Full-Dimensional Potential Energy and Dipole Moment Surfaces of

Article pubs.acs.org/JPCA

Full-Dimensional Potential Energy and Dipole Moment Surfaces of GeH4 Molecule and Accurate First-Principle Rotationally Resolved Intensity Predictions in the Infrared A.V. Nikitin,*,† M. Rey,‡ A. Rodina,§ B. M. Krishna,§ and Vl. G. Tyuterev‡ †

Laboratory of Theoretical Spectroscopy, V. E. Zuev Institute of Atmospheric Optics, SB RAS, 1, Academician Zuev Square, 634021 Tomsk, Russia ‡ Groupe de Spectrométrie Moléculaire et Atmosphérique, UMR CNRS 7331, Université de Reims, U.F.R. Sciences, B.P. 1039, 51687 Reims Cedex 2, France § Laboratory of Quantum Mechanics of Molecules and Radiative Processes, Tomsk State University, 36 Lenin Avenue, 634050 Tomsk, Russia S Supporting Information *

ABSTRACT: Nine-dimensional potential energy surface (PES) and dipole moment surface (DMS) of the germane molecule are constructed using extended ab initio CCSD(T) calculations at 19 882 points. PES analytical representation is determined as an expansion in nonlinear symmetry adapted products of orthogonal and internal coordinates involving 340 parameters up to eighth order. Minor empirical refinement of the equilibrium geometry and of four quadratic parameters of the PES computed at the CCSD(T)/aug-cc-pVQZ-DK level of the theory yielded the accuracy below 1 cm−1 for all experimentally known vibrational band centers of five stable isotopologues of 70GeH4, 72GeH4, 73 GeH4, 74GeH4, and 76GeH4 up to 8300 cm−1. The optimized equilibrium bond re = 1.517 594 Å is very close to best ab initio values. Rotational energies up to J = 15 are calculated using potential expansion in normal coordinate tensors with maximum errors of 0.004 and 0.0006 cm−1 for 74GeH4 and 76GeH4. The DMS analytical representation is determined through an expansion in symmetry-adapted products of internal nonlinear coordinates involving 967 parameters up to the sixth order. Vibration−rotation line intensities of five stable germane isotopologues were calculated from purely ab initio DMS using nuclear motion variational calculations with a full account of the tetrahedral symmetry of the molecules. For the first time a good overall agreement of main absorption features with experimental rotationally resolved Pacific Northwest National Laboratory spectra was achieved in the entire range of 700−5300 cm−1. It was found that very accurate description of state-dependent isotopic shifts is mandatory to correctly describe complex patterns of observed spectra at natural isotopic abundance resulting from the superposition of five stable isotopologues. The data obtained in this work will be made available through the TheoReTS information system.

1. INTRODUCTION Germane plays an important role in astrophysical chemistry, being one of the important components of the atmospheres of giant planets Jupiter and Saturn,1−3 where germane was detected at abundances orders of magnitude greater than their thermochemical equilibrium values in the upper tropospheres. Precise knowledge of GeH4 absorption/emission is crucial for the study of such planetary systems and can be used to understand the physical properties of their atmospheres.4 Germane molecule is also known to preferentially photodissociate into GeH2, which is a carbine-like radical that could react with methane in these atmospheres.5 The infrared absorption spectrum of germane has been of interest for a long time,6 and there have been many experimental studies of spectra in the infrared region of the XH4 type of molecules (X = C, Ge, Si).7,8 The laboratory infrared spectra of the germaniumbearing molecules have also been reported with view to understand chemistry of planetary atmospheres.9 © 2016 American Chemical Society

Line intensities are the fundamental parameters necessary for remote measurements of the abundance of molecular species and for the determination of temperature conditions of gaseous media by spectral analyses, particularly in the atmospheres of exoplanets and cool stars.10 Laboratory measurements only (in a limited temperature range) do not suffice, as the spectra must be theoretically assigned to recalculate the intensity of quantum transitions for larger temperature variations via the population of the lower state levels. Germane in a natural isotopic composition has complex infrared spectra because of the existence of five stable isotopologues,11−13 namely, 70Ge (20.84%), 72Ge (27.54%), 73Ge (7.73%), 74Ge (36.28%), and 76Ge (7.61%), which all significantly contribute to experimentally recorded spectral patterns. Received: August 1, 2016 Revised: October 20, 2016 Published: October 20, 2016 8983

DOI: 10.1021/acs.jpca.6b07732 J. Phys. Chem. A 2016, 120, 8983−8997

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The Journal of Physical Chemistry A Extrapolation of the isotopic dependence for individual line positions and intensities in a large frequency range is not a trivial task. Unlike many atmospheric molecules like methane, where calculated bands intensities of the major isotopologue dominate the opacity of the natural isotopic compositions and could produce a qualitatively good agreement with experimental spectra in strong absorption ranges, the germane calculations must necessarily include the contributions from all these stable isotopologues. This represents a major challenge for the theory, as predicted energies must be accurate enough to describe state-dependent isotopic shifts that are necessary to produce a correct shape of the absorption bands. GeH4 is a high-symmetry spherical top molecule, and this gives rise to degeneracies and quasi-degeneracies of vibration modes and to complicated resonance interactions due to intermode couplings. Complexity in the spectra due to Coriolis interaction between fundamentals ν2 and ν4 had been discussed since early experimental studies.14,15 Many weak bands (as ν1 and ν2 fundamentals and their overtone and combination bands) would be dipole-forbidden within an isolated band model but become allowed via strong coupling with other active modes in the infrared spectra like ν3 and ν4. The groundstate rotational constants were determined in ref 16, whereas “forbidden” purely rotation transitions were measured in refs 17 and 18. Infrared spectra of germane and its dimer have been also recorded in low-temperature nitrogen matrices.19 Line-by-line analyses of congested experimental spectra proved to be quite challenging14,20−23 because of accidental perturbations produced by interactions among closely lying levels and simultaneously appearing isotopic lines. Most analyses of rotationally resolved germane spectra have been focused on line positions fits, whereas systematic experimental line intensity measurements11,20,24,25 are available in very limited number of spectral interval. Germane represents a considerable interest for the fundamental molecular physics as a high-symmetry molecule exhibiting local mode effects in stretching vibrations. The corresponding local-mode models have been applied in several studies to GeH4 overtone spectra26−32 that enabled describing the observed energy trends for this sample of valence states. In this context a simplified four-dimensional dipole moment surface restricted to stretching vibrations was computed in ref 24 with the density functional theory method and then in ref 28 using CCDS(T) method with the cc-pVQZ basis set. Initial values for seven corresponding dipole moment parameters have been determined from ab initio calculations and then fitted to experimental band intensities to match low-resolution observations.24,25,28,29 Rotationally resolved line intensity measurements still remain very sparse and limited to few stretching transitions. To our knowledge no combination bend−stretch bands of germane have been assigned up to now, though the corresponding spectral ranges show quite strong absorption features in observed spectra,33 which are not yet analyzed. For purely bending bands the available GeH4 line-by-line analyses are currently limited by v2 + v4 = 2 up to 2000 cm−1 (for line positions only),23 which is in contrast to the state-of-art in methane analyses where rotationally resolved spectra for fivebending overtones been assigned and modeled34−36 beyond 6300 cm−1. Spectroscopic databanks TDS37 and STDS38 included some line positions and relative intensities (in arbitrary units) of GeH4 isotopologues calculated using effective polyad models. The latter line lists are limited to the fundamental bands, a part of this data for 74GeH4 being also incorporated to GEISA databank.39

There exist experimental Fourier-transform spectra of GeH4 at natural isotopic abundance covering the range of 700−5300 cm−1 that are included in the Pacific Northwest National Laboratory (PNNL) experimental library.33 However, the most part of these spectra has not been analyzed, and there are no theoretical predictions for line intensities that would enable (obs−calc) comparison for rotationally resolved GeH4 spectra. Empirical effective models do not permit extrapolating line intensities to lager frequency range because of lacking dipole moment parameters. Another issue is that resonance coupling parameters involved in these models are often poorly defined: a determination of such parameters from experimental levels is known to be a mathematically ill-defined problem.40,41 This could have an impact on intensity calculations because the coupling parameters are responsible for the intensity transfer among strongly interacting bands. Also, a drawback of a purely empirical approach is that every isotopic species is considered as a separate molecule, and accurate links among line parameters are difficult to establish. Recent accurate ab initio potential energy surfaces (PES) and dipole moment surfaces (DMS) for three- to six-atomic molecules (see, e.g., refs 42−67, the list being not exhaustive) together with an improvement of variational methods44,64,68−75 (and references therein) have permitted a significant progress in the understanding of their absorption/emission properties. For example, first-principle intensity predictions of methane spectra49,76,72 have made it possible to double the number of assigned rovibrational bands35 and to extend the overview77−79 and analyses for isotopic species. Exhaustive high-temperature theoretical line lists have been generated for methane80−82 and ethylene83 in the infrared for atmospheric and astrophysical applications. The challenge for accurate electronic structure calculations is that GeH4 contains larger number of electrons than above-mentioned atmospheric species, which makes computations much more demanding. In this work we present first full-dimensional PES and DMS of the germane molecule that are then applied for variational prediction of rotationally resolved vibration−rotation spectra of five stable isotopic species. The paper is structured as follows. Section 1 describes electronic structure calculations, comparison of valence PES cuts in coupled cluster and MRCI methods, equilibrium geometry determination, and contribution of relativistic effects. The construction of dense grid of 19 882 nuclear configurations for ab initio calculations and analytical PES representation with a full account of symmetry properties is outlined in Section 2. Section 3 is devoted to variational calculations of vibration and rotation levels of GeH4 isotopologues using two independent methods. A fine-tuning of four quadratic PES parameters and of the equilibrium configuration re to better match fundamental band centers and observed rotational energies is also described in this section. Ab initio 9D DMS in the molecular-bond representation is derived in Section 4. Variational predictions of GeH4 vibration−rotation intensities using our DMS are discussed in Section 5. Validation of these results against available experimental spectra is presented in Section 6. For the first time a very good agreement of ab initio intensities with observed PNNL germane spectra33 was obtained in the entire range of this library.

2. ELECTRONIC STRUCTURE CALCULATIONS AND DETERMINATION OF THE AB INITIO POTENTIAL ENERGY SURFACE To obtain accurate calculations of vibrational and rotational energy levels from the theoretical PES useful for spectroscopic 8984

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The Journal of Physical Chemistry A analysis, it is necessary to combine high-level ab initio methods with sufficiently large basis sets in electronic structure calculations. The MOLPRO suite of programs was used for all of the calculations reported here as implemented in the 2012.1 version.84 To construct the germane molecular PES, we first computed ab initio a set of electronic energies at various molecular configurations. Equilibrium geometries and vibrational frequencies were computed with both single and multireference methods and are compared. The well-established coupled cluster approach with all single and double substitutions from the Hartree−Fock reference determinant augmented by a perturbative treatment of connected triple excitations, CCSD(T),85 was found to provide good descriptions for the equilibrium geometry and frequencies, and it is primarily used in this work to calculate the PES. We used correlationconsistent polarized valence n-tuple ζ-basis sets of Dunning, Peterson, and co-workers86−89 for the calculations and specifically adapted Douglas−Kroll (DK) basis for calculations including relativistic effects.90 The quintuple ζ-basis was used to check convergence and to derive the complete best set (CBS) extrapolated energies. Energies for all electronic structure calculations were converged to 1 × 10−10 au, and for the geometry optimizations the tolerance for the gradient was further tightened from the default 3 × 10−4 to 5 × 10−5 a.u. The coupled cluster energies are compared with the MRCI +Q results to see whether a multideterminental approach would be important to describe the spectroscopically accessible regions of the potential well and whether a CCSD(T) PES would be valid in these regions. The calculations on the ground (and excited electronic states) of the molecule were performed using the multiconfigurational self-consistent-field (MCSCF) method followed by the internally contracted multireference configuration interaction (MRCI) method with the orbitals obtained from the MCSCF calculations being used to set up the CI expansion. The energy values in this work also include Davidson’s size consistency correction (denoted as MRCI+Q). The orbitals used to set up the configuration interaction expansion were obtained by singlet state-averaged MCSCF calculations in a complete active space defined by eight electrons and eight orbitals, a CAS(8,8), using Cs symmetry for the 1-d dissociation PES, with equal weights for the participating states. All dipole moments were computed as the derivative of the energy with respect to the weak external uniform electric field using the finite difference scheme with the field variation of 0.0001 au around at the zero field strength. As a first step of the DMS construction, the dipole moment values in the electronic ground state were calculated with four basis sets on the grid of 19 882 nuclear configurations. This is described in more detail in later sections. Equilibrium Geometry. The CCSD(T) method treats the triples contribution to dynamic electron correlation perturbatively and can be accurate where a single configuration is dominant. This is the case near equilibrium geometries, where a single configuration is predominant. However, a single reference description could possibly degrade for significantly large nuclear displacements. To check the range of validity of the CCSD(T) calculations we compared this with a sample of MRCI calculations. The energy-ordered natural orbitals obtained from the state-averaged CASSCF calculations were used for the MRCI calculations. Here, a valence complete active space is composed of eight electrons in eight orbitals; a CAS(8,8) is used. The energy of the equilibrium geometry and the equilibrium Ge−H

Table 1. Optimized Minimum Energy of the PES and Equilibrium Born−Oppenheimer Geometry of GeH4 at Various Levels of Theory method/basis

energy/Eh

r(Ge−H) equilibrium/Å

CCSD(T) /CVQZ CCSD(T) /ACVQZ CCSD(T)/ACV5Z CCSD(T)/CBS CCSD(T)/CVQZ-DK MRCI+Q/CVQZ MRCI+Q/CVQZ-DK MRCI + Q/ECP empirical estimation92 this work

−2077.900 812 5415 −2079.138 688 46 −2079.274 501 16 −2079.416 619 14 −2100.773 736 34 −2077.895 559 60 −2099.643 336 00 −295.881 198 50

1.540 554 46 1.524 057 29 1.523 815 18 1.516 137 53 1.540 883 30 1.533 760 75 1.545 862 01 1.5173 1.517 594

distance obtained from CCSD(T) and MRCI+Q at various basis sets is collected in Table 1. The equilibrium geometry obtained from a CCSD(T)/ CVQZ geometry optimization gives a Ge−H bond distance of 1.540 554 46 Å and the energy of −2077.900 812 5415 Eh. The corresponding MRCI+Q/CVQZ method gives a slightly longer Ge−H bond length of 1.540 883 30 Å and the energy of −2077.895 5596 Eh. Further improving the basis with CCSD(T)/ACVQZ and including more correlation gives a shorter Ge−H bond distance of 1.523 405 729 Å and the energy of −2079.138 688 46 Eh, whereas further CCSD(T)/ACV5Z, the biggest basis used here, would give a similar Ge−H bond length of 1.523 815 18 Å and the energy of −2079.274 501 16 Eh. These bond distances are closer to the empirically estimated values. These calculations are prohibitively expensive to produce a full-dimensional ab initio PES at this level of the theory. However, this allows us to use the Karton−Martin extrapolation scheme to obtain the CBS extrapolated energy (using the ACVQZ and ACV5Z bases).91 The CBS gives the equilibrium energy of −2079.416 619 14 Eh for the GeH4 system. Relativistic Effects and One-Dimensional Cut toward the Dissociation of GeH4. An account for the scalar relativistic effects has been shown to be important in earlier studies93 of germanium-containing molecules like GeC2 for better determination of the equilibrium geometry and vibrational frequencies. We also observe that the inclusion of scalar relativistic effects using the DK94 Hamiltonian with the specifically adapted CC-pCVQZ-DK basis provides a more accurate description of the GeH4 equilibrium geometry and fundamental vibrational frequencies. Relativistic corrections are estimated here using the DK Hamiltonian to fourth order, as we found that the corresponding energy contributions were converged up to the fourth order at most. The overall behavior of the potential-energy function does not vary with addition of scalar relativistic effects. However, an account for the scalar relativistic effects reduced the equilibrium bond distance of GeH4 bringing the calculated values closer to the empirically estimated values. The equilibrium geometry obtained from a CCSD(T)/CVQZ-DK geometry optimization gives the Ge−H bond distance of 1.516 137 53 Å and an energy of −2100.773 736 34 Eh. The MRCI+Q/CVQZ-DK method with a CAS(8,8) gives a Ge−H bond length of 1.533 760 75 Å and the energy of −2099.643 336 Eh. The MRCI equilibrium distance values are slightly longer with both CVQZ-DK and CVQZ basis in comparison with corresponding CCSD(T) Ge−H bond lengths. This is in line with expectations, because CCSD(T) is known to recover more dynamical correlations, 8985

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with a CCSD(T)/CVQZ-DK calculation with DK method with that of CCSD(T)/CVQZ calculation (same for MRCI method). In comparison, the energy difference obtained from both the MRCI and the CCSD(T) methods can be seen as almost equal for geometries along the 1D dissociation path (see Figure 2). This confirms that an inclusion of relativistic effect is essential for GeH4.

resulting thus to shorter equilibrium bond lengths. An increase in active space in MRCI would give shorter bond lengths but becomes computationally expensive. The use of relativistic effective core potential basis,95 ECP10MDF, in an MRCI optimization calculation gives longer equilibrium Ge−H bond lengths. Importantly, the preliminary harmonic vibrational frequencies at the optimized geometry for GeH4 with CCSDT/CVQZDK with the DK Hamiltonian showed deviations of more than 25 cm−1 from corresponding calculations at CCSD(T)/CVQZ. Similar behavior was observed with MRCI/CVQZ-DK results. The frequencies obtained including the scalar relativistic effects were now closer to observed values. This points to the importance of creating an ab inito PES and DMS that would include relativistic effects for GeH4. The Figure 1 compares a one-dimensional (1D) dissociation path of GeH4 calculated with CCSD(T) and MRCI methods

Figure 2. Scalar relativistic correction for the 1D dissociation path GeH4 → GeH3 + H. It was obtained as the difference between CCSD(T)/CVXZ-DK calculations using DK Hamiltonian and nonrelativistic CCSD(T)/CVXZ calculation, where X = T or Q. This is compared with the sum of energy contributions from massvelocity term and Darwin term.

3. SAMPLING OF AB INITIO POINTS AND ANALYTICAL POTENTIAL ENERGY SURFACE REPRESENTATION To parametrize all possible nuclear geometries of the GeH4 molecule, it is convenient to use nine polar coordinates, since they are internally built in ab initio program suites like MOLPRO.84 The internal polar coordinates are defined in a standard way via four vectors {ri⃗ } (i = 1, 2, 3, 4) linking the Ge atom with the four H atoms: four Ge−H bond lengths {r1, r2, r3, r4}, three interbond angles {q12, q13, q14}, and two torsion angles {t23, t24}. More convenient expression for the kinetic energy can be obtained in terms of mass-dependent orthogonal coordinates.50,97,98,73 Mass-dependent coordinates (2) keep the same symmetry properties as the initial polar coordinates and are defined via four vectors {ri′⃗ }:

Figure 1. 1D cut of the electronic ground A′ state PES of GeH4 for the dissociation path to GeH3 + H. The values obtained with CCSD(T)/ CVQZ-DK (red ×) are compared with MRCI+Q/CVQZ-DK method (black ●). Other Ge−H bonds are held fixed to 1.540 862 01 Å.

including the scalar relativistic effects. The Davidson corrected energy values are used in the graphs designated as MRCI + Q. The MRCI calculations give a smooth dissociation curve tending to a constant asymptote. The direct comparison of the behavior of the potential energy function along the 1D dissociation from CCSD(T) and MRCI shows a good agreement of both methods up to (E − Emin) ≈ 0.1 au that corresponds to ∼22 000 cm−1 on the wavenumber scale. This gives us confidence in using CCSD(T) method to compute PES and DMS in the range of energies covered by existing spectroscopic experiments. These calculations reveal that a single configuration is indeed dominant at and close to equilibrium positions of the GeH4 PES, and therefore CCSD(T) (with the T1 diagnostic values of ∼0.01 across different geometries) is an appropriate method for producing the ab initio PES and DMS for spectra calculations. It was instructive to compute the electronic energies in the ACVQZ basis sets including MVD (“mass velocity” + Darwin) method.96 The one-electron Darwin term, which is always positive, corrects the coulomb attraction and the mass-velocity term, which always negatively corrects the kinetic energy of the system. The contribution of energy from the sum of mass velocity and Darwin from MVD calculations in the CVQZ basis set could then be compared with the energy difference obtained

4

r ′⃗ i = ri ⃗ + d ∑ rj⃗ where d = j=1

⎞ 1⎛ M ⎜⎜ − 1⎟⎟ 4 ⎝ mGe ⎠

(1)

Here, mGe is the atomic mass of Ge atom, and M is the total mass of the molecule. In the case of GeH4 the orthogonal coordinates are very close to internal coordinates. So PES parameters in two types of coordinates are closed and could be easily converted using the fit. The following 1D elementary functions were used for radial and angular coordinates: ϕ(r ) = 1 − exp[− a(ri − re)] and ϕ(q) = cos(q) − cos(qe)

(2)

where a = 1.6 for PES and a = 1.9 for DMS. A full account of symmetry using irreducible tensor representation permits an optimum choice of the grid of nuclear 8986

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published local-mode GeH4 calculations have been limited to this stretching sample of states and excluded all bending and mixed states. This work aims at full-dimensional calculations of GeH4 rovibrational spectra. As a first step we employ two independent algorithms for variational calculations of vibrational energy levels. The so-called “6A-algorithm” (treating on a similar footing all six interbond angles) and their coupling with stretching coordinates using the exact nuclear kinetic energy operator was applied following the method described previously.73Another variational method is based on the normal mode-Eckart-frame nuclear motion Hamiltonian108 using the truncation-compression techniques109,72 of the vibrational basis sets. Both these methods72,73,109,110 are fully symmetrized according to the Td point group. Their consistency had been verified in previous works on methane.76 They also give very similar results for the GeH4 levels with the same PESs. The comparison of the observed fundamental band centers of 74GeH4 with variational ab initio values calculated from VQZ and CVQZ-DK surfaces using CCSD(T) method is given in Table 2. The three fundamentals calculated with VQZ

displacements to minimize the cost of calculations. This is essential for the molecules of the tetrahedral point group. Here we give only a brief outline of the approach, as the same method has been applied for the methane PES50 having the same symmetry properties. To this end the symmetrized S-coordinates99 are convenient for an analytical PES representation. We use symmetrized S-coordinates99 for the calculation of symmetry-adapted grid of points in the coordinate space suitable for the PES calculation following the techniques described in refs 50, 100, and 101. The approach based on the force field constants permits to find an optimal set of geometric nuclear configurations sufficient for a construction of the force field up to a certain order of expansion. The only modification of scheme of the grid construction with respect to CH450 is more shallow potential well of GeH4 with the 1D energy steps of 0, 1000, 2500, 4000, 7000, 10 000, and 12 500 cm−1. The total number of 19 882 points for the optimal grid in the nuclear configuration space was built in this way. The ninedimensional PES was constructed in a similar way to that of methane.50,73 We used the standard definition of the direct product of irreducible tensors following refs 102−104. The potential function was developed in power series of irreducible tensors

Table 2. Fundamental Band Centersa (cm−1) for 74GeH4

V (r1 , r2 , r3 , r4 , q12 , q13 , q14 , t 23 , t 24) =

∑ K iR ip(r1, r2 , r3 , r4 , q12 , q13 , q14 , t23 , t24) i

bands

obs

VQZ

obs-VQZ

CVQZ-DK

obs-CVQZ-DK

ν2 ν4 ν1 ν3

929.909 820.711 2110.702 2111.143

910.21 818.33 2092.10 2083.38

19.70 2.38 18.60 27.76

922.01 814.46 2117.03 2117.30

7.90 6.25 −6.33 −6.16

(3)

where a

Comparison of purely ab initio calculations using the VQZ and CVQZ-DK PESs with experiment.

R ip = (([SR A1]p1 × [SAE]p2 )C × ([SR F2]p3 × [SA F2]p4 )C )A1 (4)

PES were significantly underestimated, the fundamentals with CVQZ-DK accounting for relativistic contributions being considerably better. Note that in previous calculations for molecules CH4,50 SiH4,111 PH3,101 C2H4,47 CH3F,100 and other CH3X-type species (X = Cl, Br, I),112,58 the ab initio PESs at the CCSD(T)/VQZ level provided much better accuracy. To improve the calculation of band centers in germane spectra we applied the same approach of an empirical PES scaling that proved to be efficient for the methane molecule.50 Four quadratic parameters of the initial CVQZ-DK PES were slightly adjusted to match (1000), (0100), (0010), and (0001) vibration levels. The fundamental band centers of five stable GeH4 isotopologues calculated from this empirically optimized PES are shown in Table 3. Other high-order PES parameters of the initial CVQZ-DK PES remained unaltered. With this scaling of four quadratic parameters the calculation of excited vibration levels was improved both for stretching and bending levels (Table 4). It is well-known that rotational energy levels are very sensitive to the equilibrium geometry. In case of the methane molecule,50 the ab initio determined re value permitted rotational calculations accurate from 1 × 10−3 to 1 × 10−4 cm−1 up to J = 15.50,113 It is more complicated in the germane case: though various ab initio ansätze give qualitatively good agreement with the empirical re estimation of ref 92 (see Table 1), the scatter in these values is considerably larger than for CH4.50 To improve positions of rovibration lines in calculated spectra we made a fine-tuning of the CCSD(T)/CVQZ-DK equilibrium bond distance to better match rotation levels of the vibration ground state that resulted to the optimized value re = 1.517 594. Rotational levels of 70GeH4 and 74GeH4 calculated from spectra

and where p = p1 + p2 + p3 + p4 is the total power of the term. The sixth-order PES constructed in this way contains 287 expansion parameters. To improve the fit of the PES to ab initio electronic energies, selected additional higher-order terms were added: 43 angular seventh- and eighth-order parameters. Ab initio potential energies were fitted using the analytical symmetry-adapted representation (3) − (4) and the weight function employed by Schwenke and Partridge44 w(E) =

tanh(− 0.0005(E − E1) + 1.002002002) 2.002002002

(5)

This weight function decreases with energy E (expressed in cm−1) and de-emphasizes the contribution of larges grid displacements for large energies beyond E1 = 15 000 cm−1. 299 parameters (of 330 initially included) were statistically well-determined in this fit on the entire grid of all 19 882 ab initio points with the weighted RMS deviation of 0.93 cm−1.

4. VARIATIONAL CALCULATIONS OF VIBRATIONAL AND ROTATIONAL LEVELS. EMPIRICAL PES OPTIMIZATION In many previous works, the GeH4 stretching vibrational levels have been described by the local-mode theory26−32,105−107 (and references therein), based on the assumption that purely stretching vibrations (v1(A1)/v3(F2)) and their combinations could be considered separately from all other states by neglecting the bend−stretch interactions. The advantage of the local-mode theory is the simplicity of this four-dimensional model that had permitted a good fit of observed stretching levels with only few adjustable parameters.26,32 However, the 8987

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The Journal of Physical Chemistry A Table 3. Reduced on X Value Band Centersa for Five Isotopologues of GeH4 70

72

GeH4

73

GeH4

state

X

obs

calc

O−C

ν4 ν2 ν1 ν3

820. 929. 2110. 2110.

1.543 0.900 0.720 2.032

1.546 0.900 0.722 2.033

−3 0 −2 −1

obs

calc

0.717 1.575

1.119 0.904 0.712 1.577

74

GeH4

O−C

5 −2

obs

calc

0.715 1.356

0.913 0.906 0.708 1.357

76

GeH4

GeH4

O−C

obs

calc

O−C

obs

calc

O−C

0.714 0.908 0.704 1.145

−2 1 −2 −2

0.327 0.913

0.329 0.912 0.696 0.736

−2 1

7 −1

0.712 0.909 0.702 1.143

0.734

−2

Calculated with empirically optimized CVQZ-DK PES. Energy levels are in cm−1 and O−C in 0.001 cm−1. Observed vibration levels for ν1 and ν3 for 72GeH4 are taken from ref 20; those for ν2 and ν4 for 74GeH4 and 76GeH4 are from ref 21, and those for 70GeH4 are from ref 38. a

Table 4. Comparison of Experimental Vibrational Band Centers (cm−1) for Five Stable Isotopologues of GeH4 with Variational Calculations from the Optimized PES molecule

a

74

GeH4

70

GeH4

72

GeH4

73

GeH4

76

GeH4

stretching local modesa

bending (V2,V4)

obs level [ref]

calc

sym

obs-calc

(0000) (0000) (2000) (2000) (3000) (3000) (2100) (4000) (4000) (3100) (2000) (2000) (3000) (3000) (2100) (4000) (4000) (2000) (2000) (3000) (3000) (2100) (4000) (4000) (2000) (2000) (0000) (0000) (0000) (0000) (0000) (0000) (0000) (2000) (2000)

(1,1) (1,1) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,2) (0,2) (0,2) (1,1) (1,1) (2,0) (2,0) (0,0) (0,0)

1748.77723 1752.88623 4153.5632 4153.8232 6128.5832 6128.5832 6263.6732 8035.8429 8035.8429 8241.6228 4154.8032 4155.1232 6130.4332 6130.4332 6265.1832 8038.1929 8038.1929 4154.1732 4154.4632 6129.4832 6129.4832 6264.4332 8038.1929 8038.1929 4153.9232 4154.1432 1627.49523 1639.25723 1642.14223 1748.39623 1752.50323 1857.27223 1860.66723 4152.9932 4153.2432

1748.12 1752.89 4154.26 4154.51 6129.50 6129.53 6264.29 8035.44 8035.45 8242.53 4155.50 4155.81 6131.35 6131.38 6265.87 8037.82 8037.81 4154.86 4155.14 6130.40 6130.43 6265.07 8036.98 8036.98 4154.56 4154.82 1627.39 1639.37 1641.76 1747.74 1752.50 1856.77 1860.37 4153.68 4153.91

F2 F1 A1 F2 A1 F2 F2 A1 F2 F2 A1 F2 A1 F2 F2 A1 F2 A1 F2 A1 F2 F2 A1 F2 A1 F2 A1 F2 E F2 F1 A1 E A1 F2

0.66 0.00 −0.70 −0.69 −0.92 −0.95 −0.62 0.40 0.39 −0.91 −0.70 −0.69 −0.92 −0.95 −0.69 0.37 0.38 −0.69 −0.68 −0.92 −0.95 −0.64 0.38 0.37 −0.64 −0.68 0.10 −0.11 0.38 0.66 0.00 0.50 0.30 −0.69 −0.67

Note: Local mode assignments of stretching levels are given according to experimental works cited in the fourth column.

analyses of ref 20 have been reported in the TDS37 databank and later extrapolated to other isotopic species of germane via parameters available in STDS.38 More representative sample of ∼100 experimental ground-state combination differences (GSCD) in spectra of 76GeH4 was recently determined.21 For 76 GeH4 only 10 GSCD have been measured21 resulting thus in less reliable empirical determination of 74GeH4 levels. Deviations between empirically determined GS rotational levels and our variational calculations for five GeH4 isotopologues are plotted in Figure 3. Rotational levels included in STDS databank38 (based on earlier works38 and unpublished parameters)

were less precise than recent 76GeH4 measurements.21 Note that our calculations (Figure 3) show the best agreement with accurate 76 GeH4 rotational levels of ref 21.

5. DETERMINATION OF AB INITIO DIPOLE MOMENT SURFACE The dipole moment was computed as the derivative of the energy with respect to the weak external uniform electric field using the finite difference scheme with the field variation of 0.0001 au near the zero field strength. The dependence of final 8988

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three functions μi(r1⃗ , r2⃗ , r3⃗ r4⃗ ) for i = 2, 3, 4 can be obtained from μ1 by cyclic permutation (1, 2, 3, 4) of hydrogen atoms; for instance, μ2(r2⃗ , r3⃗ , r4⃗ r1⃗ ) = μ1(r1⃗ , r2⃗ , r3⃗ r4⃗ ). Such representation of μ⃗ corresponds to the body fixed frame, which depends on instantaneous positions of the nuclei. The details of symmetry properties are discussed in a previous report.49 With this choice of the body fixed frame the chargefunction μ1 of an AB4-type molecule in eq 1 has the same symmetry properties as a PES for an AB3C type of molecule (A1 represents of the C3v point group), though the full dipole moment μ⃗ transforms according to the F2 representation of the AB4 point group Td. Using the algorithm of our previous work50,101,112 a full set of 967 totally symmetric irreducible tensors up to the sixth-order expansion was constructed with MIRS computational suite of codes.110 Every tensor was represented as the sum of symmetrized products: p

p

p

p

p

p

A1 3 4 5 6 1 2 R ip = (((SRH, A1 × SRH, E) × SRH1, A1) × ((SQH, A1 × SQH, E) × STH, E))

Figure 3. Deviations of variational rotational levels computed with our optimized PES from STDS spectroscopic database values38 for germane isotopic species (green, blue, black, and purple symbols). Deviations from recent empirical determinations of ref 21 based on 100 measured GSDC for 76GeH4 and on 10 GSDC for 74GeH4 are shown by pink ▼ and red ● (see the text for more comments).

(7a)

of symmetry-adapted C3v coordinates Si. Here SRH1 stands for the axis corresponding to the selected hydrogen atom i = 1, and SRH stands for other hydrogen atoms i = 2, 3, 4, whereas the notations SQH and STH are used for angular coordinates (see ref 50 for more detail). The upper index p = p1 + p2 + p3 + p4 + p5 + p6 is the total power of the term. The 1D elementary functions (2) were used to construct the symmetry-adapted coordinates SRH, SRH1, SQH, and STH. The use of Morse elementary functions in the DMS expansion of sixth order (p = 6) permitted fitting the analytical surfaces to ab initio points somewhat better than the use of simple linear functions ϕ′(r) = ri − re. The standard definition of the direct product of irreducible tensors102,103 with the subindices corresponding to irreducible representations of the C3v point group was applied. The first step corresponds to the construction of the symmetrized powers of Si, and the second step corresponds to the coupling of the symmetrized powers of different symmetrized coordinates in irreducible balanced trees according to the algorithm of ref 104. A set of all possible trees of the totally symmetric A1 representation gives a set of the 9D expansion terms for the μ1 function. This DMS component was developed in power series of irreducible tensors (7) depending on internal coordinates

dipole moment values on the external field was not significant in the range of 0.001−0.000 05 au of the field variations. As a first step of the DMS construction, the dipole moment values in the electronic ground state were calculated with VQZ basis set on the grid of 19 882 nuclear configurations described above. The distribution of the fit deviations using our analytical DMS representation is given in Figure 4 as a function of the energy.

μ1(r1 , r2 , r3 , r4 , q12 , q13 , q14 , q23 , q24 , q34) =

∑ K iR ip(r1, r2 , r3 , r4 , q12 , q13 , q14 , q23 , q24 , q34) i

(7b)

Note that in our works on CH3F and PH3100,53 calculations of exactly the same form have been used for the PES but not for the DMS μ1 component as here. In the similar case of a pyramidal AB3-type molecule48 (symmetry group C3v) the dipole moment function μ1 has been constructed as a function invariant under a certain part of group operations only. This suggests a general rule that the charge function μi(r1⃗ , r2⃗ , r3⃗ , r4⃗ ) in the DMS representation of eq 1 should have a behavior corresponding to a lower symmetry group than the full dipole moment (6) obtained as a sum of all components. However, the advantage of the form (6)−(8) is that other DMS components could be easily obtained from the μ1 component by hydrogen permutations. These symmetry properties of the DMS have been checked by direct computation of all components in the case of the methane dipole moment.49 Note that

Figure 4. Errors of the GeH4 DMS fit with respect to the ab initio values computed with the CCSD(T)/VQZ ansatz.

As in previous works49,72,76 on the methane molecule, which has the same Td point group, we considered two forms of the analytical DMS representation. One was the molecular-bond (MB) representation114−116,48,49 in internal coordinates: μ⃗ =

∑ i = 1...4

μi ( r1⃗ , r2⃗ , r3⃗ , r4⃗ ) ei⃗

(6)

where four “charge-functions” μi(r1⃗ , r2⃗ , r3⃗ , r4⃗ ) depend on internal coordinates μ1(r1, r2, r3, r4, cos(q12), cos(q13), cos(q14), cos(q23), cos(q24), cos(q34)), and ei⃗ are the unit vectors along the Ge−Hi bonds. Because of the symmetry considerations 8989

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Figure 5. Overview intensity-stick spectrum of 74GeH4 from 0 to 9000 cm−1 from first-principles calculations. Horizontal axis: transition wavenumbers. Vertical axis: log scale line intensities (in cm−1/(molecule × cm−2)).

the number of ab initio geometries included in the fit was considerably larger than the number of fitted DMS parameters. The weight function (5) (depending on energy E in cm−1, with E1 = 10 000) was also used for the DMS fit. In total 967 parameters were determined in this fit on the entire grid of all 19 882 ab initio points with the weighted standard deviation = 0.000 02 au (dipole moment atomic units84). Figure 4 shows the VQZ DMS error distribution of the final fit. The errors (defined as ab initio dipole value minus fitted surface value) are quite small up to energies of ∼12 000−13 000 cm−1. A larger scatter of points above this range occurs because the weighting function (5) rapidly de-emphasizes energies above this threshold similarly to the case of methane molecule.44,49,50,67 We also derived the ab initio DMS in the representation of power series of normal coordinates ⎛ λ (C) ⎞1/4 qk(σC) = ⎜⎜ k 2 ⎟⎟ L−1[Sk(C)]rect = Lq−1[Sk(C)]rect ⎝ ℏ ⎠

(8a)

defined according to the standard Wilson’s normal-mode formalism117 described in detail for the Td molecules in ref 109. Here Lq is the matrix composed or the eigenvectors of the GF matrix,117 λ(C) are the eigenvectors of the GF matrix, and k [Sk(C)]rect are the symmetry coordinates in the rectilinear approximation,109 where uppercase index (C) stands for the Td symmetry type (C = A1, E, or F2). To obtain the DMS in this representation we fitted it to the ab initio values in normal coordinates using 680 parameters up to sixth order. This gives the molecular fixed Eckart frame μMFF expansion as α μαMFF =

∑ K̃ iR̃ iαp

where

Figure 6. Comparison of the GeH4 spectrum in the 750−890 cm−1 region (top) and more detailed portion of the spectra in the 825− 860 cm−1 region (bottom) between experimental PNNL database (straight) and variational calculations (upside down hanging). The spectra were simulated with the Voigt line profile at a resolution of 0.006 cm−1 at T = 296 K and P = 1 atm at natural isotopic abundance. The absorbance is given in ppm−1 m−1 units. Complex patterns resulting from the superposition of five stable isotopologues 70GeH4, 72 GeH4, 73GeH4, 74GeH4, and 76GeH4, are clearly seen in the calculated spectrum.

i

R̃ iαp = (([q1(A1)]p1 × [q2(E)]p2 )C1 × ([q3(F2)]p3 × [q4(F2)]p4 )C2 )αF2 (8b)

The standard deviation of the dipole moment fit of 0.000 02 au was obtained using the weight function (5) with E1 = 10 000 cm−1 (the same as for DMS in internal coordinates). In the Supporting Information, we provide the component-by-component DMS normal-mode expansion in an easy-to-use explicit form for 74GeH4. 8990

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Figure 7. Comparison of the GeH4 spectrum in the 860−1080 cm−1 region (top) and more detailed portion of the spectra in the 950− 1030 cm−1 region (bottom) between experimental PNNL database (straight) and variational calculations (upside down hanging). See Figure 6 caption for experimental and calculation conditions. Figure 9. (top) Comparison of the GeH4 spectrum in the 1950− 2250 cm−1 region between experimental PNNL database (straight) and variational calculations (upside down hanging). (bottom) Comparison of detailed portions of the absorption spectra in the region of between variational calculations using ab initio DMS and experimental data centered around 2111 cm−1. A careful account of all stable isotopologues 70GeH4, 72GeH4, 73GeH4, 74GeH4, and 76GeH4 is necessary to describe the observed spectrum. See Figure 6 caption for experimental and calculation conditions.

Variational Line Intensity Predictions. To compute vibration−rotation spectra accounting for all transitions that could contribute to the absorption or emission at a given temperature, we used full-dimensional quantum nuclear motion calculations with nine vibrational and three rotational coupled degrees of freedom. Line intensities Sij for rovibrational transitions vij at a temperature T are defined by sij =

Figure 8. Comparison of the GeH4 spectrum in the 1100−1950 cm−1 region between experimental PNNL database (straight) and variational calculations (upside down hanging). See Figure 6 caption for experimental and calculation conditions.

8π 310−36 I0g νije−(c2Ei)/ T (1 − e−(c2νij )/ T )R ij 3hcQ (T ) C −1

(9) −2

in standard spectroscopic units [cm /(molecule × cm )],118 where c2 = hc/k with k the Boltzmann constant, gc are the nuclear spin statistical weights, and Ei are the lower state energies expressed in inverse centimeter wavenumber units. Q(T) is the partition function, which can be expressed as a direct sum of Boltzmann factors at a given temperature

The DMSs for all Td isotopologues can be obtained using the algorithm described in ref 78. 8991

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∑ (2J + 1)gC e−(c E

2 νJ )/ T

(10)

This requires the calculation of all energy levels EvJ up to the a maximum value Emax for which Q(T) converges. In this work we computed variational energies up to Jmax = 35 that gave us the partition function values Q(80 K) = 224.27 and Q(298 K) = 1737.71 for 74GeH4. The line strength of a dipole-allowed rovibrational transition is defined as R ij = 3

∑ MM ′

i Zμ j

2

(11)

where the summation is over all magnetic sublevels of both initial and final states. The dipole moment component Zμ corresponds to the laboratory-fixed frame (LFF) projection, and |i⟩ and |j⟩ are vibration−rotational wave functions of the initial and final state of the transition. For practical calculations the ab initio dipole moment function depending on nine nuclear coordinates was expressed in the molecular fixed Eckart frame in normal coordinates (eqs 8a and 8b) and linked to the LFF frame via the direction cosines. All quantities involved in eq 11 were expressed in the irreducible tensor representation119,109,72 for the full account of tetrahedral symmetry properties of GeH4. The formalism of variational intensity calculations has been described in our previous works on the methane molecule72,76,35 that belongs to the same point group. The bird’s eye overview of rotationally resolved germane spectrum in the infrared range is given in Figure 5. This spectrum was computed using CCSD(T)/CVQZ-DK PES with four empirically optimized quadratic parameters that provided rovibrational levels and wave functions. Transition intensities result from fist-principles calculations, since no empirical corrections were applied to the ab initio DMS constructed in the previous section. The figure reveals quite strong absorption patterns belonging to stretching, bending, and mixed combination bands.

6. RESULTS AND DISCUSSION To validate our ab initio intensities we compared theoretically simulated absorption cross sections with experimental GeH4 Fourier-transform laboratory spectra available of the PNNL

Figure 11. Comparison of the GeH4 spectrum in the 2750−3200 cm−1 region (top) and more detailed portion of the spectra in the 2930− 3050 cm−1 region (bottom) between experimental PNNL database (straight) and variational calculations (upside down hanging). See Figure 6 caption for experimental and calculation conditions.

library.33 To our knowledge the PNNL data are currently the only public available experimental infrared spectra of GeH4, but no comparison of any theoretical calculations with these spectra has been published up to now. These spectra that we use for the comparisons have been recorded33 at temperature T = 25 °C and pressure of 1 atm in the form of the absorbance point-by-point files with a step of 0.06 cm−1 in the spectral range of 700− 5300 cm−1. The PNNL spectra have not been assigned. They do not thus contain line-by-line nor low state energy information and cannot be extrapolated beyond their experimental conditions. In what follows, the comparisons with PNNL spectra are given as the absorbance simulations at macroscopic experimental conditions (see the web links given below in Notes). The comparisons are given in Figures 6−13 interval by interval with an enlarged wavenumber scale. The intervals were partitioned in a way to include strong absorption features separated by relative transparency windows. To produce these detailed plots we applied the “vibrationalsub-space” (VSS) technique that permitted shifting band centers to their experimental values at the intermediate steps of variational

Figure 10. Comparison of the GeH4 spectrum in the 2365−2475 cm−1 region between experimental PNNL database (straight) and variational calculations (upside down hanging). Complex patterns resulting from the superposition of five stable isotopologues 70GeH4, 72GeH4, 73GeH4, 74 GeH4, and 76GeH4 are clearly seen in the calculated spectrum. See Figure 6 caption for experimental and calculation conditions. 8992

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Table 5. Integrated Absorbance (in PNNL units) of Natural Isotopic Composition of GeH4 at Various Spectral Ranges range/cm−1

calculated integrated absorbance

observedb integrated absorbance

ratio: calc/obs

750−900 900−1200 1200−1400b 1400−1900 1950−2350 2350−2500b 2500−3200b 3550−3950b 3950−4350

0.985 09 0.017 84 0.000 21 0.004 95 1.041 83 0.000 54 0.006 54 0.000 24 0.011 26

0.830 14 0.015 38 0.000 20 0.004 24 1.008 38 0.000 42 0.008 25 0.000 22 0.010 66

1.186 1.160 1.069 1.168 1.033 1.288 0.793 1.088 1.056

Note: In the ranges of 2350−3200 cm−1 and above 4350 cm−1 the integrated absorbance could not be reliably determined from the observed spectra due to the weakness of the bands and low signal-to-nose ratio. bThese ranges correspond to the weakest bands in the PNNL spectra. a

Figure 12. Comparison of the GeH4 spectrum in the 4000−4250 cm−1 region between experimental PNNL database (straight) and variational calculations (upside down hanging). See Figure 6 caption for experimental and calculation conditions.

that this implicitly involves contributions from lines of all the bands in the considered spectral range including weak overtones, combination, and hot bands, which may not have been analyzed in high-resolution experiments. From Table 5, it can be seen that the integrated intensity values have favorable agreement with the experimental intensities across most spectral regions in this interval-by-interval comparison. It is noted here that in some ranges corresponding to very weak bands the experimental spectra are dominated by noise and instrumental fringed. In particular, in the regions of 2350−3200 cm−1 and above 4350 cm−1 the PNNL data have low signal-to-nose ratio, and experimental spectra contain data with very high fluctuations and even have many negative absorbance values in some places. These regions in the available experimental records clearly could not be used for an accurate validation of the theory. However, the band maxima are consistent between calculations and experiment in the entire range of PNNL data.

7. CONCLUSION AND PERSPECTIVE In this work, we constructed first nine-dimensional potential and dipole moment surfaces of the germane molecules. Minor empirical refinement of the equilibrium geometry and of four quadratic parameters of the PES computed at the CCSD(T)/ aug-cc-pVQZ-DK level of the theory yielded a root-meansquare deviation of ∼0.001 cm−1 for rotational energies and a sub-wavenumber accuracy for all experimentally known vibrational band centers of five stable GeH4 isotopologues up to 8300 cm−1. Vibration−rotation line intensities of 70GeH4, 72GeH4, 73 GeH4, 74GeH4, and 76GeH4, were calculated from purely ab initio DMS using nuclear motion variational calculations with a full account of the tetrahedral symmetry of the molecule. A good overall agreement of main absorption features with experimental rotationally resolved PNNL spectra33 was achieved for the first time in the entire range of 700−5300 cm−1. The comparison with observation at natural isotopic abundance shows that the spectra of five isotopologues of GeH4 are significantly different and give non-negligent contributions to the total intensity of the bands. It is clearly seen (Figures 6, 9, 10, and 11) that considering only one isotopologue for the theoretical spectra would result in spectral patterns vastly different from those obtained from experiment across all considered regions. A good match with observations suggests that this theoretical approach

Figure 13. Comparison of the GeH4 spectrum in the 4500−5200 cm−1 region for the upper part of the PNNL filter showing the noisy part of the PNNL data (straight). Because of low signal-to-noise ratio the integrated experimental absorbance could not be reliably determined in this range. Variational calculations are given upside down hanging.

calculations as described in refs 73 and 76. In this way, all rotational calculations are done with optimized PES using ab initio DMS, but vibration−rotation resonances are better described. This improves the resonance intensity transfer among strong and weak bands. Globally, all calculations with the VQZ DMS are in a good agreement with experimental intensities, which shows excellent match for strong absorption features. A closer inspection reveals (see, e.g., Figures 6, 9, 10, and 11) that complex patterns resulting from the superposition of five stable isotopologues 70 GeH4, 72GeH4, 73GeH4, 74GeH4, and 76GeH4 had to be computed in a very self-consistent way to correctly describe the observations. Table 5 summarizes the comparisons of integrated absorbance at different regions in the GeH4 spectra between our ab initio intensity calculations and the values determined from observed PNNL spectra. The advantage of such comparison is 8993

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cross sections of C2H2, PH3, AsH3, and GeH4, with application to Saturn’s atmosphere. J. Geophys. Res. 1991, 96, 17519−17527. (2) Atreya, S. K.; Mahaffy, P. R.; Niemann, H. B.; Wong, M. H.; Owen, T. C. Composition and origin of the atmosphere of Jupiteran update,and implications for the extra solar giant planets. Planet. Space Sci. 2003, 51, 105−112. (3) Fink, U.; Larson, H. P.; Treffers, R. R. Germane in the atmosphere of Jupiter. Icarus 1978, 34, 344−354. (4) Lodders, K. Jupiter formed with more tar than ice. Astrophys. J. 2004, 611, 587−597. (5) Cook, P. A.; Ashfold, M.N. R.; Jee, Y.; Jung, K.; Harich, S.; Yang, X. Vacuum ultraviolet photochemistry of methane, silane and germane. Phys. Chem. Chem. Phys. 2001, 3, 1848−1860. (6) Steward, W. B.; Nielsen, H. H. The infrared absorption spectrum of germane. Phys. Rev. 1935, 48, 861−864. (7) Tindal, C. H.; Straley, J. W.; Nielsen, H. H. The infra-red spectra of SiH4 and GeH4. Proc. Natl. Acad. Sci. U. S. A. 1941, 27, 208−212. (8) Wilkinson, G. R.; Wilson, M. K. Infrared spectra of some MH4 molecules. J. Chem. Phys. 1966, 44, 3867−3874. (9) Carrier, W.; Osamura, Y.; Zheng, W.; Kaiser, R. I. Laboratory investigations on the infrared absorptions of germanium-bearing moleculesdirecting the identification of organo-germanium molecules in the atmospheres of jupiter and saturn. Astrophys. J. 2007, 654, 687−692. (10) Tinetti, G.; Encrenaz, T.; Coustenis, A. Spectroscopy of planetary atmospheres in our Galaxy. Astron. Astrophys. Rev. 2013, 21, 63. (11) Levin, I. W. Infrared intensities of the fundamental vibrations of GeH4 and GeD4. J. Chem. Phys. 1965, 42, 1244−1251. (12) Daunt, S. J.; Halsey, G. W.; Fox, K.; Lovell, R. J.; Gailar, N. M. High-resolution infrared spectra of v3 and 2v3 of Germane. J. Chem. Phys. 1978, 68, 1319−1321. (13) Magerl, G.; Schupita, W.; Bonek, E.; Kreiner, W. A. Observation of the isotope effect in the v2 fundamental of germane. J. Chem. Phys. 1980, 72, 395−398. (14) Lee, E.; Sutherland, G. B. B. M. A peculiarity in the infra-red absorption spectrum of germane. Math. Proc. Cambridge Philos. Soc. 1939, 35, 341−342. (15) Kreiner, W. A.; Magerl, l.G.; Furch, B.; Bonek, E. IR laser side band observations in GeH4 and CD4. J. Chem. Phys. 1979, 70, 1516− 1520. (16) Kreiner, W. A.; Opferkuch, R.; Robiette, A. G.; Turner, P. H. The ground-state rotational constants of germane. J. Mol. Spectrosc. 1981, 85, 442−448. (17) Kreiner, W. A.; Orr, B. J.; Andresen, U.; Oka, T. Measurement of the centrifugal-distortion dipole moment of GeH4 using a CO2 laser. Phys. Rev. A: At., Mol., Opt. Phys. 1977, 15, 2298−2304. (18) Ozier, I.; Rosenberg, A. The forbidden Rotational spectrum of GeH4 in the ground vibronic state. Can. J. Phys. 1973, 51, 1882−1895. (19) Kolomiitsova, T. D.; Savvateev, K. F.; Shchepkin, D. N.; Tokhadze, I. K.; Tokhadze, K. G. Infrared Spectra and Structures of SiH4 and GeH4 Dimers in Low-Temperature Nitrogen Matrixes. J. Phys. Chem. A 2015, 119, 2553−2561. (20) Lepage, P.; Champion, J. P.; Robiette, A. G. Analysisofthe v3 and v1 infrared bands of GeH4. J. Mol. Spectrosc. 1981, 89, 440−448. (21) Ulenikov, O. N.; Gromova, O. V.; Bekhtereva, E. S.; Raspopova, N. I.; Sennikov, P. G.; Koshelev, M. A.; Velmuzhova, I. A.; Velmuzhov, A. P.; Bulanov, A. D. High resolutionstudyof MGeH4 (M = 76,74) in the dyad region. J. Quant. Spectrosc. Radiat. Transfer 2014, 144, 11−26. (22) Koshelev, M. A.; Velmuzhov, A. P.; Velmuzhova, I. A.; Sennikov, P. G.; Raspopova, N. I.; Bekhtereva, E. S.; Gromova, O. V.; Ulenikov, O. N. High resolution study of strongly interacting v1(A1)/v3(F2) bands of MGeH4 (M = 76, 74). J. Quant. Spectrosc. Radiat. Transfer 2015, 164, 161−174. (23) Ulenikov, O. N.; Gromova, O. V.; Bekhtereva, E. S.; Raspopova, N. I.; Sennikov, P. G.; Koshelev, M. A.; Velmuzhova, I. A.; Velmuzhov, A. P.; Fomchenko, A. L. First high resolution ro-vibrational study of the(0200), (0101), and (0002)vibrational states of MGeH4 (M = 76,74). J. Quant. Spectrosc. Radiat. Transfer 2016, 182, 199−218.

opens a promising way for assignments and modeling of complicated germane spectra, including isotopologues, at various temperatures as well as for atmospheric and planetary applications. As a further step we plan to build effective polyad models for quasi-degenerate levels from obtained PES and DMS using accurate high-order contact transformation method implemented in the MOL_CT program suite similarly to the previous work on the methane molecule.113 This aims at combining advantages of the global and local approaches and could help introducing ab initio information to effective spectroscopic models. On the basis of the previous experience for other molecules120,121 this can provide physically meaningful initial values of the coupling parameters to extend spectra analyses for stretching, bending, and mixed bands and to study isotopic effects on the equal footing for all species. We also plan to calculate germane partition function and line list for high-temperature applications involving hot bands following the method recently reported for methane82,122 and ethylene.83 The data obtained in this work will be made available through the TheoReTS123 information system (see the web links given below in Notes) that contains ab initio born line lists and provides a user-friendly graphical interface for a fast simulation of the absorption cross sections and radiance. Further progress in the description of germane absorption/ emission in wider range and for applications under various conditions could benefit from a collaboration of spectroscopists, theoretical chemists, specialists in atmospheric modeling, and astrophysicists.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b07732. PES and DMS of germane (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +3822−491111-1208. Notes

The authors declare no competing financial interest. See also https://secure2.pnl.gov/nsd/nsd.nsf/Welcome, http://theorets.univ-reims.fr, and http://theorets.tsu.ru for additional data related to this work.

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ACKNOWLEDGMENTS The supports from Tomsk State Univ. Academic D. Mendeleev funding Program, from French National Planetology program (PNP), from LEFE Chat program of the “Institut National des Sciences de l’Univers” (CNRS), from CNRS (France), and RFBR (Russia) Grant No. 16-53-16022 in the frame of “Laboratoire International Associé SAMIA” are acknowledged. A.V.N. acknowledges the support from Champagne-Ardenne region. We have benefited from the computational resources of IDRIS/CINES computer center of France, of the ROMEO computer center Reims Champagne-Ardenne and computer center SKIF Siberia (Tomsk).

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REFERENCES

(1) Chen, F.; Judge, D. L.; Robert Wu, C. Y.; Caldwell, J.; White, H. P.; Wagener, R. High- resolution, low-temperature photo absorption 8994

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