J . Phys. Chem. 1992, 96. 1630-1640
1630
TABLE IV: Pairing Energies in Atomic Units for Some Metal Cations ion eq 31 eq 34 ref 16 0.1712 0.1281 0.0930 d4 Cr2+
Mn3+ d5
Crt Mn2+
Fe3+ d6 d'
Mnt
Fe2' Co3+ Fe' Co2+
0.1970 0.1342 0.2168 0.2480 0.1498 0.1756 0.1934 0.1162 0.1340
0.1460 0.0976 0.1729 0.1972 0.1128 0.1311 0.1441 0.0775 0.0890
0.1149 0.0806 0.1086 0.1361 0.0663 0.0873 0.1076 0.0806 0.0952
hardness for ground-state system and it depends, as pf, on the kind of process involved during derivation. The energy difference of eq 34 represents the pairing energy in concordance to eq 33, if the expansion is taken around the high-spin system and if 6, is equal to -1. In this case the relevant quantities are p; and q;, the derivatives in the direction of lower multiplicities. In Table IV are compared, for a sample of metal cations, the pairing energies calculated through eq 34, to first and second order, with those obtained by using Racah's parameters for coulomb and exchange integrals.I6 p; is obtained through eq 29 and q; by one-side numerical derivation of p; on a five-point grid with 0.01-au step. To first order in N,, the tendency within each configuration (at constant number of electrons) is preserved, indicating that p; is a qualitative measure of the tendency of metal cations to give high- or low-spin complexes. On the other hand, the second-order correction produces a better numerical concordance and gives an alternative to estimate pairing energies.
Conclusions Spin potential has a well-defined place in the mathematical structure of density functional theory. However, its physical
meaning is far from been completely established. In this work we showed some trends of I.LS along the periodic table. In particular p, may be classified as a periodic property in the same sense as chemical potential or ionization potential have that characteristic. The relation of p, with processes in which the system changes multiplicity without moving its number of electrons is established by estimating pairing energies of metal cations. This strongly supports the idea that p, is a measure of the tendency of a system to change its multiplicity. Within the context of Kohn-Sham model in the local density approximation for the exchange and correlation potential, it was found that the curve of the energy against the spin number shows an absolute minimum for ground-state multiplicity, and it has a discontinuous first derivative at that point. It is possible to say, in relation to the chemical potential obtained in a derivation trajectory at constant multiplicity, that it preserves some of the tendencies of the chemical potential calculated without that restriction.
Acknowledgment. We thank Albert0 Vela for a critical reading of the manuscript and to JOSELuis GBzquez, Andrts Cedillo, and Juvencio Robles for valuable discussions. We also acknowledge the support of the DirecciBn Adjunta de Desarrollo Cientifico of CONACYT. R.V. acknowledges CONACYT for a scholarship. Registry No. Li, 7439-93-2; Be, 7440-41-7; B, 7440-42-8; C, 744044-0; N, 17778-88-0; 0, 17778-80-2; F, 14762-94-8; Ne, 7440-01-9; Na, 7440-23-5; Mg, 7439-95-4; Al, 7429-90-5; Si, 7440-21-3; P, 7723-14-0; S,7704-34-9; CI, 22537-15-1; Ar, 7440-37-1; K, 7440-09-7; Ca, 744070-2; Sc, 7440-20-2; Ti, 7440-32-6; V, 7440-62-2; Cr, 7440-47-3; Mn, 7439-96-5; Fe, 7439-89-6; Co, 7440-48-4; Ni, 7440-02-0; Cu, 7440-50-8; Zn, 7440-66-6; Ga, 7440-55-3; Ge, 7440-56-4; As, 7440-38-2; Se, 7782-49-2; Br, 10097-32-2; Kr, 7439-90-9; Rb, 7440-17-7; Sr, 7440-24-6; Y, 7440-65-5; Zr, 7440-67-7; Nb, 7440-03-1; Mo, 7439-98-7; Tc, 7440-26-8; Ru, 7440-18-8; Rh, 7440-16-6; Pd, 7440-05-3; Ag, 7440-22-4; Cd, 7440-43-9; In, 7440-74-6; Sn, 7440-31-5; Sb, 7440-36-0; Te, 13494-80-9; I, 14362-44-8; Xe, 7440-63-3.
Theoretical Study of the Transitlon Probabilities of the Doubly-Excited States E'Z; of C2 and 222i of C2+ Pablo J. Bruna and James S. Wright* Ottawa- Carleton Chemistry Institute, Carleton University, Ottawa, Ontario, Canada K1 S 5B6 (Received: June 20, 1991)
The radiative properties of the doubly-excited states EIZB+(ui-u:) of C2and 22Z:(u$-a:) of C2+are investigatedtheoretically by using MRD-CI wave functions. Due to a heavy mixing between bonding and antibonding configurations, the E state (with a maximum contribution of 4 u: of 45%) does not show the typical features of strongly bound states (short Re and high we). Because of the mixing, the experimental Freymark system E'Z: - AlII, of C2 exhibits a moderate intensity of though. At R = 2.43 bohr (an intermediate geometry between both minima), a calculated transition moment CReterT2 0.151 au2disagrees with an experimental estimate of 2.26 f 1.13 au2 from Cooper and Nicholls. The same authors assigned of 0.13 f 0.05 au2,whereas this work and earlier data from the literature to the Mulliken band DIZ: - X'Z' of C2a CRe'e't2 indicate a higher value of 0.41 au2for that quantity. A total radiative lifetime f 0 i= 15 ns for the EIZ+state is mainly dictated state of C2' by the radiative process E A, the competitive decay E D being about 10 times slower. The 2%:(4-a:) is strongly bound, with a shortening A& = 0.35 bohr and an increase Aue = 800 cm-I (60%) relative to the X42;ground state. A lifetime s,(~~Z:) i= 28 ns results from radiative decay into both 122: and 2%" states. On the basis of the present spectroscopic data for Czt, an absorption band observed at 2.77 eV (assumed as belonging to C,) is tentatively reassigned 1%: transition of C2+, with a calculated AE, = 2.81 eV. to the 222:
-
-
-
-
1. Introduction In a series of ab initio studies on B2 compounds, the existence of stable, strongly-bound excited states arising from configurations having the antibonding 0, MO empty' has &n demonstrated.2-5 ( I ) Wright, J. S.; Bruna, (2) Bruna, P. J.; Wright, (3) Bruna, P. J.; Wright, (4) Bruna, P. J.; Wright,
P. J. Chem. Phys. Lett. 1989, 156, 533. J. S.J . Chem. Phys. 1989, 91, 1126. J. S.J . Phys. Chem. 1990, 94, 1774. J. S.J . Chem. Phys. 1990, 93, 2617.
0022-3654/92/2096-1630%03.00/0
At equilibrium, the electronic structure of such states corresponds to u:uirT, with n 4- m equal to 4, 3, and 2 for Bz, B2+,and B22+, respectively. Relative to the ground configuration, strongly-bound states are generated by double excitations u2 M02, with M 0 2 standing for any combination ui, ugau,or A,s between the species
-
( 5 ) Bruna, P. J.; Wright, J. S. J . Mol. Struct. (THEOCHEM)1990, 210,
243.
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1631
EIZZ and 22Z: States of C2 and C2+
2. TechnicaIDetails ug (weakly bonding) and K, (strongly bonding). These types of states will be designated as doubly-excited states (DES). 2.1. Basis Sets and CI Treatment. The A 0 basis set for C2+ Maximal occupation of bonding MO's in DES implies an inis based on Dunning's 5s3p18 contraction of Huzinaga's 10s6p crease in the bond strength relative to the ground state, as meaprimitive set19 for carbon atom. This set has been expanded by sured by specific spectroscopic parameters. In B2, the semidiffuse s (as= 0.04) and p ( a p= 0.034) and two d-polari3I23u3r27r3 DES, for instance, shows a shorter equilibrium bond zation functions (ad= 1.2 and 0.3, uncontracted). The final basis distance kom the X?Z;(u;ut~$ ground state (by 0.35 bohr, refs set for C2+,of triple-r and double-polarization quality, contains 2 and 5). Moreover, the net gain in bonding is manifested by an 60 contracted functions. On the other hand, the C2 molecule was increase in the vibrational frequency Awe of 712 cm-l relative to studied by using instead a 5s4p contraction,'* plus the same adwe(X32,) = 1058 cm-I. ditional functions as above (66 contracted AO's in total). Despite their short Re and high we, DES do not necessarily The CI computations were carried out with the MRD-CI exhibit large (adiabatic) dissociation energies (De). In a diabatic package of programs,20 a multireference single- and double-CI picture, DES correlate with dissociation asymptotes in which both procedure including selection and extrapolation techniques.21 The atoms are s p excited, a channel generally lying not only several contribution into the total energy of higher excitations not explicitly electronvolts higher but also above a number of more stable included in the CI treatment was estimated according to a genproducts. Hence, DES potential curves are crossed at larger eralized Langhoff-Davidson correction;22 the total energies redistances by other states of the same symmetry but of a less ported here correspond to the estimated full-CI. The DZhsubgroup bonding or even repulsive character.' Such avoided crossings of Dmhhas been used in the calculations. account for the apparent anomaly between short Re, high we versus At the CI level, all C states were studied with the S C F M O s small De commonly associated with the doubly-excited states under derived for b3Z;[u2cr~u'~t],whereas those of C2+were treated with the X4ZJu:u~ug~u] ? MO's. The lowest-lying lug and la, discussion. Another family worth investigating comprises the species C2, MO's remain doubly-occupied throughout; the corresponding C2+, and C:+.6 Since ground-state C2 corresponds to complementary higher-lying species were discarded. In C2, the X I Z ~ ( u ~ uthe ~ only ~ ~ )possible , strongly-bound DES arises from selection threshold T was fixed at 2.5 phartree for the A'II, and the excitation ut u;(lZ+). Already in 1932, Mulliken' postuD'Z: states; however, a somewhat higher Tvalue of 3.5 phartree lated the existence of 3'Z?(u2+u2). Freymark8 first identified was used for the '2; symmetry in order to keep a manageable the experimental EIZ: state as 3 BZ,: an assumption later corsecular equation size (ca. 14000) after simultaneous selection of roborated by Barsuhn's ab initio computation^.^ Recently, Zhang configurations with respect to the X, B', and E states.23 In C2+, and HansonlO reported for the E state a strongly-mixed character. the selection of configurations was carried out with T = 2.50 For C2+,with an X4Z-(u2u2ur 2 )ground state,I1-l3two DES phartree. The largest calculation (C2, '2') used 57 reference can be. generated: 'Z:(&r$ inud 211u(u~+ug~U). An ab inito configurations, which had been determine4 by making test calwork12 investigated the 222,+potential curves, which shows a deep culations a t four C-C distances. The percentage contributions local minimum at short R; its electronic structure near equilibrium, of the leading cofligurations to each state are shown as a function however, was not indicated. With respect to the 211ustates, only of geometry in Figure 2a-d. 1211,(u +K,) and 2211u(~u-ug)have been studied theoretical2.2. Radiative Data. The transition moment Reren= ly.11JzJ5 To the best of our knowledge, nothing is known about PI*$) was computed by taking P = Cer (length form of the the stability of the strongly-bound ut u,K,(~II,) structure.lS electric-dipole transition moment operator). and/or *$ Assuming that for a given system the existence of DES has been correspond to the DZhcomponents a and b in the case of spatially established, the additional question arises about their possible degenerate electronic states. Double and single primes label the experimental detection by means of one-photon ~ p e c t r o s c o p y . ~ ~ ~lower and upper electronic states, respectively, of a given band Besides restrictions imposed by symmetry and spin selection rules,I5 system (and their vibrational levels as well). another difficulty lies in the unfavorable Franck-Condon factors The total transition moment squared (~ R l e t t 2corresponds ) to (FCFs) for electronic transitions involving significant changes ARC K the parameter K taking into account all suitable comand Awe. In this connection, singly-excited states (SES) of the binations (a,bJamong components of type and *$.24 For the type u, M O are relevant since they should exhibit analogous selected C2states studied in this work, K = 1 (IZi-IZ; transitions) tendencies for Re and o,.~,~Thus, it would be of interest to gain or K = 2 (I2Z-'IIutransitions). For the doublet-manifold of C2+, some insight into electronic transitions between SES (a, ) and the K parameter is twice the above values, namely 2 (2Z:-2Z:) DES (ut ). and 4 (2Z:-zII,). The goals of this ab initio work are (a) characterization of The band strength So,d,,defined as doubly-excited states of C2 and C2+ by providing their potential curves, stabilities, and changes in electronic structure with bond ,s, 4 S ~ d ( R ) R e ~ e ' , ( R ) ~ , ( Rd) ~ 1 2 distance; and (b) determination of the radiative data (dipole transition moment Rlrl Fillator strength f&, Einstein emission is derived by using a polynomial fit to the discrete data Retet,(R) coefficients Ado",and lifetime 7",)for those dipole-allowed elecand the vibrational wave function &(R)'s generated for each electronic ~ t a t e . The ~ ~ .absorption ~~ oscillator strength fdojtcortronic transitions having a DES as upper state. responds to A study on the low-lying states of C?+, a dication isoelectronic with B2, will be published e1~ewhere.l~ f&, = Y3K'A.t!?dpSdot,
-
-
-
-
*:,
-
+
(6) Bruna, P. J.; Wright, J. S. J . Mol. Sfrucf.(THEOCHEM) 1991, 230, 213. (7) Mulliken, R. S. Reu. Mod. Phys. 1932, 4, 1. 19) Freymark, H. Ann. Phys. 1951, 8, 221. (9) Barsuhn, J. Z . Nufurforsch. 1972, A27, 1031. (10) Zhang, Y.; Hanson, D. M. J. Chem. Phys. 1987,86, 347,666; Chem. Phys. 1989, 138, 71. (11) Petrongolo, C.; Bruna, P. J.; Peyerimhoff, S. D.; Buenker, R. J. J. Chem. Phys. 1981, 74, 4594. (12) Rosmus, P.; Werner, H.-J.; Reinsch, E.-A,; Larsson, M. J. Electron Specfrosc. 1986, 41, 289. (13) Meier, J. P.; Rosslein, M. J . Chem. Phys. 1988, 88, 4614, and references therein. (14) Kraemer, W. P.; Roos, J. 0. Chem. Phys. 1987, 118, 345. (15) Weltner, W.; Van Zee, R. J. Chem. Rev. 1989, 89, 1713. (16) Herzberg, G. Molecular Spectra and Molecular Structure. I . Specfra of Diafomic Molecules; van Nostrand-Reinhold: New York, 1950.
(17) Bruna, P. J.; Wright, J. S . To be published. (18) Dunning, T. H. J . Chem. Phys. 1971, 55, 716. (19) Huzinaga, S. J . Chem. Phys. 1965, 42, 1293. ( 2 0 ) Buenker, R. J. In Proceedings of the Workshop on Quantum Chemisfry and Molecular Physics; Burton, P., Ed.; Woolongong University Press, Australia, 1980) p 151. Buenker, R. J. In Studies in Physical and Theorefical Chemistry; Vol. 21 Carbb, R., Ed.; Elsevier: Amsterdam, 1982; p 17. Buenker, R. J.; Phillips, R. A. J. Mol. Sfrucf.(THEOCHEM) 1985, 123, 291. (21) Buenker, R. J.; Peyerimhoff, S . D. Theor. Chim. Acfa 1974,35,33; 1975, 39, 217. (22) Buenker, R. J.; Peyerimhoff, S. D.; Bruna, P . J. In Compufational Organic Chemistry; Csizmadia, I. G . , Daudel, R.,Eds.; Reidel: Dordrecht, 1981; p 55. Bruna, P. J.; Peyerimhoff, S. D. Adu. Chem. Phys. 1987, 67, 1. (23) Bruna, P. J.; Wright, J. S . Chem. Phys. 1991, 157, 111. (24) Chabalowski, C. F.; Peyerimhoff, S. D.; Buenker, R. J. Chem. Phys. 1983, 81, 57.
1632 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992
Bruna and Wright
TABLE I: Spectroscopic Parameters of Selected Singlet States of C p
state
T. 0.0
X'Z: (1lZ:)
O.Oc
A'II, (I'II,)
D'Z: (1'Z:)
E'Z; (3'2:)
0.0 1.04 1.04 1.14 5.36 5.47 6.23 6.82 6.83 7.18
R.
Be
a,
we
WJ.
2.347 2.355 2.381 2.491 2.496 2.532 2.340 2.351 2.381 2.368 2.370 2.457
1.820 1.81 1.76 1.617 1.61 1.55 1.833 1.815 1.77 1.790 1.785 1.65
0.018 0.019 0.018 0.017 0.018 0.016 0.020 0.018 0.019 0.039 0.049 0.047
1855.0 1835.0 1788.6 1608.2 1615.0 1507.5 1829.6 1843.0 1757.8 1671.5 1638.0 1287.2
13.6 15.5 13.6 12.1 10.6 5.0 13.9 14.1 13.6 40.0 46.0 32.8
\ ',
\
T, in eV, R, in bohr, other parameters in cm-'. First, second, and third entries: refs 31 and 34 (exptl), this work and ref 37 (ab initio), respectively. eThe total energy is -75.7675 hartrees (FCI estimate). (1
with AEddt given in atomic units throughout. In C2, absorption from XiZ+ (or DiZ:) and A'n, is described by K' = 1 and respectivefy. In C2+,absorbing 2Z and 211 states have K' = and respectively. The Einstein coefficient Add,(in s-l) for spontaneous emission is given by A,,~,, = 2.14 19 x ~ ~ ~ o K ~ ~ A E ~ ~ ~ , ~ s ~ , ~ ~ 1.75~
In C2, K" = 1 for emission from 'Z states; in C2+,K" = and for emission from 22and *IIstates, respectively. The natural fluorescence lifetime T,, (in seconds) involves the term Eu,,A,,,ul, = T,,-' over all (dipole-allowed) transitions into lower-lying electronic states. In those cases in which a given electronic state a may decay into the lower b and c states, the partial lifetimes will be written as T ' ~E T:! and Pd T:F, respectively. The total lifetime of the upper state is given by T ~ , -=~ r',.-l + T ' ~ ~ , - ' . When available, experimental energies differences AEd,,, will be used throughout to minimize errors in the computed radiative data. 3. Selected States of Cz Molecule 3.1. Potential Curves and Electronic Structure. The electronic states studied belong to the symmetries lF;(X,B', and E), 'lI,(A), and '2:(D), with the experimental labeling given in parentheses. The corresponding spectroscopic parameters and potential curves are displayed in Table I and Figure 1, respectively. The present data reproduce the experimental results reasonably well, with mean deviations of 0.01 bohr for Re and 30 cm-I for we (Table I). Interestingly, the states X ' Z ~ ( u i u t n ~ ) , DIE:( uiuuugr:), and EIZ+(U ~ C T ~ Khave :) similar equilibrium geometries (2.36 f 0.01 boar), afthough the bonding features of the corresponding leading configurations would suggest a progressive decrease in Re from X to E through D; deviations from the ideal situation are caused by relatively large contributions of antibonding configurations. In fact, as a result of a heavy bondingantibonding mixing (see below), the E'Z: potential curve is characterized by smaller we and higher w,xe values than that of the X and D states. The largest equilibrium bond distance r ~line ) , with its smaller K, popucorresponds to A l ~ , ( u ~ u ~ u , in lation; in the range of geometries investigated, this state is essentially described by the above c o n f i g u r a t i ~ n . ~For ~ . ~DiZ:, ~ the ~K, important as the bond secondary configuration u ~ u $ ~ becomes distance increases.24 For the '2: symmetry, the following configurations are of interest: -
--
X:
u;u:7r'Y
B':
uiutui?rt
(at
E: M: T: Z:
u;u:7r:
(Ut
uiut~,$:
(d
ugu:uga4y
(ug
"2g"uQgd~g
(a,*,
["g
--- 74 T"l
0;)
[*"
Ugl
0;)
[Ut
UgTTUl
---
2.25
2.75
3.25
3 75
R (bohr)
Figure 1. Potential energy curves for selected singlet states of C1.
and A (second entry) states. Figure 2a-c shows the changes i;i the relative weights (c2)of these configurations as a function of distance for the first three IZ; states. Near equilibrium, the X and B' states arise from the corresponding configuration having the same label, in each case with c2 = 70-80%. Close to 3.0 bohr, both states are strongly mixed (Figure 2a,b); for R I 3.0 bohr, they reverse their structure (the X state arises from configuration B' while B' acquires X chara~ter).~~.~~ The doubly-excited configuration ut ui(E) makes a relatively high contribution (-20%) into the X and B' wave functions, particularly a t shorter bond distances. The significance of this double excitation for obtaining an accurate description of the C2 ground state has earlier been stressed in the l i t e r a t ~ r e . ~This ~,~~ double excitation is also the second most important contributing i ) function of B2,2J,27 to the ground-state ( X 3 Z ; , u ~ u i ~wave The composition of EIZ%+ is not simplei0 since this state is described by several configurations (Figure 2c). Near 2.0 bohr and below, it has a predominantly B' character (40-45%), followed by the E structure with 20% contribution. Around equilibrium ( R = 2.40 bohr), the E state is mainly generated by the E configuration (45%, the maximum weight attained by the doublyexcited structure), and with about 15% participation each of X, B', and Z. For larger R, the importance of configuration E diminishes rapidly, whereas that of U,K, u,?r,(Z) increase noticeably, reaches a maximum of about 30% near 2.90 bohr and decreases sharply afterwards. At larger bond distances, EiZ; has a K: T * ( M ) character. From ah configurations making up the E'Z: state, only X and E favor short bond distances, whereas the less bonding structures B', Z , and M act in the opposite direction. Due to such bonding-antibonding mixing, the E potential curve does not exhibit the properties expected for a strongly-bound, doubly-excited state (shorter Re and higher w e ) . An indirect proof of the antibonding mixture into the CI wave function of the E state is provided by the S C F potential curves of both X and E configurations, as shown in Figure 3. For the ground state, the SCF equilibrium data Re = 2.34 bohr and w e
-
-
-
[up"
ug) "gTJ
[""
-
T"l Tgl
Excitations given in parentheses are relative to the X (first entry)
( 2 5 ) Fraga, S.;Ransil, B. J. J . Chem. Phys. 1962, 36, 1127. (26) Lepetit, M . B.; Malrieu, J . P. Chem. Phys. Lett. 1990, 169, 2 8 5 . (27) Knight, L. B.; Gregory, B. W.; Cobranchi, S. T.; Feller, D.; Davidson, E. R. J . Am. Chem. SOC.1987, 109, 109.
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1633
E'Z: and 2221States of C2 and C2+
2c
20
I
E
I
22
1.8
\
3.0
3.4
R &r)
2d
2b
\
3.0
2.2
1.8
3.4
1.8
3.0
2.2
R &)
3.4
R (i%r)
Figure 2. Relative weight (c2) of relevant electronic configurations contributing to the '2' states X (2a), B' (2b), E (2c) of C2 as well to 2%: (2d) of C2+. The configurations are X(u:ut*i), B'(u:uiU:~t), E(a:o:n:), M(u:uiat?r:), T ( u , u ~ u , Tand ~ ) , Z ( a ~ a , o , . ~ ~The ~ ~configurations ). b', e, t and z of C2+are derived from B', E, T, and Z by removing an electron of the second us MO.
= 1910 cm-' are in relatively good agreement with experimental values (cf. Table I). At the monodeterminent level of approximation, the ut ui(E) excitation in fact generates a stronglybound curve, the E (SCF) state having a shorter Re (by about 0.20 bohr) and higher we (by -700 cm-') than the X (SCF) ground state. It is interesting to compare C2 with B2. Doubly-excited states of B21-3*sshowing an increase in bonding are 2311,(ut ugr,) and 3IZ+(ut rt), but not 33X;(ut ui). The latter state shows a aeavy mixture of the bonding ut ui structure with the antibonding u,r, u g r and ug ugconfigurations (the last two being called as Z and T in Cz, see above). As a result of the mixing, the equilibrium geometry of 3% (B2) compares with that of the ground state, though ~ ~ ( 3 ~is2a&ut ; ) 300 cm-' higher than w,(X3ZJ. Summing up, for B2 and C2 the ut ui excitation mainly generate the rhird electronic state of the same symmetry as the corresponding ground state; moreover, in both systems the admixture of antibonding configurations is quite large, a feature explaining why these potential curves do not show the local properties characteristic of a pure ut ui, strongly-bound state. 3.2. Radiative Data. The EIX: state may decay radiatively into DIE: and A'II,. Only the latter transition has been detected
-
-
-
-
--
-
-
experimentally.* Because of discrepancies between t h e o r e t i ~ a l ~ ~ and experimental results2*concerning the transition probabilities of the Mulliken system DIZ:-XIZ: (as well the Freymark band, see below), the D-X transition has been reinvestigated here. The values of the transition moment CRet,.t2(R) are listed in Table I1 and shown in Figure 4. Earlier experience with the theoretical estimation of the A'II, lifetime (Phillips band) under calculations similar to this indicated for R&,,,an error of about 10% (or 20%for radiative parameters). The present results are believed to have similar accuracy. The predicted transition data for C2 are collected in Tables 111-VI. 3.2.1. The EIE+-AIII, (Freymark) Band. The emission WlII, observed by Freymark* appears in the spectrum E'Zi 5.80-eV region. Cooper and Nicholls (CN) provided the corresponding transition probability data.% They reported a high value CRe,e,t2 of 2.26 f 1.13 au2, and an oscillator strengthf&,(E+A) = 0.0464. Based on their tabulated Add,coefficients, the lifetime T ' ~should be about 2.0 ns.
-
(28) Cooper, D. M.; Nicholls, R. W. J . Quant. Spectrosc. Radial. Transfer
1975, IS, 139; Spectrosc. Lett. 1976, 9, 139. Cooper, D. M. Ph.D. Disser-
tation, York University, Toronto, 1974.
1634 The Journal of Physical Chemistry, Vol. 96, No. 4, I992
Bruna and Wright
TABLE II: Calculated Total Transition Moment Square XRe,e~,2 = KR,.efr2(in mu2) for Several Transitions of C, and C2+as a Function of tbe Bond Length R (bohr) I
E
R 1.90 2.00 2.10 2.20 2.348 2.50 2.65 2.80 2.90 3.00 3.20
c2
-
E+D
A" 0.442 16 0.418 95 0.386 32 0.35475 0.320 15 0.286 83 0.262 46 0.234 70 0.21092 0.09095 0.075 57
D
1.930 57 2.31975 2.416 54 2.31280 1.968 31 1.497 28 0.960 57 0.489 92 0.253 75 0.082 66 0.082 66
-
Xb
0.323 94 0.37251 0.405 85 0.421 62 0.423 74 0.396 25 0.347 08 0.268 00 0.191 73 0.11062 0.11062
R
222:
1.95 2.05 2.10 2.20 2.348 2.50 2.65 2.80 2.95 3.10 3.25
-
c2+ 222;
122:
0.704 96 1.055 08 1.240 39 1.346 17 1.260 36 1.036 02 0.7 15 96 0.290 60 0.036 33
0.258 56 0.286 94 0.283 77 0.257 02 0.197 69 0.173 84 0.141 84 0.107 37 0.059 21 0.02097 0.009 36
"Freymark band system8 Near equilibrium, XRere,?= 2.26 f 1.13 au2 (ref 28, exptl). *Mulliken band 0.13 f 0.05 au2 (ref 28, exptl). At R = 2.3 bohr, ref 24 (ab initio) gives a value of 0.42 au2.
-
1,22n, 0.11298 0.062 86 0.040 55 0.014 14 0.002 12 0.000 81 0.000 02 0.001 16 0.015 63 0.024 83 0.01603
Near equilibrium, CRe,,"2=
TABLE III: FranckCondon Factors qddt,Oscillator Strengths fdd,, Einstein Transition Probabilities Add,(in s-l), and Radiative Lifetimes T'", (in ns) Predicted for the Freymark System E %:( v3 - A%,( v'? of C P U"
0 h
e
1
d .
3 8
B
2
3
4 R (bohd
Figure 3. SCF potential curves for the X'Z; and E'Z; states of C2.
5
ns
T'~,,
uf=O
uf= 1
u'= 2
0.635 0.581 0.268 (-1) 0.389 (8) 0.240 0.271 0.106 (-1) 0.144 (8) 0.087 0.102 0.391 (-2) 0.494 (7) 0.027 0.033 0.122 (-2) 0.144 (7) 0.008 0.001 0.369 (-3) 0.403 (6) 0.002 0.003 0.995 (-4) 0.101 (6) 16.6
0.342 0.37 1 0.137 (-1) 0.213 (8) 0.255 0.162 0.104 (-1) 0.152 (8) 0.199 0.222 0.885 (-2) 0.120 (8) 0.124 0.144 0.553 (-2) 0.701 (7) 0.052 0.065 0.235 (-2) 0.278 (7) 0.017 0.024 0.883 (-3) 0.970 (6) 16.7
0.021 0.047 0.726 (-3) 0.120 (7) 0.473 0.482 0.186 (-1) 0.288 (8) 0.151 0.047 0.584 (-2) 0.846 (7) 0.123 0.140 0.570 (-2) 0.773 (7) 0.128 0.143 0.561 (-2) 0.711 (7) 0.059 0.082 0.287 (-2) 0.340 (7) 16.9
4dd1 4dd'c fd,n
A",",,
"On" ( n ) stands for O n " X IO". bAoo= 2.7 (8), io = 2.3 ns andfoo = 0.464 (-3) according to ref 28 (exptl). 'Reference 28.
-
1.75
2.25
2.75
3.25
R (Mr)
Figure 4. Variation of the total transition moment XRere,,2(in au) with respect to the nuclear distance for selected transitions of C2.
The experimental radiative results imply that the Freymark system is of high intensity, a finding difficult to correlate with the main electronic structures of the upper and lower states. In detail, the E A transition primarily results from the double-
-
-
excitation uga, ut, which in a monodeterminant approximation leads to a strictly vanishing (electric dipole) transition moment.29 Obviously, this band system borrows its intensity from those single excitationr connecting secondary configurations of the E state with the leading configuration of the A state, namely ug a,(X), a, ug(B'), ug a,(T), and u, a,(Z) (cf. section 3.1). The data plotted in Figure 4 corroborate theoretical expectations, as shown by a relatively small (and almost constant) value r2 1.90 and 3.20 bohr. At R = 2.43 bohr, a for ~ R e t e lbetween geometry intermediate between the respective minima, the total transition moment squared is 0.151 au2, well below an experimental estimate of 2.26 k 1.13 au2.28 The latter value clearly cannot be related to the Freymark system. The present study assigns to the E-A system smaller&, and A,, values and a longer lifetime T~ than experiments (Table 111). In detail, a predicted oscillator strength& = 0.0268 amounts to about 60% of CN's estimate (0.0464), while Am = 0.389 X lo8 lies well below its experimental counterpart (2.70 X lo8). On the (computed by asother hand, the parameter T',, = (CU,,Ao,d8)-'
-
-
(29) Jaffi, H. H.; Orchin, M. In Theory and Applications of Ultrauiolet Spectroscopy; Wiley: New York, 1962.
The Journal of Physical Chemistry, Vola96, No. 4, 1992 1635
EIZ: and 222; States of C2 and Cz+
TABLE I V FranckCondon Factors qddz Oscillator Strengths f d d j , Einstein Transition Probabilities Add,(in d),and Radiative Lifetimes T'~ (in ns) Predicted for the System E 'Z:( v') - D'Z:( v") of Ct' u" "u 0 uf= 1 u"2 ur= 3 0
0.916 0.616 (-1) 0.563 (7) 0.076 0.264 (-2) 0.173 (6) 0.007 0.121 (-3) 0.534 (4) 0.001 0.116 (-4) 0.317 ( 3 )
1 2 3
4 0.284 (-6) 0.413 (1) 172.0
T"~, ns
0.080 0.848 0.100 0.688 0.425 0.375 0.189 0.631 0.398 0.034 0.589 0.250 0.007 0.780 0.209 193.1
(-2) (7)
(-1) (7) (-2) (6) (-3) (5) (-4) (4)
0.003 0.572 0.837 0.209 0.193 0.218 0.370 0.207 0.175 0.282 0.880 0.529 0.097 0.154 0.622 216.9
(-3) (5) (-1) (7) (-1) (7)
(-2) (6) (-2) (5)
4dd.
0.296 (-4) 0.522 (4) 0.025 0.330 (-2) 0.460 (6) 0.329 0.263 (-1) 0.282 (7) 0.087 0.449 (-2) 0.358 (6) 0.265 0.756 (-2) 0.428 (6) 238.0
fdd'
Add,
"See footnote a. Table 111. suming the label v" runs over the A'& state only; see section 3.2.3) lies close to 17 ns, about eight times larger than a mean lifetime of 2 ns inferred from CN's results. A lifetime close to 17 ns indicates the process E A occurs X'Z; ( 7 0 at a faster rate than that of the Phillips band A'n, = 14 ~ s ) . ~ ~ 3.2.2. The EIZ:-DIZ: Band. E'Z; can also decay into D'Z:, both states showing similar equilibrium bond distances (Table I) and energetically separated by hEoo= 1.45 eV (compared with 5.79 eV for the Freymark band). The radiative data is summarized in Table IV. Close to equilibrium, the E D transition is mainly related to the single excitation u uurwhich in general leads to sizeable transition moments?O d e a r R = 2.10 bohr, the function &&R) has a maximum of about 2.42 au2 (Figure 4). For this band, a fooof 0.0616 is predicted (Table IV). The partial lifetime T'~(E+D) increases from 172 ns ( u t = 0) to 238 ns ( u t = 3); this strong dependency of 7'u, with the vibrational level results from the large change of the transition moment with bond length (Table I1 and Figure 4). There are no other data in the literature to compare with. The intensity of the E-D system is not as high as for the E-A band, despite the larger transition moment assigned to the former (2.42 vs 0.15 au2). The reason for such behavior lies in the great difference between the transition energies. Looking for the relative Einstein coefficient A g A / A g D the , following expression holds:
--
--
(A&*/ A&') =
( M E A /aEED)3(S&A/S&D)
= (63.3) X (0.109) = 6.90
This result indicates the overwhelming influence of the transition energies upon the quotient between the Amcoefficients, the former being large enough to overcome a relative value for S, of only 0.109. As a matter of fact, the partial lifetime T',(E+A) is about ten times shorter than r",,(E+D). 3.2.3. Total Lifetime 7JE'Z:). As stated above, a given vibrational level u'(E) can decay radiatively into D'Z: or A'n,. Hence, both lower states have to be taken into account in the lifetime computation. The net lifetimes 7u'(E)predicted for a few vibrational levels are displayed in Table V. Analysis of the results indicates that the lifetime of the E state is dictated by the E A radiative process. Due to a significant difference in the values of the corresponding Einstein coefficients Aut,,(of about one order of magnitude), inclusion of the E D system into the total sum Cd,Au,,,,leads to a final 7,4E) being only 2 ns shorter than the partial lifetime T',,,,(E+A) (Tables I11 and V). For low-lying vibrational levels, the lifetime of the E state is of 15-16 ns.
-
(30) Bruna, P. J.; Wright, J. S.J . Phys. B 1990, 23, 2197s.
-
TABLE V Total Einstein Coefficient Add,(in 9-l) and Net Radiative Lifetime T~ of the E'Z: State of C2" Z"UAd"V Zd'Add? total E'Z; (DIZ:)b (A'IIJc &Addf i d ns ,
u'= 0 u'= 1 u'= 2
0.0581 (8) 0.0518 (8) 0.0461 (8)
0.6021 (8) 0.5925 (8) 0.5882 (8)
0.6602 (8) 0.6443 (8) 0.6343 (8)
15.1 15.5 15.8
'See footnote a, Table 111. bData from Table IV. CDatafrom Table 111.
3.3. The DIZ:-XIZ: (Mulliken) Band. The Mulliken system has been detected both in emission and ab~orption.~'Two experimental studies reported lifetimes 7d(D) of 14.6 f 1.5 ns32and 18.1 f 1.0 ns,33 whereas a third one (by CN) indicated a significantly longer i dof 47 n ~ . ~ ~ * ~ ~ The radiative properties studied theoretically by Chabalowski et confirmed the low range of lifetimes (=14 ns); a predicted fm = 0.0544 was also in excellent agreement with a value of 0.055 f 0.006 deduced by Smith,32but in disagreement withfm = 0.0171 from Cooper and Nicholls.28 Because of these discrepancies among earlier studies, we decided to reanalyze the Mulliken band. The results are collected in Table VI. In general terms, the prior a b initio radiative data are reproduced within 5%, the recalculated lifetimes being slightly longer by about 1 ns. In line with the results from Chabalowski et al.,24 at R = 2.40 bohr the quantity CR6$ is of about 0.41 au2 (Table VI and Figure 4), which has to be compared with 0.13 f 0.05 au2reported by CN.28 Since near equilibrium the upper and lower states are connected by the uu ugexcitation, which in general produces sizable transition moments,30 the experimental results from C N are definitely in error. As a matter of fact, Cooper and Nicholls assigned to the Mulliken and Freymark bands exactly the opposite radiative properties from that predicted by theory (as well as from separate experiments for the Mulliken system). It appears that the measurement data of these two transitions have inadvertently been mixed up by CN. This is possible because, although the structure of each band is different, their transition energies are comparable: 5.36 eV (D-X) vs 5.78 eV (E-A) (Table I). Another radiative depopulation process from DIE: may involve the lower states C 1 n , ( u 2 u u u ~ r ~at) T, = 4.25 eV and B ' l Z Z ( u ~ u ~ u at ~ ~T, t ) = 1.81 eV.23931334 The transition D C, occurring in the Moo = 9000 cm-' infrared region, may have a character near sizable transition moment because of its ug rU
-
-
-
(31) Huber, K.; Herzberg, G. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules; Van Nostrand-Reinhold: New York, 1979. (32) Smith, W. H. Astrophys. J . 1969, 156, 791. (33) Curtiss, L.; Engman, B.; Erman, P. Phys. Scr. 1976, 13, 270. (34) Douay, M.; Nietmann, R.; Bernath, P. F. J . Mol. Spectrosc. 1988, 131, 250, 261.
1636 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992
Bruna and Wright
-
TABLE VI: FranckCondon Factors qdv,,,Oscillator Strengths fdd,, Einstein Transition Probabilities Adurn(in s-'), and Radiative Lifetimes T~ (in ns) Predicted for the Mulliken System DIZ:(v') X1Z;(v'') of Czec u'I u'= 0 v'= 1 v'= 2 "u 3 v'= 4 0.995 0.997 (0) 0.546 (-1) 0.680 (8) 0.005 0.251 (-2) 0.353 (-3) 0.404 (6)
0
1
0.162 (-3) 0.601 (-5) 0.629 (4)
0.005 0.259 (-2) 0.224 (-3) 0.303 (6) 0.985 0.993 (0) 0.528 (-1) 0.658 (8) 0.010 0.394 (-2) 0.678 (-3) 0.775 (6)
0.364 (-5) 0.1 15 (-7) 0.1 10 (2)
0.421 (-3) 0.132 (-4) 0.139 (5)
0.176 (-7) 0.110 (-7) 0.952 ( I ) 14.6 14.0
0.118 (-4) 0.115 (-7) 0.1 IO (2) 15.0 14.4
2
9Jd'
0.914 (-4) 0.1 14 (-4) 0.166 (5) 0.010 0.418 (-2) 0.395 (-3) 0.533 (6) 0.975 0.990 (0) 0.510 (-1) 0.634 (8) 0.014 0.451 (-2) 0.987 (-3) 0.113 (7) 0.001 0.728 (-3) 0.241 (-4) 0.253 (5) 15.4 14.8
3
4
ns
0.375 (-7) 0.147 (-7) 0.232 (2)
0.107 (-6) 0.153 (-7) 0.257 (2)
0.265 (-3) 0.312 (-4) 0.455 (5) 0.015 0.492 (-2) 0.521 (-3) 0.702 (6) 0.967 0.989 (0) 0.493 (-1) 0.613 (8) 0.017 0.443 (-2) 0.13 1 (-2) 0.150 (7) 15.7 15.2
0.3 12 (-6) 0.147 (-7) 0.230 (2)
9ddrd
fdd'
Ad",,
0.509 (-3) 0.142 (-7) 0.206 (2) 0.019 0.496 (-2) 0.613 (-3) 0.824 (6) 0.958 0.989 (0) 0.475 (-1) 0.589 (8) 16.7 15.7
= 'See footnote a, Table 111. b A d = (0.685 f 0.07) X lo*, rd = 14.6 f 1.5 ns, and& = 0.055 f 0.006 according to Smith [ref 32, exptl]. 0.213 X lo7, i o= 46.9 ns, and& = 0.0171 according to Cooper and Nicholls [ref 28, exptl]. dReference 28. 'First line: this work. Second line: ab initio data from ref 24. fCompare with an averaged rW3= 18.1 f 1 ns from ref 33 (exptl). TABLE VII: Spectroscopic Parameters of Selected States of C2+ statec
ref
x4z,
12n,'
11 12 14 13d 11 12 14
1%; 11 12 2*n, 11 122:f 11 12 2*2;/ 12 32nj
T, 0.00 0.00 0.00 0.00 0.00 0.61 0.72 0.79 0.70 1.67 1.74 1.76 1.90 1.80 3.62 3.76 3.64 6.43 6.59 7.72
Re 2.665 2.668 2.666 2.670 2.651 2.470 2.500 2.462 2.474 2.772 2.759 2.750 2.835 2.859 2.319 2.319 2.294 2.317 2.313 1.351
Be 1.410
a, 0.016
1.410 1.407 1.427 1.646
0.017 0.017 0.0176 0.015
1.654 1.641 1.306
0.019 0.023 0.016
1.327 1.249
0.020 -0.001
1.867
0.009
1.906 1.869 1.874 1.539
0.021 0.015 0.030
we
1348 1360 1335 1330 1351 1615 1630 1624 1590 1075 1090 1103 1360 1360 1875 2330 1977 2203 2141 1245
\
w;se
13 12 14 12.1 22
I
1 3 2x;
14 9.4 15 11 7 27 16 31 28
7,in eV, Re in bohr, other parameters in cm-'. *First entry: this work. E(X42J: -75.3375 hartrees (FCI est). c4211, and 322: lie near 8.6 and 10.2 eV, respectively. dExperimental results. ' R e = 2.459 bohr according to ref 35 (exptl). /Doubly-excited states.
equilibri~m.~'We are unaware of any theoretical or experimental study of this transition. On the other hand, the system D B' appears in the visible region (A&,, = 28 000 cm-I). Both upper and lower states are u,ug. Accordingly, the related by the double-excitation A: square of the transition moment is rather small (Figure 4). The significant variation in bonding strength caused by that double excitation ( M e= 0.26 bohr and Awe = 400 cm-l) should be reflected in a broad Franck-Condon envelope. Summarizing, the two radiative processes D C and D B' might take place on a longer time scale (of the order of microseconds) than r;(D--X), An indirect proof of such an assumption is provided by the almost perfect agreement between experimental and theoretical lifetimes (Table VI), the latter being computed by summing over the X vibrational levels only. During his measurements of the lifetime of the Mulliken bands, Smith32detected a small amount of radiative cascading, with a lifetime close to 60 ns. He suspected the transition EIZZ D'Z: as a carrier of the cascading process. However, our study indicates
-
-
- -
1.75
2.25
2.75
335
1.75
R (bohr)
Figure 5. Potential energy curves of C2+of 2, and 2, symmetry.
that the lifetime of the E (section 3.2.2).
-
D band is at least 3 times longer
4. Selected States of C2+ The spectroscopic parameters are collected in Table VII, together with prior results from the literature. The potential curves are displayed in Figures 5 and 6 . The ground state corresponds to X4Z;(u:a~ag~~), which is generated from neutral X'Z; by the excitation ?ri 0,ug.6*ll Loss of two bonding A, electrons upon excitation is accompanied by an increase AR, = 0.304 bohr and a lowering Awe of about 500 cm-' (Tables I and VII). The transition B4Z;-X4Z; has recently been detected experimentally, with a reported T, = 2.44 eVI3 in good agreement with a prior MRD-CI estimate of 2.47 eV."
-
EIZ: and 222: States of C2 and C2+
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1637 TABLE VIII: Franck-Condon Factors qu,d,,Oscillator Strengths fdd,, Einstein Transition Probabilities Add, (in d),and Radiative Lifetimes T',, (in ns) Predicted for the System 22&!(v9 - l2Z;(v") of C2+.sb 0" u" 0 v'= 1 u" 2 u'= 3 0 0.992 0.003 0.005 4dd' 0.437(-1) 0.530 (-4) 0.107 (-3) 0.150 (-4) f d d , 0.151 (8) 0.219 (5) 0.517 (5) 0.839 (4) 1 0.003 0.981 0.001 0.013 0.509(-4) 0.415 (-1) 0.153 (-4) 0.272 (-3) 0.644 (4) 0.149 (5) 0.147 (8) 0.134 (6) 2 0.005 0.001 0.966 0.001 0.157 (-3) 0.194(-3) 0.387 (-1) 0.769 (-5) 0.585 (5) 0.140 (8) 0.385 (5) 0.329(4) 3 0.013 0.001 0.929 0.238 (-4) 0.355 (-3) 0.589 (-3) 0.346 (-1) 0.478 (4) 0.900 (5) 0.182(6) 0.128 (8) 4 0.002 0.019 0.026 0.237 (5) 0.970 (-4) 0.457 (-3) 0.212(-2) 0.120 (6) 0.672 (6) 0.387(3) 0.204 (5) 7'", ns 65.8 67.0 69.2 72.0
\
OSee footnote a, Table 111. bAEw = 2.81 eV.
1.75
225
2.75
1.25
1.75
R (bohr)
Figure 6. Potential energy curves of Cz+ of ground-state X42,is also shown.
nu symmetry.
The
4.1. The Doubly-Excited 122: and 222: States. Inspection of Table VI1 and Figure 5 indicates the existence of two doublet states possessing shorter Re and higher we than the ground state. These strongly-bound states are 122: ( T , = 3.62 eV) and 2 2 2 i ( T , = 6.43 eV). 4.1.1. Potential Curves and Electronic Structure. The l22:(uiu,T;) state is related to the ground configuration by the double excitation uuug r:.6*11 Its equilibrium geometry (2.319 bohr) compares with that of 2%: of C2+ (2.317 bohr) as well as with D12: of C2 (2.351 bohr). Near equilibrium, 122: has about 80% contribution of uiu,r: (a); at 2.80 bohr, it is a mixture between the structures a (20%), b (u:u,u~T:,45%), and c ( u ~ u ~ u g r U 15%); r g , at larger distances, this state acquires a predominately c character. Around 3.10 bohr, the 12Z: potential curve shows a barrier of 1.20 eV, with the maximum lying ca. 4.8 eV above the X state (the latter with De = 6.30 eV, ref 11). Since both states correlate with the first dissociation products, at larger distances another minimum in the 12Z: potential curve should exist. According to MRD-CI computations covering a wider range of geometries," this second minimum (a rather shallow one) occurs between 3.5 and 4.0 bohr. The quartet states 14Z: and 14A,, generated by the excitation ru Tg,asdoes 122Ain that region, also have rather broad potential minima." At equilibrium, 122: is described by same configuration as the ground state, whereas 222: mainly arises from the doubly-excited structure ut r;. As seen in Figure 5, the latter potential curve has a deep local minimum (see also ref 12); relative to the ground state, a shortening in the bond distance of 0.35 bohr leads to an increase of about 800 cm-I (-60%) in the vibrational frequency. 2%' correlates with the second dissociation channel C(lD,) C+(jP,), placed at 1.27 eV above ground-state products. Taking De(X42g)E 6.30 f 0.10 eV" and Te(222:) = 6.50 eV (Table VII), it follows that the 22Z: minimum lies only 1.10 eV below its dissociation limit. This feature is a good example of the general properties of doubly-excited states mentioned in the Introduction (Le., short Re, high we but small De). The 2*2: potential curve also shows a barrier (Figure 5), with its maximum occurring near 3.4 bohr and lying about 2.90 eV above the potential minimum. Approximately up to 10 vibrational levels can be accommodated in the potential well. The origin of the barrier can be understood on the basis of changes in the electronic structure of 22Z: with geometry (Figure
-
-
-
+
1.75
2.25
2.75
3.25
R (bohr)
Figure 7. Variation of the total transition moment ZR,$ (in au) with respect to the nuclear distance for selected transitions of C2+. The dotted line corresponds to the transition 2228+ 2211,.
-
2d). At short distances, the first two '2: states undergo an avoided crossing: the lowest-lying state is described by the excitation (e) ui ri,whereas 2 2 2 i arises from the (ground state) configuration u2u~ug?ri.Near equilibrium (R i= 2.35 bohr), 22Zf has about 6 5 b contribution of the doubly-excited configuration ugu s rd (e); close to the potential barrier is a mixture between e (20!6), 40%) (N.B.: the configurations t (ugufr:, 20%), and z (u~U,r~?rg, e, t, and z of C2+are derived from the corresponding E, T, and Z structures of C2discussed in section 3.1 by removing one electron of the second ugMO; see also section 5). At larger R, the 222: state becomes repulsive due to a switch into the antibonding structure +:ugri. 4.1.2. Radiative Data for the 22Ef-12Z: Band. Table VI11 summarizes the results. Because of the similar values of Re and we, the Franck-Condon parabola is practically constrained to the diagonal terms. There is no other study in the literature for this system to compare with. Theory predicts a transition energy AE, of 2.81 eV (this work) or 2.95 eV (ref 12). Since the upper and lower states are connected ug as for the E-D transition of Cz, by the single-excitation u, the corresponding transition moment should be large. The data from Table I1 and Figure 7 corroborate such expectations. At R = 2.348 bohr, we compute ZRe,et,2= 1.26 au2, to be compared
-
-
1638 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 TABLE I X Franck-Condon Factors 9ddf,Oscillator Strengths fdd,, Einstein Transition Probabilities Add, (in SI), and Radiative Lifetimes T”,,, (in ns) Predicted for the Svstem 2*2t(v9 - l*II.(vf9 of CT+aib 0.3 15 0.221 (-2) 0.654 (7) 0.330 0.243 (-2) 0.670 (7) 0.202 0.157 (-2) 0.404 (7) 0.095 0.800 (-3) 0.193 (7) 0.038 0.336 (-3) 0.755 (6) 0.009 0.132 (-3) 0.276 (6)
0
1 2 3 4 5
6
0.452 (-4) 0.879 (5) 7 0.156 (-4) 0.283 ( 5 ) Pus, ns 49.1
0.397 0.268 (-2) 0.866 ( 7 ) 0.013 0.941 (-4) 0.285 ( 6 ) 0.097 0.672 (-3) 0.191 (7) 0.18 0.133 (-2) 0.353 (7) 0.150 0.117 (-2) 0.289 (7) 0.090 0.722 (-3) 0.167 ( 7 ) 0.044 0.368 (-3) 0.795 (6) 0.020 0.178 (-3) 0.359 (6) 49.2
0.218 0.116 (-2) 0.405 (7) 0.187 0.121 (-2) 0.399 (7) 0.119 0.790 (-3) 0.245 ( 7 ) 0.005 0.425 (-4) 0.124 (6) 0.069 0.549 (-3) 0.150 (7) 0.118 0.917 (-3) 0.234 (7) 0.108 0.851 (-3) 0.203 (7) 0.074 0.600 (-3) 0.134 (7) 51.0
0.061 0.291 (-3) 0.1 11 (7) 0.307 0.184 (-2) 0.656 (7) 0.036 0.274 (-3) 0.921 (6) 0.141 0.100 (-3) 0.315 (7) 0.030 0.201 (-3) 0.596 (6) 0.010 0.180 (-4) 0.504 ( 5 ) 0.057 0.431 (-3) 0.113 (7) 0.085 0.665 (-3) 0.164 (7) (54.0)’
9L+”,,
Au,L+,
RSeefootnote a, Table 111. b A E , = 5.86 eV. coefficients for are 0.151 (6), 0.582 (5), and 0.314 ( 5 ) . dAs in b, A2,d,: 0.149 (7), 0.112
with 1.97 au2 for the E-D band system of C2. The partial lifetime T’,(222;-122:) varies from 66 ( u ’ = 0) to 72 (u’ = 3) ns. Moreover, in line with the predominance of the diagonal qdd,factors, the largest values of Add,andfdd, appear in the Au = 0 sequence. The Am coefficient and fm number are estimated to be 1.51 X lo8 and 0.0437, respectively. 4.1.3. Radiative Data for the 222:-1211, Band. Strictly speaking, 222: can be depopulated by decaying not only into 1”: but also into the lower-lying 1,2211, states as well. As seen in Table I1 and Figure 7, decay into 2211, has a rather small transition moment; besides, the transition 222: 2211, has quite unfavorable qddtfactors because of changes aR,= 0.52 bohr and Awe ii- 840 cm-l (Table VII). In practical terms, the values of AE,,, making a significant contribution to the emission Einstein coefficients (lifetimes) are expected to be about 1.0 eV smaller than a predicted AEm of 4.60 eV (Table VII). Hence, decay from 222: into 2211, should occur on the microsecond scale, or longer. By contrast, the process 222+ 1211, plays a relevant role for radiative depopulation of 2228. The corresponding results are listed in Table IX. Because tkis transition leads to AR, = 0.1 5 bohr and Awe 600 cm-I, the Franck-Condon factors are spread over a few vibrational levels. Around R = 2.40 bohr, the parameter Z&&? is ~ 0 . 1 au2, 9 which in conjunction with a relatively high transition energy (about 5.30-5.80 eV) finally leads to 2Add, of the order of lo8 S-I, that is, comparable to the values assigned 122:. A partial lifetime to the emission system 222+ T ” ~ ( X - + I I ) = 49.1 ns is slightjy shorter than T’~(Z+Z) = 65.8 ns (cf. Tables VI11 and IX). Afm number of 0.0022 is predicted for the absorption process 1211,. This transition offers the possibility to detect 222: experimentally the doubly-excited state 222: by using a combination of ionization and optical spectroscopy since the long-lived, metastable 12n, (ri)state of C2+can be generated from both XlZ:(r!) and a311,(u,ri) states of C2 by standard one-photon ionization. Our estimate Moo of 5.86 eV for this transition corroborates an earlier value of 5.83 eV reported in ref 12. Hence, an absorotion sDectrum observed bv M e i ~ ~aet l4.98 ~ ~ eV. assumed to +
+
-
+
(35) Meinel, H. Can. J . Phys. 1972, 50, 158.
TABLE X Total Einstein Coefficient Adi,,,(in s-’) and Net Radiative Lifetime 7d of the 2%: State of C2+n total
fUy
u“ = 8,9,10
0.758 (6), 0.412 (6), and 0.228 (6). eAs in b, (7), and 0.751 (6). /Upper value.
Bruna and Wright
v’= 2
0.1445 (8)
0.1961 (8)
0.3406 (8)
29.4
QSeefootnote a, Table 111. bData from Table VIII. CData from Table IX. originate on 1211, of C2+and to have a 2Z, upper state, cannot be correlated with the present 222,+ 1211, band system (see also refs 11 and 12). The net radiative lifetimes ~’”(2~2,’) involving both 12Z: and 12n, states are displayed in Table X. For the lower vibrational levels of this doubly-excited C2+state, we predict mean lifetime slightly below 30 ns. 4.1.4. Has the 222: 122: Band Already Been Observed Experimentally? A few years ago, van de Burgt and Heaven36 observed a new electronic transition (assumed to belong to C,) while studying the 193-nm photodissociation of toluene in a free-jet expansion experiment. They detected two close-lying bands at 22 334 and 22 449 cm-l (ca. 2.77 eV). A preliminary analysis of the rotational levels led to B” = 1.83 f 0.04 cm-’ and B’ = 1.93 f 0.04 cm-l. Such a large B’value (in fact, the largest rotational constant assigned to any C2 state, ref 3 1) indicates for the upper state an unusually strongly bound character, even higher than that of the absorbing state. On the basis of their experimental setup (which implied absorption from a metastable state with T longer than 7 ps) and the B”va1ue determined from the spectrum, those authors claimed the new absorption system originated in the c32:(u~u,,cg~~) state of C, (triplet counterpart of DIE:). The tight-binding upper state was tentatively assigned to the ’2: symmetry, with a u:Ug7r:3s Rydberg character. Interestingly, this configuration implies an ionic core having the same main electronic structure as 2 2 2 i ( u ~ u e of ~ 3C2+ plus an orbiting electron in the spatially long-range 3s orbital. From this point of view, the proposed structure for the upper state should indeed be strongly bound, with a rotational constant B’close to 1.87 cm-‘ as predicted for C2+ in its 222istate (Table VII). In the argument used by the experimentalists, however, an important point has been overlooked: based on Te(c32:) = 1.14 eV (exptl., ref 15) and a reported AE = 2.77 eV,36it follows that the aforementioned 2: Rydberg state should lie approximately at 3.90 eV above C2 ground state. Such a low T, contradicts earlier experimental findings indicating that the lowest-lying Rydberg species lies well above 8.0 eV.)’ This simple argument suggests that the newly detected absorption band can hardly be assigned to neutral C2. At this point, we suggest other plausible and alternative assignment of the spectrum: the 222: 122: system of C2+might be the carrier of the features observed experimentally, since: (i) A metastable character inferred by van de Burgt and Heaven36for the lower state (with T of a few microseconds) is also satisfied by 12Z: of C2+since its decay into either the ground (quartet) or the l22; and 1211, states occurs in the microsecond scale’* or longer. (ii) According to two ab initio studies (this work and ref 12), both upper and lower ionic states have large rotational constants (from 1.87 to 1.91 cm-I, Table VII), which are in relatively good agreement with their experimental counterparts (from 1.79 to 1.97 cm-I, ref 36). (iii) Most striking, there is the close resemblance between an experimental transition energy of 2.77 eV with theoretical predictions of 2.81 eV (this work) or 2.95 eV.I2 It should of general interest to reanalyze the experimental band detected in ref 36 in light of the assignment proposed above. If our assumptions are proved to be correct, then the recorded +
+
+
(36) Van de Burgt, L. J.; Heaven, M. C. J . Chem. Phys. 1987,87,4235. (37) Kirby, K.; Liu, B. J . Chem. Phys. 1979, 70, 893.
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1639
EiZ: and 2,2: States of C, and C2+
TABLE XI: Comparison between Spectroscopic Parameters and Radiative Data for the Selected States of C2 and C2+Studied in This Worko
Spectroscopic Parameters c 2
state X'Zi AQ, DE: E12;
c2+
config
Re 2.347 2.491 2.340 2.368
,'.:a
:.,.:a %a,*:
+',
we
state
config
1855 1608 1830 1672
x42,
u:.,.:
1'nu 122: 222;
Re 2.665 2.470 2.319 2.317
.:.: U ,.'.
.gd
we
1348 1615 1875 2203
Radiative Data c 2
transitionb
c2+
Rete"2 1.968 0.151 0.422
E~z;-D'L: E'Z:-A'II, D'Z:-X'Z;
7'&
fw
transitiond
172.0 16.6 14.6
0.0616 0.0268 0.0546
222;- 122: 222;- 12n,
Rerev2 1.260 0.190
#OC
fw
65.8 49.1
0.0437 0.0221
"Re and ERe,e,Fin atomic units, we in cm-' and 7 0 in ns. (eV): 1.45 (E-D), 5.79 (E-A) and 5.36 (D-X); experimental data from ref 31. 'Total SO(E'Z:) = 15.1 ns. (eV): 2.81 (2,-2,) and 5.86 (2,-nu);this work. 'Total ~ ~ ( 2 ~ 2=: )28.1 ns.
spectrum might constitute the first example in the literature for
an electronic transition involving two strongly-bound states. 4.2. The 3Q, State. As seen in Table VI1 and Figure 6, 3,II, does not show an enhanced bond strength relative to the ground state. In fact, its potential curve has a slightly shorter Re but a not too deep potential well (accommodating two vibrational levels only since the barrier is of about 0.25 eV). Near equilibrium, the strongly-bonding excitation u', ugruhas about 50% contribution, followed by ug r, and u, rg,each with -15 wt.%. Around the potential bamer at R = 3.10 bohr, this state is a heavy mixing between (p) u, rgand (9) ugr, ri,both structures being antibonding (the latter in particular). At larger distances, 3,II, arises from the q configuration exclusively.
- -.-
-
-
5. Comparison between C2 and C2+ Table XI summarizes the spectroscopic parameters and radiative data for the different electronic states and transitions discussed in the previous sections. The C2+states 1211,, 12Z:, and 2,2; are related vis 1 vis the C, states A i n u , D'Z:, and E'Z: by simply taking out one ug electron from the corresponding main electronic configurations of the neutral molecule. The weak bonding character of such MO's is reflected in the small changes in R, and we between neutral and ionic states of II, and Z,symmetry, respectively. With respect to doubly-excited 2: states, there exists a noticeable difference between the neutral and the positive ion: the heavy bonding-antibonding mixing in E'Z: (C,)versus a more pure strongly-bound, DES character in 2,Z: (C,+) is reflected in the corresponding Awe relative to the ground state: the vibrational frequency is lowered by 10% in C, but increased by 63% in C2+ (Table XI). In both carbon compounds, the largest transition moments are associated with the parallel bands of type Zl-Z:, with values of E&&?in the range from 1.2 to 2.0 au., By contrast, the transition moments squared of the Z:-II, perpendicular bands are 1 order of magnitude smaller. Accordingly, the absorption oscillator strength fo0for the parallel bands is roughly twice the value assigned to the parallel bands. The net lifetime of the 2: states is controlled by decay into both 2: and II, lower states. However, in neutral Cz the process E A is about 10 time faster than E D, so that the former transition essentially governs the radiative depopulation of the E i Z i state. The situation is somewhat different for C2+since both the parallel and perpendicular bands have comparable radiative rates (Table XI).
-
-
6. Summary and Conclusions This a b initio investigation focuses on the electronic structure and radiative properties of the doubly-excited states E'B; of Cz and 2,Z: of C2+. The accuracy in the computed transition moments and lifetimes are expected to lie within 10 and 20%, respectively.
For neutral Cz, our most relevant finding concerns the transition probabilities of the Freymark system E'Z:-AiII,. It is demonstrated that the experimental radiative results from Cooper and Nicholls28are seriously in error. In detail, they reported for this transition near equilibrium a ERete"of 2.26 f 1.13 au2, a result at variance with only 0.15 au2computed here with MRD-CI wave functions. A smaller value for this quantity is understandable since the upper E state (with a strong mixture between bonding and antibonding configurations) is mainly connected to the lower A state by double excitations, which intrinsically lead to vanishing matrix elements for the transition moment. The Freymark system borrows its intensity from a few secondary configurations making up the E'2: wave function which are singly-excited with respect to A'II,. The lifetime of the E'2' state ( ~ 1 ns) 5 is essentially governed by the radiative process I! A, the competitive decay E D being about 10 times slower. The last feature could make the experimental detection of the E D emission band somewhat difficult. A reinvestigation of the Mulliken band system DiB:-XiZ: confirms earlier theoretical findingsz4about the incorrectness of the experimental transition moment inferred by Cooper and Nicholls.z8 Again theory finds the opposite trend, namely E&&, = 0.41 au2versus 0.13 auz (CN, ref 28). A present estimate for the lifetime T~(D'Z:) of 14.6 ns, however, agrees with other e ~ p e r i m e n t a land ~ ~ ,the~retical,~ ~~ results. For C2+,the 2,2: state undoubtedly constitutes a good example of a strongly-bound excited state arising from the double excitation u', r$ The 3,II, state has a prominent contribution of the strongly-bound configuration u: ugru,though a heavy mixing with antibonding configurations prevents this state from showing a tight-binding character at equilibrium. Radiative depopulation of 2%+, with a net lifetime of about 30 ns, involves both the l22: and l2II, states. Formally, 2,Z: can be generated by uu ionization from c3Z:(u~u,ugr~)of C2. Other possibility is provided by the optical transition 2,Z: lZII,, in which the lower state is not only long-lived but also accessible from the first two states of C, through one-photon ionization. The most striking prediction by this study concerns the reassignment of a recently observed absorption transition at 2.77 eV, assumed to belong to neutral C2.36 However, the experimental information was far from beiig complete enough to allow a definite characterization either of the molecular species (neutral or ionic C,) or the multiplicity of the states involved. In fact, besides the transition energy, only the rotational constants of the upper and lower states could be derived from the spectrum, which indicated both states are strongly bound. On the basis of spectroscopic parameters derived theoretically, we suspect this new electronic transition has the positive ion C2+ as a carrier. In fact, the absorption band 2,Z: 12Z:, with a predicted AE = 2.81 eV, fulfills all the experimental features regarding the values of the transition energy and rotational
-
-
-
-
-
-
-
1640
J. Phys. Chem. 1992, 96, 1640-1648
-
constants. By contrast, a valence Rydberg transition of C2 as assumed by van de Burgt and Heaven36would place the upper Rydberg state at an extremely low energy (==4.0 eV), an assumption at variance with well-founded experimental results3' indicating that the lowest-lying Rydberg state of C2 lies above 8.0 eV.
Acknowledgment. Financial support by NSERC (Canada) is gratefully acknowledged. We also thank NSERC for a special grant to purchase and implement S U N workstations, on which the present calculations were carried out. Registry No. C,, 12070-15-4; C2+, 12595-79-8.
Structures and Energies of the Lowest Lying Singlet and Triplet States of C3H2 and C3F2. A Theoretical Study Voker Jonas, Marlis Bohme, and Gernot Frenking* Fachbereich Chemie, Universitat Marburg, Hans- Meenvein-Strasse, 0-3550Marburg. Germany (Received: July 25, 1991)
The theoretically predicted geometries for the propargylene, cyclopropenylidene,vinylidenecarbene, and cyclopropyne forms of C3H2 and C3F2 in their singlet and triplet states are reported at the MP2/6-31G(d) level of theory, using spin-restricted wave functions for the singlets and spin-unrestricted wave functions for the triplets. The calculated vibrational spectra for the different isomers have been calculated at HF/6-31G(d) and, in some cases, at MP2/6-31G(d). Improved total energies were calculated for C3H2at MP4/6-31 lG(2df) and for C3F2at MP4/6-31G(d) using the MP2/6-31G(d) optimized geometries. The relative stabilities of the C3H2isomers are predicted at MP4/6-31 lG(2df)//MP2/6-3lG(d) and for C3F2isomers at MP4/6-3 lG(d)//MP2/6-3 1G(d) corrected by zero-point energies, using spin-projected wave functions for the triplet states. To give an estimate for the accuracy of the calculated singlet-triplet gap, the energy differences between the lowest lying singlet and triplet states of CH2 and CF2 are calculated and compared with experimental values.
Introduction The study of C,H2 species is a challenge for theoretical and experimental chemists with many questions still being open. The first C3H2molecule identified by direct spectroscopic methods was triplet propargylene I t (Chart I), for which the ESR spectrum was published in 1965.l On the basis of the zero-field splitting parameters, a quasi-linear structure was postulated for It.' In spite of the observation of the IR spectra of I t and its deuterated isomers? it was very difficult to determine exactly the structure of triplet propargylene, Le., whether I t has a nonplanar C2geometry with two identical CC bonds or a planar C, structure with one short and one long CC bond. A recent analysis of the calculated and experimentally observed I R spectra suggests that I t has a C, equilibrium g e ~ m e t r y . ~ l t is not the global minimum on the C,H2 potential energy hypersurface. The energetically lowest lying isomer is singlet cyclopropenylidene 2s, which was prepared for the first time in 1984 by high-vacuum flash pyroly~is,~ 19 years after the preparation of the less stable species It. The interest in cyclopropenylidene rose considerably when it was discovered that 2s is the most abundant hydrocarbon identified in interstellar space.5 The third C3Hzisomer that has been detected experimentally is singlet vinylidenecarbene, 3se6 The preparation of 3s was achieved via photochemical conversion 2s It 39,which shows that interconversion among the three C3H2 isomers is possible.
--
(1) Bernheim, R. A.; Kempf, R. J.; Gramas, J. V.; Skell, P. S. J. Chem. Phys. 1965, 43, 196. (2) (a) Chi, F. K. Dissertation, Michigan State University, 1972. (b) Jacox, M.; Milligan, D. E. Chem. Phys. 1974, 4, 45. (3)Maier, G.; Reisenauer, H.-P.; Schwab, W.; Cirsky, P.; Spirko, V.; Hess Jr., B. A.; Schaad, L.J. J. Chem. Phys. 1989, 91, 4763. (4) Reisenauer, H.-P.; Maier, G.; Riemann, A,; Hoffmann, R. W. Angew. Chem. 1984, 96, 596;Angew. Chem., Int. Ed. Engl. 1984, 23, 641. ( 5 ) Thaddeus, P.; Vrtilek, J. M.; Gottlieb, C. A. Asrrophys. . . J . 1985, 299,
L63.
(6)Maier, G.; Reisenauer, H.-P.; Schwab, W.; CBrsky, P.; Hess Jr., B. A,; Schaad, L.J. J . Am. Chem. SOC.1987, 109, 5183.
0022-3654/92/2096- 1640$03.00/0
CHART I x-c===c=c X
/c/
x--c=c-c
\X
X /c=c\
X
X = H
1s
It
2s 2 t
X = F
5s
5t
6s
IDif t u o r o l p r o p o r g y t e n e
6t
IDif LuorolcycLopropenyLidene
x\
*/c=c=c X = H
3s
3t
4s
4t
X = F
7s
7t
8s
8t
(Dif1uoro)vinylidenecarbene
(Diftuoro)cyclopropyne
The infrared spectrum of 3s was compared with the theoretically predicted IR data, which confirmed that the newly detected isomer is the H2C=C=C species 3sS6 A number of theoretical studies have been devoted to C3H2 molecules, and the identification of experimentally observed species * ~ most ~~ has been aided by quantum mechanical c a l c ~ l a t i o n s . ~The complete theoretical study of the C3H2potential energy hypersurface is an early study by Hehre et al.,' who studied singlet and triplet states of eight different isomers using single determinant ab initio methods. More recent calculations by DeFrees and McLean* predict energies and geometries of Is, It, 2s, 39, and (7) Hehre, W. J.; Pople, J. A.; Lathan, W. A,; Radom, L.;Wasserman, E.; Wasserman, 2.R. J. A m . Chem. SOC.1976, 98, 4378. (8) DeFrees, D.J.; McLean, A. D.Astrophys. J . 1986, 308, L31.
0 1992 American Chemical Society
