# Time-dependent orbital-optimized coupled-cluster methods families for
fermion-mixtures dynamics

Haifeng Lang<EMAIL_ADDRESS>Department of Nuclear Engineering and
Management, Graduate School of Engineering, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-8656, Japan  Takeshi Sato
<EMAIL_ADDRESS>Department of Nuclear Engineering and Management,
Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-
ku, Tokyo 113-8656, Japan  Photon Science Center, Graduate School of
Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656,
Japan  Research Institute for Photon Science and Laser Technology, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 Japan

###### Abstract

Five time-dependent orbital optimized coupled-cluster (TD-ooCC) methods, of
which four can converge to the complete active space self-consistent-field
method, are presented for fermion-mixtures with arbitrary fermion kinds and
numbers. Truncation schemes maintaining the intragroup orbital rotation
invariance, as well as equations of motion of CC amplitudes and orbitals, are
derived. Present methods are compact CC-parameterization alternatives to the
time-dependent multiconfiguration self-consistent-field method for systems
consisting of arbitrarily different kinds and numbers of interacting fermions.
Theoretical analysis of applications of present methods to various chemical
systems are reported.

##  I introduction

Simulating dynamics of fermions and fermion mixtures quantitatively is
essential for understanding and predicting strong-field and ultrafast
phenomenaProtopapas, Keitel, and Knight (1997); Agostini and DiMauro (2004);
Krausz and Ivanov (2009); Gallmann, Cirelli, and Keller (2013); Nisoli  _et
al._ (2017) of molecular systems (Noticing that Hilbert spaces of vibrational
systems and photons are isomorphic to fermion mixtures with one occupation in
each kindMordovina  _et al._ (2020); Christiansen (2004); Hansen  _et al._
(2019); Christiansen (2004); Sverdrup Ofstad  _et al._ (2023)). However,
solving the time-dependent Schrödinger equation (TDSE) exactly is impractical
for even moderate size systems due to the factorial growth of computational
costs. Commonly used benchmark methods to describe such dynamics are the
multiconfiguration time-dependent Hartree-Fock (MCTDHF) method,Zanghellini
_et al._ (2003); Kato and Kono (2004); Caillat  _et al._ (2005); Nest,
Klamroth, and Saalfrank (2005); Kato and Kono (2008); Hochstuhl and Bonitz
(2011); Haxton, Lawler, and McCurdy (2012); Haxton and McCurdy (2014) as well
as its extensions for fermion-mixturesLode  _et al._ (2020); Alon, Streltsov,
and Cederbaum (2007). In these methods, the wavefunction is expanded as a full
configuration-interaction (CI) form inside an active space, of which single
particle orbitals are time-dependent. Equations of motion (EoMs) of parameters
are determined by time-dependent variational principle (TDVP). Due to the
compactness of the ansatz, the numbers of optimized time-dependent orbitals
are usually much less than the numbers of grid points in TDSE, and thus
reducing the computational cost hugely. However, the full CI expansion makes
methods still suffer from the factorial growth of the computational cost with
respect to the number of fermions. Therefore, the methods can be only applied
to moderate size systems. The more general time-dependent multiconfiguration
self-consistent-field (TD-MCSCF) methodNguyen-Dang  _et al._ (2007); Ishikawa
and Sato (2015); Anzaki, Sato, and Ishikawa (2017) further restricts
configurations in the wavefunction expansion, which bridges the gap bewteen
time-dependent Hartree-Fock and MCTDHF. One of the attractive TD-MCSCF
methods, the time-dependent complete-active-space self-consistent-field (TD-
CASSCF) method Sato and Ishikawa (2013); Li, Sato, and Ishikawa (2021),
significantly reduces the cost by assigning some occupied orbitals as cores
(which must be occupied in all configurations within the expansion), resulting
in a computational cost that scales factorially with respect to the number of
fermions in the active orbitals rather than the total number of fermions. It
is also possible to restrict configurations in the active space by the
excitation levels with respect to the reference state, for instance,
MCTDH[n]Madsen  _et al._ (2020a) and multireference MCTDH[n]Madsen  _et al._
(2020b) for the vibrational dynamics, and time-dependent restricted active
space self-consistent field (TD-RASSCF)Miyagi and Madsen (2013, 2014); Haxton
and McCurdy (2015) and time-dependent occupation-restricted multiple active
space (TD-ORMAS)Sato and Ishikawa (2015) for the electron dynamics, to achieve
the polynomial cost scalingHaxton and McCurdy (2015); Sato  _et al._ (2016);
Sawada, Sato, and Ishikawa (2016); Omiste, Li, and Madsen (2017) with a
drawback of not being size extensiveHelgaker, Jørgensen, and Olsen (2002).

Recently, the polynomial scaling and size-extensive CC parameterizationSzabo
and Ostlund (1996); Helgaker, Jørgensen, and Olsen (2002); Kümmel (2003);
Shavitt and Bartlett (2009) in the time-dependent active space gains lots of
popularity. In the CC parameterization, the ket vector is described by an
exponential truncated excitation operator acting on the reference state, while
the bra vector is not the Hermitian conjugate of the ket vector. For this
reason, the time-dependent bivariational principleArponen (1983) and
biorthogonal orbitals are also widely used in additional to the TDVP and
orthogonal orbitals. Many methods have been developed and successively
implemented, e.g., time-dependent optimized coupled-cluster method (TD-
OCC)Sato  _et al._ (2018); Pathak, Sato, and Ishikawa (2020a, b, 2021) and
orbital-adapted time-dependent coupled-clusterKvaal (2012) (OATDCC) method for
the electron dynamics, and time-dependent vibrational coupled cluster with
time-dependent modals (TDMVCC) familiesMadsen  _et al._ (2020c); Højlund,
Zoccante, and Christiansen (2024); Højlund and Christiansen (2024); Sverdrup
Ofstad  _et al._ (2023); Højlund  _et al._ (2022); Jensen  _et al._ (2023) for
the vibrational dynamics. These methods can be considered as the time-
dependent extensions of stationary OCCG. E. Scuseria and H. F. Schaefer III
(1987); Sherrill  _et al._ (1998); Krylov  _et al._ (1998); Köhn and Olsen
(2005) and nonorthogonal orbital optimized coupled cluster (NOCC)Pedersen,
Fernández, and Koch (2001).

Despite the fruitful results of CC parameterization of the active space
wavefunction, it is still less explored than the CI parameterization. First,
previous works mainly focus on either electron dynamics (one kind fermion) or
vibrational dynamics (fermion mixtures but only one occupation in each fermion
kind). A general theory of arbitrary occupations of fermion mixtures is still
unclear. Second, numerically stable and convergentKöhn and Olsen (2005); Myhre
(2018) (in the sense of converging to the CASSCF/TD-CASSCF) CC
parameterizations using orthogonal orbitals, for instance, the standard
CCHuber and Klamroth (2011); D. R. Nascimento and A. E. DePrince III (2016)
and Brueckner coupled cluster (BCC)Raghavachari  _et al._ (1990); Hampel,
Peterson, and Werner (1992); Chiles and Dykstra (1981); Handy  _et al._ (1989)
types of parameterization in the active space, are unknown. In this paper, we
advance in these directions by presenting five time-dependent orbital-
optimized coupled-cluster (TD-ooCC) methods for a system composed of
arbitrarily different kinds and numbers of interacting fermions (with particle
conservation).

This paper is organized as follows. In the theory section Sec. II, we first
analyze the convergence condition of time-dependent CC with time varying
orthogonal orbitals to the TD-CASSCF. Further, the selections of both single
CC (de)exication operators ($\hat{T}_{1}$ and $\hat{\Lambda}_{1}$, defined in
Eq. (3,4)), which determines the type of the method, and non-single CC
(de)exication operators ($\hat{T}_{0}$, $\hat{T}_{1}$, $\cdots$,
$\hat{\Lambda}_{0}$ $\hat{\Lambda}_{2}$, $\cdots$, defined in Eq. (3,4)) ,
which can reduce the computational resources potentially, in the wavefunction
ansatz of TD-ooCC methods are introduced. The equations of motion (EoMs) for
CC amplitudes and orbitals are presented. Finally, we discuss their advantages
and potential drawbacks. In the application section Sec. III, we discuss three
major applications of our methods to the chemical systems, electronic
dynamics, vibrational dynamics, and non-adiabatic dynamics. Sec. IV summarizes
our theories. The Hartree atomic units are used throughout unless otherwise
noted.

##  II Theory

###  II.1 Full coupled cluster expansion in the complete active space

Following the CASSCF method, general orthogonal time-dependent orbitals
$\mu^{m},\nu^{m},\cdots$ of fermion kind $m$ are classified into virtual
orbitals $\alpha^{m},\beta^{m},\cdots$, and occupied orbitals
$\bar{p}^{m},\bar{q}^{m},\cdots$, which can be further classified into frozen-
cores $i^{\prime\prime m},j^{\prime\prime m}\cdots$, dynamical-cores
$i^{\prime m},j^{\prime m}$, and active orbitals $t^{m},u^{m},\cdots$ that
contain hole orbitals $i^{m},j^{m},\cdots$, and particle orbitals
$a^{m},b^{m},\cdots$. Here, the hole orbitals and particle orbitals refer to
active orbitals that are occupied and unoccupied, respectively, in the
reference state. EoMs of orbitals can be formally expressed as

$i|\dot{\psi}_{\nu^{m}}\rangle=i|\psi_{\mu^{m}}\rangle
X^{\mu^{m}}_{\nu^{m}}\,,$ (1)

where $X^{\mu^{m}}_{\nu^{m}}=\braket{\psi_{\mu^{m}}}{\dot{\psi}_{\nu^{m}}}$ is
anti-Hermitian to enforce the orthonormality of orbitals
$\langle\psi_{\mu^{m}}(t)|\psi_{\nu^{m}}(t)\rangle=\delta^{\mu^{m}}_{\nu^{m}}$
at any time, and thus called the constraints of orbital rotations. There are
two types of constraints of orbitals: the first type includes pre-determined
constraints that are intra-group rotation constraints and frozen-core related
orbital constraintsCon ; the second type includes others that are variational
parameters which should be determined by the time-dependent variational
principle. In this article, we will use the Einstein notation except for
fermion kind index $m,n,\cdots$. For the convenience, we use the notations
$p^{m},q^{m},\cdots$ to label time-dependent occupied orbitals except for the
frozen-cores due to the fact that frozen-cores are pre-determineded by the
gauge choiceSato  _et al._ (2016). Additionally, $\hat{c}_{\mu^{m}}^{\dagger}$
and $\hat{c}_{\mu^{m}}$ are defined as the creation and annihilation operators
for $\psi_{\mu^{m}}$ (as well as other orbital labels).

The most general ansatz of the CC wave function with time-dependent orbital
can be expressed as

$\ket{\Psi_{\rm R}}=e^{\hat{T}}\ket{\Phi}\,,\quad\bra{\Psi_{\rm
L}}=\bra{\Phi}\hat{\Lambda}e^{-\hat{T}}\,,$ (2)

where $\ket{\Phi}$ and $\bra{\Phi}$ are the reference states, i.e., tensor
product of Slater determinants of all fermion kinds
$\ket{\Phi}=\prod_{m}\prod_{i^{\prime\prime m}i^{\prime
m}i^{m}}\hat{c}^{\dagger}_{i^{\prime\prime m}}\hat{c}^{\dagger}_{i^{\prime
m}}\hat{c}^{\dagger}_{i^{m}}\ket{0}$, and

$\displaystyle\hat{T}$
$\displaystyle=\hat{T}_{0}+\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{3}+\cdots$ (3)
$\displaystyle=\tau_{0}\hat{E}_{0}+\sum_{m}\tau_{i^{m}}^{a^{m}}\hat{E}_{i^{m}}^{a^{m}}+\sum_{m_{1}m_{2}}\tau_{i^{m_{1}}j^{m_{2}}}^{a^{m_{1}}b^{m_{2}}}\hat{E}_{i^{m_{1}}j^{m_{2}}}^{a^{m_{1}}b^{m_{2}}}$
$\displaystyle+\sum_{m_{1}m_{2}m_{3}}\tau_{i^{m_{1}}j^{m_{2}}k^{m_{3}}}^{a^{m_{1}}b^{m_{2}}c^{m_{3}}}\hat{E}_{i^{m_{1}}j^{m_{2}}k^{m_{3}}}^{a^{m_{1}}b^{m_{2}}c^{m_{3}}}+\cdots$
$\displaystyle=\tau^{\mathring{A}}_{\mathring{I}}\hat{E}^{\mathring{A}}_{\mathring{I}}=\sum_{m}\tau_{i^{m}}^{a^{m}}\hat{E}_{i^{m}}^{a^{m}}+\tau^{{A}}_{{I}}\hat{E}^{{A}}_{{I}}\,,$
$\displaystyle\hat{\Lambda}$
$\displaystyle=\hat{\Lambda}_{0}+\hat{\Lambda}_{1}+\hat{\Lambda}_{2}+\hat{\Lambda}_{3}+\cdots$
(4)
$\displaystyle=\lambda_{0}\hat{E}_{0}+\sum_{m}\lambda^{i^{m}}_{a^{m}}\hat{E}^{i^{m}}_{a^{m}}+\sum_{m_{1}m_{2}}\lambda^{i^{m_{1}}j^{m_{2}}}_{a^{m_{1}}b^{m_{2}}}\hat{E}^{i^{m_{1}}j^{m_{2}}}_{a^{m_{1}}b^{m_{2}}}$
$\displaystyle+\sum_{m_{1}m_{2}m_{3}}\lambda^{i^{m_{1}}j^{m_{2}}k^{m_{3}}}_{a^{m_{1}}b^{m_{2}}c^{m_{3}}}\hat{E}^{i^{m_{1}}j^{m_{2}}k^{m_{3}}}_{a^{m_{1}}b^{m_{2}}c^{m_{3}}}+\cdots$
$\displaystyle=\lambda_{\mathring{A}}^{\mathring{I}}\hat{E}_{\mathring{A}}^{\mathring{I}}=\sum_{m}\lambda^{i^{m}}_{a^{m}}\hat{E}^{i^{m}}_{a^{m}}+\lambda_{{A}}^{{I}}\hat{E}_{{A}}^{{I}}\,,$

Here, $\mathring{I}=ijk\cdots$ and $\mathring{A}=abc\cdots$ (can be empty)
denote hole and particle orbital strings to make the notation brevity. The
general-rank excitation (deexcitation) operators and amplitudes are
$\hat{E}^{\mathring{A}}_{\mathring{I}}$
($\hat{E}_{\mathring{A}}^{\mathring{I}}$) and
$\tau^{\mathring{A}}_{\mathring{I}}$
($\lambda_{\mathring{A}}^{\mathring{I}}$), where
${\hat{E}}^{\mu_{1}^{m_{1}}\mu_{2}^{m_{2}}\mu_{3}^{m_{3}}\cdots}_{\nu_{1}^{m_{1}}\nu_{2}^{m_{2}}\nu_{3}^{m_{3}}\cdots}=\hat{c}^{\dagger}_{\mu_{1}^{m_{1}}}\hat{c}^{\dagger}_{\mu_{2}^{m_{2}}}\hat{c}^{\dagger}_{\mu_{3}^{m_{3}}}\cdots{\hat{c}}_{\nu_{3}^{m_{3}}}{\hat{c}}_{\nu_{2}^{m_{2}}}{\hat{c}}_{\nu_{1}^{m_{1}}}$,
and $\hat{E}_{0}=\hat{I}$ for the zero-rank excitation (empty strings of
$\mathring{I}$ and $\mathring{A}$). We will also use the notation $I$ and $A$
to label non-single hole and particle orbital strings (the orbital number of
the strings is not one) in this article.

Before working on time-dependent variational principles, we stress that single
amplitudes terms $\hat{T}_{1}$ and $\hat{\Lambda}_{1}$ are included in the
ansatz due to the convenience of the theoretical analysis although they can
cause the numerical instability in practical simulations and are omitted in
previous workSato  _et al._ (2018); Köhn and Olsen (2005). Additionally, zero-
rank amplitudes $\tau_{0}$ and $\lambda_{0}$ are also explicitly included in
the ansatz to recover the CASSCF wavefucntion in the full CC expansion limit.
The normalization condition makes $\lambda_{0}\equiv 1$ conserved, and
$\tau_{0}$ has no impacts on observables but essential for the
autocorrelations.

Following the previous workPedersen, Koch, and Hättig (1999); Kvaal (2012);
Helgaker, Jørgensen, and Olsen (2002); Sato  _et al._ (2018), we begin with
the coupled-cluster Lagrangian $L$ and real action functional $S$,

$\displaystyle L$ $\displaystyle=\langle\Psi_{\rm
L}|(\hat{H}-i{\partial_{t}})|\Psi_{\rm R}\rangle\,,$ (5) $\displaystyle S$
$\displaystyle=\Re\int_{t_{0}}^{t_{1}}Ldt=\frac{1}{2}\int_{t_{0}}^{t_{1}}\left(L+L^{*}\right)dt\,.$
(6)

Here, the Hamiltonian $\hat{H}$ can be an arbitrary Hamiltonian with particle
conservations. EoMs of parameters are given by the stationary condition of the
action with respect to the variation of all parameters of left- and right-
state blackwavefunctions (with necessary constraints),

black

$\displaystyle{\color{black}2}\delta S$ $\displaystyle=$
$\displaystyle\delta\tau^{\mathring{A}}_{\mathring{I}}\left\\{\langle\Psi_{\rm
L}|[\hat{H}-i\hat{X},\hat{E}^{\mathring{A}}_{\mathring{I}}]|\Psi_{\rm
R}\rangle+i\dot{\lambda}_{\mathring{A}}^{\mathring{I}}\right\\}+\delta\lambda_{\mathring{A}}^{\mathring{I}}\left\\{\langle\Phi^{\mathring{A}}_{\mathring{I}}|e^{-\hat{T}}(\hat{H}-i\hat{X})|\Psi_{\rm
R}\rangle-i\dot{\tau}^{\mathring{A}}_{\mathring{I}}\right\\}+c.c$ (7)
$\displaystyle+$
$\displaystyle\sum_{m}\Delta^{\mu^{m}}_{\nu^{m}}\left\\{\langle\Psi_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\hat{T})-i\hat{X},\hat{E}^{\mu^{m}}_{\nu^{m}}]|\Psi_{\rm
R}\rangle+i\langle\Phi|(\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda})e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle\right.$ $\displaystyle\hskip 13.00005pt\left.-\langle\Psi_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\hat{T})-i\hat{X},\hat{E}_{\mu^{m}}^{\nu^{m}}]|\Psi_{\rm
R}\rangle^{*}+i\langle\Phi|(\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda})e^{-\hat{T}}\hat{E}_{\mu^{m}}^{\nu^{m}}|\Psi_{\rm
R}\rangle^{*}\right\\}\,.$

where $\partial^{\hskip
0.20999pt\prime}_{t}\hat{T}=\dot{\tau}^{\mathring{A}}_{\mathring{I}}\hat{E}^{\mathring{A}}_{\mathring{I}}$,
$\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda}=\dot{\lambda}_{\mathring{A}}^{\mathring{I}}\hat{E}_{\mathring{A}}^{\mathring{I}}$,
$\hat{X}=\sum_{m}X^{\mu^{m}}_{\nu^{m}}\hat{E}^{\mu^{m}}_{\nu^{m}}$,
$\hat{\Delta}=\sum_{m}\Delta^{\mu^{m}}_{\nu^{m}}\hat{E}^{\mu^{m}}_{\nu^{m}}$,
and
$\Delta^{\mu^{m}}_{\nu^{m}}=\braket{\psi_{\mu^{m}}}{\delta{\psi}_{\nu^{m}}}$.
Similar to the matrices $X^{\mu^{m}}_{\nu^{m}}$, $\Delta^{\mu^{m}}_{\nu^{m}}$
are also anti-Hermitian to enforce the orthonormality of orbitals. Meanwhile,
$\Delta^{\mu^{m}}_{i^{\prime\prime m}}\equiv 0$ due to the fact that EoMs of
frozen-cores are not determined by variational principle. Additionally, we
only consider the ansatz that satisfies the intragroup rotation invariance
including dynamical-cores-dynamical-cores, hole-hole, particle-particle, and
virtual-virtual rotation invariance, i.e., the wavefunction is invariant up to
a phase factor with these orbital rotations and the corresponding inverse
rotations of CC amplitudes. These invariances give the redundancy of
$X^{i^{\prime m}}_{j^{\prime m}}$, $X_{i^{m}}^{j^{m}}$, $X_{a^{m}}^{b^{m}}$,
and $X_{\alpha^{m}}^{\beta^{m}}$. For simplicity, we always set all of them as
zero. The details of the wavefunction ansatz that maintain the hole-hole,
particle-particle, and virtual-virtual rotation invariance will be discussed
in the Sec. II.2.2 and B.

Variations with respect to CC amplitudes, $\delta
S/\delta\lambda_{\mathring{A}}^{\mathring{I}}=0$ and $\delta
S/\delta\tau^{\mathring{A}}_{\mathring{I}}=0$ give

$\displaystyle i\dot{\tau}^{A}_{I}$
$\displaystyle=\langle\Phi^{{A}}_{{I}}|e^{-\hat{T}}(\hat{H}-i\hat{X})|\Psi_{\rm
R}\rangle\,,$ (8) $\displaystyle i\dot{\tau}_{i^{m}}^{a^{m}}$
$\displaystyle=\bra{\Phi_{i^{m}}^{a^{m}}}e^{-\hat{T}}(\hat{H}-i\hat{X})\ket{\Psi_{\rm
R}}\,,$ (9) $\displaystyle-i\dot{\lambda}^{I}_{A}$
$\displaystyle=\langle\Psi_{\rm
L}|e^{-\hat{T}}[\hat{H}-i\hat{X},\hat{E}^{A}_{I}]|\Psi_{\rm R}\rangle\,,$ (10)
$\displaystyle-i\dot{\lambda}^{i^{m}}_{a^{m}}$ $\displaystyle=\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X},\hat{E}_{i^{m}}^{a^{m}}]\ket{\Psi_{\rm R}}\,.$ (11)

Here, we separate EoMs of CC amplitudes into non-single amplitudes part and
single amplitudes part. The variation with respect to $\Delta_{a^{m}}^{i^{m}}$
gives

$\displaystyle\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X}-i(\partial_{t}^{\prime}\hat{T}),\hat{E}^{i^{m}}_{a^{m}}]\ket{\Psi_{\rm
R}}$ (12) $\displaystyle+$ $\displaystyle
i\bra{\Phi}(\partial_{t}^{\prime}\hat{\Lambda})e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}|\Psi_{\rm
R}\rangle$ $\displaystyle-$ $\displaystyle\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X}-i(\partial_{t}^{\prime}\hat{T}),\hat{E}_{i^{m}}^{a^{m}}]\ket{\Psi_{\rm
R}}^{*}$ $\displaystyle+$ $\displaystyle
i\bra{\Phi}(\partial_{t}^{\prime}\hat{\Lambda})e^{-\hat{T}}\hat{E}_{i^{m}}^{a^{m}}|\Psi_{\rm
R}\rangle^{*}=0\,,$

Substituting Eq. (8,10) into Eq. (12) yields,

$\displaystyle
i(\delta_{b^{m}}^{a^{m}}D_{i^{m}}^{j^{m}}-D_{b^{m}}^{a^{m}}\delta_{i^{m}}^{j^{m}})(\dot{\tau}_{j^{m}}^{b^{m}}+X_{j^{m}}^{b^{m}})+i\frac{1}{2}(\dot{\lambda}^{i^{m}}_{a^{m}})^{*}=$
(13) $\displaystyle
B_{i^{m}}^{a^{m}}-i\frac{1}{2}\sum_{n}\left\\{\langle\Phi_{j^{n}}^{b^{n}}|e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}|\Psi_{\rm
R}\rangle\dot{\lambda}^{j^{n}}_{b^{n}}+A^{a^{m}j^{n}}_{i^{m}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\right\\}\,,$

where

$\displaystyle A^{a^{m}j^{n}}_{i^{m}b^{n}}$ $\displaystyle=\bra{\Psi_{\rm
L}}[\hat{E}^{A}_{I},\hat{E}^{j^{n}}_{b^{n}}]\ket{\Psi_{\rm
R}}\langle\Phi^{A}_{I}|e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}\ket{\Psi_{\rm R}}$
$\displaystyle-\bra{\Psi_{\rm
L}}[\hat{E}^{A}_{I},\hat{E}^{i^{m}}_{a^{m}}]\ket{\Psi_{\rm
R}}\langle\Phi_{I}^{A}|e^{-\hat{T}}\hat{E}^{j^{n}}_{b^{n}}\ket{\Psi_{\rm R}},$
(14) $\displaystyle B_{i^{m}}^{a^{m}}$
$\displaystyle=-\frac{1}{2}\braket{\Psi_{\rm
L}}{[\hat{H},\hat{E}^{i^{m}}_{a^{m}}]}{\Psi_{\rm
R}}-\frac{1}{2}\braket{\Psi_{\rm
R}}{[\hat{H},\hat{E}^{i^{m}}_{a^{m}}]}{\Psi_{\rm L}}$
$\displaystyle-\frac{1}{2}\bra{\Psi_{\rm
L}}[\hat{E}^{A}_{I},\hat{H}]\ket{\Psi_{\rm
R}}\langle\Phi^{A}_{I}|e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}\ket{\Psi_{\rm R}}$
$\displaystyle+\frac{1}{2}\bra{\Psi_{\rm
L}}[\hat{E}^{A}_{I},\hat{E}^{i^{m}}_{a^{m}}]\ket{\Psi_{\rm
R}}\langle\Phi_{I}^{A}|e^{-\hat{T}}\hat{H}\ket{\Psi_{\rm R}}\,.$ (15)

Solving coupled equations Eq. (9,11,13), EoMs of single amplitudes and
particle-hole orbital rotations $\dot{\tau}_{i^{m}}^{a^{m}}$,
$\dot{\lambda}^{i^{m}}_{a^{m}}$, and $X_{i^{m}}^{a^{m}}$ can be obtained.
Substituting $X_{i^{m}}^{a^{m}}$ into Eq. (8,10), EoMs of non-single
amplitudes can be solved.

In the full CC expansion limit, the particle-hole orbital rotations
$X_{i^{m}}^{a^{m}}$ are redundant, and can be freely selected. For an explicit
proof, see the Appendix. B. The simplest and the most common choice is to set
them as zero. However, coupling equations Eq. (9,11,13) suggest that the
redundancy of $X_{i^{m}}^{a^{m}}$ can be transferred to
$\lambda^{i^{m}}_{a^{m}}$ or $\tau_{i^{m}}^{a^{m}}$. For instance, one can set
$\lambda_{a^{m}}^{i^{m}}\equiv 0$ as redundant parameters, and this type of
wavefunction parameterization is known as the BCCRaghavachari  _et al._
(1990); Hampel, Peterson, and Werner (1992); Chiles and Dykstra (1981); Handy
_et al._ (1989). In fact, one can arbitrarily select one of
$X_{i^{m}}^{a^{m}}$, $\tau_{i^{m}}^{a^{m}}$ and ${\lambda}^{i^{m}}_{a^{m}}$ as
redundant parameter for each kind of fermion, and set them as zero. In the
Appendix. B, more rigorous arguments of transferring redundancy and
preparation of wavefunctions are presented.

###  II.2 Wavefunction Ansatz of Time dependent orbital-optimized coupled-
cluster families

####  II.2.1 Selection of single amplitudes

In principle, the redundancy caused by including all $X_{i^{m}}^{a^{m}}$,
$\tau_{i^{m}}^{a^{m}}$, and $\lambda^{i^{m}}_{a^{m}}$ only occurs at the full
CC expansion limit. Nonetheless, our experience shows that including them all
is numerically instable even when only double (rank two) excitations are
considered, which suggests the compactness of CC wavefunction
parameterization. One has to exclude (at least) one of $X_{i^{m}}^{a^{m}}$,
$\tau_{i^{m}}^{a^{m}}$ and $\lambda^{i^{m}}_{a^{m}}$ in practical simulations.

The first method we will consider is TD-OCC (terminology adapted from the
referenceSato  _et al._ (2018)), of which the ansatz only includes
$X_{i^{m}}^{a^{m}}$. This method fails to converge to the CASSCF limitKöhn and
Olsen (2005); Myhre (2018); Højlund, Zoccante, and Christiansen (2024), but
shows an acceptable accuracy in ultrafast electron dynamicsSato  _et al._
(2018) and vibrational systemsHøjlund, Zoccante, and Christiansen (2024)
according to previous reports. Specifically, TD-OCC with double and triple
excitations (TD-OCCDT) is in nearly perfect agreement with the CASSCF results
for benchmarks of ultrafast electron dynamicsSato  _et al._ (2018). We will
also present three methods with the absence of one of $X_{i^{m}}^{a^{m}}$,
$\tau_{i^{m}}^{a^{m}}$, and $\lambda^{i^{m}}_{a^{m}}$ in the ansatz, thus all
three methods can converge to the CASSCF limit. The details of four methods
are summarized as:

(a). TD-OCC: only $X_{i^{m}}^{a^{m}}$ are included in the ansatz, or
equivalently
$\tau_{i^{m}}^{a^{m}}=\lambda^{i^{m}}_{a^{m}}=\delta\tau_{i^{m}}^{a^{m}}=\delta\lambda^{i^{m}}_{a^{m}}\equiv
0$.

(b). TD-OCCT1: $\tau_{i^{m}}^{a^{m}}$ and $X_{i^{m}}^{a^{m}}$ are included in
the ansatz, or equivalently
$\lambda^{i^{m}}_{a^{m}}=\delta\lambda^{i^{m}}_{a^{m}}\equiv 0$.

(c). TD-BCC: $X_{i^{m}}^{a^{m}}$ and $\lambda^{i^{m}}_{a^{m}}$ are included in
the ansatz, or equivalently
$\tau_{i^{m}}^{a^{m}}=\delta\tau_{i^{m}}^{a^{m}}\equiv 0$.

(d). TD-OCCX0: $\tau_{i^{m}}^{a^{m}}$ and $\lambda^{i^{m}}_{a^{m}}$ are
included in the ansatz, or equivalently
$X_{i^{m}}^{a^{m}}=\Delta_{i^{m}}^{a^{m}}\equiv 0$.

Here, the latter definition of each method can be regarded as the constraints
in the sense that some parameters are pre-determined (to be zero). We stress
that the absence of particle-hole orbital rotations $X_{i^{m}}^{a^{m}}$ of TD-
OCCX0 is only justified for the dynamics without external fields. In the
presence of external fields $X_{i^{m}}^{a^{m}}$ has to be assigned different
values to ensure the gauge invariance, akin to the treatment of frozen-
coresSato  _et al._ (2016). Additionally, we will consider the most general
hybrid form of wavefunction parameterization, i.e., the parameter selections
(a), (b), (c), or (d) is separately, and flexibly adopted for each fermion
kinds, which is named as TD-OCCH and will be discussed with all the details in
the Appendix.D.

####  II.2.2 Selection of non-single excitations

Inspired by TD-MCSCF, one can restrict the non-single excitations in TD-ooCC
to make the ansatz more compact. To ensure the intragroup rotation invariance
of particle and hole orbitals, one can select non-single excitations that
satisfy the property

$\displaystyle\hat{E}^{\mu^{m}}_{\nu^{m}}\ket{\Psi_{\rm
R}}=e^{\hat{T}}\hat{\Pi}e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}\ket{\Psi_{\rm
R}}\,,$ (16) $\displaystyle\bra{\Psi_{\rm
L}}\hat{E}^{\mu^{m}}_{\nu^{m}}=\bra{\Psi_{\rm
L}}\hat{E}^{\mu^{m}}_{\nu^{m}}e^{\hat{T}}\hat{\Pi}e^{-\hat{T}}\,,$

where
$\hat{\Pi}=\ket{\Phi_{\mathring{I}}^{\mathring{A}}}\bra{\Phi_{\mathring{I}}^{\mathring{A}}}$.
Here $\mu^{m}$ and $\nu^{m}$ should be particle or hole orbital index
simultaneously. For the details of the proof, see the Appendix. B. One of the
simplest selection scheme is the $[k]$-truncation scheme, i.e., non-single
excitation is included in the ansatz if and only if it is not larger than the
certain truncation order $k$, which has been successfully implemented in TD-
OCC.

It is also possible to assign different truncation orders for different kinds
to adapt their contribution to the dynamics. Here, we propose two possible
schemes. The first, the weighted excitation truncation scheme, is the modified
version of the truncation scheme reported in Ref. [Madsen  _et al._ , 2020d].
Define the nonnegative excitation weight and the excitation number of each
kind as $w^{m}$ and $n_{\rm ext}^{m}$, respectively. The generalized
excitation level of a given excitation amplitude is

$w=\sum_{m}n_{\rm ext}^{m}w^{m}\,,$ (17)

and all non-single excitation amplitude that satisfy $w\leq k$ should be
included in the ansatz, where $k$ is the maximum generalized excitation level.
Here, different spin fermion configurations can be regarded as different
“fermion kinds” for both unrestricted and restricted spin orbitals if the each
spin component is conserved. This truncation scheme can reduce to the scheme
in Ref. [Madsen  _et al._ , 2020d] in the vibrational system, in which $n_{\rm
ext}^{m}$ can only be 1 or 0. One can set the weights of important kinds as
relatively small numbers, even zero. When all $w^{m}=1$, the weighted
excitation truncation scheme reduces to the $[k]$-truncation scheme.

The second one, group excitation truncation scheme, is the following: (1) all
kinds of fermions can be divided into several groups A,B,C,$\cdots$, and their
truncation order of excitations can be assigned as $n^{A}_{A}$, $n^{B}_{B}$,
$n^{C}_{C}$,$\cdots$. Here, $n=1$ if only zero excitations (and single
excitations when included) are included in the ansatz. All possible non-single
excitations beyond the truncation order must be excluded while all others must
be included in the ansatz. The different spin components of a fermion can also
be regarded as different kinds if they are conserved; (2) for each type of
excitations across groups, truncation order of the total excitation and all
single groups should be assigned, for instance,
$(n_{AB},n_{AB}^{A},n_{AB}^{B})$ should be assigned for the excitation
acrossing groups $A$ and $B$, and
$(n_{ABC},n_{ABC}^{A},n_{ABC}^{B},n_{ABC}^{C})$ should be assigned for the
excitation acrossing groups $A$, $B$ and $C$. Iff both the total excitation
number and the excitation number of each group are not larger than the
corresponding truncation parameters, it is included in the ansatz; (3) $n$
must satisfy the inequalities: for all $X$, $n^{X}_{Y}\leq n^{X}_{Z}$ if the
group numbers of $Y$ is larger than $Z$. We will list two explicit examples on
the electron dynamics. We set the $\alpha$ spin and $\beta$ spin are two
different groups. The first example is
$n_{\alpha}^{\alpha}=n_{\beta}^{\beta}=n_{\alpha\beta}^{\alpha}=n_{\alpha\beta}^{\beta}=2$
and $n_{\alpha\beta}=3$, which means all double excitations of $\alpha$ and
$\beta$ spins are included, and all $2\alpha 1\beta$ and $2\beta 1\alpha$ type
triple excitations are included. The second is
$n_{\alpha}^{\alpha}=n_{\beta}^{\beta}=n_{\alpha\beta}^{\alpha}=n_{\alpha\beta}^{\beta}=1$,
and $n_{\alpha\beta}=2$, which means only $1\alpha 1\beta$ excitations are
included. For a slightly more complicated example, see Fig. (1).

Figure 1:  Illustration of the Truncation of non-single CC amplitudes (cores
and virtual orbitals are neglected). In the example of this figure, three
kinds fermions ($m_{1}$, $m_{2}$, and $m_{3}$) are classified into two groups:
Group A containing $m_{1}$ and $m_{2}$, and Group B containing $m_{3}$. We set
$n_{A}^{A}=n_{B}^{B}=n_{AB}^{B}=2$, $n_{AB}=3$, and $n_{AB}^{A}=1$. Fig. (1a)
is the reference configuration, where dashed line is the fermion surface.
Figs. (1b,c) show two of allowed CC excitations across the group, where blue
rectangles circle the excitation kinds. Fig. (1d) shows the forbidden
excitations, where red rectangles circle the excitation kinds. In the Fig.
(1d), although the total excitation and excitation of group B satisfy the
conditions, $3=n_{AB}$ and $1<n_{AB}^{B}$, the excitation is still forbidden
due to the condition violation of the group A excitation $2>n_{AB}^{A}$.

###  II.3 Equation of motions of Time dependent optimized coupled cluster
families

EoMs of single amplitudes and particle-hole orbital rotations of five methods
can be obtained by setting corresponding constraints on Eq. (9,11,13), which
will be presented in the corresponding subsubsections and the Appendix. EoMs
of other variables are the same for all five methods [Eq. (8,10) ]. EoMs of
intergroup rotations $X^{\alpha^{m}}_{p^{m}}$ and $X^{t^{m}}_{i^{\prime m}}$
are obtained from the variation with respect to $\Delta_{\alpha^{m}}^{p^{m}}$
and $\Delta_{t^{m}}^{i^{\prime m}}$,

$\displaystyle\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X},\hat{E}_{\alpha^{m}}^{p^{m}}]\ket{\Psi_{\rm
R}}+\bra{\Psi_{\rm
R}}[\hat{H}-i\hat{X},\hat{E}_{\alpha^{m}}^{p^{m}}]\ket{\Psi_{\rm L}}=0\,,$
(18) $\displaystyle\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X},\hat{E}_{t^{m}}^{i^{\prime m}}]\ket{\Psi_{\rm
R}}+\bra{\Psi_{\rm R}}[\hat{H}-i\hat{X},\hat{E}_{t^{m}}^{i^{\prime
m}}]\ket{\Psi_{\rm L}}=0\,,$

which can be further simplified as

$\displaystyle 2iX^{u^{m}}_{i^{\prime m}}$
$\displaystyle(2\delta_{u^{m}}^{t^{m}}-D_{u^{m}}^{t^{m}})=$
$\displaystyle\braket{\Psi_{\rm L}}{\hat{E}_{t^{m}}^{i^{\prime
m}}\hat{H}}{\Psi_{\rm R}}+\braket{\Psi_{\rm R}}{\hat{E}_{t^{m}}^{i^{\prime
m}}\hat{H}}{\Psi_{\rm L}}\,,$ (19) $\displaystyle 2iX^{\alpha^{m}}_{q^{m}}$
$\displaystyle D^{q^{m}}_{p^{m}}=\braket{\Psi_{\rm
L}}{\hat{E}_{\alpha^{m}}^{p^{m}}\hat{H}}{\Psi_{\rm R}}+\braket{\Psi_{\rm
R}}{\hat{E}_{\alpha^{m}}^{p^{m}}\hat{H}}{\Psi_{\rm L}}\,,$ (20)

with

$\displaystyle\rho_{q^{m}}^{p^{m}}$ $\displaystyle=\bra{\Psi_{\rm
L}}\hat{E}^{q^{m}}_{p^{m}}\ket{\Psi_{\rm R}}\,,\hskip
5.0ptD_{q^{m}}^{p^{m}}=\frac{1}{2}(\rho_{q^{m}}^{p^{m}}+(\rho^{q^{m}}_{p^{m}})^{*})$
(21)

In practical simulations, virtual orbitals are never propagated. Instead, one
can convertBeck  _et al._ (2000); Anzaki, Sato, and Ishikawa (2017) the
propagation of occupied orbitals arisen from the virtual-occupied orbital
rotations to

$\displaystyle
i\braket{\psi_{\bar{\mu}^{m}}}{\psi_{\alpha^{m}}}X^{\alpha^{m}}_{p^{m}}=(D^{-1})_{q^{m}}^{p^{m}}$
(22) $\displaystyle\times$ $\displaystyle\Big{\\{}\braket{\Psi_{\rm
L}}{\hat{E}_{\bar{\mu}^{m}}^{q^{m}}\hat{H}}{\Psi_{\rm R}}-\braket{\Psi_{\rm
L}}{\hat{E}_{\bar{o}^{m}}^{q^{m}}\hat{H}}{\Psi_{\rm
R}}\braket{\psi_{\bar{\mu}^{m}}}{\psi_{\bar{o}^{m}}}$
$\displaystyle+\braket{\Psi_{\rm
R}}{\hat{E}_{\bar{\mu}^{m}}^{q^{m}}\hat{H}}{\Psi_{\rm L}}-\braket{\Psi_{\rm
R}}{\hat{E}_{\bar{o}^{m}}^{q^{m}}\hat{H}}{\Psi_{\rm
L}}\braket{\psi_{\bar{\mu}^{m}}}{\psi_{\bar{o}^{m}}}\Big{\\}}\,,$

where $\bar{\mu}^{m}$ is the primary time-independent basis to expand orbitals
that can be discrete variables representationBeck  _et al._ (2000), grid
pointsLi, Sato, and Ishikawa (2021), etc.

####  II.3.1 TD-OCC

Setting $\tau_{i^{m}}^{a^{m}}={\lambda}^{i^{m}}_{a^{m}}\equiv 0$ in Eq. (13),

$i(\delta_{b^{m}}^{a^{m}}D_{i^{m}}^{j^{m}}-D_{b^{m}}^{a^{m}}\delta_{i^{m}}^{j^{m}})X_{j^{m}}^{b^{m}}+i\sum_{n}\frac{1}{2}A^{a^{m}j^{n}}_{i^{m}b^{n}}(X^{b^{n}}_{j^{n}})^{*}=B_{i^{m}}^{a^{m}}\,.$
(23)

In the case of one occupation of each kind of fermion and a single kind of
fermion, the method reduces to oTDMVCCHøjlund, Zoccante, and Christiansen
(2024) for vibrational systems and TD-OCCSato  _et al._ (2018) for electrons,
respectively.

####  II.3.2 TD-OCCT1

Setting ${\lambda}^{i^{m}}_{a^{m}}\equiv 0$ in Eq. (11,13),

$i\left(\rho_{j^{m}}^{i^{m}}\delta_{b^{m}}^{a^{m}}-\rho_{a^{m}}^{b^{m}}\delta_{j^{m}}^{i^{m}}\right)X^{j^{m}}_{b^{m}}=\bra{\Psi_{\rm
L}}[\hat{H},\hat{E}_{i^{m}}^{a^{m}}]\ket{\Psi_{\rm R}}\,,$ (24) $\displaystyle
i(\delta_{b^{m}}^{a^{m}}D_{i^{m}}^{j^{m}}-D_{b^{m}}^{a^{m}}\delta_{i^{m}}^{j^{m}})\dot{\tau}_{j^{m}}^{b^{m}}=$
(25) $\displaystyle
B_{i^{m}}^{a^{m}}-i(\delta_{b^{m}}^{a^{m}}D_{i^{m}}^{j^{m}}-D_{b^{m}}^{a^{m}}\delta_{i^{m}}^{j^{m}})X_{j^{m}}^{b^{m}}-i\sum_{n}\frac{1}{2}A^{a^{m}j^{n}}_{i^{m}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\,.$

Interestingly, TD-OCCT1 is mathematically equivalent to spilt OATDCC (sOATDCC,
or equivalently, sTDMVCC)Højlund and Christiansen (2024), which uses
orthogonal virtual orbitals and bi-orthogonal active orbitals without single
CC amplitudes and is reported in the vibrational systems and electron dynamics
recently. The biorthogonal orbitlas in sOATDCC have the same role as
$\hat{T}_{1}$ in the TD-OCCT1, however, they will give different
$X_{i^{m}}^{j^{m}}$ and $X_{a^{m}}^{b^{m}}$ constraints, which might give
different numerical performances. For more details, see Appendix. C. When
virtual orbitals do not exist, TD-OCCT1 and sOATDCC are equivalent to
TDMVCCMadsen  _et al._ (2020c) and OATDCCKvaal (2012) (NOCC)Pedersen,
Fernández, and Koch (2001) for vibrational systems and electron dynamics,
respectively.

####  II.3.3 TD-BCC

Setting ${\tau}^{a^{m}}_{i^{m}}\equiv 0$ in Eq. (9,13),

$\displaystyle iX_{i^{m}}^{a^{m}}+\sum_{n}i$
$\displaystyle\langle\Phi_{i^{m}}^{a^{m}}|e^{-\hat{T}}\hat{E}^{j^{n}}_{b^{n}}|\Psi_{\rm
R}\rangle
X^{j^{n}}_{b^{n}}=\langle\Phi_{i^{m}}^{a^{m}}|e^{-\hat{T}}\hat{H}|\Psi_{\rm
R}\rangle\,,$ (26) $\displaystyle
i\frac{1}{2}\left\\{\sum_{n}\langle\Phi_{j^{n}}^{b^{n}}|e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}|\Psi_{\rm
R}\rangle\dot{\lambda}^{j^{n}}_{b^{n}}+(\dot{\lambda}^{i^{m}}_{a^{m}})^{*}\right\\}=$
$\displaystyle
B_{i^{m}}^{a^{m}}-i(\delta_{b^{n}}^{a^{m}}D_{i^{m}}^{j^{n}}-D_{b^{n}}^{a^{m}}\delta_{i^{m}}^{j^{n}})X_{j^{n}}^{b^{n}}-i\sum_{n}\frac{1}{2}A^{a^{m}j^{n}}_{i^{m}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\,.$
(27)

####  II.3.4 TD-OCCX0

TD-OCCX0 can be considered as the most straightforward extension of TD-CC when
virtual orbitals are considered, due to the similarity of the parameterization
of the CC amplitudes parts and the absence of particle-hole orbital rotations.
The EoMs of the single amplitudes of TD-OCCX0 are given by Eq. (9,11). The
nonvariance of particle-hole orbital rotations makes the method not fully
variational. Therefore, the Ehrenfest theorem should be modifiedSato  _et al._
(2016) even when frozen-cores are excluded,

$\displaystyle 2i\frac{d}{dt}\braket{\hat{O}}$
$\displaystyle=\braket{\Psi_{\rm L}}{[\hat{O},\hat{H}]}{\Psi_{\rm
R}}+\braket{\Psi_{\rm R}}{[\hat{O},\hat{H}]}{\Psi_{\rm L}}$ (28)
$\displaystyle+\sum_{m}O_{t^{m}}^{u^{m}}\\{\braket{\Psi_{\rm
L}}{(\hat{H}-i\hat{X})e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}\hat{E}_{t^{m}}^{u^{m}}}{\Psi_{\rm
R}}$ $\displaystyle-\braket{\Psi_{\rm
L}}{\hat{E}_{t^{m}}^{u^{m}}e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
R}}$ $\displaystyle+\braket{\Psi_{\rm
R}}{(\hat{H}-i\hat{X})e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}\hat{E}_{t^{m}}^{u^{m}}}{\Psi_{\rm
L}}$ $\displaystyle-\braket{\Psi_{\rm
R}}{\hat{E}_{t^{m}}^{u^{m}}e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
L}}\\}\,,$

where $\bar{\hat{\Pi}}=\hat{I}-\hat{\Pi}$. Similarly, TD-OCCH is not fully
variational either when TD-OCCX0-type parameterization is considered, and its
modified Ehrenfest theorem (as well as frozen-cores including scenario) will
be presented in the Appendix. D. As a comparison, Ehrenfest theorem holds for
other three methods since they are fully variational.

When frozen orbitals are included, the modified Ehrenfest theoremsSato  _et
al._ (2016) are necessary for all five methods: two additional terms that
replace $t^{m}$, $u^{m}$ with $i^{\prime\prime m}$, $\mu^{m}$ and $\mu^{m}$,
$i^{\prime\prime m}$ in the last term of Eq. (28) should be added in case of
TD-OCCX0, and replace $t^{m}$, $u^{m}$ with $i^{\prime\prime m}$, $\mu^{m}$
and $\mu^{m}$, $i^{\prime\prime m}$ in the last term of Eq. (28) in case of
TD-BCC, TD-OCC, and TD-OCCT1. The modified Ehrenfest theorem of TD-OCCH with
frozen-cores will be reported in the Appendix. D.

###  II.4 Discussion

One of the major problems of TD-CC is that the growth of CC amplitudes could
cause numerical instabilityPedersen and Kvaal (2019); Kristiansen  _et al._
(2020), which can be mitigated by rotations of reference determinants that
could compensate for excitations and reduce the amplitude norms $\tau$ and
$\lambda$ in all five methods considered in this article. However, one must be
careful when using TD-OCCX0 since such a compensation might not be huge when
virtual orbital subspaces are relatively small due to the absence of particle-
hole orbital rotations.

Although the absence of particle-hole orbital rotations in TD-OCCX0 poses
inconveniences for observable evaluations and numerical stability, it does
save computing resources, especially for vibrational systems. Inverse matrix
operations required to solve $X_{i^{m}}^{a^{m}}$ in other methods are
unnecessary for TD-OCCX0. Comparable methods in this regard are TD-OCCT1, in
which $X_{i^{m}}^{a^{m}}$ [Eq. (24)] and $\tau_{i^{m}}^{a^{m}}$ [Eq. (25)] of
different modes do not couple with each other, and TD-OCCH with few fermion
kinds using BCC and OCC types parameterization. For the details, see the
Appendix. D. In both two methods, only several small-dimensional matrix
inverse operations are required, which is much cheaper than TD-OCC [Eq. (23)]
and TD-BCC [Eq. (26,II.3.3)] cases, requiring huge-dimensional matrix inverse
operations. However, when strong external fields exist, the advantage of TD-
OCCX0 and TD-OCCH can be dismissed due to the unfavorable resource cost of the
evaluations of position matrix elementsSato  _et al._ (2016); Sato, Teramura,
and Ishikawa (2018), which will be discussed in Sec. III.1 and Appendix. D.

##  III Applications

In this section, we will discuss three major applications of our methods that
cover most of chemical systems: electron dynamics, nuclei dynamics, and
nuclei-electron coupling dynamics.

###  III.1 Electron dynamics in a strong laser field

For a many-electron system in external semi-classical electromagnetic fields,
the Hamiltonian in the second quantization can be expressed as

$\displaystyle\quad\quad\quad\quad\hat{H}(t)=h^{\mu}_{\nu}(t)\hat{E}^{\mu}_{\nu}+\frac{1}{2}g^{\mu\gamma}_{\nu\delta}\hat{E}^{\mu\gamma}_{\nu\delta},$
(29) $\displaystyle\quad\quad\quad h^{\mu}_{\nu}(t)=\int
dx\psi^{*}_{\mu}[h_{0}+V_{\rm ext}(t)]\psi_{\nu},$ (30) $\displaystyle
g^{\mu\gamma}_{\nu\delta}=\int\int
dx_{1}dx_{2}\psi^{*}_{\mu}(x_{1})\psi^{*}_{\gamma}(x_{2})r^{-1}_{12}\psi_{\nu}(x_{1})\psi_{\delta}(x_{2}),$
(31)

where $x\\!=\\!(\boldsymbol{r},\sigma)$ labels spatial-spin coordinate. Here,
we drop off the kind lable for the simplicity (although one can treat
different spin component as different kinds and use group excitation
truncation scheme to save the computational resources). $V_{\rm ext}$ is the
electron-laser interaction term. $h_{0}$ is the one-electron field free
Hamiltonian that contains the kinetic term and Coulomb’s terms induced by
fixed nuclei. Under the framework of the electric dipole approximation,
$V_{\rm ext}=\boldsymbol{E}(t)\cdot\boldsymbol{r}$ in the length gauge (LG) or
$V_{\rm ext}=-i\boldsymbol{A}(t)\cdot\boldsymbol{\nabla}_{\boldsymbol{r}}$ in
the velocity gauge (VG), with $\boldsymbol{E}(t)$ and
$\boldsymbol{A}(t)=-\int^{t}dt^{\prime}\boldsymbol{E}(t^{\prime})$ being the
electric field and vector potential, respectively. In principle, two gauges
are equivalent in the basis-set limit (basis numbers tend to the infinity),
but inequivalent in practical numerical simulations due to different
convergence speeds of basis-sets. Empirically, VG is more suitable for ultra
strong fields scenario while LG is more suitable for other casesHan and Madsen
(2010).

When methods are fully variational, the gauge invariance of variational
methods are satisfied automatically. Otherwise, the non-variance part of
orbital rotations should be adjusted to ensure the gauge invariance. For the
methods considered in this article, FC-part of $\hat{X}$ need to be selected
as

$X_{\mu}^{i^{\prime\prime}}=i\boldsymbol{E}(t)\cdot\braket{\psi_{i^{\prime\prime}}}{\boldsymbol{r}}{\psi_{\mu}}\,,$
(32)

and the particle-hole orbital rotations of TD-OCCX0 should be

$X_{i}^{a}=i\boldsymbol{E}(t)\cdot\braket{\psi_{a}}{\boldsymbol{r}}{\psi_{i}}\,.$
(33)

in the VGSato  _et al._ (2016). These adjustments are also necessary in the
many-kind fermion systems, and the kind label should be explicitly included.
Specifically, Eq. (33) should be applied in all kinds with TD-OCCX0 type
parameterization in TD-OCCH method, see also Appendix. D.

###  III.2 Vibrational systems

The application of methods to vibrational problems is straightforward. A
vibrational system with $m$ vibrational modes is isomorphic to a $m$-specie
fermion system with only one fermion occupation in each specie. We point out
that quantum photons, which are essential for molecules in cavities, can also
be described in the same framework.

Previous works of TDVCC familiesHøjlund, Zoccante, and Christiansen (2024);
Højlund and Christiansen (2024); Højlund  _et al._ (2022); Hansen  _et al._
(2019); Madsen  _et al._ (2020c) focus mainly on the $[k]$-truncation scheme,
which is sufficient for ordinary systems. Nonetheless, for the systems with
double well potentials and vibronic coupling systems (see next subsection),
weighted excitation truncation scheme or group excitation truncation scheme is
necessary for the faithful simulations since the degree of freedom (DOF) with
double well potentials has more contribution than other DOFs to the dynamics.
A practical strategy of weighted excitation truncation scheme for such a
system is to set the DOF with double well potentials as 0 and other DOFs as 1.
Currently, special truncation schemes have been successfully implemented in
TDVCCMadsen  _et al._ (2020d) and TDMVCCJensen  _et al._ (2023); TDH , but
lacking theoretical discussion on orbital rotation invariance. The proof of
the intragroup rotations of particle and hole orbitals in the Appendix. B also
holds for the biorthogonal orbitals scenario. Therefore, we show that weighted
excitation truncation scheme can be safely used in TDVCC families without
compromising the intragroup rotations of particle and hole orbitals, which
benefits future numerical implementations. Despite existing TDVCC families and
weighted excitation truncation schemes, our work provides new ansatz, TD-
OCCT1, TD-BCC, TD-OCCX0, and TD-OCCH, and the group excitation truncation
scheme as new candidates for future implementations. Additionally, although
TD-OCCT1 and sTDMVCC are mathematically equivalent, different constraints
$X_{i^{m}}^{j^{m}}$ and $X_{a^{m}}^{b^{m}}$, as well as using orthogonal or
nonorthogonal orbitals, might give difference performances in practical
implementations. Therefore, they should be benchmarked systematically.

###  III.3 Vibronic coupling systems

There are two ways to simulate vibronic coupling systems. Traditionally, the
electronic DOF is described as a $S$-level system, thus a vibronic coupling
system is isomorphic to a vibrational system. The major differences are: 1.
almost always all $S$ orbitals of the electronic DOF are active in the
wavefunction ansatz (no virtual orbitals); 2. vibrational contributions to the
dynamics associated with different electronic levels are comparableWorth and
Cederbaum (2004); Köuppel, Domcke, and Cederbaum (1984); Bao, Raymond, and
Nooijen (2024). These make TD-OCCX0 and TD-OCC unsuitable for the vibronic
coupling simulations, and one should also avoid using OCCX0 and OCC types
parameterization for the electronic DOF in TD-OCCH. Additionally, the same
weighted excitation truncation scheme as systems with the double well
potentials (proposed in the last subsection) should be used.

In recent years, treating electronic and nuclear DOF on an equal footing has
attracted increasing attention due to advancements in theorySibaev  _et al._
(2020); Muolo  _et al._ (2020); Sasmal and Vendrell (2020, 2022); Sasmal,
Schröder, and Vendrell (2024). The five CC ansatz and two special truncation
schemes proposed in this article can be regarded as the prototype of future
advanced CC simulations.

##  IV Summary

We have successfully formulated five new orbital-optimized time-dependent
coupled-cluster methods, which are distinguished by whether single amplitudes,
$\tau_{i^{m}}^{a^{m}}$, $\lambda_{a^{m}}^{i^{m}}$ and particle-hole orbital
rotations $X_{i^{m}}^{a^{m}}$ are constraints, for arbitrary fermion mixtures.
When one of $\tau_{i^{m}}^{a^{m}}$, $\lambda_{a^{m}}^{i^{m}}$, and
$X_{i^{m}}^{a^{m}}$ is a constraint for each fermion kind $m$, the method can
converge to the CASSCF. Two advanced truncation schemes of higher-order
amplitudes which maintain the intragroup rotation invariance are introduced.
The applications to electronic dynamics, vibrational dynamics, and non-
adiabatic dynamics are also discussed. Our methods are more compact CC-
parameterization alternatives of the CI-parameterization of TD-MCSCF method,
and would shed light on the high-accuracy numerical simulations of large
systems. The numerical implementations on the electronic dynamics will be
presented in a forthcoming article.

###### Acknowledgements.

This research was supported in part by a Grant-in-Aid for Scientific Research
Grant Number JP22H05025 from the Ministry of Education, Culture, Sports,
Science and Technology (MEXT) of Japan. This research was also partially
supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number
JPMXS0118067246 and JST COI-NEXT Grant Number JPMJPF2221. This work is partly
supported by IBM-UTokyo lab.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Haifeng Lang: Conceptualization (equal); Methodology (lead); Investigation
(lead); Writing – original draft (lead); Writing – review & editing (equal).
Takeshi Sato: Conceptualization (equal); Methodology (supporting); Funding
acquisition (lead); Writing –review & editing (equal).

## DATA AVAILABILITY

The data that supports the findings of this study are available within the
article.

##  Appendix A Redundancy transfer of $X_{i^{m}}^{a^{m}}$,
$\tau_{i^{m}}^{a^{m}}$ and ${\lambda}^{i^{m}}_{a^{m}}$ in the full CC limit

In this section, we will demonstrate that the redundancy of
$X_{i^{m}}^{a^{m}}$ can be transferred to $\lambda^{i^{m}}_{a^{m}}$ or
$\tau_{i^{m}}^{a^{m}}$, and the preparation of the exact wavefunction without
$\lambda^{i^{m}}_{a^{m}}$ or $\tau_{i^{m}}^{a^{m}}$ is possible in the full CC
limit. The analysis on $\lambda^{i^{m}}_{a^{m}}$ and $\tau_{i^{m}}^{a^{m}}$
are extremely similar, for convenience, we will consider the case of all
$\lambda^{i^{m}}_{a^{m}}$ redundancy first and briefly discuss the most
general scenario.

Mathematically, the redundancy of $X_{i^{m}}^{a^{m}}$ means that the
coefficient of $\Delta_{i^{m}}^{a^{m}}$ is identically zero, i.e, Eq. (13) is
identiy regardless the choice of $X_{i^{m}}^{a^{m}}$. This can be rigorously
proved using the same method in the next section. Now we assume that the time-
dependent trajectories of $\lambda^{i^{m}}_{a^{m}}$ are arbitrarily given. Eq.
(11) becomes equations of $X_{i^{m}}^{a^{m}}$. If the solution of Eq. (11)
exists and is unique, Eq. (9,13) should hold due to the redundancy of
$X_{i^{m}}^{a^{m}}$. Further, if one treats Eq. (13) as equations of
$\dot{\tau}_{i^{m}}^{a^{m}}$ and assumes the solution exists and is unique,
the redundancy of $X_{i^{m}}^{a^{m}}$ has been successfully transferred to
$\lambda^{i^{m}}_{a^{m}}$. In general, it is very hard to prove the existence
and uniqueness of the solution even without constraint
$X_{a^{m}}^{b^{m}}=X_{i^{m}}^{j^{m}}\equiv 0$, but one can expect that the
statement is true, at least when the state is dominantly of single reference
characterMyhre (2018).

To completely exclude $\lambda^{i^{m}}_{a^{m}}$ in the time-dependent full CC
expansion, one needs to prepare the initial state with the parameterization
$\lambda^{i^{m}}_{a^{m}}(0)=0$ and use the constraint
$\dot{\lambda}^{i^{m}}_{a^{m}}\equiv 0$. Here, we provide two possible methods
to convert the parameterization of an arbitrary state $\ket{\Psi}$ into the
vanishing $\lambda^{i^{m}}_{a^{m}}$ form. The first is to prepare the initial
state as an arbitrary Slater determinant $\ket{\Phi_{0}}$, and let it evolve
from $t=0$ to $t=\pi/2$ under the Hamiltonian
$\hat{H}=-i(\ket{\Phi_{0}}\bra{\Psi}-\ket{\Psi}\bra{\Phi_{0}})$ with
$\dot{\lambda}^{i^{m}}_{a^{m}}\equiv 0$. The second is to let the state
$\ket{\Psi}$ in an arbitrary parameterization form evolve from $t=0$ to $t=1$
under the Hamiltonian $\hat{H}=0$ with
$\dot{\lambda}^{i^{m}}_{a^{m}}\equiv-\lambda^{i^{m}}_{a^{m}}(0)$.

Now we are in the position to give the most general redundancy analysis of
parameters. Assuming one of $X_{i^{m}}^{a^{m}}$, $\lambda^{i^{m}}_{a^{m}}$,
and $\tau_{i^{m}}^{a^{m}}$ is pre-determined for each kind of fermion,
equations corresponding to the variation coefficients in $\delta S$ should be
removed in coupled equations Eq. (9,11,13). If the solution of remaining
equations exists and is unique, the removing equations are identities and the
redundancy of parameters have been proved. To exclude these redundant
parameters in the time-dependent full CC expansion, one may follow the same
procedures as for the $\lambda^{i^{m}}_{a^{m}}$-redundancy scenario described
above, with the replacements with corresponding pre-determined variables
(constraints).

##  Appendix B Redundancy proof of particle-particle and hole-hole orbital
rotations

We will explicitly show the redundancy of $X_{i^{m}}^{j^{m}}$ and
$X_{a^{m}}^{b^{m}}$ from $\delta S$. For the scenarios
$\mu^{m},\nu^{m}\in\\{i^{m}\\}$, and $\mu^{m},\nu^{m}\in\\{a^{m}\\}$,

$\displaystyle\langle\Psi_{\rm L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\hat{T})-i\hat{X},\hat{E}^{\mu^{m}}_{\nu^{m}}]|\Psi_{\rm
R}\rangle$ $\displaystyle+$ $\displaystyle i\langle\Phi|(\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda})e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle$ $\displaystyle=$ $\displaystyle\langle\Psi_{\rm
L}|[\hat{H}-i\hat{X},\hat{E}^{\mu^{m}}_{\nu^{m}}]|\Psi_{\rm R}\rangle$
$\displaystyle-$ $\displaystyle
i\dot{\tau}_{\mathring{I}}^{\mathring{A}}\langle\Psi_{\rm
L}|[\hat{E}_{\mathring{I}}^{\mathring{A}},\hat{E}^{\mu^{m}}_{\nu^{m}}]|\Psi_{\rm
R}\rangle+i\dot{\lambda}^{\mathring{I}}_{\mathring{A}}\langle\Phi_{\mathring{I}}^{\mathring{A}}|e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle$ (34) $\displaystyle=$ $\displaystyle\braket{\Psi_{\rm
L}}{\hat{E}_{\mathring{I}}^{\mathring{A}}(\hat{H}-i\hat{X})}{\Psi_{\rm
R}}\langle\Phi_{\mathring{I}}^{\mathring{A}}|e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle$ $\displaystyle-$
$\displaystyle\braket{\Phi_{\mathring{I}}^{\mathring{A}}}{e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
R}}\langle\Psi_{\rm
L}|\hat{E}_{\mathring{I}}^{\mathring{A}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle\,,$ (35)

Here, time derivatives of all single amplitudes are included in Eq. (34).
Converting Eq. (34) to Eq. (35), one needs to replace all EoMs of CC
amplitudes with Eq. (8,9,10,11) and use Eq. (16). The equations are
straightforward for TD-OCCX0, and we will explain more on other methods. We
first consider TD-OCCT1. $\lambda_{a^{m}}^{i^{m}}$ in the ansatz should be
understood as a constraint $\lambda_{a^{m}}^{i^{m}}\equiv 0$. Therefore, Eq.
(34) holds. Noticing that the multiplier of $\dot{\tau}^{a^{m}}_{i^{m}}$ in
Eq. (34) vanishes, one can safely replace $\dot{\tau}^{a^{m}}_{i^{m}}$ with
Eq. (9). Additionally, Eq. (11) also ensures
$\dot{\lambda}_{a^{m}}^{i^{m}}\equiv 0$, and we have finished the proof of
equations for TD-OCCT1. For the discussion on TD-BCC, one just needs to swap
$\lambda_{a^{m}}^{i^{m}}$ and $\tau^{a^{m}}_{i^{m}}$ in the proof of TD-OCCT1.
For the TD-OCC method, both $\lambda_{a^{m}}^{i^{m}}$ and
$\tau^{a^{m}}_{i^{m}}$ should be regarded as constraints and their multipliers
in Eq. (34) are always zero, and thus equations hold. Additionally, equations
of TD-OCCH automatically hold when equations of four other methods hold.

Notice that
$\langle\Phi^{\mathring{B}}_{\mathring{J}}|\hat{E}_{\mathring{I}}^{\mathring{A}}$
is not zero iff $\mathring{A}$ and $\mathring{I}$ are substrings of
$\mathring{B}$ and $\mathring{J}$, respectively. For convenience, we use
${\mathring{B}}-{\mathring{A}}$ to express the absolute complementary string
of $\mathring{A}$ with respect to $\mathring{B}$. Expanding $\bra{\Psi_{\rm
L}}$ in Eq. (35) as
$\lambda_{\mathring{B}}^{\mathring{J}}\bra{\Phi^{\mathring{B}}_{\mathring{J}}}$,
Eq. (35) can be converted to

$\displaystyle\lambda_{\mathring{B}}^{\mathring{J}}$
$\displaystyle\big{(}\bra{\Phi^{\mathring{B}-\mathring{A}}_{\mathring{J}-\mathring{I}}}(\hat{H}-i\hat{X})\ket{\Psi_{\rm
R}}{\bra{\Phi^{\mathring{A}}_{\mathring{I}}}}e^{-\hat{T}}\hat{E}_{\nu^{m}}^{\mu^{m}}\ket{\Psi_{\rm
R}}$ (36)
$\displaystyle-{\bra{\Phi^{\mathring{A}}_{\mathring{I}}}}(\hat{H}-i\hat{X})\ket{\Psi_{\rm
R}}\bra{\Phi^{\mathring{B}-\mathring{A}}_{\mathring{J}-\mathring{I}}}e^{-\hat{T}}\hat{E}_{\nu^{m}}^{\mu^{m}}\ket{\Psi_{\rm
R}}\big{)}\,.$

Resuming the second term by swapping ${\mathring{B}}-{\mathring{A}}$,
${\mathring{J}}-{\mathring{I}}$ and ${\mathring{A}}$, ${\mathring{I}}$, one
can immediately obtain that Eq. (35) is zero, which completes the proof that
$X_{i^{m}}^{j^{n}}$ and $X_{a^{m}}^{b^{n}}$ are redundant.

Following the same discussion, one can also explicitly show that $\hat{X}$ is
redundant for TD-ooCC with $\hat{T}_{1}$ and $\hat{\Lambda}_{1}$ in the CASSCF
limit. In this case, it is easy to check that Eq. (34) and Eq. (35) also hold,
and applying the same resummation argument as the proof of redundancy of
$X_{i^{m}}^{j^{m}}$ and $X^{b^{m}}_{a^{m}}$, the proof of redundancy of
$X_{i^{m}}^{a^{m}}$ can be obtained.

##  Appendix C OATDCC and OATDCC with split orthogonal and biorthogonal
orbitals

Before presenting the equivalence proof of sTDMVCCHøjlund and Christiansen
(2024) (sOATDCC) and TD-OCCT1, we will briefly review TDVCC families,
including TDVCCHansen  _et al._ (2019), TDMVCC, restricted polar TDMVCC
(rpTDMVCC)Højlund  _et al._ (2022), orthogonal TDMVCC (oTDMVCC)Højlund,
Zoccante, and Christiansen (2024), and sTDMVCCHøjlund and Christiansen (2024).
First, we will focus on the EoMs and parameterization of singlel amplitudes of
CC coefficients and orbitals. In TDVCC, all orbitals are active, and TD-OCCX0
is equivalent to TDVCC in this case. The fast growth of norms of CC amplitudes
arisen from the absence of orbital rotations makes applications of TDVCC
limited. It is the vibrational analog of TD-CC. To resolve this problem,
orbital rotations and virtual orbitals are introduced. All other four methods,
TDMVCC, rpTDMVCC, sTDMVCC and oTDMVCC have formally same (but not equivalent
due to orthogonal/biorthogonal orbitals) CC part parameterizations and EoMs,
and their major differences are the parameterization of orbitals and their
EoMs. oTDMVCC is completely identical to TD-OCCSato, Teramura, and Ishikawa
(2018) of this article, which uses orthogonal orbitals but suffers from the
huge matrix inversion and nonconvergence to CASSCF. Instead of using
orthogonal orbitals, TDMVCC, which is the vibrational analog of OATDCC, solves
the matrix inversion and convergence problems by using biorthogonal orbitals,
but the non-orthogonality causes numerical instabilities in long time
simulations. One can force active orbitals lie in the conjugate ket and bra
spaces to mitigate instabilities, which is implemented by using orthogonal
virtual orbitals and biorthogonal active orbitals in rpTDMVCC and sTDMVCC.
EoMs of active-virtual orbital rotations are determined by the variational
principle and projection in sTDMVCC and rpTDMVCC, respectively, while EoMs of
CC amplitudes and particle-hole rotations of two methods are completely
identical. In fact, the contribution of biorthogonal active orbitals is
equivalent to the one-rank excitation in TD-OCCT1, and the TDVP ensures the
equivalence of TD-OCCT1 and sTDMVCC.

For convenience, we only consider models without dynamical-core in this
Appendix. For models with dynamical-core, the treatments of dynamical-core
orbitals are almost identical to the virtual orbitals except that they should
be evolved explicitly. We will first review the standard OATDCC, which uses
birothogonal bra and ket orbitals rather than orthogonal orbitals in TD-OCC.
The highly flexible variational space choices could cause serious numerical
instabilities arisen from the different spaces spanned by ket and bra active
orbitals in practical implementationsHøjlund  _et al._ (2022). Next, we will
review sOATDCC, i.e., OTADCC with virtual orbitals of ket and bra are
orthogonal in the Lagrangian and active orbitals are biorthogonal. Such a
parameterization forces the spaces spanned by active orbitals of ket and bra
are identical, which eliminates the numerical instability. The modifications
only appear in the intergroup rotation between active orbitals and other types
of orbitals. OATDCC and sOATDCC reviewed in this Appendix are describing
fermion mixtures with arbitrary occupations, which are more general than the
references results. When sOATDCC is applied to the vibrational problem, it is
the sTDMVCC and inherits the advantage of TDMVCCMadsen  _et al._ (2020c);
Højlund, Zoccante, and Christiansen (2024): EoMs of different vibrational
modes are separate and converge to the CASSCF. In contrast to rpTDMVCCHøjlund
_et al._ (2022), this approach is fully variational. We further prove that
this approach is equivalent to TD-OCCT1.

We define $\ket{\phi_{\mu^{m}}}$ and $\bra{\tilde{\phi}_{\mu^{m}}}$ as
biorthogonal bases of OATDCC, and their corresponding creation (annihilation)
operators are denoted as $\hat{a}_{\mu^{m}}^{\dagger}$ ($\hat{a}_{\mu^{m}}$)
and $\tilde{\hat{a}}_{\mu^{m}}^{\dagger}$ ($\tilde{\hat{a}}_{\mu^{m}}$). They
satisfy the following relations,

$\displaystyle\braket{\tilde{\phi}_{\mu^{m}}}{\phi_{\nu^{m}}}=\delta_{\mu^{m}\nu^{m}}\,,$
(37)
$\displaystyle\\{\hat{a}_{\nu^{m}}^{\dagger},\tilde{\hat{a}}_{\mu^{m}}\\}=\braket{\tilde{\phi}_{\mu^{m}}}{\phi_{\nu^{m}}}=\delta_{\mu^{m}\nu^{m}}\,.$
(38)

In the standard OATDCC, $\hat{a}_{\mu^{m}}$ and
$\tilde{\hat{a}}_{\mu^{m}}^{\dagger}$ are never used, nonetheless, they play
important roles in the sOATDCC. We also introduce the operator
$\tilde{\hat{E}}^{\mu_{1}^{m_{1}}\mu_{2}^{m_{2}}\mu_{3}^{m_{3}}\cdots}_{\nu_{1}^{m_{1}}\nu_{2}^{m_{2}}\nu_{3}^{m_{3}}\cdots}=\hat{a}^{\dagger}_{\mu_{1}^{m_{1}}}\hat{a}^{\dagger}_{\mu_{2}^{m_{2}}}\hat{a}^{\dagger}_{\mu_{3}^{m_{3}}}\cdots\tilde{\hat{a}}_{\nu_{3}^{m_{3}}}\tilde{\hat{a}}_{\nu_{2}^{m_{2}}}\tilde{\hat{a}}_{\nu_{1}^{m_{1}}}$,
and all other quantities in OATDCC are denoted as corresponding quantities in
TD-OCC with tildes, similarly. $m_{1},m_{2},\cdots,m_{3}$ can be the same
index of fermion kind. The ket and bra reference wavefunctions of OATDCC are
tensor products Slater determinants of $\ket{\phi_{\mu^{m}}}$ and
$\bra{\tilde{\phi}_{\mu^{m}}}$ for all $m$, denoted as $\ket{{\Phi}}$ and
$\bra{\tilde{\Phi}}$, respectively. Wavefunction and their time derivatives
and variations as well as the Lagrangian in OATDCC are formally identical to
the TD-OCC correspondence with tildes replacements. Due to the loss of
orthogonality,
$(\tilde{\hat{E}}_{I}^{A})^{\dagger}\neq\tilde{\hat{E}}_{A}^{I}$,
$(\tilde{X}^{\mu^{m}}_{\nu^{m}})^{*}\neq\tilde{X}_{\mu^{m}}^{\nu^{m}}$, and
$(\tilde{\Delta}^{\mu^{m}}_{\nu^{m}})^{*}\neq\tilde{\Delta}_{\mu^{m}}^{\nu^{m}}$
in general. Also, the anti-Hermicity of $\tilde{\hat{X}}$ and
$\tilde{\hat{\Delta}}$ cannot be expected. Therefore, all variables and their
complex conjugate are independent quantities in OATDCC. The action of OATDCC
is a complex functional

$\displaystyle\tilde{S}$ $\displaystyle=$
$\displaystyle\int_{t_{0}}^{t_{1}}dt\tilde{L}\,,$ (39)

and require the action to be stationary

$\displaystyle{\color{black}}\delta\tilde{S}$ $\displaystyle=$
$\displaystyle\langle\delta\tilde{\Psi}_{\rm L}|\hat{H}|\tilde{\Psi}_{\rm
R}\rangle+\langle\tilde{\Psi}_{\rm L}|\hat{H}|\delta\tilde{\Psi}_{\rm
R}\rangle$ (40) $\displaystyle-$ $\displaystyle
i\langle\delta\tilde{\Psi}_{\rm L}|\dot{\tilde{\Psi}}_{\rm
R}\rangle+i\langle\dot{\tilde{\Psi}}_{\rm L}|\delta\tilde{\Psi}_{\rm
R}\rangle=0\,.$

The non-existence of complex conjugate variables in OATDCC action makes the
action complex analytic, and ensures the existence of the solution. The
complex even dimension of the variational manifold of OATDCC ensures
uniqueness of the solution. OTADCC converges to CASSCF when all excitation and
deexcitation operators except for the single excitation and deexcitation ones
are included. We also point out that the real action choice will lead the same
solutions due to the independence of all variables and their complex
conjugates.

Working out the variations explicitly black

$\displaystyle\delta\tilde{S}$ $\displaystyle=$
$\displaystyle\delta\tilde{\tau}^{A}_{I}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{A}_{I}]|\tilde{\Psi}_{\rm
R}\rangle+i\dot{\tilde{\lambda}}^{I}_{A}\right\\}+\delta\tilde{\lambda}^{I}_{A}\left\\{\langle\tilde{\Phi}^{A}_{I}|e^{-\tilde{\hat{T}}}(\hat{H}-i\tilde{\hat{X}})e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle-i\dot{\tilde{\tau}}^{A}_{I}\right\\}$
(41) $\displaystyle+$
$\displaystyle\tilde{\Delta}^{\mu^{m}}_{\nu^{m}}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{T}})-i\tilde{\hat{X}},\tilde{\hat{E}}^{\mu^{m}}_{\nu^{m}}]|\tilde{\Psi}_{\rm
R}\rangle+i\langle\tilde{\Phi}|(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{\Lambda}})e^{-\tilde{\hat{T}}}\tilde{\hat{E}}^{\mu^{m}}_{\nu^{m}}e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle\right\\}\,,$

black EoMs of CC coefficients,

$\displaystyle i\dot{\tilde{\tau}}^{A}_{I}$ $\displaystyle=$
$\displaystyle\langle\tilde{\Phi}_{I}^{A}|e^{-\tilde{\hat{T}}}(\hat{H}-i\tilde{\hat{X}})e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle\,,$
(42) $\displaystyle-i\dot{\tilde{\lambda}}^{I}_{A}$ $\displaystyle=$
$\displaystyle\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{A}_{I}]|\tilde{\Psi}_{\rm
R}\rangle\,,$ (43)

and EoMs of orbital rotations,

$\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}}]\ket{\tilde{\Psi}_{\rm
R}}=0\,,$ (44) $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}]\ket{\tilde{\Psi}_{\rm
R}}=0\,,$ (45) $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{i^{m}}^{a^{m}}]\ket{\tilde{\Psi}_{\rm
R}}=0\,,$ (46) $\displaystyle\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{T}})-i\tilde{\hat{X}},\tilde{\hat{E}}_{a^{m}}^{i^{m}}]|\tilde{\Psi}_{\rm
R}\rangle$ $\displaystyle+$ $\displaystyle
i\langle\tilde{\Phi}|(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{\Lambda}})e^{-\tilde{\hat{T}}}\tilde{\hat{E}}_{a^{m}}^{i^{m}}e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle=0\,,$
(47)

can be obtained. Similar to the TD-OCC scenario, all intragroup rotations are
redundant, thus should be pre-selected, for instance, setting all the
redundant rotations as zero.

In principle, $\\{\ket{\phi_{p^{m}}}\\}$ and
$\\{\ket{\tilde{\phi}_{p^{m}}}\\}$ can span different spaces, which can make
the propagation unstable. To resolve this problem, we adapt real action with
the constraints that virtual orbitals are orthogonal,

$\displaystyle S_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{t_{0}}^{t_{1}}dt\left(\tilde{L}+\tilde{L}^{*}\right),$
(48)

We still use the notations of OATDCC in sOATDCC when there is no ambiguity. In
this case, corresponding ket and bra virtual orbitals are identical,
$\ket{\phi_{\alpha^{m}}}^{*}=\bra{\tilde{\phi}_{\alpha^{m}}}$,
$\hat{a}_{\alpha^{m}}=\tilde{\hat{a}}_{\alpha^{m}}$ and
$\hat{a}_{\alpha^{m}}^{\dagger}=\tilde{\hat{a}}_{\alpha^{m}}^{\dagger}$.
Therefore, $\\{\ket{\phi_{p^{m}}}\\}$ and $\\{\ket{\tilde{\phi}_{p^{m}}}\\}$
span the same space, and the following relations hold,

$\displaystyle\ket{\tilde{\phi}_{p^{m}}}=\braket{\tilde{\phi}_{q^{m}}}{\tilde{\phi}_{p^{m}}}\ket{\phi_{q^{m}}}\,,\quad\tilde{\hat{a}}_{p^{m}}^{\dagger}=\braket{\tilde{\phi}_{q^{m}}}{\tilde{\phi}_{p^{m}}}\hat{a}_{q^{m}}^{\dagger}\,,$
(49)
$\displaystyle\bra{\phi_{p^{m}}}=\bra{\tilde{\phi}_{q^{m}}}\braket{\phi_{p^{m}}}{\phi_{q^{m}}}\,,\quad\hat{a}_{p^{m}}=\braket{\phi_{p^{m}}}{\phi_{q^{m}}}\tilde{\hat{a}}_{q^{m}}\,.$

The constraints also force
$\ket{\delta\phi_{\alpha^{m}}}\equiv\ket{\delta\tilde{\phi}_{\alpha^{m}}}$ and
$\ket{\dot{\phi}_{\alpha^{m}}}\equiv\ket{\dot{\tilde{\phi}}_{\alpha^{m}}}$.
Using above relations, one could find that
$(\tilde{X}^{\alpha^{m}}_{p^{m}})^{*}$ and $\tilde{X}_{\alpha^{m}}^{p^{m}}$,
as well as $(\tilde{\Delta}^{\alpha^{m}}_{q^{m}})^{*}$ and
$\tilde{\Delta}_{\alpha^{m}}^{q^{m}}$ are not independent,

$\displaystyle(\tilde{X}^{\alpha^{m}}_{p^{m}})^{*}=-\tilde{X}_{\alpha^{m}}^{q^{m}}\braket{\phi_{p^{m}}}{\phi_{q^{m}}}\,,$
(50)
$\displaystyle\tilde{X}_{\alpha^{m}}^{p^{m}}=-(\tilde{X}^{\alpha^{m}}_{q^{m}})^{*}\braket{\tilde{\phi}_{p^{m}}}{\tilde{\phi}_{q^{m}}}\,,$
(51)
$\displaystyle(\tilde{\Delta}^{\alpha^{m}}_{p^{m}})^{*}=-\tilde{\Delta}_{\alpha^{m}}^{q^{m}}\braket{\phi_{p^{m}}}{\phi_{q^{m}}}\,,$
(52)
$\displaystyle\tilde{\Delta}_{\alpha^{m}}^{p^{m}}=-(\tilde{\Delta}^{\alpha^{m}}_{q^{m}})^{*}\braket{\tilde{\phi}_{p^{m}}}{\tilde{\phi}_{q^{m}}}\,,$
(53)
$\displaystyle(\tilde{X}^{\alpha^{m}}_{p^{m}}\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}})^{\dagger}=-\tilde{X}_{\alpha^{m}}^{p^{m}}\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}\,,$
(54)
$\displaystyle(\tilde{\Delta}^{\alpha^{m}}_{p^{m}}\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}})^{\dagger}=-\tilde{\Delta}_{\alpha^{m}}^{p^{m}}\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}\,.$
(55)

Therefore, the explicit form of the variation with independent variational
variables is black

$\displaystyle 2\delta{S}_{s}$ $\displaystyle=$
$\displaystyle\delta\tilde{\tau}^{A}_{I}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{A}_{I}]|\tilde{\Psi}_{\rm
R}\rangle+i\dot{\tilde{\lambda}}^{I}_{A}\right\\}+\delta\tilde{\lambda}^{I}_{A}\left\\{\langle\tilde{\Phi}^{A}_{I}|e^{-\tilde{\hat{T}}}(\hat{H}-i\tilde{\hat{X}})e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle-i\dot{\tilde{\tau}}^{A}_{I}\right\\}+c.c.$
(56) $\displaystyle+$
$\displaystyle\tilde{\Delta}^{p^{m}}_{q^{m}}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{T}})-i\tilde{\hat{X}},\tilde{\hat{E}}^{p^{m}}_{q^{m}}]|\tilde{\Psi}_{\rm
R}\rangle+i\langle\tilde{\Phi}|(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{\Lambda}})e^{-\tilde{\hat{T}}}\tilde{\hat{E}}^{p^{m}}_{q^{m}}e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle\right\\}\,+c.c.$
$\displaystyle+$
$\displaystyle\tilde{\Delta}^{p^{m}}_{\alpha^{m}}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{p^{m}}_{\alpha^{m}}]|\tilde{\Psi}_{\rm
R}\rangle-\braket{\phi_{q^{m}}}{\phi_{p^{m}}}\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{q^{m}}^{\alpha^{m}}]|\tilde{\Psi}_{\rm
R}\rangle^{*}\right\\}$ $\displaystyle+$
$\displaystyle\tilde{\Delta}_{p^{m}}^{\alpha^{m}}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{p^{m}}^{\alpha^{m}}]|\tilde{\Psi}_{\rm
R}\rangle-\braket{\tilde{\phi}_{p^{m}}}{\tilde{\phi}_{q^{m}}}\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{q^{m}}_{\alpha^{m}}]|\tilde{\Psi}_{\rm
R}\rangle^{*}\right\\}\,.$

Compared with Eq. (41), it is clear that the constraints do not change the
EoMs of coefficients and intergroup orbital rotations between particle and
hole orbitals. The variations with respect to $\Delta_{\alpha^{m}}^{p^{m}}$
give EoMs of intergroup rotation between active orbitals and virtual orbitals

$\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}]\ket{\tilde{\Psi}_{\rm
R}}+\bra{\tilde{\Psi}_{\rm
R}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}]\ket{\tilde{\Psi}_{\rm
L}}=0\,,$ (57) $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}}]\ket{\tilde{\Psi}_{\rm
R}}+\bra{\tilde{\Psi}_{\rm
R}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}}]\ket{\tilde{\Psi}_{\rm
L}}=0\,.$ (58)

sOATDCC is equivalent to TD-OCCT1. To see this, one just needs to prove their
manifolds are identical up to normalization factors, which corresponds to a
point transformation in Lagrange MechanicsGoldstein (1980). More explicitly
speaking, for any states $\ket{\tilde{\Psi}_{\rm R}}$ and
$\bra{\tilde{\Psi}_{\rm L}}$ parameterized via the sOATDCC formalism, one can
always find states parameterized via the TD-OCCT1 formalism $\ket{\Psi_{\rm
R}}$ and $\bra{\Psi_{\rm L}}$ that satisfy $\ket{\tilde{\Psi}_{\rm
R}}=A\ket{\Psi_{\rm R}}$, and $\bra{\tilde{\Psi}_{\rm L}}=A^{-1}\bra{\Psi_{\rm
L}}$ and vice versa. Here, $A$ is a normalization factor and only gives a
trivial total derivative in Lagrange. In both transformations, one can always
choose $\ket{\psi_{\alpha}}\equiv\ket{\phi_{\alpha}}$ and they only appear in
the time derivatives of wavefunctions.

We will first consider converting TD-OCCT1 to sOATDCC. The transformation is
given by

$\displaystyle\ket{\phi_{i^{m}}}=\ket{\psi_{i^{m}}}+\tau_{i^{m}}^{a^{m}}\ket{\psi_{a^{m}}}\,,\quad\hat{a}^{\dagger}_{i^{m}}=\hat{c}_{i^{m}}^{\dagger}+\tau_{i^{m}}^{a^{m}}\hat{c}^{\dagger}_{a^{m}}\,,$
(59)
$\displaystyle\bra{\tilde{\phi}_{a^{m}}}=\bra{\psi_{a^{m}}}-\tau_{i^{m}}^{a^{m}}\bra{\psi_{i^{m}}}\,,\quad\tilde{\hat{a}}_{a^{m}}=\hat{c}_{a^{m}}-\tau_{i^{m}}^{a^{m}}\hat{c}_{i^{m}}\,,$
(60)
$\displaystyle\ket{\phi_{a^{m}}}=\ket{\psi_{a^{m}}}\,,\quad\hat{a}^{\dagger}_{a^{m}}=\hat{c}_{a^{m}}^{\dagger}\,,$
(61)
$\displaystyle\bra{\tilde{\phi}_{i^{m}}}=\bra{\psi_{i^{m}}}\,,\quad\tilde{\hat{a}}_{i^{m}}=\hat{c}_{i^{m}}\,,$
(62)
$\displaystyle\tilde{\tau}_{I}^{A}=\tau_{I}^{A}\,,\quad\tilde{\lambda}_{A}^{I}=\lambda_{A}^{I}\,,\quad\ket{\tilde{\Psi}_{\rm
R}}=\ket{\Psi_{\rm R}}\,,\quad\bra{\tilde{\Psi}_{\rm L}}=\bra{\Psi_{\rm
L}}\,.$ (63)

These identities are ensured by the invariance of particle-hole orbital
rotation in OATDCC parameterization:

$\displaystyle\hat{a}_{p^{m}}^{\dagger}\rightarrow
e^{\tilde{\hat{T}}_{1}}\hat{a}_{p^{m}}^{\dagger}e^{-\tilde{\hat{T}}_{1}}\,,\quad\tilde{\hat{a}}_{p^{m}}\rightarrow
e^{\tilde{\hat{T}}_{1}}\tilde{\hat{a}}_{p^{m}}e^{-\tilde{\hat{T}}_{1}}\,,$
(64)
$\displaystyle\bra{\tilde{\Phi}}\rightarrow\bra{\tilde{\Phi}}e^{-\tilde{\hat{T}}_{1}}=\bra{\tilde{\Phi}}\,,\quad\ket{{\Phi}}\rightarrow
e^{\tilde{\hat{T}}_{1}}\ket{{\Phi}}\,,$ $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}\rightarrow\bra{\tilde{\Phi}}(1+e^{\tilde{\hat{T}}_{1}}\tilde{\hat{\Lambda}}e^{-\tilde{\hat{T}}_{1}})e^{-\tilde{\hat{T}}+\tilde{\hat{T}}_{1}}=\bra{\tilde{\Psi}_{\rm
L}}\,,$ $\displaystyle\ket{\tilde{\Psi}_{\rm R}}\rightarrow
e^{\tilde{\hat{T}}-\tilde{\hat{T}}_{1}}e^{\tilde{\hat{T}}_{1}}\ket{\tilde{\Phi}}=\ket{\tilde{\Psi}_{\rm
R}}\,.$

where
$\tilde{\hat{T}}_{1}=\tilde{\tau}_{i^{m}}^{a^{m}}\tilde{\hat{E}}_{i^{m}}^{a^{m}}$
is an arbitrary single excitation operator. Such a rotation invariance causes
the $\tilde{\hat{T}}_{1}$ redundancy in OATDCC and has been extensively
discussed in the supplementary material of the original OATDCC paperKvaal
(2012).

To convert sOATDCC wavefunctions to TD-OCCT1 correspondence needs two steps.
The first step is to orthonormalize $\ket{\phi_{a^{m}}}$ and
$\bra{\tilde{\phi}_{i^{m}}}$,

$\ket{\phi_{a^{m}}^{\prime}}=G_{b^{m}}^{a^{m}}\ket{\phi_{b^{m}}}\,,\quad\bra{\tilde{\phi}_{i^{m}}^{\prime}}=\tilde{G}_{j^{m}}^{i^{m}}\bra{\tilde{\phi}_{j^{m}}}\,,$
(65)

where $\ket{\phi_{a^{m}}^{\prime}}$ and $\bra{\tilde{\phi}_{i^{m}}^{\prime}}$
are orthonormalized vectors, $G$ and $\tilde{G}$ are transformation matrices.
We also denote quantities after the orthonormalization as original quantities
with primes. Dual vectors of $\ket{\phi_{a^{m}}^{\prime}}$ and
$\bra{\tilde{\phi}_{i^{m}}^{\prime}}$ are

$\ket{\phi_{i^{m}}^{\prime}}=(\tilde{G}^{-1})_{i^{m}}^{j^{m}}\ket{\phi_{j^{m}}}\,,\quad\bra{\tilde{\phi}_{a^{m}}^{\prime}}=({G}^{-1})_{a^{m}}^{b^{m}}\bra{\tilde{\phi}_{b^{m}}}\,.$
(66)

We also define general-rank transformations of $G$ and $\tilde{G}$ as
$G_{I}^{J}$ and $\tilde{G}_{A}^{B}$, which are the tensor products of single-
rank transformations of $G_{i^{m}}^{j^{m}}$ and $\tilde{G}_{a^{m}}^{b^{m}}$,
respectively. Wavefunctions as well as relevant quantities under such a
transformation are given by

$\displaystyle\tilde{\hat{E}}_{I}^{A}\rightarrow(\tilde{\hat{E}}_{I}^{A})^{\prime}=G^{A}_{B}\tilde{\hat{E}}_{J}^{B}\tilde{G}^{I}_{J}\,,$
(67)
$\displaystyle\tilde{\hat{E}}_{A}^{I}\rightarrow(\tilde{\hat{E}}_{A}^{I})^{\prime}=(\tilde{G}^{-1})_{I}^{J}\tilde{\hat{E}}_{B}^{J}({G}^{-1})_{A}^{B}\,,$
$\displaystyle\tilde{{\tau}}_{I}^{A}\rightarrow(\tilde{\tau}_{I}^{A})^{\prime}=(G^{-1})_{A}^{B}\tilde{\tau}_{J}^{B}(\tilde{G}^{-1})_{I}^{J}\,,$
$\displaystyle\tilde{{\lambda}}_{A}^{I}\rightarrow(\tilde{\lambda}_{A}^{I})^{\prime}=\tilde{G}^{I}_{J}\tilde{\lambda}_{B}^{J}{G}_{B}^{A}\,,$
$\displaystyle\ket{\tilde{\Psi}_{\rm R}}\rightarrow\ket{\tilde{\Psi}_{\rm
R}^{\prime}}=\prod_{m}(\det{(\tilde{G}_{j^{m}}^{i^{m}})})^{-1}\ket{\tilde{\Psi}_{\rm
R}}\,,$ $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}\rightarrow\bra{\tilde{\Psi}_{\rm
L}^{\prime}}=\prod_{m}\det{(\tilde{G}_{j^{m}}^{i^{m}})}\bra{\tilde{\Psi}_{\rm
L}}\,.$

In fact, if one replaces single-rank $G$ and $\tilde{G}$ with arbitrary
invertible matrices, the above transformation invariance still holds and that
is the reason of the redundancy of $\tilde{X}_{i^{m}}^{j^{m}}$ and
$\tilde{X}_{a^{m}}^{b^{m}}$ Kvaal (2012).

The second step is to express $\ket{{\Psi}_{\rm R}}$ and $\bra{{\Psi}_{\rm
L}}$ via $\ket{\tilde{\Psi}_{\rm R}^{\prime}}$ and $\bra{\tilde{\Psi}_{\rm
L}^{\prime}}$,

$\displaystyle\ket{\psi_{a^{m}}}=\ket{\phi_{a^{m}}}\,,\quad\hat{c}_{a^{m}}^{\dagger}=(\hat{a}^{\dagger}_{a^{m}})^{\prime}\,,$
(68)
$\displaystyle\bra{\psi_{i^{m}}}=\bra{\tilde{\phi}_{i^{m}}}\,,\quad\hat{c}_{i^{m}}=(\tilde{\hat{a}}_{i^{m}})^{\prime}\,,$
(69)
$\displaystyle\tau_{I}^{A}=(\tilde{\tau}_{I}^{A})^{\prime}\,,\quad\lambda_{A}^{I}=(\tilde{\lambda}_{A}^{I})^{\prime}\,,\quad\tau_{i^{m}}^{a^{m}}=\braket{\phi_{a^{m}}^{\prime}}{\phi_{i^{m}}^{\prime}}\,,$
(70) $\displaystyle\ket{\Psi_{\rm R}}=\ket{\tilde{\Psi}_{\rm
R}^{\prime}}\,,\quad\bra{\Psi_{\rm L}}=\bra{\tilde{\Psi}_{\rm L}^{\prime}}\,.$
(71)

Now we have finished the proof of the equivalence of sOATDCC and TD-OCCT1.
Although it is unnecessary for the proof of the equivalence of two methods, we
will present the explicit form of the transformation and show that EoMs of two
methods coincide for the scenario that converting TD-OCCT1 to sOATDCC since it
will give readers a straightforward understanding. Useful transformations are

$\displaystyle\tilde{\hat{E}}_{i^{m}}^{a^{m}}=\hat{E}_{i^{m}}^{a^{m}}\,,\quad\tilde{X}_{i^{m}}^{a^{m}}=X_{i^{m}}^{a^{m}}+\dot{\tau}_{i^{m}}^{a^{m}}-\tau_{j^{m}}^{a^{m}}\tau_{i^{m}}^{b^{m}}X^{j^{m}}_{b^{m}}\,,$
(72)
$\displaystyle\tilde{\hat{E}}_{i^{m}}^{j^{m}}=\hat{E}_{i^{m}}^{j^{n}}+\tau_{j^{m}}^{a^{m}}\hat{E}_{i^{m}}^{a^{m}}\,,\quad\tilde{X}_{i^{m}}^{j^{m}}=X_{a^{m}}^{j^{m}}\tau_{i^{m}}^{a^{m}}\,,$
$\displaystyle\tilde{\hat{E}}_{a^{m}}^{b^{m}}=\hat{E}_{a^{m}}^{b^{n}}-\tau_{i^{m}}^{a^{m}}\hat{E}_{i^{m}}^{b^{m}}\,,\quad\tilde{X}_{a^{m}}^{b^{m}}=-\tau_{i^{m}}^{b^{m}}X^{i^{m}}_{a^{m}}\,,$
$\displaystyle\tilde{\hat{E}}^{i^{m}}_{a^{m}}=\hat{E}^{i^{m}}_{a^{m}}-\tau_{j^{m}}^{a^{m}}\hat{E}_{j^{m}}^{i^{m}}+\tau_{i^{m}}^{b^{m}}\hat{E}_{a^{m}}^{b^{m}}-\tau_{j^{m}}^{a^{m}}\tau_{i^{m}}^{b^{m}}\hat{E}_{j^{m}}^{b^{m}}\,,$
$\displaystyle\tilde{\hat{E}}_{\alpha^{m}}^{i^{m}}=\hat{E}_{\alpha^{m}}^{i^{m}}+\tau_{i^{m}}^{a^{m}}\hat{E}_{\alpha^{m}}^{a^{m}}\,,\quad\tilde{X}_{\alpha^{m}}^{i^{m}}=X_{\alpha^{m}}^{i^{m}}\,,\quad\tilde{X}^{i^{m}}_{a^{m}}=X^{i^{m}}_{a^{m}}\,,$
$\displaystyle\tilde{\hat{E}}_{\alpha^{m}}^{a^{m}}=\hat{E}_{\alpha^{m}}^{a^{m}}\,,\quad\tilde{X}_{\alpha^{m}}^{a^{m}}=X_{\alpha^{m}}^{a^{m}}-\tau_{i^{m}}^{a^{m}}{X}_{\alpha^{m}}^{i^{m}}\,,$
$\displaystyle\tilde{\hat{E}}^{\alpha^{m}}_{i^{m}}=\hat{E}^{\alpha^{m}}_{i^{m}}\,,\quad\tilde{X}^{\alpha^{m}}_{i^{m}}=X^{\alpha^{m}}_{i^{m}}+\tau_{i^{m}}^{a^{m}}{X}^{\alpha^{m}}_{a^{m}}\,,$
$\displaystyle\tilde{\hat{E}}^{\alpha^{m}}_{a^{m}}=\hat{E}^{\alpha^{m}}_{a^{m}}-\tau_{i^{m}}^{a^{m}}\hat{E}^{\alpha^{m}}_{i^{m}}\,,\quad\tilde{X}^{\alpha^{m}}_{a^{m}}=X^{\alpha^{m}}_{a^{m}}\,,$
$\displaystyle\tilde{\hat{X}}=\hat{X}+(\partial^{\hskip
0.20999pt\prime}_{t}{\hat{T}_{1}})\,.$

where we already set $X_{i^{m}}^{j^{m}}$ and $X_{a^{m}}^{b^{m}}$ as zero. We
stress that the constraint selection of $\tilde{X}_{i^{m}}^{j^{m}}$ and
$\tilde{X}_{a^{m}}^{b^{m}}$ in sOATDCC in this case is no longer zero, which
is not a usual choice in practical simulations. It is straightforward to
verify that EoMs of active-virtual orbital rotations and CC coefficients are
satisfied. To show the EoMs of intergroup rotations of active orbitals also
hold, one notices that their EoMs can be expressed uniformly as

$\displaystyle\langle\tilde{\Psi}_{\rm L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{T}})-i\tilde{\hat{X}},\tilde{\hat{E}}_{p^{m}}^{q^{m}}]|\tilde{\Psi}_{\rm
R}\rangle$ (73) $\displaystyle+$ $\displaystyle
i\langle\tilde{\Phi}|(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{\Lambda}})e^{-\tilde{\hat{T}}}\tilde{\hat{E}}_{p^{m}}^{q^{m}}e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle=0\,,$

which also holds since it is the summation (with some weights) of

$\displaystyle\langle\Psi_{\rm L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\hat{T})-i\hat{X},\hat{E}^{q^{m}}_{p^{m}}]|\Psi_{\rm
R}\rangle$ (74) $\displaystyle+$ $\displaystyle i\langle\Phi|(\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda})e^{-\hat{T}}\hat{E}^{q^{m}}_{p^{m}}|\Psi_{\rm
R}\rangle=0\,.$

##  Appendix D EoMs of TD-OCCH

In this section, we will present the EoMs of single amplitudes and particle-
hole orbital rotations of TD-OCCH. EoMs of other parameters of the method have
already been presented in the Sec. II.3. We will use notations $m_{\rm O}$,
$m_{\rm T}$, $m_{\rm B}$, and $m_{\rm X}$ to label the fermion kinds with
parameterization of TD-OCC, TD-OCCT1, TD-BCC, and TD-OCCX0. $m$ without any
subscripts represent an arbitrary fermion kind. $X_{i^{m_{\rm X}}}^{a^{m_{\rm
X}}}$ are set as Eq. (33) in VG and zero in LG. EoMs of $\lambda^{i^{m_{\rm
X}}}_{a^{m_{\rm X}}}$ and $X_{i^{m_{\rm T}}}^{a^{m_{\rm T}}}$ are decoupled
with other EoMs

$i\left(\rho_{j^{m_{\rm T}}}^{i^{m_{\rm T}}}\delta_{b^{m_{\rm T}}}^{a^{m_{\rm
T}}}-\rho_{a^{m_{\rm T}}}^{b^{m_{\rm T}}}\delta_{j^{m_{\rm T}}}^{i^{m_{\rm
T}}}\right)X_{b^{m_{\rm T}}}^{j^{m_{\rm T}}}=\bra{\Psi_{\rm
L}}[\hat{H},\hat{E}_{i^{m_{\rm T}}}^{a^{m_{\rm T}}}]\ket{\Psi_{\rm R}}\,,$
(75)

$\displaystyle-i\dot{\lambda}^{i^{m_{\rm X}}}_{a^{m_{\rm X}}}=$ (76)
$\displaystyle\bra{\Psi_{\rm L}}[\hat{H},\hat{E}_{i^{m_{\rm X}}}^{a^{m_{\rm
X}}}]\ket{\Psi_{\rm R}}-i\left(\rho_{j^{m_{\rm X}}}^{i^{m_{\rm
X}}}\delta_{b^{m_{\rm X}}}^{a^{m_{\rm X}}}-\rho_{a^{m_{\rm X}}}^{b^{m_{\rm
X}}}\delta_{j^{m_{\rm X}}}^{i^{m_{\rm X}}}\right)X_{b^{m_{\rm X}}}^{j^{m_{\rm
X}}}\,.$

After obtaining $\dot{\lambda}^{i^{m_{\rm X}}}_{a^{m_{\rm X}}}$ and
$X_{i^{m_{\rm T}}}^{a^{m_{\rm T}}}$, one can solve the following coupled
equations to get $\dot{\lambda}^{i^{m_{\rm B}}}_{a^{m_{\rm B}}}$,
$X_{i^{m_{\rm B}}}^{a^{m_{\rm B}}}$, and $X_{i^{m_{\rm O}}}^{a^{m_{\rm O}}}$.

$\displaystyle i\frac{1}{2}\left\\{\sum_{n_{\rm B}}\langle\Phi_{j^{n_{\rm
B}}}^{b^{n_{\rm B}}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm B}}}_{a^{m_{\rm
B}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n_{\rm B}}}_{b^{n_{\rm
B}}}+(\dot{\lambda}^{i^{m_{\rm B}}}_{a^{m_{\rm B}}})^{*}+\sum_{n\in n_{\rm
B},n_{\rm O}}A^{a^{m_{\rm B}}j^{n}}_{i^{m_{\rm
B}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\right\\}+i(\delta_{b^{m_{\rm B}}}^{a^{m_{\rm
B}}}D_{i^{m_{\rm B}}}^{j^{m_{\rm B}}}-D_{b^{m_{\rm B}}}^{a^{m_{\rm
B}}}\delta_{i^{m_{\rm B}}}^{j^{m_{\rm B}}})X_{j^{m_{\rm B}}}^{b^{m_{\rm B}}}=$
$\displaystyle B_{i^{m_{\rm B}}}^{a^{m_{\rm B}}}-i\sum_{n\in n_{\rm T},n_{\rm
X}}\frac{1}{2}A^{a^{m_{\rm B}}j^{n}}_{i^{m_{\rm
B}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}-i\frac{1}{2}\sum_{n_{\rm
X}}\langle\Phi_{j^{n_{\rm X}}}^{b^{n_{\rm X}}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm
B}}}_{a^{m_{\rm B}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n_{\rm
X}}}_{b^{n_{\rm X}}}\,,$ (77)

$\displaystyle i\frac{1}{2}\left\\{\sum_{n_{\rm B}}\langle\Phi_{j^{n_{\rm
B}}}^{b^{n_{\rm B}}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm O}}}_{a^{m_{\rm
O}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n_{\rm B}}}_{b^{n_{\rm
B}}}+\sum_{n\in n_{\rm B},n_{\rm O}}A^{a^{m_{\rm O}}j^{n}}_{i^{m_{\rm
O}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\right\\}+i(\delta_{b^{m_{\rm O}}}^{a^{m_{\rm
O}}}D_{i^{m_{\rm O}}}^{j^{m_{\rm O}}}-D_{b^{m_{\rm O}}}^{a^{m_{\rm
O}}}\delta_{i^{m_{\rm O}}}^{j^{m_{\rm O}}})X_{j^{m_{\rm O}}}^{b^{m_{\rm O}}}=$
$\displaystyle B_{i^{m_{\rm O}}}^{a^{m_{\rm O}}}-i\sum_{n\in n_{\rm T},n_{\rm
X}}\frac{1}{2}A^{a^{m_{\rm O}}j^{n}}_{i^{m_{\rm
O}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}-i\frac{1}{2}\sum_{n_{\rm
X}}\langle\Phi_{j^{n_{\rm X}}}^{b^{n_{\rm X}}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm
O}}}_{a^{m_{\rm O}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n_{\rm
X}}}_{b^{n_{\rm X}}}\,,$ (78)

$\displaystyle iX_{i^{m_{\rm B}}}^{a^{m_{\rm B}}}+\sum_{n\in n_{\rm B},n_{\rm
O}}i\langle\Phi_{i^{m_{\rm B}}}^{a^{m_{\rm
B}}}|e^{-\hat{T}}\hat{E}^{j^{n}}_{b^{n}}|\Psi_{\rm R}\rangle
X^{j^{n}}_{b^{n}}=\langle\Phi_{i^{m_{\rm B}}}^{a^{m_{\rm
B}}}|e^{-\hat{T}}\hat{H}|\Psi_{\rm R}\rangle-\sum_{n\in n_{\rm T},n_{\rm
X}}i\langle\Phi_{i^{m_{\rm B}}}^{a^{m_{\rm
B}}}|e^{-\hat{T}}\hat{E}^{j^{n}}_{b^{n}}|\Psi_{\rm R}\rangle
X^{j^{n}}_{b^{n}}\,.$ (79)

Finally, $\dot{\tau}_{i^{m_{\rm T}}}^{a^{m_{\rm T}}}$ and
$\dot{\tau}_{i^{m_{\rm X}}}^{a^{m_{\rm X}}}$ can be solved

$\displaystyle i(\delta_{b^{m_{\rm T}}}^{a^{m_{\rm T}}}D_{i^{m_{\rm
T}}}^{j^{m_{\rm T}}}-D_{b^{m_{\rm T}}}^{a^{m_{\rm T}}}\delta_{i^{m_{\rm
T}}}^{j^{m_{\rm T}}})\dot{\tau}_{j^{m_{\rm T}}}^{b^{m_{\rm T}}}=B_{i^{m_{\rm
T}}}^{a^{m_{\rm T}}}$ $\displaystyle-$ $\displaystyle i(\delta_{b^{m_{\rm
T}}}^{a^{m_{\rm T}}}D_{i^{m_{\rm T}}}^{j^{m_{\rm T}}}-D_{b^{m_{\rm
T}}}^{a^{m_{\rm T}}}\delta_{i^{m_{\rm T}}}^{j^{m_{\rm T}}})X_{j^{m_{\rm
T}}}^{b^{m_{\rm T}}}-\sum_{n}\frac{i}{2}A^{a^{m_{\rm T}}j^{n}}_{i^{m_{\rm
T}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}$ $\displaystyle-$
$\displaystyle\frac{i}{2}\sum_{n\in n_{\rm B},n_{\rm
X}}\langle\Phi_{j^{n}}^{b^{n}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm T}}}_{a^{m_{\rm
T}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n}}_{b^{n}}\,,$ (80)

$i\dot{\tau}_{i^{m_{\rm X}}}^{a^{m_{\rm X}}}=\bra{\Phi_{i^{m_{\rm
X}}}^{a^{m_{\rm X}}}}e^{-\hat{T}}(\hat{H}-i\hat{X})\ket{\Psi_{\rm R}}\,.$ (81)

It is straightforward to verify that TD-OCCH reduce to other four methods when
the corresponding parameterization is used in all fermion kinds. It is also
possible to use two or three types of parameterization in TD-OCCH method, and
one needs to remove unused types of parameters in above EoMs. An attractive
parameterization is the OCCT1-OCCX0 mixed one, in which all single amplitudes
and particle-hole rotations of different kinds are decoupled. One can also
assign few kinds fermions OCC and BCC type parameterization, in which only few
of coupled equations need to be solved.

Similar to the TD-OCCX0 method, the non-variance of $X_{i^{m_{\rm
X}}}^{a^{m_{\rm X}}}$ makes the Ehrenfest theoremSato  _et al._ (2016)
modified even without frozen-cores. One only needs to account for the TD-
OCCX0-type parameterization fermion kinds

$\displaystyle 2i\frac{d}{dt}\braket{\hat{O}}$
$\displaystyle=\braket{\Psi_{\rm L}}{[\hat{O},\hat{H}]}{\Psi_{\rm
R}}+\braket{\Psi_{\rm R}}{[\hat{O},\hat{H}]}{\Psi_{\rm L}}$ (82)
$\displaystyle+\sum_{m\in m_{\rm X}}O_{t^{m}}^{u^{m}}\\{\braket{\Psi_{\rm
L}}{(\hat{H}-i\hat{X})e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}\hat{E}_{t^{m}}^{u^{m}}}{\Psi_{\rm
R}}$ $\displaystyle-\braket{\Psi_{\rm
L}}{\hat{E}_{t^{m}}^{u^{m}}e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
R}}$ $\displaystyle+\braket{\Psi_{\rm
R}}{(\hat{H}-i\hat{X})e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}\hat{E}_{t^{m}}^{u^{m}}}{\Psi_{\rm
L}}$ $\displaystyle-\braket{\Psi_{\rm
R}}{\hat{E}_{t^{m}}^{u^{m}}e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
L}}\\}\,.$

When frozen orbitals are included, two additional terms that replace $t^{m}$,
$u^{m}$ with $i^{\prime\prime m}$, $\mu^{m}$ and $\mu^{m}$, $i^{\prime\prime
m}$ in the last term of Eq. (82) should be added. Additionally, the summation
for the new terms should be for all kinds of fermions.

## References

  * Protopapas, Keitel, and Knight (1997) M. Protopapas, C. H. Keitel, and P. L. Knight, Rep. Prog. Phys. 60, 389 (1997).
  * Agostini and DiMauro (2004) P. Agostini and L. F. DiMauro, Rep. Prog. Phys. 67, 813 (2004).
  * Krausz and Ivanov (2009) F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009).
  * Gallmann, Cirelli, and Keller (2013) L. Gallmann, C. Cirelli, and U. Keller, Annu. Rev. Phys. Chem. 63, 447 (2013).
  * Nisoli  _et al._ (2017) M. Nisoli, P. Decleva, F. Calegari, A. Palacios, and F. Martín, Chem. Rev. 117, 10760 (2017).
  * Mordovina  _et al._ (2020) U. Mordovina, C. Bungey, H. Appel, P. J. Knowles, A. Rubio, and F. R. Manby, Physical Review Research 2, 023262 (2020).
  * Christiansen (2004) O. Christiansen, The Journal of chemical physics 120, 2149 (2004).
  * Hansen  _et al._ (2019) M. B. Hansen, N. K. Madsen, A. Zoccante, and O. Christiansen, The Journal of Chemical Physics 151 (2019).
  * Sverdrup Ofstad  _et al._ (2023) B. Sverdrup Ofstad, E. Aurbakken, Ø. Sigmundson Schøyen, H. E. Kristiansen, S. Kvaal, and T. B. Pedersen, Wiley Interdisciplinary Reviews: Computational Molecular Science 13, e1666 (2023).
  * Zanghellini  _et al._ (2003) J. Zanghellini, M. Kitzler, C. Fabian, T. Brabec, and A. Scrinzi, Laser Physics 13, 1064 (2003).
  * Kato and Kono (2004) T. Kato and H. Kono, Chem. Phys. Lett. 392, 533 (2004).
  * Caillat  _et al._ (2005) J. Caillat, J. Zanghellini, M. Kitzler, O. Koch, W. Kreuzer, and A. Scrinzi, Phys. Rev. A 71, 012712 (2005).
  * Nest, Klamroth, and Saalfrank (2005) M. Nest, T. Klamroth, and P. Saalfrank, J. Chem. Phys. 122, 124102 (2005).
  * Kato and Kono (2008) T. Kato and H. Kono, J. Chem. Phys. 128, 184102 (2008).
  * Hochstuhl and Bonitz (2011) D. Hochstuhl and M. Bonitz, J. Chem. Phys. 134, 084106 (2011).
  * Haxton, Lawler, and McCurdy (2012) D. J. Haxton, K. V. Lawler, and C. W. McCurdy, Phys. Rev. A 86, 013406 (2012).
  * Haxton and McCurdy (2014) D. J. Haxton and C. W. McCurdy, Phys. Rev. A 90, 053426 (2014).
  * Lode  _et al._ (2020) A. U. Lode, C. Lévêque, L. B. Madsen, A. I. Streltsov, and O. E. Alon, Reviews of Modern Physics 92, 011001 (2020).
  * Alon, Streltsov, and Cederbaum (2007) O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, The Journal of chemical physics 127 (2007).
  * Nguyen-Dang  _et al._ (2007) T. T. Nguyen-Dang, M. Peters, S.-M. Wang, E. Sinelnikov, and F. Dion, J. Chem. Phys. 127, 174107 (2007).
  * Ishikawa and Sato (2015) K. L. Ishikawa and T. Sato, IEEE J. Sel. Topics Quantum Electron 21, 8700916 (2015).
  * Anzaki, Sato, and Ishikawa (2017) R. Anzaki, T. Sato, and K. L. Ishikawa, Phys. Chem. Chem. Phys. 19, 22008 (2017).
  * Sato and Ishikawa (2013) T. Sato and K. L. Ishikawa, Phys. Rev. A 88, 023402 (2013).
  * Li, Sato, and Ishikawa (2021) Y. Li, T. Sato, and K. L. Ishikawa, Physical Review A 104, 043104 (2021).
  * Madsen  _et al._ (2020a) N. K. Madsen, M. B. Hansen, G. A. Worth, and O. Christiansen, The Journal of Chemical Physics 152 (2020a).
  * Madsen  _et al._ (2020b) N. K. Madsen, M. B. Hansen, G. A. Worth, and O. Christiansen, Journal of Chemical Theory and Computation 16, 4087 (2020b).
  * Miyagi and Madsen (2013) H. Miyagi and L. B. Madsen, Phys. Rev. A 87, 062511 (2013).
  * Miyagi and Madsen (2014) H. Miyagi and L. B. Madsen, Phys. Rev. A 89, 063416 (2014).
  * Haxton and McCurdy (2015) D. J. Haxton and C. W. McCurdy, Phys. Rev. A 91, 012509 (2015).
  * Sato and Ishikawa (2015) T. Sato and K. L. Ishikawa, Phys. Rev. A 91, 023417 (2015).
  * Sato  _et al._ (2016) T. Sato, K. L. Ishikawa, I. Břesinová, F. Lackner, and J. Burgdörfer, Phys. Rev. A 94, 023405 (2016).
  * Sawada, Sato, and Ishikawa (2016) R. Sawada, T. Sato, and K. L. Ishikawa, Phys. Rev. A 93, 023434 (2016).
  * Omiste, Li, and Madsen (2017) J. J. Omiste, W. Li, and L. B. Madsen, Phys. Rev. A 95, 053422 (2017).
  * Helgaker, Jørgensen, and Olsen (2002) T. Helgaker, P. Jørgensen, and J. Olsen,  _Molecular Electronic-Structure Theory_ (Wiley, 2002).
  * Szabo and Ostlund (1996) A. Szabo and N. S. Ostlund,  _Modern Quantum Chemistry_ (Dover, Mineola, 1996).
  * Kümmel (2003) H. G. Kümmel, Int. J. Mod. Phys. B 17, 5311 (2003).
  * Shavitt and Bartlett (2009) I. Shavitt and R. J. Bartlett,  _Many-body methods in chemistry and physics: MBPT and coupled-cluster theory_ (Cambridge university press, 2009).
  * Arponen (1983) J. Arponen, Ann. Phys. 151, 311 (1983).
  * Sato  _et al._ (2018) T. Sato, H. Pathak, Y. Orimo, and K. L. Ishikawa, The Journal of chemical physics 148 (2018).
  * Pathak, Sato, and Ishikawa (2020a) H. Pathak, T. Sato, and K. L. Ishikawa, The Journal of chemical physics 152 (2020a).
  * Pathak, Sato, and Ishikawa (2020b) H. Pathak, T. Sato, and K. L. Ishikawa, The Journal of Chemical Physics 153 (2020b).
  * Pathak, Sato, and Ishikawa (2021) H. Pathak, T. Sato, and K. L. Ishikawa, The Journal of Chemical Physics 154 (2021).
  * Kvaal (2012) S. Kvaal, J. Chem. Phys. 136, 194109 (2012).
  * Madsen  _et al._ (2020c) N. K. Madsen, M. B. Hansen, O. Christiansen, and A. Zoccante, The Journal of Chemical Physics 153 (2020c).
  * Højlund, Zoccante, and Christiansen (2024) M. G. Højlund, A. Zoccante, and O. Christiansen, The Journal of Chemical Physics 160 (2024).
  * Højlund and Christiansen (2024) M. G. Højlund and O. Christiansen, arXiv preprint arXiv:2402.11378 (2024).
  * Højlund  _et al._ (2022) M. G. Højlund, A. B. Jensen, A. Zoccante, and O. Christiansen, The Journal of Chemical Physics 157 (2022).
  * Jensen  _et al._ (2023) A. B. Jensen, M. G. Højlund, A. Zoccante, N. K. Madsen, and O. Christiansen, The Journal of Chemical Physics 159 (2023).
  * G. E. Scuseria and H. F. Schaefer III (1987) G. E. Scuseria and H. F. Schaefer III, Chem. Phys. Lett. 142, 354 (1987).
  * Sherrill  _et al._ (1998) C. D. Sherrill, A. I. Krylov, E. F. C. Byrd, and M. Head-Gordon, J. Chem. Phys. 109, 4171 (1998).
  * Krylov  _et al._ (1998) A. I. Krylov, C. D. Sherrill, E. F. C. Byrd, and M. Head-Gordon, J. Chem. Phys. 109, 10669 (1998).
  * Köhn and Olsen (2005) A. Köhn and J. Olsen, J. Chem. Phys. 122, 084116 (2005).
  * Pedersen, Fernández, and Koch (2001) T. B. Pedersen, B. Fernández, and H. Koch, The Journal of chemical physics 114, 6983 (2001).
  * Myhre (2018) R. H. Myhre, The Journal of chemical physics 148 (2018).
  * Huber and Klamroth (2011) C. Huber and T. Klamroth, J. Chem. Phys. 134, 054113 (2011).
  * D. R. Nascimento and A. E. DePrince III (2016) D. R. Nascimento and A. E. DePrince III, J. Chem. Theory Comput. 12, 5834 (2016).
  * Raghavachari  _et al._ (1990) K. Raghavachari, J. A. Pople, E. S. Replogle, M. Head-Gordon, and N. C. Handy, Chemical physics letters 167, 115 (1990).
  * Hampel, Peterson, and Werner (1992) C. Hampel, K. A. Peterson, and H.-J. Werner, Chemical physics letters 190, 1 (1992).
  * Chiles and Dykstra (1981) R. A. Chiles and C. E. Dykstra, The Journal of Chemical Physics 74, 4544 (1981).
  * Handy  _et al._ (1989) N. C. Handy, J. A. Pople, M. Head-Gordon, K. Raghavachari, and G. W. Trucks, Chemical physics letters 164, 185 (1989).
  * (61) The pre-determined constraints can be further classified into those that remove the redundancy of the wavefunction parameterization, which are intra-group orbital rotations, and those that restrict the variational space of the wavefunction, which are frozen-core related orbital rotations. Strictly speaking, the first subclass of the first type of constraint is the true constraint, the second subclass of the first type of constraint is the restriction, and the second type of constraint is the variational parameter. However, we do not distinguish these three types of constraints since “constraint” is widely used in referencesMadsen  _et al._ (2020c); Højlund, Zoccante, and Christiansen (2024); Højlund and Christiansen (2024); Sverdrup Ofstad  _et al._ (2023); Højlund  _et al._ (2022); Jensen  _et al._ (2023).
  * Pedersen, Koch, and Hättig (1999) T. B. Pedersen, H. Koch, and C. Hättig, J. Chem. Phys. 110, 8318 (1999).
  * Madsen  _et al._ (2020d) N. K. Madsen, A. B. Jensen, M. B. Hansen, and O. Christiansen, The Journal of Chemical Physics 153 (2020d).
  * Beck  _et al._ (2000) M. H. Beck, A. Jäckle, G. A. Worth, and H.-D. Meyer, Physics reports 324, 1 (2000).
  * Pedersen and Kvaal (2019) T. B. Pedersen and S. Kvaal, The Journal of chemical physics 150 (2019).
  * Kristiansen  _et al._ (2020) H. E. Kristiansen, Ø. S. Schøyen, S. Kvaal, and T. B. Pedersen, The Journal of chemical physics 152 (2020).
  * Sato, Teramura, and Ishikawa (2018) T. Sato, T. Teramura, and K. L. Ishikawa, Applied Sciences 8, 433 (2018).
  * Han and Madsen (2010) Y.-C. Han and L. B. Madsen, Physical Review A 81, 063430 (2010).
  * (69) The truncation in TDMVCC is a TDH-TDMVCC hybrid form, which is equivalent to setting the weights of the TDH kinds as infinity.
  * Worth and Cederbaum (2004) G. A. Worth and L. S. Cederbaum, Annu. Rev. Phys. Chem. 55, 127 (2004).
  * Köuppel, Domcke, and Cederbaum (1984) H. Köuppel, W. Domcke, and L. S. Cederbaum, Advances in chemical physics , 59 (1984).
  * Bao, Raymond, and Nooijen (2024) S. Bao, N. Raymond, and M. Nooijen, The Journal of Chemical Physics 160 (2024).
  * Sibaev  _et al._ (2020) M. Sibaev, I. Polyak, F. R. Manby, and P. J. Knowles, The Journal of Chemical Physics 153 (2020).
  * Muolo  _et al._ (2020) A. Muolo, A. Baiardi, R. Feldmann, and M. Reiher, The Journal of chemical physics 152 (2020).
  * Sasmal and Vendrell (2020) S. Sasmal and O. Vendrell, The Journal of Chemical Physics 153 (2020).
  * Sasmal and Vendrell (2022) S. Sasmal and O. Vendrell, The Journal of Chemical Physics 157 (2022).
  * Sasmal, Schröder, and Vendrell (2024) S. Sasmal, M. Schröder, and O. Vendrell, The Journal of Chemical Physics 160 (2024).
  * Goldstein (1980) H. Goldstein,  _Classical Mechanics (2nd ed.)_ (Addison-Wesley., 1980).