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Excited doublet and quartet states of SiP: a high level theoretical investigation
Chemical Physics 295 (2003) 195–203 www.elsevier.com/locate/chemphys
Excited doublet and quartet states of SiP: a high level theoretical investigation Levi G. dos Santos, Fernando R. Ornellas
*
Dept. de Quımica Fundamental, Instituto de Quımica, Universidade de S~ ao Paulo, Caixa Postal 26077, S~ ao Paulo SP 05513-970, Brazil Received 25 August 2003; accepted 16 September 2003
Abstract Doublet and quartet states of the SiP molecule dissociating into the four lowest dissociation channels are characterized theoretically at a high-level of correlation treatment (multireference single and double excitation configuration interaction). Potential energy curves give a global view of the manifold of possible electronic states. For selected states, dipole and transition moment functions, and transition probabilities and radiative lifetimes are also reported as well as an extensive set of spectroscopic constants. A new 2 P state offers another likely route for exploring transitions to excited vibrational states of both X2 P and A2 Rþ states. A detailed set of data for the quartet states is expected to provide valuable information for the experimental identification of these states. Ó 2003 Elsevier B.V. All rights reserved. Keywords: SiP; Multireference configuration interaction; Excited electronic states; Potential energy curves; Spectroscopic constants; Transition probability; Radiative lifetime
1. Introduction In recent studies, we characterized the ground state (X2 P) [1] and two excited states of 2 Rþ symmetry [2] of the radical SiP. In the first investigation, a very accurate multireference configuration interaction calculation was set up to definitely establish the relative order of the first two electronic states which had been a subject of discussion in previous theoretical studies [3–7], and in the second work, due to the small adiabatic excitation energy (470 cm1 ) between the two lowest states estimated in the first study, we described the next 2 Rþ state and proposed that transitions involving this state would be a promising path for the spectroscopic characterization of all three states. More recently, in a study by Jakubek et al. [8], the SiP radical was observed in the laboratory and the three lowest states characterized spectroscopically, thus verifying our predictions, and providing accurate laboratory data that can aid astrophysicists in the *
Corresponding author. Tel.: +55-11-3091-3895; fax: +55-11-38155579. E-mail address: [email protected] (F.R. Ornellas). 0301-0104/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2003.09.007
search of this species in interstellar/circumstellar gases. Since other silicon-containing diatomics like SiC, SiN, SiO, and SiS [9] had been observed in outer space, as well as the isoelectronic radical CP [10], and also the species PN [11,12], the possibility of observation of the SiP molecule has become a challenge since the cosmic abundance of Si and P atoms makes its formation very likely. In this work, guided by the same approach of our previous investigations of silicon-containing species like SiB [13], SiN [14], and SiAl [15], we extended the previous studies of SiP to include several electronic states of both doublet and quartet multiplicities that correlate with the first four dissociation channels. Besides describing new possibilities of transitions to the ground state, and providing reliable data that can guide spectroscopists to make the first observation of quartet transitions, the characterization of the quartet state correlating with the first dissociation channel makes available all the data needed for the evaluation of the rate constant by the radiative association model, commonly used by astrophysicists, thus contributing to a better understanding of the chemistry of theses species in astrophysical environments.
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2. Methods The electronic calculations carried out in this study involved basically two steps. In the first one, the aim was to assess the importance of static correlation effects and to define a common set of molecular orbitals to be used for all states; in the second step, one was mainly concerned with incorporating as much dynamic correlation as possible in the final wavefunction having in mind a balance between its dimension and computation time. Our choice of atomic basis functions was the set of augmented correlation consistent valence-polarized quadruple-zeta (aug-cc-pVQZ) functions developed by Woon and Dunning [16]. The first step consisted of state-averaged complete active space self-consistent field (CASSCF) [17,18] calculations separately for each set of doublet and quartet states, involving a total of 16 states, distributed according to the symmetry representations A1 (5), B1 (4), B2 (4), and A2 (3) of the C2v point group for the doublets, and 20 states (A1 (5), B1 (5), B2 (5), and A2 (5)) for the quartets. Averaged natural orbitals for each multiplicity were next computed using the CASSCF wavefunctions which contained all possible electronic excitations resulting from the distribution of nine electrons into 11 active orbitals ð5; 3; 3; 0Þ, which comprised the valence orbitals plus one correlating orbital for each of the A1 , B1 , B2 symmetries; core and inner-shell orbitals were kept doubly occupied. To limit the dimension of the final wavefunction, the most important configuration state functions of the CASSCF calculations were next selected on the basis of their coefficients (greater than 0.015 in magnitude for each internuclear distance) and formed the core of a zerothorder wavefunction from which the final configuration interaction wavefunction (CI) was generated as all single and double excitations from this reference set. The CI wavefunction was constructed by the internally contracted approach [19,20] implemented in the Molpro 98 suite of programs [21]. For the range of internuclear distances investigated, cubic spline fits to the calculated points generated potential energy functions used in the solution of SchroedingerÕs radial equation for the nuclear motion by the program Intensity [22], which besides calculating vibrational–rotational energies and
vibrational wavefunctions, allowed also for the calculation of radiative transition probabilities; radiative lifetimes were evaluated as the inverse of the total radiative transition probability. Standard fitting procedures used for the calculation of the spectroscopic constants and the convention employed in the transition probabilities evaluation are discussed elsewhere [23–26].
3. Results and discussion 3.1. Energetic aspects The first step in this study involved applying the Wigner–Witmer rules [27] to identify the possible molecular electronic states correlating with the first few dissociation channels associated with the electronic states of the atomic fragments Si and P. Table 1 summarizes the manifold of states allowed for the four lowest dissociation channels. Energies as a function of the internuclear distance for the 17 lowest lying states are graphically displayed in Figs. 1 and 2, for the doublets and the quartets, respectively. To avoid an excessive number of lengthy tables, only numerical values of the energy at the equilibrium distance ðRe Þ are collected in Table 2 together with the adiabatic energies ðTe Þ; the remaining values are available upon request to the authors. Compared to our previous studies where only four states (2 Rþ ð2Þ, 2 Px , 2 Py ), transforming according to the representations A1 (2), B1 (1), B2 (1) of the C2v point group symmetry, were optimized in the CASSCF calculation, the new description of the X and A states was slightly improved. However, we note that the previously denoted B2 Rþ state corresponds, in fact, in the present investigation, to the third excited state; by including all the doublet states compatible with the new active space ð5; 3; 3; 0Þ in the state-averaged CASSCF calculation, a2 P state lower in energy than our previous B2 Rþ state was found, thus calling for a renaming of the corresponding state found in the recent experiment and of our previous result. In this study, the inclusion of more states in the optimization process improved significantly our description of the second excited 2 Rþ state (now
Table 1 Low-lying electronic states of the molecule SiP, their dissociation channels, and energy separation at the dissociation limit States of separated atoms
P(4 Su ) + Si(3 Pg ) P(4 Su ) + Si(1 Dg ) P(4 Du ) + Si(3 Pg ) P(4 Su ) + Si(1 Sg ) a b
Calculated at R ¼ 9:0 a0 . Ref. [28].
DE (cm1 )
Molecular states
Rþ , 2;4 P R , 4 P, 4 D 2;4 þ R ð2Þ, 2;4 R , 4 R 2;4 4
2;4
Pð3Þ,
2;4
Dð2Þ,
2;4
U
Theoreticala
Experimentalb
0.0 6421.7 12084.5 15290.8
0.0 6298.9 11376.6 15394.4
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Table 2 Equilibrium distances ða0 Þ, energies (a.u.) at the equilibrium distance, and adiabatic excitation energies (cm1 ) for the doublet and quartet states of the molecule SiP State
Re
EðRe Þ
Te
X2 P A2 Rþ B2 P C2 R þ D2 U E2 R F2 D G2 D H2 P I2 Rþ
3.947a;b 3.777a;b 4.422 4.035a;b 4.460 4.149 4.144 4.236 4.436 3.939
0.888216 0.886096 0.798019 0.792261 0.779714 0.769141 0.764604 0.759619 0.757084 0.735856
0.0 465a; b 19,795 21,059a ;b 23,813 26,133 27,129 28,223 28,779 33,438
a4 Rþ b4 P c4 D d4 R f4 D g4 Rþ h4 R
4.141 4.440 4.150 4.166 4.958 4.471 5.410
0.817898 0.817692 0.797147 0.781930 0.722902 0.719134 0.713611
15,432 15,478 19,987 23,326 36,281 37,108 38,320
a Theor. 3.948 a0 (X); 3.783 a0 , 470 cm1 (A); 4. 063 a0 , 20,588 cm1 (C), Ref. [1,2]. b Exp. 3.926 a0 (X), 3.715 a0 (A), 395.4 cm1 ; 4.021 a0 , 21,398 cm1 (C); derived from the data in [8].
Fig. 1. Potential energy curves for the doublet states of the molecule SiP.
Fig. 2. Potential energy curves for the quartet states of the molecule SiP.
C2 Rþ ), with Re decreasing to 4.035 a0 , compared to 4.063 a0 from our previous investigation, and 4.021 a0 from experiment; as to the adiabatic excitation energy, our new value of 21,059 cm1 shows a much better agreement with the experimental result of 21,398 cm1 than the value of 20,588 cm1 derived from our previous investigation. The new B2 P state lies 19,795 cm1 higher than the ground state and due to its relatively long equilibrium distance (4.422 a0 ) transitions to higher vibrational states of both X2 P and A2 Rþ states will be favored during an emission process. Other still higherlying doublet states are located in a very congested and perturbed region with various crossings by both doublet and quartet states and certainly will pose great difficulties to be investigated experimentally. As to the lowest quartets, this work predicts the first 4 Rþ and4 P states to be practically identical in energy, with the 4 Rþ slightly lower in energy by only 0.006 eV; similarly to the two lowest doublets, their internuclear distances differ significantly, with the b4 P longer by about 0.3 a0 . The next excited quartet is a D state (Re ¼ 4:150 a0 , Te ¼ 19,987 cm1 ) which, despite the favorable Franck– Condon region, does not couple to the a4 Rþ state by dipole-allowed transition rules. Transitions between the states 4 P and 4 D are in principle allowed leading to a vibrationally excited 4 D state in an IR absorption process. The energy difference at the minimum of these two states amounts to 4509 cm1 , which can be compared to the value of 5400 cm1 of the less extensive study of Bruna et al. [4].
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Experimentally, it was observed that the m ¼ 1 level of the now C2 Rþ state is strongly perturbed and, based on the theoretical estimates of Te by Bruna et al., the authors of the experimental work suggested the b4 P state as the most likely perturber. In fact, in Bruna et al.Õs calculation the a4 Rþ and b4 P states were predicted to lie very close, at 5400 cm1 below the C2 Rþ state, the c4 D was found also to lie lower at 80 cm1 , and the d4 R state higher at 3800 cm1 , but no overall view of curve crossings was presented; we also recall that in Bruna et al.Õs [4] study the potential energy curve of the 2 P state, more extensively investigated in this work, was not reported. To help elucidate in more details the nature of the perturber, we plotted in Fig. 3 an enlarged view of the potential energy curves close to the C2 Rþ state with an indication of the location of the zero-point vibrational energy and the first excited vibrational state. Clearly, we can see that the C2 Rþ state is crossed by the B2 P state below the m ¼ 0 level ðG0 ¼ 231 cm1 ), and by the c4 D state and the b4 P very close to the m ¼ 1 level ðG1 ¼ 688 cm1 ), with the b4 P closer to this latter level. More quantitatively, the C2 Rþ state is crossed first at 93 cm1 , then at 486 cm1 , and finally at 677 cm1 above the energy minimum. This result strongly supports the inference of Jakubek et al. about being the a4 P state the unknown perturber.
Fig. 4. Dipole moment functions for selected doublet states of the molecule SiP.
3.2. Dipole moment functions, vibrational energies, averaged dipole moments, and spectroscopic constants Dipole moments as a function of the internuclear distance are displayed in Figs. 4 and 5 for selected
Fig. 5. Dipole moment functions and transition moment functions for selected quartet states of the molecule SiP.
Fig. 3. Enlarged view of the potential energy curves of the molecule SiP showing curves crossing of the doublet and quartet states and vibrational levels.
doublet and quartet states, respectively; numerical values are available upon request to the authors. We recall that a negative value of the moment corresponds to the molecular polarity Siþ P , and that at large distances
L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
they all tend to zero since neutral fragments are formed. The behavior of the dipole moment function for the states of direct relevance to the present experimental data has been discussed in our previous papers. For the new states, in general terms, we can say that they all have a smooth behavior, except in regions of avoided crossings in the potential energy curves where a corresponding abrupt variation shows up in the dipole moment function reflecting changes in the character of the wavefunction. For the new 2 P state, likely to be identified in the near future, beyond 3.6 a0 and until 5.4 a0 , this function behaves quite linearly decreasing from about 0.34 a.u. (Si Pþ ) to )0.55 a.u. (Siþ P ), with a value close to zero around the equilibrium distance. For the three lowest-lying quartets, the dipole moment function of the a4 Rþ state shows a practically constant value of about )0.15 a.u. around the equilibrium distance (4.14 a.u.), whereas for the b4 P state, it shows a steep linear decrease from a value of about +0.36 a.u., at a short distance of 3.7 a.u., then changes sign close to the equilibrium distance (4.44 a.u.) and continues its linear decrease until about )0.48 a.u., at a longer distance of 5.3 a.u.; for the c4 D state, a linear behavior around the equilibrium distance is also observed, but with a smaller slope than that of the 4 P state, from about )0.18 a.u. at 3.5 a.u. to )0.48 a.u. at 5.0 a.u. The physical relevance of these curves rests on the fact that,
199
when properly combined with the vibrational wavefunctions, one has the possibility of predicting infrared transitions from first principles. Vibrational energies differences ðDGmþ1=2 Þ and zeropoint energies, and vibrationally averaged dipole moments ðhlm iÞ are collected in Tables 3 and 4, respectively, for selected vibrational and electronic states. These averages are of direct relevance since only dipole averages are determined experimentally. Concerning the zeropoint energies listed in Table 3, notice that the value of 734 cm1 reported in [8], and derived from experimental data via a RKR potential energy curve for the A2 Rþ state, is not consistent with the known fact that G0 xe =2 ¼ 340 cm1 ; the remaining Gm seem also to be incorrectly assigned. If not a misprint, this might be indicative that the labelling of the vibrational states assigned experimentally need to be reanalyzed. For the vibrational constants listed in Table 5 and evaluated by standard fitting procedures, we have been careful to specify the number of points (vibrational spacings) and of adjustable parameters (spectroscopic constants), since the fitting is dependent on these numbers as discussed by Richards et al. [29]. Rotational constants Bm were obtained as the averages hmj16:8576=lR2 jmi, where jmi stands for the vibrational state m; these constants are collected in Table 6 for selected states, and in Table 7 are the results for the other rotational constants. Since in
Table 3 Vibrational energies differences DGmþ1=2 and zero-point energies E0 , in cm1 , for selected electronic and vibrational m
A2 Rþ
X2 P
B2 P
C2 R þ
a4 Rþ
b4 P
c4 D
d4 R
Theor.
Exper.a
Theor.
Exper.a
Theor.
Theor.
Exper.a
Theor.
Theor.
Theor.
Theor.
0 1 2 3 4 5 6 7 8 9
606.7 601.3 595.8 590.3 585.3 581.0 576.6 572.1 567.2 561.9
611.0 606.3 601.7 597.0 592.3 587.6 582.9
668.8 661.7 655.1 648.9 643.0 637.1 631.2 625.5 620.0 614.4
674.4 668.8 663.2 657.6 652.1 646.5 640.9
433.1 425.8 420.7 416.8 413.2 409.8 406.4 403.1 400.0 397.1
457.3 451.3 445.5 439.7 434.0 428.7 423.8 418.9 414.2 409.5
449.0 443.0
496.2 483.9 472.4 463.2 455.7 448.7 442.5 436.8 431.8 427.5
405.3 400.8 396.5 392.6 388.6 384.6 380.4 376.1 371.6 366.9
483.6 476.7 469.8 463.2 457.0 451.5 446.3 441.6 437.6 434.2
466.9 459.0 453.0 447.8 443.3 440.0 438.3 437.8 437.5 436.7
E0
305.4
306.7
336.7
734.0a
196.2
230.6
251.9
204.4
244.5
236.3
a
See text.
Table 4 ) for selected electronic and vibrational states of the molecule SiP Vibrationally averaged dipole moments (e A m
X2 P
A2 Rþ
B2 P
C2 Rþ
a4 Rþ
b4 P
c4 D
d4 R
0 1 2 3 4 5
)0.1747 )0.1772 )0.1800 )0.1828 )0.1856 )0.1884
)0.2672 )0.2679 )0.2687 )0.2694 )0.2700 )0.2707
)0.0147 )0.0208 )0.0266 )0.0319 )0.0369 )0.0416
)0.0956 )0.0947 )0.0937 )0.0927 )0.0917 )0.0908
)0.0810 )0.0809 )0.0809 )0.0807 )0.0804 )0.0801
)0.0591 )0.0643 )0.0697 )0.0748 )0.0798 )0.0846
)0.1452 )0.1477 )0.1508 )0.1540 )0.1570 )0.1599
)0.1974 )0.2028 )0.2081 )0.2135 )0.2191 )0.2251
200
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Table 5 Vibrational constants (cm1 ) for selected electronic states of the molecule SiP State
xe
xe x e
xe ye
X2 P
612.5(5) 612.0(5) 615.7 676.1(5) 674.8(5) 680 439.9(6) 439.0(3) 464.1(5) 462.8(5) 454.6 510.2(6) 409.3(8) 491.0(5) 474.8(5)
2.888 2.689 2.34 3.804 3.213 2.8 3.851 3.115 3.465 2.910 2.83 7.405 2.098 3.716 4.136
0.022
A2 Rþ
B2 P C2 R þ
a4 Rþ b4 P c4 D d4 R
0.066
0.151
Table 7 Rotational constants (cm1 ) for selected states of the molecule SiP State
Be
ae
ce
X2 Pð5Þ A2 Rþ ð5Þ B2 Pð6Þ C2 Rþ ð5Þ a4 Rþ ð6Þ b4 Pð8Þ c4 Dð5Þ d4 R ð5Þ
0.2629 0.2871 0.2095 0.2515 0.2390 0.2078 0.2379 0.2360
0.122E ) 02 0.146E ) 02 0.155E ) 02 0.141E ) 02 0.201E ) 02 0.153E ) 02 0.123E ) 02 0.160E ) 02
)0.350E ) 04 )0.229E ) 04 0.516E ) 04 )0.757E ) 04 0.138E ) 04 0.105E ) 04 )0.707E ) 04 )0.407E ) 04
Values in parenthesis are the numbers of Bm data used in the fitting.
0.062
0.254 0.003 0.042 0.093
Values in parenthesis indicate the number of vibrational states used in the fitting. Experimental values [8] are given in italics.
our previous studies a comparison of the results with other theoretical investigations for three low-lying states has been done in detail, here we restrict ourselves to a comparison of the present results with the recent experimentally derived data. For the ground state, our present values derived from a five spacings/three parameters (5s/3p) fitting expression resulted in 612.5, 2.888, and 0.022 cm1 for xe , xe xe , and xe ye , whereas if only two parameters are used, our new values are: 612.0 and 2.689 cm1 ; note that it represents a slight improvement for xe xe if a comparison is made with the experimental values of 615.7(6) and 2.34(11) cm1 . For the A2 Rþ state, our new results of 676.1 and 3.804 cm1 (5s/3p) represent also an improvement over our previous values (661 and 2.003 cm1 ) when compared to experiment, 680(2) and 2.8(3) cm1 . Finally, for the now C2 Rþ state, since only two quanta (449.0 and 892.0 cm1 were used to derive the experimental constants (454.6 and 2.83 cm1 , with no statistical error estimate), and since the present calculation provides a much better description of this state, we expect that our new values resulted from a (5s/3p) fitting, 464.1, 3.464, and 0.062 cm1 for xe , xe xe , and xe ye , represent a more accurate determi-
nation of these constants; for a (5s/2p) fitting, we obtained 462.8 and 2.910 cm1 for xe and xe xe , respectively. In our previous works, we had 443.6 (440.7) and 1.423 (0.700) cm1 using only two adjustable parameters and six (two) points, and 440.2, 0.253, and )0.111 cm1 , using three parameters and six points. As an indication that the present calculation represents a better description of the C2 Rþ state, we note that our
Fig. 6. Transition dipole moment functions for selected doublet states of the molecule SiP. Solid lines and j, this work; , m, and +, Ref. [2].
Table 6 Rotational constants Bm (cm1 ) for selected electronic and vibrational states of the SiP molecule m
X2 P
A2 Rþ
B2 P
C2 R þ
a4 Rþ
b4 P
c4 D
d4 R
0
0.2864 0.2958 0.2849 0.2833
0.2087
0.2071
0.2372
0.2352
0.2361 0.2341
0.2056 0.2040
0.2359 0.2344
0.2335 0.2317
3 4 5
0.2582 0.2567 0.2553
0.2817 0.2801 0.2785
0.2047 0.2036 0.2025
0.2508 0.2491 0.2493 0.2476 0.2465 0.2457 0.2437 0.2417
0.2380
1 2
0.2623 0.2644 0.2610 0.2596
0.2320 0.2302 0.2284
0.2026 0.2012 0.1998
0.2327 0.2309 0.2292
0.2299 0.2279 0.2257
Experimental values are given in italics [8].
0.2073 0.2059
L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
previous value for Te (20,588 cm1 ) changed to 21,059 cm1 , a significant improvement compared to the value derived experimentally of 21,398 cm1 .
201
explored in the near future, that is from the new B2 P state, and also from the better described C2 Rþ state to the ground and A states; for the quartets our natural choice were the c4 D–b4 P and d4 R –b4 P band systems. For these systems, the behavior of the transition moment as a function of the internuclear distance is shown in Figs. 5 and 6. Numerical values and graphs for all possible transitions are available upon request to the authors.
3.3. Transition moments, transition probabilities, and radiative lifetimes To avoid an excessive number of tables and figures, we focused our attention only on transitions likely to be
Table 8 Einstein coefficients Am0 m00 (s1 ) and Franck–Condon factors (italics) for selected doublets and bands of SiP B2 P–A2 Rþ
B2 P–X2 P m00
m0 ¼ 0
4
429.6 0.126 598.5 0.174 656.6 0.189 582.8 0.164 426.2 0.118 259.8 0.070 132.7 0.035 56.9 0.015 20.5 0.005 6.2 0.002
5 6 7 8 9 10 11 12 13 A0m
3521.7
Qm0 a a
m0 ¼ 1
m0 ¼ 2
392.0 0.119 173.6 0.053 6.3 0.002 79.9 0.021 322.7 0.088 514.4 0.139 524.6 0.139 391.3 0.102 225.7 0.058 103.6 0.026 3604.4
1.000
56.0 0.018 30.6 0.008 234.3 0.067 295.3 0.086 113.9 0.034 0.8 0.000 157.4 0.041 416.8 0.108 525.5 0.135 434.1 0.11 3680.2
1.000
0.999
m0 ¼ 0
m0 ¼ 1
m0 ¼ 2
5.9 0.004 16.0 0.011 34.6 0.028 61.1 0.055 89.4 0.091 110.0 0.127 114.8 0.151 102.4 0.153 78.4 0.135 51.8 0.103
31.1 0.018 65.0 0.043 102.3 0.076 120.9 0.101 104.0 0.098 59.0 0.063 15.4 0.019 0.1 0.000 15.0 0.023 40.3 0.072
81.6 0.044 124.5 0.076 128.8 0.089 81.4 0.063 20.8 0.018 0.6 0.001 26.9 0.03 54.5 0.068 46.8 0.067 16.8 0.027
719.8
746.5
764.9
0.999
0.990
0.909
00
Sum up to m ¼ 19.
Table 9 Einstein coefficients Am0 m00 (s1 ) and Franck–Condon factors (italics) for selected doublets and bands of SiP C2 Rþ –X2 P m00 0 1 2 3 4 5 6 Am0 Qm0
C2 Rþ –A2 Rþ m0 ¼ 0
m0 ¼ 1
m0 ¼ 2
m0 ¼ 0
452,120 0.759 109,806 0.190 21,494 0.043 2887 0.007 307 0.001 25 0.000 2 0.000
133,788 0.221 219,420 0.390 145,688 0.259 49,038 0.103 9409 0.023 1328 0.004 143 0.000
10,544 0.020 209,600 0.356 81,432 0.158 135,138 0.248 72,584 0.158 19,095 0.047 3518 0.011
586,641
558,830
532,465
1.000
1.000
1.000
m0 ¼ 1
60,936 0.103 93,196 0.232 74,743 0.268 40,611 0.206 16,430 0.116 5231 0.051 1366 0.018 292,883 1.000
m0 ¼ 2
176,510 0.228 83,227 0.163 3165 0.010 11,625 0.046 27,719 0.156 22,998 0.180 11671 0.125
258,500 0.259 5879 0.011 40,060 0.083 37,169 0.118 2884 0.014 4879 0.029 15,020 0.127
342,817
389,280
1.000
0.999
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L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
Table 10 Total Einstein coefficients Am0 (s1 ) and radiative lifetimes (ls) for selected transitions and vibrational states of SiP Am0 m0
C2 Rþ –X2 P
C2 Rþ –A2 Rþ
Total C
s(C)
0
586,641 (513,244) 558,830 (480,270) 532,465 (451,655)
292,883 (332,142) 342,817 (416,251) 389,280 (465,608)
879,524 (845,386) 901,647 (896,521) 821,745 (917,263)
1.14 (1.18) 1.11 (1.12) 1.08 (1.09)
B2 Pþ –X2 P
B2 Pþ –A2 Rþ
Total B
s(B)
3522 3604 3746
720 747 765
4 242 4 351 4 511
235.7 229.8 221.6
1 2
0 1 2
Values in parenthesis are from [2].
For transitions involving the B2 P state to either the X or A states, the abrupt changes in behavior for internuclear distances smaller than about 3.5 a.u., reflecting an avoided crossing of the B2 P state with the much higher lying H2 P excited state, are not expected to have any significant contribution to the transition probability for the lower-lying vibrational states here discussed; for the region around the equilibrium internuclear distance of the B2 P state, the moments for both the B–A and B– X transitions are indeed very small (0.01–0.02 a.u.). We recall that, in the equilibrium region, the state B2 P is described mainly by the electronic configuration . . . 9r2 3p3 4p1 , with the orbital 4p antibonding in character, a fact that accounts for its large equilibrium distance, and also for the small transition moment, which is
dominated by the integral h3pjlj4pi in the case of the B2 P–X2 P transition, and by h9rjlj4pi, for B2 P–A2 Rþ . Under this circumstance, the intensities of transitions originating from the B state are expected to be very weak. A quantitative expression of these transition probabilities in terms of the Einstein Am0 m00 coefficients is given in Table 8 for both B–X and B–A transitions, and m0 ¼ 0–2. Clearly, for m0 ¼ 0, the most intense band will correspond to m00 ¼ 6 (A0;6 ¼ 657 s1 ) and be centered around 16 149 cm1 for the B–X system, and to m00 ¼ 10 (A0;10 ¼ 115 s1 ) and centered around 12,807 cm1 for B–A, with an intensity ratio (B–X/B–A) of about 7; if fact, for both systems the intensity is spread over the adjacent states. Notice also in Table 8 that estimates of intensity ratios based on the Franck–Condon factors differ significantly from those based on the Am0 m00 coefficients. How much weaker are the intensities for transitions originating from the B state compared to those of the now C2 Rþ state? For this new description of the C2 Rþ state, and in accordance with our previous study, Table 9 shows that the (0,0) band is predicted to be the most intense one (A0;0 ¼ 452120 s1 ) for the C–X system, and the ð0; 1Þ one ðA0;1 ¼ 93196 s1 Þ for the C–A system, also with an intensity ratio of about 7. Our new prediction for the (0,0) band origin, 20,984 cm1 , compared with our previous result, 20,504 cm1 improves the agreement with the experimentally determined origin of 21,223.7 cm1 for the C2 Rþ –X2 P1=2 system. For the C–A system, the ð1; 0Þ and ð2; 0Þ bands with origins at 20,945 cm1 (exp. 21,339 cm1 ) and 21,397 cm1 (exp. 21,782 cm1 ), are also expected to be relatively intense with A1;0 ¼ 176,510 cm1 , and A2;0 ¼ 258,500 s1 . Total
Table 11 Einstein coefficients Am0 m00 (s1 ) and Franck–Condon factors for selected quartet states and bands, and radiative lifetimes (ls) for the SiP molecule d4 R –b4 P m00 0 1 2 3 4 5 6 7 8
c4 D–b4 P m0 ¼ 0 4320.0 0.134 6869.2 0.232 6139.7 0.230 4099.5 0.173 2269.3 0.109 1098.6 0.061 480.5 0.032 193.8 0.015 73.2 0.007
m0 ¼ 1
m0 ¼ 2
10942.0 0.313 4426.0 0.133 78.1 0.002 1026.4 0.039 2520.2 0.105 2675.6 0.126 1982.5 0.108 1186.1 0.075 613.0 0.046
11484.0 0.311 119.5 0.004 4408.2 0.134 2979.4 0.096 292.8 0.010 270.3 0.011 1268.9 0.059 1711.8 0.091 1483.0 0.091
m0 ¼ 0 571.4 0.098 919.2 0.197 800.7 0.222 501.2 0.186 249.4 0.130 103.4 0.079 36.5 0.044 11.0 0.023 2.7 0.011
m0 ¼ 1
m0 ¼ 2
1894.1 0.257 1034.1 0.169 106.3 0.021 41.3 0.011 206.6 0.072 239.6 0.115 167.9 0.116 86.9 0.092 35.6 0.063
2724.5 0.304 40.2 0.005 524.6 0.084 611.3 0.119 166.0 0.040 0.0 0.000 60.0 0.027 109.4 0.069 94.5 0.089
Am0
25583.1
25931.5
26122.6
3196.0
3827.6
4418.5
sm0
39.1
38.6
38.2
312.9
261.2
226.3
L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
Einstein coefficients Am0 and radiative lifetimes sm0 for selected vibrational states are collected in Table 10. Interesting to note is the fact that although the present investigation reports a much improved description of the C2 Rþ state, including the transition moment function, the corrections in the Am0 m00 coefficients occur in opposite directions making the final corrections in the total Am0 coefficients very small and not significantly different from our previous prediction. Also worth pointing out is the ratio A0 ðCÞ=A0 ðBÞ ¼ 207, which is an expected measure of the intensity ratio for emissions originating from these two states. Excited molecules in the B state are expected to live longer than those in the C state by a factor of about 200. For the quartets, since the states d4 R and c4 D have similar internuclear distances, and the transition moment involving the c4 D is higher by about 10% than that of the d4 R , the intensity for emission from m0 ¼ 0, of either state, to the lower b4 P state will be dictated mainly by the excitation energy, which varies as m3 . For both c and d states, as shown in Table 11, this study predicts the ð0; 1Þ band as the most intense: for the c4 D state, it is expected to occur close to 3985 cm1 Am0 ¼ 919 s1 , and close to 7475 cm1 Am0 ¼ 6869 s1 for the d4 R state. If emission occurs from m0 ¼ 1, the transition probability almost doubles if m00 ¼ 0 is the final state. As to the radiative lifetime, s0 is predicted to be 39.1 ls for the state b4 P, and 312.9 ls for c4 D.
4. Conclusions The investigation described in this work provides reliable theoretical data characterizing several doublet and quartet states of the molecule SiP. A new 2 P state below the recently experimentally identified2 Rþ state is expected to allow one to access vibrationally excited states of both X and A states, and thus be another path for an extended characterization of these states. For the quartets, a detailed theoretical description is presented for the first time in the literature, which is expected to be a useful guide to experimentalists to properly identify the states and assign the transitions resulting from these states. A high-lying and dense set of states is expected to pose great difficulties for a proper experimental identification due to the various crossings and perturbations.
Acknowledgements The authors acknowledge the provision of computational facilities at the Laborat orio de Computacß~ao
203
Cientıfica Avancßada (LCCA) of the University of S~ ao Paulo, and the financial support of Fundacß~ao de Amparo a Pesquisa do Estrado de S~ao Paulo (FAPESP) under contract No. 00/08920-1R. A research fellowship of the Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico (CNPq) of Brazil for F.R.O., and a graduate one for L.G.S. of Brazil are also greatly appreciated.
References [1] F.R. Ornellas, C.M. Andreazza, A.A. de Almeida, Astrophys. J. 538 (2000) 675. [2] F.R. Ornellas, Chem. Phys. Lett. 335 (2001) 420. [3] P.J. Bruna, H. Dohmann, J. Anglada, V. Krumbach, S. Peyerrimhoff, R.J. Buenker, Mol. Phys. 93 (1983) 309. [4] P.J. Bruna, H. Dohmann, S. Peyerimhoff, Can. J. Phys. 62 (1984) 1508. [5] A.D. McLean, B. Liu, G.S. Chandler, J. Chem. Phys. 97 (1992) 8459. [6] A. Boldyrev, J. Simons, J. Phys. Chem. 97 (1993) 6147. [7] A. Boldyrev, N. Gonzales, J. Simons, J. Phys. Chem. 98 (1994) 9931. [8] Z.J. Jakubek, S.G. Nakhate, B. Simard, J. Chem. Phys. 116 (2002) 6513. [9] M.C. McCarthy, C.A. Gottlieb, P. Thaddeus, Mol. Phys. 101 (2003) 697. [10] M. Guelin, J. Cernicharo, G. Paubert, B.E. Turner, Astron. Astrophys. 230 (1990) L9. [11] B.E. Turner, J. Bally, Astrophys. J. Lett. 321 (1987) L75. [12] L.M. Ziurys, Astrophys. J. Lett. 321 (1987) L81. [13] F.R. Ornellas, S. Iwata, J. Chem. Phys. 107 (1997) 6782. [14] F.R. Ornellas, S. Iwata, Chem. Phys. 232 (1998) 95. [15] P.D. Singh, G.C. Sanzovo, A.C. Borin, F.R. Ornellas, Mon. Not. R. Astron. Soc. 303 (1999). [16] D.E. Woon, T.H. Dunning, J. Chem. Phys. 98 (1993) 1358. [17] H.-J. Werner, P.J. Knowles, J. Chem. Phys. 82 (1985) 5053. [18] P.J. Knowles, H.-J. Werner, Chem. Phys. Lett. 115 (1985) 259. [19] H.-J. Werner, P.J. Knowles, J. Chem. Phys. 89 (1988) 5803. [20] P.J. Knowles, H.-J. Werner, Chem. Phys. Lett. 145 (1988) 514. [21] MOLPRO is a package of ab initio programs written by H.-J. Werner, P.J. Knowles with contributions from J. Alml€ of, R.D. Amos, A. Berning, D.L. Cooper, M.J.O. Deegan, A.J. Dobbyn, F. Eckert, S.T. Elbert, C. Hampel, R. Lindh, A.W. Lloyd, W. Meyer, A. Nicklass, K. Peterson, R. Pitzer, A.J. Stone, P.R. Taylor, M.E. Mura, P. Pulay, M. Sch€ utz, H.Stoll, T. Thorsteinsson. [22] W.T. Zemke, W.C. Stwalley, QCPE Bull. 4 (1981) 79. [23] F.R. Ornellas, J. Chem. Phys. 82 (1985) 379. [24] F.B.C. Machado, F.R. Ornellas, J. Chem. Phys. 94 (1991) 7237. [25] A.C. Borin, F.R. Ornellas, J. Chem. Phys. 98 (1993) 8761. [26] A.C. Borin, F.R. Ornellas, Chem. Phys. 184 (1994) 59. [27] G. Herzberg (Ed.), Molecular Spectra and Molecular Structure, Spectra of Diatomic Molecules, vol. I, Van Nostrand Reinhold, New York, 1950, p. 315. [28] A.A. Radzig, B.M. Smirnov, Reference Data on Atoms, Molecules, and Ions, Springer, Berlin, 1985. [29] W.G. Richards, J. Raftery, R.K. Hinkley, in: R.N. Dixon (Ed.), Theoretical Chemistry, vol. 2, The Chemical Society, London, 1973 (Chapter 1).
Excited doublet and quartet states of SiP: a high level theoretical investigation Levi G. dos Santos, Fernando R. Ornellas
*
Dept. de Quımica Fundamental, Instituto de Quımica, Universidade de S~ ao Paulo, Caixa Postal 26077, S~ ao Paulo SP 05513-970, Brazil Received 25 August 2003; accepted 16 September 2003
Abstract Doublet and quartet states of the SiP molecule dissociating into the four lowest dissociation channels are characterized theoretically at a high-level of correlation treatment (multireference single and double excitation configuration interaction). Potential energy curves give a global view of the manifold of possible electronic states. For selected states, dipole and transition moment functions, and transition probabilities and radiative lifetimes are also reported as well as an extensive set of spectroscopic constants. A new 2 P state offers another likely route for exploring transitions to excited vibrational states of both X2 P and A2 Rþ states. A detailed set of data for the quartet states is expected to provide valuable information for the experimental identification of these states. Ó 2003 Elsevier B.V. All rights reserved. Keywords: SiP; Multireference configuration interaction; Excited electronic states; Potential energy curves; Spectroscopic constants; Transition probability; Radiative lifetime
1. Introduction In recent studies, we characterized the ground state (X2 P) [1] and two excited states of 2 Rþ symmetry [2] of the radical SiP. In the first investigation, a very accurate multireference configuration interaction calculation was set up to definitely establish the relative order of the first two electronic states which had been a subject of discussion in previous theoretical studies [3–7], and in the second work, due to the small adiabatic excitation energy (470 cm1 ) between the two lowest states estimated in the first study, we described the next 2 Rþ state and proposed that transitions involving this state would be a promising path for the spectroscopic characterization of all three states. More recently, in a study by Jakubek et al. [8], the SiP radical was observed in the laboratory and the three lowest states characterized spectroscopically, thus verifying our predictions, and providing accurate laboratory data that can aid astrophysicists in the *
Corresponding author. Tel.: +55-11-3091-3895; fax: +55-11-38155579. E-mail address: [email protected] (F.R. Ornellas). 0301-0104/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2003.09.007
search of this species in interstellar/circumstellar gases. Since other silicon-containing diatomics like SiC, SiN, SiO, and SiS [9] had been observed in outer space, as well as the isoelectronic radical CP [10], and also the species PN [11,12], the possibility of observation of the SiP molecule has become a challenge since the cosmic abundance of Si and P atoms makes its formation very likely. In this work, guided by the same approach of our previous investigations of silicon-containing species like SiB [13], SiN [14], and SiAl [15], we extended the previous studies of SiP to include several electronic states of both doublet and quartet multiplicities that correlate with the first four dissociation channels. Besides describing new possibilities of transitions to the ground state, and providing reliable data that can guide spectroscopists to make the first observation of quartet transitions, the characterization of the quartet state correlating with the first dissociation channel makes available all the data needed for the evaluation of the rate constant by the radiative association model, commonly used by astrophysicists, thus contributing to a better understanding of the chemistry of theses species in astrophysical environments.
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2. Methods The electronic calculations carried out in this study involved basically two steps. In the first one, the aim was to assess the importance of static correlation effects and to define a common set of molecular orbitals to be used for all states; in the second step, one was mainly concerned with incorporating as much dynamic correlation as possible in the final wavefunction having in mind a balance between its dimension and computation time. Our choice of atomic basis functions was the set of augmented correlation consistent valence-polarized quadruple-zeta (aug-cc-pVQZ) functions developed by Woon and Dunning [16]. The first step consisted of state-averaged complete active space self-consistent field (CASSCF) [17,18] calculations separately for each set of doublet and quartet states, involving a total of 16 states, distributed according to the symmetry representations A1 (5), B1 (4), B2 (4), and A2 (3) of the C2v point group for the doublets, and 20 states (A1 (5), B1 (5), B2 (5), and A2 (5)) for the quartets. Averaged natural orbitals for each multiplicity were next computed using the CASSCF wavefunctions which contained all possible electronic excitations resulting from the distribution of nine electrons into 11 active orbitals ð5; 3; 3; 0Þ, which comprised the valence orbitals plus one correlating orbital for each of the A1 , B1 , B2 symmetries; core and inner-shell orbitals were kept doubly occupied. To limit the dimension of the final wavefunction, the most important configuration state functions of the CASSCF calculations were next selected on the basis of their coefficients (greater than 0.015 in magnitude for each internuclear distance) and formed the core of a zerothorder wavefunction from which the final configuration interaction wavefunction (CI) was generated as all single and double excitations from this reference set. The CI wavefunction was constructed by the internally contracted approach [19,20] implemented in the Molpro 98 suite of programs [21]. For the range of internuclear distances investigated, cubic spline fits to the calculated points generated potential energy functions used in the solution of SchroedingerÕs radial equation for the nuclear motion by the program Intensity [22], which besides calculating vibrational–rotational energies and
vibrational wavefunctions, allowed also for the calculation of radiative transition probabilities; radiative lifetimes were evaluated as the inverse of the total radiative transition probability. Standard fitting procedures used for the calculation of the spectroscopic constants and the convention employed in the transition probabilities evaluation are discussed elsewhere [23–26].
3. Results and discussion 3.1. Energetic aspects The first step in this study involved applying the Wigner–Witmer rules [27] to identify the possible molecular electronic states correlating with the first few dissociation channels associated with the electronic states of the atomic fragments Si and P. Table 1 summarizes the manifold of states allowed for the four lowest dissociation channels. Energies as a function of the internuclear distance for the 17 lowest lying states are graphically displayed in Figs. 1 and 2, for the doublets and the quartets, respectively. To avoid an excessive number of lengthy tables, only numerical values of the energy at the equilibrium distance ðRe Þ are collected in Table 2 together with the adiabatic energies ðTe Þ; the remaining values are available upon request to the authors. Compared to our previous studies where only four states (2 Rþ ð2Þ, 2 Px , 2 Py ), transforming according to the representations A1 (2), B1 (1), B2 (1) of the C2v point group symmetry, were optimized in the CASSCF calculation, the new description of the X and A states was slightly improved. However, we note that the previously denoted B2 Rþ state corresponds, in fact, in the present investigation, to the third excited state; by including all the doublet states compatible with the new active space ð5; 3; 3; 0Þ in the state-averaged CASSCF calculation, a2 P state lower in energy than our previous B2 Rþ state was found, thus calling for a renaming of the corresponding state found in the recent experiment and of our previous result. In this study, the inclusion of more states in the optimization process improved significantly our description of the second excited 2 Rþ state (now
Table 1 Low-lying electronic states of the molecule SiP, their dissociation channels, and energy separation at the dissociation limit States of separated atoms
P(4 Su ) + Si(3 Pg ) P(4 Su ) + Si(1 Dg ) P(4 Du ) + Si(3 Pg ) P(4 Su ) + Si(1 Sg ) a b
Calculated at R ¼ 9:0 a0 . Ref. [28].
DE (cm1 )
Molecular states
Rþ , 2;4 P R , 4 P, 4 D 2;4 þ R ð2Þ, 2;4 R , 4 R 2;4 4
2;4
Pð3Þ,
2;4
Dð2Þ,
2;4
U
Theoreticala
Experimentalb
0.0 6421.7 12084.5 15290.8
0.0 6298.9 11376.6 15394.4
L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
197
Table 2 Equilibrium distances ða0 Þ, energies (a.u.) at the equilibrium distance, and adiabatic excitation energies (cm1 ) for the doublet and quartet states of the molecule SiP State
Re
EðRe Þ
Te
X2 P A2 Rþ B2 P C2 R þ D2 U E2 R F2 D G2 D H2 P I2 Rþ
3.947a;b 3.777a;b 4.422 4.035a;b 4.460 4.149 4.144 4.236 4.436 3.939
0.888216 0.886096 0.798019 0.792261 0.779714 0.769141 0.764604 0.759619 0.757084 0.735856
0.0 465a; b 19,795 21,059a ;b 23,813 26,133 27,129 28,223 28,779 33,438
a4 Rþ b4 P c4 D d4 R f4 D g4 Rþ h4 R
4.141 4.440 4.150 4.166 4.958 4.471 5.410
0.817898 0.817692 0.797147 0.781930 0.722902 0.719134 0.713611
15,432 15,478 19,987 23,326 36,281 37,108 38,320
a Theor. 3.948 a0 (X); 3.783 a0 , 470 cm1 (A); 4. 063 a0 , 20,588 cm1 (C), Ref. [1,2]. b Exp. 3.926 a0 (X), 3.715 a0 (A), 395.4 cm1 ; 4.021 a0 , 21,398 cm1 (C); derived from the data in [8].
Fig. 1. Potential energy curves for the doublet states of the molecule SiP.
Fig. 2. Potential energy curves for the quartet states of the molecule SiP.
C2 Rþ ), with Re decreasing to 4.035 a0 , compared to 4.063 a0 from our previous investigation, and 4.021 a0 from experiment; as to the adiabatic excitation energy, our new value of 21,059 cm1 shows a much better agreement with the experimental result of 21,398 cm1 than the value of 20,588 cm1 derived from our previous investigation. The new B2 P state lies 19,795 cm1 higher than the ground state and due to its relatively long equilibrium distance (4.422 a0 ) transitions to higher vibrational states of both X2 P and A2 Rþ states will be favored during an emission process. Other still higherlying doublet states are located in a very congested and perturbed region with various crossings by both doublet and quartet states and certainly will pose great difficulties to be investigated experimentally. As to the lowest quartets, this work predicts the first 4 Rþ and4 P states to be practically identical in energy, with the 4 Rþ slightly lower in energy by only 0.006 eV; similarly to the two lowest doublets, their internuclear distances differ significantly, with the b4 P longer by about 0.3 a0 . The next excited quartet is a D state (Re ¼ 4:150 a0 , Te ¼ 19,987 cm1 ) which, despite the favorable Franck– Condon region, does not couple to the a4 Rþ state by dipole-allowed transition rules. Transitions between the states 4 P and 4 D are in principle allowed leading to a vibrationally excited 4 D state in an IR absorption process. The energy difference at the minimum of these two states amounts to 4509 cm1 , which can be compared to the value of 5400 cm1 of the less extensive study of Bruna et al. [4].
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L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
Experimentally, it was observed that the m ¼ 1 level of the now C2 Rþ state is strongly perturbed and, based on the theoretical estimates of Te by Bruna et al., the authors of the experimental work suggested the b4 P state as the most likely perturber. In fact, in Bruna et al.Õs calculation the a4 Rþ and b4 P states were predicted to lie very close, at 5400 cm1 below the C2 Rþ state, the c4 D was found also to lie lower at 80 cm1 , and the d4 R state higher at 3800 cm1 , but no overall view of curve crossings was presented; we also recall that in Bruna et al.Õs [4] study the potential energy curve of the 2 P state, more extensively investigated in this work, was not reported. To help elucidate in more details the nature of the perturber, we plotted in Fig. 3 an enlarged view of the potential energy curves close to the C2 Rþ state with an indication of the location of the zero-point vibrational energy and the first excited vibrational state. Clearly, we can see that the C2 Rþ state is crossed by the B2 P state below the m ¼ 0 level ðG0 ¼ 231 cm1 ), and by the c4 D state and the b4 P very close to the m ¼ 1 level ðG1 ¼ 688 cm1 ), with the b4 P closer to this latter level. More quantitatively, the C2 Rþ state is crossed first at 93 cm1 , then at 486 cm1 , and finally at 677 cm1 above the energy minimum. This result strongly supports the inference of Jakubek et al. about being the a4 P state the unknown perturber.
Fig. 4. Dipole moment functions for selected doublet states of the molecule SiP.
3.2. Dipole moment functions, vibrational energies, averaged dipole moments, and spectroscopic constants Dipole moments as a function of the internuclear distance are displayed in Figs. 4 and 5 for selected
Fig. 5. Dipole moment functions and transition moment functions for selected quartet states of the molecule SiP.
Fig. 3. Enlarged view of the potential energy curves of the molecule SiP showing curves crossing of the doublet and quartet states and vibrational levels.
doublet and quartet states, respectively; numerical values are available upon request to the authors. We recall that a negative value of the moment corresponds to the molecular polarity Siþ P , and that at large distances
L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
they all tend to zero since neutral fragments are formed. The behavior of the dipole moment function for the states of direct relevance to the present experimental data has been discussed in our previous papers. For the new states, in general terms, we can say that they all have a smooth behavior, except in regions of avoided crossings in the potential energy curves where a corresponding abrupt variation shows up in the dipole moment function reflecting changes in the character of the wavefunction. For the new 2 P state, likely to be identified in the near future, beyond 3.6 a0 and until 5.4 a0 , this function behaves quite linearly decreasing from about 0.34 a.u. (Si Pþ ) to )0.55 a.u. (Siþ P ), with a value close to zero around the equilibrium distance. For the three lowest-lying quartets, the dipole moment function of the a4 Rþ state shows a practically constant value of about )0.15 a.u. around the equilibrium distance (4.14 a.u.), whereas for the b4 P state, it shows a steep linear decrease from a value of about +0.36 a.u., at a short distance of 3.7 a.u., then changes sign close to the equilibrium distance (4.44 a.u.) and continues its linear decrease until about )0.48 a.u., at a longer distance of 5.3 a.u.; for the c4 D state, a linear behavior around the equilibrium distance is also observed, but with a smaller slope than that of the 4 P state, from about )0.18 a.u. at 3.5 a.u. to )0.48 a.u. at 5.0 a.u. The physical relevance of these curves rests on the fact that,
199
when properly combined with the vibrational wavefunctions, one has the possibility of predicting infrared transitions from first principles. Vibrational energies differences ðDGmþ1=2 Þ and zeropoint energies, and vibrationally averaged dipole moments ðhlm iÞ are collected in Tables 3 and 4, respectively, for selected vibrational and electronic states. These averages are of direct relevance since only dipole averages are determined experimentally. Concerning the zeropoint energies listed in Table 3, notice that the value of 734 cm1 reported in [8], and derived from experimental data via a RKR potential energy curve for the A2 Rþ state, is not consistent with the known fact that G0 xe =2 ¼ 340 cm1 ; the remaining Gm seem also to be incorrectly assigned. If not a misprint, this might be indicative that the labelling of the vibrational states assigned experimentally need to be reanalyzed. For the vibrational constants listed in Table 5 and evaluated by standard fitting procedures, we have been careful to specify the number of points (vibrational spacings) and of adjustable parameters (spectroscopic constants), since the fitting is dependent on these numbers as discussed by Richards et al. [29]. Rotational constants Bm were obtained as the averages hmj16:8576=lR2 jmi, where jmi stands for the vibrational state m; these constants are collected in Table 6 for selected states, and in Table 7 are the results for the other rotational constants. Since in
Table 3 Vibrational energies differences DGmþ1=2 and zero-point energies E0 , in cm1 , for selected electronic and vibrational m
A2 Rþ
X2 P
B2 P
C2 R þ
a4 Rþ
b4 P
c4 D
d4 R
Theor.
Exper.a
Theor.
Exper.a
Theor.
Theor.
Exper.a
Theor.
Theor.
Theor.
Theor.
0 1 2 3 4 5 6 7 8 9
606.7 601.3 595.8 590.3 585.3 581.0 576.6 572.1 567.2 561.9
611.0 606.3 601.7 597.0 592.3 587.6 582.9
668.8 661.7 655.1 648.9 643.0 637.1 631.2 625.5 620.0 614.4
674.4 668.8 663.2 657.6 652.1 646.5 640.9
433.1 425.8 420.7 416.8 413.2 409.8 406.4 403.1 400.0 397.1
457.3 451.3 445.5 439.7 434.0 428.7 423.8 418.9 414.2 409.5
449.0 443.0
496.2 483.9 472.4 463.2 455.7 448.7 442.5 436.8 431.8 427.5
405.3 400.8 396.5 392.6 388.6 384.6 380.4 376.1 371.6 366.9
483.6 476.7 469.8 463.2 457.0 451.5 446.3 441.6 437.6 434.2
466.9 459.0 453.0 447.8 443.3 440.0 438.3 437.8 437.5 436.7
E0
305.4
306.7
336.7
734.0a
196.2
230.6
251.9
204.4
244.5
236.3
a
See text.
Table 4 ) for selected electronic and vibrational states of the molecule SiP Vibrationally averaged dipole moments (e A m
X2 P
A2 Rþ
B2 P
C2 Rþ
a4 Rþ
b4 P
c4 D
d4 R
0 1 2 3 4 5
)0.1747 )0.1772 )0.1800 )0.1828 )0.1856 )0.1884
)0.2672 )0.2679 )0.2687 )0.2694 )0.2700 )0.2707
)0.0147 )0.0208 )0.0266 )0.0319 )0.0369 )0.0416
)0.0956 )0.0947 )0.0937 )0.0927 )0.0917 )0.0908
)0.0810 )0.0809 )0.0809 )0.0807 )0.0804 )0.0801
)0.0591 )0.0643 )0.0697 )0.0748 )0.0798 )0.0846
)0.1452 )0.1477 )0.1508 )0.1540 )0.1570 )0.1599
)0.1974 )0.2028 )0.2081 )0.2135 )0.2191 )0.2251
200
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Table 5 Vibrational constants (cm1 ) for selected electronic states of the molecule SiP State
xe
xe x e
xe ye
X2 P
612.5(5) 612.0(5) 615.7 676.1(5) 674.8(5) 680 439.9(6) 439.0(3) 464.1(5) 462.8(5) 454.6 510.2(6) 409.3(8) 491.0(5) 474.8(5)
2.888 2.689 2.34 3.804 3.213 2.8 3.851 3.115 3.465 2.910 2.83 7.405 2.098 3.716 4.136
0.022
A2 Rþ
B2 P C2 R þ
a4 Rþ b4 P c4 D d4 R
0.066
0.151
Table 7 Rotational constants (cm1 ) for selected states of the molecule SiP State
Be
ae
ce
X2 Pð5Þ A2 Rþ ð5Þ B2 Pð6Þ C2 Rþ ð5Þ a4 Rþ ð6Þ b4 Pð8Þ c4 Dð5Þ d4 R ð5Þ
0.2629 0.2871 0.2095 0.2515 0.2390 0.2078 0.2379 0.2360
0.122E ) 02 0.146E ) 02 0.155E ) 02 0.141E ) 02 0.201E ) 02 0.153E ) 02 0.123E ) 02 0.160E ) 02
)0.350E ) 04 )0.229E ) 04 0.516E ) 04 )0.757E ) 04 0.138E ) 04 0.105E ) 04 )0.707E ) 04 )0.407E ) 04
Values in parenthesis are the numbers of Bm data used in the fitting.
0.062
0.254 0.003 0.042 0.093
Values in parenthesis indicate the number of vibrational states used in the fitting. Experimental values [8] are given in italics.
our previous studies a comparison of the results with other theoretical investigations for three low-lying states has been done in detail, here we restrict ourselves to a comparison of the present results with the recent experimentally derived data. For the ground state, our present values derived from a five spacings/three parameters (5s/3p) fitting expression resulted in 612.5, 2.888, and 0.022 cm1 for xe , xe xe , and xe ye , whereas if only two parameters are used, our new values are: 612.0 and 2.689 cm1 ; note that it represents a slight improvement for xe xe if a comparison is made with the experimental values of 615.7(6) and 2.34(11) cm1 . For the A2 Rþ state, our new results of 676.1 and 3.804 cm1 (5s/3p) represent also an improvement over our previous values (661 and 2.003 cm1 ) when compared to experiment, 680(2) and 2.8(3) cm1 . Finally, for the now C2 Rþ state, since only two quanta (449.0 and 892.0 cm1 were used to derive the experimental constants (454.6 and 2.83 cm1 , with no statistical error estimate), and since the present calculation provides a much better description of this state, we expect that our new values resulted from a (5s/3p) fitting, 464.1, 3.464, and 0.062 cm1 for xe , xe xe , and xe ye , represent a more accurate determi-
nation of these constants; for a (5s/2p) fitting, we obtained 462.8 and 2.910 cm1 for xe and xe xe , respectively. In our previous works, we had 443.6 (440.7) and 1.423 (0.700) cm1 using only two adjustable parameters and six (two) points, and 440.2, 0.253, and )0.111 cm1 , using three parameters and six points. As an indication that the present calculation represents a better description of the C2 Rþ state, we note that our
Fig. 6. Transition dipole moment functions for selected doublet states of the molecule SiP. Solid lines and j, this work; , m, and +, Ref. [2].
Table 6 Rotational constants Bm (cm1 ) for selected electronic and vibrational states of the SiP molecule m
X2 P
A2 Rþ
B2 P
C2 R þ
a4 Rþ
b4 P
c4 D
d4 R
0
0.2864 0.2958 0.2849 0.2833
0.2087
0.2071
0.2372
0.2352
0.2361 0.2341
0.2056 0.2040
0.2359 0.2344
0.2335 0.2317
3 4 5
0.2582 0.2567 0.2553
0.2817 0.2801 0.2785
0.2047 0.2036 0.2025
0.2508 0.2491 0.2493 0.2476 0.2465 0.2457 0.2437 0.2417
0.2380
1 2
0.2623 0.2644 0.2610 0.2596
0.2320 0.2302 0.2284
0.2026 0.2012 0.1998
0.2327 0.2309 0.2292
0.2299 0.2279 0.2257
Experimental values are given in italics [8].
0.2073 0.2059
L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
previous value for Te (20,588 cm1 ) changed to 21,059 cm1 , a significant improvement compared to the value derived experimentally of 21,398 cm1 .
201
explored in the near future, that is from the new B2 P state, and also from the better described C2 Rþ state to the ground and A states; for the quartets our natural choice were the c4 D–b4 P and d4 R –b4 P band systems. For these systems, the behavior of the transition moment as a function of the internuclear distance is shown in Figs. 5 and 6. Numerical values and graphs for all possible transitions are available upon request to the authors.
3.3. Transition moments, transition probabilities, and radiative lifetimes To avoid an excessive number of tables and figures, we focused our attention only on transitions likely to be
Table 8 Einstein coefficients Am0 m00 (s1 ) and Franck–Condon factors (italics) for selected doublets and bands of SiP B2 P–A2 Rþ
B2 P–X2 P m00
m0 ¼ 0
4
429.6 0.126 598.5 0.174 656.6 0.189 582.8 0.164 426.2 0.118 259.8 0.070 132.7 0.035 56.9 0.015 20.5 0.005 6.2 0.002
5 6 7 8 9 10 11 12 13 A0m
3521.7
Qm0 a a
m0 ¼ 1
m0 ¼ 2
392.0 0.119 173.6 0.053 6.3 0.002 79.9 0.021 322.7 0.088 514.4 0.139 524.6 0.139 391.3 0.102 225.7 0.058 103.6 0.026 3604.4
1.000
56.0 0.018 30.6 0.008 234.3 0.067 295.3 0.086 113.9 0.034 0.8 0.000 157.4 0.041 416.8 0.108 525.5 0.135 434.1 0.11 3680.2
1.000
0.999
m0 ¼ 0
m0 ¼ 1
m0 ¼ 2
5.9 0.004 16.0 0.011 34.6 0.028 61.1 0.055 89.4 0.091 110.0 0.127 114.8 0.151 102.4 0.153 78.4 0.135 51.8 0.103
31.1 0.018 65.0 0.043 102.3 0.076 120.9 0.101 104.0 0.098 59.0 0.063 15.4 0.019 0.1 0.000 15.0 0.023 40.3 0.072
81.6 0.044 124.5 0.076 128.8 0.089 81.4 0.063 20.8 0.018 0.6 0.001 26.9 0.03 54.5 0.068 46.8 0.067 16.8 0.027
719.8
746.5
764.9
0.999
0.990
0.909
00
Sum up to m ¼ 19.
Table 9 Einstein coefficients Am0 m00 (s1 ) and Franck–Condon factors (italics) for selected doublets and bands of SiP C2 Rþ –X2 P m00 0 1 2 3 4 5 6 Am0 Qm0
C2 Rþ –A2 Rþ m0 ¼ 0
m0 ¼ 1
m0 ¼ 2
m0 ¼ 0
452,120 0.759 109,806 0.190 21,494 0.043 2887 0.007 307 0.001 25 0.000 2 0.000
133,788 0.221 219,420 0.390 145,688 0.259 49,038 0.103 9409 0.023 1328 0.004 143 0.000
10,544 0.020 209,600 0.356 81,432 0.158 135,138 0.248 72,584 0.158 19,095 0.047 3518 0.011
586,641
558,830
532,465
1.000
1.000
1.000
m0 ¼ 1
60,936 0.103 93,196 0.232 74,743 0.268 40,611 0.206 16,430 0.116 5231 0.051 1366 0.018 292,883 1.000
m0 ¼ 2
176,510 0.228 83,227 0.163 3165 0.010 11,625 0.046 27,719 0.156 22,998 0.180 11671 0.125
258,500 0.259 5879 0.011 40,060 0.083 37,169 0.118 2884 0.014 4879 0.029 15,020 0.127
342,817
389,280
1.000
0.999
202
L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
Table 10 Total Einstein coefficients Am0 (s1 ) and radiative lifetimes (ls) for selected transitions and vibrational states of SiP Am0 m0
C2 Rþ –X2 P
C2 Rþ –A2 Rþ
Total C
s(C)
0
586,641 (513,244) 558,830 (480,270) 532,465 (451,655)
292,883 (332,142) 342,817 (416,251) 389,280 (465,608)
879,524 (845,386) 901,647 (896,521) 821,745 (917,263)
1.14 (1.18) 1.11 (1.12) 1.08 (1.09)
B2 Pþ –X2 P
B2 Pþ –A2 Rþ
Total B
s(B)
3522 3604 3746
720 747 765
4 242 4 351 4 511
235.7 229.8 221.6
1 2
0 1 2
Values in parenthesis are from [2].
For transitions involving the B2 P state to either the X or A states, the abrupt changes in behavior for internuclear distances smaller than about 3.5 a.u., reflecting an avoided crossing of the B2 P state with the much higher lying H2 P excited state, are not expected to have any significant contribution to the transition probability for the lower-lying vibrational states here discussed; for the region around the equilibrium internuclear distance of the B2 P state, the moments for both the B–A and B– X transitions are indeed very small (0.01–0.02 a.u.). We recall that, in the equilibrium region, the state B2 P is described mainly by the electronic configuration . . . 9r2 3p3 4p1 , with the orbital 4p antibonding in character, a fact that accounts for its large equilibrium distance, and also for the small transition moment, which is
dominated by the integral h3pjlj4pi in the case of the B2 P–X2 P transition, and by h9rjlj4pi, for B2 P–A2 Rþ . Under this circumstance, the intensities of transitions originating from the B state are expected to be very weak. A quantitative expression of these transition probabilities in terms of the Einstein Am0 m00 coefficients is given in Table 8 for both B–X and B–A transitions, and m0 ¼ 0–2. Clearly, for m0 ¼ 0, the most intense band will correspond to m00 ¼ 6 (A0;6 ¼ 657 s1 ) and be centered around 16 149 cm1 for the B–X system, and to m00 ¼ 10 (A0;10 ¼ 115 s1 ) and centered around 12,807 cm1 for B–A, with an intensity ratio (B–X/B–A) of about 7; if fact, for both systems the intensity is spread over the adjacent states. Notice also in Table 8 that estimates of intensity ratios based on the Franck–Condon factors differ significantly from those based on the Am0 m00 coefficients. How much weaker are the intensities for transitions originating from the B state compared to those of the now C2 Rþ state? For this new description of the C2 Rþ state, and in accordance with our previous study, Table 9 shows that the (0,0) band is predicted to be the most intense one (A0;0 ¼ 452120 s1 ) for the C–X system, and the ð0; 1Þ one ðA0;1 ¼ 93196 s1 Þ for the C–A system, also with an intensity ratio of about 7. Our new prediction for the (0,0) band origin, 20,984 cm1 , compared with our previous result, 20,504 cm1 improves the agreement with the experimentally determined origin of 21,223.7 cm1 for the C2 Rþ –X2 P1=2 system. For the C–A system, the ð1; 0Þ and ð2; 0Þ bands with origins at 20,945 cm1 (exp. 21,339 cm1 ) and 21,397 cm1 (exp. 21,782 cm1 ), are also expected to be relatively intense with A1;0 ¼ 176,510 cm1 , and A2;0 ¼ 258,500 s1 . Total
Table 11 Einstein coefficients Am0 m00 (s1 ) and Franck–Condon factors for selected quartet states and bands, and radiative lifetimes (ls) for the SiP molecule d4 R –b4 P m00 0 1 2 3 4 5 6 7 8
c4 D–b4 P m0 ¼ 0 4320.0 0.134 6869.2 0.232 6139.7 0.230 4099.5 0.173 2269.3 0.109 1098.6 0.061 480.5 0.032 193.8 0.015 73.2 0.007
m0 ¼ 1
m0 ¼ 2
10942.0 0.313 4426.0 0.133 78.1 0.002 1026.4 0.039 2520.2 0.105 2675.6 0.126 1982.5 0.108 1186.1 0.075 613.0 0.046
11484.0 0.311 119.5 0.004 4408.2 0.134 2979.4 0.096 292.8 0.010 270.3 0.011 1268.9 0.059 1711.8 0.091 1483.0 0.091
m0 ¼ 0 571.4 0.098 919.2 0.197 800.7 0.222 501.2 0.186 249.4 0.130 103.4 0.079 36.5 0.044 11.0 0.023 2.7 0.011
m0 ¼ 1
m0 ¼ 2
1894.1 0.257 1034.1 0.169 106.3 0.021 41.3 0.011 206.6 0.072 239.6 0.115 167.9 0.116 86.9 0.092 35.6 0.063
2724.5 0.304 40.2 0.005 524.6 0.084 611.3 0.119 166.0 0.040 0.0 0.000 60.0 0.027 109.4 0.069 94.5 0.089
Am0
25583.1
25931.5
26122.6
3196.0
3827.6
4418.5
sm0
39.1
38.6
38.2
312.9
261.2
226.3
L.G. dos Santos, F.R. Ornellas / Chemical Physics 295 (2003) 195–203
Einstein coefficients Am0 and radiative lifetimes sm0 for selected vibrational states are collected in Table 10. Interesting to note is the fact that although the present investigation reports a much improved description of the C2 Rþ state, including the transition moment function, the corrections in the Am0 m00 coefficients occur in opposite directions making the final corrections in the total Am0 coefficients very small and not significantly different from our previous prediction. Also worth pointing out is the ratio A0 ðCÞ=A0 ðBÞ ¼ 207, which is an expected measure of the intensity ratio for emissions originating from these two states. Excited molecules in the B state are expected to live longer than those in the C state by a factor of about 200. For the quartets, since the states d4 R and c4 D have similar internuclear distances, and the transition moment involving the c4 D is higher by about 10% than that of the d4 R , the intensity for emission from m0 ¼ 0, of either state, to the lower b4 P state will be dictated mainly by the excitation energy, which varies as m3 . For both c and d states, as shown in Table 11, this study predicts the ð0; 1Þ band as the most intense: for the c4 D state, it is expected to occur close to 3985 cm1 Am0 ¼ 919 s1 , and close to 7475 cm1 Am0 ¼ 6869 s1 for the d4 R state. If emission occurs from m0 ¼ 1, the transition probability almost doubles if m00 ¼ 0 is the final state. As to the radiative lifetime, s0 is predicted to be 39.1 ls for the state b4 P, and 312.9 ls for c4 D.
4. Conclusions The investigation described in this work provides reliable theoretical data characterizing several doublet and quartet states of the molecule SiP. A new 2 P state below the recently experimentally identified2 Rþ state is expected to allow one to access vibrationally excited states of both X and A states, and thus be another path for an extended characterization of these states. For the quartets, a detailed theoretical description is presented for the first time in the literature, which is expected to be a useful guide to experimentalists to properly identify the states and assign the transitions resulting from these states. A high-lying and dense set of states is expected to pose great difficulties for a proper experimental identification due to the various crossings and perturbations.
Acknowledgements The authors acknowledge the provision of computational facilities at the Laborat orio de Computacß~ao
203
Cientıfica Avancßada (LCCA) of the University of S~ ao Paulo, and the financial support of Fundacß~ao de Amparo a Pesquisa do Estrado de S~ao Paulo (FAPESP) under contract No. 00/08920-1R. A research fellowship of the Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico (CNPq) of Brazil for F.R.O., and a graduate one for L.G.S. of Brazil are also greatly appreciated.
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