1

# Coupled-Cluster Theory Revisited
Part II: Analysis of the single-reference Coupled-Cluster equations

Mihály A. Csirik<EMAIL_ADDRESS>Hylleraas Centre for Quantum
Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033
Blindern, N-0315 Oslo, Norway ()  and  Andre Laestadius
<EMAIL_ADDRESS>Department of Computer Science, Oslo Metropolitan
University, P.O. Box 4 St. Olavs plass, NO-0130 Oslo, Norway ()

(Date: Received June 22, 2022. Accepted November 30, 2022.)

###### Abstract.

In a series of two articles, we propose a comprehensive mathematical framework
for Coupled-Cluster-type methods. In this second part, we analyze the
nonlinear equations of the single-reference Coupled-Cluster method using
topological degree theory. We establish existence results and qualitative
information about the solutions of these equations that also sheds light of
the numerically observed behavior. In particular, we compute the topological
index of the zeros of the single-reference Coupled-Cluster mapping. For the
truncated Coupled-Cluster method, we derive an energy error bound for
approximate eigenstates of the Schrödinger equation.

###### Key words and phrases:

quantum mechanics, many-body problem, quantum chemistry, electronic structure,
coupled-cluster theory, nonlinear analysis, topological degree, Brouwer degree

###### 1991 Mathematics Subject Classification:

81V55, 81-08, 81-10, 47H11

##  1\. Introduction

The present article is concerned with the analysis of the single-reference
Coupled-Cluster (SRCC) equations and its truncated variants. It is known that
the _truncated_ CC equations have a large number of solutions, some of which
exhibit unphysical behavior. Recall that the truncated CC equations are _not_
equivalent to the truncated CI equations (see Section 2.3 of Part I), hence,
there is no obvious connection with the Schrödinger equation in
general—truncated CC solutions can be very far from the desired Full CI
solutions. Also, the SRCC method is typically unreliable for treating
degenerate states.

Since the early days of the CC method, researchers in quantum chemistry have
been interested in understanding the complicated behavior of the (truncated)
CC equations [44, 45, 30, 20, 22, 19, 32, 17, 18, 21]. This line of
investigation mainly consisted in somehow “connecting” the truncated CC
solutions to the untruncated ones, thereby identifying which truncated
solutions can be considered “physical” and which should be discarded as
“unphysical”. The comprehensive investigation by K. Kowalski and P. Piecuch
[32] classifies solutions based on a certain homotopy. While the universality
of this approach is unclear (see Remark 4.30 below), [32] demonstrates without
doubt that the solutions of the truncated CC equations exhibit rather
complicated behavior, which signals the need for a deeper analytical
investigation. Since the CC equations are a system of quartic polynomial
equations, the complete set of solutions can be computed by numerical means
(Kowalski and Piecuch used the HOMPACK software [40]). However, in our
analysis we view the CC equations as an abstract nonlinear equation and
disregard their polynomial structure (this route was also taken by the seminal
[37] and its follow-up works, discussed below). The main reason for this is
that system is _very large_ , and this seems prohibitive for the application
of algebraic approaches.

In this article, we analyze the CC equations and the homotopy of Kowalski and
Piecuch using a standard tool of nonlinear analysis, topological degree
theory. In order to do this, we need to restrict ourselves to the finite-
dimensional case (Remark 3.1). The present article differs in flavor from the
previous mathematical investigations, which where more standard “numerical
analysis” approaches. However, a closer look at the qualitative properties of
the (truncated) CC equations seems necessary, that, in addition to the ground-
state also describes excited states.

###  1.1. Previous work

Similarly to Part I, our approach is based on the analysis of the single-
reference CC method by R. Schneider [37]. In that seminal work, a thorough
description of the basic building blocks of the method, namely excitation- and
cluster operators, and their algebraic and functional-analytic properties are
given. Under certain assumptions, the CC equations (a nonlinear system of
equations consisting of quartic polynomials) are formulated in terms of a
locally strongly monotone and locally Lipschitz operator defined on an
appropriate space. Under further hypotheses, this establishes local existence
and uniqueness of a (Galerkin projected) ground-state solution of the equation
and moreover _quasi-optimality_ of the projected CC solution (Theorem 5.8
ibid.). Perhaps the most important contribution of [37] is a quadratic energy
error estimate (Theorem 6.3 ibid.). It is worth emphasizing that Schneider’s
analysis is a local one: _“[…] experience indicates that, in general, it
cannot be expected that strong monotonicity always holds, or the constants
might be extremely bad. Therefore we expect to get local existence results at
best.”_ (ibid. p. 30)

Schneider’s original analysis was carried out in the finite-dimensional case
only. This was remedied in two subsequent articles by T. Rohwedder [35] and
then by both of them [36]. The article [35] establishes important technical
tools and rigorously proves that the untruncated CC problem is equivalent to
the Full CI problem in the infinite-dimensional case, i.e. to essentially the
Schrödinger equation. Using the said tools, the subsequent paper [36] also
establishes local uniqueness and existence of a solution to the truncated CC
equations in a neighborhood of the untruncated CC solution, under certain
assumptions. Further, [36] also extends the energy error estimates of [37] to
the infinite-dimensional case.

This line of investigation was continued by S. Kvaal and A. L. in [24] for the
Extended CC (ECC) method based on the “bivariational principle” [1]. In this
case, local strong monotonicity for the ECC mapping can be established so that
quasi-optimality follows along similar lines as previously done by Schneider
and Rohwedder. In the ECC theory, the traditional CC theory is recovered as a
special case.

Furthermore, the local strong monotonicity-based analysis was applied to a
variant of the traditional CC method by F. M. Faulstich et al. [12], namely to
the Tailored Coupled-Cluster (TCC) method. The TCC approach splits the
computational task of evaluating the ground state energy into two parts:
solving for the statically correlated wave function on a complete active
space, and then on top of that accounting for the dynamical correlation using
the CC method. Numerical investigations based on [12] were conducted in [13].

Finally, we mention the survey article [23] for more details on the use of
local strong monotonicity-based methods in the analysis of CC methods.

###  1.2. Outline

While the knowledge of the notations and results of Part I is not strictly
necessary for this Part II, some of the results in Section 4 do employ
_excitation graphs_ to a certain extent, introduced in Section 3.2 of Part I.
To make the present work as self-contained as possible, we define all the
necessary concepts without the use of Part I; in other words, in the
“traditional” second-quantized way.

In Section 2, we describe the setting of the quantum-mechanical problems the
CC theory is aimed at. In Section 3 we state a few results that we use from
finite-dimensional topological degree theory.

The analysis of the single-reference CC (SRCC) method begins in Section 4.
Basic properties of the SRCC mapping are discussed in Section 4.1. After this,
the local properties of the SRCC mapping, such as strong monotonicity and
topological index in both the non-degenerate-, and in the degenerate case are
considered in Section 4.2. We also look at the complex SRCC mapping in Section
4.3.

In Section 4.4, an important class of homotopies is defined that can be used
for proving the existence of a solution to the truncated SRCC mappings. In
Section 4.5 a homotopy is considered that was invented specifically to connect
CC methods of different truncation levels. We prove an existence result and
calculate the topological index of the homotopy. Finally, we derive an energy
error estimate in Section 4.6 using the results of Appendix B.

##  2\. Background

The usual notation $B(a,r)$ is used for the open ball of radius $r$ and center
$a$, also $B^{*}(a,r)=B(a,r)\smallsetminus\\{a\\}$ denotes the punctured ball.

The spectrum of a linear operator $A$ is written $\sigma(A)$, the elements of
its discrete spectrum as $\mathcal{E}_{n}(A)$, where $n=0,1,2,\ldots$, if $A$
is bounded from below. We use the usual notation $[A,B]=AB-BA$ for the
commutator. The (conjugate) transpose of $A$ is denoted as $A^{\dagger}$. For
normed spaces $V$ and $W$, the symbol $\mathcal{L}(V,W)$ denotes normed space
of _bounded_ linear mappings $V\to W$ endowed with the operator norm
$\|\cdot\|_{\mathcal{L}(V,W)}$. Furthermore, $V^{*}$ denotes the (continuous)
dual space. As usual, $[a,b]$ denotes the (closed) line segment between
$a,b\in V$.

###  2.1. Second quantization

In this section, we first briefly review the framework of second quantization.
The notations mainly follow [25] and [39]. Next, we formulate the Schrödinger
Hamiltonian in second quantization. Finally, we introduce the excitation-, and
cluster operators and cluster amplitude spaces through the use of creation-,
and annihilation operators.

####  2.1.1. Fermionic Fock space

For simplicity, and because the Hamiltonian we will be considering is spin-
independent, we neglect spin and consider $\mathfrak{h}=L^{2}(\mathbb{R}^{3})$
as the _one-particle Hilbert space_. The tensor powers of $\mathfrak{h}$ are
denoted by $\mathfrak{h}^{N}=\bigotimes^{N}\mathfrak{h}$ and antisymmetric
powers of $\mathfrak{h}$ are denoted by
$\mathfrak{h}_{a}^{N}=\bigwedge^{N}\mathfrak{h}$ for any $N\geq 0$, where we
introduced the notation $\mathfrak{h}^{0}=\mathbb{C}$ and
$\mathfrak{h}_{a}^{0}=\mathbb{C}$. The _fermionic Fock space_ is then the
Hilbert space given by the infinite direct sum
$\mathfrak{F}_{a}:=\bigoplus_{N\geq
0}\mathfrak{h}_{a}^{N}=\mathfrak{h}_{a}^{0}\oplus\mathfrak{h}_{a}^{1}\oplus\mathfrak{h}_{a}^{2}\oplus\ldots$,
in other words

$\mathfrak{F}_{a}=\Big{\\{}\Psi=(\psi^{0},\psi^{1},\ldots):\psi^{N}\in\mathfrak{h}_{a}^{N}\,(N\geq
0),\;\|\Psi\|_{\mathfrak{F}}^{2}:=\sum_{N\geq
0}\|\psi^{N}\|_{\mathfrak{h}^{N}}^{2}<\infty\Big{\\}},$

endowed with the inner product
$\langle\Psi_{1},\Psi_{2}\rangle_{\mathfrak{F}}=\sum_{N\geq
0}\langle\psi_{1}^{N},\psi_{2}^{N}\rangle_{\mathfrak{h}^{N}}$, for any
$\Psi_{1}=(\psi_{1}^{0},\psi_{1}^{1},\ldots)\in\mathfrak{F}_{a}$ and
$\Psi_{2}=(\psi_{2}^{0},\psi_{2}^{1},\ldots)\in\mathfrak{F}_{a}$. The _vacuum
state_ is the distinguished element $\Omega=(1,0,\ldots)\in\mathfrak{F}_{a}$.
The space $\mathfrak{h}_{a}^{N}$ can be identified as a subspace of
$\mathfrak{F}_{a}$, in this context it is called the _$N$ -particle sector of
$\mathfrak{F}$_. For $\Psi_{1}\in\mathfrak{h}_{a}^{N_{1}}$ and
$\Psi_{2}\in\mathfrak{h}_{a}^{N_{2}}$ define
$\Psi_{1}\wedge\Psi_{2}\in\mathfrak{h}_{a}^{N_{1}+N_{2}}$ via

$\Psi_{1}\wedge\Psi_{2}(\mathbf{x}_{1},\ldots,\mathbf{x}_{N_{1}+N_{2}})=C_{N_{1},N_{2}}\sum_{\sigma\in\mathfrak{S}_{N_{1}+N_{2}}}(\operatorname{sgn}\,\sigma)\Psi_{1}(\mathbf{x}_{\sigma(1)},\ldots,\mathbf{x}_{\sigma(N_{1})})\Psi_{2}(\mathbf{x}_{\sigma(N_{1}+1)},\ldots,\mathbf{x}_{\sigma(N_{1}+N_{2})}),$

with the normalization constant
$C_{N_{1},N_{2}}=(N_{1}!N_{2}!(N_{1}+N_{2})!)^{-1/2}$. Here,
$\mathfrak{S}_{n}$ denotes the permutation group of $\\{1,\ldots,n\\}$ and
$\operatorname{sgn}\sigma$ the sign of the permutation $\sigma$.

Let $\\{\varphi_{p}\\}_{p=1}^{\infty}\subset\mathfrak{h}$ be an orthonormal
basis and define the _Slater determinant_
$\Phi_{\alpha}:=\varphi_{\alpha_{1}}\wedge\ldots\wedge\varphi_{\alpha_{N}}\in\mathfrak{h}_{a}^{N}$
for every multiindex $\alpha\in\mathbb{N}_{0}^{N}$ with
$\alpha_{1}<\ldots<\alpha_{N}$. Then $\\{\Phi_{\alpha}\\}_{\alpha}$ is an
orthonormal basis of $\mathfrak{h}_{a}^{N}$.

####  2.1.2. Creation-, and annihilation operators

For fixed $\varphi\in\mathfrak{h}$, the (fermionic) _creation operator_
$a^{\dagger}(\varphi):\mathfrak{F}_{a}\to\mathfrak{F}_{a}$ is defined as
$a^{\dagger}(\varphi)\Psi=\varphi\wedge\Psi$ for any
$\Psi\in\mathfrak{h}_{a}^{N}$ and extended boundedly and linearly to the whole
space. Also, for fixed $\varphi\in\mathfrak{h}$, define the (fermionic)
_annihilation operator_ $a(\varphi):\mathfrak{F}_{a}\to\mathfrak{F}_{a}$ as
the adjoint of $a^{\dagger}(\varphi)$, i.e.
$\langle\Psi_{1},a^{\dagger}(\varphi)\Psi_{2}\rangle_{\mathfrak{F}}=\langle
a(\varphi)\Psi_{1},\Psi_{2}\rangle_{\mathfrak{F}}$ for all
$\Psi_{1},\Psi_{2}\in\mathfrak{F}_{a}$. It is an easy calculation to show that
$(a(\varphi)\Psi)(\mathbf{x}_{1},\ldots,\mathbf{x}_{N-1})=N^{-1/2}\int\varphi(\mathbf{x})\Psi(\mathbf{x},\mathbf{x}_{1},\ldots,\mathbf{x}_{N-1})\,\mathrm{d}\mathbf{x}$
for all $\Psi\in\mathfrak{h}_{a}^{N}$ ($N\geq 1$) and $a(\varphi)\Omega=0$.
The _canonical anticommutation relations (CAR)_

$\displaystyle
a^{\dagger}(\varphi)a(\varphi^{\prime})+a(\varphi^{\prime})a^{\dagger}(\varphi)$
$\displaystyle=\langle\varphi,\varphi^{\prime}\rangle_{\mathfrak{h}}I_{\mathfrak{F}_{a}},$
$\displaystyle
a^{\dagger}(\varphi)a^{\dagger}(\varphi^{\prime})+a^{\dagger}(\varphi^{\prime})a^{\dagger}(\varphi)$
$\displaystyle=0,$ $\displaystyle
a(\varphi)a(\varphi^{\prime})+a(\varphi^{\prime})a(\varphi)$
$\displaystyle=0,$

hold true for any $\varphi,\varphi^{\prime}\in\mathfrak{h}$, where
$I_{\mathfrak{F}_{a}}:\mathfrak{F}_{a}\to\mathfrak{F}_{a}$ denotes the
identity map on $\mathfrak{F}_{a}$. Since both $a^{\dagger}(\varphi)$ and
$a(\varphi)$ are linear and bounded in $\varphi$, it is enough to specify them
on an orthonormal basis $\\{\varphi_{p}\\}_{p\geq 1}\subset\mathfrak{h}$, i.e.
$a_{p}^{\dagger}:=a^{\dagger}(\varphi_{p})$ and $a_{p}:=a(\varphi_{p})$
($p\geq 1$) completely determines the families
$\\{a^{\dagger}(\varphi)\\}_{\varphi\in\mathfrak{h}}$ and
$\\{a(\varphi)\\}_{\varphi\in\mathfrak{h}}$. It is then clear that any Slater
determinant $\Phi_{\alpha}\in\mathfrak{h}_{a}^{N}$ can be “created” from the
vacuum state as $\Phi_{\alpha}=a_{\alpha_{1}}^{\dagger}\cdots
a_{\alpha_{N}}^{\dagger}\Omega$.

####  2.1.3. First-, and second quantization of one-, and two-body operators

Let $h:\mathfrak{h}\to\mathfrak{h}$ be a linear operator. Define its _first
quantization_ as $\sum_{1\leq j\leq
N}h_{j}:\mathfrak{h}_{a}^{N}\to\mathfrak{h}_{a}^{N}$ where

$h_{j}=I\otimes\ldots\otimes I\otimes\underbrace{h}_{\text{$j$th}}\otimes
I\otimes\ldots\otimes I.$

Here, $\mathbbold{1}$ denotes the identity on $\mathfrak{h}$. The _second
quantization_ of $h$ is defined as $0\oplus\bigoplus_{N\geq 1}\sum_{1\leq
j\leq N}h_{j}:\mathfrak{F}_{a}\to\mathfrak{F}_{a}$.

Analogously, one can define the first quantization of a two-body operator
$W:\mathfrak{h}_{a}^{2}\to\mathfrak{h}_{a}^{2}$ as $\sum_{1\leq i,j\leq
N}W_{ij}:\mathfrak{h}_{a}^{N}\to\mathfrak{h}_{a}^{N}$, where

$W_{ij}=I\otimes\ldots\otimes I\otimes\underbrace{W}_{\text{$i$th}}\otimes
I\ldots I\otimes\underbrace{W}_{\text{$j$th}}\otimes I\otimes\ldots\otimes I.$

The second quantization of $W$ is $0\oplus 0\oplus\bigoplus_{N\geq
2}\sum_{1\leq i<j\leq N}W_{ij}:\mathfrak{F}_{a}\to\mathfrak{F}_{a}$. Both the
one-, and the two-body operators can be expressed using creation-, and
annihilation operators as follows.

######  Theorem 2.1.

Let $\\{\varphi_{p}\\}_{p\geq 1}\subset\mathfrak{h}$ be an orthonormal basis.
Then

$0\oplus\bigoplus_{N\geq 1}\sum_{1\leq j\leq N}h_{j}=\sum_{p,q\geq 1}\langle
h\varphi_{p},\varphi_{q}\rangle a_{p}^{\dagger}a_{q}.$

and

$0\oplus 0\oplus\bigoplus_{N\geq 2}\sum_{1\leq i<j\leq
N}W_{ij}=\sum_{\begin{subarray}{c}1\leq p\leq q\\\ 1\leq r\leq
s\end{subarray}}\langle
W\varphi_{p}\wedge\varphi_{q},\varphi_{r}\wedge\varphi_{s}\rangle
a_{p}^{\dagger}a_{q}^{\dagger}a_{r}a_{s}.$

###  2.2. Schrödinger Hamiltonian in second-quantized form

For convenience and in accordance to Part I, we define the $N$-particle
fermionic spaces

$\mathfrak{L}^{2}:=\mathfrak{h}_{a}^{N}=\bigwedge^{N}L^{2}(\mathbb{R}^{3}),\quad\text{and}\quad\mathfrak{H}^{1}:=\mathfrak{L}^{2}\cap
H^{1}(\mathbb{R}^{3N}),$

endowed with usual inner products $\langle\cdot,\cdot\rangle$ and
$\langle\cdot,\cdot\rangle_{\mathfrak{H}^{1}}$, and norms $\|\cdot\|$ and
$\|\cdot\|_{\mathfrak{H}^{1}}$, see Section 2.1 in Part I. We also set
$\mathfrak{H}^{2}:=\mathfrak{L}^{2}\cap H^{2}(\mathbb{R}^{3N})$, and
$\mathfrak{H}^{-1}:=(\mathfrak{H}^{1})^{*}$ as usual. Henceforth, we employ
the same convention as explained after Remark 2.1 in Part I.

Recall (see Section 2.2 of Part I) the Schrödinger Hamiltonian
$\mathcal{H}:\mathfrak{L}^{2}\to\mathfrak{L}^{2}$, which is the self-adjoint
operator

$\mathcal{H}=\sum_{1\leq i\leq
N}\Bigg{[}-\frac{1}{2}\triangle_{\mathbf{x}_{i}}+V(\mathbf{x}_{i})\Bigg{]}+\sum_{1\leq
i<j\leq N}w(\mathbf{x}_{i}-\mathbf{x}_{j}),$

with domain $D(\mathcal{H})=\mathfrak{H}^{2}$, where
$V,w:\mathbb{R}^{3}\to\mathbb{R}$ are Kato class potentials: $V,w\in
L^{3/2}(\mathbb{R}^{3})+L^{\infty}_{\varepsilon}(\mathbb{R}^{3})$ and $w$ is
even. The quadratic form corresponding to $\mathcal{H}$ is denoted as
$\mathcal{E}(\Psi)=\langle\mathcal{H}\Psi,\Psi\rangle$, and has form domain
$Q(\mathcal{E})=\mathfrak{H}^{1}$. Recall that there is a constant $M>0$, such
that

$\langle\mathcal{H}\Psi,\Phi\rangle\leq
M\|\Psi\|_{\mathfrak{H}^{1}}\|\Phi\|_{\mathfrak{H}^{1}}$ (2.1)

for all $\Psi,\Phi\in\mathfrak{H}^{1}$. Thus, $\mathcal{H}$ can be extended to
a bounded mapping $\mathfrak{H}^{1}\to\mathfrak{H}^{-1}$, which we denote with
the same symbol. We say that $\Psi\in\mathfrak{H}^{1}$ and
$\mathcal{E}\in\mathbb{R}$ satisfy the _weak Schrödinger equation_ if
$\langle\mathcal{H}\Psi,\Phi\rangle=\mathcal{E}\langle\Psi,\Phi\rangle$ for
all $\Phi\in\mathfrak{H}^{1}$.

According to Theorem 2.1, $\mathcal{H}$ can also be given in the second
quantized form

$\mathcal{H}=\sum_{p,q\geq
1}h_{pq}a_{p}^{\dagger}a_{q}+\sum_{\begin{subarray}{c}1\leq p\leq q\\\ 1\leq
r\leq s\end{subarray}}W_{pq,rs}a_{p}^{\dagger}a_{q}^{\dagger}a_{r}a_{s},$

where
$h_{pq}=\frac{1}{2}\langle\bm{\nabla}\varphi_{p},\bm{\nabla}\varphi_{q}\rangle+\langle
V\varphi_{p},\varphi_{q}\rangle$ and $W_{pq,rs}=\langle
W\varphi_{p}\wedge\varphi_{q},\varphi_{r}\wedge\varphi_{s}\rangle$.

####  2.2.1. Truncation of the orbital basis

Our analysis of the CC method is restricted to the finite-dimensional case due
to reasons described later (Remark 3.1). Let $K\geq N$ and assume that an
$L^{2}$-orthonormal _(spin-)orbital set_ $\\{\varphi_{p}\\}_{1\leq p\leq
K}\subset H^{1}(\mathbb{R}^{3})$ is given. We define

$\mathfrak{B}_{K}=\\{\Phi_{\alpha}\in\mathfrak{H}^{1}:1\leq\alpha_{1}<\ldots<\alpha_{N}\leq
K\\},$

as in Section 2.1 of Part I. Then $\mathfrak{B}_{K}$ is
$\mathfrak{L}^{2}$-orthonormal. Set

$\mathfrak{H}^{1}_{K}=\operatorname{Span}\mathfrak{B}_{K}\subset\mathfrak{H}^{1}.$

We define $\mathcal{H}_{K}:\mathfrak{H}_{K}^{1}\to(\mathfrak{H}_{K}^{1})^{*}$
to be the projection of $\mathcal{H}:\mathfrak{H}^{1}\to\mathfrak{H}^{-1}$
onto the finite-dimensional subspace
$\mathfrak{H}_{K}^{1}\subset\mathfrak{H}^{1}$, i.e.
$\langle\mathcal{H}_{K}\Phi,\Psi\rangle=\langle\mathcal{H}\Phi,\Psi\rangle$
for all $\Phi,\Psi\in\mathfrak{H}_{K}^{1}$. Then $\Psi\in\mathfrak{H}^{1}_{K}$
and $\mathcal{E}\in\mathbb{R}$ are said to satisfy the _projected Schrödinger
equation_ if

$\langle\mathcal{H}_{K}\Psi,\Phi\rangle=\mathcal{E}\langle\Psi,\Phi\rangle\quad\text{for
all}\quad\Phi\in\mathfrak{H}_{K}^{1}.$

For a convergence theory for this eigenvalue problem, see e.g. [41, Theorem
5.11]. Vaguely speaking, if $\bigcup_{K\geq N}\\{\varphi_{p}\\}_{1\leq p\leq
K}$ is dense in $H^{1}(\mathbb{R}^{3})$, then convergence of the projected
eigenvalues to the discrete spectrum of $\mathcal{H}$ is guaranteed as
$K\to\infty$ (see [37]).

From the computational standpoint, orbital basis truncation is always
necessary when solving the Schrödinger equation. Of course, the spectral
structure changes (the essential spectrum disappears in finite dimensions),
and this means that we replaced the original problem with one having different
qualitative properties. However, the CI and CC methods are mainly used to
approximate the ground-state-, or the first few excited energies (and other
quantities related to the lowest eigenstates). With a sufficiently good choice
the orbitals $\\{\varphi_{p}\\}_{1\leq p\leq K}$, these energies can be
approximated rather well in a lot of situations. Therefore, it is safe to
regard the rest of the projected spectrum as “junk”, and simply ignore it. We
can conclude that the restriction to finite dimensions allows for a meaningful
theory that is still relevant to CC methods.

###  2.3. The Hartree–Fock method

In practice, the CC method usually takes the HF orbitals as an input and
builds up correction based on them, as described in Section 2.3 of Part I.
Here, we collect the basic facts about the HF method that will be used later
on.

The HF method111a.k.a. Self-Consistent Field (SCF) method is based on the
minimiziation of the energy over Slater determinants. It is fairly easy to see
by direct calculation that the energy
$\mathcal{E}(\Phi)=\langle\mathcal{H}\Phi,\Phi\rangle$ of a Slater determinant
$\Phi=\varphi_{1}\wedge\ldots\wedge\varphi_{N}\in\mathfrak{H}^{1}$, with
$\\{\varphi_{j}\\}_{j=1}^{N}\subset H^{1}(\mathbb{R}^{3})$ being
$L^{2}$-orthonormal is given by

$\displaystyle\mathcal{E}(\varphi_{1},\ldots,\varphi_{N}):=\mathcal{E}(\Phi)$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{N}\int_{\mathbb{R}^{3}}|\bm{\nabla}\varphi_{i}|^{2}+\sum_{i=1}^{N}\int_{\mathbb{R}^{3}}V|\varphi_{i}|^{2}$
$\displaystyle+\frac{1}{2}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\Bigg{[}\sum_{i,j=1}^{N}|\varphi_{i}(\mathbf{x})|^{2}|\varphi_{j}(\mathbf{y})|^{2}-\Bigg{|}\sum_{i=1}^{N}\varphi_{i}(\mathbf{x})\overline{\varphi_{i}(\mathbf{y})}\Bigg{|}^{2}\Bigg{]}w(\mathbf{x}-\mathbf{y})\,\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y},$

see e.g. [7]. Hence the task is to determine the _HF energy_

$\mathcal{E}_{\mathrm{HF}}=\mathcal{E}(\Phi_{\mathrm{HF}})=\min_{\begin{subarray}{c}\varphi_{1},\ldots,\varphi_{N}\in
H^{1}(\mathbb{R}^{3})\\\
\langle\varphi_{i},\varphi_{j}\rangle=\delta_{ij}\end{subarray}}\mathcal{E}(\varphi_{1},\ldots,\varphi_{N}),$
(2.2)

along with a _HF minimizer_ (or _HF determinant_) $\Phi_{\mathrm{HF}}$. The
existence of a HF minimizer $\Phi_{\mathrm{HF}}$ is guaranteed for the case of
positive ions and neutral atoms and molecules (corresponding to the electronic
molecular Hamiltonian).

######  Theorem 2.2.

[27] If $N<Z+1$, then there exists a minimizer $\Phi_{\mathrm{HF}}$ to Eq.
2.2.

Following the seminal work [27], much more has been discovered about the
mathematical structure of the HF energy functional [28, 14, 26]. As usual, a
minimizer satisfies the corresponding Euler–Lagrange equations (which are
called _Hartree-Fock equations_ in this context). In practice, it is this
system of nonlinear integro-differential equations which is discretized and
solved. To describe the HF equations, we make a definition that will be
convenient for later purposes. Fix
$\Phi=\varphi_{1}\wedge\ldots\wedge\varphi_{N}$, where
$\\{\varphi_{p}\\}_{p=1}^{N}\subset H^{1}(\mathbb{R}^{3})$ is
$L^{2}$-orthonormal. Define the lower semibounded, self-adjoint operator
$F_{\Phi}:L^{2}(\mathbb{R}^{3})\to L^{2}(\mathbb{R}^{3})$ with domain
$D(F_{\Phi})=H^{2}(\mathbb{R}^{3})$ via the instruction

$\displaystyle(F_{\Phi}\psi)(\mathbf{x})$
$\displaystyle=-\frac{1}{2}\triangle\psi(\mathbf{x})+V(\mathbf{x})\psi(\mathbf{x})+\left(\sum_{i=1}^{N}\int_{\mathbb{R}^{3}}w(\mathbf{x}-\mathbf{y})|\varphi_{i}(\mathbf{y})|^{2}\right)\psi(\mathbf{x})-\sum_{i=1}^{N}\left(\int_{\mathbb{R}^{3}}w(\mathbf{x}-\mathbf{y})\overline{\varphi_{i}(\mathbf{y})}\psi(\mathbf{y})\,\mathrm{d}\mathbf{y}\right)\varphi_{i}(\mathbf{x})$

for all $\psi\in D(F_{\Phi})$ and all $\mathbf{x}\in\mathbb{R}^{3}$. The
operator $F_{\Phi}$ is called the _mean-field operator_.222It is sometimes
called the Fock operator, but we reserve that name for its $N$-particle
version, i.e. its first quantization. The form domain of $F_{\Phi}$ is
$H^{1}(\mathbb{R}^{3})$. The essential spectrum of $F_{\Phi}$ is
$[0,+\infty)$. We summarize the basic properties of the mean-field operator in
the next theorem. Let

$\mu_{n}(F_{\Phi})=\min_{\begin{subarray}{c}\psi_{1},\ldots,\psi_{n}\in
H^{1}(\mathbb{R}^{3})\\\
\langle\varphi_{i},\varphi_{j}\rangle=\delta_{ij}\end{subarray}}\;\max_{\begin{subarray}{c}\psi\in\operatorname{Span}\\{\psi_{1},\ldots,\psi_{n}\\}\\\
\|\psi\|=1\end{subarray}}\langle F_{\Phi}\psi,\psi\rangle$

denote the min-max values of $F_{\Phi}$.

######  Theorem 2.3.

Assume that there exists a HF minimizer
$\Phi_{\mathrm{HF}}=\varphi_{1}\wedge\ldots\wedge\varphi_{N}$.

  1. (i)

(Hartree-Fock equations) There exists a unitary matrix
$\mathbf{U}\in\mathbb{C}^{N\times N}$ so that $\widetilde{\varphi}_{i}$ are
eigenfunctions of $F_{\Phi}$ corresponding to its $N$ lowest eigenvalues
$\lambda_{1}\leq\ldots\leq\lambda_{N}$,

$F_{\Phi}\widetilde{\varphi}_{j}=\lambda_{j}\widetilde{\varphi}_{j},\quad\text{for
all}\quad j=1,\ldots,N,$ (2.3)

where
$(\widetilde{\varphi}_{1},\ldots,\widetilde{\varphi}_{N})=\mathbf{U}(\varphi_{1},\ldots,\varphi_{N})$.

  2. (ii)

(Aufbau principle) If $\mu_{N+1}(F_{\Phi})$ is an eigenvalue of $F_{\Phi}$,
then $\lambda_{N}=\mu_{N}(F_{\Phi})<\mu_{N+1}(F_{\Phi})\leq 0$.

###### Proof.

See, for example [4]. ∎

The eigenvalue $\lambda_{N}$ corresponds to the _highest occupied molecular
orbital_ (HOMO) and $\lambda_{N+1}$ to the _lowest unoccupied molecular
orbital_ (LUMO). Their difference,
$\varepsilon_{\mathrm{min}}:=\lambda_{N+1}-\lambda_{N}$ is called the _HOMO-
LUMO gap_ , which is an important quantity in quantum chemistry [2].

The first quantization of the mean-field operator $F_{\Phi}$ is called the
_Fock operator_ $\mathcal{F}_{\Phi}:\mathfrak{L}^{2}\to\mathfrak{L}^{2}$ with
domain $D(\mathcal{F}_{\Phi})=\bigwedge^{N}D(F_{\Phi})=\mathfrak{H}^{2}$,

$\mathcal{F}_{\Phi}=F_{\Phi}\otimes I\otimes\ldots\otimes
I+\ldots+I\otimes\ldots\otimes I\otimes F_{\Phi}.$

Henceforth we omit $\Phi$ from the notation, and let
$\mathcal{F}:=\mathcal{F}_{\Phi}$ whenever $\Phi$ is clear from the context.
It is immediate that the HF determinant $\Phi_{0}:=\Phi_{\mathrm{HF}}$ is an
eigenfunction of $\mathcal{F}$,

$\mathcal{F}\Phi_{0}=\Lambda_{0}\Phi_{0},\quad\text{with}\quad\Lambda_{0}=\sum_{i=1}^{N}\lambda_{i}.$

The Fock operator gives rise to a splitting of the molecular Hamiltonian

$\mathcal{H}=\mathcal{F}+\mathcal{W},\quad\text{where}\quad\mathcal{W}:=\mathcal{H}-\mathcal{F}$
(2.4)

is called the _fluctuation operator_.

For the rest of the section, we consider the finite-dimensional case. In
practice, the Galerkin projection of the Hartree–Fock equations Eq. 2.3 are
solved to obtain the orbitals $\\{\varphi_{i}\\}_{i=1}^{N}$. Since the mean-
field operator $F_{\Phi}$ is self-adjoint, its eigenfunctions can be used to
extend these orbitals to an orthonormal basis
$\\{\varphi_{i}\\}_{i=1}^{K}\subset H^{1}_{K}(\mathbb{R}^{3})$. Similarly to
$\mathcal{H}_{K}$ we introduce the projected versions of $\mathcal{F}$ and
$\mathcal{W}$, denoted as $\mathcal{F}_{K}$ and $\mathcal{W}_{K}$. In the
orbital basis $\\{\varphi_{i}\\}_{i=1}^{K}$, the Fock operator takes the
diagonal form $\mathcal{F}_{K}=\sum_{i=1}^{K}\lambda_{i}a_{i}^{\dagger}a_{i}$.

Furthermore, if
$\Phi_{\alpha}=\varphi_{\alpha_{1}}\wedge\ldots\wedge\varphi_{\alpha_{N}}$
with $\alpha_{1}<\ldots<\alpha_{N}$ that is obtained from
$\Phi_{0}=\varphi_{1}\wedge\ldots\wedge\varphi_{N}$ by swapping $r$ orbitals
$\varphi_{I_{j}}$ with $\varphi_{A_{j}}$ where $I_{j}\in\\{1,\ldots,N\\}$,
$A_{j}\not\in\\{1,\ldots,N\\}$ ($j=1,\ldots,r\leq N$), then

$\mathcal{F}_{K}\Phi_{\alpha}=\Lambda_{\alpha}\Phi_{\alpha},\quad\text{with}\quad\Lambda_{\alpha}=\sum_{i=1}^{N}\lambda_{\alpha_{i}}=\Lambda_{0}+\varepsilon_{\alpha},\quad\varepsilon_{\alpha}=\sum_{j=1}^{r}\lambda_{A_{j}}-\lambda_{I_{j}}.$
(2.5)

We would like to emphasize that in the finite-dimensional case, it is possible
to have $\lambda_{N}\leq 0<\lambda_{N+1}$.333The addition of diffuse functions
of the form $e^{-\beta|\mathbf{x}|^{2}}$ for $\beta\ll 1$ can significantly
reduce the HOMO-LUMO gap $\varepsilon_{\min}$.

Unfortunately, in the infinite-dimensional case, it might not be possible to
construct a complete eigenbasis $\\{\varphi_{i}\\}_{i=1}^{\infty}\subset
H^{1}(\mathbb{R}^{3})$ for the mean-field operator $F_{\Phi}$.

###  2.4. Excitation-, and cluster operators

We now briefly recall the basic properties excitation-, and cluster operators
using the second-quantized language. Alternatively, the reader may consult
Section 3.2 of Part I. Here, we introduce the concepts in arbitrary dimensions
for completeness, although we only need the finite-dimensional case in the
sequel.

Suppose that
$\Phi_{0}:=\Phi_{\mathrm{HF}}=\varphi_{1}\wedge\ldots\wedge\varphi_{N}$ is a
HF minimizer for some orthonormal $\\{\varphi_{p}\\}_{p=1}^{N}$, as discussed
in the preceding section. The wavefunction $\Phi_{0}$ is called the _reference
determinant_ or simply the _reference_. The functions
$\\{\varphi_{p}\\}_{p=1}^{N}$ are called _occupied orbitals_ and are extended
to a complete orthonormal basis of $H^{1}_{K}(\mathbb{R}^{3})$ with the
_virtual orbitals_ $\\{\varphi_{p}\\}_{p=N+1}^{K}$ (here, $K=\infty$ is
allowed). According to Section 2.1.2, any Slater determinant $\Phi_{\alpha}$
may be obtained from the Fock vacuum via a string of creation operators, in
particular $\Phi_{0}=a_{1}^{\dagger}\cdots a_{N}^{\dagger}\Omega$. Here, and
henceforth, the symbol $0$ denotes $(1,\ldots,N)$ as multiindex.

As in the preceding section, we may partition the indices
$\alpha_{1}<\ldots<\alpha_{N}$ into occupied-, and virtual ones by letting
$I_{r+1}<\ldots<I_{N}$ be the $\alpha_{j}$’s which lie in $\\{1,\ldots,N\\}$
and $A_{1}<\ldots<A_{r}$ be the $\alpha_{j}$’s which lie in
$\\{N+1,N+2,\ldots\\}$, where $r$ is the number of virtual indices in
$\alpha$, also called the _rank of $\alpha$_. Now let $I_{1}<\ldots<I_{r}$ be
given by
$\\{I_{1},\ldots,I_{r}\\}=\\{1,\ldots,N\\}\smallsetminus\\{I_{r+1},\ldots,I_{N}\\}$,
then we can write

$\Phi_{\alpha}=a_{A_{1}}^{\dagger}a_{I_{1}}\cdots
a_{A_{r}}^{\dagger}a_{I_{r}}\Phi_{0},$

since the string of creation and annihilation operators acting on $\Phi_{0}$
clearly swaps the occupied orbital $\varphi_{I_{j}}$ with $\varphi_{A_{j}}$ in
$\Phi_{0}$ for all $j=1,\ldots,r$, yielding precisely $\Phi_{\alpha}$. Then
$X_{\alpha}=a_{A_{1}}^{\dagger}a_{I_{1}}\cdots a_{A_{r}}^{\dagger}a_{I_{r}}$
is called an _excitation operator_. We say that the Slater determinant
$\Phi_{\alpha}$ (or the multiindex $\alpha$), is _singly-_ (S), _doubly-_ (D),
_triply-_ (T) etc. _excited_ if the rank of $\alpha$ is 1, 2, 3, etc.

This way we constructed a family of bounded operators $X_{\alpha}$, where
$\alpha$ runs over all possible _nonzero_ multiindices. The adjoint of an
excitation operator $X_{\alpha}$, denoted $X_{\alpha}^{\dagger}$, is called a
_de-excitation operator_. Notice that both excitation-, and de-excitation
operators conserve the particle number.

######  Theorem 2.4.

The following properties hold true.

  * (i)

The product of (de-)excitation operators is an (de-)excitation operator, or,
zero.

  * (ii)

(commutativity) $X_{\alpha}X_{\beta}=X_{\beta}X_{\alpha}$ and
$X_{\alpha}^{\dagger}X_{\beta}^{\dagger}=X_{\beta}^{\dagger}X_{\alpha}^{\dagger}$.

  * (iii)

(nilpotency) $X_{\alpha}^{2}=0$ and $(X_{\alpha}^{\dagger})^{2}=0$.

Linear combinations of excitation operators are called _cluster operators_ ,
i.e. operators of the form $C=\sum_{\alpha\neq 0}c_{\alpha}X_{\alpha}$.
Clearly, there is a one-to-one correspondence between $C$ and the so-called
_cluster amplitude_ $(c_{\alpha})_{\alpha\neq 0}$. We use the standard
convention that capital letters $C,T,S,\ldots$ denote cluster operators and
small letters $c,t,s,\ldots$ their corresponding (unique) cluster amplitudes.
Clearly, there is a one-to-one correspondence between functions
$\Psi\in\mathfrak{H}^{1}$ with $\langle\Psi,\Phi_{0}\rangle=0$ and the cluster
operators $C_{\Psi}$ defined as

$C_{\Psi}=\sum_{\alpha\neq 0}c_{\alpha}X_{\alpha},\quad\text{where}\quad
c_{\alpha}=\langle\Psi,\Phi_{\alpha}\rangle.$ (2.6)

######  Theorem 2.5.

[35] Fix $\Psi\in\mathfrak{H}^{1}$, such that $\langle\Psi,\Phi_{0}\rangle=0$.
Then the following properties hold true.

  * (i)

The vector space of cluster operators extended with the identity is a closed,
commutative, nilpotent subalgebra of $\mathcal{L}(\mathfrak{H}^{1})$. In
particular, $T^{N+1}=0$ for any $T$.

  * (ii)

The cluster operator $C_{\Psi}$ satisfies
$C_{\Psi}\in\mathcal{L}(\mathfrak{H}^{1},\mathfrak{H}^{1})$. Furthermore,
there is a constant $b>0$ independent of $\Psi$ such that
$\|\Psi\|_{\mathfrak{H}^{1}}\leq\|C_{\Psi}\|_{\mathcal{L}(\mathfrak{H}^{1},\mathfrak{H}^{1})}\leq
b\|\Psi\|_{\mathfrak{H}^{1}}$.

  * (iii)

$C_{\Psi}^{\dagger}\in\mathcal{L}(\mathfrak{H}^{1},\mathfrak{H}^{1})$, and
there is a constant $b^{\prime}>0$ independent of $\Psi$ such that
$\|C_{\Psi}^{\dagger}\|_{\mathcal{L}(\mathfrak{H}^{1},\mathfrak{H}^{1})}\leq
b^{\prime}\|\Psi\|_{\mathfrak{H}^{1}}$, and there cannot be a uniform lower
bound in terms of $\|\Psi\|_{\mathfrak{H}^{1}}$.

  * (iv)

$C_{\Psi}$ can be extended to
$\mathcal{L}(\mathfrak{H}^{-1},\mathfrak{H}^{-1})$.

According to [35, Lemma 5.2], for any cluster operator
$C\in\mathcal{L}(\mathfrak{H}^{1},\mathfrak{H}^{1})$, there exists a cluster
operator $T\in\mathcal{L}(\mathfrak{H}^{1},\mathfrak{H}^{1})$, such that
$I+C=e^{T}$. This $T$ is uniquely given by $\log(I+C)$. We would like to point
out, that the exponential representation is only true in the _untruncated
case_ , i.e. when _all_ cluster amplitudes $(t_{\alpha})$ and all excitation
operators $X_{\alpha}$ are considered. If some truncation is in effect (see
below), the set $\\{e^{T}\Phi_{0}\\}$ is unknown.

###  2.5. Cluster amplitude spaces

As mentioned in the previous section, the linear combination coefficients
$t=(t_{\alpha})$ of a cluster operator expansion $T=\sum_{\alpha\neq
0}t_{\alpha}X_{\alpha}$ are called cluster amplitudes. If some _truncation
scheme_ is present, some of the $t_{\alpha}$’s and $X_{\alpha}$’s will be
missing (again we refer to Section 3.5 of Part I for the precise formulation).
This truncation procedure is crucial for a feasible implementation of the CC
method, and most commonly only the singly- (S), or singly and doubly- (SD), or
the singly, doubly and triply (SDT) excited amplitudes are kept.

The vector space of (possibly truncated) cluster amplitudes form a subspace of
$\ell^{2}$. We define the _(cluster) amplitude space_

$\mathbb{V}=\\{t\in\ell^{2}:\|T\Phi\|_{\mathfrak{H}^{1}}<\infty\\},$

endowed with the inner product $\langle t,s\rangle_{\mathbb{V}}=\langle
T\Phi_{0},S\Phi_{0}\rangle_{\mathfrak{H}^{1}}$. Analogously to
$\mathfrak{H}^{1}\subset\mathfrak{L}^{2}\subset\mathfrak{H}^{-1}$, the spaces
$\mathbb{V}\subset\ell^{2}\subset\mathbb{V}^{*}$ form a Gelfand triple, so
Remark 2.1 of Part I applies. To $\mathbb{V}$, there corresponds the
_functional amplitude space_

$\mathfrak{V}=\\{T\Phi_{0}\in\mathfrak{H}^{1}:t\in\mathbb{V}\\}.$

In accordance to the concepts developed in Part I, we denote the _full_ , i.e.
untruncated amplitude space and the corresponding functional amplitude space
by $\mathbb{V}(G^{\mathrm{full}})$ and $\mathfrak{V}(G^{\mathrm{full}})$,
respectively.

######  Remark 2.6.

In the case $K<\infty$, and assuming that HOMO-LUMO gap satisfies
$\varepsilon_{\min}>0$ (no matter how small), following [37], we introduce the
norm

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|t\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2}:=\sum_{\alpha\neq
0}\varepsilon_{\alpha}|t_{\alpha}|^{2}\quad\text{for all}\quad
t\in\mathbb{V},$

where $\varepsilon_{\alpha}$ denotes the eigenvalues of the Fock operator
$\mathcal{F}_{K}$, see Eq. 2.5. It was shown in [37] that under certain
assumptions, the constants in the norm equivalence
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\sim\|\cdot\|_{\mathbb{V}}$
are independent of $K$. Finally, we also define the norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
on $\mathfrak{V}$ via
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T\Phi_{0}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}:={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|t\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
for any $t\in\mathbb{V}$.

##  3\. Topological degree theory

In this section, we give a short introduction to the basic notions of _finite-
dimensional_ topological degree theory, and state the results that we will
need in the forthcoming analysis of the CC method. While topological degree
theory is well-known in the field of nonlinear analysis and dynamical systems,
it is not so ubiquitous in mathematical physics (a nice application is
furnished by Ginzburg–Landau theory, see e.g. [9, Section 2.4]). Therefore, an
inclusion of a brief summary of this important and powerful tool in the
present article seems justified.

Here, we mainly follow the set of notes by Dinca and Mawhin [9] (which was
released as a book recently [10]), where the proofs may be found. The concept
of the topological (a.k.a. mapping-, or Brouwer-) degree of a mapping goes
back to Kronecker, Poincaré and Brouwer; Leray and Schauder extended the
concept to infinite dimensions and demonstrated its usefulness in the theory
of partial differential equations. (See [9, Chapter 23] for a fascinating
historical perspective.) The standard textbooks on the topic are [8, 43, 29].

######  Remark 3.1.

Unfortunately, there cannot be a “topological degree theory” for general
continuous mappings in infinite dimensions. The reason for this, roughly
speaking, is that such a theory would imply the Brouwer fixed point theorem,
but that does not hold in Hilbert space as pointed out by Kakutani [11,
Example 5.1.7]. There are many generalizations of topological degree theory to
infinite dimensions, all requiring some stringent assumptions on the class of
mappings considered [11, 29, 38, 31].

###  3.1. Construction and basic properties

The first result gives an axiomatic characterization of the topological degree
as the _unique_ additive homotopy invariant that can be attached to continuous
mappings (up to normalization).

######  Theorem 3.2 (Existence and uniqueness of the degree).

There exists a unique integer-valued function

$(\mathcal{A},D,z)\mapsto\deg(\mathcal{A},D,z),$

where $D\subset\mathbb{R}^{n}$ is an open, bounded and nonempty subset,
$\mathcal{A}\in C(\overline{D},\mathbb{R}^{n})$ and
$z\not\in\mathcal{A}(\partial D)$, such that the following properties hold
true:

  1. (i)

(Normalization) If $z\in D$, then we have $\deg(\mathrm{id},D,z)=1$.

  2. (ii)

(Additivity) Let $D_{1},D_{2}\subset D$ be disjoint open subsets such that
$z\not\in\mathcal{A}(\overline{D}\smallsetminus(D_{1}\cup D_{2}))$. Then
$\deg(\mathcal{A},D,z)=\deg(\mathcal{A},D_{1},z)+\deg(\mathcal{A},D_{2},z)$.

  3. (iii)

(Homotopy invariance) If $\mathcal{K}\in
C(\overline{D}\times[0,1],\mathbb{R}^{n})$ and $z\not\in\mathcal{K}(\partial
D\times[0,1])$, then $\lambda\mapsto\deg(\mathcal{K}(\cdot,\lambda),D,z)$ is
constant.

Among the many useful properties of the degree we highlight the following few.

######  Corollary 3.3.

  1. (i)

(Excision property) Let $D\subset\mathbb{R}^{n}$ be an open, bounded and
$D^{\prime}\subset D$ open. If $z\not\in\mathcal{A}(\overline{D}\smallsetminus
D^{\prime})$, then $\deg(\mathcal{A},D,z)=\deg(\mathcal{A},D^{\prime},z)$.

  2. (ii)

(Existence property) If $z\not\in\mathcal{A}(\overline{D})$, then
$\deg(\mathcal{A},D,z)=0$. Said differently, if $\deg(\mathcal{A},D,z)\neq 0$
for some open, bounded $D\subset\mathbb{R}^{n}$ such that
$z\not\in\mathcal{A}(\partial D)$, then there is a solution $u\in D$ to
$\mathcal{A}(u)=z$.

  3. (iii)

(Additivity property) Suppose that $\\{D_{j}\\}\subset D$ is a sequence of
open and disjoint sets. If
$z\not\in\mathcal{A}(\overline{D}\smallsetminus\bigcup D_{j})$, then
$\deg(\mathcal{A},D_{j},z)=0$ for all but finitely many $j$, and
$\deg(\mathcal{A},D,z)=\sum_{j}\deg(\mathcal{A},D_{j},z)$.

The following formula is essential for the practical computation of the
degree.

######  Proposition 3.4.

Let $\mathcal{A}\in C(\overline{D},\mathbb{R}^{n})\cap
C^{1}(D,\mathbb{R}^{n})$ such that $z\not\in\mathcal{A}(\partial D)$ and
$\det\mathcal{A}^{\prime}(u)\neq 0$ for all $u\in\mathcal{A}^{-1}(z)$. Then
$\deg(\mathcal{A},D,z)=\sum_{u\in\mathcal{A}^{-1}(z)}\operatorname{sgn}\det\mathcal{A}^{\prime}(u)$.

Based on this, we call a point $u$ _non-degenerate_ if
$\det\mathcal{A}^{\prime}(u)\neq 0$ and _degenerate_ otherwise. When talking
about an _isolated solution_ $u$ of $\mathcal{A}$, i.e. a point
$u\in\mathcal{A}^{-1}(z)$ such that there is a ball $B(u,r)$ with
$\mathcal{A}^{-1}(z)\cap B(u,r)=\\{u\\}$, the concept of the _index_ is
useful.

######  Definition 3.5.

Let $\mathcal{A}\in C(\overline{D},\mathbb{R}^{n})$ and let $u$ be an isolated
solution. Then the _(topological) index_ of $\mathcal{A}$ at $u$ is defined as
$i(\mathcal{A},u)=\deg(\mathcal{A},B(u,r),z)$, where $\mathcal{A}^{-1}(z)\cap
B(u,r)=\\{u\\}$.

It follows from the excision property that $i(\mathcal{A},u)$ is independent
of $r$ (for sufficiently small values of $r>0$), so the definition makes
sense.

######  Proposition 3.6.

  1. (i)

Let $\mathcal{A}\in C(\overline{D},\mathbb{R}^{n})$ such that
$z\not\in\mathcal{A}(\partial D)$ and $\mathcal{A}^{-1}(z)$ is finite. Then
$\deg(\mathcal{A},D,z)=\sum_{u\in\mathcal{A}^{-1}(z)}i(\mathcal{A},u)$.

  2. (ii)

Let $u\in\mathbb{R}^{n}$ be a zero of $\mathcal{A}\in
C(\overline{D},\mathbb{R}^{n})$, where $D$ an open neighborhood of $u$. If
$\mathcal{A}$ is differentiable at $u$ with $\det\mathcal{A}^{\prime}(u)\neq
0$, then $i(\mathcal{A},u)=\operatorname{sgn}\det\mathcal{A}^{\prime}(u)$.

The topological degree (resp. index) can be naturally extended to continuous
mappings of type $\mathcal{A}:X\to Y$, where $X$ and $Y$ are $n$-dimensional
oriented topological vector spaces and the degree (resp. index) is independent
of the choice of the basis. We refer the reader to [9, Chapter 6] for details.

######  Theorem 3.7.

[10, Theorem 1.3.1] ([9, Theorem 6.3.1]) Let $\mathcal{A}:D\to Y$ be
continuous, where $X$ and $Y$ are $n$-dimensional topological vector spaces
and $D\subset X$ is an open neighborhood of $u\in X$. Let
$h:X\to\mathbb{R}^{n}$ and $g:Y\to\mathbb{R}^{n}$ be linear homeomorphisms.
Suppose that $\mathcal{A}$ is differentiable at $u$ and that
$\mathcal{A}^{\prime}(u):X\to Y$ is invertible. Then
$i(\mathcal{A},u)=\operatorname{sgn}\det g\mathcal{A}^{\prime}(u)h^{-1}$.

The following result says the degree is stable under (almost all) small
perturbations of the right-hand side of $\mathcal{A}(u)=z$, and that the
degree provides a lower bound for the number of solutions of the perturbed
equation.

######  Theorem 3.8.

[8, Corollary 7.4] Let $D\subset\mathbb{R}^{n}$ be a bounded open set and let
$\mathcal{A}:D\to\mathbb{R}^{n}$ be a $C^{1}$ mapping. If
$z\not\in\mathcal{A}(\partial D)$, then there is a $\delta>0$ such that if
$z^{\prime}\in B(z,\delta)\smallsetminus E_{z,\delta}$, then

$\deg(\mathcal{A},D,z)=\deg(\mathcal{A},D,z^{\prime}),$

where $E_{z,\delta}\subset\\{w\in
B(z,\delta):\det\mathcal{A}^{\prime}(u)=0,\;\mathcal{A}(u)=w\\}$. Furthermore,
$\mathcal{A}^{-1}(z^{\prime})$ consists of a finite number $m$ of points,
where $|\deg(\mathcal{A},D,z)|\leq m$ and $m\equiv\deg(\mathcal{A},D,z)$ (mod
2).

The proof is based on Sard’s theorem, which says that $E_{z,\delta}$ has
$n$-dimensional Lebesgue measure zero. Next, we recall a tool that is very
useful for the computation of the degree in the degenerate case.

######  Theorem 3.9.

[9, Theorem 6.4.2] Let $X$ and $Z$ be $n$-dimensional topological vector
spaces. Let $L:X\to Z$ be a linear mapping with $\ker L\neq\\{0\\}$ and
$V\subset Z$ be a vector space such that $Z=V\oplus\operatorname{ran}L$. Let
$D\subset X$ open and bounded, $r:\overline{D}\to V$ continuous with
$0\not\in\mathcal{A}(\partial D)$, where $\mathcal{A}=L+r$. Then, for each
invertible linear map $J:\ker L\to V$, and each projector $P$ with
$\operatorname{ran}P=\ker L$, the relation

$\deg(\mathcal{A},D,0)=i(L+JP,0)\deg(J^{-1}r|_{\ker L},D\cap\ker L,0)$

called _Leray’s second reduction formula_ holds true.

######  Corollary 3.10.

Let $L:X\to Z$ be a linear mapping with $\ker L\neq\\{0\\}$ and let $Q$ be a
projector with $\ker Q=\operatorname{ran}L$. Also, let
$\mathcal{N}:D\times[0,1]\to Z$ be continuous, with $D\subset X$ open and
bounded. Furthermore, assume the following hold true.

  1. (i)

$Lu+\lambda\mathcal{N}(u,\lambda)\neq 0$ for all $u\in\partial D$ and
$\lambda\in(0,1]$.

  2. (ii)

$Q\mathcal{N}(u,0)\neq 0$ for each $u\in\partial D$.

Then

$\deg(L+\mathcal{N}(\cdot,1),D,0)=i(L+Q)\deg(Q\mathcal{N}(\cdot,0)|_{\ker
L},D\cap\ker L,0).$

###### Proof.

We adapt the the proof of [10, Theorem 2.2.3]. Define the homotopy

$\mathcal{A}(u,\lambda)=Lu+(1-\lambda)Q\mathcal{N}(u,\lambda)+\lambda\mathcal{N}(u,\lambda).$

Fix $\lambda\in(0,1]$. Projecting the equation $\mathcal{A}(u,\lambda)=0$ with
$Q$ and $I-Q$ (i.e. onto the complementary spaces $\operatorname{ran}Q$ and
$\operatorname{ran}(I-Q)=\ker Q=\operatorname{ran}L$), we get that
$\mathcal{A}(u,\lambda)=0$ is equivalent to the system
$Q\mathcal{N}(u,\lambda)=0$, $Lu+\lambda(I-Q)\mathcal{N}(u,\lambda)=0$. But
this is equivalent to $Lu+\lambda\mathcal{N}(u,\lambda)=0$. By assumption (i),
$\mathcal{A}(u,\lambda)\neq 0$ for all $u\in\partial D$ and $\lambda\in(0,1]$.
Further, $\mathcal{A}(u,0)=0$ is equivalent to the system $u\in\ker L$,
$Q\mathcal{N}(u,0)=0$, hence by assumption (ii), $\mathcal{A}(u,0)\neq 0$ for
all $u\in\partial D$. Using the homotopy invariance and Leray’s second
reduction formula with $V=\ker L$, $J=I$ and $r=\mathcal{N}(\cdot,0)$, we get

$\deg(\mathcal{A}(\cdot,1),D,0)=\deg(\mathcal{A}(\cdot,0),D,0)=i(L+Q,0)\deg(Q\mathcal{N}(\cdot,0)|_{\ker
L},D\cap\ker L,0).\qed$

###  3.2. Orientation-preserving mappings

An important class of mappings for which the degree behaves rather nicely is
the following.

######  Definition 3.11.

Let $U\subset\mathbb{R}^{n}$ be open.

  1. (i)

A linear map $L:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is said to be _orientation-
preserving_ if $\det L\geq 0$.

  2. (ii)

A mapping $\mathcal{A}\in C^{2}(U,\mathbb{R}^{n})$ is said to be _strictly
orientation-preserving in $U$_ if

    1. (a)

$\mathcal{A}^{\prime}(u)$ is orientation-preserving for all $u\in U$, and

    2. (b)

the set $\\{u\in U:\det\mathcal{A}^{\prime}(u)=0\\}$ is nowhere dense in $U$.

  3. (iii)

A mapping $\mathcal{A}\in C(U,\mathbb{R}^{n})$ is said to be _orientation-
preserving_ if for every $V\subset U$ open with $\overline{V}\subset U$ is
compact, there exists a sequence $\\{\mathcal{A}_{j}\\}$ such that
$\mathcal{A}_{j}$ is strictly orientation preserving for all $j$ and
$\mathcal{A}_{j}\to\mathcal{A}$ uniformly on $V$.

######  Theorem 3.12.

[10, Theorem 7.2.1] ([9, Theorem 19.3.1]) Let $U\subset\mathbb{R}^{n}$ be
open, $D\subset U$ an open bounded set with $\overline{D}\subset U$ and let
$\mathcal{A}:U\to\mathbb{R}^{n}$ be orientation-preserving. If
$z\not\in\mathcal{A}(\partial D)$, then the following hold true.

  1. (i)

$\deg(\mathcal{A},D,z)\geq 0$.

  2. (ii)

$\deg(\mathcal{A},D,z)>0$ if and only if $z\in\mathcal{A}(D)$.

  3. (iii)

If $\deg(\mathcal{A},D,z)=1$, then $\mathcal{A}^{-1}(z)\cap D$ is connected.

  4. (iv)

If $\mathcal{A}(D)$ contains a point of some component $C$ of
$\mathbb{R}^{n}\smallsetminus\mathcal{A}(\partial D)$, then
$C\subset\mathcal{A}(D)$.

The preceding notions are related to the well-known monotone-type mappings.

######  Theorem 3.13.

[10, Proposition 7.2.1] ([9, Proposition 19.4.1]) Let $U\subset\mathbb{R}^{n}$
be open, and suppose that $\mathcal{A}:U\to\mathbb{R}^{n}$ is monotone, i.e.

$\langle\mathcal{A}(u)-\mathcal{A}(v),u-v\rangle_{\mathbb{R}^{n}}\geq 0$

for all $u,v\in U$. Then $\mathcal{A}$ is orientation-preserving.

###  3.3. Holomorphic mappings

Next, we consider the complex case.

######  Definition 3.14.

Let $U\subset\mathbb{C}^{n}$ be open. A complex mapping
$\mathcal{A}:U\to\mathbb{C}^{n}$ is said to be _holomorphic at
$a\in\mathbb{C}^{n}$_ if there is a $\mathbb{C}$-linear mapping
$L_{a}:\mathbb{C}^{n}\to\mathbb{C}^{n}$ and a mapping
$r_{a}:U\smallsetminus\\{a\\}\to\mathbb{C}^{n}$ with $r_{a}=o(1)$, such that

$\mathcal{A}(a+h)=\mathcal{A}(a)+L_{a}h+r_{a}(h)$

for all $h\in\mathbb{C}$ such that $a+h\in U$. In this case,
$L_{a}=\mathcal{A}^{\prime}(a)$, where
$\mathcal{A}^{\prime}(a)h=h_{1}\partial_{z_{1}}\mathcal{A}(a)+\ldots+h_{n}\partial_{z_{n}}\mathcal{A}(a)$
and $\partial_{z_{k}}\mathcal{A}(a)$ denotes the $k^{\text{th}}$ complex
partial derivative of $\mathcal{A}$ at $a$.

We now remind the reader of some elementary linear algebra [34]. Every complex
vector space $V$ is also vector space over $\mathbb{R}$. This space will be
denoted as $V_{\mathbb{R}}$ and called the _realification_ of $V$. The
realification of a linear operator $A:V\to W$ over $\mathbb{C}$ is the linear
map $A_{\mathbb{R}}:V_{\mathbb{R}}\to W_{\mathbb{R}}$. If
$\\{e_{1},\ldots,e_{n}\\}$ is a basis in $V$, then
$\\{e_{1},\ldots,e_{n},ie_{1},\ldots,ie_{n}\\}$ is a basis in
$V_{\mathbb{R}}$. Further, if $\\{e_{1}^{\prime},\ldots,e_{m}^{\prime}\\}$ is
a basis in $W$, and $A=B+iC$ with some real matrices $B$ and $C$, then the
matrix of $A_{\mathbb{R}}$ with respect to the bases
$\\{e_{1},\ldots,e_{n},ie_{1},\ldots,ie_{n}\\}$ and
$\\{e_{1}^{\prime},\ldots,e_{n}^{\prime},ie_{1}^{\prime},\ldots,ie_{n}^{\prime}\\}$
is

$\begin{pmatrix}B&-C\\\ C&B\end{pmatrix}.$

The determinant of the realification $A_{\mathbb{R}}$ obeys the following
important rule:

$\det A_{\mathbb{R}}=|\det A|^{2}.$ (3.1)

This motivates that we define the realification of a mapping
$\mathcal{A}:U\to\mathbb{C}^{n}$ with
$\mathcal{A}=(\mathcal{A}_{1},\ldots,\mathcal{A}_{n})$ as
$\mathcal{A}_{\mathbb{R}}:U\to\mathbb{R}^{2n}$ via

$\mathcal{A}_{\mathbb{R}}(x,y)=(\operatorname{Re}\mathcal{A}_{1}(z),\operatorname{Im}\mathcal{A}_{1}(z),\ldots,\operatorname{Re}\mathcal{A}_{n}(z),\operatorname{Im}\mathcal{A}_{n}(z)),$

where $x=(x_{1},\ldots,x_{n})$, $y=(y_{1},\ldots,y_{n})$ and $z=x+iy$. Notice
that we do not explicitly denote the realification of the set $U$.

######  Definition 3.15.

Let $U\subset\mathbb{C}^{n}$ be open, $\mathcal{A}:U\to\mathbb{C}^{n}$ be a
holomorphic mapping and $D\subset U$ a domain. The _degree of $\mathcal{A}$ in
$D$_ is defined as the degree of the realification,
$\deg(\mathcal{A},D,z):=\deg(\mathcal{A}_{\mathbb{R}},D,z)$.

It can be proved using Eq. 3.1, that if $\mathcal{A}:U\to\mathbb{C}^{n}$ is
holomorphic, then
$\det\mathcal{A}_{\mathbb{R}}^{\prime}(x,y)=|\\!\det\mathcal{A}^{\prime}(z)|^{2}$
for all $z\in U$. This innocent-looking identity has striking consequences.

######  Theorem 3.16.

[9, Section 19.5] Let $U\subset\mathbb{C}^{n}$ be open,
$\mathcal{A}:U\to\mathbb{C}^{n}$ be a holomorphic mapping and $D\subset U$
bounded and open. Then the following statements hold true.

  1. (i)

$\mathcal{A}_{\mathbb{R}}:U\to\mathbb{R}^{2n}$ is orientation-preserving. In
particular, $\deg(\mathcal{A},D,z)\geq 0$ and $\deg(\mathcal{A},D,z)>0$ if and
only if $z\in\mathcal{A}(D)$.

  2. (ii)

Let $\overline{D}\subset U$ and $z\not\in\mathcal{A}(\partial D)$. If
$\deg(\mathcal{A},D,z)=k$, then $\mathcal{A}(u)=z$ has at most $k$ solutions
in $D$. If $\deg(\mathcal{A},D,z)=1$, then $\mathcal{A}(u)=z$ has a unique
solution in $D$.

  3. (iii)

Let $\zeta\in D$ be a solution of $\mathcal{A}(\zeta)=z$ and denote by $C$ the
connected component of $\mathcal{A}^{-1}(z)$ containing $\zeta$. Then _either_
$\zeta$ is an isolated solution of $\mathcal{A}(\zeta)=z$ (and hence
$C=\\{\zeta\\}$) _or_ $C\cap G\neq\emptyset$ for any neighborhood $G$ of
$\partial D$.

  4. (iv)

Let $\overline{D}\subset U$ and $z\not\in\mathcal{A}(\partial D)$. Then
$\deg(\mathcal{A},D,z)=1$ if and only if there is a unique non-degenerate
solution $\zeta\in D$ of $\mathcal{A}(\zeta)=z$.

  5. (v)

[42, Theorem 42 (b)] The number of zeros in $D$ is finite.

From (iv) we can conclude that if $\zeta$ is a degenerate isolated solution of
$\mathcal{A}(\zeta)=z$, then necessarily $i(\mathcal{A},\zeta)\geq 2$. Further
in the holomorphic case, Theorem 3.8 can sharpened as follows.

######  Theorem 3.17.

Let $\mathcal{A}:D\to\mathbb{C}^{n}$ be a holomorphic mapping and
$D\subset\mathbb{C}^{n}$ bounded and open. If $z\not\in\mathcal{A}(\partial
D)$, then there is a $\delta>0$ such that if $z^{\prime}\in
B(z,\delta)\smallsetminus E$, then

$\deg(\mathcal{A},D,z)=\deg(\mathcal{A},D,z^{\prime}),$

where $E\subset\\{w\in
B(z,\delta):\det\mathcal{A}^{\prime}(u)=0,\;\mathcal{A}(u)=w\\}$ has measure
0. Furthermore, $\mathcal{A}^{-1}(z^{\prime})$ consists of a finite number $m$
of points, where $m=\deg(\mathcal{A},D,z)$.

###### Proof.

From Theorem 3.8, we have $\deg(\mathcal{A},D,z)\leq m$ and the converse
inequality $m\leq\deg(\mathcal{A},D,z^{\prime})=\deg(\mathcal{A},D,z)$ follows
from Theorem 3.16 (ii). ∎

The following result is concerned with holomorphic extensions.

######  Theorem 3.18.

[42, Theorem 45.] Let $\mathcal{A}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a real
analytic mapping that can be extended to a holomorphic mapping
$\widetilde{\mathcal{A}}:\mathbb{C}^{n}\to\mathbb{C}^{n}$. Let
$D\subset\mathbb{R}^{n}$ be an open bounded set such that
$0\not\in\mathcal{A}(\partial D)$. Let

$D_{\varepsilon}=\\{x\in\mathbb{C}^{n}:\operatorname{Re}x\in
D,\quad|\operatorname{Im}x|<\varepsilon\\}.$

Then, for sufficiently small $\varepsilon>0$,

$|\deg(\mathcal{A},D,0)|\leq\deg(\widetilde{\mathcal{A}},D_{\varepsilon},0),\quad\deg(\mathcal{A},D,0)=\deg(\widetilde{\mathcal{A}},D_{\varepsilon},0)\mod
2.$

##  4\. Analysis of the SRCC method

As stressed earlier, we consider the finite-dimensional case only, by which we
mean that the cardinality of the orbital basis $\\{\varphi_{p}\\}_{p=1}^{K}$,
$K$ is finite. Recall the definition of $\mathfrak{H}^{1}_{K}$ from Section
2.2 and the interpretation of the projected Hamiltonian from Section 2.2.1.

###  4.1. Definitions and basic properties

Let $\mathbb{V}:=\mathbb{V}(G)$ be the real amplitude space corresponding to
some consistent excitation graph $G$ (see Section 3.5 of Part I). In
particular, one can take $\mathbb{V}$ to be the S, D, SD, SDT, etc. truncated
amplitude space, or the full amplitude space, see Section 2.5. Let
$\mathfrak{V}$ be the corresponding functional amplitude space, i.e. the set
of wavefunctions of the form $T\Phi_{0}$ for any $t\in\mathbb{V}$, where $T$
is the cluster operator corresponding to $t$—this convention will be used
throughout.

Define the mapping $\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$ via the
instruction

$\langle\mathcal{A}(t),s\rangle:=\langle
e^{-T}\mathcal{H}_{K}e^{T}\Phi_{0},S\Phi_{0}\rangle,$ (4.1)

for any $t,s\in\mathbb{V}$. Then, $\mathcal{A}$ is well-defined, because Eq.
4.1 can be rewritten as
$\langle\mathcal{A}(t),s\rangle=\langle\mathcal{H}_{K}e^{T}\Phi_{0},e^{-T^{\dagger}}S\Phi_{0}\rangle=\langle\mathcal{H}e^{T}\Phi_{0},e^{-T^{\dagger}}S\Phi_{0}\rangle$,
and here $e^{T}\Phi_{0}\in\mathfrak{H}^{1}_{K}$ and
$e^{-T^{\dagger}}S\Phi_{0}\in\mathfrak{H}^{1}_{K}$.

Recall the definition (see (2.6) or Theorem 4.4 of Part I) of the CC energy,

$\mathcal{E}_{\mathrm{CC}}(t):=\langle
e^{-T}\mathcal{H}_{K}e^{T}\Phi_{0},\Phi_{0}\rangle=\langle\mathcal{H}_{K}e^{T}\Phi_{0},\Phi_{0}\rangle,$

where the second equality follows from $(e^{-T})^{\dagger}\Phi_{0}=\Phi_{0}$.
Note that $\mathcal{E}_{\mathrm{CC}}(t)\in\mathbb{R}$ since the amplitude
space $\mathbb{V}$ is assumed to be real. The similarity-transformed
Hamiltonian occurs often in the forthcoming discussion, so we introduce the
notation

$\mathcal{H}_{K}(t):=e^{-T}\mathcal{H}_{K}e^{T}:\mathfrak{H}_{K}^{1}\to(\mathfrak{H}_{K}^{1})^{*},$
(4.2)

which is a bounded map for any fixed cluster amplitude $t\in\mathbb{V}$.
Furthermore, for a given bounded map
$\mathcal{T}:\mathfrak{H}_{K}^{1}\to(\mathfrak{H}_{K}^{1})^{*}$, we define the
operator $\mathcal{T}_{\mathfrak{V}}:\mathfrak{V}\to\mathfrak{V}$ via
$\langle\mathcal{T}_{\mathfrak{V}}\Psi,\Psi^{\prime}\rangle=\langle\mathcal{T}\Psi,\Psi^{\prime}\rangle$
for all $\Psi,\Psi^{\prime}\in\mathfrak{V}$. For instance, we will often
encounter the projected similarity-transformed Hamiltonian
$\mathcal{H}_{K}(t)_{\mathfrak{V}}$.

We will also use a notation analogous to Eq. 4.2 for the similarity-
transformed fluctuation operator $\mathcal{W}_{K}$ (see Eq. 2.4), i.e.
$\mathcal{W}_{K}(t)=e^{-T}\mathcal{W}_{K}e^{T}$. In our finite-dimensional
setting, the similarity-transformed Fock operator can be given explicitly as

$e^{-T}\mathcal{F}_{K}e^{T}=\mathcal{F}_{K}+[\mathcal{F}_{K},T],\quad\text{and}\quad[\mathcal{F}_{K},X_{\alpha}]=\varepsilon_{\alpha}X_{\alpha},$
(4.3)

for any $t\in\mathbb{V}$, see e.g. [12, Lemma 15]. In particular,

$[[\mathcal{H}_{K}(t),U],V]=[[\mathcal{W}_{K}(t),U],V],$ (4.4)

for any $t,u,v\in\mathbb{V}$.

The following simple observation shows the equivalence of the (strong)
Schrödinger equation with the Full CC method.

######  Lemma 4.1.

Assume that the Slater determinant basis satisfies
$\Phi_{\alpha}\in\mathfrak{H}_{K}^{2}$. Suppose that
$\mathbb{V}=\mathbb{V}(G^{\mathrm{full}})$, and that $\mathcal{A}(t_{*})=0$.
Then the function $\Psi=(c_{0}I+C)\Phi_{0}\in\mathfrak{H}_{K}^{2}$ satisfies
the Schrödinger equation $\mathcal{H}_{K}\Psi=\mathcal{E}\Psi$ if and only if
$e^{-T_{*}}(c_{0}I+C)=r_{0}I+R$, where ($c_{0}=r_{0}$ and)

$\left.\begin{aligned}
\mathcal{E}_{\mathrm{CC}}(t_{*})r_{0}+\langle\mathcal{H}_{K}(t_{*})R\Phi_{0},\Phi_{0}\rangle&=\mathcal{E}r_{0}\\\
\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}R\Phi_{0}&=\mathcal{E}R\Phi_{0}\end{aligned}\right\\}$
(4.5)

Furthermore,

$\sigma(\mathcal{H}_{K})=\\{\mathcal{E}_{\mathrm{CC}}(t_{*})\\}\cup\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}).$
(4.6)

###### Proof.

Using the splitting $\operatorname{Span}\\{\Phi_{0}\\}\oplus\mathfrak{V}$, the
similarity-transformed Hamiltonian is block upper triangular in the Slater
basis,

$\mathcal{H}_{K}(t_{*})=\begin{pmatrix}\mathcal{E}_{\mathrm{CC}}(t_{*})&\langle\mathcal{H}_{K}(t_{*})\,\cdot\,,\Phi_{0}\rangle\\\
0&\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}\end{pmatrix},$

due to $\langle\mathcal{H}_{K}(t_{*})\Phi_{0},\Phi_{\alpha}\rangle=0$. The
proof now follows by noting that the eigenvalues of $\mathcal{H}_{K}(t_{*})$
and $\mathcal{H}_{K}$ are the same, and the eigenvectors of
$\mathcal{H}_{K}(t_{*})$ are of the form $e^{-T_{*}}\Phi$, where
$\mathcal{H}_{K}\Phi=\mathcal{E}^{\prime}\Phi$. Formula Eq. 4.6 follows by the
fact that the spectrum of a block triangular matrix is the union of the
spectra of the blocks in the diagonal. ∎

We distinguish two cases. Obviously, $r_{0}\neq 0$ and $R=0$ is a solution to
the system Eq. 4.5 if and only if
$\mathcal{E}_{\mathrm{CC}}(t_{*})=\mathcal{E}$. In this case,
$\Psi=e^{T_{*}}\Phi_{0}$ is a solution.

If, however, $\mathcal{E}_{\mathrm{CC}}(t_{*})\neq\mathcal{E}$, then $R$
cannot be 0 (because that would imply $\Psi=0$). In this case,
$\Psi=(r_{0}I+R)e^{T_{*}}\Phi_{0}$, where
$r_{0}=\langle\mathcal{H}_{K}(t_{*})R\Phi_{0},\Phi_{0}\rangle/(\mathcal{E}-\mathcal{E}_{\mathrm{CC}}(t_{*})).$
Note that it is possible to have $r_{0}=0$, in which case
$\langle\Psi,\Phi_{0}\rangle=0$. We return to this latter case in Remark 4.16
and discuss the case $\mathcal{E}_{\mathrm{CC}}(t_{*})=\mathcal{E}$ below.

######  Remark 4.2.

If $\mathcal{E}_{\mathrm{CC}}(t_{*})=\mathcal{E}$, then Eq. 4.5 reduces to

$\left.\begin{aligned}
\langle\mathcal{H}_{K}(t_{*})R\Phi_{0},\Phi_{0}\rangle&=0\\\
\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}R\Phi_{0}&=\mathcal{E}R\Phi_{0}\end{aligned}\right\\}$

Suppose that this system has $\mu$ linearly independent solutions
$R_{1},\ldots,R_{\mu}$. Then it is easy to see, using
$\Psi=(r_{0}I+R_{\mu})e^{T_{*}}\Phi_{0}$, that the wavefunctions

$\\{e^{T_{*}}\Phi_{0},R_{1}e^{T_{*}}\Phi_{0},\ldots,R_{\mu}e^{T_{*}}\Phi_{0}\\}$

span the eigenspace $\ker(\mathcal{H}_{K}-\mathcal{E})$. In particular, we
have $\dim\ker(\mathcal{H}_{K}-\mathcal{E})=\mu+1$. Note also that in this
case $\sigma(\mathcal{H}_{K})=\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$.

Let us now recall that the CC equation $\mathcal{A}(t_{*})=0$ can be cast in a
form that closely resembles the CI eigenvalue problem (see (2.3) of Part I)
(although it is _not_ equivalent to it in general).

######  Lemma 4.3.

[37, Theorem 5.6] Let $G$ be excitation complete, and let
$\mathbb{V}=\mathbb{V}(G)$ be the corresponding amplitude space. Then the
“linked” CC equation $\mathcal{A}(t_{*})=0$ is equivalent to the “unlinked”
(a.k.a. “energy-dependent”) CC equation

$\langle\mathcal{H}_{K}e^{T_{*}}\Phi_{0},S\Phi_{0}\rangle=\mathcal{E}_{\mathrm{CC}}(t_{*})\langle
e^{T_{*}}\Phi_{0},S\Phi_{0}\rangle\quad\text{for all}\quad s\in\mathbb{V}.$
(4.7)

###### Proof.

We have

$\displaystyle\langle(\mathcal{H}_{K}-\mathcal{E}_{\mathrm{CC}}(t_{*}))e^{T_{*}}\Phi_{0},S\Phi_{0}\rangle=\langle
e^{-T_{*}}(\mathcal{H}_{K}-\mathcal{E}_{\mathrm{CC}}(t_{*}))e^{T_{*}}\Phi_{0},(e^{T_{*}})^{\dagger}S\Phi_{0}\rangle$
$\displaystyle=\langle
e^{-T_{*}}(\mathcal{H}_{K}-\mathcal{E}_{\mathrm{CC}}(t_{*}))e^{T_{*}}\Phi_{0},\Pi_{\mathfrak{V}}(e^{T_{*}})^{\dagger}S\Phi_{0}\rangle+\langle
e^{-T_{*}}(\mathcal{H}_{K}-\mathcal{E}_{\mathrm{CC}}(t_{*}))e^{T_{*}}\Phi_{0},\underbrace{\Pi_{\Phi_{0}}(e^{T_{*}})^{\dagger}S\Phi_{0}}_{\mathrm{const}\cdot\Phi_{0}}\rangle$
$\displaystyle=\langle
e^{-T_{*}}\mathcal{H}_{K}e^{T_{*}}\Phi_{0},\Pi_{\mathfrak{V}}(e^{T_{*}})^{\dagger}S\Phi_{0}\rangle,$

where second term on the right-hand side of the penultimate equality vanishes
by the definition of $\mathcal{E}_{\mathrm{CC}}(t_{*})$. The proof is
completed by recalling that
$\Pi_{\mathfrak{V}}(e^{T_{*}})^{\dagger}:\mathfrak{V}\to\mathfrak{V}$ is
surjective due to Proposition 3.30 of Part I. ∎

The “unlinked” form is less useful in practice, because the expansion of
$\mathcal{H}_{K}e^{T}$ does not terminate like the Baker–Campbell–Hausdorff
series

$\mathcal{H}_{K}(t)=\sum_{j=0}^{4}\frac{1}{j!}[\mathcal{H}_{K},T]_{(j)}.$
(4.8)

More generally, the _doubly_ 444The doubly similarity-transformation above
differs from the one considered in Arponen’s Extended CC (ECC) theory [1].
similarity-transformed Hamiltonian
$\mathcal{H}_{K}(t+s)=e^{-S}\mathcal{H}_{K}(t)e^{S}$ can also be expanded
using the Baker–Campbell–Hausdorff series but in this case

$\mathcal{H}_{K}(t+s)=\sum_{j=0}^{2N}\frac{1}{j!}[\mathcal{H}_{K}(t),S]_{(j)},$
(4.9)

i.e. the series terminates at $2N$. To see this, simply note that
$[\mathcal{H}_{K}(t),S]_{(2N+1)}$ consists of terms of the form
$S^{i}\mathcal{H}_{K}(t)S^{k}$, where $i+k=2N+1$, so $i,k\geq N+1$, which,
using the nilpotency of the cluster operators implies that all terms for
$j\geq 2N+1$ vanish.

###  4.2. Local properties—real case

Next, we look at the local behavior of the CC mapping
$\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$ for general (real) amplitude spaces
$\mathbb{V}$. For fixed $t\in\mathbb{V}$, define the _modified similarity-
transformed Hamiltonian_ ,

$\widehat{\mathcal{H}_{K}}(t):=\mathcal{H}_{K}(t)-\sum_{\alpha\in\Xi(G)^{c}}\langle\mathcal{H}_{K}(t)\Phi_{0},\Phi_{\alpha}\rangle
X_{\alpha},$ (4.10)

where $\Xi(G)^{c}=\Xi(G^{\mathrm{full}})\smallsetminus\Xi(G)$ and the _set of
excitations_ $\Xi(G)$ was defined in (3.3) of Part I. For example, if SD
truncation is in effect, then $\Xi(G)$ consists of all singly-, and doubly
excited multiindices (see Section 2.4). Also, if $\mathbb{V}$ is the full
amplitude space, then $\Xi(G)$ consists of all multiindices $\alpha\neq 0$.

######  Definition 4.4.

The amplitude space $\mathbb{V}(G)$ is said to be _rank-regular_ , if

$\langle X_{\alpha}\Phi_{\beta},\Phi_{\gamma}\rangle=0\quad\text{for all
$\beta,\gamma\in\Xi(G)$ and $\alpha\in\Xi(G)^{c}$.}$

We immediately get that
$\widehat{\mathcal{H}_{K}}(t)_{\mathfrak{V}}=\mathcal{H}_{K}(t)_{\mathfrak{V}}$
if $\mathbb{V}(G)$ is rank-regular. The next proposition shows that the
truncated subgraphs typically used in practice—such as ones coming from S, D,
SD, SDT, etc. truncations—are rank-regular. We refer the reader to Section 3.2
of Part I for details on truncated subgraphs of the excitation graph.

######  Proposition 4.5.

Suppose that the excitation graph $G$ is a rank-truncated subgraph of the form
$G=G(1,2,\ldots,\rho)$, for some $\rho=1,\ldots,N$ or $G=G(\mathrm{D})$. Then
$\mathbb{V}(G)$ is rank-regular.

###### Proof.

The set $\Xi(G)^{c}$ consists of elements of rank $\rho+1,\ldots,N$ (or
empty), so that $\langle X_{\alpha}\Phi_{\beta},\Phi_{\gamma}\rangle=0$ for
all $\operatorname{rk}\alpha\not\in\\{1,2,\ldots,\rho\\}$ and all
$\operatorname{rk}\beta,\operatorname{rk}\gamma\in\\{1,2,\ldots,\rho\\}$, due
to the fact that $X_{\alpha}\Phi_{\beta}$ is of rank
$\operatorname{rk}(\alpha)+\operatorname{rk}(\beta)\not\in\\{1,2,\ldots,\rho\\}$.
The proof of the case $G=G(\mathrm{D})$ is similar. ∎

Next, we compute the derivative of the mapping $\mathcal{A}$, which explains
the definition of $\widehat{\mathcal{H}_{K}}(t)$ and of rank-regularity.

######  Lemma 4.6.

Let $t_{*}$ be a zero of $\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$. Then the
derivative
$\mathcal{A}^{\prime}(t_{*})\in\mathcal{L}(\mathbb{V},\mathbb{V}^{*})$ is
given by

$\langle\mathcal{A}^{\prime}(t_{*})u,v\rangle=\langle(\widehat{\mathcal{H}_{K}}(t_{*})-\mathcal{E}_{\mathrm{CC}}(t_{*}))U\Phi_{0},V\Phi_{0}\rangle$
(4.11)

for all $u,v\in\mathbb{V}$.

###### Proof.

The derivative
$\mathcal{A}^{\prime}:\mathbb{V}\to\mathcal{L}(\mathbb{V},\mathbb{V}^{*})$ is
readily computed as

$\displaystyle\left.\frac{d}{dh}\langle\mathcal{A}(t+hu),v\rangle\right|_{h=0}$
$\displaystyle=\left.\frac{d}{dh}\langle
e^{-T-hU}\mathcal{H}_{K}e^{T+hU}\Phi_{0},V\Phi_{0}\rangle\right|_{h=0}$
$\displaystyle=\left.\langle
e^{-T-hU}(\mathcal{H}_{K}U-U\mathcal{H}_{K})e^{T+hU}\Phi_{0},V\Phi_{0}\rangle\right|_{h=0}$
$\displaystyle=\langle
e^{-T}(\mathcal{H}_{K}U-U\mathcal{H}_{K})e^{T}\Phi_{0},V\Phi_{0}\rangle,$

so using the commutativity of the cluster operators, we get

$\langle\mathcal{A}^{\prime}(t)u,v\rangle=\langle[\mathcal{H}_{K}(t),U]\Phi_{0},V\Phi_{0}\rangle$
(4.12)

for all $t,u,v\in\mathbb{V}$. Expanding
$U^{\dagger}V\Phi_{0}\in\mathfrak{H}_{K}^{1}$ in the
$\mathfrak{L}^{2}$-orthonormal basis
$\\{\Phi_{\alpha}\\}_{\alpha}\subset\mathfrak{H}^{1}_{K}$,

$U^{\dagger}V\Phi_{0}=\sum_{\alpha}\langle
U\Phi_{\alpha},V\Phi_{0}\rangle\Phi_{\alpha},$

we obtain using $\langle\mathcal{H}_{K}(t_{*})\Phi_{0},\Phi_{\alpha}\rangle=0$
for all $\alpha\in\Xi(G)$,

$\displaystyle\langle U\mathcal{H}_{K}(t_{*})\Phi_{0},V\Phi_{0}\rangle$
$\displaystyle=\langle\mathcal{H}_{K}(t_{*})\Phi_{0},U^{\dagger}V\Phi_{0}\rangle$
$\displaystyle=\mathcal{E}_{\mathrm{CC}}(t_{*})\langle
U\Phi_{0},V\Phi_{0}\rangle+\sum_{\alpha\in\Xi(G)^{c}}\langle\mathcal{H}_{K}(t_{*})\Phi_{0},\Phi_{\alpha}\rangle\langle
X_{\alpha}U\Phi_{0},V\Phi_{0}\rangle$

for all $u,v\in\mathbb{V}$. Inserting this into Eq. 4.12 with $t=t_{*}$, we
obtain the stated formula. ∎

As we noted in Section 1.1, previous analyses of the CC mapping assumed the
local strong monotonicity at a zero $t_{*}$, i.e. that there is a $\delta>0$
and a constant $C_{\mathrm{SM}}(t_{*},\delta)>0$ such that

$\langle\mathcal{A}(t)-\mathcal{A}(s),t-s\rangle\geq
C_{\mathrm{SM}}(t_{*},\delta)\|t-s\|_{\mathbb{V}}^{2},\quad\text{for all}\quad
t,s\in B_{\mathbb{V}}(t_{*},\delta).$ (4.13)

The following elementary theorem makes the observations in [36] more precise.

######  Theorem 4.7.

Let $t_{*}\in\mathbb{V}$ be a zero of
$\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$.

  1. (i)

If $\mathcal{A}$ is strongly monotone in $B_{\mathbb{V}}(t_{*},\delta)$ for
some $\delta>0$, then there exists $\delta^{\prime}>0$ such that
$\mathcal{A}^{\prime}(t_{*}+u)$ is $\mathbb{V}$-coercive for all
$\|u\|_{\mathbb{V}}<\delta^{\prime}$ with some constant $0<\gamma\leq
C_{\mathrm{SM}}(t_{*},\delta)$, i.e.

$\langle\mathcal{A}^{\prime}(t_{*}+u)v,v\rangle\geq\gamma\|v\|_{\mathbb{V}}^{2}\quad\text{for
all}\quad v\in\mathbb{V}\;\text{and}\;\|u\|_{\mathbb{V}}<\delta^{\prime}.$
(4.14)

  2. (ii)

Conversely, if Eq. 4.14 holds with $u=0$, then Eq. 4.13 holds true with
$\delta>0$ chosen so that $C_{\mathrm{SM}}(t_{*},\delta):=\gamma-
M_{\delta}\delta>0$, where

$M_{\delta}=\sup_{\|\zeta\|_{\mathbb{V}}\leq\delta}\|\mathcal{A}^{\prime\prime}(t_{*}+\zeta)\|_{\mathcal{L}(\mathbb{V}\times\mathbb{V},\mathbb{V}^{*})}.$
(4.15)

###### Proof.

To see (i), fix $\delta^{\prime}>0$ and $\|u\|_{\mathbb{V}}<\delta^{\prime}$
and write for any $\|r\|_{\mathbb{V}}<\delta^{\prime}<\delta$,

$\displaystyle C_{\mathrm{SM}}(t_{*},\delta)\|r-u\|_{\mathbb{V}}^{2}$
$\displaystyle\leq\langle\mathcal{A}(t_{*}+r)-\mathcal{A}(t_{*}+u),r-u\rangle\leq\langle\mathcal{A}^{\prime}(t_{*}+u)(r-u),r-u\rangle+\frac{1}{2}M_{\delta}\|r-u\|_{\mathbb{V}}^{3}.$

This implies

$(C_{\mathrm{SM}}(t_{*},\delta)-M_{\delta}\delta^{\prime})\|r-u\|_{\mathbb{V}}^{2}\leq\langle\mathcal{A}^{\prime}(t_{*}+u)(r-u),r-u\rangle.$

Any vector $v\in\mathbb{V}$ can be expressed as $v=\alpha(r-u)$ for some
$\alpha>0$ and $\|r\|_{\mathbb{V}}<\delta^{\prime}$, from which
$\mathbb{V}$-coercivity follows with
$\gamma=C_{\mathrm{SM}}(t_{*},\delta)-M_{\delta}\delta^{\prime}$, by choosing
$\delta^{\prime}$ sufficiently small.

Next, to prove (ii), write the Taylor expansions of $\mathcal{A}$ at $t_{*}$,

$\displaystyle\mathcal{A}(t_{*}+r)$
$\displaystyle=\mathcal{A}^{\prime}(t_{*})r+\mathcal{R}_{2}(t_{*};r),$
$\displaystyle\mathcal{A}(t_{*}+r^{\prime})$
$\displaystyle=\mathcal{A}^{\prime}(t_{*})r^{\prime}+\mathcal{R}_{2}(t_{*};r^{\prime}),$

for any $\|r\|_{\mathbb{V}},\|r^{\prime}\|_{\mathbb{V}}<\delta$ for some
$\delta>0$, from which we obtain

$\displaystyle\langle\mathcal{A}(t_{*}+r)-\mathcal{A}(t_{*}+r^{\prime}),r-r^{\prime}\rangle$
$\displaystyle=\langle\mathcal{A}^{\prime}(t_{*})(r-r^{\prime}),r-r^{\prime}\rangle+\langle\mathcal{R}_{2}(t_{*};r)-\mathcal{R}_{2}(t_{*};r^{\prime}),r-r^{\prime}\rangle$
$\displaystyle\geq\gamma\|r-r^{\prime}\|_{\mathbb{V}}^{2}+\langle\mathcal{R}_{2}(t_{*};r)-\mathcal{R}_{2}(t_{*};r^{\prime}),r-r^{\prime}\rangle.$

Using the intermediate value inequality, we have

$\displaystyle\|\mathcal{R}_{2}(t_{*};r)-\mathcal{R}_{2}(t_{*};r^{\prime})\|_{\mathbb{V}^{*}}$
$\displaystyle\leq M_{r,r^{\prime}}\|r-r^{\prime}\|_{\mathbb{V}},$

where

$\displaystyle M_{r,r^{\prime}}$
$\displaystyle=\max_{\xi\in[r,r^{\prime}]}\|\partial_{2}\mathcal{R}_{2}(t_{*};\xi)\|=\max_{\xi\in[r,r^{\prime}]}\|\mathcal{A}^{\prime}(t_{*}+\xi)-\mathcal{A}^{\prime}(t_{*})\|$
$\displaystyle\leq\Big{(}\max_{\xi\in[r,r^{\prime}]}\max_{\zeta\in[0,\xi]}\|\mathcal{A}^{\prime\prime}(t_{*}+\zeta)\|_{\mathcal{L}(\mathbb{V}\times\mathbb{V},\mathbb{V}^{*})}\Big{)}\delta$
$\displaystyle=\Big{(}\sup_{\|\zeta\|\leq\delta}\|\mathcal{A}^{\prime\prime}(t_{*}+\zeta)\|_{\mathcal{L}(\mathbb{V}\times\mathbb{V},\mathbb{V}^{*})}\Big{)}\delta=M_{\delta}\delta$

This implies
$\langle\mathcal{A}(t_{*}+r)-\mathcal{A}(t_{*}+r^{\prime}),r-r^{\prime}\rangle\geq(\gamma-
M_{\delta}\delta)\|r-r^{\prime}\|_{\mathbb{V}}^{2}$ for all
$\|r\|_{\mathbb{V}},\|r^{\prime}\|_{\mathbb{V}}<\delta$. Setting $t=t_{*}+r$
and $s=t_{*}+r^{\prime}$ proves the claim. ∎

######  Remark 4.8.

Let $\mathbb{V}^{0}\subset\mathbb{V}$ be a subspace and consider the
_projected CC mapping_ $\mathcal{A}^{0}:\mathbb{V}^{0}\to(\mathbb{V}^{0})^{*}$
via

$\langle\mathcal{A}^{0}(t^{0}),s^{0}\rangle_{\mathbb{V}^{*}\times\mathbb{V}}=\langle\mathcal{A}(t^{0}),s^{0}\rangle_{\mathbb{V}^{*}\times\mathbb{V}}\quad\text{for
all}\quad t^{0},s^{0}\in\mathbb{V}^{0}.$

Clearly, if $\mathcal{A}$ is strongly monotone on
$B_{\mathbb{V}}(t_{*},\delta)$ with a constant $C_{\mathrm{SM}}>0$ at a zero
$t_{*}$, then $\mathcal{A}^{0}$ is strongly monotone on
$B_{\mathbb{V}^{0}}(t_{*}^{0},\sqrt{\delta^{2}-\|t_{*}^{\perp}\|_{\mathbb{V}}^{2}})$
provided $\|t_{*}^{\perp}\|_{\mathbb{V}}^{2}$ is small enough, with the same
constant $C_{\mathrm{SM}}$, where we have set
$t_{*}^{0}=\Pi_{\mathbb{V}^{0}}t_{*}$ and
$t_{*}^{\perp}=(I-\Pi_{\mathbb{V}^{0}})t_{*}$. Here,
$\Pi_{\mathbb{V}^{0}}:\mathbb{V}\to\mathbb{V}$ denotes the
$\ell^{2}$-orthogonal projector onto $\mathbb{V}^{0}$. Note that $t_{*}^{0}$
is _not_ , in general, a zero of $\mathcal{A}^{0}$, hence the preceding
theorem is not applicable to $\mathcal{A}^{0}$.

######  Remark 4.9.

The quantity $M_{\delta}$ contains the second derivative of $\mathcal{A}$. It
is easy to see that

$\langle\mathcal{A}^{\prime\prime}(t)(u,v),w\rangle=\langle[[\mathcal{H}_{K}(t),U],V]\Phi_{0},W\Phi_{0}\rangle$

for all $u,v,w\in\mathbb{V}$. Using Eq. 4.4, we have
$\langle\mathcal{A}^{\prime\prime}(t)(u,v),w\rangle=\langle[[\mathcal{W}_{K}(t),U],V]\Phi_{0},W\Phi_{0}\rangle$,
so that $\mathcal{A}^{\prime\prime}(t)$ only involves the fluctuation operator
$\mathcal{W}_{K}$.

######  Remark 4.10 (Perturbative regime).

Let $\mathbb{V}(G)$ be rank-regular, and consider the case when $t_{*}\approx
0$, which is the case considered in [37, 36]. Then roughly speaking, we have
$\mathcal{H}_{K}(t_{*})\approx\mathcal{H}_{K}$. Note that

$\langle\mathcal{A}^{\prime}(t_{*})r,r\rangle=\langle(\mathcal{H}_{K}-\mathcal{E}_{\mathrm{CC}}(t_{*}))R\Phi_{0},R\Phi_{0}\rangle+\mathcal{O}(\|t_{*}\|_{\mathbb{V}}).$

Consequently, if

$\langle(\mathcal{H}_{K}-\mathcal{E}_{\mathrm{CC}}(t_{*}))R\Phi_{0},R\Phi_{0}\rangle\geq
c(t_{*})\|r\|_{\mathbb{V}}^{2},$

where $c(t_{*})>0$, then local strong monotonicity holds with constant
$C_{\mathrm{SM}}=c(t_{*})-M^{\prime}\|t_{*}\|_{\mathbb{V}}-2M_{\delta}\delta$
for $t_{*}$ sufficiently close to 0. In [36, Lemma 3.5], it is shown that such
a $c(t_{*})$ exists under the assumption that $\mathcal{H}_{K}$ has a spectral
gap and that $\Phi_{0}$ is a sufficiently good approximation of the ground
state $e^{T_{*}}\Phi_{0}$ (i.e. that $t_{*}$ is sufficiently close to 0).

######  Proposition 4.11.

If $\mathcal{A}$ is locally strongly monotone at a zero $t_{*}$, then $t_{*}$
is non-degenerate.

###### Proof.

Suppose that $\ker\mathcal{A}^{\prime}(t_{*})\neq\\{0\\}$ and that
$\mathcal{A}$ is locally strongly monotone near $t_{*}$. Then for any $0\neq
r\in\ker\mathcal{A}^{\prime}(t_{*})$ sufficiently close to 0, we have

$C_{\mathrm{SM}}(\delta)\|r\|_{\mathbb{V}}^{2}\leq\langle\mathcal{A}(t_{*}+r),r\rangle=\frac{1}{2}\langle\mathcal{A}^{\prime\prime}(t_{*})(r,r),r\rangle+o(\|r\|_{\mathbb{V}}^{4}).$

Rescaling $r$ by $\alpha>0$ small, and letting $\alpha\to 0$ we obtain that
$C_{\mathrm{SM}}=0$, a contradiction. ∎

######  Remark 4.12.

When applying topological degree theory, we will view
$\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$ as a mapping
$\mathbb{R}^{n}\to\mathbb{R}^{n}$ by identifying $\mathbb{V}$ and
$\mathbb{V}^{*}$ with $\mathbb{R}^{n}$. Following [10, Section 1.3], we fix a
basis $\\{\tau_{\alpha}\\}_{\alpha\in\Xi(G)}$ of $\mathbb{V}$ and define the
linear homeomorphism $h:\mathbb{V}\to\mathbb{R}^{n}$ with

$\mathbb{V}\ni
t=\sum_{\alpha\in\Xi(G)}\widehat{t}_{\alpha}\tau_{\alpha}\mapsto
h(t)=\sum_{\alpha\in\Xi(G)}\widehat{t}_{\alpha}e_{\alpha}\in\mathbb{R}^{n},$

where $\\{e_{\alpha}\\}_{\alpha\in\Xi(G)}$ is the standard (ordered) basis in
$\mathbb{R}^{n}$. Also, fix a basis
$\\{\tau_{\alpha}^{*}\\}_{\alpha\in\Xi(G)}$ of $\mathbb{V}^{*}$ and define the
linear homeomorphism $g:\mathbb{V}^{*}\to\mathbb{R}^{n}$ analogously. Then

$\widehat{\mathcal{A}}:=g\circ\mathcal{A}\circ
h^{-1}:\mathbb{R}^{n}\to\mathbb{R}^{n}$

gives the desired mapping. Now suppose that two other bases
$\\{\widetilde{\tau}_{\alpha}\\}_{\alpha\in\Xi(G)}\subset\mathbb{V}$ and
$\\{\widetilde{\tau}^{*}_{\alpha}\\}_{\alpha\in\Xi(G)}\subset\mathbb{V}^{*}$
are given and let $\widetilde{h}:\mathbb{V}\to\mathbb{R}^{n}$ and
$\widetilde{g}:\mathbb{V}^{*}\to\mathbb{R}^{n}$ be the corresponding linear
homeomorphism. But then

$g^{-1}\circ g\circ\mathcal{A}\circ h^{-1}\circ
h=\mathcal{A}=\widetilde{g}^{-1}\circ\widetilde{g}\circ\mathcal{A}\circ\widetilde{h}^{-1}\circ\widetilde{h},$

which implies
$\widetilde{\mathcal{A}}:=\widetilde{g}\circ\mathcal{A}\circ\widetilde{h}^{-1}=m\circ\widehat{\mathcal{A}}\circ\widetilde{m}$,
where $m=\widetilde{g}\circ g^{-1}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and
$\widetilde{m}=h\circ\widetilde{h}^{-1}:\mathbb{R}^{n}\to\mathbb{R}^{n}$.
Using [10, Lemma 1.3.1] ([9, Lemma 6.1.1]), we obtain

$\deg(\widetilde{\mathcal{A}},\widetilde{h}(D),\widetilde{g}(0))=(\operatorname{sgn}\det
m)(\operatorname{sgn}\det\widetilde{m})\deg(\widehat{\mathcal{A}},h(D),g(0))$

for any open and bounded set $D\subset\mathbb{V}$ with
$0\not\in\mathcal{A}(\partial D)$. We can conclude that the topological degree
is independent of the choice of the basis if $\mathbb{V}$ and $\mathbb{V}^{*}$
are oriented the same.

Next, we determine the topological index of a zero of $\mathcal{A}$. The fact
that the topological index of $t_{*}$ is related to its CC energy
$\mathcal{E}_{\mathrm{CC}}(t_{*})$ and the eigenvalues of the operator
$\widehat{\mathcal{H}_{K}}(t_{*})_{\mathfrak{V}}$ is interesting on its own
right.

######  Theorem 4.13 (Index formula for SRCC—non-degenerate case).

Let $t_{*}$ be a zero of the CC mapping
$\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$. Then $t_{*}$ is non-degenerate if
and only if
$\mathcal{E}_{\mathrm{CC}}(t_{*})\not\in\sigma(\widehat{\mathcal{H}_{K}}(t_{*})_{\mathfrak{V}})$,
and in this case $t_{*}$ is an isolated zero and the topological index of
$\mathcal{A}$ at $t_{*}$ is given by

$i(\mathcal{A},t_{*})=(-1)^{\nu},$

where

$\nu=|\\{j:\mathcal{E}_{j}(\widehat{\mathcal{H}_{K}}(t_{*})_{\mathfrak{V}})\in\mathbb{R},\;\mathcal{E}_{j}(\widehat{\mathcal{H}_{K}}(t_{*})_{\mathfrak{V}})<\mathcal{E}_{\mathrm{CC}}(t_{*})\\}|.$

###### Proof.

It is trivial to see that if $t_{*}$ is non-degenerate, then it is isolated:
assume that $\ker\mathcal{A}^{\prime}(t_{*})=\\{0\\}$ and write

$\langle\mathcal{A}(t_{*}+r),\mathcal{A}^{\prime}(t_{*})r\rangle_{\mathbb{V}^{*}}=\|\mathcal{A}^{\prime}(t_{*})r\|_{\mathbb{V}^{*}}^{2}+o(\|r\|_{\mathbb{V}}^{3}),$
(4.16)

for all $\|r\|_{\mathbb{V}}=\varepsilon$, where $\varepsilon>0$ is
sufficiently small. This implies that $\mathcal{A}(t_{*}+r)\neq 0$ for all
$0<\|r\|_{\mathbb{V}}<\varepsilon$.

We can apply Theorem 3.7 with the mappings $h:\mathbb{V}\to\mathbb{R}^{n}$ and
$g:\mathbb{V}^{*}\to\mathbb{R}^{n}$ defined in Remark 4.12. Using the
notations of the said remark and Eq. 4.11, we have

$\displaystyle\langle\widehat{\mathcal{A}}^{\prime}(h^{-1}(t_{*}))e_{\alpha},e_{\beta}\rangle_{\mathbb{R}^{n}}$
$\displaystyle=\langle
g\mathcal{A}^{\prime}(t_{*})h^{-1}(e_{\alpha}),e_{\beta}\rangle_{\mathbb{R}^{n}}=\langle\mathcal{A}^{\prime}(t_{*})h^{-1}(e_{\alpha}),g^{\dagger}(e_{\beta})\rangle$
$\displaystyle=\langle(\widehat{\mathcal{H}_{K}}(t_{*})-\mathcal{E}_{\mathrm{CC}}(t_{*}))U_{\alpha}\Phi_{0},V_{\beta}\Phi_{0}\rangle,$

where $u_{\alpha}=h^{-1}(e_{\alpha})$ and $v_{\beta}=g^{\dagger}(e_{\beta})$
and $\alpha,\beta\in\Xi(G)$. Here, $g^{\dagger}:\mathbb{R}^{n}\to\mathbb{V}$
is the adjoint of $g$. Therefore, using an appropriate basis transformation

$i(\mathcal{A},t_{*})=\operatorname{sgn}\det\widehat{\mathcal{A}}^{\prime}(h^{-1}(t_{*}))=\operatorname{sgn}\Bigg{(}\prod_{j\geq
0}(\mathcal{E}_{j}(\widehat{\mathcal{H}_{K}}(t_{*})_{\mathfrak{V}})-\mathcal{E}_{\mathrm{CC}}(t_{*}))\Bigg{)}.$

The proof is completed by noting that the elements of the matrix
$\widehat{\mathcal{H}_{K}}(t_{*})_{\mathfrak{V}}$ are real, so its complex
eigenvalues come in conjugate pairs, hence only real eigenvalues contribute to
the product above. ∎

######  Proposition 4.14.

If $\mathcal{A}$ is locally strongly monotone near a zero $t_{*}$, then we
have $i(\mathcal{A},t_{*})=1$.

###### Proof.

We have that in particular
$\widehat{\mathcal{A}}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is monotone near
$h(t_{*})$, so according to Theorem 3.13, $\widehat{\mathcal{A}}$ is
orientation-preserving near $h(t_{*})$. But then Theorem 3.12 implies that
$i(\mathcal{A},t_{*})=i(\widehat{\mathcal{A}},h(t_{*}))>0$. ∎

Using the “unlinked” form, we can determine the topological index in the Full
CC case. Recall that the eigenvalues $\mathcal{E}_{n}(\mathcal{H}_{K})$,
$n=0,1,\ldots$, are assumed to be increasingly ordered.

######  Theorem 4.15 (Index formula for FCC—non-degenerate case).

Let $\mathbb{V}=\mathbb{V}(G^{\mathrm{full}})$ and assume that
$t_{*}\in\mathbb{V}$ is a zero of the FCC mapping
$\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$. Then
$e^{T_{*}}\Phi_{0}\in\mathfrak{H}^{2}$ is an (intermediately normalized)
eigenfunction corresponding to some non-degenerate eigenvalue
$\mathcal{E}_{\nu}(\mathcal{H}_{K})$ if and only if $t_{*}$ is non-degenerate,
and in this case $i(\mathcal{A},t_{*})=(-1)^{\nu}$.

###### Proof.

First, note that $\mathbb{V}(G^{\mathrm{full}})$ is rank-regular so
$\widehat{\mathcal{H}_{K}}(t)_{\mathfrak{V}}=\mathcal{H}_{K}(t)_{\mathfrak{V}}$.
We have $\mathcal{E}_{\mathrm{CC}}(t_{*})=\mathcal{E}_{\nu}(\mathcal{H}_{K})$
by the equivalence of FCC and FCI (see Theorem 2.2 of Part I).

According to Lemma 4.1, we have that $e^{T_{*}}\Phi_{0}$ is a non-degenerate
intermediately normalized eigenfunction if and only if
$\mathcal{E}_{\mathrm{CC}}(t_{*})\not\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$.
In fact,
$\mathcal{E}_{\mathrm{CC}}(t_{*})\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$
if and only if there exists $R\Phi_{0}\in\mathfrak{V}$ nonzero, such that

$\langle
e^{-T_{*}}\mathcal{H}_{K}e^{T_{*}}R\Phi_{0},S\Phi_{0}\rangle=\mathcal{E}_{\mathrm{CC}}(t_{*})\langle
R\Phi_{0},S\Phi_{0}\rangle,$

for all $s\in\mathbb{V}$. Since $\mathbb{V}(G^{\mathrm{full}})$ is excitation
complete, according to Lemma 4.3 the preceding equation is equivalent to

$\langle\mathcal{H}_{K}Re^{T_{*}}\Phi_{0},S\Phi_{0}\rangle=\mathcal{E}_{\mathrm{CC}}(t_{*})\langle
Re^{T_{*}}\Phi_{0},S\Phi_{0}\rangle,$ (4.17)

for all $s\in\mathbb{V}$. But this precisely means that the FCI eigenstate
$e^{T_{*}}\Phi_{0}$ is degenerate, because $Re^{T_{*}}\Phi_{0}$ is another
eigenvector corresponding to the same eigenvalue
$\mathcal{E}_{\mathrm{CC}}(t_{*})$.

We also conclude from Eq. 4.6 that
$\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})=\sigma(\mathcal{H}_{K})\smallsetminus\\{\mathcal{E}_{\mathrm{CC}}(t_{*})\\}$.
Applying Theorem 4.13, we obtain that $t_{*}$ is non-degenerate and
$i(\mathcal{A},t_{*})=(-1)^{\nu}$. ∎

It is worth noting that, in the FCC case, the zero $t_{*}$ representing the
intermediately normalized, non-degenerate _ground state_ (i.e.
$\mathcal{E}_{\mathrm{CC}}(t_{*})=\mathcal{E}_{0}(\mathcal{H}_{K})$) has
$i(\mathcal{A},t_{*})=1$. Note that this is not necessarily true in the
truncated case. While the CC method is most commonly aimed at the ground
state, it can also be used to find other intermediately normalized
eigenfunctions as well. Furthermore, it can also be used to obtain
eigenfunctions which are orthogonal to the reference $\Phi_{0}$ according to
the remark below.

######  Remark 4.16.

The _Equation-of-Motion Coupled-Cluster_ (EOM-CC) method [15] is aimed at
calculating _excited_ energies and states (i.e.
$\mathcal{E}_{n}(\mathcal{H}_{K})$ for $n>0$, and the corresponding
eigenvectors) based on a CC ground-state solution. This is done in two steps.
Let $\mathbb{V}=\mathbb{V}(G^{\mathrm{full}})$. Firstly, a conventional CC
calculation determines the ground state $\Psi=e^{T_{*}}\Phi_{0}$ such that
$\mathcal{A}(t_{*})=0$, i.e. $\mathcal{H}_{K}\Psi=\mathcal{E}\Psi$. Secondly,
the targeted excited state is of the form
$\Psi_{\mathrm{ex}}=(r_{0}I+R)e^{T_{*}}\Phi_{0}$, where $R$ is a cluster
operator, see Lemma 4.1. We have

$\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}R\Phi_{0}=\mathcal{E}_{\mathrm{ex}}R\Phi_{0}.$
(4.18)

In other words, we need to solve the eigenproblem of the projected similarity-
transformed Hamiltonian $\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}$. Furthermore,
similarly to the proof of Lemma 4.3, it is easy to see that
$\mathcal{A}(t_{*})=0$ implies

$\langle
R\mathcal{H}_{K}(t_{*})\Phi_{0},S\Phi_{0}\rangle=\mathcal{E}_{\mathrm{CC}}(t_{*})\langle
R\Phi_{0},S\Phi_{0}\rangle\quad\text{for all}\quad s\in\mathbb{V}.$

Subtracting this from Eq. 4.18, we get the “commutator form” of the EOM-CC
equation:

$\langle[\mathcal{H}_{K}(t_{*}),R]\Phi_{0},S\Phi_{0}\rangle=\Delta\mathcal{E}\langle
R\Phi_{0},S\Phi_{0}\rangle,\quad\text{where}\quad\Delta\mathcal{E}=\mathcal{E}_{\mathrm{ex}}-\mathcal{E},$
(4.19)

and $\mathcal{E}=\mathcal{E}_{\mathrm{CC}}(t_{*})$ is the ground-state energy
as given by the CC method. Let $\mathbb{V}$ be an arbitrary amplitude space.
Recalling the expression Eq. 4.12 for $\mathcal{A}^{\prime}(t)$, we can
rephrase the EOM-CC equation Eq. 4.19 as _the weak eigenvalue problem for_
$\mathcal{A}^{\prime}(t_{*}):\mathbb{V}\to\mathbb{V}^{*}$ (cf. [16, Section
13.6.3]), i.e.

$\langle\mathcal{A}^{\prime}(t_{*})r_{j},s\rangle=\Delta\mathcal{E}_{j}\langle
r_{j},s\rangle,$ (4.20)

for $j=1,\ldots,J$555Here, $R_{j}$ is not to be confused with the rank-
decomposition (3.6) of Part I. and $s\in\mathbb{V}$ is arbitrary.666A similar
relation holds if $t_{*}$ does not represent the ground state. Notice that the
$\Delta\mathcal{E}_{j}$’s are in general complex. Using Theorem 4.13 we can
obtain the following. Suppose that
$\Delta\mathcal{E}_{1},\ldots,\Delta\mathcal{E}_{\mu}$ are given by Eq. 4.20
and are all nonzero. Then

$i(\mathcal{A},t_{*})=(-1)^{\nu},\quad\nu=|\\{j:\Delta\mathcal{E}_{j}\in\mathbb{R},\;\Delta\mathcal{E}_{j}<0\\}|.$
(4.21)

Due to the _nonvariational property_ of truncated CC (see Section 2.3 of Part
I), it is not _a priori_ clear whether the (real) excited energies are higher
than the ground-state energy, i.e. whether $\Delta\mathcal{E}_{j}>0$.
Therefore, Eq. 4.21 quantifies this nonvariational property through the
topological index $i(\mathcal{A},t_{*})$.

Next, we draw a connection between the degeneracy of a zero $t_{*}$ and the
Fock-splitting Eq. 2.4 of the Hamiltonian. Define
$\omega_{0}(t_{*})=\langle\mathcal{W}_{K}(t_{*})\Phi_{0},\Phi_{0}\rangle$,
which is the CC correction to the lowest eigenvalue $\Lambda_{0}$ of
$\mathcal{F}$, so that the CC energy at $t_{*}$ is obtained as
$\mathcal{E}_{\mathrm{CC}}(t_{*})=\Lambda_{0}+\omega_{0}(t_{*})$.

######  Proposition 4.17.

Let $\mathbb{V}(G)$ be a rank-regular amplitude space and $t_{*}$ a zero of
$\mathcal{A}$. Define the linear operator
$\mathcal{Q}(t_{*}):\mathfrak{V}\to\mathfrak{V}$ via its matrix in the Slater
determinant basis as

$[\mathcal{Q}(t_{*})]_{\alpha\beta}=\varepsilon_{\alpha}\delta_{\alpha\beta}+\sum_{\gamma\in\Xi(G)}t_{*,\gamma}\varepsilon_{\gamma}\langle
X_{\gamma}\Phi_{\beta},\Phi_{\alpha}\rangle\quad\text{for
all}\quad\alpha,\beta\in\Xi(G).$

Then
$\omega_{0}(t_{*})\not\in\sigma(\mathcal{Q}(t_{*})+\mathcal{W}_{K}(t_{*})_{\mathfrak{V}})$
is equivalent to
$\mathcal{E}_{\mathrm{CC}}(t_{*})\not\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$,
i.e. to the fact that $t_{*}$ is a non-degenerate zero of $\mathcal{A}$.

###### Proof.

We have using Eq. 4.3,

$\displaystyle\mathcal{E}_{\mathrm{CC}}(t_{*})$ $\displaystyle=\langle
e^{-T_{*}}\mathcal{F}_{K}e^{T_{*}}\Phi_{0},\Phi_{0}\rangle+\langle\mathcal{W}_{K}(t_{*})\Phi_{0},\Phi_{0}\rangle$
$\displaystyle=\langle\mathcal{F}_{K}\Phi_{0},\Phi_{0}\rangle+\langle[\mathcal{F}_{K},T_{*}]\Phi_{0},\Phi_{0}\rangle+\langle\mathcal{W}_{K}(t_{*})\Phi_{0},\Phi_{0}\rangle$
$\displaystyle=\Lambda_{0}+\langle\mathcal{W}_{K}(t_{*})\Phi_{0},\Phi_{0}\rangle=\Lambda_{0}+\omega_{0}(t_{*}).$

Similarly,

$\displaystyle\langle\mathcal{H}_{K}(t_{*})\Phi_{\beta},\Phi_{\alpha}\rangle$
$\displaystyle=\langle\mathcal{F}_{K}\Phi_{\beta},\Phi_{\alpha}\rangle+\langle[\mathcal{F}_{K},T_{*}]\Phi_{\beta},\Phi_{\alpha}\rangle+\langle\mathcal{W}_{K}(t_{*})\Phi_{\beta},\Phi_{\alpha}\rangle$
$\displaystyle=(\Lambda_{0}+\varepsilon_{\alpha})\delta_{\alpha\beta}+\sum_{\gamma\in\Xi(G)}t_{*,\gamma}\varepsilon_{\gamma}\langle
X_{\gamma}\Phi_{\beta},\Phi_{\alpha}\rangle+\langle\mathcal{W}_{K}(t_{*})\Phi_{\beta},\Phi_{\alpha}\rangle.$

Then, in the Slater determinant basis

$\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}=\Lambda_{0}I+\mathcal{Q}(t_{*})+\mathcal{W}_{K}(t_{*})_{\mathfrak{V}}.$

Hence,
$\mathcal{E}_{\mathrm{CC}}(t_{*})\not\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$
is equivalent to
$\omega_{0}(t_{*})\not\in\sigma(\mathcal{Q}(t_{*})+\mathcal{W}_{K}(t_{*})_{\mathfrak{V}})$,
which finishes the proof. ∎

We now consider the case of a degenerate zero. Clearly, if
$r\in\ker\mathcal{A}^{\prime}(t_{*})\neq\\{0\\}$, we have to consider higher-
order terms of the Taylor polynomial of $\mathcal{A}$ at $t_{*}$,

$\mathcal{A}(t_{*}+r)=\mathcal{A}^{\prime}(t_{*})r+\frac{1}{2}\mathcal{A}^{\prime\prime}(t_{*})(r,r)+\mathcal{R}_{3}(t_{*};r),$

where
$\mathcal{A}^{\prime\prime}(t_{*}):\mathbb{V}\times\mathbb{V}\to\mathbb{V}^{*}$
is a bounded bilinear mapping. Here, we only consider the second-order
information.

Assume from now on that $\mathbb{V}$ is rank-regular, so that
$\widehat{\mathcal{H}_{K}}(t)_{\mathfrak{V}}=\mathcal{H}_{K}(t)_{\mathfrak{V}}$.
Suppose that
$\mathcal{E}_{\mathrm{CC}}(t_{*})\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$
and that $R_{1}\Phi_{0},\ldots,R_{\mu}\Phi_{0}\in\mathfrak{V}$ are the _right_
eigenvectors of $\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}$ corresponding to
$\mathcal{E}_{\mathrm{CC}}(t_{*})$,

$\langle\mathcal{H}_{K}(t_{*})R_{j}\Phi_{0},S\Phi_{0}\rangle=\mathcal{E}_{\mathrm{CC}}(t_{*})\langle
R_{j}\Phi_{0},S\Phi_{0}\rangle\quad\text{for all $j=1,\ldots,\mu$ and all
$s\in\mathbb{V}$.}$

Also, suppose that $L_{1}\Phi_{0},\ldots,L_{\mu}\Phi_{0}\in\mathfrak{V}$ are
the _left_ eigenvectors of $\mathcal{H}_{K}(t_{*})_{\mathfrak{V}}$
corresponding to $\mathcal{E}_{\mathrm{CC}}(t_{*})$,

$\langle\mathcal{H}_{K}(t_{*})^{\dagger}L_{j}\Phi_{0},S\Phi_{0}\rangle=\mathcal{E}_{\mathrm{CC}}(t_{*})\langle
L_{j}\Phi_{0},S\Phi_{0}\rangle\quad\text{for all $j=1,\ldots,\mu$ and all
$s\in\mathbb{V}$.}$

The corresponding right-, and left eigenspaces are

$\displaystyle W_{R}$
$\displaystyle=\ker\mathcal{A}^{\prime}(t_{*})=\operatorname{Span}\\{r_{1},\ldots,r_{\mu}\\},$
$\displaystyle W_{L}$
$\displaystyle=\ker\mathcal{A}^{\prime}(t_{*})^{\dagger}=\operatorname{Span}\\{\ell_{1},\ldots,\ell_{\mu}\\},$

and let $Q:\mathbb{V}\to\mathbb{V}$ be the orthogonal projector onto $W_{L}$.
Further, define $\widehat{Q}:\mathfrak{V}\to\mathfrak{V}$ via
$\langle\widehat{Q}U\Phi_{0},V\Phi_{0}\rangle=\langle Qu,v\rangle$ for all
$u,v\in\mathbb{V}$. We introduce the mapping
$\mathcal{B}:\mathbb{V}\to\mathbb{V}^{*}$ via

$\langle\mathcal{B}(t),s\rangle=\frac{1}{2}\langle\widehat{Q}[[\mathcal{H}_{K}(t_{*}),T],T]\Phi_{0},S\Phi_{0}\rangle=\frac{1}{2}\langle\widehat{Q}[[\mathcal{W}_{K}(t_{*}),T],T]\Phi_{0},S\Phi_{0}\rangle,$
(4.22)

that is, the $Q$-projection of
$\frac{1}{2}\mathcal{A}^{\prime\prime}(t_{*})(t,t)$. In the second equality,
we used Eq. 4.4. Also, note that $\mathcal{B}$ is homogeneous of degree 2,
i.e. $\mathcal{B}(\alpha t)=\alpha^{2}\mathcal{B}(t)$. The next theorem
follows esentially from Leray’s second reduction formula (Theorem 3.9).

######  Theorem 4.18 (Index formula for SRCC—degenerate case).

Let $t_{*}$ be zero of the CC mapping
$\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$. Suppose that
$\mathcal{E}_{\mathrm{CC}}(t_{*})\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$
and let $W_{R}$, $W_{L}$ and $Q$ be as above. Assume that

$\mathcal{B}(t)\neq 0\quad\text{for all}\quad t\in\partial
B_{\mathbb{V}}(0,1).$ (4.23)

Then, $t_{*}$ is an isolated zero and the topological index of $\mathcal{A}$
at $t_{*}$ is given by

$i(\mathcal{A},t_{*})=i(\mathcal{A}^{\prime}(t_{*})+Q,0)\,i(\mathcal{B}|_{W_{R}},0).$

###### Proof.

First, we prove that $t_{*}$ is isolated. When
$r\not\in\ker\mathcal{A}^{\prime}(t_{*})$ and small, it follows that
$\mathcal{A}(t_{*}+r)\neq 0$ similarly to Eq. 4.16. If, however
$r\in\ker\mathcal{A}^{\prime}(t_{*})$, then we may write

$\langle\mathcal{A}(t_{*}+r),\mathcal{A}^{\prime\prime}(t_{*})(r,r)\rangle_{\mathbb{V}^{*}}=\langle\underbrace{\mathcal{A}^{\prime}(t_{*})r}_{0},\mathcal{A}^{\prime\prime}(t_{*})(r,r)\rangle_{\mathbb{V}^{*}}+\frac{1}{2}\|\mathcal{A}^{\prime\prime}(t_{*})(r,r)\|_{\mathbb{V}^{*}}^{2}+\mathcal{O}(\|r\|_{\mathbb{V}}^{5})$

for all $r\in B(0,\varepsilon)$ for sufficiently small $\varepsilon>0$.
Condition Eq. 4.23 implies that $\mathcal{A}(t_{*}+r)\neq 0$ for all $r\in
B_{\mathbb{V}}^{*}(0,\varepsilon)$.

Next, we apply Corollary 3.10 with the choice
$D=B_{\mathbb{V}}(0,\varepsilon)$, $L=\mathcal{A}^{\prime}(t_{*})$ and

$\langle\mathcal{N}(t,\lambda),s\rangle=\sum_{k=2}^{2N}\frac{\lambda^{k-2}}{k!}\langle[\mathcal{H}_{K}(t_{*}),T]_{(k)}\Phi_{0},S\Phi_{0}\rangle,$

where we used Eq. 4.9. Because
$\operatorname{ran}Q=\ker\mathcal{A}^{\prime}(t_{*})^{\dagger}$, it follows
that

$\ker
Q=(\operatorname{ran}Q)^{\perp}=(\ker\mathcal{A}^{\prime}(t_{*})^{\dagger})^{\perp}=\operatorname{ran}\mathcal{A}^{\prime}(t_{*})=\operatorname{ran}L.$

Moreover, since $t_{*}$ is an isolated zero, it is possible to choose
$\delta>0$ so that the equation

$\lambda^{-1}\mathcal{A}(t_{*}+\lambda
t)=\mathcal{A}^{\prime}(t_{*})t+\lambda\mathcal{N}(t,\lambda)=0$

does not admit a solution $t\in\partial B_{\mathbb{V}}(0,\delta)$ for any
$\lambda\in(0,1]$.

Note that $Q\mathcal{N}(t,0)=\mathcal{B}(t)\neq 0$ for all $t\in
B_{\mathbb{V}}^{*}(0,\delta)$ by assumption Eq. 4.23 and the homogenity of
$\mathcal{B}$. We see that conditions (i) and (ii) of Corollary 3.10 are
satisfied and the result follows. ∎

The preceding theorem reduces the computation of the index to a low-
dimensional problem but the zero is still degenerate. In fact, since

$\langle\mathcal{B}^{\prime}(t)u,v\rangle=\frac{1}{2}\langle\widehat{Q}([[\mathcal{H}_{K}(t_{*}),U],T]+[[\mathcal{H}_{K}(t_{*}),T],U])\Phi_{0},V\Phi_{0}\rangle,$

we have that $t=0$ is a degenerate zero of $\mathcal{B}|_{W_{R}}$ and by
assumption the only zero. Therefore, we need to apply Theorem 3.8 to determine
$i(\mathcal{B}|_{W_{R}},0)$.

######  Corollary 4.19.

For a degenerate, isolated zero $t_{*}$ of $\mathcal{A}$, $\dim W_{R}=1$, and
for which Eq. 4.23 holds, we have $i(\mathcal{A},t_{*})=0$.

###### Proof.

Let $W_{R}=\operatorname{Span}\\{r\\}$ and
$W_{L}=\operatorname{Span}\\{\ell\\}$. We apply Theorem 3.8 to the mapping
$\mathcal{B}|_{W_{R}}$ with $D=B_{\mathbb{V}}(0,\varepsilon)$, $\varepsilon>0$
arbitrary. Fix $z^{\prime}=\eta\ell\in\operatorname{ran}Q=W_{L}$ such that
$0<|\eta|<\delta$. Since $t=cr\in W_{R}$, the equation
$\mathcal{B}(t)=z^{\prime}$ in $B_{\mathbb{V}}(0,\varepsilon)\cap W_{R}$ is
equivalent to finding $0\neq c\in\mathbb{R}$ such that

$\frac{c^{2}}{2}\langle(\mathcal{H}_{K}(t_{*})-\mathcal{E}_{\mathrm{CC}}(t_{*}))R^{2}\Phi_{0},L\Phi_{0}\rangle=\eta\langle
L\Phi_{0},L\Phi_{0}\rangle.$ (4.24)

Note that the inner product on the left-hand side is nonzero by assumption Eq.
4.23. Choose $\eta$ to be of opposite sign as the inner product on the left-
hand side. Then there are no solutions $c$, so
$i(\mathcal{B}|_{W_{R}},t_{*})=0$ and therefore $i(\mathcal{A},t_{*})=0$ by
Theorem 4.18. ∎

######  Corollary 4.20.

Let $t_{*}$ be an isolated zero of the CC mapping
$\mathcal{A}:\mathbb{V}\to\mathbb{V}^{*}$. Suppose that
$\mathcal{E}_{\mathrm{CC}}(t_{*})\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$
and let $W_{R}$ as above. Assume that $\ker\mathcal{B}^{\prime}(t)=\\{0\\}$
for all $0\neq t\in W_{R}$. If $\mu:=\dim W_{R}$ is odd, then
$i(\mathcal{A},t_{*})=0$ and if $\mu$ is even, then $i(\mathcal{A},t_{*})$ is
even. In particular, under the above hypotheses, $i(\mathcal{A},t_{*})$ cannot
be $\pm 1,\pm 3,\pm 5,\ldots$

###### Proof.

Let $z^{\prime}\in B_{\mathbb{V}}^{*}(0,\delta)\cap W_{L}$. Then the equation
$\mathcal{B}(t)=z^{\prime}$ for $0\neq t\in W_{R}$ is equivalent to

$\frac{1}{2}\langle[[\mathcal{H}_{K}(t_{*}),T],T]\Phi_{0},L\Phi_{0}\rangle=\langle
Z^{\prime}\Phi_{0},L\Phi_{0}\rangle,$

for all $\ell\in W_{L}$. Let $\mathcal{T}$ denote the set of solutions $t$ of
the preceding equation (which can be empty). By Theorem 3.8, $|\mathcal{T}|=m$
for some $m$ finite. Notice that $\mathcal{T}$ is closed under the operation
$t\mapsto-t$, so $m$ is even (we also used that $0\not\in\mathcal{T}$), and
let

$\mathcal{T}=\\{t_{1},\ldots,t_{\frac{m}{2}},-t_{1},\ldots,-t_{\frac{m}{2}}\\}$

using some appropriate indexing. From the linearity of
$t\mapsto\mathcal{B}^{\prime}(t)$, we get via Theorem 3.7,

$\displaystyle\deg(\mathcal{B}|_{W_{R}},B_{\mathbb{V}}(0,r)\cap
W_{R},z^{\prime})$
$\displaystyle=\sum_{i=1}^{\frac{m}{2}}\operatorname{sgn}\det
g\mathcal{B}^{\prime}(t_{i})h^{-1}+\operatorname{sgn}\det
g\mathcal{B}^{\prime}(-t_{i})h^{-1}$ (4.25)
$\displaystyle=(1+(-1)^{\mu})\sum_{i=1}^{\frac{m}{2}}\operatorname{sgn}\det
g\mathcal{B}^{\prime}(t_{i})h^{-1},$

for some sufficiently large $r>0$. ∎

We close this section with two remarks.

######  Remark 4.21.

  1. (i)

Note that, according to Theorem 3.8, a zero $t_{*}$ of topological index 0 is
“numerically unstable”, because one could miss zeros altogether if the
equations are solved with finite precision arithmetic, even in the FCC case.
Therefore, the degenerate zeros of the SRCC mapping $\mathcal{A}$ are not
robust in general. We have already seen that in the FCC case, the CC energy
$\mathcal{E}_{\mathrm{CC}}(t_{*})$ of a degenerate zero $t_{*}$ is a
degenerate eigenvalue $\mathcal{E}$ of the Hamiltonian (see Remark 4.2). Thus,
we can conclude that the SRCC method is in general unsuitable for finding
degenerate eigenstates and eigenvalues—an empirical fact that is well known
among the practitioners of the SRCC method.

  2. (ii)

Proposition 4.14 and the preceding calculations imply that any approach that
stipulates the local strong monotonicity of $\mathcal{A}$ near $t_{*}$ can
only provide an incomplete description of the SRCC method.

###  4.3. Local properties—complex case

We now discuss what happens when _complex_ amplitude spaces are considered
instead. We explain the complex case in detail, because the differences from
the real case are somewhat subtle. Assume that $\mathbb{V}$ is a complex
amplitude space and let $\mathbb{V}^{*}$ denote its anti-dual. It is clear
that $t\mapsto\langle\mathcal{A}(t),s\rangle$ is a (complex) polynomial for
fixed $s\in\mathbb{V}$, hence with the appropriate identifications,
$\mathcal{A}_{\mathbb{C}}:\mathbb{V}\to\mathbb{V}^{*}$ is a holomorphic
mapping, where we used the subscript $\mathbb{C}$ to highlight the
difference.777Note that the realification of $\mathcal{A}_{\mathbb{C}}$,
$(\mathcal{A}_{\mathbb{C}})_{\mathbb{R}}$ does _not_ equal $\mathcal{A}$ (they
are mappings of different type:
$(\mathcal{A}_{\mathbb{C}})_{\mathbb{R}}:\mathbb{R}^{2n}\to\mathbb{R}^{2n}$).
Of course, a real zero $t_{*}\in\mathbb{V}$ to $\mathcal{A}(t_{*})=0$ is
automatically a “complex” zero: $\mathcal{A}_{\mathbb{C}}(t_{*})=0$. Further,
using the fact that the Hamiltonian is real (by which we mean
$\langle\mathcal{H}_{K}\Phi_{\alpha},\Phi_{\beta}\rangle\in\mathbb{R}$),
$\mathcal{A}_{\mathbb{C}}(t_{*})=0$ if and only if
$\mathcal{A}_{\mathbb{C}}(\overline{t}_{*})=0$. Also,
$\mathcal{E}_{\mathrm{CC}}(\overline{t})=\overline{\mathcal{E}_{\mathrm{CC}}(t)}$.

From Theorem 3.16 (i) we immediately get that
$\deg(\mathcal{A}_{\mathbb{C}},U,0)\geq 0$ for every bounded open
$U\subset\mathbb{V}$. In particular, $i(\mathcal{A}_{\mathbb{C}},t_{*})\geq 0$
for every isolated zero $t_{*}$. Notice that, even if $t_{*}$ is a zero of
both $\mathcal{A}$ and $\mathcal{A}_{\mathbb{C}}$, its real and complex
indices $i(\mathcal{A},t_{*})$ and $i(\mathcal{A}_{\mathbb{C}},t_{*})$ may
differ; for instance we know that $i(\mathcal{A},t_{*})$ can have a sign,
while $i(\mathcal{A}_{\mathbb{C}},t_{*})$ cannot. Also, Theorem 3.16 (ii)
implies that $i(\mathcal{A},t_{*})\geq 2$ for an isolated, degenerate zero
$t_{*}$. Moreover, the following is true.

######  Theorem 4.22.

If $t_{*}\in\mathbb{V}$ is a _real_ isolated zero of both $\mathcal{A}$ and
$\mathcal{A}_{\mathbb{C}}$, then

$|i(\mathcal{A},t_{*})|\leq i(\mathcal{A}_{\mathbb{C}},t_{*}),\quad
i(\mathcal{A},t_{*})\equiv i(\mathcal{A}_{\mathbb{C}},t_{*})\mod 2.$

###### Proof.

The result follows from Theorem 3.18 with the choice
$D=B_{\mathbb{V}}(t_{*},\delta)$. ∎

For simplicity, we assume from now on that $\mathbb{V}$ is rank-regular
(Definition 4.4), so that
$\widehat{\mathcal{H}_{K}}(t)_{\mathfrak{V}}=\mathcal{H}_{K}(t)_{\mathfrak{V}}$.
Adapting the proofs of Theorem 4.13 and Theorem 4.18 in the complex case, we
have

######  Theorem 4.23 (Index formula for SRCC—complex case).

Let $t_{*}$ be an isolated zero of
$\mathcal{A}_{\mathbb{C}}:\mathbb{V}\to\mathbb{V}^{*}$. Then the following
hold true.

  1. (i)

The zero $t_{*}$ is non-degenerate if and only if
$\mathcal{E}_{\mathrm{CC}}(t_{*})\not\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$,
and in this case $i(\mathcal{A}_{\mathbb{C}},t_{*})=1$.

  2. (ii)

Suppose that
$\mathcal{E}_{\mathrm{CC}}(t_{*})\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$
and that the second-order regularity assumption Eq. 4.23 holds true. Then

$i(\mathcal{A}_{\mathbb{C}},t_{*})=i(\mathcal{B}|_{W_{R}},0).$

  3. (iii)

Suppose that
$\mathcal{E}_{\mathrm{CC}}(t_{*})\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$
and that the second-order regularity assumption Eq. 4.23 holds true. Assume
further, that $\ker\mathcal{B}^{\prime}(t)=\\{0\\}$ for all $0\neq
T\Phi_{0}\in W_{R}$. Then $i(\mathcal{A}_{\mathbb{C}},t_{*})=m\geq 2$, where
$m$ is the number of solutions $0\neq R\Phi_{0}\in W_{R}$ to the equation

$\langle(\mathcal{H}_{K}(t_{*})-\mathcal{E}_{\mathrm{CC}}(t_{*}))R^{2}\Phi_{0},L\Phi_{0}\rangle=\langle
Z\Phi_{0},L\Phi_{0}\rangle\quad(L\Phi_{0}\in W_{L})$

for any $Z\Phi_{0}\in W_{L}\cap B^{*}(0,\delta)$ for sufficiently small
$\delta>0$.

###### Proof.

For (i), it is enough to note that
$\ker\mathcal{A}_{\mathbb{C}}^{\prime}(t_{*})\neq\\{0\\}$ follows via the same
calculation as in the proof of Theorem 4.13. Part (ii) follows since
$i(\mathcal{A}_{\mathbb{C}}^{\prime}(t_{*})+Q)=1$. For the proof of (iii),
notice that Eq. 4.25 now reads

$\displaystyle\deg(\mathcal{B}|_{W_{R}},B_{\mathbb{V}}(0,r)\cap W_{R},z)$
$\displaystyle=\sum_{i=1}^{m}\operatorname{sgn}|\det
g\mathcal{B}^{\prime}(t_{i})h^{-1}|^{2}=m.\qed$

######  Corollary 4.24.

For a degenerate, isolated zero $t_{*}$ of $\mathcal{A}_{\mathbb{C}}$, $\dim
W_{R}=1$, and for which Eq. 4.23 holds, we have
$i(\mathcal{A}_{\mathbb{C}},t_{*})=2$.

###### Proof.

The proof is analogous to that of Corollary 4.19, but Eq. 4.24 always has
exactly two nonzero complex solutions $c$. ∎

We close this section with a few remarks.

######  Remark 4.25.

  1. (i)

Luckily, for a real zero $t_{*}\in\mathbb{V}$, the condition
$\mathcal{E}_{\mathrm{CC}}(t_{*})\not\in\sigma(\mathcal{H}_{K}(t_{*})_{\mathfrak{V}})$
is formally the same as in the real case, therefore a non-degenerate real zero
is automatically a non-degenerate zero of $\mathcal{A}_{\mathbb{C}}$.

  2. (ii)

The degeneracy of a complex zero $t_{*}$ manifests itself in numerical
computations as follows. Suppose the hypotheses of Theorem 4.23 (iii) hold
true. Combining this with Theorem 3.17, we get that the perturbed equation
$\mathcal{A}(t)=z^{\prime}$ has _exactly_ $m$ solutions for almost all
$z^{\prime}\in\mathbb{V}$ sufficiently close to zero. This is in contrast with
the real case, when one might completely “lose” solutions when the index is
zero (Remark 4.21 (i)). The appearance of multiple complex zeros in degenerate
situations was conjectured based on numerical observations in [33, 32].

  3. (iii)

If the Hamiltonian is real (see above), then the index of the complex
conjugate zero is the same:
$i(\mathcal{A}_{\mathbb{C}},t_{*})=i(\mathcal{A}_{\mathbb{C}},\overline{t}_{*})$.

  4. (iv)

The classical _Bézout theorem_ states that if a polynomial system

$\left.\begin{aligned} P_{1}(x_{1},\ldots,x_{d})&=0\\\
P_{2}(x_{1},\ldots,x_{d})&=0\\\ \ldots&\\\
P_{n}(x_{1},\ldots,x_{d})&=0\end{aligned}\right\\}$

has a finite number of zeros in $\mathbb{C}^{d}$, then the number of zeros
(counting multiplicities) is at most $\Delta=\Delta_{1}\cdots\Delta_{d}$
(called the _Bézout number_), where $\Delta_{k}$ denotes the degree of
$P_{k}$. As remarked earlier, according to the Baker–Campbell–Hausdorff
expansion Eq. 4.8, for the polynomials constituting the system
$\mathcal{A}(t)=0$ there holds $\Delta_{1}=\ldots=\Delta_{d}=4$, hence the
Bézout number of the CC equations is $\Delta=4^{d}$, where $d=\dim\mathbb{V}$.
This is typically a huge number. However, it is known that the Bézout number
often grossly overestimates the number of zeros. In fact, it was observed
numerically that the number of zeros for the (truncated) CC equations is much
less than the Bézout number [32].

###  4.4. Continuation of solutions

In this section we discuss how solutions of different CC methods can be
“connected” in a systematic way. The idea is not new to this field (and
certainly not new to nonlinear analysis, see e.g. [43]), and it has been a
subject of both theoretical and numerical investigations in the CC literature,
as we have already mentioned in the introduction.

The main theoretical tool we use to describe the aformentioned connection is a
specific type of homotopy.

######  Definition 4.26.

Let $\mathbb{V}^{1}$ be an amplitude space with direct sum decomposition
$\mathbb{V}^{1}=\mathbb{V}^{0}\oplus\mathbb{V}^{\angle}$. Let
$\mathcal{A}^{j}:\mathbb{V}^{j}\to(\mathbb{V}^{j})^{*}$ be continuous mappings
for $j=0,1$. A continuous map
$\mathcal{K}:\mathbb{V}^{1}\times[0,1]\to(\mathbb{V}^{1})^{*}$ is said to be
an _admissible homotopy_ , if

  1. (i)

$\langle\mathcal{K}(t^{1},0),s^{0}\rangle=\langle\mathcal{A}^{0}(t^{0}),s^{0}\rangle$
for all $t^{1}\in\mathbb{V}^{1}$, $s^{0}\in\mathbb{V}^{0}$, and

  2. (ii)

$\mathcal{K}(\cdot,1)=\mathcal{A}^{1}$.

Furthermore, an admissible homotopy
$\mathcal{K}:\mathbb{V}^{1}\times[0,1]\to(\mathbb{V}^{1})^{*}$ is said to be
_faithful_ , if for every $t_{**}^{0}\in\mathbb{V}^{0}$ such that
$\mathcal{A}^{0}(t_{**}^{0})=0$, there exists
$t_{**}^{\angle}\in\mathbb{V}^{\angle}$ so that with
$t_{**}^{1}=t_{**}^{0}+t_{**}^{\angle}\in\mathbb{V}^{1}$, there holds
$\mathcal{K}(t_{**}^{1},0)=0$.

######  Example 4.27.

When $\mathcal{A}^{0}$ is the projection of $\mathcal{A}:=\mathcal{A}^{1}$
onto $\mathbb{V}^{0}$, a simple admissible homotopy can be given by

$\langle\mathcal{K}(t^{1},\lambda),s^{1}\rangle=\langle\mathcal{A}(t^{0}+\lambda
t^{\angle}),s^{0}\rangle+\langle\mathcal{A}(t^{1}),s^{\angle}\rangle,$ (4.26)

for all $t^{1}\in\mathbb{V}^{1}$, $s^{1}\in\mathbb{V}^{1}$ and
$\lambda\in[0,1]$ (cf. Eq. 4.34).

Let $\mathbb{V}^{1}=\mathbb{V}(G^{\mathrm{full}})$ and $\mathcal{A}^{1}$ be
the FCC mapping, $\mathbb{V}^{0}$ some rank-truncated space and
$\mathcal{A}^{0}$ the truncated CC mapping. This case is particularly
important due to the equivalence of FCC and FCI (see Theorem 4.4 of Part I),
so existence of a solution to the FCI problem (essentially the Schrödinger
equation) can be exploited to infer the existence of a _truncated_ CC
solution. Furthermore, the topological index of the CC solution can be
determined by the results of Section 4.2 and the homotopy invariance of the
topological degree can be used relate these quantities in certain situations
(see Eq. 4.29 below).

The _zero set_ of an (admissible) homotopy $\mathcal{K}$ is defined as

$\mathcal{Z}(\mathcal{K})=\\{(t^{1}_{*},\lambda)\in\mathbb{V}^{1}\times[0,1]:\mathcal{K}(t_{*}^{1},\lambda)=0\\}.$
(4.27)

We omit $\mathcal{K}$ from the notation $\mathcal{Z}(\mathcal{K})$ whenever it
is clear from the context. The $\lambda$-sections of $\mathcal{Z}$ are denoted
as
$\mathcal{Z}_{\lambda}=\\{t_{*}^{1}\in\mathbb{V}^{1}:(t_{*}^{1},\lambda)\in\mathcal{Z}\\}$.
Clearly, $(\mathcal{Z}(\mathcal{K}))_{1}=(\mathcal{A}^{1})^{-1}(0)$ for _any_
admissible homotopy $\mathcal{K}$. Furthermore,
$\Pi_{\mathbb{V}^{0}}(\mathcal{Z}(\mathcal{K}))_{0}=(\mathcal{A}^{0})^{-1}(0)$
for any faithful, admissible homotopy $\mathcal{K}$. We will sometimes
explicitly label $t_{*}^{1}$ with the $\lambda$ to which it corresponds as
$t_{*}^{1}(\lambda)$.

Recall that any topological space can be partitioned into a family of
_connected components_ , which are maximal (w.r.t. set inclusion), closed
connected sets. We are interested in the connected components of
$\mathcal{Z}(\mathcal{K})$. The Leray–Schauder continuation principle [10,
Theorem 2.1.3] immediately implies the following.

######  Theorem 4.28.

Suppose that $\mathcal{D}\subset\mathbb{V}^{1}\times[0,1]$ is a bounded open
set and let $\mathcal{K}:\mathbb{V}\times[0,1]\to\mathbb{V}^{*}$ be an
admissible homotopy such that

$\mathcal{K}(t^{1},\lambda)\neq 0\quad\text{for
all}\quad(t^{1},\lambda)\in\partial\mathcal{D}.$ (4.28)

Then the following statements hold true.

  1. (i)

$\deg(\mathcal{K}(\cdot,0),\mathcal{D}_{0},0)=\deg(\mathcal{A}^{1},\mathcal{D}_{1},0)=:d$.

  2. (ii)

If $d\neq 0$, then there is a connected component $\mathcal{C}$ of
$\mathcal{Z}$, such that

$\mathcal{C}\cap(\mathcal{Z}_{j}\times\\{j\\})\neq\emptyset\quad\text{for}\quad
j=0,1.$

This general theorem can be used to prove existence results, which essentially
consists of establishing the boundary condition Eq. 4.28. Furthermore, note
that (i) implies that

$\sum_{t_{**}^{1}\in\mathcal{Z}_{0}\cap\mathcal{D}_{0}}i(\mathcal{K}(\cdot,0),t_{**}^{1})=\sum_{t_{*}^{1}\in\mathcal{Z}_{1}\cap\mathcal{D}_{1}}i(\mathcal{A}^{1},t_{*}^{1})=d.$
(4.29)

This is a necessary condition for zeros in $\mathcal{Z}_{0}$ and
$\mathcal{Z}_{1}$ be in the same bounded connected component.

######  Remark 4.29.

In the case when $\mathcal{D}$ is possibly _unbounded_ , one can invoke [10,
Theorem 2.1.4] to obtain the following statement: If $\mathcal{Z}_{0}$ is
bounded and $\deg(\mathcal{K}(\cdot,0),D,0)\neq 0$ for some bounded open set
$D\supset\mathcal{Z}_{0}$, then there is a connected component $\mathcal{C}$
of $\mathcal{Z}$ intersecting $\mathcal{Z}_{0}\times\\{0\\}$ which either
intersects $\mathcal{Z}_{1}\times\\{1\\}$ or is unbounded. The unboundedness
of $\mathcal{C}$ has been observed numerically for a specific type of homotopy
and its significance is discussed in the next remark.

######  Remark 4.30.

In [32], a specific type of admissible homotopy (essentially Eq. 4.26, which
will be discussed in Section 4.5 below) is used as the basis to distinguish
“physical” truncated solutions from “unphysical” ones. Roughly speaking, they
call a truncated solution “physical” if it shares a _bounded_ connected
component with an FCC solution (although they require more regularity of the
connected component in question). They also found that some FCC (resp.
truncated) solutions cannot be “connected” to any truncated (resp. FCC)
solution. While this approach to the classification of solutions seems
attractive at first, it has a serious conceptual drawback: it is unclear why
one particular homotopy is preferred over the (infinitely many) others. For
instance, it is conceivable that an admissible homotopy classifies a truncated
solution as “unphysical” while another homotopy classifies it as “physical”.
Moreover, their homotopy itself does not admit an obvious physical
interpretation. We will not pursue this line of thought any further in this
work, and view homotopies as purely mathematical devices.

As an example of an admissible homotopy, we consider the _linear homotopy_.
Let $\mathbb{V}^{0}\subset\mathbb{V}^{1}$ be any subspace (a typical choice is
based on some truncation: $\mathbb{V}^{0}=\mathbb{V}(G(1,\ldots,\rho))$) and
$\mathbb{V}^{\angle}:=(\mathbb{V}^{0})^{\perp}$, where the orthogonal
complement is taken with respect to the $\mathbb{V}$-inner product. Also, to
emphasize that $\mathbb{V}^{\angle}$ is actually given by the
$\mathbb{V}$-orthogonal complement of $\mathbb{V}^{0}$, we shall write
$t^{1}=t^{0}+t^{\perp}$ for some unique $t^{0}\in\mathbb{V}^{0}$ and
$t^{\perp}\in(\mathbb{V}^{0})^{\perp}$, for any $t^{1}\in\mathbb{V}^{1}$. Let
$\mathcal{K}_{\mathrm{L}}:\mathbb{V}^{1}\times[0,1]\to(\mathbb{V}^{1})^{*}$ be
given by

$\langle\mathcal{K}_{\mathrm{L}}(t^{1},\lambda),s^{1}\rangle=(1-\lambda)\left(\langle\mathcal{A}^{0}(t^{0}),s^{0}\rangle+\alpha|\langle
t^{\perp}-u^{\perp},s^{\perp}\rangle_{\mathbb{V}}|\right)+\lambda\langle\mathcal{A}^{1}(t^{1}),s^{1}\rangle$

for any $t^{1},s^{1}\in\mathbb{V}^{1}$ and $\lambda\in[0,1]$, and some fixed
constants $\alpha>0$ and $u^{\perp}\in(\mathbb{V}^{0})^{\perp}$. Then, clearly
$\mathcal{K}_{\mathrm{L}}(\cdot,1)=\mathcal{A}^{1}$. Further,
$\mathcal{K}_{\mathrm{L}}(t^{1}_{**},0)=0$ is equivalent to
$\mathcal{A}^{0}(t_{**}^{0})=0$ and $t_{**}^{\perp}=u^{\perp}$. Therefore,
$\mathcal{K}_{\mathrm{L}}$ is a faithful, admissible homotopy. Also,
$\mathcal{K}_{\mathrm{L}}$ has the following trivial property: if
$t_{*}^{1}\in\mathbb{V}^{1}$ is such that $\mathcal{A}^{0}(t_{*}^{0})=0$ and
$\mathcal{A}^{1}(t_{*}^{1})=0$, then
$\mathcal{K}_{\mathrm{L}}(t_{*}^{1},\lambda)=0$ for all $\lambda\in[0,1]$.

As a simple application of the linear homotopy, we give a variant of the
existence result [36, Theorem 4.1].

######  Theorem 4.31.

Suppose that $t_{*}^{1}\in\mathbb{V}^{1}$ is a zero of $\mathcal{A}^{1}$. Let
$\varkappa=\|t_{*}^{\perp}\|_{\mathbb{V}}$. Assume the following hold true.

  1. (i)

$\mathcal{A}^{j}:\mathbb{V}^{j}\to\mathbb{V}^{j}$ is $C^{1}$.

  2. (ii)

$L^{0}\varkappa=\sup_{\|u^{0}\|_{\mathbb{V}}=1}\langle\mathcal{A}^{0}(t_{*}^{0})-\mathcal{A}^{1}(t_{*}^{1}),u^{0}\rangle$
for some $L^{0}<\infty$.

  3. (iii)

$\mathcal{A}^{j}$ is strongly monotone in
$B_{\mathbb{V}^{j}}(t_{*}^{j},\delta_{j})$ with constant
$C_{\mathrm{SM}}^{j}=C_{\mathrm{SM}}^{j}(\delta_{j})>0$ for $j=0,1$.

  4. (iv)

$\varkappa<\delta\frac{C_{\mathrm{SM}}^{0}}{L^{0}}$, where
$\delta=\min\\{\delta_{0},\delta_{1}\\}$

Then there exists a unique $t_{**}^{0}\in
B_{\mathbb{V}^{0}}(t_{*}^{0},\delta)$ such that
$\mathcal{A}^{0}(t_{**}^{0})=0$. Furthermore,
$i(\mathcal{K}_{\mathrm{L}}(\cdot,0),t_{**}^{1})=i(\mathcal{A}^{0},t_{*}^{0})=1$.

###### Proof.

Set $\alpha:=C_{\mathrm{SM}}^{0}$ and $u^{\perp}:=t_{*}^{\perp}$ in the
definition of $\mathcal{K}_{\mathrm{L}}$. We prove that the boundary condition
$\mathcal{K}_{\mathrm{L}}(t^{1},\lambda)\neq 0$ holds true for all
$t^{1}\in\partial B_{\mathbb{V}}(t_{*}^{1},\delta)$ and $\lambda\in[0,1]$.
Write $t^{1}=t^{1}_{*}+r^{1}$, where $\|r^{1}\|_{\mathbb{V}}=\delta$ and

$\langle\mathcal{K}_{\mathrm{L}}(t^{1}_{*}+r^{1},\lambda),r^{1}\rangle=:(1-\lambda)A_{0}+\lambda
A_{1}.$

Here,

$\displaystyle A_{0}$
$\displaystyle=\langle\mathcal{A}^{0}(t_{*}^{0}+r^{0}),r^{0}\rangle+C_{\mathrm{SM}}^{0}\langle
t_{*}^{\perp}+r^{\perp}-t_{*}^{\perp},r^{\perp}\rangle_{\mathbb{V}}$
$\displaystyle=\langle\mathcal{A}^{0}(t_{*}^{0}+r^{0})-\mathcal{A}^{0}(t_{*}^{0}),r^{0}\rangle+\langle\mathcal{A}^{0}(t_{*}^{0})-\mathcal{A}^{1}(t_{*}^{1}),r^{0}\rangle+C_{\mathrm{SM}}^{0}\|r^{\perp}\|_{\mathbb{V}}^{2}$
$\displaystyle\geq
C_{\mathrm{SM}}^{0}\|r^{0}\|_{\mathbb{V}}^{2}-L^{0}\varkappa\|r^{0}\|_{\mathbb{V}}+C_{\mathrm{SM}}^{0}\|r^{\perp}\|_{\mathbb{V}}^{2}$
$\displaystyle\geq(C_{\mathrm{SM}}^{0}\|r^{1}\|_{\mathbb{V}}-L^{0}\varkappa)\|r^{1}\|_{\mathbb{V}}=(C_{\mathrm{SM}}^{0}\delta-L^{0}\varkappa)\delta>0,$

where we in the last step used that
$\|r^{1}\|_{\mathbb{V}}^{2}=\|r^{0}\|_{\mathbb{V}}^{2}+\|r^{\perp}\|_{\mathbb{V}}^{2}$.
Furthermore,

$\displaystyle A_{1}$
$\displaystyle=\langle\mathcal{A}^{1}(t^{1}_{*}+r^{1}),r^{1}\rangle=\langle\mathcal{A}^{1}(t^{1}_{*}+r^{1})-\mathcal{A}^{1}(t^{1}_{*}),r^{1}\rangle\geq
C_{\mathrm{SM}}^{1}\|r^{1}\|_{\mathbb{V}}^{2}>0.$

Using Proposition 4.14 and the uniqueness of the zero $t_{*}^{1}$ in
$B_{\mathbb{V}}(t_{*}^{1},\delta)$,

$\deg(\mathcal{K}_{\mathrm{L}}(\cdot,1),B_{\mathbb{V}}(t_{*}^{1},\delta),0)=\deg(\mathcal{A}^{1},B_{\mathbb{V}}(t_{*}^{1},\delta),0)=1.$

Since we have already seen that $\mathcal{K}_{\mathrm{L}}(t^{1},\lambda)\neq
0$ for all $t^{1}\in\partial B_{\mathbb{V}}(t_{*}^{1},\delta)$ and
$\lambda\in[0,1]$, the homotopy invariance of the degree can be applied to get

$\deg(\mathcal{K}_{\mathrm{L}}(\cdot,0),B_{\mathbb{V}}(t_{*}^{1},\delta),0)=1.$

Using the existence property of the degree Corollary 3.3 (ii), there exists
$t^{0}_{**}\in\mathbb{V}^{0}$ such that $(t^{0}_{**},t_{*}^{\perp})\in
B_{\mathbb{V}^{1}}(t_{*}^{1},\delta)$ and
$\langle\mathcal{K}_{\mathrm{L}}(t_{**}^{0},0),s^{0}\rangle=\langle\mathcal{A}^{0}(t_{**}^{0}),s^{0}\rangle=0$
for all $s^{0}\in\mathbb{V}^{0}$, which is what we wanted to prove. Uniqueness
follows form the local strong monotonicity of $\mathcal{A}^{0}$. ∎

In the special case when $\mathcal{A}^{0}$ is given as a projection of
$\mathcal{A}^{1}$ (see Remark 4.8), we obtain the following.

######  Corollary 4.32.

Suppose that $t_{*}^{1}\in\mathbb{V}^{1}$ is a zero of $\mathcal{A}^{1}$. Let
$\varkappa=\|t_{*}^{\perp}\|_{\mathbb{V}}$ and suppose the following hold
true.

  1. (i)

$L^{0}\varkappa=\sup_{\|u^{0}\|_{\mathbb{V}}=1}\langle\mathcal{A}^{1}(t_{*}^{0})-\mathcal{A}^{1}(t_{*}^{1}),u^{0}\rangle$
for some $L^{0}<\infty$.

  2. (ii)

$\mathcal{A}^{1}$ is strongly monotone in
$B_{\mathbb{V}^{1}}(t_{*}^{1},\delta)$ with constant $C_{\mathrm{SM}}>0$.

  3. (iii)

$\varkappa<\delta\frac{C_{\mathrm{SM}}}{C_{\mathrm{SM}}+L^{0}}$.

Then there exists a unique $t_{**}^{0}\in
B_{\mathbb{V}^{0}}(t_{*}^{1},\delta-\varkappa)$ such that
$\mathcal{A}^{1}(t_{**}^{0})=0$.

###### Proof.

It is enough to recall that
$\mathcal{A}^{0}=\Pi_{\mathbb{V}^{0}}\mathcal{A}^{1}|_{\mathbb{V}^{0}}$ is
strongly monotone with constant $C_{\mathrm{SM}}$ in

$B_{\mathbb{V}^{0}}(t_{*}^{0},\sqrt{\delta^{2}-\varkappa^{2}})\supset
B_{\mathbb{V}^{0}}(t_{*}^{0},\delta-\varkappa).$

Applying Theorem 4.31 with $C_{\mathrm{SM}}^{0}=C_{\mathrm{SM}}$ and
$\delta_{0}=\delta-\varkappa$ gives the result. ∎

Condition (i) certainly holds, since the CC mapping is locally Lipschitz (see
[37, 36]), but the constant $L^{0}$ here is possibly smaller than the usual
Lipschitz constant (which, in turn, is bounded from below by the strong
monotonicity constant).

###  4.5. Kowalski–Piecuch homotopy

In this section, we consider another homotopy that can be used to analyze the
connection between the solutions to CC methods of different rank-truncation
levels. The approach was pioneered by the chemists K. Kowalski and P. Piecuch
[32], who conducted a comprehensive numerical study based on this idea.888They
call it the “$\beta$–nested equations”, $\beta$ being the homotopy parameter.
They considered complex amplitudes and the following discussion is easily
extended to that case.

Assumption. Let $\mathbb{V}^{1}=\mathbb{V}^{0}\oplus\mathbb{V}^{\angle}$ be an
_$\ell^{2}$ -orthogonal_ direct sum decomposition of a real amplitude space
$\mathbb{V}^{1}=\mathbb{V}(G^{1})$, where $\mathbb{V}^{0}$ is an amplitude
space corresponding to some lower truncation level. More precisely, assume
that there is a rank $\rho\geq 1$, such that $\mathbb{V}^{0}$ contains all
amplitudes with rank $\leq\rho$ and
$\mathbb{V}^{\angle}:=(\mathbb{V}^{0})^{\perp_{\ell^{2}}}$ contains all
amplitudes with rank $>\rho$. In the notations introduced in Part I,

$\mathbb{V}^{0}=\mathbb{V}(G^{0})=\mathbb{V}(G(1,\ldots,\rho))\cap\mathbb{V}^{1},\,\,\,\text{and}\,\,\,\mathbb{V}^{\angle}=\mathbb{V}(G^{\perp})=\mathbb{V}(G(\rho+1,\ldots,N))\cap\mathbb{V}^{1},$

where $G^{0}=G(1,\ldots,\rho)\cap G^{1}$ and $G^{\perp}=G(\rho+1,\ldots,N)\cap
G^{1}$. Hence, any $t^{1}\in\mathbb{V}^{1}$ may be uniquely decomposed as
$t^{1}=t^{0}+t^{\angle}$, where $t^{0}\in\mathbb{V}^{0}$,
$t^{\angle}\in\mathbb{V}^{\angle}$ and $\langle
t^{0},t^{\angle}\rangle_{\ell^{2}}=0$.

Let $\mathbb{V}=\mathbb{V}(G^{\mathrm{full}})$ be the full amplitude space and
suppose that $\mathcal{A}:\mathbb{V}^{1}\to(\mathbb{V}^{1})^{*}$ is the SRCC
mapping Eq. 4.1. Write

$\displaystyle\langle\mathcal{A}$ $\displaystyle(t^{1}),s^{1}\rangle=\langle
e^{-T^{1}}\mathcal{H}_{K}e^{T^{1}}\Phi_{0},S^{0}\Phi_{0}\rangle+\langle
e^{-T^{1}}\mathcal{H}_{K}e^{T^{1}}\Phi_{0},S^{\angle}\Phi_{0}\rangle$ (4.30)
$\displaystyle=\langle(e^{-T^{0}}+e^{-T^{0}}(e^{-T^{\angle}}-I))\mathcal{H}_{K}(e^{T^{0}}+e^{T^{0}}(e^{T^{\angle}}-I))\Phi_{0},S^{0}\Phi_{0}\rangle+\langle
e^{-T^{1}}\mathcal{H}_{K}e^{T^{1}}\Phi_{0},S^{\angle}\Phi_{0}\rangle$
$\displaystyle=\langle\mathcal{A}^{0}(t^{0}),s^{0}\rangle+\langle
e^{-T^{0}}(e^{-T^{\angle}}-I)\mathcal{H}_{K}e^{T^{0}}(e^{T^{\angle}}-I)\Phi_{0},S^{0}\Phi_{0}\rangle$
$\displaystyle\quad+\langle
e^{-T^{0}}(e^{-T^{\angle}}-I)\mathcal{H}_{K}e^{T^{0}}\Phi_{0},S^{0}\Phi_{0}\rangle+\langle
e^{-T^{0}}\mathcal{H}_{K}e^{T^{0}}(e^{T^{\angle}}-I)\Phi_{0},S^{0}\Phi_{0}\rangle+\langle
e^{-T^{1}}\mathcal{H}_{K}e^{T^{1}}\Phi_{0},S^{\angle}\Phi_{0}\rangle$
$\displaystyle=\langle\mathcal{A}(t^{0}),s^{0}\rangle+\langle
e^{-T^{0}}\mathcal{H}_{K}e^{T^{0}}(e^{T^{\angle}}-I)\Phi_{0},S^{0}\Phi_{0}\rangle+\langle\mathcal{A}(t^{1}),s^{\angle}\rangle,$

where in the last step the second and the third terms of the penultimate
expression vanish because

$(e^{-T^{\angle}}-I)^{\dagger}S^{0}\Phi_{0}=-(T^{\angle})^{\dagger}S^{0}\Phi_{0}-\frac{1}{2}\left((T^{\angle})^{\dagger}\right)^{2}S^{0}\Phi_{0}-\ldots=0,$

due to the definition of the spaces $\mathbb{V}^{0}$ and
$\mathbb{V}^{\angle}$. Note that this last relation would not hold in the case
$\mathbb{V}^{0}=\mathbb{V}(G(\mathrm{D}))\cap\mathbb{V}^{1}$,
$\mathbb{V}^{1}=\mathbb{V}(G^{\mathrm{full}})$, which is actually excluded by
assumption.

Motivated by the preceding calculation in Eq. 4.30, we define the
_Kowalski–Piecuch homotopy_
$\mathcal{K}_{\mathrm{KP}}:\mathbb{V}^{1}\times[0,1]\to(\mathbb{V}^{1})^{*}$
via the instruction

$\displaystyle\langle\mathcal{K}_{\mathrm{KP}}(t^{1},\lambda),s^{1}\rangle$
$\displaystyle:=\langle\mathcal{H}_{K}(t^{0})\Phi_{0},S^{0}\Phi_{0}\rangle+\langle\mathcal{H}_{K}(t^{1})\Phi_{0},S^{\angle}\Phi_{0}\rangle+\lambda\langle\mathcal{H}_{K}(t^{0})(e^{T^{\angle}}-I)\Phi_{0},S^{0}\Phi_{0}\rangle$
(4.31)

for all $t^{1},s^{1}\in\mathbb{V}^{1}$ and $\lambda\in[0,1]$. Here, the
relation $\mathcal{K}_{\mathrm{KP}}(t^{1}_{*},\lambda)=0$ is the same as the
system [32, eqs. (90)–(91) and (93)].

It is obvious that $\mathcal{K}_{\mathrm{KP}}$ is an admissible homotopy.
However, it is unclear whether it is faithful or not. In fact,
$\mathcal{K}_{\mathrm{KP}}(t^{1}_{**},0)=0$ is equivalent to the system

$\displaystyle\langle\mathcal{H}_{K}(t_{**}^{0})\Phi_{0},S^{0}\Phi_{0}\rangle$
$\displaystyle=0,$ (4.32)
$\displaystyle\langle\mathcal{H}_{K}(t_{**}^{0}+t_{**}^{\angle})\Phi_{0},S^{\angle}\Phi_{0}\rangle$
$\displaystyle=0,$ (4.33)

for all $s^{1}\in\mathbb{V}^{1}$. In other words, the usual SRCC equation
$\mathcal{A}(t_{**}^{0})=0$, Eq. 4.32, is augmented with an additional
equation for $t_{**}^{\angle}$, Eq. 4.33, which, in turn, depends on
$t_{**}^{0}$. It is not obvious at all that Eq. 4.33 has a solution
$t_{**}^{\angle}$ for a given zero $t_{**}^{0}$. The extensive numerical
evidence in [32] clearly indicates that Eq. 4.33 admits a solution in various
circumstances.

Before stating our existence result we recast the KP homotopy into a more
convenient form, which we already encountered in Eq. 4.26.

######  Lemma 4.33.

The following formula holds true:

$\langle\mathcal{K}_{\mathrm{KP}}(t^{1},\lambda),s^{1}\rangle=\langle\mathcal{A}(t^{0}+\lambda
t^{\angle}),s^{0}\rangle+\langle\mathcal{A}(t^{1}),s^{\angle}\rangle,$ (4.34)

for all $t^{1},s^{1}\in\mathbb{V}^{1}$ and $\lambda\in[0,1]$.

###### Proof.

It is enough to prove that

$\langle\mathcal{K}_{\mathrm{KP}}(t^{1},\lambda),s^{1}\rangle=\langle
e^{-T^{0}}\mathcal{H}_{K}e^{T^{0}+\lambda
T^{\angle}}\Phi_{0},S^{0}\Phi_{0}\rangle+\langle
e^{-T^{1}}\mathcal{H}_{K}e^{T^{1}}\Phi_{0},S^{\angle}\Phi_{0}\rangle,$ (4.35)

because $(e^{-\lambda T^{\angle}})^{\dagger}S^{0}\Phi_{0}=S^{0}\Phi_{0}$. To
see Eq. 4.35, note that in the expansion $e^{T^{0}+\lambda
T^{\angle}}=e^{T^{0}}(I+\lambda
T^{\angle}+\frac{\lambda^{2}}{2!}(T^{\angle})^{2}+\ldots+\frac{\lambda^{N}}{N!}(T^{\angle})^{N})$
the quadratic and higher-order terms do not contribute because the excitation
rank of $\mathcal{H}_{K}(T^{\angle})^{k}\Phi_{0}$ exceeds $\rho$ for
$k=2,\ldots,N$, due to the fact that $\mathcal{H}_{K}$ is a two-body operator.
∎

To formulate the existence result, suppose that $t_{*}^{1}\in\mathbb{V}^{1}$
is a non-degenerate zero of $\mathcal{A}$. Since $\mathbb{V}^{1}$ is assumed
to be finite-dimensional, it is always possible to choose an $\alpha>0$ so
that

$\langle(\mathcal{A}^{\prime}(t_{*}^{1})+\alpha
I)r^{1},r^{1}\rangle\geq\gamma_{\alpha}\|r^{1}\|^{2}_{\mathbb{V}}\quad\text{for
all}\quad r^{1}\in\mathbb{V}^{1},$ (4.36)

for some $\gamma_{\alpha}>0$. This is also true in the complex case with a
“$\operatorname{Re}$” added to the left-hand side.

Define the operator $\Theta_{\alpha}:\mathbb{V}\to\mathbb{V}$ via999The
authors learned this trick from [6, 5].

$\langle\mathcal{A}^{\prime}(t_{*}^{1})u^{1},\Theta_{\alpha}v^{1}\rangle=\langle(\mathcal{A}^{\prime}(t_{*}^{1})+\alpha
I)u^{1},v^{1}\rangle\quad\text{for all}\quad u^{1},v^{1}\in\mathbb{V}^{1}.$
(4.37)

Then $\Theta_{\alpha}$ is well-defined, as long as
$\ker\mathcal{A}^{\prime}(t_{*}^{1})=\\{0\\}$, i.e. that $t_{*}^{1}$ is non-
degenerate.

######  Theorem 4.34 (Existence for KP).

Let $t_{*}^{1}\in\mathbb{V}^{1}$ be a non-degenerate zero of $\mathcal{A}$.
Suppose the following.

  1. (i)

With
$\theta_{0}=\|\Pi_{0}(\Theta_{\alpha}-I)\Pi_{0}\|_{\mathcal{L}(\mathbb{V})}^{2}$
and
$\theta_{\angle}=\|\Pi_{0}(\Theta_{\alpha}-I)\Pi_{\angle}\|_{\mathcal{L}(\mathbb{V})}^{2}$,
there holds

$\eta:=(1-g)\frac{\gamma_{\alpha}-\frac{1}{2}M_{\delta}\|\Theta_{\alpha}\|_{\mathcal{L}(\mathbb{V})}\delta}{\Delta(t_{*}^{1})+M_{\delta}\delta}-\frac{1}{2}\max\\{\varepsilon+2(1+\varepsilon^{-1})\theta_{0},2(1+\varepsilon^{-1})\theta_{\angle}\\}>\frac{1}{2},$

for some $\varepsilon>0$ and $\delta>0$, where $M_{\delta}$ was defined in Eq.
4.15. Here,

$\Delta(t_{*}^{1})=\max_{\begin{subarray}{c}\|u^{0}\|_{\mathbb{V}}=1\\\
\|v^{\angle}\|_{\mathbb{V}}=1\end{subarray}}|\langle(\mathcal{H}_{K}+[(T_{*}^{1})_{1}^{\dagger},\mathcal{H}_{K}])U^{0}\Phi_{0},V^{\angle}\Phi_{0}\rangle|.$

Also, $\alpha>0$ and $\gamma_{\alpha}>0$ satisfy Eq. 4.36 and $0<g<1$ is such
that $|\langle t^{0},t^{\angle}\rangle_{\mathbb{V}}|\leq
g\|t^{0}\|_{\mathbb{V}}\|t^{\angle}\|_{\mathbb{V}}$ for all
$t^{1}\in\mathbb{V}^{1}$.

  2. (ii)

With $\varkappa=\|t_{*}^{\angle}\|_{\mathbb{V}}$, there holds

$\varkappa<\frac{2\sqrt{\eta}-\sqrt{2}}{2-\sqrt{2}+2\sqrt{\eta}}\delta.$
(4.38)

Let
$D=\\{t_{*}^{1}+r^{1}\in\mathbb{V}^{1}:\|r^{0}\|_{\mathbb{V}}^{2}+\|r^{\angle}\|_{\mathbb{V}}^{2}<\frac{1}{2}(\delta-\varkappa)^{2}\\}$.
Then, for any $\lambda\in[0,1)$, there exists $t_{**}^{1}(\lambda)\in D$ such
that $\mathcal{K}_{\mathrm{KP}}(t_{**}^{1}(\lambda),\lambda)=0$. Furthermore,
$\deg(\mathcal{K}_{\mathrm{KP}}(\cdot,\lambda),D,0)\equiv d\neq 0$ for all
$\lambda\in[0,1]$. In particular, there exists $t_{**}^{1}\in D$ such that
$\mathcal{A}(t_{**}^{0})=0$.

For the proof, see Appendix A.

######  Remark 4.35.

  1. (i)

A crucial difference between the linear homotopy and the KP homotopy is that