Laser-coolable polyatomic molecules with heavy nuclei
Recently a number of diatomic and polyatomics molecules has been identified as a prospective systems for Doppler/Sisyphus cooling. Doppler/Sisyphus cooling allows to decrease the kinetic energy of molecules down to microkelvin temperatures with high efficiency and then capture them to molecular traps, including magneto-optical trap. Trapped molecules can be used for creation of molecular fountains and/or performing controlled chemical reactions, high-precision spectra measurements and a multitude of other applications. Polyatomic molecules with heavy nuclei present considerable interest for the search for “new physics” outside of Standard Model and other applications including cold chemistry, photochemistry, quantum informatics etc. Herein we would like to attract attention to radium monohydroxide molecule (RaOH) which is on the one hand an amenable object for laser cooling and on the other hand provides extensive possibilities for searching for -odd and -odd effects. At the moment RaOH is the heaviest polyatomic molecule proposed for direct cooling with lasers.
pacs:31.30.-i, 37.10.Mn, 12.15.Mm, 21.10.Ky
The scheme of molecular Doppler cooling (first displayed in DiRosa (2004)) proved to be efficient, robust and applicable to a large class of molecular species. Recently highly-efficient cooling and magneto-optical trapping of diatomic molecules was demonstrated for SrF Barry et al. (2014), YO Hummon et al. (2013) and CaF Hemmerling et al. (2016); Anderegg et al. (2017). The experiments on Doppler/Sisyphus cooling on BaH Tarallo et al. (2016), MgF Xu et al. (2016) and some other molecules and their cations are currently conducted by different groups. Recently we have proposed a number of polyatomic molecules containing light atoms, which may be expected to be suitable for Doppler/Sisyphus cooling Isaev and Berger (2016). The estimate for Doppler limit temperature for considered molecules (CaOH, CaNC, MgCH, CaCH and chiral MgCHDT) showed that it is quite realistic to reach the submillikelvin temperatures with a large class of polyatomic species (see also recent proposal by I. Kozyryev et al Kozyryev et al. (2016) for a large class of polyatomic radicals amenable for laser cooling). In 2017 J. Doyle’s team at Harvard University has reported the first observation of fast Sisyphus laser cooling of SrOH to the temperature of 750 Kozyryev et al. (2017). To our knowledge the only competitive method currently allowing to cool polyatomic (four- and bigger multi-atomic molecules) to submillikelvin temperatures is optoelectrical Sisyphus cooling method Prehn et al. (2016); Zeppenfeld et al. (2012). The latter method, though having great potential, relies on strong Stark interaction with external field and is expected to be most effective when applied to molecules with rather large dipole moments. The method of molecular laser cooling can be applied both to dipolar and non-dipolar species and, besides, is directly related to very well-studied, efficient and robust atomic Doppler cooling technique. On the one hand electronic structure of heavy-atom molecules especially favors quasi-diagonal structure of Franck-Condon (FC) matrix between ground and excited electronic states Isaev et al. (2010) due to more diffuse valence electronic orbital in comparison with light-atom homologies. On the other hand the molecular species containing heavy atoms present considerable interest for the experiments devoted to search for new physics outside the Standard Model Titov et al. (2006); Klemperer et al. (1993); Kozyryev and Hutzler (2017), including search for axion-like-particle candidates for dark matter Stadnik and Flambaum (2014) and search for parity violation in chiral molecules (see e.g Berger and Klessinger (1997)). Herein we apply the principles for identifying the laser-coolable polyatomic species introduced in Isaev and Berger (2016) to the field of heavy-atom compounds and propose the triatomic molecule RaOH as a candidate for Doppler cooling and for search for space parity violating (-odd ) and also for both space parity and time-reversal violating (-odd ) interactions with polyatomic molecules. In the first part of the article we calculate molecular spectroscopic parameters for this molecule and estimate the completeness of the cooling loop by calculating the FC matrix for vibronic transitions between first electronic excited and ground states. Then we apply recently developed original methodology for calculations of electric transition dipole moments between elaborate electronic levels in fully-relativistic framework. Finally, we use the earlier developed by us approach to estimate -odd and -odd parameters of effective molecular spin hamiltonian in the framework of Zero Order Regular Approximation (ZORA) Isaev and Berger (2012, 2014) (see also Berger et al. (2005); Berger and van Wüllen (2005)).
Relativistic electronic structure calculations
Equilibrium molecular parameters
As it was pointed out in Isaev et al. (2010) the crucial prerequisite for the closed cooling transition loop in molecules (required for the quasi-diagonality of the FC matrix) is the conservation of a similar molecular geometry for both ground and first excited electronic states. This happens for e.g. transition of the valence electron between non-bonding orbitals being presented in the leading electronic configuration for both ground end excited electronic states. In Isaev and Berger (2016) we have proposed a simple scheme for obtaining the molecular structures of polyatomic species which contain non-bonding single-occupied molecular orbitals. The basic idea is to make a substitution of the halogen in the series of MX molecules (M is the metal and X is the halogen) to pseudo-halogen or functional groups. In our consideration in Isaev and Berger (2016) the heaviest nucleus in a molecule was calcium (with the charge number =20) and, thus we were able to safely neglect the relativistic effects in those electronic structure calculations. For light element compounds such an approximation indeed does not lead to noticeable changes in molecular parameters (see e.g. Dyall and Fægri (2007)).
In the opposite case of heavy element compounds one has to estimate the influence of relativistic effects on FC factors. FC factors are known to be strongly dependent on displacements in equilibrium geometries and thus are sensitive to both scalar and spin-dependent relativistic effects, if such effects are different for the considered electronic states (in our case the ground and first excited electronic states). We started our study from calculation of equilibrium structures and normal modes for both ground and first excited electronic states of RaOH using spin-averaged relativistic effective core potential (arecp) with 10 valence electrons treated explicitly for Ra and with all-electron description for O and H. Multiconfiguration Self-Consistent Field method (MCSCF) is used for ARECP calculations with 17 valence electrons correlated and with the basis sets of triple-zeta quality from Molpro program package library (see supplementary material for details). The equilibrium structures and normal mode frequencies are provided in Table 1. It can be seen that RaOH has essentially linear geometry in both ground (, according to irreps of group) and first excited () electronic states. Very weak Renner -Teller effect in RaOH detected within the spin-orbit-free approximation is in accord with similar effects found in light homologue compound CaOH Theodorakopoulos et al. (1999); Li and Coxon (1996) and should be fully suppressed due to strong spin-orbit interaction in heavy-atom compounds. We also would like to emphasize that in the vibrational analysis we did not restrict symmetry of vibrational modes to the totally-symmetric one. To estimate the influence of the relativistic spin-orbit interaction on relative displacements in equilibrium geometries we computed the properties for the states of interest with scalar-relativistic four-component spin-free (SF) Dyall (1994) and fully-relativistic four-component Dirac-Coulomb (DC) Hamiltonians. In four-component calculations we used Fock-Space Relativistic Coupled Cluster with Single and Double excitations (FS-RCCSD) method to account for electron correlations Visscher et al. (2001). In this approach augmented ANO-RCC basis set on Ra and double zeta plus polarization quality basis sets by Alrichs et al on O and H (see supplementary material) were used. Thus we could also directly check the stability of the calculated FC-factors in relation to changing of the method for electron correlation accounting. It follows from ARECP/MCSCF calculations that the main interest present the vibrations localized on the Ra-OH bond. Thus one has to consider first the cross-section of the potential energy surface along Ra-OH bond (while relative positions of O and H nuclei can be fixed). The bond angle Ra-O-H is expected to be very close to due to above-mentioned suppression of Renner-Teller effect by spin-orbit interaction. Nevertheless to additionaly confirm this we have also explicitly calculated dependence of the energies of the ground and first excited electronic states on Ra-O-H bond angle (see Fig 1) in the framework of DC/RCCSD approach. One can see from Table 1, that the changes in displacements of the equilibrium geometries for ground and excited electronic states of RaOH are rather minor, while the method of accounting for electronic correlations is changed and simultaneously the spin-orbit interaction is taken into account. To check directly the influence of dynamic electronic correlations and the spin-orbit interaction on FC factors we have calculated these factors for , and transition of the quasimolecule RaX. Here , 0, 1 and 2 are the vibrational quantum numbers for the first excited electronic state (with prime) and ground electronic state (without prime) correspondingly; mass of quasinucleus X is equal to the mass of OH group (17 a.m.u.), while internuclear potential is equivalent to that between Ra and O nuclei from either ARECP/MCSCF or SF/FS-RCCSD or DC/FS-RCCSD calculations (see supplementary material). The calculations of FC factors were performed analogously to those from Isaev et al. (2010) and the results are presented in Table I in supplementary material. Similarly to Isaev and Berger (2016) we are using here the sum of the three largest FC-factors in order to estimate the completeness of the cooling loop. One can see from the Table I that for every calculation the sum of the three largest FC-factors is greater than 0.99, thus it is reasonable to expect highly-closed cooling transition in RaOH.
Transition dipole moments
Another prerequisite for the efficient molecular Doppler cooling is strongly allowed dipole transition between working electronic states. To estimate radiative lifetimes for the first excited state on the FS-RCCSD level, we evaluated the transition electric dipole moment (TDM) between this state and the ground electronic state using the finite-filed method. The TDM component values were derived from the finite-difference approximation for the derivative matrix elements as
Here is the applied uniform electric field strength and and stand for left and right eigenvectors of the field-dependent non-Hermitian FS-RCC effective Hamiltonian acting in the field-free () model space. The chosen numerical differentiation step size atomic units corresponded approximately to the center of the interval on the logarithmic scale in which the dependence of the resulting TDM values on the step size was negligible. Although the calculations involve only the effective Hamiltonian eigenvectors, i. e. the model space projections of many-electron wavefunctions, the resulting transition moment approximations implicitly incorporate the bulk of the contributions from the remainder (outer-space) part of these wavefunctions Zaitsevskii and Pychtchev (1998). The computational procedure has been implemented within the DIRAC15 program package dir (). The lifetimes of several vibrational levels are estimated according to (see Matsushita et al. (1987)):
where is the radiative lifetime of the vibrational level (in ); is the sum of the squares of transition dipole moment moduli between one component of the initial electronic state and both components of the final state (in atomic units); is the FC-factor for the vibronic transition ; and is the energy interval between vibronic levels (in wavenumbers). The TDM value is evaluated at the equilibrium geometry of the ground electronic state of RaOH (which is nearly coinciding with that of the excited state) from DC/FS-RCCSD calculations. For we obtain the lifetime of the excited electronic state about 40 which corresponds to Doppler limit temperature .
We have also calculated the dependence of the transition dipole moment (TDM) on the internuclear distance Ra-O, to check the behavior of TDM near the equilibrium Ra-O distance and thus control possible Herzberg-Teller contribution to the vibrational spectrum (see Fig. 2). This contribution is proportional to the derivation of the TDM on the internuclear distance Ra-O.
To demonstrate the influence of the spin-dependent effects we provide on the plot with TDMs from DC/FS-RCCSD calculations also the spin-orbit-free limit of the total TDMfrom SF/FS-RCCSD calculations and the longitudinal component , which vanishes at the scalar relativistic limit. Rather slow and nearly linear variation of the TDM near the equilibrium geometries of the ground and first excited electronic states in RaOH justifies the use of the approximation (2) for radiative lifetimes.
|X ()||2.38||A ()||2.35|
|Normal modes||Normal modes|
|X ()||4243||A ()||4248|
|FC factors 0.90501110, 0.09222221 RaOH stretching, 0.00253331 RaOH stretching|
|X ()||2.30||A ()||2.30|
|A ()||2.29||A ()||2.29|
Quasirelativistic calculations of P- and P,T-odd parameters
We have previously derived the expressions for matrix elements of the -odd and -odd operators in quazirelativistic Zero-Order Regular Approximation (ZORA) framework (see Isaev and Berger (2012) and references therein) and related them to the corresponding effective operators in polyatomic molecules with open shells. Our previous calculations of -odd nuclear spin-dependent and -odd properties for diatomic molecules within Generalized Hartree-Fock (GHF) approach Isaev and Berger (2012) were later confirmed in fully-relativistic four-component Dirac-Hartree-Fock (DHF) calculations Borschevsky et al. (2012, 2013). It turned out that the difference between values of the -odd parameters in ZORA and four-component calculations does not exceed 5% (see Borschevsky et al. (2013)). High-precision FS-RCC calculations of -odd and -odd molecular parameters for RaF molecule Kudashov et al. (2014) are also in a good agreement with our previous estimates of influence of electronic correlation on these parameters.
|Scalar -odd interaction|
|Scalar -odd / hyperfine P-even interaction|
|Nuclear spin-dependent -odd interaction|
|Nuclear spin-dependent -odd /|
|hyperfine P-even interaction|
|Scalar -odd term|
|Scalar -odd interaction|
Here we briefly outline the fully-relativistic (four-component) operators describing nuclear spin-dependent -odd ( ) and scalar -odd electron-nucleus interactions and provide the corresponding expression within ZORA in Table 2 (omitting the response terms). For detailed discussion on -odd and -odd terms in quasirelativistic approximation see e.g. Berger et al. (2005); Nahrwold and Berger (2009) and Isaev and Berger (2012, 2013). We use atomic units unless other is stated explicitly.
The nuclear spin-dependent -odd and scalar -odd operators in four-component formalism can be written as:
where a.u. is Fermi’s constant of the weak interaction, is an effective parameter describing interactions for nucleus (caused both by the nuclear anapole moment Zel’dovich (1957, 1958); Flambaum and Khriplovich (1980) and by weak electron-nucleon interactions Novikov et al. (1977)), is a dimensionless coupling constant describing scalar -odd interaction for nucleus , and are the spin and normalized nuclear density distribution of nucleus , respectively, and are the Dirac -matrices and is a vector of Dirac’s -matricies. The summation goes over all nuclei and electrons . In Table 2, is the weak charge of nucleus , , where is the number of neutrons in nucleus , the nuclear charge, the Weinberg parameter, for which we employ the numerical value , and is the magnetic vector potential from the point-like nuclear magnetic moments with , being the gyromagnetic ratio, is the anticommutator and is the commutator. The ZORA factor is also used, , where is the model potential (with additional damping Liu et al. (2002)) proposed in van Wüllen (1998), which alleviates the gauge-dependence of ZORA.
Our ZORA/GHF values (obtained for equilibrium geometry from ARECP RaOH calculations) for is 1.410 Hz and for is 15410 Hz. These values are rather close to the values of the corresponding parameters for RaF from Isaev and Berger (2013); Kudashov et al. (2014). This is not surprising, taking into the consideration that the main contribution to the regarded parameters in both cases comes from the unpaired electron located on non-bonding molecular orbital centered on Ra.
We propose RaOH as a polyatomic molecule containing heavy nucleus, particularly suitable for highly effective cooling with lasers. Using recently developed method for computation of transition matrix elements we calculate transition dipole moments for cooling transitions for RaOH and estimate Doppler limit temperature. For the first time -odd and -odd properties are calculated for triatomic molecule with open electronic shells. Some other polyatomic molecules and ions with heavy nuclei may be also proposed for direct cooling with lasers according to the the scheme suggested by us in previous articles.
See supplementary material for the details on electronic structure calculations (basis sets, methods etc) and extended sets of molecular parameters.
We are indebted to Prof. R. Berger for providing us with the code hotFCHT for calculations of Franck-Condon factors in polyatomic molecules. T. I. is especially grateful to R. Berger for numerous insightful discussions during joint work on calculations of -odd and -odd properties for molecules with open electronic shells and implementing the routines for calculations of these properties withing tm2c code. We are thankful for Dr. Yu. V. Lomachuk and Dr. Yu. A. Demidov for technical help. Financial support by RFBR (grants N 16-02-01064 and N 16-03-00766) and computer time provided by the Center for Scientific Computing (CSC) Frankfurt is gratefully acknowledged.
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