ARTICLE pubs.acs.org/JPCA
Excited Electronic States, Transition Probabilities, and Radiative Lifetimes of CAs: A Theoretical Contribution and Challenge to Experimentalists Ana Paula de Lima Batista, Antonio Gustavo Sampaio de Oliveira-Filho, and Fernando R. Ornellas* Instituto de Química, Departamento de Química Fundamental, Universidade de S~ao Paulo, Av. Lineu Prestes, 748, S~ao Paulo, S~ao Paulo 05508-000, Brazil ABSTRACT: High-level CASSCF/MRCI calculations with a quintuple-ζ quality basis set are reported by characterizing for the first time a manifold of electronic states of the CAs radical yet to be investigated experimentally. Along with the potential energy curves and the associated spectroscopic constants, the dipole moment functions for selected electronic states as well as the transition dipole moment functions for the most relevant electronic transitions are also presented. Estimates of radiative transition probabilities and lifetimes complement this investigation, which also assesses the effect of spinorbit interaction on the A 2Π state. Whenever pertinent, comparisons of similarities and differences with the isovalent CN and CP radicals are made.
1. INTRODUCTION The CN radical is a species of considerable experimental and theoretical interest that has been the subject of numerous investigations in both terrestrial laboratories and extraterrestrial sources. Electrical discharges and flames are a major source of CN for local experiments, but it has also been found in environments like interstellar clouds, 1,2 stellar atmospheres,3 and comets,4 for example, and studied by a variety of spectroscopic techniques.510 Its two major band systems known as red (A 2Π r X 2Σ+) and violet (B 2Σ+ r X 2Σ+) make of CN an important probe species to help understand the carbonnitrogen chemistry under various physical conditions. Since the first communication by Herzberg,11 in the mid 1930s, on the recording of a new band system that he assigned to the radical CP, isovalent to CN, this species has also been the focus of various experimental investigations, although much less extensively than CN. Also, a candidate to be found in extraterrestrial sources, its presence in a carbon star has been investigated by Guelin et al.12 The analysis of the A 2ΠιX 2Σ+ band system, and a review of existing studies on CP has been reported by Bernath and collaborators.13,14 Arsenic is the next element in the nitrogen family that can also lead to a carbon-containing diatomic isovalent to both CN and CP, namely the species CAs. Although expected to show similar AX, BX, and AB band systems and be potentially accessible to spectroscopic investigations, surprisingly and to be best of our knowledge, not a single experimental reference to this radical has as yet been found in the literature. The toxicity of arsenic compounds might be the main motive behind this lack of data. Even on the theoretical side, the number of studies on CN, CP, and CAs follows a decreasing trend. Whereas for CN, several high level investigations have been reported on the ground state and on the low-lying electronic states,1522 significantly less was found on CP,19,2327 and just a single investigation on CAs, which appeared in a study of a series of diatomic systems containing atoms of groups 14, 15, and 16, was found in the literature;19 in this study, values are reported on various spectroscopic constants for the ground state (2Σ+) and the excited 2Π state, besides the dipole moment of the ground state. r 2011 American Chemical Society
A reliable characterization of energy profiles showing potential energy curves (PEC) for CAs extending until the dissociation limit and covering a significant number of excited states is certainly warranted. In this study, using high-level ab initio approaches, we report such a description by providing spectroscopic data for a manifold of states, besides computing transition moments and transition probabilities for the major band systems, and radiative lifetimes. It is our hope that this study can also motivate and guide further experimental investigations and help spectroscopists analyze their recorded spectra.
2. METHODS The direct application of WignerWitmer rules28 shows us that correlating with the lowest-lying dissociation channel, C (3Pg) + As (4Su), there are two doublet and two quartet (2,4Π and 2,4 + Σ ) states, and that the second channel, C (1Dg) + As (4Su), higher in energy 10 193 cm1, correlates only with quartets. The next channel, C (3Pg) + As (2Du), lies very close (10 754 cm1) and can be associated with the dissociation limit of nine other doublets and quartets, namely, 2,4Σ+(2), 2,4Σ, 2,4Π(3), 2,4Δ(2), 2,4 Φ. In this investigation, after some test calculations, we have included all states with Te smaller than about 35 000 cm1, thus accounting for the first excited 2Σ+ and 2Δ states, and the second excited 2Π state. Making this choice, we will be able to characterize band systems similar to the major ones of CN and CP. For the expansion of the one-particle space, correlation consistent atomic basis sets of quintuple-ζ quality, namely, aug-ccpV5Z for carbon29 and the aug-cc-pV5Z-PP set for As,30 briefly aV5Z, have been employed. We note that the arsenic set contains a small-core (10 electrons) energy-consistent relativistic pseudopotential (PP), namely ECP10MDF, that essentially replaces the 1s2p inner core by an energy-consistent PP that was optimized in a multiconfigurational DiracHartreeFock Received: May 13, 2011 Revised: June 24, 2011 Published: June 27, 2011 8399
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calculation. To account for static correlation effects, a stateaveraged complete active space self-consistent (SA-CASSCF)31,32 calculation involving the distribution of nine electrons in a (4, 2, 2, 0) active space was carried out. Following a common practice of using Abelian subgroups to classify the orbitals and electronic states, this notation means that one is working with the C2v point group and that of all the eight molecular orbitals in the active space, four of them transform as the irreducible representation A1, two according to B1, and the other two according to B2. This step generated reference spaces of dimensions 616 (320) for A1 (Σ+, Δ) states of doublet (quartet) multiplicities, 588 (336) for B1,2 (Πx,y), and 560 (352) for A2 (Σ, Δ). On top of the CASSCF wave function, taken as a zeroth-order reference space, dynamic correlation effects were introduced via the configuration interaction (CI) approach by allowing all single and double excitations from that reference space. The reduction of the CI space to a computationally feasible one was accomplished by the internally contracted multireference configuration interaction (iCMRCI) approach33,34 implemented in the Molpro-2009 suite of programs.35 The dimensions of the CI space ranged from about 1.7 million (quartets) to 2.7 million (doublets) configuration state functions. We further note that the molecular orbitals used to construct the N-particle basis are averaged natural orbitals obtained by the diagonalization of an averaged density matrix for each spin symmetry. Spinorbit interactions were taken into account within the interacting states method with the spinorbit matrix elements calculated at the CASSCF/aV5Z level of theory, with the oneand two-electron BreitPauli operator as implemented on the Molpro-2009 package.36 The spinorbit eigenstates were obtained by diagonalizing the matrix representation of the Hel + HSO operator, constructed by the replacement of the diagonal elements of Hel + HSO matrix by the corresponding MRCI + Q/ aV5Z values. Vibrational energies were obtained as solutions of the Schr€odinger radial equation by the Numerov method, as implemented in the Program Intensity,37 using energies corrected by the Davidson estimate of missing higher excitations needed to reach the full configuration interaction limit.38,39 Using the rovibrational energies and standard fitting procedures, a whole set of spectroscopic constants were obtained as described in previous studies.4043 Radiative transition probabilities, expressed as the Einstein spontaneous emission coefficients, involved first the evaluation of the transition dipole moment function, D(R), as a function of the internuclear distance, and next the numerical calculation of the transition dipole moment operator matrix element, Dv0 ;v00 ¼ hv0 jDðRÞjv00 i where |v0 æ and |v00 æ represent the upper and lower state vibrational wave functions. The computation of the transition moment follows the definition given by Whiting et al.,44 and a concise presentation of the conversion factors between the transition moment and other dynamical variables is given by Larsson.45 For the case of rotationless potentials, as used in this work, the Einstein Av0 v0 0 coefficients (in s1) is given by Av0 ;v00 ¼ 7:2356 106 jDv0 ;v00 j2 v3v0 ;v00
ð2 δ0, Λ0 þΛ00 Þ ð2 δ0, Λ0 Þ
Table 1. Excitation Energies, Te (cm1), Equilibrium Distances (a0), Vibrational and Rotational Constants (cm1), and Dissociation Energies (kcal mol1)a Re X Σ
2 +
a
Te
Be
ωe
ωexe
De
3.192
0 0.5521 (9)
A 2Π 3.373
5371 0.5115 (8)
866.1 (8)
4.788
0.0087
89.52
a 4Σ+ 3.519 17 803 0.4698 (8)
722.3 (8)
4.912 0.0013
53.98
b 4Π
557.1 (10) 3.606 0.0278
37.60
c 4Δ 3.515 24 089 0.4753 (10) B 2Σ+ 3.509 25 740 0.4728 (9)
720.9 (12) 4.849 0.0019 684.1 (12) 5.517 0.0267
64.25 80.59
C 2Π 3.828 27 072 0.3970 (8)
565.4 (8)
59.08
d 4Σ 3.517 29 027 0.4704 (9)
706.9 (12) 5.303 0.0089
50.13
D 2Π 3.858 31 425 0.3910 (10)
540.6 (9)
8.262
0.2483
46.71
E 2Δ
682.2 (8)
5.544
0.0017
42.45
3.824 23 528 0.4021 (8)
3.552 32 917 0.4655 (8)
1009.8 (18) 6.561
ωeye
4.793
0.0691 104.87
0.0500
Figures in parentheses are the number of fitting points.
where νv0 v00 is the transition energy, and (2 δ0,Λ0 +Λ00 )/ (2 δ0,Λ0 ) is the degeneracy factor.46,47 The multiplication constant is the appropriate factor to express the probabilities in s1, when the transition moment is given in units of e Å, and the energy in cm1. Radiative lifetimes were evaluated as the inverse of the total Einstein Av0 coefficients, which implies summing the transition probabilities to all lower vibrational states in all lower electronic states.
3. RESULTS AND DISCUSSION A. Potential Energy Curves, Vibrational Energies, and Spectroscopic Constants for the Λ + S States. In Table 1
we have collected a set of spectroscopic constants of the ten lowest-lying Λ + S electronic states investigated in this work, thus accounting for the two doublets and the two quartets correlating with the first dissociation channel, two other quartets dissociating into the second channel, and four other doublets with the third channel. In the case of the doublets, this choice essentially includes the X, A, and B states needed for a comparison with the known transition bands of the isovalent CN and CP molecules. A graphical display of these PECs is shown in Figure 1. In the case of the radical CAs, the ground state is of 2Σ+ symmetry followed closely by the A 2Π state higher in energy (Te) by 5371 cm1; for CN and CP this energy gap amounts to 92437 and 6974 cm1 (average of 6895 and 7053 cm1 for the Ω = 3/2 and 1/2 states, ref 48), respectively. For these two states, the equilibrium bond distances are predicted to be 3.192 a0 (1.689 Å) and 3.373 a0 (1.785 Å), respectively. At the equilibrium distance, the ground state can be represented by the electronic configuration ... 9σ2 4π4 10σ1 (c02 ∼ 0.77) whereas for the state A 2Π, we have ... 9σ2 4π3 10σ2 (c02 ∼ 0.82). Here 10σ is mainly a weakly bonding combination of an sp hybrid on C with the 4pz orbital on As, and 4π can be basically represented by 0.21 (2px,yC) + 0.77 (4px,yAs); as reflected in the longer equilibrium bond distance of the A 2Π state, the orbital 4π is more strongly bonding than 10σ. For the sake of completeness, we note that the bond distances in AsN (4π4 10σ2, X 1Σ+), AsO (4π4 10σ2 5π1, X 2Π), and AsF (4π4 10σ2 5π2, X 3Σ) are 1.618, 1.624, and 1.736 Å, respectively.48 The strength of this π bond is readily seen from the relative weight of the px,y orbital of N, P, and As; as the atomic number increases, the contribution of the heavier atom is much larger, reflecting an increase in the energy difference between the p orbital energies of the two bonding atoms. Without the inclusion 8400
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Table 2. Vibrational Energy Spacings ΔGv+1/2 and ZeroPoint Energies (cm1) for Selected Electronic States of the Molecule CAs v X 2Σ+ A 2Π B 2Σ+ C 2Π D 2Π E 2Δ a 4Σ+ b 4Π c 4Δ d 4Σ
Figure 1. Potential energy curves of the lowest-lying doublet and quartet Λ + S states of CAs.
of spinorbit effects, the bond dissociation energy (De) of CAs is estimated to be 104.9 kcal mol1, indicating a weaker bond than that in CN with De = 168.5 kcal mol1, and the one in CP with De = 123.5 kcal mol1.48,49 In the case of Te, our value practically matches the only other existing estimate of 5388 cm1 made by Kalcher,19 whereas for the equilibrium distances his values are slightly longer, 1.696 and 1.792 Å, respectively. In this context, it is also interesting to note that as one replaces the carbon atom in CAs by Si and Ge, the bonding character of the highest occupied σ orbital increases more than that of the π orbital and, as a consequence, the configuration π3σ2 leads to a higher stability than π4σ1 in SiAs and GeAs than in CAs, thus leading to an inversion in the ground state symmetry. Whereas in CAs we found the X 2Σ+ state lower in energy by 5371 cm1 than the A 2 Π, studies by Kalcher19 have located the 2Σ+ state higher in energy than the 2Π by 750 and 1113 cm1, respectively, in SiAs and GeAs. Concerning the harmonic frequencies in both SiAs and GeAs, a significant reduction for both states relative to the one in CAs was also estimated by Kalcher: SiAs, 479 cm1 (X 2Π), 527 cm1 (A 2Σ+); GeAs, 334 cm1 (X 2Π), 367 cm1 (A 2Σ+). The next set of excited states has quartet multiplicity with Te and Re values of 17 803 cm1 and 3.519 a0 (1.862 Å) for the state a 4Σ+ (4π3 10σ1 5π1, c02 ∼ 0.79); 23 528 cm1 and 3.824 a0 (2.024 Å) for b 4Π (4π2 10σ2 5π1, c02 ∼ 0.92); and 24 089 cm1 and 3.515 a0 (1.860 Å) for c 4Δ (4π3 10σ1 5π1, c02 ∼ 0.79). One should note that the excitation of a 4π electron to the antibonding 5π (mostly localized on the As atom) is reflected in an increase in the equilibrium bond distance of the above quartet states as well of the associated doublets derived from the same nominal electronic configuration. The region between about 24 000 and 35 000 cm1 is relatively dense of states with several doubletdoublet, doubletquartet, and quartetquartet PEC crossings, thus expected to give rise to a multitude of perturbations in the electronic spectra; their Re and Te values are listed in Table 1. Their nominal configurations are: C 2Π and D 2Π,
0
997
857
673
557
525
671
712
550
711
696
1
984
847
662
549
511
660
703
542
702
686
2
972
838
650
541
498
649
693
535
692
675
3
961
828
639
533
486
638
683
527
682
664
4 5
949 939
819 810
627 615
526 519
477 468
627 616
673 663
519 511
673 663
653 642
6
928
800
603
513
461
605
653
503
653
631
7
918
791
591
506
456
594
643
494
644
620
E0
502
432
342
282
271
340
360
278
358
352
essentially a mixture of 4π2 10σ2 5π1 and 4π1 10σ2 5π2 with c02 ∼ 0.60; d 4Σ (4π3 10σ1 5π1, c02 ∼ 0.81), and E 2Δ (4π3 10σ1 5π1, c02 ∼ 0.79). Here we also note that the wide energy separation between the 4Σ+ and 4Σ states can be traced to a major contribution of the exchange integral (4π+5π+|4π5π) where the charge distribution is mainly located on the As atom. Of particular interest for a comparison with CN is the state B 2Σ+ (4π3 10σ1 5π1, c02 ∼ 0.69), with Re = 3.509 a0 (1.857 Å) and Te = 25 740 cm1. In CN, the B X transitions (Te = 32 858 cm1) give rise to what is known as the violet band systems, whereas in CAs it is expected to occur in the blue region of the visible spectrum. The AX transitions in CN (Te = 9243 cm1), associated with the red band systems, in the case of CAs should arise in the infrared region (Te = 5371 cm1), and the BA (ΔTe = 20 369 cm1) ones in CAs should appear in the green region. One important difference to point out, however, is that the perturbation of the B state by the a 4Σ+ state is not expected to occur, because it lies much lower in CAs than in CN, and that due to the b 4Π state should occur closer to v0 = 0; the B state is also expected to be perturbed by the C 2Π close to v0 = 23. In CN the B 2Σ+ state arises from the excitation 4σ2 5σ0 1π4 f 4σ1 5σ1 1π4, whereas in CAs we have 4π4 10σ1 5π0 f 4π3 10σ1 5π1. This latter spatial distribution, with a consequent reduction in the repulsion of the electrons with parallel spins, along with the 5π occupation, result in an extra molecular stability. It is interesting also to note the avoided crossings between the X 2Σ+ and B 2Σ+ states, and the A 2Π and C 2Π states. For the X and A states, the energy difference and internuclear distance at which these two same symmetry potential energy curves approach mostly closely are 5651 cm1 and 5.224 a0, respectively; for the A and C states, we have 7347 cm1 and 5.125 a0. One half of this energy difference is the diabatic interaction matrix element and the R-value of the closest approach is the location of the crossing point between the diabatic potential energy curves. This information could provide a basis for a transformation to a diabatic representation. In Table 2 we have collected the zero-point energies and the vibrational energies differences (ΔGv+1/2) for selected electronic and vibrational states; the derived vibrational constants listed in Table 1 were obtained by fitting these energy differences to a standard expression. Although not an usual practice in the literature, it is worth-remembering that the determination of both experimental and theoretical constants is dependent on the number of vibrational spacings and also on the number of fitting parameters,50 and whence more meaningful comparisons could be made by explicitly stating the number of bands and parameters used in the fitting either the theoretical or experimental data. 8401
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Table 3. Rotational Constants Bv (cm1) for Selected Vibrational States of the Molecule CAs v
X 2Σ+
A 2Π
B 2Σ+
C 2Π
D 2Π
E 2Δ
a 4Σ+
b 4Π
c 4Δ
d 4Σ
0
0.5693
0.5096
0.4706
0.3954
0.3908
0.4592
0.4678
0.3964
0.4693
0.4682
1
0.5652
0.5058
0.4663
0.3919
0.3895
0.4546
0.4636
0.3927
0.4652
0.4636
2
0.5611
0.5021
0.4619
0.3887
0.3880
0.4501
0.4594
0.3889
0.4608
0.4591
3
0.5570
0.4983
0.4574
0.3857
0.3865
0.4454
0.4552
0.3851
0.4566
0.4545
4
0.5529
0.4945
0.4528
0.3828
0.3850
0.4408
0.4510
0.3814
0.4524
0.4499
5
0.5488
0.4908
0.4482
0.3801
0.3832
0.4361
0.4468
0.3775
0.4481
0.4453
6
0.5447
0.4870
0.4435
0.3773
0.3812
0.4314
0.4425
0.3736
0.4439
0.4406
7
0.5406
0.4831
0.4387
0.3747
0.3789
0.4265
0.4382
0.3697
0.4396
0.4360
Figure 2. Dipole moment functions for selected states of the molecule CAs.
Concerning the rotational constants Bv, collected in Table 3, they were estimated as the average values Æv|16.8576/μR2|væ, where |væ stands for the vibrational state. B. Dipole Moment Functions. In Figure 2 we have displayed the dipole moment as a function of the internuclear distance for selected doublet and quartet states. For these states, the dipole moment is always positive and tends to zero at large distances as the molecule dissociates into neutral fragments. A positive value in this context implies the Asδ+Cδ polarity for the molecule. This notation is consistent with the IUPAC recommendation of writing first the least electronegative element, and according to the Pauling electronegative scale, C (2.55) and As (2.18), this radical would be properly named monoarsenic carbide; however, reference to this molecule so far in the literature has represented it as CAs, and we will keep this latter notation. Because no avoided crossings occur for these low-lying states, these functions show a smooth behavior, most of them being linear around their equilibrium distances, with the ground state showing the least rate of change of the dipole with the distance. It is interesting also to notice in Figure 2 that, close to the equilibrium distance, the polarity of the molecule in the state A 2Π is of the same sign and about twice as large as the one in the ground state, similarly to
Figure 3. Transition dipole moments of selected electronic transitions and the spinorbit coupling constant of the A 2Π state as a function of the internuclear distance of the molecule CAs.
that reported by de Brouckere and Feller26,27 for the X and A states of CP. However, these results for CAs and CP should be contrasted with the ones reported by Knowles et al. on CN, showing that the dipole moments of the A state are about an order of magnitude smaller than those of the ground state, and also have an opposite polarity. Although plots of the dipole moment versus the internuclear distance should illustrate the changes in polarity of the molecule, of direct relevance to experimental testing is the vibrationally averaged dipole moments which are collected in Table 4. C. Transition Moments, Transition Probabilities, and Radiative Lifetimes. Transition dipole moments for the three major doublet transitions, and the b 4Πa 4Σ+ quartet one calculated as a function of the internuclear distance are shown in Figure 3. Transition probabilities for emission as expressed by the Einstein Av0 v00 coefficients, FranckCondon factors (qv0 v00 ), excitation energies (Tv0 v0 0 ), and a sum check on these factors are collected in Tables 57 for the doublet states; total transition probabilities (Av0 ) and radiative lifetimes (τv0 ) are summarized in Table 8. We note that although only the most relevant transitions are shown, the sum was sufficiently extended to have the rule 8402
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Table 4. Vibrationally Averaged Dipole Moments (e Å) for Selected Electronic and Vibrational States of the Molecule CAs v
X 2Σ+
A 2Π
B 2Σ+
a 4Σ+
b 4Π
c 4Δ
0
0.269 89
0.464 25
0.314 99
0.195 59
0.337 99
0.179 29
1
0.270 62
0.460 28
0.318 97
0.200 25
0.332 07
0.181 30
2
0.271 36
0.456 34
0.322 82
0.204 81
0.325 93
0.183 34
3
0.272 11
0.452 33
0.326 55
0.209 39
0.319 74
0.185 25
4 5
0.272 85 0.273 59
0.448 30 0.444 23
0.330 13 0.333 52
0.213 94 0.218 41
0.313 47 0.307 08
0.187 05 0.188 79
6
0.274 34
0.440 09
0.336 69
0.222 81
0.300 54
0.190 44
Table 6. Einstein Emission Av0 v00 (s1) Coefficients, FranckCondon Factors (in Italics), and Transition Energies Tv0 v00 (cm1, in Parentheses) for the Transition B 2Σ+A 2Π of the CAs Molecule v00 0
1
2
Table 5. Einstein Emission Coefficients Av0 v0 0 (s1), FranckCondon Factors (in Italics), and Transition Energies Tv0 v0 0 (cm1, in Parentheses) for the Transition A 2ΠX 2Σ+ of the CAs Molecule v00 0
1
2
3
4
5
Qv0 a a
v0 = 2
v0 = 3
v0 = 4
3
v0 = 5
v0 = 0
v0 = 1
v0 = 2
v0 = 3
v0 = 4
v0 = 5
97 069
72 198
28 621
7866
1635
261
0.537
0.328
0.107
0.024
0.004
0.000
(20 279)
(20 952)
(21 614)
(22 264)
(22 903)
(23 530)
46 368
12 909
63 588
51 175
22 077
6524
0.333
0.073
0.302
0.201
0.071
0.017
(19 422)
(20 095)
(20 757)
(21 407)
(22 046)
(22 673)
11 508 0.105
41 609 0.312
453 0.004
33 553 0.165
56 568 0.234
37 115 0.127
(18 575)
(19 248)
(19 910)
(20 560)
(21 199)
(21 826)
1849
21 426
22 409
10 527
9746
46 531
0.021
0.205
0.176
0.072
0.048
0.202
(17 737)
(18 410)
(19 072)
(19 722)
(20 361)
(20 988)
212
5409
24 646
6745
19 454
326
v0 = 0
v0 = 1
5101
10 128
10 797
8262
5119
2737
0.003
0.066
0.248
0.056
0.137
0.001
0.267
0.319
0.219
0.114
0.050
0.020
(5301)
(6158)
(7005)
(7843)
(8671)
(9490)
(16 909) 18
(17 582) 859
(18 244) 9463
(18 894) 21 379
(19 533) 264
(20 160) 19 986
3698
483
1563
7271
10 827
10 217
0.000
0.014
0.122
0.227
0.002
0.148
0.387
0.027
0.054
0.160
0.162
0.108
(16 090)
(16 763)
(17 425)
(18 075)
(18 714)
(19 341)
(4304)
(5161)
(6008)
(6846)
(7674)
(8493)
1
94
2036
12 657
14 379
1520
974
998
2504
178
1772
7287
0.000
0.002
0.034
0.172
0.161
0.014
0.241 (3320)
0.117 (4177)
0.155 (5024)
0.006 (5862)
0.043 (6690)
0.118 (7509)
(15 280)
(15 953)
(16 615)
(17 265)
(17 904)
(18 531)
1.000
1.000
1.000
1.000
1.000
1.000
109
1005
6
2042
1803
0
0.084
0.281
0.001
0.140
0.072
0.000
(2348)
(3205)
(4052)
(4890)
(5718)
(6537)
4
206
571
285
734
2528
0.018
0.185
0.182
0.039
0.056
0.111
(1387)
(2244)
(3091)
(3929)
(4757)
(5576)
0 0.003
11 0.059
229 0.241
186 0.068
671 0.105
47 0.004
(438)
(1295)
(2142)
(2980)
(3808)
(4627)
1.000
1.000
1.000
1.000
1.000
1.000
Qv0 = Σv1500 =0qv0 v00 .
satisfied. From the data in these tables we can see how the predictions of relative intensitites based on the FranckCondon factors can be corroborated by those based on the emission coefficients. For instance, for the A state, Table 5 shows that emissions from v0 = 0 are expected to have their intensities spread over v00 = 02, with a maximum at v00 = 1 according to the FranckCondon factors; however, the emission coefficient is largest for the (0, 0) transition, with T00 equal to 5301 cm1. For v0 = 1, 2, both predictions agree on the (1, 0) and (2, 0) being the most intense ones. For the AX system, the emission coefficients are about an order of magnitude smaller than the ones obtained for CN, and the radiative lifetimes are therefore expected to be about an order of magnitude longer, 101 μs (τ0) for CAs versus 11.2 μs (τ0) for CN;18 for the state A 2Π (v0 = 0) of CP, τ0 = 138 μs was calculated in ref 27. For the BA system, the (0, 0), (1, 0), and (2, 1) transitions are predicted to be the strongest ones according to the values of Av0 v00 ; from v0 = 1, the FranckCondon factors predict both (1, 0) and (1, 2) to
4
5
6
Qv0 a a
Qv0 = Σv1500 =0qv0 v00 .
have about the same intensities. For this system, T00, T10, and T20 are estimated to be 20 279, 20 952, and 21 614 cm1, thus falling in the green region of the visible spectrum. The transition probabilities for the BA system of CAs are also about a factor of 10 smaller than the ones in CN, however, for CAs, the B state is expected to be perturbed by the b 4Π close to v0 = 0, and by the C 2 Π close to v0 = 2 . In the case of the BX system, the (0, 2) and (0, 3) transitions are about equally strong as measured by the Av0 v00 coefficients, with the (0, 2) slightly stronger, with T02 and T03 equal to 23 599 and 22 627 cm1, respectively, thus being expected to fall in the blue-violet region of the visible spectrum; this order of intensity is, however, reversed if based on the qv0 v0 0 values. From v0 = 1, the (1, 1) band is predicted to be the most intense, and from v0 = 2, the (2, 0). On the basis of the total Einstein Av0 coefficients listed in Table 8, the radiative lifetimes for the B state are also predicted to be about 14 times longer than the corresponding ones for CN; for instance, τ0 = 831 ns for CAs and 61 ns for CN. For both A and B states, the radiative lifetimes decrease as v0 increases, as is also the prediction for CN,18 and for the A 2Π state in CP. Concerning transitions involving quartet states, as seen in the potential energy curves in Figure 1 and in the transition dipole moment function in Figure 2, the b 4Πa 4Σ+ system offers a possibility of experimental characterization of these states in the infrared region. The challenge is thus presented for the experimentalists. D. Potential Energy Curves and Spectroscopic Constants for Relativistic States Ω. To complement the above description, we present in Figure 4 an overview of some of the low-lying relativistic states (Ω). So besides the scalar relativistic effects for 8403
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The Journal of Physical Chemistry A
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Table 7. Einstein Emission Av0 v00 (s1) Coefficients and FranckCondon Factors (in Italics), and Transition Energies Tv0 v00 (cm1, in Parentheses) for the Transition B 2Σ+X 2Σ+ of the CAs Molecule v00 0
1
2
3
4
5
6
7
Qv0 a a
Qv = 0
v0 = 0
v0 = 1
v0 = 2
v0 = 3
v0 = 4
v0 = 5
56 533
214 490
430 780
609 370
683 300
648 460
0.027
0.087
0.148
0.178
0.170
0.139
(25 580)
(26 253)
(26 915)
(27 565)
(28 204)
(28 831)
168 140
337 690
283 500
93 054
49
97 906
0.107
0.179
0.127
0.036
0.000
0.026
(24 583)
(25 256)
(25 918)
(26 568)
(27 207)
(27 834)
237 100 0.200
173 540 0.121
5713 0.003
79 194 0.038
223 300 0.092
200 280 0.071
(23 599)
(24 272)
(24 934)
(25 584)
(26 223)
(26 850)
209 950
13 680
78 649
147 980
35 061
16 712
0.238
0.013
0.060
0.094
0.019
0.007
(22 627)
(23 300)
(23 962)
(24 612)
(25 251)
(25 878)
130 050
20 066
109 780
9028
53 192
126 260
0.201
0.025
0.111
0.008
0.037
0.074
(21 666) 59 538
(22 339) 71 997
(23 001) 24 674
(23 641) 37 801
(24 290) 80 463
(24 917) 4331
0.127
0.122
0.034
0.042
0.074
0.003
(20 717)
(21 390)
(22 052)
(22 702)
(23 341)
(23 968)
20 837
73 490
3087
64 073
1917
50 292
0.063
0.173
0.006
0.096
0.002
0.050
(19 778)
(20 451)
(21 113)
(21 763)
(22 402)
(23 029)
5700
42 810
32 998
14 874
29 224
38 996
0.025 (18 850)
0.144 (19 523)
0.086 (20 185)
0.031 (20 835)
0.048 (21 474)
0.052 (22 101)
1.000
1.000
1.000
0.999
0.992
0.937
Table 9. Equilibrium Distances (a0), Excitation Energies Te (cm1), Dissociation Energies De (kcal mol1), Zero-Point Energies E0, and Fundamental Frequencies (cm1) for the Lowest-Lying Relativistic States of the Molecule CAsa Re
Σv1500 =0qv0 v00 .
Table 8. Total Einstein Av0 (BoundBound) Coefficients and Radiative Lifetimes of Various Vibrational Levels of the B 2Σ+ and A 2Π States of CAs τv0 (μs) v0
Figure 4. Potential energy curves for selected relativistic states of CAs.
AX
BX
0
9 886
1
12 830
2 3 4 5
Te
E0
ωe
ωe x e
Deb 104.69
X 2Σ+1/2
3.193
0
501
1004.7 (8)
5.767
A 2Π3/2
3.373
5 022
432
865.9 (6)
4.512
90.38
A 2Π1/2
3.374
5 782
433
870.0 (6)
5.453
88.25
a 4Σ+1/2
3.520
17 823
359
721.2 (5)
4.666
53.94
a 4Σ+3/2
3.520
17 830
359
721.4 (5)
4.618
53.95
Values in parentheses refer to the number of fitting points. b D e approximated by the difference E(15) E(R e). a
τv0 (ns)
BA
total (B)
A
B
889 322
314 050
1 203 372
101
831
970 739
309 024
1 279 763
78
781
15 688
1 047 984
303 060
1 351 044
64
740
18 437
1 119 828
296 586
1 416 414
54
706
21 089 23 641
1 186 812 1 250 064
289 882 283 274
1 476 694 1 533 338
48 42
677 652
As introduced through the pseudopotential basis set, the curves shown in Figure 4 now takes into account also the spinorbit interactions; in Table 9 we have collected the associated spectroscopic constants. As expected, changes are very minor, with the exception of the 2Π state for which an energy splitting of 761 cm1 was determined at the equilibrium distances of the 3/2 and 1/2 states, a value about 15% larger than that obtained by Kalcher19 at the complete active space averaged coupled pair functional (CAS-ACPF) level of theory, 659 cm1. The negative of this number is the spinorbit coupling constant. Just to check how much the inclusion of dynamic correlation affects the above
estimate of the spinorbit coupling constant, we also carried out a single-point calculation at the equilibrium distance of the A 2Π state at the CASSCF/MRCI/aV5Z level of theory, mixing all the states as in the CASSCF calculation, and obtained 731 cm1. The variation of the spinorbit constant with the internuclear distance for the A 2Π state calculated at the CASSCF/aV5Z level of theory is shown in Figure 3, and close to the equilibrium distance one can see that it does not vary so much. As a further assessment of the accuracy of the spinorbit constant, we also evaluated this constant for the species AsO (X 2Π) at the same level of theory as used for CAs. Our result of 923 cm1 is about 10% smaller than the experimental one,48 1026 cm1, and this is the expected accuracy for CAs at this level of theory. Changes in the equilibrium distances are also very small, just a few mÅ. For the relativistic states, the excitation energies Tv0 v0 0 for transitions in the A 2Π1/2X 2Σ+1/2 system increase by about 8%, with T00 = 5714 cm1, compared to 5301 cm1 without the inclusion of spinorbit effects, for instance. Thus our previous Λ + S results for the radiative lifetimes of the A 2Π state can be taken as upper bounds to the true ones. In the case of the relativistic states B 2 + Σ 1/2 and C 2Π1/2, there is an avoided crossing close to 3.8 a0, 8404
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The Journal of Physical Chemistry A and thus for v0 > 1 the yet to be observed transitions should reflect this perturbation effect. But for the B 2Σ+ (v0 = 0)X 2Σ+ (v00 ) transitions, we do not expect any significant change in the transition energies T0v0 0 and in the total transition probability, A0. On the other hand, for the B 2Σ+ (v0 = 0)A 2Π (v00 ) transitions, the excitation energies should be slightly smaller and the predicted radiative lifetime about 510% longer.
4. CONCLUSIONS The results presented in this investigation provide reliable theoretical evidence characterizing the lowest-lying electronic states of the radical species CAs. Because it is as yet experimentally unknown, the potential energy curves and the derived spectroscopic constants will certainly be useful guides to the planning of its experimental detection and data analysis, especially for those states where non-negligible perturbations effects are expected to be manifested in the spectra. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT A.P.L.B. and A.G.S.O-F. express their gratitute to Fundac-~ao de Amparo a Pesquisa do Estado de S~ao Paulo (FAPESP) for doctoral fellowships, and F.R.O. acknowledges the academic support of the Conselho Nacional de Desenvolvimento Científico e Tecnologico (CNPq) of Brazil. ’ REFERENCES (1) Lambert, D. L.; Sheffer, Y.; Crane, P. Astrophys. J. 1990, 359, L19. (2) Black, J. H.; van Dishoeck, E. F. Astrophys. J. 1988, 331, 986. (3) Lambert, D. L.; Brown, J. A.; Hinkle, K. H.; Johnson, H. R. Astrophys. J. 1984, 284, 223. (4) Johnson, J. R.; Fink, U.; Larson, H. P. Astrophys. J. 1983, 270, 769. (5) Dixon, T. A.; Woods, R. C. J. Chem. Phys. 1977, 67, 3956. (6) Prasad, C. V. V.; Bernath, P. F.; Frum, C.; Engleman, R., Jr. J. Mol. Spectrosc. 1992, 151, 459. (7) Ram, R. S.; Wallace, L.; Bernath, P. F. J. Mol. Spectrosc. 2010, 263, 82. (8) Ram, R. S.; Davies, S. P.; Wallace, L.; Engleman, R.; Appadoo, D. R.T.; Bernath, P. F. J. Mol. Spectrosc. 2006, 237, 225. (9) Davies, P. B.; Hamilton, P. A. J. Chem. Phys. 1982, 76, 2127. (10) Ito, H.; Ozaki, Y.; Suzuki, K.; Kondow, T.; Kuchitsu, K. J. Chem. Phys. 1992, 96, 4195. (11) Herzberg, G. Nature 1930, 126, 132. (12) Guelin, M.; Cernicharog, J.; Paubert, G.; Turner, B. E. Astron. Astrophys. 1990, 230, L9. (13) Ram, R. S.; Tam, S.; Bernath, P. F. J. Mol. Spectrosc. 1992, 152, 89. (14) Ram, R. S.; Bernath, P. F. J. Mol. Spectrosc. 1987, 122, 282. (15) Schaefer, H. F.; Heil, T. G. J. Chem. Phys. 1971, 54, 2573. (16) Ito, H.; Ozaki, Y.; Nagata, T.; Kondow, T.; Kuchitsu, K.; Takatsuka, K.; Nakamura, H.; Osamura, Y. Chem. Phys. 1985, 98, 81. (17) Bauschlicher, C. W., Jr.; Langhoff, S. R.; Taylor, P. R. Astrophys. J. 1988, 332, 531. (18) Knowles, P. J.; Werner, H.-J.; Hay, P. J.; Cartwright, D. C. J. Chem. Phys. 1988, 89, 7334. (19) Kalcher, J. Phys. Chem. Chem. Phys. 2002, 4, 3311. (20) Thøgersen, L.; Olsen, J. Chem. Phys. Lett. 2004, 393, 36. (21) Wang, J. K.; Wu, Z. S. Chin. Phys. B 2008, 17, 2919.
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