# Aufbau Suppressed Coupled Cluster Theory for Electronically Excited States

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

###### Abstract

We introduce an approach to improve single-reference coupled cluster theory in
settings where the Aufbau determinant is absent from or plays only a small
role in the true wave function. Using a de-excitation operator that can be
efficiently hidden within a similarity transform, we create a coupled cluster
wave function in which de-excitations work to suppress the Aufbau determinant
and produce wave functions dominated by other determinants. Thanks to an
invertible and fully exponential form, the approach is systematically
improvable, size consistent, size extensive, and, interestingly, size
intensive in a granular way that should make the adoption of some ground state
techniques such as local correlation relatively straightforward. In this
initial study, we apply the general formalism to create a state-specific
method for orbital-relaxed singly excited states. We find that this approach
matches the accuracy of similar-cost equation-of-motion methods in valence
excitations while offering improved accuracy for charge transfer states. We
also find the approach to be more accurate than excited-state-specific
perturbation theory in both types of states.

††preprint: AIP/123-QED

##  I Introduction

Coupled cluster (CC) theory Bartlett and Musiał (2007); Čížek (1966); Čížek
and Paldus (1971); Shavitt and Bartlett (2009) offers highly accurate
treatments of electron correlation and is particularly effective in single-
reference (SR) settings in which the true wave function is dominated by a
single determinant. With strong formal properties like size extensivity and
size consistency that configuration interaction (CI) theory lacks, Helgaker et
al. (2013); Crawford and Schaefer III (2007) CC can reliably achieve exquisite
accuracy in SR settings, as evidenced for example by the sub-$k_{B}T$ errors
of the perturbative-triples-corrected singles and doubles theory, CCSD(T).
Raghavachari et al. (1989); Watts et al. (1993); Thomas et al. (1993);
Helgaker et al. (1997); Bak et al. (2001) By looking a little closer at the
details of CC theory, one can identify two important factors driving its
success in SR settings: the accuracy of its reference and the conditioning of
its working equations. Indeed, when the true wave function is not dominated by
a single determinant, SR-CC approaches often fail dramatically, Köhn et al.
(2013); Lyakh et al. (2012); Lischka et al. (2018) and the construction of
more multi-reference (MR) approaches can lead to ill-conditioned equations.
Van Voorhis and Head-Gordon (2001); Neuscamman et al. (2009, 2010, 2010);
Yanai et al. (2012)

In thinking about MR-CC, it can be useful to recognize that most approaches
fall into one of three categories: Jeziorski-Monkhorst (JM), internal
contraction, and extended SR. Köhn et al. (2013); Lischka et al. (2018); Lyakh
et al. (2012) JM methods use multiple determinants both to improve the
reference wave function and as separate reference points for constructing
cluster operators. As the latter practice can lead to an ill-conditioned
situation where there are more variables than available projective equations,
JM methods typically choose and impose additional sufficiency conditions.
Hubač and Neogrády (1994); Pittner et al. (1999); Mahapatra and Chattopadhyay
(2010, 2011); Mahapatra et al. (1998, 1999); Hanrath (2005) Internally
contracted approaches also employ multiple determinants in the reference wave
function, but not in the construction of the cluster operator. Instead, they
act a single cluster operator on the reference, which achieves a natural match
between the number of variables and projective equations but creates
challenges related to nonterminating expansions and ill-conditioned overlap
matrices. Hanauer and Köhn (2011, 2012); Evangelista and Gauss (2011); Datta
et al. (2011); Li (2004) Extended SR methods retain a single determinant
reference but extend other parts of the formalism to make it more amenable to
MR settings, such as by identifying and including subsets of higher
excitations. Piecuch et al. (1999); Oliphant and Adamowicz (1991); Shen and
Piecuch (2012); Bauman et al. (2017); Adamowicz et al. (2000); Lyakh et al.
(2008)

The extended SR approaches retain many of the advantages of SR-CC but also
have challenges. Like SR-CC, they have relatively straightforward working
equations, at least compared to those of MR-CC. However, they can face
difficulty in selecting the reference determinant, for example in cases where
the dominant determinant changes along a reaction pathway. Köhn et al. (2013)
Further, a too-simple reference can necessitate large excitation amplitudes
that lead to large and unwelcome contributions from the higher nonlinear terms
in the CC expansion. In the present study, we address the large amplitude
issue directly so that relatively simple SR working equations can be usefully
employed in a wider range of settings.

Specifically, we seek to avoid the need for large excitation amplitudes even
when the true wave function has a small or zero Aufbau contribution by
including a de-excitation operator that suppresses the Aufbau determinant
within the CC expansion. Possible applications of this Aufbau suppressed CC
(ASCC) approach include strongly correlated ground states as well as state-
specific treatments of both weakly and strongly correlated excited states.
Unlike our previous approach of removing the Aufbau determinant through a
pseudo-projection operator, Tuckman and Neuscamman (2023) ASCC employs an
invertible operator, making it more general and avoiding the need to
perturbatively correct for projected-out pieces of the correlation treatment.
Although the ASCC formalism has the potential to be useful in many areas, we
focus in this initial study on singly excited states, where it offers a route
to orbital-relaxed, excited-state-specific CC treatments atop spin-pure
reference functions.

One long-running challenge even in singly excited states has been to fully
capture the effects of post-excitation orbital relaxations. Subotnik (2011)
Many widely used approaches, such as time-dependent density functional theory
(TD-DFT), Runge and Gross (1984); Burke et al. (2005); Casida and Huix-
Rotllant (2012) equation-of-motion CC (EOM-CC), Rowe (1968); Stanton and
Bartlett (1993); Krylov (2008) and linear response CC (LR-CC), Monkhorst
(1977); Dalgaard and Monkhorst (1983); Sekino and Bartlett (1984); Koch et al.
(1990); Koch and Jørgensen (1990); Rico and Head-Gordon (1993); Koch et al.
(1994); Sneskov and Christiansen (2012) rely on linear response and have only
a limited ability to account for orbital relaxations or the new correlation
effects created when an electron is transfered between different regions of a
molecule. These issues are known to limit accuracy in charge transfer (CT),
Rydberg, core, and double excitations. Tozer and Handy (1998); Casida et al.
(1998); Casida and Salahub (2000); Tozer and Handy (2003); Sobolewski and
Domcke (2003); Dreuw et al. (2003); Dreuw and Head-Gordon (2004); Mester and
Kállay (2022); Kozma et al. (2020); Izsák (2020) State-specific approaches
seek to improve matters by more fully tailoring their treatments to the needs
of the excited state in question and have been developed for single
determinant theories, Ziegler et al. (1977); Kowalczyk et al. (2011); Gilbert
et al. (2008); Besley et al. (2009); Barca et al. (2018); Carter-Fenk and
Herbert (2020); Burton and Wales (2020) CI theory, Liu et al. (2012); Liu and
Subotnik (2014); Shea and Neuscamman (2018); Shea et al. (2020); Hardikar and
Neuscamman (2020); Kossoski and Loos (2022, 2023); Burton (2022) perturbation
theory, Clune et al. (2020, 2023) CC theory, Mayhall and Raghavachari (2010);
Lee et al. (2019); Kossoski et al. (2021); Marie et al. (2021); Rishi et al.
(2023); Tuckman and Neuscamman (2023) complete active space self consistent
field theory, Tran et al. (2019); Tran and Neuscamman (2020); Hanscam and
Neuscamman (2022) and DFT. Kowalczyk et al. (2011, 2013); Hait and Head-Gordon
(2020, 2021); Zhao and Neuscamman (2019); Levi et al. (2020) In this context,
ASCC offers a route to systematically improvable, state-specific CC treatments
with good spin symmetry and working equations that closely mirror the ground
state theory.

Through its fully exponential form, ASCC ensures its predictions are size
consistent, size extensive, and, perhaps most interestingly, size intensive at
a granular level. As we will see, the mathematics for electrons far from the
excitation simplify to those of ground state SR-CC in a way that guarantees
intensivity not just in the final result, but also in the details of the
working equations. When combined with the use of excited-state-specific
reference orbitals from excited state mean field (ESMF) theory, Shea and
Neuscamman (2018); Shea et al. (2020); Hardikar and Neuscamman (2020); Clune
et al. (2023) this granular intensivity should make it relatively
straightforward to eventually incorporate local correlation treatments,
Riplinger and Neese (2013); Riplinger et al. (2013, 2016); Saitow et al.
(2017); Guo et al. (2018) which may benefit from increased excited state
specificity. Frank and Hättig (2018); Helmich and Hättig (2011); Helmich and
Haettig (2013) In this initial study, however, we will focus on introducing
the general ASCC framework and investigating its efficacy in singly excited
states.

##  II Theory

###  II.1 A Challenge for Single-Reference Coupled Cluster

In the standard single-reference approach, Bartlett and Musiał (2007);
Helgaker et al. (2013); Crawford and Schaefer III (2007); Shavitt and Bartlett
(2009) the coupled cluster (CC) equations are motivated by and derived from an
exponential ansatz

$\displaystyle\ket{\Psi_{\mathrm{CC}}}=e^{\hat{T}}\ket{\phi_{0}}=\left(1+\hat{T}+\frac{1}{2}\hat{T}^{2}+\dots\right)\ket{\phi_{0}}$
(1)

involving an excitation operator $\hat{T}$ and the closed-shell Aufbau
determinant $\ket{\phi_{0}}$. If one allows $\hat{T}$ to contain all orders of
excitation, then the theory is exact within the one-electron basis, but in
practice $\hat{T}$ is usually truncated to produce a polynomial cost approach.
For example, CCSD includes only single and double excitations in $\hat{T}$.
Within the CI expansion that the CCSD wave function corresponds to, this
choice gives it the ability to set the coefficients on the singly and doubly
excited determinants however it wants while, unlike CI, still achieving a size
extensive energy. Though CCSD has much more limited control over the
coefficients on the triply and higher excited determinants, this is of little
concern so long as two conditions are met: (1) those highly excited terms are
small in the exact wave function and (2) the magnitude of $\hat{T}$ itself is
small so that its higher order powers do not create unduly large triply or
higher excited determinants. When the first condition is not met (as is
typically true in strongly correlated systems), single-reference CC struggles.
Köhn et al. (2013); Cooper and Knowles (2010) However, what happens if one
naively applies single-reference CC in a case where only the second condition
is not met?

Table 1: Example of a simple singly excited state. Determinant | Label | FCI Coefficient
---|---|---
Aufbau | A | $c_{A}=$ | 0.01
HOMO-X$\rightarrow$LUMO+X | X | $\quad c_{X}=$ | 0.68
HOMO-Y$\rightarrow$LUMO+Y | Y | $c_{Y}=$ | 0.09
HOMO${}^{2}\rightarrow$LUMO2 | D | $c_{D}=$ | 0.03
All others |  | very small

For example, consider the relatively simple FCI wave function shown in Table
1, which might arise as one of a molecule’s low-lying excited states. Being
dominated by single excitations, this state would be well treated by many
excited state theories, but what would happen if we were to try to naively
apply single-reference CC to it directly, that is to say without using linear
response theory? Due to the fact that the CC expansion puts a coefficient of 1
on the Aufbau determinant, the amplitude within $\hat{T}$ for the
HOMO-X$\rightarrow$LUMO+X single excitation would have to be huge — something
like $t_{X}\approx 0.68/0.01=68$ — in order to get the right ratio of single-
to-Aufbau in our expansion. The others would have to be large as well:
$t_{Y}\approx 0.09/0.01=9$ and $t_{D}\approx 0.03/0.01=3$. At this point, if
we ignore normalization and look only through linear order in the expansion,
we’d be doing a good job at matching FCI. However, the nonlinear terms are now
a disaster, as $\hat{T}$ is not small. Consider the $\hat{T}^{5}$ term in the
expansion, which leads to a hextuply excited determinant with a coefficient on
the order of $t_{X}^{2}t_{Y}^{2}t_{D}/5!\approx 9000$. These types of terms
now dominate the wave function, and so instead of creating a good
approximation of FCI, we’ve created a terrible mess. In sum, when trying to
apply single-reference CC directly to a state in which the Aufbau coefficient
is small relative to some others, we get in trouble regardless of whether we
keep $\hat{T}$ small or make it large. If we keep it small, the Aufbau will be
too large relative to the other determinants, but, if we make it large, the
nonlinear terms will get out of hand.

This issue is why, of course, approaches other than a direct application of
single-reference CC are typically taken for this type of state. Options
include using a non-Aufbau reference, using multiple references, and applying
equation-of-motion or linear response theory. Although these approaches all
have their merits, and, depending on the details, may work quite well for the
example above, they also involve drawbacks. The difficulty of capturing
orbital relaxations in linear response theory would be problematic if the
state above were a charge transfer excitation. Using an alternative single-
determinant reference could work well for the $c_{X}$ and $c_{Y}$ coefficients
used in the example, but would become challenging if molecular geometry
changes caused $c_{X}$ and $c_{Y}$ to gradually exchange magnitudes, as it
would then be difficult to choose which reference to use. Were the $c_{D}$
coefficient to grow large, both of these approaches would become more
challenging. At that point, one may be motivated to move away from SR
approaches, with all the complication that that entails. In this study, we
will instead explore an alternative path that, by suppressing the Aufbau
determinant, provides a systematically improvable and state-specific framework
in which a formal SR can be maintained even when non-Aufbau determinants
dominate the true FCI wave function.

###  II.2 Aufbau Suppression

To suppress the coefficient on the Aufbau determinant in the expanded CC wave
function, let us augment our ansatz as follows.

$\displaystyle\ket{\Psi_{\mathrm{ASCC}}}$
$\displaystyle=e^{-\hat{S}^{\dagger}}e^{\hat{T}}\ket{\phi_{0}}$
$\displaystyle=\left(1+\hat{T}-\hat{S}^{\dagger}\hat{T}+\frac{1}{2}\hat{T}^{2}+\dots\right)\ket{\phi_{0}}$
(2)

Here $\hat{S}$ is a (hopefully simple) excitation operator that will be chosen
such that the $-\hat{S}^{\dagger}\hat{T}\ket{\phi_{0}}$ term and any similar
higher-order terms produce additional copies of the Aufbau determinant that
can partially or fully cancel out the zeroth-order $\ket{\phi_{0}}$ term.
Basically, by de-exciting with $\hat{S}^{\dagger}$ after exciting with
$\hat{T}$, we can get back to Aufbau and, provided the amplitude coefficients
within $\hat{S}$ and $\hat{T}$ are chosen carefully, can thereby suppress the
overall coefficient on the Aufbau determinant via cancellation.

Before discussing whether or not this approach can be practical, let us first
emphasize its strong formal properties. As the ansatz remains exponential, it
will still product-factorize and thus maintain size consistency. Even better,
so long as our approach to energy evaluation and optimization does not
introduce any unlinked terms, it will remain extensive. The use of the
invertible operator $\exp(-\hat{S}^{\dagger})$ guarantees that systematic
improvability is also still present: if $\hat{T}$ contains all orders of
excitations, then $\exp(\hat{T})\ket{\phi_{0}}$ can describe
$\exp(\hat{S}^{\dagger})\ket{\Psi_{\mathrm{FCI}}}$ just as well as it can
describe $\ket{\Psi_{\mathrm{FCI}}}$. Finally, with the ability to suppress
the Aufbau determinant, this approach can describe wave functions dominated by
non-Aufbau determinants while keeping the magnitude of $\hat{T}$ modest, which
should improve our chances of achieving an accurate, state-specific
description while limiting $\hat{T}$ to low orders of excitation.

To make this approach practical, we first convert the central CC eigenvalue
equation

$\displaystyle\hat{H}\ket{\Psi_{\mathrm{ASCC}}}=E\ket{\Psi_{\mathrm{ASCC}}}$
(3)

into a similarity transformed form in which $\hat{S}^{\dagger}$ has been
wrapped around the Hamiltonian.

$\displaystyle\bar{H}e^{\hat{T}}\ket{\phi_{0}}$
$\displaystyle=Ee^{\hat{T}}\ket{\phi_{0}}$ (4) $\displaystyle\bar{H}$
$\displaystyle=e^{\hat{S}^{\dagger}}\hat{H}e^{-\hat{S}^{\dagger}}$ (5)

The key question question now becomes: what forms for $\hat{S}$ will make for
a computationally efficient set of similarity-transformed equations while
delivering the desired Aufbau suppression? We strongly suspect that different
choices for $\hat{S}$ may be optimal for different types of states and that
there will be workable choices for doubly excited states and some classes of
strongly correlated ground state. In this study, however, we will begin our
exploration of ASCC theory by choosing an $\hat{S}$ made entirely of single
excitation operators, which turns Eq. (5) into a simple if non-standard
$O(N^{5})$ transformation of the one- and two-electron integrals. While this
choice may be useful in multiple settings, we focus here on singly excited
states, where a particularly simple form for our singles-only $\hat{S}$ proves
useful.

###  II.3 Singly Excited States

Imagine a singly excited state in which one component of the excitation is
more important than the others, such as for example the wave function in Table
1. To start with, let us consider the largest configuration state function
(CSF) within this wave function on its own before we get to worrying about the
smaller components or the weak correlation details. We write this CSF as

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

where the presence or absence of a bar on an index denotes opposite electron
spin, and $h$ and $p$ are the indices corresponding to the “hole” and
“particle” spatial orbitals that are singly occupied in this CSF. Adopting the
terminology of extended single-reference CC, Oliphant and Adamowicz (1991) we
will refer to $\ket{\psi_{0}}$ as our reference (the major part of the state
around which we will fill in the details), while we will refer to the Aufbau
determinant $\ket{\phi_{0}}$ as the formal reference (the state acted upon by
$\exp(-\hat{S}^{\dagger})\exp(\hat{T})$ in our ansatz definition). By choosing
our operator $\hat{S}$ as the single excitation that excites our formal
reference to our reference (and whose adjoint de-excites the other way),

$\displaystyle\hat{S}\equiv\frac{1}{\sqrt{2}}$
$\displaystyle\left(\hat{a}^{\dagger}_{p}\hat{a}_{h}+\hat{a}^{\dagger}_{\bar{p}}\hat{a}_{\bar{h}}\right)$
(7) $\displaystyle\hat{S}\ket{\phi_{0}}$ $\displaystyle=\ket{\psi_{0}}$ (8)
$\displaystyle\hat{S}^{\dagger}\ket{\psi_{0}}$
$\displaystyle=\hat{S}^{\dagger}\hat{S}\ket{\phi_{0}}=\ket{\phi_{0}}$ (9)

we set ourselves up to achieve Aufbau suppression via excitation to and de-
excitation from our reference CSF. In particular, if we initialize our
$\hat{T}$ operator to the very simple singles and doubles form

$\displaystyle\hat{T}_{\mathrm{init}}=\hat{S}-\frac{1}{2}\hat{S}^{2}$ (10)

and note that $\hat{S}^{3}=0$, then our ASCC ansatz is initialized to our
reference CSF $\ket{\psi_{0}}$.

$\displaystyle
e^{-\hat{S}^{\dagger}}e^{\hat{T}_{\mathrm{init}}}\ket{\phi_{0}}$
$\displaystyle=e^{-\hat{S}^{\dagger}}e^{\hat{S}-\frac{1}{2}\hat{S}^{2}}\ket{\phi_{0}}$
(11)
$\displaystyle=\left(1-\hat{S}^{\dagger}+\frac{1}{2}(\hat{S}^{\dagger})^{2}\right)\left(1+\hat{S}\right)\ket{\phi_{0}}$
(12)
$\displaystyle=\left(1+\hat{S}-\hat{S}^{\dagger}\hat{S}\right)\ket{\phi_{0}}$
(13) $\displaystyle=\hat{S}\ket{\phi_{0}}$ (14) $\displaystyle=\ket{\psi_{0}}$
(15)

As intended, the Aufbau determinant has been suppressed, leaving us with a
wave function in which our reference CSF is dominant. In other words, we have
reached a qualitatively correct starting point for describing many singly
excited states. This was achieved while maintaining the Aufbau determinant as
our formal reference and while maintaining relatively modest amplitudes sizes
within $\hat{T}$. Indeed, the $1/\sqrt{2}$ and $-1/2$ amplitudes hiding inside
$\hat{T}_{\mathrm{init}}$ are much smaller than the values of 68 and 9 seen in
our example in Section II.1. Whether they are small enough to allow for an
accurate fleshing out of the remaining wave function details is a matter for
numerical tests, which we explore in Section III. While the attention thus far
has been focused on states dominated by a single CSF, this formalism naturally
extends to include states with multiple dominant CSFs as well by including
additional excitation operators in $\hat{S}$. While increasing the number of
excitation operators in $\hat{S}$ requires additional components in
$\hat{T}_{\mathrm{init}}$ to exactly reconstruct the reference – for example,
a handful of triples amplitudes and a single quadruple amplitude in the two
CSF case – this subset of amplitudes remains relatively modest in size so long
as the number of dominant CSFs in a state remains small, as is often the case.
Having successfully built up our reference starting from our formal reference,
we now turn our attention to adding and optimizing the details needed for a
robust correlation treatment.

###  II.4 Filling in the Details

To think through which additional amplitudes should be enabled to take us from
$\hat{T}_{\mathrm{init}}$ to a $\hat{T}$ more in line with that used in CCSD,
it is useful to separate this operator into two pieces.

$\displaystyle\hat{T}=\hat{T}_{P}+\hat{T}_{NP}$ (16)

We will group into $\hat{T}_{P}$ all amplitudes whose excitation operators
contain one or more of our “primary” indices $h$ and $p$ that are present
within our chosen $\hat{S}$ from Eq. 7. All other excitation operators will be
grouped into the “non-primary” $\hat{T}_{NP}$. Noting that $\hat{T}_{NP}$ thus
commutes with $\hat{S}^{\dagger}$, we can rearrange our ansatz as

$\displaystyle\ket{\Psi}$
$\displaystyle=e^{\hat{T}_{NP}}e^{-\hat{S}^{\dagger}}e^{\hat{T}_{P}}\ket{\phi_{0}}$
(17)

in which we see that the contributions from $\hat{T}_{NP}$ will behave much
the same as would those in multi-reference CC. Indeed, $\exp(\hat{T}_{NP})$
can be formally understood as acting on the Aufbau-suppressed, multi-
determinant form set up by
$\exp(-\hat{S}^{\dagger})\exp(\hat{T}_{P})\ket{\phi_{0}}$, even if,
operationally, we group $\exp(-\hat{S}^{\dagger})$ with $\hat{H}$ for
computational expediency. Recognizing this parallel to multi-reference CC and
aiming for a weak correlation treatment of similar quality to CCSD, we choose
$\hat{T}_{NP}$ to contain all single and double excitations that have no
primary indices.

In choosing what to include in $\hat{T}_{P}$, we want to ensure that we have
all singles and doubles relative to our reference $\ket{\psi_{0}}$. Although
this choice goes a bit beyond that, we start by including all singles and
doubles that have at least one primary index. We also include all triple
excitations that both a) contain at least three primary indices and b) contain
a primary single excitation. The inclusion of these triples is not a cost
concern, as they are only a small $O(ov^{2})$ slice of all the triples.
Putting it all together, our excitation operator is

$\displaystyle\hat{T}=\hat{T}_{NP}+\hat{T}_{P}=\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{3^{\prime}}$
(18)

in which $\hat{T}_{3^{\prime}}$ contains only the small slice of triples
described above.

One final detail to take care of is how, when appropriate, to allow some
contribution from the Aufbau determinant to survive. One approach, which seems
expedient especially when the contribution is expected to be small, is to
simply let the amplitude optimization modify the $1/\sqrt{2}$ coefficients on
the primary single excitation within $\hat{T}_{\mathrm{init}}$ to values
slightly above or below $1/\sqrt{2}$ so that the cancellation effect is no
longer perfect and a small contribution from Aufbau remains. In the present
study, this is the approach we will take. We will mention, though, that if one
wants the flexibility to allow larger Aufbau contributions, one option would
be to simply scale down the $\hat{S}^{\dagger}$ operator via

$\ket{\Psi}=e^{-(1-s_{0})\hat{S}^{\dagger}}e^{\hat{T}}\ket{\phi_{0}}.$ (19)

Setting $s_{0}=1$ recovers standard SR-CC, while values in between 0 and 1
provide varying amounts of Aufbau suppression. Again, in this study we set
$s_{0}=0$ and instead rely on the optimization of the amplitudes in $\hat{T}$
to reintroduce Aufbau contributions where appropriate.

###  II.5 Optimizing the Amplitudes

To determine the amplitudes within $\hat{T}$ and ultimately evaluate the
energy, we take the usual projective approach of standard CC theory. Bartlett
and Musiał (2007); Helgaker et al. (2013); Crawford and Schaefer III (2007);
Shavitt and Bartlett (2009)

$\displaystyle E$
$\displaystyle=\bra{\phi_{0}}e^{-\hat{T}}e^{\hat{S}^{\dagger}}\hat{H}e^{-\hat{S}^{\dagger}}e^{\hat{T}}\ket{\phi_{0}}$
(20) $\displaystyle 0$
$\displaystyle=\bra{\phi_{\mu}}e^{-\hat{T}}e^{\hat{S}^{\dagger}}\hat{H}e^{-\hat{S}^{\dagger}}e^{\hat{T}}\ket{\phi_{0}}$
(21)

As in SR-CC, $\ket{\phi_{\mu}}=\hat{T}_{\mu}\ket{\phi_{0}}$ are the individual
determinants reached by acting the individual excitation operators
$\hat{T}_{\mu}$ within $\hat{T}$ on the Aufbau determinant. The value of using
an invertible exponential form to achieve Aufbau suppression becomes clear
here, where it ensures that only connected and therefore linked terms appear
in the working equations, thereby guaranteeing size extensivity. Goldstone
(1957); Bartlett and Purvis (1978); Shavitt and Bartlett (2009) In practice,
we set up our working equations by first performing the MO integral
transformation needed to form the one- and two-electron integrals of the
similarity transformed $\bar{H}$ from Eq. (5), in terms of which our energy
and amplitude equations now directly mirror SR-CC theory.

$\displaystyle E$
$\displaystyle=\bra{\phi_{0}}e^{-\hat{T}}\bar{H}e^{\hat{T}}\ket{\phi_{0}}$
(22) $\displaystyle 0$
$\displaystyle=\bra{\phi_{\mu}}e^{-\hat{T}}\bar{H}e^{\hat{T}}\ket{\phi_{0}}$
(23)

We should point out that some care in implementation is needed, as the two-
electron integrals within $\bar{H}$ lack the full 8-fold permutational
symmetry that they had in $\hat{H}$, but importantly maintain the 4-fold
symmetry necessary for grouping together terms as in the ground state theory.
Aside from that, the working equations are now identical to those of a
standard SR-CC theory in which $\hat{T}$ is chosen according to Eq. (18).
Thanks to the inclusion of only a small slice of the triples, these equations
have the same $O(o^{2}v^{4})$ asymptotic scaling as CCSD.

With working equation so similar to standard CC, we have for now simply
followed the standard recipe for solving the amplitude equations.
Specifically, we use a quasi-Newton iterative solver in which we approximate
the Jacobian as the diagonal of the similarity transformed Fock operator,
leading the amplitude updates to take a familiar form.

$t_{\mu}^{new}=t_{\mu}^{old}-\frac{R_{\mu}}{\Delta_{\mu}}$ (24)

The residual $R_{\mu}$ is the right hand side of Eq. (23), and $\Delta_{\mu}$
is the virtuals-minus-occupieds difference of the similarity transformed Fock
matrix diagonal entries corresponding to the $\hat{T}_{\mu}$ excitation. As we
are not in the canonical basis, the Fock matrix is not diagonal, and so we are
making a somewhat more aggressive approximation to the Jacobian than in
standard single-reference CC. Nonetheless, when paired with DIIS acceleration,
Pulay (1980) this update scheme achieves tight convergence
($|R_{\mu}|<10^{-10}$ a.u.) in all of our results.

As a final note regarding amplitude optimization, we should point out that
ASCC delivers what one might call iteration-by-iteration intensivity. In
addition to being size extensive, size consistent, and (for excited states)
size intensive thanks to its exponential ansatz, the adoption of the same
iterative solver used by the ground state theory causes amplitudes on
molecular fragments far away from the fragment bearing the excitation to match
those of CCSD at every step of the optimization. This essentially means that
the ways in which the amplitude equations differ from the ground state theory
are “localized” around the excitation itself. Distant parts of the system
experience essentially identical mathematics as in CCSD, down to the level of
individual amplitude updates. We note that this property should make it
relatively straightforward to adapt many ground state techniques, such as
local correlation methods, for use in ASCC.

###  II.6 Primary CSF and Orbital Basis

So far, we have not specified how we choose the MO basis or the primary CSF
that we use to define $\hat{S}$ in this single-excitation-focused incarnation
of ASCC. Although many choices are possible in principle, including CASSCF,
DFT, TD-DFT, CIS, selected CI, and even EOM-CCSD, in this study we elect to
use excited state mean field theory (ESMF)Shea and Neuscamman (2018); Shea et
al. (2020); Hardikar and Neuscamman (2020) to generate an initial guess,
because it provides a state specific, orbital relaxed, and spin-pure singly
excited reference function. By transforming the ESMF wavefunction to its
transition orbital pair (TOP) orbital basis,Clune et al. (2020) which shares
many similarities with the natural transition orbital (NTO) basis,Martin
(2003) the full ESMF wavefunction is compressed into a smaller set of
determinants without loss of information, which can then be truncated to
include only the most important CSFs for an initial guess. Once we have
truncated to these primary CSFs, we separately re-canonicalize the non-primary
parts of the occupied and virtual spaces so as to make the Fock matrix used in
the quasi-Newton amplitude update as close to diagonal as possible without
modifying the truncated ESMF reference. While this truncated ESMF serves its
role relatively well for the singly excited states in this initial study, the
most effective methods for generating ASCC’s initial guess and orbital basis
will likely vary by application, and so it will be important to explore the
possibilities more systematically in the future.

##  III Results

Table 2: Excitation Energies in eV for single-CSF statesa from the QUEST
benchmark.

Molecule | State | ESMP2 | ASCC | EOM-CCSD | Referenceb
---|---|---|---|---|---
water | $1^{1}\text{B}_{1}$ | $7.59$ | $7.50$ | $7.45$ | $7.53$
| $1^{1}\text{A}_{2}$ | $9.37$ | $9.27$ | $9.21$ | $9.32$
| $2^{1}\text{A}_{1}$ | $9.95$ | $9.86$ | $9.86$ | $9.94$
hydrogen sulfide | $1^{1}\text{B}_{1}$ | $6.02$ | $6.12$ | $6.13$ | $6.10$
| $1^{1}\text{A}_{2}$ | $6.08$ | $6.28$ | $6.34$ | $6.29$
ammonia | $1^{1}\text{A}_{2}$ | $6.41$ | $6.42$ | $6.46$ | $6.48$
| $1^{1}\text{E}$ | $8.05$ | $8.03$ | $8.03$ | $8.08$
| $2^{1}\text{A}_{1}$ | $9.51$ | $9.65$ | $9.65$ | $9.68$
| $2^{1}\text{A}_{2}$ | $10.29$ | $10.45$ | $10.38$ | $10.41$
hydrogen chloride | $1^{1}\Pi$ | $7.71$ | $7.82$ | $7.86$ | $7.82$
dinitrogen | $1^{1}\Pi_{g}$ | $8.77$ | $9.64$ | $9.49$ | $9.41$
carbon monoxide | $1^{1}\Pi$ | $8.23$ | $8.66$ | $8.67$ | $8.57$
| $2^{1}\Sigma^{+}$ | $10.64$ | $11.19$ | $11.17$ | $10.94$
| $3^{1}\Sigma^{+}$ | $11.22$ | $11.48$ | $11.71$ | $11.52$
| $2^{1}\Pi_{u}$ | $11.52$ | $11.87$ | $11.97$ | $11.76$
ethylene |  $1^{1}$B3u | $7.23$ | $7.22$ | $7.33$ | $7.31$
|  $1^{1}$B1u | $7.79$ | $7.88$ | $8.04$ | $7.93$
|  $1^{1}$B1g | $7.92$ | $7.91$ | $8.01$ | $8.00$
formaldehyde | $1^{1}\text{A}_{2}$ | $4.00$ | $3.94$ | $4.02$ | $3.99$
| $1^{1}\text{B}_{2}$ | $7.35$ | $7.11$ | $7.04$ | $7.11$
| $2^{1}\text{B}_{2}$ | $8.29$ | $8.08$ | $7.99$ | $8.04$
| $2^{1}\text{A}_{2}$ | $8.90$ | $8.72$ | $8.61$ | $8.65$
| $1^{1}\text{B}_{1}$ | $9.24$ | $9.27$ | $9.37$ | $9.29$
thioformaldehyde | $1^{1}\text{A}_{2}$ | $1.98$ | $2.16$ | $2.32$ |  $2.26$c
| $1^{1}\text{B}_{2}$ | $5.85$ | $5.85$ | $5.84$ | $5.83$
| $2^{1}\text{A}_{1}$ | $6.17$ | $6.61$ | $6.75$ | $6.51$
methanimine | $1^{1}\text{A}_{d}$ | $5.09$ | $5.22$ | $5.31$ | $5.25$
acetaldehyde | $1^{1}\text{A}_{d}$ | $4.33$ | $4.30$ | $4.36$ | $4.34$
cyclopropene | $1^{1}\text{B}_{1}$ | $6.36$ | $6.77$ | $6.78$ |  $6.71$d
| $1^{1}\text{B}_{2}$ | $6.49$ | $6.86$ | $6.88$ | $6.82$
diazomethane | $1^{1}\text{A}_{2}$ | $2.72$ | $2.97$ | $3.23$ | $3.09$
| $1^{1}\text{B}_{1}$ | $5.03$ | $5.31$ | $5.43$ | $5.35$
| $2^{1}\text{A}_{1}$ | $5.29$ | $5.84$ | $5.90$ | $5.79$
formamide | $1^{1}\text{A}_{d}$ | $5.66$ | $5.62$ | $5.71$ | $5.70$
| $2^{1}\text{A}_{p}$ | $6.95$ | $6.73$ | $6.83$ | $6.67$
| $4^{1}\text{A}_{p}$ | $7.57$ | $7.40$ | $7.41$ | $7.29$
ketene | $1^{1}\text{A}_{2}$ | $3.59$ | $3.84$ | $3.97$ | $3.84$
| $1^{1}\text{B}_{1}$ | $5.68$ | $5.93$ | $5.94$ | $5.88$
| $2^{1}\text{A}_{2}$ | $6.89$ | $7.10$ | $7.15$ | $7.08$
nitrosomethane | $1^{1}\text{A}_{d}$ | $2.02$ | $2.04$ | $2.00$ | $1.99$
streptocyanine-C1 | $1^{1}\text{B}_{2}$ | $6.48$ | $7.19$ | $7.22$ | $7.14$

Single CSF Statistics
---
|  | ESMP2 | ASCC | EOM-CCSD |
MSEe $\pm$ Std. Dev. |  | -0.13$\pm$0.23 | 0.01 $\pm$ 0.08 | 0.05$\pm$0.09 |
MUEf $\pm$ Std. Dev. |  |  -0.21$\pm$0.16 | 0.06 $\pm$ 0.05 | 0.08$\pm$0.06 |
Max Error |  | 0.66 | 0.25 | 0.24 |

aStates where one ESMF singular value is $>$0.2. bThe QUEST benchmark Loos et
al. (2018) reference was exFCI unless noted otherwise. cCCSDTQ reference.
dCCSDT reference. eMean signed error (MSE). fMean unsigned error (MUE).

###  III.1 Computational Details

For the single- and two-CSF QUEST tests, the EOM-CCSD calculations were
performed with PySCF, Sun (2015); Sun et al. (2018, 2020) while for the charge
transfer tests, EOM-CCSD and $\delta$-CR-EOM-CC(2,3),A Piecuch et al. (2006);
Piecuch and Włoch (2005); Piecuch et al. (2015, 2002); Kowalski and Piecuch
(2004); Piecuch et al. (2009); Fradelos et al. (2011) calculations were
performed with GAMESS. Schmidt et al. (1993); Barca et al. (2020) These
calculations, as well as ASCC calculations, did not use the frozen core
approximation. We iterate ASCC until the maximum amplitude equation residual
was no larger than $10^{-10}$, and for all other methods we utilize the
default convergence settings. Geometries for the charge transfer tests were
adapted from the cc-pVDZ geometries in the NIST Computational Chemistry
Comparison and Benchmark Database Johnson III (2022) and can be found in the
Supporting Information (SI). Other geometries are from the QUEST #1 excitation
energy benchmark setLoos et al. (2018). All calculations were performed in the
aug-cc-pVDZ basis, with the exception that, in the charge transfer tests, we
removed augmentation from the hydrogen atoms. Any CSF with an ESMF singular
value greater than 0.2 was included in the truncated ESMF reference for both
ASCC and ESMP2.

###  III.2 Single-CSF QUEST Tests

As an initial investigation of ASCC, we evaluate its accuracy on a set of
small molecule excitations from the QUEST #1 excitation energy benchmark. Loos
et al. (2018) Of the 56 total states, ESMF was able to provide a good initial
guess for 48 of them, with 8 others showing irreconcilable state mixing or no
convergence within ESMF. Specifically, ESMF spuriously mixed large amounts of
other states into the Rydberg states of dinitrogen and the totally symmetric
excited states of formaldehyde, while it did not converge for one singly and
one doubly excited state in nitrosomethane. We have treated the 48 states with
the present ASCC approach, leaving the other 8 for future study with non-ESMF
starting points.

Looking first at the states dominated by a single CSF, the results for ASCC
are compared to those of EOM-CCSD, extrapolated FCI (exFCI), high level CC,
and ESMP2 Shea and Neuscamman (2018); Clune et al. (2020, 2023) (a state-
specific 2nd order perturbation theory also built atop ESMF) in Table 2. We
note that the reference values make use of the frozen core approximation while
our calculations do not, but the energetic differences resulting from this
approximation are rather small ($\sim$0.01 eV),Loos et al. (2018) so these
excitation energies still make excellent reference values.

As one might expect, ASCC produces a mean unsigned error (MUE) which is
significantly lower than ESMP2’s as well as a much smaller standard deviation.
Though the more robust correlation treatment in ASCC is likely responsible for
the majority of this difference, it is worth emphasizing ASCC’s ability to
adjust the reference in the presence of the discovered details of the
correlation. Such adjustments can occur both through $\hat{T}_{1}$-based
orbital relaxations Thouless (1960) as well as through adjustments to the size
of other singly excited CSFs. These abilities are both missing in the ESMP2
treatment and thus result in relatively large discrepancies between these two
theories when the ESMF reference provides less reliable depictions of the
target states, such as in the $3^{1}\Sigma^{+}$ and $2^{1}\Pi$ states of
carbon monoxide where ESMF erroneously mixes in small amounts of the
$2^{1}\Sigma^{+}$ and $1^{1}\Pi$ states respectively. This issue appears to be
responsible for ASCC’s accuracy advantage over ESMP2 in these states, where
ASCC shows errors similar to those in other states while ESMP2 shows unusually
large errors. This improved robustness against initial guess quality is a
favorable feature, especially considering that, in states where ESMF produces
an even less accurate mixing of CSFs, it could be necessary to instead employ
an alternative reference that may offer less in the way of state-specific
orbital relaxation.

Turning to a comparison with EOM-CCSD, ASCC produces a very slightly lower
MUE, but, given the statistical spreads, it’s more correct to say that the two
methods showed about the same overall accuracy in these one-CSF QUEST tests.
The more salient difference appears to be that EOM-CCSD typically errors
slightly high, while ASCC appears to error high and low in roughly equal
amounts. Given EOM-CCSD’s well earned reputation for accuracy in small
molecule single excitations, it is encouraging to see comparable accuracy from
ASCC in such tests. This observation in hand, we now turn to charge transfer,
where we would expect to see larger differences between state-specific and
linear response methods.

###  III.3 Charge Transfer Tests

To investigate ASCC in a context where post-excitation orbital relaxations are
expected to be more important, we applied it to a set of 22 different charge
transfer excitations, which consist of both intramolecular and intermolecular
excitations characterized by significant dipole shifts. Although we are
currently limited to studying relatively small systems by the pilot nature of
our ASCC implementation, these states already offer some clarity about the
method’s advantages in a CT context. TOP orbital plots for each of these
excitations are depicted in Figure 1(a), and excitation energy errors relative
to $\delta$-CR-EOM-CC(2,3),A, Piecuch et al. (2006); Piecuch and Włoch (2005);
Piecuch et al. (2015, 2002); Kowalski and Piecuch (2004); Piecuch et al.
(2009); Fradelos et al. (2011) an $N^{7}$ scaling, size intensivity corrected,
perturbative triples EOM-CC method, are shown in Figure 1(b). Note that we
elect to use the “A” variant of $\delta$-CR-EOM-CC(2,3) over others due to its
better-centered mean unsigned error. Piecuch et al. (2015)

(a) TOPs for particle and hole orbitals. Hole orbitals are in the grey
background while particle orbitals are in the white. All possible combinations
of hole and particle orbitals are considered. To save space, hole and particle
orbitals are shown separately on chloride intermolecular systems.

(b) Excitation energy errors in eV relative to $\delta$-CR-EOM-CC(2,3),A.
Figure 1: Charge transfer orbitals (a), results (b), and summary statistics.

Summary Statistics for Charge Transfer States
---
| ESMP2 | ASCC | EOM-CCSD
MSEa $\pm$ Std. Dev. | -0.09 $\pm$ 0.18 | 0.07 $\pm$ 0.10 | 0.26 $\pm$ 0.13
MUEb $\pm$ Std. Dev. |  -0.15 $\pm$ 0.14 | 0.07 $\pm$ 0.10 | 0.26 $\pm$ 0.13
Max Error | 0.56 | 0.37 | 0.64

aMean signed error (MSE). bMean unsigend error (MUE).

####  III.3.1 Intramolecular Charge Transfer

Starting with lithium fluoride (LiF), we examine transfers into its lithium 2s
orbital (p1) from the fluorine off-axis (h1) and on-axis (h2) p orbitals. In
both of these excitations, ASCC and EOM-CCSD produce excitation energies in
close alignment to the reference values, while ESMP2 produces somewhat larger
errors. While EOM-CCSD often errs high for charge transfer excitations, in
lithium fluoride, much like for hydrogen chloride in the QUEST set, the charge
transfer distance is rather short and there are not too many electrons to
correlate. Evidently, EOM-CCSD’s linear response based approximations work
well in this context.

In chloroethene (CH3Cl), we examined transfers from both the out-of-plane lone
pair (h1) and the in-plane lone pair (h2) to the $\pi^{*}$ orbital (p1). In
these excitations, EOM-CCSD produces $\sim$0.25 eV errors, which are larger
than any of EOM-CCSD’s errors in states considered up till now. For the h1p1
excitation, which has the smaller of the two dipole shifts, ASCC errors a bit
higher than EOM-CCSD, while ESMP2 performs comparably to the reference method.
However, for the h2p1 excitation, whose dipole shift and CT character are more
pronounced, ASCC improves on EOM-CCSD with an error slightly smaller than that
of ESMP2. We thus start to see what will become a pattern: ASCC’s performance
relative to EOM-CCSD is better in states with more pronounced CT character.

This pattern continues in acrolein (C3H4O), where we examine transfers into a
$\pi^{*}$ orbital (p1) from an oxygen lone pair (h1) and from a $\pi$ orbital
(h2). The h1p1 excitation has the larger dipole shift and more significant
charge transfer character, with the h1 oxygen lone pair delocalizing over the
$\pi$ system. ASCC’s excitation energy error for h1p1 is less than half the
size of EOM-CCSD’s, with ESMP2’s error intermediate between the two. The h2p1
state has a smaller dipole shift and less charge transfer character,
essentially shifting the electron within the $\pi$ system. All three methods
produce comparable and sizable errors for this state, and although it is not
entirely clear why this is, it does continue the CT accuracy pattern in which
ASCC has an advantage when CT is more pronounced.

####  III.3.2 Intermolecular Charge Transfer

Turning now to states with even stronger CT character, we find that ASCC has a
clear accuracy advantage over EOM-CCSD in intermolecular CT. We begin with a
transfer from ammonia to difluorine in which the molecules are separated by 6
Å. The lowest-lying CT excitation in this system is from the nitrogen lone
pair (h1) to difluorine’s $\sigma^{*}$ orbital (p1). EOM-CCSD’s excitation
energy error of 0.6 eV is its largest error across all states tested in this
study. ESMP2 also produces a large error ($\sim$0.5 eV), though it errors low
while EOM-CCSD errors high. ASCC, in contrast, produces an excitation energy
in close agreement with the reference method.

As seen in Figure 1(b)(b), this pattern is repeated throughout our
intermolecular CT tests: EOM-CCSD errors high by 0.2 eV or more, ESMP2 errors
low by up to 0.3 eV, and ASCC stays consistently within 0.1 eV (and usually
0.05 eV) of the reference. The lack of full orbital relaxation in EOM-CCSD
likely explains most of the difference, although we suspect that its neutral-
ground-state-based correlation treatment of the ionic CT state also plays a
role. Aside from NH3 — F2, our intermolecular charge transfer tests excite an
electron from a chloride anion to a nearby $\pi^{*}$ orbital of dinitrogen (N2
— Cl-), carbon monoxide (CO — Cl-), or ethylene (C2H4 — Cl-) at a separation
distance of 4 Å. Some of these examples are artificial in that, in a complete
basis, they would only exist as resonances,Schulz (1973) but in the finite
basis we are using they still offer meaningful tests of how closely methods
come to matching the energies of the $\delta$-CR-EOM-CC(2,3),A CT states. The
relatively large polarizability of the chlorine atom and the transfer of a
full electron across a significant distance make orbital relaxation effects
important in these states. EOM-CCSD can only approximate these relaxations
through its $\hat{R}_{2}$ operator, and its errors, which are significantly
larger than in the one-CSF QUEST tests of Section III.2, suggest that this
approximation is reaching its limit. In contrast, ASCC benefits from explicit
orbital relaxations both from the ESMF reference and its own $\hat{T}_{1}$
operator as well as from a correlation treatment optimized in the presence of
the relaxed orbitals.

###  III.4 Two-CSF QUEST Tests

In this initial study of ASCC, we explore its performance in two-CSF states
via a minimal extension of the one-CSF approach in order to determine whether
further extensions are warranted. In this minimal extension, we update the
$\hat{S}$ operator to contain the four individual single excitation operators
(two alpha and two beta) that create the two-CSF state, with their
coefficients within $\hat{S}$ set by the ESMF wave function. We extend
$\hat{T}$ by adding the $O(1)$ subset of triples and the single quadruple that
flesh out a (4,4) active space encompassing the two-CSF reference, and by
extending the definition of a primary orbital for the purposes of constructing
$\hat{T}_{3^{\prime}}$ to include both CSFs’ hole and particle orbitals. We
then set the initial amplitudes within $\hat{T}_{\mathrm{init}}$ so that the
initial ASCC wave function is equal to the two-CSF truncated ESMF state.

Optimizing this minimally extended two-CSF ASCC ansatz, we find that further
extensions will be needed in order for ASCC’s systematic improvability to
produce accurate results in multi-CSF excited states. Table 3 reveals errors
on the order of 1 eV, which stands in stark contrast to the more accurate
results from ESMP2 and EOM-CCSD. Alongside these large energy errors, we see
that some of the initially nonzero amplitudes from $\hat{T}_{\mathrm{init}}$ —
which are important for creating the reference $\ket{\psi_{0}}$ from the
Aufbau determinant $\ket{\phi_{0}}$ — change substantially during
optimization. To give a sense of scale, we can look at N2, where, during
optimization, the sum of squares of the primary CSFs’ coefficients in the CI
expansion of the wave function changed from 1.0 to 0.897 and 0.843 for the
two-CSF $1^{1}\Sigma^{-}_{u}$ and $1^{1}\Delta_{u}$ states, respectively,
which contrasts sharply with the change from 1.0 to 0.9995 seen in the one-CSF
$1^{1}\Pi_{g}$ state, which was typical of the other one-CSF states as well.
Given that the idea of a reference is that further changes should be minor,
this implies that either the reference was poor to begin with or that
artifacts arising from truncating $\hat{T}$ have led the optimization to make
erroneously large modifications to these amplitudes. With ESMP2 using the same
reference and performing much better than this minimal extension of one-CSF
ASCC, we think that the reference is not the issue. Instead, we suspect that
this minimal extension must be missing components of the full $\hat{T}$
operator that were more important than we had guessed.

Table 3: Excitation Energies in eV for two-CSF statesa from QUEST benchmark.

Molecule | State | ESMP2 | ASCC | Frozen ASCC | EOM-CCSD |  Referenceb
---|---|---|---|---|---|---
dinitrogen | $1^{1}\Sigma_{u}^{-}$ | $9.78$ | $10.90$ | $9.54$ | $10.20$ | $10.05$
| $1^{1}\Delta_{u}$ | $10.32$ | $11.66$ | $10.34$ | $10.61$ | $10.43$
carbon monoxide | $1^{1}\Sigma^{-}$ | $9.92$ | $11.12$ | $9.89$ | $10.10$ | $10.05$
| $1^{1}\Delta$ | $10.17$ | $11.37$ | $10.08$ | $10.21$ | $10.16$
acetylene | $1^{1}\Sigma_{u}^{-}$ | $6.83$ | $7.99$ | $6.91$ | $7.27$ | $7.20$
| $1^{1}\Delta_{u}$ | $7.18$ | $8.52$ | $7.43$ | $7.57$ | $7.51$
formamide | $3^{1}\text{A}_{p}$ | $6.66$ | $9.38$ | $8.06$ | $7.72$ | $7.64$

Multi CSF Statistics |
---|---
|  | ESMP2 | ASCC | Frozen ASCC | EOM-CCSD |
MSEc $\pm$ Std. Dev. |  |  -0.31 $\pm$ 0.32 |  1.13 $\pm$ 0.32 |  -0.11 $\pm$ 0.28 |  0.09 $\pm$ 0.05 |
MUEd $\pm$ Std. Dev. |  |  -0.31 $\pm$ 0.32 |  1.13 $\pm$ 0.32 |  -0.23 $\pm$ 0.18 |  0.09 $\pm$ 0.05 |
Max Error |  | 0.98 | 1.74 | 0.51 | 0.18 |

a States where two ESMF singular values are $>$0.2. b Reference from QUEST
benchmarkLoos et al. (2018) exFCI results unless stated otherwise. cMean
signed error (MSE). dMean unsigned error (MUE).

What might these missing amplitudes be? One possibility can be spotted by
noting that our minimal extension has created an ansatz whose nonlinear terms
will, if we expand the wave function out into a CI basis, produce some
strangely large quadruply excited determinants. For example, labeling our two
primary excitations by the hole and particle indices $h_{1}$, $h_{2}$,
$p_{1}$, and $p_{2}$, the $O(1)$-sized singles amplitude $t_{h_{1}}^{p_{1}}$
can combine in a quadratic term with the
$t_{\bar{h}_{1}h_{2}k}^{\bar{p}_{1}b~{}c}$ triples amplitude to produce a
quadruply excited determinant with a coefficient roughly as large as that of a
CCSD doubles amplitude (because this triple is doubly excited relative to our
reference). There will be $O(ov^{2})$ of these concerningly large quadruples,
and each of them contributes to the amplitude equation for
$t_{h_{1}\bar{h}_{1}}^{p_{1}\bar{p}_{1}}$, which plays an important role in
setting up the reference. Were we to explicitly include the corresponding
$O(ov^{2})$ set of explicit quadruples within $\hat{T}$, this effect would
presumably be mitigated. Although we do not think that this further extension
would change the theory’s asymptotic scaling, we leave a more complete study
of how different extensions of ASCC’s cluster operator would improve accuracy
in two- and multi-CSF states for future work.

Here, we attempt to answer the simpler question of whether accuracy is
improved by preventing the optimization of the weak correlation part of the
cluster operator from modifying the reference. To do so, we break the
optimization into two stages. First, we only optimize the amplitudes within
the (4,4) active space, starting from the ESMF-based guess. We then freeze
those amplitudes and optimize the rest so as to prevent the reference from
being influenced by the spurious terms discussed above. As seen in Table 3,
this “Frozen ASCC” approach reduces errors considerably, although they are
still larger than in the single-CSF states. These results suggest that, if the
inclusion of small slices of higher excitations within $\hat{T}$ prevents
spurious alterations to the reference, accuracy should be improved without
having to invoke a somewhat arbitrary freezing.

Thanks to ASCC’s systematic improvability, we expect that adding key higher
amplitudes will ultimately lead to even greater accuracy improvements.
Further, since the number of added amplitudes should be small, we anticipate
that they will not affect asymptotic scaling. Finally, we note that adding
these amplitudes will not create mismatches between the number of amplitudes
and equations, as one can simply add the corresponding projections as well.
This contrasts with the need for sufficiency conditions in JM methods. In the
one-CSF regime, such conditions can be avoided via two-determinant CC’s spin-
locking approach, Balková and Bartlett (1992); Lutz et al. (2018) but it is
not obvious how JM methods would avoid them in two-CSF states.

##  IV Conclusion

We have presented a new coupled cluster framework in which a de-excitation
operator is used to efficiently suppress the the Aufbau determinant within the
wave function expansion. This approach allows extended single-reference
methods to treat states in which the Aufbau determinant is small or absent
without resorting to large amplitude values and the problems they create.
Thanks to its fully exponential form, the approach is systematically
improvable, size consistent, size extensive, and, for excited states, achieves
a granular size intensivity in which the equations for electrons far from the
excitation simplify into the ground state equations. In initial testing on
singly excited states, we find that a particularly simple CCSD-like version of
the theory works well in single-CSF and especially charge transfer states, but
that a more sophisticated version will be needed for two-CSF states.

Looking forward, there are many exciting avenues to explore in Aufbau
suppressed coupled cluster. An obvious direction is to determine which
additional excitations must be included to achieve accurate results for multi-
CSF excited states. Is it only the small slice of quadruples that we flagged
above, or will other amplitudes be important as well? One could ask a similar
question for doubly excited states, which, if one chooses the MO basis the
right way, can usually be made to look like linear combinations of Aufbau,
singles, and doubles. In that case, as well as in some strongly correlated
ground states, it will be worth considering whether modifications to the de-
excitation operator would be helpful in addition to considering the inclusion
of higher excitation operators. Other directions might seek to exploit the way
that the theory’s granular size intensivity gives it a strong formal
relationship with ground state coupled cluster. This connection could be
helpful in adding local correlation treatments and an analogue of the ground
state’s “perturbative triples” correction, since one can intuit that, apart
from the amplitudes directly involved in the excitation, these additions can
be expected to take a similar form as in the ground state. In summary, we are
excited to explore the possibilities created by Aufbau suppression.

##  V Acknowledgements

This work was supported by the National Science Foundation’s CAREER program
under Award Number 1848012. Calculations were performed using the Savio
computational cluster resource provided by the Berkeley Research Computing
program at the University of California, Berkeley and the Lawrencium
computational cluster resource provided by the IT Division at the Lawrence
Berkeley National Laboratory. 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.

##  VI References

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##  VII Supporting Information

##  S1 Charge Transfer Excitation Energies

Table S1: Vertical excitation energies in eV for charge transfer systems Molecule | State | ESMP2 | ASCC | EOM-CCSD |  $\delta$-CR-EOM-CC(2,3),A
---|---|---|---|---|---
lithium fluoride | h1p1 | $6.60$ | $6.39$ | $6.36$ | $6.36$
| h2p1 | $7.01$ | $6.84$ | $6.82$ | $6.82$
chloroethylene | h1p1 | $7.01$ | $7.31$ | $7.22$ | $6.98$
| h2p1 | $7.51$ | $7.81$ | $7.92$ | $7.67$
acrolein | h1p1 | $3.78$ | $3.81$ | $3.92$ | $3.62$
| h2p1 | $6.16$ | $6.86$ | $6.88$ | $6.49$
ammonia $\rightarrow$ difluorine | h1p1 | $7.62$ | $8.21$ | $8.82$ | $8.18$
chloride $\rightarrow$ dinitrogen | h1p2 | $5.88$ | $5.97$ | $6.18$ | $5.90$
| h1p1 | $5.68$ | $5.87$ | $6.19$ | $5.86$
| h3p1 | $5.78$ | $5.97$ | $6.29$ | $5.98$
| h2p2 | $5.93$ | $6.05$ | $6.30$ | $5.98$
| h2p1 | $5.72$ | $5.98$ | $6.30$ | $5.98$
| h3p2 | $5.91$ | $6.03$ | $6.30$ | $5.98$
chloride $\rightarrow$ carbon monoxide | h1p1 | $5.16$ | $5.20$ | $5.39$ | $5.18$
| h1p2 | $5.22$ | $5.27$ | $5.44$ | $5.23$
| h3p1 | $5.24$ | $5.27$ | $5.46$ | $5.26$
| h2p1 | $5.19$ | $5.27$ | $5.47$ | $5.27$
| h2p2 | $5.32$ | $5.34$ | $5.52$ | $5.32$
| h3p2 | $5.30$ | $5.35$ | $5.53$ | $5.32$
chloride $\rightarrow$ ethylene | h1p1 | $5.08$ | $5.33$ | $5.55$ | $5.33$
| h2p1 | $5.25$ | $5.48$ | $5.69$ | $5.44$
| h3p1 | $5.16$ | $5.48$ | $5.71$ | $5.46$

##  S2 Molecular Geometries

The supporting information for the QUEST benchmark contains geometries for the
molecules from the QUEST set. The remaining charge transfer geometries are
reported below in angstroms.

lithium fluoride

    F   0.0000000    0.0000000    0.3968040
    Li  0.0000000    0.0000000   -1.1904110

chloroethylene

    C   0.0000000    0.7588320    0.0000000
    C   1.2921560    1.0327720    0.0000000
    Cl -0.6259060   -0.8573720    0.0000000
    H  -0.7694050    1.5061990    0.0000000
    H   2.0399880    0.2605710    0.0000000
    H   1.6168930    2.0589350    0.0000000

acrolein

    C  -0.1533010   -0.7559140    0.0000000
    C   0.0000000    0.7250820    0.0000000
    C   1.2220650    1.2941010    0.0000000
    O  -1.2237110   -1.3289000    0.0000000
    H   0.8134290   -1.3220750    0.0000000
    H  -0.9232470    1.3153200    0.0000000
    H   1.3568600    2.3802520    0.0000000
    H   2.1300570    0.6780920    0.0000000

ammonia $\rightarrow$ difluorine

    N   0.0000000    0.0000000    0.1277920
    H   0.0000000    0.9318900   -0.2981820
    H   0.8070400   -0.4659450   -0.2981820
    H  -0.8070400   -0.4659450   -0.2981820
    F   0.0000000    0.0000000    6.1277920
    F   0.0000000    0.0000000    7.5597920

chloride $\rightarrow$ dinitrogen

    N   0.0000000    0.0000000   -0.5560000
    N   0.0000000    0.0000000    0.5560000
    Cl  0.0000000    4.0000000    0.0000000

chloride $\rightarrow$ carbon monoxide

    C   0.0000000    0.0000000   -0.5690000
    O   0.0000000    0.0000000    0.5690000
    Cl  0.0000000    4.0000000    0.0000000

chloride $\rightarrow$ ethylene

    C   0.0000000    0.0000000    0.6726230
    C   0.0000000    0.0000000   -0.6726230
    H   0.9341310    0.0000000    1.2459530
    H  -0.9341310    0.0000000    1.2459530
    H  -0.9341310    0.0000000   -1.2459530
    H   0.9341310    0.0000000   -1.2459530
    Cl  0.0000000    4.0000000    0.0000000