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Su Ning, Yu Shujuan, Li Weiyan, Yang Shiping, Chen Yanjun. Probing the structure of multi-center molecules with odd–even high harmonics. Chinese Physics B, 2018, 27(5): 054213  
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Probing the structure of multi-center molecules with odd–even high harmonics
Su Ning1, Yu Shujuan1, 2, †, Li Weiyan3, ‡, Yang Shiping2, Chen Yanjun1, §
College of Physics and Information Technology, Shaan’xi Normal University, Xi’an 710119, China
College of Physics and Information Engineering, Hebei Normal University, Shijiazhuang 050024, China
School of Mathematics and Science, Hebei GEO University, Shijiazhuang 050031, China

 

† Corresponding author. E-mail: yushujuan1129@163.com liweiyanhb@126.com chenyanjun@snnu.edu.cn

Abstract

We study high-order harmonic generation (HHG) from multi-center asymmetric linear molecules numerically and analytically. Our simulations show that odd and even HHG spectra of the asymmetric multi-center system respond differently to the change of the molecular structure. Specifically, when the internuclear distances between these nuclei of the molecule have a small change, the odd spectra usually do not change basically, but the even spectra differ remarkably. Based on this phenomenon, a simple procedure is proposed to probe the positions of these nuclei with odd–even HHG. Our results shed light on attosecond probing of the structure of multi-center molecules using HHG.

PACS: 42.65.Ky;32.80.Rm
Keyword:odd-even harmonics;multi-center asymmetric molecules;molecular structure
1. Introduction

In the past two decades, high-order harmonic generation (HHG)[1,2] from atoms and molecules has been a hot subject in both experimental and theoretical studies of strong laser–matter interaction. This interest in HHG partly arises from the important applications of HHG in attosecond science,[3,4] including attosecond probing of the electron structure and dynamics.[5,6] The HHG process can be well understood by a semiclassical three-step model:[7] (i) ionization of the active electron by tunneling, (ii) propagation of the electron in the laser field, and (iii) recombination of the freed electron into the bound state with the emission of a high-energy photon.

Previous studies on HHG are mainly focused on atoms and diatomic molecules. For symmetric diatomic molecules, such as [8–11] and N2,[12–15] due to the central symmetry of the potential, only odd harmonics are emitted. The phenomenon is similar to the atomic case. In particular, the HHG spectrum of symmetric diatomic molecules also shows a striking minimum, which is absent for the atomic spectrum and has been identified as arising from the effect of two-center interference when the rescattering electron returns to and recombines with these two atomic centers of the molecule.[8] This minimum has attracted great interest in recent years.[16–24] It can be used to probe the internuclear distance of the diatomic molecule[25–28] and judge the sign of the dipole phase in the molecular orbital tomography procedure.[12] For asymmetric diatomic molecules, such as HeH2+,[29–38] CO,[39–43] and NO,[44] due to symmetry breaking, both odd and even harmonics are emitted. It has been shown that these odd and even harmonics possess different spectral properties[45–48] and carry different information of the target,[31,49–51] and therefore need to be studied separately.

The odd–even HHG from asymmetric diatomic molecules has shown promising applications in ultrafast measurements. For instance, using odd–even HHG, one can probe the electron dynamics of asymmetric molecules,[52–56] image the asymmetric orbital,[43] and probe the nuclear dynamics of asymmetric molecules.[57] The degree of orientation for asymmetric molecules can also be calibrated with odd–even HHG.[58,59]

Recently, attention is increasingly turning to the interaction of the laser field with polyatomic molecules (such as CH4,[60–62] CH2Cl2,[63] C60,[64] and SF6[65]) and solids.[66–72] For symmetric polyatomic molecules such as CO2, the orbital symmetries can be complex with multiple nodal planes, and multielectron dynamics can become significant with several orbitals close to the highest occupied molecular orbital.[27] For asymmetric polyatomic molecules such as HCCl,[73] OCS,[50] and N2O,[74] due to symmetry breaking, the situation is more complex and some new effects can emerge. To probe the structure of multi-center asymmetric molecules with odd–even HHG, a detailed theoretical study on this issue is needed.

In this paper, we study odd–even HHG from multi-center asymmetric linear molecules and with diverse internuclear distances at different orientation angles numerically and analytically. The time-dependent Schrödinger equation (TDSE) of the simple single-electron systems can be solved exactly at present. Our simulations show that the emission of odd and even harmonics from these asymmetric systems depends differently on the change of the internuclear distance. Specifically, as this distance between two neighboring nuclei has a small change, odd spectra do not change usually, while even spectra differ remarkably. As a result, the positions of interference minima in odd spectra for the multi-center systems are also insensitive to the internuclear distance, different from two-center cases. We show that the phenomena are associated with different HHG mechanisms of odd versus even harmonics. With the sensitivity of even HHG spectra to the internuclear distance, a procedure is proposed to probe this distance using the relative yields of odd versus even harmonics.

2. Theoretical methods
2.1. Numerical procedure

The Hamiltonian of the molecular system studied here is (in atomic units of ). We assume the molecular axis is located in the xoy plane and the laser field is linearly polarized along a direction parallel to the x axis. For linear molecules and , the soft-core potential used here has the following form: for and for . Here , , , and are the vectors between the origin and different nucleus, and Z is the effective charge which is adjusted to obtain the desired ground-state energy. Similarly, in our comparison studies, the soft-core potential for diatomic molecules has the following form of . Here, is the vector between these two nuclei of . ξ is the smoothing parameter which is used to avoid the Coulomb singularity. For comparison, we have used the value of ξ = 0.5 for all cases of multi-center molecules explored in the paper. The main results associated with properties of odd–even HHG from multi-center asymmetric molecules discussed in the paper are not sensitive to the choice of the parameter ξ. with is the external electric field, and is the unit vector along the x axis. f(t) is the envelope function. E and ω0 are the amplitude and the frequency of the external electric field, respectively. In our calculations, we use a ten-cycle laser pulse which is linearly ramped up for three optical cycles and then kept at a constant intensity for four additional cycles and finally linearly ramped down for three optical cycles. Numerically, the TDSE of is solved numerically using the spectral method.[75] We work with a grid size of a.u. for the x and y axes, respectively. Details of our TDSE calculations can be seen in Ref. [76].

The sketches of the coordinate systems of and used in our two-dimensional (2D) simulations are presented in Figs. 1 and 2. Once the TDSE wave function is obtained, the coherent part of the HHG spectrum parallel to the laser polarization can be evaluated using

where ω is the emitted-proton frequency.

Fig. 1. (color online) A sketch of the molecular geometry and the coordinate system used in the 2D simulations for . The polarization direction of the laser field is along the x axis. The molecular axis is located in the xoy plane with an angle θ to the x axis and the center of mass of the molecular system agrees with the origin of the coordinate system o. , , and denote the positions of these three nuclei of to the origin, respectively. , , and denote the positions of the electron to these three nuclei and denotes that to the origin. X1 and X2 are the internuclear distances between two adjacent nuclei.
Fig. 2. (color online) Same as Fig. 1 but for . , , , and denote the positions of these four nuclei of to the origin, respectively. , , , and denote the positions of the electron to these four nuclei and denotes that to the origin. X1, X2, and X3 are the internuclear distances between two adjacent nuclei.

For comparison, the ground states of the linear molecules with different molecular parameters studied here have the same ionization potential of a.u.

2.2. Analytical description

In Ref. [31], it has been shown that for asymmetric diatomic molecules, there is a close relation between odd–even HHG spectra and odd–even bound-continuum-transition dipoles. This relation reflects the main HHG route for the emission of one odd or even harmonic from asymmetric molecules.[43] The main odd (even) HHG route is associated with the rescattering electron ionizing from the gerade component of the asymmetric orbital and recombining with the gerade (ungerade) component of the asymmetric orbital. A detailed discussion on the odd–even HHG routes can be found in Refs. [32] and [54]. We now explore the relation for multi-center cases. For comparison, the diatomic cases are also discussed here.

We assume that the wave functions of the ground states for , and molecules have the forms of , , and , respectively. Here, and is the ionization potential of the ground state. is the normalization factor and rn with denotes the distance of the electron to the n-th nucleus of the multi-center system. With approximating the continuum state using the plane wave, the odd bound-continuum-transition dipole at the angle θ (i.e., the angle between the molecular axis and the laser polarization), which is mainly responsible for the emission of odd harmonics along the laser polarization , can be written as[31,32]

with
and
for the cases of , , and molecules, respectively. Similarly, the even bound-continuum-transition dipole which is related to the emission of even harmonics along the laser polarization can be written as
with
and
for the cases of and molecules, respectively. Here, is the effective momentum of the continuum state , with that considers the Coulomb acceleration.[12] is the energy of the continuum state .

Note, due to the selection rule in the integral of the dipole, the odd dipole is associated with the gerade component of the ground state and the even dipole is associated with the ungerade one . The qualitative relationship between the corresponding dipoles and the HHG power spectra along the laser polarization is[12,43]

Here, is the spectral amplitude of the continuum electron which is not sensitive to the angle θ.[12] In the following, we will compare odd–even HHG spectra to the corresponding odd–even dipoles.

3. Results and discussion
3.1. Comparisons of HHG spectra and dipoles

In Fig. 3, we plot the odd harmonic spectra and relevant dipoles from two-center symmetric molecules with the laser intensity and the wavelength λ = 600 nm for different internuclear distances R at different orientation angles θ. First of all, from Fig. 3, we can observe that as the internuclear distance changes, the odd spectra at an angle θ differ remarkably from each other. Secondly, at a certain angle, the relative yields of odd harmonics for different internuclear distances in the HHG plateau region are well predicted by relevant dipoles, as described by Eq. (4). Thirdly, the HHG spectra and the corresponding dipoles show a striking minimum arising from two-center interference effects[8] in the HHG plateau region. The position of this minimum shifts toward higher harmonic orders remarkably as the internuclear distance decreases. As the position of the minimum is very sensitive to the internuclear distance, it is possible to judge the positions of the nuclei through this minimum in the spectra. This ultrafast probing procedure with HHG is meaningful especially for a chemical reaction where the positions of the nuclei of molecules can change rapidly in the attosecond scale. However, the situation is different for multi-center linear molecules, as shown in Figs. 4 and 5.

Fig. 3. (color online) Comparisons of odd HHG spectra of Eqs. 1(a) and 1(b) and dipoles of Eqs. 2(c) and 2(d) for 2D linear model molecules with the same ionization potential of a.u. but diverse internuclear distances of X = 2 a.u. (solid black curves), X = 1.9 a.u. (dash-dotted red curves), and X = 1.7 a.u. (dotted blue curves) at θ = 30° (the left column) and θ = 40° (right). The laser parameters used are and λ = 600 nm.
Fig. 4. (color online) Comparisons of odd HHG spectra of Eqs. 1(a) and 1(b) and dipoles of Eqs. 2(c) and 2(d) for 2D linear model molecules with the same ionization potential of a.u., but diverse internuclear distances of a.u. and a.u. (solid black curves), a.u. and a.u. (dash-dotted red curves), and a.u. and a.u. (dotted blue curves) at θ = 30° (left) and θ = 40° (right). The laser parameters used are and λ = 600 nm.
Fig. 5. (color online) Comparisons of odd HHG spectra of Eqs. 1(a) and 1(b) and dipoles of Eqs. 2(c) and 2(d) for 2D linear model molecules with the same ionization potential of a.u., but diverse internuclear distances of a.u., a.u., and a.u. (solid black curves), a.u., a.u., and a.u. (dash-dotted red curves), and a.u., a.u., and a.u. (dotted blue curves) at θ = 30° (left) and θ = 40° (right). The laser parameters used are and λ = 1200 nm.

In Figs. 4 and 5, we show the odd spectra and relevant dipoles for linear and at different internuclear distances and different angles. For comparison, the whole internuclear distances of the model systems are fixed at a certain value with a.u. for and a.u. for . The laser parameters used in Fig. 4 for are the same as in Fig. 3. For , the energy gap between the ground state and the first excited state is smaller and the system is easier to ionize than the case of . So in Fig. 5, we have used the laser parameters of and λ = 1200 nm.

One can observe that the odd spectra in the first row of Fig. 4 also show a striking minimum. However, as the internuclear distance has a small change, the position of the minimum as well as the relative yields of harmonics do not change basically. These phenomena are also well predicted by relevant dipoles in the second row of Fig. 4. Results in Fig. 5 for linear are similar to the cases of linear , where the spectra are not very sensitive to the change of the internuclear distance.

The results in Figs. 4 and 5 imply that as the minimum in odd spectra for multi-center linear molecules is not sensitive to the internuclear distance, it cannot be used as a tool to probe the positions of the nuclei in an ultrafast time scale.

One of the important characteristics for the HHG from asymmetric molecules is the emission of even harmonics. Next, we turn to even spectra for multi-center asymmetric linear molecules. In Fig. 6, we show the even spectra and relevant even dipoles for asymmetric linear molecules at different internuclear distances and orientation angles. Different from the odd cases in Fig. 4, one can observe from Fig. 6 that the even spectra differ remarkably as the internuclear distances change, in agreement with the predictions of relevant dipoles. On the whole, the even spectra are stronger with the increase of the molecular asymmetry (i.e., the increase of the difference between the distances X1 and X2). Results of even spectra for linear are similar to those in Fig. 6. So we do not present them here.

Fig. 6. (color online) Comparisons of even HHG spectra of Eqs. 1(a) and 1(b) and dipoles of Eqs. 3(c) and 3(d) for 2D linear model molecules with the same ionization potential of a.u., but diverse internuclear distances of a.u and a.u (solid black curves), a.u and a.u. (dash-dotted red curves), and a.u and a.u. (dotted blue curves) at θ = 30° (left) and θ = 40° (right). The laser parameters used are and λ = 600 nm.

From the results in Figs. 4–6, we conclude that for multi-center asymmetric linear molecules, the odd spectra are not sensitive to the molecular structure, nor even spectra. However, in the situation, it remains possible to probe the structure of multi-center asymmetric linear molecules in the ultrafast time scale, with the combination of odd and even HHG spectra. Let us discuss this point in more detail. The exact difference between odd and even cases shown above arises from different generation mechanisms of odd versus even harmonics. As discussed in Ref. [43], the emission of odd harmonics is mainly associated with the gerade component of the asymmetric orbital in the recombination process and the emission of even harmonics is mainly associated with the ungerade one. For multi-center asymmetric linear molecules, when one center of the multi-center system has a small change at position, the gerade component of the asymmetric orbital does not change basically and the ungerade component differs remarkably from the previous one. This difference is mapped in the odd–even HHG spectra, providing the possibility of probing the structural change of the multi-center system with odd–even HHG. Next, we explore the potential ultrafast probing procedure with odd–even harmonics.

In Fig. 7, we plot the odd–even HHG spectra from asymmetric linear molecules with a.u. and a.u. at different orientation angles in the first row. For comparison, odd and even harmonics are presented using solid and short-dashed lines, respectively. From Fig. 7, we can see that the relative yields of odd versus even harmonics presented here show a clear dependence on the orientation angle. For example, in Fig. 7(a), the yields of even harmonics are lower than odd ones for low and high harmonic orders. The situation reverses for intermediate harmonic orders, with the odd–even HHG spectra showing two clear intersections. As the orientation angle increases, the positions of the intersections shift towards higher harmonic orders, as shown in Fig. 7(b). The characteristics are also reproduced by the relevant dipoles shown in the second row of Fig. 7. We use dashed arrows to indicate the intersections located at harmonic orders with energy near or higher than (about H40). For these harmonic orders, the influence of the Coulomb effect on the continuum electron is smaller and the plane-wave approximation for the continuum electron in Eq. (4) is expected to be more applicable.

Fig. 7. (color online) Comparisons of odd–even HHG spectra of Eqs. 1(a) and 1(b) and dipoles of Eqs. (2), (3c), and (3d) for 2D linear model molecules with a.u. and a.u. at θ = 30° (left) and θ = 40° (right). The dashed arrows indicate the intersection points of odd–even spectra and dipoles. The laser parameters used are and λ = 600 nm.

In Fig. 8, we show results for other internuclear distances with a small asymmetry in comparison with the case of Fig. 7. On the whole, the odd–even HHG spectra also show some striking intersections, the positions of which agree with the predictions of relevant odd–even dipoles. The calculated odd–even harmonic spectra and relevant dipoles for asymmetric linear also show similar phenomena as those presented in Figs. 7 and 8. Based on the above results, we propose the probing of the structure of multi-center asymmetric molecules using the intersections of odd versus even HHG spectra.

Fig. 8. (color online) Same as Fig. 7, but a.u. and a.u.

It should be stressed that at smaller orientation angles, the intersections of odd–even HHG spectra are located at lower harmonic orders near the threshold, where the influence of the parent ions on the continuum electron is stronger. In this case, equation (4) based on the plane-wave approximation is not applicable. So we focus our discussion on these two larger angles of θ = 30° and 40° here. The proposed procedure discussed below is also more applicable for larger angles.

3.2. Probing procedure

Here, all of the terms and are functions of the internuclear distances X1 and X2 (see the expressions below Eqs. (2) and (3)). The solution of the above equation gives the predictions of these distances X1 and X2. Similarly, one can reconstruct the bond lengths of linear asymmetric molecules using the following expressions:

Here, ω1, ω2, and ω3 are also the harmonic energy corresponding to the intersections of odd–even HHG spectra at three angles of θ1, θ2, and θ3, respectively. Note, for a certain angle, the odd and even HHG spectra can have two intersections located at higher harmonic orders (see Figs. 7 and 8). Both of the intersections can be used in the reconstruction. In our simulations, we change the orientation angles from θ = 20° to θ = 60° and average the bond lengthes reconstructed with odd–even HHG at different intersections and different angles. Relevant results are presented in Fig. 9.

Fig. 9. (color online) Comparisons of the exact results (big-black spheres, ACC) and the reconstructed results of Eqs. (5) and (6) (small-red spheres, REC) for geometry of linear molecules ((a)–(f)) and ((g)–(i)) with the same ionization potential of a.u. and diverse internuclear distances. The laser parameters used are and λ = 600 nm for , and and λ = 1200 nm for .

In the first row of Fig. 9, we show the results for with a.u. In the second row, those are for with a.u. and in the third row for with a.u. One can observe from Fig. 9, for all of the cases, the reconstruction procedure based on Eqs. (5) and (6) basically reproduces the internuclear distances of the multi-center molecules, implying the applicability of this procedure. We have validated our above discussions with performing simulations at other laser and molecular parameters.

4. Conclusions

In summary, we have studied the HHG from multi-center asymmetric linear molecules and . We have shown that different from symmetric diatomic molecules such as , the odd HHG spectra, especially the interference minima in the odd spectra of multi-center asymmetric molecules are not very sensitive to the change of the internuclear distance, so do the even spectra. The phenomena reflect the fact that when the position of one of the atomic centers of the multi-center system has a small change, the gerade component of the asymmetric orbital (which is mainly responsible for the emission of odd harmonics in the recombination process) does not change basically, but the ungerade component (which is mainly responsible for the emission of even harmonics) differs remarkably from the previous one. In this situation, the intersection of odd versus even HHG spectra can be used as a sensitive tool to probe the structure of multi-center molecules in an ultrafast time scale. We expect that our studies on the HHG of these simple systems will shed light on other multi-center molecules with more complex symmetries.

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