SpinâOrbit Effects in Closed-Shell Heavy and Superheavy Element

Article pubs.acs.org/JPCA

Spin−Orbit Effects in Closed-Shell Heavy and Superheavy Element Monohydrides and Monofluorides with Coupled-Cluster Theory Dong-Dong Gao,† Zhanli Cao,† and Fan Wang*,†,‡ †

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610064, P. R. China Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Sichuan University, Chengdu 610064, P. R. China

‡

ABSTRACT: Bond lengths and force constants of a set of closed-shell sixth-row and superheavy element monohydrides and monofluorides are calculated in this work. Kramers restricted coupled-cluster approaches (KR-CC) with spin−orbit coupling (SOC) included at the self-consistent field (SCF) level as well as CC approaches with SOC included in post-SCF treatment (SOC-CC) are employed in calculations. Recently published relativistic effective core potentials are employed, and highly accurate results for superheavy element molecules are achieved with KR-CCSD(T). SOC effects on bond lengths and force constants of these molecules are investigated. Effects of electron correlation are shown to be affected by SOC to a large extent for some superheavy element molecules. Bond lengths and force constants with SOC-CC agree very well with those of KR-CC for most of the sixth-row element molecules. As for superheavy element molecules, SOC-CCSD is able to afford results that are in good agreement with those of KR-CCSD except for 111F, while the error of SOC-CCSD(T) is more pronounced. Large error would be encountered with SOC-CC approaches for molecules when both SOC and electron correlation effects are sizable.

I. INTRODUCTION With the successful synthesis of element 117 in 2010 by the Joint Institute for Nuclear Research,1,2 the periodic table for the first seven periods has been completed. However, it is extremely difficult to investigate chemical and physical properties of the heaviest elements experimentally because of their short lifetime and strong radiation. In addition, these heaviest elements are produced in a “one atom a time” manner and sophisticated and expensive experiments are required to obtain their properties.3 Properties of these heaviest elements are thus obtained mainly from theoretical predictions.3,4 In fact, theoretical study of these heaviest elements also provides insight on chemical trends and periodicity of elements in the periodic table. Relativistic effects5−9 play a critical role in properties of these heaviest elements, and they have been shown to be even more important than electron correlation. The Dirac−Coulomb− Breit Hamiltonian is the basis to describe relativistic effects in electronic structure calculations. Breit interaction will affect properties of valence electrons. For example, its effect on the ionization potential of Au already reaches 0.02 eV.10 On the © XXXX American Chemical Society

other hand, this term is neglected in most practical calculations.11 Wave function of the Dirac equation is a fourcomponent complex function, and calculations based on Dirac−Coulomb or Dirac−Coulomb−Breit Hamiltonian are thus rather demanding. Approximate treatments on relativistic effects are usually employed, and they include the zeroth-order regular approximation,12 the Douglas−Kroll−Hess transformation,13 the exact two-component method (X2C),6,14 etc. One of the most popular methods in dealing with relativistic effects in practical calculations is the use of relativistic effective core potentials (RECPs).15 Relativistic effects can be divided into scalar relativistic effect and spin−orbit coupling. Computational effort of scalar relativistic calculations is similar to that of nonrelativistic calculations when the untransformed twoelectron operator is used, while it is usually much more involved to calculate spin−orbit coupling (SOC) effects. Received: December 7, 2015 Revised: February 1, 2016

A

DOI: 10.1021/acs.jpca.5b11948 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

and harmonic frequency of 117H50 with this SOC-CC approach agree very well with those of KR-CC. Furthermore, a study by Kim et al. on monohydrides of 6p-block elements60 also indicates that the SOC-CC approach provides total energies and bond lengths that are in good agreement with those of the KR-CC method. To further investigate the performance of the SOC-CC approach for cases with even stronger SOC effects, they scaled the SOC operator of the 6pblock elements by a factor of 3, and good agreement with KRCC can still be achieved. High accuracy of this SOC-CC approach in describing SOC effects stems from the fact that T1 amplitudes can account for orbital relaxation effectively. One would thus expect other SOC-CC approaches, such as SOCCCS or SOC-CC2,61,62 can also describe SOC effects reliably. In this work, we propose to evaluate performance of these SOC-CC approaches in treating SOC effects of a series closedshell superheavy element monohydrides and monofluorides. Results on sixth-row element molecules will also be given for comparison. Relativistic effects will be accounted for using the most recently developed RECPs for superheavy elements which model the Dirac−Coulomb−Breit Hamiltonian.63 Average error of the RECPs for a set of closed-shell monohydrides and monofluorides at MP2 level is reported to be 0.003 Å on bond lengths and 2.86 N/m on force constants compared with results based on Dirac−Coulomb−Breit Hamiltonian.64 CCSD(T) results reported in this work will thus provide highly accurate estimates on properties of these superheavy element compounds. Furthermore, spin−orbit coupling effects on bond lengths and force constants of these closed-shell molecules will also be investigated. This paper is organized in the following manner: basic theory and computational details are given in section II. Results of SOC-CC and KR-CC methods as well as discussion of these results are provided in section III, and conclusions are given in section IV.

Both relativistic effects and electron correlation need to be considered to achieve reliable prediction of properties of superheavy elements. Moreover, relativistic effects and electron correlation are nonadditive and they should be treated simultaneously. Density functional theory (DFT)16−18 is presently the most popular approach in treating electron correlation because of its compromise between accuracy and efficiency. Properties of superheavy elements have been studied extensively with relativistic DFT.11,19,20 However, accuracy of the achieved results depends on the chosen exchangecorrelation functional, and it is not easy to improve results systematically. Theoretical calculations on superheavy elements using methods such as configuration interaction (CI),21 multireference CI,22,23 and MP2 for electron correlation have also been reported.24−28 On the other hand, electron correlation, particularly dynamic correlation, can be described with high accuracy using coupled-cluster theory (CC).29−31 The coupled-cluster method at the singles and doubles level (CCSD)32 augmented by a perturbative treatment of triple excitations (CCSD(T))33 is nowadays the “gold standard” of quantum chemistry. Theoretical studies of the heaviest elements based on relativistic CC method have also been published.11,25−28,34−39 In relativistic CC calculations with SOC effects, SOC is usually included in self-consistent field (SCF) calculations, and CC calculations are carried out based on the obtained spinors.40−45 Kramers restriction on the spinors is usually imposed in these SCF and CC calculations, and such approaches will be denoted as KR-HF and KR-CC, respectively. Computational effort of KR-CCSD is shown to be 32 times more expensive than that of scalar relativistic or nonrelativistic CCSD for systems with a 2-fold symmetry element.40 It is worth noting that calculating SOC effects perturbatively based on equation-of-motion CC (EOM-CC) formalism has also been reported.46−49 However, the perturbative approach cannot be applied to systems with strong SOC effects, such as the heaviest elements. Recently, we developed a coupled-cluster approach with SOC at the CCSD and CCSD(T) levels (SOC-CC), in which SOC is included only in solving the CC equations.50,51 A similar approach has been pioneered by Chang and Pitzer52 in CI calculations and proposed for CC approach by Eliav et al.53 It is applicable to relativistic Hamiltonians where SOC operator can be separated out and represented by a one-electron operator. The SOC-CC method is rather efficient because of the use of real molecular spin−orbitals, real two-electron integrals, time-reversal symmetry, and single point group symmetry. Calculations show that a SOC-CCSD calculation for a closed-shell molecule is about 10−12 times more expensive than that of a spin-adapted scalar relativistic or nonrelativistic CCSD calculation regardless of symmetry of the investigated molecule.50,51 Besides total energy, analytical first- 54 and second-order derivatives55 have also been implemented for SOC-CCSD and SOC-CCSD(T) using RECPs. This SOC-CC approach is applied mainly to closedshell systems, and a convergence problem rises up for openshell systems with spatial degeneracy. Alternatively, open-shell states are calculated with equation-of-motion coupled-cluster theory for excited states,56 ionized states,57 and electronattached states,58 as well as double-ionized states59 based on SOC-CCSD. SOC is neglected in calculating molecular orbitals in SOCCC. However, our previous calculations show that bond length

II. BASIC THEORY AND COMPUTATIONAL DETAILS Basic equations used in KR-CC and SOC-CC are the same as those in nonrelativistic CC calculations, except that the Fock matrix is nondiagonal in SOC-CC. The wave function is represented by eT|Φ0⟩ in CC calculations, where Φ0 is the reference determinant and the cluster operator T is defined as the following in CCSD:65 T=

∑ tiaaa+ai + i,a

1 4

∑ a,b,i,j

tijabaa+aiab+aj (1)

tai and tab ij in eq 1 are cluster amplitudes; indices i, j, k,... represent occupied orbitals in the reference determinant; and a, b, c,... denote indices of virtual orbitals. Cluster amplitudes are determined based on the following equations in CCSD:

⟨Φia|e−T HeT |Φ0⟩ = 0

(2)

⟨Φijab|e−T HeT |Φ0⟩ = 0

(3)

where Φai and Φab ij are singly and doubly excited determinants with respect to the reference wave function Φ0, respectively. Computational effort in solving CCSD equations scales as N6, where N represents system size. Total energy in CCSD is thus B


Article

The Journal of Physical Chemistry A ECCSD = ⟨Φ0|e−T HeT |Φ0⟩ = Eref +

CC2 is an approximation to CCSD based on perturbation theory. The Hamiltonian is partitioned into Fock operator and a perturbation U: H = F+U, and F is treated as zeroth order, while U is treated as the first order. T1 amplitudes are set to zeroth order, and T2 amplitudes are approximated to be correct through first order in CC2. Total energy in CC2 is thus correct to second order in U. Total energy and singles equation in CC2 are the same as those in CCSD, while the doubles equation reads:61

∑ tiafia i,a

1 + 4

∑ ∑ (tijab + tiat jb − tibt ja)⟨ij||ab⟩ ij

(4)

ab

where Eref is energy of the reference wave function and f ia is the Fock matrix element. One can readily see that the second term on the right-hand side of eq 4 will be zero if the Hartree−Fock wave function of the corresponding Hamiltonian is chosen as the reference wave function, as in the case of KR-CCSD. On the other hand, Eref will be equal to the Hartree−Fock energy of the corresponding scalar-relativistic Hamiltonian and the second term on the right-hand side of eq 4 comes from SOC in the case of SOC-CCSD. Things are somewhat more complicated in calculating energy correction due to triples in SOC-CCSD(T). To avoid an iterative solution for triple equations which scale as N7, or to refrain from using complex spinors, the occupied−occupied and virtual−virtual blocks of the SOC matrix are neglected in SOC-CCSD(T).50 Energy correction due to triples is calculated with the following equations:66 1 36

ΔE T = +

1 4

+ ⟨Φijab|exp( −T1)H exp(T1)|Φ0⟩ = 0

ijk

abc

(5)

abc

abc abc ab Dijk tijk = P(abc)P(ijk)[∑ tijae⟨bc||ek⟩−∑ tim ⟨mc||jk⟩] e

m

(6) abc abc Dijk t ijk ̅

=

P(abc)P(ijk)tia⟨bc||jk⟩

(7)

tia = (t ia̅ ̅ )*,

where the cyclic permutation operator is given by P(pqr )f (p , q , r ) = f (p , q , r ) + f (q , r , p) + f (r , p , q)

tijab = (t ia̅j̅ b̅ ̅ )*,

(8)

t ia̅ = −(tia ̅ )* tija ̅ b ̅ = (t iab̅j ̅ )*,

tijab ̅ = −(t ia̅j̅ b̅ )*,

and the orbital-energy denominator is defined as abc Dijk = fii + f jj + fkk − faa − fbb − fcc

(10)

Computational effort in solving CC2 equations scales as N5, and CC2 results are rather close to those of MP2 except that results of CC2 may deteriorate for some systems with strong correlation.68,69 The fact that the singles equation is not approximated at all in CC2 renders its ability to describe SOC with high accuracy in SOC-CC2 formalism. Performance of SOC-CC2 will be evaluated by comparing its results on the heaviest elements with those of KR-MP2. In implementation of the SOC-CC methods, time-reversal symmetry and spatial symmetry are exploited to reduce computational effort.50,51 We are dealing with closed-shell molecules, and their wave functions can always be chosen to be invariant with respect to time reverse. The reference wave function employed in SOC-CC is the Hartree−Fock wave function of the corresponding scalar relativistic Hamiltonian and is thus also time reversible. This indicates that the cluster amplitude T is commutable with the time-reversal operator. Different spin cases of the cluster amplitudes are thus related by the following equations:50

∑ ∑ tijkabc[(tijkabc)* + ( t ijk̅ abc)*]Dijkabc

∑ ∑ tijkabcfkc (tijab)* ijk

⟨Φijab|[F , T2] + [[F , T1], T2]|Φ0⟩

(11)

tijab̅ = −(t ia̅j̅ b ̅ )*,

tijab̅ ̅ = (t ia̅j̅ b)*

(12)

where indices with a bar are β spin orbitals and indices without a bar denote α spin orbitals. These relations are employed to calculate certain spin cases of cluster amplitudes with CC equations, while the other spin cases of cluster amplitudes are determined either with eqs 11 and 12 or with permutational symmetry. It should be noted that similar relations also hold in KR-CC calculations,40 where ψp and ψp̅ are a Kramers pair with ψp̅ = Kψp and K is the time-reversal operator. To make use of spatial symmetry in KR-CC calculations, double point group symmetry has to be exploited because the molecular spinors transform according to Fermion irreducible representations (IR) of the double point group. On the other hand, molecular orbitals in SOC-CC calculations are classified based on IRs of the single point group. Wave functions of systems with evennumber electrons transform according to boson IRs of the corresponding double point group, which are also IRs of the single point group. This enables one to exploit single point group symmetry, or only boson IRs of the double point group in the implementation. Details of exploitation of spatial symmetry in SOC-CC calculations can be found in ref 51. Relativistic effects for the superheavy elements are treated with the latest published RECPs developed by the Stuttgart− Cologne group.63,64 The 92 inner shell electrons with n ≤ 5 are included in the pseudopotential core. Two-electron Breit

(9)

The last term in eq 5 is relevant only in SOC-CCSD(T) calculations, and similar to the second term on the right-hand side of eq 4, it also comes directly from the SOC matrix. Neglecting the occupied−occupied and virtual−virtual blocks of the SOC matrix in calculating triples would not result in large error when SOC effects are not strong, whereas this may not be the case for the heaviest elements. CCS equations are obtained by simply setting the doubleexcitation amplitudes in eqs 1−4 to zero, and computational effort of CCS scales as N4 if the integral transformation step is not considered. The CCS method is equivalent to the Hartree− Fock method if the Hartree−Fock wave function is adopted as the reference wave function Φ0. On the other hand, the SOCCC method is able to describe SOC effects reliably because T1 amplitudes can account for orbital relaxation due to SOC effectively. One would thus expect SOC-CCS to afford results that closely resemble those with KR-HF. This is has been confirmed for a set of closed-shell 6p-block element diatomic molecules,60,62 and its performance on the heaviest elements will be investigated in the present work. It is worth noting that it is possible to fully recover KR-HF results with properly chosen T1 amplitudes using the CCS wave function.67 C


D

a

0.000 0.000

1.515 (−0.001) 1.512 (−0.001)

−0.54 −0.28

321.69 (2.47)

317.58 (2.73)

361.35 (1.45)

260.05 (4.92)

321.97 (2.75)

387.76 (13.47) 318.12 (3.27)

259.56 (4.43)

255.13 359.90 374.29 314.85 319.22

ke

0.000 0.000

1.600 (−0.021) 1.537 (−0.016) 1.562 (−0.019) 1.562 (−0.019) 1.600 (−0.021) 1.543 (−0.016) 1.562 (−0.019) 1.562 (−0.019)

1.621 1.559 1.553 1.581 1.581

re

ke

0.26 0.27

279.95 (33.57) 323.34 (35.89) 297.15 (37.10) 293.19 (37.58) 279.70 (33.32) 319.22 (36.69) 297.41 (37.36) 293.46 (37.84)

246.38 282.53 287.45 260.05 255.62

HgH+

0.000 0.001

1.867 (−0.028) 1.846 (−0.026) 1.863 (−0.024) 1.865 (−0.024) 1.866 (−0.029) 1.847 (−0.026) 1.863 (−0.024) 1.866 (−0.023)

1.895 1.873 1.872 1.887 1.889

re

ke Scalar Relativistic 118.73 122.10 121.76 116.24 114.75 Spin−Orbit 123.63 (4.90) 124.31 (2.55) 118.40 (2.16) 116.24 (1.49) 123.97 (5.24) 124.31 (2.21) 118.23 (1.99) 116.07 (1.32) Err. −0.17 −0.17

TIH

0.000 0.002

1.789 (−0.021) 1.785 (−0.017) 1.800 (−0.014) 1.805 (−0.013) 1.788 (−0.022) 1.786 (−0.018) 1.800 (−0.014) 1.807 (−0.011)

1.810 1.804 1.802 1.814 1.818

re

ke

−0.81 −2.17

164.18 (−11.60)

172.54 (−7.54)

184.21 (−5.03)

196.70 (0.64)

166.35 (−9.43)

173.35 (−6.73)

186.09 (−4.00)

197.13 (1.07)

196.06 189.24 190.09 180.08 175.78

PbH+

−0.001 0.000

1.709 (0.026) 1.699 (0.023) 1.713 (0.027) 1.716 (0.026) 1.709 (0.026) 1.702 (0.024) 1.712 (0.026) 1.716 (0.026)

1.683 1.678 1.676 1.686 1.690

re

ke

0.72 −0.23

266.52 (−40.22) 263.03 (−33.59) 244.94 (−37.84) 236.61 (−38.74) 265.02 (−41.72) 258.81 (−34.65) 245.66 (−37.12) 236.38 (−38.97)

306.74 293.46 296.62 282.78 275.35

AtH

Effects of SOC are listed in parentheses. Error of SOC-CCSD and SOC-CCSD(T) with respect to KR-CCSD and KR-CCSD(T), respectively, are also listed.

SOCCSD SOCCSD(T)

SOC-CCSD(T)

SOC-CCSD

SOC-CC2

SOC-CCS

KR-CCSD(T)

KR-CCSD

1.515 (−0.001) 1.512 (−0.001) 1.569 (−0.003) 1.494 (0.000)

1.569 (−0.003) 1.512 (0.029)

KR-HF

KR-MP2

1.572 1.494 1.483 1.516 1.513

HF CC2 MP2 CCSD CCSD(T)

re

AuH

Table 1. Bond Lengths (in Å) and Force Constants (ke, in N·m−1) for Closed-Shell Monohydrides of Au, Hg+, TI, Pb+, At, and Rn+a

0.000 0.000

1.688 (0.016) 1.683 (0.015) 1.691 (0.017) 1.696 (0.022) 1.689 (0.017) 1.684 (0.015) 1.691 (0.017) 1.696 (0.022)

1.672 1.669 1.668 1.674 1.674

re

ke

0.97 −0.95

274.85 (−31.08) 261.54 (−30.08) 251.47 (−33.90) 243.50 (−43.13) 274.08 (−31.85) 259.56 (−30.75) 252.44 (−32.93) 242.55 (−43.08)

305.93 290.31 291.62 285.37 285.63

RnH+

The Journal of Physical Chemistry A Article


E

a

0.47 −0.98

326.62 (9.83)

332.16 (4.84)

327.23 (8.65)

281.93 (5.17)

327.60 (4.66)

287.96 (11.20) 361.18 (11.84) 331.69 (4.37)

276.76 318.58 349.34 327.32 316.79

ke

0.000 0.001

1.898 (−0.011) 1.881 (−0.012) 1.892 (−0.012) 1.902 (−0.014) 1.898 (−0.011) 1.903 (−0.005) 1.892 (−0.012) 1.903 (−0.013)

1.909 1.908 1.893 1.904 1.916

re

ke

−0.79 −0.60

381.92 (25.11)

370.38 (−23.15) 423.45 (23.49)

463.77 (21.22)

382.52 (25.71)

424.24 (24.28)

431.57 (22.72)

464.39 (21.84)

442.55 393.53 408.85 399.96 356.81

HgF+

0.000 0.001

2.083 (−0.007) 2.088 (−0.005)

2.079 (−0.012) 2.090 (−0.006) 2.083 (−0.007) 2.087 (−0.006) 2.079 (−0.012) 2.100 (0.009)

2.091 2.091 2.096 2.090 2.093

re

ke

Err. −0.13 0.03

232.31 (2.40)

219.04 (−17.27) 234.74 (2.71)

234.76 (9.75)

232.28 (2.37)

234.87 (2.84)

228.34 (2.49)

Scalar Relativistic 225.01 236.31 225.85 232.03 229.91 Spin−Orbit 234.83 (9.82)

TIF

0.000 0.001

1.957 (−0.004) 1.965 (−0.003)

1.946 (−0.006) 1.964 (−0.005) 1.957 (−0.004) 1.964 (−0.004) 1.946 (−0.006) 1.986 (0.021)

1.952 1.965 1.969 1.961 1.968

re

ke

−1.12 −0.69

405.23 (−1.12)

351.18 (−74.46) 421.09 (0.92)

445.08 (5.79)

405.92 (−0.43)

422.21 (2.04)

407.15 (0.52)

445.64 (6.35)

439.29 425.64 406.63 420.17 406.35

PbF+

0.002 0.003

2.011 (0.050) 2.041 (0.050) 2.029 (0.046) 2.042 (0.059) 2.018 (0.057) 2.055 (0.069) 2.031 (0.048) 2.045 (0.062)

1.961 1.986 1.991 1.983 1.983

re

ke

−3.90 −7.77

331.63 (−95.47) 297.50 (−135.53) 307.32 (−74.27) 286.15 (−95.44) 318.17 (−108.93) 269.19 (−112.68) 303.42 (−78.17) 278.38 (−103.21)

427.10 381.87 433.03 381.59 381.59

AtF


SOCCSD SOCCSD(T)

SOC-CCSD(T)

SOC-CCSD

SOC-CC2

SOC-CCS

KR-CCSD(T)

KR-CCSD

0.000 0.000

1.965 (−0.008) 1.888 (−0.009) 1.916 (−0.008) 1.915 (−0.003) 1.964 (−0.009) 1.910 (−0.019) 1.916 (−0.008) 1.915 (−0.009)

KR-HF

KR-MP2

1.973 1.929 1.897 1.924 1.924


re

AuF

Table 2. Bond Lengths (in Å) and Force Constants (ke, in N·m−1) for Closed-Shell Monofluorides of Au, Hg+, TI, Pb+, At, and Rn+a

−0.001 0.003

1.923 (0.027) 1.966 (0.030) 1.955 (0.027) 1.971 (0.045) 1.925 (0.029) 1.975 (0.042) 1.956 (0.029) 1.974 (0.047)

1.896 1.933 1.936 1.927 1.927

re

ke

−8.09 −7.51

481.73 (−130.88) 367.74 (−94.97) 387.95 (−142.44) 339.35 (−191.04) 474.80 (−137.82) 353.58 (−80.75) 379.86 (−150.53) 331.84 (−198.55)

612.62 434.33 462.71 530.39 530.39

RnF+



F

a

−0.32 −0.95

448.57 (−24.35) 473.60 (−22.27) 437.27 (−28.68) 428.02 (−28.70) 444.03 (−28.89) 448.69 (−27.91) 436.95 (−29.00) 427.07 (−29.65)

472.92 476.60 495.87 465.95 456.72

ke

0.000 0.000

1.528 (0.001)

1.525 (0.000)

1.528 (−0.004) 1.525 (0.002)

1.525 (−0.000) 1.528 (0.001)

1.528 (−0.004) 1.519 (0.001)

1.532 1.523 1.518 1.525 1.527

re

ke

−0.61 −0.30

397.32 (−5.69) 399.02 (−3.38) 390.79 (−4.56)

407.01 (−4.36) 399.63 (−2.77) 391.09 (−4.26) 414.48 (7.15)

413.86 (6.53)

407.33 403.01 411.37 402.40 395.35

112H+

−0.005 0.006

1.716 (−0.279) 1.734 (−0.219) 1.745 (−0.235) 1.756 (−0.224) 1.657 (−0.338) 1.721 (−0.231) 1.740 (−0.160) 1.762 (−0.218)

1.995 1.952 1.953 1.980 1.980

re

ke

re

Scalar Relativistic 95.59 1.903 107.32 1.887 107.16 1.886 97.09 1.903 96.64 1.907 Spin−Orbit 150.34 1.722 (54.75) (−0.181) 141.44 1.748 (34.28) (−0.138) 131.07 1.761 (33.98) (−0.142) 122.44 1.781 (25.80) (−0.126) 179.48 1.696 (83.89) (−0.207) 143.94 1.747 (36.62) (−0.140) 130.19 1.766 (33.10) (−0.137) 113.61 1.873 (16.97) (−0.034) Err. −0.88 0.005 −8.83 0.092

113H ke

−7.40 −82.59

160.16 (−10.37) 141.81 (−17.89) 48.48 (−107.17)

149.21 (−10.49) 131.07 (−24.58) 190.98 (20.25)

166.06 (−6.28)

196.70 (25.97)

170.73 170.53 172.34 159.70 155.65

114H+

−0.002 0.009

1.965 (0.179) 1.907 (0.136) 1.946 (0.163) 1.950 (0.163) 1.983 (0.197) 1.919 (0.148) 1.944 (0.161) 1.959 (0.172)

1.786 1.771 1.771 1.783 1.787

re

ke

0.55 −8.89

147.35 (−115.92) 165.84 (−100.43) 143.83 (−106.42) 138.73 (−105.49) 135.32 (−127.95) 158.21 (−104.82) 144.38 (−105.87) 129.84 (−114.38)

263.27 263.03 266.27 250.25 244.22

117H


SOCCSD SOCCSD(T)

SOCCCSD(T)

SOC-CCSD

SOC-CC2

SOC-CCS

KR-CCSD(T)

KR-CCSD

0.000 0.000

1.519 (0.008) 1.500 (0.007) 1.513 (0.009) 1.515 (0.009) 1.521 (0.010) 1.508 (0.008) 1.513 (0.009) 1.515 (0.009)

KR-HF

KR-MP2

1.511 1.500 1.493 1.504 1.506


re

111H

Table 3. Bond Lengths (in Å) and Force Constants (ke, in N·m−1) for Closed-Shell Monohydrides of 111, 112+, 113, 114+, 117, and 118+a

−0.002 0.011

1.901 (0.131) 1.887 (0.124) 1.904 (0.134) 1.913 (0.139) 1.878 (0.108) 1.929 (0.168) 1.902 (0.132) 1.924 (0.150)

1.770 1.761 1.763 1.770 1.774

re

ke

1.61 −13.38

148.65 (−104.03)

173.15 (−88.45)

183.46 (−83.56)

186.88 (−88.47)

162.03 (−90.65)

171.54 (−88.26)

190.50 (−77.52)

193.49 (−81.86)

275.35 267.02 268.02 259.80 252.68

118H+



G

a

1.827 (−0.014) 1.845 (−0.009) 1.851 (−0.008) 1.865 (−0.007) 1.826 (−0.015) 1.859 (−0.007) 1.852 (−0.007) 1.870 (−0.002) 0.001 0.005

−23.36 −90.47

1.841 1.866 1.854 1.859 1.872

re

539.43 (−96.12) 498.45 (−90.82) 485.44 (−82.01) 445.39 (−78.44) 540.22 (−95.33) 413.20 (−141.17) 462.08 (−105.37) 354.92 (−168.91)

635.55 554.37 589.27 567.45 523.83

ke

ke

−1.21 −35.97

455.71 (−33.31)

543.49 (3.16)

532.27 (1.11)

684.93 (48.40)

491.68 (2.66)

544.70 (4.37)

570.04 (11.55)

690.74 (54.21)

636.53 531.16 558.49 540.33 489.02

112F+

0.001 0.001

2.180 (−0.062) 2.187 (−0.037) 2.184 (−0.045) 2.189 (−0.053) 2.179 (−0.063) 2.204 (−0.031) 2.185 (−0.044) 2.190 (−0.052)

2.242 2.235 2.224 2.229 2.242

re

202.89 (37.84) 201.82 (22.66) Err. −0.32 −0.18 0.002 0.012

2.063 (−0.020) 2.089 (0.000)

2.037 (−0.046) 2.076 (−0.008) 2.061 (−0.022) 2.077 (−0.012) 2.035 (−0.048) 2.134 (0.047)

2.083 2.087 2.084 2.083 2.089

re

ke Scalar Relativistic 176.68 186.69 195.26 165.05 179.16 Spin−Orbit 197.21 (20.53) 211.08 (15.82) 203.21 (38.16) 202.00 (22.84) 193.55 (16.87) 187.83 (1.14)

113F ke

−6.37 −37.08

194.49 (−168.46) 337.74 (−27.22) 273.66 (−82.10)

327.33 (−37.26) 344.11 (−20.85) 310.74 (−45.02) 379.99 (14.00)

380.29 (14.30)

365.99 362.95 364.59 364.96 355.76

114F+

0.001 0.006

2.216 (0.139) 2.207 (0.116) 2.208 (0.120) 2.214 (0.117) 2.235 (0.158) 2.219 (0.124) 2.208 (0.120) 2.220 (0.123)

2.077 2.095 2.091 2.088 2.097

re

ke

−0.61 −7.61

242.12 (−140.46) 247.97 (−102.02) 244.60 (−110.32) 237.26 (−97.70) 222.12 (−160.46) 227.89 (−120.75) 243.99 (−110.93) 229.65 (−105.31)

382.58 348.64 349.99 354.92 334.96

117F


SOCCSD SOCCSD(T)

SOC-CCSD(T)

SOC-CCSD

SOC-CC2

SOC-CCS

KR-CCSD(T)

KR-CCSD

0.005 0.019

1.874 (0.017) 1.881 (0.022) 1.887 (0.020) 1.898 (0.021) 1.878 (0.021) 1.902 (0.030) 1.892 (0.025) 1.917 (0.040)

KR-HF

KR-MP2

1.857 1.872 1.859 1.867 1.877


re

111F

Table 4. Bond Lengths (in Å) and Force Constants (ke, in N·m−1) for Closed-Shell Monofluorides of 111, 112+, 113, 114+, 117, and 118+a

0.001 0.017

2.091 (0.086) 2.104 (0.068) 2.102 (0.074) 2.102 (0.060) 2.096 (0.091) 2.118 (0.086) 2.103 (0.075) 2.119 (0.077)

2.005 2.032 2.036 2.028 2.042

re

ke

−4.63 −40.75

401.72 (−93.64) 356.33 (−49.59) 363.84 (−54.81) 363.84 (−16.40) 402.98 (−92.38) 335.74 (−65.66) 359.21 (−59.44) 323.09 (−57.15)

495.36 401.40 405.92 418.65 380.24

118F+



Article

The Journal of Physical Chemistry A interaction, finite nucleus effects, and leading order quantum electrodynamics effects are taken into consideration in adjusting parameters in RECPs.63,64 Gaunt interaction has been shown to increase bond length by 0.008 Å and decreases force constant by 5 N/m for 113H. This means error in neglecting Breit interaction may be even larger than error of RECPs in describing relativistic effects for some molecules. The valence quadruple-ζ basis set developed together with the RECPs suitable for calculations of SOC, i.e., (12s11p9d3f2g)/ [5s7p5d3f2g] for elements 111 and 112 and (12s11p9d3f2g)/ [6s7p5d3f2g] basis set for elements 113−118,64 is adopted. On the other hand, the basis set suitable for scalar relativistic calculations for these superheavy elements64 is employed when SOC is not considered. As for the sixth-row elements, the ECP60MDF pseudopotentials developed by the same groups70−72 are used, and the def2-QZVP basis set with extension for two-component calculations73 in order to describe split of (n − 1)p orbitals is employed. Uncontracted aug-cc-pVQZ basis for H and aug-cc-pVQZ basis74 for F are adopted. KR-MP2 and KR-CC calculations are performed with the DIRAC program package.75 KR-HF and SOC-CC as well as scalar relativistic calculations are carried out with a locally modified CFOUR program package.76 All electrons not treated via ECPs are correlated in calculations. Equilibrium bond lengths and force constants for these closed-shell molecules are calculated by fitting total energies at seven points with a space of 0.05 Å around the equilibrium bond length with a polynomial function up to fourth order in KR-MP2, KR-CC, and SOC-CC2 calculations. For some molecules with large force constants, seven points with a space of 0.02 Å are used. On the other hand, results with KR-HF, SOC-CCS, SOCCCSD, and SOC-CCSD(T) are obtained based on available analytical energy gradient.

of HgH+ is probably due to participation of 6p1/2 spinor in chemical bonding. Reduction in bond length of AuF by SOC is somewhat larger than that of AuH, while the SOC effect on bond length and force constant of HgF+ is slightly smaller than that of HgH+. On the other hand, electron correlation reduces bond lengths of AuH and AuF by about 0.05−0.06 Å and increases force constants of these two molecules accordingly by about 40−60 N/m. However, increase in the force constant of HgH+ by electron correlation is rather moderate although bond length is reduced by 0.04 Å. The force constant of HgF+ at the CCSD(T) level is about 80 N/m smaller that that at the HF level while bond length at the KR-CCSD(T) level is only 0.004 Å longer than that at the KR-HF level. Contrary to AuH and AuF, SOC increases bond lengths and decreases force constants of 111H and 111F. This may be due to contribution of 6d5/2 spinor on bonding. Furthermore, the SOC effect on bond length and force constant in 111F is larger than that in 111H, similar to the case in AuH and AuF. On the other hand, SOC effects on 112H+ and 112F+ are quite small, possibly due to a cancellation of effects between 7p1/2 spinor and 6d5/2 spinor. Electron correlation effect on bond lengths and force constants of 111H and 112H+ is rather small, while it increases bond lengths by about 0.02 and 0.04 Å and decreases force constants by 95 N/m and 200 N/m for 111F and 112F+, respectively, when SOC is present. Bond length of 111H is almost the same as that of AuH, and 111F has a shorter bond length than AuF at KR-CCSD(T) level, while bond lengths of 112H+ and 112F+ are about 0.04 Å shorter than those of HgH+ and HgF+, respectively. Force constants of 111H, 111F, 112H+, and 112F+ are much larger than those of their corresponding sixth-row molecules. In fact, these four molecules have the largest force constant among the superheavy molecules investigated in this work. Furthermore, 111H has a slightly shorter bond length and a larger force constant than 112H+, while 111F has a longer bond length and smaller force constant than 112F+. Bond lengths of TlH and PbH+ are reduced by SOC because of participation of 6p1/2 spinor on chemical bonding. Reduction in bond lengths by SOC is smaller for TlF and PbF+ compared with TlH and PbH+. This could be because fluoride has a larger electron affinity, and electron population of 6p1/2 will thus be diminished in monofluorides compared with that in monohydrides. Alternatively, this can also be explained by a more ionic nature of monofluorides.3 Moreover, SOC effects on bond lengths of TlH and TlF are somewhat larger than those of PbH+ and PbF+. In fact, reduction in bond length of PbH+ and PbF+ by SOC is even smaller than that of HgH+ and HgF+, respectively. Force constants of TlH, TlF, PbH+, and PbF+ are not affected much by SOC. SOC effects on bond lengths of 111H and 112H+ are −0.22 Å and −0.13 Å, respectively, at the KR-CCSD(T) level, and this large effect originates from sizable contraction of 7p1/2 spinor. In fact, change in bond length due to SOC is the largest for 111H among all the molecules studied here. Reduction in bond length by SOC is more remarkable for 111H than for 112H+. Similar to the cases of Tl and Pb+, the effect of SOC on bond length for 111F and 112F+ is much smaller than that for 111H and 112H+ because of a more ionic nature of the monofluorides. Correlation effects on bond lengths and force constants of 113 and 114 monohydrides and monofluorides are minor at the scalar relativistic level. On the other hand, electron correlation increases bond lengths of 113H, 114H+, and 114F+ by about 0.04−0.06 Å when SOC is present. This indicates that electron correlation is affected by SOC pronouncedly. Electron correlation and SOC should be

III. RESULTS AND DISCUSSION Bond lengths and force constants for AuH, HgH+, TlH, PbH+, AtH, and RnH+ as well as corresponding monofluorides with KR-HF, KR-MP2, KR-CCSD, KR-CCSD(T), and various SOC-CC approaches are listed in Tables 1 and 2, together with results from scalar relativistic calculations for comparison. Results for superheavy element monohydrides and monofluorides are listed in Tables 3 and 4, respectively. Data in parentheses in these tables are effects of SOC on bond lengths and force constants. Errors of SOC-CCSD and SOC-CCSD(T) with respect to KR-CCSD and KR-CCSD(T), respectively, are also present. One can see from Tables 3 and 4 that results of KR-HF and KR-MP2 are almost the same as those in ref 64, in which the same ECPs and the same basis functions for superheavy elements are used. Results on monohydrides of KRCCSD and KR-CCSD(T) also agree well with those CC methods based on the Dirac−Coulomb Hamiltonian.11 This demonstrates the reliability of the present KR-HF, KR-MP2, and KR-CC calculations. A. KR-CC Results. We will first address effects of spin−orbit coupling on bond lengths and force constants of these closedshell molecules by comparing scalar relativistic CCSD(T) and KR-CCSD(T) results. Effects of electron correlation will also be estimated from the difference between HF and CCSD(T) results. SOC effects on AuH are rather small as one would expect because contribution to bonding mainly comes from 6s orbitals of Au. On the other hand, reduction in bond length and increase in force constant for HgH+ by SOC is more pronounced than that for AuH. More significant SOC effect H


Article


difference between SOC-CCSD(T) and KR-CCSD(T) is not necessarily larger than that between SOC-CCSD and KRCCSD. It is just that changes in energy difference with respect to bond lengths are more significant at the CCSD(T) level than those at the CCSD level. Agreement in bond lengths or force constants between SOC-CCSD and KR-CCSD or between SOC-CCSD(T) and KR-CCSD(T) depends on the change in their energy difference with respect to bond lengths. One can see from Tables 1 and 2 that results of SOC-CCS and SOC-CCSD agree very well with those of KR-HF and KRCCSD, respectively, for sixth-row element molecules. Difference in bond lengths between SOC-CCSD and KR-CCSD is at most 0.002 Å. SOC-CCSD(T) also provides results that are in rather good agreement with those of KR-CCSD(T) for these molecules, although their error is slightly larger than that of SOC-CCSD. On the other hand, error on force constants of AtF and RnF+ with SOC-CCSD and SOC-CCSD(T) is a little bit more significant. This is probably because both SOC effects and electron correlation are quite pronounced for these two molecules. In fact, SOC effects on bond lengths and force constants of these two molecules are the most remarkable among the investigated sixth-row closed-shell molecules. As for the results of SOC-CC2 and KR-MP2, their difference is around 0.001−0.006 Å for monohydrides except for AuH, while their difference reaches 0.009−0.028 Å for monofluorides. This is consistent with the fact that CC2 is more sensitive to electron correlation, as one can see from these tables that electron correlation is more pronounced for monofluorides than that for monohydrides. As for superheavy element molecules, agreement between results of SOC-CCS and KR-HF is related to the magnitude of SOC effects. Their difference on bond length reaches 0.059 Å for 113H where SOC effects reduce bond length by more than 0.2 Å, while it is only 0.001 Å for 113F where the SOC effect on bond length is more moderate. Agreement between bond lengths of SOC-CC2 and those of KR-MP2 is rather good for supherheavy element monohydrides except for 118H+. On the other hand, their difference is more pronounced for monofluorides. Results of SOC-CCSD agree well with those of KR-CCSD in most cases, although SOC effects are pronounced for these superheavy element compounds. Error in bond lengths with SOC-CCSD compared with KR-CCSD is smaller than 0.005 Å. Difference in force constants between SOC-CCSD and KR-CCSD reaches 6−8 N/m for 114H+, 114F+, and 118F+, while SOC-CCSD underestimates force constant of 111F by 23 N/m compared with KR-CCSD result. On the other hand, bond lengths with SOC-CCSD(T) are within 0.02 Å from those of KR-CCSD(T) in most cases, except for 114H+, where their difference reaches 0.09 Å. In fact, if the occupied−virtual block of SOC is also omitted in energy correction due to triples, i.e., neglecting the last term of eq 5, SOC-CCSD(T) results will be even worse. This demonstrates that SOC effects on triple excitation amplitudes are rather important for these superheavy element molecules. Force constants with SOC-CCSD(T) are underestimated by 83 N/m for 114H+. This large deviation is again related to both large SOC effects and electron correlation effects of this molecule. SOC-CCSD(T) also underestimates force constants of superheavy element monofluorides by about 40 N/m for 112F+, 114F+, and 118F+, possibly due to more pronounced electron correlation effects on force constants. On the other hand, the force constant for 111F with SOC-CCSD(T) is 90 N/m smaller than that with KR-CCSD(T). Both SOC-CCSD and

calculated at the same time to achieve a reliable description of these molecules. Reduction in bond length is usually accompanied by an increase in force constant. SOC decreases bond lengths of Tl, Pb+, 113, and 114+ monohydrides and monofluorides. However, force constants are decreased by SOC for Pb+ and 114+ monohydrides and monofluorides at the CCSD(T) level, although they are increased by SOC at the HF level. This again shows complicated interaction between electron correlation and SOC. Force constants of these four superheavy molecules are smaller than their corresponding 6pblock molecules at scalar relativistic level, and this also holds when SOC is present except for 113H. SOC increases bond lengths of AtH and RnH+ by about 0.026 and 0.022 Å at the CCSD(T) level, respectively, which is attributed to expansion of the 6p3/2 spinor. Increase in bond lengths of AtF and RnF+ by SOC is more pronounced than that of AtH and RnH+. On the other hand, SOC increases bond length of 117H and 118H+ by 0.16 and 0.14 Å, respectively, at the KR-CCSD(T) level, while they are 0.12 and 0.06 Å for 117F and 118F+, respectively. Force constants of these molecules are reduced by SOC accordingly. Electron correlation on bond lengths of 117H and 118H+ is rather small at scalar relativistic level, while difference in bond lengths between KR-HF and KR-CCSD(T) for these two molecules is a little bit more pronounced. On the other hand, electron correlation increases bond lengths of 117F and 118F+ by 0.02 and 0.04 Å at the scalar relativistic level, while its effect is minor when SOC is present. Bond lengths of 117H, 118H+, 117F, and 118F+ are larger than their corresponding 6p-block molecules and force constants are smaller except for 118F+ when SOC is present. The force constant of 118F+ is 24 N/m larger than that of RnF+, although the bond length of 118F+ is 0.13 Å longer than that of RnF+ at the KR-CCSD(T) level. This is due to the fact that the effect of SOC on the force constant of 118F+ is much smaller than that of RnF+. B. SOC-CC Results. In SOC-CC calculations, SOC effects are taken into account in post-SCF calculations. Results of SOC-CC for superheavy element molecules become unreasonable when the inner (n − 1)s2(n − 1)p6 electrons are kept frozen, while KR-CC approach can still provide reasonable results. This means SOC effects of inner shell electrons are highly important to achieve reliable results for superheavy element molecules. The total energy of SOC-CCSD and SOCCCSD(T) is within 0.1−0.3 mHartree of KR-CCSD and KRCCSD(T) energy for sixth-row element molecules, respectively, and their difference is about 1−3 mHartree for superheavy element molecules around equilibrium bond lengths. These data are consistent with those obtained in ref 60, and these results show that SOC-CCSD and SOC-CCSD(T) are able to describe the SOC effect on energy reasonably even for superheavy element molecules. However, the difference in total energy between SOC-CCSD and KR-CCSD changes with respect to bond length. In most cases, their differences increase with bond length when SOC-CCSD energy is lower than KRCCSD energy, while differences tend to be smaller at longer bond length if SOC-CCSD energy is higher. Bond lengths with SOC-CCSD will thus be longer than those with KR-CCSD, and force constants will be consequently smaller in most cases. The same rule also holds for SOC-CCSD(T). In SOC-CCSD(T), the occupied−occupied and virtual−virtual blocks of the SOC matrix are neglected in calculating energy correction due to triples. This indicates SOC-CCSD(T) will have a larger error in describing SOC effects than SOC-CCSD. However, energy I


The Journal of Physical Chemistry A SOC-CCSD(T) underestimate the force constant of 111F considerably. It is not easy to understand this large discrepancy, and this may be related to its large force constant as well as to a significant participation of 6d5/2 spinors on bonding.

■

ACKNOWLEDGMENTS

■

REFERENCES

We thank the National Nature Science Foundation of China (Grants 21473116 and 21273155) for financial support.

IV. CONCLUSION In this work, bond lengths and force constants for a series of closed-shell sixth-row and superheavy element monohydrides and monofluorides are calculated. KR-HF, KR-MP2, and KRCC approaches with SOC included in the SCF step as well as SOC-CC methods with SOC included in th post-SCF treatment are employed in calculations. Relativistic effects for the superheavy elements are dealt with the most recently developed RECPs, where effects of two-electron Breit interaction, finite nucleus effects, and leading order quantum electrodynamics effects are considered. Results of KR-CCSD(T) provide highly accurate estimates of properties of these superheavy element molecules. Scalar relativistic calculations are also carried out to investigate SOC effects on bond lengths and force constants of these molecules. SOC effects on bond lengths of these molecules can be rationalized by contribution of p1/2 or p3/2 spinors on chemical bonding. It would be helpful to develop bond analysis methods based on spinors for superheavy element molecules to understand SOC effects on the chemical bond. Decrease in bond length is usually accompanied by an increase in the force constant. However, the force constants of 114H+ and 114F+ at the KR-CCSD(T) level are decreased by SOC although their bond lengths are also reduced by SOC. Furthermore, the SOC effect on force constants of these two molecules at the SCF level is even qualitatively opposite to that at the CCSD(T) level. Effects of electron correlation on bond lengths and force constants of 113H and 114H+ are minor at the scalar relativistic level, while they are sizable when SOC is present. On the other hand, effects of electron correlation on bond lengths and force constants of 114F+, 117F, and 118F are pronounced in scalar relativistic calculations, while they are insignificant with SOC included. These results show that SOC has a large impact on electron correlation for superheavy element molecules. Calculations with SOC-CC approach indicate that the SOC effects of inner shell electrons are critical to achieve reliable results for superheavy element molecules. SOC-CC methods provide highly accurate results for most of the sixth-row element molecules. Bond lengths and force constants with SOC-CCSD for superheavy element molecules also agree rather well with those of KR-CCSD, except for 111F. Error of SOC-CCSD(T) is larger than that of SOC-CCSD because of a further approximation in calculating energy correction due to triples. In fact, SOC effects on triples are rather important to obtain reliable results. Bond lengths with SOC-CCSD(T) for superheavy element molecules are in reasonable agreement with those of KR-CCSD(T) except for 114H+, while error on force constants with SOC-CCSD(T) is more pronounced. When both SOC effects and electron correlation effects are significant, the error of SOC-CC methods will be somewhat larger.

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*Phone: 86-15828332921. E-mail: [email protected]. Notes

The authors declare no competing financial interest. J


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