THE JOURNAL OF CHEMICAL PHYSICS 140, 044319 (2014)
Nonadiabatic calculations of ultraviolet absorption cross section of sulfur monoxide: Isotopic effects on the photodissociation reaction Sebastian O. Danielache,1 Suzuki Tomoya,1 Alexey Kondorsky,2,3 Ikuo Tokue,4 and Shinkoh Nanbu1 1
Department of Materials and Life Sciences, Faculty of Science & Technology, Sophia University, Chiyoda Ku, Tokyo 102-8554, Japan 2 P. N. Lebedev Physical Institute of Russian Academy of Science, Leninsky pr., 53, Moscow, 119991, Russia 3 Moscow Institute of Physics and Technology (State University), Institutsky per., 9, Dolgoprudny Moscow region, 141700, Russia 4 Department of Chemistry, Faculty of Science, Niigata University, Ikarashi, Niigata 950-2181, Japan
(Received 22 September 2013; accepted 6 January 2014; published online 29 January 2014) Ultraviolet absorption cross sections of the main and substituted sulfur monoxide (SO) isotopologues were calculated using R-Matrix expansion technique. Energies, transition dipole moments, and nonadiabatic coupling matrix elements were calculated at MRCI/AV6Z level. The calculated absorption cross section of 32 S16 O was compared with experimental spectrum; the spectral feature and the absolute value of photoabsorption cross sections are in good agreement. Our calculation predicts a long lived photoexcited SO* species which causes large non-mass dependent isotopic effects depending on the excitation energy in the ultraviolet region. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862429] I. INTRODUCTION
Sulfur monoxide (SO) is a molecular system that it has been studied for many years. This diatomic molecule is suitable for theoretical calculations which have been used to explore many of its spectroscopic properties. Experimental studies are however less abundant since SO is highly reactive and unstable under most experimental conditions. The implications of SO photochemistry extends over a wide range of areas such as interstellar chemistry,1 molecular clouds,2 and the photochemistry of Jupiter’s satellites.3 In the terrestrial Archean atmosphere the photodissociation of SO may play a significant role, because oxygen is estimated to have been 10−5 times presents atmospheric levels and therefore ultraviolet light permeated throughout the entire atmosphere.4 In this scenario SO photodissociation is a candidate to influence the non-mass dependent (NMD) signal recorded in the geological record.5 These geochemical systems make necessary the study of SO photochemistry induced by its ultraviolet spectra and the isotopic fractionations during photodissociation. Numerical modeling of Earth’s Archean atmosphere4,6, 7 makes use of photodissociation rate constants presented in the literature,8, 9 and these values have been determined by indirect methods which may have compromised its reliability, especially at wavelengths below 200 nm where signal-to-noise ratios are frequently close to detection limits. Theoretical and experimental studies of numerous SO electronic states such as a1 , b1 + , A3 , B3 − and their electronic transitions from the ground state X3 − have been reported in the literature since the 1930s. The first report on the spectroscopic properties of SO was presented by Emmett in 193210 where the B3 − -X3 − transition was first recorded. This and other transitions have been further studied by experimental methods11–17 where several
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predissociation pathways have been reported. The latest theoretical studies include the calculation of potential energy curves and spectroscopic constants for the C3 -X3 − electronic state at CASSCF/cc-pVQZ level,18 and the A3 − X3 − transition calculated at CASSCF/cc-pV5Z.19 Liu et al.15 recorded the spectra of the B3 − -X3 − transition above the first dissociation limit using degenerated four wave mixing spectra and complimented their experiments with calculations of quintets and repulsive curves at MRCI/AVQZ level. Yu and Bian20 have produced an extensive theoretical study of this system at icMRI+Q with Douglas-Kroll scalar relativistic correction and aug-cc-pV5Z basis sets. Several of these studies have reported radiative lifetimes obtained by time independent frameworks such as the Einstein’s A and B constants; however, there is no attempt in the literature to calculate ultraviolet absolute absorption cross sections. The theoretical bases of R-matrix applied to solve scattering problems of electron impacts with molecules has been applied widely;21 however, this methodology has rarely been used to study photoexcitation and photo-dissociation mechanisms. Despite the abundance of calculations at high theoretical level there is no report in the literature to theoretically calculate ultraviolet absorption cross sections relevant to the study of photochemistry under the geochemical conditions presented above, nor has been there any attempt to take into account the nonadiabatic transitions between the triplet states contributing to the photo-dissociation reaction. We set out to calculate ultraviolet absorption cross sections of sulfur isotopologues 32–34,36 SO by ab initio methodology and conclude on the isotopic effects produced by the ultraviolet absorption spectra. The organization of this paper is as follows. Sections II A and II B present the methodology used during the ab initio calculations and the R-Matrix expansion, respectively. Section III A discusses the calculated potential
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energy curves, transition dipole moments, and nonadiabatic coupling matrix elements. Section III B presents the results of the R-matrix expansion and Sec. III C presents and discusses the isotopic effects and their associated photodissociation dynamics. II. COMPUTATIONAL DETAILS
J. Chem. Phys. 140, 044319 (2014)
|(t, j ) = e−i H t |(0, j ). ˆ
Here the wave function, | = (| 1 ,. . . | Np )T , is a vector with respect to electronic state index, Np is the number of ˆ electronic states involved, e−iHt is the evolution operator for ˆ system Hamiltonian, H, and j is index of electronic state of the initial wave function given by
A. Ab initio energies and dipole moments
The employed procedure has been widely reported in theoretical studies of ultraviolet spectra produced by electronic excitations.14 This procedure consists of a mapping separation of the SO atomic nuclei and at each step the MOs were determined by complete active space self-consistent field (CASSCF) calculations, once the MOs were obtained, multireference configuration interaction (MRCI) calculations were performed by using the diffusion-function-augmented, correlation-consistent, polarized-valence sextuple-zeta (AV6Z also known as aug-cc-pV6Z) basis functions22 resulting in a total of 382 contracted functions. The 12 electrons occupied valance space was constructed with eight molecular orbitals which were assigned as active space. The six core orbitals were fully optimized but constrained to be doubly occupied in all configurations. All calculations have been performed in the Abelian C2v (a1 b1 b2 a2 ) subgroup of C∞v point group and active spaces for the singlet, triplet, and quintet spin symmetries were assigned as (3333), (3334), and (1112), respectively. Then an MRCI calculation was carried out to include the effects of dynamical correlation and Davidson correction to account for contributions of quadruple excitations to the correlation energy. Out of these calculations potential energies for the ground electronic state (X3 − ), the electronically excited singlet (a1 , b1 + , c1 − , d1 , e1 , f1 , g1 , and 21 + ), triplet (A3 , A3 + , A3 , C3 , C3 , B3 − , and 33 − ) and quintet states (5 , 5 − , and 5 ) were obtained, then the transition dipole moments between these states were evaluated from the CI wave functions. The total number of configurations after the internal contraction was 3.4 × 106 for A1 symmetries and 4.5 × 106 for A2 . The final number of the grid-data was 77 points between 2.0 and 20.0 a0 , where a0 is the Bohr radius. In order to determine the potential energy curves (PECs) and the dipole moment functions, the interpolant cubic non-periodic Spline method was used. All of the calculations have been performed using the MOLPRO 2006.1 package on the PRIMEQUEST 580 system of the Research Institute for Information Technology at Kyushu University and the PRIMEQUEST system of the Research Center for Computational Science of the Okazaki National Institute. B. Calculation of photoabsorbtion spectrum
1. Expression for photoabsorbtion spectra
The photoabsorbtion spectrum is calculated as a Fourier transform of the autocorrelation function: ⎡∞ ⎤ Np 1 ⎣ eiEt (0, j ) | (t, j )dt ⎦ , (1) S(E) = Re π j =1 0
(2)
|j (0, j ) = δjj μjg |g ,
(3)
where |g is the ground vibrational state of the ground electronic state and μjg is the transition dipole moment matrix element between the ground and excited electronic state, j. To calculate Eq. (1) the wave function, |(t), is expanded in terms of eigen-functions of discrete and continuum spectra of ˆ as the Hamiltonian, H, −iEn t |(t, j ) = an (j )e |ψ n + dE blE (j )e−iEt |ψ lE , n
l
(4) where l is quantum number to distinguish degenerate continuum states and Hˆ |ψ n = En |ψ n , Hˆ |ψ lE = E|ψ lE .
(5)
The expansion coefficients are given by an (j ) = (0, j ) | ψ n , blE (j ) = (0, j ) | ψ lE .
(6)
Using orthogonal properties of eigen-functions of discrete and continuum states ψ n | ψ n = δnn , ψ lE | ψ l E = δll δ(E − E ).
(7)
The expression for photoabsorbtion spectra could be finally obtained in the form Np 2 2 S(E) = |an (j )| δ(E − En ) + |blE | . (8) j =1
n
l
In actual calculations δ-function is substituted by a resonant-like function of finite width. 2. R-matrix expansion
To find eigen-functions for a system of multiple adiabatic potentials with nonadiabatic, coupling (Eq. (5)), we have adopted the R-matrix expansion technique23–25 and have developed an effective and numerically stable procedure to find the discrete state eigen-functions for such a system. For the system under consideration, Eq. (5) reads
1 d2 − I + V(x) − EI 2m dx 2 d 1 d 1 M(x) + M(x) ψ E (x) = 0, (9) − m dx 2m dx where m is reduced mass of the system, I is unity matrix, V(x) is diagonal matrix of adiabatic potentials and M(x) is the bidiagonal matrix of nonadiabatic, coupling elements, so that M(x) = – (M(x))T .
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For a fixed interval [xi , xf ], Eq. (9) could be rewritten in symmetric form as ˆ + V(x) − EI) − U]ψ ˆ E (x) = [L ˆ + D]ψ ˆ E (x), [(K
where
⎡ ⎤ W1,1 c1 ⎢ ⎥ ⎢ d = ⎣ ... ⎦ , W = ⎣ ... cNb WNb ,1
(10)
where the symmetrized kinetic operator is and
←
2 ˆ = − 1 I d + Lˆ = d I d , K 2 2m dx dx 2m dx the Bloch operator is 1 d d 1 Lˆ = I , δ(x − xf ) − δ(x − xi ) 2m dx 2m dx
(11)
⎤ W1,Nb ⎥ ... ⎦ , WNb ,Nb
(18)
ˆ + V(x) − U)π ˆ j (x)dx. πj (x)(K
(19)
⎡
xf
Wj,j =
(12)
Using the basis defined by Eq. (15), the solution of Eq. (10) could be written as ψ E (x) =
(13)
ˆ = M(x) [δ(x − xf ) − δ(x − xi )]. D (14) 2m To represent solution of Eq. (10) at [xi , xf ], we introduce the basis set of Nb functions, ϕ n (x), which are orthonormal on that interval and satisfy the eigen-problem
Nb
cnj πj (x),
j =1
Nb
(15)
|cnj |2 = 1.
(16)
j =1
Wdn = εn dn ,
Rff (E)
Rf i (E)
Rif (E)
Rii (E)
(17)
=
(20)
˜ x,f (E)ψ E (xf ) + R ˜ x,i (E)ψ E (xf ) ψ E (x) = R ˜ x,f (E)M(xf )ψ E (xf ) + R ˜ x,i (E)M(xi )ψ E (xi ), +R (21) where ψ (xz ) stands for wave function derivative at point xz and ˜ x,z (E) = R
Nb
(1 − 2δiz )
n=1
This reduces solution of Eq. (15) to a simple algebraic eigen-problem of dimensionality (Np Nb ) × (Np Nb )
Cn (E)ϕ n (x).
After calculating coefficients, Cn (E), the solution could be written as
To solve Eq. (15), the functions, ϕ n (x), are expanded in terms of DVR-functions {π j (x)}26, 27 as φ n (x) =
Nb n=1
←
ˆ n (x) = 0. ˆ + V(x) − εn I) − U]ϕ [(K
...
xi
and coupling operators are ˆ = d M(x) − M(x) d , U dx 2m 2m dx
... ...
˜ ff (E)M(xf ) I−R ˜ if (E)M(xf ) −R
The calculation of {ϕ n (x)} requires the diagonalization of a matrix of dimensionality (Np Nb ) × (Np Nb ), which could be very time-consuming because large Nb is required to represent the solution for the whole interval [xi , xf ]. In order to overcome this difficulty, we use division-by-sector scheme.23 The whole interval [xi , xf ] is divided into a large number of short sectors so that the basis size, Nb , required representing R-matrix for each sector is small enough. The R-matrix which relates wave function and its derivative at borders
ϕ n (x)ϕ Tn (xz ) . 2m(En − E)
(22)
Finally common R-matrix expansion could be obtained as
ψ E (xf ) ψ E (xi )
=
Rff (E)
Rf i (E)
Rif (E)
Rii (E)
ψ E (xf ) , ψ E (xi )
(23)
where
˜ f i (E)M(xi ) −R ˜ ii (E)M(xi ) I−R
−1
˜ ff (E) R ˜ if (E) R
˜ f i (E) R . ˜ ii (E) R
(24)
of different sectors are calculated using simple recursion formula.23 3. Wave functions of continuum spectra
Wave function of continuum spectra in an asymptotic region presents a sum of incoming and outgoing waves. Since all the potentials grow infinitely at the left side of the interval (x < xi ) and become flat at the right side (x ≥ xf ), the wave function could be written as
⎧ m ⎪ ⎨ 2π|pj,f (x)| [δjj0 eipj (xf )(x−xf ) − Sjj0 ,ff e−ipj (xf )(x−xf ) ], ψjj0 E (x) = ⎪ ⎩ − 2π|pmj,i (x)| Sjj0 ,if e−ipj (xi )(x−xi ) ,
x ≥ xf x ≤ xi
,
(25)
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where j and j0 are the electronic state indices of wave function itself and of its incoming wave, respectively, and 2m(E − Vj (x)), E ≥ V (x) , pj,z (x) = i(δiz − δf z ) 2m(Vj (x) − E), E < V (x) (26) where the side index z = i, f. Rewriting Eq. (23) using the asymptotic expression of Eq. (25). Fjj0 ,zz0 − Gjj ,zz Sj j0 ,z z0
introduced, so that (l) Rff (E) ψ E (xs ) = ψ E (xi ) R(l) if (E)
ψ E (xf ) ψ E (xs )
j z
j z
(27) the S-matrix coefficients could be obtained as
−1
˜ Sjj0 ,zz0 = Gjj0 ,zz0 − Rjj ,zz (E)Gj j0 ,z z0
Fjj ,zz0 = δzz δjj
R(l) ii (E)
ψ E (xf ) . ψ E (xs )
[Pz (E)]jj = (δiz − δf z )δjj 2m(Vj (xz ) − E).
(31)
(32)
(33)
(34)
Using Eqs. (31)–(34), a couple of independent relations between ψ E (xs ) and ψ E (xs ) could be established by considering intervals [xi , xs ] and [xs , xf ], respectively: (l) ψ (l) E (xs ) = Q (E)ψ E (xs ),
(35)
(r) ψ (r) E (xs ) = Q (E)ψ E (xs ),
(36)
(l)
(28)
(r)
where m , 2π pj,z (xz )
F˜jj ,zz = ipj,z (xz )Fjj ,zz , (29)
Gjj ,zz = δzz δjj
R(l) if (E)
,
where the side index z = i, f and
j z
where
=
R(l) f i (E)
ψ E (xi )
R(l) ii (E)
R(l) ff (E)
ψ E (xs )
ψ E (xz ) = Pz (E)ψ E (xz ),
j z
˜ Fjj0 ,zz0 − Rjj ,zz (E)Fj j0 ,z z0 ,
In the asymptotic region
j z
˜ j j ,z z Sj j0 ,z z0 , G = Rjj ,zz (E) F˜j j0 ,z z0 −
R(l) f i (E)
m ˜ jj ,zz = −ipj,z (xz )Gjj ,zz . , G 2π |pj,z (xz )| (30)
After S-matrix coefficients have been calculated, the wave function of the continuum state can be calculated for the whole interval [xi , xf ] using Eqs. (21)–(24) and values of wave function and its derivatives at the borders by Eq. (25). 4. Wave functions of discrete spectra
In order to calculate discrete state wave function, we need to get eigen-energy, En , and values of wave functions, ψ, and their derivatives, ψ , at some point xs . The main difficulty comes from the fact that the difference between the energies of nonadiabatically, coupled potentials could be large. If the energy is not equal to eigen-energy, the R-matrix expansion of the wave function would present a linear combination including infinitesimally small and infinitesimally large coefficients. This could spoil the numerical stability of calculations due to rounding errors. To overcome this difficulty, we have introduced the following procedure to find the wave functions of discrete spectra. The energies of discrete states of the group of nonadiabatically, coupled adiabatic states lies between the potential minimum and dissociation energy of the lowest adiabatic state of the group. Consider xs is coordinate of potential minimum of the lowest electronic state inside the interval [xi , xf ]. Two R-matrixes for intervals [xi , xs ] and [xs , xf ] could be
Q(l) (E) = R(l) f i (E)Pi (E) (l) −1 (l) ×[I − R(l) ii (E)Pi (E)] Rif (E) + Rff (E),
(37)
Q(r) (E) = R(r) if (E)Pf (E) (r) −1 (r) ×[I − R(r) ff (E)Pf (E)] Rf i (E) + Rii (E). (38)
If the energy, E, is equal to the energy of some discrete state, the wave function and its derivative calculated using intervals [xi , xs ] and [xs , xf ] should coincide so that an equation for ψ E (xs ) could be obtained as [Q(r) (E) − Q(l) (E)]ψ E (xs ) = 0.
(39)
This equation is solvable if det[Q(r) (E) − Q(l) (E)] = 0,
(40)
which is an equation for discrete state eigen-energy. The above expressions allow introducing a numerically stable procedure to calculate energy and wave function of discrete state as follows: (i) starting from the minimal potential energy Emin = Vlowest (xs ) search the energy range up until the root of Eq. (40), En , would be found. (ii) Solve Eq. (39) for E = En using singular value decomposition. (iii) Use the values of wave function derivatives found ψ (xs ) to find correspondent values of the wave function. To reduce numerical errors and improve calculation-stability, both sets of values, ψ (l) (xs ) and ψ (r) (xs ), should be calculated using Eq. (35) and Eq. (36), respectively, with same values of ψ (xs ). (iv) Use ψ (l) (xs ) and ψ (xs ) to construct wave function at the interval [xi , xs ] and ψ (r) (xs ) and ψ (xs ) at the interval [xs , xf ]. (v) Continue searching the next energy level until the dissociation energy of the lowest potential would be reached. The design of Eqs. (35)–(40) and the calculation procedure described above
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FIG. 1. Potential energy curves for the triplet states of the SO molecule calculated at MRCI/AV6Z level. All energies are relative to minimum of the ground state, X3 − .
prevent rounding errors to affect the calculation of energy and wave function of a discrete state. The final spectrum was constructed by Boltzmann averaging of all rovibrational states at 298 K.28 III. RESULTS AND DISCUSSION A. Potential energy curves and spectroscopic constants
Figure 1 shows the potential energy curves of SO triplet states correlating to the lowest (S(3 P) + O(3 P)) and the first excited (S(1 D) + O(3 P)) dissociation channels calculated at MRCI/aug-cc-pV6Z level. The computed grid data points include increasing steps of 0.04 a0 for atomic distances between 2.07 and 4.67 a0 and 0.094 a0 for atomic distances between 4.67 and 5.57 a0 , PECs for singlet and quintet states are provided as supplementary material (Ref. 29). The states included in the calculations of the absorption spectra are presented and discussed here, while the other calculated states are presented in the supplementary material.29 Table I TABLE I. Theoretical spectroscopic constants of calculated isotopologues in the ground state (X3 − ), compared to available experimental data (round brackets). Isotopologue 32 SO 33 SO 34 SO 36 SO a b
ωe (cm−1 )
ωe χ e (cm−1 )
Be (cm−1 )
1095.24 (1150.695)a 1089.50 1084.09 1074.07
5.630 5.630 5.630 5.630
0.7208 (0.717949)b 0.7135 (0.710707)b 0.7067 (0.703911)b 0.6941
Experimental data reported by Clerbaux and Colin.29 Experimental data reported by Tiemann.28
summarizes spectroscopic constants for the ground state of 32–34,36 SO isotopologues and compares them to values in the literature.30, 31 These constants include the harmonic and anharmonic vibrational constants (ωe and ωe χ e ), and the rotational constant (Be ). The values of dissociation energy (De ) were obtained by comparing the potential energy at the equilibrium geometry with the energy at the dissociation limit which was defined at the largest internuclear distance (20 a0 ) in this work. The energy curves included in the calculation of the absorption spectra are A3 , C3 , C3 , B3 − , and 33 − . The 33 − , C3 , and B3 − states connect to the dissociation channel S(1 D) + O(3 P), while A3 and C3 connect to the dissociation channel S(3 P) + O(3 P). The dissociation energy presented in the oldest literature15, 18, 19, 32 requires an empirical shift of about 0.24 eV, this inaccuracy has been solved by calculations at MRCI/aug-cc-pV5Z level reported by Yu and Bian.20 Our calculations present a dissociation energy of 5.359 eV for the ground state and an adiabatic ionization potential of 10.473 eV, and both values are in good agreement with the latest literature20 and experimental data.33 The difference between the values presented in this report and the ones presented by Yu and Bian20 is that we do not make use of the complete basis sets (CBS) extrapolation. An amplified view of the PECs contributing to the spectra is embedded in Figure 2, the most relevant exited states to the photodissociation mechanism present several crossing and avoided crossing within the energy range corresponding to the ultraviolet spectrum. In order to account for these interactions, nonadiabatic, coupling matrix elements (NACME) were added to the calculations carried out by the MOLPRO program and included in the theoretical framework of the R-matrix formulation presented in II B part 2 “R-matrix expansion.” Figure 2 presents the NACMEs calculated at the CASSCF
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FIG. 2. Nonadiabatic, coupling matrix elements (NACME) coefficients related to the C3 -C3 and the 33 − -B3 − states (panel (b)). The maxima of the coupling elements correspond to the avoided curve-crossing regions showed panel (a).
level, the matrix elements for C3 -C3 shows a maxima at 3.8 a0 , this maxima coincides with the geometries of the avoided coupling of this system. In this figure, a smaller maxima is also presented at 4.3 a0 for the 33 − -B3 − coupling elements. Of these two pseudo-curve-crossings, the first one is more important, because the exchange between C3 and C3 leads to the different dissociation channels. Transition dipole moments (TDMs) between the ground state (X3 − )
and the five lower excited states are presented in Figure 3. The A-X transitions replicate quite well the ones reported by Fulscher et al.34 and the calculations of Borin and Ornellas19 for the A-X, B-X, C-X, C -X, and 3-X transitions present similar spectral features to our results. However, our TDM of the 33 − -X3 − transition shows a discontinuity around 3.2 a0 , and also this behavior has been observed at the calculations of Yu and Bian.20 Since the methodology of Yu and Bian20
0.4
0.2
[a.u.]
0.0
-0.2
3 −
3
A Π-X Σ 3 3 − C Π-X Σ 3 3 − C' Π-X Σ 3 − 3 − B Σ -X Σ 3 − 3 − 3 Σ -X Σ
-0.4
-0.6
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
R(S-O) [ 0] FIG. 3. Transition dipole moments (in a.u.) for the lowest-lying triplet states A3 , C3 , C3 , B3 − , and 33 − coupled to the ground state, X3 − .
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3
3 −
A Π−X Σ ,
3 −
3 −
B Σ −X Σ ,
3 −
3
3
C Π−X Σ ,
3 −
C' Π−X Σ ,
3 −
3 −
3 Σ −X Σ
2
10
1
10
0
10
-1
10
-3
10
-4
10
σ x10
-17
2
(cm )
-2
10
-5
10
-6
10
-7
10
-8
10
-9
10
-10
10
-11
10
-12
10
-13
10
160
170
180
190
200
210
220
230
240
Wavelength (nm) FIG. 4. Photoabsorption cross sections for excitation of the different triplet states A3 , C3 , C3 , B3 − , and 33 − sates for the zero-point vibrational level with total angular momentum, J = 0.
differs from us in the active space and basis sets, this discontinuity is not a random artifact but is due to the mixing electronic structure. We have also confirmed that the curvature of the PEC of the 33 − state correlates with the character of this TDM at around R = 3.2 a0 in Fig. 1 (around 3.2 a0 in Fig. 3). On the other hand, we consider that the electronic transition involving the 33 − state are only reached by the excitation energy which is too high to be significant for an ultraviolet absorption spectrum, furthermore we found that the contributions from the 33 − state are 10 orders of magnitude smaller than the ones from the main contributing state (B3 − ) as shown in Fig. 4. An inquiry into how higher excited states may affect the 33 − state would have to include Rydberg states, because the photoionization would become relevant and in order to do this exploring a proper account of electron correlation during the photoionization is needed and beyond the scope of this work.
B. R-Matrix expansion
The R-Matrix expansion was carried out for all considered electronic states and nonadiabatic, transitions for 36 rotational states for all sulfur isotopologues. Figure 4 shows the partial contribution to the total absorption spectra at each allowed transition for the zero-point vibrational level with total angular momentum, J = 0, of the 32 SO isotopologue. The B3 − -X3 − transition represents the largest contribution to the spectra connecting to the S(1 D) + O(3 P) channel. This result is counter intuitive because the A3 -X3 − transition is located at a lower energy and connects to the S(3 P) + O(3 P) channel; however, the absorption cross sections from this transition is several orders of magnitude smaller than the B3 − X3 − transition. Since the B3 − state is electronically cou-
pled with the 33 − state, the nonadiabatic transition would strongly affect the photodissociation process. The C3 − X3 − transition should also be taken into account, because it has a significant contribution to the spectra and also has large NACME coefficients with the C3 state. The C3 − -X3 − transition dipole moments are large near the Franck-Condon region; however, the calculated absorption cross sections from these transitions are relatively small. The origin of this discrepancy can be attributed to the large excitation energy of the C3 state especially near the Franck-Condon region. The contributions from the A3 -X3 − and 33 − -X3 − transitions to the final spectra are several orders of magnitude smaller and therefore negligible. Figure 5 shows the contribution to the total absorption cross section of each rotational state as a function of wavelength. The Boltzmann distribution at 298 K shows that the 16th to 20th quantum number of the X3 − state makes the largest contributions and since the energy difference between states are about 20 cm−1 , the changes in temperature may result in significant differences in the absorption spectra. The final theoretical spectra is a contribution from the absorption bands of numerous rovibrational progression and from 5 different electronically exited states, all these partial contributions were combined by means of a Boltzmann distribution function calculated at 298 K. Figure 6 displays a comparison between theoretical and experimental spectra of 32 SO, from it can be clearly recognized that the spectral features in the progression, the resonant state, and the continuous band calculated in this study reproduces those of the experimental spectra of Phillips.9 In order to improve comparison with experimental data, the theoretical spectra was uplifted by a factor of 4, this difference should be taken with precaution and not necessarily in detriment of the theoretical spectra since the comparison is based on only one unconfirmed experimental report.9 The experimental devise used in the report by
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J. Chem. Phys. 140, 044319 (2014)
32
SO2
-19
x10 8.0
-19
2
(cm ) section s s o r C Partial
x10 7.0
-19
x10 6.0
-19
x10 5.0
-19
x10 4.0
-19
x10 3.0
-19
x10 2.0
-19
x10 1.0
0.0
36 4 3 2 Ro 3 30 8 2 6 ta 2 t
ion al Q
24
22
20
18 ua 16 nt 14 um 12 10 Nu m be r
8
6
4
2
25 0
24 0
23 0
22 0
21 0
16 17 0 0 18 19 0 20 0 m) 0 (n
e av W
g len
th
FIG. 5. R-Matrix calculations for the lowest 36 rotational quantum numbers and all calculated electronic states per wavelength for the 32 SO2 isotopologue. Continuous Band
Progressions
Resonant States
-17
3.0x10
theoretical experimental
2
Absorption Cross Section (cm )
-17
2.5x10
-17
2.0x10
-17
1.5x10
-17
1.0x10
-18
5.0x10
190
200
210
220
230
Wavelength (nm) FIG. 6. Calculated 32 SO2 spectra compared to experimental spectra reported by Phillips et al.10 For the sake of comparison of the spectral features, the theoretical cross sections have been uplifted by a factor of 4. The theoretical spectrum is divided into 3 segments namely continuous band, resonant states, and progressions.
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J. Chem. Phys. 140, 044319 (2014) -17
3.5x10
32
SO SO 34 SO 36 SO
-17
33
2
Absorption Cross Section (cm )
3.0x10
-17
2.5x10
-17
2.0x10
-17
1.5x10
-17
1.0x10
-18
5.0x10
190
195
200
205
210
215
220
225
230
Wavelength (nm) FIG. 7. Comparison of calculated absorption cross sections of sulfur isotopologues calculated at 298 K for the lowest vibrational and 36 rotational states.
Phillips9 generates SO radicals by reacting CS2 with atomic oxygen in a discharge-flow system followed by indirect measurement of absorption cross sections. This indirect measurement methodology requires the subtraction of SO2 absorption cross sections to the recorded spectra and a set of procedures to determine the concentration of the generated SO radical to obtain absolute values of SO absorption cross sections. It is important to point out that the wavelength axis of the spectra has not been modified and this is a significant achievement since shifting the energy access is a common technique in this type of calculations. The peak to valley amplitude ratio is also somewhat different and the theoretical spectra drops to 0 cm2 at valleys in the progression below 204.8 nm, these differences can be accounted by the FWHM (0.01 nm) of the Gaussian function employed to convolute the final spectra and by incrementing the number of rotational states in the theoretical calculation. A more systematic analysis of the convolution mechanism needs to take into account several broadening effects in order to produce a synthetic spectra better suited to a comparison with experimental data. The interplay of the considered electronically exited states produce an absorption spectra with a photodissociation threshold at 199.5 nm for the S(1 D) + O(3 P) channel and at 231.3 nm for the S(1 P) + O(3 P) channel, where the segment of the spectra between 231.3 and 199.46 nm represents a clear progression of the vibrational states. The 199.46–191 nm region shows the resonant states connecting to the S(1 D) + O(3 P) channel, while the wavelengths below 191 nm belong to the continuous band. This implies that the progression segment of the spectra is expected to create a long lived exited state SO*. These results are supported by the experimental measurements of Elks and Western,17 where they reported that the fluorescence lifetimes of the vibrational states in B3 − are in the order of 30–50 ns. Thus, the combination of different exit-channels at different energies should be taken
into account when considering the photochemistry of planetary atmospheres.
C. Isotopic effects
The spectra of isotopically substituted x SO (X = 32, 33, 34, and 36) isotopologues are presented in Figure 7. The expected red shifting of the 33,34,36 S isotopologues respect to 32 SO is initiated at the band origin and linearly increments toward the high energy shoulder corresponding to energy differences at the vibrational progressions of the electronically exited states, in this case the B3 − state. The isotopic effect does not produce any significant change on the general shape of the spectrum; however, there is a tendency of the heavier isotopologues to show lower peaks in the progression region. The isotopic effect at each wavelength was calculated with Eq. (41); x σi x , (41) εi = 1000 × ln σi32 where σi32 are the photoabsorption cross sections for the 32 SO isotopologue at wavelength i, and σix χ = 33, 34, and 36 are the photoabsorption cross sections for the heavier isotopologues. For a discussion on the definition of mass dependency, non-mass dependency (NMD) and their formulations see Röckman et al.35 Figure 8 displays the progression, resonant, and continuum segments (panel (d)) and the calculated ε33 and ε34 isotopic effects at each segment (panels (a)–(c)). At wavelengths below 191 nm, the continuum segment of the spectrum shows a strong mass dependent distribution (panel (a), Fig. 8). The segment of the spectra dominated by resonant states presents (191–199 nm) shows large NMD effects (panel (b), Fig. 8) and the progression segment (>199 nm) presents mass dependent distribution with significant deviations (panel
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50
ε33 ε36
40 30 20
J. Chem. Phys. 140, 044319 (2014)
Continuous Band
1600 1400 1200
0
1000
-10
800
ε33 & ε36 (per mil)
ε33 & ε36 (per mil)
10
-20 -30 -40 -50 -60
6000
400 200 0
-400
-90
-600
-40
-17
3.2x10
-20
ε34 (per mil)
0
20
40
-6000
(c)
-500
0
500
ε34 (per mil)
1000
1500
-8000 -6000
-4000
-2000
0
2000
ε34 (per mil)
4000
6000
Progressions
Resonant States
Continuous Band
0
(b)
-1200 -60
2000
-4000
-1000
-120
Progressions
-2000
-800
(a)
-110
ε33 ε36
4000
600
-80
-100
2
8000
-200
-70
Absorption Cross Section (cm )
Resonant States
ε33 ε36
1800
ε33 & ε36 (per mil)
044319-10
-17
2.8x10
(d)
-17
2.4x10
-17
2.0x10
-17
1.6x10
-17
1.2x10
-18
8.0x10
-18
4.0x10
0.0
180
185
190
195
200
205
210
215
220
225
230
Wavelength (nm) FIG. 8. Calculated spectra for 32 SO2 (panel (d)) divided in continuous band, resonant states and vibrational progression sections (see text) and calculated isotopic fractionations at each segment (panels (a)–(c)). The solid lines in panels (a)–(c) represent the mass dependent lines for the 33 SO (red) and 34 SO (blue) isotopologues.
(c), Fig. 8). The potential consequences of these results for photochemistry of planetary and stellar systems are important since solar fluxes can have a large variability triggering different isotopic effects at different opacity conditions. Supplementary materials S3 and S4,29 present theoretical isotopic constants and NMD constants as a function of wavelengths.
IV. CONCLUSIONS
We report ultraviolet absorption cross sections calculated from newly obtained energy surfaces at MRCI/AV6Z level and by implementation of R-Matrix propagation methodology where nonadiabatic effects were taken into account. Calculated vibrational and rotational constants reproduce experimental data available in the literature and our theoretical spectra also reproduce well the only experimental report in the literature. The analysis of the electronically exited states contributing to our spectrum shows that a continuous band, resonant states and the vibrational progression are predominant depending on the excitation energy. Additionally we show how depending on the incident photoexcitation energy the photodissociated fragments vary between S(1 D) and S(1 P) and the possibility of a long lived photoexcited SO* species. Sulfur isotopic effects are also investigated in this report and we show that NMD effects are found to be either large or small depending on the wavelength of the absorption spectra.
The implications of the results presented in this report are significant to stellar and planetary atmospheres where strong reduced states of the sulfur cycle can produce large SO pools product of photodissociation of SO2 . Our results also suggest that the isotopic imprint initiated by the photodissociation of SO2 36, 37 is further extended by the photodissociation of SO.
ACKNOWLEDGMENTS
The authors would like to thank Yuichiro Ueno for his constructive comments. A part of this study was supported by a Grant in Aid for Young Scientists (B) (No. 12835147) and Grant in Aid for Scientific Research (S) (Nos. 12941651 and 23224013) from MEXT, Japan. A.K. thanks the Russian Foundation for Basic Research (Grant No. 13-08-01193). 1 S.
S. Prasad and W. T. Huntress, Jr., Astrophys. J. 260, 590 (1982).
2 A. Nilsson, P. Bergman, and A. Hjalmarson, Astron. Astrophys. Suppl. Ser.
144, 441 (2000). Y. Zolotov and B. Fegley, Icarus 434, 431 (2000). 4 Y. Ueno, M. S. Johnson, S. O. Danielache, C. Eskebjerg, A. Pandey, and N. Yoshida, Proc. Natl. Acad. Sci. U.S.A. 106, 14784 (2009). 5 A. L. Zerkle, M. W. Claire, S. D. Domagal-Goldman, J. Farquhar, and S. W. Poulton, Nat. Geosci. 5, 359 (2012). 6 A. A. Pavlov and J. F. Kasting, Astrobiology 2, 27 (2002). 7 J. F. Kasting, Origins Life Evol. Biospheres 20, 199 (1990). 8 P. Warneck, F. F. Marmo, and J. O. Sullivan, J. Chem. Phys. 40, 1132 (1964). 3 M.
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F. Phillips, J. Phys. Chem. 85, 3994 (1981). M. Emmett, Phys. Rev. 41, 167 (1932). 11 R. Colin, Can. J. Phys. 47, 979 (1969). 12 M. A. A. Clyne and J. P. Liddy, J. Chem. Soc., Faraday Trans. 2 78, 1127 (1982). 13 K. Yamasaki, F. Taketani, S. Tomita, K. Sugiura, and I. Tokue, J. Phys. Chem. A 107, 2442 (2003). 14 K. Yamasaki, S. Tomita, T. Hatano, F. Taketani, and I. Tokue, Chem. Phys. Lett. 413, 231 (2005). 15 C.-P. Liu, N. L. Elliott, C. M. Western, Y.-P. Lee, and R. Colin, J. Mol. Spectrosc. 238, 213 (2006). 16 T. Hatano, S. Watanabe, H. Fujii, I. Tokue, and K. Yamasaki, J. Phys. Chem. A 111, 1200 (2007). 17 J. M. F. Elks and C. M. Western, J. Chem. Phys. 110, 7699 (1999). 18 A. C. Borin and F. R. Ornellas, Chem. Phys. 247, 351 (1999). 19 A. C. Borin and F. R. Ornellas, Chem. Phys. Lett. 322, 149 (2000). 20 L. E. Yu and W. Bian, J. Comput. Chem. 32, 1577 (2011). 21 M. Tashiro, J. Chem. Phys. 132, 134306 (2010). 22 A. K. Wilson, T. van Mourik, and T. H. Dunning, J. Mol. Struct.: THEOCHEM 388, 339 (1996). 23 B. I. Schneider and R. B. Walker, J. Chem. Phys. 70, 2466 (1979). 24 K. L. Baluja, P. G. Burke, and L. A. Morgan, Comput. Phys. Commun. 27, 299 (1982). 10 V.
J. Chem. Phys. 140, 044319 (2014) 25 G.
V. Mil’nikov, H. Nakamura, and J. Horáˇcek, Comput. Phys. Commun. 135, 278 (2001). 26 D. O. Harris, G. G. Engerholm, and W. D. Gwinn, J. Chem. Phys. 43, 1515 (1965). 27 A. S. Dickinson, J. Chem. Phys. 49, 4209 (1968). 28 G. Herzberg, Molecular Spectra and Molecular Structure: Infrared and Raman Spectra of Polyatomic Molecules (D. Van Nostrand Company, Inc. Princeton, New Jersey, 1991). 29 See supplementary material at http://dx.doi.org/10.1063/1.4862429 for potential energy plots of singlet and triplet states and plots of calculated 33,34,36ε, and 34,36E values. 30 E. Tiemann, J. Mol. Spectrosc. 51, 316 (1974). 31 C. Clerbaux and R. Colin, J. Mol. Spectrosc. 165, 334 (1994). 32 C. P. Archer, J. M. F. Elks, and C. M. Western, J. Chem. Phys. 112, 6293 (2000). 33 K. Norwood and C. Y. Ng, J. Chem. Phys. 91, 2898 (1989). 34 M. P. Fülscher, M. Jaszunski, B. O. Roos, and W. P. Kraemer, J. Chem. Phys. 96, 504 (1992). 35 T. Rockmann, C. Brenninkmeijer, G. Saueressig, P. Bergamaschi, J. Crowley, H. Fischer, and P. Crutzen, Science 281, 544 (1998). 36 S. O. Danielache, C. Eskebjerg, M. S. Johnson, Y. Ueno, and N. Yoshida, J. Geophys. Res. 113, D17314, doi:10.1029/2007JD009695 (2008). 37 I. Tokue and S. Nanbu, J. Chem. Phys. 132, 024301 (2010).
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The Journal of Chemical Physics is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material is subject to the AIP online journal license and/or AIP copyright. For more information, see http://ojps.aip.org/jcpo/jcpcr/jsp
Nonadiabatic calculations of ultraviolet absorption cross section of sulfur monoxide: Isotopic effects on the photodissociation reaction Sebastian O. Danielache,1 Suzuki Tomoya,1 Alexey Kondorsky,2,3 Ikuo Tokue,4 and Shinkoh Nanbu1 1
Department of Materials and Life Sciences, Faculty of Science & Technology, Sophia University, Chiyoda Ku, Tokyo 102-8554, Japan 2 P. N. Lebedev Physical Institute of Russian Academy of Science, Leninsky pr., 53, Moscow, 119991, Russia 3 Moscow Institute of Physics and Technology (State University), Institutsky per., 9, Dolgoprudny Moscow region, 141700, Russia 4 Department of Chemistry, Faculty of Science, Niigata University, Ikarashi, Niigata 950-2181, Japan
(Received 22 September 2013; accepted 6 January 2014; published online 29 January 2014) Ultraviolet absorption cross sections of the main and substituted sulfur monoxide (SO) isotopologues were calculated using R-Matrix expansion technique. Energies, transition dipole moments, and nonadiabatic coupling matrix elements were calculated at MRCI/AV6Z level. The calculated absorption cross section of 32 S16 O was compared with experimental spectrum; the spectral feature and the absolute value of photoabsorption cross sections are in good agreement. Our calculation predicts a long lived photoexcited SO* species which causes large non-mass dependent isotopic effects depending on the excitation energy in the ultraviolet region. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862429] I. INTRODUCTION
Sulfur monoxide (SO) is a molecular system that it has been studied for many years. This diatomic molecule is suitable for theoretical calculations which have been used to explore many of its spectroscopic properties. Experimental studies are however less abundant since SO is highly reactive and unstable under most experimental conditions. The implications of SO photochemistry extends over a wide range of areas such as interstellar chemistry,1 molecular clouds,2 and the photochemistry of Jupiter’s satellites.3 In the terrestrial Archean atmosphere the photodissociation of SO may play a significant role, because oxygen is estimated to have been 10−5 times presents atmospheric levels and therefore ultraviolet light permeated throughout the entire atmosphere.4 In this scenario SO photodissociation is a candidate to influence the non-mass dependent (NMD) signal recorded in the geological record.5 These geochemical systems make necessary the study of SO photochemistry induced by its ultraviolet spectra and the isotopic fractionations during photodissociation. Numerical modeling of Earth’s Archean atmosphere4,6, 7 makes use of photodissociation rate constants presented in the literature,8, 9 and these values have been determined by indirect methods which may have compromised its reliability, especially at wavelengths below 200 nm where signal-to-noise ratios are frequently close to detection limits. Theoretical and experimental studies of numerous SO electronic states such as a1 , b1 + , A3 , B3 − and their electronic transitions from the ground state X3 − have been reported in the literature since the 1930s. The first report on the spectroscopic properties of SO was presented by Emmett in 193210 where the B3 − -X3 − transition was first recorded. This and other transitions have been further studied by experimental methods11–17 where several
0021-9606/2014/140(4)/044319/11/$30.00
predissociation pathways have been reported. The latest theoretical studies include the calculation of potential energy curves and spectroscopic constants for the C3 -X3 − electronic state at CASSCF/cc-pVQZ level,18 and the A3 − X3 − transition calculated at CASSCF/cc-pV5Z.19 Liu et al.15 recorded the spectra of the B3 − -X3 − transition above the first dissociation limit using degenerated four wave mixing spectra and complimented their experiments with calculations of quintets and repulsive curves at MRCI/AVQZ level. Yu and Bian20 have produced an extensive theoretical study of this system at icMRI+Q with Douglas-Kroll scalar relativistic correction and aug-cc-pV5Z basis sets. Several of these studies have reported radiative lifetimes obtained by time independent frameworks such as the Einstein’s A and B constants; however, there is no attempt in the literature to calculate ultraviolet absolute absorption cross sections. The theoretical bases of R-matrix applied to solve scattering problems of electron impacts with molecules has been applied widely;21 however, this methodology has rarely been used to study photoexcitation and photo-dissociation mechanisms. Despite the abundance of calculations at high theoretical level there is no report in the literature to theoretically calculate ultraviolet absorption cross sections relevant to the study of photochemistry under the geochemical conditions presented above, nor has been there any attempt to take into account the nonadiabatic transitions between the triplet states contributing to the photo-dissociation reaction. We set out to calculate ultraviolet absorption cross sections of sulfur isotopologues 32–34,36 SO by ab initio methodology and conclude on the isotopic effects produced by the ultraviolet absorption spectra. The organization of this paper is as follows. Sections II A and II B present the methodology used during the ab initio calculations and the R-Matrix expansion, respectively. Section III A discusses the calculated potential
140, 044319-1
© 2014 AIP Publishing LLC
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energy curves, transition dipole moments, and nonadiabatic coupling matrix elements. Section III B presents the results of the R-matrix expansion and Sec. III C presents and discusses the isotopic effects and their associated photodissociation dynamics. II. COMPUTATIONAL DETAILS
J. Chem. Phys. 140, 044319 (2014)
|(t, j ) = e−i H t |(0, j ). ˆ
Here the wave function, | = (| 1 ,. . . | Np )T , is a vector with respect to electronic state index, Np is the number of ˆ electronic states involved, e−iHt is the evolution operator for ˆ system Hamiltonian, H, and j is index of electronic state of the initial wave function given by
A. Ab initio energies and dipole moments
The employed procedure has been widely reported in theoretical studies of ultraviolet spectra produced by electronic excitations.14 This procedure consists of a mapping separation of the SO atomic nuclei and at each step the MOs were determined by complete active space self-consistent field (CASSCF) calculations, once the MOs were obtained, multireference configuration interaction (MRCI) calculations were performed by using the diffusion-function-augmented, correlation-consistent, polarized-valence sextuple-zeta (AV6Z also known as aug-cc-pV6Z) basis functions22 resulting in a total of 382 contracted functions. The 12 electrons occupied valance space was constructed with eight molecular orbitals which were assigned as active space. The six core orbitals were fully optimized but constrained to be doubly occupied in all configurations. All calculations have been performed in the Abelian C2v (a1 b1 b2 a2 ) subgroup of C∞v point group and active spaces for the singlet, triplet, and quintet spin symmetries were assigned as (3333), (3334), and (1112), respectively. Then an MRCI calculation was carried out to include the effects of dynamical correlation and Davidson correction to account for contributions of quadruple excitations to the correlation energy. Out of these calculations potential energies for the ground electronic state (X3 − ), the electronically excited singlet (a1 , b1 + , c1 − , d1 , e1 , f1 , g1 , and 21 + ), triplet (A3 , A3 + , A3 , C3 , C3 , B3 − , and 33 − ) and quintet states (5 , 5 − , and 5 ) were obtained, then the transition dipole moments between these states were evaluated from the CI wave functions. The total number of configurations after the internal contraction was 3.4 × 106 for A1 symmetries and 4.5 × 106 for A2 . The final number of the grid-data was 77 points between 2.0 and 20.0 a0 , where a0 is the Bohr radius. In order to determine the potential energy curves (PECs) and the dipole moment functions, the interpolant cubic non-periodic Spline method was used. All of the calculations have been performed using the MOLPRO 2006.1 package on the PRIMEQUEST 580 system of the Research Institute for Information Technology at Kyushu University and the PRIMEQUEST system of the Research Center for Computational Science of the Okazaki National Institute. B. Calculation of photoabsorbtion spectrum
1. Expression for photoabsorbtion spectra
The photoabsorbtion spectrum is calculated as a Fourier transform of the autocorrelation function: ⎡∞ ⎤ Np 1 ⎣ eiEt (0, j ) | (t, j )dt ⎦ , (1) S(E) = Re π j =1 0
(2)
|j (0, j ) = δjj μjg |g ,
(3)
where |g is the ground vibrational state of the ground electronic state and μjg is the transition dipole moment matrix element between the ground and excited electronic state, j. To calculate Eq. (1) the wave function, |(t), is expanded in terms of eigen-functions of discrete and continuum spectra of ˆ as the Hamiltonian, H, −iEn t |(t, j ) = an (j )e |ψ n + dE blE (j )e−iEt |ψ lE , n
l
(4) where l is quantum number to distinguish degenerate continuum states and Hˆ |ψ n = En |ψ n , Hˆ |ψ lE = E|ψ lE .
(5)
The expansion coefficients are given by an (j ) = (0, j ) | ψ n , blE (j ) = (0, j ) | ψ lE .
(6)
Using orthogonal properties of eigen-functions of discrete and continuum states ψ n | ψ n = δnn , ψ lE | ψ l E = δll δ(E − E ).
(7)
The expression for photoabsorbtion spectra could be finally obtained in the form Np 2 2 S(E) = |an (j )| δ(E − En ) + |blE | . (8) j =1
n
l
In actual calculations δ-function is substituted by a resonant-like function of finite width. 2. R-matrix expansion
To find eigen-functions for a system of multiple adiabatic potentials with nonadiabatic, coupling (Eq. (5)), we have adopted the R-matrix expansion technique23–25 and have developed an effective and numerically stable procedure to find the discrete state eigen-functions for such a system. For the system under consideration, Eq. (5) reads
1 d2 − I + V(x) − EI 2m dx 2 d 1 d 1 M(x) + M(x) ψ E (x) = 0, (9) − m dx 2m dx where m is reduced mass of the system, I is unity matrix, V(x) is diagonal matrix of adiabatic potentials and M(x) is the bidiagonal matrix of nonadiabatic, coupling elements, so that M(x) = – (M(x))T .
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J. Chem. Phys. 140, 044319 (2014)
For a fixed interval [xi , xf ], Eq. (9) could be rewritten in symmetric form as ˆ + V(x) − EI) − U]ψ ˆ E (x) = [L ˆ + D]ψ ˆ E (x), [(K
where
⎡ ⎤ W1,1 c1 ⎢ ⎥ ⎢ d = ⎣ ... ⎦ , W = ⎣ ... cNb WNb ,1
(10)
where the symmetrized kinetic operator is and
←
2 ˆ = − 1 I d + Lˆ = d I d , K 2 2m dx dx 2m dx the Bloch operator is 1 d d 1 Lˆ = I , δ(x − xf ) − δ(x − xi ) 2m dx 2m dx
(11)
⎤ W1,Nb ⎥ ... ⎦ , WNb ,Nb
(18)
ˆ + V(x) − U)π ˆ j (x)dx. πj (x)(K
(19)
⎡
xf
Wj,j =
(12)
Using the basis defined by Eq. (15), the solution of Eq. (10) could be written as ψ E (x) =
(13)
ˆ = M(x) [δ(x − xf ) − δ(x − xi )]. D (14) 2m To represent solution of Eq. (10) at [xi , xf ], we introduce the basis set of Nb functions, ϕ n (x), which are orthonormal on that interval and satisfy the eigen-problem
Nb
cnj πj (x),
j =1
Nb
(15)
|cnj |2 = 1.
(16)
j =1
Wdn = εn dn ,
Rff (E)
Rf i (E)
Rif (E)
Rii (E)
(17)
=
(20)
˜ x,f (E)ψ E (xf ) + R ˜ x,i (E)ψ E (xf ) ψ E (x) = R ˜ x,f (E)M(xf )ψ E (xf ) + R ˜ x,i (E)M(xi )ψ E (xi ), +R (21) where ψ (xz ) stands for wave function derivative at point xz and ˜ x,z (E) = R
Nb
(1 − 2δiz )
n=1
This reduces solution of Eq. (15) to a simple algebraic eigen-problem of dimensionality (Np Nb ) × (Np Nb )
Cn (E)ϕ n (x).
After calculating coefficients, Cn (E), the solution could be written as
To solve Eq. (15), the functions, ϕ n (x), are expanded in terms of DVR-functions {π j (x)}26, 27 as φ n (x) =
Nb n=1
←
ˆ n (x) = 0. ˆ + V(x) − εn I) − U]ϕ [(K
...
xi
and coupling operators are ˆ = d M(x) − M(x) d , U dx 2m 2m dx
... ...
˜ ff (E)M(xf ) I−R ˜ if (E)M(xf ) −R
The calculation of {ϕ n (x)} requires the diagonalization of a matrix of dimensionality (Np Nb ) × (Np Nb ), which could be very time-consuming because large Nb is required to represent the solution for the whole interval [xi , xf ]. In order to overcome this difficulty, we use division-by-sector scheme.23 The whole interval [xi , xf ] is divided into a large number of short sectors so that the basis size, Nb , required representing R-matrix for each sector is small enough. The R-matrix which relates wave function and its derivative at borders
ϕ n (x)ϕ Tn (xz ) . 2m(En − E)
(22)
Finally common R-matrix expansion could be obtained as
ψ E (xf ) ψ E (xi )
=
Rff (E)
Rf i (E)
Rif (E)
Rii (E)
ψ E (xf ) , ψ E (xi )
(23)
where
˜ f i (E)M(xi ) −R ˜ ii (E)M(xi ) I−R
−1
˜ ff (E) R ˜ if (E) R
˜ f i (E) R . ˜ ii (E) R
(24)
of different sectors are calculated using simple recursion formula.23 3. Wave functions of continuum spectra
Wave function of continuum spectra in an asymptotic region presents a sum of incoming and outgoing waves. Since all the potentials grow infinitely at the left side of the interval (x < xi ) and become flat at the right side (x ≥ xf ), the wave function could be written as
⎧ m ⎪ ⎨ 2π|pj,f (x)| [δjj0 eipj (xf )(x−xf ) − Sjj0 ,ff e−ipj (xf )(x−xf ) ], ψjj0 E (x) = ⎪ ⎩ − 2π|pmj,i (x)| Sjj0 ,if e−ipj (xi )(x−xi ) ,
x ≥ xf x ≤ xi
,
(25)
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where j and j0 are the electronic state indices of wave function itself and of its incoming wave, respectively, and 2m(E − Vj (x)), E ≥ V (x) , pj,z (x) = i(δiz − δf z ) 2m(Vj (x) − E), E < V (x) (26) where the side index z = i, f. Rewriting Eq. (23) using the asymptotic expression of Eq. (25). Fjj0 ,zz0 − Gjj ,zz Sj j0 ,z z0
introduced, so that (l) Rff (E) ψ E (xs ) = ψ E (xi ) R(l) if (E)
ψ E (xf ) ψ E (xs )
j z
j z
(27) the S-matrix coefficients could be obtained as
−1
˜ Sjj0 ,zz0 = Gjj0 ,zz0 − Rjj ,zz (E)Gj j0 ,z z0
Fjj ,zz0 = δzz δjj
R(l) ii (E)
ψ E (xf ) . ψ E (xs )
[Pz (E)]jj = (δiz − δf z )δjj 2m(Vj (xz ) − E).
(31)
(32)
(33)
(34)
Using Eqs. (31)–(34), a couple of independent relations between ψ E (xs ) and ψ E (xs ) could be established by considering intervals [xi , xs ] and [xs , xf ], respectively: (l) ψ (l) E (xs ) = Q (E)ψ E (xs ),
(35)
(r) ψ (r) E (xs ) = Q (E)ψ E (xs ),
(36)
(l)
(28)
(r)
where m , 2π pj,z (xz )
F˜jj ,zz = ipj,z (xz )Fjj ,zz , (29)
Gjj ,zz = δzz δjj
R(l) if (E)
,
where the side index z = i, f and
j z
where
=
R(l) f i (E)
ψ E (xi )
R(l) ii (E)
R(l) ff (E)
ψ E (xs )
ψ E (xz ) = Pz (E)ψ E (xz ),
j z
˜ Fjj0 ,zz0 − Rjj ,zz (E)Fj j0 ,z z0 ,
In the asymptotic region
j z
˜ j j ,z z Sj j0 ,z z0 , G = Rjj ,zz (E) F˜j j0 ,z z0 −
R(l) f i (E)
m ˜ jj ,zz = −ipj,z (xz )Gjj ,zz . , G 2π |pj,z (xz )| (30)
After S-matrix coefficients have been calculated, the wave function of the continuum state can be calculated for the whole interval [xi , xf ] using Eqs. (21)–(24) and values of wave function and its derivatives at the borders by Eq. (25). 4. Wave functions of discrete spectra
In order to calculate discrete state wave function, we need to get eigen-energy, En , and values of wave functions, ψ, and their derivatives, ψ , at some point xs . The main difficulty comes from the fact that the difference between the energies of nonadiabatically, coupled potentials could be large. If the energy is not equal to eigen-energy, the R-matrix expansion of the wave function would present a linear combination including infinitesimally small and infinitesimally large coefficients. This could spoil the numerical stability of calculations due to rounding errors. To overcome this difficulty, we have introduced the following procedure to find the wave functions of discrete spectra. The energies of discrete states of the group of nonadiabatically, coupled adiabatic states lies between the potential minimum and dissociation energy of the lowest adiabatic state of the group. Consider xs is coordinate of potential minimum of the lowest electronic state inside the interval [xi , xf ]. Two R-matrixes for intervals [xi , xs ] and [xs , xf ] could be
Q(l) (E) = R(l) f i (E)Pi (E) (l) −1 (l) ×[I − R(l) ii (E)Pi (E)] Rif (E) + Rff (E),
(37)
Q(r) (E) = R(r) if (E)Pf (E) (r) −1 (r) ×[I − R(r) ff (E)Pf (E)] Rf i (E) + Rii (E). (38)
If the energy, E, is equal to the energy of some discrete state, the wave function and its derivative calculated using intervals [xi , xs ] and [xs , xf ] should coincide so that an equation for ψ E (xs ) could be obtained as [Q(r) (E) − Q(l) (E)]ψ E (xs ) = 0.
(39)
This equation is solvable if det[Q(r) (E) − Q(l) (E)] = 0,
(40)
which is an equation for discrete state eigen-energy. The above expressions allow introducing a numerically stable procedure to calculate energy and wave function of discrete state as follows: (i) starting from the minimal potential energy Emin = Vlowest (xs ) search the energy range up until the root of Eq. (40), En , would be found. (ii) Solve Eq. (39) for E = En using singular value decomposition. (iii) Use the values of wave function derivatives found ψ (xs ) to find correspondent values of the wave function. To reduce numerical errors and improve calculation-stability, both sets of values, ψ (l) (xs ) and ψ (r) (xs ), should be calculated using Eq. (35) and Eq. (36), respectively, with same values of ψ (xs ). (iv) Use ψ (l) (xs ) and ψ (xs ) to construct wave function at the interval [xi , xs ] and ψ (r) (xs ) and ψ (xs ) at the interval [xs , xf ]. (v) Continue searching the next energy level until the dissociation energy of the lowest potential would be reached. The design of Eqs. (35)–(40) and the calculation procedure described above
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J. Chem. Phys. 140, 044319 (2014)
FIG. 1. Potential energy curves for the triplet states of the SO molecule calculated at MRCI/AV6Z level. All energies are relative to minimum of the ground state, X3 − .
prevent rounding errors to affect the calculation of energy and wave function of a discrete state. The final spectrum was constructed by Boltzmann averaging of all rovibrational states at 298 K.28 III. RESULTS AND DISCUSSION A. Potential energy curves and spectroscopic constants
Figure 1 shows the potential energy curves of SO triplet states correlating to the lowest (S(3 P) + O(3 P)) and the first excited (S(1 D) + O(3 P)) dissociation channels calculated at MRCI/aug-cc-pV6Z level. The computed grid data points include increasing steps of 0.04 a0 for atomic distances between 2.07 and 4.67 a0 and 0.094 a0 for atomic distances between 4.67 and 5.57 a0 , PECs for singlet and quintet states are provided as supplementary material (Ref. 29). The states included in the calculations of the absorption spectra are presented and discussed here, while the other calculated states are presented in the supplementary material.29 Table I TABLE I. Theoretical spectroscopic constants of calculated isotopologues in the ground state (X3 − ), compared to available experimental data (round brackets). Isotopologue 32 SO 33 SO 34 SO 36 SO a b
ωe (cm−1 )
ωe χ e (cm−1 )
Be (cm−1 )
1095.24 (1150.695)a 1089.50 1084.09 1074.07
5.630 5.630 5.630 5.630
0.7208 (0.717949)b 0.7135 (0.710707)b 0.7067 (0.703911)b 0.6941
Experimental data reported by Clerbaux and Colin.29 Experimental data reported by Tiemann.28
summarizes spectroscopic constants for the ground state of 32–34,36 SO isotopologues and compares them to values in the literature.30, 31 These constants include the harmonic and anharmonic vibrational constants (ωe and ωe χ e ), and the rotational constant (Be ). The values of dissociation energy (De ) were obtained by comparing the potential energy at the equilibrium geometry with the energy at the dissociation limit which was defined at the largest internuclear distance (20 a0 ) in this work. The energy curves included in the calculation of the absorption spectra are A3 , C3 , C3 , B3 − , and 33 − . The 33 − , C3 , and B3 − states connect to the dissociation channel S(1 D) + O(3 P), while A3 and C3 connect to the dissociation channel S(3 P) + O(3 P). The dissociation energy presented in the oldest literature15, 18, 19, 32 requires an empirical shift of about 0.24 eV, this inaccuracy has been solved by calculations at MRCI/aug-cc-pV5Z level reported by Yu and Bian.20 Our calculations present a dissociation energy of 5.359 eV for the ground state and an adiabatic ionization potential of 10.473 eV, and both values are in good agreement with the latest literature20 and experimental data.33 The difference between the values presented in this report and the ones presented by Yu and Bian20 is that we do not make use of the complete basis sets (CBS) extrapolation. An amplified view of the PECs contributing to the spectra is embedded in Figure 2, the most relevant exited states to the photodissociation mechanism present several crossing and avoided crossing within the energy range corresponding to the ultraviolet spectrum. In order to account for these interactions, nonadiabatic, coupling matrix elements (NACME) were added to the calculations carried out by the MOLPRO program and included in the theoretical framework of the R-matrix formulation presented in II B part 2 “R-matrix expansion.” Figure 2 presents the NACMEs calculated at the CASSCF
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J. Chem. Phys. 140, 044319 (2014)
FIG. 2. Nonadiabatic, coupling matrix elements (NACME) coefficients related to the C3 -C3 and the 33 − -B3 − states (panel (b)). The maxima of the coupling elements correspond to the avoided curve-crossing regions showed panel (a).
level, the matrix elements for C3 -C3 shows a maxima at 3.8 a0 , this maxima coincides with the geometries of the avoided coupling of this system. In this figure, a smaller maxima is also presented at 4.3 a0 for the 33 − -B3 − coupling elements. Of these two pseudo-curve-crossings, the first one is more important, because the exchange between C3 and C3 leads to the different dissociation channels. Transition dipole moments (TDMs) between the ground state (X3 − )
and the five lower excited states are presented in Figure 3. The A-X transitions replicate quite well the ones reported by Fulscher et al.34 and the calculations of Borin and Ornellas19 for the A-X, B-X, C-X, C -X, and 3-X transitions present similar spectral features to our results. However, our TDM of the 33 − -X3 − transition shows a discontinuity around 3.2 a0 , and also this behavior has been observed at the calculations of Yu and Bian.20 Since the methodology of Yu and Bian20
0.4
0.2
[a.u.]
0.0
-0.2
3 −
3
A Π-X Σ 3 3 − C Π-X Σ 3 3 − C' Π-X Σ 3 − 3 − B Σ -X Σ 3 − 3 − 3 Σ -X Σ
-0.4
-0.6
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
R(S-O) [ 0] FIG. 3. Transition dipole moments (in a.u.) for the lowest-lying triplet states A3 , C3 , C3 , B3 − , and 33 − coupled to the ground state, X3 − .
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J. Chem. Phys. 140, 044319 (2014)
3
3 −
A Π−X Σ ,
3 −
3 −
B Σ −X Σ ,
3 −
3
3
C Π−X Σ ,
3 −
C' Π−X Σ ,
3 −
3 −
3 Σ −X Σ
2
10
1
10
0
10
-1
10
-3
10
-4
10
σ x10
-17
2
(cm )
-2
10
-5
10
-6
10
-7
10
-8
10
-9
10
-10
10
-11
10
-12
10
-13
10
160
170
180
190
200
210
220
230
240
Wavelength (nm) FIG. 4. Photoabsorption cross sections for excitation of the different triplet states A3 , C3 , C3 , B3 − , and 33 − sates for the zero-point vibrational level with total angular momentum, J = 0.
differs from us in the active space and basis sets, this discontinuity is not a random artifact but is due to the mixing electronic structure. We have also confirmed that the curvature of the PEC of the 33 − state correlates with the character of this TDM at around R = 3.2 a0 in Fig. 1 (around 3.2 a0 in Fig. 3). On the other hand, we consider that the electronic transition involving the 33 − state are only reached by the excitation energy which is too high to be significant for an ultraviolet absorption spectrum, furthermore we found that the contributions from the 33 − state are 10 orders of magnitude smaller than the ones from the main contributing state (B3 − ) as shown in Fig. 4. An inquiry into how higher excited states may affect the 33 − state would have to include Rydberg states, because the photoionization would become relevant and in order to do this exploring a proper account of electron correlation during the photoionization is needed and beyond the scope of this work.
B. R-Matrix expansion
The R-Matrix expansion was carried out for all considered electronic states and nonadiabatic, transitions for 36 rotational states for all sulfur isotopologues. Figure 4 shows the partial contribution to the total absorption spectra at each allowed transition for the zero-point vibrational level with total angular momentum, J = 0, of the 32 SO isotopologue. The B3 − -X3 − transition represents the largest contribution to the spectra connecting to the S(1 D) + O(3 P) channel. This result is counter intuitive because the A3 -X3 − transition is located at a lower energy and connects to the S(3 P) + O(3 P) channel; however, the absorption cross sections from this transition is several orders of magnitude smaller than the B3 − X3 − transition. Since the B3 − state is electronically cou-
pled with the 33 − state, the nonadiabatic transition would strongly affect the photodissociation process. The C3 − X3 − transition should also be taken into account, because it has a significant contribution to the spectra and also has large NACME coefficients with the C3 state. The C3 − -X3 − transition dipole moments are large near the Franck-Condon region; however, the calculated absorption cross sections from these transitions are relatively small. The origin of this discrepancy can be attributed to the large excitation energy of the C3 state especially near the Franck-Condon region. The contributions from the A3 -X3 − and 33 − -X3 − transitions to the final spectra are several orders of magnitude smaller and therefore negligible. Figure 5 shows the contribution to the total absorption cross section of each rotational state as a function of wavelength. The Boltzmann distribution at 298 K shows that the 16th to 20th quantum number of the X3 − state makes the largest contributions and since the energy difference between states are about 20 cm−1 , the changes in temperature may result in significant differences in the absorption spectra. The final theoretical spectra is a contribution from the absorption bands of numerous rovibrational progression and from 5 different electronically exited states, all these partial contributions were combined by means of a Boltzmann distribution function calculated at 298 K. Figure 6 displays a comparison between theoretical and experimental spectra of 32 SO, from it can be clearly recognized that the spectral features in the progression, the resonant state, and the continuous band calculated in this study reproduces those of the experimental spectra of Phillips.9 In order to improve comparison with experimental data, the theoretical spectra was uplifted by a factor of 4, this difference should be taken with precaution and not necessarily in detriment of the theoretical spectra since the comparison is based on only one unconfirmed experimental report.9 The experimental devise used in the report by
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32
SO2
-19
x10 8.0
-19
2
(cm ) section s s o r C Partial
x10 7.0
-19
x10 6.0
-19
x10 5.0
-19
x10 4.0
-19
x10 3.0
-19
x10 2.0
-19
x10 1.0
0.0
36 4 3 2 Ro 3 30 8 2 6 ta 2 t
ion al Q
24
22
20
18 ua 16 nt 14 um 12 10 Nu m be r
8
6
4
2
25 0
24 0
23 0
22 0
21 0
16 17 0 0 18 19 0 20 0 m) 0 (n
e av W
g len
th
FIG. 5. R-Matrix calculations for the lowest 36 rotational quantum numbers and all calculated electronic states per wavelength for the 32 SO2 isotopologue. Continuous Band
Progressions
Resonant States
-17
3.0x10
theoretical experimental
2
Absorption Cross Section (cm )
-17
2.5x10
-17
2.0x10
-17
1.5x10
-17
1.0x10
-18
5.0x10
190
200
210
220
230
Wavelength (nm) FIG. 6. Calculated 32 SO2 spectra compared to experimental spectra reported by Phillips et al.10 For the sake of comparison of the spectral features, the theoretical cross sections have been uplifted by a factor of 4. The theoretical spectrum is divided into 3 segments namely continuous band, resonant states, and progressions.
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J. Chem. Phys. 140, 044319 (2014) -17
3.5x10
32
SO SO 34 SO 36 SO
-17
33
2
Absorption Cross Section (cm )
3.0x10
-17
2.5x10
-17
2.0x10
-17
1.5x10
-17
1.0x10
-18
5.0x10
190
195
200
205
210
215
220
225
230
Wavelength (nm) FIG. 7. Comparison of calculated absorption cross sections of sulfur isotopologues calculated at 298 K for the lowest vibrational and 36 rotational states.
Phillips9 generates SO radicals by reacting CS2 with atomic oxygen in a discharge-flow system followed by indirect measurement of absorption cross sections. This indirect measurement methodology requires the subtraction of SO2 absorption cross sections to the recorded spectra and a set of procedures to determine the concentration of the generated SO radical to obtain absolute values of SO absorption cross sections. It is important to point out that the wavelength axis of the spectra has not been modified and this is a significant achievement since shifting the energy access is a common technique in this type of calculations. The peak to valley amplitude ratio is also somewhat different and the theoretical spectra drops to 0 cm2 at valleys in the progression below 204.8 nm, these differences can be accounted by the FWHM (0.01 nm) of the Gaussian function employed to convolute the final spectra and by incrementing the number of rotational states in the theoretical calculation. A more systematic analysis of the convolution mechanism needs to take into account several broadening effects in order to produce a synthetic spectra better suited to a comparison with experimental data. The interplay of the considered electronically exited states produce an absorption spectra with a photodissociation threshold at 199.5 nm for the S(1 D) + O(3 P) channel and at 231.3 nm for the S(1 P) + O(3 P) channel, where the segment of the spectra between 231.3 and 199.46 nm represents a clear progression of the vibrational states. The 199.46–191 nm region shows the resonant states connecting to the S(1 D) + O(3 P) channel, while the wavelengths below 191 nm belong to the continuous band. This implies that the progression segment of the spectra is expected to create a long lived exited state SO*. These results are supported by the experimental measurements of Elks and Western,17 where they reported that the fluorescence lifetimes of the vibrational states in B3 − are in the order of 30–50 ns. Thus, the combination of different exit-channels at different energies should be taken
into account when considering the photochemistry of planetary atmospheres.
C. Isotopic effects
The spectra of isotopically substituted x SO (X = 32, 33, 34, and 36) isotopologues are presented in Figure 7. The expected red shifting of the 33,34,36 S isotopologues respect to 32 SO is initiated at the band origin and linearly increments toward the high energy shoulder corresponding to energy differences at the vibrational progressions of the electronically exited states, in this case the B3 − state. The isotopic effect does not produce any significant change on the general shape of the spectrum; however, there is a tendency of the heavier isotopologues to show lower peaks in the progression region. The isotopic effect at each wavelength was calculated with Eq. (41); x σi x , (41) εi = 1000 × ln σi32 where σi32 are the photoabsorption cross sections for the 32 SO isotopologue at wavelength i, and σix χ = 33, 34, and 36 are the photoabsorption cross sections for the heavier isotopologues. For a discussion on the definition of mass dependency, non-mass dependency (NMD) and their formulations see Röckman et al.35 Figure 8 displays the progression, resonant, and continuum segments (panel (d)) and the calculated ε33 and ε34 isotopic effects at each segment (panels (a)–(c)). At wavelengths below 191 nm, the continuum segment of the spectrum shows a strong mass dependent distribution (panel (a), Fig. 8). The segment of the spectra dominated by resonant states presents (191–199 nm) shows large NMD effects (panel (b), Fig. 8) and the progression segment (>199 nm) presents mass dependent distribution with significant deviations (panel
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Danielache et al.
50
ε33 ε36
40 30 20
J. Chem. Phys. 140, 044319 (2014)
Continuous Band
1600 1400 1200
0
1000
-10
800
ε33 & ε36 (per mil)
ε33 & ε36 (per mil)
10
-20 -30 -40 -50 -60
6000
400 200 0
-400
-90
-600
-40
-17
3.2x10
-20
ε34 (per mil)
0
20
40
-6000
(c)
-500
0
500
ε34 (per mil)
1000
1500
-8000 -6000
-4000
-2000
0
2000
ε34 (per mil)
4000
6000
Progressions
Resonant States
Continuous Band
0
(b)
-1200 -60
2000
-4000
-1000
-120
Progressions
-2000
-800
(a)
-110
ε33 ε36
4000
600
-80
-100
2
8000
-200
-70
Absorption Cross Section (cm )
Resonant States
ε33 ε36
1800
ε33 & ε36 (per mil)
044319-10
-17
2.8x10
(d)
-17
2.4x10
-17
2.0x10
-17
1.6x10
-17
1.2x10
-18
8.0x10
-18
4.0x10
0.0
180
185
190
195
200
205
210
215
220
225
230
Wavelength (nm) FIG. 8. Calculated spectra for 32 SO2 (panel (d)) divided in continuous band, resonant states and vibrational progression sections (see text) and calculated isotopic fractionations at each segment (panels (a)–(c)). The solid lines in panels (a)–(c) represent the mass dependent lines for the 33 SO (red) and 34 SO (blue) isotopologues.
(c), Fig. 8). The potential consequences of these results for photochemistry of planetary and stellar systems are important since solar fluxes can have a large variability triggering different isotopic effects at different opacity conditions. Supplementary materials S3 and S4,29 present theoretical isotopic constants and NMD constants as a function of wavelengths.
IV. CONCLUSIONS
We report ultraviolet absorption cross sections calculated from newly obtained energy surfaces at MRCI/AV6Z level and by implementation of R-Matrix propagation methodology where nonadiabatic effects were taken into account. Calculated vibrational and rotational constants reproduce experimental data available in the literature and our theoretical spectra also reproduce well the only experimental report in the literature. The analysis of the electronically exited states contributing to our spectrum shows that a continuous band, resonant states and the vibrational progression are predominant depending on the excitation energy. Additionally we show how depending on the incident photoexcitation energy the photodissociated fragments vary between S(1 D) and S(1 P) and the possibility of a long lived photoexcited SO* species. Sulfur isotopic effects are also investigated in this report and we show that NMD effects are found to be either large or small depending on the wavelength of the absorption spectra.
The implications of the results presented in this report are significant to stellar and planetary atmospheres where strong reduced states of the sulfur cycle can produce large SO pools product of photodissociation of SO2 . Our results also suggest that the isotopic imprint initiated by the photodissociation of SO2 36, 37 is further extended by the photodissociation of SO.
ACKNOWLEDGMENTS
The authors would like to thank Yuichiro Ueno for his constructive comments. A part of this study was supported by a Grant in Aid for Young Scientists (B) (No. 12835147) and Grant in Aid for Scientific Research (S) (Nos. 12941651 and 23224013) from MEXT, Japan. A.K. thanks the Russian Foundation for Basic Research (Grant No. 13-08-01193). 1 S.
S. Prasad and W. T. Huntress, Jr., Astrophys. J. 260, 590 (1982).
2 A. Nilsson, P. Bergman, and A. Hjalmarson, Astron. Astrophys. Suppl. Ser.
144, 441 (2000). Y. Zolotov and B. Fegley, Icarus 434, 431 (2000). 4 Y. Ueno, M. S. Johnson, S. O. Danielache, C. Eskebjerg, A. Pandey, and N. Yoshida, Proc. Natl. Acad. Sci. U.S.A. 106, 14784 (2009). 5 A. L. Zerkle, M. W. Claire, S. D. Domagal-Goldman, J. Farquhar, and S. W. Poulton, Nat. Geosci. 5, 359 (2012). 6 A. A. Pavlov and J. F. Kasting, Astrobiology 2, 27 (2002). 7 J. F. Kasting, Origins Life Evol. Biospheres 20, 199 (1990). 8 P. Warneck, F. F. Marmo, and J. O. Sullivan, J. Chem. Phys. 40, 1132 (1964). 3 M.
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