Spontaneous radiative properties of the excited electronic states of TiO

J. Quant. Spectrosc. Radiat. Transfer Vol. 48, No. 2, pp. 147-152, 1992

0022-4073/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press Ltd

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SPONTANEOUS

RADIATIVE

EXCITED

ELECTRONIC

PROPERTIES STATES

OF

OF

THE

TiO

J. SCHAMPS, J. M. SENNESAL, a n d P. CARETTE Laboratoire de Dynamique Molfculaire et Photonique, Unit6 de Recherche Assocife au C.N.R.S. No. 779, Universit6des Scienceset Technologiesde Lille,U.F.R. de Physique---l~timentP5, 59655VilleneuveD'Ascq Cedex, France (Received 19 December 1991)

Abstract--Ab initio electronic wavefunctions have been used to calculate and interpret absolute

transition probabilities and oscillator strengths for all of the transitions involving the observed valence states of TiO. Spontaneous radiative lifetimes have been derived from the calculated probabilities and compared to related experimental data. The transitions are shown to be primarily atomically-forbidden transitions allowed in the molecule by orbital polarization.

1. I N T R O D U C T I O N

Among the numerous molecules containing a transition metal atom, TiO is one for which the radiative dynamics has been widely studied experimentally. This interest is justified by the astrophysical importance of the TiO spectrum that is known to be one of the main features in the absorption spectra of M-type stars. 1-3 These studies have provided numerous results about the radiative properties of TiO. Radiative properties of the excited states of a molecule can be characterized by several quantities such as (squared) transition moments, Einstein coefficients, oscillator strengths, radiative lifetimes, etc. All of these quantities are related as follows: 4 the first three are proportional to each other and are typical of a given transition; the radiative lifetime is characteristic of a given energy level and is inversely proportional to the sum of all the probabilities of the spontaneous emission transitions starting from this level. The determination of these quantities may be carried out by using at least three different methods. The first involves their derivation from absolute intensity measurements of individual lines or bands. The second is based on comparisons of equivalent widths deduced from absorption coefficients. Both of these methods are suitable for the determination of individual transition probabilities. They may be considered to be indirect methods in the sense that they require other, independently determined data such as number densities or other known absolute transition probabilities. The third method is a direct procedure since it is time-resolved and yields lifetimes directly. It involves recording the exponential decay of the emission immediately after selective excitation of a given transition by a short laser pulse. Whenever the third method can be used, i.e. in spectral regions where lasers operate with sufficiently short pulses, this pulse-decay method is easy to implement. Moreover, it is the most accurate procedure since the dynamical behaviour of the state is derived directly from the experimental data without resorting to more or less ill-determined external quantities. For TiO, it was used previously in order to determine lifetimes of the v = 0, 1, 2 vibrational levels of the C3A state 5 and also those of individudl rotational lines in the v -- 2 level of the D = 3 component? It was used by Feinberg and Davis to determine the lifetime of the v = 0 level of the c1¢ state. 7,8 Quite recently, this same method was used by Simard and Hackett for the lifetime of the v = 0 level of the E2H state 9 and by us for the lifetime of the v = 0 level of the B3II~ state. ~° For three other states (A 3¢, B3/-/', b I/-/), as well as for the c3A and c~¢ states transition probabilities were derived from indirect methods, using either absolute intensities ~-~4 or absorption-coefficient 15-17 measurements. 147

148

J. SCHAMPS et al

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Spontaneous radiative properties of excited electronic states of TiO

149

The present paper deals with ab initio calculations originally carried out for the purpose of interpreting the measured values of the lifetimes in terms of electronic configuration changes. However, the good agreement we obtained between experimental and calculated lifetimes in the case of the B3II, C3A and a l ~ states motivated us to extend the ab initio study to other TiO states that are more difficult to study experimentally for the purpose of predicting the magnitudes of their lifetimes. 2. C A L C U L A T I O N S OF T R A N S I T I O N P R O B A B I L I T I E S A N D O S C I L L A T O R S T R E N G T H S IN TiO In a previous paper, two of us (JMS, JS) published the results of ab initio SCF-CI calculations of the energies and wavefunctions of the low-lying electronic states of TiO. t8 Population analyses were also carried out in this paper. These showed that the molecule is ionic and that, roughly speaking, the titanium ion seems to have a propensity to keep a structure as close as possible to the Ti ÷ (3d 2 4s) structure of the ground state in all of the calculated excited states, in spite of the promotion of an electron into a different orbital. This structure retention is made possible by molecular polarization and configuration interaction that partly compensate for the gain or loss of a 3d electron when an excited state is created and thus leads to (3d2Ms °5) or (3d~54s~5) gross structures of the Ti ÷ ion. A good procedure for ascertaining the reliability of the wavefunctions responsible for these electronic populations is to calculate the lifetimes of the TiO states since it is known that calculations of transition moments are very sensitive to the accuracy of the wavefunctions used. The fundamental elements required for lifetime calculations of a given state are the electronic transition dipole moments between this state and all of the other states lying at lower energies. 4 For instance, for the B3FI-X3A transition, the electronic transition moment is

Re(B-X) = (X3AI 2-~/2(/~X+ ip>.)IB3H). The calculations of all relevant transition moments have been performed at a single distance, namely 3.1 Bohr, which is close to the calculated equilibrium internuclear separation in the ground state (we neglect the variation of Re over the internuclear separation). Wavefunctions built on different sets of SCF molecular orbitals for different states were used so as to keep the advantages of rapidly converging CI. The results are given in Table 1, together with the electronic transition probabilities (Einstein coefficients A in sec-~) and oscillator strengthsf(dimensionless) derived from the transition moments using the following definitions 4 (in SI units):

A = (16rt 3/3hEoc 3)v 31Re 12g(A ',A "), f = (87z2me/3he2)v[Re[2g(A',A"), where v is the frequency (in Hz) of the electronic transition. This calculation assumes that the potential curves are sufficiently parallel to each other to allow us to consider that the vibrational intensity is confined to the Av = 0 sequence. The g(A',A ") factor is equal to unity except for 1-I-S transitions for which it is equal to 2. Since the CI wavefunctions have been found to be concentrated in a single configuration, ~8 a qualitative interpretation of the magnitude of the calculated transition moments can be obtained in terms of electron jumps between the predominant configurations ~s of the states involved in the observed transitions. In view of this, it is useful to recall that the 9a molecular orbital is a titanium 4s orbital polarized by some 4p character, while the three 3d(Ti) molecular components are the 16 (100% 3d6), 3n (between 75 and 95% 3dn, depending on the states, plus some 4p~r polarization) and the 10a (hardly more than 50% 3dtr plus considerable 4pa polarization) orbitals. ~8 From this point of view, the TiO electronic transitions can be classified into three groups according to the order of magnitude of their transition moments reported in Table 1. The first group includes the transitions with large moments. All of these arise as the result of an electron jump from one of the 4p-polarized 3d orbitals (i.e. either 4n or 10a) towards the 4p-polarized 4s orbital (i.e. 9tr) of titanium. The transitions belonging to this group are the emission from b~H towards d12~+, those from A3~, B31-I, and C3A towards X3A, and those from C)~ and

150

J. SCHAMPSet al

f~A towards a ~A. It is somewhat puzzling that the ~H-a ~A transition involving the ~H state which is isoconfigurational with cl~, A3~ and B311, has not yet been spectroscopically detected in the 2 -~ 600 nm spectral region where it is expected to occur. ]8 The second group is made up of two transitions with smaller transition probabilities: the E3II-X3A transition and its singlet analog, the bJFl-a~A transition. These both arise from a predominantly 3dz~-3d6 transition. Such a transition would violate Laporte's rule AE-- _+ 1; it appears in the molecule owing to secondary polarization effects in the 3dzr orbital and, to a lesser extent, to configuration interaction. The third group is the group with the transitions having a negligible calculated probability (transition moments lower than 0.1 a.u.). In first approximation, these transitions would require two-electron jumps to occur, which are forbidden on account of the monoelectronic nature of the dipole-moment operator. Therefore, it is not surprising that they have not been observed spectroscopically, especially since they would take place between excited states (CaA~E311,

c3A__.)A 3~, flA._.)b Ill).

3. C A L C U L A T I O N S OF T H E R A D I A T I V E L I F E T I M E S OF E X C I T E D S T A T E S OF TiO Once the A transition probabilities have been determined, it is easy to calculate the lifetime of each electronic state k by using the definition4

where the summation extends over all of the transitions to lower electronic states. For most of these states, it was sufficient to include only one term in the summation, i.e. the term corresponding to emission towards the lowest state of the same multiplicity (either X3A or a ~A); the other channels give completely negligible contributions, as may be verified from Table I. The only exception was the b ~/-/state, for which two radiative channels (towards a ~A and d~S +) had to be considered, the dominant channel being the one towards dIZ ÷ (96% of the deexcitation lifetime). To be complete, it should be noted that previous ab initio calculations by Bauschlicher et al 2° indicate the presence of the 3Z - state of the (8e 2 3~ 4 162) configuration at 12,800 cm-). Since the B3_FI state can radiate towards this 31;- state via a 4~---)16 electron jump, this could have been a second deexcitation channel for B3II. But the corresponding transition moment is expected to be rather weak compared with other 4rc-l~ transition moments such as the E 3 H - X 3 A or b~II-a~A transitions given in Table 1. Furthermore, the energy distance is so small (,-, 3000 cm-1) that this radiative channel is quite negligible compared to B3H-X3A. The calculated lifetimes derived from the transition moments of Table 1 are given in Table 2. Whenever possible, these are compared with experimental values. The reported experimental values were either taken from direct measurements using the pulse-decay method or else were derived from reported values of absolute intensities or absorption coefficients assuming a single-channel deexcitation whenever this was indicated by the results of the calculations, as discussed above. On account of the disparity and generally poor accuracy of the experimental values, it must be concluded that the ab initio calculated lifetimes are fairly satisfactory for the short-lived states. For instance, in the case of the Bail state, the two now available experimental values, I°.]6 which were measured by using independent methods, are both about 44 nsec while our calculated value is about 56 nsec. For the c3A (at least at v = 0) 5"13and c ~ states, 8 experimental and ab initio determinations are even closer (see Table 2). This is also true for the A 3~ state if one retains Davis et al's value, 16 which appears to be the most reliable experimental value. However, marked discrepancies are observed for the long-lived states. The lifetime of the b ~H state is calculated to be more than twice too long (455 nsec instead of 192 nsec). Even worse for the E3H state, its lifetime is calculated to be three times too long compared to the recent pulse-decay measurement of 770 _+ 40 nsec. 9 This observation is ascribed to the well known difficulty of calculating small transition dipole moments. In this case more accurate wavefunctions are desirable. In any case, the consistently good agreement obtained for the short-lived states, when comparisons are made with experimental values, gives us confidence in the reliability of ab initio lifetime values found for other short-lived states.

Spontaneous radiative properties of excited electronicstates of TiO

151

Table 2. Calculated lifetimes(nsec) of the observedexcitedelectronicstates of TiO and comparisons with experiments (the experimental values given in italics are lifetimes derived from indirect measurements of transition moments or oscillator strengths; values not in italics are direct measurements of lifetimes using the pulse-decay method) State E311

A3¢

B31I C3A

Tcalc (i0-9s) 2 300

56.2

38.7

co

die+

oo

b~

770 + 40 (v=O)

55.2

aIA

Texp (10-9s)

455

c~¢

15.0

f~A

41.6

Ref. 9

24 + 10

(v=O) (v=l) 15 _+ 5 (v=l)

13

51 +- 9

16

44 -+ 2

(v=l)

10

43 + 14 (v=0)

16

13

37 -+ 9

(v=O)

5

31 + 6

(v=O)

13

29 -+ 7

(v=1)

5

21 +- 4

(v=l)

13

28 -+ 7 29 + I

(v=2) (v--2)

5

18.5 +- 3

(u=2)

16

192 +- 55 (v=0)

16

17.5 + I

(v--0)

6

8

Thus, the as yet unmeasured lifetime of the f~A state is predicted from the present calculations to be 42 + 10 nsec. Because of the usual selection rules AS = 0 and AA = 0 ( _ 1), the a~A and dl~ + states have no way of radiating at the level of approximation used in this work. They probably depopulate via collisions and through weak spin-orbit contamination of their wavefunctions by interacting triplet states and should have radiative lifetimes several orders of magnitude longer than the other states considered here. 4. C O N C L U S I O N S The radiative properties calculated here are in fairly satisfactory agreement with the experimental values for the short-lived states of TiO but discrepancies by factors of two or three are observed for the lifetimes of the long-lived b 1 i / a n d E3F1 states. Since there is no reason to question the reliability of the experimental data, this result seems to indicate that the wavefunctions used in our calculations should be improved for calculating long lifetimes with some confidence. On the other hand, for short lifetimes, the need for accurate descriptions of the configuration interaction parts of the wavefunctions is much less crucial. In this case, the value of the transition moment calculated at the monoconfigurational approximation, i.e. using the predominant configuration of each of the two transiting states is large and configuration interaction can only introduce negligible change in the SCF value. The main transitions observed in the titanium atom (neutral or singly-ionized) are due to 4 p - 4 s electron jumps and essentially fall in the u.v. 2~ The present calculations of transition probabilities,

152

J. SCHAMPSet al

oscillator strengths and lifetimes in TiO provide a coherent set of data from which emerges the conclusion that none of the transitions of the TiO molecule observed in the visible (i.e. below 25,000 cm-1) have analogs in the titanium atom spectrum. These molecular transitions would be forbidden in the atom limit since they violate the At~ = +1 selection rule. Most of these, i.e. those with the greatest oscillator strengths are primarily 3d-4s electron jumps, except for the E3/-/-X3A and f H - a ~ A transitions that are 3d-3d transitions. Molecular symmetry introduces 4p-polarization into these 3d and 4s orbitals. This fact not only makes the molecular transitions possible by bringing a part of 4p-4s or 4p-3d character into the transition moments but it also gives rise to some of them attaining quite large values that are more typical of a plainly allowed transition than of a transition that would be atomically forbidden, although it is allowed by molecular-polarization effects. The property of giving a secondary effect the appearance of a primary effect shows how large the 4p-4s transition moment is. This result should prompt spectroscopists to explore again the u.v. spectrum of TiO and to look for molecular transitions analogous to the atomic 4p-4s transitions. If the upper states are not predissociated, one would predict these transitions to be the most intense in the entire TiO spectrum. REFERENCES C. C. Kiess, Publ. Astr. Soc. Pacif. 60, 252 (1948). P. W. Merrill, A. J. Deutsch, and P. C. Keenan, Astrophys. J. 136, 21 (1962). H. Machara and Y. Yamashito, Publ. Astr. Soc. Japan 28, 135 (1976). E. E. Whiting and R. W. Nicholls, Astrophys. J. Suppl. 27, 1 (1974). R. E. Steele and C. Linton, J. Molec. Spectrosc. 69, 66 (1978). J. Feinberg and S. P. Davis, J. Molec. Spectrosc. 69, 445 (1978). J. Feinberg, M. G. Bilal, S. P. Davis, and J. G. Phillips, Appl. Lett. 17, 147 (1976). J. Feinberg and S. P. Davis, J. Molec. Spectrosc. 65, 264 (1977). B. Simard and P. A. Hackett, J. Molec. Spectrosc. 148, 127 (1991). P. Carette and J. Schamps, J. Molec. Spectrosc., in press (1992). W. Zyrnicki, JQSRT 15, 575 (1975). M. L. Price, K. G. Sulzmann, and S. S. Penner, JQSRT 11, 427 (1971). M. L. Price, K. G. Sulzmann, and S. S. Penner, JQSRT 14, 1273 (1974). C. Linton and R. W. Nicholls, JQSRT 10, 311 (1970). S. A. Golden, JQSRT 7, 225 (1967). S. P. Davis, J. E. Littleton, and J. G. Phillips, Astrophys. J. 309, 449 (1986). J. G. Collins and T. D. Fay Jr., JQSRT 14, 1259 (1974). J. M. Sennesal and J. Schamps, Chem. Phys. 114, 37 (1987). K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 4. Constants of Diatomic Molecules, Van Nostrand-Reinhold, Princeton, NJ (1979). 20. C. W. Bauschlicher, P. S. Bagus, and C. J. Nelin, Chem. Phys. Lett. 101, 229 (1983). 21. C. E. Moore, "Atomic Energy Levels," NBS Circular 467, U.S. Govt Printing Office, Washington, DC (1952). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.