Theoretical calculations of the tunneling ionization rate of CF₄ were carried out by WFAT (Tolstikhin et al., 2011). The tunneling ionization rate is expressed as (Madsen et al., 2012) Γ β , γ = | G 00 β , γ | 2 W 00 F . (7) The structure factor G ₀₀(β, γ) describes the dependence on the molecular orientation relative to the laser electric field F defined by the Euler angles (α, β, γ) (Zare, 1988). The field factor W ₀₀(F) is given as W 00 F = ϰ 2 4 ϰ 2 F 2 / ϰ − 1 ⁡ exp − 2 ϰ 3 3 F , (8) which defines the dependence on the field strength F. Here, ϰ = − 2 E 0 , with E ₀ being the energy of the molecular orbital from which the electron is ionized, and the subscript 00 refers to the dominant ionization channel (Tolstikhin et al., 2011).

The HOMO (1t ₁) and HOMO-1 (4t ₂) of CF₄ are both triply degenerate (Figure 3). The Stark interaction with the ionizing field removes the degeneracy. Tunneling ionization occurs from eigenorbitals of the operator –( μ ⋅ F ) within each degenerate subspace, where μ is the electric dipole moment of the considered orbital (Kraus et al., 2015). We denote these eigenorbitals as ϕ _A, ϕ _B, and ϕ _C. The three eigenorbitals are the particular linear combinations of the three degenerate HOMOs shown in Figure 3, which diagonalize the Stark term ( μ ⋅ F ). The structure factors G ₀₀(β, γ) incorporating the effect of the dipole for the eigenorbitals are calculated using the integral representation of the WFAT (Dnestryan and Tolstikhin, 2016; Madsen et al., 2017; Dnestryan et al., 2018) implemented by means of the GAMESS package with a polarization consistent basis set at the pc-4 level (Jensen, 2001).

Figures 6A–C show the squared norms of the structure factors |G ₀₀(β, γ)|² of the three eigenorbitals, ϕ _A, ϕ _B, and ϕ _C, in the subspace of HOMO (E ₀ = –18.66 eV), where the orbitals are labeled with A, B, and C in the ascending order of the dipole, μ _A < μ _B < μ _C. The orbital energy in the field to the first order is given as E 0 , i F = E 0 − μ i ⋅ F , (9) where i = A,B,C. Figure 7 shows the energy of eigenorbitals calculated using Eq. 9 at four different molecular orientations with respect to F. The structure factors for HOMO in Figures 6A–C show that the largest contribution to the tunneling ionization comes from eigenorbital ϕ _B because the field factor W ₀₀(F) is common for ϕ _A, ϕ _B, and ϕ _C (see Eq. 7). Each orbital has nodes along the C-F axis, which appear as the minima in the respective structure factors. The nodes remain visible in the sum of |G ₀₀(β, γ)|² in Figure 6D.

The squared norms of the structure factors |G ₀₀(β, γ)|² of HOMO-1 (E ₀ = –19.44 eV) are shown in Figures 6E–H. The eigenorbital ϕ _A having the highest energy among the three orbitals has the largest contributions to the sum in Figure 6H. Figure 6H shows that the tunneling ionization is enhanced by the electric field pointing from F to C when the three degenerated eigenorbitals are equally populated. Because eigenenergy E _0,A of ϕ _A at (β, γ) = (124°, 314°) is slightly smaller than that at (β, γ) = (54°,134°), the large structure factors for the former orientation indicate that the shape of the molecular orbital is more important in determining the tunneling rate in the present case than the increase in the effective ionization potential by the Stark interaction with the dipole moment.

If the breaking of each of the four C-F bonds after ionization occurs with an equal probability, the angular distribution of the F fragment in the laboratory frame can be expressed as follows (Zare, 1988): P θ s , ϕ s = ∫ 0 2 π d α ∫ 0 π ⁡ sin ⁡ β d β ∫ 0 2 π d γ P mol α , β , γ f θ m , ϕ m , (10) where (θ _s , ϕ _s ) and (θ _m , ϕ _m ) are the spherical angles with respect to the laboratory and molecular frame, respectively, and f(θ _m , ϕ _m ) is the angular distribution of the fragment ion in the molecular frame. The orientation distribution of the molecular ion formed in the ω-2ω laser fields in the laboratory frame may be expressed as P mol α , β , γ = 1 8 π 2 1 − exp − ∫ − ∞ + ∞ Γ s α , β , γ , F t d t , (11) where Γ _s (α, β, γ, F(t)) represents the tunneling rate in the ω-2ω laser field F(t) of Eq. 2 for molecular orientation defined by the Euler angles (α, β, γ) relative to the Z-axis of the laboratory frame (see Figure 2). It can be expressed by |G ₀₀(β, γ)|² and W ₀₀(F) as follows: Γ s α , β , γ , F t = | G 00 β , γ | 2 W 00 | F t | F t ≥ 0 | G 00 π − β , γ + π | 2 W 00 | F t | F t < 0 . (12) When the ionization probability is sufficiently smaller than unity, Eq. 11 reduces to P mol α , β , γ = 1 8 π 2 ∫ − ∞ + ∞ Γ s α , β , γ , F t d t . (13) The angular distribution P _mol(α, β, γ) can be expanded by the rotation matrices D q ′ q k ( R ) as follows: P mol α , β , γ = 1 8 π 2 ∑ k , q , q ′ a q ′ q k D q ′ q k ∗ α , β , γ . (14) Here, the coefficients a q ′ q k are given as follows: a q ′ q k = 2 k + 1 ∫ P mol α , β , γ D q ′ q k R d Ω . (15) The angular distribution of the fragment ion can be expressed using the spherical harmonics Y _jm (θ _m , ϕ _m ): f θ m , ϕ m = ∑ j , m b j m Y j m θ m , ϕ m , (16) b j m = ∫ 0 2 π d ϕ m ∫ 0 π ⁡ sin θ m d θ m Y j m ∗ θ m , ϕ m f θ m , ϕ m . (17) Thus, we have P θ s , ϕ s = ∑ k , q , q ′ a q ′ q k b k q 2 k + 1 Y k q ′ θ s , ϕ s . (18) Under the axial recoil approximation, the angle distribution f(θ _m , ϕ _m ) may be expressed as follows: f θ m , ϕ m = 1 sin θ m δ θ m − θ m 0 δ ϕ m − ϕ m 0 , (19) with ( θ m 0 , ϕ m 0 ) = (54.7°, 45°) for CF₄ in T _d symmetry. By substituting to Eq. 17, we have b j m = Y j m ∗ θ m 0 , ϕ m 0 = − 1 m Y j − m θ m 0 , ϕ m 0 , (20) from which we obtain an expression for the fragment angular distribution as follows: P θ s , ϕ s = P θ s = 1 4 π ∑ k c k P k cos θ s , (21) c k = 1 2 k + 1 ∑ q a 0 q k Y k q ∗ θ m 0 , ϕ m 0 . (22)

Figure 8 shows the fragment angular distributions obtained for the relative phase ϕ = 0 of the ω - 2ω pulse (I _ω+2ω = 1.4 × 10¹⁴ W/cm² and I _2ω /I _ω = 0.23). The calculated fragment yields for HOMO-1 is larger than that of HOMO under the present experimental conditions (F _ω = 0.057 a.u. and F _2ω = 0.027 a.u.).

This is attributed to the large structure factor |G ₀₀|² for HOMO-1 (Figure 6H), which is about 10 times larger than |G ₀₀|² for HOMO (Figure 6D), because of the small difference between the ionization potentials of these orbitals (∼1 eV) giving rise to the relatively small field factor ratio of W ₀₀(1t ₁)/W ₀₀(4t ₂) ∼ 3. The angular distribution calculated for each HOMO exhibits characteristic structures associated with the nodes of the molecular orbitals. The total fragment distribution carries the nodal pattern with a larger ionization probability on the larger amplitude side of the ω-2ω laser fields. In contrast, the angular distribution of HOMO-1 is more directional along the laser polarization direction, consistent with the fragment ion image and the KER spectra in Figures 5A,B, where the ionization from HOMO-1 contributes more to the parallel component than to the perpendicular one.

The yields of the F fragment in a finite acceptance angle θ ₀ around 0° and 180° can be expressed as follows: Y + θ 0 ϕ = 2 π ∫ 0 θ 0 P θ s sin θ s d θ s , (23) Y − θ 0 ϕ = 2 π ∫ π − θ 0 π P θ s sin θ s d θ s . (24) The asymmetry parameters defined by Eq. 5 are calculated using Eqs 23, 24, where θ ₀ = 45° compared with the experimental results. The asymmetry parameter A _F(ϕ) thus obtained for HOMO shows a clear dependence on the relative phase ϕ between the ω and 2ω laser fields. The asymmetry parameter for HOMO (Figure 9A) is positive at ϕ = 0, showing that tunneling ionization is more efficient when the larger amplitude side of the laser fields points from C to F. In contrast, the parameter for HOMO-1 exhibits the opposite dependence with negative values at ϕ = 0. The difference originates essentially from the shape of the eigenorbitals dominating the tunneling ionization of the respective MOs.

Figure 9B plots the experimental asymmetry parameter A _F(ϕ) for the counterpart fragment F produced by the dissociative ionization (Eq. 3), which is obtained from the asymmetry parameter for C F 3 + by A F ( ϕ ) = − A CF 3 + ( ϕ ) . It is compared with the corresponding asymmetry parameter calculated with the contributions from the two orbitals, where the angular distribution is given as P(θ _s ) = P _HOMO(θ _s ) + P _HOMO-1(θ _s ). The obtained amplitude of A ₀ = 0.1 is slightly larger than the experimental results. The small experimental amplitude might be attributed to the contribution from HOMO-2 (1e), located ∼1.4 eV below the HOMO-1. The B²E state of C F 4 + produced by the tunneling ionization from 1e has a lifetime of 10^–10–10^–12 s (Maier and Thommen, 1980). This lifetime is longer than or comparable with the molecular rotational periods and could form an isotropic fragment distribution, which reduces the asymmetry of the fragmentation. Figure 9B plots the asymmetry parameter A _F(ϕ) obtained at a higher field intensity 3.0 × 10¹⁴ W/cm² with a similar intensity ratio of I _2ω /I _ω = 0.25. The increase in the field intensity resulted in a small amplitude A ₀ ∼ 0.04, while the amplitude of the calculated results remained essentially the same. Because the relative contribution from the B²E state is expected to increase by an increase in the field intensity, the experimental results support the involvement of the B²E state in the dissociative ionization.

Interestingly, the calculated asymmetry parameter in Figure 9B has an opposite phase dependence to the experimental results, showing that the dissociative tunneling ionization of CF₄ in the ω-2ω laser fields cannot be explained by the angular distribution of the tunneling ionization from the HOMO and HOMO-1 alone, although the F (or C F 3 + ) fragments are promptly ejected on the repulsive potentials of the X²T₁ and A²T₂ states after the tunneling ionization (Figure 4). The present experimental results show a marked contrast to those obtained by recent studies on the dissociative ionization of CF₄ in circularly polarized laser fields (35 fs, 0.8 × 10¹⁴ W/cm², 1,035 nm) (Fujise et al., 2022). The recoil-frame photoelectron angular distribution (RFPAD) showed that the dissociative tunneling ionization occurs more efficiently when the electric field points from F to C than the opposite, which is consistent with the prediction by WFAT for the tunneling ionization (see also Figure 8).

Previous studies on spatially oriented OCS showed that the tunneling ionization yields exhibit different angular dependence in linearly polarized and circularly polarized laser fields (Holmegaard et al., 2010; Hansen et al., 2012) as in the present case, where the tunneling ionization is enhanced at different directions of the applied electric fields in the molecular frame. For circularly polarized fields, a significant enhancement of tunneling ionization was observed when the electric fields were applied from C to S along the molecular axis, while the linearly polarized fields favor the tunneling ionization from the direction perpendicular to the axis. The discrepancy was discussed in terms of electron rescattering and the involvement of electronic excitation (Hansen et al., 2012), as well as orbital modification (Murray et al., 2010) and multielectron effects (Majety and Scrinzi, 2015) in the ionization process. These effects can, in principle, be involved in the present case of CF₄ to explain the deviation between the experimental and theoretical results in Figure 9B. Furthermore, Figure 7 suggests that the energy shifts of eigenorbitals formed by the Stark interaction becomes large enough to induce mixing between HOMO and HOMO-1, for example, at a field intensity F ≥ 0.06 a.u. in the molecular orientation in Figure 7B. This would result in additional polarization (field-induced deformation) of the ionizing orbitals, which affects the ionization rate (Matsui et al., 2021) but is not considered in the calculation of the structure factors in Figure 6.

Because the directional ejection of the fragments involves both ionization and fragmentation, post-ionization interaction with the laser fields (Endo et al., 2019; Endo et al., 2022) is another important factor to consider. The post-ionization interaction in ω-2ω laser fields has been extensively studied with H 2 + (Ray et al., 2009; Wanie et al., 2015). The dissociative ionization shows a clear dependence on the relative phase ϕ. The H⁺ ejection direction is determined by the quantum interference between the pathways associated with excitation and deexcitation between the 1sσ _g and 2pσ _u states of H 2 + by absorption or emission of ω and 2ω photons. This results in the spatial asymmetry of H⁺ ejection dependent on both phase ϕ and KER. The quantum interference effect can also manifest itself in circularly polarized laser fields when the tunneling electron is detected in coincidence with H⁺ (Wu et al., 2013). It appears as the distortion of the molecular-frame photoelectron angular distribution (MFPAD). As for CF₄, the RFPADs recorded for the dissociative ionization in Eq. 3 in circularly polarized fields exhibited clear dependences on both the helicity of circularly polarized laser fields and the KER ¹. The observed results are interpreted in terms of the laser-induced coupling between the electronic states, depending on the phase of the rotating electric fields in the molecular frame. The coupling between the ground state X²T₁ and the excited state A²T₂ through non-adiabatic population transfer in the alternating laser electric fields was suggested as a possible dynamics contributing to the helicity dependence. In the present case of the two-color laser fields consisting of 800 and 400 nm for ω and 2ω, the energy differences between the states and the A²T₂ and B²E states are close to the photon energy of hν = 1.5 and 3.1 eV of the present ω and 2ω fields (see Figure 4), which further facilitates such coupling to modify the asymmetry of the fragmentation through quantum interferences.