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Non-orthonormal basis calculations of the dipole transition moment for the phillips system (A 1Πu → X 1Σ+g) in C2. Theoretical lifetime of the A 1Πu state
Chemical Physics 112 (1987) 319-324 North-Holland, Amsterdam
319
NON-ORTHONORMAL BASIS CALCULATIONS OF THE DIPOLE TRANSITION FOR THE PHILLIPS SYSTEM (A ‘II” + X ‘2,‘) IN C,. THEORETICAL LIFETIME OF THE A ‘l-I, STATE G. THEODORAKOPOULOS,
I.D. PETSALAKIS,
MOMENT
C.A. NICOLAIDES
Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, 48 Vas. Constantinou Ave., 116/3.5 Athens, Greece
and R.J. BUENKER Theoretische Ch,emie, Berg&he
Universiriit Wuppertal, D-5600 Wuppertai I, FRG
Received 22 September 1986
Non-orthonormal basis calculations have been carried out on the A’& + X “2: transition moment of C2 as a function of C-C internuclear distance. Two types of A0 basis set and different sizes of CI vectors were employed in order to examine the convergence of the theoretical results. Published ab initio potential energy curves and the transition moments of the present work were combined to calculate radiative lifetimes of vibrational levels of the A’II, state. The resulting lifetimes are in agreement with other theoretical values, obtained with orthonormal basis calculations, which however, are significantly lower than those of the most recent experiments.
1. Introdu~on Electronic transitions in C!, have been observed in astrophysical as weIl as in terrestrial sources [l-14]. The Phillips system, involving the A’II,,X “ZZp*transition, is used for determination of the C, abundance in interstellar clouds [7] (where the Swan system, yet stronger in the laboratory, is not detected) and in the carbon stars. For this reason, it is required to have accurate lifetimes and oscillator strengths, which, for this system are still to be established [I]. Recent experimental work employing the techniques of excimer laser photolysis (ELP) [1] obtained lifetimes for the vibrational levels of A’II, which are much larger than those of earIier shock tube experiments [2] but close to those of other direct measurements of the lifetimes [3,4]. The most recent experimental work [5] employing the laser-induced fluorescence technique (LIF) updates the u’ = 0 and u’ = 3 data of ELP fl]. In this way, the experimental lifetime of u’ = 0,
7(O), (for example) is given as 18.5 + 3.0 ps by LIF [5], 13.4 it 2.5 ps by ELP while the shock tube work [2] gave 7.4 p.s. Theoretical calculations on the Phillips system of C, [9,10] give lifetimes significantly lower than the direct measurements [1,3-51, with ~(0) of 11.1 [9] and 10.7 [lo] p.s. The theoretical fW values [9-111 are larger than foe of ref. [5] by about a factor of 2, while they are in agreement with some &, values obtained in earlier experiments ([1,6] and references therein). The discrepancy between the theoretica results and the experimental ones is rather large, in view of the excellent agreement between theory [12] and experiment [13,14] for the Swan system in C,. As an explanation, it has been suggested that the experimental values might require corrections for the effects of cascading, due to allowed transitions from other upper states into A ‘II, [lo]. One might also try to do more work on the theoretical side. The theoretical works of van Dishoeck [9] and ChabaIowski et al. [lo] employ the
0301-01~/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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G. Theodorakopoulos et al. / Calculations on the Phillips system in C,
MRD CI method [15,16] with large A0 basis sets including polarization and diffuse functions. In ref. [lo] the MO of the D ‘Z: state is the common one-electron basis while in ref. [9] different (but common to both states) MO basis sets are employed in order to examine the effect on the calculated transition moments. Pouilly et al. [ll] have carried out CI calculations with a Slater-type A0 basis set and also work with a common oneelectron basis for the X ‘2: and A’II, states. It is not expected that another, orthonormal basis theoretical work would yield greatly different results (&20%) from the above. However, it would be interesting to know the results of nonorthonormal basis calculations (NON) for the transition moments of Al&,--X ‘Ep’, as a further test of the stability of the theoretical values, within the Born-Oppenheimer approximation. In a NON calculation, two different MO basis sets are used for the two states, ground and excited, for an optimum and concise description of each state, in the philosophy of a state-specific theory of electronic states [17]. We have developed a method for NON calculations with one- and two-electron operators [18-201. As a starting point it has the corresponding orbitals transformation [21] and the generalized Slater-Condon rules of Ring et al. [22]. Our method has been formulated for general LCAO MO CI wavefunctions and is completely general. We employ the NON method with MRD CI vectors where the reference spaces, the selection thresholds and the number of roots solved for may be varied, to best account for the requirements of a given problem, without having to deal with very large CI expansions. Convergence calculations are used to evaluate the stability of the results obtained with relatively small CI vectors. An alternative method for NON calculations, currently described in the literature [23], is based on the use of complete active space vectors for the electronic states.
lb:,lbz, (‘A,). For internuclear distance rc_c longer than about 2.8 bohr [9,10] two other configurations become important, 3atlb:, and 3ailb& (giving a$:). However, it has been found that at R,_, > 2.8 bohr the transition moments are very small and have a negligible effect on the calculated lifetimes of the A’III, state [9,10]. Thus it is sufficient to use the MO of the r4 configuration for the ground state calculation throughout the range of internuclear distances calculated from 2.0 to 3.2 bohr, although this is correct only for the first few vibrational levels (u” = O-4) [9]. In order to account for the X ‘El-2 ‘xp’ interaction it is necessary to calculate both states together using all the important reference configurations for both states [9,10]. We calculate the first three roots of ‘As symmetry, since the ‘Aa component of the first ‘As state is also in the same energy region
2. Method of calculation
The non-orthonormal basis program developed by the authors [19,20] was used for the calculation of R elell. Details of the method have been given elsewhere [20]. In order to facilitate comparison with experi-
The calculations have been carried out in D,, symmetry. In this point group, the . . -7~~ main configuration of the ground state is given by
WI.
The excited state, A ‘II”, represented by &~a is calculated with the MO of configuration . . . lb:,3a,lb, (lb”) for one-electron basis. Some preliminary calculations have been carried out with the (9s5p/5s3p) carbon basis set of Dunning [24]. Two d functions (exponents 1.2 and 0.3) for polarization and one s and one p (exponents 0.023 and 0.021, respectively) diffuse function were added to the carbon A0 basis for most of the calculations. The MRD CI programs were employed to calculate CI vectors of various sizes for the two states of interest. Large reference spaces (see also ref. [lo]) were required to obtain a contribution of over 90% of the reference set in the CI vectors, over the R,_, distance from 2.0 to 3.2 bohr. As in ref. [lo], the length form of the dipole operator was used to calculate the electronic transition moment integral R,,,,, (with primed quantities refering to the upper state and doubly primed to the lower state) at each ‘c-c value:
321
G. Theodorakopoulos et al, / Calculations on the Phillips system in C,
mental quantities it is necessary to calculate lifetimes and oscillator strengths for transitions between vibrational levels of the two electronic states. The potential energy curves of Chabalowski et al. [lo] were used to calculate vibrational wavefunctions { xU,, x,,,} for the two states. Then, the line strength is given by
s“‘U”
=
21(x,&-,)
l~e~,~.(~c-c)lX,~(~c-c))/2~ (2)
where the factor 2 accounts for the degeneracy of the ‘l&, state [lo]. It should be noted that the notation of ref. [lo] is followed, the real form of the dipole operator is employed, which in this case is equivalent to the form CJx,+iy,)/fi, used by van Dishoeck [9]. The coefficients A,,,, for spontaneous emission are given by A “I”,’=
2.1419 x 10” A E,?,, (au) S,,“,, (2 - ~OJdW
+ 1)
(3)
(with 2 - So,,, = 2 for this system) and the corresponding absorption oscillator strengths are given by
(with 2 - So A,, = 1). Finally, the radiative lifetimes of the u’ levels, of the A ‘II,, state are given by
r(d) = ($A..n,,)-l
ments. For example, at rc_c of 2.348 bohr, calculations with the smaller A0 basis set gave Ret,,, values of 0.3781, 0.4154 and 0.4237 for vectors of about 50, 500 and 1000 configurations, respectively. The effect of adding polarization and diffuse functions to the A0 basis set is to decrease the calculated transition moment. R,,,, at re is calculated as 0.3827 with vectors of size between 900 and 1000 configurations and the larger A0 basis set. Orthonormal basis transition moments were also calculated at re, for comparison, employing in turn the MO of X ‘xl state, those of the A’Il, state and those of the D ‘2: state as the one-electron basis set. The sizes of these MRD CI calculations were about 4000 and 6000 configurations for A0 basis sets (i) (smaller basis) and (ii) (larger basis) respectively. The results on the orthonormal basis calculations are summarized in table 1. As shown in table 1, the use of different MO and A0 basis sets has a moderate effect on the calculated orthonormal basis transition moments, in agreement with the results of van Dishoeck [9]. Non-orthonormal basis calculations of the transition moment integral have been carried out, for several rc _ c values, with the large A0 basis set and CI vectors of about 500 and 1000 configurations for each state. The results are given in table 2 along with NON calculations employing the small A0 basis and 500 configuration vectors. As shown in table 2, the calculated Rerep, values decrease with increasing rc_c in agreement with the previous calculations [9,10]. The present results do not support assumptions of a constant R e,e,, or increasing R,,,,, with rc_c [3]. The potential energy curves for the A’III, and lx+ states of C, calculated by Chabalowski et al. [lG (including the full-C1 correction to the ex-
s.
3. Results and discussion Several test calculations have been carried out at the ground-state equilibrium distance ( re = 2.348 bohr [25]) in order to obtain convergence of the results with respect to the size of the CI vectors and to estimate A0 basis effects. It was found that the size of the CI has an effect on the transition moment for vectors up to 400-500 configurations. Calculations with larger vectors do not give significantly different transition mo-
Table 1 Orthonormal basis transition moments for the A’&-X transition at rc_c of 2.348 bohr MO basis
A0 basis (i)
A0 basis (ii)
0.4094 0.4100 0.4207
0.4020 0.3783 0.4072
x lx+ AG D ‘I$”
‘Ep’
G. Theodorakopoulos et al. / Calculations on the Phillips system in C,
322
Table 2 Non-orthonormal basis transition moment for the Phillips system of C, as a function of the internuclear distance rc_c rc_c (bohr)
R&m a)
R;.,n ‘)
RF,,,,, 4
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
0.4384 0.4315 0.4048 0.3565 0.2964 0.1973 0.0276 0.0761
0.4154 0.4049 0.3817 0.3604 0.1938 0.0250 0.0388 0.0354
0.4098 0.4005 0.3809 0.3372 0.1618 0.0247 0.0406 0.0768
a) Superscript a denotes small A0 basis and 5OOconfiguration vectors; superscript b denotes large A0 basis and 500~configuration vectors; superscript c denotes large A0 basis and lOOO-configuration vectors.
trapolated energy [26] were used for vibrational calculations [27]. The curve fittings gave the following results: For the X ‘xl state re = 2.364 bohr with experimental value [25] of 2.3480 bohr, we= 1848 cm-’ (exp. 1854.7) and B, = 1.80 cm-’ (exp.1.82). For the AllI, state, re = 2.519 bohr (exp. 2.491), w, = 1548 cm-’ (exp. 1608.4) and B, = 1.58 cm-’ (exp. 1.61). In table 3, AE,,,, are listed, corresponding to the energy differences from a vibrational level u’ of the A ‘II” state to u” of X ‘xp’ , for the first six vibrational levels for each state. The theoretical AE, (or qx,) is 8124 cm-’ or 0.037014 au, while the experimental value is 8268 cm-’ or 0.037671 au. The difference between the experimental and theoretical AE, of about 0.017 eV is within the Table 3 Energy differences, A E,,,,, (cm -‘) u”
0 1 2 3 4 5 6 7 8 9
‘) for the AlIf,-X
errors of the calculations. Thus, ‘we ‘choose to work with the theoretical potential energy curves since the RKR curves employed in ref. [9], where the theoretical errors were of the order of 0.1 eV, are only reliable for internuclear distances between 2.1 and 2.7 bohr [9]. Line strengths (eq. (2)) were calculated using the vibrational wavefuno tions obtained above and the transition moments of table 2, where a polynomial fit of the calculated R ese#e values gives Rereer(rc_c). The coefficients for spontaneous emission, A,#“PS(eq.(3)), obtained with the transition moments listed under RC,t,,, (i.e. the large A0 basis and large CI vectors), are listed in table 4, for the first nine vibrational levels of each state. The corresponding lifetimes of the vibrational levels of AlIT, are given in table 5 (column c) along with the lifetimes obtained with the other two sets of transition moments of table 2 and along with some literature values for comparison. The lifetimes calculated with the smaller basis set (i.e. without polarization or diffuse functions), given in column a in table 5, are close to those obtained with the larger basis set (columns b and c). Thus, the effect of changing the A0 basis is only about 1 us on the calculated lifetimes. There is virtually no difference in the lifetimes obtained with 500-configuration vectors and those obtained with lOOO-configuration vectors (columns b and c, respectively, in table 5). This is very gratifying and it seems to be generally true for non-orthonormal basis calculations of transition moments in our work [19,20] as well as in the work of others [28].
‘Ep’ transition in Cz
V’ 0
1
2
3
4
5
0.81243(4) 0.63089(4) 0.45261(4) 0.27766(4) 0.10608(4)
0.%493(4) 0.78339(4) O&0512(4) 0.43016(4); 0.25858(4) 0.90406(3)
0.11151(5) 0.93361(4) 0.75533(4) 0.58038(4) 0.40879(4) 0.24062(4) 0.75848(3)
0.12631(5) 0.10815(5) 0.90324(4) 0.72829(4) 0.55671(4) 0.38853(4) 0.22376(4) 0.62284(3)
0.14087(5) 0.12271(5) 0.10488(5) 0.87389(4) 0.70230(4) 0.53413(4) 0.36936(4) 0.20788(4) 0.49349(3)
0.15519(5) 0.13704(5) 0.11921(5) 0.10171(5) 0.84556(4) 0.67739(4) 0.51262(4) 0.35114(4) 0.19261(4) 0.3582q3)
‘) The theoretical values are given.
G.~~orakop~i~
et al. /
Table 4 Theoretical spontaneous emission probabilities. A,,‘u,. (s-t)
Calculations on the Phil&p
system
323
in C,
‘) for the Phillips system of C,
0
1
2
3
4
5
6
7
8
9
0
X55(4)
9.06(4)
1 2 3 4 5 6 7 8 9
2.66(4) 3.7?(3) 1.33(2) 1.88(-l)
3*40(l) 1.23(4) 4.36(3) 2.20(2) 2.42(- 1)
7.744) 3.33(4) 1.21(4) 2.243) 3.12(3) 2.27(2) 1.75(--l)
4.83(4) 7.77(4) X33(3) 1.75(4) 1.58(O) 1.71(3) 1.87(2) ssq-2)
2.52(4) 8.10(4) 4.19(4) 2.63(3) 1.28(4)
1.16(4) 5.91(4) 7.87(4) 1.17(4) 1.16(4} 5.95(3) 2.13(3) 2X(2)
4.86(3) 3.43(4)
1.85(3) 1.83(4) 6.10(4) 8.11(4) 2.63(4) 2.65(3) 1.48(4) 5.97(l) 2.20(3) 7.18ili
6.48(2) 8.50(3) 3.86(4) 7.81(4) 6.50(4) 7.42(3) 9.36(3) 1.03(4) 2.37(2) 1.64(3\
2.12(2) 3.56(3) 2.14(4) 5.93(4) 8.16(4) 4.19(4) 2.70(2) 1.42(4) 5.7113) 8.39i2j
9.90(2) 7.46(2) 1.35(2) 2.79(- 2)
9.040) 4.51(- 3)
8.04(4) 5.42(4) 3.28(2) 1.62(4) 1.56{3) 2.49(3) 5.81(l) 5.7Gli
‘) Using the non-orthonormal basis transition moments calculated with the large A0 basis and lGOO-configuration vector (i.e. R& of table 3). The numbers in parentheses are the power of ten to multipIy the numbers outside the parentheses.
the A0 basis employed has some effect and the procedure followed for the selection of the configurations is also important for the calculation of the transition moment [19], it is not neeessary to use very large CI expansions in order to obtain converged results. The lifetimes calculated in the present work are in good agreement with the previous theoretical orthonormal basis results [9,10] and also significantly lower than the experimental of ref. [1,3,5] (also given in table 5). Thus, the discrepancy between theory and experiment remains. Let us examine how our theoretical values might be altered by possible improvements in the theoretical treatment. The size of the CI vectors is not expected to While
Table 5 Lifetimes ( ps) of the vibrational levels (u’) of the A ‘a, d
0 1 2 3 4 5 6 7 8 9
Present work *
make any significant difference on the calculated lifetimes. Improvement of the s and p sets of the A0 basis is not expected to have large effects. In fact, van Dishoeck [9] obtained with a smaller s and p set similar lifetimes as with her larger A0 basis set. Nonetheless, adding more polarization functions (d and f) to the A0 basis set might have a non-negligible effect on the lifetimes. Chabalowski et al. {lo] used one more d function than in the present work, without obtaining different results from ours. Even though the above considerations are reasonable, it cannot be concluded that making use of an even larger aud more flexible A0 basis than that used here would not affect the theoretical results to some significant extent.
of C,
Other theoretical
Experimental
a
b
c
ref. [lo]
ref. [9]
refs. [l,S]
10.60 8.17 6.80 5.90 5.28 4.79 4.44 4.17 3.95 3.81
11.04 9.13 7.71 6.70 6.00 5.44 5.03 4.71 4.46 4.28
11.64 9.30 7.79 6.77 6.05 5.52 5.12 4.80 4.55 4.73
10.7 8.3 6.9 6.0 5.3 4.9 4.5 4.3 4.0 3.8
11.1 9.0 7.6 6.6 5.9 5.3 4.9 4.6 4.3 4.0
13.4j, 2.5,18.5 + 3.0 15.0&4.0 14.4* 2.0 12.0&-2.0,11.4&2.0 10.7 j, 2.0 7.9 f 2.0 7.9 + 1.5 6.7 + 1.0 6.8 f 1.0
a> See footnote of table 2 for meaning of a, b and c.
ref. [3]
12.1+ 2.0 11.0+ 1.1 10.2+ 1.3 12.5 + 1.3
ref. [2] 7.4 6.3 5.5 5.2 5.2 4.3 4.2 3.8 3.6 3.5
324
G. Theodorakopoulos et al. / Calculations on the Phillips system in C,
However, it does not appear that the theoretical values would be altered to the extent of reaching the latest experimental lifetimes of the A ‘II, state of C,. The use of multiconfiguration wavefunctions might also improve the theoretical results since for the ground state, multiconfiguration orbitah would be more appropriate. Analogous discrepancies to those for the Iifetimes between theory and experiments exist also for the f,,,,, values (where u’ refers to levels of the A state and u” to those of the X state). We obtain foe= 2.52 x 10-3, fro = 2.92 x 10-3, fol = 2.00 x 10-3, fro = 1.86 x 1O-3 and fez = 0.55 x 10-3. may be calculated using the data (Additional fuJopt of tables 3 and 4 and eqs. (3) and (4).) From the literature, theoretical foe,,, values are: foe = 2.7 X low3 [9,10] and 2.5 X low3 [ll]; fol = 1.84 X 10T3, fi, = 2.84 x 10-3, fzo = 1.67 x 1O-3 and fo2= 0.42 X lop3 [9]. Some experimental values for f, are 1.5 X 10e3 [5], 1.38 X 10e3 [1,14], 2.3 X 10e3 [3], 1.5 x 1O-3 [3], 2.5 x 1O-3 [6], 3.94 x 1O-3 [2], while astronomical observations obtain for foe = 1.41 x 1O-3 [8] and 1.5 x 1O-3 [29], fro= 1.38 x 1O-3 [8] and fol= 1.12 x 1O-3 [8]. In conclusion, non-orthonormal basis transition moments have been calculated for the Phillips system of C,, in an attempt to resolve existing differences between experiments and theory on the lifetimes and oscillator strengths. The results of the present work support the previous theoretical values. As the origin of the discrepancy between experiment and theory is still not clear, it appears that further work, experimental and theoretical, is required on this system.
References [l] W. Bauer, K.H. Becker, C. Hubrich, R. Meuser and J.
Wildt, Astrophys. J. 296 (1985) 758. [2] D.M. Cooper and R.W. Nicholls, Spectros. Letters 9 (1976) 139; J. Quant. Spectry. Radiative Transfer 15 (1975) 139. [3] P. Erman, D.L. Lambert, M. Larsson and B. Mannfors, Astrophys. J. 253 (1982) 983. [4] J.R. McDonald, A.P. Baronavski and V.M. Donnelly, Chem. Phys. 33 (1978) 161. [5] W. Bauer, K.H. Becker, M. Bielefeld and R. Meuser, Chem. Phys. Letters 123 (1986) 35.
[6] F. Roux, D. Cemy and J. dIncan, Astrophys. J. 204 (1976) 904. [7] L.M. Hobbs and B. Campbell, Astrophys. J. 254 (1982) 108. [8] J.W. Brault, L. Debouille, N. Grevesse, G. Roland, A.J. SauvaI and L. Testerman, Astron. Astrophys. 108 (1982) 201. [9] E.F. van Dishoeck, Chem. Phys. 77 (1983) 277. [lo] CF. Chabalowski, SD. Peyerimhoff and R.J. Buenker, Chem. Phys. 81 (1983) 57. Ill] B. Pouilly, J.M. Robbe, J. Schamps and E. Roueff, J. Phys. B 16 (1983) 437. [12] CF. Chabalowski, R.J. Buenker and SD. Peyerimhoff, Chem. Phys. Letters 83 (1981) 441. [13] S.P. Davis, W.M. Handen Smith, J.W. Bra&, R. Pecyner and J. Wagner, Astrophys. J. 287 (1984) 455. [14] G. Stark and S.D. Davis, Z. Physik A 321 (1985) 75. [15] R.J. Buenker, S.D. Peyerimhoff, Theoret. Chim. Acta 35 (1974) 33; 39 (1975) 217; R.J. Buenker, S.D. Peyerimhoff and W. Butscher, Mol. Phys. 35 (1978) 771. [16] R.J. Buenker, in: Studies in physical and theoretical chemistry, Vol. 21. Current aspects of quantum chemistry, ed. R. Carbo (Ehevier, Amsterdam, 1982) pp. 17-34. [17] CA. Nicolaides, in: Advanced theories and computational approaches to the electronic structure of molecules, cd. C.F. Dykstra (Reidel, Dordrecht, 1984) p. 161. [18] I.D. Pet&&is, G. Theodorakopoulos, C.A. Nicolaides, R.J. Buenker and S.D. Peyerimboff, J. Chem. Phys. 81 (1984) 3161; I.D. Petsahtkis, G. Theodorakopoulos, CA. Nicolaides and R.J. Buenker, J. Chem. Phys. 81 (1984) 5952. (191 G. Theodorakopoulos, I.D. Petsalakis, C.A. Nicolaides and R.J. Buenker J. Chem. Phys. 82 (1985) 912; Chem. Phys. 100 (1985) 331. [21] A.T. Amos and G.G. HalI, Proc. Roy. Sot. A 263 (1961) 483. [22] H.F. King, R.E. Stanton, H. Kim, R.E. Wyatt and R.G. Parr, J. Chem. Phys. 47 (1967) 1936. [23] B.H. Lengsfield, J.A. Jafri, D.H. Phillips and C.W. Bauschlicher, J. Chem. Phys. 74 (1981) 6849. (241 T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. [25] K.P. Huber and G. Herzberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand-Reinhold, New York, 1979). [26] S.R. Langhoff and E.R. Davidson, Intern. J. Quantum Chem. 8 (1974) 61. [27] R.J. Buenker, S.D. Peyerimhoff and M. Peric, Chem. Phys. Letters 42 (1976) 383; M. Peric, R. Runau, J. Romelt and S.D. Peyerimhoff, J. Mol. Spectry. 78 (1979) 309. [28] M. Larsson and P.E.M. Siegbahn, J. Chem. Phys. 79 (1983) 2270. [29] AC. Danks and D.L. Lambert, Astron. Astrophys.
(1983) 188.
124
319
NON-ORTHONORMAL BASIS CALCULATIONS OF THE DIPOLE TRANSITION FOR THE PHILLIPS SYSTEM (A ‘II” + X ‘2,‘) IN C,. THEORETICAL LIFETIME OF THE A ‘l-I, STATE G. THEODORAKOPOULOS,
I.D. PETSALAKIS,
MOMENT
C.A. NICOLAIDES
Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, 48 Vas. Constantinou Ave., 116/3.5 Athens, Greece
and R.J. BUENKER Theoretische Ch,emie, Berg&he
Universiriit Wuppertal, D-5600 Wuppertai I, FRG
Received 22 September 1986
Non-orthonormal basis calculations have been carried out on the A’& + X “2: transition moment of C2 as a function of C-C internuclear distance. Two types of A0 basis set and different sizes of CI vectors were employed in order to examine the convergence of the theoretical results. Published ab initio potential energy curves and the transition moments of the present work were combined to calculate radiative lifetimes of vibrational levels of the A’II, state. The resulting lifetimes are in agreement with other theoretical values, obtained with orthonormal basis calculations, which however, are significantly lower than those of the most recent experiments.
1. Introdu~on Electronic transitions in C!, have been observed in astrophysical as weIl as in terrestrial sources [l-14]. The Phillips system, involving the A’II,,X “ZZp*transition, is used for determination of the C, abundance in interstellar clouds [7] (where the Swan system, yet stronger in the laboratory, is not detected) and in the carbon stars. For this reason, it is required to have accurate lifetimes and oscillator strengths, which, for this system are still to be established [I]. Recent experimental work employing the techniques of excimer laser photolysis (ELP) [1] obtained lifetimes for the vibrational levels of A’II, which are much larger than those of earIier shock tube experiments [2] but close to those of other direct measurements of the lifetimes [3,4]. The most recent experimental work [5] employing the laser-induced fluorescence technique (LIF) updates the u’ = 0 and u’ = 3 data of ELP fl]. In this way, the experimental lifetime of u’ = 0,
7(O), (for example) is given as 18.5 + 3.0 ps by LIF [5], 13.4 it 2.5 ps by ELP while the shock tube work [2] gave 7.4 p.s. Theoretical calculations on the Phillips system of C, [9,10] give lifetimes significantly lower than the direct measurements [1,3-51, with ~(0) of 11.1 [9] and 10.7 [lo] p.s. The theoretical fW values [9-111 are larger than foe of ref. [5] by about a factor of 2, while they are in agreement with some &, values obtained in earlier experiments ([1,6] and references therein). The discrepancy between the theoretica results and the experimental ones is rather large, in view of the excellent agreement between theory [12] and experiment [13,14] for the Swan system in C,. As an explanation, it has been suggested that the experimental values might require corrections for the effects of cascading, due to allowed transitions from other upper states into A ‘II, [lo]. One might also try to do more work on the theoretical side. The theoretical works of van Dishoeck [9] and ChabaIowski et al. [lo] employ the
0301-01~/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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G. Theodorakopoulos et al. / Calculations on the Phillips system in C,
MRD CI method [15,16] with large A0 basis sets including polarization and diffuse functions. In ref. [lo] the MO of the D ‘Z: state is the common one-electron basis while in ref. [9] different (but common to both states) MO basis sets are employed in order to examine the effect on the calculated transition moments. Pouilly et al. [ll] have carried out CI calculations with a Slater-type A0 basis set and also work with a common oneelectron basis for the X ‘2: and A’II, states. It is not expected that another, orthonormal basis theoretical work would yield greatly different results (&20%) from the above. However, it would be interesting to know the results of nonorthonormal basis calculations (NON) for the transition moments of Al&,--X ‘Ep’, as a further test of the stability of the theoretical values, within the Born-Oppenheimer approximation. In a NON calculation, two different MO basis sets are used for the two states, ground and excited, for an optimum and concise description of each state, in the philosophy of a state-specific theory of electronic states [17]. We have developed a method for NON calculations with one- and two-electron operators [18-201. As a starting point it has the corresponding orbitals transformation [21] and the generalized Slater-Condon rules of Ring et al. [22]. Our method has been formulated for general LCAO MO CI wavefunctions and is completely general. We employ the NON method with MRD CI vectors where the reference spaces, the selection thresholds and the number of roots solved for may be varied, to best account for the requirements of a given problem, without having to deal with very large CI expansions. Convergence calculations are used to evaluate the stability of the results obtained with relatively small CI vectors. An alternative method for NON calculations, currently described in the literature [23], is based on the use of complete active space vectors for the electronic states.
lb:,lbz, (‘A,). For internuclear distance rc_c longer than about 2.8 bohr [9,10] two other configurations become important, 3atlb:, and 3ailb& (giving a$:). However, it has been found that at R,_, > 2.8 bohr the transition moments are very small and have a negligible effect on the calculated lifetimes of the A’III, state [9,10]. Thus it is sufficient to use the MO of the r4 configuration for the ground state calculation throughout the range of internuclear distances calculated from 2.0 to 3.2 bohr, although this is correct only for the first few vibrational levels (u” = O-4) [9]. In order to account for the X ‘El-2 ‘xp’ interaction it is necessary to calculate both states together using all the important reference configurations for both states [9,10]. We calculate the first three roots of ‘As symmetry, since the ‘Aa component of the first ‘As state is also in the same energy region
2. Method of calculation
The non-orthonormal basis program developed by the authors [19,20] was used for the calculation of R elell. Details of the method have been given elsewhere [20]. In order to facilitate comparison with experi-
The calculations have been carried out in D,, symmetry. In this point group, the . . -7~~ main configuration of the ground state is given by
WI.
The excited state, A ‘II”, represented by &~a is calculated with the MO of configuration . . . lb:,3a,lb, (lb”) for one-electron basis. Some preliminary calculations have been carried out with the (9s5p/5s3p) carbon basis set of Dunning [24]. Two d functions (exponents 1.2 and 0.3) for polarization and one s and one p (exponents 0.023 and 0.021, respectively) diffuse function were added to the carbon A0 basis for most of the calculations. The MRD CI programs were employed to calculate CI vectors of various sizes for the two states of interest. Large reference spaces (see also ref. [lo]) were required to obtain a contribution of over 90% of the reference set in the CI vectors, over the R,_, distance from 2.0 to 3.2 bohr. As in ref. [lo], the length form of the dipole operator was used to calculate the electronic transition moment integral R,,,,, (with primed quantities refering to the upper state and doubly primed to the lower state) at each ‘c-c value:
321
G. Theodorakopoulos et al, / Calculations on the Phillips system in C,
mental quantities it is necessary to calculate lifetimes and oscillator strengths for transitions between vibrational levels of the two electronic states. The potential energy curves of Chabalowski et al. [lo] were used to calculate vibrational wavefunctions { xU,, x,,,} for the two states. Then, the line strength is given by
s“‘U”
=
21(x,&-,)
l~e~,~.(~c-c)lX,~(~c-c))/2~ (2)
where the factor 2 accounts for the degeneracy of the ‘l&, state [lo]. It should be noted that the notation of ref. [lo] is followed, the real form of the dipole operator is employed, which in this case is equivalent to the form CJx,+iy,)/fi, used by van Dishoeck [9]. The coefficients A,,,, for spontaneous emission are given by A “I”,’=
2.1419 x 10” A E,?,, (au) S,,“,, (2 - ~OJdW
+ 1)
(3)
(with 2 - So,,, = 2 for this system) and the corresponding absorption oscillator strengths are given by
(with 2 - So A,, = 1). Finally, the radiative lifetimes of the u’ levels, of the A ‘II,, state are given by
r(d) = ($A..n,,)-l
ments. For example, at rc_c of 2.348 bohr, calculations with the smaller A0 basis set gave Ret,,, values of 0.3781, 0.4154 and 0.4237 for vectors of about 50, 500 and 1000 configurations, respectively. The effect of adding polarization and diffuse functions to the A0 basis set is to decrease the calculated transition moment. R,,,, at re is calculated as 0.3827 with vectors of size between 900 and 1000 configurations and the larger A0 basis set. Orthonormal basis transition moments were also calculated at re, for comparison, employing in turn the MO of X ‘xl state, those of the A’Il, state and those of the D ‘2: state as the one-electron basis set. The sizes of these MRD CI calculations were about 4000 and 6000 configurations for A0 basis sets (i) (smaller basis) and (ii) (larger basis) respectively. The results on the orthonormal basis calculations are summarized in table 1. As shown in table 1, the use of different MO and A0 basis sets has a moderate effect on the calculated orthonormal basis transition moments, in agreement with the results of van Dishoeck [9]. Non-orthonormal basis calculations of the transition moment integral have been carried out, for several rc _ c values, with the large A0 basis set and CI vectors of about 500 and 1000 configurations for each state. The results are given in table 2 along with NON calculations employing the small A0 basis and 500 configuration vectors. As shown in table 2, the calculated Rerep, values decrease with increasing rc_c in agreement with the previous calculations [9,10]. The present results do not support assumptions of a constant R e,e,, or increasing R,,,,, with rc_c [3]. The potential energy curves for the A’III, and lx+ states of C, calculated by Chabalowski et al. [lG (including the full-C1 correction to the ex-
s.
3. Results and discussion Several test calculations have been carried out at the ground-state equilibrium distance ( re = 2.348 bohr [25]) in order to obtain convergence of the results with respect to the size of the CI vectors and to estimate A0 basis effects. It was found that the size of the CI has an effect on the transition moment for vectors up to 400-500 configurations. Calculations with larger vectors do not give significantly different transition mo-
Table 1 Orthonormal basis transition moments for the A’&-X transition at rc_c of 2.348 bohr MO basis
A0 basis (i)
A0 basis (ii)
0.4094 0.4100 0.4207
0.4020 0.3783 0.4072
x lx+ AG D ‘I$”
‘Ep’
G. Theodorakopoulos et al. / Calculations on the Phillips system in C,
322
Table 2 Non-orthonormal basis transition moment for the Phillips system of C, as a function of the internuclear distance rc_c rc_c (bohr)
R&m a)
R;.,n ‘)
RF,,,,, 4
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
0.4384 0.4315 0.4048 0.3565 0.2964 0.1973 0.0276 0.0761
0.4154 0.4049 0.3817 0.3604 0.1938 0.0250 0.0388 0.0354
0.4098 0.4005 0.3809 0.3372 0.1618 0.0247 0.0406 0.0768
a) Superscript a denotes small A0 basis and 5OOconfiguration vectors; superscript b denotes large A0 basis and 500~configuration vectors; superscript c denotes large A0 basis and lOOO-configuration vectors.
trapolated energy [26] were used for vibrational calculations [27]. The curve fittings gave the following results: For the X ‘xl state re = 2.364 bohr with experimental value [25] of 2.3480 bohr, we= 1848 cm-’ (exp. 1854.7) and B, = 1.80 cm-’ (exp.1.82). For the AllI, state, re = 2.519 bohr (exp. 2.491), w, = 1548 cm-’ (exp. 1608.4) and B, = 1.58 cm-’ (exp. 1.61). In table 3, AE,,,, are listed, corresponding to the energy differences from a vibrational level u’ of the A ‘II” state to u” of X ‘xp’ , for the first six vibrational levels for each state. The theoretical AE, (or qx,) is 8124 cm-’ or 0.037014 au, while the experimental value is 8268 cm-’ or 0.037671 au. The difference between the experimental and theoretical AE, of about 0.017 eV is within the Table 3 Energy differences, A E,,,,, (cm -‘) u”
0 1 2 3 4 5 6 7 8 9
‘) for the AlIf,-X
errors of the calculations. Thus, ‘we ‘choose to work with the theoretical potential energy curves since the RKR curves employed in ref. [9], where the theoretical errors were of the order of 0.1 eV, are only reliable for internuclear distances between 2.1 and 2.7 bohr [9]. Line strengths (eq. (2)) were calculated using the vibrational wavefuno tions obtained above and the transition moments of table 2, where a polynomial fit of the calculated R ese#e values gives Rereer(rc_c). The coefficients for spontaneous emission, A,#“PS(eq.(3)), obtained with the transition moments listed under RC,t,,, (i.e. the large A0 basis and large CI vectors), are listed in table 4, for the first nine vibrational levels of each state. The corresponding lifetimes of the vibrational levels of AlIT, are given in table 5 (column c) along with the lifetimes obtained with the other two sets of transition moments of table 2 and along with some literature values for comparison. The lifetimes calculated with the smaller basis set (i.e. without polarization or diffuse functions), given in column a in table 5, are close to those obtained with the larger basis set (columns b and c). Thus, the effect of changing the A0 basis is only about 1 us on the calculated lifetimes. There is virtually no difference in the lifetimes obtained with 500-configuration vectors and those obtained with lOOO-configuration vectors (columns b and c, respectively, in table 5). This is very gratifying and it seems to be generally true for non-orthonormal basis calculations of transition moments in our work [19,20] as well as in the work of others [28].
‘Ep’ transition in Cz
V’ 0
1
2
3
4
5
0.81243(4) 0.63089(4) 0.45261(4) 0.27766(4) 0.10608(4)
0.%493(4) 0.78339(4) O&0512(4) 0.43016(4); 0.25858(4) 0.90406(3)
0.11151(5) 0.93361(4) 0.75533(4) 0.58038(4) 0.40879(4) 0.24062(4) 0.75848(3)
0.12631(5) 0.10815(5) 0.90324(4) 0.72829(4) 0.55671(4) 0.38853(4) 0.22376(4) 0.62284(3)
0.14087(5) 0.12271(5) 0.10488(5) 0.87389(4) 0.70230(4) 0.53413(4) 0.36936(4) 0.20788(4) 0.49349(3)
0.15519(5) 0.13704(5) 0.11921(5) 0.10171(5) 0.84556(4) 0.67739(4) 0.51262(4) 0.35114(4) 0.19261(4) 0.3582q3)
‘) The theoretical values are given.
G.~~orakop~i~
et al. /
Table 4 Theoretical spontaneous emission probabilities. A,,‘u,. (s-t)
Calculations on the Phil&p
system
323
in C,
‘) for the Phillips system of C,
0
1
2
3
4
5
6
7
8
9
0
X55(4)
9.06(4)
1 2 3 4 5 6 7 8 9
2.66(4) 3.7?(3) 1.33(2) 1.88(-l)
3*40(l) 1.23(4) 4.36(3) 2.20(2) 2.42(- 1)
7.744) 3.33(4) 1.21(4) 2.243) 3.12(3) 2.27(2) 1.75(--l)
4.83(4) 7.77(4) X33(3) 1.75(4) 1.58(O) 1.71(3) 1.87(2) ssq-2)
2.52(4) 8.10(4) 4.19(4) 2.63(3) 1.28(4)
1.16(4) 5.91(4) 7.87(4) 1.17(4) 1.16(4} 5.95(3) 2.13(3) 2X(2)
4.86(3) 3.43(4)
1.85(3) 1.83(4) 6.10(4) 8.11(4) 2.63(4) 2.65(3) 1.48(4) 5.97(l) 2.20(3) 7.18ili
6.48(2) 8.50(3) 3.86(4) 7.81(4) 6.50(4) 7.42(3) 9.36(3) 1.03(4) 2.37(2) 1.64(3\
2.12(2) 3.56(3) 2.14(4) 5.93(4) 8.16(4) 4.19(4) 2.70(2) 1.42(4) 5.7113) 8.39i2j
9.90(2) 7.46(2) 1.35(2) 2.79(- 2)
9.040) 4.51(- 3)
8.04(4) 5.42(4) 3.28(2) 1.62(4) 1.56{3) 2.49(3) 5.81(l) 5.7Gli
‘) Using the non-orthonormal basis transition moments calculated with the large A0 basis and lGOO-configuration vector (i.e. R& of table 3). The numbers in parentheses are the power of ten to multipIy the numbers outside the parentheses.
the A0 basis employed has some effect and the procedure followed for the selection of the configurations is also important for the calculation of the transition moment [19], it is not neeessary to use very large CI expansions in order to obtain converged results. The lifetimes calculated in the present work are in good agreement with the previous theoretical orthonormal basis results [9,10] and also significantly lower than the experimental of ref. [1,3,5] (also given in table 5). Thus, the discrepancy between theory and experiment remains. Let us examine how our theoretical values might be altered by possible improvements in the theoretical treatment. The size of the CI vectors is not expected to While
Table 5 Lifetimes ( ps) of the vibrational levels (u’) of the A ‘a, d
0 1 2 3 4 5 6 7 8 9
Present work *
make any significant difference on the calculated lifetimes. Improvement of the s and p sets of the A0 basis is not expected to have large effects. In fact, van Dishoeck [9] obtained with a smaller s and p set similar lifetimes as with her larger A0 basis set. Nonetheless, adding more polarization functions (d and f) to the A0 basis set might have a non-negligible effect on the lifetimes. Chabalowski et al. {lo] used one more d function than in the present work, without obtaining different results from ours. Even though the above considerations are reasonable, it cannot be concluded that making use of an even larger aud more flexible A0 basis than that used here would not affect the theoretical results to some significant extent.
of C,
Other theoretical
Experimental
a
b
c
ref. [lo]
ref. [9]
refs. [l,S]
10.60 8.17 6.80 5.90 5.28 4.79 4.44 4.17 3.95 3.81
11.04 9.13 7.71 6.70 6.00 5.44 5.03 4.71 4.46 4.28
11.64 9.30 7.79 6.77 6.05 5.52 5.12 4.80 4.55 4.73
10.7 8.3 6.9 6.0 5.3 4.9 4.5 4.3 4.0 3.8
11.1 9.0 7.6 6.6 5.9 5.3 4.9 4.6 4.3 4.0
13.4j, 2.5,18.5 + 3.0 15.0&4.0 14.4* 2.0 12.0&-2.0,11.4&2.0 10.7 j, 2.0 7.9 f 2.0 7.9 + 1.5 6.7 + 1.0 6.8 f 1.0
a> See footnote of table 2 for meaning of a, b and c.
ref. [3]
12.1+ 2.0 11.0+ 1.1 10.2+ 1.3 12.5 + 1.3
ref. [2] 7.4 6.3 5.5 5.2 5.2 4.3 4.2 3.8 3.6 3.5
324
G. Theodorakopoulos et al. / Calculations on the Phillips system in C,
However, it does not appear that the theoretical values would be altered to the extent of reaching the latest experimental lifetimes of the A ‘II, state of C,. The use of multiconfiguration wavefunctions might also improve the theoretical results since for the ground state, multiconfiguration orbitah would be more appropriate. Analogous discrepancies to those for the Iifetimes between theory and experiments exist also for the f,,,,, values (where u’ refers to levels of the A state and u” to those of the X state). We obtain foe= 2.52 x 10-3, fro = 2.92 x 10-3, fol = 2.00 x 10-3, fro = 1.86 x 1O-3 and fez = 0.55 x 10-3. may be calculated using the data (Additional fuJopt of tables 3 and 4 and eqs. (3) and (4).) From the literature, theoretical foe,,, values are: foe = 2.7 X low3 [9,10] and 2.5 X low3 [ll]; fol = 1.84 X 10T3, fi, = 2.84 x 10-3, fzo = 1.67 x 1O-3 and fo2= 0.42 X lop3 [9]. Some experimental values for f, are 1.5 X 10e3 [5], 1.38 X 10e3 [1,14], 2.3 X 10e3 [3], 1.5 x 1O-3 [3], 2.5 x 1O-3 [6], 3.94 x 1O-3 [2], while astronomical observations obtain for foe = 1.41 x 1O-3 [8] and 1.5 x 1O-3 [29], fro= 1.38 x 1O-3 [8] and fol= 1.12 x 1O-3 [8]. In conclusion, non-orthonormal basis transition moments have been calculated for the Phillips system of C,, in an attempt to resolve existing differences between experiments and theory on the lifetimes and oscillator strengths. The results of the present work support the previous theoretical values. As the origin of the discrepancy between experiment and theory is still not clear, it appears that further work, experimental and theoretical, is required on this system.
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124