Theoretical approaches for ionization rates of

3. Reaction and Ionization of Polyatomic Molecules

1.2 Ionization Rate of Molecules in Intense

1.2.1 Theoretical approaches for ionization rates of

Theoretical methods have been developed for calculating the ionization rates of atoms or molecules in intense laser fields for many years. The quantum theoretical calculations of the ionization rates for atoms are readily available now-a-days. But, considering molecules with the additional degrees of freedom in nuclear vibration and rotation, and nonspherically symmetric electron orbitals,ab initio calculations for the ionization rates in intense laser fields are extremely difficult in solving time-dependent Schrửdinger equation or very computationally demanding in numerical procedures, and therefore, theoretical approaches need to be developed with some approximations. In general, two of most common assumptions for the case of atoms are single-active electron (SAE) and strong-field approximation (SFA). In the SAE approximation only a single electron is considered to move in the potential created by the nucleus and the remaining electrons of the atom which are frozen in their ground state orbital.18,19 In the SFA, it is assumed that the electronic continuum wavefunction is coupled much strongly to the field than to the residual ion so that the Coulomb field can be neglected after ionization.20,21

On the other hand, various approaches such as Keldysh,5PPT theory,22 KFR,21 and ADK23 are established for the case of atoms for years.

In general, one can take these atomic-like models in the first-order approximation and compare the ionization rates of molecules with respect to these atoms that have nearly identical binding energies since the ionization rate depends critically on the ionization potential of the atom or molecule.

Because SAE and SFA play their role for molecules in intense laser fields,24 as in the case of atoms, several theoretical methods are also developed based on the SAE approximation, following the tunneling theory (MO-ADK16) or KFR theory with SFA (g-KFR,25,26,27) in which Born–Oppenheimei approximation and molecular orbital are also adapted. These calculations are widely applied for interpreting and/or comparing the observations although in the further calculations it is necessary to include the effects involved with vibronic movement and interference from many atom-centers of molecules. We have outlined it here only to put these approaches of the theories in context and a more complete and detailed review may be found in the references cited here, the book of Grossmann28 and other related chapters in the books of this series.

Hamiltonian for the atoms in laser fields is described by HˆF(r , t)= − ¯h2∇2

2m +V(r )−dãF (t) (1.4) wheredis the dipole moment,F (t) is electric field given by

F (t) =Fcosωt. (1.5) Thus, the time-dependent Schrửdinger equation is written as

ih¯ ∂

∂t(r , t)=HF(r , t)(r , t). (1.6) Omitting the potential term in the Hamiltonian, its solution should be in the form of,

ψp(r , t)=exp i

¯ h

[p−eA(t)] ã r − 1 2m

t

0

dt[p−eA(t )]2

(1.7) This is the Volkov function withpthe momentum of the released electrons.6

According to Keldysh,5 the total wavefunction is a sum of the wavefunctions of the ground-state and of free electrons, i.e.,

(r , t)=ψg(r )exp

−i

¯ hEgt

+

d3pc p(t)ψ

p(r , t), (1.8) c

p(t)= i

¯ h

t

0

dtψp(r , t)|dãF|ψg(r ) exp

−i

¯ hEgt

cosωt, (1.9) whereψg(r )is the ground-state wavefunction, Eg is the eigenenergy of ground state,F is the amplitude of the incoming optical wave. When the ground-state is assigned as the 1sstate for a hydrogen-like atom,

ψg(r )= 1 πa3 exp

−r a

, a= a0

Z, (1.10)

the Keldysh theory gives the photoionization rate of the ground-state hydrogen atom in the dipole approximation as

k(F ) = 2

¯ h2 lim

T→∞Re

d3p (2πh)¯ 3

T

0

dtcos(ωt)cos(ωT )

×V0∗

p+eF ω sinωT

V0

p+eF ω sinωt

×exp



i

¯ h

t

T

dτ



I0+ 1 2m

p+eF ω sinωτ

2







 (1.11)

where the ionization potentialI0is the 1sstate energy of the hydrogen-like atom,

I0 = −Eg = Z2e2

2a0 (1.12)

and

V0(p)=8i(πa3)1/2eh¯Fã∇p

1+p2a2

¯ h2

−2

. (1.13)

Furthermore by using residue theorem, the formula of the atomic photoion- ization rate can be given as:

k=4 2I0ω

¯ h

γ

1+γ2 3/2

N(γ, ω, I0,I˜0, B, C)

×exp

−2I˜0

¯ hω

sinh−1γ −γ 1+γ2 1+2γ2

, (1.14)

whereN(γ, ω, I0,I˜0, B, C)is a pre-exponential factor and I˜0=I0+Up, Up= e2F2

4mω2. (1.15)

Faisal29 and Reiss,21,30 established the formula for calculating the rates from the velocity gauge Keldysh theory. This KFR theory is simple and practical in calculating the rates than its original Keldysh theory, and therefore to be used frequently. Similarly, the ionization continuum is treated by Volkov function and the general Bessel functionJN is employed for calculating the integrals in KFR theory, then the ionization rates is formulated as

kH(ω)=

d3p

(2πh)¯ 3χˆ∗1s(p, a0)χˆ1s(p, a0) p2

2m+IH

× ∞ N=−∞

JN+nJN∗

p2/(2m)+ ˜IH−Nhω¯ +iε

− JN−nJN∗

p2/(2m)+ ˜IH−Nhω¯ +iε

(1.16) In Keldysh or KFR theory, the formulas are established based on a hydrogen-like atomic model and thus, only atomic ionization can be calculated. For molecular ionization, the case is more complex. Lin and his collaborators generalized the KFR theory and applied it to the case of molecular ionization by combining the theory with molecular orbital (MO) theory and Born–Oppenheimer approximation.25,27 Thus, this g- KFR theory can take into account many electron features (i.e., to reduce

to the one-electron problem) by molecular orbital method and treat the effect of nuclear motion (through the vibrational overlap integral) by Born–

Oppenheimer approximation.

In theg-KFR approach it is assumed that the ground electronic state of molecule or molecular cation is well described in terms of molecular orbitals obtained from ab initio calculation and for the ionized state the electron wave function is described by the Volkov continuum state. Then, the total electronic wave function of the molecule or molecular cation is expressed as

M(r, R, t)=ψg(r, R)exp

−i

¯ hEgt

+

d3p

(2πh)¯ 3cp(t)ψp(r, R, t)exp

−i

¯ hEpt

, where r refers to electronic and R to nuclear coordination, ψg(r, R), ψp(r, R, t) is the neutral molecular wavefunction and molecular cation wavefunction, respectively.

ψg(r, R)= χ1s(1)α(1)χ1s(2)β(2)ã ã ãχHOMO(Ne−1)

×α(Ne−1)χHOMO(Ne)β(Ne) withNeis the number of electrons, and

ψp(r, R)=c1χ1s(1)α(1)χ1s(2)β(2)ã ã ãχHOMO(Ne−1)

×α(Ne−1)χp(Ne)β(Ne)

+c2χ1s(1)α(1)χ1s(2)β(2)ã ã ãχp(Ne−1)

×α(Ne−1)χHOMO(Ne)β(Ne) withc1 =c2= −1/√

2. Fori=Ne−1,Ne, χHOMO(i)=

Nn

j=1

bj,2pχj,2p(ri−Rj)

χ

p(i)=exp i

¯ h

pãri− 1 2m

t

−∞dt(p−eA(t ))2

. (1.17)

Therefore, by using similar treatment of Keldysh and KFR theories and under the assumption that the ionization only takes place from the HOMO, the photoionization rate constant can be formulated,25,31 given as

k(F ) =2πS2

Ne

j,j=1

cjcj∗

d3p

(2π)3χˆj(p) χˆj∗(p)

× p2

2me +I0

2 JN

eF ã p

meω2 ,Up 2ω

2

×cos

pãRj − Rj ∞

N=−∞

δ

I0+Up+ p2

2me −Nω

=

N

2πS2

Ne

j,j=1

cjcj∗

d3p

(2π)3χˆj(p) χˆ∗j(p)

× p2

2me +I0

2 JN

eF ã p meω2 ,Up

2ω

2

ìcos(pã(Rj− Rj))δ

I0+Up+ p2

2me −Nω

=

N

k(N) (1.18)

withJN is the generalized Bessel function,cj the coefficients of the linear combination of atomic orbitals-molecular orbital,S =√

2 for the closed shell parent molecule or molecular cation, andS=1 for the open shell. The g-KFR theory has been widely used to diatomic and polyatomic molecules.

On the other hand, Ammosov, Delone and Krainov32 developed the PPT theory22 for treating arbitrary states of hydrogen atoms in intense electromagnetic fields to the ionization rates for arbitrary atoms

w= 3e

π 3/2

Z2 3n∗3

2l+1 2n∗−1

4eZ3 (2n∗−1) n∗3F

2n∗−3/2

exp

−2Z3 3n∗3F

, (1.19)

with e = 0.71828. . . , n∗ andl∗ the effective quantum numbers. In this atomic ADK theory, the major improvement is to modify the radial wave function of the outermost electron in the asymptotic region where tunneling occurs and therefore the theory is an extension of the PPT only for hydrogen atoms to more complex atomic system. However, for a molecular system, the calculation for ionization rates is even complicated since multi-centre problem has to be treated. Based on the similar consideration on the asymptotic feature of electronic wave functions and symmetric feature,16 Tong et al. expressed the molecular electronic wave functions in the asymptotic region in terms of summations of spherical harmonics in a one- center expansion,16

ψm(r)=

l

ClFl(r)Ylm(ˆr),

with a normalized coefficient Cl for insuring the wave function in the asymptotic region can be expressed as

Fl(r→ ∞)≈rZc/κ−1e−κr, withZc the effective Coulomb charge,κ =

2Ip, andIp the ionization potential for the given valence orbital. They realized the ADK theory calculation for the ionization rates of diatomic molecules with an arbitrary Euler angleR with respect to the low frequency ac field direction (non- aligned) is

w(F, R)= 3F

πκ3

1/2

m

B2(m) 2|m||m|!

1 κ2Zc/κ−1

2κ3 F

2Zc/κ−|m|−1

e−2κ3/3F (1.20) where, ifDml ,m(R) is the rotation matrix, one has

B(m)=

l

ClDlm,m(R)Q(lm ),

Q(lm)=(−1)m

(2l+1)(l+ |m|)! 2(l− |m|)! .

This MO-ADK method was generalized to nonlinear polyatomic molecules33,34 previously and extended to multi-electron cases by Brabec et al.35and Zhaoet al.36