Orbital-optimized coupled-cluster methods are very helpful for theoretical predictions of the molecular properties of challenging chemical systems, such as excited states. In this research, an efficient implementation of the equation-of-motion orbital-optimized coupled-cluster doubles method with the density-fitting (DF) approach, denoted by DF-EOM-OCCD, is presented. The computational cost of the DF-EOM-OCCD method for excitation energies is compared with that of the conventional EOM-OCCD method. Our results demonstrate that DF-EOM-OCCD excitation energies are dramatically accelerated compared to EOM-OCCD. There are almost 17-fold reductions for the $C_{5} H_{12}$ molecule in an aug-cc-pVTZ basis set with the RHF reference. This dramatic performance improvement comes from the reduced cost of integral transformation with the DF approach and the efficient evaluation of the particle-particle ladder (PPL) term, which is the most expensive term to evaluate. Further, our results show that the DF-EOM-OCCD approach is very helpful for the computation of excitation energies in open-shell molecular systems. Overall, we conclude that our new DF-EOM-OCCD implementation is very promising for the study of excited states in large-sized challenging chemical systems.

1 INTRODUCTION

Orbital-optimized (OO) many-body perturbation theory and coupled-cluster (CC) methods are helpful for computations of molecular properties of chemical systems with challenging electronic structures.^1-29 OO variants of various perturbation theory and CC methods have been reported.^{1-3, 5, 13, 14, 17-19, 21-24, 28, 30-34} Perturbative triples excitation corrections for the orbital-optimized coupled-cluster doubles (OCCD) method have also been considered.^{6, 15, 20, 29, 35, 36} Previous studies demonstrated that the OO-MP and OO-CC methods provide accurate results for molecular properties in the case of molecular systems with problematic electronic structures, such as free radicals,^{10, 17, 18, 23, 37-39} symmetry-breaking problems,^{2, 13, 14, 17} transition states,^{10, 37-39} bond-breaking problems,^{15, 20, 29, 40} weak interactions in open-shell systems,^{22, 24, 32, 33, 41} direct computations of ionization potentials and electron affinities.^42-44

The density-fitting (DF) approximation is one of the most popular techniques employed for the decomposition of the electron repulsion integrals (ERIs).^{22, 23, 28, 29, 32-34, 45-59} The DF approach enables one to express the four-dimensional ERIs in terms of three-dimensional DF factors. The DF approximation is very helpful to the reduced I/O time. For the OO methods, the DF approach was applied to the second-order perturbation theory (OMP2),^{10, 22, 23} the third-order perturbation theory, and the MP2.5 model (OMP3 and OMP2.5),^{33, 34} the linearized coupled-cluster doubles (OLCCD) approach,³² and the coupled-cluster doubles (OCCD) method.^{28, 29}

Excitation energy computations (EEs) in open-shell chemical systems require robust theoretical models and include further challenges compared to ground-state energy computations. The Equation of motion coupled-cluster singles and doubles method (EOM-CCSD) provides reliable results for the excited-state properties of most molecular systems.^60-78 The accuracy of EOM-CCSD has been reported to be in the range of 0.1–0.2 eV.^{62, 65} However, for open-shell molecular systems, a more robust method, such as OCCD, is more computationally demanding.^{2-8, 28, 29}

The conventional OCCD approach, employing the four-index ERIs, needs more integral transformation procedures, resulting in the extensive usage of I/O procedures. This problem has been addressed in our recent study,²⁸ where energy and analytic gradients for the OCCD method with the DF approach were reported. In this research, the EOM formalism is implemented for the OCCD method with the DF approach to extend the applicability of the EOM-OCCD method to large-size chemical systems. The equations reported have been implemented in a new computer code, written by present authors, and added to the MacroQC software.⁷⁹ Our new implementation includes both restricted and unrestricted Hartree-Fock (RHF and UHF) references. The computational time of our DF-EOM-OCCD method is compared with that of the EOM-OCCD, denoted EOM-OD in the Q-chem 5.3 software.⁸⁰ The DF-EOM-CCD method is applied to a test set for excitation energies.

2 THEORETICAL APPROACH

2.1 DF-CCD energy and amplitude equations

The correlation energy for the CCD method can be written as follows:

Δ E = ⟨ 0 | e^{- {\hat{T}}_{2}} Ĥ_{N} e^{{\hat{T}}_{2}} | 0 ⟩,

(1)

where

Ĥ_{N}

is the normal-ordered Hamiltonian operator,^{81, 82}

| 0 ⟩

is the reference determinant, and

{\hat{T}}_{2}

is the cluster double excitation operator:

{\hat{T}}_{2} = \frac{1}{4} \sum_{i, j}^{o c c} \sum_{a, b}^{v i r} t_{i j}^{a b} â^{†} {\hat{b}}^{†} ĵ î,

(2)

where

â^{†}

and

î

are the creation and annihilation operators and

t_{i j}^{a b}

is a double excitation amplitude,

o c c

and

v i r

denote occupied and virtual orbitals, respectively.

The DF-CCD correlation energy can be written explicitly as follows:

Δ E = \frac{1}{4} \sum_{i, j}^{o c c} \sum_{a, b}^{v i r} t_{i j}^{a b} \sum_{Q}^{N_{a u x}} (b_{i a}^{Q} b_{j b}^{Q} - b_{i b}^{Q} b_{j a}^{Q}),

(3)

where

b_{i a}^{Q}

is a MO basis DF factor.

The DF-CCD amplitude equation can be written as

⟨ Φ_{i j}^{a b} | e^{- {\hat{T}}_{2}} Ĥ_{N} e^{{\hat{T}}_{2}} | 0 ⟩ = 0,

(4)

where

Φ_{i j}^{a b}

is a doubly-excited Slater determinant. Details of our DF-CCD implementation can be found in our previous studies.^55-57

2.2 DF-CCD- $Λ$ Lagrangian

To perform orbital optimization for the DF-CCD method, it is convenient^{83, 84} to define a Lagrangian (

L

), which is also called the DF-CCD-

Λ

functional, as follows:

L = ⟨ 0 | (1 + {\hat{Λ}}_{2}) e^{- {\hat{T}}_{2}} Ĥ e^{{\hat{T}}_{2}} | 0 ⟩,

(5)

where

Ĥ

is the Hamiltonian operator and

{\hat{Λ}}_{2}

is the CC double de-excitation operator,

{\hat{Λ}}_{2} = \frac{1}{4} \sum_{i, j}^{o c c} \sum_{a, b}^{v i r} λ_{a b}^{i j} î^{†} ĵ^{†} \hat{b} â,

(6)

where

λ_{a b}^{i j}

is a double de-excitation amplitude.

The

Λ_{2}

-amplitude equations can be obtained from References 55, 56, 85-89:

⟨ 0 | (1 + {\hat{Λ}}_{2}) [e^{- {\hat{T}}_{2}} Ĥ e^{{\hat{T}}_{2}} - E] | Φ_{i j}^{a b} ⟩ = 0,

(7)

where

E

is the CCD total energy. The explicit form of DF-CCD

Λ_{2}

amplitude equations was reported in our previous studies.^55-57

2.3 Orbital-optimization for the DF-CCD wave function

Following our previous formulations^{13-15, 17-19, 24, 32, 33, 41, 42} for the OO methods, transformations of the molecular orbital (MO) coefficients can be performed with the help of a unitary operator^90-93 as follows:

C (κ) = C^{(0)} e^{K},

(8)

where

C^{(0)}

and

C (κ)

are the old and new MO coefficient matrices, respectively,

e^{\hat{K}}

is the MO rotation operator, and

\hat{K}

is defined as follows:

\hat{K} = \sum_{p, q} K_{p q} {\hat{p}}^{†} \hat{q} = \sum_{p > q} κ_{p q} ({\hat{p}}^{†} \hat{q} - {\hat{q}}^{†} \hat{p}),

(9)

where

κ_{p q}

is an MO rotation parameter.

The DF-OCCD Lagrangian with explicit parametrization for the orbital rotations can be written as,^{13, 28, 29}

L (κ) = ⟨ 0 | (1 + {\hat{Λ}}_{2}) e^{- {\hat{T}}_{2}} Ĥ^{κ} e^{{\hat{T}}_{2}} | 0 ⟩,

(10)

where,

Ĥ^{κ} = e^{- \hat{K}} Ĥ e^{\hat{K}} .

(11)

Derivatives of

L (κ)

with respect to

κ

defines the MO gradient and Hessian, as follows:

w_{p q} = {\frac{\partial L}{\partial κ_{p q}}|}_{κ = 0},

(12)

A_{p q, r s} = {\frac{\partial^{2} L}{\partial κ_{p q} \partial κ_{r s}}|}_{κ = 0} .

(13)

Then, the

L (κ)

can be cast into the following form using the second-order approximation:

L^{(2)} (κ) = L^{(0)} + κ^{†} w + \frac{1}{2} κ^{†} A κ,

(14)

where

w

is the MO gradient vector,

A

is the MO Hessian matrix, and

κ

is the MO rotation vector. Minimization of the second-order Lagrangian yields the following equation for the MO rotation parameters:

κ = - A^{- 1} w .

(15)

One- and two-particle density matrices (OPDM and TPDM) and the generalized Fock matrix (GFM) are required for the orbital-optimization procedure.^{22, 23} The OPDM, TPDM, and GFM are decomposed into reference, correlation, and separable parts as follows:^{22, 23}

\begin{array}{c} γ_{p q} = γ_{p q}^{r e f} + γ_{p q}^{c o r r}, \end{array}

(16)

\begin{array}{c} Γ_{p q}^{Q} = Γ_{p q}^{Q (r e f)} + Γ_{p q}^{Q (c o r r)} + Γ_{p q}^{Q (s e p)}, \end{array}

(17)

\begin{array}{c} F_{p q}^{Q} = F_{p q}^{Q (r e f)} + F_{p q}^{Q (c o r r)} + F_{p q}^{Q (s e p)} . \end{array}

(18)

where

γ

Γ

, and

F

are OPDM, TPDM, and GFM, respectively. The superscripts

(r e f)

(s e p)

, and

(c o r r)

refer to the reference, separable, and correlation contributions.

The reference and separable parts of OPDM, TPDM, and GFM, as well as the Fock matrix, are computed with the JKFIT basis set, while the correlation contributions and amplitude equations are evaluated with the RI auxiliary basis set. All the details were explained extensively in our previous studies.^{22, 23}

2.4 The EOM-CCD model

In the EOM-CCD framework, the target excited-state wave functions are written as follows:

| Ψ_{R} ⟩ = \hat{R} e^{{\hat{T}}_{2}} | 0 ⟩,

(19)

⟨ Ψ_{L} | = ⟨ 0 | e^{- {\hat{T}}_{2}} \hat{L},

(20)

where

\hat{R}

and

\hat{L}

are linear excitation and de-excitation operators, respectively. For EOM-CCD

\hat{R} = {\hat{R}}_{1} + {\hat{R}}_{2}

{\hat{R}}_{1} = \sum_{i a} r_{i}^{a} {â^{†} î},

(21)

{\hat{R}}_{2} = \frac{1}{4} \sum_{i j a b} r_{i j}^{a b} {â^{†} {\hat{b}}^{†} ĵ î},

(22)

where

r_{i}^{a}

and

r_{i j}^{a b}

are the single and double excitation amplitudes, respectively.

For the ground state we have the following Schrödinger equation:

Ĥ e^{{\hat{T}}_{2}} | 0 ⟩ = E e^{{\hat{T}}_{2}} | 0 ⟩,

(23)

hence, left multiplying Equation (23) by

e^{- {\hat{T}}_{2}}

we obtain:

\bar{H} | 0 ⟩ = E | 0 ⟩,

(24)

where

\bar{H} = e^{- {\hat{T}}_{2}} Ĥ e^{{\hat{T}}_{2}}

Further, the normal ordered

\bar{H}

can be written as follows:

\bar{H} = \hat{H} + ⟨ 0 | Ĥ | 0 ⟩,

(25)

hence, we define:

\hat{H} = e^{- {\hat{T}}_{2}} Ĥ_{N} e^{{\hat{T}}_{2}} = {(Ĥ_{N} e^{{\hat{T}}_{2}})}_{c},

(26)

where subscript

c

means that only connected diagrams should be included. Therefore, we may re-write Equation (24) as follows

\hat{H} | 0 ⟩ = Δ E | 0 ⟩,

(27)

where

Δ E

is the ground-state CC correlation energy.

The excited state eigenvalue equation can be written as follows:

\hat{H} \hat{R} | 0 ⟩ = Δ E_{R} \hat{R} | 0 ⟩,

(28)

where

Δ E_{R}

is the excited state CC correlation energy. The excitation energy can be written as,

ω = E_{R} - E = Δ E_{R} - Δ E .

(29)

After some algebra we obtain the EOM-CCD equation as follows:

{(\hat{H} \hat{R} | 0 ⟩)}_{C} = ω \hat{R} | 0 ⟩ .

(30)

Equation (30) is equivalent to the following matrix eigenvalue equation for EOM-CCD:

(\begin{matrix} 0 & H_{0 S} & H_{0 D} \\ H_{S 0} & H_{S S} & H_{S D} \\ 0 & H_{D S} & H_{D D} \end{matrix}) (\begin{matrix} R_{0} \\ R_{1} \\ R_{2} \end{matrix}) = ω (\begin{matrix} R_{0} \\ R_{1} \\ R_{2} \end{matrix}) .

(31)

We note that the

H_{S 0}

term is zero for the EOM-CCSD model since it corresponds to the singles amplitude equation. However,

H_{S 0}

is not zero for the CCD model.

We solve Equation (31) iteratively with the Davidson algorithm.^94-97 Hence, we need to introduce so-called

σ

vector as follows:

σ_{I} = \sum_{J} H_{I J} R_{J} .

(32)

where

I, J = 0, S, D

2.5 DF-EOM-CCD intermediates

2.5.1 DF-EOM-CCD 3-index intermediates

One-and three-index intermediates that are used for EOM-CCD are defined as follows:

T_{i a}^{Q} = \sum_{j}^{o c c} \sum_{b}^{v i r} t_{i j}^{a b} b_{j b}^{Q},

(33)

r^{Q} = \sum_{m}^{o c c} \sum_{e}^{v i r} r_{m}^{e} b_{m e}^{Q},

(34)

r_{i a}^{Q} = \sum_{e}^{v i r} r_{i}^{e} b_{a e}^{Q},

(35)

r_{i j}^{Q} = \sum_{e}^{v i r} r_{i}^{e} b_{j e}^{Q},

(36)

r_{a i}^{Q} = \sum_{m}^{o c c} r_{m}^{a} b_{i m}^{Q},

(37)

r_{a b}^{Q} = \sum_{m}^{o c c} r_{m}^{a} b_{m b}^{Q},

(38)

R_{i a}^{Q} = \sum_{m}^{o c c} \sum_{e}^{v i r} r_{i m}^{a e} b_{m e}^{Q} .

(39)

2.5.2 $F$ intermediates

Density-fitted

F

and

F

intermediates are:^{55, 56}

F_{m i} = f_{m i} + \sum_{Q}^{N_{a u x}} \sum_{e}^{v i r} T_{i e}^{Q} b_{m e}^{Q},

(40)

F_{a e} = f_{a e} - \sum_{Q}^{N_{a u x}} \sum_{m}^{o c c} T_{m a}^{Q} b_{m e}^{Q},

(41)

F_{m e} = f_{m e},

(42)

where

f_{p q}

is the MO basis Fock matrix.

2.5.3 $W$ intermediates

W

intermediates, with the DF approximation, are:^{55, 56}

W_{m n i j} = {⟨ m n | | i j ⟩}_{D F} + \sum_{e}^{v i r} \sum_{f}^{v i r} t_{i j}^{e f} {⟨ m n | e f ⟩}_{D F},

(43)

\begin{array}{lcr} W_{m b e j} & = & {⟨ m b | | e j ⟩}_{D F} + \frac{1}{2} \sum_{Q}^{N_{a u x}} T_{j b}^{Q} b_{m e}^{Q} \\ - \frac{1}{2} \sum_{n}^{o c c} \sum_{f}^{v i r} t_{j n}^{b f} {⟨ e m | n f ⟩}_{D F} . \end{array}

(44)

2.5.4 Four-index intermediates

The

𝒲

intermediates are defined as follows:^{55, 56}

𝒲_{m b e j} = {⟨ m b | | e j ⟩}_{D F} - \sum_{n}^{o c c} \sum_{f}^{v i r} t_{n j}^{b f} {⟨ m n | | e f ⟩}_{D F},

(45)

\begin{array}{lcr} 𝒲_{m b i j} = {⟨ m b | | i j ⟩}_{D F} - \sum_{e}^{v i r} t_{i j}^{b e} F_{m e} \\ + \sum_{e, f}^{v i r} t_{i j}^{e f} {⟨ m b | e f ⟩}_{D F} \\ + P_(i j) \sum_{n}^{o c c} \sum_{e}^{v i r} t_{j n}^{b e} {⟨ m n | | i e ⟩}_{D F}, \end{array}

(46)

𝒱_{i j m n} = \sum_{e, f}^{v i r} r_{i j}^{e f} {⟨ m n | e f ⟩}_{D F},

(47)

𝒱_{i j a m} = \sum_{e, f}^{v i r} r_{i j}^{e f} {⟨ a m | e f ⟩}_{D F},

(48)

R_{m n i j} = P_(m n) \sum_{Q}^{a u x} r_{i m}^{Q} b_{j n}^{Q},

(49)

R_{m b i f} = P_(m b) \sum_{Q}^{a u x} r_{i m}^{Q} b_{b f}^{Q},

(50)

R_{m b i j} = \sum_{e}^{v i r} r_{i}^{e} 𝒲_{m b e j},

(51)

where

P_{\pm} (p q)

is defined by

P_{\pm} (p q) = 1 \pm 𝒫 (p q),

(52)

with

𝒫 (p q)

acts to permute the indices

p

and

q

2.5.5 Two-index intermediates

Two-index intermediates are defined as follows:

R_{i m} = \sum_{e}^{v i r} r_{i}^{e} F_{m e},

(53)

X_{i j} = \sum_{Q}^{a u x} \sum_{e}^{v i r} (R_{i e}^{Q} - r_{e i}^{Q}) b_{j e}^{Q} + \sum_{Q}^{a u x} b_{i j}^{Q} r^{Q},

(54)

X_{a b} = \sum_{Q}^{a u x} \sum_{m}^{o c c} (R_{m a}^{Q} + r_{m a}^{Q}) b_{m b}^{Q} - \sum_{Q}^{a u x} b_{a b}^{Q} r^{Q} .

(55)

2.6 $Δ t_{i}^{a}$ intermediate

Δ t_{i}^{a}

intermediate are defined as follows:

Δ t_{i}^{a} = f_{i a} + \sum_{m}^{o c c} \sum_{e}^{v i r} t_{i m}^{a e} F_{m e} + \sum_{Q}^{a u x} \sum_{m}^{o c c} T_{m a}^{Q} b_{m i}^{Q} + \sum_{Q}^{a u x} \sum_{e}^{v i r} T_{i e}^{Q} b_{a e}^{Q} .

(56)

2.7 DF-EOM-CCD $σ$ equations

The DF-EOM-CCD

σ_{0}

equation can be written as:

σ_{0} = \sum_{i}^{o c c} \sum_{a}^{v i r} r_{i}^{a} F_{i a} + \frac{1}{4} \sum_{i j}^{o c c} \sum_{a b}^{v i r} r_{i j}^{a b} ⟨ i j | | a b ⟩ .

(57)

With the DF approximation DF-EOM-CCD

σ_{i}^{a}

equation can be written as:

\begin{array}{lcr} σ_{i}^{a} = Δ t_{i}^{a} r_{0} + \sum_{e}^{v i r} r_{i}^{e} F_{a e} - \sum_{m}^{o c c} r_{m}^{a} F_{m i} \\ + \sum_{m}^{o c c} \sum_{e}^{v i r} r_{m}^{e} 𝒲_{m a e i} + \sum_{m}^{o c c} \sum_{e}^{v i r} r_{i m}^{a e} F_{m e} \\ - \sum_{Q}^{a u x} \sum_{m}^{o c c} b_{i m}^{Q} R_{m a}^{Q} + \sum_{Q}^{a u x} \sum_{e}^{v i r} R_{i e}^{Q} b_{a e}^{Q} . \end{array}

(58)

With the DF approximation DF-EOM-CCD

σ_{i j}^{a b}

equation can be written as:

\begin{array}{lcr} σ_{i j}^{a b} = P_(a b) \sum_{e}^{v i r} r_{i j}^{e b} F_{a e} - P_(i j) \sum_{m}^{o c c} r_{m j}^{a b} F_{m i} \\ + \frac{1}{2} \sum_{e, f}^{v i r} r_{i j}^{e f} {⟨ a b | | e f ⟩}_{D F} - P_(a b) \sum_{m}^{o c c} r_{m}^{a} 𝒲_{m b i j} \\ + \frac{1}{2} \sum_{m, n}^{o c c} t_{m n}^{a b} (R_{m n i j} - R_{m n j i} + 𝒱_{i j m n}) \\ + \frac{1}{2} \sum_{m, n}^{o c c} r_{m n}^{a b} W_{m n i j} \\ + P_(i j) P_(a b) \sum_{m}^{o c c} \sum_{e}^{v i r} r_{i m}^{a e} 𝒲_{m b e j} \\ - P_(i j) P_(a b) \sum_{m}^{o c c} \sum_{e}^{v i r} t_{i m}^{a e} R_{m b j e} \\ + P_(i j) P_(a b) \sum_{Q}^{a u x} r_{i a}^{Q} b_{j b}^{Q} \\ - P_(i j) \sum_{m}^{o c c} t_{m j}^{a b} R_{i m} \\ + P_(a b) \sum_{e}^{v i r} X_{a e} t_{i j}^{b e} + P_(i j) \sum_{m}^{o c c} X_{i m} t_{j m}^{a b} . \end{array}

(59)

We would like to note that for the canonical CCD model, the terms including off-diagonal Fock matrix elements vanish because of the Brillioun theorem. However, the DF-OCCD orbitals are not canonical; hence, the off-diagonal Fock terms should be kept in all equations.

3 RESULTS AND DISCUSSION

The efficiency of the EOM-OCCD⁵ and DF-EOM-OCCD methods were compared for an alkanes set. For the alkanes set, Dunning's correlation-consistent polarized valence triple- $ζ$ basis set (cc-pVTZ) was employed.^{98, 99} The cc-pVTZ-JKFIT⁵⁰ and cc-pVTZ-RI¹⁰⁰ auxiliary basis sets were employed for the reference and correlation energies, respectively, as the fitting basis sets for cc-pVTZ.

Results from the DF-EOM-CCD, DF-EOM-OCCD, DF-EOM-CCSD,¹⁰¹ EOM-OCCD,⁸⁰ and EOM-CCSD(fT)⁸⁰ methods were obtained for a set of closed-shell molecules for comparison of the excitation energies. The geometries of the closed-shell molecules considered were optimized at the B3LYP/cc-pVTZ level. For the excitation energy computations, Dunning's augmented correlation-consistent polarized valence triple- $ζ$ (aug-cc-pVTZ) basis set was used.^{98, 99} The aug-cc-pVTZ-JKFIT⁵⁰ and aug-cc-pVTZ-RI¹⁰⁰ auxiliary basis sets were used for reference and correlation energies, respectively.

Moreover, a set of open-shell molecules were considered to assess the performance of the DF-EOM-CCD, DF-EOM-OCCD, DF-EOM-CCSD,¹⁰¹ and conventional EOM-OCCD⁸⁰ methods. The geometries and the reference excitation energies, from the MRCISDT+Q level,^102-106 of the open-shell set were taken from Li and Liu.¹⁰⁷ For the open-shell system, the aug-cc-pVTZ basis set was employed, along with the aug-cc-pVTZ-JKFIT⁵⁰ and aug-cc-pVTZ-RI¹⁰⁰ auxiliary basis sets.

All excitation energy computations were performed for the lowest five roots and all electrons are correlated in all computations.

3.1 Efficiency of DF-EOM-OCCD

A set of alkanes is considered to investigate the efficiency of the EOM-OCCD and DF-EOM-OCCD methods. The conventional EOM-OCCD computations were performed with the Q-chem 5.3 package.⁸⁰ The total computational time for the EOM-OCCD and DF-EOM-OCCD methods are presented graphically in Figure 1, while separate timings for OO-CCD and EOM parts are depicted in Figures 2 and 3. Timing computations were carried out for a single root with a $1 0^{- 7}$ energy and $1 0^{- 7}$ EOM eigenvalue convergence tolerances on a single node (1 core) Intel(R) Xeon(R) CPU E5-2620 v4 $@$ 2.10 GHz computer (memory $\sim$ 64 GB). RHF versions of the EOM-OCCD and DF-EOM-OCCD codes were used in timing computations. For the EOM-OCCD code of Q-chem 5.3; MEM_TOTAL 64,000, MEM_STATIC 2000, and CC_MEMORY 51,200 options were used.

Details are in the caption following the image — **FIGURE 1**
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Wall-time (in min) for computations of excitation energies for the C $_{n}$ H $_{2 n + 2}$ ( $n$ = 1–5) set from the DF-EOM-OCCD and EOM-OCCD (from Q-Chem⁸⁰) methods with the cc-pVTZ basis set. All computations were performed with a $1 0^{- 7}$ energy convergence tolerance on a single node (1 core) Intel(R) Xeon(R) CPU E5-2620 v4 $@$ 2.10 GHz computer (memory $\sim$ 64 GB).

The DF-EOM-OCCD method dramatically reduces the computational cost compared to the conventional EOM-OCCD, there are 16.9-fold reductions in the computational time compared to EOM-OCCD for the largest member ( $C_{5} H_{12}$ ) of the alkanes set (Figure 1). For the $C_{5} H_{12}$ molecule, the time required to converge the OO-CCD wavefunction for the OD method is 23-fold more expensive compared to DF-OCCD (Figure 2). Further, for the $C_{5} H_{12}$ molecule the EOM time is 1130.8 (EOM-OCCD) and 162.5 (DF-EOM-OCCD) min; hence, there are 7-fold reductions in the computational time compared to EOM-OCCD (Figure 3). The dramatic difference between the computational cost of DF-EOM-OCCD and EOM-OCCD mainly arises from the efficiency of the DF integral transformation procedure as well as reduced I/O time. In the orbital optimization part one needs to transform ERIs at each iteration, which is substantially accelerated with the DF algorithm; hence, we obtain great time improvements for this part.²⁸ In the EOM part, we do not have integral transformation; however, the evaluation of the particle-particle ladder (PPL) term of EOM can be dramatically improved with the DF approach as illustrated in our recent paper.¹⁰¹

3.2 Accuracy of DF-EOM-OCCD

In this section, we consider a test set to assess the accuracy of the DF-EOM-OCCD method. For this purpose, we start with the closed-shell test set. Excitation energies (in eV) for the test set considered from the DF-EOM-CCD, DF-EOM-OCCD, EOM-OCCD, DF-EOM-CCSD, and EOM-CCSD(fT) methods with the aug-cc-pVTZ basis set are reported in Table 1. Errors of methods considered are illustrated graphically in Figure 4, while the mean absolute errors (MAEs) with respect to EOM-CCSD(fT) are depicted in Figure 5. The MAE values for the lowest three excitation energies are 0.19 (DF-EOM-CCD), 0.22 (DF-EOM-OCCD), 0.22 (DF-EOM-CCSD), and 0.27 (EOM-OCCD) eV, while those for the lowest excitation energies are 0.20 (DF-EOM-CCD), 0.26 (DF-EOM-OCCD), 0.26 (DF-EOM-CCSD), and 0.30 (EOM-OCCD) eV. Further, the maximum absolute errors ( $Δ_{m a x}$ ) are 0.33 (DF-EOM-CCD), 0.30 (DF-EOM-OCCD), 0.29 (DF-EOM-CCSD), and 0.34 (EOM-OCCD) eV for the lowest excitation energies. Hence, the performance of the considered methods is very similar as expected. Especially, the results of DF-EOM-OCCD and DF-EOM-CCSD are virtually the same. Minor differences observed between the DF-EOM-OCCD and EOM-OCCD are attributed to the usage of DF integrals, and the presence of $Δ t_{i}^{a}$ in our implementation, which is neglected in the previous study of Krylov et al.⁵

TABLE 1. Excitation energies for the first three excited state (in eV) of the closed-shell set from the DF-EOM-CCD, DF-EOM-OCCD, EOM-OCCD, DF-EOM-CCSD and EOM-CCSD(fT) methods with the aug-cc-pVTZ basis set.

Set entry	Molecule	DF-EOM-CCD^a	DF-EOM-OCCD^a	EOM-OCCD^b	DF-EOM-CCSD^c	EOM-CCSD(fT)^b
1	Acetamide	5.63	5.77	5.84	5.78	5.50
		6.63	6.69	6.74	6.69	6.41
		6.74	6.73	6.76	6.73	6.48
2	Acetone	4.42	4.52	4.58	4.53	4.24
		6.56	6.61	6.65	6.59	6.36
		7.54	7.58	7.62	7.56	7.37
3	Cyclopropene	6.83	6.80	6.80	6.79	6.50
		6.91	6.92	6.92	6.92	6.62
		7.08	7.04	7.03	7.05	6.81
4	E-butadiene	6.36	6.40	6.44	6.37	6.16
		6.51	6.45	6.76	6.46	6.47
		6.82	6.77	6.92	6.77	6.64
5	Ethene	7.53	7.48	7.46	7.50	7.24
		8.08	8.09	8.12	8.09	7.90
		8.17	8.13	8.18	8.14	7.96
6	Formaldehyde	3.95	4.04	4.08	4.06	3.81
		7.15	7.22	7.26	7.21	7.08
		8.07	8.11	8.15	8.10	7.98
7	Formamide	5.55	5.68	5.75	5.69	5.43
		6.93	6.93	6.94	6.94	6.68
		6.94	7.00	7.04	7.00	6.71

^a This work.
^b Computations were performed with the Q-Chem program.⁸⁰
^c Computations were performed with the MacroQC program.⁷⁹

Next, we continue with the open-shell test set where the OCCD methods provide a more robust wavefunction compared with CCSD.^{2, 13, 28} Excitation energies (in eV) for the open-shell considered DF-EOM-CCD, DF-EOM-OCCD, EOM-OCCD, DF-EOM-CCSD, and MRCISDT+Q methods with the aug-cc-pVTZ basis set are reported in Table 2. Errors of methods considered are illustrated graphically in Figure 6, while the mean absolute errors (MAEs) with respect to MRCISDT+Q are depicted in Figure 7. The MAE values for the lowest three excitation energies are 0.19 (DF-EOM-CCD), 0.14 (DF-EOM-OCCD), 0.18 (DF-EOM-CCSD), and 0.21 (EOM-OCCD) eV, while those for the lowest excitation energies are 0.19 (DF-EOM-CCD), 0.07 (DF-EOM-OCCD), 0.13 (DF-EOM-CCSD), and 0.07 (EOM-OCCD) eV. Further, for the lowest three excitation energies the $Δ_{m a x}$ values are 0.33 (DF-EOM-CCD), 0.30 (DF-EOM-OCCD), 0.29 (DF-EOM-CCSD), and 0.34 (EOM-OCCD) eV, while those are 0.99 (DF-EOM-CCD), 0.99 (DF-EOM-OCCD), 0.98 (DF-EOM-CCSD), and 1.30 (EOM-OCCD) eV for the lowest excitation energies. Hence, results of DF-EOM-OCCD are in very good agreement with the MRCISDT+Q method. These results demonstrate that the DF-EOM-OCCD method is very helpful for the excited state studies of open-shell molecules. The $Δ_{m a x}$ value of 0.99 eV is obtained for the second excited state of the ${NH}_{2}$ molecule. If we omit this value from the set, $Δ_{m a x}$ value dramatically decreases to 0.14 eV for the DF-EOM-OCCD method. The large deviation observed for the mentioned state is attributed to the high multireference character of the second-excited state. Hence, except for the excited states with high multireference character, the DF-EOM-OCCD performs very well for open-shell species.

TABLE 2. Excitation energies for the first three excited states (in eV) of the open-shell set from the DF-EOM-CCD, DF-EOM-OCCD, EOM-OCCD, EOM-CCSD and MRCISD+Q¹⁰⁷ methods with the aug-cc-pVTZ basis set.

Molecule	DF-EOM-CCD^a	DF-EOM-OCCD^a	EOM-OCCD^b	DF-EOM-CCSD^c	MRCISD+Q^d
$C_{2} H_{3}$	3.51	3.17	3.17	3.27	3.03
	4.53	4.44	4.44	4.28	4.74
	5.07	4.84	4.84	4.76	5.36
${CH}_{2} N$	4.22	3.96	3.96	4.07	3.84
	4.39	4.40	4.40	4.30	4.31
	4.87	4.66	4.66	4.71	4.48
${CH}_{2} O^{+}$	3.61	3.72	3.72	3.37	3.71
	5.19	5.26	5.27	4.91	5.26
	5.79	5.75	5.77	5.65	5.60
${CH}_{3}$	5.99	5.91	5.89	5.89	5.85
	7.04	6.99	6.99	7.02	6.99
	7.04	6.99	6.99	7.02	7.14
${ClO}_{2}$	3.33	3.26	3.28	3.24	3.25
	3.35	3.29	3.32	3.27	3.28
	3.58	3.57	3.62	3.66	3.62
${NH}_{2}$	2.14	2.12	2.12	2.12	2.11
	7.53	7.53	7.53	7.52	6.54
	7.81	7.76	7.75	7.74	7.78
${NO}_{2}$	2.89	2.95	2.98	2.82	2.81
	3.16	3.20	3.28	3.23	3.24
	3.60	3.60	4.96	3.68	3.66

^a This work.
^b Computations were performed with the Q-Chem program.⁸⁰
^c Computations were performed with the MacroQC program.⁷⁹
^d MRCISD+Q results taken from Reference 107.

4 CONCLUSIONS

In this research, a new implementation of the density-fitted orbital-optimized coupled-cluster doubles (DF-EOM-OCCD) method has been presented. The computational time of the DF-EOM-OCCD excitation energies has been compared with that of the EOM-OCCD (from Q-chem 5.3 package⁸⁰). The DF-EOM-OCCD method significantly reduces the computational cost compared to the conventional EOM-OCCD, there are almost 17-fold reductions for the $C_{5} H_{12}$ molecule in a cc-pVTZ basis set with the RHF reference. These cost savings come from the accelerated evaluation of molecular integral transformations as well as our efficient algorithm for the evaluation of the particle-particle ladder (PPL) term.¹⁰¹ For the $C_{5} H_{12}$ molecule the convergence of the orbital-optimized CC state is 23-fold faster with our DF-EOM-OCCD method. The reason for this dramatic improvement is mainly due to the efficiency of the DF integral transformation procedure, which is performed at each iteration. For the $C_{5} H_{12}$ molecule, the EOM part of DF-EOM-OCCD is 7-fold faster than that of EOM-OCCD. The main reason behind this cost saving is the efficiency of our DF-based PPL algorithm.¹⁰¹

Moreover, the performance of the DF-EOM-OCCD method has been investigated for excitation energies of closed- and open-shell molecular systems. For the closed-shell set considered, the mean absolute errors (MAEs) in the lowest excitation energies, with respect to the EOM-CCSD(fT), are 0.20 (DF-EOM-CCD), 0.26 (DF-EOM-OCCD), 0.26 (DF-EOM-CCSD), and 0.30 (EOM-OCCD) eV. Hence, the performances of the methods considered are similar for the closed-shell set. Further, for the open-shell set considered the MAE values in the lowest excitation energies, with respect to the MRCISDT+Q, are 0.19 (DF-EOM-CCD), 0.07 (DF-EOM-OCCD), 0.13 (DF-EOM-CCSD), and 0.07 (EOM-OCCD) eV. Hence, the orbital-optimized methods, DF-EOM-OCCD and EOM-OCCD, provide improved results compared with EOM-CCD and EOM-CCSD.

We conclude that the DF-EOM-OCCD method is a very promising computational tool for the excited state studies of challenging molecular systems. Our present efficient implementation may allow large-scale chemical applications that are not computationally affordable with the conventional EOM-OCCD method.

ACKNOWLEDGMENTS

This research was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK-118Z916). The computations reported in this paper were performed at TÜBİTAK UKAKBİM High-Performance and Grid Computing Center (TRUBA resources).

Open Research

DATA AVAILABILITY STATEMENT

The data that supports the findings of this study are available in the supplementary material of this article.

Supporting Information

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Volume45, Issue32

December 15, 2024

Pages 2969-2978

Equation-of-motion orbital-optimized coupled-cluster doubles method with the density-fitting approximation: An efficient implementation

Abstract

1 INTRODUCTION

2 THEORETICAL APPROACH

2.1 DF-CCD energy and amplitude equations

2.2 DF-CCD- $Λ$ Lagrangian

2.3 Orbital-optimization for the DF-CCD wave function

2.4 The EOM-CCD model

2.5 DF-EOM-CCD intermediates

2.5.1 DF-EOM-CCD 3-index intermediates

2.5.2 $F$ intermediates

2.5.3 $W$ intermediates

2.5.4 Four-index intermediates

2.5.5 Two-index intermediates

2.6 $Δ t_{i}^{a}$ intermediate

2.7 DF-EOM-CCD $σ$ equations

3 RESULTS AND DISCUSSION

3.1 Efficiency of DF-EOM-OCCD

3.2 Accuracy of DF-EOM-OCCD

4 CONCLUSIONS

ACKNOWLEDGMENTS

Open Research

DATA AVAILABILITY STATEMENT

Supporting Information

REFERENCES

Figures

References

Information

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Equation-of-motion orbital-optimized coupled-cluster doubles method with the density-fitting approximation: An efficient implementation

Abstract

1 INTRODUCTION

2 THEORETICAL APPROACH

2.1 DF-CCD energy and amplitude equations

2.2 DF-CCD- Λ Lagrangian

2.3 Orbital-optimization for the DF-CCD wave function

2.4 The EOM-CCD model

2.5 DF-EOM-CCD intermediates

2.5.1 DF-EOM-CCD 3-index intermediates

2.5.2 F intermediates

2.5.3 W intermediates

2.5.4 Four-index intermediates

2.5.5 Two-index intermediates

2.6 Δ t i a intermediate

2.7 DF-EOM-CCD σ equations

3 RESULTS AND DISCUSSION

3.1 Efficiency of DF-EOM-OCCD

3.2 Accuracy of DF-EOM-OCCD

4 CONCLUSIONS

ACKNOWLEDGMENTS

Open Research

DATA AVAILABILITY STATEMENT

Supporting Information

REFERENCES

Figures

References

Related

Information

2.2 DF-CCD- $Λ$ Lagrangian

2.5.2 $F$ intermediates

2.5.3 $W$ intermediates

2.6 $Δ t_{i}^{a}$ intermediate

2.7 DF-EOM-CCD $σ$ equations