## Final Report Summary - RICHMOL (Optical activity of molecules with rotational chirality: Theoretical study of the novel effect)

Molecular chirality is central to many areas of physical, chemical and biological research such as spectroscopy, symmetry and reaction dynamics, fundamental physics, evolution of biologically important molecules, as well as drug research in pharmacology. Chiral molecules have identical physical properties unless they interact with another chiral entity such as light or for example a chiral nanostructure. There is currently significant interest in laser-governed manipulation of internal and external degrees of freedom of chiral molecules, ultimately aimed at resolving and controlling chiral dynamics on very short time scales. The ability to precisely modulate and monitor the weights of different parity components in the enantiomeric state will lead to significant breakthroughs in the engineering of optical devices, stereochemistry and fundamental physics. Theoretical predictions of quantum dynamics of chiral molecules in the presence of external electromagnetic fields and the relationship with the desired chiroptical properties are incredibly important for the successful interplay between theory and experiment.

The chirality and optical activity induced by extreme molecular rotations in statically non-chiral molecules is a novel and unexplored research field that could offer exciting opportunities for new optical-control devices, or for advancing fundamentally important measurements of parity violation. The phenomenon of rotationally induced chirality is closely connected to the effect of rotational energy clustering exhibited by some polyatomic molecules at high rotational excitation. Phosphine (PH3) is an excellent example of a four-atomic molecule which at high rotational excitation forms well separated near degenerate rotational cluster states that exhibit properties of a chiral molecule. Theoretical analysis shows that in such states, PH3 rotates about a localization axis which is approximately parallel to one of the P-H bonds, either in a clockwise (R) or anti-clockwise (L) sense. This is a analogous to a molecule with static chirality: the R- and L-forms are indistinguishable from each other in the same way as the static enantiomeric forms of a conventional chiral molecule. Because of the local-mode character, the rotation of the molecule is nearly orthogonal to and is not destabilized by the Coriolis-type interactions with the vibrational modes. The six orientations of such stable rotations are characterized by the insuperable energy barriers with very slow and tunable tunnelling between them.

The overall goal of the project is twofold: (i) To develop a theoretical constructor for performing accurate numerical experiments involving the manipulation and control of molecules using the external electromagnetic (laser) fields; (ii) Using the computational tools developed to perform theoretical study of the rotationally induced chirality, which also includes techniques to exert extreme rotational excitations on polyatomic molecules and methods to detect the dynamically induced chirality.

To achieve the first objective, we have developed highly sophisticated quantum mechanical approaches TROVE and RichMol, that take into account all major electronic, nuclear motion and external field effects in small polyatomic molecules with near spectroscopic accuracy. The variational approach TROVE has been developed in the host group of the University College London and was one of the first methodologies based on the black-box paradigm in rotational-vibrational calculations for general non-rigid polyatomic molecules. In this project, we have significantly improved the computational efficiency of TROVE by implementing the rotational-vibrational Hamiltonian expressed in curvilinear (valence) internal coordinates with respect to the Eckart frame [2]. This new development allowed us to considerably accelerate the convergence of the variational results with respect to the basis set size. Thus, the exponentially growing cost of variational calculations was greatly reduced.

TROVE approach provides field-free molecular rotational-vibrational energies and wave functions. To model the nuclear dynamics of molecules in the presence of external electromagnetic fields, we have developed the new method RichMol. It uses the field-free molecular solutions of TROVE as basis and solves the time-dependent Schroedinger equation for molecule in the presence of external field perturbations. RichMol is currently the only tool for quantum mechanical modelling of the molecule-field interactions for an arbitrary polyatomic molecule capable of meeting the high accuracy demands of modern spectroscopic experiments. The development of this program took two thirds of the total project time and hence more publications relating to the project will appear in the following few months. The first published application of the RichMol is the theoretical proposal of a novel robust experimental method for detecting chirality, that offers high sensitivity to even tiny enantiomeric excess and is applicable to molecules with short-lived chirality [1] (more details are below).

To study the effect of rotationally induced chirality we have selected two molecular candidates, PH3 and PF3, that exhibit stable rotational energy clustering at relatively low energies. For these molecules we have first performed extensive electronic structure calculations of the full-dimensional potential energy surfaces and electric dipole moment and polarizability surfaces. The effect of rotational clustering was explored by solving the nuclear motion problem variationally for states of extreme rotational excitation, up to J=80 for PH3 and up to J=210 for PF3. Fig. 1 shows the computed rotational density function (probability density function integrated over all degrees of freedom except two spherical angles) of PH3 in the excited rotational state (J=50). The rotational density function is used to characterise rotation of molecule in the laboratory-fixed frame. Six maxima (red islands) correspond to six positions of the rotational axis, each localised along one of the P-H molecular bonds. The density plot can be explained as if molecule rotates around one of its bonds in a clockwise or anti-clockwise manner. To separate the clockwise and anti-clockwise rotating molecules, i.e. two enantiomeric forms of PH3, we have studied the effect of the external field. The results are demonstrated on Fig. 2: the two enantiomeric forms exhibit the low-field and high-field seeking behaviour in the external electric field and thus can be spatially separated by applying the inhomogeneous electric fields. The manuscript with the results for PH3 and PF3 molecules is now in preparation.

To induce and exert control over the extreme rotational excitations in molecules, we have performed theoretical study of the rotational-vibrational dynamics of PH3 and PF3 subject to the chirped laser excitation technique called optical centrifuge. The optical centrifuge consists of a powerful ultrafast laser pulses whose linear polarisation axis rotates with gradually increasing angular frequency. Experiments of optical centrifuge have only been performed on diatomic and recently triatomic molecules. The results of theoretical simulations of the rotational-vibrational dynamics of PH3 in optical centrifuge are shown on Fig. 3. The plot with rotational state population dynamics shows the excitation of PH3 from the ground J=0 rotational state to rotational states with the extreme values of J>30. The quantum number M of projection of angular momentum onto a laboratory-fixed axis indicates that rotational motion has a unidirectional character. The rotational cluster states start to form after the exposure time of about 170 ps, as seen from the plot of the rotational density function. The manuscript presenting the results of calculations together the optimised laser field parameters is in preparation.

Having proposed methods to dynamically induce chirality in PH3 and PF3 and separate the chiral enantiomers, we were left with the problem of detecting the respective chiroptical signal. Recently, few ground breaking experimental methods to investigate chiral molecules in the gas phase have been developed. Among them, the most accurate and promising method is the three-wave mixing technique, that rely on probing the three different electric dipole moment projections of chiral molecule via three different microwave excitations. The resulting transient microwave emission has a Pi phase difference for two different enantiomers, which is measured with the phase sensitive microwave instrumentation. This method, however, cannot be straightforwardly applied to rotational chiral enantiomers of PH3 and PF3, which is due to fact that molecules in such states possess only two dipole projections, one of which is very tiny. To circumvent such limitation of the three-wave mixing technique, we proposed a novel scheme by involving the non-zero off-diagonal anisotropic polarizabilities of the enantiomers [1]. We have theoretically demonstrated that by exciting a chiral molecule with pair of linearly polarised intense laser pulses with tilted mutual polarisations, the enantiomer-specific Pi phase shift is induced in the emitted microwave signals, required for the differentiation of enantiomers. As an example, Fig. 4 shows the scheme of the two-pulse technique and the calculated time evolutions of the electric dipole moment for two short-lived chiral enantiomers of HSOH molecule [1].

The preliminary results of calculations of the two-pulse technique applied to PH3 molecule in the rotational chiral state at J=50 are shown on Fig. 5. It is clearly seen that the time oscillations of the electric dipole moment of two enantiomeric forms differ by a Pi phase shift, which can be amplified and detected with the phase sensitive microwave spectroscopy. Currently, work is in progress to optimise the laser field parameters to increase the intensity of the dipole transient emission. The manuscript presenting the results of calculations is in preparation.

The developed theoretical tools TROVE and RichMol have incredible potential which has been explored in several other applications as a result of collaboration with other research groups.

The developments made in TROVE during the course of the current project are extensively used by the ERC funded project ExoMol, which uses TROVE for spectroscopic refinement and high-energy spectral simulations of molecules important in the spectroscopy of atmospheres of exoplanets and cool stars (e.g. methane, acetylene, ammonia) [3-6,8-10].

In collaboration with the group of Per Jensen (Wuppertal, Germany) we studied the rovibrational and temperature effects on the hyperfine coupling of CH3 radical [7] and at the moment perform accurate simulations of the infra-red and Raman spectra for this molecule.

As a result of collaboration with the group of Per Jensen and Stephen Schlemmer (Cologne, Germany) we have developed a new accurate semi-classical method that simplifies the calculations of the highly excited rotational states of polyatomic molecules. This approach is shown to be very efficient in application to the rotational cluster states described above (in preparation).

In collaboration with Mike Tarbutt (Imperial College, London) we theoretically proposed and tested a new Stark deceleration method that exploits the idea of periodic population transfer between the low-field and high-field seeking states using the Stark-chirped rapid adiabatic passage excitation (the manuscript is in preparation).

In collaboration with the ExoMol group we have performed extensive calculations of the Raman and electric quadrupole spectroscopic line lists for ammonia molecule at room temperature (submitted to Journal of Chemical Physics) as well as terahertz field-induced alignment and orientation dynamics of ammonia and phosphine molecules (submitted to Scientific Reports).

Overall, we have made significant progress within each objective and expect to meet all project goals over the longer term. The project has been a success, leading to significant improvements of the existing variational methodology TROVE and development of the new time-dependent approach RichMol to model the molecule-field interactions with spectroscopic precision.

A number of successful collaborative projects over a range of different topics proves the importance and considerable impact of the current project on the field of contemporary molecular physics.

We note, that a proposal submitted for EPSRC with the aim to continue the development of the present theoretical approach with applications in topics of cavity cooling of molecules, chirality induced by extreme rotational excitation, and laser control over the large-amplitude internal and rotational motions of non-rigid molecules.

The chirality and optical activity induced by extreme molecular rotations in statically non-chiral molecules is a novel and unexplored research field that could offer exciting opportunities for new optical-control devices, or for advancing fundamentally important measurements of parity violation. The phenomenon of rotationally induced chirality is closely connected to the effect of rotational energy clustering exhibited by some polyatomic molecules at high rotational excitation. Phosphine (PH3) is an excellent example of a four-atomic molecule which at high rotational excitation forms well separated near degenerate rotational cluster states that exhibit properties of a chiral molecule. Theoretical analysis shows that in such states, PH3 rotates about a localization axis which is approximately parallel to one of the P-H bonds, either in a clockwise (R) or anti-clockwise (L) sense. This is a analogous to a molecule with static chirality: the R- and L-forms are indistinguishable from each other in the same way as the static enantiomeric forms of a conventional chiral molecule. Because of the local-mode character, the rotation of the molecule is nearly orthogonal to and is not destabilized by the Coriolis-type interactions with the vibrational modes. The six orientations of such stable rotations are characterized by the insuperable energy barriers with very slow and tunable tunnelling between them.

The overall goal of the project is twofold: (i) To develop a theoretical constructor for performing accurate numerical experiments involving the manipulation and control of molecules using the external electromagnetic (laser) fields; (ii) Using the computational tools developed to perform theoretical study of the rotationally induced chirality, which also includes techniques to exert extreme rotational excitations on polyatomic molecules and methods to detect the dynamically induced chirality.

To achieve the first objective, we have developed highly sophisticated quantum mechanical approaches TROVE and RichMol, that take into account all major electronic, nuclear motion and external field effects in small polyatomic molecules with near spectroscopic accuracy. The variational approach TROVE has been developed in the host group of the University College London and was one of the first methodologies based on the black-box paradigm in rotational-vibrational calculations for general non-rigid polyatomic molecules. In this project, we have significantly improved the computational efficiency of TROVE by implementing the rotational-vibrational Hamiltonian expressed in curvilinear (valence) internal coordinates with respect to the Eckart frame [2]. This new development allowed us to considerably accelerate the convergence of the variational results with respect to the basis set size. Thus, the exponentially growing cost of variational calculations was greatly reduced.

TROVE approach provides field-free molecular rotational-vibrational energies and wave functions. To model the nuclear dynamics of molecules in the presence of external electromagnetic fields, we have developed the new method RichMol. It uses the field-free molecular solutions of TROVE as basis and solves the time-dependent Schroedinger equation for molecule in the presence of external field perturbations. RichMol is currently the only tool for quantum mechanical modelling of the molecule-field interactions for an arbitrary polyatomic molecule capable of meeting the high accuracy demands of modern spectroscopic experiments. The development of this program took two thirds of the total project time and hence more publications relating to the project will appear in the following few months. The first published application of the RichMol is the theoretical proposal of a novel robust experimental method for detecting chirality, that offers high sensitivity to even tiny enantiomeric excess and is applicable to molecules with short-lived chirality [1] (more details are below).

To study the effect of rotationally induced chirality we have selected two molecular candidates, PH3 and PF3, that exhibit stable rotational energy clustering at relatively low energies. For these molecules we have first performed extensive electronic structure calculations of the full-dimensional potential energy surfaces and electric dipole moment and polarizability surfaces. The effect of rotational clustering was explored by solving the nuclear motion problem variationally for states of extreme rotational excitation, up to J=80 for PH3 and up to J=210 for PF3. Fig. 1 shows the computed rotational density function (probability density function integrated over all degrees of freedom except two spherical angles) of PH3 in the excited rotational state (J=50). The rotational density function is used to characterise rotation of molecule in the laboratory-fixed frame. Six maxima (red islands) correspond to six positions of the rotational axis, each localised along one of the P-H molecular bonds. The density plot can be explained as if molecule rotates around one of its bonds in a clockwise or anti-clockwise manner. To separate the clockwise and anti-clockwise rotating molecules, i.e. two enantiomeric forms of PH3, we have studied the effect of the external field. The results are demonstrated on Fig. 2: the two enantiomeric forms exhibit the low-field and high-field seeking behaviour in the external electric field and thus can be spatially separated by applying the inhomogeneous electric fields. The manuscript with the results for PH3 and PF3 molecules is now in preparation.

To induce and exert control over the extreme rotational excitations in molecules, we have performed theoretical study of the rotational-vibrational dynamics of PH3 and PF3 subject to the chirped laser excitation technique called optical centrifuge. The optical centrifuge consists of a powerful ultrafast laser pulses whose linear polarisation axis rotates with gradually increasing angular frequency. Experiments of optical centrifuge have only been performed on diatomic and recently triatomic molecules. The results of theoretical simulations of the rotational-vibrational dynamics of PH3 in optical centrifuge are shown on Fig. 3. The plot with rotational state population dynamics shows the excitation of PH3 from the ground J=0 rotational state to rotational states with the extreme values of J>30. The quantum number M of projection of angular momentum onto a laboratory-fixed axis indicates that rotational motion has a unidirectional character. The rotational cluster states start to form after the exposure time of about 170 ps, as seen from the plot of the rotational density function. The manuscript presenting the results of calculations together the optimised laser field parameters is in preparation.

Having proposed methods to dynamically induce chirality in PH3 and PF3 and separate the chiral enantiomers, we were left with the problem of detecting the respective chiroptical signal. Recently, few ground breaking experimental methods to investigate chiral molecules in the gas phase have been developed. Among them, the most accurate and promising method is the three-wave mixing technique, that rely on probing the three different electric dipole moment projections of chiral molecule via three different microwave excitations. The resulting transient microwave emission has a Pi phase difference for two different enantiomers, which is measured with the phase sensitive microwave instrumentation. This method, however, cannot be straightforwardly applied to rotational chiral enantiomers of PH3 and PF3, which is due to fact that molecules in such states possess only two dipole projections, one of which is very tiny. To circumvent such limitation of the three-wave mixing technique, we proposed a novel scheme by involving the non-zero off-diagonal anisotropic polarizabilities of the enantiomers [1]. We have theoretically demonstrated that by exciting a chiral molecule with pair of linearly polarised intense laser pulses with tilted mutual polarisations, the enantiomer-specific Pi phase shift is induced in the emitted microwave signals, required for the differentiation of enantiomers. As an example, Fig. 4 shows the scheme of the two-pulse technique and the calculated time evolutions of the electric dipole moment for two short-lived chiral enantiomers of HSOH molecule [1].

The preliminary results of calculations of the two-pulse technique applied to PH3 molecule in the rotational chiral state at J=50 are shown on Fig. 5. It is clearly seen that the time oscillations of the electric dipole moment of two enantiomeric forms differ by a Pi phase shift, which can be amplified and detected with the phase sensitive microwave spectroscopy. Currently, work is in progress to optimise the laser field parameters to increase the intensity of the dipole transient emission. The manuscript presenting the results of calculations is in preparation.

The developed theoretical tools TROVE and RichMol have incredible potential which has been explored in several other applications as a result of collaboration with other research groups.

The developments made in TROVE during the course of the current project are extensively used by the ERC funded project ExoMol, which uses TROVE for spectroscopic refinement and high-energy spectral simulations of molecules important in the spectroscopy of atmospheres of exoplanets and cool stars (e.g. methane, acetylene, ammonia) [3-6,8-10].

In collaboration with the group of Per Jensen (Wuppertal, Germany) we studied the rovibrational and temperature effects on the hyperfine coupling of CH3 radical [7] and at the moment perform accurate simulations of the infra-red and Raman spectra for this molecule.

As a result of collaboration with the group of Per Jensen and Stephen Schlemmer (Cologne, Germany) we have developed a new accurate semi-classical method that simplifies the calculations of the highly excited rotational states of polyatomic molecules. This approach is shown to be very efficient in application to the rotational cluster states described above (in preparation).

In collaboration with Mike Tarbutt (Imperial College, London) we theoretically proposed and tested a new Stark deceleration method that exploits the idea of periodic population transfer between the low-field and high-field seeking states using the Stark-chirped rapid adiabatic passage excitation (the manuscript is in preparation).

In collaboration with the ExoMol group we have performed extensive calculations of the Raman and electric quadrupole spectroscopic line lists for ammonia molecule at room temperature (submitted to Journal of Chemical Physics) as well as terahertz field-induced alignment and orientation dynamics of ammonia and phosphine molecules (submitted to Scientific Reports).

Overall, we have made significant progress within each objective and expect to meet all project goals over the longer term. The project has been a success, leading to significant improvements of the existing variational methodology TROVE and development of the new time-dependent approach RichMol to model the molecule-field interactions with spectroscopic precision.

A number of successful collaborative projects over a range of different topics proves the importance and considerable impact of the current project on the field of contemporary molecular physics.

We note, that a proposal submitted for EPSRC with the aim to continue the development of the present theoretical approach with applications in topics of cavity cooling of molecules, chirality induced by extreme rotational excitation, and laser control over the large-amplitude internal and rotational motions of non-rigid molecules.