Role of excited states in molecular alignment-dependent ionization

Shilin Hu; Zheng Shu; Mingqing Liu; Li Guo; Xiaolei Hao; Jing Chen; Chaohong Lee

doi:10.1364/OE.26.032225

1. Introduction

Investigations of excited states have attracted considerable attentions in intense-field physics recently, because excited states play important roles in the generation of fascinating strong-field phenomena. One typical example is the effect of excited state on ionization behavior. It is found that charge-resonance-enhanced ionization occurs for extended molecules due to the strong coupling of charge states (1σ_g and 1σ_u) for $H_{2}^{+}$ and H₂ [1, 2], and the above distinct feature has been observed experimentally [3]. For asymmetric molecule HeH²⁺, the coupling of the ground state and the excited state shows dependence on the carrier-envelope phase of few-cycle pulses, which has an important impact on the enhanced ionization process [4]. Recently, it is shown that the excited states are related to the very low or near-zero energy structure in the photoelectron spectrum in intense laser fields [5, 6]. For high-energy structures in photoelectron spectra, it is also found that the excited states have an important impact on the resonance-like enhancement in high-order above threshold ionization [7, 8]. Another example is the role of excited states in high-order harmonic generation (HHG). Atoms (molecules) in excited state are prepared to increase HHG yields and extend the cutoff frequency of the harmonic spectrum compared with systems in ground state [9–11]. The excited orbital has a great impact on the dip and the cutoff of the harmonic spectra for stretched $H_{2}^{+}$ in intense laser fields [12, 13]. Recently, it is predicted that the resonant population of excited states during the laser pulse plays a role in fractional-order harmonics and spectral redshift [14, 15], and the aforementioned effect of ionization from excited states and recombination to the ground state has been verified experimentally [16].

Due to improvements in molecular orientation and alignment technology [17], study of the orientation effect on strong-field ionization has attracted significant attention lately. The molecular tunneling ionization probability shows dependence on the alignment of the molecular axis with regard to the laser field for small internuclear distance, and the alignment-dependent ionization probability can be used to reconstruct the contour of the spatial electron distribution of the active orbital, which stimulates a number of experimental studies on molecular angular-dependent ionization [18, 19]. However, for CO₂, the narrow ionization distribution and its maximum ionization at alignment angle 45^° are observed in experiments, while the maximum ionization shows up at alignment angle near 25^° for the theoretical result [19]. For OCS, the experimental observation of the peak ionization probability appears at alignment angle 90^°, while the theory predicts at 45^° [20]. Considerable efforts have been devoted to addressing the above problems, but the underlying mechanisms behind their anomalous alignment-dependent ionization are still under debate [21–25].

In theory, the methods of time-dependent Schrödinger equation (TDSE) and strong-field approximation (SFA) are widely used to investigate angular-dependent ionization of molecules with small internuclear distances [26–30]. For TDSE, although it is time-consuming and requires large computation resources, the roles of coulomb potential and excited states are taken into account. For SFA, the contribution of excited states to the evolution of the system is neglected, and the length-gauge SFA is more appropriate to study molecular alignment-dependent ionization with respect to velocity-gauge SFA [31, 32]. Recently, oscillation of the ionization yields is shown for $H_{2}^{+}$ as a function of internuclear distances in a high-frequency laser field parallel to the molecular axis, and the above peculiar feature disappears for the polarization direction of the laser field perpendicular to the molecular axis, which is traced to the high-energy electron two-center interference [33]. Afterwards, it is demonstrated that the molecular ionization probability shows oscillatory behavior with respect to the alignment angle for large internuclear separation due to the low-energy electron interference [34].

As mentioned above, excited states have a significant impact on the ionization dynamics and HHG, while a detailed investigation of the effect of excited states on the molecular angular-dependent ionization still lacks, which is in need for further understanding on the experimental findings, for instance, imaging electron molecular orbitals via ionization and HHG [18, 19, 35]. In this paper, we study the alignment-dependent ionization of $H_{2}^{+}$ by a three-dimensional TDSE and SFA, and our calculations show that the excited states play an important role in the orientation-dependent ionization of $H_{2}^{+}$ . Atomic units (a.u.) are used unless otherwise indicated.

2. Theoretical methods

2.1. I. Time-dependent Schrödinger equation

Within the dipole approximation, the full-dimensional TDSE of $H_{2}^{+}$ in the length gauge is given in the following form

i \frac{\partial}{\partial t} Ψ (r, t) = [H_{1} (r) - r \cdot E (t)] Ψ (r, t) .

For a given magnetic quantum number m, the detailed expression of H₁(r) is given by [36, 37]

H_{1} (r) = - \frac{1}{2 r^{2}} \frac{\partial}{\partial r} r^{2} \frac{\partial}{\partial r} + \frac{1}{2 r^{2}} [\frac{\partial}{\partial ξ} (1 - ξ^{2}) \frac{\partial}{\partial ξ} - \frac{m^{2}}{1 - ξ^{2}}] - \sum_{λ = 0}^{\infty} \frac{r_{<}^{λ}}{r_{>}^{λ + 1}} P_{λ} (ξ) - \sum_{λ = 0}^{\infty} {(- 1)}^{λ} \frac{r_{<}^{λ}}{r_{>}^{λ + 1}} P_{λ} (ξ) .

Here r_> (r_<) denotes the bigger(smaller) one of $(r, \frac{R}{2})$ , and ξ = cos θ(θ is the angle between r and the internuclear distance R). P_λ(ξ) are the Legendre polynomials, and E(t) indicates the time-varying electric field. We assume that the polarization vector of the field lies in the plane x-z (the z axis is parallel to the molecular axis), and the laser-molecule interaction term is written as

H_{t} = - r E (t) (ξ \cos β + \sqrt{1 - ξ^{2}} \sin β \cos φ),

where β is the alignment angle between the polarization direction of the laser field and the molecular axis, and the field-free Schrödinger equation of

H_{2}^{+}

is

H_{1} (r) ψ_{j} (r, ξ, φ) = ϵ_{j} ψ_{j} (r, ξ, φ) .

The wave function is expanded by B-spline as

ψ_{j} (r, ξ, φ) = \frac{1}{\sqrt{2 π}} \sum_{μ v} C_{μ v}^{j} \frac{B_{μ}^{k} (r)}{r} B_{v}^{k} (ξ) {(1 - ξ^{2})}^{\frac{| m |}{2}} e^{i m φ},

where k=7, and the details of B-splines can be found in Refs. [36, 38]. Diagonalizing the Hamiltonian in Eq. (4), we obtain the eigenvalues ϵ_j and the corresponding eigenfunction ψ_j , and the time-varying wave function is expanded by eigenfunctions

Ψ (r, t) = \sum_{j} C_{j} (t) ψ_{j} (r, ξ, φ) .

Substituting the above time-dependent wave function into Eq. (1), we propagate it by Crank-Nicolson method [36–38]. A cos^1/8 absorber function is used near the boundary to reduce spurious reflection. At the end of the pulse, the ionization probability is defined as P_ion = 1 − Σ_j |C_j |² with ϵ_j < 0. To study the effect of a specific excited state on the ionization dynamics, we include or exclude it in the basis expansion artificially, which has been used to analyse the contribution of 1σ_u state to the enhanced ionization of $H_{2}^{+}$ subjected to intense laser fields recently [39].

In the present work, the vector potential is $A (t) = \frac{E_{0}}{ω} ε \sin^{2} (π t / t_{\max}) \cos ω t$ with unit vector ε, 0 < t < t_max , and the time-dependent electric field is defined via $E (t) = - \frac{\partial A (t)}{\partial t}$ . E₀ is the peak electric field, and t_max and ω are the duration and the frequency of the laser pulse, respectively. In the following calculation, $H_{2}^{+}$ is exposed to a three-cycle laser pulse of frequency ω = 0.057 a.u. (λ = 800 nm), I₀=10¹⁴ W/cm² is used as the unit of laser intensity, and the time step is 0.1 a.u. The internuclear distance R=2 a.u. is assumed, and the radial box is truncated at r_max =90 a.u. 100 radial B-splines and 20 angular B-splines are adopted, the magnetic quantum numbers are kept from m=−6 to m=6, and convergence is reached with the above settings. The initial state is 1π_u unless otherwise indicated.

2.2. II. Modified strong-field approximation

In this work, we also study the alignment-dependent ionization of $H_{2}^{+}$ adopting strong-field approximation (SFA) [31, 40–45]. Within the modified SFA, the corresponding wave function in the present paper can be assumed to be [43]

Ψ (r, t) = e^{i I_{p} t} [a_{0} (t) | 0 > + a_{1} (t) | 1 > + \int d P b (P, t) | P >],

where |0> and |1> denote

ψ_{1 π_{u}}

and

ψ_{2 σ_{g}}

, respectively.

\begin{matrix} | 0 > = c_{1} [ϕ_{2 p_{x}} (r + R / 2) + ϕ_{2 p_{x}} (r - R / 2) + 0.1 (ϕ_{2 p_{x}}^{'} (r + R / 2) + ϕ_{2 p_{x}}^{'} (r - R / 2))], \\ | 1 > = c_{2} [ϕ_{2 s} (r + R / 2) + ϕ_{2 s} (r - R / 2)] . \end{matrix}

Here c₁ and c₂ are the normalization factors. Then the equations of a₀(t) and a₁(t) are given by

\begin{matrix} i {\dot{a}}_{0} (t) = - E (t) [a_{0} (t) d_{00} + a_{1} (t) d_{01}], \\ i {\dot{a}}_{1} (t) = - E (t) [a_{0} (t) d_{10} + a_{1} (t) d_{11}] + (I_{p 1} - I_{p 2}) a_{1} (t) . \end{matrix}

Here the time-varying electric field is defined as $E (t) = - \frac{\partial A (t)}{\partial t}$ , and the vector potential is the same as that aforementioned in TDSE calculations. d_ij (i, j=0, 1) represent the transition matrices between the bound states, and I_p₁ and I_p₂ indicate the ionization potentials of 1π_u and 2σ_g states, respectively.

For the SFA theory in the length gauge, the single ionization rate of a molecule in the state |s> (s=0, 1) under the linearly polarized laser field is written as

W_{s} = 2 π N_{e} \sum_{n = n_{0}}^{\infty} \int | T_{p n}^{s} |^{2} δ (E_{p} + I_{p} - n ω) d P,

with

T_{p n}^{s} = \frac{1}{T} \int_{0}^{T} d t \exp [i (\frac{E_{0}^{2}}{8 ω^{3}} \sin 2 ω t + \frac{E_{0} e \cdot P}{ω^{2}} \sin ω t)] \times \exp [i (\frac{P^{2}}{2} + I_{p} + U_{p}) t] E_{0} \sin ω t \tilde{ϕ} [P + A (t)] a_{s} (t) .

e is the unit vector, N_e indicates the number of equivalent electrons in the valence orbital, and n₀ is the minimum number of photons needed to release the electron. E_p =U_p + P²/2 indicates the quasienergy, where

U_{p} (= E_{0}^{2} / 4 ω^{2})

represents the ponderomotive energy, and P²/2=nω−U_p −I_p indicates the kinetic energy of the emitted electron after absorbing n photons for the ionization potential I_p . In the following calculations, we employ

\tilde{ϕ} [P + A (t)] = 2 \cos [(P + A (t)) \cdot R / 2] d_{j} [P + A (t)]

for undressed modified molecular SFA [31, 43, 44], and d_j [P+A(t)] (j = 2p_x ,

2 p_{x}^{'}

or 2s) indicate the atomic dipole moment of the 2p_x ,

2 p_{x}^{'}

, and 2s orbitals for the initial states of 1π_u and 2σ_g, respectively. The total ionization probability is defined as W = ΣW_s (s=0, 1). The 2p_x orbital is given by

ϕ_{2 p_{x}} (r) = k^{\frac{5}{2}} \frac{r}{\sqrt{π}} \exp (- k r) \sin θ \cos φ

with

k = 1.5 \sqrt{2 I_{p 1}}

and I_p₁=0.428 a.u. (

ϕ_{2 p_{x}}^{'} (r) = {k^{'}}^{\frac{5}{2}} \frac{r}{\sqrt{π}} \exp (- k^{'} r) \sin θ \cos φ

with

k^{'} = 0.4 \sqrt{2 I_{p 1}}

), and the 2s orbital is written as

ϕ_{2 s} (r) = \frac{1}{\sqrt{π}} k^{\frac{5}{2}} (1 - k r) \exp (- k r)

with

k = 1.35 \sqrt{2 I_{p 2}}

and I_p₂=0.36 a.u. In this work, Eqs. (9) and (10) are calculated numerically [34, 45].

3. Results and discussion

We present the energy of the bound states for various values of magnetic quantum numbers m=0 and 1 in Table I, which is in a good accordance with those in Ref. [26]. We also show the contour plot of the electronic wave function for 1σ_g, 1σ_u, 2σ_g and 1π_u states in Fig. 1. It is demonstrated that the charge distribution of σ orbital is parallel to the molecular axis, while the electron cloud of 1π_u orbital is distributed perpendicularly to the internuclear axis.

Table 1. The eigenvalue (a.u.) of the bound states for magnetic quantum numbers m=0 and 1, and the internuclear separation R=2 a.u.

View Table

Fig. 1 Two-dimensional distribution of the wave function for different orbitals. (a): 1σ_g; (b): 1σ_u; (c): 2σ_g; (d): 1π_u .

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In order to visualize orientation effect, the ionization probability is normalized to its maximum value unless otherwise indicated. Figure 2 displays the angular dependence of ionization yields for $H_{2}^{+} (1 π_{u})$ calculated by different theoretical methods. Fig. 2(a) shows the results obtained by TDSE, which has been checked by our previous numerical procedure [34]. For 0.3I₀, it can be seen that the ionization probability increases from 0^° to 50^°, while a quick decrease of the ionization yield is present between 60^° and 90^°. For 0.5I₀, the peak ionization yield shows up at the alignment angle 40^°. In Fig. 2(b), we depict the alignment-dependent ionization of $H_{2}^{+}$ obtained by SFA calculation (2σ_g state is not taken into account in Eq. (7)), and it is found that the ionization yields increase with ascending alignment angle for 0.3I₀ and 0.5I₀ due to the spacial distribution of 1π_u perpendicular to the molecular axis in Fig. 1(d). The normalized ionization probability of 0.3I₀ is close to those of 0.5I₀ due to the small difference of the above two laser intensities. Clearly the location of peak ionization probability in Fig. 2(b) is different from that in Fig. 2(a). It is well known that the effect of excited states is not taken into account for SFA, which maybe account for the discrepancy of the alignment-dependent ionization obtained by TDSE and SFA calculations in Fig. 2.

Fig. 2 Alignment-dependent ionization probabilities of $H_{2}^{+}$ for the initial state 1π_u as a function of the laser intensity calculated by different methods. (a): TDSE; (b): SFA without considering 2σ_g state in Eq. 7 (see text). I₀=10¹⁴W/cm².

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Now let’s turn to the maximum ionization probability, which occurs at different alignment angles for TDSE and SFA in Fig. 2. In Table 1, the energy gap between 2σ_g and 1π_u states is about 0.068 a.u., which is close to the photon energy 0.057 a.u. (λ = 800 nm) of the laser field, so a strong coupling of single photon transition may occur between the 2σ_g and 1π_u states. Fig. 3 depicts the time-dependent populations of 1π_u , 2σ_g and other excited states as a function of time with different alignment angles for 0.3I₀ and 0.5I₀, respectively. In general, the populations of the 2σ_g state grow with increasing alignment angles, since the coupling between 1π_u state and 2σ_g state becomes stronger with increasing alignment angle. Consequently, 2σ_g state plays an important role in the ionization of $H_{2}^{+} (1 π_{u})$ when the alignment angle is large. In addition, it can be found in Fig. 3 that the total populations of other excited states are even considerably larger than that of the 2σ_g state and also increase when the alignment angle raises from 30^° to 90^° for a specific laser intensity. In Fig. 4, we also present the time-dependent populations of 2σ_u and 3σ_g states with increasing alignment angle for different laser intensities. It is found that the time-varying probabilities of 2σ_u and 3σ_g decrease as a function of alignment angle for a specific laser intensity, because the coupling of 2σ_g state and other σ states becomes weak with increasing alignment angle. It should be noted that the population of 2σ_u state is negligible in Figs. 4(c) and (f), since the coupling of 2σ_g and 2σ_u states is zero for β=90°. In Figs. 4(c) and 4(f), the time-dependent probabilities of 3σ_g states are much larger than those of 2σ_u states due to the coupling between 1π_u and 3σ_g states.

Fig. 3 Time-varying probabilities of 1π_u and 2σ_g, and other excited states calculated by TDSE with increasing alignment angle β for different laser intensities. (a)–(c): 0.3I₀, (d)–(f): 0.5I₀. T = 2π/ω and I₀=10¹⁴W/cm².

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Fig. 4 Time-varying probabilities of 2σ_u and 3σ_g states obtained by TDSE as a function of alignment angles β for different laser intensities. (a)–(c): 0.3I₀, (d)–(f): 0.5I₀. T = 2π/ω and I₀=10¹⁴W/cm².

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To see the significance of the 2σ_g state in the alignment-dependent ionization of $H_{2}^{+}$ more clearly, we presents the ionization yields obtained by removing the contribution of the 2σ_g state artificially from the TDSE simulation in Fig. 5 for 0.3I₀ and 0.5I₀. Because there is no coupling between 2σ_g and 1π_u states at alignment angle β = 0^°, the alignment-dependent ionization yields are studied from β = 10^° to β = 90^° in Fig. 5(a). The ionization probabilities increase monotonously as a function of the alignment angle, which is qualitatively consistent with the result of the SFA calculation shown in Fig. 2(b) but different from those in Fig. 2(a). In Figs. 5(b) and 5(c), we also depict the time-dependent probabilities of other excited states as a function of the alignment angles for 0.3I₀ and 0.5I₀, respectively. It is shown that the probabilities in Fig. 5(b) and 5(c) are much smaller with respect to those shown in Fig. 3, which indicates that the other excited states are populated mainly via transition from the 2σ_g state but not directly from the 1π_u state. Therefore, the 2σ_g state has an important impact on the alignment dependence of the ionization for $H_{2}^{+}$ exposed to different laser fields.

Fig. 5 Simulations of $H_{2}^{+}$ via TDSE by excluding 2σ_g state in the basis expansion with the initial state of 1π_u. (a): Alignment-dependent ionization probability. (b) and (c): Time-varying probabilities of other excited states with increasing alignment angles for different laser intensities. T = 2π/ω and I₀=10¹⁴W/cm².

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To shed more physical insights into the ionization dynamics of $H_{2}^{+}$ in different laser fields, we show the alignment-dependent ionization probability obtained by the SFA theory with initial state 1π_u and 2σ_g state taken into account (see Eq. (7)) in Fig. 6. In Figs. 6(a) and 6(b), the alignment-dependent ionization probabilities of 1π_u and 2σ_g states are displayed, respectively. In Fig. 6(a), the maximal ionization yield from 1π_u appears at alignment angle β=50^° and 30^° for laser intensities of 0.3I₀ and 0.5I₀, respectively, which is different from that in Fig. 2(b). The distribution of the electron cloud of 1π_u is perpendicular to the molecular axis (see Fig. 1(d)) and the depletion of 1π_u state is not taken into account, therefore the ionization yield increases with ascending alignment angles as shown in Fig. 2(b). However, the coupling of 1π_u and 2σ_g states becomes stronger when the alignment angle increases, hence the population of the 1π_u state drops with increasing alignment angle (see Fig. 7). So the ionization yield does not increase monotonically like that in Fig. 2(b). In addition, since the coupling between the 1π_u and 2σ_g states becomes stronger with increasing laser intensity for a specific alignment angle, the population of the 1π_u state decreases faster for 0.5I₀ than that of 0.3I₀ (see Fig. 7), so the alignment angle corresponding to the peak ionization yield shifts to lower angle when the laser intensity increases. Moreover, the ionization yield of the 2σ_g also increases as a function of the alignment angle (see Fig. 6(b)), though its wavefunction distributes along the molecular axis as shown in Fig. 1(c). This can be attributed to that the coupling between the 1π_u and 2σ_g states becomes stronger when the alignment angle increases, leading to fast increasing population of the 2σ_g state with ascending alignment angle (see Fig. 7).

Fig. 6 Alignment-dependent ionization probabilities calculated by SFA with increasing laser intensities for the initial state 1π_u (2σ_g state is taken into account in Eq. (7)). (a): The ionization yields of 1π_u state. (b): The ionization probabilities of 2σ_g state. (c): Sum of the ionization probabilities of 1π_u and 2σ_g states. I₀=10¹⁴W/cm².

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Fig. 7 Time-dependent probabilities of 1π_u and 2σ_g states calculated by SFA with increasing alignment angles for different laser intensities (2σ_g state is taken into account in Eq. (7)). (a)–(c): 0.3I₀; (d)–(f): 0.5I₀. T = 2π/ω and I₀=10¹⁴W/cm².

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In Fig. 6(c), the total alignment-dependent ionization yield of the 1π_u and 2σ_g states are present, and it can be seen that the peak ionization probability shifts from β=60^° to 40^° with increasing laser intensity, which is in a reasonable agreement with the TDSE simulation in Fig. 2(a). It is worthwhile mentioning that the coupling of the 1π_u and 2σ_g states are much stronger than that obtained by the TDSE calculation (see Fig. 3). Since the other excited states are not taken into account in the modified SFA, the 2σ_g state can be considered to play the role of all the excited states (comparing Fig. 7 with Fig. 3), which gives rise to the qualitative agreement on the positions of the peaks in the alignment-dependent ionization probabilities calculated by the TDSE and modified SFA theory. However, it should be noted that the ionization probability calculated by the SFA at angle larger than 60^° is noticeably higher than that of the TDSE calculation. This discrepancy can be attributed to that, since only the 2σ_g excited state is included, the population of the 1π_u state will be retrieved after two optical cycles via Rabi oscillation between these two states in the SFA calculation (e. g., see Figs. 7(b) and (c)) while in the TDSE with all the excited states considered, large portion of the population of the electron will be pumped to the other excited state but not transfers back to the 1π_u state after it is pumped to the 2σ_g state (e. g., see Figs. 3(b) and (c)). Consequently, the ionization channel of 1π_u → 2σ_g → other excited states (such as 2σ_u and 3σ_g states in Fig. 4)→ionization gives rise to the decrease of ionization after β = 50^° and β = 40^° for 0.3I₀ and 0.5I₀ in Fig. 2(a), respectively.

4. Conclusions

In summary, we investigate the angular-dependent ionization of H⁺ ₂ subjected to few-cycle laser fields using a three-dimensional TDSE based on single-center method, B-splines in spherical coordinates and Crank-Nicolson propagation method. It is found that the peak ionization probability occurs at alignment angles of β=50^° and 40^° for intensities of 0.3I₀ and 0.5I₀ (I₀=10¹⁴W/cm²) for the initial state 1π_u, respectively. However, the ionization yield increases monotonically as a function of the alignment angle if the 2σ_g state is excluded in the basis expansion, which is qualitatively in agreement with the calculation of SFA for the initial state 1π_u without considering the contribution of the 2σ_g state. When the 2σ_g state is taken into account in the modified SFA, it gives peak ionization probability at β=60^° and 40^° for intensities of 0.3I₀ and 0.5I₀, in qualitative agreement with the complete TDSE calculation. Our analysis indicates that a part of electron is directly released from the 1π_u orbital by the laser field, and another portion of the electron firstly transits from 1π_u to the 2σ_g state via single-photon transition and transits to the other excited states thereafter, then the electron is ionized by the laser field from the excited states. The maximal ionization yields at alignment angles β=50^° and 40^° for 0.3I₀ and 0.5I₀ can be attributed to the sum of contributions from different ionization channels. Our work demonstrates that the excited states have an important impact on molecular alignment-dependent ionization in the laser field, which may be related to the peculiar angular dependence of ionization dynamics for CO₂ and OCS molecules in the strong laser pulses [19, 20].

Funding

National Key Research and Development program of China (2016YFA0401100); National Natural Science Foundation of China (NSFC) (11774361, 11804405, 11425414, 11504215); Fundamental Research Fund of Sun Yat-Sen University (20187100031610008).

Acknowledgments

This work was supported by the National Key R&D program of China (No. 2016YFA0401100) and the NSFC of China (Grants No. 11774361, No. 11334009, No. 11425414, No. 11374197, No. 11804405, and No. 11504215), and the Fundamental Research Fund of SUN YAT-SEN UNIVERSITY (No. 20187100031610008).

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state	Present	[26]
1σ_g	−1.10261	−1.1026342145
1σ_u	−0.667532	−0.66753439220
2σ_g	−0.360862	−0.36086487534
2σ_u	−0.255412	−0.25541316509
3σ_g	−0.235776	−0.23577762883
1π_u	−0.428766	−0.42877181990
1π_g	−0.226699	−0.22669962664
2π_u	−0.200862	−0.20086482991

Role of excited states in molecular alignment-dependent ionization

Abstract

1. Introduction

2. Theoretical methods

2.1. I. Time-dependent Schrödinger equation

2.2. II. Modified strong-field approximation

3. Results and discussion

4. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (7)

Tables (1)

Equations (11)

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