The anomalous behavior of the Zeeman anticrossing spectra
of à 1Au acetylene: Theoretical considerations
George Vacek, C. David Sherrill, Yukio Yamaguchi, and Henry F. Schaefer III
The Center for Computational Quantum Chemistry, The University of Georgia, Athens, Georgia 30602
~Received 1 September 1995; accepted 23 October 1995!
P. Dupré, R. Jost, M. Lombardi, P. G. Green, E. Abramson, and R. W. Field have observed
anomalous behavior of the anticrossing density in the Zeeman anticrossing ~ZAC! spectra of gas
phase à 1Au acetylene in the 42 200 to 45 300 cm21 energy range. To best explain this result, they
hypothesize a large singlet–triplet coupling due to the existence of a linear isomerization barrier
connecting a triplet-excited cis- and trans-acetylene in the vicinity of the studied energy range
~;45 500 cm21!. Theoretically such a linear stationary point, however, must have two different
degenerate bending vibrational frequencies which are either imaginary or exactly zero. Neither case
has yet been experimentally detected. Here, we have studied the two lowest-lying linear
3
triplet-excited-state stationary points of acetylene, 3 S 1
u and D u , to see if they fit Dupré et al.’s
hypothesis. We have completed geometry optimization and harmonic vibrational frequency analysis
using complete-active-space self-consistent field ~CASSCF! wave functions as well as determined
energy points at those geometries using the second-order configuration interaction ~SOCI! method.
Harmonic vibrational analyses of both stationary points reveal two different doubly degenerate
vibrational modes with imaginary vibrational frequencies ~or negative force constants! indicating
that they are indeed saddle points with a Hessian index of four. At the DZP SOCI//CASSCF level
of theory with zero-point vibrational energy ~ZPVE! correction, the 3 S 1
u stationary point lies 35 840
cm21 above the ground state of acetylene. This is much too low in energy to contribute to the ZAC
spectral anomaly. At the same level of theory with ZPVE correction, the 3 D u stationary point lies
44 940 cm21 above the ground state consistent with Dupré et al.’s hypothesis. Several solutions to
the anomalous ZAC spectra are discussed. We propose that the anomaly may also be due to coupling
with a nearly linear structure on the T 3 surface of acetylene. © 1996 American Institute of Physics.
@S0021-9606~96!00705-5#
INTRODUCTION
Gas phase acetylene has been the focus of a great
amount of spectroscopic, photochemical, and theoretical research. Due to the large degree of attention it has received,
combined with the relative simplicity of the system, C2H2 is
perhaps one of the best understood tetra-atomic molecular
systems. Of course, there still remain many unanswered
questions, one of which relates to the anomalous behavior of
the Zeeman anticrossing ~ZAC! density in the spectra of gas
phase à 1Au trans-bent acetylene as a function of the excitation energy, as reported by Dupré, Jost, Lombardi, Green,
Abramson, and Field ~DJLGAF!.1 In the 42 200– 45 300
cm21 energy range above the rotationless zero-point level of
X̃ 1 S 1
g , they observed an unexpectedly large coupling of the
S 1 levels with the S 0 levels. Since u – g interactions are electronically forbidden, S 1 – S 0 interactions could at best be vibronically allowed. Their results clearly imply a two-step
model where S 1 and S 0 couple via triplet levels. They believe that the key to understanding the increase in S 1 anticrossing densities is an increase in the S 1 – T i and T i – S 0
coupling strengths ~i51, 2, or 3, most probably! which
would occur near the top of a T i isomerization barrier ~for
clarity see Fig. 2 of Demoulin’s work2 and Fig. 10 of
DJLGAF!.1 Specifically they hypothesize a linear isomerization barrier between cis-bent and trans-bent triplet acetylenes
located at an energy near the S 1 n 38 5 3 level in the vicinity of
1774
J. Chem. Phys. 104 (5), 1 February 1996
;45 500 cm21. S 1 @trans-bent Ã# can only interact strongly
with a cis-bent T i near the top of such a cis–trans isomerization barrier due to Franck–Condon overlap between the
vibrational levels of the excited S 1 state and the triplet state.
The nature of the large singlet–triplet coupling of the Ã
state of acetylene was further characterized by a series of
studies by Dupré, Green, and Field.3–5 First Dupré and
Green3 mapped the dependence of the energy levels on magnetic field, obtaining a Lande g factor near that expected for
a pure triplet state. Then Dupré, Green, and Field4 analyzed
results obtained by quantum beat spectroscopy of gaseous
acetylene, determining that inter-triplet couplings are sufficiently strong that the mediating triplet in S 1 – T i – S 0 may
not be pure but rather mixed-triplet levels. Finally, Dupré5
used Fourier transform techniques to statistically analyze the
Zeeman anticrossing spectra of à and was able to determine
the fundamental quantities driving the energy transfers of
intersystem crossing. All three followup studies to the original DJLGAF work seem to be either consistent with or supportive of the linear cis–trans isomerization barrier hypothesis. Furthermore, both Abe and Hayashi’s6 study of
fluorescence quenching rates and Drabbels, Heinze and
Meerts’s7 recent study of laser-induced fluorescence ~LIF!
spectra of specific vibronic bands of à acetylene are also
compatible with the hypothesis of DJLGAF.
Thus these experimental works seem to establish the ex-
0021-9606/96/104(5)/1774/5/$10.00
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Vacek et al.: Spectra of à 1Au acetylene
istence of a linear barrier connecting cis- and trans-bent triplet acetylenes. If so, these rather indirect experimental probes
have detected a remarkable point on the C2H2 potential energy surface. Following the arguments of East, Johnson, and
Allen,8 for a tetra-atomic molecule with bent equilibria lying
lower in energy than the optimum linear configuration: if
both cis- and trans-bent minima exist, then characteristically
either the transition state for cis–trans isomerization is nonlinear and the linear stationary point is a higher-order saddle
point, or the linear structure is a transition state for the cis–
trans isomerization and the interconversion of equivalent cis
and trans structures. In the improbable second case, the linear structure is a bifurcating transition state which has eigenvalues exactly equal to zero for both cis- and trans-bending
vibrational modes. A hypothetical discussion of the properties of such a transition state can be found in a paper by
Valtazanos and Ruedenberg.9
This leaves only a few possibilities for a linear triplet
acetylene state. First, it could be a true minimum on a tripletexcited surface with no cis- nor trans-bent minima. Second,
it could be a true transition state on a triplet excited surface
with only a cis or trans minimum, but not both. Third, it
could be not the true transition state for the cis–trans isomerization but rather a saddle point with two different degenerate imaginary bending vibrational frequencies, i.e., a saddle
point with Hessian index of four. Fourth, it could be the true
transition state and have two different degenerate bending
vibrational frequencies which are exactly zero. The first and
second possibilities, although interesting, are not expected to
occur on any of the T 1 , T 2 , or T 3 surfaces, since they are
theoretically predicted to have both cis- and trans-bent
minima.2,10,11 If those two possibilities would occur on the
T 4 ~or higher! surface, they would be expected to lie above
the necessary energy range, and regardless would not fit
DJLGAF’s criteria that the linear state connect cis- with
trans-bent minima. The third and fourth possibilities could
occur on the T 1 , T 2 , or T 3 surfaces, would satisfy the other
criteria for enhanced S 1 – T i – S 0 coupling, and would be the
first compelling experimental evidence of a higher-order
saddle point or bifurcation. In this work we will study the
linear stationary points on the three lowest-lying triplet
3
excited-state surfaces of acetylene, 3 S 1
u (T 1 ) and D u ~T 2
and T 3 !, to see whether they are consistent with DJLGAF’s
hypothesis. Note that we will use T 2/3 when refering to the
3
D u linear stationary point since the potential energy surfaces
are identical there. When refering to any other point on those
surfaces we will specify either T 2 or T 3 .
THEORETICAL METHODS
The basis set was a fairly standard DZP consisting of a
Huzinaga–Dunning–Hay double-z set12–14 of contracted
Gaussian functions designated (9s5 p/4s2p) for carbon and
(4s/2s) for hydrogen augmented with a single set of polarization functions with orbital exponents a d ~C!
5a p ~H!50.75. The molecular structures were fully optimized to residual Cartesian and internal coordinate gradients
less than 1026 a.u. using analytic gradient techniques for the
1775
3 1
3
TABLE I. Theoretically predicted properties of the X̃ 1 S 1
g , S u , and D u
states of acetylene at their respective DZP CASSCF optimized geometries.
Bond lengths are given in Ångstroms, harmonic vibrational frequencies in
cm21, total energies in Hartrees and zero-point vibrational energies ~ZPVE!
given in cm21.
Geom params
r e ~C–C!
r e ~C–H!
Harm vib freq
v1 ( s 1
g )
v2 ( s 1
g )
v3 ( s 1
u )
v4 ( p g )
v5 ( p u )
Energies
CASSCF
SOCI//CASSCF
ZPVE
S 0 (X̃) 1 S 1
g
T 1 3S 1
u
T 2/3 3 D u
~expt in parenth!
1.2124 ~1.2026!a
1.0605 ~1.0622!a
1.3448
1.0577
1.3389
1.0582
3648
1612
3602
1080i
1117i
3651
1637
3603
932i
928i
276.746 049
276.935 232
4431
276.713 343
276.893 834
4446
3652
2068
3569
472
649
~3495!b
~2008!b
~3415!b
~624!b
~747!b
276.929 597
277.104 630
5766
a
A. Baldacci, S. Ghersetti, S. C. Hurlock, and K. N. Rao, J. Mol. Spectrosc.
59, 116 ~1976!.
b
G. Strey and I. M. Mills, J. Mol. Spectrosc. 59, 103 ~1976!.
complete active space self-consistent field ~CASSCF! wave
function. The active space included the p- and p*-occupied
molecular orbitals ~MOs! and all the virtual MOs. The total
number of configurations were 16 923 for S 0 , 23 193 for T 1 ,
and 21 009 for T 2/3. At the optimized geometries, energies
were obtained using the method of second-order configuration interaction ~SOCI!.15 The shape driven graphical unitary
group approach16 was used to obtain SOCI wave functions,
with a full CI treatment within an active space and all single
and double excitations out of that active space ~but the two
lowest core MOs were kept doubly occupied in all configurations. With the SOCI wave function, the active space for
full CI included the three lowest-energy MOs, whether occupied or virtual, in the irreducible representations of the pand p*-occupied MOs within a D 2h -symmetry constraint
~that is, MOs in the B 2u , B 3u , B 2g , and B 3g irreps if the
molecular axis is considered as the z axis!. The total number
of configurations were 2 265 768 for S 0 , 4 151 785 for T 1 ,
and 4 135 896 for T 2/3. In all cases energy points were obtained using the CASSCF wave function as a reference wave
function for further correlation. Harmonic vibrational frequencies were determined at the optimized geometries by the
method of finite differences. All calculations were performed
using the PSI suite of codes17 on IBM RS6000 workstations.
The CASSCF capabilities of PSI have been recently added
by one of us ~Y.Y.!.
RESULTS
The DZP CASSCF optimized geometries of the linear
stationary points on the S 0 , T 1 , and T 2 potential energy surfaces are presented in Table I along with harmonic vibrational frequencies for those structures and total energies. In
the case of S 0 , established experimental values of the geometric parameters and harmonic vibrational frequencies are
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Vacek et al.: Spectra of à 1Au acetylene
1776
3 1
TABLE II. Theoretical predictions for the S 0 (X̃) 1 S 1
g – T 1 S u and the
21 a
3
(S 0 )X̃ 1 S 1
g – T 2/3 D u acetylene energy separations in cm .
3 1
S 0 (X̃) 1 S 1
g – T1 Su
CASSCF
SOCI//CASSCF
a
3
S 0 (X̃) 1 S 1
g – T 2/3 D u
Te
T0
Te
T0
40 280
37 180
38 950
35 840
47 460
46 260
46 140
44 940
SOCI energies were determined at CASSCF optimized geometries. In all
cases, the DZP CASSCF zero-point vibrational energy corrections were
used to determine T 0 values.
also presented for comparison. Table II contains theoretically
determined T e and T 0 energy splittings between the ground
state and the linear stationary points. Figure 1 depicts the
true transition state for cis–trans isomerization on the T 1
potential energy surface of acetylene, reproduced here from a
previous paper by Vacek et al.18
DISCUSSION
For the S 0 ground state of acetylene our DZP CASSCF
prediction of 1.2124 Å for r e ~C–C! is closer to the experimental value19 of 1.2026 Å than those determined with a
comparable basis set and the CISD ~1.2131 Å!,10 CISDT
~1.2165!,10 CCSD ~1.2198 Å!,20 or CCSD~T! ~1.2252 Å!20
methods of electron correlation. This is partly fortuitous
since it is well known that those methods with increased
inclusion of electron correlation tend to overestimate bond
lengths when used with a DZP basis set and are much more
accurate when balanced with a more complete basis set.21
Likewise, our prediction of 1.0605 Å for r e ~C–H! compares
better with the experimental value19 of 1.0622 Å than the
other methods @1.0691,10 1.0699,10 1.0689,20 and 1.0702,20
respectively#. Our predictions for the bond lengths for the
other structures should be reliable as well. Our predictions
for harmonic vibrational frequencies do not fare as well as
those for geometrical parameters. In general we see more
accurate results than for the HF level of theory, but not so
accurate as the other methods of electron correlation10,20 with
the same basis set. However, they certainly remain qualita-
FIG. 1. Predicted Structure of C s -symmetry transition state for the 3 B u
trans-bent↔3 B 2 cis-bent acetylene isomerization reaction taken from G.
Vacek, J. R. Thomas, B. J. DeLeeuw, Y. Yamaguchi, and H. F. Schaefer, J.
Chem. Phys. 98, 4766 ~1993!.
tively accurate, which should suffice for our purpose of
merely classifying the nature of the linear triplet stationary
points. Harmonic vibrational analysis of both stationary
points reveals two different degenerate bending vibrational
modes with imaginary vibrational frequencies ~or negative
force constants! indicating that they are indeed stationary
points of Hessian index four.
In our previous study of isomerization reactions on the
T 1 potential-energy hypersurface of triplet vinylidene and
triplet acetylene, while searching for a possible cis–trans
isomerization transition state via rotation about the C–C
bond, a linear stationary point was found.18 It was characterized as a stationary point with two imaginary vibrational
frequencies, although the single-reference methods used
were not perfectly suitable for the study of such a linear
triplet state. Here, we have confirmed that those results are
correct. Our previous study found the true transition state
between cis-bent ã 3 B 2 acetylene and trans-bent b̃ 3 B u
acetylene to be a C s -symmetry inversion transition state. A
depiction of that transition state is reproduced in Fig. 1. With
regards to the DJLGAF hypothesis, however, the
C s -symmetry transition state lies much too low in energy at
about 34 300 cm21. Furthermore, another T 1 transition state
between ã 3 B 2 vinylidene and b̃ 3 B u trans-bent acetylene is
energetically reasonable at 47 300 cm21 but is expected to
have no local enhancement of the Franck–Condon overlap
with the lowest singlet state due to a C 1 -symmetry geometric
configuration far from linearity. Considering our previous
results18 along with our current results in Table II ~ZPVE
corrected DZP SOCI//CASSCF! which place the linear T 1
saddle-point structure at 35 840 cm21 with respect to the
rotationless zero-point vibrational level of X̃ 1 S 1
g , we must
conclude that there is no stationary point on the T 1 surface of
acetylene which would satisfy the necessary criteria of
DJLGAF’s hypothesis.
Although results for the low-symmetry candidates for
the cis–trans isomerization on the T 2 PES are not reported
upon here, we have examined the linear T 2/3 stationary point
which is focal to DJLGAF’s hypothesis. As shown in Table
II, at the ZPVE corrected DZP SOCI//CASSCF level of
theory, the linear T 2/3 structure lies at 44 940 cm21 with respect to the rotationless zero-point vibrational level of
X̃ 1 S 1
g . DJLGAF proposed that the linear state which would
enhance their S 1 – T – S 0 coupling must lie at an energy near
the S 1 n 38 5 3 level in the vicinity of 45 500 cm21. So energetically, the 3 D u structure is a good candidate to satisfy
DJLGAF’s hypothesis. Likewise it should be consistent with
all the other criteria for enhanced state coupling. As shown in
Table I, and mentioned above, the linear 3 D u stationary point
has two different degenerate imaginary vibrational frequencies. To our knowledge, compelling evidence has never been
presented for the experimental observation of a stationary
point of Hessian index four. While our energetic result for
the T 2/3 linear structure shows remarkable agreement with
the specified energy of DJLGAF’s hypothesized linear barrier, it would be naive to immediately conclude that this
structure is responsible for the ZAC anomaly. First, considering the margin of uncertainty in the DZP SOCI//CASSCF
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Vacek et al.: Spectra of à 1Au acetylene
result ~on the order of magnitude of 1000 cm21! and the
roughness of the experimental expectation, that the two values should match to under 60 cm21 is certainly fortuitous.
But more importantly, the DJLGAF results are sufficiently
indirect that they could not distinguish between a truly linear
higher-order saddle point and other possibilities, such as a
nearly linear true transition state. Therefore we must continue by considering additional possibilities.
Another possible solution to the ZAC spectral anomaly
would be to attribute it to a true transition state on the T 2
surface of C2H2 . To date there has been no accurate theoretical study of the transition states on the T 2 surface of C2H2 .
Part of the reason for this is that the RHF SCF wave functions of the T 2 acetylene minima have instability indices of
two with regards to MO rotation.10 Specifically, this means
that although the totally symmetric properties of the c̃ 3 A u
and d̃ 3 A 2 can be reliably studied with standard singlereference methods, the properties in C 2 or C s symmetry may
be unreliable. Thus it may be unreasonable to study the T 2
cis–trans isomerization transition state along a C 2 -symmetry
bond rotation or C s -symmetry in-plane inversion coordinate
in a manner analogous to our study of the T 1 surface18 and
expect accurate results. We are currently pursuing such a
study to determine the extent of the unreliability.22 Certainly
multireference methods should be able to overcome the
problem of RHF SCF MO instability. Also, the equation-ofmotion coupled cluster method in the singles and doubles
approximation ~EOM! should prove reliable for evading the
problem. One might expect results as accurate as those of
Stanton and co-workers for the S 1 surface of C2H2 .23,24
Without presenting specific data for the T 2 transition
states, we can make several qualitative predictions nonetheless. As with the T 1 surface, the transition state between
vinylidene and acetylene would have a structure not only
nonlinear, but highly nonplanar,25 allowing us to rule it out.
That leaves only the transition state for cis–trans isomerization on that surface of acetylene. We have shown here by
characterizing the linear structure as a higher-order saddle
point that the true transition state is nonlinear, fitting the
expectations raised in the introduction. The most reasonable
nonlinear structures would be those resulting from
C 2 -symmetry bond rotation or C s -symmetry in-plane inversion. The inversion should be the more energetically favorable process of the two since it does not require any bond
breakage. The transition state structure for inversion isomerization on the T 2 acetylene surface would look qualitatively
like the one on the T 1 surface as shown in Fig. 1, although
slightly cis-bent rather than slightly trans-bent following
Hammond’s Postulate26 and the fact that the cis-bent 3 A 2
reaction endpoint is higher in energy than the trans-bent
3
A u endpoint.10 Such a half-linear structure would fit the
Franck–Condon
criteria
for
the
enhanced
S1
~trans!–T~cis!–S 0 ~linear! coupling that DJLGAF have observed, perhaps even better than a perfectly linear structure.
Energetically, such a structure must lie above the d̃ 3 A 2
minimum and below the linear 3 D u second-order saddle
point. We place these limits roughly at 37 170 cm21 @at the
DZP CCSD~T!//CISD level of theory#10 and 46 260 cm21 ~at
1777
the DZP SOCI//CASSCF!, respectively. Although that is admittedly a very broad energy window, early results suggest
that the transition state should lie near 40 500 cm21 @40 130
cm21 for DZP CCSD~T!//CISD,22 40 900 cm21 for DZP
EOM#.27 Again, this lies too low energetically compared to
the necessary value of 45 500 cm21, even allowing for uncertainty in both the theoretical and experimental values.
Considering that T i surface with i.3 would be too highlying in energy to suit our needs,28 the only remaining area
of consideration should be the T 3 surface of C2H2 . The T 3
surface of acetylene is even more opaque to ab initio theoretical study than the T 2 surface. We hope to undertake a
studies of that surface in the future to obtain quantitatively
accurate results,27,29 but for now must limit ourselves to a
more qualitative discussion. The T 3 component of the T 2/3
linear 3 D u second-order saddle point should be stabilized
relatively little along the H–C–C bending coordinates. For
instance, in the early work of Demoulin,2 while the T 2 root
of 3 D u was stabilized by ;13 600 cm21 to a trans-bent bond
angle of 135°, the T 3 root was only stabilized by ;5600
cm21 to a trans-bent angle of 160°. Since the linear 3 D u
second-order saddle point still remains at an energetic upper
bound for the trans- or cis- ~or possibly gauche-! bent T 3
acetylene minima, and the true transition state between them,
we expect them all to lie between roughly 40 000 and 45 000
cm21 above the rotationless zero-point vibrational level of
X̃ 1 S 1
g . Furthermore, since the trans- or cis- ~or gauche-!
minima would have large H–C–C bond angles ~.160°! an
appropriate transition state connecting them, akin to the halflinear transition states of the T 1 and T 2 surfaces, should be
very nearly linear. This T 3 transition state seems a very reasonable candidate for understanding the DJLGAF ZAC
anomaly,1 especially considering the possibility for mixed
triplet level contributions as shown by the follow up work by
Dupré, Green and Field.4 Such S 1 – T 3 – T 2 – S 0 coupling also
appears consistent with the LIF spectra of à acetylene by
Ochi and Tsuchiya.30
CONCLUSION
We have studied the lowest-lying linear triplet-excited3
state stationary points of acetylene, 3 S 1
u and D u , doing
geometry optimization and harmonic vibrational frequency
analysis using the DZP CASSCF level of theory as well as
energies at those geometries using SOCI, to see if they fit the
Dupré, Jost, Lombardi, Green, Abramson, and Field ~DJLGAF! hypothesis1 of the existence of a linear isomerization
barrier connecting a triplet-excited cis- and trans-acetylene
in the vicinity of ;45 500 cm21. Harmonic vibrational
analysis of both stationary points reveals two different degenerate vibrational modes with imaginary vibrational frequencies ~or negative force constants! indicating that they are
indeed higher-order saddle points. Zero-point vibrational energy ~ZPVE! corrected DZP SOCI//CASSCF energies place
21
3
the 3 S 1
u and D u saddle-points 35 840 and 44 940 cm
1 1
above the rotationless zero-point vibrational level of X̃ S g .
Our results show that no feature on the T 1 surface of
acetylene could fit DJLGAF’s hypothesis. Likewise there is
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Vacek et al.: Spectra of à 1Au acetylene
no suitable stationary point on the T 2 surface of acetylene,
except possibly the linear 3 D u saddle-point itself. Although
we present no new data on the equilibria or transition states
of the T 3 surface, we provide compelling arguments to show
that they could match all necessary criteria presented by
DJLGAF. We propose that the most likely structure for understanding the Zeeman anticrossing anomaly of DJLGAF is
either the linear 3 D u saddle-point or a ‘‘half-linear’’ transition state connecting nonlinear equilibria on the T 3 surface
of acetylene.
ACKNOWLEDGMENTS
This research was supported by the United States Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Fundamental Interactions
Branch, Grant No. DE-FG05-94ER14428. The work done in
this research was supported in part by graduate fellowships
from the United States Department of Defense for G.V. and
the United States National Science Foundation for C.D.S.
G.V. would like to thank Patrick Dupré, Bob Field and
Wesley Allen for discussions. G.V. would further like to
thank all the members of the CCQC, past and present, for an
interesting and fun graduate career.
1
P. Dupré, R. Jost, M. Lombardi, P. G. Green, E. Abramson, and R. W.
Field, Chem. Phys. 152, 293 ~1991!.
D. Demoulin, Chem. Phys. 11, 329 ~1975!.
3
P. Dupré and P. G. Green, Chem. Phys. Lett. 212, 555 ~1993!.
4
P. Dupré, P. G. Green, and R. W. Field, Chem. Phys. 196, 211 ~1995!.
5
P. Dupré, Chem. Phys. 196, 239 ~1995!.
6
H. Abe and H. Hayashi, Chem. Phys. Lett. 206, 337 ~1993!.
7
M. Drabbels, J. Heinze, and W. L. Meerts, J. Chem. Phys. 100, 165
~1994!.
8
A. L. L. East, C. S. Johnson, and W. D. Allen, J. Chem. Phys. 98, 1299
~1993!.
9
P. Valtazanos and K. Ruedenberg, Theor. Chim. Acta 69, 281 ~1986!.
2
10
Y. Yamaguchi, G. Vacek, and H. F. Schaefer, Theor. Chim. Acta 86, 97
~1993!.
11
H. Liska and A. Karpfen, Chem. Phys. 102, 77 ~1986!.
12
S. Huzinaga, J. Chem. Phys. 42, 1293 ~1965!.
13
T. H. Dunning, J. Chem. Phys. 53, 2823 ~1970!.
14
T. H. Dunning and P. J. Hay, Modern Theoretical Chemistry, Vol. 3, edited
by H. F. Schaefer ~Plenum, New York, 1977!, pp. 1–27.
15
H. F. Schaefer, Ph. D. thesis, Department of Chemistry, Stanford University, 1969.
16
P. Saxe, D. J. Fox, H. F. Schaefer, and N. C. Handy, J. Chem. Phys. 77,
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17
PSI 2.0.8, C. L. Janssen, E. T. Seidl, G. E. Seuseria, T. P. Hamilton, Y.
Yamaguchi, R. Remington, Y. Xie, G, Vacek, C. D. Sherrill, T. D. Crawford, J. T. Fermann, W. D. Allen, B. R. Brooks, G. B. Fitzgerald, D. J. Fox,
J. F. Gaw, N. C. Handy, W. D. Laidig, T. J. Lee, R. M. Pitzer, J. E. Rice,
P. Saxe, A. C. Scheiner, and H. F. Schaefer ~PSITECH, Inc., Watkinsville,
GA, 1994!.
18
G. Vacek, J. R. Thomas, B. J. DeLeeuw, Y. Yamaguchi, and H. F. Schaefer,
J. Chem. Phys. 98, 4766 ~1993!.
19
A. Baldacci, S. Ghersetti, S. C. Hurlock, and K. N. Rao, J. Mol. Spectrosc.
59, 116 ~1976!.
20
J. R. Thomas, B. J. DeLeeuw, G. Vacek, and H. F. Schaefer J. Chem. Phys.
98, 1336 ~1993!.
21
J. R. Thomas, B. J. DeLeeuw, G. Vacek, T. D. Crawford, Y. Yamaguchi,
and H. F. Schaefer, J. Chem. Phys. 99, 403 ~1993!.
22
C. D. Sherrill, G. V. Vacek, Y. Yamaguchi, and H. F. Schaefer, J. Chem.
Phys. ~submitted!.
23
J. F. Stanton, C.-H. Huang, and P. G. Szalay, J. Chem. Phys. 101, 356
~1994!.
24
J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 3001 ~1994!.
25
L. B. Harding, J. Am. Chem. Soc. 103, 7469 ~1981!.
26
G. S. Hammond, J. Am. Chem. Soc. 77, 334 ~1955!.
27
G. Vacek, J. F. Stanton, and J. Gauss ~in progress!.
28
3
For instance, the T 4 3 S 2
u – T 2/3 D u energy splitting should be approxi21
mately the same as that of T 2/3 3 D u – T 1 3 S 1
u , ;9000 cm , as shown in
21
this work, placing T 4 3 S 2
at
;54
000
cm
.
Furthermore,
the T 4 linear
u
structure should be only slightly stabilized by cis- or trans-bending. See
also Demoulin, Ref. 2.
29
C. D. Sherrill, Y. Yamaguchi, G. Vacek, and H. F. Schaefer ~in preparation!.
30
N. Ochi and S. Tsuchiya, Chem. Phys. 152, 319 ~1991!.
J. Chem. Phys., Vol. 104, No. 5, 1 February 1996
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