Density-functional calculation for the tunnel ionization rate of ...

PHYSICAL REVIEW A 75, 062507 共2007兲

Density-functional calculation for the tunnel ionization rate of hydrocarbon molecules 1

T. Otobe1 and K. Yabana2

Kansai Photon Science Institute, JAEA, Kyoto 619–0215, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan 共Received 15 March 2007; published 15 June 2007兲

2

The tunnel ionization rate under a static intense electric field is calculated for some hydrocarbon molecules, acetylene, ethylene, and benzene, employing the ab initio density-functional theory. The ionization rate is found to reflect the properties of the highest occupied molecular orbital, in agreement with the description of the molecular Ammosov-Delone-Krainov theory. Many-electron effects beyond a single-electron approximation are discussed. The screening effect on the ionization rate is quantitatively evaluated. DOI: 10.1103/PhysRevA.75.062507

PACS number共s兲: 31.15.Ew, 33.80.Eh, 31.15.Fx

I. INTRODUCTION

Atoms and molecules under intense and ultrashort-pulse laser fields has been attracting substantial interest 关1兴. Among the various mechanisms induced by intense laser field, it is of fundamental importance to understand the ionization mechanism since most measurements and manipulations of molecules are achieved for ionized systems. Ionization itself is an interesting physical process showing a crossover of the mechanism from perturbative multiphoton absorption to tunnel and field ionization. Ionization is also a process which is followed by various interesting secondary processes induced by the rescattering of electrons. The ionization of rare gas atoms has now been well understood and described by analytical approaches such as the Ammosov-Delone-Krainov 共ADK兲 theory 关2兴 and the Keldysh-Faisal-Reiss 共KFR兲 theory 关3–5兴. These theories have been successfully extended to the ionization of diatomic molecules 关6–9兴. It has been realized that the orbital symmetry of the highest occupied molecular orbital 共HOMO兲 plays an important role in the ionization of diatomic molecules. For example, the ionization rate of the O2 molecule is substantially suppressed compared with other atoms and molecules 关10,11兴. This suppression is attributed to the ␲* character of the HOMO of O2 molecules. Recent measurement clearly indicate that the orbital symmetry is manifest in the ionization 关12兴. These analytical approaches for the molecular ionization, extensions of ADK and KFR theories and their variants, commonly assume that each electron moves independently in the ionization 共single-electron approximation兲. The ionization process is also treated by a certain perturbation theory employing the orbital wave function in the absence of the external electric field. Experimental and theoretical studies are now extending to larger molecules. For organic molecules, systematic suppression of the ionization rate has been reported 关13,14兴. Recent measurement for organic amines 关15兴, however, suggests that the ADK model still gives a reasonable description for the ionization rate of this large molecule. For fullerene molecules, ionized molecules up to C6012+ has been reported 关16兴, where a strong suppression of the saturation irradiance was observed. For the ionization of large molecules, the significance of many-electron effects beyond the single-electron approxima1050-2947/2007/75共6兲/062507共8兲

tion has been discussed. In metal clusters, the suppressed ionization has been attributed to the screening effect 关17兴. Nonadiabatic multielectron dynamics have also been discussed 关18兴. For the ionization of C60, theories incorporating a strong screening effect have been employed to explain the suppressed ionization 关16,19兴. The dipole electric polarization is expected to induce the giant quadrupole vibration of the fullerene cage 关16,20兴. On the other hand, there is an argument that the destructive interference among atomic orbitals alone can explain the suppression 关21兴. We have recently developed a first-principle computational approach based on the density-functional theory to describe the ionization of molecules by the intense laser field 关22兴. In this approach, we treat the tunnel ionization process in atoms and molecules under a static intense electric field. Our approach does not include any adjustable parameter, and has been successful to describe the ionization of diatomic molecules including the suppression of O2. Our approach also does not employ perturbation theory and takes into account many-electron effects beyond the single-electron approximation such as the static screening effect. Therefore, we expect that our formalism has the potential to provide an unambiguous picture for the ionization mechanism of molecules under intense field. On the other hand, we ignore the time variation of the electric field. We also ignore the effect of the nuclear motion during the irradiation of the pulse laser. Therefore, our approach will be useful for ionization under ultrashort-pulse laser 共a few tens of femtosecond and shorter兲 and under intense electric field where the tuunel ionization is dominated. The effects of finite frequency and nuclear motion can be incorporated if one solves the electronic dynamics in real time. The analyses in this direction is now under progress and will be reported in the future. In this paper, we analyze tunnel ionization rate of several hydrocarbon molecules, acetylene, ethylene, and benzene under the static intense electric field. We will try to elucidate changes in the ionization mechanism as the molecular size increases. The organization of the present article is as follows. In Sec. II, we present our formalism to calculate the ionization rate in the Kohn-Sham framework. In Sec. III, we show our results for the acetylene, ethylene, and benzene molecules. In Sec. IV, a summary will be presented.

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PHYSICAL REVIEW A 75, 062507 共2007兲

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II. FORMALISM

ជ · rជ − iW共r兲其␾ 共rជ兲 = ⑀ ␾ 共rជ兲, 兵h关n共rជ兲兴 + eE i i i

共1兲

where h关n共rជ兲兴 is the self-consistent Kohn-Sham Hamiltonian ជ · rជ is the external dipole calculated with the wall potential, eE potential, and −iW共r兲 is the absorbing potential. The orbital wave function, ␾i共rជ兲, is set to zero outside the spatial region with an absorbing potential. The calculated ionization rate in the above prescription includes effects beyond the single-electron approximation. In particular, the screening effect which is a key concept bridging the optical property between small and extended systems is taken into consideration. If we replace the electron density, n共rជ兲, with the ground state density, n0共rជ兲, in the above calculation, we obtain the ionization rate ignoring the screening effect. Later we will show a comparison of the ionization rate with or without the screening effect. The choice of the exchange-correlation potential is a crucial issue for quantitative description of the ionization rate. It is known that the local density approximation 共LDA兲 overestimates the ionization rate substantially, due to the wrong asymptotic behavior of the potential as well as the fact that the HOMO energy is much smaller than the ionization potential. In our previous analysis 关22兴, we employed the selfinteraction correction in the Krieger, Li, and Iafrate scheme 关26兴, which overcomes the difficulty to some extent. We will employ the same one in the present study. To express orbital wave functions, a real-space grid method in the three-dimensional Cartesian coordinate is employed. For each Cartesian coordinate, the following coordinate transformation is achieved:

0 .0 0 6

I o n iz a t io n r a t e [ 1 / f s ]

We employ the formalism and the computational method presented in 关22兴, which we briefly recapitulate here. We employ the density-functional theory 共DFT兲 to calculate the ionization rate. An argument to justify the use of DFT to calculate the ionization rate is presented in 关22兴 based on the time-dependent density-functional theory, which is an extension of DFT for electron dynamics 关23,24兴. The calculation of the ionization rate is achieved in the following two steps. First we solve the Kohn-Sham equation for a molecule under an external dipole field. At this stage of the calculation, we also place a wall potential outside the molecule so that the ionization by the dipole field is prevented. This step provides us with the self-consistent potential. We then utilize this self-consistent potential to calculate the Gamow state solution, which is a static solution of the Kohn-Sham equation with the outgoing boundary condition. The orbital energies of the Gamow states are complex. The ionization rate, W, can be obtained from the imaginary part of the orbital energies, ⑀i, through W = −共2 / ប兲兺i Im ⑀i. In practical calculations, we introduce an absorbing potential outside the barrier which mimics the outgoing boundary condition. We thus solve the following equation,

0 .0 0 5 0 .0 0 4 0 .0 0 3 0 .0 0 2 0 .0 0 1 0

2 0

4 0

6 0

8 0

A n g le [ d e g r e e ]

FIG. 1. Ionization rate of acetylene as a function of the angle between the molecular axis and the electric field. The laser irradiance is set at 5 ⫻ 1013 W / cm2.

x=

kt

冋

t 1 + 共k − 1兲 a sinh共t/a兲

册

共2兲

n,

with a = 10 Å, k = 10, and n = 2. The coordinate, t, is then discretized with uniform grid spacing. In the calculations below, a grid spacing of ⌬t = 0.2 Å is used. We employ all the grid points inside a sphere of radius 30 Å and have a vanishing boundary condition for ␾i outside the spherical box. This procedure produces grid points which are fine enough to describe molecular orbitals and which cover a spatial region wide enough to describe the ionization rate accurately. As for the absorbing potential, we employed a potential with a linear radial dependence 关22,25兴, − iW共r兲 =

冦

共0 ⬍ r ⬍ R兲,

0 − iW0

r−R ⌬R

共R ⬍ r ⬍ R + ⌬R兲,

冧

共3兲

where R is set outside the barrier region. The height W0共 ⬎0兲 and the thickness ⌬R are determined from the conditions that the electrons coming into the region r ⬎ R should be absorbed as completely as possible. We employ R = 15 Å and ⌬R = 15 Å. III. RESULTS AND DISSCUSION

We report calculations of the ionization rate for three hydrocarbon molecules: acetylene, ethylene, and benzene. They all have ␲ orbitals as the HOMO. They differ by their shapes, axial and planar ones, and the number of valence electrons. We expect to elucidate characteristic features of the ionization mechanisms by comparing these molecules. A. Acetylene

The acetylene molecule has four ␲ electrons in the degenerate HOMO. In our calculation, the HOMO energy of acetylene is 11.5 eV, which is in good agreement to the measured ionization potential, 11.4 eV. We first show the ionization rate as a function of the angle between the molecular axis and the direction of the electric field. Figure 1 shows the calculated result. The figure clearly shows that the ionization rate has a maximum when the electric field is applied per-

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DENSITY-FUNCTIONAL CALCULATION FOR THE TUNNEL…

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FIG. 2. The ionization rate of acetylene is decomposed into contributions of Kohn-Sham orbitals. The height represents the ionization rate while the horizontal position indicates the orbital energy which is shifted by Stark effect. The open circles indicate that they are ␴ orbitals, while the open squares indicate the ␲ orbitals. The laser irradiance is set at 5 ⫻ 1013 W / cm2, and the direction of the electric field is perpendicular to the molecular axis.

pendicular to the molecular axis. This can be understood from the ␲-orbital character of the HOMO. The ␲ orbital has a nodal line along the molecular axis, and the electron distribution is suppressed along the molecular axis. In other words, there is a centrifugal barrier potential along the molecular axis because the ␲ electron has azimuthal quantum number, l = 1, and rotates around the molecular axis. This potential prevents ionization parallel to the molecular axis. The angle dependence of the ionization rate has been discussed from the measurement of the fragmentation process, + + C2H2+ 2 → CH + CH , and has been analyzed with the molecular ADK theory in Ref. 关27兴. The angle-dependent ionization rate given in 关27兴 shows a dip at 0 degrees, in coincidence with our result. Our result shows an almost flat rate in the angle region of 45 to 90 degrees, while the result of Ref. 关27兴 shows a maximum around 40 degrees and shallow dip at 90 degrees. We next show the decomposition of the ionization rate into orbitals in Fig. 2. The figure clearly shows that only the HOMO contributes to the ionization. There are two degenerate HOMO’s in acetylene with ␲ character. By applying the electric field perpendicular to the molecular axis, the degeneracy is lifted: one ␲ orbital which extends in the direction of the electric field shows a larger ionization rate than the other which extends to the direction perpendicular to the electric field. The ionization rate as a function of the laser irradiance is shown in Fig. 3. The open circles indicated as “include screening” are the calculated value of the ionization rate where the direction of the electric field is perpendicular to the molecular axis. For comparison, we indicate the result of ADK theory by the dotted line. We adopt a standard ADK formula for atoms, assuming a p orbital employing the measured ionization potential of acetylene. The figure shows that the absolute values of the ionization rate as well as the laser irradiance dependence are very close to each other between our first-principle calculation and the ADK theory. We note that the calculated rate of the ionization will decrease slightly after averaging over the molecular orientation. We previously reported that the present framework explains the suppression of the ionization in the O2 molecule in

FIG. 3. The ionization rate of the acetylene molecule is shown by open circles connected with a solid line as a function of the laser irradiance. The direction of the electric field is set perpendicular to the molecular axis. The filled circles connected with a dashed line represent the ionization rate when we ignore the screening effect. The ionization rate by ADK theory is shown by the dotted curve.

comparison with ADK theory. Let us consider why the acetylene molecule does not show a suppressed ionization rate, although both molecules have HOMO of ␲ character. There are two differences between O2 and acetylene. First, the ionization of the O2 molecule is suppressed in two directions, parallel and perpendicular to the molecular axis reflecting the ␲* character of the HOMO, while in acetylene only one direction, parallel to the molecular axis, is suppressed. Second, the HOMO’s of O2 is occupied by two electrons, while the HOMO of acetylene is occupied by four electrons. This fact brings in a factor of 2 difference in the ionization rate. Finally, we discuss the screening effect in this molecule. As explained in Sec. II, our calculation includes the screening effect, one of the important many-body effects beyond the single-electron approximation, in constructing the selfconsistent potential included in the Kohn-Sham Hamiltonian, h关n共rជ兲兴. We can examine the significance of the screening effect by employing the self-consistent potential in the absence of the external electric field, namely, the self-consistent potential in the ground state. In Fig. 3, the ionization rate without the screening effect is indicated by filled circles. The difference between filled and open circles shows the effect of the screening on the ionization rate. In this small molecule acetylene, the screening effect is found to be rather small, suppressing the rate by less than a factor of 2. B. Ethylene

Ethylene is a planar molecule with two electrons in the HOMO of ␲ character. The calculated orbital energy of the HOMO is 10.7 eV, which is close to the measured ionization potential, 10.5 eV. Figure 4 shows the angular dependence of the tunnel ionization rate of the ethylene. The angle is measured from the molecular axis. The magnitude of the electric field is set at the laser irradiance of 5 ⫻ 1013 W / cm2. For this planar molecule, we calculate the angle dependences in the two planes. The filled circles show the angle dependence of the ionization rates when the electric field lies in the molecular plane, and the open circles show the angle dependence when the electric field lies in the plane which includes the molecular axis and is perpendicular to the molecular plane.

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16 14 12 Binding energy [eV]

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-1

FIG. 4. Ionization rate of the ethylene molecule as a function of the angle between the molecular axis and the direction of the electric field. The filled circles show the ionization rate when the direction of the electric field is within the molecular plane, while the open circles show the ionization rate when the electric field is in a plane which includes the molecular axis and is perpendicular to the molecular plane. The magnitude of the electric field corresponds to the laser irradiance of 5 ⫻ 1013 W / cm2.

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FIG. 5. Ionization rate from each Kohn-Sham orbital in ethylene under the electric field corresponding to the irradiance 5 ⫻ 1013 W / cm2. The height represents the ionization rate while the horizontal position indicates the orbital energy which is shifted by Stark effect. In the upper 共lower兲 panel, the direction of the electric field is perpendicular 共parallel兲 to the molecular plane where the ionization rate is maximum 共minimum兲. The square indicates the ␲ orbital, while circles indicate ␴ orbitals.

9.24 eV. Because the computational cost is substantially large for this molecule, we calculated the ionization rate for limited directions and laser irradiances. The ionization rate is calculated for seven directions of the electric field in the xz plane, from 0 to 90 degrees with a step of 15 degrees. 0 degrees is for the direction of the electric field in the molecular plane, while 90 degrees is for the electric field perpendicular to the molecular plane. Figure 8 shows the angle dependence of the ionization rate. The ionization rate shows a maximum around 30– 45 degrees. This reflects the HOMO character of the benzene molecule. The HOMO of benzene is the ␲ orbital and is twofold degenerate 10 Ionization rate [1/fs]

The figure clearly shows that the ionization rate has a maximum when the electric field is applied perpendicular to the molecular plane. The ionization is suppressed along the molecular axis, and in the direction in the molecular plane and perpendicular to the molecular axis. This angle dependence is again in agreement with an intuitive expectation from the property of the HOMO. The ionization rate is large in the direction where the HOMO extends spatially. This result is also consistent with the results of molecular strongfield approximation 关8,28兴 and the results of generalized Keldysh theory 关9兴. Figure 5 shows the contribution of each molecular orbital to the ionization rate when the electric field is applied perpendicular 共upper兲 and parallel 共lower兲 to the molecular plane. As seen from the upper panel of Fig. 5, the ionization rate is dominated by the HOMO which is the ␲ orbital located in the carbon atoms. This is similar to the case of acetylene. We have also found that the ionization in the direction within the molecular plane where the ionization of the ␲ orbital is suppressed, the HOMO-1 ␴ orbital makes a dominant contribution to the ionization. Figure 6 shows the ionization rate as a function of the irradiance of the laser field. The direction of the electric field is perpendicular to the molecular plane where the ionization rate is maximum. The open circles are our results including the screening effect, while the filled circles neglect the screening effect. The calculation in atomic ADK theory assuming a p orbital is also shown. As in the case of the acetylene molecule, the calculated ionization rate is very close to the result of the ADK theory. The screening effect suppresses the ionization rate by about a factor of 2 for this molecule.

0

W it h s c r e e n in g W it h o u t s c r e e n in g A D K th e o ry

-1

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2x10

C. Benzene

Benzene is a medium size molecule with 6␲ and 24␴ electrons in the valence orbitals. We define the direction of the benzene molecule as shown in Fig. 7. The HOMO energy without the laser field is calculated to be 9.0 eV, which is slightly smaller than the measured ionization potential,

13

13

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3x10 4x10 5x10 2 Laser irradiance [W/cm ]

6x10

13

FIG. 6. Ionization rate of ethylene as a function of laser irradiance. The direction of the electric field is perpendicular to the molecular plane. The open circles indicate the ionization rate including the screening effect, while the filled circles neglect the screening effect. The dotted curve indicates the result of ADK theory.

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FIG. 7. Definition of the directions for the benzene molecule.

with two nodal planes, one in the molecular plane and the other perpendicular to the molecular plane. The ionization is suppressed when the electric field is parallel to the nodal planes 共0 and 90 degrees in Fig. 8兲. This angle dependence is consistent with the results by Kjeldsen et al. 关28兴 with the molecular ADK theory, and is similar to the case of the O2 molecule with the ␲* HOMO. We show the contribution of each molecular orbital to the ionization rate in Fig. 9 for several directions of the electric field. The three figures indicated by 0, 45, and 90 degrees are for the cases shown in Fig. 8. The figure indicated with y axis corresponds to the case where the electric field is in the molecular plane and is perpendicular to the direction connecting two carbon atoms. The squares show the ionization rates of ␲ orbitals, while the circles show those of ␴ orbitals. The level density is rather high in this molecule, and several orbitals contribute to the ionization. At 45 degrees where the angle dependence of the ionization rate shows a maximum, the ␲ HOMO makes a dominant contribution and the contributions of other orbitals are an order of magnitude smaller than the HOMO. Since the total ionization rate is dominated by the contribution of this angle, the ionization rate averaged over molecular orientation should be dominated by the HOMO. However, at molecular orientations where the ionization rate of the HOMO becomes small 共the directions in the molecular plane and the direction perpen-

Ionization rate [1/fs]

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16 14 12 10 Binding energy [eV]

1 6

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H 6

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( Y a x is )

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8

FIG. 9. Ionization rate of each molecular orbital is shown for four directions of electric field. The three figures indicated by 0, 45, and 90 degrees are for the cases shown in Fig. 8 where the electric field is in the xz plane. The figure with y axis indicates the case where the electric field is in the molecular plane and is perpendicular to the direction connecting two carbon atoms.

0.1 8 7 6 5 4 3

2

0

15

30 45 60 Angle [degree]

75

90

FIG. 8. Angle dependence of the ionization rate of the benzene molecule. The direction of the electric field is set in the xz plane, which is perpendicular to the molecular plane and includes the line connecting two carbon atoms. The angle is measured between the direction of the electric field and the direction connecting two carbon atoms. The laser irradiance is set at 4 ⫻ 1013 W / cm2.

dicular to the molecular plane兲, the HOMO-1 and even deeper-bound orbitals contribute substantially to the ionization. Therefore, contributions other than the HOMO should be detectable through careful analysis of measurements in which we are controlling the relative orientation between the molecular axis and the laser polarization. In Fig. 10, the ionization rates of the HOMO 共doubly degenerate ␲ orbitals, filled circle兲, and that of the HOMO-1 共doubly degenerate ␴

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Carbon atom Hydrogen atom

3

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75

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FIG. 10. Angle dependences of the ionization rate of HOMO 共␲ orbital, filled circles兲 and HOMO-1 共␴ orbital, open circles兲 are plotted as a function of the angle between the molecular plane and the electric field.

orbitals, open circles兲 are shown for several angles in the xz plane. The ionization rate of the HOMO-1 shows a maximum when the electric field lies within the molecular plane and is comparable to the contribution of the HOMO at this angle. Kjeldsen et al. 关28兴 reported that the ionization is maximum around 45 degrees, in agreement with our result. They also report that the ionization rate has a minimum when the electric field is parallel to the molecular plane. However, our result shows that the ionization rate has a minimum when the electric field is perpendicular to the molecular plane. This difference is considered to originate from the contribution of ␴ orbitals, which is not considered in 关28兴. As seen in Fig. 9, many orbitals are doubly degenerate and the ionization rates of the degenerate orbitals are, in general, not equal. To obtain insight into the ionization mechanism, we show the two-dimensional electron distribution plot in Fig. 11 for the HOMO 共left panels兲 and the HOMO-1 共right panels兲. The electron density integrated over the z direction is plotted in the xy plane. The electric field is applied in the x direction 共0 degrees in Figs. 8 and 10兲. The upper 共lower兲 figures correspond to the orbitals with larger 共smaller兲 ionization rate. In the figure, the electric field is applied from the right to the left, and the ionization is sensitive to the electron density around the atom at the rightmost position. For ␲ orbitals 共left panels兲, the electron density around the carbon atom at the rightmost position is different between the two orbitals. The orbital of the upper panel, which shows the larger ionization rate, has more electrons around the carbon atom at the rightmost position. For ␴ orbitals 共right panels兲, the electron density in the hydrogen atom at the rightmost position is larger in the upper panel. Again, this explains the larger ionization rate of the orbital described in the upper panel. The electron density in Fig. 11 shows another interesting feature that the number of electrons around the hydrogen atom in the leftmost position is much larger than that in the rightmost position. This movement of the electrons works to substantially suppress the ionization rate. To see this electron movement by the electric field more clearly, we show the electron distribution in the direction of the electric field in Fig. 12, where the electron density is integrated over the other two directions perpendicular to the electric field. The orbitals of the HOMO 共␲兲 and the

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FIG. 11. 共Color兲 Electron densities integrated over the z direction are plotted in the xy plane for the ␲ HOMO and ␴ HOMO-1 of the benzene molecule. Both orbitals are doubly degenerate and the degeneracy is removed by the electric field. The upper panels show the electron densities of the orbitals with a larger ionization rate, while the lower panels show the electron densities of the orbitals with smaller ionization rate. The irradiance of the electric field is set at 4 ⫻ 1013 W / cm2.

HOMO-1 共␴兲 are shown for two cases, the electric field parallel to the molecular plane 共upper panels兲, and the angle of 45 degrees between the molecular plane and the electric field where the ionization rate has a maximum. The solid 共dashed兲 curves are the electron distribution of the orbital with the larger 共smaller兲 ionization rate. The figures clearly show that the electron density shifts towards the downstream direction of the electric field, opposite to the direction which screens the external field. In other words, the electrons in the HOMO and HOMO-1 orbitals contribute to enhance 共antiscreen兲 the external electric field. Thus the number of electrons around the potential barrier decreases and the ionization rate is suppressed. This behavior of the HOMO may be explained as follows: By applying the intense electric field, the self-consistent potential around the barrier region is lowered. Then, the electron motion becomes fast in this region, and the staying probability of electrons around this area becomes smaller. We have an experience that the movement of electrons in the HOMO and other orbitals around the HOMO towards the downstream side of the electric field occurs commonly for large molecules with a wide HOMO–lowest-unoccupiedmolecular-orbital 共LUMO兲 gap. This shift of electrons as well as the screening effect are not included in the singleelectron approximation and are important for a quantitative understanding of the ionization process in large molecules.

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FIG. 12. The electron density distribution along the direction of the electric field and integrated over two directions perpendicular to the electric field. The left two panels are for HOMO and the right two panels are for HOMO-1. The electric field is applied in the x direction which is in the molecular plane for the upper two panels, and 45 degrees from the molecular plane for the lower two panels. The laser irradiance is set at 4 ⫻ 1013 W / cm2.

Finally we mention the screening effect in benzene. We calculate the ionization rate without the screening effect at two angles, 45 and 90 degrees, and it is shown in Fig. 8. The screening effect in benzene is stronger than that in the other two molecules, and suppresses the ionization rate by about a factor of 3. Thus the significance of the screening effect gradually increases as the molecular size increases.

field. Our computational scheme is a first-principles method in the density-functional theory. The self-interaction correction is taken into account to obtain a reliable ionization rate. We confirm the significance of the character of the HOMO to the absolute value and the angle dependence of the ionization rate. Reflecting the ␲ orbital character of the HOMO, the ionization rate of C2H2 has a maximum when the electric field is perpendicular to the molecular axis. For C2H4, the ionization rate has a maximum when the electric field is perpendicular to the molecular plane. These results are consistent with previously reported results by semianalytical methods such as molecular ADK theory. In the benzene molecule, the ionization rate is still dominated by the HOMO. However, when the electric field is applied parallel to the molecular plane where the ionization from the HOMO is suppressed, the ionization from deeperbound ␴ orbitals becomes significant and comparable to that of the HOMO. We have found that the effects beyond the single-electron approximation become significant in medium size molecules. We observe two effects beyond the single-electron approximation. One is the dielectric screening effect. As is expected, it becomes more significant as the molecular size increases. The other is the antiscreening behavior of electrons in the HOMO and other orbitals around the HOMO. By applying the electric field, the electron distribution in these orbitals shows movement downward in the electric field. This movement decreases the electron density around the potential barrier and suppresses the ionization rate.

ACKNOWLEDGMENTS

We have investigated the tunnel ionization rate of some medium-size hydrocarbon molecules under a intense electric

The numerical calculations are performed on the supercomputers at the Kansai Photon Science Institute, Japan Atomic Energy Agency 共JAEA兲. This work is supported in part by NAREGI Nanoscience Project and by the Grand-inAid for Scientific Research 共Grants No. 18540366, No. 18036002, and No. 19019002兲, MEXT, Japan.

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