# An Excited-State-Specific Projected Coupled-Cluster Theory

Harrison Tuckman  Department of Chemistry, University of California, Berkeley,
California 94720, USA  Eric Neuscamman<EMAIL_ADDRESS>Department of
Chemistry, University of California, Berkeley, California 94720, USA

###### Abstract

We present an excited-state-specific coupled-cluster approach in which both
the molecular orbitals and cluster amplitudes are optimized for an individual
excited state. The theory is formulated via a projection of the traditional
coupled-cluster wavefunction that allows correlation effects to be introduced
atop an excited state mean field starting point. The approach shares much in
common with ground state CCSD, including size consistency and an $N^{6}$ cost
scaling. Preliminary numerical tests show that the method can improve over
excited-state-specific second order perturbation theory in valence, charge
transfer, and Rydberg states.

Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
CA, 94720, USA

## 1 Introduction

Mean field methods such as Hartree-Fock (HF) theory 1 are useful for creating
qualitatively correct depictions of a molecules’ electronic structure, but the
finer details of electron correlation must also be addressed in order to
achieve quantitative energy predictions. When working with ground state
molecules near their equilibrium geometries, wavefunction methods
traditionally capture this correlation by hierarchically applying perturbation
based and/or coupled-cluster (CC) based corrections to the HF wavefunction. 2,
3, 4, 5 An especially noteworthy part of this hierarchy is coupled-cluster
with singles, doubles, and perturbative triples (CCSD(T)),6 which is often
referred to as the “gold standard” thanks to its ability to deliver
sub-$\mathrm{k_{B}}$T accuracy at polynomial cost in a variety of chemically
important settings. 7, 8, 9 In practice, of course, wavefunction methods are
often eschewed in favor of the more computationally efficient density
functional theory (DFT), 10, 11, 12, 13 but, one way or another, accurate
chemical predictions rely on correctly accounting for the effects of electron
correlation.

Achieving quantitative predictions in electronically excited states becomes
more complicated, as methods must now contend with open-shell spin-
recombination, post-excitation orbital relaxations, and the effects of these
on the details of electron correlation. In practice, many of the most widely
used excited state methods such as time-dependent DFT (TD-DFT)14, 15, 16,
equation-of-motion CC (EOM-CC),17, 18, 19 and linear response CC (LR-CC)20, 21
approach excited states by way of linear response theory. While this approach
captures open-shell singlet character and has a long record of success, it is
not without its difficulties. Typically, the linear response approximation
requires the assumption that key aspects of the electronic structure are
shared between ground and excited states. For example, all states share one
cluster operator in EOM-CC. Likewise, the widely used adiabatic approach to
TD-DFT retains the same orbital shapes for all spectator electrons, regardless
of how substantially an excitation has changed the electrostatic environment
that these electrons sit in. These assumptions can be problematic in
situations involving large and thus nonlinear post-excitation relaxations,
such as many charge transfer, Rydberg, core, and double excitations. 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 Linear response methods can
also struggle in situations where the ground state is not well described by
mean field theory, as they can inherit the ground state difficulties even if
the structure of the excited state is simple. While there are many different
corrections that can be added to these methods in order to address specific
shortcomings – e.g. long range corrections in TD-DFT 37, 38 – the various
challenges facing linear response methods have prevented the community from
coalescing around a unifying “gold standard” and have left excited state
modeling at a distinct disadvantage relative to ground states.

Motivated in part by the limitations of linear response, there has been much
recent work on excited-state-specific methods. Whether studying single
determinant wavefunctions, 39, 40, 41, 42, 43 configuration interaction (CI)
wavefunctions, 44 CC wavefunctions, 45, 46, 47 or DFT functionals, 48, 49, 50
the theme of locating higher energy, state-specific excited state and open-
shell roots is becoming increasingly prevalent across the field. Recognizing
that, in the ground state, CCSD is a crucial stepping stone towards CCSD(T) as
well as a useful method in its own right, our focus here is to construct a
CCSD-like excited-state-specific CC theory atop an excited state mean field
(ESMF) starting point 51, 52, 53 that, like HF in the ground state, has
already taken care of state-specific orbital relaxations. Besides, with an
$N^{5}$ cost ESMF-based second order perturbation theory (ESMP2) already
established, 54 the longstanding pattern in ground state correlation methods
makes it seem natural to expect an $N^{6}$ cost CC method to provide the next
step in a hierarchy of excited-state-specific correlation methods.

Creating this CC method poses immediate challenges, however, due to the
inherently multireference nature of excited states, and so it is prudent to
consider the pros and cons of existing approaches to multireference CC theory.
Such methods typically fall into one of three categories: the Jeziorski-
Monkhort (JM) based methods, the internal contraction (ic) based methods, and
the single-reference based methods. 55, 56, 57 While research continues into
notable multi-reference coupled-cluster methods which make use of the
Jeziorski-Monkhorst ansatz such as State-Universal (SU)-MRCC,58, 59, 60
Brilluoin-Wigner (BW)-MRCC,61, 62 single root (sr)-MRCC,63, 64 Mukherjee
(Mk)-MRCC,65, 66 MRexpT,67 and two-determinant (TD)-CC,68, 69, 70 as well as
methods which make use of internally contracted schemes such as multiple
flavors of ic-MRCC,71, 72, 73 partially internally contracted (pIC)-MRCC,74
and block correlated CC,75 these methods are not without their difficulties.
In general, the state-universal JM methods can face numerical instabilities
due to the intruder state problem, 57 while the state-specific approaches have
an inherent mismatch between the number of cluster amplitudes and residual
equations. 55, 56 Furthermore, both of these methods lack an invariance to
orbital rotations within the active space,76, 77 making the final energy
dependent on the choice of orbitals. Meanwhile, the internally contracted
methods must contend with both overlap matrices that can introduce their own
numerical instabilities and, in some cases, approximate truncations of
nonterminating BCH expansions. 55, 56, 57, 78, 79

To avoid these challenges and maintain as much of the convenience of single-
reference CC as possible, we will therefore attempt to tackle multi-reference
character via an approach similar to the “little t” and “little tq” methods,
80, 81, 82, 83 which rely on a selective inclusion of only a subset of triples
and/or quadruples. The subset in question has often been identified by
defining an active space, but more recent work has also explored more general
approaches. 84, 85 In a manner similar to the $N^{5}$ variant of ESMP2, 54 we
propose to use the ESMF starting point to automatically identify a subset of
triples to include, thus limiting the number of triples in a way that makes an
$N^{6}$ scaling possible and avoids user-defined active spaces. With the
introduction of a projection operator that suppresses the Aufbau configuration
while maintaining size consistency, and with the help of a substantial dose of
automated algebra, we find that this approach leads to a fully excited-state-
specific, CCSD-like theory for adding CC-style correlation to an ESMF starting
point.

## 2 Theory

### 2.1 Conventional Ground State Coupled-Cluster

While there exist a plethora of resources illustrating ground state coupled-
cluster theory in extensive detail,86, 87, 5 we wish to provide a quick
summary here so that we may draw parallels between our excited state method
and the ground state theory. Although CC theory is at its heart an
approximation to the many-body energy, it can be motivated by a wavefunction
that is defined by operating the exponential of a cluster operator, $\hat{T}$,
on a reference, $\ket{\phi_{0}}$, which is most often the Hartree-Fock
determinant,

$\ket{\psi}=e^{\hat{T}}\ket{\phi_{0}}=\left(1+\hat{T}+\frac{1}{2}\hat{T}^{2}+\dots\right)\ket{\phi_{0}}.$
(1)

One of the challenges in applying this exponential operator is that obtaining
a variational solution for the energy necessitates either exponentially
scaling computational cost or an approximate truncation of the exponential
expansion. Therefore, what is often done in practice is to utilize a
projective approach where cluster amplitudes are determined by satisfying the
eigenvector condition for the similarity transformed Hamiltonian in a
chemically relevant subspace. For example, in CC with single and double
excitations (CCSD), $\hat{T}$ is truncated to include only $\hat{T}_{1}$ and
$\hat{T}_{2}$ and projections are taken against the reference determinant,
$\ket{\phi_{0}}$, and all singly and doubly excited determinants,
$\ket{\phi_{i}^{a}}$ and $\ket{\phi_{ij}^{ab}}$, in order to define an
approximate (and non-variational) energy equation and a set of amplitude
equations that can be solved to determine $\hat{T}_{1}$ and $\hat{T}_{2}$.86,
87, 5

$\displaystyle E_{CCSD}$
$\displaystyle=\bra{\phi_{0}}e^{-(\hat{T}_{1}+\hat{T}_{2})}\hat{H}e^{\hat{T}_{1}+\hat{T}_{2}}\ket{\phi_{0}},$
(2) $\displaystyle 0$
$\displaystyle=\bra{\phi_{i}^{a}}e^{-(\hat{T}_{1}+\hat{T}_{2})}\hat{H}e^{\hat{T}_{1}+\hat{T}_{2}}\ket{\phi_{0}},$
(3) $\displaystyle 0$
$\displaystyle=\bra{\phi_{ij}^{ab}}e^{-(\hat{T}_{1}+\hat{T}_{2})}\hat{H}e^{\hat{T}_{1}+\hat{T}_{2}}\ket{\phi_{0}}.$
(4)

Here and in the rest of this paper, $i,j,k,\dots$ represent occupied spin
orbitals while $a,b,c,\dots$ represent virtual spin orbitals relative to the
reference. Solving the above equations produces the energy and cluster
amplitudes, which can in turn be used to evaluate other properties.5

### 2.2 A Suitable Reference and the Usefulness of a Projector

As is the case for the ground state, choice of a reference is the first step
in our CC method. For studying electronically excited states, we elect to use
the Excited State Mean Field (ESMF) method, 51, 52, 53 which is an excited
state analog to the mean field HF reference, as a starting point. In a spin
orbital basis, the ESMF wavefunction can, if we neglect the possibility of the
Aufbau determinant participating in the excited state, be expressed as

$\ket{\psi_{ESMF}}=e^{\hat{X}}\left(\sum_{ia}c_{ia}\hat{a}_{a}^{\dagger}\hat{a}_{i}\ket{\phi_{0}}\right)=\sum_{ia}c_{ia}\hat{a}_{a}^{\dagger}\hat{a}_{i}\ket{\phi^{*}_{0}}.$
(5)

Here $e^{\hat{X}}$ is an orbital rotation operator, $\hat{a}_{a}^{\dagger}$
and $\hat{a}_{i}$ are second quantization creation and destruction operators
respectively, and $\ket{\phi^{*}_{0}}$ is the Aufbau determinant in the ESMF
orbital basis (which is optimized for the excited state of interest).
Immediately it is evident that our reference is a multi-determinant
wavefunction, which can complicate CC methods.57, 55 We can mitigate the
extent of the multireference nature of our wavefunction by performing a
singular value decomposition on the CI coefficients of the ESMF wavefunction
and transforming our basis to a transition orbital pair (TOP) basis. 54 In the
TOP basis, which shares many similarities with the natural transition orbital
basis,88 each occupied index becomes paired with a virtual index, thereby
reducing the number of determinants in our ESMF wavefunction. We can further
reduce the number of determinants by truncating the ESMF wavefunction after a
specified threshold. Although we will discuss generalizations below, in this
initial investigation we choose to employ the most aggressive truncation
scheme possible by truncating after the first singular value. For now, we have
a (renormalized) truncated ESMF wavefunction, $\ket{\psi_{0}}$,

$\ket{\psi_{0}}=\frac{1}{\sqrt{2}}\left(\ket{{\phi^{*}}_{h}^{p}}+\ket{{\phi^{*}}_{\bar{h}}^{\bar{p}}}\right).$
(6)

Here any index with a bar denotes the opposite electron spin as compared to
the index without a bar, and $h$ and $p$ are the special paired indices which
correspond to the orbital that is primarily excited out of in the occupied
space and the orbital that is primarily excited into for a particular excited
state. Because these orbitals are so important for describing the excited
state of interest, we will henceforth refer to them as the hole orbitals and
the particle orbitals respectively. It is worth pointing out that our
truncated ESMF starting point, $\ket{\psi_{0}}$, bears many similarities to
the reference in TD-CC,68, 69, 70 as both wavefunctions represent the simplest
possible state capable of describing open shell singlet excited states. As we
will explain presently, however, our formulation differs significantly from
TD-CC due to its “little t” structure and its projection operator.

Following the general strategy of CCSDt 81, 80 but using the ESMF TOP orbital
basis, we apply $\mathrm{exp}(\hat{T})$ to the Aufbau determinant,
$\ket{\phi^{*}_{0}}$, and then attempt to reconstruct the ESMF starting point,
$\ket{\psi_{0}}$, through proper initialization of our $\hat{T}$ operator.
However, applying the exponential ansatz directly to the Aufbau determinant
creates an immediate issue for excited states. Looking at the Taylor expansion
of the exponential ansatz in equation 1, we see that our wavefunction would
have large a contribution from the Aufbau determinant regardless of our
initialization of $\hat{T}$, which is neither typical in excited states nor
present in our desired ESMF starting point, $\ket{\psi_{0}}$. We address this
issue by introducing a projection operator $\hat{P}$ into the definition of
our excited state projected coupled-cluster (ESPCC) wavefunction.

$\displaystyle\ket{\Psi}$ $\displaystyle=\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}$
(7) $\displaystyle\hat{P}$
$\displaystyle=2-\hat{a}_{h}^{\dagger}\hat{a}_{h}-\hat{a}_{\bar{h}}^{\dagger}\hat{a}_{\bar{h}}$
(8)

Here $\hat{a}_{h}^{\dagger}$/$\hat{a}_{\bar{h}}^{\dagger}$ and
$\hat{a}_{h}$/$\hat{a}_{\bar{h}}$ are the creation and destruction operators
for the hole orbitals that are excited out of in the ESMF starting point, and
the 2 comes from the total number of these orbitals (one alpha and one beta).
$\hat{P}$ deletes determinants in which the hole orbital is doubly occupied,
and so in particular deletes the Aufbau determinant. While there are many
potential choices for the projection operator that could achieve similar ends,
we chose this particular form for its simplicity and its ability to retain a
size consistent energy (see Section 2.6). We note that by deviating slightly
from a rigid value of 2 in equation 8, we could introduce the flexibility to
allow some Aufbau contribution, which does occur for excited states in the
same symmetry representation as the ground state. One could also consider
multi-orbital generalizations of $\hat{P}$ that could work with a less
truncated ESMF starting point, but for simplicity in this initial study we
stick to the projector in equation 8. It’s worth emphasizing that the details
of the hole orbitals in the projection operator are completely specified by
the ESMF procedure. To help organize the wavefunctions presented so far, we
have compiled them in Table 1.

Wavefunction | Description | Equation
---|---|---
$\ket{\phi_{0}}$ | RHF | —
$\ket{\phi^{*}_{0}}$ | ESMF Aufbau | 5
$\ket{\psi_{ESMF}}$ | ESMF | 5
$\ket{\psi_{0}}$ | Truncated ESMF | 6
$\ket{\Psi}$ | ESPCC | 7
Table 1: Presented here is a brief summary of the wavefunctions discussed in
this paper, a short description of what they represent, and the relevant
equation where they are defined.

### 2.3 The Cluster Operator

As is the case in other single-reference-based CC methods aimed at strongly
multireference situations, 80, 81, 82, 83 a choice must be made as to which
excitation operators are included in the wavefunction. In order to produce an
analog to ground state CCSD, we wish to include the excitations necessary to
produce all determinants that are a single or double excitation away from the
truncated ESMF starting point, as illustrated in Figure 1. To do so, we
require a subset of triple excitations relative to the Aufbau determinant,
which leads to the following form for our cluster operator,

Figure 1: Presented here are all determinants within two excitations of one of
the two determinants comprising the truncated ESMF starting point. Note that
the Aufbau determinant is intentionally missing, as it is removed by our
projector. The orbital space can be divided into four distinct regions: the
closed space, hole space, particle space, and virtual space. The closed and
hole spaces together form the set of orbitals which are occupied in the Aufbau
determinant, while the particle and virtual spaces combine to form the set of
unoccupied orbitals for the Aufbau determinant. Furthermore, the hole and
particle TOP orbitals comprise the set of orbitals which are primarily excited
out of and into respectively for the excited state of interest and are
determined by an SVD of the ESMF starting point. The entire set of
determinants is also partitioned according to where they appear in the current
theory. Those determinants under the linear heading either contain a hole in
the hole space but not a TOP or contain multiple holes in the hole space, and
thus are treated primarily by excitation operators appearing at linear order
in the Taylor expansion of the exponential ansatz. In contrast, those under
the nonlinear heading contain a single TOP with no additional holes in the
hole space, and therefore see large contributions from the quadratic order
$\hat{T}^{2}$ terms. Finally, those under the ESMF/ESMP2 heading are missed in
the current implementation because they contain completely filled hole
orbitals and thus are destroyed by our projector, though their contribution is
later estimated and corrected for via both the ESMF and ESMP2 theories. The
determinant marked by an ‘*’ corresponds to the spin flipped TOP, which while
included in the theory has its amplitude fixed to achieve intermediate
normalization.

$\hat{T}=\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{\tilde{3}},$ (9)

where $\hat{T}_{\tilde{3}}$ represents only a small subset of all possible
triples, specifically those which include a hole-particle TOP. Note that,
despite the fact that not all double excitations off the Aufbau determinant,
$\ket{\phi^{*}_{0}}$, are within the first order interacting space of our
reference, $\ket{\psi_{0}}$, all of the double excitation operators must be
included in order to ensure size consistency, as will be illustrated in
Section 2.6. As stated above, we will initialize our CC ansatz such that it
reconstructs the truncated ESMF wavefunction in equation 6. This can be done
by working in the ESMF TOP basis and setting the $\hat{T}_{1}$ amplitudes
corresponding to the primary TOP ($t_{h}^{p}$ and $t_{\bar{h}}^{\bar{p}}$)
equal to $\frac{1}{\sqrt{2}}$. As is evident in the Taylor series expansion in
equation 1, this creates a large doubles contribution via the
$\hat{T}_{1}^{2}$ term, but it can be eliminated by initializing the
corresponding amplitude within $\hat{T}_{2}$ to $-\frac{1}{2}$. Fortunately,
after making this adjustment, all higher order terms cancel out perfectly such
that we are able to exactly reconstruct the truncated ESMF state as the
initial guess for our CC wavefunction $\ket{\Psi}$.

The introduction of the projector in equation 7 adds an extra layer of
complexity that is not present in the ground state theory. In the Taylor
series expansion of our wavefunction, components of $\hat{T}$ that contain a
hole index survive the projector and appear at linear order, whereas the
linear order appearances of components without the special hole index are
eliminated by the projector. Thus, these components only contribute to
$\ket{\Psi}$ via $\hat{T}^{2}$ or higher order terms in which they appear
alongside a component with the special hole index. We expect that $t_{h}^{p}$
and $t_{\bar{h}}^{\bar{p}}$ will be the largest components of $\hat{T}_{1}$,
so the largest contributions from excitation operators that do not contain a
hole index will appear inside $\hat{T}^{2}$, where they will show up in cross
terms with $t_{h}^{p}$ and $t_{\bar{h}}^{\bar{p}}$. So, for a cluster
amplitude such as $t_{jk}^{bc}$ that contains no hole index, the primary
contribution will be to the part of Hilbert space projected to by
$\bra{\phi_{hjk}^{pbc}}$ and $\bra{\phi_{\bar{h}jk}^{\bar{p}bc}}$. In stark
contrast to ground state CCSD, the projection of Hilbert space usually
associated with this amplitude, $\bra{\phi_{jk}^{bc}}$, is actually
unimportant, as it is more than two excitations away from the truncated ESMF
starting point. In the amplitude equations below, we will project with
$\bra{\phi_{hjk}^{pbc}}+\bra{\phi_{\bar{h}jk}^{\bar{p}bc}}$ and then ensure
that the corresponding projection,
$\bra{\phi_{hjk}^{pbc}}-\bra{\phi_{\bar{h}jk}^{\bar{p}bc}}$, is satisfied by
enforcing $t_{hjk}^{pbc}=-t_{\bar{h}jk}^{\bar{p}bc}$, which has the added
benefit of avoiding any potential redundancy in the amplitudes. It is worth
noting that, unlike the sufficiency conditions often introduced in Jeziorski-
Monkhort based MRCC methods, the amplitude equations resulting from this
system of linear combinations of projections are uniquely satisfied when the
individual projections are satisfied, and thus satisfy the proper residual
problem.55, 56, 57 Furthermore, while it might be tempting to instead simply
remove any parts of $\hat{T}$ that only contribute through $\hat{T}^{2}$ and
higher terms, these components are critically important for maintaining size
consistency, as will be detailed in Section 2.6.

Lastly, we set the theory’s definition of intermediate normalization by fixing
at $\frac{1}{\sqrt{2}}$ the $t_{h}^{p}$ and $t_{\bar{h}}^{\bar{p}}$ amplitudes
used to construct the ESMF starting point. This is analogous to how the Aufbau
coefficient is always 1 in the expanded CCSD wavefunction. Here, our choice of
intermediate normalization ensures a unique solution to the forthcoming
amplitude equations and also creates a nice parallel to the ground state
theory,

$\displaystyle\braket{\psi_{0}}{\Psi}=\bra{\psi_{0}}\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}=1,$
(10)

which matches the CCSD expression,

$\displaystyle\braket{\phi_{0}}{\psi_{CCSD}}=\bra{\phi_{0}}e^{\hat{T}}\ket{\phi_{0}}=1.$
(11)

### 2.4 The Energy and Amplitude Equations

With an excited-state-appropriate CC wavefunction to get us started, we now
invoke the usual CC procedure to produce an approximate, non-variational
energy equation as well as a set of amplitude equations. To obtain the energy
equation, we project against the truncated ESMF reference, $\ket{\psi_{0}}$:

$E\equiv\frac{\bra{\psi_{0}}e^{-\hat{T}}\hat{H}\ket{\Psi}}{\bra{\psi_{0}}e^{-\hat{T}}\ket{\Psi}}=\frac{\bra{\psi_{0}}e^{-\hat{T}}\hat{H}\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}}{\bra{\psi_{0}}e^{-\hat{T}}\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}}.$
(12)

With the intermediate normalization introduced in the previous section, the
denominator in this expression evaluates to 1. Similarly, the amplitude
equations are defined by taking projections against all possible excited
determinants no more than two excitations from the truncated ESMF starting
point, as are illustrated in Figure 1:

$\displaystyle\bra{{\phi^{*}}_{i}^{a}}e^{-\hat{T}}\left(\hat{H}-E\right)\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}$
$\displaystyle=0,$ (13)
$\displaystyle\bra{{\phi^{*}}_{ij}^{ab}}e^{-\hat{T}}\left(\hat{H}-E\right)\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}$
$\displaystyle=0,$ (14)
$\displaystyle\bra{{\phi^{*}}_{hjk}^{pbc}}e^{-\hat{T}}\left(\hat{H}-E\right)\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}$
$\displaystyle=0.$ (15)

Note that we do not have amplitude equations for all singles, doubles, and
especially triples, only those in the “linear” and “nonlinear” categories in
Figure 1. While satisfying the remaining singles and doubles amplitude
equations under the “ESMF/ESMP2” category of Figure 1 would make for an ideal
parallel to ground state CCSD, with the simple projector scheme that we use
here, our wavefunction does not have the flexibility to access these regions
of Hilbert space, although we anticipate that more general choices of the
projector may resolve this issue in the future. For now, the energetic
contributions of these missing terms will be estimated and corrected for by
taking the corresponding terms’ energy contributions from untruncated ESMF and
ESMP2. Finally, with all singles and doubles present in $\hat{T}$ despite the
fact that we only include singles and doubles bra states within two
excitations of our ESMF starting point in equations 13 and 14, it may seem at
first glance that we have a mismatch between the number of amplitudes and
amplitude equations. However, as will be illustrated in the following section,
the seemingly extra singles and doubles amplitudes go along with the single-
TOP-containing doubles and triples amplitude equations, respectively, leaving
us with the exact same number of amplitude equations as we have amplitudes to
solve for.

### 2.5 Solving the Amplitude Equations

As mentioned in Section 2.3, there exist two distinct groups of excitation
operators: those which contain a hole orbital as one of their indices and
those that do not. Those which do survive the projection operator and
contribute at linear order in the Taylor expansion of the wavefunction, while
those which do not are destroyed by the projector and therefore first appear
at quadratic order where they must be paired with an excitation that does
contain a hole index. As a result, we find it effective to use different
approaches for these two distinct classes of amplitudes in the iterative
solution of the amplitude equations.

To solve the amplitude equations, we make use of a quasi-Newton iterative
scheme reminiscent of that often used in ground state CC.87 To calculate the
approximate Jacobian, we begin by approximating the Hamiltonian as only the
diagonal portion of the Fock operator constructed from the ESMF density
matrix. It is worth noting that during the ESMF optimization the orbitals were
rotated such that they are no longer in a canonical regime. 51, 52, 53
However, we expect that the Fock matrix should remain diagonally dominant, so
we ignore the off-diagonal terms. For a given cluster amplitude containing
either a hole index without a TOP or multiple hole indices, $t_{\mu}$, we
obtain a diagonal approximate Jacobian which leads to an update scheme very
similar to that in the ground state theory:

$t^{new}_{\mu}=t^{old}_{\mu}-\frac{\bra{\phi^{*}_{\mu}}e^{-\hat{T}}(\hat{H}-E)e^{\hat{T}}\ket{\phi^{*}_{0}}}{N_{\mu}\left(\Delta_{\mu}+(E-E_{0})\right)}.$
(16)

Here $\bra{\phi^{*}_{\mu}}$ corresponds to the Hilbert subspace resulting from
the action of $\hat{T}_{\mu}$ on $\ket{\phi_{0}^{*}}$, $\Delta_{\mu}$ is the
orbital energy difference (using the diagonal elements of our Fock operator)
corresponding to the $\hat{T}_{\mu}$ excitation, $N_{\mu}$ is the number of
hole indices in the cluster amplitude, and
$E_{0}=\bra{\phi_{0}^{*}}\hat{H}\ket{\phi_{0}^{*}}$.

For the cluster amplitudes that lack a hole index, the update is a little less
conventional and is paired with the update for the related one-higher-
excitation amplitude that contains the hole-particle TOP. As discussed in
Section 2.3, the most important contribution from a no-hole cluster amplitude
$t_{\nu}$ occurs in the terms inside $\hat{T}^{2}$ in which it partners with
$t_{h}^{p}$ or $t_{\bar{h}}^{\bar{p}}$ in an overall excitation that is one
higher than $t_{\nu}$ itself. Thus, we update $t_{\nu}$ in conjunction with
the amplitudes $t_{\omega}$ and $t_{\bar{\omega}}$, which are like $t_{\nu}$
but contain an additional TOP excitation. For example, if $t_{\nu}$ is
$t_{jk}^{bc}$, then $t_{\omega}$ is $t_{hjk}^{pbc}$ and $t_{\bar{\omega}}$ is
$t_{\bar{h}ij}^{\bar{p}ab}$.

$\displaystyle t^{new}_{\nu}$
$\displaystyle=t^{old}_{\nu}-\frac{\bra{\phi^{*}_{\omega}}e^{-\hat{T}}(\hat{H}-E)e^{\hat{T}}\ket{\phi^{*}_{0}}+\bra{\phi^{*}_{\bar{\omega}}}e^{-\hat{T}}(\hat{H}-E)e^{\hat{T}}\ket{\phi^{*}_{0}}}{(t_{h}^{p}+t_{\bar{h}}^{\bar{p}})\Delta_{\nu}},$
(17) $\displaystyle t^{new}_{\omega}$
$\displaystyle=t^{old}_{\omega}-\frac{\bra{\phi^{*}_{\omega}}e^{-\hat{T}}(\hat{H}-E)e^{\hat{T}}\ket{\phi^{*}_{0}}-\bra{\phi^{*}_{\bar{\omega}}}e^{-\hat{T}}(\hat{H}-E)e^{\hat{T}}\ket{\phi^{*}_{0}}}{2\left(\Delta_{\omega}+(E-E_{0})\right)},$
(18) $\displaystyle t^{new}_{\bar{\omega}}$ $\displaystyle=-t^{new}_{\omega}$
(19)

Crucially, only two of these three amplitudes are independent variables
($t_{\bar{\omega}}$ being defined as the negative of $t_{\omega}$ as discussed
in Section 2.3), which is why we are able to arrive at this style of iterative
update from just two amplitude equations (the ones involving
$\bra{\phi^{*}_{\omega}}$ and $\bra{\phi^{*}_{\bar{\omega}}}$) without
encountering a linear dependency. Also note that, in our present formalism,
$t_{h}^{p}$ and $t_{\bar{h}}^{\bar{p}}$ are fixed at $1/\sqrt{2}$ by
intermediate normalization.

In addition to the above approach to iterative amplitude updates, we choose to
employ a two phase optimization procedure. While a two phase procedure is not
strictly necessary, we find that it assists in achieving rapid and tight
convergence of the residual equations. In the first phase, the two “higher
order” excitation amplitudes $t_{\omega}$ and $t_{\bar{\omega}}$ are held
fixed at zero, leaving only the amplitude associated with equation 17,
$t_{\nu}$, to change. After this subset of residual equations is sufficiently
converged, the remaining amplitudes associated with equations 18 and 19 are
introduced and all the residual equations are iterated until the final
convergence is achieved. Finally, to accelerate the convergence of our method
we utilize a DIIS implementation 89 which is turned on after the first few
iterations, and whose history is reset when transitioning from phase I to
phase II of the iterative scheme.

### 2.6 Size Consistency

Size consistency is a central property of coupled-cluster methods which we
will review here in order to show how it is maintained in the present theory.
Imagine that there are two noninteracting molecular fragments, one of which is
in its excited state, molecular fragment $A$, and another which is in its
ground state, molecular fragment $B$. Under these circumstances, our
wavefunction maintains the usual product factorizability due to the local
nature of the projector in equation 8 (indeed, $\hat{P}$, which we will write
here as $\hat{P}_{A}$, only acts on A):

$\ket{\Psi}=\hat{P}_{A}e^{\hat{T}_{A}+\hat{T}_{B}}\ket{\phi^{*}_{A}}\otimes\ket{\phi_{B}}=\left(\hat{P}_{A}e^{\hat{T}_{A}}\ket{\phi^{*}_{A}}\right)\otimes
e^{\hat{T}_{B}}\ket{\phi_{B}}.$ (20)

Note that this factorization would not work if we employed the intuitive
projector $\hat{P}=1-\ket{\phi_{0}}\bra{\phi_{0}}$, as it acts globally on
both fragments $A$ and $B$. Due to the product factorization shown in equation
20, the energy expression in equation 12 decomposes into

$\frac{\bra{{\psi_{0}}_{A}}e^{-\hat{T}_{A}}\hat{H}_{A}\hat{P}_{A}e^{\hat{T}_{A}}\ket{\phi^{*}_{A}}}{\bra{{\psi_{0}}_{A}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{\phi_{A}}}+\bra{\phi_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{\phi_{B}}=E_{A}+E_{B}=E,$
(21)

where the first term in this equation is the excited state energy for fragment
$A$ and the second term is the ground state energy expression for fragment
$B$. This energy expression shows great promise of size consistency, but in
order to guarantee size consistency in practice it must be shown that the
iterative procedure produces the correct ground state amplitudes for fragment
$B$, and thus the correct ground state CCSD energy expression for fragment
$B$. It can be shown that in a size consistent regime where the $i,h,a,p$
indices in the projection belong to fragment $A$ and the remaining $j,k,b,c$
indices belong to fragment $B$, equations 14 and 15 factorize to produce

$\displaystyle
t_{i}^{a}\bra{{\phi_{j}^{b}}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{\phi_{B}}$
$\displaystyle=0,$ (22) $\displaystyle
t_{h}^{p}\bra{{\phi_{jk}^{bc}}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{\phi_{B}}$
$\displaystyle=0.$ (23)

Note that, for at least the cluster amplitudes which construct our ESMF
reference, $t_{h}^{p}$/$t_{\bar{h}}^{\bar{p}}$, the $t_{i}^{a}$ amplitudes
cannot be zero. Therefore, to satisfy these amplitude equations, the fragment
$B$ portion of the equations must be zero, which is exactly equivalent to the
ground state CCSD amplitude equations for fragment $B$. While this is
sufficient for proving size consistency, we can make an even stronger claim.
By plugging these expressions into the phase I iterative method in equation
17, we find that the iterative update for cluster amplitudes on fragment $B$
is given by

$t^{new}_{\nu}=t^{old}_{\nu}-\frac{\bra{{\phi_{\nu}}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{\phi_{B}}}{\Delta_{\nu}},$
(24)

which is exactly equivalent to the ground state CCSD iterative update. In
other words, at any iteration during phase I of this method, the cluster
amplitudes that are noninteracting with the excitation will be exactly
equivalent to their counterparts in a ground state CCSD calculation (see
Supporting Information for more details). We would likewise expect that
amplitudes that interact only weakly with the excitation will behave very
similarly as in CCSD, creating a strong connection between the ground and
excited state theories.

## 3 Results

### 3.1 Computational Details

To implement this theory, we first derived by hand the tensor contractions
that would be necessary for the energy and amplitude equations if we were
using a full $\hat{T}_{3}$ operator containing all triples. We then applied an
automatic code generator to split each term involving triples into the various
sub-terms corresponding to the $O(N^{4})$ subset of the triples that are
actually included in our $\hat{T}_{\tilde{3}}$. This guarantees an $N^{4}$
memory footprint, and, as discussed in the next section, leads to an
implementation with an $N^{6}$ cost scaling.

For all of the forthcoming results, the EOM-CCSD, CR-CC(2,3),90, 91 and
$\delta$-CR-EOM-CC(2,3),D92, 93, 94, 95 calculations were performed with
GAMESS, 96, 97 while the CISD and FCI calculations were performed with
PySCF.98, 99, 100 To obtain the timing and scaling data, calculations were
performed on a single core on the same machine to ensure consistency. No
frozen core approximation was utilized on any of these methods. For our
method, we iterate until the maximum amplitude equation residual was no larger
than $10^{-10}$, and for all other methods we utilize the default convergence
settings. For the “ESMF/ESMP2” terms in Figure 1, we run an $N^{5}$-cost ESMP2
calculation, 54 extract the energy contribution for these terms, and add it to
our CC energy to form our corrected ESPCC energy.

### 3.2 Scaling Analysis

With the inclusion of a subset of triples amplitudes, one might begin to
wonder whether the scaling of this method may be worse than its ground state
counterpart. However, due to the transformation to the TOP orbital basis and
the truncation of the ESMF wavefunction, we find that the asymptotic scaling
remains equivalent to CCSD at $O(N^{6})$. In order to verify this scaling, we
collected a series of timing data in order to analyze the scaling contribution
of each term in the current implementation. To collect this data, we timed the
contribution from each term in the theory on a system of spatially separated,
minimal basis H2 molecules ranging from 5 to 25 molecules. The timing data for
each term was then plotted on a log-log plot and fit to a least squares linear
regression. The slope resulting from each regression was conservatively
rounded up to the nearest whole number and binned according to scaling. These
results are reported along with the minimum and mean coefficient of
correlation for the linear regression, $R$, in Table 2. It is worth noting
that the maximum slope for any regression was 5.829, thereby confirming our
proposed $O(N^{6})$ scaling. Furthermore, each of the linear regressions for
the $N^{6}$ scaling terms was manually reviewed to ensure that the plots did
indeed appear well fit by the regression. For a number of the lower scaling
terms, the coefficient of correlation indicates a lower quality for the linear
regression. Because many of these terms have timings on the order of
milliseconds, we suspect that a combination of the overhead for the matrix
multiplication functions or the random variations in processor performance may
be obfuscating the exact linear relationship. However, we have no reason to
believe they would affect the method’s asymptotic $N^{6}$ scaling. Note that,
while some optimization in the form of tensor contraction ordering was done to
ensure an $O(N^{6})$ scaling, no other optimization was performed for this
implementation. For example, we have not yet attempted any factorization of
the amplitude equations nor have we consolidated the terms which are related
through simple permutations of each other, and so the number of terms in Table
2 is unnecessarily large. With a more careful implementation, we suspect that
a large reduction in the total number of terms and wall time will be possible.

| $1$ | $N$ | $N^{2}$ | $N^{3}$ | $N^{4}$ | $N^{5}$ | $N^{6}$
---|---|---|---|---|---|---|---
Number of terms | 4 | 17953 | 7047 | 5622 | 3899 | 2381 | 293
Min R | -0.074 | 0.002 | 0.888 | 0.921 | 0.855 | 0.875 | 0.996
Mean R | -.056 | 0.878 | 0.977 | 0.988 | 0.990 | 0.993 | 0.999
Table 2: Presented are the total number of terms belonging to each asymptotic
scaling group along with the minimum and mean correlation coefficients for the
linear regression, $R$. The high $R$ values in combination with a manual
review of the linear fits on the higher scaling terms confirms that the
presented theory asymptotically scales at the same $N^{6}$ scaling as ground
state CCSD.

### 3.3 Small basis testing

| FCI | CR-CC(2,3) | RHF | MP2 | CCSD
---|---|---|---|---|---
H6 (6-31G) | $-3.437\,120$ | – | $-3.356\,782$ | $-3.411\,371$ | $-3.437\,119$
H2O (6-31G) | $-76.120\,137$ | – | $-75.984\,322$ | $-76.112\,169$ | $-76.118\,663$
H2O (cc-pVDZ) | – | $-76.242\,856$ | $-76.027\,022$ | $-76.230\,220$ | $-76.239\,548$
CH2O (cc-pVDZ) | – | $-114.221\,280$ | $-113.877\,084$ | $-114.195\,345$ | $-114.211\,046$
CH2CO (cc-pVDZ) | – | $-152.214\,501$ | $-151.740\,890$ | $-152.181\,539$ | $-152.196\,705$
NH3 F2 (cc-pVDZ) | – | $-255.507\,183$ | $-254.879\,868$ | $-255.468\,537$ | $-255.493\,049$
Ne (aug-cc-pVTZ) | – | $-128.825\,887$ | $-128.533\,273$ | $-128.819\,179$ | $-128.820\,401$
Table 3: Ground state energies in Hartrees.  | FCI |  $\delta$-CR-EOM-CC(2,3),D | ESMF | ESMP2 | ESPCC | Corrected ESPCC | EOM-CCSD
---|---|---|---|---|---|---|---
H6 (6-31G) | $-2.935\,923$ | – | $-2.880\,453$ | $-2.919\,699$ | $-2.935\,911$ | $-2.935\,919$ | $-2.935\,910$
H2O (6-31G) | $-75.803\,955$ | – | $-75.692\,508$ | $-75.794\,302$ | $-75.801\,109$ | $-75.801\,860$ | $-75.806\,267$
H2O (cc-pVDZ) | – | $-75.937\,963$ | $-75.747\,005$ | $-75.923\,046$ | $-75.923\,188$ | $-75.932\,350$ | $-75.935\,170$
CH2O (cc-pVDZ) | – | $-114.071\,520$ | $-113.752\,709$ | $-114.040\,002$ | $-114.034\,620$ | $-114.056\,952$ | $-114.054\,292$
CH2CO (cc-pVDZ) | – | $-152.075\,519$ | $-151.610\,512$ | $-152.042\,782$ | $-152.031\,461$ | $-152.057\,364$ | $-152.044\,487$
NH3 F2 (cc-pVDZ) | – | $-255.166\,706$ | $-254.593\,956$ | $-255.137\,512$ | $-255.139\,057$ | $-255.146\,565$ | $-255.133\,689$
Ne (aug-cc-pVTZ) | – | $-127.094\,052$ | $-126.754\,857$ | $-127.069\,362$ | $-127.015\,682$ | $-127.082\,043$ | $-127.083\,667$
Table 4: Excited state energies in Hartrees.  Figure 2: The excitation energy
errors for (from left to right) EOM-CCSD, ESMF, ESMP2, ESPCC, and Corrected
ESPCC on a small set of test molecules. For the two leftmost molecules in the
smallest basis, the reference energy was determined via FCI, while for all
remaining molecules the reference energy was calculated via $\delta$-CR-EOM-
CC(2,3),D. Bars marked with an ‘*’ extend beyond the limits of the plot,
precise numbers can be found in the Supporting Information.

For the initial evaluation of ESPCC, we studied its performance on H6 and H2O
in a 6-31G basis. With these small molecules in a small basis, we are able to
compare the errors of this method directly with FCI. The absolute ground and
excited state energies are reported in Tables 3 and 4, respectively, while the
excitation energy errors are reported in Figure 2. For H6, we have one H2 in
the center with a slightly stretched bond surrounded by two other H2 with
equilibrium bond lengths approximately 3Å away. We see that EOM-CCSD and ESPCC
both with and without the ESMP2-based correction almost perfectly match both
the absolute and excitation FCI energies, while ESMF and ESMP2 have a little
more difficulty, presumably due to their less complete correlation treatments.

For H2O, our method and EOM-CCSD perform comparably for absolute energies, but
the sign of the error is very important. The ground state CCSD calculation
errors high for the ground state energy, so EOM-CCSD erring low for the
excited state energy causes it to perform worse than ESMP2 and both the
uncorrected and corrected versions of ESPCC for the excitation energy. This
cancellation of errors is important when determining excitation energies, and
results in our method performing almost an order of magnitude better than EOM-
CCSD in this case. Although further study is needed, it would make sense if
the present theory, in which ground and excited state are treated on a much
more equal footing, had better error cancellation tendencies.

### 3.4 Larger basis testing

With promising results in our initial assessment, we now test these methods in
the slightly larger cc-pVDZ basis on the excited states of H2O, formaldehyde
(CH2O), ketene (CH2CO), and ammonia difluorine (NH3 F2). Of these excitations,
the ammonia difluorine charge transfer excitation is especially noteworthy as
state-specific orbital relaxations should be particularly important. Finally,
we also test these methods on Ne in an aug-cc-pVTZ basis in order to
investigate performance in a Rydberg excitation. Because both the molecule
size and basis size have grown, using FCI as a reference is no longer
convenient. Therefore, we have used the triples corrected, $N^{7}$ scaling CR-
CC(2,3) and $\delta$-CR-EOM-CC(2,3),D as our new ground and excited references
respectively. The absolute ground and excited state energies are reported in
Tables 3 and 4, respectively, while the excitation energies errors are
reported in Figure 2.

For H2O, EOM-CCSD has both a better absolute and excitation energy as compared
to ESPCC. However, in formaldehyde we see the opposite trend, with
ESMP2-corrected ESPCC slightly better than EOM-CCSD for both the absolute and
excitation energies. For ketene, we observe the largest discrepancy, where our
method results in a significantly lower absolute energy than EOM-CCSD which
results in an error just under a hundredth of an eV for corrected ESPCC versus
an error on the order of tenths of eV for EOM-CCSD. It is worth noting for all
of these molecules in this larger basis, the magnitude of the ESMP2 correction
for the few terms incompatible with our current projector has become much
larger and is now crucial for accuracy.

In the ammonia difluorine system, we study the lowest lying charge transfer
which is from the ammonia lone pair to the difluorine. While methods such as
EOM-CCSD have the ability to partially relax the orbitals after an excitation,
in situations like this one where the charge density shifts dramatically, this
partial relaxation may not be enough and can result in significant errors. In
contrast, ESPCC builds a CC ansatz atop an already orbital relaxed, excited-
state-specific starting point. This contrast manifests itself in our results,
where ESPCC achieves a better absolute energy than EOM-CCSD which results in
an excited state energy error a third that of EOM-CCSD.

Finally, we compare these methods in the 2s to 3p Rydberg excitation of Ne.
While we may expect EOM-CCSD to face similar difficulties here as in charge
transfer, it performs slightly better than ESPCC. It is worth noting that in
this particular excited state, the size of the ESMP2 based correction is much
larger than any previous molecule ($\sim$2 eV). It is possible that the error
associated with the ESMP2 based correction may be responsible for the bulk of
ESPCC’s error here, which provides motivation for developing more general
versions of ESPCC that do not need to rely on an ESMP2 correction for these
few terms.

### 3.5 Size Consistent Energies

To test whether this method is size consistent as we expect, we calculate the
excitation energy of H2O in the presence of a variable number of very well
separated He atoms. The results for these calculations on a variety of methods
are shown in Table 5. As we can see, most of the methods presented produce the
exact same excitation energy regardless of the number of noninteracting He
atoms that are introduced. The exception is CISD, which has well known size
consistency issues. Furthermore, because the ground state energy is calculated
using CCSD, this implies that our method is correctly matching the ground
state energy for the He atoms and that it is both size consistent and
intensive in the excited state. Though one may begin to wonder whether the
ESMP2 based correction affects the size consistency of our method in any way,
it is worth noting that this correction involves orbitals localized only on
the fragment that is being excited, and thus does not affect the energy on any
other fragment.

He atoms | ESMF | ESMP2 | ESPCC | Corrected ESPCC | EOM-CCSD | CISD
---|---|---|---|---|---|---
0 | 7.620 | 8.359 | 8.609 | 8.359 | 8.283 | 11.600
1 | 7.620 | 8.359 | 8.609 | 8.359 | 8.283 | 12.404
2 | 7.620 | 8.359 | 8.609 | 8.359 | 8.283 | 13.198
3 | 7.620 | 8.359 | 8.609 | 8.359 | 8.283 | 13.983
4 | 7.620 | 8.359 | 8.609 | 8.359 | 8.283 | 14.760
5 | 7.620 | 8.359 | 8.609 | 8.359 | 8.283 | 15.528
Table 5: Excitation energies (eV) across multiple methods for cc-pVDZ water in
the presence of a variable number of noninteracting He atoms. These results
corroborate the proof of size consistency presented earlier for this method,
as well as illustrate how large of an influence a lack of size consistency
like that in CISD has on the quality of excitation energies.

## 4 Conclusion

We have presented an approach to excited-state-specific coupled-cluster theory
that applies a simple projector to a single-reference coupled-cluster
expansion containing singles, doubles, and some triples in order to build a
CCSD-like wavefunction atop an excited state mean field starting point. Like
its ground state cousin, this ESPCC approach is size-consistent, has an
$N^{6}$ cost, and can be optimized by a DIIS-accelerated diagonal Jacobian
approximation. For amplitudes that do not interact with the excitation, the
working equations simplify to match those of CCSD, while amplitudes that are
only weakly coupled to the excitation see slightly modified CCSD-like working
equations. With a correction from excited-state-specific perturbation theory
to account for correlation terms incompatible with the simple projector used
in this initial exploration of ESPCC, excitation energies are within 0.2 eV of
high-level benchmarks in tests that include valence states, a Rydberg state,
and a charge transfer state. This compares favorably with the performance of
ESMP2 and EOM-CCSD on the same states.

In the future, it will be especially interesting to explore more general forms
for the projection operator in order to both eliminate the reliance on
perturbation theory corrections and admit a more full-fledged ESMF starting
point. Given that many of the doubles amplitudes experience working equations
that are little different than those in CCSD, it will also be interesting to
explore whether, at least for these amplitudes, straightforward analogues to
ground state perturbative triples corrections can be formulated. Finally, a
reimplementation in spatial rather than spin orbitals that exploits
factorization and permutation symmetry should allow the theory to be tested on
a much wider range of molecules and excited states.

This work was supported by the National Science Foundation’s CAREER program
under Award Number 1848012. Calculations were performed using the Berkeley
Research Computing Savio cluster. H.T. acknowledges that this material is
based upon work supported by the National Science Foundation Graduate Research
Fellowship Program under Grant No. DGE 2146752. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the National Science
Foundation.

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## 5 Supporting Information

## S1 Relevant Proofs

### S1.1 Intermediate Normalization

$\displaystyle\braket{\psi_{0}}{\Psi}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\bra{{\phi^{*}}_{h}^{p}}\hat{P}e^{\hat{T}}\ket{\phi_{0}^{*}}+\bra{{\phi^{*}}_{\bar{h}}^{\bar{p}}}\hat{P}e^{\hat{T}}\ket{\phi_{0}^{*}}\right)$
(S1)
$\displaystyle=\frac{1}{\sqrt{2}}\left(\bra{{\phi^{*}}_{h}^{p}}\hat{P}(1+\hat{T}+\dots)\ket{\phi_{0}^{*}}+\bra{{\phi^{*}}_{\bar{h}}^{\bar{p}}}\hat{P}(1+\hat{T}+\dots)\ket{\phi_{0}^{*}}\right)$
(S2)
$\displaystyle=\frac{1}{\sqrt{2}}\left(t_{h}^{p}+t_{\bar{h}}^{\bar{p}}\right)=1$
(S3)

$\displaystyle\bra{\psi_{0}}e^{-\hat{T}}\ket{\Psi}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\bra{{\phi^{*}}_{h}^{p}}e^{-\hat{T}}\hat{P}e^{\hat{T}}\ket{\phi_{0}^{*}}+\bra{{\phi^{*}}_{\bar{h}}^{\bar{p}}}e^{-\hat{T}}\hat{P}e^{\hat{T}}\ket{\phi_{0}^{*}}\right)$
(S4)
$\displaystyle=\frac{1}{\sqrt{2}}\left(\bra{{\phi^{*}}_{h}^{p}}(1-\hat{T}+\dots)\hat{P}(1+\hat{T}+\dots)\ket{\phi_{0}^{*}}\right.$
(S5)
$\displaystyle\quad\left.+\bra{{\phi^{*}}_{\bar{h}}^{\bar{p}}}(1-\hat{T}+\dots)\hat{P}(1+\hat{T}+\dots)\ket{\phi_{0}^{*}}\right)$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\bra{{\phi^{*}}_{h}^{p}}\hat{P}(1+\hat{T}+\dots)\ket{\phi_{0}^{*}}+\bra{{\phi^{*}}_{\bar{h}}^{\bar{p}}}\hat{P}(1+\hat{T}+\dots)\ket{\phi_{0}^{*}}\right.$
(S6)
$\displaystyle\quad\left.-t_{h}^{p}\bra{{\phi^{*}}_{0}}\hat{P}(1+\hat{T}+\dots)\ket{\phi_{0}^{*}}-t_{\bar{h}}^{\bar{p}}\bra{{\phi^{*}}_{0}}\hat{P}(1+\hat{T}+\dots)\ket{\phi_{0}^{*}}\right)$
$\displaystyle=\frac{1}{\sqrt{2}}\left(t_{h}^{p}+t_{\bar{h}}^{\bar{p}}\right)=1$
(S7)

### S1.2 Size Consistency of Energy

$\displaystyle\frac{\bra{\psi_{0}}e^{-\hat{T}}\hat{H}\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}}{\bra{\psi_{0}}e^{-\hat{T}}\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}}=\frac{1}{\sqrt{2}}\left(\bra{{\phi^{*}}_{h}^{p}}e^{-\hat{T}}\hat{H}\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}+\bra{{\phi^{*}}_{\bar{h}}^{\bar{p}}}e^{-\hat{T}}\hat{H}\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}\right)$
(S8)
$\displaystyle=\frac{1}{\sqrt{2}}\left[\left(\bra{{\phi^{*}_{A}}_{h}^{p}}\otimes\bra{\phi^{*}_{B}}\right)\left(e^{-\hat{T}_{A}}\otimes
e^{-\hat{T}_{B}}\right)\left(\hat{H}_{A}+\hat{H}_{B}\right)\hat{P}_{A}\left(e^{\hat{T}_{A}}\otimes
e^{\hat{T}_{B}}\right)\left(\ket{{\phi_{A}^{*}}}\otimes\ket{{\phi_{B}^{*}}}\right)\right.$
(S9)
$\displaystyle\quad+\left.\left(\bra{{\phi^{*}_{A}}_{\bar{h}}^{\bar{p}}}\otimes\bra{\phi^{*}_{B}}\right)\left(e^{-\hat{T}_{A}}\otimes
e^{-\hat{T}_{B}}\right)\left(\hat{H}_{A}+\hat{H}_{B}\right)\hat{P}_{A}\left(e^{\hat{T}_{A}}\otimes
e^{\hat{T}_{B}}\right)\left(\ket{{\phi_{A}^{*}}}\otimes\ket{{\phi_{B}^{*}}}\right)\right]$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\bra{{\phi^{*}_{A}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{H}_{A}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{\phi^{*}_{B}}e^{-\hat{T}_{B}}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}\right.$
(S10)
$\displaystyle\quad+\bra{{\phi^{*}_{A}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{\phi^{*}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
$\displaystyle\quad+\bra{{\phi^{*}_{A}}_{\bar{h}}^{\bar{p}}}e^{-\hat{T}_{A}}\hat{H}_{A}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{\phi^{*}_{B}}e^{-\hat{T}_{B}}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
$\displaystyle\quad+\left.\bra{{\phi^{*}_{A}}_{\bar{h}}^{\bar{p}}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{\phi^{*}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}\right]$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\bra{{\phi^{*}_{A}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{H}_{A}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}+\bra{{\phi^{*}_{A}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{\phi^{*}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}\right.$
(S11)
$\displaystyle\quad\left.+\bra{{\phi^{*}_{A}}_{\bar{h}}^{\bar{p}}}e^{-\hat{T}_{A}}\hat{H}_{A}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}+\bra{{\phi^{*}_{A}}_{\bar{h}}^{\bar{p}}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{\phi^{*}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}\right]$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\bra{{\phi^{*}_{A}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{H}_{A}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}+\bra{{\phi^{*}_{A}}_{\bar{h}}^{\bar{p}}}e^{-\hat{T}_{A}}\hat{H}_{A}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\right]$
(S12)
$\displaystyle\quad+\bra{\phi^{*}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
$\displaystyle=\bra{{{\psi_{0}}_{A}}}e^{-\hat{T}_{A}}\hat{H}_{A}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}+\bra{\phi^{*}_{B}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}=E_{A}+E_{B}=E$
(S13)

### S1.3 Size Consistency of Amplitude Equations

$\displaystyle\bra{{\phi^{*}}_{hj}^{pb}}e^{-\hat{T}}\left(\hat{H}-E\right)\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}$
(S14)
$\displaystyle=\left(\bra{{\phi_{A}^{*}}_{h}^{p}}\otimes\bra{{\phi_{B}^{*}}_{j}^{b}}\right)\left(e^{-\hat{T}_{A}}\otimes
e^{-\hat{T}_{B}}\right)\left(\hat{H}_{A}+\hat{H}_{B}-E\right)\hat{P}_{A}\left(e^{\hat{T}_{A}}\otimes
e^{\hat{T}_{B}}\right)\left(\ket{{\phi_{A}^{*}}}\otimes\ket{{\phi_{B}^{*}}}\right)$
(S15)
$\displaystyle=\bra{{\phi_{A}^{*}}_{h}^{p}}e^{-\hat{T}_{A}}\left(\hat{H}_{A}-E\right)\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{{\phi_{B}^{*}}_{j}^{b}}e^{-\hat{T}_{B}}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S16)
$\displaystyle\quad+\bra{{\phi_{A}^{*}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{{\phi_{B}^{*}}_{j}^{b}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
$\displaystyle=\bra{{\phi_{A}^{*}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{{\phi_{B}^{*}}_{j}^{b}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S17)
$\displaystyle=\bra{{\phi_{A}^{*}}_{h}^{p}}\left(1-\hat{T}_{A}+\dots\right)\hat{P}_{A}\left(1+\hat{T}_{A}+\dots\right)\ket{{\phi_{A}^{*}}}\otimes\bra{{\phi_{B}^{*}}_{j}^{b}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S18)
$\displaystyle=\left[\bra{{\phi_{A}^{*}}_{h}^{p}}\hat{P}_{A}\left(1+\hat{T}_{A}\right)\ket{{\phi_{A}^{*}}}-t_{h}^{p}\bra{{\phi_{A}^{*}}}\hat{P}_{A}\left(1+\hat{T}_{A}\right)\ket{{\phi_{A}^{*}}}\right]\otimes\bra{{\phi_{B}^{*}}_{j}^{b}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S19)
$\displaystyle=\left[t_{h}^{p}{\bra{{\phi_{A}^{*}}_{h}^{p}}\hat{P}_{A}\ket{{\phi_{A}^{*}}_{i}^{a}}}-t_{h}^{p}{\bra{{\phi_{A}^{*}}}\hat{P}_{A}\ket{{\phi_{A}^{*}}}}\right]\otimes\bra{{\phi_{B}^{*}}_{j}^{b}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S20)
$\displaystyle=t_{h}^{p}\bra{{\phi_{B}^{*}}_{j}^{b}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S21)

$\displaystyle\bra{{\phi^{*}}_{hjk}^{pbc}}e^{-\hat{T}}\left(\hat{H}-E\right)\hat{P}e^{\hat{T}}\ket{\phi^{*}_{0}}$
(S22)
$\displaystyle=\left(\bra{{\phi_{A}^{*}}_{h}^{p}}\otimes\bra{{\phi_{B}^{*}}_{jk}^{bc}}\right)\left(e^{-\hat{T}_{A}}\otimes
e^{-\hat{T}_{B}}\right)\left(\hat{H}_{A}+\hat{H}_{B}-E\right)\hat{P}_{A}\left(e^{\hat{T}_{A}}\otimes
e^{\hat{T}_{B}}\right)\left(\ket{{\phi_{A}^{*}}}\otimes\ket{{\phi_{B}^{*}}}\right)$
(S23)
$\displaystyle=\bra{{\phi_{A}^{*}}_{h}^{p}}e^{-\hat{T}_{A}}\left(\hat{H}_{A}-E\right)\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{{\phi_{B}^{*}}_{jk}^{bc}}e^{-\hat{T}_{B}}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S24)
$\displaystyle\quad+\bra{{\phi_{A}^{*}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{{\phi_{B}^{*}}_{jk}^{bc}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
$\displaystyle=\bra{{\phi_{A}^{*}}_{h}^{p}}e^{-\hat{T}_{A}}\hat{P}_{A}e^{\hat{T}_{A}}\ket{{\phi_{A}^{*}}}\otimes\bra{{\phi_{B}^{*}}_{jk}^{bc}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S25)
$\displaystyle=\bra{{\phi_{A}^{*}}_{h}^{p}}\left(1-\hat{T}_{A}+\dots\right)\hat{P}_{A}\left(1+\hat{T}_{A}+\dots\right)\ket{{\phi_{A}^{*}}}\otimes\bra{{\phi_{B}^{*}}_{jk}^{bc}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S26)
$\displaystyle=\left[\bra{{\phi_{A}^{*}}_{h}^{p}}\hat{P}_{A}\left(1+\hat{T}_{A}\right)\ket{{\phi_{A}^{*}}}-t_{h}^{p}\bra{{\phi_{A}^{*}}}\hat{P}_{A}\left(1+\hat{T}_{A}\right)\ket{{\phi_{A}^{*}}}\right]\otimes\bra{{\phi_{B}^{*}}_{jk}^{bc}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S27)
$\displaystyle=\left[t_{h}^{p}{\bra{{\phi_{A}^{*}}_{h}^{p}}\hat{P}_{A}\ket{{\phi_{A}^{*}}_{h}^{p}}}-t_{h}^{p}{\bra{{\phi_{A}^{*}}}\hat{P}_{A}\ket{{\phi_{A}^{*}}}}\right]\otimes\bra{{\phi_{B}^{*}}_{jk}^{bc}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S28)
$\displaystyle=t_{h}^{p}\bra{{\phi_{B}^{*}}_{jk}^{bc}}e^{-\hat{T}_{B}}\hat{H}_{B}e^{\hat{T}_{B}}\ket{{\phi_{B}^{*}}}$
(S29)

## S2 Vertical Excitation Energies

| FCI |  $\delta$-CR-EOM-CC(2,3),D | ESMF | ESMP2 | ESPCC | Corrected ESPCC | EOM-CCSD
---|---|---|---|---|---|---|---
H6 (6-31G) | 13.6383 | – | 12.962 | 13.380 | 13.639 | 13.6383 | 13.6386
H2O (6-31G) | 8.604 | – | 7.941 | 8.650 | 8.641 | 8.621 | 8.501
H2O (cc-pVDZ) | – | 8.297 | 7.620 | 8.359 | 8.609 | 8.359 | 8.283
CH2O (cc-pVDZ) | – | 4.075 | 3.384 | 4.227 | 4.801 | 4.193 | 4.266
CH2CO (cc-pVDZ) | – | 3.782 | 3.548 | 3.776 | 4.497 | 3.792 | 4.142
NH3 F2 (cc-pVDZ) | – | 9.265 | 7.780 | 9.008 | 9.633 | 9.428 | 9.779
Ne (aug-cc-pVTZ) | – | 47.126 | 48.393 | 47.615 | 49.109 | 47.303 | 47.259
Table S1: The excitation energies in eV for each molecule on a small set of
test molecules.

## S3 Molecular Geometries

All geometries are reported in Bohr.

H6

    H  0.000000000 -0.900000000  0.000000000
    H  0.000000000  0.900000000  0.000000000
    H  7.000000000  0.200000000  0.000000000
    H  8.400000000  0.200000000  0.050000000
    H -7.000000000 -0.200000000  0.050000000
    H -8.400000000 -0.200000000  0.000000000

Water

    O  0.000000000  0.000000000 -0.112851412
    H  0.000000000  1.425999575  0.970077421
    H  0.000000000 -1.425999575  0.970077421

Formaldehyde

    C  0.000000000  0.000000000 -1.126692899
    O  0.000000000  0.000000000  1.111413933
    H  0.000000000  1.746571764 -2.224070962
    H  0.000000000 -1.746571764 -2.224070962

Ketene

    C  0.000000000  0.000000000 -2.433659698
    C  0.000000000  0.000000000  0.034004982
    O  0.000000000  0.000000000  2.198008243
    H  0.000000000  1.764688153 -3.424475103
    H  0.000000000 -1.764688153 -3.424475103

Ammonia $\rightarrow$ Difluorine

    N  0.000000000 -0.443921233 -6.652831844
    H  0.000000000  1.331763699 -7.379272802
    H  1.537789200 -1.331763699 -7.379272802
    H -1.537789200 -1.331763699 -7.379272802
    F  0.000000000 -0.443921233  4.685524903
    F  0.000000000 -0.443921233  7.379272802

Neon

    Ne  0.000000000  0.000000000  0.000000000