27 卷 8 期 2008. 8
结 构 化 学（JIEGOU HUAXUE）Chinese J. Struct. Chem. Dedicated to Professor Zhang Qianer on the occasion of his 80th birthday
Vol. 27, No.8 967~974
Benchmark Calculations on the Atomization Enthalpy, Geometry and Vibrational Frequencies of UF6 with Relativistic DFT Methods① XIAO Hai
LI Jun②
(Institute of Theoretical and Computational Chemistry, Department of Chemistry & Key Laboratory of Organic Optoelectronics and Molecular Engineering of Ministry of Education, Tsinghua University, Beijing 100084, China) ABSTRACT
Benchmark calculations on the molar atomization enthalpy, geometry, and
vibrational frequencies of uranium hexafluoride (UF6) have been performed by using relativistic density functional theory (DFT) with various levels of relativistic effects, different types of basis sets, and exchange-correlation functionals. Scalar relativistic effects are shown to be critical for the structural properties. The spin-orbit coupling effects are important for the calculated energies, but are much less important for other calculated ground-state properties of closed-shell UF6. We conclude through systematic investigations that ZORA- and RECP-based relativistic DFT methods are both appropriate for incorporating relativistic effects. Comparisons of different types of basis sets (Slater, Gaussian, and plane-wave types) and various levels of theoretical approximation of the exchange-correlation functionals were also made. Keywords: UF6, atomization enthalpy, relativistic effect, density functional theory
1 INTRODUCTION
molecule with an octahedral structure has been extensively studied by using various experimental
Theoretical prediction of physicochemical pro-
techniques, thus building an ideal platform of ex-
perties of actinide compounds is challenging to
perimental data for benchmark of theoretical me-
computational chemistry because accurate theoreti-
thods[1]. Theoretical study of UF6 molecule has a
cal calculations of the complicated electronic struc-
long history[2
tures of actinidecomplexes require simultaneous
calculations of the physicochemical properties of
consideration of electron correlation effects, scalar
UF6 have only emerged recently. For example, de
relativistic and spin-orbit coupling effects, and high-
Jong et al. studied relativistic effects on the elec-
ly accurate one-electron basis sets. Unlike in light-
tronic structure and atomization energy of UF6 via
element chemistry, actinide chemistry suffers from
Dirac-Fock CI calculations[6]. Hay and Martin per-
the lack of gas-phase experimental data to appraise
formed benchmark calculations on geometry and
theoretical methodologies. Due to its important role
vibrational frequencies of UF6 with large-core ECPs
in separating and enriching uranium isotopes in
for uranium[7]; García-Hernández et al. studied rela-
atomic energy and other nuclear technology, UF6
tivistic effects on geometry and vibrational fre-
~ 5]
, although quantitative relativistic
Received 10 June 2008; accepted 18 July 2008 ① This work was supported by NKBRSF (2006CB932305, 2007CB815200) and NNSFC (20525104). The calculations were partially performed using an HP Itanium2 cluster at Tsinghua National Laboratory for Information Science and Technology ② Corresponding author. Li Jun, PhD, Cheung Kong Professor of Chemistry. E-mail:
[email protected]
968
XIAO H. et al.: Benchmark Calculations on the Atomization Enthalpy, Geometry and Vibrational Frequencies of UF6 with Relativistic DFT Methods
quencies of UF6 with two-component DFT methods [8]
No.8
generalized gradient approach (GGA), and hybrid
based on Douglas-Kroll-Hess (DKH) approach ;
GGA, a number of new exchange-correlation func-
Batista et al. performed benchmark calculations on
tionals have been developed in recent years, in-
U-F bond dissociation energy, geometry and vibra-
cluding meta-GGA, hybrid meta-GGA, and others. It
tional frequencies of UF6 to compare large- and
is thus important to evaluate the performance of
[9]
small-core ECPs for uranium . Rösch and co-
these new generations of DFT methods on calcu-
workers carried out detailed theoretical investi-
lating the properties of actinide compounds. In this
gations of the heat of formation of UF6 and UF5
[10]
.
article we performed systematic theoretical calcu-
Despite these theoretical investigations providing
lations on the molar atomization enthalpy, geometry,
important insight on various aspects of accurately
and vibrational properties to assess the performance
calculating the geometry, vibrational frequencies,
of various exchange-correlation functionals in acti-
and energetics, no systematic investigation has been
nide chemistry.
carried out on the performance of various ex-
Considering a system at a local minimum of
change-correlation functionals on the properties of
geometry R0 , its ground energy, as a function of
UF6. In addition to the local density approach (LDA),
geometric variable R , can be expanded as,
v v dE E ( R ) = E ( R0 ) + v dR
v
v
v v 1 d 2E v ( R − R0 ) + v 2 dR 2 R0
v R0
v v ( R − R0 ) 2 + K
(1),
where the first term on the right-hand-side is
de[11]. These calculations employed the local density
implicitly related to atomization enthalpy or bond
approach (i.e. Slater-Vosko-Wilk-Nusair[12]) and ge-
dissociation energy etc., while the second and third
neralized gradient approaches with Becke-Lee-
terms explicitly correspond to local minimum geo-
Yang-Parr (BLYP)[13], Perdew-Wang 91 (PW91)[14],
metry and vibrational frequencies, respectively. The
and Perdew-Burke-Ernzerhof (PBE)[15] XC func-
higher-order terms are small and negligible under
tionals. Uncontracted all-electron triple-zeta STO
normal chemical conditions. Therefore, we perfor-
basis sets with two polarization functions were used
med systematic benchmark calculations on the molar
for U and F. Scalar relativistic (SR) effects and spin-
atomization enthalpy, geometry, and vibrational fre-
orbit (SO) coupling effects were taken into account
quencies of UF6, employing various levels of re-
by using zeroth order regular approximation (ZO-
lativistic DFT methods, basis sets, and exchange-
RA)[16]. Tight numerical integration accuracy of 10-8
correlation (XC) functionals, to shed light on
was used throughout, with the convergence thre-
choosing appropriate theoretical methodology for
sholds set at 10-5 Hartree/Å for Cartesian gradients
calculating properties of uranium-containing com-
during geometry optimizations and at 10-8 for energy
pounds. We have chosen to use relativistic DFT
iterations during self-consistent field (SCF) calcu-
methods because they are efficient in treating large
lations. For the open-shell calculations of atoms,
molecules and convenient in handling electron cor-
noncollinear relativistic approach was used when SO
relation and relativistic effects.
coupling effect was included. The Gaussian type orbital (GTO) basis sets cal-
2
CALCULATIONAL DETAILS
culations were performed by using NWChem program[17]. In addition to LDA and GGA functionals
The theoretical calculations with Slater type
mentioned above, meta-GGA functional Tao-Per-
orbital (STO) basis sets were performed by using the
dew-Staroverov-Scuseria (TPSS)[18], hybrid-GGA
Amsterdam Density Functional (ADF2007.01) co-
functionals B3PW91[19], B3LYP[20] and PBEh[21],
2008
Vol. 27
结
构
化
学（JIEGOU HUAXUE）Chinese
hybrid-meta-GGA functionals TPSSh[22] and M06[23]
J.
969
Struct. Chem.
in which all species are treated as ideal gases and the
, and Hartree-Fock (HF) method were also
thermal contributions of vibration, rotation and
used in the benchmark calculations. Small-core,
translation of the atoms are omitted. The enthalpy
relativistic energy-consistent pseudopotential (RE-
was computed under standard condition (1 atm.) at
CP), augmented by spin-orbit operators, was used
298.15 K.
2X
for U, where the 32 “valence” electrons are varia-
Inasmuch as theoretical determinations of atomic
tionally treated. The Stuttgart basis set of triple-zeta
reference energies using DFT methods are generally
quality was used for U, with the default (12s11p-
cumbersome for heavy elements[29], we used an
10d8f)/[8s7p6d4f] contraction[24] and two g-type
approach that has been recently proposed by
polarization functions added (α = 1.52399155,
Johnson, Dickson, and Becke to calculate atomic
[25]
. Dunning’s all-electron augmen-
reference energy of uranium atom[30]. In this ap-
ted correlation-consistent basis set aug-cc-pVTZ has
proach, the configuration energies of all possible
0.3745056345)
[26]
been used for F
. Two levels of relativistic DFT
electron occupations in real orbitals were calculated
methodologies were used, i.e., quasi-relativistic (QR)
and occupations with the lowest total energies were
DFT and spin-orbit coupled DFT (SODFT) methods,
used to reproduce the multiplet energies of complex
and the convergence criteria are comparable to those
angular momentum eigenstates. In the NR, SR and
used in the ADF calculations.
QR calculations we adopted a D2h symmetry in
We also performed supercell DFT calculations
calculating atomic energies to differentiate dege-
using plane-wave (PW) basis sets and the Vienna
nerate real orbitals. Through numerous calculations
[27]
Ab-initio Simulation Package (VASP 4.6.31)
1
.
we find that the electron configurations d x2 − y 2 -
Only LDA, PW91 and PBE were employed with this
(ag) f z 3 (b1u) f z ( x2 − y 2 ) (b1u) f y ( 3 x2 − y 2 ) (b2u) and
1
1
1
code. The projector augmented wave (PAW) me-
p1z (b1u) produce the lowest reference energies for U
thod[28] was applied to describe the core electrons
and F atoms, respectively. However, we adopted C1
and to incorporate scalar relativistic effects impli-
symmetry for spin-orbit calculations of open-shell
citly, i.e., quasi-relativistically (QR). The plane-
atoms in the ADF and NWChem codes, so average-
wave basis sets with cutoff energy of 400 eV were
of-configuration (AOC) spin-orbit results of atoms
used to expand the valence electronic wavefunction
were obtained. Although we constrained D2h sym-
2
6
2
3
1
metry on atoms by using cuboid supercells in VASP,
and F 2s 2p . The convergence criteria for electronic
only AOC results were obtained due to the lack of
with valence configurations of U 6s 6p 7s 5f 6d 2
5
-6
SCF and geometry optimization were set at 10 and
occupation control in this code.
-4
10 eV/Å, respectively. The sum of electronic energy calculated by
3 RESULTS AND DISCUSSION
quantum chemistry methods and thermal contributions from vibration, rotation and translation render the thermodynamic internal energy. For the ato-
3. 1
Comparison of relativistic methods
For heavy-element compounds relativistic effects
mization reaction of UF6 below,
play an important role in their geometries and
UF6 (g) = U (g) + 6F (g)
electronic structures. Fig. 1 depicts the AOC energy
The molar atomization enthalpy (∆H0) can be obtai-
levels of U atom calculated using non-relativistic (NR), scalar relativistic (SR), and spin-orbit (SO)
ned as follows:
relativistic methods. The SR effects stabilize the 7s
∆H = EU + 6 EF − ( EUF6 + Evib + Erot + Etrans ) + 6 RT 0
(2)
orbital (direct relativistic effects) and destabilize the 5f and 6d orbitals (indirect relativistic effects). The SO effects split the 5f and 6d orbitals into two
XIAO H. et al.: Benchmark Calculations on the Atomization Enthalpy, Geometry and Vibrational Frequencies of UF6 with Relativistic DFT Methods
970
No.8
components, with the energetic splitting for 5f larger
gies, while the NR formalism incorrectly weakens
than that for 6d. From Table 1, the inclusion of SR
the U-F bonding because of erroneous radial
effects improves the description of structural and
contractions and energy lowering of the 5f and 6d
ground-state vibrational properties of UF6. For
orbitals (Fig. 1). In fact, from Fig. 1, the NR for-
example, the mean absolute errors (MAE) of cal-
malism predicts a wrong ground-state electron con-
culated vibrational frequencies decrease more
figuration for U due to the significant stabilization of
-1
from non-relativistic (NR) calcula-
5f orbitals from SR to NR case. When SO coupling
tions to SR calculations. All the methods over-
effects are further included in the calculations, the
estimate the molar atomization enthalpy partially
calculated energies and molar atomization enthalpy
due to the DFT difficulty in accurately calculating
are improved, while little improvement is shown in
the multiplet energy of U ground state. However, the
the structural parameters due to the absence of the
molar atomization enthalpy from the NR calcula-
first-order SO coupling effect in the closed-shell UF6.
tions agrees better with experiments than that from
In general, SO coupling has important effects on
the SR and SO calculations, which is caused by
energies but few effects on ground-state geometry
cancellation of two errors: these LDA and GGA XC
and vibrational frequencies because of the quasi-
functionals generally overestimate the binding ener-
atomic nature of SO coupling[8].
than 40 cm
Fig. 1.
Average-of-configuration (AOC) atomic orbital energy levels of U calculated with various relativistic effects and PW91 functional. NR, SR, and SO denote non-relativistic, scalar-relativistic, and spin-orbit, respectively. Table 1. Calculated Molar Atomization Enthalpies (∆H0), U-F Bond Lengths (R), and Vibrational Frequencies (ν) with Various Levels of Relativistic Methods
XC
∆H0 / kJ/mol-1
|δ∆H0/∆H0|
R/Å
ν / cm-1
|δR/R|%
%
A1g
Eg
T1u
T1u
T2g
T2u
NR LDA
3641
15.4
1.995
0.1
600
447
593
132
188
69
55
BLYP
3081
2.3
2.055
3.0
546
414
540
125
176
99
76
PW91
3093
1.9
2.031
1.8
558
415
557
128
181
88
72
PBE
3097
1.8
2.030
1.7
565
423
558
128
183
89
68
LDA
4076
29.2
1.994
0.1
657
541
628
171
182
133
10
BLYP
3360
6.5
2.040
2.2
610
507
581
173
185
131
28
PW91
3490
10.6
2.021
1.3
627
519
598
169
185
132
21
PBE
3497
10.9
2.021
1.3
625
516
595
174
185
133
21
LDA
3994
26.6
1.989
0.4
664
550
634
170
183
134
12
BLYP
3372
6.9
2.035
2.0
615
513
586
173
187
133
25
ZORA-SR
ZORA-SO
2008
Vol. 27
结
构
化
学（JIEGOU HUAXUE）Chinese
J.
971
Struct. Chem.
PW91
3504
11.1
2.016
1.0
631
524
602
170
186
133
18
PBE
3485
10.5
2.017
1.1
633
525
601
175
186
134
17
LDA
4062
28.8
1.988
0.4
664
547
629
172
185
132
10
BLYP
3385
7.3
2.034
1.9
613
508
579
176
188
131
27
PW91
3524
11.7
2.015
1.0
631
519
602
186
198
130
15
PBE
3504
11.1
2.015
1.0
630
518
594
176
187
132
20
LDA
4043
28.2
1.982
0.7
666
548
633
174
186
133
10
BLYP
3408
8.1
2.029
1.7
615
509
583
177
189
133
25
PW91
3545
12.4
2.009
0.7
633
521
600
177
188
133
17
PBE
3525
11.8
2.010
0.7
632
520
598
177
188
134
18
Expt.a
3154(~10)
—
1.996(8)
—
667(1)
534(1)
626(1)
186(1)
200(1)
143(2)
—
QR/RECP
SODFT/RECP
a.
Experimental data are taken from Refs. [32], [33] and [34].
Meanwhile, by comparing the calculation results
The two traditional quantum chemistry basis sets,
from RECP- and ZORA-based relativistic methods
STO and GTO, have their respective merits: STOs
(Table 1), one can see that these relativistic methods
are closer to the actual orbitals in radial forms, thus
both give reasonable results for the properties con-
requiring less BFs to achieve completeness; while
sidered, indicating that it is appropriate to incor-
GTO basis sets greatly reduce the computational
porate relativistic effects via either ZORA Hami-
costs because of the well-known advantages in
ltonian or RECP.
molecular integrals. Although PW basis set type is
3. 2
primarily designed for periodic systems, it performs
Comparison of basis set types
For the evaluation of different types of basis sets,
well for the calculations of aperiodic UF6 molecule
completeness and computational efficiency are two
when using large supercell. PW basis set has several
of the most important aspects. From Table 2, cal-
advantages over STO and GTO basis sets because it
culations with three types of basis sets (STO, GTO,
does not suffer from the basis set superposition
and PW) all demonstrate good performance. With
errors (BSSE) and the linear dependence problem.
STO using 339 basis functions (BFs), GTO using
When combined with pseudopotentials or the PAW
381 BFs, and PW using millions of BFs, the com-
method, PW basis set is probably more promising in
putational costs for calculations with three different
calculating large actinide systems, such as nano-
types of basis sets are all comparable to each other.
clusters, surfaces, and crystals.
Table 2. XC
∆H0 / kJ/mol-1
Calculated Molar Atomization Enthalpies (∆H0), U-F Bond Lengths (R), and Vibrational Frequencies (ν) with Various Basis Set Types
|δ∆H0/∆H0|
R/Å
ν / cm-1
|δR/R|%
%
A1g
Eg
T1u
T1u
T2g
T2u
ZORA-SR (STO) LDA
4076
29.2
1.994
0.1
657
541
628
171
182
133
PW91
3490
10.6
2.021
1.3
627
519
598
169
185
132
10 21
PBE
3497
10.9
2.021
1.3
625
516
595
174
185
133
21
LDA
4062
28.8
1.988
0.4
664
547
629
172
185
132
10
PW91
3524
11.7
2.015
1.0
631
519
602
186
198
130
15
PBE
3504
11.1
2.015
1.0
630
518
594
176
187
132
20
LDA
4033
27.9
1.996
0.0
661
547
629
169
177
126
13
PW91
3618
14.7
2.023
1.4
629
523
597
173
182
129
21
PBE
3555
12.7
2.023
1.4
628
522
596
173
183
130
21
Expt.a
3154(~10)
—
1.996(8)
—
667(1)
534(1)
626(1)
186(1)
200(1)
143(2)
—
QR/RECP (GTO)
QR/PAW (PW)
a. Experimental data are taken from Refs.
XIAO H. et al.: Benchmark Calculations on the Atomization Enthalpy, Geometry and Vibrational Frequencies of UF6 with Relativistic DFT Methods
972
3. 3
No.8
properties of UF6 in our benchmark calculations,
Comparison of XC functionals
Since the SO coupling effects are less important in
with only 2% relative errors in general. However,
ground-state properties of UF6, we performed RE-
hybrid-meta-GGA functional (TPSSh) does not seem
CP-based QR calculations with GTO basis sets for
to perform as well as hybrid-GGA functionals, but
comparing the performance of the new generation of
the relative errors around 2% are already within the
XC functionals implemented in NWChem. The
accuracy limit of approximate XC functionals in
calculated molar atomization enthalpy, U-F bond
DFT for such a simple closed-shell compound UF6.
length, and vibrational frequencies and their relative
Other benchmark calculations indeed show that
or absolute errors are shown in Table 3. As expected,
hybrid-meta-GGA functionals possess obvious im-
all the DFT techniques perform better than the Har-
provements over hybrid-GGA functionals[22, 23].
tree-Fock (HF) method due to the lack of Coulomb
In summary, as far as accuracy is concerned, hy-
electron correlation effects in the latter. Though
brid density functionals should be the best choice in
higher order wavefunction-based electron corre-
applying DFT methods for actinide complexes like
lation methods, such as multi-reference (MR) con-
UF6. Overall the PBEh and TPSSh functionals
figuration interaction (CI) and coupled cluster me-
perform reasonably well for the properties inves-
thods, can handle complicated electronic correlation
tigated and can be used for future theoretical in-
effects in actinides, they are usually too expensive
vestigations of actinide compounds. However, the
for large actinide complexes. With much less com-
use of hybrid functionals raises computational cost
putational effort, DFT methods can account for the
due to the evaluation of HF exchange term. In
electron correlation effects reasonably well.
addition, hybrid functionals are not implemented or
In Table 3, the performances of different levels of
quite expensive in many computational chemistry
DFT theoretical approximations in predicting the
codes using Slater or plane-wave basis functions
properties of UF6 are shown. As the basic appro-
(e.g., ADF and VASP). Therefore, GGA functionals,
ximation of XC functionals in density functional
such as PW91 and PBE, are still a good choice in
theory, LDA usually overestimates the binding
computational actinide chemistry, and the LDA
energies. Consequently, the molar atomization en-
functional provides good geometries and can be used
thalpy calculated by using LDA method is in great
as reference for comparing the XC functionals. With
error even though the LDA geometry is reasonable.
these benchmark calculations, one can investigate
From Table 3, GGA functionals remedy this defect
the geometries, electronic structures, and spec-
reasonably well, and the meta-GGA functional (TP-
troscopic properties of other complicated actinide
SS) further improves the results. Compared to pure
complexes. We have recently used these DFT
density functionals (LDA, GGA, meta-GGA), hy-
methods to investigate a UO6 molecule that has an
brid density functionals (B3LYP, B3PW91, PBEh,
extremely high +XII oxidation state and the results
M06-2X) show excellent accuracy in predicting
are presented in a recent publication[31].
Table 3
XC
∆H0 / kJ/mol-1
Calculated Molar Atomization Enthalpies (∆H0), U-F Bond Lengths (R), and Vibrational Frequencies (ν) with Various XC Functionals |δ∆H0/∆H0| %
R/Å
|δR/R|%
ν / cm-1 A1g
Eg
T1u
T1u
T2g
T2u
QR/RECP (GTO) HF
1987
37.0
1.976
1.0
746
540
657
199
222
143
25
LDA
4062
28.8
1.988
0.4
664
547
629
172
185
132
10
BLYP
3385
7.3
2.034
1.9
613
508
579
176
188
131
27
PW91
3524
11.7
2.015
1.0
631
519
602
186
198
130
15
2008
Vol. 27
结
构
化
学（JIEGOU HUAXUE）Chinese
J.
973
Struct. Chem.
PBE
3504
11.1
2.015
1.0
630
518
594
176
187
132
20
TPSS
3381
7.2
2.013
0.9
638
524
602
173
184
129
18
B3PW91
3210
1.8
1.994
0.1
671
535
624
188
198
140
2
B3LYP
3186
1.0
2.006
0.5
660
530
614
185
198
139
5
PBEh
3186
1.0
1.987
0.5
683
539
632
186
199
141
5
TPSSh
3258
3.3
2.001
0.3
658
532
617
178
190
133
8
M06-2X
3142
0.4
1.981
0.8
711
552
650
198
214
152
20
3154(~10)
—
1.996(8)
—
667(1)
534(1)
626(1)
186(1)
200(1)
143(2)
—
a
Expt.
a. Experimental data are taken from Refs.
incorporating relativistic effects and handling elec-
4
CONCLUSION
tron correlation at the same time. For the selection of different types of basis sets, Slater, Gaussian and
We have performed benchmark calculations on
plane-wave basis functions are all good options for
the molar atomization enthalpy, geometry, and
computational investigations of actinide complexes.
vibrational frequencies of UF6 using relativistic DFT
For various approximations of DFT exchange-
methods with various exchange-correlation func-
correlation functionals, we have demonstrated that
tionals. In general, the scalar relativistic effects are
hybrid density functionals show excellent perfor-
important for structural properties of actinide com-
mance for predicting properties of UF6, while GGA
plexes, while the SO coupling effects are much less
functionals, such as PW91 and PBE, can produce
important for calculated ground-state geometries and
satisfactory results with less computational costs.
vibrational frequencies of closed-shell UF6. We have
These results provide theoretical guidance in selec-
also shown that the ZORA- and RECP-based re-
ting exchange-correlation functionals for calculating
lativistic DFT methods are both appropriate for
actinide complexes.
REFERENCES (1)
Batista, E. R.; Martin, R. L.; Hay, P. J. J. Chem. Phys. 2004, 121, 11104– 11111.
(2)
Peralta, J. E.; Batista, E. R.; Scuseria, G. E.; Martin, R. L. J. Chem. Theory Comput. 2005, 1, 612–616.
(3)
Straka, M.; Hrobárik, P.; Kaupp, M. J. Am. Chem. Soc. 2005, 127, 2591–2599.
(4)
Gagliardi, L.; Willetts, A.; Skylaris, C. K.; Handy, N. C.; Spencer, S.; Ioannou, A. G.; Simper, A. M. J. Am. Chem. Soc. 1998, 120, 11727–11731.
(5)
Schreckenbach, G. Inorg. Chem. 2000, 39, 1265–1274.
(6)
de Jong, W. A.; Nieuwpoort, W. C. Int. J. Quantum Chem. 1996, 58, 203–216.
(7)
Hay, P. J.; Martin, R. L. J. Chem. Phys. 1998, 109, 3875–3881.
(8)
García-Hernández, M.; Lauterbach, C.; Krüger, S.; Matveev, A.; Rösch, N. J Comput. Chem. 2002, 23, 834–846.
(9)
Batista, E. R.; Martin, R. L.; Hay, P. J.; Peralta, J. E.; Scuseria, G. E. J. Chem. Phys. 2004, 121, 2144–2150.
(10)
Moskaleva, L. V.; Matveev, A. V.; Krüger, S.; Rösch N. Chem. Eur J. 2005, 629–634.
(11)
ADF2007.01, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, http://www.scm.com. Fonseca Guerra, C.; Snijders, J. G.; te Velde, G.; Baerends, E. J. Theor. Chem. Acc. 1998, 99, 391–403. te Velde, G.; Bickelhaupt, F. M.; van Gisbergen, S. J. A.; Fonseca Guerra, C.; Baerends, E. J.; Snijders, J. G.; Ziegler, T. J. Comput. Chem. 2001, 22, 931–967.
(12)
Slater, C. J. Quantum Theory of Molecules and Solids, Volume 4. McGraw-Hill: New York 1974. Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200–1211.
(13)
Becke, A. D. Phys. Rev. A 1988, 38, 3098–3100. Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785–789.
(14)
Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Phys. Rev. B 1992, 46, 6671–6687.
(15)
Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865–3868.
974 (16)
XIAO H. et al.: Benchmark Calculations on the Atomization Enthalpy, Geometry and Vibrational Frequencies of UF6 with Relativistic DFT Methods
No.8
Chang, C.; Pelissier, M.; Durand, P. Phys. Scr. 1986, 34, 394–404. van Lenthe, E.; Baerends, E. J.; Snijders, J. G. J. Chem. Phys. 1993, 99, 4597–4610.
(17)
Bylaska, E. J.; de Jong, W. A.; Kowalski, K.; Straatsma, T. P.; Valiev, M.; Wang, D.; Aprà, E.; Windus, T. L.; Hirata, S.; Hackler, M. T.; Zhao, Y.; Fan, P. D.; Harrison, R. J.; Dupuis, M.; Smith, D. M. A.; Nieplocha, J.; Tipparaju, V.; Krishnan, M.; Auer, A. A.; Nooijen, M.; Brown, E.; Cisneros, G.; Fann, G. I.; Früchtl, H.; Garza, J.; Hirao, K.; Kendall, R.; Nichols, J. A.; Tsemekhman, K.; Wolinski, K.; Anchell, J.; Bernholdt, D.; Borowski, P.; Clark, T.; Clerc, D.; Dachsel, H.; Deegan, M.; Dyall, K.; Elwood, D.; Glendening, E.; Gutowski, M.; Hess, A.; Jaffe, J.; Johnson, B.; Ju, J.; Kobayashi, R.; Kutteh, R.; Lin, Z.; Littlefield, R.; Long, X.; Meng, B.; Nakajima, T.; Niu, S.; Pollack, L.; Rosing, M.; Sandrone, G.; Stave, M.; Taylor, H.; Thomas, G.; van Lenthe, J.; Wong, A.; Zhang, Z. NWChem, A Computational Chemistry Package for Parallel Computers, Version 5.0. Pacific Northwest National Laboratory: Richland, Washington 99352–0999, USA 2006.
(18)
Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Phys. Rev. Lett. 2003, 91, 146401.
(19)
Becke, A. D. J. Chem. Phys. 1993, 98, 5648–5652.
(20)
Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623–11627.
(21)
Adamo, C.; Barone, V. J. Chem. Phys. 1999, 110, 6158–6170.
(22)
Staroverov, V. N.; Scuseria, G. E.; Tao, J.; Perdew, J. P. J. Chem. Phys. 2003, 119, 12129–12137.
(23)
Zhao, Y.; Truhlar, D. G. Theor. Chem. Acc. 2008, 120, 215–241.
(24)
Küchle, W.; Dolg, M.; Stall, H.; Preuss, H. J. Chem. Phys. 1994, 100, 7535– 7542.
(25)
Pyykkö, P.; Runeberg, N.; Straka, M.; Dyall, K. G. Chem. Phys. Lett. 2000, 328, 415–419.
(26)
Kendall, R. A.; Dunning Jr., T. H.; Harrison, R. J. J. Chem. Phys. 1992, 96, 6796–6806.
(27)
VASP code, http://cms.mpi.univie.ac.at/vasp/. (a) Kresse, G.; Hafner, J. Phys. Rev. B 1993, 48, 13115–13118. (a) Kresse, G.; Furthmüller, J. Phys. Rev. B 1996, 54, 11169–11186. (b) Kresse, G.; Furthmüller, J. Comput. Mater. Sci. 1996, 6, 15–50.
(28)
(a) Blöchl, P. E. Phys. Rev. B 1994, 50, 17953–17979. (b) Kresse, G.; Joubert, D. Phys. Rev. B 1999, 59, 1758–1775.
(29)
Baerends, E. J.; Branchadell, S. M. Chem. Phys. Lett. 1997, 265, 481–489.
(30)
Johnson, E. R.; Dickson, R. M.; Becke, A. D. J. Chem. Phys. 2007, 126, 184104.
(31)
Xiao, H.; Li, J. Phys. Chem. Chem. Phys. 2008, Submitted.
(32)
Hildenbrand, D. L.; Lau, K. H. J. Chem. Phys. 1991, 94, 1420–1425.
(33)
Kimura, M.; Schomaker, V.; Smith, D. W. J. Chem. Phys. 1968, 48, 4001–4012.
(34)
McDowell, R. S.; Asprey, L. B.; Paine, R. T. J. Chem. Phys. 1974, 61, 3571–3580.
