Double ionization potential equation-of-motion coupled-cluster approach with full inclusion of 4-hole–2-particle excitations and three-body clusters

Gururangan, Karthik; Dutta, Achintya Kumar; Piecuch, Piotr

doi:10.1063/5.0253059

The double ionization potential (DIP) equation-of-motion (EOM) coupled-cluster (CC) method with a full treatment of 4-hole–2-particle (4h–2p) correlations and triply excited clusters, abbreviated as DIP-EOMCCSDT(4h–2p), and its approximate form called DIP-EOMCCSD(T)(a)(4h–2p) have been formulated and implemented in the open-source CCpy package available on GitHub. The resulting codes work with both nonrelativistic and spin-free scalar-relativistic Hamiltonians. By examining the DIPs of a few small molecules, for which accurate reference data are available, we demonstrate that the DIP-EOMCCSDT(4h–2p) and DIP-EOMCCSD(T)(a)(4h–2p) approaches improve the results obtained using the DIP-EOMCC methods truncated at 3h–1p or 4h–2p excitations on top of the CC calculations with singles and doubles.

The single-reference coupled-cluster (CC) theory^1–5 and its equation-of-motion (EOM) extensions to electronically excited^6–10 and electron attached and ionized states^11–40 have become preeminent methods of quantum chemistry. In this Communication, we focus on the double ionization potential (DIP) EOMCC framework, which allows one to directly determine the ground and excited states of doubly ionized molecular species and which is useful in many applications, such as Auger electron spectroscopy,^41–44 singlet–triplet gaps in biradicals,^33–36 and strong-field-induced chemical reactivity.⁴⁵

In the DIP-EOMCC formalism, the ground (μ = 0) and excited (μ > 0) states of the target (N − 2)-electron system are expressed as

| Ψ_{μ}^{(N - 2)} 〉 = R_{μ}^{(- 2)} | Ψ^{(N)} 〉,

(1)

where the doubly ionizing operator

R_{μ}^{(- 2)} = \sum_{n = 0}^{M_{R}} R_{μ, (n + 2) h - n p},

(2)

with R_{μ,(n+2)h–np} representing its (n + 2)-hole–n-particle [(n + 2)h–np] components, removes two electrons from the CC ground state

| Ψ^{(N)} 〉 = e^{T} | Φ 〉

(3)

of the underlying N-electron species, in which

T = \sum_{n = 1}^{M_{T}} T_{n}

(4)

is the cluster operator and |Φ⟩ is the reference determinant that serves as a Fermi vacuum. The many-body components of the T and

R_{μ}^{(- 2)}

operators are given by

T_{n} = \sum_{\begin{array}{c} i_{1} < \dots < i_{n} \\ a_{1} < \dots < a_{n} \end{array}} t_{a_{1} \dots a_{n}}^{i_{1} \dots i_{n}} a^{a_{1}} \dots a^{a_{n}} a_{i_{n}} \dots a_{i_{1}}

(5)

and

R_{μ, (n + 2) h - n p} = \sum_{\begin{array}{c} k_{1} < \dots < k_{n} < i < j \\ c_{1} < \dots < c_{n} \end{array}} r_{c_{1} \dots c_{n}}^{i j k_{1} \dots k_{n}} (μ) a^{c_{1}} \dots a^{c_{n}} a_{k_{n}} \dots a_{k_{1}} a_{j} a_{i},

(6)

respectively, where, as usual, indices i, j, … (a, b, …) denote the spinorbitals that are occupied (unoccupied) in |Φ⟩ and a^p (a_p) represents the fermionic creation (annihilation) operator associated with the spinorbital |p⟩. M_R and M_T control the DIP-EOMCC theory level, with M_R = N − 2 defining exact calculations and M_R < N − 2 and M_T ≤ N leading to DIP-EOMCC approximations.

The DIP-EOMCC approaches that have been implemented so far include the basic DIP-EOMCCSD(3h–1p) method,^{28,29,31,33–36} defined by M_R = 1 and M_T = 2, which describes 2h and 3h–1p correlations on top of the CC calculations with singles and doubles (CCSD),^46–49 its higher-level DIP-EOMCCSD(4h–2p) extension^35,36 corresponding to M_R = 2 and M_T = 2, which also describes 4h–2p correlations on top of CCSD, and the DI-EOMCCSDT scheme³¹ corresponding to M_R = 1 and M_T = 3, which accounts for 2h and 3h–1p correlations on top of the CC calculations with singles, doubles, and triples (CCSDT).^50,51 While all of these methods and the various approximations to them aimed at reducing computational costs are useful tools for determining DIP energies in molecular systems and singlet–triplet gaps in certain biradicals, our recent numerical tests, including the DIPs of a few small molecules examined in this Communication, indicate that the explicit incorporation of 4h–2p correlations, needed to achieve a highly accurate description,^35,36 may not be well balanced with the CCSD treatment of the underlying N-electron species, resulting in some cases in loss of accuracy compared to the basic DIP-EOMCCSD(3h–1p) level.

The main goal of this work is to enrich the existing arsenal of DIP-EOMCC methods and, in particular, address the above-mentioned concerns by examining the high-level DIP-EOMCC approach with a full treatment of both 4h–2p and T₃ correlations, corresponding to M_R = 2 and M_T = 3 and abbreviated as DIP-EOMCCSDT(4h–2p). The present study describes the efficient formulation and computer implementation of full DIP-EOMCCSDT(4h–2p) and its less expensive DIP-EOMCCSD(T)(a)(4h–2p) approximation, including the factorized and programmable expressions that define them. We illustrate the performance of both methods, as coded for nonrelativistic and spin-free scalar-relativistic Hamiltonians in the open-source CCpy package⁵² available on GitHub, by calculating the vertical DIPs of H₂O, CH₄, BN, Cl₂, Br₂, and HBr and comparing the results with those obtained using the DIP-EOMCCSD(3h–1p) and DIP-EOMCCSD(4h–2p) approaches and the reliable, high-accuracy, reference data.

The key step of any DIP-EOMCC calculation is a diagonalization of the similarity-transformed Hamiltonian

{\bar{H}}_{N} = e^{- T} H_{N} e^{T} = {(H_{N} e^{T})}_{C}

associated with the N-electron CC ground state in the relevant (N − 2)-electron subspace of the Fock space corresponding to the content of the ionizing operator

R_{μ}^{(- 2)}

⁠. Here, subscript C designates the connected operator product and H_N = H − ⟨Φ|H|Φ⟩ = F_N + V_N is the electronic Hamiltonian in the normal-ordered form relative to |Φ⟩, with F_N and V_N representing its Fock and two-electron interaction components. In the case of the cluster and ionizing operators defined by Eqs. and , respectively, the basis states that span the (N − 2)-electron subspace of the Fock space used in the DIP-EOMCC calculations are |Φ_ij⟩ = a_ja_i|Φ⟩ and

| Φ_{i j k_{1} \dots k_{n}}^{c_{1} \dots c_{n}} 〉 = a^{c_{1}} \dots a^{c_{n}} a_{k_{n}} \dots a_{k_{1}} a_{j} a_{i} | Φ 〉

⁠, n = 1, …, M_R. Assuming that M_R ≤ M_T (a condition required for retaining size intensivity of the resulting double ionization energies^35,36,38), the DIP-EOMCC eigenvalue problem is given by

{({\bar{H}}_{N, open} R_{μ}^{(- 2)})}_{C} | Φ 〉 = ω_{μ}^{(N - 2)} R_{μ}^{(- 2)} | Φ 〉,

(7)

where

{\bar{H}}_{N, open}

refers to the diagrams of

{\bar{H}}_{N}

containing external fermion lines and

ω_{μ}^{(N - 2)} = E_{μ}^{(N - 2)} - E_{0}^{(N)}

is the vertical DIP energy representing the difference between the total energy of the ground (μ = 0) or excited (μ > 0) state of the (N − 2)-electron target system, denoted as

E_{μ}^{(N - 2)}

⁠, and the ground-state CC energy of the underlying N-electron species,

E_{0}^{(N)} = ⟨ Φ | H | Φ ⟩ + ⟨ Φ | {\bar{H}}_{N} | Φ ⟩

⁠. In practice, including this work, the solutions of Eq. are obtained using the Hirao–Nakatsuji generalization⁵³ of the Davidson diagonalization algorithm⁵⁴ to non-Hermitian Hamiltonians.

In the DIP-EOMCCSDT(4h–2p) approach developed in this work, the CCSDT similarity-transformed Hamiltonian, designated as

{\bar{H}}_{N}^{(CCSDT)}

⁠, is diagonalized in the (N − 2)-electron subspace of the Fock space spanned by |Φ_ij⟩,

| Φ_{ijk}^{c} 〉

⁠, and

| Φ_{ijkl}^{c d} 〉

⁠. The programmable factorized expressions for the required projections of the left-hand side of Eq. , with T truncated at T₃ and

R_{μ}^{(- 2)}

truncated at R_μ,4h–2p, as implemented in CCpy, are

\begin{align} ⟨ Φ_{i j} | {({\bar{H}}_{N, o p e n}^{(CCSDT)} R_{μ}^{(- 2)})}_{C} | Φ ⟩ & = A^{i j} [- {\bar{h}}_{m}^{i} r^{m j} (μ) + \frac{1}{4} {\bar{h}}_{m n}^{i j} r^{m n} (μ) \\ + \frac{1}{2} {\bar{h}}_{m}^{e} r_{e}^{ijm} (μ) - \frac{1}{2} {\bar{h}}_{m n}^{if} r_{f}^{mjn} (μ) \\ + \frac{1}{8} {\bar{h}}_{m n}^{e f} r_{e f}^{ijmn} (μ)], \end{align}

(8)

\begin{align} ⟨ Φ_{ijk}^{c} | {({\bar{H}}_{N, o p e n}^{(CCSDT)} R_{μ}^{(- 2)})}_{C} | Φ ⟩ & = A^{ijk} [\frac{1}{2} {I^{'}}^{i e} (μ) t_{e c}^{j k} - \frac{1}{2} {\bar{h}}_{c m}^{k i} r^{m j} (μ) \\ + \frac{1}{6} {\bar{h}}_{c}^{e} r_{e}^{ijk} (μ) - \frac{1}{2} {\bar{h}}_{m}^{k} r_{c}^{ijm} (μ) \\ + \frac{1}{4} {\bar{h}}_{m n}^{i j} r_{c}^{mnk} (μ) + \frac{1}{2} {\bar{h}}_{c m}^{k e} r_{e}^{ijm} (μ) \\ + \frac{1}{6} {\bar{h}}_{m}^{e} r_{c e}^{ijkm} (μ) - \frac{1}{4} {\bar{h}}_{m n}^{k f} r_{c f}^{ijmn} (μ) \\ + \frac{1}{12} {\bar{h}}_{c n}^{e f} r_{e f}^{ijkn} (μ) + \frac{1}{12} I^{e f} t_{efc}^{ijk}], \end{align}

(9)

and,

\begin{align} ⟨ Φ_{ijkl}^{c d} | {({\bar{H}}_{N, o p e n}^{(CCSDT)} R_{μ}^{(- 2)})}_{C} | Φ ⟩ & = A^{ijkl} A_{c d} [\frac{1}{12} {\bar{h}}_{d c}^{l e} r_{e}^{ijk} (μ) - \frac{1}{4} {\bar{h}}_{d m}^{l k} r_{c}^{ijm} (μ) \\ - \frac{1}{12} I_{m}^{ijk} (μ) t_{c d}^{m l} + \frac{1}{4} I_{c}^{ije} (μ) t_{e d}^{k l} \\ + \frac{1}{24} {\bar{h}}_{d}^{e} r_{c e}^{ijkl} (μ) - \frac{1}{12} {\bar{h}}_{m}^{i} r_{c d}^{mjkl} (μ) \\ + \frac{1}{16} {\bar{h}}_{m n}^{i j} r_{c d}^{mnkl} (μ) + \frac{1}{96} {\bar{h}}_{c d}^{e f} r_{e f}^{ijkl} (μ) \\ + \frac{1}{6} {\bar{h}}_{d m}^{l e} r_{c e}^{ijkm} (μ) + \frac{1}{12} I^{i e} (μ) t_{ecd}^{jkl} \\ + \frac{1}{8} I_{e}^{ijm} (μ) t_{cde}^{klm} + \frac{1}{12} I_{c}^{ef k} (μ) t_{efd}^{ijl}], \end{align}

(10)

where we use the Einstein summation convention over repeated upper and lower indices and

A^{p q} = A_{p q} = 1 - (p q)

⁠,

A^{pqr} = A^{p / q r} A^{q r}

⁠, and

A^{pqrs} = A^{p / q r s} A^{qrs}

⁠, with

A^{p / q r} = 1 - (p q) - (p r)

and

A^{p / q r s} = 1 - (p q) - (p r) - (p s)

⁠, are index antisymmetrizers. The expressions for the one-body

({\bar{h}}_{p}^{q})

and two-body

({\bar{h}}_{p q}^{r s})

components of the similarity-transformed Hamiltonian as well as the additional intermediates entering Eqs. and are provided in Tables I and II. Equations – imply that the diagonalization step of DIP-EOMCCSDT(4h–2p) has computational costs identical to those characterizing DIP-EOMCCSD(4h–2p), which scale as

n_{o}^{4} n_{u}^{4}

or

N^{8}

⁠, where n_o (n_u) is the number of occupied (unoccupied) orbitals in |Φ⟩ and

N

is a measure of the system size. However, the overall computational effort associated with the DIP-EOMCCSDT(4h–2p) approach is considerably higher than that of DIP-EOMCCSD(4h–2p) since in the DIP-EOMCCSDT(4h–2p) case, one also has to solve the CCSDT equations for the underlying N-electron species, which involve

n_{o}^{3} n_{u}^{5}

steps, as opposed to the much less expensive

n_{o}^{2} n_{u}^{4}

(N^{6})

steps of CCSD.

TABLE I.

One- and two-body components of the similarity-transformed Hamiltonian expressed in terms of the one-, two-, and three-body cluster amplitudes, $t_{a}^{i}$ ⁠, $t_{a b}^{i j}$ ⁠, and $t_{abc}^{ijk}$ ⁠, respectively, and matrix elements of the Fock and two-electron interaction operators, denoted as $f_{p}^{q} \equiv ⟨ p | f | q ⟩$ and $v_{p q}^{r s} \equiv ⟨ p q | v | r s ⟩ - ⟨ p q | v | s r ⟩$ ⁠. In the case of the DIP-EOMCCSDT(4h–2p) method, the $t_{a}^{i}$ ⁠, $t_{a b}^{i j}$ ⁠, and $t_{abc}^{ijk}$ values are obtained with CCSDT. In the case of the DIP-EOMCCSD(T)(a)(4h–2p) approach, they are replaced by their ${\tilde{t}}_{a}^{i}$ ⁠, ${\tilde{t}}_{a b}^{i j}$ ⁠, and ${\tilde{t}}_{abc}^{ijk}$ counterparts corresponding to the ${\tilde{T}}_{1}$ ⁠, ${\tilde{T}}_{2}$ ⁠, and ${\tilde{T}}_{3}$ operators defined by Eqs. (11)–(13).

Component of ${\bar{H}}_{N}$	Expressiona
${\bar{h}}_{m}^{e}$	$f_{m}^{e} + v_{i m}^{a e} t_{e}^{m}$
${\bar{h}}_{j}^{i}$	$f_{j}^{i} + {\bar{h}}_{j}^{e} t_{e}^{i} + v_{j m}^{i e} t_{e}^{m} + \frac{1}{2} v_{j n}^{e f} t_{e f}^{i n}$
${\bar{h}}_{a}^{b}$	$f_{a}^{b} - {\bar{h}}_{m}^{b} t_{a}^{m} + v_{a m}^{b e} t_{e}^{m} - \frac{1}{2} v_{m n}^{b f} t_{a f}^{m n}$
${\bar{h}}_{m n}^{e f}$	$v_{m n}^{e f}$
${\bar{h}}_{a m}^{e f}$	$v_{a m}^{e f} - v_{m n}^{f e} t_{a}^{n}$
${\bar{h}}_{m n}^{i e}$	$v_{m n}^{i e} + v_{m n}^{f e} t_{f}^{i}$
${\bar{h}}_{a b}^{e f}$	$v_{a b}^{e f} + \frac{1}{2} v_{m n}^{e f} τ_{a b}^{m n} - A_{a b} v_{a m}^{e f} t_{b}^{m}$
${\bar{h}}_{m n}^{i j}$	$v_{m n}^{i j} + \frac{1}{2} v_{m n}^{e f} τ_{e f}^{i j} + A^{ij} v_{n m}^{j e} t_{e}^{i}$
${\bar{h}}_{a m}^{i e}$	$v_{a m}^{i e} + v_{a m}^{f e} t_{f}^{i} - {\bar{h}}_{n m}^{i e} t_{a}^{n} + v_{m n}^{e f} t_{a f}^{i n}$
${\bar{h}}_{a m}^{i j}$	$v_{a m}^{i j} + {\bar{h}}_{m}^{e} t_{a e}^{i j} - {\bar{h}}_{n m}^{i j} t_{a}^{n} + \frac{1}{2} v_{a m}^{e f} t_{e f}^{i j} + A^{i j} ({\bar{h}}_{m n}^{jf} t_{a f}^{i n} + {χ^{'}}_{a m}^{i e} t_{e}^{j}) + v_{m n}^{e f} t_{aef}^{ijn}$
${\bar{h}}_{a b}^{i e}$	$v_{a b}^{i e} - {\bar{h}}_{m}^{e} t_{a b}^{i m} + v_{a b}^{e f} t_{f}^{i} + \frac{1}{2} {\bar{h}}_{m n}^{i e} t_{a b}^{m n} - A_{a b} (χ_{a m}^{i e} t_{b}^{m} - v_{b n}^{e f} t_{a f}^{i n}) - v_{m n}^{e f} t_{abf}^{imn}$
${χ^{'}}_{a m}^{i e}$	$v_{a m}^{i e} + \frac{1}{2} v_{a m}^{e f} t_{e}^{i}$
$χ_{a m}^{i e}$	${\bar{h}}_{a m}^{i e} + \frac{1}{2} {\bar{h}}_{n m}^{i e} t_{a}^{n}$
$τ_{a b}^{i j}$	$t_{a b}^{i j} + A^{i j} t_{a}^{i} t_{b}^{j}$

^a

In each expression, summation is carried out over repeated upper and lower indices.

TABLE II.

Intermediates entering Eqs. (9) and (10) that are introduced in order to evaluate the contributions to the DIP-EOMCCSDT(4h–2p) and DIP-EOMCCSD(T)(a)(4h–2p) equations due to the three- and four-body components of the similarity-transformed Hamiltonian.

Intermediate	Expressiona
I′^ie(μ)	$\frac{1}{2} {\bar{h}}_{m n}^{i e} r^{m n} (μ) - \frac{1}{2} {\bar{h}}_{m n}^{f e} r_{f}^{inm} (μ)$
I^ie(μ)b	$\frac{1}{2} {\bar{h}}_{m n}^{i e} r^{m n} (μ) - \frac{1}{2} {\bar{h}}_{m n}^{f e} r_{f}^{inm} (μ) - {\bar{h}}_{m}^{e} r^{i m} (μ)$
I^ef(μ)	$\frac{1}{2} {\bar{h}}_{m n}^{e f} r^{m n} (μ)$
$I_{m}^{ijk} (μ)$	$A^{ijk} [\frac{1}{2} {\bar{h}}_{n m}^{k e} r_{e}^{ijm} (μ) - \frac{1}{2} {\bar{h}}_{n m}^{i k} r^{nj} (μ) + \frac{1}{12} {\bar{h}}_{m n}^{e f} r_{e f}^{ijkn} (μ)]$
$I_{c}^{ije} (μ)$	${\bar{h}}_{c m}^{f e} r_{e}^{ijm} (μ) + \frac{1}{2} I^{f e} (μ) t_{c f}^{i j} + A^{i j} [{\bar{h}}_{c m}^{i e} r^{mj} (μ) + \frac{1}{2} {\bar{h}}_{n m}^{i e} r_{c}^{njm} (μ)] - \frac{1}{2} {\bar{h}}_{m n}^{e f} r_{c f}^{ijmn} (μ)$
$I_{e}^{ijm} (μ)$ c	${\bar{h}}_{m n}^{e f} r_{f}^{ijn} (μ) - A^{i j} {\bar{h}}_{n m}^{j e} r^{i n} (μ)$
$I_{c}^{ef k} (μ)$ d	$\frac{1}{2} {\bar{h}}_{m n}^{e f} r_{c}^{mnk} (μ) + {\bar{h}}_{c m}^{e f} r^{k m} (μ)$

^a

In each expression, summation is carried out over repeated upper and lower indices.

^b

In the expression for I^ie(μ) used in DIP-EOMCCSD(T)(a)(4h–2p), ${\bar{h}}_{m n}^{i e}$ and ${\bar{h}}_{m}^{e}$ are replaced by $v_{m n}^{i e}$ and $f_{m}^{e}$ ⁠, respectively.

^c

In the expression for $I_{e}^{ijm} (μ)$ used in DIP-EOMCCSD(T)(a)(4h–2p), ${\bar{h}}_{m n}^{j e}$ is replaced by $v_{m n}^{j e}$ ⁠.

^d

In the expression for $I_{c}^{e f k} (μ)$ used in DIP-EOMCCSD(T)(a)(4h–2p), ${\bar{h}}_{c m}^{e f}$ is replaced by $v_{c m}^{e f}$ ⁠.

Given the high computational costs of the DIP-EOMCCSDT(4h–2p) method, we also consider the more practical DIP-EOMCCSD(T)(a)(4h–2p) scheme, in which we adopt the Møller–Plesset (MP) partitioning of the Hamiltonian and, following Ref. 55, incorporate the leading T₃ correlation effects by correcting the T₁ and T₂ clusters obtained with CCSD using the formulas

{\tilde{T}}_{1} | Φ 〉 = T_{1} | Φ 〉 + D_{1}^{- 1} {(V_{N} {\tilde{T}}_{3})}_{C} | Φ 〉

(11)

and

{\tilde{T}}_{2} | Φ 〉 = T_{2} | Φ 〉 + D_{2}^{- 1} {(H_{N} {\tilde{T}}_{3})}_{C} | Φ 〉,

(12)

where D₁ and D₂ are the usual MP denominators for singles and doubles and

{\tilde{T}}_{3} | Φ 〉 = D_{3}^{- 1} {(V_{N} T_{2})}_{C} | Φ 〉

(13)

is the lowest-order approximation to T₃, with D₃ representing the MP denominator for triples. Once

{\tilde{T}}_{1}

⁠,

{\tilde{T}}_{2}

⁠, and

{\tilde{T}}_{3}

are determined via Eqs. –, the resulting CCSD(T)(a) similarity-transformed Hamiltonian, constructed using the recipe described in Ref. 55, is diagonalized in the same way as

{\bar{H}}_{N}^{(CCSDT)}

in DIP-EOMCCSDT(4h–2p) with the help of Eqs. –. By eliminating the need for performing CCSDT calculations, the most expensive steps of DIP-EOMCCSD(T)(a)(4h–2p) scale as

n_{o}^{4} n_{u}^{4}

rather than

n_{o}^{3} n_{u}^{5}

⁠.

To illustrate the performance of the DIP-EOMCCSDT(4h–2p) and DIP-EOMCCSD(T)(a)(4h–2p) methods, as implemented for nonrelativistic and spin-free scalar-relativistic Hamiltonians in the open-source CCpy package,⁵² we applied them, along with their DIP-EOMCCSD(3h–1p) and DIP-EOMCCSD(4h–2p) counterparts, available in CCpy as well, to the vertical DIPs of two sets of small molecules, for which highly accurate reference data can be found in the literature. The first set, which is the primary focus of our discussion below, consisted of the H₂O, CH₄, and BN molecules, as described by the aug-cc-pVTZ basis set,^56,57 where we compared our DIP-EOMCC DIPs corresponding to the lowest singlet and triplet states of the ${(H_{2} O)}^{2 +}$ ⁠, ${(C H_{4})}^{2 +}$ ⁠, and (BN)²⁺ dications with their near-exact counterparts obtained in Ref. 58 with the configuration interaction (CI) approach using perturbative selection made iteratively, abbreviated as CIPSI,^59–61 extrapolated to the full CI limit. We also calculated the vertical DIPs of the Cl₂, Br₂, and HBr molecules corresponding to the triplet ground states and low-lying singlet states of ${(C l_{2})}^{2 +}$ ⁠, ${(B r_{2})}^{2 +}$ ⁠, and (HBr)²⁺, comparing our DIP-EOMCC results with the reliable experimental data reported in Refs. 62–64. In this case, to obtain insights into the convergence of our calculated DIP values with the basis set, we used the cc-pVTZ and cc-pVQZ bases.^56,65,66 Following Ref. 58, the equilibrium geometries of the ground-state H₂O, CH₄, and BN molecules used in our DIP-EOMCC computations were taken from Ref. 67, where they were optimized with the CC3/aug-cc-pVTZ approach. The equilibrium bond lengths of Cl₂, Br₂, and HBr were taken from Ref. 68. In setting up and solving the DIP-EOMCC eigenvalue problems for the ${(H_{2} O)}^{2 +}$ ⁠, ${(C H_{4})}^{2 +}$ ⁠, (BN)²⁺, ${(C l_{2})}^{2 +}$ ⁠, ${(B r_{2})}^{2 +}$ ⁠, and (HBr)²⁺ target species and executing the preceding CC computations for their neutral parents, we used the restricted Hartree–Fock (RHF) orbitals of the respective H₂O, CH₄, BN, Cl₂, Br₂, and HBr molecules. The relevant RHF reference determinants and transformed one- and two-electron integrals were generated with the PySCF code,^69,70 with which CCpy is interfaced. The scalar-relativistic effects included in our DIP-EOMCC calculations for the DIPs of Cl₂, Br₂, and HBr were handled using the SFX2C-1e spin-free exact two-component approach of Ref. 71, as implemented in PySCF, and the lowest-energy orbitals correlating with the chemical cores of non-hydrogen atoms were frozen in post-RHF steps.

The results of our DIP-EOMCCSD(3h–1p), DIP-EOMCCSD(4h–2p), DIP-EOMCCSD(T)(a)(4h–2p), and DIP-EOMCCSDT(4h–2p) computations for the vertical DIPs of H₂O, CH₄, and BN corresponding to the lowest singlet and triplet states of the ${(H_{2} O)}^{2 +}$ ⁠, ${(C H_{4})}^{2 +}$ ⁠, and (BN)²⁺ dications are summarized in Table III. The vertical DIPs of Cl₂, Br₂, and HBr corresponding to the ground and low-lying excited states of ${(C l_{2})}^{2 +}$ ⁠, ${(B r_{2})}^{2 +}$ ⁠, and (HBr)²⁺ resulting from our DIP-EOMCCSD(3h–1p), DIP-EOMCCSD(4h–2p), DIP-EOMCCSD(T)(a)(4h–2p), and DIP-EOMCCSDT(4h–2p) calculations are reported in Table IV. In much of the remaining discussion, we focus on the DIPs of H₂O, CH₄, and BN shown in Table III, where we compare our DIP-EOMCC data with their near-full-CI counterparts determined with the help of CIPSI in Ref. 58. While brief remarks about our DIP-EOMCC calculations for the DIPs of Cl₂, Br₂, and HBr summarized in Table IV are given here as well, a more thorough analysis of these computations is provided in the supplementary material. As shown in Table III, the vertical DIPs obtained in the DIP-EOMCCSD(3h–1p) calculations using the aug-cc-pVTZ basis set are characterized by significant errors relative to their near-exact counterparts resulting from the CIPSI extrapolations performed in Ref. 58, which are 0.77 and 0.61 eV for the X ³B₁ and a ¹A₁ states of ${(H_{2} O)}^{2 +}$ ⁠, 0.32 and 0.31 eV for the X ³T₁ and a ¹E states of ${(C H_{4})}^{2 +}$ ⁠, and 0.44 and 0.35 eV for the X ³Σ⁻ and a ¹Δ states of (BN)²⁺. For all of the calculated states of ${(H_{2} O)}^{2 +}$ ⁠, ${(C H_{4})}^{2 +}$ ⁠, and (BN)²⁺ considered in Table III and for the majority of states of ${(C l_{2})}^{2 +}$ ⁠, ${(B r_{2})}^{2 +}$ ⁠, and (HBr)²⁺ examined in Table IV, especially when a larger cc-pVQZ basis set is employed, the DIP-EOMCCSD(3h–1p) computations overestimate the corresponding DIPs of H₂O, CH₄, BN, Cl₂, Br₂, and HBr relative to the reference data. In both sets of molecular examples shown in Tables III and IV, the poor quality of the DIPs obtained with the DIP-EOMCCSD(3h–1p) approach seems to be a consequence of neglecting the 4h–2p component of $R_{μ}^{(- 2)}$ ⁠.

TABLE III.

Vertical DIP energies, in eV, of H₂O, CH₄, and BN corresponding to the lowest singlet and triplet states of the dicationic species obtained in the DIP-EOMCCSD(3h–1p), DIP-EOMCCSD(4h–2p), DIP-EOMCCSD(T)(a)(4h–2p), and DIP-EOMCCSDT(4h–2p) calculations, alongside the high-level benchmark DIPs taken from Ref. 58 resulting from the CIPSI computations extrapolated to the exact, full CI, limit. The DIP-EOMCC calculations used the RHF orbitals of the respective neutral systems. All calculations employed the aug-cc-pVTZ basis set and the frozen-core approximation was assumed in all post-RHF steps.

Molecule	Dication state	CCSD(3h–1p)a	CCSD(4h–2p)b	CCSD(T)(a)(4h–2p)c	CCSDT(4h–2p)d	CIPSIe
H₂Of	X ³B₁	41.06	40.04	40.26	40.27	40.29
H₂Of	a ¹A₁	42.04	41.18	41.39	41.40	41.43
CH₄g	X ³T₁	38.59	38.10	38.26	38.27	38.27
CH₄g	a ¹E	39.29	38.80	38.96	38.97	38.98
BNh	X ³Σ⁻	34.17	33.12	33.76	33.74	33.73
BNh	a ¹Δ	35.33	34.31	34.98	34.98	34.98

Molecule	Dication state	CCSD(3h–1p)a	CCSD(4h–2p)b	CCSD(T)(a)(4h–2p)c	CCSDT(4h–2p)d	CIPSIe
H₂Of	X ³B₁	41.06	40.04	40.26	40.27	40.29
H₂Of	a ¹A₁	42.04	41.18	41.39	41.40	41.43
CH₄g	X ³T₁	38.59	38.10	38.26	38.27	38.27
CH₄g	a ¹E	39.29	38.80	38.96	38.97	38.98
BNh	X ³Σ⁻	34.17	33.12	33.76	33.74	33.73
BNh	a ¹Δ	35.33	34.31	34.98	34.98	34.98

^a

The DIP-EOMCCSD(3h–1p) approach.

^b

The DIP-EOMCCSD(4h–2p) approach.

^c

The DIP-EOMCCSD(T)(a)(4h–2p) approach.

^d

The DIP-EOMCCSDT(4h–2p) approach.

^e

The results of CIPSI calculations extrapolated to the exact, full CI, limit, taken from Ref. 58.

^f

The O–H bond length and the H–O–H bond angle in the C_2v-symmetric ground-state H₂O, optimized using CC3/aug-cc-pVTZ and taken from Ref. 67, are 0.9591 Å and 103.2°, respectively.

^g

The C–H bond length in the T_d-symmetric ground-state CH₄, optimized using CC3/aug-cc-pVTZ and taken from Ref. 67, is 1.0879 Å.

^h

The equilibrium B–N bond length in the ground-state BN, optimized using CC3/aug-cc-pVTZ and taken from Ref. 67, is 1.2765 Å.

TABLE IV.

Vertical DIP energies, in eV, of Cl₂, Br₂, and HBr corresponding to the experimental data reported in Refs. 62–64 obtained in the DIP-EOMCCSD(3h–1p), DIP-EOMCCSD(4h–2p), DIP-EOMCCSD(T)(a)(4h–2p), and DIP-EOMCCSDT(4h–2p) calculations using the cc-pVTZ and cc-pVQZ basis sets. Scalar-relativistic effects were incorporated using the SFX2C-1e methodology of Ref. 71. All DIP-EOMCC calculations used the RHF orbitals of the respective neutral diatomics and the frozen-core approximation was assumed in all post-RHF steps.

Molecule		CCSD(3h–1p)a		CCSD(4h–2p)b		CCSD(T)(a)(4h–2p)c		CCSDT(4h–2p)d
Molecule	Dication state	TZ	QZ	TZ	QZ	TZ	QZ	TZ	QZ	Experimente
Cl₂f	$X^{3} Σ_{g}^{-}$	31.28	31.57	30.58	30.82	30.82	31.12	30.84	31.13	31.13
	a ¹Δ_g	31.78	32.06	31.12	31.35	31.36	31.64	31.37	31.66	31.74
	$b^{1} Σ_{g}^{+}$	32.16	32.45	31.51	31.74	31.74	32.03	31.76	32.05	32.12
	$c^{1} Σ_{u}^{-}$	33.22	33.52	32.56	32.82	32.79	33.09	32.80	33.11	32.97
Br₂g	$X^{3} Σ_{g}^{-}$	28.47	28.68	27.92	28.12	28.12	28.35	28.13	28.37	28.53
	a ¹Δ_g	28.89	29.10	28.38	28.56	28.57	28.79	28.58	28.81	28.91
	$b^{1} Σ_{g}^{+}$	29.21	29.42	28.71	28.90	28.90	29.13	28.91	29.14	29.38
	$c^{1} Σ_{u}^{-}$	29.93	30.16	29.43	29.63	29.61	29.85	29.62	29.87	30.3
HBrh	X ³Σ⁻	32.70	32.93	32.29	32.51	32.42	32.67	32.43	32.69	32.62
	a ¹Δ	34.06	34.25	33.69	33.86	33.82	34.02	33.83	34.04	33.95
	b ¹Σ⁺	35.31	35.50	34.95	35.12	35.07	35.27	35.09	35.28	35.19

Molecule		CCSD(3h–1p)a		CCSD(4h–2p)b		CCSD(T)(a)(4h–2p)c		CCSDT(4h–2p)d
Molecule	Dication state	TZ	QZ	TZ	QZ	TZ	QZ	TZ	QZ	Experimente
Cl₂f	$X^{3} Σ_{g}^{-}$	31.28	31.57	30.58	30.82	30.82	31.12	30.84	31.13	31.13
	a ¹Δ_g	31.78	32.06	31.12	31.35	31.36	31.64	31.37	31.66	31.74
	$b^{1} Σ_{g}^{+}$	32.16	32.45	31.51	31.74	31.74	32.03	31.76	32.05	32.12
	$c^{1} Σ_{u}^{-}$	33.22	33.52	32.56	32.82	32.79	33.09	32.80	33.11	32.97
Br₂g	$X^{3} Σ_{g}^{-}$	28.47	28.68	27.92	28.12	28.12	28.35	28.13	28.37	28.53
	a ¹Δ_g	28.89	29.10	28.38	28.56	28.57	28.79	28.58	28.81	28.91
	$b^{1} Σ_{g}^{+}$	29.21	29.42	28.71	28.90	28.90	29.13	28.91	29.14	29.38
	$c^{1} Σ_{u}^{-}$	29.93	30.16	29.43	29.63	29.61	29.85	29.62	29.87	30.3
HBrh	X ³Σ⁻	32.70	32.93	32.29	32.51	32.42	32.67	32.43	32.69	32.62
	a ¹Δ	34.06	34.25	33.69	33.86	33.82	34.02	33.83	34.04	33.95
	b ¹Σ⁺	35.31	35.50	34.95	35.12	35.07	35.27	35.09	35.28	35.19

^a

The DIP-EOMCCSD(3h–1p) approach.

^b

The DIP-EOMCCSD(4h–2p) approach.

^c

The DIP-EOMCCSD(T)(a)(4h–2p) approach.

^d

The DIP-EOMCCSDT(4h–2p) approach.

^e

The experimentally determined DIP values taken from Ref. 62 for Cl₂, Ref. 63 for Br₂, and Ref. 64 for HBr.

^f

The equilibrium Cl–Cl bond length in the ground-state Cl₂, taken from Ref. 68, is 1.987 Å.

^g

The equilibrium Br–Br bond length in the ground-state Br₂, taken from Ref. 68, is 2.281 Å.

^h

The equilibrium H–Br bond length in the ground-state HBr, taken from Ref. 68, is 1.414 Å.

Indeed, when R_μ,4h–2p is included in $R_{μ}^{(- 2)}$ via the DIP-EOMCCSD(4h–2p) approach, the errors in the lowest triplet and singlet DIPs of the water molecule relative to the extrapolated CIPSI data obtained with DIP-EOMCCSD(3h–1p), of 0.77 and 0.61 eV, reduce in the DIP-EOMCCSD(4h–2p) calculations to 0.25 and 0.25 eV, respectively. The analogous errors characterizing the DIP-EOMCCSD(3h–1p) computations for methane, which are 0.32 and 0.31 eV, reduce to 0.17 and 0.18 eV, respectively, when the DIP-EOMCCSD(4h–2p) method is employed. Unfortunately, the DIP-EOMCCSD(4h–2p) approach does not always improve the DIP-EOMCCSD(3h–1p) results. For example, the DIP values obtained in the DIP-EOMCCSD(4h–2p) calculations for BN are less accurate than their DIP-EOMCCSD(3h–1p) counterparts, increasing the 0.44 and 0.35 eV errors in the DIP-EOMCCSD(3h–1p) data for the X ³Σ⁻ and a ¹Δ states of (BN)²⁺ relative to their extrapolated CIPSI values to 0.61 and 0.67 eV, respectively, when DIP-EOMCCSD(3h–1p) is replaced by DIP-EOMCCSD(4h–2p). In general, the DIPs of H₂O, CH₄, and BN resulting from the DIP-EOMCCSD(4h–2p) calculations reported in Table III lie substantially below the corresponding near-full-CI extrapolated CIPSI data. Similar remarks apply to the DIPs of Cl₂, Br₂, and HBr shown in Table IV. This behavior of DIP-EOMCCSD(4h–2p) can be attributed to the imbalance between the high-level 4h–2p treatment of double ionization and the lower-level CCSD description of the neutral species.

The results of our DIP-EOMCCSDT(4h–2p) calculations, in which the similarity-transformed Hamiltonian of CCSD is replaced by its CCSDT counterpart, allowing us to treat the N- and (N − 2)-electron species in a more accurate and balanced manner, confirm this observation. As shown in Table III, the DIPs of H₂O, CH₄, and BN become much more accurate when we move from the DIP-EOMCCSD(4h–2p) approach to its higher-level DIP-EOMCCSDT(4h–2p) counterpart. The errors in the DIPs associated with the lowest triplet and singlet dicationic states of H₂O, CH₄, and BN relative to the extrapolated CIPSI reference data, which are 0.25 and 0.25 eV for ${(H_{2} O)}^{2 +}$ ⁠, 0.17 and 0.18 eV for ${(C H_{4})}^{2 +}$ ⁠, and 0.61 and 0.67 eV for (BN)²⁺ in the DIP-EOMCCSD(4h–2p) case, reduce to the minuscule 0.02 and 0.03 eV for ${(H_{2} O)}^{2 +}$ ⁠, 0.00 and 0.01 eV for ${(C H_{4})}^{2 +}$ ⁠, and 0.01 and 0.00 eV for (BN)²⁺, respectively, when the DIP-EOMCCSDT(4h–2p) method is employed. The same is generally true when examining the DIPs of Cl₂, Br₂, and HBr shown in Table IV, where the DIP-EOMCCSDT(4h–2p) approach offers similar improvements. The only exceptions are the $c^{1} Σ_{u}^{-}$ state of ${(C l_{2})}^{2 +}$ and the three states of (HBr)²⁺, for which the DIP-EOMCCSD(4h–2p)/cc-pVQZ DIPs are already very accurate. We conclude by pointing out that the DIP-EOMCCSD(T)(a)(4h–2p) method, which offers significant savings in the computational effort compared to full DIP-EOMCCSDT(4h–2p), reproduces the DIPs of H₂O, CH₄, and BN reported in Table III as well as the DIPs of Cl₂, Br₂, and HBr considered in Table IV resulting from the parent DIP-EOMCCSDT(4h–2p) calculations to within 0.02 eV. While we will continue testing the DIP-EOMCCSD(T)(a)(4h–2p) approach against the DIP-EOMCCSDT(4h–2p) and other high-accuracy data, its excellent performance in this study is encouraging.

In summary, we presented the fully factorized and programmable equations defining the DIP-EOMCCSDT(4h–2p) approach and the perturbative approximation to it, abbreviated as DIP-EOMCCSD(T)(a)(4h–2p). We incorporated the resulting computer codes into the open-source CCpy package available on GitHub. We applied the DIP-EOMCCSDT(4h–2p) and DIP-EOMCCSD(T)(a)(4h–2p) methods and their DIP-EOMCCSD(3h–1p) and DIP-EOMCCSD(4h–2p) predecessors to the vertical DIPs of H₂O, CH₄, and BN, as described by the aug-cc-pVTZ basis set, and Cl₂, Br₂, and HBr, using the cc-pVTZ and cc-pVQZ bases and the spin-free two-component SFX2C-1e treatment of the scalar-relativistic effects. We demonstrated that with the exception of the higher-lying $c^{1} Σ_{u}^{-}$ state of ${(B r_{2})}^{2 +}$ ⁠, the DIP values computed with DIP-EOMCCSDT(4h–2p) are not only in generally good agreement with the available high-accuracy theoretical or experimental reference data, but also more accurate than those obtained with DIP-EOMCCSD(4h–2p), which uses CCSD instead of CCSDT to construct the underlying similarity-transformed Hamiltonian. The DIP-EOMCCSD(T)(a)(4h–2p) method, which avoids the most expensive steps of DIP-EOMCCSDT(4h–2p), turned out to be similarly effective, recovering the vertical DIPs of H₂O, CH₄, BN, Cl₂, Br₂, and HBr obtained with its DIP-EOMCCSDT(4h–2p) parent to within 0.02 eV. Our future plans include the development of nonperturbative ways of reducing the costs of the DIP-EOMCCSDT(4h–2p) calculations through the active-space treatments of CCSDT^72–74 and 4h–2p amplitudes^35,36 and the use of frozen natural orbitals, combined with Cholesky decomposition and density fitting techniques, which will also be useful in improving our description of relativistic effects following the four-component methodology of Ref. 75.

SUPPLEMENTARY MATERIAL

See the supplementary material for the more detailed analysis of the DIPs of Cl2, Br2, and HBr resulting from the DIP-EOMCC calculations reported in Table IV.

This work has been supported by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (Grant No. DE-FG02-01ER15228 to P.P). A.K.D. acknowledges support from SERB-India under the CRG (Project No. CRG/2022/005672) and MATRICS (Project No. MTR/2021/000420) schemes. We acknowledge Dr. Jun Shen for inspecting Eqs. (8)–(10).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Karthik Gururangan: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (lead). Achintya Kumar Dutta: Conceptualization (equal); Data curation (supporting); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Software (supporting); Validation (supporting); Writing – review & editing (supporting). Piotr Piecuch: Conceptualization (equal); Data curation (supporting); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Software (supporting); Supervision (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (lead).

DATA AVAILABILITY

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

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Author(s)

Double ionization potential equation-of-motion coupled-cluster approach with full inclusion of 4-hole–2-particle excitations and three-body clusters

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Double ionization potential equation-of-motion coupled-cluster approach with full inclusion of 4-hole–2-particle excitations and three-body clusters