Theoretical study of the electronic transition moments for the d3Πg-a3Πu (SWAN) and e3Πg-a3Πu (Fox-Herzberg) bands in C2

CHEMKXL PHYSICS LETTERS

Volume 83, number 3

THEORETICAL

1 Novemkr 1981

STUDY OF THE ELECTRONIC TRANSITION MOMENTS

FOR THE d 3KIg-a311U (SWAN) Gary F. CHABALOWSKI,

AND e 3fIg-a 311u (FOX-HERZBERG)

BANDS IN C,

Robert J. BLJENKER and S&rid D. PEYERIMHOFF

LelrrstuhljUt Theoretzsche Chemie, UniberszMt Bonn, D-5300 Bonn 1. West Germany and Lehrstuhl $2 Theoretzsche Chemre. Universitat-Gesamthochschule Wuppertal. iUS

Wuppertal I. West Gemtany

Recewed 20 July 1981

Cl calculations for the Swan and Fox-Henberg band intenutles IIIC2 are reported and excellent agreement with eupervalues is found. The X IReI’ values vary strongly with bond drstance and in ~ertarn C;IWS az~ found to depend ngruticantl~ on the length of the Cl eupmslon.

imental

l_ Introduction The electromc transitions m the C, molecule have been observed m several terrestrial and astrophysical sources [I ,Z J, emission from comets and absorption in the atmosphere of R- and N-types stars bemg prime examples. The bands are produced III the laboratory

by various mechanisms includmg shock-tilbe experiments, combustion processes and arc chscharges between carbon electrodes. The C2 band systems are reported to be of significance III reducing the amount of radiative heat reachmg the heat shield of a space vehcle upon h&speed reentry into the planetary atmosphere_ Consequently there have been many attempts to determine the mtensity of these bands, especudly the square of the electronic dipole transition moments Z IReI’ determinmg the probability for an electronic tram&on. The purpose of the present study 1s to investigate the electronic tranntion moment for the Swan d 317g-a311u and Fox-Herzberg (FH) e 311g-a 311u band systems using extended ab imtio configuration mteractioa (CI) methods. The calculation of 2 IReI2 1s performed as a tinction of the internuclear carbon-carbon distance. A strong dependence on this vanable is expected because the 311g states involved exhiiit considerable change in composition with CC stretch due to various avoided crossings [S] _ No such curve has yet been published for the FH system. 0 009-2614/81/0000-0000/~

02.75 Q 1981 North-Holland

An earher study of the Swan band system by Zeitz, Peyerimhoff and Buenker (ZPB) [4] employing CI methods gave a shape for the plot of Z IR,$ versus bond length wkuch agreed with two experimenta! counterparts, whereas the absolute value at rcc = 2.44 bohr was found to he tzlgher (5.2 in the calculaeons compared to 352 au in the expedment). An-

other Independent theoretical study of the Swan bands by Arnold and Langhoff (AL) [S] emplomg sirrular CI methods as ZPB but Slater-type basis funct~onsgave essentially the same dependence of X @,(2 upon the internuclear distance as the curve calculated

by ZPB but led to ccnslstently lower values for tfais quantity, in better agreement with the expenmental results. It IS thus a further goal of the present work to analyze and explam these discrepancies_

2. Method of calculation Open-shell HF SCF calculations are performed for the configuration

in the reduced D,

spatial symmerry for bond lengths varying from r = 2.0 to 33 (all calculated values are reported in atomic units unless otherwise specified) at increments ofdr =O.l,and one additional cakulation is performed at r=35 bohr. The carbon A0 basis set 441

Volume 83. number 3

CHEhIICAL

PHYSICS

corwsts of a (9s. Sp) set of pnnutive carteslan gausslans contracted to [Ss, 3p] accordmg to the scheme of

Dunrung [6] and contams two uncontracted d functtons with exponents CY~ (d)= 12, aZ(d)= 035 located at each of the carbon atoms. At the center of the bond two uncontracted s functions with al(s) = 1.0, and a2(s) = 0 02, and two uncontracted p functions w.;h a,(p) = 0.70, and a,@) = 0.018 are added so that the total A0 basts ctksts of 60 gausnan-type orbltals (CTOs) and Includes polanzatlon as weU as

1 November 1981

LETTERS

a 3 li, bands under consideration presslon

(3ilgl c

er,$lI,,)

I

are even

I

by the ex-

* ,

functions_

in whxh Z,er, is the length form of the electronic dipole moment operator summed over all electrons The factor of six occurs because (a) both the 3 l$ and 3KIu states are m-fold degenerate but (b) only tranauons between respective components are allowed and (c) all those SIX moments are equal. Hence only one

The Cl treatment employed ISof the multl-reference doubleexcltatlon (TURDCI) type [7] wnh con-

such component IScaicuiated m the present work and Its m-fold value can be directly compared urlth E we]‘,

and enerfl extrapolation [Sj _ state functions (or symmetryadapted functions SAFs) are thereby generated as smgle and double excitations relative to a set of reference configurations and are directly mcluded m the secular equatton tf their interactlon with the set of reference configurations IS larger than a predetermmed threshold T. The energy obtamed by dlagonahzmg the harmltontan matrtx correspondmg to tlus threshold value wdl be referred to as E(T). The effect of those SAFs not directly mcluded m tie secular problem IS extrapolated m the standard manner [8] and the estimate of the energy for zero threshold IS referred to as E(MRD Cl). In ad&non the energy correspondmg to a full CI is estunated from perturbation energy usmg the expression

I e the quantlry

hffuse

Rydberg-type

figuration

selectlon

The configuratlon

E(full +

CI est.) =E(MRD

(

CI)

ref

I-CC; P

)

[E(MRD

CI) - E(ref)]

,

(1)

m whch E(ref) 1s the energy resultmg from the configurations in the reference set alone Thts expresslon IS analogous to the equation denved for the energy contnbutions of htgher-than-double excitation spectes [9]

andhubeen

successt-diy

employedmnumerous

typIcally

reported

by

experimental-

1sts

FmaUy, the oscdlator the entire band as

I

strengths

f, =f AE (311,1Cer,13fl,) I

are obtamed

I

* .

3 Potential energy surfaces and characterization 3.1. 77le a311u state

The number of reference configurattons employed for generatmg the lowest 3KIu state is held constant at five for all bond lengths treated (table l), leading to a total of 71357 SAFs. A threshold of T = 3& is used throughout, which leads to a space of 6011 SAFs at a bond length of rCC = 2.4 bohr m the unmelate nelghborhood of the potenttal curve muumum, the number of selected configurations remains somewhat above 6000 in the area between rcc = 2.0 and 3.1 bohr and decreases only shghly to around 5400 III the regon between 3.2 and 3.5 bohr. The potential curves thus obtained for the entue MRD Cl space are plotted in fig. 1; the lowest calculated energy for the a 3n,

For all electromc states calculated m the present work a core of two doubly occupied MOs IS

state is thereby E(MRD CI) = -75.74544 rcc = 2 5 bohr (E(T = 3ph) = -75.73413

mamtamed throughout, corresponding to the carbon 1s tnnershell orbltals; likewise the two virtual MOs with the highest orbital energies corresponding to the inner-shell complements are elinunated enttrely from the CL The square of the electronic transltlon &pole mo-

which pomt the full CI estimate IS -75.77065 llus bond length is III reasonable agreement

442

311Uand FH e 311g-

of

electronic states

cases [ lo]_

ments for the Swan d 3lI,-a

for

hat-tree at hat-tree) at hartree. (without

explictt potential curve-fitting procedures) with the minimum found at r, = 2.479 bohr experimental [ 111. The calculated ground&ate frequency of 1627 cm-l is also tn accord with the measured value of 1641 cm-l.

Volume

83, number

1 Nowmber

CAEMICAL PHYSICS LElTERS

3

19Bf

Table I Gwen are the reference config~attons, the total number of SAFs generated and Technical deta.ds of the CI cakuiat~ons undertaken the SAFs explintly employed at the threshold 7-1. Selecaon ISundertaken wth respect to one root IIIthe 3&, space and with respect to two ro5ts for ‘lTg

3%l

12/.ih

108294

3ng

The state descnphon clearly remains that of

EIMRD-Cl1

1tJ; la~24.@o; iIrL ‘?TUY35$

fhcwtreet l

for the entxre bond length regzon 2.0 to 35 bohr, whereby the contribution of the dominant term is reduced from 86%atrcc = 2.0 bohr to 78% at 35 bohr, while that of the reference sef on a Fefc$ basis still remams between 0.87 and 0.89 for these distances. These fmdmgs for the ?ffu composition are in accord with resuks reported earlier [45 1.

J

-7s LO -

-75 50 -

-75 60-

3500-6000

1

The two 3fIg states are believed to be adequately described by seven reference configurations (table 1) over the entlZe bond-length range of interest in the presentwork_ThecharacteroFthe si&cantly

-7570

thereby,

d ‘Fig state changs

so that the contribution

of the

_.2uz2au In& 17r,3O,2 con&p&ion decreases steadrly from c$ = 0.8 I at ra = 2.0 bohr to C$ = 1

20

ZL

Fig 1. Calculated potent& ande3iIgstatesofC2.

28

32 36 rCC Ibohr)

)

energy curves for the a3 n,, d 3~g

0.55 at rcc = 2.8 bohr,ci = 0.40 at rcc = 29 bohr and fiially to c$ = 0.01 at rW = 35 bohr.At the same time the coefficient for the compIemen_ configuration _.2@&2a,2I7r& ‘R;u3iF&_& increases steadily from+ z = 0.02 (ra = 2.0 bohr) to cs .= 0.16 (r- = ZSbohcJ, c; = 0.26 (pS = 29 bohr) andc; = 443

Volume 83, number 3

0.72 at rcc = 3.5. This change in state descnphon 1s consistent with the results of earher calculations [3-S] _ The total number of SAFs treated m this symmetry 15 108294, whereby 5822 resultmg from a threshold of T, = 12 fi are duectly mcluded III the secular equation at rcc = 2.4 bohr, leadmg to an energy E(T) = -75 -62943, E@lRD Cl) = -75 65625 and a full CI estmate of -75 -68439 hartree The approumate eqtibnum bond length (agarn without a curve-firtmg procedure)

of 2.3 shouId be compared

with rhe

euper-

value of re = 2392 bohr and the calculatsd = 1735 cm-l with the experunental 1788 cm-l In $1, tl le entlre calculated curve IS also contamed fig 1. From a techrucal pomt of hew 1~should be remarked that the number of selected SAFs at the g~~en threshold of 12 I.rh vanes from 6059 at rcc = 3.0 to 1375 at I- = 3 0 bohr unental

The second 3fIp state 1s prlmarlly reference configuration

descrrbed

the con5guraUon

fmdmg is not surprlsmg m h&t of the proxmuty of the various 3TIg states at small distances 131, but tlus

aspect of the calculations gated smce the pnme

at larger mternuclear bormg the mmunum

_%~2o~

has not been further LnvestL-

rntelest in the present study lies

separations,

1-e the region nelgh-

of the potential

curve.

properties

The calculated fransltlon energy between mmuna of the Swan band system IS AE,(full

.

1 nz,20g

lx&\ with

cz=O17(r,,=2 lbohr).O07-(rCC=30)and000 zt 3 5 bohr The change m descnptlon of the two 3ilg states under conslderatlon m the present work as well as that of the order two 311s states also ansmg from .M,n~3c~~ and 2c&r~30~n~ electronic configuratlons can be seen very clearly from the non-adlabatlc coupimg matrLu elements (4O la/&,, 1@fi) between these four states wluch have been calculated (m a smaller A0 basis) III earher work [3]. The computed potenrlal energy curve for the e 311g state is also contalned II-Ifig 1. Its lowest enerm 1s E(T = 12 PII) = -75 52846 hartree and E(full CT est ) = -75 59139 hartree. Smce the extrapolation procedure at the threshold T = 13 m leads to larger fluctuatlons than deslrable for a good descnption of the very shallow part of the e 311g state, an even smaller threshold, namely T2 = 6.5 ph, LSemployed m tlus instance, which places the e 311g nunimum at rcc = 220 bohr in agreement with the expenmental bond length of 2901 444

This

by the

wth cz values of 0.69 (rCc = 3.1 bohr), 0.78 (r 3.0 bohr) and 0.83 (rCc = 3 5 bohr) At smaller%o~d lengths there IS dn rmportant secondary contnbutlon from

bohr Ill]. The correspondmg energes are E(T = 6.5ph) = -75 54196, E@lRD CI)= -75.56590 (compared to -75 56493 derived from T= 12 &) and E(fuU CT est.) = -75 -59280 hartree. The corresponding techmcal data are also contained m table l_ Finally it should be noted that for small bond lengths (rcC < 2 0 bohr) an mteratitlon between the second and tird 3 Ils states becomes apparent, whereby the configuration 2u~20~ 1 7rUux17ruy3ug 17rW is the mun component of the thud lowest 311g state and the second IS prrmanly described by _20~2a, II& lx,,,30,4u,_

4 Transition

3.3. i%e e 311g scare

zag27u&r”,171”,,3Qg17r~, I

1 November 1981

CHEhUCAL PHYSICS LETTERS

the curve

CI est.) = 2 35 eV, thus value increases by 54 cm-l If the calculated zero-point energes are also conndered and compares favorably with the expenmental O-O value of 2.40 eV [I?,131 ThecalculatedXIR,I* valuesofeq.(?)are bond plotted m fig 2 as a function of the mternuclear distance rcc There LSquite good agreement between the current MRD CT results (curve A) and the theoretlcal data of AL [S], as well as the expenmental results represented by curves I [12] and II [Z ,131. Furthermore it IS seen that the present MRD CI curve 15 everywhere lower than that obtamed m the ZPB calculation, although It runs parallel to the latter for all practical considerations; the curve labelled C has been taken from ZPB as that corresponding to the lowest value of XI&l* at rCc = 24. The present reductlon in i$ IReI2 at this distance is = 15%. The present calculated value for foO IS 0.03 137 compared to the earher 0.037 [4] and the experunentally denved 0.025 [131-

In the ZPB work two dtiferent-sized basis sets were used, contauung 44 and 72 AOs, but it was found that X IR, 1’ changed little from one set to the other. Hence A0 basis-set dfferences can probably be disregarded

Volume 83, number 3

1 November 1981

CHEMICAL PHYSiCS LEITEX.5 as a sigruficant

cause for the lowering in ZiR, I2 values. A more important distinctron IS seen in the lengths of the CI expansions ILLthe two treatments; whereas the ZPB study always employs approximately 1000-2000 SAFs the present treats 5000-6300 SAPS for the 3fSU and 3500-6000 SAFs for the 3KIg states, dependmg on the mternucIear separation_ To check the dependence of Z: [R,I* on the number of SAPS treated explicitly, three additional Ci calculattons were carried out at rcc = 3.4 and variable threshold values empioyed, leading to secular equations of the order ooFroughly

trvely. The

6UoQ and 9000

respec-

fig. 2 by T’, T” and Tz, whereas the ongmal value (curve At ) corresponds to Ti _The Z [R, I2 walues do depend fatrfy strongly on the Lengtfi of tie CI eupansion, showing a steady decrease from 5.3 to 4.6 a~ 2.40 bohr as the secular equatton order increaser. It

z-o1

fCKM3.200Q,

results are m tabie 2 and arz mlicated in

o-

2%

2L

2%

32

36

seems that the relatively short CI expanston accounts for most of the discrepancy observed between the onguzaf ZPB results and those of AL and the present

w

rcc lbohrl

treatment Fmally the last curve in fig 2 labelled As results from CI calcuiatrons employrng the smaliest threshold pair T’, at bond lengths 2-1) 2_7,2 8,3_9,3_0 and 3-1 b&r, i-e_ wavefunctton expansions contaming appro_timately 9000 terms. The change from At (~11th order 5000-6000) to A;2 is relahvely minor, as is the step co curve B of AL derived from secular equation orders of roughly 6000 in an ST0 A0 basrs. It should be noted that the total energrcs m the AL work are somewhat higher than m the present treatment, 1 e_ E(T) = -75.709665 for aSKI, and -75.616456 ford 3fIg at = 2.4790 bohr, wrth energres extrapolated at zero ‘cc

Erg 2 Compmon of various cues showing the dependence of the electronrc tranntion moment C IRei UI the Swan bands (d 3 rig-a 3 I&J of C2 as a funcnon of the C-C separanon. The curves derived from experunent are iabelled I (ref [I 1). U (refs 12,13]) and Lff (ref. [ 161) and are nornaked to the mrrvtmal Y&_W of Xl&# = 3 57 at 2 44 bohr. The curwesderrved from ab m&o cakuhtrons are B (ref (5 ]) and C Cref. [4]) and Al and A2 UI the present work In the Grst (A!) a tRreshoId IS employed wluch paves nse to a CI expansion length of = 6000 terms, wlulc curve A? results from = 9000 terms, correspondmg to a smaller tieshold Tz. At rcc = 2 40 bohr values resukmg from YZUIOJS other thresholds are also HIdicated (table 2)

Table 2 Magrutude of the z l&l2 value for the Swan bands as a function af the CI erpannon length at ‘CC = 2 JO. The threshold (m Mh) employed and the number of SAFs treated are gaven exphcldy

threshofd

EfTI

threshold

no.ofSAFs

EW)

48 T” = 2s Y-1= 3

-75 68431 -75 70158 -75.73083

72 39 i2

10% 2030 5822

-7557693 -7559907 -7542943

5.33 5.16 1.74

65

9073

-75 63987

G&5

T”=

{see

T2=

table 1) 14

-75.73564

-__I_

CHEhUCAL

Volume 83, riumber 3

PHYSICS

threshold reported asE(MFLD CI) = -75.722372 for a311u and -75.631861 for d3I$ at the same distance. The correspondmg values in the present study are mterpolated for tlus internuclear distance to be E(T1) = -75 -73 (a 311,), -75 63 (d 3 I&) and E(MRD CI) = -75.77 for a 3 ll, and -75.68 hartree for d 311 . In summary, the varrous theoretical results For G IR, 12 agree welI with each other and the cause of the previous discrepancy has been clarified. Furthermore, these theorehcal results descnbe the most recent experimental data extremely well, especially gtven the rather large variations and error hnuts m the measured findmgs. 4 2. 77ie Fox-Herzberg

band vstem

The experimental transition energy between the e 3 IIs(u’ = 0) and a 3 II, (u” = 0) states IS reported [ 111 to be 25 12 A or 4.935 eV from emission studtes Tlus

2mo

.

z’L

2.*

.

3.*

.

rcc

.

lbohrl

i6

-

Frg 3 Calculated cume for the dependence of the electronic trannhon moment r lR,I’ III the Fox-Herzberg bands (e 3 Cig-a 3 U,) of C2 as a functton of the C-C separzuon. The values are obtamed from two ddferent calculations, respondmg to CI expanuon lengths of = 6000 (threshold and = 9000 (threshold T2) 446

corTt )

LEJI-TERS

1 November

1981

value compares wah the energy difference AE(fulI CI est.) = 4 88 eV between the respective potential miruma of the a 3 IIs curves. The computed Z IR, 12 values are plotted m fig. 3 as a function of rcc_ Agam the calculatrons have been carned out for two different thresholds (Tl = 12 and T2 = 6 5 a) m the most critical region in order to assess the change in Z IRei upon increasing the CI expansion from 6000 to 9000, and smular changes as rn the correspondmg quanttty for the Swan bands are found thereby, the absolute values are consrderably smaller for the FH bands. A duect comparison with a curve derived from experiment IS not possible, but the lower and upper hmtts of ZIRe120f0.45 20.10 [14] and ZIRe12 =O 55 au [ 15) derived from the Au = 0 series from two shocktube ennssron experunents are seen to straddle nicely the computed curve u-t fig. 3 The caiculated values corresponding most closely to those expenmental values are ZIlRe12 = 0.5 for T, and 0.51 for T2 at rcC = 2.9 bohr (Iowest calculated energy for e 311s) It IS noteworthy that m the FH system strong transitions have also been seen for Av = -2 at 4.468 eV [ 11. Assummg vertical transttions, this AE corresponds most closely to that around rcc = 2 8 bohr, where rlE(full CI est ) = 4-483 eV. The calculated Z l&l2 value at thrs distance IS 0.38 for tlueshold T, and 0 37 au from threshold T2, and matches quote sattsfactonly wrth the expertmental Z]ReIz = 0.40 +O.lO [l] for tlus transitron energy. Svnidov et al. [15] reported a value for the upper limrt of the oscrhator strength obtamed from an absorption spectrum. Theu value of& < 0.011 can be compared wnh the results obtamed from the su-nple formula of eq. (3) by employmg the vertical AE, (0 215 hartree) and the B IReI (0.08 au) value at the a 3 fI, eqmhbnum bond length - I e. without summation over fan , assummg a vibrational overlap of one which also leads to f = 0.01 l_ Thus agam the experimental evidence is in complete accord with the present theoretical results. Cooper and NrchoUs [ 171 derrved absolute mtensrty data on the basis of a synthetrc spectrum analysis for the FH bands from shockexcrted C,. Under the assumption of a constant Z IReI = 0.40 they derived band strengths Sv,u., (which control the band oscdIatar strengths and band Ernstem coefficients) as

s U’V”

=

C

IRe 12qv’v” 9

cfimic~

Voiume 83, number 3

PHYSICS LETTERS

1 November 1981

Table 3 Comparison of calculated rntenslhes between various viiratlonal levels m the FH system on the basu of the present pootenbalenergy and tranntion moment data wth those of ref. [ 171 as SVnV-based on a constant ~IR,1* behavior; the latter values are in parentheses d’

V’ = 0

--

ii’= 1

V’ = 2

Y’ = 3

“’ = 4

0’ = 5

0

544 (8.684)

2.34-3 (3 57-3)

5.16-3 (B-08-3)

0 81-2 (l-34-2)

1.04-2 (l-81-2)

1.14-2

1

5 04-3 (6 84-3)

148-2 (2.04-2)

2.20-2 (3 30-2)

2.19-2 (3 86-2)

1~58-2 (3 58-2)

0X4-2 (2.56-2)

2

2.08-2 (2 45-2)

3.93-2 (4 68-2)

3.27-2 (4 52-2)

1.34-2 (2 73-2)

1.32-3 (9 40-3)

9.04 (8.564)

3

5 21-2 (5 32-2)

5.35-2 (5.32-2)

1.43-2 (1.84-2)

1 804 (2.99-I)

10.1-3 (5 96-3)

1.54-2 (l-50-2)

4

8 74-2 (7 92-2)

3 36-2 (2 61-2)

2 20-2 (1 86-2)

1.03-l (0 856-l)

3.01-2 (2 85-2)

2.18-t (2 66-l) 1 73-3 (5.72-3)

0 47-2 (: -45-2)

5

S-44 (3 564) 2 78-2 (2 45-2) 4.72-2 (3.77-2)

3 2-3 (3 37-3)

148-2 (0 804-2)

2.47-2 (2.01-2)

1.99-2 (1 33-2)

1.24-2 (l-02-2)

3 22-2 (2.72-2)

2.7-3 (7 64-3)

0.604 (8 644)

3 86-2 (3.46-2)

5.22-3 (6 88-3)

14.0-3 (4.48-3)

1 94-2 (2.04-2)

9.18-3 (5 80-3) 1.93-2

27.8-3 (X33-3)

(3.1 S-2)

(7.36-3)

6

3 36-3 (8 344) 8 94-3 (1 34-2)

8 57-2 (7 00-2)

7

4 30-2 (4 88-2)

8 9 10

_

-- -

where qvnuIIare the Franck-Condon factors. However, the electronrc transItron moment 1s far from constant according to the present results. Based on the present calculated potentlal energy curves we performed a vrbrational treatment and calculated the mtenslty between varrous vrbrational levels involved in the FH system, the tranntron moment data berng fitted by a polynomial and drrectly employed m the mtegrand [18,19], results for s,.,,,

= 6 IR,,,,~,.,,,

= 6

12

Sx,,l(rCC)R,~,,,(‘CC)x.‘(‘CC)d’CC ’

I I are compared in table 3 with those of ref. [I 7] _me values for the primarily vertical transrtrons from the upper state minimum are somewhat larger in the present calculations because this area corresponds to the peak in the X IRei curve, while the intensities involving predominantly the transtion moment at smaller values of rCc are smaller rn the present case than in ref. [17],

which employs a constant of the measured spectrum data is desirable* _

73.24 (8.4-S)

2.1-3

value of Z JR, 12. paralysis m the light of the present

5. Conciusion An ab ~ILIO HF SCF plus MRD CI treatment of the Swan and Fou-Herzberg band systems in C2 has generated values for the electronic transrtion moments Z IR, 12 whrch are in good agreement with experimental and theoretical data. The Z IReI value-s calculated for the Swan band system are moderately dependent upon the length of the CI expansion and this feature accounts for the = 14% higher Z IR, $ values obtained rn an earlier MRD CI study. The transitron moments are an order of magnitude larger for the Swan bands. The Z IReI curve of the Swan band falls off rapidly with * The calculated Franck-Condon factors and oscillator strengthsf,‘,” can be obtained from the authors on request.

447

Volume 83. number 3

larger bond !engths, whde that of the FH bands has m~mum at appro~a~e~y the pomt of largest mteraction between the d 3 IIg and e 3 iIg states, as measured by the wavefunction composltlon or the nonahabatlc couplmg matril element [3] between the two states. Consistency m agreement between the theoretxal and e~penment~ results suggests first that the computed Z lR,ja curve for the FH bands can be employed as a reakk functional form of the electromc transk

its

tlon moment for evaluation of C2 spectral data, and more importantly. that CI calculations type with moderate13

1 November

CHJZMICAL PHYSICS LEl’TERS

large CI ewpansron

of the MRDCI length are a

very efftctent means for obtammg such properttes, whxh m&t be rzlatlvely difficult to dense from experunent alone

Acknowledgement The authors thank the Deutsche Forschungsgememschaft for fmancml support w~thrn :he framework of the SFB 42 The serl’lces and computer time made available by the fWR2 a: the Umiernty of Bonn have been essentml to this study and are gratefully acknowledged

References [ 1 j D hi Cooper and R-W. Nxholls, J Quant Spectry Radlattve Tr3nsfer 15 (1975) 139 [2] T Tatarcz) k. E H Ftnk and K H Becker, Chem Phys Letters 40 (1976) 126

1981

I[31 G. Hxsch, P J. Bruna. R J Buenker and S D Pevertmhoff_ Chem

Phys. 45 (1980)

335.

i41 M. Zeltz, S_D. Peyenmhoff

and R 3 Buenker,

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