david j. wilson et al- rovibrational states and vibrational intensities of the chi^2 a1 state of...

8/3/2019 David J. Wilson et al- Rovibrational states and vibrational intensities of the chi^2 A1 state of He2C^3+

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Rovibrational states and vibrational intensities of the v state of2A1

He2

C3

David J. Wilson, Sudarko, Jason M. Hughes and Ellak I. von Nagy-Felsobuki*

Department of Chemistry, T he University of Newcastle, Callaghan, NSW 2308, Australia.

E-mail: chvo=cc.newcastle.edu.au

Received 15th February 1999, Accepted 21st April 1999

The all-electron CCSD(T)/cc-pCVTZ level of theory was used to generate a 63 point discrete potential energyhypersurface for the s electronic state of The optimized geometry was of symmetry with a2A

1He

2C3`. C

2vbond length of 1.218 and an included bond angle of 107.9. Dissociation products were also examined.R

ChHeA

A Pade (4,5) potential function, employing a SimonsParrFinlan expansion variable, was used in subsequentcalculations. The t to the discrete ab initio surface yielded a (s2)1@2 value of 1.35] 10~5 The potentialE

h.

function was embedded in an EckartWatson rovibrational Hamiltonian, which was solved variationally. Thefull anharmonic fundamental frequencies for the breathe, bend and asymmetric stretch vibrations were

1199.2, 673.4 and 1411.9 cm~1 respectively. Vibrational intensities were calculated using the variationalwavefunctions and a dipole moment function generated from an all-electron QCISD/aug-cc-pVTZ 43 pointdipole moment surface.

1 Introduction

Accurate potential energy surfaces (PES) have been con-structed for a variety of rare gas systems from experimentallydetermined properties.1 More recently, the focus has been onthe ab initio construction of discrete potential energy and elec-tric dipole moment surfaces for use in rovibration calcu-lations.2 This is particularly important for helides, sincehelium is considered the most inert of elements and so itschemistry has so far been largely probed by theoretical calcu-lations.3

Helium has been known to form a number of weakly stablecomplexes (which are bound by van der Waals or dispersionforces). However, the formation of stable helides has domi-nated recent interest. Frenking and Cremer3 have reviewedthe last sixty-ve years of helium chemistry with a particularemphasis on the structures, stabilities and bonding character-istics of these species. They found that isoelectronic reasoningalone is not reliable in predicting whether a helide is stable.4They have adopted a three-step method to assess the bondingcharacteristics (which includes an examination of the degree ofinteraction using a donoracceptor model and analysis of theelectron density distribution of the molecule). Alternatively,

Radom and co-workers5h6 have employed a qualitativemolecular orbital (MO) method to describe trends evident inthe bonding and stability of helide ions.

Theoretical studies of helide ions have continued unabated.Recently, Nowek and Leszczynski7 have examined the struc-ture of HeHCO` and Hughes and von Nagy-Felsobuki thestructure of HHeXn` (where X \ C, N, O and n \ 1,2).8 Morepertinent to this work is the investigation of Radom and co-workers6 who have studied the electronic structure of the 2A

1ground state of They have noted that its electronicHe2C3`.

structure is similar to that of the state of (i.e.3B1

He2

C2`having one electron in the anti-bonding MO). So far no3a

1rovibrational structure or spectroscopic constants of the s 2A1state of have been calculated, even though it is pre-He

2C3`

dicted to be stable.6As an extension of our earlier work on the potential energy

functions and rovibrational structure of we wish toHe2

C2`,9

detail an ab initio discrete potential and electric dipolemoment surface, potential energy functions, rovibrationalstructure and vibrational band intensities of the s state of2A

1Although the constructed potential and subsequentHe2C3`.

calculations are speculative, the motivation for this work is toencourage and possibly assist in its spectroscopic detection.

2 Computational procedure

The computational method adopted in this investigation hasbeen outlined by von Nagy-Felsobuki and co-workers else-where.2,9 Briey, the strategy involves using an appropriate level of ab initio electronic theory to construct a discretepotential energy hypersurface from which a potential functionis constructed and embedded into an EckartWatson rovibra-tional Hamiltonian, which is solved variationally. The deni-tion of the Eckart framework has been given elsewhere.2Similarly, a dipole moment function is constructed from a dis-crete ab initio dipole moment surface. By employing the varia-tional vibrational wavefunctions, vibrational intensities arethen calculated.

The rectilinear rovibrational Hamiltonian used in this workis suited for calculations near the potential energy minimum.2The electron correlation model chosen was the all-electronunrestricted CCSD(T) method10 within the GAUSSIAN 94suite of programs.11 The correlation-consistent polarized core-valence triple-zeta (cc-pCVTZ) basis set of Dunning12 wasemployed for all calculations.13

The electron correlation model chosen for the discretedipole moment calculations was the all-electron QCISD (ref.14) within GAUSSIAN 94.11 This model was chosen due tothe unavailability of the CCSD(T) method for dipole momentcalculations within GAUSSIAN 94. Dunnings augmented cc-pCVTZ (denoted aug-cc-pCVTZ)12,15 basis set was employedsince it has been shown16 that diuse polarization functionsare necessary for accurate determination of dipole moments.

A three-dimensional discrete rectilinear normal coordinate

PES was constructed by calculating electronic energies coin-cident with the quadrature points required by the potentialenergy integrator.2 A number of dierent potential functions

Phys. Chem. Chem. Phys., 1999, 1, 29072912 2907


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were investigated, including the related Dunham, SimonParrFinlan (SPF) and Ogilvie expansion variable.2 A poten-tial energy function in terms of a power series or Padeapproximants expansion is well known within the BornOppenheimer approximation for a triatomic molecule.2 It wasassumed that the energy functional was well behaved andamenable to a (s2)1@2 analysis. However, the magnitude of(s2)1@2 is not necessarily a good indicator of the best t to asurface, since higher-order power series (or Padeapproximants) often produce singularities within the region of

interest. Hence, the singular value decomposition (SVD)method was used to identify near rank deciencies so as toeradicate singularities within the t.

The rovibrational Hamiltonian used in this work was the tco-ordinate Hamiltonian devised by Carney and co-workers.17,18 The basic approach used for the variationalsolution has been detailed by von NagyFelsobuki and co-workers elsewhere.2,9 The full rovibrational Hamiltonian canbe written in matrix notation as,17,18

HijRV \ E

iSST

ij]

1

2SAT

ijP

x2 ]

1

2SBT

ijP

y2 ]

1

2SCT

ijP

z2

]1

2SDT

ij(P

xP

y]P

yP

x) ]

i

HSFT

ijP

z(1)

where is the ith pure vibrational eigen-energy and areEi

SSTij@

the overlap vibration matrix elements. The vibrationally-averaged rotational constants are labeled andSAT

ij@ , SBT

ij@

where the prime indicates the incorporation of theSCTij@

multiplying constant of12

.The rovibrational wavefunction was constructed from a

linear combination of products of three-dimensional vibra-tional wavefunctions and symmetric-top rotor functions. Toobtain a rovibrational Hamiltonian matrix representationcontaining real elements the plus and minus com-(H

ijRV),

binations of the regular symmetric-top eigenfunctions ofCarney and co-workers were employed.17,18 Finally, thediagonalization of yielded the rovibrational eigen-H

ijRV

functions and eigen-energies.The spectroscopic constants (such as fundamental fre-quencies and anharmonic constants) were obtained using aleast-squares t to the rotational energy levels of the rst vevibrational states in the usual manner.19,20

A 43 point discrete ab initio electric dipole moment surfacewas calculated using the same quadrature points as the PES(within the QCISD

AE/aug-cc-pCVTZ ansatz). All pointswere calculated within the Eckart framework. The discretesurface was tted to a dipole moment function of form,21

k(R1, R

2, a) \; C

ijk(Q

1)i(Q

2)j(Q

3)k (2)

where and andQ1,2

\ (Ri[ R

ie) Q

3\ (a[ a

e) R

1e\ R

2e \

and The analytical function was dened1.218 A ae

\ 107.9.

in the range and 54.1\a\ 170.5. It0.919\Ri\ 1.536 Ashould be noted for and for bykx

, Cijk

\ Cjik

kz

, Cijk

\ [Cjiksymmetry.

3 Results and discussion

For the state of the structural parameters2A1

He2C3` r

e, h

eusing the CCSD(T)

AE/cc-pCVTZ level of theory were 1.218107.9 respectively. This is in excellent agreement with theA ,

results of Radom and co-workers,6 who have calculatedvalues of 1.163 107.9 and 1.220 108.1 at the MP2/6-A , A ,31G* and QCISD(T)/6-311G(MC)** levels of theory respec-tively.

Radom and co-workers6 have noted that the electronicstructure of the state of is similar to that of the2A

1He

2C3`

state of having one electron in the anti-3B1He

2C2`, 3a1bonding MO. However, they determined the bond length of

the state of to be slightly larger (i.e. MP2/6-31G*2A1

He2C3`

yielded 1.163 as compared with 1.155 While theA A ).increased charge of the ion was expected to increaseHe

2C3`

the participation of the and orbitals of carbon in the2pz

2pxbonding, it was also expected to increase electrostatic repul-

sion, thus leading to a net increase in the bond length. Thegreater contribution of the carbon 2p orbitals could be seen inthe calculated MOs where, for example, the contribution ofthe orbital of carbon and the 1s orbitals of helium to the2p

xMO of are comparable in magnitude. The1b2

(2A1) He

2C3`

onset of sp hybridisation could also be seen in the MO,3a1which had almost equal contributions from the 2s and 2p

zorbitals. Calculations employing the CCSD(T)AE/cc-pCVTZlevel of theory support these conclusions.

The possible atomic dissociation products were also exam-ined at the CCSD(T)

AE/cc-pCVTZ level of theory (and arepresented in Table 1) with the corresponding dissociationreactions given by,

He2C3` ] 2He ] C3` D

0\ 1044.3 kJ mol~1 (3)

] He` ] He ] C2` D0

\ [1203.7 kJ mol~1 (4)

] 2He` ] C` D0

\ [1179.4 kJ mol~1 (5)

] He2` ] C2` D

0\ [1428.0 kJ mol~1 (6)

The thermodynamically most stable dissociation products ofare neutral helium and C3` (reaction (3)). The otherHe

2C3`

atomization reactions, (4) and (5), are both determined to beexothermic. Similar to the analysis given by Frenking et al.4on other helide species, (3) is more stable due to heterolyticCHe bond breakage as compared with (4) or (5), wherehomolytic CHe bond breakage occurs. The dissociation ener-gies of reactions of (4) and (5) are quite similar, thus indicatingthat would not dissociate to the helium ion. The non-He

2C3`

atomization reaction (6) is highly exothermic with the forma-tion of the molecular ion The signicance of theHe

2`.

formation reaction, given by reaction (3), is that it highlights apossible mechanism for the formation of in inter-He

2C3`

stellar gas clouds. This is further emphasized by the realiza-

tion that helium is the second most abundant element ininterstellar space and that neutral and cationic carbonaceousmaterial have also been identied in interstellar space.22

Table 1 Dissociation products of state of calculated at the CCSD(T)

AE/cc-pCVTZ level of theory2A1

He2C3`

Product State Energy/ Eh

Zero-point energy/kJ mol~1

He2C3` 2A

1[40.9766 20.8595

HeC3` 2&` [37.8649 3.2129HeC2` 1&` [39.4117 4.5564He

2` 2&

g` [4.9886 10.2217

He` 2S [1.9990 He 1S [2.9004 C3` 2S [34.7702 C2` 1S [36.5278 C` 2P [37.4199

2908 Phys. Chem. Chem. Phys., 1999, 1, 29072912


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Table 2 The SPF Pade (4,5) potential function (in Eh)a

Variable Numerator Denominator Variable Denominator

1 3.3426 ] 2 8.2501 [ 2 o15 ] o

25 3.7175 [ 7

o1

]o2

8.7154 ] 1 6.3787 [ 2 o35 2.0410 [ 7

o3

5.3408 ] 1 5.5963 [ 2 o14o

2]o

24o

11.4719-7

o12 ]o

22 3.3061 ] 1 3.4745 [ 2 o

14o

3]o

24o

31.2562 [ 7

o32 2.1555 ] 1 2.5068 [ 2 o

1o

34 ]o

2o

34 3.2992 [ 8

o1o

21.4378 ] 1 1.8854 [ 2 o

13o

22 ]o

12o

23 1.3458 [ 8

o2o

3]o

1o

39.7091 ] 0 1.3109 [ 2 o

13o

32 ]o

23o

32 1.0985 [ 8

o13 ]o23 6.1830 ] 0 1.0934 [ 2 o12o33 ]o22o33 7.0471 [ 9o33 5.6696 ] 0 6.1133 [ 3 o

13o

2o

3]o

1o

23o

32.4518 [ 9

o12o

2]o

22o

14.2823 ] 0 2.9246 [ 3 o

1o

2o

33 8.2888 [ 10

o12o

3] o

22o

32.9408 ] 0 1.2470 [ 3 o

12o

22o

33.1682 [ 10

o1o

32 ] o

2o

32 2.2120 ] 0 4.5874 [ 3 o

12o

2o

32 ] o

1o

22o

32 4.8342 [ 11

o1o

2o

31.9383 ] 0 1.0164 [ 4

o14 ]o

24 1.6251 ] 0 2.6342 [ 5

o34 7.6791 [ 1 1.5403 [ 5

o13o

2] o

23o

16.0457 [ 1 1.2869 [ 5

o13o

3] o

23o

35.5314 [ 1 8.9015 [ 6

o1o

33 ] o

2o

33 3.9085 [ 1 4.9255 [ 6

o12o

22 2.9604 [ 1 1.5761 [ 6

o12o

32 ] o

22o

32 2.1197 [ 1 1.4344 [ 6

o12o

2o

3]o

1o

22o

31.6952 [ 1 7.3924 [ 7

o1o

2o

32 1.5728 [ 1 6.6102 [ 7

(s2)1@2 1.35 [ 5a The singular values and were zeroed to ensure no oscillatory behaviour or singularities in the integration region. Notation of 1.35 [ 5p

51v53p

55represents 1.35]10~5.

The construction of a PES electronic grid was achieved byimposing a limit of^0.5 on bond distortion in each normalAvibrational mode. Discrete points were selected to representthe PES in the region of equilibrium geometry such that thedensity of points was a maximum at the equilibrium geometryand decreased radially (in all normal modes of vibration).Coupling of modes was incorporated by the inclusion of o-diagonal points. From a sparse number of points, a prelimi-nary surface was constructed to generate 8000 quadrature

points (20 along each t coordinate). Of these, only 63 pointson the energy hypersurface were selected for the ab initio elec-tronic calculations. The calculated discrete potential energysurface of the state of is available on request.2A

1He

2C3`

The most physically acceptable surface with the lowest(s2)1@2 value was the Pade (4,5) potential function employingthe SPF expansion variable, with the singular values p

51h53,55

Available as supplementary material (SUP 57546, 9 pp.) depositedwith the British Library. Details are available from the EditorialOffice.

zeroed. This hypersurface is illustrated in Fig. 1 together withthe distribution of the 63 points within the vibrational coordi-nate scheme. Table 2 gives the coefficients of the potential t.The (s2)1@2 value of 1.35] 10~5 is sufficiently small forE

hreliable determination of spectroscopic parameters.Solution of the one-dimensional vibrational Hamiltonian

(i.e. uncoupled mode approximation) was achieved using 1000nite-elements within the domains of 2.0t

1[[1.5 a

0, a

0],

2.0 and 2.0 The variationalt2[[1.0 a

0, a

0] t

3[[1.0 a

0, a

0].

solutions of the three-dimensional vibrational Hamiltonianare given in Table 3 in terms of the lowest-lying fteen vibra-tional band origins for the state of The vibra-2A

1He

2C3`.

tional Hamiltonian used in this work incorporates fullmechanical anharmonicity.2 As a consequence the assignmentof the vibrational band origins is no longer simple, sincemixing may occur for congurational basis functions of thesame irreducible representation.2 Thus in Table 3 the percent-age weights of each dominant conguration are highlighted.

As a comparison, the CCSD(T)

AE/cc-pCVTZ(GAUSSIAN 94) harmonic fundamental frequencies (l

1, l

2,

Table 3 Assignment of the vibrational band origins (in cm~1)

Vibrationall1l2l3a Symmetry Percentage weightb band origin

1 000 a1

97.2 0.02 010, 100 a

179.2, 13.0 673.4

3 100, 010 a1

75.3, 16.1 1199.24 020, 110 a

161.4, 18.7 1337.4

5 001 b1

89.3 1411.96 110, 020, 200 a

136.3, 28.1, 19.6 1853.4

7 030, 120 a1

46.5, 18.6 1994.68 011, 101 b

169.4, 11.0 2053.7

9 200, 110 a1

58.3, 25.4 2365.010 030, 210, 120 a

134.3, 21.9, 16.3 2514.2

11 101, 011, 201 b1

55.0, 16.8, 13.7 2554.312 040, 130, 120 a

134.8, 15.4, 9.8 2647.9

13 021, 111, 121 b1

52.7, 14.3, 8.3 2692.814 002, 102 a

174.9, 17.0 2780.8

15 120, 300, 210 a1 31.4, 24.5, 17.5 3015.7a The harmonic frequencies are : cm~1 ; cm~1 ; cm~1. b Percentagel

1(a

1) \ 1275.8 l

2(a

1) \ 714.3 l

3(b

1) \ 1497.4 weight \ MC

ij2/& C

ij2N1@2]100.2



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Fig. 1 Two-dimensional constant potential energy plots for P(4,5)surface with SVD analysis and with singular values andp

51~53p

55zeroed: (a) breathe mode vs. asymmetric stretch mode (b)(t1) (t

3) ;

breathe mode vs. bend mode (c) bend mode vs. asym-(t1) (t

2) ; (t

2)

metric stretch mode The discrete ab initio points are overlayed as(t3).

open circles.

were calculated to be 1275.8, 714.3, and 1497.4 cm~1,l3

)whereas the variational solution which incorporates fullmechanical anharmonicity determined the corresponding fre-quencies to be 1199.2, 673.4, and 1411.9 cm~1) respectively.Comparison of these models yields a harmonic: anharmonicratio of 0.94 which is consistent with the usual ratio of 0.93between harmonic and experimental frequencies.23 The varia-tional solution yielded a zero-point energy of 20.426 kJmol~1.

The rotational constants dened by eqn. (1) were calculated

using the vibrational wavefunctions determined from thethree-dimensional vibrational Hamiltonian, employing athird-order expansion of the Watson operator.2 The groundstate values ofSAT, SBT, SCT, SDT and SFT are 6.931, 2.114,1.604, 0.2502 and 3.101 cm~1 respectively. Comparison can bemade with the ground state rotational constants of SAT, SBTand SCT determined from the harmonic approximation inGAUSSIAN 94,11 which yield values of 6.820, 2.171 and 1.647cm~1 respectively and are in excellent agreement with theanharmonic calculation.

A 43 point discrete dipole moment surface was constructedusing a subset of the quadrature points selected for the PES.The points culled were at large displacements from the equi-librium geometry. The calculated 43 point discrete dipolemoment surface of the state of is available on2A

1He

2C3`

request. The expansion coefficients of the dipole moment func-tion (see eqn. (2)) are given in Table 4. The permanent dipolemoment was determined to be 1.9012 D (i.e. the coeffi-l

eC

000cient of the dipole moment expansion). Vibrational bandcentres, Einstein coefficients and band strengths were deter-mined from the variational solution. The values for theseJ \ 0 transition probabilities are given in Table 5.

The limiting case for the rotational levels determined isMullikens prolate symmetric-top. This is reected by Raysasymmetry parameter,19 which is ca. [0.82 for the rst vevibrational states. Assignment of vibrational states was madewithin this framework. The variationally calculated rotationaleigen-energies up to J \ 5 for the low-lying vibrational stateshave been calculated and included as supplementary material.

To ensure convergence of these calculated eigen-energies anumber of truncated basis sets were used. The mean dierencebetween calculated rotational levels employing ve and tenvibrational eigenfunctions was found to be only 0.001 cm~1.

For an asymmetric top molecule, the rotational constantmatrices A, B, and C (see eqn. (1)) correspond to andB

xx, B

yyrespectively. From the least-squares t of the diagonal ele-Bzzments of the rotational constants spanned by the 20 lowest-

lying vibrational wavefunctions, the rotational constants Bxxand were calculated. Table 6(A

0, A

e), B

yy(B

0, B

e) B

zz(C

0, C

e)

yields the tted rotational constants.The calculated rotational constants from the previously

determined ab initio force elds using the rovibrational statesis also included in the supplementary material. It is important

to note that these calculations neglect spinrotation inter-actions. Moreover, the quartic centrifugal distortion constantsare calculated at the equilibrium geometry.

Acknowledgements

wish to acknowledge the CPU allocation on the super-Wecomputer facilities at ANU and the workstation facilities ofthe University of Newcastle. D.J.W. and J.M.H. wish toacknowledge an Australian University of Newcastle Post-graduate Research award. Sudarko wishes to acknowledge thesupport of the Australian Agency for InternationalDevelopment.



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Table 4 Expansion coefficients of the dipole moment function (in ea0)a

kx

Component (Cijk

\ Cjik

) kz

Component (Cijk

\ [Cjik

)

C000

0.74843640 C100

0.10921143C

001[0.00251369 C

1010.00050049

C002

[0.00006074 C102

0.00002069C

0030.00000048 C

1030.00000052

C004

[0.00000001 C200

0.05934982C

100[0.09894549 C

201[0.00000487

C101

[0.00281472 C202

0.00004554C

102[0.00000695 C

210[0.05642942

C103 [0.00000017 C211 [0.00310182C110

[0.17629001 C300

[0.25475903C

111[0.00220606 C

301[0.00444261

C112

0.00003280 C310

[0.26141078C

200[ 0.06116171 C

400[0.12453506

C201

[0.00336675 C

202[0.00000019

C210

0.04614517 C

2110.00359625

C220

0.07876697 C

3000.08392556

C301

0.01082978 C

3100.28616985

C400

0.19405557 (s2)1@2 \ 1.81]10~2 (s2)1@2\ 1.42]10~4

a See eqn. (2) of text and ref. 19 for further details.

Table 5 Vibrational band centres, square dipole moment matrix elements, Einstein coefficients,a band strengthsa and radiative lifetimes

j,i lji

/cm~1 kji2/D2 A

ji/s~1 B

ji/1016 cm3 erg~1 s~1 S

ji/cm molecule~1 q/sb

1,0 673.4 2.70 [ 03 0.258 5.07 [ 01 6.93 [ 19 3.362,0 1199.2 1.10 [ 03 0.595 2.07 [ 01 5.23 [ 19 1.593,0 1337.4 1.60 [ 05 0.012 2.98 [ 03 8.42 [ 21 1.604,0 1411.9 2.00 [ 03 1.758 3.75 [ 01 1.12 [ 18 0.565,0 1853.4 1.20 [ 06 0.002 2.25 [ 04 8.80 [ 22 1.036,0 1994.5 1.10 [ 06 0.003 2.08 [ 04 8.78 [ 22 0.977,0 2053.7 1.50 [ 06 0.004 2.79 [ 04 1.21 [ 21 0.508,0 2365.0 4.70 [ 08 0.000 8.82 [ 06 4.41 [ 23 1.059,0 2514.2 1.50 [ 07 0.001 2.78 [ 05 1.48 [ 22 0.7410,0 2554.3 5.30 [ 05 0.276 9.94 [ 03 5.36 [ 20 0.56

a Calculated at 300 K. Notation of 2.70 [ 03 represents 2.70]10~3. b Lifetime of the upper state.

Table 6 Fitted rotational constants (in cm~1)

Rotational constants Rotational constants

A0

6.9307 a1A [1.3746

Ae

7.8546 a2A [0.6928

B0

2.1141 a1B 0.1622

Be

1.8956 a2B 0.0819

C0

1.6041 a1C 0.1307

Ce

1.5627 a2C 0.0666

References

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sible Computational Chemistry Environment Basis Set Database,



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Version 1.0, developed and distributed by the Molecular ScienceComputing Facility, Environmental and Molecular SciencesLaboratory, which is part of the Pacic Northwest Laboratory,P. O. Box 999, Richland, Washington, 99352, USA, and fundedby the U.S. Department of Energy. The Pacic Northwest Labor-atory is a multiprogram laboratory operated by Battelle Memo-rial Institute for the U.S. Department of Energy under ContractNo. DE-AC06-76RLO1830. Contact D. Feller, K. Schuchardt orD. Jones for further information.

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david j. wilson et al- rovibrational states and vibrational intensities of the chi^2 a1 state of...

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