A theoretical study of the electronic transition moment for the C2 swan band system

A THEORETICAL STUDY OF THE ELECTRONIC TRANSITION MOMENT FOR THE Cz SWAN BAND SYSTEM

Computational

Chemistry

J. 0.

ARNOLD

Group,

Ames

and S. R. LANGHoFFt

Research

Center,

NASA,

Moffett

Field,

CA 94035.

USA

(Received 27 October 1977) Abstract-Self-consisient-field plus configuration-interaction calculations have been performed on the a311, and d’II, states of Cz. Good agreement is obtained between the theoretical potential energy curves and those obtained by a Klein-Dunham analysis of measured molecular constants. The sum of the squares of the theoretical transition moments EIR$ between these states at 2.44 bohr is 4.12, a.u. which is in good agreement with the range of values of 3.3-3.6 a.u. obtained from shock tube measurements. The computed variation of the ZlR,[* with internuclear distance is in remarkably good agreement with the relative measurements by Danylewych and Nicholls.

I. INTRODUCTION

band system of C2 (a’& - d311,) lies in the wavelength range between 340 to 785 nm and is observed in a wide variety of astrophysical and terrestrial sources. This system is often prominent in the emission spectra from the heads of comets and is an important”’ absorption feature in the spectra of the sun and.type R and N “Carbon” stars. These bands are often observed in the emission spectra from carbon arcs”’ and combustion processes’2’ in the presence of carbonaceous species. The C2 Swan band system also plays an important role in studies of planetary entry physics. Specifically, the C2 Swan bands would be quite prominent in the emission from bow-shock-heated flow fields surrounding vehicles entering the atmospheres of Mars and Venus at speeds in the 5-l 1 kmlsec range. Also, if carbonaceous heat shields are employed for the very high speed (45-50 kmlsec) entry into the Jovian atmosphere, the C2 formed in the vehicle boundary layers would strongly absorb the intense radiation from the shock layer. This photoabsorption occurs mainly in the Swan bands and has the effect of significantly reducing the amount of radiative heating experienced by the vehicle. The Swan band system of C2 has been the subject of many previous studies, and the values of the electronic transition moment obtained by various experimenters are quite discordant. We believe that the present results provide strong evidence as to the actual value of the electronic transition moment for the C2 Swan band system and its variation with internuclear distance. Thus, the present results increase the certainty with which one can predict the absolute absorption or emission intensities for the Cz Swan band system. Furthermore, the work described herein serves to illustrate the current utility ab inilio methods have in providing reliable molecular data for nontrivial systems. These data for the C2 Swan band system are the first results from a program underway at Ames Research Center to provide additional information on the low-lying electronic states of CZ needed to predict accurately the absorption spectra of this molecule under the conditions expected for the boundary layers which would be formed on Jupiter entry probes. THE

SWAN

2. WAVEFUNCTIONS

AND AND

POTENTIAL d?I, STATES

CURVES

FOR

THE

&I.

The atomic basis used in our study on C2 consists of a set of 46 Slater-type orbitals (STOs) which is specified in Table 1. This set consists of the “accurate” s, p atomic carbon basis defined in Ref. (3), augmented with polarization functions and 10d orbitals so that the configuration interaction calculations could be extended into the S molecular orbital space. The one- and two-electron integrals were evaluated with our version of the BISON Code.“’ tNational

Research

Council

Postdoctoral

Associate

461

462

J. 0. ARNOLD and S. R. LANGHOFF Table I. Slater-type orbital basis set used on each atomic center for the current study on Cz,

The Hartree-Fock equations were solved for the closed-shell la,210U22a,22aU217rU4 configurationt Both the canonical and internally consistent self-consistent-field (ICSCF)‘4’ molecular orbitals were used in subsequent configuration-interaction (CI) calculations to determine the a311Uand d311, potential curves and the electronic transition dipole moment of the Swan band system. The spin-adapted configuration state functions (CSFs) that were included in the final wavefunctions were chosen on the basis of their contribution to the Rayleigh-Schrodinger perturbation theory estimate (‘) of the correlation energy using a procedure equivalent to the Al, method outlined by GERSHGORNand SHAVIIT.@’Through a series of small CI calculations, we first determined a reference list that contained all of the CSFs that have a large coefficient (~0.09) in the wavefunctions of the two lowest 311Uand 311sstates at internuclear distances between 2.17 and 3.80 bohr. The reference lists represent the zerothorder approximations to the wavefunctions for those four low-lying states and contain the principal correlation effects and corrections needed to compensate for using molecular orbitals optimized for the closed-shell configuration. The 311Ureference list consisted of 21 CSFs, and the a311, state was dominated (coef = 0.9) at all internuclear distances by the llr, +3u8 excitation. The 311sreference list consisted of 24 CSFs, and the d311, state was dominated at small internuclear distances by the 2u,, 1P, + 3o,* double excitation and at larger internuclear distances by the l?rU2+30,17r~ double excitation. Hence, the character of the d3?rgwavefunction changes considerably over the range of internuclear distances considered. At the larger R values, the two lowest states of 3118symmetry begin to mix, and an avoided crossing between these states appears likely near 4 bohr. This point and a detailed discussion of the asymptotic behavior of the potential curves will be given in a future publication. The final list of CSFs consisted of all the configurations that are formed by single and double excitations from any of the CSFs in the reference list which contribute more than 50 microhartrees (kh) to the correlation energy estimate for either of the two lowest states of the given symmetry. This resulted in matrices containing about 6000 CSFs. The energies (E) obtained using the 50-ph threshold were then extrapolated to a zero-energy threshold using the relation E’= E,,+AE’, AE’=(E-E&(1+

perturbation thrown away perturbation kept >’

(1)

where E’ is the extrapolated energy and EO is the zeroth-order energy of the reference list. The extrapolation serves to lower each curve nonuniformly as a function of internuclear separation by amounts ranging from 0.35 to 0.60 eV. The lowering is greater at larger R values because the wavefunctions are not as compactly described by just a few CSFs. We have also extrapolated the CI energies for R = 2.479 bohr using the procedure developed by BUENKER and PEYERIMHOFF.(‘) Calculations were carried out at thresholds of 10 and 20 ph in addition to 50 CL h. The use of these energies and eqn (8) of Ref. (7) gives extrapolated energies that agreed to within 0.02 eV of those obtained from eqn (1). Table 2 contains the values of E’ and E for both states at each of the internuclear distances considered in the present study. The energies in parenthesis for the calculations at 2.17 and 2.6 tAl1 configurations referred to in the text below are defined with respect to excitations from this configuration.

A theoretical Table

2. Eigenvalues

study of the electronic

and sum of squares

transition

moment

for the Cz Swan

of electronic transition moments Cr Swan band system.”

T-

III&(’

463

band system

= ~.O~(\Y”‘“~(IZ~~~~‘~~)(*

for the

R (bohr)

Paramererb 2.1700 E’(a311u) -75.677180

2.2000

2.3480

2.4790

2.6000

2.9007

3.2000

3.8000

-75.686508

-75.715073

-75.722372

-75.719085 (-75.726269)

-75.689020

-75.649012

-75.574478

(-75.682644)

E(a3nu)

-75.663917 (-75.670210)

-75.673562

-75.702261

-75.709665

-75.706003 (-75.712690)

-75.675140

-75.635609

-75.559000

E'(d311g)

-75.606278 (-75.612220)

-75.613320

-75.631956

-75.631861

-75.622271 (-75.630469)

-75.589137

-75.559745

-75.526050

E (d3$)

-75.593152 (-75.598700)

-75.600132

-75.617990

-75.616456

-75.605906 (-75.612677)

-75.563454

-75.532711

-75.503347

5.825 (5.5217)

5.634

3.8711

3.131 (3.234)

1.475

0.005

0.582

~lRe12

4.747

Results in parenrheses obtained with internally consistent - self-consistent field molecular E' = extrapolated energy for threshold in perturbation selection = 0 hartree E = energy for threshold in perturbation selection = 50 uh

orbitals.

bohr were obtained using the ICSCF orbitals, and are somewhat lower than the corresponding values for the canonical orbitals. This result is consistent with an earlier observation’4’ that the use of ICSCF orbitals leads to faster convergence in the CI expansion. These ICSCF calculations were performed to test the sensitivity of the transition moment matrix elements to a unitary transformation among the molecular orbitals. In the limit of a full CI, the two sets of calculations should yield identical energies and transition moments. The theoretical potential energy curves are plotted in Fig. 1 and are referenced to zero at the internuclear separation of 2.479 bohr for the a311, state. These results are compared with similarly prepared Klein-Dunham potential energy curves, which were obtained”’ from experimental data by our colleague at York University, Dr. Lucy Danylewych. The agreement between the experimental and theoretical results is excellent in both the shape of the potential curves and the excitation energy.

-

-

-

PRESENT REF.

2

THEORY

EXPERIMENT,

KLEIN-DUNHAM,

8

3

4

R, bohr

Fig. 1. Comparison of computed SCF+CF potential energy curves with curves. The theoretical energy curves are referenced to zero at the minimum experimental ones.

experimental of the a%.

Klein-Dunham curve, as are the

The spectroscopic constants for the present theoretical curves were obtained from a least-squares fit to a Morse curve and are oe = 1630.90 and 0~~ = 12.26 for the a311, state and o, = 1778.94 and ode = 26.02 for the d311, state. These data compare well with the experimental results”’ of 1641.35, 11.67, 1788.22 and 16.44, respectively.

J. 0.

464 3. TRANSITION

MOMENT

ARNOLD and S. R. LANCHOFF RESULTS

FOR

THE

C2 SWAN

BAND

SYSTEM

Our calculated values for the sum of the squares of the electronic transition moments, 81R,1*= 6.0((~43n~I~~iJ~d’n~)12, for the CZ Swan band system are plotted in Fig. 2 and compiled in Table 2. The factor of six in the above definition accounts for the summation over allowed transitions,“’ while the summation in the dipole moment operator zi, runs over the 12 electrons for the Cz molecule. These results were computed from the wavefunctions determined with the threshold of 50 ph. The transition moment results at 2.17 and 2.60 bohr using the ICSCF orbitals are also included in parenthesis in Table 2. These results are within 5% of the corresponding ones using the canonical orbitals. The relative insensitivity of the results to a unitary transformation among the molecular orbitals is supporting evidence that the CI expansions have converged. 10.0

r

l-

2 $

;+?; :::::::

\

.:.:.:. :.:.:.

1.0 -

E w

--

PRESENT THEORY

-

EXPERIMENT :

0 RANGE OF OTHER WORK

2.0

2.2

2.4

2.6 R, bohr

2.8

3.0

REF 23 11 12 10 13.22

3.2

Fig. 2. Variation of the sum of the squares of the electronic transition moments 6.01(VI”‘““IZz,(Yd’“~)IZ with internuclear distance R for the Cz Swan band system. The relative ments of Ref. (23) were normalized to the present theoretical results at 2.44 bohr.

IJR$

=

measure-

As can be seen from Table 2, the value of the SlR,12becomes quite small at an internuclear distance of 3.2 bohr and rises to a value of 0.582 at 3.8 bohr. This behavior is a consequence of the mixing of the two lowest states of 311s symmetry at the larger R values, which was described above. The transition moment data at 3.2 and 3.8 bohr were not plotted in Fig. 2 because of this rapid variation at the larger internuclear distances. Our data for the XIRe12are compared with measured results”““’ in Fig. 2. These experimental data are compiled in Table 2 and were extracted from Refs. (10) and (11). By interpolating our results along the dashed curve in Fig. 2, we find our result at 2.44 bohr for the 21R,12 to be 4.12 a.u., which is in good agreement with the shock-tube measurements obtained by FAIRBAIRN (I*) of 3.36 -t 1.20 a .u ., ARNOLD(“) of 3.56 2 0.50 a.u., and by COOPER and NICHOLLS”~’ of 3.52 + 0.50 a.u. These data were obtained from the (0,O) band region of the spectrum which corresponds to 2.44 bohr in the r-centroid approximation. The remaining experimental data from Table 3 lie below these results by as much as a factor of six and are represented in Fig. 2 by the shaded bar. Reviews and discussions of these experimental results are contained in Refs. (lo), (11) and (22). Also plotted on Fig. 2 with the solid line is the relative variation of the 2jRJ’ determined by DANYLEWYCH and NICHOLL~‘*~’ from a synthetic spectrum analysis of data from a stable electric discharge. We have normalized their relative data to the theoretical value of 4.12 at an internuclear separation of 2.44 bohr. As can be seen from the figure, our variation of the 21Re12

A theoretical study of the electronic transition moment for the CZ Swan band system

465

Table 3. Comparison of present theoretical results with measurements of the values of the sum of the electronic transition moments squared for the Av = 0 sequenceof the Swan system. rIae12* ExperimenCer(~f3 present

Method

results

COOPER and NICHOLLS (10) ARNOLD (11) FAIRBAIRN (12) "ARRINGTON (13) SVIRIWV (14) SVIRIWV (15) HICKS (16) "AGEN (17) BARGER (18)

et al. et al. et al.

et al.

JElRJEH0MX.Eand SCHWENKER FINK and WELGE WRNTINK

et al.

(20) (21)

LEACH and VELGHE

(22)

(19)

Theoretical SCF+CI Shack-tube Shock-tube Shock-tube Shock-cube Shock-tube Shock-tube King furnace King furnace Carbon effusion furnace Laser excitation Phase-shift Laser excitation Laser excitation

atomic units 4.12 3.5220.50 3.5eo.50 3.36t1.20 2.95kO.92 2.24f0.81 2.85tO.20 2.64 0.71~0.40 0.02 0.6OiO.02 2.24t0.70 1.35 2.0 to.20

with internuclear distance compares quite well with that measured by Danylewych and Nicholls over a wide range of internuclear distances. These two sets of data appear to be somewhat at odds with that reported in Ref. (1 1), but those data were obtained over a fairly narrow range of internuclear distance and thus probably are not definitive. 4. SUMMARY Large-scale SCF+ CI calculations have been performed for the a311, and d311, states of Cz. The theoretical potential curves agree well with the experimental Klein-Dunham ones in terms of both shape and excitation energy. The theoretical calculations give a value of 81&1*= 4.12 a.u. at 2.44 bohr that is in agreement with the upper end of the range of experimental data (3.3-3.6 au.). The computed variation of the 8)&l* with internuclear separation agrees well with the measurements by Danylewych and Nicholls. On the basis of our experience with C2 and the transition moments for other molecules, e.g. C10,‘24’SiO,‘*” and CO,@j’we feel that large CI calculations employing near Hartree-Fock quality Slater basis sets are capable of producing transition moments as reliable as those from high-quality laboratory experiments. Acknowledgements-The authors wish to thank their colleague, Dr. LUCY DANYLEWYCH,for providing the Klein-Dunham potential curves used for comparison with our theoretical results. We also wish to thank LINDA SHARBAUGH and JIM DIX for their expert programming assistance.

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466

J. 0. ARNOLDand S. R. LANCHOFF

20. E. H. FINK and K. H. WELGE. .I. Chem. Phys. 46. 4315 (1967). 21. T. WENTINK,JR., E. MARRAM,L. ISAAC~~Nand R. SPINDLER,Experimental determination of molecular oscillator strengths. (AVCO Corporation) Air Force Weapons Laboratory Rept. AFWL-TR67.30, Vol. I (Nov. 1%7). 22. S. LEACHand M. VELCHE,JQSRT 16. 861 (1%7). 23. L. L. DANYLEWYCH and R. w. NICHOLLS,Proc Roy. Sot. Ser. A 339, 213 (1974). 24. J. 0. ARNOLD.E. E. WHITINGand S. R. LANGHOFF,1. Chem. Phvs. 66.4459 (1977) 25. S. R. LANGHOFFand J. 0. ARNOLD,J. Chem. Phys. to be published. 26. D. M. COOPERand S. R. LANGHOFF,J. Chem. Phys. to be published.