A global potential energy surface and dipole moment surface for silane
Alec Owens, Sergei N. Yurchenko, Andrey Yachmenev, Walter Thiel
AA global potential energy surface and dipole moment surface for silane
Alec Owens,
1, 2, a) Sergei N. Yurchenko, Andrey Yachmenev, and Walter Thiel Max-Planck-Institut f¨ur Kohlenforschung, Kaiser-Wilhelm-Platz 1,45470 M¨ulheim an der Ruhr, Germany Department of Physics and Astronomy, University College London, Gower Street,WC1E 6BT London, United Kingdom (Dated: 17 August 2018)
A new nine-dimensional potential energy surface (PES) and dipole moment sur-face (DMS) for silane have been generated using high-level ab initio theory. ThePES, CBS-F12 HL , reproduces all four fundamental term values for SiH withsub-wavenumber accuracy, resulting in an overall root-mean-square (rms) error of0 .
63 cm − . The PES is based on explicitly correlated coupled cluster calculations withextrapolation to the complete basis set limit, and incorporates a range of higher-leveladditive energy corrections to account for core-valence electron correlation, higher-order coupled cluster terms, and scalar relativistic effects. Systematic errors in com-puted intra-band rotational energy levels are reduced by empirically refining theequilibrium geometry. The resultant Si-H bond length is in excellent agreement withprevious experimental and theoretical values. Vibrational transition moments, abso-lute line intensities of the ν band, and the infrared spectrum for SiH includingstates up to J = 20 and vibrational band origins up to 5000 cm − are calculatedand compared with available experimental results. The DMS tends to marginallyoverestimate the strength of line intensities. Despite this, band shape and structureacross the spectrum are well reproduced and show good agreement with experiment.We thus recommend the PES and DMS for future use. a) Electronic mail: [email protected] a r X i v : . [ phy s i c s . c h e m - ph ] A ug . INTRODUCTION The infrared (IR) absorption spectrum of silane (SiH ) was first documented over eightyyears ago. Since then numerous high-resolution spectroscopic studies of SiH and its iso-topomers have followed, including astronomical observation of rotation-vibration transitionsaround the carbon star IRC +10216 and in the atmospheres of Jupiter and Saturn. Inindustry silane gas is used extensively in the semiconductor manufacturing process and forthe production of solar cells.Despite its industrial and astrophysical importance, very few rigorous theoretical studieshave been carried out. Martin, Baldridge, and Lee computed an accurate quartic forcefield for silane based on CCSD(T) [coupled cluster with all single and double excitations anda perturbational estimate of connected triple excitations] calculations using the correlationconsistent quadruple zeta basis set, cc-pVQZ, plus an additional high-exponent d -function (denoted as cc-pVQZ+1 in Ref. 8). Minor empirical refinement of the four diagonal quadraticconstants produced a force field of spectroscopic quality ( ± − when reproducing thefundamental frequencies) applicable for several isotopomers of silane.The resultant force field was subsequently used to calculate vibrational energy levels ofSiH , SiH D, SiHD , and SiH D by means of canonical Van-Vleck perturbation theory(CVPT). When compared to results of a variational four-dimensional stretch model, full-dimensional CVPT calculations were necessary to accurately describe certain stretch levelsas they incorporated the effects of Fermi resonance. The importance of treating Fermiinteractions to compute vibrational energies of silane was also highlighted previously usingan algebraic approach. The use of stretch-only models has generally been successful in describing stretchingovertones and corresponding band intensities however. This is because of thepronounced local mode behaviour of silane, the effects of which have been documentedexperimentally in a series of papers by Zhu et al.
It is only at higher energies (above12 000 cm − ) that the rotational structure of the | (cid:105) and | (cid:105) stretch eigenstates canno longer be analysed in a local mode description due to vibrational resonances. Forintensity calculations, even a small treatment of bending motion can improve the descriptionof intensities compared to stretch-only models (an overview of previously computed abinitio dipole moment surfaces for silane can be found in Ref. 28).2he motivation for the present work is that SiH (henceforth labelled as SiH ) is atarget molecule of the ExoMol project, which is creating a comprehensive database ofall molecular transitions deemed necessary to model exoplanet and other hot atmospheres.Although unlikely, SiH has already been considered in the context of biosignature gases onrocky exoplanets. At present there is no coverage of SiH in several of the popular spectroscopic databases. The PNNL spectral library is an exception, covering the range of 600 to 6500 cm − at aresolution of around 0 .
06 cm − for temperatures of 5, 25, and 50 ◦ C. The Spherical Top DataSystem (STDS) is another valuable resource for spectral information on silane. However,some of the measured transitions and intensities are from unpublished work which makes ithard to verify the methods used and subsequently the reliability of the data.It is our intention to construct a global nine-dimensional potential energy surface (PES)and dipole moment surface (DMS) for silane. To do this we employ state-of-the-art electronicstructure calculations to generate the respective surfaces. After fitting the ab initio datawith suitable analytic representations, the quality of the PES and DMS will be tested bymeans of variational calculations of the infrared spectrum.The paper is structured as follows: In Sec. II the ab initio calculations and analyticrepresentation of the PES are presented. Similarly, in Sec. III the electronic structurecalculations and analytic representation of the DMS are detailed. Pure rotational energies,the equilibrium Si-H bond length, vibrational J = 0 energy levels, absolute line intensitiesof the ν band, and an overview of the rovibration spectrum up to J = 20 are calculatedand compared against available experimental data in Sec. IV. We offer concluding remarksin Sec. V. II. POTENTIAL ENERGY SURFACEA. Electronic structure calculations
Focal-point analysis is used to represent the total electronic energy as E tot = E CBS + ∆ E SR + ∆ E CV + ∆ E HO (1)The energy at the complete basis set (CBS) limit E CBS was computed using the explicitlycorrelated F12 coupled cluster method CCSD(T)-F12b (Ref. 38) with the F12-optimized3orrelation consistent polarized valence basis sets, cc-pVTZ-F12 and cc-pVQZ-F12. Cal-culations were carried out in the frozen core approximation and used the diagonal fixedamplitude ansatz 3C(FIX) with a Slater geminal exponent value of β = 1 . a − . For theresolution of the identity (RI) basis and the two density fitting (DF) basis sets, we employedthe corresponding OptRI, cc-pV5Z/JKFIT, and aug-cc-pwCV5Z/MP2FIT auxiliarybasis sets (ABS), respectively. All calculations were carried out with MOLPRO2012 un-less stated otherwise.A parameterized two-point formula, E C CBS = ( E n +1 − E n ) F Cn +1 + E n , proposed by Hill etal. was used to extrapolate to the CBS limit. For the coefficients F Cn +1 , which are specificto the CCSD-F12b and (T) components of the total CCSD(T)-F12b energy, we employedvalues of F CCSD − F12b = 1 . F (T) = 1 . calculated in the larger basis set was used.The scalar relativistic (SR) correction ∆ E SR was computed using the second-orderDouglas-Kroll-Hess approach at the CCSD(T)/cc-pVQZ-DK level of theory in thefrozen core approximation. The spin-orbit interaction was not considered as for light,closed-shell molecules it can be safely ignored in spectroscopic calculations. The core-valence (CV) electron correlation correction ∆ E CV was calculated at theCCSD(T)-F12b level of theory in conjunction with the F12-optimized correlation con-sistent core-valence basis set cc-pCVTZ-F12. The same ansatz and ABS as in the frozencore approximation computations were used, however we set β = 1 . a − . The (1 s ) orbitalof Si was frozen for all-electron calculations.To estimate the higher-order (HO) correction ∆ E HO we used the hierarchy of cou-pled cluster methods such that ∆ E HO = ∆ E T + ∆ E (Q) . Here the full triples contri-bution is ∆ E T = (cid:2) E CCSDT − E CCSD(T) (cid:3) , and the perturbative quadruples contribution is∆ E (Q) = (cid:2) E CCSDT(Q) − E CCSDT (cid:3) . Calculations were carried out in the frozen core approxi-mation at the CCSD(T), CCSDT, and CCSDT(Q) levels of theory using the general coupledcluster approach as implemented in the MRCC code interfaced to CFOUR. The fulltriples computation utilized the correlation consistent triple zeta basis set, cc-pVTZ(+d forSi), whilst the perturbative quadruples computation employed the double zeta basisset, cc-pVDZ(+d for Si).The contribution from the diagonal Born-Oppenheimer correction (DBOC) was com-4uted with all electrons correlated (bar the (1 s ) orbital of Si) using the CCSD method as implemented in CFOUR with the aug-cc-pCVDZ basis set. A preliminary analysis ofthe DBOC on the vibrational energy levels showed no improvement overall when comparedagainst experimental values. Given that inclusion of the DBOC means the PES becomesapplicable only for SiH and no other isotopologues, the correction was not included.In generating a high-level ab initio PES for silane we have opted for a more pragmaticapproach. Obtaining tightly converged energies with respect to basis set size for the HLcorrections is less important, particularly for the CV and HO contributions which are com-putationally more demanding. Since the CV and HO corrections usually enter the electronicenergy with opposing sign, we have calculated them together utilizing smaller basis sets.Although independently the separate corrections are not fully converged, this error is com-pensated for when considering their sum. This is illustrated through one-dimensional cutsof the PES in Fig. (1), most noticeably in the bending cut.The global grid was built in terms of nine internal coordinates; four Si-H bond lengths r , r , r , r , and five ∠ (H j -Si-H k ) interbond angles α , α , α , α , and α , where j and k label the respective hydrogen atoms. The Si-H stretch distances ranged from 0 . ≤ r i ≤ .
95 ˚A for i = 1 , , , ≤ α jk ≤ ◦ where jk = 12 , , , ,
24. All terms in Eq. (1) were calculated on a grid of 84 002 geometrieswith energies up to hc ·
50 000 cm − ( h is the Planck constant and c is the speed of light). Atevery grid point the coupled cluster energies were extrapolated to the CBS limit, and eachHL correction was calculated and added to the total electronic energy.The HL corrections have been computed at each grid point which is in fact time-effectiveat the levels of theory chosen for the electronic structure calculations. The alternative isto design reduced grids for each correction, fit a corresponding analytic representation andapply the resulting form to the global grid of geometries by interpolation (see Refs. 59 and60 for examples of this strategy). Although this alternative is computationally less intensive,achieving a satisfactory description of each HL correction requires careful consideration andmay not be trivial; any such problems are avoided in our present approach.5 = HO (D / T)Z
X = HO (T / Q)Z
X = CV TZ X = CV QZ r (SiH) / ˚A E X / c m − TZ +HO (D / T)Z
X = CV QZ +HO (T / Q)Z r (SiH) / ˚A X = HO (D / T)Z
X = HO (T / Q)Z
X = CV TZ X = CV QZ α (H SiH ) / deg E X / c m − TZ +HO (D / T)Z
X = CV QZ +HO (T / Q)Z α (H SiH ) / deg FIG. 1. One-dimensional cuts of the CV, HO, and CV+HO corrections for different sizes of basisset. For CV the subscript TZ(QZ) refers to calculations with the cc-pCVTZ-F12(cc-pCVQZ-F12)basis set. For HO the subscript (D/T)Z refers to calculations with the cc-pVDZ and cc-pVTZ basissets for the perturbative quadruples and full triples, respectively. Likewise the (T/Q)Z subscriptcorresponds to the cc-pVTZ and cc-pVQZ basis sets.
B. Analytic representation
The analytic representation chosen for the present study has previously been used formethane.
For the stretch coordinates, ξ i = 1 − exp (cid:0) − a ( r i − r ref ) (cid:1) ; i = 1 , , , a = 1 .
47 ˚A − and the reference equilibrium structural parameter r ref = 1 . ξ = 1 √
12 (2 α − α − α − α − α + 2 α ) (3)6 = 12 ( α − α − α + α ) (4) ξ = 1 √ α − α ) (5) ξ = 1 √ α − α ) (6) ξ = 1 √ α − α ) (7)The potential function (maximum expansion order of i + j + k + l + m + n + p + q + r = 6), V ( ξ , ξ , ξ , ξ , ξ , ξ , ξ , ξ , ξ ) = (cid:88) ijk... f ijk... V ijk... (8)contains the terms V ijk... = { ξ i ξ j ξ k ξ l ξ m ξ n ξ p ξ q ξ r } T d (M) (9)which are symmetrized combinations of different permutations of the coordinates ξ i , andtransform according to the T d (M) molecular symmetry group. They are found by solvingan over-determined system of linear equations in terms of the nine coordinates given above.A total of 287 symmetrically unique terms were derived up to sixth order of which only104 were employed for the final PES. The corresponding expansion parameters f ijk... weredetermined from a least-squares fitting to the ab initio data. Weight factors of the form, w i = tanh (cid:104) − . × ( ˜ E i −
15 000) (cid:105) + 1 . . × N ˜ E ( w ) i (10)were used in the fit. Here ˜ E ( w ) i = max( ˜ E i ,
10 000), where ˜ E i is the potential energy at the i th geometry above equilibrium and the normalization constant N = 0 . − ). The final fitted PES required 106 expansion parameters and employed Watson’srobust fitting scheme, which reduces the weights of outliers and improves the fit at lowerenergies. A weighted root-mean-square (rms) error of 1 .
77 cm − was obtained for energiesup to hc ·
50 000 cm − .Note that geometries with r i ≥ .
30 ˚A for i = 1 , , , > . and so the corresponding weights were reduced by several orders of magnitude.Although the coupled cluster method is not completely accurate at these points, by includingthem the PES maintains a reasonable shape towards dissociation. In subsequent calculationswe refer to this PES as CBS-F12 HL . The CBS-F12 HL expansion parameter set is providedin the supplementary material along with a FORTRAN routine to construct the PES. II. DIPOLE MOMENT SURFACEA. Electronic structure calculations
The electric dipole moment is equal to the first derivative of the electronic energy withrespect to external electric field strength. For each of the X , Y , and Z Cartesian coordinateaxes with origin at the Si nucleus, an external electric field with components ± .
005 a.u.was applied and the dipole moment components µ X , µ Y , and µ Z computed by means ofthe central finite difference scheme. Calculations were carried out at the CCSD(T)/aug-cc-pVTZ(+d for Si) level of theory in the frozen core approximation using MOLPRO2012. Thesame nine-dimensional grid as used for the PES with energies up to hc ·
50 000 cm − wasemployed. B. Analytic representation
To represent the dipole moment surface (DMS) analytically it is necessary to transformto a suitable molecule-fixed xyz coordinate system. For the present study we utilize thesymmetrized molecular bond (SMB) representation for XY molecules. We first defineunit vectors along the four Si-H bonds, e i = r i − r | r i − r | ; i = 1 , , , r is the position vector of the Si nucleus, and r i is that of the respective H i atom.Three symmetrically independent reference vectors which span the F representation areformed, n = 12 ( e − e + e − e ) (12) n = 12 ( e − e − e + e ) (13) n = 12 ( e + e − e − e ) (14)Using these the ab initio dipole moment vector µ can be expressed as µ = µ x n + µ y n + µ z n (15)8ere µ α ( α = x, y, z ) are the dipole moment functions (also of F symmetry) which take theform µ α ( ξ , ξ , ξ , ξ , ξ , ξ , ξ , ξ , ξ ) = (cid:88) ijk... F ( α ) ijk... µ F α,ijk... (16)The expansion terms µ F α,ijk... = { ξ i ξ j ξ k ξ l ξ m ξ n ξ p ξ q ξ r } F α (17)are symmetrized combinations of different permutations of coordinates ξ i , and span the F α representation of the T d (M) molecular symmetry group (see Ref. 61 for more detail). Asixth order expansion was employed in terms of the coordinates, ξ i = ( r i − r ref ) exp (cid:0) − β ( r i − r ref ) (cid:1) ; i = 1 , , , (cid:0) − β ( r i − r ref ) (cid:1) prevents the expansion from diverging at large values of r i . Our DMSfitting employed the parameters r ref = 1 . β = 1 . − .The expansion coefficients F ( α ) ijk... for all three components α = x, y, z were determinedsimultaneously through a least squares fitting to the ab initio data. Again weight factorsof the form given in Eq. (10) were used which favor energies below hc ·
15 000 cm − . Thefitting required 283 parameters and reproduced the ab initio data with a weighted rms errorof 0 .
001 D for energies up to hc ·
50 000 cm − . The expansion parameter set for the DMS isprovided in the supplementary material along with a FORTRAN routine to construct thecorresponding analytic representation. IV. RESULTSA. Equilibrium bond length and pure rotational energies
Since rotational energies are highly dependent on the molecular geometry through themoments of inertia, we first refine the Si-H reference equilibrium structural parameter r ref before we proceed to extensive rovibrational energy level calculations. Thereby, the accuracyof the computed intra-band rotational wavenumbers can be significantly improved. Two iterations of a nonlinear least-squares fit to the experimental J ≤ r ref = 1 . r eq = 1 . r (Si − H) =1 . and an ab initio value of r (Si − H) = 1 . Note that before the refinement the original ab initio bond length of the CBS-F12 HL PES was r eq ab initio = 1 . J ≤ . − . We thereforeexpect the true Si-H equilibrium bond length to be very close to the value r eq = 1 . B. Vibrational J = 0 energies To calculate rovibrational energy levels, transition frequencies and corresponding in-tensities we use the variational nuclear motion code TROVE. Here we only summarizethe key aspects of our calculations. Details of the general methodology can be found inRefs. 69, 73, and 74.The rovibrational Hamiltonian was represented as a power series expansion around theequilibrium geometry in terms of the coordinates given in Eqs. (2) to (7), and was constructednumerically using an automatic differentiation method. The kinetic and potential energyoperators were truncated at 6th and 8th order, respectively, which is sufficient for ourpurposes. For a discussion of the associated errors of such a scheme see Refs. 73 and 74.Note that atomic mass values were employed in the subsequent TROVE calculations.The vibrational basis set was generated using a multi-step contraction scheme. For SiH the polyad number P = 2( n + n + n + n ) + n + n + n + n + n ≤ P max (19)controls the size of the basis set and does not exceed a predefined maximum value P max . For J = 0 vibrational energy level calculations we set P max = 14. Here the quantum numbers n k for k = 1 , . . . , φ n k , which are obtained bysolving a one-dimensional Schr¨odinger equation for each vibrational mode by means of theNumerov-Cooley method. The normal modes of silane are classified by the symmetry species, A , E , and F . Of A symmetry is the non-degenerate symmetric stretching mode ν (2186 .
87 cm − ). The doubly10 ABLE I. Comparison of calculated and experimental J ≤ − )for SiH . The observed ground state energy levels are from Ref. 36. J K
Sym. Experiment Calculated Obs − calc0 0 A F E F A F F A E F F E F F F A A E F F F degenerate asymmetric bending mode ν (970 .
93 cm − ) has E symmetry. Whilst of F sym-metry are the triply degenerate modes; the asymmetric stretching mode ν (2189 .
19 cm − ),and the asymmetric bending mode ν (913 .
47 cm − ). The values in parentheses are theexperimentally determined values from Ref. 36. To be of spectroscopic use we map thevibrational quantum numbers n k of TROVE to the normal mode quantum numbers v k com-monly used. For SiH the vibrational states are labelled as v ν + v ν + v ν + v ν wherev i counts the level of excitation. 11n Table II the computed vibrational energies using the CBS-F12 HL PES are listed againstall available experimental data up to 8500 cm − . The four fundamental frequencies are allreproduced with sub-wavenumber accuracy, resulting in an overall rms error of 0 .
63 cm − and a mean-absolute-deviation (mad) of 0 .
57 cm − . Altogether the 49 experimental levelsare reproduced with a rms error of 1 .
33 cm − and mad of 1 .
07 cm − . Note that energies areconverged to 0 .
01 cm − or better (the majority are converged to orders of magnitude lower),except for the two levels at 8347 .
86 cm − which are converged to within 0 .
02 cm − . Thiswas confirmed by performing a complete vibrational basis set extrapolation with values of P max = { , , } (see Refs. 60 and 77 for further details). TABLE II: Comparison of calculated and experimental J = 0 vibrational term values (in cm − )for SiH . The zero-point energy was computed to be 6847 .
084 cm − .Mode Sym. Experiment Calculated Obs − calc Ref. ν F ν E ν A ν F ν E ν + ν F ν + ν F ν A ν E ν A ν F ν F ν A ν F ν F ν + 2 ν E ν + 2 ν F ν + 2 ν A ν + 2 ν F ν + 2 ν E ν + 2 ν A ABLE II: (
Continued )Mode Sym. Experiment Calculated Obs − calc Ref.2 ν + ν F ν + ν F ν + ν F ν E ν A ν A ν + ν F a ν + ν F a ν + ν E a ν + ν F a ν + ν A a ν + ν F a ν + ν F a ν + ν E a ν A b ν + ν F ν A c ν E ν F c ν + 2 ν A d ν F d ν A c ν + ν F ν + 2 ν E c ν F c ν F c ν + 3 ν A d ν + 3 ν F da Originally attributed to Ref. 36, but unable to confirm value independently. b Originally attributed to Ref. 24. c Originally attributed to Ref. 78. d Originally attributed to Refs. 21–23. ABLE II: (
Continued )Mode Sym. Experiment Calculated Obs − calc Ref. Of the 35 term values up to 3153 .
60 cm − , the energy of 32 levels is underestimatedby the CBS-F12 HL PES. This can be explained by the residual errors of the ν and ν fundamentals, which largely dictates the accuracy of the subsequent combination bandsand overtones. Above 3153 .
60 cm − computed energy levels are consistently higher thanexperiment which is a result of overestimating the ν and ν fundamentals. Despite this, theperformance of the CBS-F12 HL PES is extremely encouraging, especially considering thatfor vibrational J = 0 energy levels the PES can be regarded as an ab initio surface.Experimental values for stretching overtones above 8500 cm − are available. How-ever, the corresponding values in TROVE are harder to identify given the increased densityof states at higher energies. Highly excited modes also show slower convergence with re-spect to vibrational basis set size. Thus, to obtain reasonably well converged energies wouldrequire calculations with P max = 16 or greater, which is currently unachievable with thecomputational resources available to us.As an aside in Table III we show the effect of the empirical refinement of the equilibriumgeometry on the fundamental frequencies. Results computed using the ab initio bond length(overall rms error of 0 .
57 cm − ) are marginally better which is to be expected. In the refinedgeometry PES the shape of the original ab initio PES has been altered by shifting itsminimum, resulting in a poorer representation of vibrational energies. For spectral analysisan improved description of rotational structure is more desirable however, as vibrationalband position can be easily corrected at a later stage. C. Vibrational transition moments
The vibrational transition moment is defined as, µ if = (cid:115) (cid:88) α = x,y,z |(cid:104) Φ ( f )vib | ¯ µ α | Φ ( i )vib (cid:105)| (20)where | Φ ( i )vib (cid:105) and | Φ ( f )vib (cid:105) are the initial and final state vibrational eigenfunctions respectively,and ¯ µ α is the electronically averaged dipole moment function along the molecule-fixed axis14 ABLE III. Comparison of the computed fundamental term values (in cm − ) with the refined and ab initio equilibrium geometry.Mode Sym. Experiment a Refined eq. (A)
Ab initio eq. (B) Obs-calc (A) Obs-calc (B) ν A ν E ν F ν F a See Table II for experimental references. α = x, y, z . In Table IV we list computed vibrational transition moments from the vi-brational ground state. Calculations used the CBS-F12 HL PES and a polyad number of P max = 12 which ensured converged results.Experimentally determined transitions moments have only been derived for the ν (2189 .
19 cm − ) and ν (913 .
47 cm − ) modes. Fox and Person using earlier band intensitymeasurements found µ ν = 0 . ±
4% D and µ ν = 0 . ±
7% D. The reliability ofthe intensity data has however been questioned. In other work, Cadot determined atransition moment of µ ν = 0 . ±
3% D. Whilst a value of µ ν = 0 .
247 D was quoted inRef. 4 but attributed to unpublished results.Although the experimental situation is not entirely clear, the computed TROVE transi-tion moments of µ ν = 0 . µ ν = 0 . µ ν and µ ν however. Experimentally derived transition moments for the other levels of silane could helpclarify previous results and assist future theoretical benchmarking.It is worth nothing that if we use the values from Ref. 80 and compare the ratio µ exp ν /µ exp ν = 0 .
599 with µ TROVE ν /µ TROVE ν = 0 . ABLE IV. Calculated vibrational transition moments (in Debye) and frequencies (in cm − ) fromthe vibrational ground state for SiH . Only levels of F symmetry are accessible from the groundstate in IR absorption.Mode Sym. Experiment a Calculated µ if ν F ν F ν + ν F ν F ν F ν F ν + 2 ν F ν + ν F ν + ν F ν + ν F ν + ν F ν + ν F ν F - 3609.08 0.4741E-34 ν F - 3638.92 0.1892E-4 ν + 3 ν F - 3677.72 0.6075E-3 ν + 3 ν F - 3704.01 0.5424E-3 ν + 3 ν F - 3707.66 0.2098E-42 ν + 2 ν F - 3758.50 0.1628E-32 ν + 2 ν F - 3767.13 0.5799E-43 ν + ν F - 3810.86 0.2432E-33 ν + ν F - 3827.61 0.3848E-3 ν + ν F ν F ν F ν + ν F ν F ν + 3 ν F a See Table II for experimental references. . Absolute line intensities of the ν band To simulate absolute absorption intensities we use the expression, I ( f ← i ) = A if πc g ns (2 J f + 1) exp ( − E i /kT ) Q ( T ) ν if (cid:20) − exp (cid:18) − hcν if kT (cid:19)(cid:21) , (21)where A if is the Einstein-A coefficient of a transition with frequency ν if between an initialstate with energy E i , and a final state with rotational quantum number J f . Here k is theBoltzmann constant, T is the absolute temperature, and c is the speed of light. The nuclearspin statistical weights are g ns = { , , , , } for states of symmetry { A , A , E, F , F } ,respectively. The partition function Q ( T ) was estimated using, Q ( T ) ≈ Q rot ( T ) × Q vib ( T ).For tetrahedral molecules the rotational partition function is given as, Q rot ( T ) = 43 π / (cid:18) BhckT (cid:19) − / exp (cid:18) Bhc kT (cid:19) (22)where for SiH we use a ground state rotational constant of B = 2 . T = 296 K, Q rot = 1447 . Q vib = 1 . resulting in Q = 1527 . ν band measured the absolute line intensities ofnumerous P-branch transitions up to J = 16 at 296 K. Line intensities were recorded at aresolution of 0 . − and were given an estimated experimental measurement accuracyof 10%. To validate our DMS and to a lesser extent the PES, in Table V we comparefrequencies and absolute line intensities of over 100 transitions from Ref. 89. The results arealso illustrated in Fig. (2). TABLE V: Comparison of calculated and observed frequencies (in cm − ) and absolute line inten-sities (in cm/molecule) for transitions between the ν and ground vibrational state. To quantifythe error in the computed line intensity we use the percentage measure, %[(obs − calc) / obs].Γ (cid:48) J (cid:48) K (cid:48) Γ (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) ν obs ν calc ∆ obs − calc I obs I calc % (cid:2) obs − calcobs (cid:3) F F E E A A F F F F F F an Helden et al.TROVE Wavenumber (cm − ) A b s o l u t e i n t e n s i t y ( c m / m o l ec u l e ) − ) % [ ( o b s - c a l c ) /o b s ] FIG. 2. Absolute line intensities of the ν band for transitions up to J = 16 (left) and thecorresponding residuals (cid:0) % (cid:2) obs − calcobs (cid:3)(cid:1) (right) when compared with measurements from van Helden et al. . TABLE V: ( Continued )Γ (cid:48) J (cid:48) K (cid:48) Γ (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) ν obs ν calc ∆ obs − calc S obs S calc % (cid:2) obs − calcobs (cid:3) E E F F A A F F E E F F F F A A F F F F F F A A F F F F E E F F E E F F F F A A ABLE V: (
Continued )Γ (cid:48) J (cid:48) K (cid:48) Γ (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) ν obs ν calc ∆ obs − calc S obs S calc % (cid:2) obs − calcobs (cid:3) F F F F F F F F E E F F E E F F E E A A F F A A F F F F F F F F E E
10 3 2131.274 2131.402 -0.128 8.116E-20 8.571E-20 -5.61 F F
10 3 2131.298 2131.424 -0.126 9.629E-20 1.082E-19 -12.38 A A
10 1 2131.302 2131.439 -0.137 4.663E-20 6.417E-20 -37.62 F F
10 2 2131.315 2131.445 -0.130 5.899E-21 7.399E-21 -25.42 F F
10 1 2131.340 2131.467 -0.127 2.822E-20 3.382E-20 -19.86 F F
10 1 2131.381 2131.512 -0.131 1.117E-20 1.499E-20 -34.22 E E
10 5 2131.399 2131.527 -0.128 6.337E-21 9.306E-21 -46.85 F F
10 4 2131.594 2131.678 -0.084 6.694E-21 3.753E-21 43.93 F F
10 3 2131.600 2131.764 -0.164 1.449E-20 1.615E-20 -11.48 A A
10 4 2131.629 2131.796 -0.167 1.534E-19 1.616E-19 -5.31 A A
10 1 2131.672 2131.826 -0.154 1.876E-19 1.952E-19 -4.06 F
10 4 F
11 2 2125.142 2125.281 -0.139 1.315E-20 1.212E-20 7.82 E
10 1 E
11 3 2125.162 2125.302 -0.140 2.551E-20 2.212E-20 13.30 F
10 4 F
11 4 2125.194 2125.333 -0.139 1.512E-20 1.610E-20 -6.46 E
10 1 E
11 1 2125.249 2125.389 -0.140 8.867E-21 1.032E-20 -16.41 F
10 4 F
11 2 2125.312 2125.441 -0.129 1.016E-19 1.011E-19 0.58 E
10 2 E
11 3 2125.340 2125.467 -0.127 5.186E-20 5.236E-20 -0.97 F
10 1 F
11 2 2125.348 2125.481 -0.133 1.369E-20 1.531E-20 -11.88 F
10 3 F
11 3 2125.362 2125.488 -0.126 9.684E-20 1.020E-19 -5.32 ABLE V: (
Continued )Γ (cid:48) J (cid:48) K (cid:48) Γ (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) ν obs ν calc ∆ obs − calc S obs S calc % (cid:2) obs − calcobs (cid:3) A
10 4 A
11 1 2125.809 2125.973 -0.164 1.579E-19 1.712E-19 -8.44 E
10 4 E
11 1 2125.851 2126.025 -0.174 3.963E-20 4.194E-20 -5.82 F
11 1 F
12 4 2119.300 2119.431 -0.131 9.978E-21 1.400E-20 -40.30 A
11 2 A
12 4 2119.331 2119.461 -0.130 1.160E-19 1.440E-19 -24.15 F
11 3 F
12 5 2119.389 2119.515 -0.126 6.041E-20 6.883E-20 -13.94 A
11 3 A
12 3 2119.414 2119.540 -0.126 1.131E-19 1.477E-19 -30.63 F
11 1 F
12 2 2119.440 2119.571 -0.131 1.284E-20 1.634E-20 -27.32 F
11 3 F
12 1 2119.449 2119.576 -0.127 7.866E-21 1.017E-20 -29.29 F
11 2 F
12 2 2119.508 2119.635 -0.127 1.635E-20 2.204E-20 -34.82 F
12 1 F
13 1 2114.154 2114.321 -0.167 4.479E-20 5.868E-20 -31.03 E
12 5 E
13 1 2114.169 2114.352 -0.183 2.707E-20 3.394E-20 -25.38 F
12 1 F
13 1 2114.179 2114.349 -0.170 4.882E-20 5.374E-20 -10.08 F
12 1 F
13 2 2114.187 2114.373 -0.186 3.173E-20 4.001E-20 -26.09 A
12 1 A
13 5 2114.252 2114.453 -0.201 4.283E-20 5.208E-20 -21.59 F
12 3 F
13 5 2114.259 2114.457 -0.198 2.253E-20 2.647E-20 -17.51 F
12 4 F
13 1 2114.263 2114.463 -0.200 2.538E-20 2.854E-20 -12.45 A
12 4 A
13 2 2114.309 2114.506 -0.197 3.990E-20 4.713E-20 -18.13 F
12 3 F
13 2 2114.354 2114.554 -0.200 2.886E-21 3.277E-21 -13.55 E
13 2 E
14 7 2108.308 2108.486 -0.178 2.272E-20 2.725E-20 -19.96 F
13 1 F
14 1 2108.321 2108.499 -0.178 3.210E-20 3.888E-20 -21.14 A
13 5 A
14 6 2108.343 2108.545 -0.202 5.088E-20 5.941E-20 -16.77 F
13 2 F
14 2 2108.349 2108.544 -0.195 2.889E-20 3.389E-20 -17.30 A
13 2 A
14 1 2108.354 2108.535 -0.181 5.234E-20 5.969E-20 -14.04 F
13 2 F
14 3 2108.392 2108.590 -0.198 2.090E-20 2.445E-20 -17.00 F
13 2 F
14 5 2108.482 2108.694 -0.212 1.629E-20 1.955E-20 -20.03 F
13 3 F
14 1 2108.501 2108.711 -0.210 1.259E-20 1.537E-20 -22.06 E
13 4 E
14 3 2108.510 2108.721 -0.211 9.767E-21 1.165E-20 -19.31 A
14 3 A
15 5 2101.289 2101.420 -0.131 5.038E-20 5.580E-20 -10.74 F
14 4 F
15 4 2101.294 2101.420 -0.126 9.089E-21 7.565E-21 16.77 F
14 2 F
15 2 2101.310 2101.440 -0.130 5.713E-21 6.863E-21 -20.13 F
14 3 F
15 4 2101.345 2101.472 -0.127 1.368E-20 1.974E-20 -44.37 F
14 5 F
15 4 2101.369 2101.496 -0.127 2.202E-20 2.588E-20 -17.52 A
14 4 A
15 4 2101.397 2101.523 -0.126 3.615E-20 5.131E-20 -41.95 E
14 2 E
15 1 2101.445 2101.569 -0.124 2.451E-21 3.595E-21 -46.67 A
15 4 A
16 0 2096.608 2096.799 -0.191 2.530E-20 3.021E-20 -19.42 ABLE V: (
Continued )Γ (cid:48) J (cid:48) K (cid:48) Γ (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) ν obs ν calc ∆ obs − calc S obs S calc % (cid:2) obs − calcobs (cid:3) E
15 2 E
16 1 2096.658 2096.850 -0.192 9.113E-21 1.064E-20 -16.71 F
15 2 F
16 3 2096.686 2096.897 -0.211 1.131E-20 1.532E-20 -35.38 E
15 6 E
16 7 2096.743 2096.963 -0.220 8.317E-21 9.454E-21 -13.67 F
15 3 F
16 1 2096.772 2096.994 -0.222 9.262E-21 1.085E-20 -17.14 F
15 7 F
16 2 2096.802 2097.017 -0.215 9.495E-21 1.206E-20 -26.98
Due to the computational demands of calculating higher rotational excitation (rovibra-tional matrices scale linearly with J ), calculations were performed with P max = 10. Con-vergence tests were carried out up to J = 6 for P max = 12. The corresponding transitionfrequencies showed a consistent correction of around ∆( P max = 12) = − . − . Thiscorrection was applied to all computed frequencies listed in Table V. For the correspondingintensities, the 1 ← J (cid:48) ← J (cid:48)(cid:48) ) transitions possessed a convergence correction of the order10 − . The magnitude of this correction showed a linear relationship with increasing J , fromwhich we estimate that for the 15 ←
16 transitions the correction would be of the order10 − . The respective intensities therefore have an error of at most 1%. We are confidentthat the results in Table V are sufficiently converged to reliably evaluate the DMS and PES.Around one third of the calculated absolute line intensities are within the estimatedexperimental measurement accuracy of 10%. However, as is best seen by the residuals plottedin Fig. (2), nearly all of the computed line intensities are larger than the correspondingexperimental values. We suspect this is due to the electronic structure calculations andthe use of only a triple-zeta basis set, aug-cc-pVTZ(+d for Si), to generate the DMS. Alarger (augmented) correlation consistent basis set and possibly the inclusion of additionalhigher-level corrections (such as those incorporated for the PES) would most likely reducethe strength of computed line intensities. Despite this, Fig. (2) shows that the ν band iswell reproduced. Computed frequencies are on average larger by 0 . − . − across alltransitions. This more or less systematic error can be attributed to the minor empiricalrefinement of the equilibrium Si-H bond length.21 . Overview of rotation-vibration spectrum As a final test of the PES and DMS, in Fig. (3) we have simulated the rotation-vibrationspectrum of SiH for transitions up to J = 20 at 296 K. A polyad number of P max = 10was employed. Transition frequencies and corresponding intensities were calculated for a5000 cm − frequency window with a lower state energy threshold of 5000 cm − . To simulatethe spectrum a Gaussian profile with a half width at half maximum of 0 .
135 cm − was chosenas this appears to closely match the line shape used by the PNNL spectral library. Theexperimental PNNL silane spectrum, also shown in Fig. (3), is at a resolution of around0 .
06 cm − . It was measured at a temperature of 25 ◦ C with the dataset subsequently re-normalized to 22.84 ◦ C (296 K). Note that the PNNL spectrum is of electronics grade silanegas which is composed of SiH (92 . SiH (4 . SiH (3 . SiH cross-sections by 0 .
922 to provide a reliablecomparison.The computed TROVE intensities are marginally stronger but overall there is good agree-ment with the experimental PNNL results. Even with P max = 10 which does not give fullyconverged transition frequencies both band shape and position appear reliable. Of coursethere are shortcomings in our simulations which we will now discuss.Some of the band structure is undoubtedly lost as we have not considered SiH or SiH , and by only computing transitions up to J = 20 the spectrum is unlikely to becomplete at room temperature. There may also be minor errors arising from the use of aGaussian profile to model the line shape. More desirable would be to fit a Voigt profilewhich incorporates instrumental factors. The largest source of error, as discussed before, islikely to be the electronic structure calculations. For the purposes of modelling exoplanetatmospheres however, we expect that the level of theory employed to compute the DMS issufficient. The features of the SiH spectrum are clear and identifiable as seen in Fig. (3).Note that in Fig. (3) the ν (2189 .
19 cm − ) band is stronger than the ν (913 .
47 cm − )band. This is contrast to the vibrational transition moments where µ ν > µ ν . If howeverwe plot absolute line intensities up to J = 20 as shown in Fig. (4), the ν band is indeedstronger than the ν band. The behaviour displayed in Fig. (3) is caused by the use of a lineprofile to model the spectrum. 22 NNLTROVEWavenumber (cm − ) C r o ss - s ec t i o n ( c m / m o l ec u l e ) FIG. 3. Overview of simulated SiH rotation-vibration spectrum up to J = 20. Note that theexperimental PNNL spectrum is composed of SiH (92 . SiH (4 . SiH (3 . V. CONCLUSIONS
High-level ab initio theory has been used to generate global potential energy and dipolemoment surfaces for silane. The quality of the PES is reflected by the achievement ofsub-wavenumber accuracy for all four fundamental frequencies. Combination and overtonebands are also consistently reproduced which confirms that the level of ab initio theory usedto generate the PES is adequate. Minor empirical refinement of the equilibrium geometryof SiH produced an Si-H bond length in excellent agreement with previous experimentaland theoretical results. The rotational structure of vibrational bands was improved as aresult of the refinement. Ultimately though, to achieve sub-wavenumber accuracy for allrotation-vibration energy levels a rigorous empirical refinement of the PES is necessary. A new ab initio
DMS has been computed and utilized to simulate the infrared spectrum ofSiH . Absolute line intensities are marginally overestimated and we suspect this behaviour23 ROVE
Wavenumber (cm − ) A b s o l u t e i n t e n s i t y ( c m / m o l ec u l e ) FIG. 4. Overview of absolute line intensities of SiH up to J = 20. can be resolved by using a larger basis set for the electronic structure calculations whencomputing the DMS. Overall however, band shape and structure across the spectrum displaygood agreement with experiment. The PES and DMS presented in this work will be usedto compute a comprehensive rovibrational line list applicable for elevated temperatures aspart of the ExoMol project. ACKNOWLEDGMENTS
This work was supported by ERC Advanced Investigator Project 267219, and FP7-MC-IEF project 629237.
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