DOI: 10.2478/s11534-006-0029-7
Research article
CEJP 4(4) 2006 448–460
Towards relativistic ECP / DFT description of
chemical bonding in E112 compounds: spin-orbit and
correlation effects in E112X versus HgX (X=H, Au)
Andréi Zaitsevskii1∗ , Elena Rykova2, Nikolai S. Mosyagin3 ,
Anatoly V. Titov3
1
Chemistry Department, Moscow State University,
Moscow 119992, Russia
2
Photochemistry Center, Russian Academy of Sciences,
Moscow 117421, Russia
3
Petersburg Nuclear Physics Institute,
Gatchina, St.-Petersburg district 188300, Russia
Received 13 March 2006; accepted 5 July 2006
Abstract: The relativistic effective core potential (RECP) approach combined with the spinorbit DFT electron correlation treatment was applied to the study of the bonding of eka-mercury
(E112) and mercury with hydrogen and gold atoms. Highly accurate small-core shape-consistent
RECPs derived from Hartree–Fock–Dirac–Breit atomic calculations with Fermi nuclear model
were employed. The accuracy of the DFT correlation treatment was checked by comparing the
results in the scalar-relativistic (spin-orbit-free) limit with those of high level scalar-relativistic
correlation calculations within the same RECP model. E112H was predicted to be slightly more
stable than its lighter homologue (HgH). The E112-Au bond energy is expected to be ca. 25-30 %
weaker than that of Hg-Au. The role of correlations and magnetic (spin-dependent) interactions
in E112-X and Hg-X (X=H, Au) bonding is discussed. The present computational procedure can
be readily applied to much larger systems and seems to be a promising tool for simulating E112
adsorption on metal surfaces.
c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.
Keywords: Superheavy elements, relativistic electronic structure calculations, effective core
potentials
PACS (2006): 31.10.+z, 31.15.Ar, 31.15.Ew, 31.25.Nj, 33.15.-e, 31.30.Jv, 33.15.Fm
∗
E-mail: [email protected]
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449
Introduction
Quantitative theoretical studies of interactions of superheavy elements (SHEs) with large
molecules, clusters and solids are of primary importance for detecting SHEs and understanding their chemical properties. At present, relativistic density functional theory
(RDFT) appears to be the most appropriate tool for such studies. Several all-electron versions of a fully relativistic four-component spinor based DFT have been developed (see
[1–3] and references therein) and their applications to relatively large SHE-containing
systems have been reported [4–6]. However, the description of relativistic effects using two-component molecular (pseudo)spinors and relativistic effective core potentials
(RECP) seems to be a more promising approach. The RECP model allows one to exclude a large number of chemically inactive shells from molecular calculations and explicitly treat only valence and outermost core electrons. Furthermore, the oscillations of
valence spinors in heavy-atom cores are usually smoothed. As a result, the number of
primitive basis functions can be dramatically reduced; this is particularly important for
the evaluation of two-electron integrals and electronic densities in studying actinide and
SHE compounds. The RECP method implies that well-developed nonrelativistic calculation techniques are used; however, both scalar (spin-averaged) and spin-dependent (spinorbit etc.) relativistic effects are taken into account by means of effective one-electron
operators. Finally, in RECP/DFT calculations, the interactions with excluded core electrons can be extracted ab initio from the solutions of atomic Hartree–Dirac–Fock–Breit
(HDFB) equations with the Fermi nuclear charge distribution. In contrast, in all-electron
RDFT an approximate functional is used to simulate exchange and Breit interactions
(large for the electrons localized in the core region). Quite recently, a simple and efficient algorithm to solve the Kohn-Sham equations for fully unrestricted two-component
spinors within the RECP model (spin-orbit DFT or SO DFT) has been implemented
and successfully tested in calculations for heavy metal compounds [7, 8]. In addition
to computational efficiency, an advantage of the SO DFT technique as applied to SHE
compounds is that the spin-scalar and spin-dependent terms of the effective one-electron
equations are treated on an equal footing without a substantional increase in the computational cost. Most pseudopotential-based high-level relativistic correlation approaches
with explicit description of electron-electron interactions (e.g., see [9–11]), to the contrary,
suffer from favouring spin-independent effects when spin-restricted basis sets and spinorbit-free references are used to minimize the expenses of the correlation treatment. The
interference of spin-orbit couplings and correlation effects is properly taken into account
independently of the complexity of a system under consideration; this feature appears to
be of importance when the SO DFT method is used for computing the SO corrections to
accurate scalar-relativistic eigenenergies. These advantages may be counterbalanced by
a rough approximate description of correlations in terms of conventional simple exchange
functionals; recent numerical experiments [8] have demonstrated a strong dependence of
the SO DFT estimates of molecular properties on the employed form of the functional.
However, corresponding errors can be reduced by the appropriate choice of the funcUnauthenticated
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tional, carefully validated using the results of accurate ab initio correlation calculations
on relatively simple systems and/or at the spin-scalar level.
One of the main shortcomings of the conventional DFT formulations is related to
their formal single-determinantal nature. It is usually supposed that the 6d − 7s quasidegeneracies in atomic SHEs [12] might block the applicability of single-reference methods
to E112-containing molecules; however, in numerous cases of practical importance, the
effects of these quasidegeneracies should be associated with single substitutions in the
determinantal wavefunctions and can thus be in large part recovered within the onedeterminantal approach, provided that one-electron molecular spinors are determined
in a fully unrestricted manner and correlation effects are incorporated in effective oneelectron equations. These conditions are satisfied for the SO DFT procedure [7]. It is
also worth noting that the use of unrestricted (and eventually symmetry-broken) solutions ensures the proper description of dissociation limits (i.e. the exact coincidence of
the computed molecular energy at the infinite interatomic separation with the sum of
atomic energy estimates).
In this paper we report pilot applications of the SO DFT method within the frame
of the shape-consistent small-core RECP approximation to the description of bonding
in simple eka-mercury (E112) compounds, E112H and E112Au, discuss the accuracy of
the DFT correlation treatment and compare the results with those for analogous Hg
compounds. E112 is expected to have unique chemical properties due to the closed-shelllike nature of its atomic ground state and strong relativistic contraction of the doubly
occupied 7s-shell (see [13] and references therein); in 1975, Pitzer [14] put forward a
hypothesis of the relative chemical inertness of 112 as similar to that of heavy rare gases.
As was demonstrated in recent calculations on the E112H molecule [15], correlations
and spin-orbit couplings can play a crucial role in the E112-X bonding. An accurate
description of E112 interactions with coinage metal atoms (and especially with Au) is
essential for using RECP/SO DFT in studies of E112 adsorption for detection purposes
[16, 17]. The all-electron RDFT calculations on E112Au and E112Aun have been reported
in Refs. [4, 5]; however, their results can be affected by severe basis set limitations and
simplified treatment of some relativistic contributions, so that studies of these systems
by alternative techniques are desirable. Note also that the importance of the chemical
identification of E112 has increased in the last years because of the controversial data on
the detection of the long-lived 283 112 isotope [18, 19].
2
Method of calculations
2.1 Relativistic effective core potentials
We employed shape-consistent semilocal RECPs for the 60-electron cores of Au and Hg
and the 92-electron core of E112. The RECPs for mercury and eka-mercury were taken as
the valence-shell parts of the generalized relativistic effective core potentials (GRECPs)
from Refs. [20, 21]. The details of the GRECP approach [22] and the GRECP generation
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procedure [20, 21, 23, 24] can be found elsewhere. Let us recall that the GRECP operator
contains non-local (separable) terms in addition to semilocal terms of the conventional
RECP [25]. These new terms imply that different potentials are used for the valence
and Rydberg (e.g. ns np, n = 6 for Au and Hg and n = 7 for E112) and for the
outer-core ((n − 1)s (n − 1)p) electrons with the same angular quantum numbers. The
GRECPs are readily converted into semilocal valence RECPs via removing the nonlocal terms and attributing the same “valence” potentials to both valence and outer-core
shells. Let us notice that the valence potentials are generated for the nodal pseudospinors
using the singularity smoothing technique [22] in inverting Hartree-Fock-like equations.
The resulting RECPs can now be used in conventional codes for electronic structure
calculations; however, their accuracy will be generally somewhat lower than that of the
GRECPs. An analysis of the accuracy of the valence RECPs for Au, Hg and E112 used
in this paper can be found in [26]. Here only note that the errors of the RECPs used are
small enough and can not seriously affect the results obtained.
Following the method described in Refs. [20, 21, 23, 24], we have generated the requisite valence RECP for the Au atom from the solution of atomic HFDB equations with the
Fermi nuclear model for the reference state averaged over the nonrelativistic configuration
5d9 6s0.6 6p0.4 .
2.2 Computational details
The spin-orbit (relativistic) and spin-orbit-free (scalar-relativistic) DFT calculations were
performed using the nwchem code [7]. Accurate ground-state solutions of scalar-relativistic
problems defined by spin-averaged RECPs were also obtained by the coupled cluster
method with the spin-unrestricted Hartree-Fock (UHF) reference, including singles and
doubles, UCCSD, and additionally non-iterative triples, UCCSD(T), implemented in the
gaussian program package [27]. We tried to incorporate spin-dependent effects approximately into the results of the scalar calculation by means of a simple additive scheme,
i.e. by evaluating the spin-orbit correction to the total electronic energy ΔESO as the
difference of SO DFT and spin-orbit-free DFT energies and adding the obtained function ΔESO (r) (where r is the internuclear distance) to UCCSD (UCCSD(T)) potential
curves. In what follows, the UCCSD+SO (UCCSD(T)+SO) labels will be used for these
“spin-orbit-corrected” results. To estimate the risk of the inadequacy of single-reference
approaches associated with the 6d − 7s quasidegeneracy in E112, a multireference version of the approximate quadratic coupled cluster method (MR AQCC) [28–30] with the
complete-active-space reference was applied to E112H and E112Au; the active spaces
comprised all orbitals correlating with 1s H, 6d, 7s E112 and 5d, 6s Au at the dissociation limits. All outer core and valence electrons (19 for Au and 20 for Hg and E112) were
correlated.
Primitive Gaussian orbital bases for heavy atoms were essentially those proposed in
Refs. [31, 32]; a few innermost functions were replaced by those appropriate for the present
form of pseudopotentials. In CCSD, CCSD(T) and MR AQCC calculations the most loUnauthenticated
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calized s, p, and d functions were contracted and two sets of g orbitals were added; the
sizes of the resulting bases were [7s7p7d4f 2g] for Hg and E112 and [7s7p6d4f 2g] for Au.
For the hydrogen atom, the contracted (7s4p2d)/[5s4p2d] basis [33] was chosen. Significant basis set superposition errors (BSSEs) appearing in coupled-cluster-type approaches
implying the use of identity resolution to compute correlation energies were eliminated
by introducing conventional counterpoise corrections; much smaller BSSEs in DFT calculations employing numerical quadratures to evaluate exchange-correlation contributions
and also in Hartree–Fock ones were neglected.
Vibrational constants were derived from a few lowest eigenenergies of the rovibrational
Hamiltonian obtained by numerically solving the radial equation; mass number 283 was
assumed for E112.
3
Results and discussion
3.1 E112H vs. HgH
A scalar-relativistic DFT approximation with the exchange-correlation functional [34]
usually referred to as becke98 rather accurately reproduces the results of high-level
scalar-relativistic correlation calculations of E112H, while some other popular forms of
hybrid functionals (b3lyp [35] and pbe0 [36]) seem to overshoot the binding energy
(Fig. 1). Good agreement between the ground-state properties for the HgH molecule
computed at the SO DFT / becke98 level with those derived from spin-orbit-corrected
coupled cluster (UCCSD(T)+SO) results and the experimental data (Table 1) provides
an additional (though obviously insufficient per se) justification of the reliability of the
becke98 correlation treatment. In what follows, only this exchange-correlation functional was used. In spite of using restricted spin-orbitals in MR AQCC calculations, a
single configuration was found to dominate strongly in the wavefunction expansion in the
whole range of internuclear separations under study (r ≥ 2.6 a.u.).
Spin-orbit-free calculations predict a much smaller dissociation energy for E112H than
for its lighter analogue HgH. The incorporation of spin-orbit interactions dramatically
changes the shape of the potential curve (Fig. 2), considerably increasing the bond
energy (0.62 eV at the SO DFT level) and reducing the bond length. Note that the dissociation energy (De ) estimates derived from the coupled-cluster potential energy functions
with DFT spin-orbit corrections (Fig. 3) both at the UCCSD+SO and UCCSD(T)+SO
levels virtually coincide with the SO DFT estimate. The De value obtained in Ref.[15] by
the Fock-space relativistic coupled cluster (RCCSD) method [13] with a spin-restricted
closed-shell vacuum state and approximate incorporation of contributions from higher
cluster amplitudes with the help of spin-orbit CI [44] corrections is somewhat smaller
(0.42 eV). This non-negligible difference is at least partly explained by the basis set limitations, neglect of correlations with the 6s, 6p shells of E112, and approximate accounting
for multiple excitations in [15]. An analysis of the results presented in the Table 1 suggests that the more accurate RCC De estimate with a larger basis set and 19 electrons
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E(r)-E(∞), eV
0.1
0.2
0.3
0.4
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-0.2
-0.1
0
ROHF
UHF
DFT / becke98
DFT / b3lyp
DFT / pbe0
UCCSD
UCCSD(T)
MR AQCC
E112H
1.4
1.6
r, A
1.8
2
Fig. 1 Scalar-relativistic approximations for the E112H potential energy function.
correlated should be expected larger than De for HgH. It is worth noting that the use
of different forms for shape-consistent RECPs (semilocal in this work and generalized in
[15]) do not significantly contribute to the discrepancies: replacing the present RECP
by its counterpart optimized to reproduce core orbitals (i.e. by another component of
GRECP [15]) leads to a very slight (ca. 0.015 eV) increase in the SO DFT dissociation
energy, which can be neglected in the context of our research.
3.2 HgAu and E112Au
The calculations within the scalar-relativistic and spin-orbit Hartree–Fock approximation
predict a very weakly bound ground state for HgAu and an unbound one for E112Au;
the chemical bonding in these molecules is mainly due to correlation effects (Fig. 4). Let
us notice that the convergence of the cluster expansion for the correlation energy of both
systems appears to be slow, as follows from the rather large difference between UCCSD
and UCCSD(T) results. At the same time, the results of MR AQCC calculations do not
reveal any particular importance of using multireference approaches. As in the previous
case, the DFT/becke98 correlation treatment seems to be justified by comparing the
results of DFT and coupled-cluster calculations in the scalar-relativistic approximation
(Fig. 4). As one might expect, spin-orbit stabilization in E112Au is much more significant
than in HgAu; however, its role is incomparable with that in E112H. The latter fact is
obviously related to the rather large size of the Au atom and the rapid decrease of the
spin-orbit correction to the interaction energy (Fig. 3) with an increase in interatomic
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Table 1 Equilibrium bond lengths, dissociation energies and vibrational constants for
HgH, E112H, HgAu and E112Au.
re , A
De , eV
ωe , cm−1
re , A
HgH
scalar UHF a)
1.755
0.23
scalar DFT a)
1.762
0.40
1.743
0.42
UCCSD a)
a)
1.748
0.40
UCCSD(T)
a)
MR AQCC
scalar GRECP CCSD b)
1.739
0.29
SO UHF a)
1.742
0.45
SO DFT a)
1.725
0.47
UCCSD+SO a)
a)
UCCSD(T)+SO
1.730
0.45
c)
1.723
0.39
SO MRCI
b)
GRECP RCCSD
1.709
0.35
1.738 d)
0.41 d)
GRECP RCCSD + h.exc. b)
Exptl. [37–39]
1.738±0.003 0.46
2.736
2.699
2.673
0.48
0.39
0.49
2.63
2.711
2.713
2.677
2.653
2.67
2.59
0.58
0.39
0.51
0.43
0.53
0.50
1.03
0.55
ωe , cm−1
E112H
1292
1361
1313
1353
1394
1363
1445
1575
1395 d)
1403±18
1.804
1.787
1.797
1.800
1.746
1.621
1.651
1.637
1.643
0.14
0.12
0.13
0.16
−0.03
0.35
0.62
0.62
0.62
1878
1766
1813
1784
1.638
1.662 e)
0.36
0.42 e)
1859
1800 e)
HgAu
scalar DFT a)
UCCSD a)
UCCSD(T) a)
MR AQCC a)
MP2 f )
CCSD(T) f )
SO-DFT a)
UCCSD+SO a)
UCCSD(T)+SO a)
RDFT g) [4]
RDFT h) [6]
RDFT g) [6]
De , eV
E112Au
100
103
111
103
104
108
116
100
2.913
2.936
2.868
2.908
0.21
0.15
0.22
0.18
63
55
67
60
2.774
2.763
2.727
2.73
2.65
0.36
0.30
0.39
0.27
0.93
0.41
83
84
95
74
a)
Present work; b) Ref.[15], 13 electrons correlated; c) Ref. [40]; d) RCCSD + scalar triples (T); e) corrections for higherorder excitations were derived from spin-orbit multireference CI data; f ) scalar relativistic calculations with energy-adjusted
RECP, Ref. [41]; g) with exchange-correlation functional [42, 43]; h) local density functional approximation.
distance. One might thus suppose that extra strong spin-orbit bond stabilization can
occur only in hydrides and maybe in E112 compounds with some other light elements.
As a consequence, E112Au is notably less stable than HgAu. This finding is in accordance
with the results of all-electron RDFT studies [4, 6]. The dissociation energy estimates for
HgAu by the present SO DFT and RDFT with gradient-corrected exchange-corelation
functional [42, 43] agree very well. For E112Au both SO DFT and UCCSD(T) with spinUnauthenticated
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scalar UHF
scalar DFT
SO DFT
SO UHF
UCCSD+SO
UCCSD(T)+SO
scalar UHF
SO UHF
SO UHF
SO DFT
UCCSD+SO
UCCSD(T)+SO
-0.4
E(r)-E(∞), eV
-0.2
0
0.2
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-0.6
HgH
1.4
1.6
1.8
2
E112H
1.4
r, A
1.6
1.8
2
0
Fig. 2 Calculated potential energy functions for HgH and E112H.
HgAu
ΔESO(r)-ΔESO(∞), eV
-0.4
-0.2
HgH
E112Au
-0.6
E112H
re
1.5
2
2.5
r, A
3
3.5
Fig. 3 Spin-orbit contributions to atom-atom interaction energies (ΔESO (r)−ΔESO (∞))
as functions of the internuclear separation.
orbit correction predict somewhat larger De (0.36–0.39 eV) than Ref. [4]; at the same
time, the agreement between our De values and that from gradient-corrected RDFT
calculations [6] is nearly quantitative.
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E(r)-E(∞), eV
-0.2
0
456
-0.4
scalar UHF
scalar DFT
UCCSD
UCCSD(T)
MR AQCC
SO UHF
SO DFT
UCCSD(T)+SO
HgAu
2.5
3
3.5
2.5
E112Au
3
3.5
r, A
Fig. 4 Calculated potential energy functions of HgAu and E112Au.
4
Conclusions
The spin-orbit DFT method employing the shape-consistent relativistic small-core effective potential model was applied to the description of the ground electronic states of the
E112H, HgH, E112Au, and HgAu molecules. The good accuracy of the DFT correlation
treatment with the hybrid exchange-correlation functional of Schmider and Becke [34]
(becke98) is demonstrated by comparing the results in the spin-orbit-free limit with
their counterparts obtained by high-precision scalar-relativistic calculations using extensive orbital basis sets. Taking into account a low cost of DFT computations, the chosen
procedure appears to be a valuable tool for studies of interactions of E112 atoms with
rather large clusters of coinage metals.
Although the scalar-relativistic approximation predicts a very weak E112-H bonding,
the bond energy in E112H (ca. 0.6 eV) exceeds that in HgH due to an extra strong
spin-orbit stabilization. The incorporation of spin-orbit couplings also markedly (by ca.
0.15 eV) increases the estimate for E112Au dissociation energy. The E112-Au bond, in
agreement with all-electron RDFT results [4, 6], is found to be weaker than the Hg-Au
one (De =0.36–0.39 eV for E112Au versus 0.51–0.53 eV for HgAu). The E112-X bond
in both molecules studied (X=H and Au) seems to be strong enough for rejecting the
hypothesis on the rare-gas-like chemical behavior of E112 [14].
Acknowledgements
We are grateful to V.Pershina for attracting our attention to the problem. The present
work is supported by the RFBR (grant Nos. 06–03–32346 and 06–03–33060). Thanks are
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due to Kinetic Technologies Ltd. for providing computer facilities.
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