Chemical Physics 76 (1983) 175-184 North-Holland Publishing Compnny

175

THE RADIATIVE LIFETIME OF THE All-I STATE OF CH * CALCULATED FROM LONG CAS SCF EXPANSIONS

Received 26 November 1982

A theoretical study of the A’IT-X’s_ transition of CH _ is presented. Radiative lifetimes (5 ) for lhr A JWW and Ihe /,,.,..(A-X) oscillator strengths are calcula~cd. using long mutually non-orthogonal CXS SCF expansions. The convergence of these properties is studied wiih respect to the size of the active orbital space and the basis WI. IL is found that very laqc active spaces and basis sets are needed for obtaining reliably convrr_ecm results. The best calculared vslus for Y( ~7. = 0) of S50 ns nnd for& of 5.45~ 10-j are in good agreement with the resuhs from the most recent experimcnrsl invcs@iion. S15 I 2% ns and

(5.66iO.20)~ 10e3 respectively. In the calculation use has heen made of csperimen~al excitation energies snd porential curves. Compared IO the previously most accepted lifetime rhs present theorcticsl and the recent s+perimentsl values demand xn increase of the interstellar CH+ abundance.

1. Introduction

The CH+ radical plays a key role in gas-phase interstellar chemistry. It was the first observed ion in the interstellar space and was together with CH and CN observed already prior to 1940. Today more than fifty molecules have been observed in interstellar clouds [l], in most cases by means of radiotelescopes. In the formation of these interstel- lar molecules the CH and CHi radicals are be- lieved to be of great importance and the de- termination of the CH and CH’ abundances is therefore of fundamental significance. These mole- cules were studied via their optical absorption spectra, which are observed when they fall on the line of sight to a hot star. Thus, laboratory de- terminations (experimental and theoretical) of the oscillator strengths for the transitions between the ground and first excited sfates are needed in urdcr to determine the abundances_ The oscillator strength of the CH A%-X ‘I7 transition is now

well established [2.3]. while there is still a consider- able spread among the various experimental and theoretical results reported in the literature for the A’l7-X ‘2’ transition of CH-. The main reason for this is that measurements on molecular ions present special type of problems not encountered in measurements on neutral species. In general, the radiative lifetime of an excited state is studied by observation of the fluorescence following the peri- odic excitation of the particular stare being studied. It is thus important that the molecules remain kvithin the obser\-ation region for a rime period Lvhich is much longer than the radiative lifetime. When the studied molecules are charged. the mut- ual repulsion tends to drive the molecules out of the viewing region on a timescale comparable to the radiative lifetime. This effect results in an apparent lifetime which is shorter than the true radiarive lifesrime. While rhc errors in rhs results for the first two reported measurements of the A’n state lifetime [4.5] ( T( L?’ = 0) = 70 ns) proba-

0301-0104/S3/0000-0000/$03.00 6 1983 North-Holland

176 M. Larsso~t. P.&.Af. Siegbaim / Radiative lifetime of the A tll state of CH +

Table I

Experimental and theoretical results for the lifetime of the A’II stale (ns)

Experimental

HFD

ref. [IO]

laser

ref. [ 121

Theoretical

Cl

ref_ [S]

pal. prop.

ref. [l l]

Cl CAS SCF

ref. [ 131

0 63Ok 50 s15*25 720 S98 1025 850 1 7502 60 845 711 1175 9S.5

2 850*70 953 845 1355 1145

bly were due to spectral blending of the A-X A-X transition were also performed with the sec- system of CH’ by rest-gas emission. the spread in ond-order polarization propagator method [ 111. in subsequently reported results was due to the very good agreement with the experimental results space-charge effects [6,7]. Nevertheless, also the in ref. [IO]. Since the basis set used in ref. [ 111 is

longer of these measured lifetimes, r(u’ = 0) = 408 smaller than what was used in ref. [S], and the _+ 30 ns obtained with the high frequency deflec- same set of orbitals were used for all states. the

tion technique, was almost 50% shorter than the calculations in ref. [8] must, however. be con- lifetime obtained from the accurate Cl calculations sidered as slightly more accurate. Apart from the

in ref. [S], where the transition moment was calcu- small difference between the CI results [S] and the

lated over non-orthogonal orbitals using large ba- results in refs. [lO,l 11, the situation thus seemed

sis sets. A full discussion of the space-charge ef- satisfactory. Recently, however. a measurement fects was given in ref. 191. where it was shown that was performed of the radiative lifetime of the A the Coulomb repulsion between the CH+ ions state with the ions confined in a quadrupole ion

present in the target region of the HFD apparatus trap and excited with a laser [12] yielding a 30% could be quenched by the introduction of low-en- higher value for T(U’ = 0) than found in ref. [lo]. ergy electrons_ A remeasurement of the CH’ A- Due to the long confinement time (10 ms), the state lifetime with the HFD technique and charge problem with space-charge effects was avoided. neutralization [IO] brought the experimental life- The electronic transition moment of the A-X tran-

time into reasonable agreement with the theoreti- sition was also recently calculated with CI meth- cal value [8]. Theoretical calculations of the CH+ ods [ 131 yielding a 20% smaller transition moment

Table 2 Oscillalor srrengLhs( X 103) for :he A-X transition

Experimental laser

ref. [12]

Theoretical

CI

ref. [8] pal. prop.

ref. [I l]

CI

ref. [13]

CAS SCF

5.66iO.20 6.67 7.43 4.48 5.45 1.00 1.76 0.82 1.08 4.41 4.26 2.80 3.31 0.83 1.20 0.72 0.88

0.68 1.70 0.75 1.00 0.05 0.20 0.29 2.93 3.27 2.05 2.46

2.81 2.78 1.78 2.11

around the equilibrium internuclear distance than the previous CI calculation [S]. In ref. [13] the same set of orbitals was used for both states, however. All results discussed in this section are compiled in tables 1 and 2. It should be noted that the interstellar abundance of CH+ is directly re- lated to the A-X oscillator strength_ In view of the large discrepancies still remaining among the vari- ous experimental and theoretical results despite considerable efforts. we have decided to make a renewed experimental and theoretical investigation of the CH+ A-X transition. In this report only the theoretical results will be presented. The experi- mental part of this work is not yet completed and will be published in a forthcoming paper [14]. It should, however, be mentioned that we plan to connect an ion trap similar to that of ref. [ 121 to the target region of the HFD apparatus and re- measure the A-state lifetime. The theoretical calcu- lations in this work were performed in the frame- work of thk complete active space SCF method and will be described in the next sections_

2. Details of the calculations

Part of the purpose of the present study of CH- was to make a detailed investigation of the convergence properties of the transition moment. With only four valence electrons CH’ is a very suitable candidate for such a study since highly accurate results should be obtainable_ The CAS SCF method [I51 was used for the wavefunction calculations with active orbital sets ranging from

Basis set used for CH-

small to very large. Basis sets from medium size to large including f functions were also tested_ The technical details of the calculations are presented in this section.

The basis sets used in the calculations are de- scribed in table 3. The smallest basis set. basis 4. is the standard Dunning-Huzinaga lOs6p set con- tracted to 5s4p [16.17]. with a single d function with exponent 1.0 for carbon. For hydrogen a 5s set [ 171 \vas contracted to 3s and a p function with exponent 0-S was added. X larger 1157~ set con- tracted to 6s5p Leas then constructed for carbon from the lOs6p set by replacing the two outermost s functions by three. with esponents 0.60. 0.25 and 0.10. and replacing the three outermost p func- tions by four. with esponcnts 0.65. 0.35. 0.15 and 0.05. This carbon sp basis \vas used in the rest of the basis sets. A similarly constructed basis set was used recently for Nz and was shown to perform very well [ 181. In basis B and D a hydrogen 9s set [19] was contracted to 4s. and three p functions were added with csponenrs 2.12. 0.77 and 0.28 optimized for H, in ref. [20]_ from where also the d exponent 1.0 used in basis D was taken. In basis C the 10s basis [17] contracted 10 6s with three functions. 0.93. 0.38 and 0.16. recently used for

BH 1211 was used instead. Two sets of d functions were used on carbon in the basis sets B. C and D with exponents 1.0 and O-3_ In the largest basis D

a set of f functions with exponent 0-S was also

added. The 3s combination of the d functions and

the 4p combination oi the f functions were finally

deleted from the basis sets.

The wavefunctions used for the transition mo-

Basis set Atom Uncomracwd Comrxted C, funcrions C, functions H, functions Ii, funclions

A C 10.6 5-3 1.0

H 5 3 OS B C 11.7 6.5 I .0.0.3 _

H 9 1 1.12.0.77.0.7s

C C 11.7 6.5 1.0.0.3

H 10 6 0.93.0.X.0.16

D C 11.7 6.5 1.0.0.3 OS

H 9 4 1.12.0.77.0.23 1.0

178 M. Larsson, P. E. M. Siegbohn / Radiorive liferime of the A ’ Il axe oj CH +

Table 4

Active spaces used in the CAS SCF-calculations

Active space

A

B

C

D E F G H 1 K L M

3o.ln 40.1~ (C,, included) 30.2a 40.2~ (C,, included) 30,3sr 40.2~ 50.2n 7a.35; 60.4.x 70,4T 60.35;,16 60.4n.28

ments were calculated using the CAS SCF method. Several sets of active spaces were tried and these are all listed in table 4. The smallest possible ac- tive space which could be used. includes the 20. 3a and 4a orbitals which are necessary for describing the X ‘Z+ ground-state curve including the dissoci- ated products, and the lrr orbital which is occupied in the A’II state. The la orbital, which has to be the same for the two states (see below) was taken to be the optimal orbital for the X state. This active space is called A in table 4. In active space B the carbon Is orbital was also included in the active space. The active spaces were then gradually increased by adding additional u and 71 orbitals until finally in active spaces L and M also S orbitals were added. The largest calculation included six u orbitals, four n orbitals and two 6 orbitals. In order to reduce the number of config- urations obtained when four electrons are distrib- uted among these orbitals. the number of electrons in the u orbitals was minimized to one and the number of electrons in the T and 6 orbitals maxi- mized to three and two respectively_ Since exact linear symmetry is not automatically obtained when Cartesian gaussian basis sets are used, a mixing between u orbitals and S orbitals will nor- mally occur. In order to make the reduction of the configuration space well defined the u-6 orbital mixing was prevented. This was done by simply eliminating the corresponding parameters from the orbital optimization procedure. It should be noted

that the invariance of the energy with respect to active orbital rotations is not destroyed by the configuration space reduction even though a com- plete CI is no longer performed. This procedure gave 1769 configurations for the ‘Z+ state and 1744 configurations for the ‘n state. The calcula- tion was carried out only with the largest basis set D. A calculation of the full potential curves with this combination of active space and basis set would have been too tedious. Thus. for this pur- pose basis set B and an active space containing six u orbitals, three v orbitals and one 6 orbital were used. No constraint was put on the number of electrons in the different symmetries and the num- ber of configurations was 881 for the ‘Z+ state and 784 for the ‘I7 state. The potential curves obtained in these calculations are shown in fig. 1 and the energy values are given in table 7.

The transition moments were evaluated over the optimal non-orthogonal orbitals using a recently suggested method [22]. which is particularly well suited for CAS SCF expansions. The main idea which is used in this method is that wavefunctions which are described by a complete CI expansion in a set of orbitals, like in the CAS SCF method, are

I I I

CH+

Fig. 1. Calculated potential curves for the X Ix+ and rhe A’II

states and the transition moment curve between the states. 0 marks our curve and * marks the results from the Cl calcula- tions in ref. [S].

Table 5

Transition moments and energies for different hasis sets and active spaces at R = 2.10 au

Basis set Active

space

Transition

moment Ener_qv

A’IZ _I E

(cm-‘)

A A 0.2664 - 37.S4493 - 37.97 175 ‘7529

B 0.2683 - 37.84532 - 37.97 1 ss 2’7773 C 0.2932 - 37.85 IS3 -37.98815 29911

D 0.2950 -37X5228 - 37.98537 29562

E 0.3025 - 37.55559 - 37.99059 29622

F 0.2938 - 37.45667 - 37.997 12 30x20

G 0.3006 -37X6446 - 38.00262 30317

B A 0.2187 -37.85010 - 37.97523 27613

G 0.2479 - 37.87009 - 38.0088 1 3OU1

H 0.2653 - 37.88046 - 38.01635 79519

L 0.2769 - 37.89472 -38.01522 26441

C H 0.2628 - 37.88036 - 38.01631 ‘9S10 D H 0.2695 - I 37.ss10 -33fi.01967 30196

I 0.2703 - 37.58725 - 38.01763 28611

K 0.2654 - 37.88442 - 3R.02056 29876 M*=’ 0.2706 - 37.86864 -38.01952 33111

M 0.2834 - 37.90249 -3S.01952 25679

sxp. 1121 - 0.2902 25104

a’ Ground state orbitals for both states.

invariant to rotations of these orbitals. This makes it possible to go over to a corresponding orbital basis [23] which in turn makes the transition-mo- ment evaluation as easy as in the orthogonal case. In a recent paper on the CN red system [24] this method has been adopted for the use of Gelfand states and unitary group methods for the evalua- tion of the matrix elements [25]. A drawback of the method is that the inactive orbitals have to be the same for both states, which means that only true core orbitals can be used as inactive orbitals. For CH+ the use of different carbon Is orbitals for the X and A states was tested in a few cases.

3. Results

The results from the calculations combining different basis sets with different active spaces are collected in table 5. The quantity entered in the column (‘E’]x-]‘ff,) as “exp” is not directly mea- surable, but follows instead from the analysis pre- sented in section 4. Some interesting conclusions may be drawn from the results in table 5 and these are presented in this section.

The transition moment and excitation energy are sensitive to the choice of basis set and active space. In fact certain combinations of a medium- size basis set lvith a small active space gave perfect agreement \vith the experimental transition mo- ment. but poor agreement with the escitarion en- ergy_ An example of this is the calculation with basis A and active orbital space C. which gave the best result for ths transition moment \vith an error of only 0.003 au compared to experiments. The error for the excitation energy was on the other hand an order of magnitude larger than what was obtained in the best calculation. Other combina- tions gave a reversed picture. The basis set effects are not easily saturated_ Particularly striking is the large difference in the transition moment of 0.05 au. corresponding to a lifetime difference of 300 ns. in going from basis A to basis B. The inclusion of an f function on the carbon atom makes only- a minor contribution to the transition moment. how- ever. which is similar to what has recently been found for the CH radical [26]. The sensitivity of the results to the choice of basis se: for CH- and CH is quite different from what was obtained for the CN molecule [24]. where changes in the basis

180 M Lmsson. P. E.hf. Siegbahn / Radiarioe liferime of rhe A I I7 srarc of CH +

sets gave very small effects on the calculated tran- sition moments.

The inclusion of S orbitals is important both for the transition moment and the excitation energy. Compare, for example, the difference in results with active orbital space I with space M for the largest basis set D. The only difference in the active spaces is that two S orbitals have been added in the latter case. The change in transition moment of 0.013 au corresponds to a lifetime difference of 80 ns. It is also clear that the contri- bution to the transition moment from the first S orbital is an order of magnitude larger than from the second. The difference in energy between the two states was 25679 cm- ’ when two S orbitals are included in the active space, 26262 cm-’ when the second is excluded (this value is not in table 5) and 28611 cm: ’ when both S orbitals are excluded. This shows that the first S orbital is by far most important also for the excitation energy. It also shows that there is no reason to add more S orbitals to the active space.

The effect of including the carbon 1s orbital in the active space can be seen in comparisons be- tween active spaces A and B and between C and D for basis set A. The contribution is as expected small, only around 0.002 au, and can probably fairly safely be added to the results for the largest calculations with basis set D.

Two additional calculations were finally per- formed to check the necessity of performing the transition moment evaluation with the accuracy it

Table 6 Transition moments for different thresholds of the CI coeffi- ciems. Active space M and basis set D was used

No. of configurations

X A

Threshold Transition moment

12 10 0.05 0.3027 54 66 0.01 0.2867

LOO 118 0.005 0.2854 290 381 0.001 0.2835 800 1072 0.0001 0.28337 1 980 1254 0.00005 0.283370

1336 1569 o.Ocim1 0.283368 1769 1744 0 0.283368

has been done here. With the large active orbital space M it could be argued that the use of differ- ent orbitals for the two states would not be neces- sary. Clearly if all orbitals were included in the active space the value of the transition moment would be independent of orbital choice. If the ground state orbitals are used for both states with active space M and basis set D a decrease in the transition moment of 0.013 au is. however, ob- tained (M* in tabIe 5). This corresponds to an increase in the lifetime of the A state of 80 ns. which could be compared to the final error in the computed lifetime of = 35 ns. In the second inves- tigation the transition moment was calculated using only those CI coefficients which were larger than a certain threshold. If this procedure would work with a reasonably high threshold it would be very useful particularly for larger molecules. The results with different thresholds are presented in table 6. The convergence of the transition moment is in- deed quite fast. A threshold for the CI coefficients of 0.001 gives nearly perfect agreement with the final transition moment even though only 15% of the CI coefficients are kept. Even a threshold of 0.01 gives satisfactory results.

As a final comment it should be noted that in most of the calculations a symmetry restriction was introduced by averaging the density matrices of n, and I?,, symmetries. When S orbitals were added a similar procedure would require averaging also over the A symmetries, which is technically slightly more complicated. Since the program was not yet modified to do this no averaging was performed in these cases. Test calculations with and without averaging for the smaller active spaces gave very similar results, however.

4. Discussion

The calculated electronic transition moments obtained with various combinations of active spaces and basis sets at R = 2.10 au given in

table 5 are not directly comparable with experi- mental quantities such as radiative lifetimes and absorption oscillator strengths. To calculate these quantities the transition moment must be evaluated over the full potential curves. These results, ob-

Table 7

Transition moments and energies with basis set B and active

space L for different internuclear distances

R Transition Energy

(au) moment A’Il

1.60 0.3953 - 37.80598

1.70 0.3729 - 37.83901

1.80 0.3495 - 37.86211

1.90 0.3256 - 37.87788

2.00 0.3013 - 37.88826

2.10 0.2769 - 37.89472

2.20 0.2529 - 37.89836

2.30 0.2293 - 37.90001

2.40 0.2065 - 37.90027

2.50 0.1848 - 37.89960

2.60 0.1643 - 37.89833

2.75 0.1368 - 37.89578

3.00 0.0978 - 37.89120

3.50 0.0454 - 37.88357

4.50 0.0063 - 37.87544

6.00 0.0014 - 37.87073

20.00 0.0000 - 37.86859

Energy x ‘x+

- 37.94094

-37.97185

- 37.99247 - 38.OCl537 - 38.01246

-38.01522

-38.01476

- 38.01194 - 38.00742

- 38.00170

-37.99518

- 37.98494

- 37.967 19

- 37.93454

- 37.89237

- 37.87249

- 37.86880

tained with basis set B and active space L are given in table 7 and fig. 1. Also included in fig. 1 is the transition moment curve from ref. [8]. If interaction between electronic and vibrational mo- tion is neglected, the absorption oscillator strength and the transition probability (in s- ‘) can be expressed as

f,.,,. = 3.038 x 10%[(2 - 6,.x.+,..)/(2 - St&]

x ](u’]R,(r)]u”)]‘. (1)

A “‘$’ = 2.026 x lo-“v’ [(Z - &+&(2 - S,.^-)]

x ](u’]R,(r)]v”)]‘. (2)

where Y is the transition energy in cm-‘. R,(r) is the electronic transition moment at the inter- nuclear distance r in au and ]u’),]u”) are the vibrational wavefunctions for the upper and lower states respectively. The following procedure was followed in order to evaluate (1) and (2): The vibrational wavefunctions were obtained from the experimental potential curves constructed with the Klein-Dunham method [27] based on recently derived spectroscopic data [28.29]. Since the tran- sition probabilities are sensitive to the transition

energies due to the third-power term. the accurate experimental results were used also for these quan- tities. A spline fit (see ref. [ 111) was made for the calculated transition moments R,(r) in table 6 and the integral was evaluated numerically. The radiative lifetime of the A’fl( u’ = 0) level, defined as ~+a = (2,.._&...)- ‘_ was summed to 900 ns. which is 10% longer than the most recent experi- mental result [ 121. -q_” = 815 & 25 ns. The next step was to use the transition moments calculated at three internuclear distances 2.10. 2.75 and 3.50 au. with the largest basis set D and the extended active space M to scale the transition moments in table 7. To these new transition moments. the ef- fect of 0.002 au of including the C,, orbital in the active space was added at all internuclear dis- tances_ A new calculation of transition probabili- ties was performed with these adjustments and we finally arrived at a value of ~..._c = 850 ns. which is almost within the experimental error bars. To make a comparison with earlier theoretical calculations possible. a similar analysis was performed with the tabulated transition moments in refs. 18.131. while the results from the polarization propagator calcu- lation were taken directly from table IV of ref. [ll]. These results are collected in table 1. The absorption oscillator strengths for the hitherto ob- served bands of the X-X transition are listed in table 2. The experimental value is obtained using our calculated transition moment curve as a func- tion of internuclear distance scaled to fit ~~.-__a = 815 ns. Our calculated result for_&,, is within 4% of the experimental value.

It is interesting to compare our calculation with earlier theoretical results. In the recent CI calcula- tion [13]. which was mainly designed to study many states simultaneously. too small transition moments were obtained and the corresponding radiative lifetimes for the different u levels of the A state are all around 20% too long. A probable reason for this is that the same set of orbitals were used for both states in the transition moment calculations_ The decrease of the transition mo- ment when ground state orbitals are used for both states can be seen also in our table 5. in a compari- son between the M and M* results. The results of the polarization propagator calculation [ 1 l] using a 30 ST0 basis set are in close agreement with the

earlier experimental results obtained with the HFD technique [lo]. However, in view of the very con- vincing laser measurement using an ion-trap. it must be concluded thal the propagator calculation gave too large transition moments. In a more recent paper the transition moment was calculated at on< internuclear distance (2.137 au) with. vari- ous different approaches including the polariza- tion propagator method and with a 45 ST0 basis

set [30]. Although not explicitly tabulated, it ap- pears that the transition moment was 10% smaller than the earlier calculation with the smaller basis set [ 1 I]. The more recent value is in good agree-

ment with our calculations. Different orbitals were used for the two states in large Cl calculations in

ref. [8] and the results are in reasonable agreement with the most accurate experiment. It should be noted. however. that a 20% too large transition moment at ‘; is compensated by a too fast decrease of the transition moment towards longer distances which substantially increases the radiative lifetime of particularly the u= 0 level compared to our results. Our foe oscillator strength is 5.45 x lo-’ compared to 6.67 x IO-’ in ref. [8] and the experi- mental value of 5.66 X IO-“. This fairly large dif- ference must be explained by a too small reference space for the CI calculations and a correspond- ingly too limited orbital optimiza,tion in ref. [8]. The two other calculations [ 11,131 give a variation of the transition moment with internuclear dis- tance which is more similar to our results. The better agreement between our curve and the prop- agator curve from ref. [l l] than between our curve and the CI curve from ref. [S] is somewhat surpris- ing and is probably to some extent a result of compensating errors in the propagator calculation. Since the propagator calculation does not describe

. _ a proper dlssoclation of the two states, a curve which gets worse for longer distances is expected. This is certaiflly the case for situations where the transition moment does not go to zero for large distances. In CH t the transition moment goes very fast to zero, however, and should thus be a favora- ble case for the propagator method. There is un- fortunately no experimental determination of the transition moment curve. The f,Jj& ratio can, however, be derived from CH’ containing inter- stellar clouds. For example, equivalent width mea-

Table 8

Theoretical and experimental spectroscopic conStan& for the

X I\-+ 5181C

This work Saxon er al. Experiments

I131 I2Sl

‘; 1.1284 1.1291 1.1309

BC 14.2372 14.2009 14.1766

ac 0.523 1 0.4806 0.4939

YC - 0.0037 - 0.002 1 0.0023

DJ x lo>) 1.27 1.22 1.373

/3,(X IO31 - 0.094 0.026 -0.0193

UC 2873.4 286 1 .O 2857.6 C.& .I-, 64.1 59.0 59.3

WC -1; - 0.568 - 0.083 0.226

AC,,, 2743.4 2142.9 2739.7

A%,, 2610.2 2625.3 2623.0

G,, 2473.5 2507.9 2507.8

D&V) 3.8 I 3.96 4.080 n’

-’ Ref. 129).

surements of the R(0) line of the (0. 0) (X = 4232) and (I. 0) (A = 3957) bands from the interstellar cloud in front of & Oph were used in ref. [8] to derive f,O/fOO = 0.6 1. The following ratios are ob- tained from table 2: 0.67 [8], 0.57 [ll]. 0.625 [13] and 0.61 (CAS SCF). Recent remeasurements of equivalent widths from the same astrophysical

source gave 0.628 [3 11. Spectroscopic constants were finally derived

from a rovibrational analysis of the potential curves shown in fig. 1. These data are listed in tables 8 and 9. Due to perturbations in the A’n state it is not straightforward to fit the experimental energy levels of this state to a power series in u [28]. Apart from the re and T, values, which thus must be taken with some caution, only directly observed quantities are therefore tabulated for the A state. We also performed a similar analysis of the poten- tial curves reported in ref. [13]. The agreement

between the ground state CAS SCF curve and the experimental curve is reasonable, but not as close as the CI curve in ref. [ 131. On the other hand, our A’i7 state curve is somewhat better than the corre- sponding CI curve. None of these curves are how- ever as good as the A-state curve in ref. 1321 (from which the transition moments in ref. [8] were calculated), although the limited number of inter-

Table 9

Theoretical and experimental spectroscopic consmms for the

A’Il state

This work

‘c 1.255 1.262 1.235 .J’

To 24660 25059 23608.3

T, 25257 25655 24124.4

%/, 1465.1 1401.6 1641.3

=3/z 1207.1 1170.6 1433.3

+,z 96S.9 965.9 1241.1

D&w 0.76 0.82 1.159h’

n’ The rotational ccmstant given in ref. [ZS] does nor fir a power

series in u due to perturbations. The tabulated re wlue was obtained from d16.85763/&. where B, was obtained from B, and B,.

” Ref. [29].

nuclear distances used to calculate this potential curve should have introduced uncertainties in. for example. the position of the equilibrium inter- nuclear distance 1111. As a general conclusion it seems that a CI calculation, which includes the major part of the dynamical correlation energy through the inclusion of all double excitations from the leading reference states. gives better curves than the present type of CAS SCF calcula- tion_

5. Conclusions

Large CAS SCF calculations have been per- formed to determine the radiative lifetime of the A’II state of CH’. For the transition-moment curve between the A and X state. accurate results

are obtained if large sp basis sets are used with substantial d polarization on carbon and if 6 orbitals are included in the active space. To obtain even higher accuracy f functions on carbon and d functions on hydrogen were included. The impor- tance of different orbital optimization for the t\vo states was illustrated by a calculation which gave an 80 ns longer lifetime even for the largest active space used here if ground state orbitals were used also for the A state. The rather different results obtained here and in the lsrge CI calculations in

ref. [S] also point towards the importance of orbital optimization. For obtaining highly accurate poten- tial curves regular multi-reference CI calculations should. however. be preferred. In the present calculations as much information as possible has been taken from spectroscopic measurements_ Par- ticularly for the excitation energies. which enter with the third power in the expression for the lifetime. even very accurate calculations can never compete with experiments when they are availabie. The most valuable information is therefore ob- tained by a combination of esperimental results and calculations. The finally obtained lifetime and lowest transition moment are almost within the error bars of the recent laser experiment. Sum- marizing. very large calculations are nesded in order to get a converged value for the _A state lifetime of CH-. but reasonable results can be obtained already in less extensive trrfatment3.

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