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Journal of Molecular Structure: THEOCHEM 815 (2007) 119–125
Ab initio study of the torsional motion in tolane
Dong Xu a, Andrew L. Cooksy b,*
a Computational Science Research Center, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-1245, USAb Department of Chemistry, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-1030, USA
Received 12 December 2006; received in revised form 19 March 2007; accepted 22 March 2007Available online 30 March 2007
Abstract
Accurate prediction of the torsional barrier height in tolane is achieved by systematically extrapolating to the Dunning complete basisset limit at the MP2 level with a spin-component-scaled correction. The zero-point energy correction is calculated at the B98/cc-pVDZlevel based on a benchmark test using the experimental data for benzene. The final calculated barrier height is 202 cm�1, in agreementwith the observed value. The correct barrier height enables the vibrational energy levels and other spectroscopic properties to be deter-mined accurately through numerical integration of the torsional Schrodinger equation. This study provides a nearly complete compu-tational solution to the torsional problem in tolane and may aid the exploration of torsional motions in similar molecules.� 2007 Elsevier B.V. All rights reserved.
Keywords: Tolane; Diphenylacetylene; Torsion; Barrier height; Vibrational Schrodinger equation
1. Introduction
A great deal of experimental and theoretical researchhas been devoted to tolane (diphenylacetylene,
C C ) and its derivatives. As the elementary
framework of the poly(phenylethynylene) (PPE) family,tolane is a molecule of fundamental importance.
Materials based on tolane and its derivatives have awide range of applications. Their traditional role as build-ing blocks in polymer synthesis [1,2] has been extended tothe aromatic core group of the main classes of liquid crystalcompounds [3]. The large liquid crystalline phase rangesand high clearing temperatures demonstrated by tolanederivatives contribute to favorable optical behavior whentolane or a related moiety is incorporated into liquidcrystalline compounds. In addition, they are used in newmetal-containing liquid crystals as ligands for metal–alkynecomplexes [4]. A photochemistry study showed the uniquestructure of tolane moieties results in high thermal stability
0166-1280/$ - see front matter � 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.theochem.2007.03.028
* Corresponding author. Tel.: +1 619 594 5571; fax: +1 619 594 4634.E-mail address: [email protected] (A.L. Cooksy).
when homogeneous photoalignment of thin tolane polymerfilms is induced by light irradiation [5].
Perhaps the most interesting application of tolane andits derivatives is the development of the ‘‘p-way’’ or‘‘molecular wire’’-like architectures [6–8]. Experimentsdemonstrate that tolane derivatives with alkyl thiol groupsdisplay negative differential resistance (NDR), which isconsistent with the concept of substantial ‘‘through mole-cule’’ conductance that can be influenced by applied poten-tial [9–11]. These systems could be used to store one bitcurrent flow by the torsional motion of the benzene rings[12]. The storage retention time for this class of moleculesappears to depend on the specific side groups and theirlocations [13]. The interaction between the p-conjugatedsystems and metal d electron systems may lead to an evengreater potential for tolane and its derivatives to be used ascircuit elements along with nanoparticles and nanotubes inmolecular electronics [14]. Applications in molecular elec-tronics have led to new technologies, such as the synthesisof a prototype single-molecule diode [15] and the DNAhairpins in which photoinduced charge transfer is believedto take place through one-dimensional array of p-stackedbase pairs in the linked duplex DNA. Substantial interesthas been spurred due to its relevance to oxidative damage
Table 1Previous computational results for the torsional barrier height in tolane
Methods Barrier height (cm�1) Source
LMP2/6-311G** 224 [27]MP2/6-311G** 128 [28]MP2/6-31G* 184 [29]HF/6-311G** 148 [28]HF/3-21G 147 [30]B3LYP/6-31+G* 284 [26]B3PW91/6-311G** 301 [21]AM1 77 [25]MM3 �0 [26]INDO barrier at 0� [23]CNDO barrier at 0� [24]Exp. 202 [22]
120 D. Xu, A.L. Cooksy / Journal of Molecular Structure: THEOCHEM 815 (2007) 119–125
and repair [16,17]. Tolane has also received increasingattention in optoelectronics (e.g., organic light emittingdiodes, OLED) [2,18–20].
The origin of many of these developments can beascribed to the extended linear p-systems of tolane andits derivatives, and consequently to the relative intramolec-ular orientation of the aromatic rings. The extended p-elec-tron conjugation functions as a natural electricalconductor, and the torsional angle(s) between the phenylrings works as the electric current controller [21]. Tolane,the smallest oligomer in its class, serves as a perfect para-digm for modeling these types of interactions.
Okuyama et al. [22] carried out the most thoroughexperimental study of the torsional motion in tolane, prob-ing the vibronic states by fluorescence spectroscopy of asupersonic free jet. Their work identified transitions involv-ing levels up to v = 15 for the torsional motion and pre-dicted a torsional barrier of 202 cm�1 (0.58 kcal mol�1),with the coplanar configuration of D2h symmetry beingthe torsional minimum and the perpendicular D2d structurethe maximum (Fig. 1).
Many computational studies have investigated the tor-sional motion in tolane. These include semi-empiricalINDO [23] and CNDO [24] calculations, which yieldedthe opposite conclusion that the perpendicular geometryis more stable than the coplanar geometry, and a more suc-cessful AM1 calculation by Ferrante et al. [25]. A recentMM3 calculation found a nearly flat potential energy curve[26]. The small torsional barrier in tolane poses a challengenot only to semi-empirical methods, but also to ab initiocalculations because the size of the molecule prohibits thehigh levels of theory and large basis sets needed for thislevel of precision. Saebø et al. [27] performed LMP2 calcu-lations with the 6-311G** basis set and obtained a 224 cm�1
(0.64 kcal mol�1) barrier height. DFT methods such asB3LYP and B3PW9 were also employed in previous stud-ies. These previously calculated barrier heights and theirrespective sources are listed in Table 1.
The data suggest that an appropriate quantum compu-tational treatment is required to reproduce the observedbarrier. The current study presents an accurate calculation
Fig. 1. The coplanar and perpendicular geometries of tolane.
of the torsional potential energy function in tolane, andintegration of the corresponding Schrodinger equation topredict the torsional energy levels.
2. Computational method
2.1. Ab initio PES determination
All the electronic structure calculations were performedusing Gaussian 03 [31]. The coplanar and perpendiculargeometries were optimized using the Hartree–Fock (HF)and second order Møller–Plesset (MP2) [32,33] levels usingboth Pople-type basis sets (6-31G, 6-311G series) [34] andDunning’s correlation consistent basis sets (cc-pVDZ, cc-pVTZ and cc-pVQZ) [35–37]. To improve efficiency, theinitial geometries were first optimized by the HF methodwith smaller basis sets. All the geometry optimizations werecarried out using the Berny algorithm in redundant internalcoordinates. Full geometry optimizations (save for freezingthe dihedral angle) were carried out at coplanar and per-pendicular configurations for all calculations in this work.The dihedral angle was frozen to map out the relaxed PES.Use of symmetry was disabled in all calculations.
2.2. SCS-MP2 (spin-component-scaled) correction
The relatively high computational cost of MP2 and itsneed for large atomic orbital basis sets to obtain goodresults are significant deficiencies [38], reducing the upperlimit on system size. MP2 also underestimates the contribu-tions from spin-paired electrons, which can result in poorenergy prediction. To address these drawbacks, we firstapplied Grimme’s SCS-MP2 correction technique [39] toall the MP2 calculations. Grimme [39] showed that MP2energies can be systematically improved by separate scalingof the antiparallel-spin (S) and parallel-spin (T) compo-nents of the MP2 correlation energy:
EC ¼ ES þ ET ð1ÞThis method was termed ‘‘spin-component scaled’’ MP2,or SCS-MP2, and, denoting the scaling factors as pS andpT, the modified correlation energy is simply:
D. Xu, A.L. Cooksy / Journal of Molecular Structure: THEOCHEM 815 (2007) 119–125 121
EC;SCS ¼ pSES þ pT ET ð2ÞIn this study, Grimme’s optimized scaling factors pS = 6/5and pT = 1/3 were used for antiparallel-spin (S) and paral-lel-spin (T) correlations. For a given basis set, Grimme’sapproach showed clear statistical improvements over tradi-tional MP2, in some cases achieving results comparable toQCISD or QCISD(T) quality [39].
2.3. Extrapolation to the complete basis set (CBS) limit
In this work, we applied the extrapolation schemesformulated by Wilson and Dunning [40] to estimate theSCS-MP2 energy at the CBS limit. Recently, Jung andHead-Gordon applied similar techniques to extrapolatethe antiparallel-spin component in their SOS-MP2treatment of dispersion interactions [41].
First, the Hartree–Fock energy at the CBS limit,-EHF(1), is calculated through a 3-point extrapolationusing the following Dunning–Feller [37,40,42,43] exponen-tial functional:
EHFð1Þ ¼ EHFðnÞ � Ae�C�n ð3Þwhere n represents the cardinal number of the basis set(e.g., n = 2 for cc-pVDZ), EHF(n) is the Hartree–Fock en-ergy at the cc-pVnZ basis set, and A and C are constantswhich are determined along with EHF(1) via a least-squares fit to EHF(2), EHF(3) and EHF(4).
The extrapolation of the MP2 correlation energy to thecomplete basis set (CBS) limit has been well studied[38,40,44–49]. A detailed review by Martin summarizesthe formulation of these extrapolations by exponentialand power law expressions [45]. We have tested extrapola-tions of the SCS correction using several schemes, includ-ing expressions obtained by Helgaker et al. [38,44],Martin [45,47,48], Wilson and Dunning [40], the MP2 totalenergy extrapolation of de Lara-Castells [49] and the sin-gle-point extrapolation of Varandas [46]. The scheme ofWilson and Dunning [40] is described below.
The antiparallel-spin (S) and parallel-spin (T) contribu-tions of the MP2 correlation energy ES(1) and ET(1) atthe CBS limit are obtained from the following functionalforms:
ESðnÞ ¼ ESð1Þ þ Bðlmax þ dÞm ð4Þ
ET ðnÞ ¼ ET ð1Þ þ B0
ðlmax þ dÞm ð5Þ
where ES(n) and ET(n) are the antiparallel-spin (S) and par-allel-spin (T) contributions of the MP2 correlation energyat cc-pVnZ basis set, lmax is the maximum angular momen-tum represented in each basis set, d is an angular momen-tum offset and m = 3 or 4. Here we use Dunning’sempirical numbers for these parameters: d = 0, m = 4,lmax = 2 when n = 2 and lmax = 3 when n = 3. Thus,ES(1), ET(1) and the constants B, B 0 can be calculatedusing ES(2), ES(3), ET(2) and ET(3), respectively.
The final SCS-MP2 energy ESCS-MP2(1) is availablefrom
ESCS-MP2ð1Þ ¼ EHFð1Þ þ ECð1Þwhere EC(1) = pSES(1) + pTET(1). All energies used inthe CBS extrapolation are calculated from full geometryoptimizations, at the corresponding level of theory and ba-sis set, at the coplanar and perpendicular configurations.
Because the MP2 correlation energy Ec is the simple sumof the antiparallel-spin and parallel-spin contributions (Eq.(1)), it is reasonable that the same extrapolation procedurescan be applied to the individual spin components as well.However, it is not clear that the coefficients and power lawsare the same for the two energy contributions. In particu-lar, Kutzelnigg and Morgan [50] showed that the leadingterm in the asymptotic energy expression was proportionalto (l + 1/2)�4 for the singlet state of a two-electron atom,whereas the leading term in the triplet state expression isproportional to (l + 1/2)�6. Prompted by this result, theschemes of Helgaker et al. and Martin were modeled a sec-ond time by assuming separate power laws for the antipar-allel and parallel spin contributions, with the parallel spincomponent treated as triplet-like and having a power lawdiffering by a factor of (l + 1/2)�2. The predicted barrierheights are remarkably insensitive to the selection ofextrapolation scheme, varying over a range of less than15 cm�1 among all of these methods. The results cited inthe remainder of this work are obtained following the Wil-son–Dunning method described above.
2.4. Calculation of the torsional energy levels
The torsional Schrodinger equation was written asdescribed by Okuyama et al. [22]. The torsional energy lev-els and wave functions for the independent twisting modesare solutions of the one-dimensional Schrodinger equation:
� �h2
2I red
d2
d/2þ V ð/Þ
� �wð/Þ ¼ Ewð/Þ ð6Þ
where V(/) is a periodic potential for the torsional motionand Ired is the reduced moment of inertia. Since the rota-tional constant B ¼ �h=2I red, the above equation can bewritten as:
�Bd2
d/2þ V ð/Þ
� �wð/Þ ¼ Ewð/Þ: ð7Þ
V(/) is represented by the following cosine function:
V /ð Þ ¼ 1
2V 2 1� cos 2/ð Þ ð8Þ
where V2 is the twofold barrier height for the internal rota-tion. The experimental barrier height 202 cm�1 was ob-tained by fitting the observed spectroscopic transitionfrequencies to Eq. (7) [22]. In the current study, we calcu-late the torsional barrier by applying the ab initio proce-dures described in Sections 2.1–2.3; the Schrodingerequation was then solved numerically in an effort to pro-
Table 2Calculation results for tolane torsional barrier height
Methods Barrier (cm�1) SCS-MP2 barrier (cm�1)
HF/6-31G 169.1HF/6-31G* 152.2HF/6-31G** 151.0HF/6-311G** 142.7HF/6-311++G �190.8HF/6-311++G** �11.4MP2/6-31G 31.4 39.8MP2/6-31G* 170.4 140.8MP2/6-31G** 176.9 147.4MP2/6-311G** 170.8 139.3MP2/6-311++G 222.9 118.6MP2/6-311++G** 186.7 127.5HF/cc-pVDZ 151.7HF/cc-pVTZ 154.9HF/cc-pVQZ 161.6HF/CBS 167.4MP2/cc-pVDZ 223.6 183.1MP2/cc-pVTZ 237.5 192.3MP2/CBS 252.6 206.2
Exp. 202.0
122 D. Xu, A.L. Cooksy / Journal of Molecular Structure: THEOCHEM 815 (2007) 119–125
vide a computational solution to the tolane torsionalproblem.
The one-dimensional Schrodinger equation (6) is a lin-ear, second-order and self-adjoint ordinary differentialequation (ODE). The Numerov process is the ‘‘canonical’’method for the integration of the ODEs of this type [51]. Itis similar to the Runge–Kutta method, but achieves higherorder of accuracy without increasing computational cost.Eq. (6) is also a two-point boundary value problem. Oneeffective numerical treatment is the shooting method [52]in which the initial guess of the dependent variable (thetrial energy eigenvalue) is chosen at one boundary; Eq.(6) is then integrated by the Numerov process, arriving atthe other boundary (a classical turning point in this case).The discrepancy from the desired boundary value is mini-mized by adjusting the trial energy eigenvalue via a rootfinding procedure (bisection method). The trial eigenvalueeventually converges to the correct torsional energy thatzeros the discrepancy between the two boundary points.If integrating the differential equation is pictured as follow-ing the trajectory of a shot from gun to target, then pickingthe initial condition virtually corresponds to aiming. Theshooting method provides an intuitive approach to takinga set of ‘‘ranging’’ shots that allow us to systematicallyimprove our ‘‘aim’’ and eventually hit the target.
Using a combined Numerov process and shootingmethod [53], the numerical integration of the torsionalSchrodinger equation was carried out on 5000 point gridsover the range [p/2, 3p/2] for the torsional angles /, usingthe benzene rotational constant to estimate a torsionalrotational constant B = 0.379 cm�1 and the barrier heightpredicted by SCS-MP2 calculations in the Dunning com-plete basis set limit.
3. Results and discussion
3.1. Optimized geometry
All calculations except the HF/6-311++G and HF/6-311++G** correctly predicted the coplanar geometry asthe potential energy minimum. An imaginary frequencywas found at the optimized geometry when the dihedralangle was fixed to 90�, indicating the perpendicular geom-etry is a transition state. These results agree with the exper-imental findings.
3.2. Torsional barrier calculations
The results of preliminary calculations on the torsionalbarrier as a function of basis set and level are shown inTable 2. HF calculations with Pople-type basis sets showa trend of smaller basis sets producing better results, withdiffuse functions leading to the wrong minimum-energygeometry. However, HF results using Dunning’s correla-tion consistent basis sets show that the accuracy improvesas the basis set size increases. The subsequent MP2 calcula-tions show improvement over HF results using the same
basis set, when the SCS correction is applied using the cor-relation consistent basis set.
After systematically extrapolating to the complete basisset (CBS) limit in combination with SCS-MP2 correction,we obtain a barrier height of 206 cm�1, in excellent agree-ment with Okuyama’s analysis. The level of agreement iscertainly fortuitous, because even coupled cluster calcula-tions predict relative isomer energies to only within a fewtenths of a kcal mol�1 or about a hundred cm�1. The gen-erally good agreement with experiment across several levelsof theory, including HF, may be peculiar to the particularlylow barrier height of the tolane torsional potential energy.
3.3. Zero-point energy correction
Table 3 shows the zero-point energy differences (DZPEs)between the 90� transition state and the 0� energy mini-mum, predicted by different methods using the cc-pVDZbasis set. MP2 calculates a relatively large positive valueof 64.8 cm�1 for DZPE, perhaps for the same reason thatuncorrected MP2/cc-pVnZ levels of theory generally over-estimate the barrier height. The SCS-MP2 correction can-not be applied to analytical frequency calculations, andin order to better estimate the DZPE, we carried out fre-quency calculations on benzene. The 30 vibrational modesof benzene were compared side by side with the experimen-tal frequencies reported by Goodman et al. [54]. The resultsare listed in Table 4. The order of accuracy in terms of thetotal ZPE for benzene is B98 > B3LYP > MP2 > HF. Thesame order follows when comparing the maximum erroramong all vibrational modes. B98 leads the pack with only0.9% error in total ZPE and 3.5% in maximum error acrossall 30 vibrational modes. The maximum error for MP2 is12.5%, which is as high as HF. The maximum error comesfrom mode 19, a C–C bond stretching mode with B2u sym-
Tab
le3
Ben
zen
efr
equ
ency
calc
ula
tio
nb
ench
mar
kte
sta
Mo
de
Sym
met
ryE
xp.
mp
2/cc
-pV
DZ
Err
or
(%)
HF
/cc-
pV
DZ
Err
or
(%)
B3L
YP
/cc-
pV
DZ
Err
or
(%)
B98
/cc-
pV
DZ
Err
or
(%)
1E
(2u)
398
400.
90.
744
9.8
13.0
414.
14.
141
1.5
3.4
2E
(2u)
398
400.
90.
744
9.8
13.0
414.
84.
241
2.0
3.5
3E
(2g)
607.
260
5.6
�0.
366
0.8
8.8
617.
71.
761
4.8
1.3
4E
(2g)
607.
260
5.6
�0.
366
0.8
8.8
617.
81.
761
5.4
1.4
5A
(2u)
674
633.
9�
6.0
752.
211
.669
0.5
2.4
690.
52.
56
B(2
g)
707
686.
0�
3.0
771.
39.
172
3.1
2.3
719.
11.
77
E(1
g)
847.
185
8.9
1.4
948.
312
.086
5.5
2.2
863.
72.
08
E(1
g)
847.
185
8.9
1.4
948.
312
.086
5.8
2.2
864.
12.
09
E(2
u)
967
953.
7�
1.4
1080
.711
.898
6.6
2.0
984.
11.
810
E(2
u)
967
953.
7�
1.4
1085
.812
.398
6.7
2.0
984.
31.
811
B(2
g)
990
961.
9�
2.8
1089
.910
.110
13.6
2.4
1010
.52.
112
A(1
g)
994.
410
07.0
1.3
1089
.99.
610
18.6
2.4
1014
.32.
013
B(1
u)
1010
1017
.00.
711
16.5
10.5
1021
.91.
210
16.6
0.7
14E
(1u)
1038
.310
61.3
2.2
1130
.78.
910
58.4
1.9
1056
.31.
715
E(1
u)
1038
.310
61.3
2.2
1130
.78.
910
58.8
2.0
1056
.61.
816
B(2
u)
1149
.711
63.1
1.2
1188
.23.
311
61.9
1.1
1160
.10.
917
E(2
g)
1177
.811
90.7
1.1
1273
.08.
111
86.3
0.7
1185
.30.
618
E(2
g)
1177
.811
90.7
1.1
1273
.08.
111
86.5
0.7
1185
.60.
719
B(2
u)
1309
.414
72.4
12.5
1341
.02.
413
55.6
3.5
1352
.13.
320
A(2
g)
1367
1358
.9�
0.6
1477
.48.
113
64.1
�0.
213
60.3
�0.
521
E(1
u)
1494
1504
.90.
716
26.9
8.9
1506
.10.
815
02.5
0.6
22E
(1u)
1494
1505
.00.
716
27.0
8.9
1506
.50.
815
02.7
0.6
23E
(2g)
1607
1649
.72.
717
87.3
11.2
1644
.62.
316
40.7
2.1
24E
(2g)
1607
1649
.72.
717
87.3
11.2
1645
.02.
416
41.5
2.1
25B
(1u)
3166
.332
09.4
1.4
3328
.85.
131
66.3
0.0
3163
.5�
0.1
26E
(2g)
3174
3220
.51.
533
40.6
5.2
3176
.40.
131
73.8
0.0
27E
(2g)
3174
3220
.51.
533
40.6
5.2
3176
.60.
131
74.0
0.0
28E
(1u)
3181
.132
36.9
1.8
3359
.85.
631
92.8
0.4
3190
.30.
329
E(1
u)
3181
.132
36.9
1.8
3359
.85.
631
93.0
0.4
3190
.60.
330
A(1
g)
3191
3246
.81.
833
71.8
5.7
3203
.60.
432
01.1
0.3
ZP
E21
770.
922
061.
31.
323
424.
17.
622
009.
71.
121
968.
90.
9
Max
.er
ror
12.5
13.0
4.2
3.5
aF
req
uen
cies
are
incm�
1.
D. Xu, A.L. Cooksy / Journal of Molecular Structure: THEOCHEM 815 (2007) 119–125 123
Fig. 2. The tolane torsional barrier and energy levels. The quantum statesare doubly degenerate for v 6 12, but split for v P 13.
Table 4Observed and calculated torsional transitions of tolane (cm�1)
vf ‹ vi Obsvd.a Okuyama et al.a Calc.b
2 ‹ 0 c 34 33.9 33.83 ‹ 1 33 33.0 33.04 ‹ 0 66 66.0 66.05 ‹ 1 65 64.3 64.36 ‹ 0 97 96.3 96.37 ‹ 1 94 93.6 93.78 ‹ 0 125 124.6 124.79 ‹ 1 121 120.8 120.910 ‹ 0 150 150.6 150.711 ‹ 1 144 145.4 145.512 ‹ 0 173 173.4/173.6 173.613 ‹ 1 164 165.9/166.9 166.0/166.814 ‹ 0 189 190.5/193.8 190.6/193.815 ‹ 1 177 179.8/186.7 179.8/186.7
a Ref. [22].b This Work.c 8.7 cm�1 (v = 0) and 25.8 cm�1 (v = 1) are calculated in this work.
124 D. Xu, A.L. Cooksy / Journal of Molecular Structure: THEOCHEM 815 (2007) 119–125
metry. In the MP2 calculation, the frequency at mode 19 ispredicted higher than mode 20, an in-plane C–H bondbending mode with A2g symmetry, in clear disagreementwith experiment. The same error was also observed byGoodman et al. in their MP2/6-311** calculations [54].We believe this same effect to some extent explain the dis-crepancies between MP2 and other ZPE results. Since theperpendicular geometry is generally more rigid than thecoplanar geometry in tolane, it is expected that the correctDZPE has a small negative value. Both DFT results showthe correct sign, �10.8 cm�1 from B3LYP and �4.0 cm�1
from B98. Based on the satisfactory performance ofB3LYP and B98 in benzene benchmark test, we believethe B98 DZPE (�4.0 cm�1) for tolane provides the mostreliable value. Because most of the ZPE scaling factorsare very close to 1 and the predicted DZPE is small, ZPEscaling does not greatly affect the final barrier height. Thefinal value, with the B98 ZPE correction, is 202 cm�1.
3.4. Determination of torsional energy levels
The second goal in this study is to precisely reproducethe experimentally measured torsional vibrational levels.The first column of Table 4 shows the energy levelsobserved from v = 0 to the highest vibrational stateobserved v = 15, which lies near the top of the barrier.The energy levels are represented in the form of energygaps as a result of the spectroscopy experimental measure-ment. For even-quantum levels, the values are measuredrelative to the v = 0 state, for odd-quantum levels, thevalues are measured relative to the v = 1 state. The secondcolumn is Okuyama et al.’s calculation based on the Ham-iltonian matrix diagonalization technique developed byLewis et al. [55]. The third column shows the energy levelscalculated by our numerical integration method. The threesets of data are in excellent agreement on all torsional lev-els. The data also illustrate that the energy levels are
approximately doubly degenerate near the bottom of thepotential well, with observable splittings of about 1 cm�1
appearing at values of v greater than 12 as tunnelingthrough the barrier becomes significant. These splittingsare not observed directly by experiment. They would beresolvable in the work of Okuyama et al. [22] only in therelatively weak and broad T 3
15ðv0 ¼ 3Þ and T 414ðv0 ¼ 4Þ tran-
sitions. Lacking a detailed characterization of the torsionalmodes in the upper electronic state, it is not certain thatboth tunneling components are excited simultaneously.Furthermore, the nuclear spin statistics of the torsionalstates slightly favor the observed tunneling component.
Our solver also predicts the fundamental frequency forground state tolane to be 17.4 cm�1. The final potentialenergy curve and torsional energy levels are plotted inFig. 2.
4. Conclusion
A nearly complete computational solution to the tor-sional motion problem in the tolane molecule is presentedin this study. Combining the Dunning complete basis setextrapolation technique [40] with the SCS-MP2 correction[39], the torsional barrier height is predicted to high accu-racy. Frequency benchmark calculations on benzene wereused to predict the zero-point energy correction for tolane.A subsequent numerical solution to the torsional Schro-dinger equation over the computed PES reproduces theexperimental vibrational energy levels and other spectro-scopic properties, some of which are unavailable fromexperiments. The computational results are in excellentagreement with the observed data. It must be emphasized,however, that the absolute error in the barrier height is lowacross several computational methods, and that the appar-ent precision of our final value is certainly somewhat fortu-itous. Nevertheless, this work demonstrates the tractability
D. Xu, A.L. Cooksy / Journal of Molecular Structure: THEOCHEM 815 (2007) 119–125 125
and potential efficacy of these computational tools in treat-ing the torsional dynamics of this and similar systems.
Acknowledgements
This work is supported by NSF Grant CHE-0216563, theBlasker Fund of San Diego Foundation, and the Computa-tional Science Research Center at San Diego State Univer-sity. The authors thank Prof. Michael Bromley of the latterinstitution for illuminating discussions, and Prof. DaniellMattern of the University of Mississippi for introducing usto the system, and Ed Harris for preliminary calculations.
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