Matrix element | $=$
---|---|---
Hx1 | $\langle^{2}\Psi_{0}|\hat{H}|^{2}\Psi_{0}\rangle$ | $\sum_{n=1}^{k}(h_{nn}+F_{nn})+\frac{1}{2}(h_{00}+F_{00})-\frac{1}{4}J_{00}:=E_{0}$
Hx2 | $\langle^{2}\Psi_{0}|\hat{H}|^{2}\Psi_{\bar{i}}^{\bar{0}}\rangle$ | $F_{i0}+\frac{1}{2}(i0|00)$
Hx3 | $\langle^{2}\Psi_{0}|\hat{H}|^{2}\Psi_{0}^{j^{\prime}}\rangle$ | $F_{0j^{\prime}}-\frac{1}{2}(0j^{\prime}|00)$
Hx4 | $\langle^{2}\Psi_{0}|\hat{H}|\Psi_{i}^{j^{\prime}}\rangle$ | $F_{ij^{\prime}}-\frac{1}{2}(i0|0j^{\prime})$
Hx5 | $\langle^{2}\Psi_{0}|\hat{H}|\Psi_{\bar{i}}^{\bar{j^{\prime}}}\rangle$ | $F_{ij^{\prime}}+\frac{1}{2}(i0|0j^{\prime})$
Hx6 | $\langle^{2}\Psi_{0}|\hat{H}|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$ | $(i0|0j^{\prime})$
Hx7 | $\langle^{2}\Psi_{\bar{k}}^{\bar{0}}|\hat{H}|^{2}\Psi_{\bar{i}}^{\bar{0}}\rangle$ | $\delta_{ik}(E_{0}+F_{00}+\frac{1}{2}J_{00})-F_{ik}-(ik|00)+\frac{1}{2}(i0|0k)$
Hx8 | $\langle^{2}\Psi_{\bar{k}}^{\bar{0}}|\hat{H}|^{2}\Psi_{0}^{j^{\prime}}\rangle$ | $(0k|0j^{\prime})$
Hx9 | $\langle^{2}\Psi_{\bar{k}}^{\bar{0}}|\hat{H}|\Psi_{i}^{j^{\prime}}\rangle$ | $(0k|ij^{\prime})$
Hx10 | $\langle^{2}\Psi_{\bar{k}}^{\bar{0}}|\hat{H}|\Psi_{\bar{i}}^{\bar{j^{\prime}}}\rangle$ | $\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]+(0k|ij^{\prime})-(0j^{\prime}|ik)$
Hx11 | $\langle^{2}\Psi_{\bar{k}}^{\bar{0}}|\hat{H}|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$ | $\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]-(0j^{\prime}|ik)$
Hx12 | $\langle^{2}\Psi_{0}^{l^{\prime}}|\hat{H}|^{2}\Psi_{0}^{j^{\prime}}\rangle$ | $\delta_{j^{\prime}l^{\prime}}(E_{0}-F_{00}+\frac{1}{2}J_{00})+F_{j{{}^{\prime}}l^{\prime}}-(00|j{{}^{\prime}}l^{\prime})+\frac{1}{2}(j^{\prime}0|0l^{\prime})$
Hx13 | $\langle^{2}\Psi_{0}^{l^{\prime}}|\hat{H}|\Psi_{i}^{j^{\prime}}\rangle$ | $-\delta_{j^{\prime}l^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]+(ij^{\prime}|l^{\prime}0)-(i0|l^{\prime}j^{\prime})$
Hx14 | $\langle^{2}\Psi_{0}^{l^{\prime}}|\hat{H}|\Psi_{\bar{i}}^{\bar{j^{\prime}}}\rangle$ | $(ij^{\prime}|l^{\prime}0)$
Hx15 | $\langle^{2}\Psi_{0}^{l^{\prime}}|\hat{H}|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$ | $\delta_{j^{\prime}l^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]+(i0|l^{\prime}j^{\prime})$
Hx16 | $\langle\Psi_{k}^{l^{\prime}}|\hat{H}|\Psi_{i}^{j^{\prime}}\rangle$ | $\delta_{ik}\delta_{j^{\prime}l^{\prime}}E_{0}+\delta_{ik}[F_{j^{\prime}l^{\prime}}-\frac{1}{2}(j^{\prime}0|0l^{\prime})]-\delta_{j^{\prime}l^{\prime}}[F_{ik}-\frac{1}{2}(i0|0k)]+(ij^{\prime}|l^{\prime}k)-(ik|l^{\prime}j^{\prime})$
Hx17 | $\langle\Psi_{k}^{l^{\prime}}|\hat{H}|\Psi_{\bar{i}}^{\bar{j^{\prime}}}\rangle$ | $(ij^{\prime}|kl^{\prime})$
Hx18 | $\langle\Psi_{k}^{l^{\prime}}|\hat{H}|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$ | $-\delta_{j^{\prime}l^{\prime}}(0k|i0)$
Hx19 | $\langle\Psi_{\bar{k}}^{\bar{l^{\prime}}}|\hat{H}|\Psi_{\bar{i}}^{\bar{j^{\prime}}}\rangle$ | $\delta_{ik}\delta_{j^{\prime}l^{\prime}}E_{0}+\delta_{ik}[F_{j^{\prime}l^{\prime}}+\frac{1}{2}(j^{\prime}0|0l^{\prime})]-\delta_{j^{\prime}l^{\prime}}[F_{ik}+\frac{1}{2}(i0|0k)]+(ij^{\prime}|l^{\prime}k)-(ik|l^{\prime}j^{\prime})$
Hx20 | $\langle\Psi_{\bar{k}}^{\bar{l^{\prime}}}|\hat{H}|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$ | $\delta_{ik}(j^{\prime}0|0l^{\prime})$
Hx21 | $\langle\Psi_{\bar{k}0}^{\bar{0}l^{\prime}}|\hat{H}|\Psi_{\bar{i}0}^{\bar{0}j^{\prime}}\rangle$ | $\delta_{ik}\delta_{j^{\prime}l^{\prime}}E_{0}+\delta_{ik}[F_{j^{\prime}l^{\prime}}+\frac{1}{2}(j^{\prime}0|0l^{\prime})]-\delta_{j^{\prime}l^{\prime}}[F_{ik}-\frac{1}{2}(i0|0k)]-(ik|l^{\prime}j^{\prime})$

## Appendix B - XCIS Hamiltonian matrix elements in the basis of spin-adapted
states

Table 9: Matrix elements for the XCIS Hamiltonian $\mathrm{\mathbf{H_{\textrm{XCIS}}^{s}}}$ rotated into the basis of spin-adapted states. Those matrix elements which are the same as those in Table 8 are omitted. As expected there is zero interaction between doublet and quartet states. # | Matrix element | $=$
---|---|---
Hs1 | $\langle^{2}\Psi_{0}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$ | $\sqrt{2}F_{ij^{\prime}}$
Hs2 | $\langle^{2}\Psi_{0}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{j^{\prime}}\rangle$ | $\frac{3}{\sqrt{6}}(i0|0j^{\prime})$
Hs3 | $\langle^{2}\Psi_{0}|\hat{H}|^{4}\Psi_{i}^{j^{\prime}}\rangle$ | 0
Hs4 | $\langle^{2}\Psi_{\bar{k}}^{\bar{0}}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$ | $\frac{\delta_{ik}}{\sqrt{2}}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]+\sqrt{2}(0k|ij^{\prime})-\frac{1}{\sqrt{2}}(0j^{\prime}|ik)$
Hs5 | $\langle^{2}\Psi_{\bar{k}}^{\bar{0}}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{j^{\prime}}\rangle$ | $\frac{3}{\sqrt{6}}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]-\frac{3}{\sqrt{6}}(0j^{\prime}|ik)$
Hs6 | $\langle^{2}\Psi_{\bar{k}}^{\bar{0}}|\hat{H}|^{4}\Psi_{i}^{j^{\prime}}\rangle$ | 0
Hs7 | $\langle^{2}\Psi_{0}^{l^{\prime}}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$ | $-\frac{\delta_{j^{\prime}l^{\prime}}}{\sqrt{2}}[F_{i0}-\frac{1}{2}(i0|00)]+\sqrt{2}(ij^{\prime}|l^{\prime}0)-\frac{1}{\sqrt{2}}(i0|l^{\prime}j^{\prime})$
Hs8 | $\langle^{2}\Psi_{0}^{l^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{j^{\prime}}\rangle$ | $\frac{3}{\sqrt{6}}\delta_{j^{\prime}l^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]+\frac{3}{\sqrt{6}}(i0|j^{\prime}l^{\prime})$
Hs9 | $\langle^{2}\Psi_{0}^{l^{\prime}}|\hat{H}|^{4}\Psi_{i}^{j^{\prime}}\rangle$ | 0
Hs10 | $\langle^{2\mathrm{S}}\Psi_{k}^{l^{\prime}}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{j^{\prime}}\rangle$ | $\delta_{ik}\delta_{j^{\prime}l^{\prime}}E_{0}+\delta_{ik}F_{j^{\prime}l^{\prime}}-\delta_{j^{\prime}l^{\prime}}F_{ik}-(ik|l^{\prime}j^{\prime})+2(ij^{\prime}|kl^{\prime})$
Hs11 | $\langle^{2\mathrm{S}}\Psi_{k}^{l^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{j^{\prime}}\rangle$ | $\frac{\sqrt{3}}{2}[\delta_{ik}(j^{\prime}0|0l^{\prime})-\delta_{j^{\prime}l^{\prime}}(0k|i0)]$
Hs12 | $\langle^{2\mathrm{S}}\Psi_{k}^{l^{\prime}}|\hat{H}|^{4}\Psi_{i}^{j^{\prime}}\rangle$ | 0
Hs13 | $\langle^{2\mathrm{T}}\Psi_{k}^{l^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{j^{\prime}}\rangle$ | $\delta_{ik}\delta_{j^{\prime}l^{\prime}}E_{0}+\delta_{ik}[F_{j^{\prime}l^{\prime}}+(j^{\prime}0|0l^{\prime})]-\delta_{j^{\prime}l^{\prime}}[F_{ik}-(i0|0k)]-(ik|l^{\prime}j^{\prime})$
Hs14 | $\langle^{2\mathrm{T}}\Psi_{k}^{l^{\prime}}|\hat{H}|^{4}\Psi_{i}^{j^{\prime}}\rangle$ | 0
Hs15 | $\langle^{4}\Psi_{k}^{l^{\prime}}|\hat{H}|^{4}\Psi_{i}^{j^{\prime}}\rangle$ | $\delta_{ik}\delta_{j^{\prime}l^{\prime}}E_{0}+\delta_{ik}[F_{j^{\prime}l^{\prime}}-\frac{1}{2}(j^{\prime}0|0l^{\prime})]-\delta_{j^{\prime}l^{\prime}}[F_{ik}+\frac{1}{2}(0k|i0)]-(ik|l^{\prime}j^{\prime})$

## Appendix C - XCIS Hamiltonian matrix elements in the basis of spin-adapted
_plus_ and _minus_ states

Table 10: Exact matrix elements for the spin-pure XCIS Hamiltonian
$\mathrm{\mathbf{H_{\textrm{XCIS}}^{p}}}$ in the basis of _plus_ and _minus_
combinations of spin-adapted states _without_ applying the approximations of
alternant hydrocarbons and PPP theory. As expected there is zero interaction
between doublet and quartet states.

# | Matrix element | $=$
---|---|---
Hp1 | $\langle^{2}\Psi_{0}|\hat{H}|^{2}\Psi_{0}\rangle$ | $E_{0}$
Hp2 | $\langle^{2}\Psi_{0}|\hat{H}|^{2}\Psi_{0i}^{-}\rangle$ | $\frac{1}{\sqrt{2}}\bigl{[}F_{i0}+\frac{1}{2}(i0|00)-F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)\bigr{]}$
Hp3 | $\langle^{2}\Psi_{0}|\hat{H}|^{2}\Psi_{0i}^{+}\rangle$ | $\frac{1}{\sqrt{2}}\bigl{[}F_{i0}+\frac{1}{2}(i0|00)+F_{0i^{\prime}}-\frac{1}{2}(0i^{\prime}|00)\bigr{]}$
Hp4 | $\langle^{2}\Psi_{0}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{i^{\prime}}\rangle$ | $\sqrt{2}F_{ii^{\prime}}$
Hp5 |  $\langle^{2}\Psi_{0}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $F_{ij^{\prime}}-F_{ji^{\prime}}$
Hp6 |  $\langle^{2}\Psi_{0}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $F_{ij^{\prime}}+F_{ji^{\prime}}$
Hp7 | $\langle^{2}\Psi_{0}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$ | $\frac{3}{\sqrt{6}}(i0|0i^{\prime})$
Hp8 |  $\langle^{2}\Psi_{0}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $\frac{\sqrt{3}}{2}[(i0|0j^{\prime})+(j0|0i^{\prime})]$
Hp9 |  $\langle^{2}\Psi_{0}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $\frac{\sqrt{3}}{2}[(i0|0j^{\prime})-(j0|0i^{\prime})]$
Hp10 | $\langle^{2}\Psi_{0}|\hat{H}|^{4}\Psi_{i}^{i^{\prime}}\rangle$ | 0
Hp11 |  $\langle^{2}\Psi_{0}|\hat{H}|^{4}\Psi_{ij}^{-}\rangle$, $i\neq j$ | 0
Hp12 |  $\langle^{2}\Psi_{0}|\hat{H}|^{4}\Psi_{ij}^{+}\rangle$, $i\neq j$ | 0
Hp13 | $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{2}\Psi_{0i}^{-}\rangle$ | $\delta_{ik}(E_{0}+\frac{1}{2}J_{00})+\frac{F_{i^{\prime}k^{\prime}}-F_{ik}}{2}-\frac{1}{2}[(00|i^{\prime}{k^{\prime}})+(ik|00)+(0i|0k^{\prime})+(0k|0i^{\prime})]$
|  | $+\frac{1}{4}[(i0|0k)+(i^{\prime}0|0k^{\prime})]$
Hp14 | $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{2}\Psi_{0i}^{+}\rangle$ | $\delta_{ik}F_{00}-\frac{F_{ik}+F_{i^{\prime}{k^{\prime}}}}{2}+\frac{1}{2}[(00|i^{\prime}{k^{\prime}})-(ik|00)-(0i|0k^{\prime})+(0k|0i^{\prime})]$
|  | $+\frac{1}{4}[(i^{\prime}0|0k^{\prime})-(i0|0k)]$
Hp15 | $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{i^{\prime}}\rangle$ | $\frac{\delta_{ik}}{2}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]+\frac{\delta_{i^{\prime}k^{\prime}}}{2}[F_{i0}-\frac{1}{2}(i0|00)]+(0k|ii^{\prime})-(ii^{\prime}|k^{\prime}0)+\frac{1}{2}[(i0|k^{\prime}i^{\prime})$
|  | $-(0i^{\prime}|ik)]$
Hp16 |  $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $\frac{1}{2\sqrt{2}}\bigl{\\{}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]-\delta_{jk}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]+\delta_{k^{\prime}j^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]$
|  | $-\delta_{k^{\prime}i^{\prime}}[F_{j0}-\frac{1}{2}(j0|00)]\bigr{\\}}+\frac{1}{2\sqrt{2}}\bigl{[}-(0j^{\prime}|ik)+(0i^{\prime}|jk)+(i0|k^{\prime}j^{\prime})$
|  | $-(j0|k^{\prime}i^{\prime})\bigr{]}+\frac{1}{\sqrt{2}}\bigl{[}(0k|ij^{\prime})-(0k|ji^{\prime})-(ij^{\prime}|k^{\prime}0)+(ji^{\prime}|k^{\prime}0)\bigr{]}$
Hp17 |  $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $\frac{1}{2\sqrt{2}}\bigl{\\{}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]+\delta_{jk}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]+\delta_{k^{\prime}j^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]$
|  | $+\delta_{k^{\prime}i^{\prime}}[F_{j0}-\frac{1}{2}(j0|00)]\bigr{\\}}+\frac{1}{2\sqrt{2}}\bigl{[}-(0j^{\prime}|ik)-(0i^{\prime}|jk)+(i0|k^{\prime}j^{\prime})$
|  | $+(j0|k^{\prime}i^{\prime})\bigr{]}+\frac{1}{\sqrt{2}}\bigl{[}(0k|ij^{\prime})+(0k|ji^{\prime})-(ij^{\prime}|k^{\prime}0)-(ji^{\prime}|k^{\prime}0)\bigr{]}$
Hp18 | $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$ | $\frac{\sqrt{3}}{2}\bigl{\\{}\delta_{ik}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]-(0i^{\prime}|ik)-\delta_{i^{\prime}k^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]-(i0|i^{\prime}k^{\prime})\bigr{\\}}$

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# | Matrix element | $=$
---|---|---
Hp19 |  $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $\frac{3}{2\sqrt{6}}\bigl{\\{}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]-(0j^{\prime}|ik)+\delta_{jk}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]-(0i^{\prime}|jk)$
|  | $-\delta_{j^{\prime}k^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]-(i0|j^{\prime}k^{\prime})-\delta_{i^{\prime}k^{\prime}}[F_{j0}-\frac{1}{2}(j0|00)]-(j0|i^{\prime}k^{\prime})\bigr{\\}}$
Hp20 |  $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $\frac{3}{2\sqrt{6}}\bigl{\\{}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]-(0j^{\prime}|ik)-\delta_{jk}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]+(0i^{\prime}|jk)$
|  | $-\delta_{j^{\prime}k^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]-(i0|j^{\prime}k^{\prime})+\delta_{i^{\prime}k^{\prime}}[F_{j0}-\frac{1}{2}(j0|00)]+(j0|i^{\prime}k^{\prime})\bigr{\\}}$
Hp21 | $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{4}\Psi_{i}^{i^{\prime}}\rangle$ | 0
Hp22 |  $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{4}\Psi_{ij}^{-}\rangle$, $i\neq j$ | 0
Hp23 |  $\langle^{2}\Psi_{0k}^{-}|\hat{H}|^{4}\Psi_{ij}^{+}\rangle$, $i\neq j$ | 0
Hp24 | $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{2}\Psi_{0i}^{+}\rangle$ | $\delta_{ik}(E_{0}+\frac{1}{2}J_{00})+\frac{F_{i^{\prime}k^{\prime}}-F_{ik}}{2}+\frac{1}{2}[-(00|i^{\prime}k^{\prime})-(ik|00)+(0i|0k^{\prime})$
|  | $+(0k|0i^{\prime})]+\frac{1}{4}[(i^{\prime}0|0k^{\prime})+(i0|0k)]$
Hp25 | $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{i^{\prime}}\rangle$ | $\frac{\delta_{ik}}{2}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]-\frac{\delta_{i^{\prime}k^{\prime}}}{2}[F_{i0}-\frac{1}{2}(i0|00)]+(0k|ii^{\prime})+(ii^{\prime}|k^{\prime}0)$
|  | $-\frac{1}{2}[(i0|k^{\prime}i^{\prime})+(0i^{\prime}|ik)]$
Hp26 |  $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $\frac{1}{2\sqrt{2}}\bigl{\\{}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]-\delta_{jk}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]-\delta_{k^{\prime}j^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]$
|  | $+\delta_{i^{\prime}k^{\prime}}[F_{j0}-\frac{1}{2}(j0|00)]\bigr{\\}}+\frac{1}{2\sqrt{2}}[-(0j^{\prime}|ik)+(0i^{\prime}|jk)-(i0|k^{\prime}j^{\prime})$
|  | $+(j0|k^{\prime}i^{\prime})]+\frac{1}{\sqrt{2}}[(0k|ij^{\prime})-(0k|ji^{\prime})+(ij^{\prime}|k^{\prime}0)-(ji^{\prime}|k^{\prime}0)]$
Hp27 |  $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $\frac{1}{2\sqrt{2}}\bigl{\\{}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]+\delta_{jk}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]-\delta_{k^{\prime}j^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]$
|  | $-\delta_{i^{\prime}k^{\prime}}[F_{j0}-\frac{1}{2}(j0|00)]\bigr{\\}}+\frac{1}{2\sqrt{2}}[-(0j^{\prime}|ik)-(0i^{\prime}|jk)-(i0|k^{\prime}j^{\prime})$
|  | $-(j0|k^{\prime}i^{\prime})]+\frac{1}{\sqrt{2}}[(0k|ij^{\prime})+(0k|ji^{\prime})+(ij^{\prime}|k^{\prime}0)+(ji^{\prime}|k^{\prime}0)]$
Hp28 | $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$ | $\frac{\sqrt{3}}{2}\bigl{\\{}\delta_{ik}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]-(0i^{\prime}|ik)+\delta_{i^{\prime}k^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]+(i0|i^{\prime}k^{\prime})\bigr{\\}}$
Hp29 |  $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $\frac{3}{2\sqrt{6}}\bigl{\\{}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]-(0j^{\prime}|ik)+\delta_{jk}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]-(0i^{\prime}|jk)$
|  | $+\delta_{j^{\prime}k^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]+(i0|j^{\prime}k^{\prime})+\delta_{i^{\prime}k^{\prime}}[F_{j0}-\frac{1}{2}(j0|00)]+(j0|i^{\prime}k^{\prime})\bigr{\\}}$
Hp30 |  $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $\frac{3}{2\sqrt{6}}\bigl{\\{}\delta_{ik}[F_{0j^{\prime}}+\frac{1}{2}(0j^{\prime}|00)]-(0j^{\prime}|ik)-\delta_{jk}[F_{0i^{\prime}}+\frac{1}{2}(0i^{\prime}|00)]+(0i^{\prime}|jk)$
|  | $+\delta_{j^{\prime}k^{\prime}}[F_{i0}-\frac{1}{2}(i0|00)]+(i0|j^{\prime}k^{\prime})-\delta_{i^{\prime}k^{\prime}}[F_{j0}-\frac{1}{2}(j0|00)]-(j0|i^{\prime}k^{\prime})\bigr{\\}}$
Hp31 | $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{4}\Psi_{i}^{i^{\prime}}\rangle$ | 0
Hp32 |  $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{4}\Psi_{ij}^{-}\rangle$, $i\neq j$ | 0
Hp33 |  $\langle^{2}\Psi_{0k}^{+}|\hat{H}|^{4}\Psi_{ij}^{+}\rangle$, $i\neq j$ | 0
Hp34 | $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{S}}\Psi_{i}^{i^{\prime}}\rangle$ | $\delta_{ik}(E_{0}+F_{i^{\prime}k^{\prime}}-F_{ik})-(ik|k^{\prime}i^{\prime})+2(ii^{\prime}|kk^{\prime})$

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# | Matrix element | $=$
---|---|---
Hp35 |  $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $\frac{1}{\sqrt{2}}[\delta_{ik}F_{j^{\prime}k^{\prime}}-\delta_{j^{\prime}k^{\prime}}F_{ik}-(ik|k^{\prime}j^{\prime})+2(ij^{\prime}|kk^{\prime})-\delta_{jk}F_{i^{\prime}k^{\prime}}$
|  | $+\delta_{i^{\prime}k^{\prime}}F_{jk}+(jk|k^{\prime}i^{\prime})-2(ji^{\prime}|kk^{\prime})]$
Hp36 |  $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $\frac{1}{\sqrt{2}}[\delta_{ik}F_{j^{\prime}k^{\prime}}-\delta_{j^{\prime}k^{\prime}}F_{ik}-(ik|k^{\prime}j^{\prime})+2(ij^{\prime}|kk^{\prime})+\delta_{jk}F_{i^{\prime}k^{\prime}}$
|  | $-\delta_{i^{\prime}k^{\prime}}F_{jk}-(jk|k^{\prime}i^{\prime})+2(ji^{\prime}|kk^{\prime})]$
Hp37 | $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$ | $\delta_{ik}[(i^{\prime}0|0k^{\prime})-(0k|i0)]$
Hp38 |  $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $\frac{3}{2\sqrt{6}}[\delta_{ik}(j^{\prime}0|0k^{\prime})-\delta_{j^{\prime}k^{\prime}}(0k|i0)+\delta_{jk}(i^{\prime}0|0k^{\prime})-\delta_{i^{\prime}k^{\prime}}(0k|j0)]$
Hp39 |  $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $\frac{3}{2\sqrt{6}}[\delta_{ik}(j^{\prime}0|0k^{\prime})-\delta_{j^{\prime}k^{\prime}}(0k|i0)-\delta_{jk}(i^{\prime}0|0k^{\prime})+\delta_{i^{\prime}k^{\prime}}(0k|j0)]$
Hp40 | $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{4}\Psi_{i}^{i^{\prime}}\rangle$ | 0
Hp41 |  $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{4}\Psi_{ij}^{-}\rangle$, $i\neq j$ | 0
Hp42 |  $\langle^{2\mathrm{S}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{4}\Psi_{ij}^{+}\rangle$, $i\neq j$ | 0
Hp43 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{-}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{-}\rangle$, $i\neq j$, $k\neq l$ | $\frac{E_{0}}{2}(\delta_{ik}\delta_{j^{\prime}l^{\prime}}-\delta_{jk}\delta_{i^{\prime}l^{\prime}}-\delta_{il}\delta_{j^{\prime}k^{\prime}}+\delta_{jl}\delta_{i^{\prime}k^{\prime}})+\frac{1}{2}(\delta_{ik}F_{j^{\prime}l^{\prime}}$
|  | $-\delta_{jk}F_{i^{\prime}l^{\prime}}-\delta_{il}F_{j^{\prime}k^{\prime}}+\delta_{jl}F_{i^{\prime}k^{\prime}}-\delta_{j^{\prime}l^{\prime}}F_{ik}+\delta_{i^{\prime}l^{\prime}}F_{jk}+\delta_{j^{\prime}k^{\prime}}F_{il}$
|  | $-\delta_{i^{\prime}k^{\prime}}F_{jl})+\frac{1}{2}[-(ik|l^{\prime}j^{\prime})+2(ij^{\prime}|kl^{\prime})+(jk|l^{\prime}i^{\prime})-2(ji^{\prime}|kl^{\prime})$
|  | $+(il|k^{\prime}j^{\prime})-2(ij^{\prime}|lk^{\prime})-(jl|k^{\prime}i^{\prime})+2(ji^{\prime}|lk^{\prime})]$
Hp44 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{-}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{+}\rangle$, $i\neq j$, $k\neq l$ | $\frac{E_{0}}{2}(\delta_{ik}\delta_{j^{\prime}l^{\prime}}+\delta_{jk}\delta_{i^{\prime}l^{\prime}}-\delta_{il}\delta_{j^{\prime}k^{\prime}}-\delta_{jl}\delta_{i^{\prime}k^{\prime}})+\frac{1}{2}(\delta_{ik}F_{j^{\prime}l^{\prime}}$
|  | $+\delta_{jk}F_{i^{\prime}l^{\prime}}-\delta_{il}F_{j^{\prime}k^{\prime}}-\delta_{jl}F_{i^{\prime}k^{\prime}}-\delta_{j^{\prime}l^{\prime}}F_{ik}-\delta_{i^{\prime}l^{\prime}}F_{jk}+\delta_{j^{\prime}k^{\prime}}F_{il}$
|  | $+\delta_{i^{\prime}k^{\prime}}F_{jl})+\frac{1}{2}[-(ik|l^{\prime}j^{\prime})+2(ij^{\prime}|kl^{\prime})-(jk|l^{\prime}i^{\prime})+2(ji^{\prime}|kl^{\prime})$
|  | $+(il|k^{\prime}j^{\prime})-2(ij^{\prime}|lk^{\prime})+(jl|k^{\prime}i^{\prime})-2(ji^{\prime}|lk^{\prime})]$
Hp45 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{-}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$, $k\neq l$ | $\frac{3}{2\sqrt{6}}[\delta_{ik}(i^{\prime}0|0l^{\prime})-\delta_{i^{\prime}l^{\prime}}(0k|i0)-\delta_{il}(i^{\prime}0|0k^{\prime})+\delta_{i^{\prime}k^{\prime}}(0l|i0)]$
Hp46 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{-}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{-}\rangle$, $i\neq j$, $k\neq l$ | $\frac{\sqrt{3}}{4}[\delta_{ik}(j^{\prime}0|0l^{\prime})-\delta_{j^{\prime}l^{\prime}}(0k|i0)+\delta_{jk}(i^{\prime}0|0l^{\prime})-\delta_{i^{\prime}l^{\prime}}(0k|j0)$
|  | $-\delta_{il}(j^{\prime}0|0k^{\prime})+\delta_{j^{\prime}k^{\prime}}(0l|i0)-\delta_{jl}(i^{\prime}0|0k^{\prime})+\delta_{i^{\prime}k^{\prime}}(0l|j0)]$
Hp47 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{-}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$, $i\neq j$, $k\neq l$ | $\frac{\sqrt{3}}{4}[\delta_{ik}(j^{\prime}0|0l^{\prime})-\delta_{j^{\prime}l^{\prime}}(0k|i0)-\delta_{jk}(i^{\prime}0|0l^{\prime})+\delta_{i^{\prime}l^{\prime}}(0k|j0)$
|  | $-\delta_{il}(j^{\prime}0|0k^{\prime})+\delta_{j^{\prime}k^{\prime}}(0l|i0)+\delta_{jl}(i^{\prime}0|0k^{\prime})-\delta_{i^{\prime}k^{\prime}}(0l|j0)]$
Hp48 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{-}|\hat{H}|^{4}\Psi_{i}^{i^{\prime}}\rangle$, $k\neq l$ | 0
Hp49 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{-}|\hat{H}|^{4}\Psi_{ij}^{-}\rangle$, $i\neq j$, $k\neq l$ | 0
Hp50 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{-}|\hat{H}|^{4}\Psi_{ij}^{+}\rangle$, $i\neq j$, $k\neq l$ | 0

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# | Matrix element | $=$
---|---|---
Hp51 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{+}|\hat{H}|^{2\mathrm{S}}\Psi_{ij}^{+}\rangle$, $i\neq j$, $k\neq l$ | $\frac{E_{0}}{2}(\delta_{ik}\delta_{j^{\prime}l^{\prime}}+\delta_{jk}\delta_{i^{\prime}l^{\prime}}+\delta_{il}\delta_{j^{\prime}k^{\prime}}+\delta_{jl}\delta_{i^{\prime}k^{\prime}})+\frac{1}{2}(\delta_{ik}F_{j^{\prime}l^{\prime}}+\delta_{jk}F_{i^{\prime}l^{\prime}}$
|  | $+\delta_{il}F_{j^{\prime}k^{\prime}}+\delta_{jl}F_{i^{\prime}k^{\prime}}-\delta_{j^{\prime}l^{\prime}}F_{ik}-\delta_{i^{\prime}l^{\prime}}F_{jk}-\delta_{j^{\prime}k^{\prime}}F_{il}-\delta_{i^{\prime}k^{\prime}}F_{jl})$
|  | $+\frac{1}{2}[-(ik|l^{\prime}j^{\prime})+2(ij^{\prime}|kl^{\prime})-(jk|l^{\prime}i^{\prime})+2(ji^{\prime}|kl^{\prime})-(il|k^{\prime}j^{\prime})$
|  | $+2(ij^{\prime}|lk^{\prime})-(jl|k^{\prime}i^{\prime})+2(ji^{\prime}|lk^{\prime})]$
Hp52 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{+}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$, $k\neq l$ | $\frac{3}{2\sqrt{6}}[\delta_{ik}(i^{\prime}0|0l^{\prime})-\delta_{i^{\prime}l^{\prime}}(0k|i0)+\delta_{il}(i^{\prime}0|0k^{\prime})-\delta_{i^{\prime}k^{\prime}}(0l|i0)]$
Hp53 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{+}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{-}\rangle$, $i\neq j$, $k\neq l$ | $\frac{\sqrt{3}}{4}[\delta_{ik}(j^{\prime}0|0l^{\prime})-\delta_{j^{\prime}l^{\prime}}(0k|i0)+\delta_{jk}(i^{\prime}0|0l^{\prime})-\delta_{i^{\prime}l^{\prime}}(0k|j0)$
|  | $+\delta_{il}(j^{\prime}0|0k^{\prime})-\delta_{j^{\prime}k^{\prime}}(0l|i0)+\delta_{jl}(i^{\prime}0|0k^{\prime})-\delta_{i^{\prime}k^{\prime}}(0l|j0)]$
Hp54 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{+}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$, $i\neq j$, $k\neq l$ | $\frac{\sqrt{3}}{4}[\delta_{ik}(j^{\prime}0|0l^{\prime})-\delta_{j^{\prime}l^{\prime}}(0k|i0)-\delta_{jk}(i^{\prime}0|0l^{\prime})+\delta_{i^{\prime}l^{\prime}}(0k|j0)$
|  | $+\delta_{il}(j^{\prime}0|0k^{\prime})-\delta_{j^{\prime}k^{\prime}}(0l|i0)-\delta_{jl}(i^{\prime}0|0k^{\prime})+\delta_{i^{\prime}k^{\prime}}(0l|j0)]$
Hp55 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{+}|\hat{H}|^{4}\Psi_{i}^{i^{\prime}}\rangle$, $k\neq l$ | 0
Hp56 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{+}|\hat{H}|^{4}\Psi_{ij}^{-}\rangle$, $i\neq j$, $k\neq l$ | 0
Hp57 |  $\langle^{2\mathrm{S}}\Psi_{kl}^{+}|\hat{H}|^{4}\Psi_{ij}^{+}\rangle$, $i\neq j$, $k\neq l$ | 0
Hp58 | $\langle^{2\mathrm{T}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{i}^{i^{\prime}}\rangle$ | $\delta_{ik}[E_{0}+F_{i^{\prime}k^{\prime}}+(i^{\prime}0|0k^{\prime})-F_{ik}+(i0|0k)]-(ik|k^{\prime}i^{\prime})$
Hp59 |  $\langle^{2\mathrm{T}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{-}\rangle$, $i\neq j$ | $\frac{1}{\sqrt{2}}\bigl{\\{}\delta_{ik}[F_{j^{\prime}k^{\prime}}+(j^{\prime}0|0k^{\prime})]-\delta_{j^{\prime}k^{\prime}}[F_{ik}-(i0|0k)]+\delta_{jk}[F_{i^{\prime}k^{\prime}}$
|  | $+(i^{\prime}0|0k^{\prime})]-\delta_{i^{\prime}k^{\prime}}[F_{jk}-(j0|0k)]-(ik|k^{\prime}j^{\prime})-(jk|k^{\prime}i^{\prime})\bigr{\\}}$
Hp60 |  $\langle^{2\mathrm{T}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{+}\rangle$, $i\neq j$ | $\frac{1}{\sqrt{2}}\bigl{\\{}\delta_{ik}[F_{j^{\prime}k^{\prime}}+(j^{\prime}0|0k^{\prime})]-\delta_{j^{\prime}k^{\prime}}[F_{ik}-(i0|0k)]-\delta_{jk}[F_{i^{\prime}k^{\prime}}$
|  | $+(i^{\prime}0|0k^{\prime})]+\delta_{i^{\prime}k^{\prime}}[F_{jk}-(j0|0k)]-(ik|k^{\prime}j^{\prime})+(jk|k^{\prime}i^{\prime})\bigr{\\}}$
Hp61 | $\langle^{2\mathrm{T}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{4}\Psi_{i}^{i^{\prime}}\rangle$ | 0
Hp62 |  $\langle^{2\mathrm{T}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{4}\Psi_{ij}^{-}\rangle$, $i\neq j$ | 0
Hp63 |  $\langle^{2\mathrm{T}}\Psi_{k}^{k^{\prime}}|\hat{H}|^{4}\Psi_{ij}^{+}\rangle$, $i\neq j$ | 0
Hp64 |  $\langle^{2\mathrm{T}}\Psi_{kl}^{-}|\hat{H}|^{2\mathrm{T}}\Psi_{ij}^{-}\rangle$, $i\neq j$, $k\neq l$ | $\frac{E_{0}}{2}(\delta_{ik}\delta_{j^{\prime}l^{\prime}}+\delta_{jk}\delta_{i^{\prime}l^{\prime}}+\delta_{il}\delta_{j^{\prime}k^{\prime}}+\delta_{jl}\delta_{i^{\prime}k^{\prime}})+\frac{1}{2}\bigl{(}\delta_{ik}[F_{j^{\prime}l^{\prime}}$
|  | $+(j^{\prime}0|0l^{\prime})]+\delta_{jk}[F_{i^{\prime}l^{\prime}}+(i^{\prime}0|0l^{\prime})]+\delta_{il}[F_{j^{\prime}k^{\prime}}+(j^{\prime}0|0k^{\prime})]$
|  | $+\delta_{jl}[F_{i^{\prime}k^{\prime}}+(i^{\prime}0|0k^{\prime})]-\delta_{j^{\prime}l^{\prime}}[F_{ik}-(i0|0k)]$