Two-Component Relativistic Equation-of-Motion Coupled-Cluster

Article pubs.acs.org/JPCA

Two-Component Relativistic Equation-of-Motion Coupled-Cluster Methods for Excitation Energies and Ionization Potentials of Atoms and Molecules Published as part of The Journal of Physical Chemistry virtual special issue “Mark S. Gordon Festschrift”. Yoshinobu Akinaga*,† †

VINAS Co., Ltd., Osaka 530-0003, Japan

Takahito Nakajima*,‡ ‡

RIKEN Advanced Institute for Computational Science, Kobe 650-0047, Japan ABSTRACT: Two-component relativistic equation-of-motion coupled-cluster methods are developed and implemented. Scalarrelativistic and spin−orbit effects are taken into account through a two-component scheme in both Hartree−Fock and correlation calculations. Excitation energies and spin−orbit splittings of atoms and diatomic molecules, and ionization potentials of OsO4 are reported. The advantage of the present two-component scheme is illustrated particularly for heavy-element systems.

1. INTRODUCTION Relativistic atomic and molecular calculations are becoming more and more popular during recent years in quantum chemistry.1−3 Numerous works have emerged that focused on relativistic effects upon various molecular properties such as ground-state geometry, excited-state geometry, excitation spectra, and ionization spectra.4−13 Practically speaking, relativistic effects in quantum chemical calculations can be divided into two; the scalar-relativistic effect and the spin−orbit effect. The former can be easily incorporated into the conventional nonrelativistic scheme via, e.g., the Douglas− Kroll transformation.14−17 Full consideration of spin−orbit interaction is more elaborate,18−21 requiring two-component molecular spinors instead of the conventional one-component molecular orbitals in the nonrelativistic or scalar-relativistic framework. Both effects need to be treated appropriately for accurate reproduction of molecular properties for heavyelement systems. A large number of relativistic calculations in the literature treated the spin−orbit effect as a perturbation, with a zeroth-order wave function obtained through nonrelativistic or scalar-relativistic calculation. Although such strategy has been shown to work well for systems with light elements, more rational treatment is required for heavy-element systems. In this work, we implement and benchmark a fully twocomponent framework where the spin−orbit effect is taken into account from the beginning; that is, the reference molecular spinors are calculated through two-component spin−orbit selfconsistent field (SOSCF) calculations followed by twocomponent spin−orbit correlation calculations. We adopt the two-component spin−orbit coupled-cluster (SOCC) approach for the correlation calculation. There are several advantages in © XXXX American Chemical Society

the CC method, for example, controllability of accuracy, sizeconsistency, and applicability to a wide range of electronic properties in ground and excited states.22,23 Various relativistic CC calculations have been conducted for both atoms and molecules.5−13,24,25 In the present work we focus on calculation of excitation energies and ionization potentials by twocomponent equation-of-motion CC (EOM-CC) methods.26−30 Two-component relativistic EOM-CC calculations of excitation and ionization spectra are still limited in the literature.8−11 We adopt a code generation technique in the present implementation. The rest of the paper is organized as follows. Theoretical background and implementation details are described in section 2. Section 3 gives the numerical results and discussions for selected atomic and molecular systems. The conclusions are given in section 4.

2. THEORETICAL DETAILS Because CC theory is well documented and several excellent reviews are available, we do not repeat details of the CC and related methods here. Instead, the reader is referred to review articles and books.22,23 Only the fundamental equations in ground-state CC and EOM-CC theories are described in the next two subsections. The following conventions are adopted: indices i, j, k, ... and a, b, c, ... stand for occupied and virtual spin orbitals, respectively; symbols Ψ and Φ indicate correlated and single determinant wave functions, respectively. Received: October 31, 2016 Revised: January 6, 2017

A

DOI: 10.1021/acs.jpca.6b10921 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A 2.1. Coupled-Cluster Theory. A ground-state CC wave function |ψCC⟩ is parametrized using the exponential ansatz such as |ΨCC⟩ = exp(T̂ )|Φ⟩ ≡ exp(T1̂ + T2̂ + T3̂ + ···)|Φ⟩

Tn̂ =

be solved using, e.g., the generalized Davidson’s method for non-Hermitian matrices.38 The EOM-CC method for excitation energies can be easily generalized to ionization potentials and electron affinities. For ionization potentials, for example, one of the particle indices is dropped in each R̂ n operator. Hereafter, we adopt the word “EOM” for excitation energies and “IP-EOM” for ionization potentials. 2.3. Two-Component Method. In the two-component framework, molecular orbitals are no longer spin eigenfunctions. Instead, they are two-component spinors, which have both the α and β spin components,

(1)

†

abc ··· † ̂ † ̂̂ ̂ tijk ··· {a ̂ b c ̂ ··· ··· kj i }

∑ ijk ⋯ abc ⋯

(2)

† †̂ †

where {â b ĉ ··· ···kĵ î }̂ is an n-body excitation operator in the normal-order form and |Φ⟩ is a reference determinant (e.g., the abc··· Hartree−Fock wave function). The amplitudes tijk··· are determined such that |ψCC⟩ fulfills the electronic Schrödinger equation in the subspace spanned by up to n-tuply excited determinants. The resulting CC amplitude equations can be written in a general form abc ··· ̂ ⟨Φijk ··· |HN

exp(T1̂ + T2̂ + T3̂ ···)|Φ⟩C = 0

⎛ψ α⎞ ⎜ p⎟ ψp = ⎜ ⎟ ⎜ψ β ⎟ ⎝ p⎠

(3)

The Fock matrix elements have imaginary components, and so do the molecular spinor coefficients. Accordingly, the twocomponent SOCC equations need to be solved with complex arithmetic. Double point group and time-reversal symmetry are not exploited in the present implementation. We note that, however, for the systems investigated in this work, it is possible to make the CC amplitudes and two-electron integrals real by making use of symmetric properties. Although the exact treatment of spin−orbit interaction requires one- and twoelectron considerations, in this study we adopt the screenednuclear spin−orbit (SNSO) model within the no-pair approximation,39 which is a simplified one-electron model with approximate consideration of two-electron spin−orbit interaction. The scalar-relativistic effects are considered with the third-order Douglas−Kroll (DK3) Hamiltonian40,41 with the finite-nucleus model. 2.4. Implementation. One of the problems in practical implementation of CC methods is that complexity of CC equations grows rapidly as the order of CC approximation increases. CC response equations and EOM-CC equations are even more complex because they involve, in addition to T̂ n, other types of operators such as de-excitation, thereby requiring considerable effort for implementation when higher-order excitations are considered. Several works have been devoted to develop automatic code-generation techniques such as the tensor-contraction-engine (TCE) by Hirata42 and the stringbase diagrammatic method by Kállay and Surján.43 We adopt the scheme by Kállay and Surján for the development of our SOCC programs, making a few modifications such that the level 3 BLAS library for a matrix−matrix multiplication can be employed instead of the level 2 matrix-vector multiplication library, which was suggested in the original work. The present implementation is made into the NTChem program package.44 MPI parallelization is not applied in the current implementation.

where Ĥ N is the molecular electronic Hamiltonian in the normal-order form and the subscript C means only the connected terms are retained. The correlation energy is obtained by projection onto the reference determinant, CC Ecorr = ⟨Φ|HN̂ exp(T1̂ + T2̂ + T3̂ + ···)|Φ⟩C

{tai }

ECC corr

(4)

{tab ij }.

Note that is a function of only and CC theory becomes exact if the cluster operator T̂ involves all T̂ n with n = 1, 2, ..., Nelec. In practice, T̂ is truncated at a certain level, typically two-body excitation. 2.2. Equation-of-Motion Coupled-Cluster Theory for Electronic Excitation Energies and Ionization Potentials. One of the great features of CC theory is its applicability to various molecular electronic properties. The most prominent example is excited-state calculation through the EOM-CC, CC linear response31−35 and SAC-CI36,37 methods, which are more or less equivalent to each other to some extent. The EOM-CC wave function of an excited state is constructed by acting a linear excitation operator R̂ upon the ground-state CC wave function, (k ) |Ψk⟩ = R̂ |ΨCC⟩ (k )

= (R1̂

(k )

(k )

+ R̂ 2 + R3̂ + ···)exp(T1̂ + T2̂ + T3̂ + ···)|Φ⟩ (5)

where R̂ (k) n is for the kth excited state and its amplitudes are defined in a way similar to that for T̂ n operators. Usually, R̂ (k) contains operators of the same excitation levels as the groundstate T̂ operators. Amplitudes of R̂ (k) are determined by n projecting the Schrödinger equation for |ψk⟩ onto excitation manifolds in a fashion similar to that for ground-state CC theory. The resulting equations can be written in the form (k) (k) ̂ ̂ ⟨Φμ|e−T HN̂ eT (R1̂ + R̂ 2 + ···)|Φ⟩C = ωkrμ(k)

(7)

(6)

3. RESULTS AND DISCUSSIONS 3.1. Atoms. We calculate spectra of several neutral and charged atoms with EOM-SOCCSD and EOM-SOCCSDT methods. We use relativistic all-electron valence triple-ζ polarization basis sets DK3-Gen-TK/NOSeC-V-TZP augmented by diffuse functions.45−49 3.1.1. Group 11 Cations. The ground-state configuration of the group 11 cations (Cu+, Ag+, and Au+) is nd10(n + 1)s0. The nd → (n + 1)s excitation leads to 3D and 1D states in LS coupling scheme. The 3D states are reorganized into 3D3, 3D2,

which is the right eigenequation of the CC Jacobian matrix. Here, {Φμ} span the excitation manifolds. The eigenvalues {ωk} and right-eigenvectors r(k) provide excitation energies and R̂ n amplitudes, respectively. Due to non-hermiticity of CC Jacobian, the right- and left-eigenvectors are not mutually complex conjugate. The diagrams representing the EOM-CC right equations can be derived from the diagrams for the ground-state CC equations by replacing one of the T̂ n vertices by R̂ n with the same shape. All the distinct diagrams generated in this way consist of the left-hand-side of eq 6. Equation 6 can B


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The Journal of Physical Chemistry A

Table 1. d → s Excitation Energies and Energy Splittings of Group 11 Cations Calculated with EOM-SOCC Methodsa excitation energy Cu

+

EOM-SOCCSD

EOM-SOCCSDT

exptld

EOM-SOCCSD

EOM-SOCCSDT

exptld

D3 D2 3 D1 1 D2 MAEb Maxc 3 D3 3 D2 3 D1 1 D2 MAEb Maxc 3 D3 3 D2 3 D1 1 D2 MAEb Maxc

2.003 2.126 2.285 2.570 0.700 0.716 4.581 4.785 5.178 5.472 0.231 0.274 1.710 2.078 3.339 3.662 0.095 0.155

2.364 2.481 2.646 2.893 0.350 0.363 4.711 4.908 5.309 5.579 0.108 0.144 1.859 2.190 3.497 3.726 0.029 0.054

2.719 2.833 2.975 3.256

−0.097 0.025 0.185 0.470 0.012 0.022 −0.188 0.017 0.410 0.703 0.039 0.101 −0.448 −0.081 1.180 1.504 0.048 0.120

−0.096 0.022 0.186 0.433 0.011 0.020 −0.185 0.012 0.413 0.682 0.042 0.104 −0.438 −0.107 1.200 1.429 0.028 0.045

−0.089 0.025 0.167 0.448

3

3

Ag+

Au+

splitting

state

atom

4.855 5.051 5.323 5.709

1.865 2.187 3.443 3.672

−0.159 0.037 0.309 0.695

−0.423 −0.101 1.155 1.384

a

The energy splittings are relative to the J-averaged triplet energy. A total of 16 electrons are correlated. Spin−orbit integrals are calculated using SNSO. Values are in eV. bMean absolute error. cMaximum absolute error. dFrom ref 50.

Table 2. s → p Excitation Energies and Energy Splittings of Group 12 Atoms Calculated with EOM-SOCC Methodsa excitation energy atom Zn

state

EOM-SOCCSD

EOM-SOCCSDT

exptl

P0 P1 3 P2 1 P1 MAEb Maxc 3 P0 3 P1 3 P2 1 P1 MAEb Maxc 3 P0 3 P1 3 P2 1 P1 MAEb Maxc

3.871 3.895 3.946 5.763 0.109 0.135 3.624 3.689 3.826 5.360 0.100 0.120 4.806 4.987 5.417 6.805 0.096 0.139

3.944 3.969 4.020 5.786 0.048 0.062 3.624 3.688 3.825 5.286 0.119 0.131 4.791 4.971 5.407 6.720 0.070 0.124

4.006 4.030 4.078 5.796

3 3

Cd

Hg

splitting d

3.734 3.801 3.946 5.417

4.667 4.886 5.461 6.704

EOM-SOCCSD

EOM-SOCCSDT

exptld

−0.050 −0.025 0.025 1.842 0.026 0.100 −0.134 −0.069 0.068 1.602 0.018 0.059 −0.400 −0.219 0.211 1.600 0.084 0.114

−0.051 −0.026 0.026 1.792 0.014 0.050 −0.133 −0.069 0.068 1.529 0.007 0.014 −0.402 −0.222 0.214 1.527 0.064 0.112

−0.048 −0.024 0.024 1.742

−0.140 −0.073 0.072 1.543

−0.514 −0.295 0.280 1.523

a

The energy splittings are relative to the J-averaged triplet energy. A total of 12 electrons are correlated. Spin−orbit integrals are calculated using SNSO. Values are in eV. bMean absolute error. cMaximum absolute error. dFrom ref 50.

and 3D1 states by the presence of spin−orbit interaction, whereas the 1D state becomes 1D2. Table 1 shows the calculated excitation energies and energy splittings for the low-lying states of the group 11 cations. The energy splitting is defined as the energy difference relative to the average of 3D3, 3 D2, and 3D1 energies. Experimental values taken from ref 50 are also given. The basis sets are augmented by sp diffuse functions. It is known that accurate calculation of excitation energies of group 11 atoms necessitates the semivalence electrons to be correlated.51−53 Our preliminary calculations indicate the same

trend; when only 10 electrons in the valence nd shell are correlated, the excitation energies are systematically overestimated, giving rise to more than 0.4 eV errors even at the EOM-SOCCSDT level. Substantial improvement is obtained when additional six electrons in the np shell are correlated as well. The exception is Cu+, for which the mean-absolute-errors (MAE) in the calculated excitation energies are unacceptably large at both EOM-SOCCSD and EOM-SOCCSDT levels. Such a large error in Cu+ excitation energies was also found in the work of Wang et al.51 It was argued that the strong multiconfiguration character of the Cu+ ground state is C


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The Journal of Physical Chemistry A Table 3. Spin−Obrit Splittings of Group A Doublet Radicals Calculated with IP-EOM-SOCC Methodsa SNSO OH SH SeH

AMFI

basis set

IP-EOM-SOCCSD

IP-EOM-SOCCSDT

IP-EOM-SOCCSD

IP-EOM-SOCCSDT

ref 59b

ref 61c

exptld

Sapporo-VTZ cc-pCVTZ Sapporo-VTZ cc-pCVTZ Sapporo-VTZ cc-pCVTZ

157.59 158.96 411.16 407.05 1784.00 1702.35

156.03 157.58 403.02 399.19 1747.49 1673.62

135.74 136.87 375.09 376.74 1720.69 1717.08

134.36 135.67 367.33 369.20 1685.29 1687.80

132.34

136.72

139.20

366.13

372.37

377.00

1650.09

1680.40

1764.40

Spin−orbit integrals are calculated using SNSO or AMFI. The AMFI integrals are first calculated with ANO-RCC basis sets, and then projected onto the molecular basis sets. Values are in cm−1. All electrons are correlated for OH and SH. For SeH, core electrons up to the 3p shell are frozen. b IP-EOM-CCSD with cc-pCVTZ basis sets. Spin−orbit integrals are approximated with the SOMF (spin−orbit mean-field) method. cIP-EOMCCSD with the AMFI ansatz and ANO basis sets. dFor OH and SH, from ref 54. For SeH, from ref 55. a

electrons is quite enough for accurate reproduction of excitation energies of Hg, and enlarging the correlation space has virtually no effect upon the calculated excitation energies. This is possibly because the reference orbitals in our work are generated through two-component SCF calculation, whereas the work in ref 51 employed scalar-relativistic one-component orbitals. This implies the importance of using a fully twocomponent scheme for heavy elements. 3.2. Molecules. 3.2.1. Spin−Orbit Splitting of Doublet Radicals. We compute spin−orbit splittings of two groups of doublet radicals. Group A consists of group 16 hydrides (OH, SH, and SeH), and Group B involves halogen oxides (ClO and BrO). Spin−orbit splittings are calculated by means of IPEOM-SOCCSD and IP-EOM-SOCCSDT using ground-state wave functions optimized for the closed-shell electron attached systems. Table 3 gives the spin−orbit splittings of hydride radicals calculated by the IP-EOM-SOCCSD and IP-EOMSOCCSDT methods. Table 4 gives IP-EOM-SOCCSD results for halogen oxide radicals. The experimental values are taken from refs 54−56 and 57.

responsible for this problem. Our results support this argument, i.e., when the correlation level is increased from EOMSOCCSD to EOM-SOCCSDT the Cu+ excitation energies are improved significantly. This is because the static correlation, which is not involved in the SOCCSD wave function, is partly taken into account at the SOCCSDT level through triple excitation amplitudes. We confirmed further increasing the correlated electrons has virtually no effect upon the excitation energies at the EOMSOCCSD level. Using a more extended DK3-Gen-TK/ NOSeC-V-QZP45−49 basis set (the g functions are omitted) augmented by sp diffuse functions leads to no improvement. In contrast to the excitation energies, the energy splittings are reproduced quite accurately even at the EOM-SOCCSD level. Accuracy of energy splittings is rather insensitive to the level of correlation, basis set quality, and the number of correlated electrons. The energy splittings of Au+ are exceptional, where the errors are substantially reduced at the EOM-SOCCSDT level. 3.1.2. Group 12 Atoms. Table 2 shows the calculated EOMSOCC excitation energies and splittings of the low-lying four levels of neutral Zn, Cd, and Hg atoms. The ground-state configuration is nd10(n + 1)s2. The low-lying states originate from valence ns → np excitation, leading to 3P and 1P in LS coupling. The 3P states split into 3P0, 3P1, and 3P2 states by the presence of spin−orbit interaction, whereas the 1P state becomes the 1P1 state. The basis sets are augmented by spdf diffuse functions. Unlike the group 11 cations, effect of the semivalence nsnp electrons upon the excitation energies is found negligible. We therefore concentrate on the results obtained by correlating the valence nd10(n + 1)s2 electrons. In contrast to group 11 cations, EOM-SOCCSD gives reasonable estimation for both excitation energies and spin− orbit splittings of the group 12 atoms. Inclusion of triple excitations results in only minor improvements. In the case of Zn, the triple excitations improve excitation energies of the lowest 3P0, 3P1, and 3P2 states, which are underestimated at the EOM-SOCCSD level. The 1P1 state is almost unchanged. For Cd and Hg, the main effect of triples appears in the excitation energy of 1P1 while the triplet energies are unaffected. The same systems were studied by Wang et al.51 by means of EOM-CCSD using pseudopotentials. The spin−orbit interaction was included in the post Hartree−Fock treatment. The tendency in their results roughly agrees with the present findings. The case of Hg is an exception, where the work in ref 51 claims that correlating the semivalence electrons is critical. In contrast, our results indicate that correlating only valence 12

Table 4. Spin−Orbit Splittings of Group B Doublet Radicals Calculated with IP-EOM-SOCCSDa ClO BrO

basis set

IP-EOM-SOCCSDb

Sapporo-VTZ cc-pCVTZ Sapporo-VTZ cc-pCVTZ

343.90 340.70 978.10 928.24

SOMFc

exptld 320.30

308.25 975.40 905.84

Spin−orbit integrals are calculated using SNSO. Values are in cm−1. All electrons are correlated. cIP-EOM-CCSD with cc-pCVTZ basis sets. From ref 61. dFor ClO, from ref 54. For BrO, from refs 56 and 57.

a b

Several calculations have been conducted for these systems.58−61 The recently published values by Epifanovsky et al.61 obtained by IP-EOM-CCSD with a spin−orbit mean-field (SOMF) approximation are given in Tables 3 and 4. We use two kinds of basis functions, Sapporo-DKH3-VTZ-201262 and cc-pCVTZ.63−67 The latter was also employed in ref 61. Epifanovsky et al.61 calculated spin−orbit splittings of OH, SH, and SeH by means of IP-EOM-CCSD with perturbative treatment of spin−orbit interaction. Compared to their results, the present IP-EOM-SOCCSDT result for SeH with SapporoDKH3-VTZ-2012 basis sets agree to the experimental value quite well. In contrast, however, the results for the light systems (OH and SH) are less accurate. Using more extended basis sets (Sapporo-DKH3-2012-VQZ62) or relativistic correlation-conD


Article

The Journal of Physical Chemistry A sistent basis sets (cc-pVTZ-DK,63,68 aug-cc-pVTZ-DK,63,68,69 aug-cc-pCVTZ-DK63,66,68,69) we obtain 162.14, 157.09, 165.54, and 167.67 cm−1 for OH, respectively, with 18−28 cm−1 deviations from experiment. The discrepancy with the experiment in OH and SH is caused by the inadequate spin−orbit operator. For such lightelement systems, it is important to consider two-electron spin− orbital interactions. It seems that SNSO makes a too drastic approximation to the two-electron spin−orbit interactions for describing light-element systems such as OH and SH, resulting to severe overestimation of spin−orbit splittings. A better choice for spin−orbit operator may be the AMFI (atomic mean-field integral) approach,70 where the two-electron contribution is treated in a mean-field approximation. Klein and Gauss59 used the AMFI ansatz together with atomic natural orbital (ANO) basis sets to calculate spin−orbit splittings of diatomic systems. Thus, we also adopt AMFI to the present calculations of OH, SH, and SeH. Spin−orbit integrals are calculated by the AMFI ansatz within the no-pair approximation using relativistic ANO-type basis sets (ANORCC).71,72 The calculated spin−orbit integrals are projected onto the molecular basis sets. The AMFI code73 in the Dalton program74,75 was interfaced with NTChem. The results are given Table 3. The present IP-EOM-SOCCSD results with the AMFI/ANO-RCC ansatz are in excellent agreement with the experiments. In the heavy-element system SeH, the agreement is better than the previous works. This illustrates advantage of the present fully two-component approach. In this work, the reference spinors are optimized to the electron-attached systems, rather than neutral OH, SH, and SeH. To obtain spinors optimized to neutral systems, a restricted-open or multiconfigurational approach is needed due to the degeneracy of the highest molecular spinors of the present radicals, which is beyond the scope of the present work. Mück et al.58 applied Mukkherjee’s multireference CC theory for computing spin−orbit splittings of these systems. For the halogen oxide radicals, excellent agreement between theory and experiment is achieved for BrO with the relativistic Sapporo basis sets. This also illustrates advantage of the present two-component approach for systems involving heavy elements. 3.2.2. Excitation Energies of Transition Metal Hydrides. In this section we discuss the EOM-SOCC results for excited states of AuH and TlH molecules. We use the DK3-Gen-TK/ NOSeC-V-TZP basis sets for Au and Tl, and Sapporo-DZP basis set for H. Diffuse functions of s and p types are added for both the metals and hydrogen. Equilibrium internuclear distance and vibrational frequency are evaluated by five-point polynomial fitting with 0.01 a0 displacements. Table 5 gives the calculated vertical excitation energies of AuH at the experimental structure (RAu−H = 1.524 Å). The

optimized internuclear distance and vibrational frequency for the ground and excited 0+(II) states are given in Table 6. Table 6. Ground and Excited-State Properties of AuH Calculated with EOM-SOCC Methods: Vertical Excitation Energy (Te), Equilibrium Bond Distance (Re), and Harmonic Frequency (ω)a state +

0 (I)

0+(II)

EOM-SOCCSDb

EOM-SOCCSDTb

exptlc

0− 1 0+(II)

2.947 2.963 3.529

2.995 3.009 3.570

n/a n/a 3.430

EOM-SOCCSDb

EOM-SOCCSDTb

exptlc

Te (eV) Re (Å) ω (cm−1) Te (eV) Re (Å) ω (cm−1)

1.503 2513.8 3.567 1.616 2042.1

1.503 2515.1 3.608 1.62 1964.4

1.524 2305 3.43 1.672 1692

a Spin−orbit integrals are calculated using SNSO. bA total of 18 electrons are correlated. cFrom ref 54.

We have examined several choices of frozen-core orbitals, and similar values are obtained for the ground-state bond distance, differing from each other by approximately 0.01 Å at most. The present results are fairly close to the experimental value 1.524 Å, with the largest deviation smaller than 0.02 Å. The outer-core electrons have only a very minor effect upon molecular structure. We also observe EOM-SOCCSD and EOM-SOCCSDT give virtually the same bond distances. The excited-state vibrational frequency is substantially overestimated for both EOM-SOCCSD and EOM-SOCCSDT. The same was reported in the work by Witek et al.76 The reason may be lack of g functions in the Au basis set. Excitation energies of AuH strongly depend on the choice of core orbitals. Correlating only 12 electrons, which are from Au 5d6s and H 1s shells, significantly overestimates the excitation energies at the EOM-SOCCSD level. Reasonable estimation is obtained when additional six electrons are correlated as well. The results for the TlH molecule are listed in Tables 7 and 8. The experimental values are taken from refs 77 and 78. The Table 7. Vertical Excitation Energies (eV) of TlH Calculated with EOM-SOCC Methods at the Experimental Geometry (RTl−H = 1.870 Å)a state −

0 0+(II) 1(I) 2 1(II)

EOM-SOCCSDb

EOM-SOCCSDTb

exptlc

2.112 2.172 2.269 2.610 2.961

2.117 2.175 2.272 2.609 2.964

n/a 2.200 n/a n/a 3.000

a

Spin−orbit integrals are calculated using SNSO. bA total of 14 electrons are correlated. cFrom ref 78.

same system was studied by several authors.79−81 In contrast to AuH, correlating valence 14 electrons is quite enough for describing the ground and low-lying excited states of this molecule; the differences from the more extended 26-electron correlation case are 0.006 Å for the ground-state bond distance, and of the order of 10−3 eV for the excitation energies at the EOM-SOCCSD and EOM-SOCCSDT levels. For excitation energies of the TlH molecule, experimental data are available for the 0+(II) and 1(II) states. For 0+(II), both the EOM-SOCCSD and EOM-SOCCSDT excitation energies achieve excellent agreement with the experimental value. The 1(II) excitation energies by EOM-SOCCSD and

Table 5. Vertical Excitation Energies (eV) of AuH Calculated with EOM-SOCC Methods at the Experimental Geometry (RAu−H = 1.524 Å)a state

property

a Spin−orbit integrals are calculated using SNSO. bA total of 18 electrons are correlated. cFrom ref 54.

E


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exhibits much smaller spin−orbit splitting. The present IPEOM-SOCCSD calculation with 54 active electrons gives quite reasonable estimation of experimentally observed peak positions A, B, C, and D. The ionization potentials corresponding to peak E are significantly overestimated. This is because these states involve higher-order correlation and orbital relaxation effects.88 Table 9 also gives the energy of the molecular spinor, which is responsible for ionization in each ionic state. It is observed that Koopmans’ theorem incorrectly predicts the order of ionization peaks B and C. The Kohn−Sham (KS) spinors obtained with the B3LYP exchange−correlation functional, in contrast, exhibit the correct order of spinor energies, implying importance of electron correlation in these ionization states. Because the IP-EOM-SOCCSD method does not give reasonable estimation for the ionization potentials corresponding to the peak E, we examine IP-EOM-SOCCSDT. To reduce computational cost, we use smaller basis sets, Sapporo-DKH3DZP-2012 for Os92 and Dunning’s DZ for O.93 In addition, the number of correlated electrons is reduced to 24. In the IPEOM-SOCCSDT calculation of OsO4, a serious convergence problem occurred in diagonalizing the SOCC Jacobian. We thus adopt a loose energy convergence threshold 10−5 hartree. For the purpose of checking for improvements from IP-EOMSOCCSD, this level of convergence threshold is sufficient. Table 10 shows the calculated IP-EOM-SOCCSD and IP-

Table 8. Ground and Excited-State Properties of TlH Calculated with EOM-SOCC Methods: Vertical Excitation Energy (Te), Equilibrium Bond Distance (Re), and Harmonic Frequency (ω)a state +

0 (I)

0+(II)

property

EOM-SOCCSDb

EOM-SOCCSDTb

exptlc

Te (eV) Re (Å) ω (cm−1) Te (eV) Re (Å) ω (cm−1)

1.856 1454.2 2.169 1.781 1410.9

1.858 1442.9 2.169 1.799 1288.8

1.87 1391 2.2 1.86 n/a

a Spin−orbit integrals are calculated using SNSO. bA total of 14 electrons are correlated. cFrom refs 77 and 78.

EOM-SOCCSDT are less satisfactory, though the error is fairly small. 3.3. Ionization Potentials of OsO4. We apply the present IP-EOM-SOCCSD method to molecular ionization potentials of OsO4. Ionization spectra of the OsO4 molecule have been known to exhibit a clear splitting due to spin−orbit interaction,82 and a number of computational studies on ionization potentials as well as excitation energies of OsO4 have been published.83−88 Nakajima et al.88 successfully reproduced the OsO4 ionization spectra using CASPT2 combined with the RESC (relativistic elimination of small component) scheme,89,90,40 where the spin−orbit interaction was taken into account at the RESC-CASPT2 stage. In the present twocomponent IP-EOM-SOCC calculation, the spin−orbit interaction is considered from the beginning. We use the DK3-GenTK/NOSeC-V-TZP basis set augmented by sp diffuse functions for Os, and aug-cc-pVDZ63 for O. The molecular structure is assumed to have the Td symmetry and the Os−O bond distances are set to the experimental value, 1.711 Å.91 Table 9 displays ionization potentials and ionization intensities of OsO4 calculated by IP-EOM-SOCCSD with 54

Table 10. Ionization Potentials of OsO4 (eV) Calculated with IP-EOM-SOCCSD and IP-EOM-SOCCSDTa ionization potential

Table 9. Ionization Potentials of OsO4 (eV) Calculated with IP-EOM-SOCCSDa spinor energy state 1U′ 1E′ 2U′ 1E″ 2E′ 2E″ 3U′ 4U′

ionization potential (ionization intensity) 12.41 12.47 13.23 13.50 14.86 17.30 17.35 17.92

(0.908) (0.907) (0.907) (0.907) (0.902) (0.888) (0.884) (0.881)

HF

KS

14.21 14.56 15.45 15.37 17.13 19.94 19.87 20.45

10.25 10.27 10.94 11.26 12.59 14.63 14.65 15.16

exptl 12.35 12.35 13.14 13.54 14.66 16.40 16.40 16.80

state

IP-EOM-SOCCSD

IP-EOM-SOCCSDT

1U′ 1E′ 2U′ 1E″ 2E′ 2E″ 3U′ 4U′

12.24 12.30 13.03 13.34 14.52 17.27 17.35 17.91

12.08 12.09 12.73 13.07 14.22 15.74 15.74 15.74

exptlb 12.35 12.35 13.14 13.54 14.66 16.40 16.40 16.80

(A) (A) (B) (C) (D) (E) (E) (E)

a A total of 24 electrons are correlated. Spin−orbit integrals are calculated using SNSO. bFrom ref 82.

b

(A) (A) (B) (C) (D) (E) (E) (E)

a

EOM-SOCCSDT ionization potentials under the conditions described above. Compared with the case of IP-EOMSOCCSD, ionization potentials for peak E calculated by IPEOM-SOCCSDT are significantly reduced by considering the triple excitation amplitudes. The obtained IP-EOM-SOCCSDT ionization potentials are totally underestimated in comparison with the experimental values. This is possibly due to insufficient basis function space.

electrons correlated. Also given are the experimentally observed peak positions. Ionization intensities are evaluated from the IPEOM R̂ amplitudes on the basis of the monopole approximation. The experimentally observed separation of peaks B and C, which has been understood as a consequence of spin−orbit splitting of Os atomic orbitals, is clearly reproduced in the present IP-EOM-SOCCSD results. These states originate from the 2T2 state in the spin-free domain. The 2T1 state

4. CONCLUSIONS Relativistic spin−orbit equation-of-motion coupled-cluster methods are implemented in a fully two-component manner, where reference spinors are taken from two-component spin− orbit SCF calculation. The scalar-relativistic effect and spin− orbit interaction are considered via DK3 transformation and approximate one-electron models, namely, SNSO and AMFI, respectively. Excitation energies and ionization potentials of atoms and molecules are calculated via the EOM scheme with full consideration of up to triple excitations. The results agree well with experiments, particularly for those systems involving

A total of 54 electrons are correlated. Ionization intensities are given in parentheses. Spin−orbit integrals are calculated using SNSO. bFrom ref 82.

F


Article


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heavy elements, illustrating the advantage of the present fully two-component approach.

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AUTHOR INFORMATION

Corresponding Authors

*Y. Akinaga. E-mail: [email protected]. *T. Nakajima. E-mail: [email protected]. ORCID

Yoshinobu Akinaga: 0000-0002-3495-1758 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Next-Generation Supercomputer project and the FLAGSHIP2020 within the priority study5 (Development of new fundamental technologies for high-efficiency energy creation, conversion/storage and use) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. This work was also supported by FOCUS Establishing Supercomputing Center of Excellence.

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Two-Component Relativistic Equation-of-Motion Coupled-Cluster

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