Tunnelling splitting in the phosphine molecule
TTunnelling splitting in the phosphine molecule
Clara Sousa-Silva, Jonathan Tennyson, and Sergey N. Yurchenko Department of Physics and Astronomy, University College London, London WC1E 6BT,UK (Dated: 9 November 2018)
Splitting due to tunnelling via the potential energy barrier has played a significant role in the study ofmolecular spectra since the early days of spectroscopy. The observation of the ammonia doublet led toattempts to find a phosphine analogous, but these have so far failed due to its considerably higher barrier. Fulldimensional, variational nuclear motion calculations are used to predict splittings as a function of excitationenergy. Simulated spectra suggest that such splittings should be observable in the near infrared via overtonesof the ν bending mode starting with 4 ν .The umbrella mode in ammonia provides a textbookexample of tunnelling splitting. That the inversion ofpyramidal NH should lead to an observable splittingof the energy levels was first theoretically predicted in1932 and then detected using microwave spectroscopyin 1934. The subtle effects of this tunnelling on theenergy levels of ammonia are now well-studied. As adirect analogue of ammonia, phosphine can also be ex-pected to display splitting of its energy levels due to thetunnelling effect. However, splitting in PH is yet tobe observed, despite multiple attempts spread over morethan 80 years. Although otherwise similar to ammo-nia, phosphine has a larger mass and a much higher andwider barrier which makes for a much smaller splittingof the energy levels . Ab initio calculations of the energybarrier for phosphine range from 12 270 cm − to 12 560cm − , while the only empirical estimate gave a slightlylower value of 11 030 cm − . Experimentally, the infrared spectrum of PH has beenwell studied (see Table 1 of Sousa-Silva et al. and therecent study by Devi et al. ). While most of this workhas concentrated on the region below 3500 cm − , whereour calculations suggest the tunnelling splitting is verysmall, Ulenikov et al. report observed spectra between1750 and 9200 cm − and clearly demonstrate that PH spectra can be observed at higher frequencies.Of all the possible phosphine modes, the tunnellingeffect should be most prominent in the symmetric bend-ing mode, ν , as it is the mode most strongly associatedwith the height of the pyramid formed with the phospho-rous atom on top. In ammonia the analogous ν mode isknown as the inversion mode. Figure 1 shows schemati-cally the relationship between this mode and the barrierto tunnelling for the phosphine molecule.The ExoMol group works on constructing comprehen-sive line lists for modelling the atmospheres of hot bodiessuch as cool stars and exoplanets . As part of this workwe have computed two line lists for PH in its groundelectronic state. The more accurate of these line lists,called SAlTY, contains 16 billion transitions between9.8 million energy levels and it is suitable for simulatingspectra up to temperatures of 1500 K. It covers wavenum-bers up to 10 000 cm − and includes all transitions toupper states with energies below hc ·
18 000 cm − androtational excitation up to J = 46. FIG. 1. Splitting of the energy levels for phosphine, showingthe splitting for the ground state and the vibrational excita-tions up to v = 10 in the bending band ν . The PH line lists were computed by the variationalsolution of the Schr¨odinger equation for the rotation-vibration motion employing the nuclear-motion program TROVE . The line lists were computed using C (M)symmetry, considering phosphine as a rigid moleculeand thus with the potential barrier between the twosymmetry-equivalent minima effectively set to infinity.Consequently, it originally neglected the possibility of atunnelling mode.Tunnelling can be considered a chemical reaction andas such it is very sensitive to the shape of the po-tential energy surface (PES) . The SAlTY line listused a spectroscopically-refined version of the ab initio (CCSD(T)/aug-cc-pV(Q+d)Z) potential energy surface(PES). The value of splitting in various vibrationalstates as well as the intensity of the inversion-rotationand inversion-ro-vibrational lines can be computed byadapting the procedure used to simulate the phosphinespectrum to work with D h (M) symmetry. D h (M) isthe permutation inversion group for ammonia, since it is a r X i v : . [ phy s i c s . c h e m - ph ] S e p much less rigid molecule than phosphine. TROVE is used to compute differences between splitenergy levels. Here we employ the same refined PESas that used to compute the SAlTY line list to pre-dict the splitting between states of A and A symmetryfor J = 0, considering a zero point energy of 5232.26cm − . The potential energy function and the kinetic en-ergy operator are expanded (six and eight orders, respec-tively) in terms of the five non-linearized internal coor-dinates (three stretching and two deformational bend-ing) around a symmetric one-dimensional non-rigid ref-erence configuration represented by the inversion mode.The vibrational basis functions are obtained in a two-step contraction approach as described by Yurchenko etal. The stretching ( ν and ν ) primitive basis function | v str (cid:105) ( v str = 0 . . .
7) are obtained using the Numerov-Cooley method.
Harmonic oscillators are used as ba-sis functions for the bending ( ν ) primitive, | v bend , with (cid:105) ( v bend = 0 . . . ν inversion mode, primitivebasis functions, | v inv (cid:105) , are used. These were also gener-ated with the Numerov-Cooley method, with v inv ≤ − for the planar localminimum with P–H bonds at 1.3611 ˚A, refined from an ab initio value of 11353.6 cm − for the local minimum at1.3858 ˚A. Both the ab initio and refined barrier heightsare extrapolated values from the potential parameters inthe PES. These values are somewhat lower than the pre-vious, lower-level ab initio estimates but are in reasonableagreement with the empirical estimate of Weston. To help assess the uncertainty in our predicted split-tings, calculations were made using two different PESsurfaces, pre and post refinement, corresponding to thesurfaces used to calculate the phosphine line list at roomtemperature, and the complete SAlTY line list, re-spectively. Even though the refined PES resulted in asignificant improvement in the accuracy of the overallphosphine line list, the predicted splittings agreed com-pletely up to four significant figures.Table I shows how the predicted splittings change as afunction of ν excitation. Splittings in the ground stateare known to be extremely small (our calculations sug-gest about 10 − cm − ) but increase significantly as the ν mode is excited. All splitting predictions are con-verged to within 40 % up to 7 ν ; use of even larger inver-sion basis sets ( v inv >
64) became numerically unstable.Our tunnelling splitting for the overtones of ν aresomewhat larger than those predicted by previous one-dimension studies. This could have been anticipatedas it is well-known that the treatment of tunnellingwhich considers all dimensions of the problem leads toincreases in the magnitude of the splitting (or faster tun-nelling). Additionally, our lower value for the barrierheight also contributes to the larger predicted splittingvalues.Any observation of the tunnelling splitting in PH hasto consider a number of factors. First this splitting hasto be distinguished from the hyperfine structure. The TABLE I. Calculated splitting for the ground state (GS), fun-damental and excited bands of the bending mode ν .. Band Energy Splitting (cm − GS 0.000 ≤ − ν × − ν × − ν × − ν ν ν ν ν ν hyperfine splitting in PH has been observed to beless than 1 MHz or 4 × − cm − and should not increasesignificantly with vibrational excitation. Consequently,the splitting due to inversion should be distinguishablefrom the hyperfine splitting for all bands associated withvibrational excitation to 4 ν and higher. Besides, thenuclear statistics of PH as a D h (M) symmetry moleculeshould be also taken into account. For example, as in thecase of NH , the ro-vibrational states of the A (cid:48) and A (cid:48)(cid:48) symmetries have zero nuclear statistical weights g ns andthus forbidden, with g rmns = 8, 8, 4 and 4 for A (cid:48) , A (cid:48)(cid:48) , E (cid:48) and E (cid:48)(cid:48) , respectively. FIG. 2. Transition moments (Debye, log scale) for excitationsfrom the ground vibrational states up to 16 000 cm − . Tran-sitions associated with ν mode excitations are highlighted. Due to their reasonably large energy splittings, promis-ing regions of possible detection are those of the sym-metric bending bands, 6 ν ( ≈ − ) and 7 ν ( ≈ − ). Their splittings are predicted to be approxi-mately 0.003 cm − and 0.02 cm − for 6 ν and 7 ν , respec-tively. Our calculations suggest that most intense lines inthis band have intensities of about 10 − cm/molecule at T = 296 K and should be easily observable with moderninstruments. Figure 2 summarises the dipole transitionmoments to various vibrational states from the vibra-tional ground state; transitions to the ν overtone seriesassociated with the tunnelling motion are highlighted.However, detection will also depend on the locationof the splitting transitions as it may be difficult to dis-tinguish the ν bands in regions of the spectrum thatare strongly populated by other bands. Figure 3 high-lights the spectroscopic regions where splitting could bedetected in the context of the surrounding spectrum forthe 4 ν , 5 ν , 6 ν and 7 ν bands. In this context, the po-sitions of the 4 ν and 5 ν bands appear to be particularlypromising for investigation, since they can be mostly iso-lated from the surrounding stronger bands.Figure 4 shows the predicted spectra in the region ofthe strongest transitions for 4 ν and 5 ν bands, com-paring spectra when the molecule is allowed to undergoinversion and when tunnelling is not permitted. Addi-tionally, Figure 5 shows how the 6 ν and 7 ν bands willbe harder to detect amongst the surrounding bands, de-spite having much larger splitting values.Our calculations show that the ν overtones displaysplittings of a magnitude that should be resolvable withmodern experiments. We therefore hope that the the-oretical predictions of phosphine tunnelling shown herewill be validated with experimental detection in the nearfuture. Simulated spectra for other regions and/or condi-tions can be provided by the authors to aid this process. ACKNOWLEDGMENTS
This work is supported by ERC Advanced InvestigatorProject 267219. We would like to thank Laura McKem-mish, Ahmed Al-Refaie, Jack D. Franklin and WilliamAzubuike for their support and advice. R. P. Bell,
The tunnel effect in chemistry (Springer, 1980) pp.51–76. D. M. Dennison and G. E. Uhlenbeck, Phys. Rev. , 313 (1932). C. E. Cleeton and N. H. Williams, Phys. Rev. , 234 (1934). A. G. Csaszar and T. Furtenbacher, Phys. Chem. Chem. Phys. , 1092 (2016). N. Wright and H. M. Randall, Phys. Rev. , 391 (1933). R. E. Stroup, R. A. Oetjen, and E. E. Bell, J. Opt. Soc. Am. , 1096 (1953). P. Helminger and W. Gordy, Phys. Rev. , 100 (1969). P. B. Davies, R. M. Neumann, S. C. Wofsy, and W. Klemperer,J. Chem. Phys. , 3564 (1971). A. G. Maki, R. L. Sams, and W. B. Olson, J. Chem. Phys. ,4502 (1973). V. ˇSpirko and D. Papouˇsek, Mol. Phys. , 791 (1978). S. P. Belov, A. V. Burenin, L. I. Gershtein, A. F. Krupnov, V. N.Markov, A. V. Maslovsky, and S. M. Shapin, J. Mol. Spectrosc. , 184 (1981). FIG. 3. Contrast between the (top to bottom) 4 ν , 5 ν , 6 ν and 7 ν bands and the neighbouring transitions at T = 296 K.The SAlTY absorption intensities (cm/molecule) are com-puted using a C (M) model with PH as a rigid molecule,i.e. neglecting the inversion splitting. FIG. 4. Comparison between predicted phosphine spectrawithout (red) and with (blue and green) the inclusion of tun-nelling motion, for the strongest transitions in the 4 ν andthe 5 ν overtone bands. The ro-vibrational splitting is esti-mated using the pure vibrational values from Table I. TheSAlTY line list is used to simulate absorption intensities fora temperature of 296 K. P. Schwerdtfeger, L. J. Laakkonen, and P. Pyykk¨o, J. Chem.Phys. , 6807 (1992). R. E. Weston Jr, J. Am. Chem. Soc. , 2645 (1954). C. Sousa-Silva, S. N. Yurchenko, and J. Tennyson, J. Mol. Spec-trosc. , 28 (2013). V. Malathy Devi, I. Kleiner, R. L. Sams, L. R. Brown, D. C.Benner, and L. N. Fletcher, J. Mol. Spectrosc. , 11 (2014). O. N. Ulenikov, E. S. Bekhtereva, V. A. Kozinskaia, J. J. Zheng,S. G. He, S. M. Hu, Q. S. Zhu, C. Leroy, and L. Pluchart, J.Mol. Spectrosc. , 295 (2002). J. Tennyson and S. N. Yurchenko, Mon. Not. R. Astron. Soc. , 21 (2012). C. Sousa-Silva, A. F. Al-Refaie, J. Tennyson, and S. N.Yurchenko, Mon. Not. R. Astron. Soc. , 2337 (2015). S. N. Yurchenko, W. Thiel, and P. Jensen, J. Mol. Spectrosc. , 126 (2007). R. I. Ovsyannikov, W. Thiel, S. N. Yurchenko, M. Carvajal, andP. Jensen, J. Chem. Phys. , 044309 (2008). S. N. Yurchenko, R. J. Barber, and J. Tennyson, Mon. Not. R.Astron. Soc. , 1828 (2011). B. Noumeroff, “M´ethode nouvelle de la d´etermination des orbiteset le calcul des ´eph´em´erides en tenant compte des perturbations,”in
Trudy Glavnoi Rossiiskoi Astrofizicheskoj Observatorii , Vol. 2
FIG. 5. Comparison between predicted phosphine spectrawithout (red) and with (blue and green) the inclusion of tun-nelling motion, for the strongest transitions in the 6 ν and7 ν overtone bands. The ro-vibrational splitting is estimatedusing the pure vibrational values from Table I. The SAlTYline list is used to simulate absorption intensities for a tem-perature of 296 K. (Moscow, Gosudarsvennoe Izdatel’stvo, 1923) pp. 188–259. J. W. Cooley, Math. Comp. , 363 (1961). W. Klopper, M. Quack, and M. A. Suhm, Chem. Phys. Lett. , 35 (1996). H. S. P. M¨uller, J. Quant. Spectrosc. Radiat. Transf. , 335(2013). G. Cazzoli and C. Puzzarini, J. Mol. Spectrosc.239