JOURNAL OF CHEMICAL PHYSICS
VOLUME 117, NUMBER 22
8 DECEMBER 2002
An L-shaped equilibrium geometry for germanium dicarbide „GeC2…?
Interesting effects of zero-point vibration, scalar relativity,
and core–valence correlation
Levent Sari
Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia 30602-2525
Kirk A. Peterson
Department of Chemistry, Washington State University, Richland, Washington 99352
Yukio Yamaguchi and Henry F. Schaefer IIIa)
Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia 30602-2525
共Received 11 July 2002; accepted 12 September 2002兲
The ground state potential energy surface of the GeC2 molecule has been investigated at highly
correlated coupled cluster levels of theory. Large basis sets including diffuse functions and functions
to describe core correlation effects were employed in order to predict the true equilibrium geometry
for GeC2. Like the much-studied valence isoelectronic SiC2, the linear ( 1 兺 ⫹ ), L-shaped ( 1 A ⬘ ), and
T-shaped structures ( 1 A 1 ) must be investigated. The L-shaped Cs geometry is found to have real
harmonic vibrational frequencies along every internal coordinate, and the linear stationary point has
an imaginary vibrational frequency along the bending mode at every level of theory employed. The
T-shaped geometry is found to have an imaginary vibrational frequency along the asymmetric
stretching mode. At the coupled cluster with single and double excitations and perturbative triple
excitations 关CCSD共T兲兴/correlation consistent polarized valence quadrupole-␨ 共cc-pVQZ兲 level, the
nonrelativistic classical relative energies of the T-shaped and linear structures with respect to the
L-shaped minimum are 0.1 and 2.8 kcal/mol, respectively. Including zero-point vibrational energy,
scalar relativistic, and core-valence corrections, the T-L energy separation is shifted to 0.4 kcal/mol
and the relative energy between the L-shaped and linear structures is still 2.8 kcal/mol. All
nonrelativistic and relativistic computations predict that the L-shaped ( 1 A ⬘ ) structure is most
favored for the ground state. The linear structure is predicted to be a transition state, as the case of
SiC2. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1518966兴
I. INTRODUCTION
The group IV diatomics and triatomics such as CSi,
CSi2, SiC2, GeC, Ge2C, GeC2, and SnC have held a growing
interest for both experimental and theoretical researchers,
mainly due to their extraordinary electronic structures and
potential usages in optoelectronic and semiconductor
applications.1– 6 The inconsistency in the electronic structures
of these periodically related species gave rise to incorrect
conclusions about the ground state structures, some of which
had been held for many years. One reason for encountering
these surprising results is that C, Si, Ge, and Sn have different core sizes, which play a significant role in the minimum
energy geometries and in the extent of relativistic energy
corrections. Also, the absence of occupied d electrons for C
and Si causes an unbalanced competition for the valence
electrons compared to Ge or Sn. There have been many cases
reported in the literature showing this inconsistency. For in5
3
stance, the ground state of C2 7 is 1 兺 ⫹
g , that of CSi is 兿 ,
⫺
8
and for Ge2 it is 3 兺 g . Further, C3 has a linear structure9 in
a兲
Electronic mail: [email protected]
0021-9606/2002/117(22)/10008/11/$19.00
10 1
its ground state ( 1 兺 ⫹
g ), Si3 has a triangular structure ( A 1 ),
while GeSi2 is predicted4 to have a triplet ( 3 B 2 ) rather than
singlet ground state.
The most discussed molecule of this kind is SiC2. Once
evidence was found for SiC2 in carbon-rich stars11 in 1926, it
became a focus of interest. However, until 1982, experimental and theoretical studies seemed to show that SiC2 has a
linear structure in its ground state ( 1 兺 ⫹ ). In 1982,
Bondybey12 carried out a time-resolved laser-induced fluorescence spectroscopy experiment, and he concluded that linear ground state structure is not a priori obvious. The first ab
initio study of SiC2 was published by Green13 in 1983. They
excluded the possibility of the CSiC isomer, and concluded
that the linear 1 兺 ⫹ state is the ground state. However, in
1984, Grev and Schaefer14 carried out CISD/DZP studies and
reported that the T-shaped 1 A 1 state is, in fact, lower in energy than the linear structure by 0.4 kcal/mol. Simultaneously, Smalley and co-workers15 concluded that the linear
1 ⫹
兺 structure is not consistent with the observed rotational
spectra. In 1988, Shepherd and Graham16 performed Fourier
transform infrared spectroscopy 共FTIR兲 experiments and
confirmed the cyclic T-shaped geometry for the ground state
( 1 A 1 ).
10008
© 2002 American Institute of Physics
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J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
Germanium dicarbide
The analogous GeC2 molecule was detected by
Schmude, Gingerich, and Kingcade6 in a mass spectrometry
experiment in 1995. They reported enthalpies of formation
for GeC2 as well as Ge2C, Ge2C2, and Ge3C. However, they
assumed the structure of GeC2 to be T-shaped, considering
the Si analog. Of course, the mass spectroscopic experiments
did not provide any information concerning the geometry of
the global minimum and the electronic structure. The only
theoretical study to date was reported by Li et al.4 in 2001.
They performed DFT calculations on Am Bn 共A,B⫽Si,Ge,C
and m⫹n⬍10). For GeC2, they excluded the possibility of a
linear structure, stating that AB2 binary clusters of group IV
elements have extremely low stabilities for linear geometries. Li and co-workers reported that GeC2, Ge2C, SiC2,
Si2C, and SiGe2 all have T-shaped geometries in their ground
states.
Because there is no experimental data pertinent to the
structure and energetics of the GeC2 system, as well as no
high level theoretical study, we aimed to study the ground
state electronic structure of this elusive molecule by employing highly correlated coupled-cluster theories in conjunction
with substantial basis sets. Especially, we wanted to see the
effects of relativistic and core–valence correlations on the
structure and energetics of the system due to the existence of
the Ge atom. The primary motivation here is that the ground
state potential energy surface is extremely flat, like SiC2, and
even 2–3 kcal/mol energy is sufficient to invert the Ge atom
around the molecule. Second, the main differences between
the Si and Ge atoms are 共a兲 the relatively large core of Ge,
which makes relativity more important; and 共b兲 the 3d electrons of Ge which gives rise to additional core–valence correlations with the 4s and 4p electrons. Therefore, the Ge
atom sometimes behaves differently from the Si atom, and in
turn, the ground state potential energy surface of GeC2 might
be different from that of SiC2, due to the fact that very small
energy differences determine the shape of the surface.
II. ELECTRONIC STRUCTURE CONSIDERATIONS
The three different geometries that have been studied in
this work are given in Fig. 1. The ground electronic state of
GeC2 has accordingly the following electronic configurations: Linear C ⬁ v symmetry,
关 core兴共 10␴ 兲 2 共 11␴ 兲 2 共 12␴ 兲 2 共 13␴ 兲 2 共 4 ␲ 兲 4 ⇒ 1 兺 ⫹ ;
共1兲
cyclic C 2 v symmetry 共T-shaped geometry兲,
关 core兴共 9a 1 兲 2 共 10a 1 兲 2 共 5b 2 兲 2
⫻ 共 4b 1 兲 2 共 11a 1 兲 2 共 12a 1 兲 2 ⇒ 1 A 1 ;
共2兲
bentC s symmetry 共L-shaped geometry兲,
关 core兴共 13a ⬘ 兲 2 共 14a ⬘ 兲 2 共 15a ⬘ 兲 2
⫻ 共 5a ⬙ 兲 2 共 16a ⬘ 兲 2 共 17a ⬘ 兲 2 ⇒ 1 A ⬘ .
共3兲
10009
FIG. 1. The three different geometries of GeC2 studied in this work.
In the above equations, 关core兴 denotes the 16 core 共Ge:
1s-, 2s-, 2p-, 3s-, 3p-, and 3d-like and C: 1s-like兲 orbitals. One of the 4␲ molecular orbitals 共MOs兲 of linear GeC2
becomes the 4b 1 MO for the T-shaped structure and represents the ␲ bonding. The 5a ⬙ MO of the L-shaped structure
corresponds to the ␲ molecular orbital. The in-plane ␲ MO
in the L-shaped structure is mixed with other MOs of the
same symmetry. The 10␴ and 11␴ MOs of the linear structure, 9a 1 , 10a 1 , and 5b 2 MOs of the T-shaped, and 13a ⬘
and 14a ⬘ MOs of the L-shaped arrangement correspond to
the ␴共C–C兲⫹␴共C–Ge兲 and ␴共C–C兲⫺␴共C–Ge兲 bonds. The
12␴ and 13␴ orbitals of the linear structure, 11a 1 and 12a 1
orbitals for T-shaped, and 16a ⬘ and 17a ⬘ orbitals for
L-shaped are associated with the lone pairs of the C and Ge
atoms, respectively.
III. THEORETICAL APPROACH
SCF 共restricted open shell兲 wave functions have been
used for the zeroth-order descriptions of the ground state for
each geometry 共T-shaped, linear, and L-shaped兲. The configuration interaction with single and double excitations
共CISD兲, the coupled cluster with single and double excitations 共CCSD兲,17 and the CCSD with perturbative triple excitations 关CCSD共T兲兴18 methods have been employed to include
correlation effects. The 11 lowest-lying MOs 共Ge: 1s-, 2s-,
2 p-, 3s- 3p-like and C: 1s-like兲 were frozen and the two
highest-lying virtual MOs were deleted with all correlated
levels with the TZ2P⫹diff and TZ3P共2f兲 basis sets. Only the
11 lowest-lying MOs were frozen for the correlation-
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10010
Sari et al.
J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
TABLE I. Total nonrelativistic energies 共in hartree兲, bond distances 共in Å兲, dipole moments 共in Debye兲, and harmonic vibrational frequencies 共in cm⫺1兲 for
the T-shaped geometry ( 1 A 1 symmetry兲.
TZ2P⫹diff SCF
TZ3P共2f兲 SCF
cc-pVTZ SCF
cc-pVQZ SCF
TZ2P⫹diff CISD
TZ3P共2f兲 CISD
cc-pVTZ CISD
cc-pVQZ CISD
TZ2P⫹diff CCSD
TZ3P共2f兲 CCSD
cc-pVTZ CCSD
cc-pVQZ CCSD
TZ2P⫹diff CCSD共T兲
TZ3P共2f兲 CCSD 共T兲
cc-pVTZ CCSD共T兲
aug-cc-pVTZ CCSD 共T兲
cc-pVQZ CCSD共T兲
cc-pwCVTZ CCSD共T兲
Total energy
␮e
r e 共CC兲
r e 共GeC兲
⌰
␻ 1 (a 1 )
␻ 2 (a 1 )
␻ 3 (b 2 )
⫺2150.974 221
⫺2150.979 296
⫺2151.015 394
⫺2151.023 467
⫺2151.385 447
⫺2151.464 142
⫺2151.412 295
⫺2151.482 824
⫺2151.442 661
⫺2151.532 887
⫺2151.465 256
⫺2151.545 978
⫺2151.470 570
⫺2151.564 938
⫺2151.495 462
⫺2151.506 748
⫺2151.579 612
⫺2151.843 234
3.70
3.67
3.60
1.2536
1.2525
1.2539
1.2519
1.2641
1.2609
1.2658
1.2600
1.2743
1.2718
1.2755
1.2702
1.2825
1.2804
1.2842
1.2834
1.2789
1.2842
1.9431
1.9354
1.9375
1.9350
1.9379
1.9212
1.9237
1.9150
1.9467
1.9289
1.9296
1.9211
1.9555
1.9373
1.9375
1.9371
1.9291
1.9363
37.64
37.75
37.76
37.75
38.07
38.31
38.41
38.41
38.21
38.50
38.60
38.61
38.29
38.59
38.70
38.69
38.72
38.73
1938
1943
1938
1943
1850
1878
1855
1874
1781
1802
1793
1808
1725
1744
1735
1739
1750
1738
665
667
669
667
665
677
679
678
650
662
668
665
633
646
652
640
649
644
283i
259i
264i
257i
215i
166i
144i
154i
172i
95i
43i
73i
170i
80i
36
38
38i
105i
3.48
3.28
3.25
consistent basis sets 共cc-pVXZ兲. In the correlated relativistic
calculations, core and core–valence correlation effects were
explicitly included.
Four basis sets, TZ2P⫹diff, TZ3P共2f兲, cc-pVTZ, and ccpVQZ, were used at the SCF, CISD, and CCSD levels, for all
three structures. The triple-␨ 共TZ兲 valence basis for germanium is obtained from Schäfer, Huber, and Alhrichs19 with
the contraction scheme (17s12p6d/6s5p2d). The TZ basis
set for carbon is from Dunning’s contraction20 of Huzinaga’s
primitive Gaussian set,21 with the contraction scheme
(10s6p/5s3p). The detailed descriptions of these basis sets
were given in our recent study of HCGe.22 To obtain more
reliable results for the L-shaped and T-shaped geometries,
which are the main candidates for the ground state, an augcc-pVTZ basis set was also included at the CCSD共T兲 level.
All correlation-consistent basis sets were obtained from the
EMSL basis set library.23
The cc-pwCVTZ basis set for germanium is constructed
here from the cc-pVTZ set by adding two s functions with
orbital exponents 5.6565 and 0.9693, two p functions with
orbital exponents 3.1638 and 1.4818, two d functions with
␣⫽2.9448 and 1.0686, two f functions 共5.2610 and 1.3303兲,
and one g function 共1.5394兲. This new basis set, designated
correlation-consistent polarized weighted core–valence
triple-␨ 共cc-pwCVTZ兲, is weighted so that core–valence correlation effects are stressed. This choice is due to the fact that
we employ the mass-velocity and Darwin 共MVD兲 contributions for recovering the relativistic effects. This perturbative
method for recovering relativistic effects couples with the
correlation treatment 共especially correlation effects involving
electrons near the nuclei兲 because it is based on the Breit–
Pauli relativistic Hamiltonian.24,25 Therefore, we tried to
avoid overemphasizing the core–core correlation when employing the cc-pwCVTZ basis set. Pure angular momentum d
and f functions were used throughout.
The geometry optimizations and harmonic vibrational
frequency evaluations for all three structures were performed
using analytic first and second derivative methods26,27 at the
SCF level. At the CISD level, optimizations were performed
using gradients, whereas for the frequency calculations fivepoint numerical differentiation of the total energies was used.
At all coupled-cluster levels, both geometry optimizations
and harmonic vibrational frequency evaluations were carried
out using five-point numerical differentiation of the total energies. In the relativistic optimizations and frequency calculations, the same five-point procedure was used with the total
relativistic energies 共nonrelativistic⫹MVD correction兲. Cartesian forces at optimized geometries were required to be
less than 10⫺7 hartree/bohr in all geometry optimizations.
In the evaluation of the relativistic energy corrections,
the one-electron Darwin term, which is always positive, and
the mass–velocity term, which is always negative, were
evaluated using first-order perturbation theory.28,29 The Darwin term corrects the Coulomb attraction, and the mass–
velocity term corrects the kinetic energy of the system. This
level of relativistic treatment gives adequate results for germanium compounds 共and other atoms up to Z⫽40) compared to methods such as Dirac–Hartree–Fock 共DHF兲 and
the use of relativistic effective core potentials 共RECP兲.30,31
Throughout our study, all computations were carried out
using the PSI 2.0.8 program package,32 except the evaluation
for relativistic effects which were performed using the ACES
33
II package. IBM RS/6000 workstations, an IBM SP2, and
PCs were used.
IV. RESULTS AND DISCUSSION
A. Nonrelativistic results
We present the equilibrium geometries, total energies,
dipole moments evaluated at some levels, and harmonic vibrational frequencies for the T-shaped geometry ( 1 A 1 ) in
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J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
Germanium dicarbide
10011
TABLE II. Total nonrelativistic energies 共in hartree兲, bond distances 共in Å兲, dipole moments 共in Debye兲, and harmonic vibrational frequencies 共in cm⫺1兲 for
the linear geometry ( 1 兺 ⫹ symmetry兲.
Total energy
␮e
r e 共CC兲
r e 共GeC兲
␻ 1 共␴兲
␻ 2 共␲兲
␻ 3 共␴兲
TZ2P⫹diff SCF
TZ3P共2f兲 SCF
cc-pVTZ SCF
⫺2150.972 623
⫺2150.978 020
⫺2151.013 818
5.41
5.38
1.2596
1.2594
1.2608
1.7592
1.7549
1.7547
2033
2040
2039
92i
72i
68i
699
702
703
cc-pVQZ SCF
TZ2P⫹diff CISD
TZ3P共2f兲 CISD
⫺2151.021 455
⫺2151.380 541
⫺2151.459 335
1.2589
1.2690
1.2667
1.7543
1.7652
1.7548
2037
1968
1993
83i
92i
57i
702
675
687
cc-pVTZ CISD
cc-pVQZ CISD
⫺2151.406 247
⫺2151.476 538
1.2712
1.2658
1.7559
1.7483
1988
1996
54i
26i
685
693
TZ2P⫹diff CCSD
TZ3P共2f兲 CCSD
cc-pVTZ CCSD
⫺2151.437 311
⫺2151.527 896
⫺2151.458 566
1.2786
1.2771
1.2801
1.7757
1.7648
1.7644
1911
1932
1933
102i
72i
73i
654
665
667
cc-pVQZ CCSD
TZ2P⫹diff CCSD共T兲
⫺2151.539 271
⫺2151.467 811
1.2754
1.2901
1.7562
1.7901
1942
1839
41i
93i
674
622
TZ3P共2f兲 CCSD共T兲
cc-pVTZ CCSD共T兲
cc-pVQZ CCSD共T兲
cc-pwCVTZ CCSD共T兲
⫺2151.562 810
⫺2151.491 380
⫺2151.575 353
⫺2151.839 143
1.2885
1.2916
1.2866
1.2925
1.7784
1.7780
1.7692
1.7749
1863
1864
1873
1857
50i
60i
12i
79i
634
636
643
633
5.35
5.32
5.12
Table I. Similar properties for the linear structure ( 1 兺 ⫹ ) are
given in Table II, and those for the L-shaped geometry ( 1 A ⬘ )
may be seen in Table III.
1. Geometries
Table I indicates that the T-shaped geometry was optimized at every level of theory. The C–C bond length r e (CC)
systematically increases with the inclusion of correlation effects. At the SCF level it is ⬃1.25 Å, at the CISD level
⬃1.26 Å, at the CCSD level ⬃1.27 Å, and at the CCSD共T兲
level ⬃1.28 Å. However, the same trend is not observed for
the Ge–C bond distance. As seen in Table I, the basis set
dependence is more apparent than the effects of correlation
for r e 共GeC兲. This distance changes between 1.96 and 1.92 Å
depending on the basis set choice. For instance, with the
TZ2P⫹diff basis set r e 共GeC兲 is 1.943 Å at the SCF level,
1.938 Å at the CISD level, 1.947 Å at the CCSD level, and
1.956 Å at the CCSD共T兲 level, which distances are reasonably close to each other. At the highest nonrelativistic level
of theory, CCSD共T兲/cc-pVQZ, r e 共CC兲 is predicted to be
1.279 Å and r e (GeC) is 1.929 Å. The linear geometry ( 1 兺 ⫹ )
has similar C–C bond distances to the T-shaped structure. As
TABLE III. Total nonrelativistic energies 共in hartree兲, bond distances 共in Å兲, dipole moments 共in Debye兲, and harmonic vibrational frequencies 共in cm⫺1兲 for
the L-shaped geometry ( 1 A ⬘ symmetry兲.
Total energy
␮e
r e 共CC兲
r e 共GeC兲
⌰
␻ 1 (a ⬘ )
␻ 2 (a ⬘ )
␻ 3 (a ⬘ )
162
TZ2P⫹diff SCF
⫺2150.977 849
4.40
1.2615
1.7822
100.53
1916
878
TZ3P共2f兲 SCF
⫺2150.982 279
4.37
1.2609
1.7796
99.49
1919
877
159
cc-pVTZ SCF
cc-pVQZ SCF
TZ2P⫹diff CISD
TZ3P共2f兲 CISD
⫺2151.018 486
⫺2151.026 371
⫺2151.386 991
⫺2151.464 491
4.27
1.2622
1.2602
1.2710
1.2677
1.7807
1.7795
1.7988
1.7920
99.31
99.20
93.19
90.86
1914
1915
1823
1845
877
875
836
838
165
160
162
143
cc-pVTZ CISD
cc-pVQZ CISD
⫺2151.412 824
⫺2151.483 414
1.2718
1.2664
1.8006
1.7918
88.87
89.11
1828
1843
824
829
136
143
TZ2P⫹diff CCSD
TZ3P共2f兲 CCSD
cc-pVTZ CCSD
⫺2151.443 616
⫺2151.533 135
⫺2151.465 276
1.2812
1.2780
1.2783
1.8116
1.8085
1.8422
92.14
88.41
80.95
1754
1775
1774
810
800
735
141
103
59
cc-pVQZ CCSD
TZ2P⫹diff CCSD共T兲
⫺2151.546 061
⫺2151.472 036
1.2748
1.2924
1.8167
1.8107
84.37
98.15
1782
1695
769
813
85
121
TZ3P共2f兲 CCSD共T兲
cc-pVTZ CCSD共T兲
aug-cc-pVTZ CCSD共T兲
⫺2151.565 544
⫺2151.495 586
⫺2151.506 847
1.2909
1.2927
1.2922
1.7978
1.8087
1.8071
97.76
92.28
93.04
1718
1703
1704
830
813
813
100
86
81
cc-pVQZ CCSD共T兲
cc-pwCVTZ CCSD共T兲
⫺2151.579 785
⫺2151.843 890
1.2874
1.2941
1.8024
1.7995
91.43
95.18
1713
1703
809
819
96
116
4.01
4.00
3.80
3.41
3.70
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10012
J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
presented in Table II, the trend in r e (CC) with the inclusion
of correlation effects parallels that for the T-shaped geometry. However, the linear Ge–C bond length is about 0.15–
0.20 Å shorter than that of the T-shaped structure at every
level of theory. Our most reliable nonrelativistic values for
the r e (CC) and r e (GeC) distances at the linear stationary
point geometry are 1.287 and 1.769 Å, respectively.
The L-shaped ( 1 A ⬘ ) equilibrium geometry has a slightly
longer C–C bond distance than the T-shaped and linear structures, by about 0.01 Å. On the other hand, the L-shaped
Ge–C bond distance falls between that of the T-shaped and
linear geometries. The effects of electron correlation on the
L-shaped geometry are similar to those found for the
T-shaped structure. The primary difference is an increase in
the C–C bond distance. An important point is that the bond
angle in the L-shaped geometry is very much dependent on
both the theoretical model and basis set. As shown in Table
III, larger basis sets produce smaller bond angles. However,
as we include electron correlation, no regular trend was observed. Although the CISD and CCSD Ge–C–C angles are
smaller than the SCF values, the CCSD共T兲 angles are larger
than both CISD and CCSD. This is not very surprising because, as will be discussed later, the potential energy surface
is extremely flat, and even 2–3 kcal/mol of energy is enough
to invert the Ge atom with respect to the C–C bond.
Nielsen et al.34 published a significant theoretical 共ab
initio兲 paper on SiC2 in 1997. Their predictions for the C–C
bond distances in the T-shaped, linear, and L-shaped structures are very similar to our predictions for GeC2. The Ge–C
bond distance is about 0.10, 0.06, and 0.10 Å longer than the
analogous Si–C distances in the T-shaped, linear, and
L-shaped geometries, respectively. Nielsen and co-workers
reported that they could not locate an L-shaped stationary
point at several levels of theory. Here we find an L-shaped
GeC2 stationary point at every level of theory. As is now well
known, the true ground state geometry of SiC2 is T-shaped,
and the experimental structural parameters35 are r 0 (CC)
⫽1.269 Å, r 0 (SiC)⫽1.832 Å, while the C–Si–C bond angle
is 40.4°. The predicted geometrical parameters for the
T-shaped GeC2 are r e (CC)⫽1.279 Å, r e (GeC)⫽1.929 Å,
and ⌰ e (CGeC)⫽38.7° at the CCSD共T兲/cc-pVQZ level of
theory. However, as we will discuss in the next section, unlike SiC2 the T-shaped geometry of GeC2 is found to be a
transition state rather than the global minimum.
2. Harmonic vibrational frequencies
The T-shaped geometry ( 1 A 1 ), which is the true global
minimum for SiC2, is found to be a stationary point having
an imaginary harmonic vibrational frequency, for the asymmetric stretching mode ␻ 3 (b 2 ). As presented in Table I, although the imaginary value is getting smaller as correlation
effects are included, our CCSD共T兲 GeC2 ␻ 3 (b 2 ) predictions
are still imaginary, except with the cc-pVTZ and aug-ccpVTZ basis sets, for which real frequencies of 36 and 38
cm⫺1 were obtained, respectively. However, the cc-pVQZ
CCSD共T兲 method yields an imaginary ␻ 3 (b 2 ) of 38i cm⫺1,
and the core–valence correlation-consistent basis set 共ccpwCVTZ兲 produced an imaginary frequency of 105i cm⫺1.
Interestingly, as discussed later, relativistic corrections turn
Sari et al.
the real ␻ 3 (b 2 ) values for the cc-pVTZ and aug-cc-pVTZ
basis sets into imaginary values. Therefore, considering that
almost all nonrelativistic levels 共except the two just mentioned兲, the core–valence correlated correlation-consistent
basis set at the CCSD共T兲 level, and all relativistic computations, produce imaginary values for the asymmetric stretching ␻ 3 (b 2 ) mode, the T-shaped GeC2 is predicted to be a
transition state between the two L-shaped geometries. Although our theoretical results support this conclusion, we
realize that new experiments are necessary in this respect,
because the system is elusive and the surface is extraordinarily flat.
The C–C ␻ 1 (a 1 ) stretching vibrational frequency of the
T-shaped structure decreases with more sophisticated treatments of correlation effects, whereas the Ge–C stretching
␻ 2 (a 1 ) frequency is mainly dependent on the basis set
choice, similar to the Ge–C bond length. At the highest nonrelativistic level of theory, CCSD共T兲/cc-pVQZ, T-shaped frequencies of 1750, 649, and 38i cm⫺1 are predicted for the
␻ 1 (a 1 ), ␻ 2 (a 1 ), and ␻ 3 (b 2 ) modes, respectively.
The linear structure has an imaginary harmonic vibrational frequency for the degenerate bending mode ␻ 2 ( ␲ ).
This imaginary frequency, as well as the C–C stretching frequency ␻ 1 ( ␴ ), decreases as we improve the level of theory.
The linear Ge–C stretching harmonic vibrational frequency
␻ 3 ( ␴ ) is slightly higher at the SCF, CISD, and CCSD levels
than the values for the corresponding mode of the T-shaped
␻ 2 (a 1 ), whereas at the CCSD共T兲 level ␻ 3 ( ␴ ) is slightly
lower than ␻ 2 (a 1 ). Although some ab initio and DFT
methods34,36 give a spurious real vibrational frequency for
the bending mode of SiC2 in the linear structure, it has been
concluded both from experiment and theory14 –16,34 that the
linear geometry of SiC2 is a transition state to the cyclic
T-shaped structure. In contrast, we have obtained imaginary
frequencies for the bending ␻ 2 ( ␲ ) mode of GeC2 at all levels of theory 共Table II兲. Therefore, the linear geometry of
GeC2 is a transition state. However, as mentioned above, this
transition state cannot connect two T-shaped structures because the T-shaped geometry is also predicted to be a transition state. Consequently, the linear geometry of GeC2 should
be a transition state between the two equivalent L-shaped
geometries.
The L-shaped geometry was successfully optimized at
every level of theory, and real values are obtained for all
harmonic vibrational frequencies, as may be observed in
Table III. All three vibrational frequencies decrease with
higher level treatments of electron correlation, with the exception that the CCSD共T兲 values for the Ge–C stretching
mode ( ␻ 2 ) and Ge–C–C bending mode ( ␻ 3 ) are slightly
higher than those predicted by CCSD. At the highest nonrelativistic level, CCSD共T兲/cc-pVQZ, frequencies of 1713, 809,
and 96 cm⫺1 are predicted for the C–C stretching ( ␻ 1 ),
Ge–C stretching ( ␻ 2 ), and Ge–C–C bending ( ␻ 3 ) modes,
respectively. Because we did not encounter any imaginary
vibrational frequencies, and found this L-shaped geometry to
lie energetically lowest 共see Sec. IV A 4 below兲, the
L-shaped geometry is predicted to be the ground state equilibrium geometry. As will be discussed later, the inclusion of
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J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
Germanium dicarbide
10013
FIG. 2. The relative energies of the T-shaped ( 1 A 1 ) and linear ( 1 兺 ⫹ ) geometries with respect to the L-shaped ( 1 A ⬘ ) geometry.
core–valence correlation and relativistic corrections supports
this conclusion.
3. Dipole moments
Some predicted dipole moments are presented in Tables
I–III. For all three geometries, the molecule has a significant
dipole moment. At the CISD/cc-pVTZ level of theory, values
of 3.28, 5.12, and 3.80 Debye are predicted for the T-shaped,
linear, and L-shaped geometries, respectively. At the
T-shaped geometry, the Ge atom has a positive Mulliken
charge of 0.50, and each C atom has a negative charge of
⫺0.25. For the linear geometry, the Ge atom has a positive
Mulliken charge of 0.46 and the adjacent C atom has a negative charge of ⫺0.45. The second C atom in the linear geometry is essentially neutral in this oversimplified picture. At
the L-shaped geometry, the Ge atom again is predicted to
have a positive charge of 0.34, the neighboring C atom has a
negative charge of ⫺0.20, and the other C atom has the
remaining negative charge of ⫺0.14. These Mulliken atomic
populations were predicted at the CISD/cc-pVTZ level of
theory.
4. Energetics
At all nonrelativistic levels of theory, the L-shaped geometry is found to be lower in energy than the T-shaped and
linear structures. This supports our earlier conclusion from
the vibrational analysis that the L-shaped structure is the global minimum for the ground state of GeC2. The relative energies of the T-shaped and linear geometries with respect to
the L-shaped structure are plotted in Figs. 1 and 2. The relative energy of the T-shaped geometry decreases as we in-
crease the level of theory. At the CCSD共T兲/cc-pVQZ level,
the T–L energy difference is only 0.11 kcal/mol.
It is necessary to include the zero-point vibrational energy 共ZPVE兲 correction to obtain a quantum mechanical energy separation. However, due to the extremely flat and sensitive potential energy surface of the GeC2 system, the ZPVE
corrections should be evaluated cautiously. According to the
study by Grev, Janssen, and Schaefer,37 the ZPVE estimated
from SCF harmonic vibrational frequencies is usually overestimated by few tenths of a kcal/mol. Thus, they recommended a scaling factor of 0.91 for determination of ZPVEs.
With the CCSD共T兲 method, a scaling factor of 0.95 would be
a reasonable estimate to obtain the ZPVE correction from
CCSD共T兲 harmonic vibrational frequencies. At the
CCSD共T兲/cc-pVQZ level, the T-shaped isomer is predicted
to be a transition state, the ZPVE correction for the T–L
energy difference being ⫺0.31 共⫺0.30兲 kcal/mol. However,
the T-shaped isomer is predicted to be a minimum at the
CCSD共T兲/cc-pVTZ and CCSD共T兲/aug-cc-pVTZ levels.
The ZPVE corrections for the T–L energy difference at these
two levels are ⫺0.26 共⫺0.24兲 kcal/mol and ⫺0.25 共⫺0.23兲
kcal/mol, respectively. The above ZPVE corrections in parentheses are the scaled values. Since the T-shaped structure
is a reasonable candidate for the global minimum, it should
be treated as if it were a minimum. Thus, employing the
scaled ZPVE correction of ⫺0.23 kcal/mol and classical
relative energy of 0.11 kcal/mol at the CCSD共T兲/cc-pVQZ
level, the nonrelativistic quantum mechanical T–L energy
separation becomes ⫺0.12 kcal/mol.
An energy difference this small suggests the need for
highly precise experimental work or more advanced theoretical methods 共perhaps some kind of extrapolation technique
Downloaded 20 Nov 2002 to 134.121.44.19. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
10014
Sari et al.
J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
to the full CI limit兲. The basis set convergence at the CISD,
CCSD, and CCSD共T兲 levels does not absolutely guarantee
our assignment of the L-shaped geometry to the equilibrium
geometry 共see Fig. 2兲. However, the basis set convergence
appears satisfactory, and the convergence seems to be approached near the cc-pVQZ basis set. The cc-pVQZ T–L
energy differences are very slightly greater than the cc-pVTZ
results at the correlated levels 关by 0.038 kcal/mol higher with
CISD, 0.039 kcal/mol with CCSD, and 0.031 kcal/mol with
CCSD共T兲兴. This observation may suggest that larger basis
sets will not significantly lower the relative energy. As a
result, although all of our nonrelativistic and relativistic 共see
below兲 calculations put the T-shaped geometry above the
L-shaped structure, we are concerned that a definitive conclusion about the energetics of these two geometries is difficult due to the extremely flat nature of the ground state potential energy surface.
The energy relative to the global minimum is more evident for the linear structure. As displayed in Fig. 2, the SCF
method puts the linear geometry about 3 kcal/mol, CISD and
CCSD about 4 kcal/mol, and CCSD共T兲 2.5–3.0 kcal/mol
above the L-shaped structure. At the CCSD共T兲 level, except
for one basis set, TZ3P共2f兲, the other seven basis sets produced values between 2.5 and 3.0 kcal/mol for the relative
energy of the linear geometry. The best nonrelativistic classical energy separation is predicted to be 2.78 kcal/mol at the
CCSD共T兲/cc-pVQZ level and the quantum mechanical energy difference 共with the scaled ZPVE correction兲 to be 2.64
kcal/mol. Therefore, the linear structure is certainly not the
true global minimum.
B. Effects of core–valence correlation
As described in Sec. III, the cc-pwCVTZ basis set documented here for Ge was used at the CCSD共T兲 level to consider core–valence electron correlation effects. The results
for this level of theory, CCSD共T兲/cc-pwCVTZ, are included
in Tables I–III. The predictions with this basis set should be
compared with those for the cc-pVTZ basis set in order to
deduce the differential effects of Ge core–valence correlations. This is because the only difference between the two
basis sets is the usage of the core–valence correlated version
of the normal cc-pVTZ basis sets for the Ge atom 共see the
above theoretical procedures for more information兲.
The most obvious effect of the inclusion of Ge core–
valence correlation was seen to be on the asymmetric stretching mode ␻ 3 (b 2 ) of the T-shaped geometry and on the bending harmonic vibrational frequencies of the linear and
L-shaped geometries. This is expected because movement of
the Ge atom in the molecular plane requires very little energy, and all these harmonic vibrations are associated with
the motion of the Ge atom. As noted above, the CCSD共T兲/
cc-pVTZ level of theory predicts a real harmonic vibrational
frequency of 36 cm⫺1 for the asymmetric stretching mode
␻ 3 (b 2 ) of the T-shaped geometry, although most other methods predict imaginary values. As seen in Table I, the inclusion of the core–valence correlation replaces the real frequency with an imaginary value of 105i cm⫺1. This is quite
important because this core–valence correlation effect very
much supports our earlier finding that the T-shaped geometry
of GeC2 is a transition state. The situation is similar in the
linear case. As presented in Table II, our result for the bending harmonic vibration ␻ 2 ( ␲ ) with the normal cc-pVTZ basis set 关at the CCSD共T兲 level兴 is 60i cm⫺1 and the inclusion
of the core–valence correlation produces a larger imaginary
value of 79i cm⫺1. This confirms our earlier prediction that
the linear geometry is a transition state. The inverse effect is
seen for the L-shaped geometry 共see Table III兲. The Ge–C–C
bending harmonic vibrational frequency ( ␻ 3 ) becomes more
real, 116 cm⫺1 共the cc-pVTZ prediction is 86 cm⫺1兲. In other
words, the core–valence correlation effects associated with
the Ge atom in the L-shaped geometry decrease the likelihood that the L-shaped structure is a transition state.
The effects of the core–valence correlation on the energetics of the GeC2 system are also important. The relative
energy of the T-shaped geometry with the cc-pwCVTZ basis
set is 0.41 kcal/mol, and it is larger by 0.33 kcal/mol relative
to that 共0.078 kcal/mol兲 with the cc-pVTZ basis set. This
stabilization of the L-shaped geometry with respect to the
T-shaped structure also suggests the L-shaped to be the favored choice for the ground state geometry. On the other
hand, the energy difference between the linear and L-shaped
geometries is 2.98 kcal/mol with the cc-pwCTZ basis set and
2.64 kcal/mol with the cc-pVTZ basis set. Therefore, the
relative energy of the linear structure increases with the inclusion of core–valence correction as well, by a similar
amount of 0.34 kcal/mol with respect to the L-shaped geometry.
C. Effects of relativistic corrections
The relativistic optimizations are computationally demanding. They require nine single point and nine numerical
first derivative CCSD共T兲 calculations for the T-shaped and
linear geometries, and 13 single points and 13 numerical first
derivative CCSD共T兲 calculations for the L-shaped geometry
just for one optimization cycle. Therefore, we focused more
attention on the T-shaped geometry. This is also because the
nonrelativistic results for the L-shaped and linear geometries
are definitive, whereas those for the L-shaped and T-shaped
geometries are not. As discussed earlier, the T–L energy difference is small, and two of our nonrelativistic levels of
theory gave real frequencies for the asymmetric stretching
vibration ␻ 3 (b 2 ) of the T-shaped geometry 共see Table I兲.
Therefore, we carried out relativistic optimizations and frequency evaluations for the T-shaped geometry with four different basis sets at the CCSD共T兲 level. For the linear geometry we employed three different basis sets, whereas only one
basis set was used for the L-shaped geometry, due to the high
computational cost 关a total of 52 single point and 52 numerical first derivative CCSD共T兲 calculations in C s symmetry
were performed for the relativistic optimization of the
L-shaped geometry with one basis set兴. We present the effects of relativistic corrections in Table IV for the T-shaped,
in Table V for the linear, and Table VI for the L-shaped
geometries.
The effects of relativity on the geometries are somewhat
different among the three structures. The Ge–C bond shortens by ⬃0.003 Å in the T-shaped geometry, by ⬃0.005 Å in
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J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
Germanium dicarbide
10015
TABLE IV. The effects of relativistic corrections, namely mass–velocity and one-electron Darwin terms, on the geometry, energetics, and harmonic
vibrational frequencies of the T-shaped geometry ( 1 A 1 symmetry兲.
CCSD共T兲/cc-pVTZ
Total energy
r e 共CC兲
r e 共GeC兲
⌰
␻ 1 (a 1 )
␻ 2 (a 1 )
␻ 3 (b 2 )
CCSD共T兲/aug-cc-pVTZ
Total energy
r e 共CC兲
r e 共GeC兲
⌰
␻ 1 (a 1 )
␻ 2 (a 1 )
␻ 3 (b 2 )
CCSD共T兲/cc-pVQZ
Total energy
r e 共CC兲
r e 共GeC兲
⌰
␻ 1 (a 1 )
␻ 2 (a 1 )
␻ 3 (b 2 )
CCSD共T兲/cc-pwCVTZ
Total energy
r e 共CC兲
r e 共GeC兲
⌰
␻ 1 (a 1 )
␻ 2 (a 1 )
␻ 3 (b 2 )
Nonrelativistic
Relativistic
⌬
⫺2151.495 462
1.2842
1.9375
38.70
1735.1
652.2
35.5
⫺2172.748 839
1.2841
1.9341
38.78
1734.4
649.3
75.7i
⫺21.253 377
⫺0.0001
⫺0.0034
⫹0.08
⫺0.7
⫺2.9
35.5→75.7i
⫺2151.506 748
1.2834
1.9371
38.69
1739.0
648.9
37.8
⫺2172.760 546
1.2833
1.9335
38.76
1736.1
646.3
82.4i
⫺21.253 798
⫺0.0001
⫺0.0036
⫹0.07
⫺2.9
⫺2.6
37.8→82.4i
⫺2151.579 612
1.2789
1.9291
38.72
1750.1
648.7
38.0i
⫺2172.834 445
1.2790
1.9261
38.78
1747.8
647.7
74.7i
⫺21.254 833
⫹0.0001
⫺0.0030
⫹0.06
⫺2.3
⫺1.0
38.0i→74.7i
⫺2151.843 234
1.2842
1.9363
38.73
1737.5
644.3
105.0i
⫺2173.098 334
1.2841
1.9336
38.79
1734.1
640.4
131.1i
⫺21.255 100
⫺0.0001
⫺0.0027
⫹0.06
⫺3.4
⫺3.9
105.0i→131.1i
TABLE V. The effects of relativistic corrections, namely mass–velocity and one-electron Darwin terms, on the geometry, energetics, and harmonic vibrational
frequencies of the linear geometry ( 1 兺 ⫹ symmetry兲.
CCSD共T兲/cc-pVTZ
Total energy
r e 共CC兲
r e 共GeC兲
␻ 1 共␴兲
␻ 2 共␲兲
␻ 3 共␴兲
CCSD共T兲/cc-pVQZ
Total energy
r e 共CC兲
r e 共GeC兲
␻ 1 共␴兲
␻ 2 共␲兲
␻ 3 共␴兲
CCSD共T兲/cc-pwCVTZ
Total energy
r e 共CC兲
r e 共GeC兲
␻ 1 共␴兲
␻ 2 共␲兲
␻ 3 共␴兲
Nonrelativistic
Relativistic
⌬
⫺2151.491 380
1.2916
1.7780
1864.1
60.4i
635.8
⫺2172.745 452
1.2914
1.7723
1863.0
52.7i
634.2
⫺21.254 072
⫺0.0002
⫺0.0057
⫺1.1
60.4i→52.7i
⫺1.6
⫺2151.575 353
1.2866
1.7692
1873.3
11.8i
643.4
⫺2172.830 886
1.2864
1.7643
1872.9
34.3
642.7
⫺21.255 533
⫺0.0002
⫺0.0049
⫺0.4
11.8i→34.3
⫺0.7
⫺2151.839 143
1.2925
1.7749
1856.9
79.2i
632.6
⫺2173.094 758
1.2923
1.7703
1854.8
88.4i
630.2
⫺21.255 627
⫺0.0002
⫺0.0046
⫺2.1
79.2i→88.4i
⫺2.4
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10016
J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
TABLE VI. The effects of relativistic corrections, namely mass–velocity
and one-electron Darwin terms, on the geometry, energetics, and harmonic
vibrational frequencies of the L-shaped geometry ( 1 A ⬘ symmetry兲.
CCSD共T兲/cc-pVTZ
Total energy
r e 共CC兲
r e 共GeC兲
⌰
␻ 1 (a ⬘ )
␻ 2 (a ⬘ )
␻ 3 (a ⬘ )
Nonrelativistic
Relativistic
⌬
⫺2151.495 586
1.2927
1.8087
92.28
1703.0
812.7
86.1
⫺2172.749 386
1.2937
1.7960
96.38
1705.9
827.4
106.7
⫺21.253 800
⫹0.0010
⫺0.0127
⫹4.1
⫹2.9
⫹14.7
⫹20.6
the linear case, and by ⬃0.013 Å in the L-shaped geometry.
The C–C bond distance decreases very slightly for the
T-shaped and linear geometries, whereas it elongates by
⬃0.001 Å for the L-shaped geometry. The relativistic effects
on the L-shaped geometry are considerably larger than those
for the other two structures.
For the T-shaped geometry, the asymmetric stretching
vibrational frequency ␻ 3 (b 2 ) becomes more imaginary when
relativity is considered. The two real frequencies, 36 and 38
cm⫺1 obtained at the CCSD共T兲/cc-pVTZ and CCSD共T兲/augcc-pVTZ levels, shifted to imaginary values of 76i and 82i
cm⫺1, respectively, as presented in Table IV. At the highest
level of theory, CCSD共T兲/cc-pVQZ, the imaginary value of
38i cm⫺1 becomes 75i cm⫺1 for the same mode. This shift is
in the same direction as the effects of core–valence correlation 共see the previous section兲. Therefore, depending on the
relativistic and core–valence correlated results, we can say
that both of these corrections tend to move the Ge atom in
Sari et al.
the T-shaped geometry toward the L-shaped geometry. This
interesting effect of relativity may be attributed to the fact
that the valence electrons, which mainly contribute to the
chemical bonding, also display direct relativistic effects due
to their core-penetrating natures as well as indirect relativistic effects.
The main relativistic effects for the linear structure are
observed for the bending harmonic vibrational frequency
␻ 2 ( ␲ ), as shown in Table V. The CCSD共T兲/cc-pVTZ basis
set produces a lower imaginary frequency, whereas the
CCSD共T兲/cc-pwCVTZ generates a higher imaginary value
compared to the corresponding nonrelativistic imaginary
bending ␻ 2 ( ␲ ) frequencies. The CCSD共T兲/cc-pVQZ basis
set shifts the small imaginary value of 12i cm⫺1 to a real
value of 34 cm⫺1 when relativity is included. This shift
should not be taken too seriously because the relativistic
CCSD共T兲/cc-pwCVTZ level, which has a core–valence correlation, presents a larger imaginary bending frequency, as
mentioned above. For the L-shaped structure, the GeC
stretching ( ␻ 2 ) and Ge–C–C bending ( ␻ 3 ) frequencies increase by 15 and 21 cm⫺1, respectively, upon the inclusion of
the relativity.
The relativistic effects on the energetics of the system
are significant. The relative energy of the T-shaped geometry
increases by 0.27 kcal/mol at the CCSD共T兲/cc-pVTZ, 0.23
kcal/mol at the CCSD共T兲/cc-pVQZ, and 0.27 kcal/mol at the
CCSD共T兲/cc-pwCVTZ levels of theory. These energy shifts
are important because the nonrelativistic energy T–L differences at the CCSD共T兲 levels are very close to zero 共see Fig.
2兲. Unlike the T-shaped geometry the linear geometry is stabilized with respect to the L-shaped structure, by 0.17, 0.21,
FIG. 3. The effects of relativistic corrections on the relative energies of the T-shaped ( 1 A 1 ) and linear ( 1 兺 ⫹ ) geometries with respect to the L-shaped ( 1 A ⬘ )
geometry.
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J. Chem. Phys., Vol. 117, No. 22, 8 December 2002
Germanium dicarbide
10017
FIG. 4. The proposed potential energy
surface for the ground state of GeC2 at
the CCSD共T兲/cc-pVQZ level with the
inclusion of zero-point vibrational
energy, scalar relativistic, and core–
valence corrections.
and 0.07 kcal/mol, at the CCSD共T兲/cc-pVTZ, CCSD共T兲/ccpVQZ, and CCSD共T兲/cc-pwCVTZ levels, respectively.
These relativistic effects on the energetics of the system are
plotted in Fig. 3.
In Sec. IV A 4, the nonrelativistic classical and ZPVEcorrected T–L energy separations at the CCSD共T兲/ccpCVQZ level of theory were determined to be 0.11 and
⫺0.12 kcal/mol, respectively. Assuming the additivity of
core–valence correlation effects of ⫹0.33 kcal/mol in Sec.
IV B and the relativistic effects of ⫹0.23 kcal/mol, the relativistic quantum mechanical T–L energy difference is predicted to be ⫹0.44 kcal/mol. On the other hand, the nonrelativistic classical and quantum mechanical energy separations between the linear and L-shaped structures at the
CCSD共T兲/cc-pCVQZ level of theory were found to be 2.78
and 2.64 kcal/mol, respectively. Including core–valence correlation effects of ⫹0.34 kcal/mol and relativistic effects of
⫺0.21 kcal/mol, the relativistic quantum mechanical energy
difference between the linear and L-shaped structures is predicted to be ⫹2.77 kcal/mol. The final predicted shape of the
ground state potential energy surface at the relativistic
CCSD共T兲/cc-pVQZ level of theory is depicted in Fig. 4.
V. CONCLUDING REMARKS
The ground state potential energy surface of the experimentally observed GeC2 molecule reflects the same kind of
elusive structure as that of the much discussed SiC2. Although the T- and L-shaped geometries are very close in
energy, and a conclusive result is difficult to reach due to the
very flat nature of the potential energy surface, the L-shaped
structure is predicted to be the global minimum. The scalar
relativistic corrections 共mass–velocity and one-electron Darwin terms兲 and core–valence corrections are found to stabi-
lize the L-shaped geometry over the T-shaped structure. All
of our nonrelativistic and relativistic computations predict
the L-shaped structure to be lower in energy than the
T-shaped geometry, which is the equilibrium geometry for
SiC2. The linear structure is predicted to be a transition state,
as is the case with SiC2. However, this transition state connects the two L-shaped geometries, not the two T-shaped
structures. The predicted shape of the ground state potential
energy surface is given in Fig. 4. Finally, it is important to
note that new experimental work will be necessary to come
to definitive conclusions.
ACKNOWLEDGMENT
This research was supported by the U.S. National Science Foundation, Grant No. CHE-0136186.
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