Equation of motion coupled cluster method for electron attached states with spin–orbit coupling

Chemical Physics Letters 531 (2012) 236–241

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Equation of motion coupled cluster method for electron attached states with spin–orbit coupling Dong-Dong Yang a,b, Fan Wang b,⇑, Jingwei Guo a a b

Key Laboratory of Chemical Laser, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, PR China College of Chemistry, Sichuan University, Chengdu 610064, PR China

a r t i c l e

i n f o

Article history: Received 25 October 2011 In final form 7 February 2012 Available online 15 February 2012

a b s t r a c t We report implementation of the equation of motion coupled-cluster method for electron attachment (EOMEA-CC) based on a previously developed CC approach with spin–orbit coupling included in postHartree–Fock treatment at the CC singles and doubles level (CCSD). Time-reversal symmetry is exploited by dealing with one partner of a Kramers pair explicitly and determining the other one through time reversal. This EOMEA-CCSD approach has been applied to calculate properties of some open-shell atoms and molecules containing heavy elements and is found to be able to describe SOC effects with reasonable accuracy. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction In our previous works, we developed a coupled-cluster (CC) method [1–4] for closed-shell systems with spin–orbit coupling (SOC) [5,6]. By including SOC in post-Hartree–Fock (HF) treatment, this approach has been shown to be rather efficient due to the use of real molecular spin–orbitals and to be accurate because of a proper treatment of orbital relaxation via single excitations in the cluster operator. The computational effort of this approach at the CC singles and doubles level (CCSD) is about 10–15 times that of a closed-shell spin-adapted CCSD calculation without SOC [5,6], while this factor is 32 when SOC is included at the HF level [7]. The accuracy of this approach, as shown in our previous work, has also been verified recently by Kim et al. [8]. This approach has been implemented at the CCSD level [9] as well as at the CCSD level augmented by a perturbative treatment of triple excitations (CCSD(T)) [10] for ground state energy [5,6], analytic first [6,11] and second order derivatives [12] calculations using relativistic effective core potentials (ECPs) [13]. In fact, SOC usually has a more pronounced effect on open-shell systems than on closed-shell systems. This SOC-CC approach could also be applied straightforwardly to open-shell systems. However, convergence problem in solving the CC equations with SOC will show up if the ground state of the system without spin–obit coupling is spatially degenerate. To deal with open-shell systems with SOC, we resort to the equation of motion (EOM) coupled-cluster approach [14]. With the EOM-CC approach, a proper closed-shell reference state is first chosen and the open-shell state can be reached by exciting ⇑ Corresponding author. Fax: +86 28 85407797. E-mail addresses: [email protected], [email protected] (F. Wang). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2012.02.014

electrons from occupied orbitals to virtual orbitals [15], removing electrons from the reference state [16], or injecting electrons to this reference state [17]. Besides these states, double ionized or double electron attached states can also be calculated based on EOM-CC [18]. An advantage of this approach is that the wavefunction of the target state is free from spin-contamination. This is particularly important in treating open-shell molecules with SOC since SOC will mix states with different spin. Otherwise, artificial coupling due to SOC between states which are not pure spin states could show up and thus results in spurious level splitting. The EOM approach for excitation energy (EOMEE) and ionization energy (EOMIP) at the CCSD level based on our SOC-CC approach has been implemented previously [19]. In this work, we report implementation of the EOM approach for electron affinities (EOMEA) at the CCSD level based on our previous CC approach with SOC. It should be noted that IPs and EAs can also be calculated easily using the EOMEE approach by including a Gaussian basis function with a nearly zero exponent and calculating excitations from or to this orbital [20]. However, the computational effort for solving the equation of motion in EOMEE-CCSD is of N6, while it is only of N5 in EOMIP-CCSD or EOMEA-CCSD, where N is a measure of system size. Calculating IPs or EAs with EOMIP or EOMEA method will thus be much more efficient than that with the EOMEE method. On the other hand, the available EOMEE code could also be used to validate the EOMIP or EOMEA code. A closely related CC approach in treating electron excitation, ionization or electron attachment is the Fock–Space (FS) CC approach [21], which gives rise to the same IPs or EAs as the EOMIP or EOMEA method. FS-CC approach with SOC has also been implemented previously, but mainly at the level where SOC is included in Hartree–Fock or Dirac–Fock calculations [22] where four-component relativistic Hamiltonian is employed.

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It has been well established that the EOMEE-CCSD approach could provide accurate excitation energies for states dominant by single excitations, but fails to do so for states with significant double excitation character [23]. This rule also applies to EOMIP-CCSD and EOMEA-CCSD approaches, where states mainly from simply removing or attaching one electron with respect to the reference state can be calculated rather accurately. On the other hand, states which can only be reached by removing or adding one electron and exciting another one from occupied to virtual orbital at the same time are generally poorly described. It can thus be seen that although both EOMIP-CCSD and EOMEA-CCSD approaches with closed-shell reference can treat states such as s1 or r1, but the p3 or p5 states can only be calculated with reasonable accuracy based on EOMIP-CCSD approach, while the p1 or p1 states can only be treaded using the EOMEA-CCSD method reliably. Another issue worth mentioning is the dependence of the results on the orbitals used in calculations. It is shown by Nooijen et al. [17] that EAs from the EOMEA-CCSD have a more significant dependence on the chosen orbitals than IPs from the EOMIP-CCSD method. In the present approach with SOC, we always use orbitals from the HF wavefunction of the closed-shell reference state. Fortunately, dependence of excitation energies, i.e. difference between EAs for higher electronattached states and the energy-lowest electron-attached state, from EOMEA-CCSD is not as sensitive on the orbitals [17] and we could thus expect reasonable excitation energies from the EOMEA-CCSD approach with SOC. This Letter is organized in the following manner: the basic equations for the EOMEA-CCSD approach with SOC and implementation details are discussed in Section 2. In Section 3, we present results for ionization potentials, excitation energies as well as energy splitting due to SOC of Cu, Ag, Au, Ga, In and Tl as well as the ground and excited state equilibrium bong length, harmonic frequencies and adiabatic excitation energies of PbF and HgH. 2. Basic equations and implementation To solve the EOMEA-CCSD equations for EAs, CCSD equations for the reference state must first be solved. In the present CC approach, spin–orbit coupling is included in post-HF treatment and the reference Slater determinant adopted is the HF wavefunction of the Hamiltonian without including spin–orbit coupling. CC equations for this approach are thus similar to those in nonrelativistic CC approaches with a non-HF reference determinant [24,25]. The CCSD wavefunction for the reference state is written as eT jU0 i, where U0 is the reference Slater determinant and T is the cluster operator defined in the following:

T¼

X

t ai aþa ai þ

i;a

1 X ab þ þ t a ai ab aj ; 4 a;b;i;j ij a

ð1Þ

where i, j, . . .(a, b, . . .) are indices for occupied (virtual) spin orbitals. The amplitudes t ai and tab ij in the cluster operator T are determined using the following equations in CCSD: T T < Uai jeT HeT jU0 >¼ 0; < Uab ij je He jU0 >¼ 0;

ð2Þ

where Uai and Uab ij are single and double excitation determinants with respect to U0 . In our CC approach with SOC, the SOC operator is included in the Hamiltonian in Eq. (2). The wavefunction corresponding to electron-attached states with respect to the reference state in EOMEA-CCSD is represented by ReT jU0 i with the electronattaching operator R defined as:

R¼

X a

r a aþa þ

1 X ab þ þ r a ai ab ; 2 a;b;i i a

ð3Þ

where ra and rab are r-amplitudes to be determined in i EOMEA-CCSD. It should be noted that there exist other choices for

the operator R, where either higher or lower excitation levels [26] are included. In this work, we use the form of the R operator presented in Eq. (3). By inserting the wavefunction of the electron-attached state to the Schrödinger equation, we can arrive at the following EOMEA-CCSD equations using Eq. (2) and the fact that the R operator is commutable with the cluster operator T:

 RjU0 >¼ Eea r a ; < Uab j½H;  RjU0 >¼ Eea r ab ; < Ua j½H; i i

ð4Þ

 is the effective Hamiltonian defined as H  ¼ eT HeT , Eea is where H the electron affinity, Ua and Uab are the electron-attached determii nant with respect to U0 defined in the following: þ þ jUa >¼ aþa jU0 >; jUab i >¼ aa ai ab jU0 > :

ð5Þ

In our approach, SOC operator should also be included in the Hamiltonian in Eq. (4). Detailed algebraic expressions for Eq. (4) can be found in Ref. [17], where some intermediates Fpq and Wpqrs are introduced to facilitate implementation and reduce computational effort. These intermediates are actually the one- and two [27]: electron parts of the effective Hamiltonian H

eT HeT ¼

X

F pq aþp aq þ

p;q

1X W pqrs aþp aþq as ar þ :::: 4 pq;rs

ð6Þ

In the EOMEA-CCSD approach for closed-shell system without SOC,  ab one only need to calculate the r-amplitudes of spin cases {ra, rab i ; ri }, where indices with a bar refer to b spin orbitals, and those without a bar denote a spin orbitals. On the other hand, when SOC is included, much more spin cases for r-amplitudes have to be considered since SOC will mix states with different spin. In the present work, we choose to solve the following spin cases for r-amplitudes: {ra,     b b ab ab ab a a ra ; r ab i ; ri ; r i ; ri ; r i ; ri } and other spin cases are determined using permutation symmetry. Furthermore, the r-amplitudes are complex number when SOC is present while they are real without SOC. As for spin cases for the t-amplitudes and intermediates used in the implementation, our previous work can be referred to [5]. The EOMEA-CCSD approach with SOC for electron affinities of closed-shell systems is implemented based on our SOC-CCSD program through an interface to the CFOUR program package [28]. In our previous implementation for the SOC-CC approach, time-reversal symmetry is exploited to reduce computational effort [5,7]. For systems with even number electrons, one can always require the wavefunction to be invariant under time reversal if the Hamiltonian is invariant under time reversal. This will result in some kind of relations between different spin cases for the t-amplitudes and the r-amplitudes in EOMEE equation [5,7] with closed-shell state as reference, which has been exploited to reduce computational effort. On the other hand, for systems with odd number electrons, an eigenfunction of the system and its time-reversed state are always orthogonal and energy-degenerate, i.e. Kramers degeneracy. For the wavefunction of an electron attached state represented by ReT jU0 i satisfying Eq. (4), it can easily be shown that its timereversed partner R0 eT jU0 i also satisfies Eq. (4) with the same electron affinity. The r-amplitudes in the R operator and those in the R0 operator satisfy the following relations:

r0a ¼ ðr a Þ ; r 0a ¼ ðr a Þ ; r

0ab i

r

0aib



¼

 ðraib Þ ;

¼

ðrab Þ ; i

r

0ab i

¼

 ðr aib Þ ;

¼

 ðr ab i Þ ;



r

0ab i

ð7Þ r

 0ai b

¼

ðriba Þ ;

r

 0ab i

¼

ðr bi a Þ ; ð8Þ

where the fact that the cluster operator T and the reference Slater determinant are invariant under time reversal has been used. These relations are adopted to exploit time-reversal symmetry for electron-attached states in our implementation and will be discussed below. The EOMEA-CCSD equation in Eq. (4) is equivalent to calculate  the eigenvalues and eigenvectors of the effective Hamiltonian H

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D.-D. Yang et al. / Chemical Physics Letters 531 (2012) 236–241

in the space spanned by {Ua ; Uab i } and Davidson’s algorithm [29] with slight modification for non-Hermitian matrix [30] is adopted to calculate the lowest several or some specified electron-attached states. In the EOMEE approach with SOC for closed-shell system,  can be made real [19] by imposing the matrix representation of H time-reversal symmetry on trial vectors: the matrix element  W2 i will be invariant under time-reverse if W1, W2 and H  hW1 jHj are all invariant under time-reverse, which means this matrix element is real since its time-reverse is simply its complex conjugate. However, in the EOMEA and EOMIP approach, the matrix  cannot be made real for general case since representation of H the wavefunctions are not invariant under time-reverse. In fact, time-reversal symmetry can still be exploited in EOMEA and EOMIP calculations to reduce computational effort. In Davidson’s algorithm, the most time-consuming part is calculating the left hand side (l.h.s.) of Eq. (4) for a given trial R operator. When this is done, the l.h.s. of Eq. (4) for the time-reversal of the trial R operator can be determined easily using the same relations as in Eqs. (7) and (8) since the effective Hamiltonian is invariant under time reversal. Using this fact and Kramers degeneracy, the computational effort can be reduced in solving Eq. (4), since only one partner of a Kramers pair has to be considered explicitly and its time-reversal can be calculated with ease based on Eqs. (7) and (8).

3. Results To demonstrate the applicability and accuracy of the EOMEACCSD approach with SOC for electron affinities and excitation energies, we calculated the ground and some excited states of Cu, Ag, Au, Ga, In and Tl atoms as well as HgH and PbF. The ground state of these atoms or molecules is open shell with one unpaired electron while the ground state of the corresponding cations is closedshell. In the calculations, the ground cationic state is used as reference and the neutral state for these systems can thus be reached by

attaching one more electron to the cationic state with EOMEACCSD. The lowest EAs for these cations are thus also IPs for the neutral species. In the present work, the SOC operator is taken from relativistic ECPs and the energy-consistent pseudopotentials developed by the Stuttgart/Cologne groups [31] are adopted for the involving atoms except for H and F. To be more specific, the ECP10MDF for Cu [32] and Ga [33], the ECP28MDF for Ag [32] and In [33], the ECP60MDF for Au [32], Hg [32], Tl [34] and Pb [33] are employed. As for the basis set, it has been found that basis functions without a proper description for variation of the innershell orbitals (the (n  1)p and (n  1)d orbitals) of heavy elements are not suitable for calculations with SOC due to the splitting of the inner-shell p and d orbitals [12,35–37]. In the present work, the ccpwCVQZ basis set [38,39] developed to describe core-valence correlation are employed for heavy elements, which has been shown to be able to provide accurate description of SOC [12]. To achieve reliable energies also for Rydberg states, this basis set is augmented with a set of diffuse functions [38,40,41]. For H and F, the cc-pVQZ basis set [42] is used. In all the calculations, all electrons (those not treated via ECPs) except for the 1s electrons of F are included in CCSD and EOMEA-CCSD calculations. The calculated IPs, excitation energies and splittings due to SOC for Cu, Ag and Au are listed in Table 1, together with experimental results [43] for comparison. It can be seen from Table 1 that excitation energies for Cu, Ag and Au obtained in this work are generally in good agreement with experimental data with an error of less than 0.15 eV even for transitions to some high lying Rydberg states, except for the states corresponding to the (n  1)d9ns2 configuration. The (n  1)d10 state are chosen as reference state for Cu, Ag and Au and the (n  )d9ns2 state can only be reached from the reference state by adding one electron to the ns orbital and exciting another one from the (n  )d orbital to the ns orbital at the same time, which is a double excitation in EOMEA calculation. It is thus understandable that these states cannot be calculated with reasonable accuracy using EOMEA-CCSD. On the other hand, the

Table 1 IPs, excitation energies and splittings due to SOC for Cu, Ag and Au (units: eV). Configuration

State

Present work EEa

Cu 3d104s 3d94s2 3d104p 3d105s 3d105p Ag 4d105s 4d105p

2

S1/2 D5/2 2 D3/2 2 P1/2 2 P3/2 2 S1/2 2 P3/2 2 P1/2 2

S1/2 P1/2 P3/2 2 D5/2 2 D3/2 2 S1/2 2 P1/2 2 P3/2 2

4d106s 4d106p Au 5d106s 5d96s2

S1/2 D5/2 D3/2 2 P1/2 2 P3/2 2 S1/2 2 P1/2 2 P3/2 2

5d106p 5d107s 5d107p a

Excitation energy.

EEa

Splitting

0.031

0.012

0.253 0.031

0.000

7.500 3.656 3.765 7.718 8.231 5.233 5.977 6.009

7.576 3.664 3.778 3.750 4.304 5.276 5.988 6.013

0.109 0.513

0.032

0.114 0.554

0.025

9.139 5.044 6.570 4.697 5.191 6.716 7.490 7.600

1.526 0.494

0.110

IP 7.726

1.389 1.642 3.786 3.817 5.348 6.123 6.123

0.280

2

2

IP 7.507

6.549 6.829 3.731 3.762 5.190 6.048 6.036

2

2

4d95s2

Experiment work [43] Splitting

9.226 1.136 2.658 4.632 5.105 6.755 7.443 7.529

1.522 0.473

0.086

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D.-D. Yang et al. / Chemical Physics Letters 531 (2012) 236–241 Table 2 IPs, excitation energies and splittings due to SOC for Ga, In and Tl (units: eV). Configuration

State

Present work EE

Ga 4s24p

Experiment work [43] Splitting

IP

2

P1/2 P3/2 S1/2 2 P1/2 2 P3/2 2 D3/2 2 D5/2 2

2

2

4s 5s 4s25p 4s24d In 5s25p

P1/2 P3/2 2 S1/2 2 P1/2 2 P3/2 2 D3/2 2 D5/2

5s26s 5s26p 5s25d Tl 6s26p

P1/2 P3/2 2 S1/2 2 P1/2 2 P3/2 2 D3/2 2 D5/2

6s27s 6s27p 6s26d

Splitting

0.013 0.001

0.014 0.001

5.782 0.267 3.027 3.946 3.982 4.192 4.196

5.786 0.274 3.022 3.945 3.982 4.078 4.081

0.036 0.004

0.037 0.003

6.079 0.921 3.258 4.214 4.333 4.566 4.582

6.108 0.966 3.283 4.235 4.359 4.478 4.488

0.119 0.016

IP 5.999

0.102 3.073 4.097 4.111 4.312 4.313

2 2

EEa

5.969 0.100 3.064 4.078 4.091 4.443 4.444

2 2

a

a

0.124 0.010

Excitation energy.

Table 3 Spectroscopic constants for the ground state and excited states of PbF and HgH molecules.

xe

re

PbF SOC-EOMEA X12G1/2 X22G3/2 P A(2 +) P B(2 +) EOMEA 2 G P A2 + P B2 + CCSD 2 G 2P+

d e

Calc.

Exp.

Ref.

Calc.

Exp.

Ref.

2.054 2.032 2.150 1.971

2.058a 2.034a 2.160a 1.976a

2.070b 2.047b 2.152b 1.982b

514 536 406 620

503a 529a 395a 606a

494b 519b 388b 601b

0.98 2.86 4.43

1.02a 2.80a 4.42a

0.97b 2.86b 4.30b

3.00

3.05c

3.44

3.51a 3.51c

3.06d (2.98)e 3.51d (3.43)e

2.040 2.167 1.976

532 394 619

2.21 3.84

2.040 2.175

534 386

2.18

1.580

A22G3/2

1.577

2

c

Ref.

A12G1/2

G CCSD 2P+

b

Exp.

1.706

2

a

Calc.

HgH SOC-EOMEA P X(2 +)

EOMEA 2P+

G

Te (eV)

(cm1)

0

(Å A)

1.766a 1.741c 1.601a 1.583c 1.579a 1.581c

1.730d (1.702)e 1.582d (1.578)e 1.579d (1.576)e

1583 2091 2108

1203a 1385c 1939a 2068c 2069a 2091c

1424d (1597)e 2065d (2100)e 2083d (2117)e

1.720 1.578

1545 2103

3.20

1.747 1.576

1372 2118

3.28

Ref. [44]. Results from Ref. [46]. Note that bond length and harmonic frequencies are those based on RASCI, while excitation energies are vertical excitation energies from FS-CCSD. Ref. [45]. FS-CCSD results with corrections from triples in Ref. [47]. FS-CCSD results in Ref. [47].

calculated splittings of the states due to SOC are rather accurate compared with experimental data even for the (n  )d9ns2 configuration due to error cancellation, which may be because that errors

in energies of states origin from a same electron configuration are similar. This may indicate that electron correlation effects on SOC splitting for this configuration are not important. Another

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D.-D. Yang et al. / Chemical Physics Letters 531 (2012) 236–241

interesting aspect related to the (n  )d9ns2 states is that excitation energies for these states based on present EOMEA-CCSD calculations are 4–5 eV larger than the experimental results, while one would generally encounter an error of about 1–2 eV for double excitations in EOMEA-CCSD [26]. This large discrepancy may stem from the multiconfiguration nature of these states, which is not able to be captured adequately with EOMEA-CCSD approach. As for the IPs, the calculated IPs are always underestimated and it is about 0.2 eV smaller than the experimental data for Cu, while they are about 0.08 eV smaller for Ag and Au. The calculated results and experimental data for Ga, In and Tl are listed in Table 2. It can be seen from this table that the calculated IPs and excitation energies for these elements agree much better with experimental data that those for the IB block elements except for the excitation energies to the ns2(n + 1)d1 states. The calculated IPs for Ga, In and Tl are only about 0.03 eV smaller than the experimental data, and the deviation of the calculated excitation energies from experimental results are usually less than 0.03 eV except for the ns2(n + 1)d1 states. Even for the ns2(n + 1)d1 states, excitation energies are only overestimated by about 0.12 eV. As for the splittings of states due to SOC for these atoms, the agreement between the calculated ones and experimental data are always better than 0.01 eV, except for the splitting between the 2 P1/2 and 2P3/2 states of the 6p1 configuration for Tl, where the difference is about 0.04 eV. From these results, one can conclude that the EOMEA-CCSD approach could provide accurate estimates for excitation energies and IPs with SOC for these atoms if the target state can be reached mainly by attaching one electron to the virtual orbital of the reference state. The fact that the IPs are always underestimated for these systems may be attributed to that the reference state, i.e. the ionized state are better described than the ground state of the electron attached state in the EOMEA-CCSD approach. The equilibrium bond length, harmonic frequencies for the ground and some excited states as well as adiabatic excitation energies of PbF and HgH are listed in Table 3 together with available experimental data [44,45]. Previous results based on FS-CCSD with relativistic ECP or Dirac–Fock are also listed [46,47] for comparison. In addition, we also present results from EOMEA-CCSD calculations without SOC for these states as well as CCSD results without SOC for the ground and the lowest excited states based on spin-unrestricted HF reference state for comparison. It should be noted that due to different symmetry between the ground state and the lowest excited state for HgH and PbF without SOC, they can be calculated directly with CC approaches by specifying a proper occupation number. It can be seen from this table that EOMEACCSD results for PbF with SOC agree rather well with experimental data. The calculated bond lengths are always smaller than experimental data with a difference of less than 0.005 Å except for the P A(2 +) state, the frequencies are about 10–15 cm1 larger than experimental data and differences between calculated adiabatic excitation energies and experimental values are about 0.05 eV. As for results without SOC for PbF, the EOMEA-CCSD results compare well the CCSD results. It can be seen from this table that SOC has pronounced effects on properties of these states. Splitting between the 2P1/2 and 2P3/2 states amounts to about 1 eV and the 0 difference in bond lengths of these two states is larger than 0.02 Å A. Even for P the A(2 +) state, SOC changes bond length and adiabatic excitation energy significantly. The equilibrium bond length and harmonic frequencies listed in Table 3 from Ref. [46] are those based on Dirac–Fock restricted active space configuration interaction calculations, while excitation energies are actually vertical excitation energies at bond length of 2.06 Å with Dirac–Fock FS-CCSD approach. It can be seen that our results agree better with experimental data than those in Ref. [46], which could be related to the larger basis set employed in present calculations.

For HgH, it can be seen from this table that the effects of SOC is much smaller than those in PbF, where the difference in bond length between the 2P1/2 and 2P3/2 states is 0.003 Å and energy difference between these two states is about 0.4 eV. The agreement between the experimental and calculated results for the excited states is quite good. In addition, our results also agree very well with those based on FS-CCSD using a different relativistic ECP. However, large discrepancy between the calculated and experimental results in bond length and vibrational frequencies exists for the ground state of the HgH. In fact even the EOMEA-CCSD and the CCSD results without SOC differ significantly for the ground state of HgH and CCSD results agree much better with experimental results. By inspecting the r-amplitudes corresponding to this state in EOMEA-CCSD calculation, double excitation character for this state is in fact insignificant. This large difference may be related to the reference state used in EOMEA-CCSD calculations and further investigation is required to clarify this point. According to previous calculated results listed in this table, one can also see that triple excitations have more pronounced effects on the ground state than on other excited states of HgH. 4. Conclusion In this work, we present implementation of the EOMEA-CCSD method based on a previously developed CC approach where SOC is included in post-HF treatment to calculate properties of the ground and some excited states of open-shell systems. In the EOMEA-CCSD approach, the closed-shell cationic state is chosen as reference and the target state can be reached through electron attachment. With this reference, the target state is free from spin-contamination which is critical in calculating splitting of degenerate states since SOC will mix states with different spin. Time-reversal symmetry has been exploited in the implementation, where only one partner of a Kramers pair is considered explicitly and the other state can be determined with ease through time reversal. This approach has been applied to some open-shell atoms and molecules containing heavy elements and generally satisfactory results have been achieved except for those states with significant double excitation character. In fact, reasonable energy splitting can still be obtained even for double excitation states. For the ground state of HgH, somewhat larger difference between the calculated results and the experimental ones is observed and this difference may be related to the chosen reference state. Acknowledgments We thank the National Nature Science Foundation of China (Grant No. 20973116) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry for financial support. References [1] J. Gauss, in: P.V.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P.A. Kollmann, H.F. Schaefer, P.R. Schreiner (Eds.), Encyclopedia of Computational Chemistry, Wiley and Sons, New York, 1998, p. 615. [2] P.R. Taylor, in: B.O. Roos (Ed.), Lecture Notes in Quantum Chemistry II, European Summer School in Quantum Chemistry, Springer-Verlag, Berlin, Heidelberg, 1994, p. 125. [3] R.J. Bartlett, M. Musial, Rev. Mod. Phys. 79 (2007) 291. [4] R.J. Bartlett, in: D.R. Yarkony (Ed.), Modern Electronic Structure Theory (vol. 2, Chapter 16), World Scientific, Singapore, 1995, p. 1047. [5] F. Wang, J. Gauss, C. van Wüllen, J. Chem. Phys. 129 (2008) 064113. [6] Z. Tu, D. Yang, F. Wang, J. Guo, J. Chem. Phys. 135 (2011) 034115. [7] L. Visscher, K.G. Dyall, T.J. Lee, Int. J. Quant. Chem. Symp. 29 (1995) 411. [8] I. Kim, Y.C. Park, H. Kim, Y.S. Lee, accepted by Chem. Phys. . [9] G.D. Purvis, R.J. Bartlett, J. Chem. Phys. 76 (1982) 1910.

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