From chiral laser pulses to femto- and attosecond electronic chirality flips in achiral molecules

Chirality is an important topic in biology, chemistry and physics. Here we show that ultrashort circularly polarized laser pulses, which are chiral, can be fired on achiral oriented molecules to induce chirality in their electronic densities, with chirality flips within femtoseconds or even attoseconds. Our results, obtained by quantum dynamics simulations, use the fact that laser pulses can break electronic symmetry while conserving nuclear symmetry. Here two laser pulses generate a superposition of three electronic eigenstates. This breaks all symmetry elements of the electronic density, making it chiral except at the periodic rare events of the chirality flips. As possible applications, we propose the combination of the electronic chirality flips with Chiral Induced Spin Selectivity.

The main text already documents chirality flips in NaK in the 1 1 1 11 1  1 2 superposition states by selective snapshots of the electronic density, cf.Fig. 2 of the main text.
Here we support the documentation by additional snapshots in Supplementary Figures 1-3.Note for better view of the chirality flips, the snapshots in Fig. 2 of the main text are rotated around the z-axis by certain angles.Specifically, the five snapshots from top to bottom in Fig. 2a of the main text are rotated by 188.7, 188.0-30, 188.0, 188.0+30, and 187.3 degrees, respectively, while the five snapshots from top to bottom in Fig. 2b of the main text are rotated by 147.4,49.2, 49.2, 49.2, and -49.0 degrees, respectively.To show all snapshots in the same coordinates frame, we plot snapshots that are all rotated around the z-axis by the same angle in Supplementary Figures 1 and   2. Finally, we show snapshots in the original coordinates frame, namely none of the snapshots is rotated, in Supplementary Figure 3, together with the corresponding snapshots of the twodimensional reduced density ( , , ) ( , , , ) x y t r z t dz Our step-by-step presentation in this section reminds of various properties of the underlying electronic Hamilton operator, and of the electronic eigenfunctions, the many-electron probability density, and the electronic density at the level of full configuration interaction (full-CI).The final result eqn.(S66), will be obtained at the end of this Section.

2.1: Molecular orbitals of oriented heteronuclear diatomic molecules
We The corresponding molecular orbitals (MOs) ( , , ) mn rz  are eigenfunctions of the one-electron where (1) () T r and (1) ( ; ) VR r are the operators of the kinetic and potential energies of the electron, () with electron mass e m .The potential energy (1) ( ; ) VR r is approximated as the sum of the Coulomb interactions of the electron with the two nuclei.Interactions of nuclear and electronic spins are neglected.

2.2: Slater determinants for oriented heteronuclear diatomic molecules
The electronic eigenfunctions of oriented heteronuclear molecules AB such as NaK with N electrons are described in terms of Slater determinants for N molecular spin-orbitals 12 , ,..., By convention, they are listed with ascending order of the labels for the quantum numbers, 12 ...
where is the anti-symmetrization operator, ) with normalized sum of all even ν (( 1) 1) P    permutations of the molecular spin-orbitals; the sum '  is over all permutations except the identity ("1").The relation (S13) for the molecular spin-orbitals implies the complex conjugation of the Slater determinants, q q q q q q q q q q q q (S18) The second eqn.(S18) exploits the fact that is Hermitian and a projection operator, The overlap integral (S18) is equal to zero if the two sets of spin-orbitals labeled 1 ,..., The sum !' 2 N v  does not contribute, due to the orthogonality of the spin-orbitals.To summarize, Slater determinants are orthonormal, , ,..., , ,..., ... .In analogy to the molecular spin-orbitals () l  q , we use the label k for the adequate set of quantum numbers.
The electronic Hamilton operator can be written as   (1) el el,el nu,nu 1 ( ) ( ) with Coulomb interactions between the electrons labeled i and j , and the Coulomb interaction / , The electronic eigenfunctions are, therefore, not only eigenfunctions of the electronic Hamilton operator, eqn.(S22), but also of z L , , 0, 1, 2,...
As a consequence, the wavefunctions k  can be written in full configuration interaction (full-CI) as sums of Slater determinants In the present application, we focus on singlet states ( 0 S  , s M0  ).Eqn. (S30) then implies that for even numbers N of electrons, the numbers of  and  spin-orbitals in the Slater determinants are equal to each other.
The present application focusses on the ground state 1 1   and two pairs of degenerate excited states  as well as The degenerate wavefunctions are complex conjugate to each other, in accord with the complex conjugation of the spin-orbitals and the Slater determinants with opposite angular momentum quantum numbers, cf.eqns.(S13), (S17) The orthonormality of the Slater determinants implies that the electronic eigenfunctions (S31)-(S33) are normalized according to The complex conjugation of the degenerate eigenfunctions (S34) implies that they have the same many-electron probability densities, The expressions for the many-electron densities, eqn.(S37) at the full-CI level are formidable.We shall now show, however, that they yield rather simple results for the related electronic densities.
For a quantum state k  , the electronic density at the position r can be obtained as the mean value of the density operator ˆ() The resulting electronic densities for 0 k  , 1 and Let us now consider the integrals in expression (S40    q q r r q q q q r r q q q q q q r r q q q ' ! 2 ( )] .
Moreover, any permutations v P of the orbitals '' q q r r q q q q r r q q q r r q q (S43) The equality of all spin-orbitals but one, eqn.(S42) implies that the corresponding angular orbital quantum numbers must also be the same, , ,..., , ,..., , ,..., , ,..., .
Likewise, the corresponding spin quantum numbers must be equal to each other, implies that the angular momentum and spin quantum numbers of the spin-orbitals () j lj  q and ' ()  q must also be the same, ' jj mm  , ' ,, .
Inserting the explicit forms of the spin-orbitals (S12) then yields The electronic densities are thus evaluated as The result (S50) confirms chemical intuition that means the electronic densities of the eigenstates 1   and    rr (S52) The total number of electrons can be obtained by the integration of the electronic density, For the present application, the electronic densities ( ), 0 k rk   , 1 , 2  of NaK are illustrated in Fig. 1 of the main text.Analogous results can be derived for all eigenstates of oriented heteronuclear diatomic molecules.Since the wave function does not depend on an overall phase, we may set 0 0   for the ground state 0 k  .The corresponding many-electron probability densities are sums of time-independent (ti) and time-dependent (td) contributions which consist of three diagonal plus three off-diagonal terms,

2.5: The time-dependent electronic wave functions and the many-electron probability densities and the electronic densities of the
The many-electron probability densities are normalized, This is a consequence of the normalization of the coefficients (S55) and the orthonormality of the eigenfunctions, eqn.(S36).
By analogy with eqn.(S40), the electronic densities in full-CI are obtained as the corresponding mean values of the density operator in eqn.(S39).Accordingly, they consist of a sum of a timeindependent and a time-dependent contributions with corresponding three diagonal plus three off- For the first three diagonal time-independent terms of eqn.(S58), we can adapt the results (S50) for the electronic densities of the eigenstates Similar to eqn.(S49), the integration in eqn.(S59) can be simplified,  ( , , )  ( , , ) Different from the previous case (S46), however, the angular momentum quantum numbers of the two sets of spin-orbitals for the 1 1   and the .
The time-dependent phase 11 t   in the expression (S56) for the many-electron probability density is thus supplemented by the angular phase   in the corresponding term for the electronic density, yielding the total phase 11 t       ; this is an important intermediate result which has enormous consequences for the final result, cf.eqn.(S66) below.
Likewise, integration of the N -electron term * 01   in eqn. (S56) yields the complementary complex conjugate contribution 11 . Adding the two contributions yields the first time-dependent off-diagonal term .
By analogy with the derivation for the first time dependent contribution (S63) to the electronic densities of the 1      superposition states, we obtain for the second and third contributions 0,2 ... ... 2 ( , ) ( , ) , .
The time-dependent electronic density of the 1

 
,ti ,td     In practice, the matrix elements (S76) for j or 1 k  or 2  are evaluated by decomposing the eigenfunctions into x-and y-components All the electronic properties are calculated by MOLPRO  Pt are in panels (c) and (d), respectively.Equivalent results are obtained for all  combinations of the polarizations of the laser pulses.Apparently, the ( ') Pt approach asymptotic values for times '

, 1 TSupplementary Figure 3 :.
case of ( ) circular polarizations of the laser pulses.All the figures are rotated around the z-axis by 49.2 degrees such that the middle panel, which is for 1 tT   , has its symmetry plane perpendicular to the paper plane.The first row from left to right is for 10  , and 1 Tt    .The last row from left to right is for case of ( ) circular polarizations of the laser pulses.All the figures are rotated around the z-axis by 188.0 degrees such that the middle panel, which is for 1 tT   , has its symmetry plane perpendicular to the paper plane.Similar to Supplementary Figure 1, the times of neighboring snapshots in each row from left to right increase by t  , while in each column from top to bottom they increase by 7 t  .Non-rotated version for the snapshots shown in Fig. 2 of the main text with corresponding two-dimensional reduced density ( , , ) x y t  The times of the five snapshots are the same as for Fig. 2 of the main text.

) Supplementary Note 2 :
Derivation of equation (2) of the main text for the time evolution of the electronic density of an oriented heteronuclear diatomic molecule such as NaK prepared   superposition states.
consider heteronuclear diatomic molecules AB such as NaK with N electrons.The molecules are oriented along the laboratory z-axis, with nucleus A pointing to positive values of z.The nuclear center of mass is at the origin.The nuclei have cylindrical symmetry v C  .This suggests to use cylindrical coordinates ( , , ) rz  r for the electronic positions.The corresponding combined set of spatial and electron spin coordinates is written as ( , ) s  qr.
each other, due to the orthogonality of the molecular spin-orbitals (S14).For equal sets of spin-orbitals, eqn.(S18) is evaluated by means of eqn.(S16),

. 3 :
Electronic eigenfunctions for the 1   , Electronic eigenfunctions of oriented heteronuclear diatomic molecules AB such as NaK are obtained as solutions of the electronic Schrödinger equation

B
Qe at distance R. It commutes with the z-component of the angular momentum operator   , 11 to the condition that the angular momentum quantum numbers j m of the spin-orbitals j l  must sum up to the chosen value of M, valued.Since the electronic Hamiltonian does not depend on electron spins, it also commutes with the electron spin operators.The total electron spin S and its z-component z S yield, therefore, also good quantum numbers.In particular, the magnetic spin quantum numbers , sj m must sum up to the chosen total value s

1 
1,0), (-1,1,0) as well as (1,2,0), (-1,2,0), respectively.Here E N is the quantum number for different values of energies.For convenience, the five states are labeled 0 k  and 1 , 1 k    as well as 2 , 2 k    , respectively.The energy of the ground state is set equal to zero, 0 0 k E   .The energies of the excited states are degenerate; they are written 1 E as well as 2 E .Accordingly, the full-CI expansion for the 1 qq would yield zero integrals in expression (S41), again due to the orthonormality of the spin orbitals.Hence, we are left with

2 
have cylindrical symmetry v C they do not depend on  .Moreover, the equivalence (S38) implies that the electronic densities of the degenerate states 1 k  and 2 k  are equal to each other,


S54)with full-CI representations (S31)-(S33) of the electronic eigenfunctions k depend on the  combinations of the circularly right (  ) or left (  ) polarized laser pulses.In any case, the angular momentum quantum numbers of the spin-orbitals () Inserting the explicit forms of the spin-orbitals (S12) then yields

11
  superposition states can be obtained by inserting eqns.(S63) -(S65) into (S58).It is a sum of time-independent and timedependent contributions which consist of three diagonal plus three off-diagonal terms, of this Section, eqn.(S66), is adapted as eqn.(2) in the main text.It sheds important light on the assignment of chirality or non-chirality of the electronic density.All timeindependent contributions and also the time-dependent contribution for the special case of two laser pulses with the same  or  circular polarizations depend on r and z , but not on  : they have an infinite number of vertical symmetry planes and many other symmetry elements, i. e. they have v C  symmetry, i. e. they are achiral.In contrast, the first two terms of the time dependent contributions depend on , rz and  .The -dependence is in the arguments of the two cosine-, respectively.This breaks all symmetry elements, that means it causes 1 C symmetry and makes the electronic density chiral, except for the rare events at n tT  when the two arguments of the cosine functions are co-incidentally equal to zero or to integer ( n  ) multiples of  , for the same value of  called n  : This leaves the electronic density with a mirror plane at the angle n  and makes it achiral at the exclusive instants n T .Gratifyingly, if the third term of the time-dependent contribution depends on  , then the argument of its cosine function is automatically also equal to zero or to n , at the same time n T and for the same angle n  .As a resume, the first two terms of the time dependent contributions to the electronic density are decisive: they make the electronic density chiral for all times, except at the rare events n tT  .

Supplementary Figure 4 : 1 
The values of the corresponding maximum intensities and the photon energies are Electric fields of two circularly polarized laser pulses for excitation of the oriented NaK molecule from the electronic ground state 1 Panels (a), (c) and (b), (d) are for two cases with the same ( ) or with opposite ( ) polarizations.In panels (a) and (b), the x-and y-components of the electric fields are shown by red and blue lines.The laser parameters are the same for both cases, as specified in eqns.(S80), (S81).As example, the corresponding electric fields of the circularly polarized laser pulses for two cases " " and " " with the same (right,  and again right,  ) and with opposite (left,  and right,  ) polarizations are shown in panels (a) and (b) of Supplementary Figure 4.The resulting population dynamics ( ') k 2  are 12the effect of the laser pulses is negligible.Hence it is reasonable to define the end of the pulses at time 12 is considered as the beginning of the (quasi-)field-free time evolution of the system, 0 t  .The parameters of the laser pulses (S80) are chosen to prepare the oriented NaK molecule in the , 1 , 2  , obtained by the quantum dynamics simulations are listed in Supplementary Table1.Supplementary Table1also has the amplitudes and the phases of the the TDSE (S69) into the set of time-dependent differential equations for the coefficients (S75), now it is converted into the corresponding set of coupled nuclear TDSEs for the nuclear reduced mass.Eqns.(S83)accountfor the laserdipole couplings which induce transitions between states j and k mediated by R-dependentThe coupled nuclear TDSEs (S83) are solved numerically by means of the Split operator method12.The propagation starts at the beginning of the laser pulses ( ' i tt  ) when the oriented NaK is in the electronic and nuclear ground state, with nuclear ground state wavefunction fse tt  converting i t   .The resulting nuclear wavefunctions yield the probabilities of occupying electronic state k ,