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TD-CI Simulation of the Strong-Field Ionization of Polyenes

Jason A. Sonk and H. Bernhard Schlegel*

Department of Chemistry, Wayne State University, Detroit, Michigan 48202, United States

ABSTRACT: Ionization of ethylene, butadiene, hexatriene, and octate-

traene by short, intense laser pulses was simulated using the time-dependent

single-excitation configuration-interaction (TD-CIS) method and Klamroth’s

heuristic model for ionization (J. Chem. Phys. 2009, 131, 114304). The

calculations used the 6-31G(d,p) basis set augmented with up to three sets

of diffuse sp functions on each heavy atom as well as the 6-311++G(2df,2pd)

basis set. The simulations employed a seven-cycle cosine pulse (ω = 0.06 au,

760 nm) with intensities up to 3.5 × 1014W cm−2(Emax= 0.10 au) directed

along the vector connecting the end carbons of the linear polyenes. TD-CIS

simulations for ionization were carried out as a function of the escape

distance parameter, the field strength, the number of states, and the basis set

size. With a distance parameter of 1 bohr, calculations with Klamroth’s

heuristic model reproduce the expected trend that the ionization rate

increases as the molecular length increases. While the ionization rates are too high at low intensities, the ratios of ionization rates

for ethylene, butadiene, hexatriene, and octatetraene are in good agreement with the ratios obtained from the ADK model. As

compared to earlier work on the optical response of polyenes to intense laser pulses, ionization using Klamroth’s model is less

sensitive to the number of diffuse functions in the basis set, and only a fraction of the total possible CIS states are needed to

model the strong field ionizations.

■INTRODUCTION

Strong field chemistry encompasses the study of interactions

between atoms/molecules and intense laser pulses. The

strength of the electric field of these laser pulses is comparable

to the electric field sampled by valence electrons. A variety of

effects can be observed as a result of these interactions (for

recent advances, see refs 1−3). Because the electric field of the

laser field approaches that of the field binding the valence

electrons, the interactions of the electrons with the field cannot

be treated perturbatively. Numerical simulations are needed to

model the nonlinear behavior of the electron density. In this

Article, we examine the behavior of a series of polyenes subject

to a short, intense 760 nm laser pulse. With increasing length

and degree of conjugation, the ionization rates of these systems

change considerably.4−8Time-dependent configuration inter-

action and the heuristic model of Klamroth and co-workers9are

used to examine the trends in ionization rates for ethylene,

butadiene, hexatriene, and octatetraene.

The electron dynamics of a few electron atoms and

diatomics, such as H2+and H2, have been successfully modeled

using highly accurate methods (see refs 10,11, and references

therein). These methods, however, cannot readily be used to

study the electron dynamics of larger, many electron systems.

Two approximate methods are available for many-electron

systems: (1) real time integration of the time-dependent

Hartree−Fock or density functional equations (rt-TD-HF and

rt-TD-DFT methods)12−16and (2) time-dependent config-

uration interaction (TD-CI). Previously, we have used time-

dependent Hartree−Fock and TD-CIS methods to simulate the

response of CO2, polyenes, and polyacenes and their cations to

short, intense laser pulses.17−23Klamroth, Saalfrank, and co-

workers9,24−33have used TD-CI to study dipole switching,

pulse shaping, ionization, dephasing, and dissipation. In the

present work, we choose to use the TD-CIS approach to

simulate the ionization of a series of polyenes.

Our earlier studies18on linear polyenes examined the

electronic excitation of conjugated molecules by short, intense

laser pulses. The amount of nonadiabatic excitation was found

to increase with the length of the polyene. In recent studies,23,34

we have looked at the number of excited states and the size of

the basis set needed in TD-CIS simulations to describe the

excited-state populations after the laser pulse. We found that a

large number of states (∼300−500) and a basis set augmented

with three sets of diffuse functions were needed to model the

response to the laser pulse. These studies did not examine

ionization. For few-electron systems, grid-based methods with

absorbing boundary conditions can be used to calculate

accurate ionization rates.35−37Mukamel38,39and co-workers

have simulated π electron dynamics in octatetraene with a

semiempirical Hamiltonian and have modeled ionization

saturation intensities in a multielectron system in a finite one-

dimensional box. For larger systems, Klamroth and co-workers9

have developed a heuristic approach to model ionization using

TD-CI and standard atom centered Gaussian basis sets. For

states above the ionization potential, the ionization rate is

assumed to be proportional to the speed of the excited electron

Received:

Revised:

Published: June 4, 2012

March 12, 2012

May 17, 2012

Article

pubs.acs.org/JPCA

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divided by a characteristic escape distance. This model is

appropriate in the high field case where above-threshold

ionization is dominant. Our goal is to see how well this model

applies to the ionization rates of a series of linear polyenes and

to examine the effect of the basis set size, number of states, and

escape distance parameter on the ionization rates.

■METHODS

The time-dependent Schrödinger equation (TDSE) in atomic

units is

Ψ

d

=

̂

Ψ

i

t

t

H tt

d ( )

( ) ( )

(1)

The wave function can be expanded in terms of the ground

state |φ0⟩ and excited states |φi⟩ of the time-independent, field-

free Hamiltonian.

∑

=

s

ψΨ= | ⟩

t C t

s

( ) ( )

n

s

0

(2)

For the full solution of the TDSE, the sum in eq 2 extends over

all bound states and the continuum. For practical applications,

the sum needs to be restricted to a suitable subset of states. In

the present work, we include only the ground state and the

singly excited states.

∑

s

i a

,

ψφ=

a s ( )

i

a

i

a

CIS

(3)

The amplitudes, ai

by diagonalizing the corresponding field-free Hamiltonian

matrix of the time-independent Schrodinger equation.

a(s), and excitation energies, ωs, are obtained

ψ ω ψ

s

ψ ψ

r

δ

̂| ⟩ =

s

0

| ⟩ ⟨ | ⟩ =

,

s

H

s

rs

(4)

Inserting eq 2 into eq 1 and multiplying from the left by ⟨φi|

reduces the time-dependent Schrödinger equation to a set of

coupled differential equations for the time-dependent coef-

ficients:

∑

s

=

i

C t

d ( )

d

t

H t C t

( ) ( )

rs

r

s

(5)

This can be integrated numerically using a unitary transform

approach:

+ Δ=− + ΔΔ

ttitttt

CHC

() exp[( /2) ] ( )

(6)

In the dipole approximation, the matrix elements of the field-

dependent Hamiltonian in eqs 5 and 6 can be expressed in

terms of the field-free energies, ωs, transition dipole moments,

Drs, and the electric field, e(t):

ψψ

ωδ

ψψ ψ ψ

r

r

= ⟨ | ̂ | ⟩ = ⟨ | ̂| ⟩ + ⟨ | ̂| ⟩·

+

D e

( )

s rs rs

=·

H t

rs

H tHt

t

e

( ) ( )( )

rsrss

0

(7)

Practical considerations limit the total number of states that

can be used. Increasing the number of states included until no

further change is seen in the simulation is one means of

determining whether the number of states is adequate.

Typical molecular electronic structure calculations use atom

centered basis functions. Because continuum functions are not

usually included in these calculations, the TD-CI simulations

cannot model ionization directly. Klamroth and co-workers9

formulated a heuristic method to model ionization. For states

above the ionization potential (IP), the energy is modified by

adding an imaginary component (i/2) Γnto the excited-state

energy, where Γnis the estimated ionization rate for that excited

state.

i

2

In the following calculations, the Γnterm was added to states

above the experimental IPs listed in Table 1. Vertical IPs

calculated by UHF and Koopman’s theorem are ca. 1.1 and 0.2

eV lower, respectively; nevertheless, the results using the UHF

IPs are similar to those obtained with the experimental IPs. The

ionization rate, Γn, for a state is obtained by summing

contributions from the excited determinants that form the

excited state. The ionization rate for an electron in an excited

determinant is estimated from the velocity of the electron in the

virtual orbital divided by an escape distance parameter, d. In

turn, the velocity of the electron is proportional to the square

root of its orbital energy, εa.

∑

d

i a

,

where |ai

involving an excitation from orbital i to orbital a describing

state s.

The present study uses a linearly polarized and spatially

homogeneous external field:

ω

r ttt

eE

( , ) ( )sin()

ωω→−Γ

sss

(8)

ε

Γ =

s

||

a s

i

( )

a

a

2

(9)

a(s)|2is the probability amplitude for the determinant

φ≈+

(10)

This is a good approximation for the laser field, because typical

wavelengths are much larger than molecular dimensions. The

present simulations use a cosine envelope for the laser pulse.

π

g ttn

( ) 1/2cos[2 /()]/2

τ=+

(11)

τ=−

0

≤

0and

≤

t

=<>

t g t

(

nt nt

ttnt

EE

E

( )/2)for0

( )for

max

(12)

where τ = 2π/ω is the period and n is the number of cycles.

Table 1. Linear Polyenes Used in the Current Study, Their Experimentally Determined Ionization Potentials, Total Number of

CIS Excited States, and Maximum Number of States Used in the Current Study

total number of CIS states for the 6-31 n+

G(d,p) basis

maximum number of excited states used for

6-31 n+ G(d,p)

experimental ionization potential (eV)

10.513845

9.07245

8.4246

7.7947

n = 1n = 2n = 3n = 1n = 2n = 3

ethylene

butadiene

hexatriene

octatetraene

288

957

2016

3465

336

1111

2320

3969

378

1254

2592

4431

288

957

999

800

336

999

999

800

378

999

999

800

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The CIS calculations were carried out with the development

version of the Gaussian software package.40For this study

ethylene, trans 1,3-butadienene, all trans 1,3,5-hexatriene, and

all trans 1,3,5,7-octatetraene were optimized at the HF/6-

31G(d,p) level of theory. Excited-state calculations were carried

out with the 6-31 n+ G(d,p) basis set. The 6-31 n+ G(d,p) basis

has one set of five d functions on the carbons, one set of p

functions on the hydrogens, and n sets of diffuse s and p

functions on all carbons (n = 1, 2, and 3, with exponents of

0.04380, 0.01095, 0.0027375). Some additional calculations

were carried out with the 6-311++G(2df,2pd) basis set. A

seven-cycle cosine pulse with ω = 0.06 au (760 nm) was used in

the simulations. The length of the pulse is about 18 fs, and the

simulation is allowed to run for an additional 6 fs after the

pulse. For maximal effect, the field was directed along the long

axis of the molecule, specifically along the vector connecting the

end carbons. Practical considerations in the calculation of

excited to excited-state transition dipoles limited the simu-

lations to ca. 1000 states for butadiene and hexatriene, and 800

states for octatetraene. The total number of singly excited states

and the maximum number of states used in the simulations for

each molecule are listed in Table 1. Mathematica41was used to

integrate the TD-CI equations and analyze the results. The TD-

CI integrations were carried out with a step size of 0.05 au (1.2

as).

Figure 1. Ionization rates for the excited states of (a) ethylene, (b) butadiene, (c) hexatriene, and (d) octatetraene, using the 6-31 1+ G(d,p) (blue),

6-31 2+ G(d,p) (red), and 6-31 3+ G(d,p) (green) basis sets and a distance parameter of d = 1 bohr in eq 9.

Figure 2. Density of states for (a) ethylene, (b) butadiene, (c) hexatriene, and (d) octatetraene, found using the 6-31 1+ G(d,p) (blue), 6-31 2+

G(d,p) (red), and 6-31 3+ G(d,p) (green) basis sets, all CIS excited states, and a distance parameter d = 1 bohr.

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■RESULTS AND DISCUSSION

In earlier studies,23we examined various levels of theory and

basis sets to help determine what one should consider when

simulating the response of a molecule to an intense laser pulse.

For systems that cannot directly model ionization, TD-CIS

calculations of butadiene needed up to 500 excited states

computed with the 6-31G(d,p) basis set with three additional

sets of diffuse sp functions to describe the optical response to a

three-cycle, 760 nm pulse with an intensity of ca. 1014W cm−2.

In a similar vein, this Article looks at the effect of basis set size

and the number of states on the ionization rate of a set of linear

polyenes using Klamroth’s heuristic model.

Table 1 lists the linear polyenes used in the present study,

along with their experimentally determined ionization

potentials. Also indicated in the table are the total number of

singly excited states available for a given basis set and the

maximum number used in the simulations. The excited-state

ionization rates, Γn, computed with eq 9 and an escape distance

parameter of d = 1 bohr are shown in Figure 1 for ethylene,

butadiene, hexatriene, and octatetraene. The general trend is Γn

increases as the energies of the states increase, but there are

large fluctuations in the value of Γn. The higher energy states

usually involve excitation to higher energy virtual orbitals,

which result in larger values of Γn. However, some of the higher

excited states involve excitations from low lying occupied

orbitals to low lying virtual orbitals, yielding smaller values of

Γn. Larger basis sets generate more states at lower energy and

more low values of Γn. Changing the distance parameter shifts

these curves up or down by the appropriate factor, but does not

change the shape of the plots.

The ionization rate or lifetime of the excited states leads to a

broadening of the excited-state energies. The energy can be

represented by a normalized Lorentzian with a width of Γn.

Summing over all of the CIS states for a given basis set yields

the density of states plots shown in Figure 2 for a distance

parameter d = 1 bohr. The blue, green, and red curves

correspond to n = 1, 2, 3 for the 6-31 n+ G(d,p) basis. The

vertical dashed lines indicate the ionization potential, and the

solid vertical lines are drawn at 20 eV above the ionization

potential. For higher energies, the density of states converges

nicely into a broad continuum-like feature for all three basis sets

for each molecule. Increasing the number of diffuse functions

from n = 1 to n = 3 primarily affects the states within ca. 20 eV

of the ionization potential and corresponds to an increasing

number of low-lying pseudocontinuum states. For a distance

parameter of d = 10 bohr (not shown), the widths of the states

are reduced by a factor of 10, and more structure is seen in the

10−30 eV range. As in the d = 1 bohr case, the density of states

at higher energies for d = 10 is the same for 1, 2, and 3 sets of

diffuse functions.

In the heuristic model, the ionization rate depends on three

factors. The probability amplitude and the molecular orbital

energies are determined by the calculation, but the escape

distance parameter d must be determined empirically. The

ionization rate Γndepends inversely on d. Klamroth and co-

workers found the loss of norm of their systems reached a

maximum near d = 1. Figure 3 shows the loss of norm of the

population as a function of d for the four linear polyenes with

each of the 6-31 n+ G(d,p) basis sets using the maximum

number of states listed in Table 1. For ethylene, the peak in the

loss of norm is near d = 1. For the longer polyenes, the peak

becomes broader, extending to larger values of d, corresponding

to smaller values of Γn.

The trends in Figure 3 can be understood by using

perturbation theory to describe the time-dependent behavior

of a simple two-state problem. Let the lower state have an

energy of 0, the upper state an energy of ω − iΓ/2. If a

perturbation causes the states to interact, the loss of population

depends on Im(1/(ω − iΓ/2)) = (Γ/2)/(ω2+ (Γ/2)2) as well

as on the magnitude of the perturbation. The loss of population

is proportional to Γ for small values of Γ, reaches maximum for

Γ/2 = ω, and goes to zero for large Γ. Thus, the maximum

ionization rate for given state occurs when Γn/2 is equal to the

excitation energy of the state. Ethylene has relatively few states

that interact with the ground state under the influence of the

laser field, and strong ionization occurs near d = 1. For longer

polyenes, there are more states that interact with the ground

Figure 3. Loss of norm as a function of the distance parameter d (in bohr) for (a) ethylene, (b) butadiene, (c) hexatriene, and (d) octatetraene,

using the 6-31 1+ G(d,p) (blue), 6-31 2+ G(d,p) (red), and 6-31 3+ G(d,p) (green) basis sets for Emax= 0.05 au.

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state. Because these states are lower in energy for longer

polyenes, smaller values of Γnand hence larger values of d will

also cause strong ionization. Because all of the polyenes ionize

strongly for d = 1, this is the primary value used for additional

analyses.

Figure 4 shows the seven-cycle 760 nm cosine pulse and the

time evolution of the norm of the wave functions for ethylene,

butadiene, hexatriene, and octatetraene during the pulse. The

norm of ethylene decreases the least, reaching ca. 0.30 by the

end of the pulse, while the norm for octatetraene decreases

nearly to zero just after the maximum in the pulse. The

instantaneous ionization rate shown in Figure 4c can be

obtained from the derivative of the norm with respect to time.

Alternatively, the instantaneous ionization rate can be

calculated by multiplying the value of Γnfor a state by its

population and summing over all of the states. Early in the

pulse, when the intensities are low, it is already apparent that

the ionization rate is greatest for octatetraene and least for

ethylene. Toward the end of the pulse, the ionization rate for

ethylene is still significant, whereas the rate for octatetraene is

nearly zero. This reversal of the trend in the instantaneous rates

is because the population of octatetraene is very small during

the last few cycles of the pulse but the population of ethylene is

still fairly large.

Inspection of the ionization rates for the individual states

provides some insight into the dependence of the total

ionization rate on the escape distance parameter. Instantaneous

ionization rates for ethylene and hexatriene are shown in Figure

5 as a function of state energy and time. As expected, the

populations of the excited states and hence ionization rates for

these states peak when the laser field peaks and the polarization

of the electronic distribution is the greatest. For each of the

polyenes, the ionization is dominated by a relatively small

number of excited states in the range of 0− 20 eV above the IP.

There are many more states in this range that contribute only

weakly to the ionization but are needed to treat the polarization

of the electron cloud in the simulation. Even for calculations of

the static polarizability in the sum-over-states formalism, states

up to 20 eV above the IP are needed to get within 3% of the

correct values.

Figure 4. (a) Electric field for a seven-cycle 760 nm cosine pulse with

intensity of 0.88 × 1014W cm−2(Emax= 0.05 au), (b) time evolution

of the wave function norm during the pulse, and (c) instantaneous

ionization rate for ethylene (blue), butadiene (red), hexatriene

(green), and octatetraene (black) with a distance parameter d = 1

bohr using the 6-31 1+ G(d,p) basis set.

Figure 5. Instantaneous ionization rates as a function of state number

and time for (a) ethylene and (b) hexatriene. The simulations used a

seven-cycle 760 nm cosine pulse with Emax= 0.05 atomic units, and

employed a distance parameter d = 1 bohr, 288 excited states for

ethylene, and 999 excited states for hexatriene computed with the 6-31

1+ G(d,p) basis.

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