Download time-dependent density functional theoretical - Prof. Shih

In: Theoretical and Computational Developments in Modern
Density Functional Theory
ISBN 978-1-61942-779-2
c 2012 Nova Science Publishers, Inc.
Editor: A. K. Roy, pp. 1-34
Chapter 1
T IME - DEPENDENT DENSITY FUNCTIONAL
THEORETICAL METHODS FOR TREATMENT OF
MANY- ELECTRON MOLECULAR SYSTEMS IN
INTENSE LASER FIELDS
Dmitry A. Telnov1∗, John T. Heslar2† and Shih-I Chu2,3‡§
1
Department of Physics, St. Petersburg State University,
St. Petersburg 198504, Russia
2
Center for Quantum Science and Engineering and Department of Physics,
National Taiwan University, Taipei 10617, Taiwan
3
Department of Chemistry, University of Kansas,
Lawrence, Kansas 66045, USA
October 2012
PACS 33.80.Rv, 42.65.Ky, 32.80.Rm Keywords: time-dependent density functional theory, multielectron molecules, multiphoton ionization, high-order harmonic generation.
∗
E-mail address: [email protected]
E-mail address: [email protected]
‡
E-mail address: [email protected]
§
Corresponding author
†
2
Telnov, Heslar & Chu
ABSTRACT
In this chapter, we review the recent new developments in time-dependent density functional (TDDFT) methods for treatment of dynamics of many-electron molecules in intense
laser fields. We discuss some recent development of TDDFT with optimized effective potential (OEP) and self-interaction-correction (SIC) for many-electron systems which allows
the use of orbital-independent single-particle local potential with correct asymptotic bahavior. We also describe several numerical techniques recently developed for efficient and
high-precision treatment of the time-dependent Kohn-Sham equations. The usefulness of
these procedures is illustrated by a few case studies of multiphoton processes in diatomic
molecules, including multiphoton ionization and high-order harmonic generation.
TDDFT methods for molecular systems in laser fields
1.
3
INTRODUCTION
In recent years, the density-functional theory (DFT) has become a widely used formalism
for electron structure calculations of atoms, molecules, and solids [1–5]. The DFT is based
on the earlier fundamental work of Hohenberg and Kohn [6] and Kohn and Sham [7]. In the
Kohn-Sham DFT formalism [7], the electron density is decomposed into a set of orbitals,
leading to a set of one-electron Schrödinger-like equations to be solved self-consistently.
The Kohn-Sham equations are structurally similar to the Hartree-Fock equations but include in principle exactly the many-body effects through a local exchange-correlation (xc)
potential. Thus DFT is computationally much less expensive than the traditional ab initio
many-electron wavefunction approaches and this accounts for its great success for large
systems. However, the DFT is well developed mainly for the ground-state properties only.
The treatment of excited states and time-dependent processes within the DFT is much less
developed.
The essential element of DFT is the input of the xc energy functional whose exact
form is unknown. The simplest approximation for the xc-energy functional is through the
local spin-density approximation (LSDA) [1, 8] of homogeneous electronic gas. A severe
deficiency of the LSDA is that the xc potential decays exponentially and does not have the
correct long-range behavior. As a result, the LSDA electrons are too weakly bound and for
negative ions even unbound. More accurate forms of the xc-energy functionals are available
from the generalized gradient approximation (GGA), [9–12] which takes into account the
gradient of electron density. However, the xc potentials derived from these GGA energy
functionals suffer similar problems as in LSDA and do not have the proper long-range
Coulombic tail −1/r either. Thus while the total energies of the ground states of atoms
and molecules predicted by these GGA density functionals [9–12] are reasonably accurate,
the excited-state energies and the ionization potentials obtained from the highest occupied
orbital energies of atoms and molecules are far from satisfactory, typically 40 to 50% too
low [13]. The problem of the incorrect long-range behavior of the LSDA and GGA energy
functionals can be attributed to the existence of the self-interaction energy [13, 14].
For proper treatment of atomic and molecular dynamics such as collisions or multiphoton ionization processes etc., the conventional ground-state DFT is not sufficient. It is the
time-dependent density functional theory (TDDFT) which extends the concept of the stationary DFT to the time-dependent domain. The rigorous formulation of TDDFT is due to
the Runge-Gross theorem [15]. For any interacting many-particle quantum system subject
to a given time-dependent potential, all physical observables are uniquely determined by
knowledge of the time-dependent density and the state of the system at any instant in time
[15, 16]. In particular, if the time-dependent potential is turned on at some time t0 and the
system has been in its ground state until t0 , all observables are unique functionals of only
the density. In this case the initial state of the system at time t0 will be a unique functional
of the ground state density itself, i.e. of the density at t0 . This unique relationship allows
one to derive a computational scheme in which the effect of the particle-particle interaction
is represented by a density-dependent single-particle potential, so that the time evolution
of an interacting system can be investigated by solving a time-dependent auxiliary singleparticle problem. Additional simplifications can be obtained in the linear response regime
[17, 18]. In the last several years there has been a considerable effort and success in the use
4
Telnov, Heslar & Chu
of linear response theory to the study of excitation energies [18–21], frequent-dependent
multipole polarizabilities [17, 22, 23], optical spectra of molecules, clusters, and nanocrystals [24–26], and autoionizing resonances [13], etc.
In this chapter, we review some new developments in TDDFT beyond the linear response regime, recently carried out for accurate and efficient nonperturbative treatment of
multiphoton dynamics and nonlinear optical processes of molecules in intense and superintense laser fields. In the remaining part of this section, we remind the basic concepts and
theorems of DFT and TDDFT. In Sec. 2., we briefly describe the time-dependent optimized
effective potential method and its simplified version, the time-dependent Krieger-Li-Iafrate
approximation. In Sec. 3., we present the TDDFT approaches for treatment of multiphoton
processes in diatomic molecules. Atomic units are used in this chapter.
1.1.
The Hohenberg-Kohn Theorem
Consider a system of N interacting electrons in a non-degenerate ground state associated
with an external potential v(r). The ground state density, ρ(r), uniquely determines the
potential v(r) [6]. Since with v(r) the full Hamiltonian is known, now ρ(r) completely
determines all properties of the system such as the ground state eigenfunction, response
functions, thermal properties, etc.
The Hohenberg-Kohn variational theorem states there exists a functional F [ρ0 (r)] for
all non-degenerate ground state densities ρ0 (r) such that, for a given v(r), the quantity
Z
Ev(r) [ρ0 (r)] = d3 r v(r)ρ0 (r) + F [ρ0 (r)]
(1)
has its unique minimum for the correct ground-state density, ρ0 (r) = ρ(r), associated with
v(r). The physical meaning of F [ρ0 (r)] is
F [ρ0 (r)] ≡ hΨρ0 |T + U |Ψρ0 i
(2)
where Ψρ0 is the multielectron ground state wave function associated with ρ0 (r), and T and
U are the kinetic energy and Coulomb repulsion operators.
M. Levy [27] and E. Lieb [28] have independently shown by specific example that not
all well-behaved functions ρ(r) can be realized as ground state densities. However, this
so-called v-representability problem has, so far, caused no practical difficulty. The same
authors have also pointed out that F [ρ0 (r)] can be defined as
F [ρ0 (r)] ≡ minhΨρ0 |T + U |Ψρ0 i,
Ψρ0
(3)
where the minimum is taken over all normalized antisymmetric wave-functions Ψρ0 which
give rise to a density ρ0 (r). This “constrained search” definition of F [ρ0 (r)] is valid for a
larger class of densities, ρ0 (r), than those which are physical, non-degenerate ground states.
1.2.
Kohn-Sham Equations
The Hohenberg-Kohn variational principle can be recast in the form of exact single particle
self consistent equations, similar to the Hartree equations [7]:
1 2
− ∇ + veff (r) ψj (r) = j ψj (r),
(4)
2
TDDFT methods for molecular systems in laser fields
ρ(r) =
N
X
|ψj (r)|2 ,
5
(5)
j=1
Z
veff (r) = v(r) +
ρ(r 0 ) 3 0
d r + vxc (r).
|r − r 0 |
(6)
Here vxc (r) is the local exchange-correlation (xc) potential, defined as
δExc [ρ(r)]
,
δρ(r)
Z
Z
1
ρ(r)ρ(r 0 )
3
Exc [ρ(r)] = F [ρ(r)] −
d r d3 r0
− Ts [ρ(r)],
2
|r − r 0 |
vxc (r) =
(7)
(8)
where Ts [ρ(r)] is the kinetic energy of non-interacting electrons with the ground state density ρ(r).
The exact functional form of Exc [ρ(r)] is unknown. However, a number of approximate
functionals are available varying in their complexity and quality of the results produced.
Solution of the equations (4) for the given Exc [ρ(r)] yields ρ(r) and allows calculation of
the ground state energy
E=
N
X
j=1
1.3.
1
j −
2
Z
3
d r
Z
ρ(r)ρ(r 0 )
d r
−
|r − r 0 |
3 0
Z
d3 r vxc (r)ρ(r) + Exc [ρ(r)].
(9)
The Runge-Gross Theorem
The Runge-Gross theorem is the central theorem of TDDFT. It proves that there is a oneto-one correspondence between the external time-dependent potential, v(r, t), and the electronic density, ρ(r, t), for many-body systems evolving from a fixed initial state [15]. The
latter condition is important; it reflects the fact that the original time-dependent Schrödinger
equation is an initial value problem unlike the the time-independent Schrödinger equation
which is a boundary value problem. For the system evolving from its ground state, however, the only required quantity is the electronic density, since it determines the ground
state wave function according to the ordinary Hohenberg-Kohn theorem. The Runge-Gross
theorem actually proves that no two potentials, v(r, t) and v 0 (r, t), can produce the same
time-dependent density if they differ by more than a purely time-dependent function c(t).
The proof of this theorem is much more involved than the proof of the Hohenberg-Kohn
theorem; the details can be found elsewhere [15, 29].
Just like the time-independent theory, we can introduce an auxiliary system of noninteracting electrons subject to an external local potential veff (r, t). The Runge-Gross theorem can be applied to the non-interacting system as well. Then the potential veff (r, t) is
unique and determined by the condition that the density of the non-interacting system is the
same as the density of the original interacting system. This is a time-dependent Kohn-Sham
scheme, and the Kohn-Sham spin-orbitals obey the time-dependent Schrödinger equation
with the potential veff (r, t):
∂
1 2
i ψj (r) = − ∇ + veff (r, t) ψj (r).
(10)
∂t
2
6
Telnov, Heslar & Chu
As in the Kohn-Sham scheme for the ground-state (Eq. (6)), the time-dependent Kohn-Sham
potential veff (r, t) can be written as the sum of three terms,
Z
veff (r, t) = v(r, t) +
ρ(r 0 , t) 3 0
d r + vxc (r, t).
|r − r 0 |
(11)
where the first term is due to the interaction with the nuclei and external fields, and the
second term represents the classical electron-electron repulsion. The third term, the timedependent xc potential, includes in principle all non-trivial multielectron effects. The xc
potential is a functional of the time-dependent electron density but its explicit form is unknown. This functional form can be very complicated and non-local both in space and time
domains. In practice, various approximations can be used to represent the time-dependent
xc potential, and the results of the calculations depend strongly on the quality of the approximate xc potential.
2.
TDDFT WITH OEP/KLI-SIC FOR MANY-ELECTRON
SYSTEMS IN LASER FIELDS
In this section, we discuss TDDFT with optimized effective potential (OEP) and selfinteraction correction (SIC) for nonperturbative treatment of many-electron quantum systems in intense laser fields [30]. The steady-state OEP method [31, 32] has become a
practical tool in DFT after the work of Kriger, Li, and Iafrate (KLI) who suggested a simple
yet accurate procedure for determination of OEP by a set of linear equations [33, 34]. This
method was extended to the time domain [35] but the original formulation was not very efficient computationally since it involved the construction of Hartree-Fock-like non-local potential at each time step. The advantage of the time-dependent OEP/KLI-SIC approach [30]
(TDOEP/KLI-SIC) is that it allows the construction of self-interaction-free time-dependent
local OEP. This greatly facilitates the study of time-dependent processes of many-electron
quantum systems in strong fields.
We start from the quantum mechanical action of a many-electron system interacting
with an external field [30, 35]:
A[{ψnσ }] =
−
Nσ Z
XX
t1
σ n=1 −∞
Z
X Z t1
dt
σ
Z
dt
∂
1
∗
ψnσ
(r, t) i + ∇2 ψnσ (r, t)dr
∂t 2
ρσ (r, t)[vn (r) + vext (r, t)]dr
−∞
Z
Z Z
ρ(r, t)ρ(r 0 , t)
1 t1
drdr 0
−
dt
2 −∞
|r − r 0 |
− Axc [{ψnσ }],
(12)
P
where ψnσ (r, t) are the time-dependent spin-orbitals, N = σ Nσ is the total number of
electrons, vn (r) is the electron-nucleus Coulomb interaction, vext (r, t) describes the coupling of the electron to the external laser fields, and Axc [{ψnσ }] is the exchange-correlation
TDDFT methods for molecular systems in laser fields
7
(xc) action functional. The electron spin-densities ρσ (r, t) are defined as follows:
ρσ (r, t) =
Nσ
X
|ψnσ (r, t)|2 ,
(13)
n=1
and the total electron density ρ(r, t) is obtained by summation of the spin-densities:
X
ρ(r, t) =
ρσ (r, t).
(14)
σ
The spin orbitals satisfy the one-electron Schrödinger-like equation:
1 2
∂
i ψnσ (r, t) = − ∇ + Vσ (r, t) ψnσ (r, t),
∂t
2
(15)
where Vσ (r, t) will be the time-dependent OEP (TDOEP) if we choose the set of spinorbitals {ψnσ } which render the total action functional A[{ψnσ }] stationary:
δA[{ψnσ }]
= 0.
δVσ (r, t)
(16)
Generally, Vσ (r, t) contains the memory effect. It depends not only on the density at
the time moment t but also on the densities at preceding times. However, if we use the
following explicit SIC expression for the exchange-correlation (xc) action functional [30],
Z t1
Axc [{ψnσ }] =
dtExc [ρ↑ (r, t), ρ↓ (r, t)]
−
−∞
Nσ Z t1
XX
dt {J[ρnσ ] + Exc [ρnσ , 0]} ,
(17)
σ n=1 −∞
then the memory term vanishes identically. Similar results are obtained as long as one uses
an explicit Exc form (such as that in LSDA or GGA) of energy functional and the adiabatic
approximation. The use of the SIC form in Eq. (17) removes the spurious self-interaction
terms in conventional TDDFT and results in a proper long-range asymptotic potential. Another major advantage of this procedure is that only local potential is required to construct
the orbital-independent OEP. This facilitates considerably the numerical computation.
TDOEP can be expressed as follows:
Vσ (r, t) = vn (r) + vext (r, t) +
where
VσSIC (r, t) =
δJ[ρ]
+ VσSIC (r, t),
δρσ (r, t)
h
io
X ρnσ (r, t) n
SIC
vnσ (r, t) + V nσ (t) − v nσ (t) ,
ρσ (r, t)
n
vnσ (r, t) =
−
δExc [ρ↑ (r, t), ρ↓ (r, t)] δJ[ρnσ (r, t)]
−
δρσ (r, t)
δρnσ (r, t)
δExc [ρnσ (r, t), 0]
,
δρnσ (r, t)
(18)
(19)
(20)
8
Telnov, Heslar & Chu
and
SIC
V nσ (t) = hψnσ |VσSIC (r, t)|ψnσ i,
(21)
v nσ (t) = hψnσ |vnσ (r, t)|ψnσ i.
(22)
Equations (15) and (18) are to be solved self-consistently. Note that since the exact form of
Vxc,σ (r, t) is unknown, the adiabatic approximation is often used in the TDDFT calculations:
Vxc,σ (r, t) = Vxc,σ [ρσ ]|ρσ =ρσ (r,t) .
(23)
Finally Eq. 15 is an initial value problem and the initial wave function can be determined
by
ψnσ (r, t)t=0 = φnσ (r) exp(−inσ t)|t=0 ,
(24)
where, φnσ (r) and nσ are the eigenfunction and eigenvalue of the time-independent KohnSham equation (with OEP/KLI-SIC) for the static case [13].
3.
3.1.
TDDFT FOR MULTIPHOTON PROCESSES IN DIATOMIC MOLECULES
Time-Dependent Generalized Pseudospectral Method for Numerical Solution of Self-Interaction-Free TDDFT Equations in Two-Center Systems
In the following, we extend the time-dependent generalized pseudospectral (TDGPS) procedure to the numerical solution of the time-dependent equations in two-center multielectron
systems. Consider the solution of the time-dependent one-electron Kohn–Sham equations
(10). In the spin-polarized theory, the spin-orbitals ψnσ (r, t) corresponding to different
spin projections σ satisfy the equations with different effective potentials veff,σ (r, t):
∂
1 2
i ψnσ (r, t) = − ∇ + veff,σ (r, t) ψnσ (r, t),
∂t
2
(25)
n = 1, 2, ..., Nσ .
The time-dependent effective potential veff,σ (r, t) is a functional of both electron spin densities ρ↑ (r, t) and ρ↓ (r, t). The potential veff,σ (r, t) can be written in the general form
veff,σ (r, t) = vn (r) + vH (r, t) + vxc,σ (r, t) + vext (r, t)
(26)
where vn (r) is the electron interaction with the nuclei,
vn (r) = −
Z1
Z2
−
|R1 − r| |R2 − r|
(27)
with Z1 and Z2 being the charges of the nuclei, and R1 and R2 being the positions of the
nuclei (which are assumed to be fixed at their equilibrium positions); vH (r, t) is the Hartree
potential due to electron-electron Coulomb interaction,
Z
ρ(r 0 , t)
vH (r, t) = d3 r0
.
(28)
|r − r 0 |
TDDFT methods for molecular systems in laser fields
9
The potential vext (r, t) in Eq. (26) describes the interaction with the laser field. Using the
dipole approximation and the length gauge, it can be expressed as follows:
vext (r, t) = (F (t) · r).
(29)
Here F (t) is the electric field strength of the laser field, and the linear polarization is assumed. For the laser pulses with the sine-squared envelope, one has:
F (t) = F0 sin2
πt
sin ω0 t
T
(30)
where T and ω0 denote the pulse duration and the carrier frequency, respectively; F0 is the
peak field strength.
The wave functions and operators are discretized with the help of the generalized pseudospectral (GPS) method in prolate spheroidal coordinates [36–39]. The prolate spheroidal
coordinates ξ, η, and ϕ are related to the Cartesian coordinates x, y, and z as follows [40]:
p
x = a (ξ 2 − 1)(1 − η 2 ) cos ϕ,
p
(31)
y = a (ξ 2 − 1)(1 − η 2 ) sin ϕ,
z = aξη
(1 ≤ ξ < ∞,
−1 ≤ η ≤ 1).
In Eq. (31), we assume that the molecular axis is directed along the z axis, and the nuclei are
located on this axis at the positions −a and a, so the internuclear separation R = 2a. For the
unperturbed molecule, the projection m of the angular momentum onto the molecular axis is
conserved, and the exact spin orbitals have factors (ξ 2 −1)|m|/2 (1−η 2 )|m|/2 which are nonanalytical at nuclei for odd |m|. Straightforward numerical differentiation of such functions
could result in significant loss of accuracy. Therefore different forms of the kinetic energy
operators have been suggested for even and odd m [37, 41]. However, for the molecules
in the linearly polarized laser field with arbitrary orientations of the molecular axis, the
projection of the electron angular momentum onto the molecular axis is not conserved any
longer. In this case, we apply a full 3D discretization with respect to the coordinates ξ, η,
and ϕ. For ξ and η, we use the GPS discretization with non-uniform distribution of the grid
points; for ϕ, the Fourier grid (FG) method [42] with uniform spacing of the grid points
is more appropriate. To take care of the possible singularities at the nuclei, we use special
mapping transformations of the coordinates ξ and η [43] which make the wave functions
analytic at the nuclei for both even and odd projections of the angular momentum. The
discretized kinetic energy operator takes the form of the matrix Tijk;i0 j 0 k0 :

Tijk;i0 j 0 k0

(ξ)
(η)
(ϕ)
0
0
T
δ
+
T
δ
T 0 δii0 δjj 0
1
ii0 jj
jj 0 ii

= 2 q
δkk0 + 2 kk
2)
2a
2
2
2
2
(ξ
−
1)(1
−
η
i
j
(ξi − ηj )(ξi0 − ηj 0 )
(ξ)
(η)
(ϕ)
(32)
where the partial matrices Tii0 , Tjj 0 , and Tkk0 related to the coordinates ξ, η, and ϕ,
respectively, have quite simple expressions [43].
The time-dependent Kohn–Sham equations (25) are solved by means of the splitoperator method with spectral expansion of the propagator matrices [36–38, 44]. We em-
10
Telnov, Heslar & Chu
ploy the following split-operator, second-order short-time propagation formula:
i
b0
ψnσ (r, t + ∆t) = exp − ∆t H
2
1
× exp −i∆t V (r, t + ∆t)
2
i
b 0 ψnσ (r, t) + O((∆t)3 ).
× exp − ∆t H
2
(33)
b 0 is the unperturbed electronic Hamiltonian which
Here ∆t is the time propagation step, H
includes the kinetic energy and the the effective potential before the laser field switched on,
b 0 = − 1 ∇2 + veff,σ (r, 0).
H
2
(34)
The potential V (r, t) describes the interaction with the laser field and can be expressed as
follows:
V (r, t) = veff,σ (r, t) − veff,σ (r, 0).
(35)
It contains the direct interaction with the field vext (r, t) (29) as well as terms due to the
variation of the density. For the field polarized under the angle γ with respect to the molecular axis, the direct interaction can be expressed as follows, using the prolate spheroidal
coordinates:
vext (ξ, η, ϕ, t) = aF (t) ξη cos γ
(36)
p
+ (ξ 2 − 1)(1 − η 2 ) cos ϕ sin γ .
Note that Eq. (33) is different from the conventional split-operator techniques [45, 46],
where Ĥ0 is usually chosen to be the kinetic energy operator and V̂ the remaining Hamiltonian depending on the spatial coordinates only. The use of the energy-representation in
Eq. (33) allows the explicit elimination of the undesirable fast-oscillating high-energy components and speeds up considerably
the
time propagation [36, 44, 47]. For the given ∆t,
i
b
the propagator matrix exp − 2 ∆t H0 is time-independent and constructed only once from
b 0 before the propagation process
the spectral expansion of the unperturbed Hamiltonian
H
1
starts. The matrix exp −i∆t V (r, t + 2 ∆t) is time-dependent and must be calculated at
each time step. However, for interaction with the laser field in the length gauge, this matrix
is diagonal (as any multiplication by the function of the coordinates in the GPS and FG
methods), and its calculation is not time-consuming.
3.2.
Exploration of the Underlying Mechanisms for High Harmonic Generation of H2 in Intense Laser Fields
In this section we show an application of the TDOEP/KLI-SIC procedure to the study of
high-order harmonic generation of H2 in intense pulsed laser fields. First we discuss the
field-free electronic structure calculations using the steady-state OEP/KLI-SIC procedure
TDDFT methods for molecular systems in laser fields
11
[13, 36], and the GPS procedure [36, 44] is extended to discretize the molecular Hamiltonian in the prolate spheroidal coordinates. For H2 , the calculated ground-state energy is
−1.1336 a.u. (using LSDA exchange energy functional only) and −1.1828 a.u. (including
both LSDA exchange and correlation energy functionals), the latter is within 1% of the exact value of −1.174448 a.u. If the GGA energy functional such as that of BLYP [1] is used,
the calculated ground-state energy is improved to −1.17444 a.u.
Consider now H2 molecules subject to an intense laser field with the wavelength
1064 nm, sin2 pulse shape, and 20 optical cycles in pulse length, linearly polarized along
the molecular axis. The time-dependent xc potential is constructed by means of the timedependent OEP/KLI-SIC procedure using the adiabatic LSDA exchange and correlation
energy functional. In the following, we shall focus our discussion on the HHG process of
H2 molecules from the ground vibrational state with the internuclear separation R fixed at
the equilibrium distance (R = 1.4 a. u.).
The solution of the TDOEP/KLI-SIC equation is performed by means of the TDGPS
method described above. To explore the detailed spectral and temporal structure of HHG
and the underlying mechanisms in different energy regimes, one can perform the timefrequency analysis by means of the wavelet transform [48, 49] of the induced dipole,
Z
AW (t0 , ω) = d(t)Wt0 ,ω (t)dt ≡ dω (t),
(37)
√
with the wavelet kernel Wt0 ,ω (t) = ωW (ω(t − t0 )). For the harmonic emission, a natural
choice of the mother wavelet is given by the Morlet wavelet [49]
√
2
2
W (x) = (1/ τ )eix e−x /2τ .
(38)
Fig. 1 shows the modulus of the time-frequency profiles of H2 (at R = 1.4a0 ) in (1064 nm,
20 o.c., sin2 pulse shape, and 1014 W/cm2 ) laser fields, revealing striking and vivid details
of the spectral and temporal structures. Several salient features are noticed. (a) First, for
the lowest few harmonics, the time profile (at a given frequency) shows a smooth function
of the driving laser pulse. This is an indication that the multiphoton mechanism dominates
this lower harmonic regime. In this regime, the probability of absorbing N -photons is
roughly proportional to I N , and I (laser intensity) is proportional to E(t)2 . (b) Second, the
smooth time profile is getting shorter (in time duration) and broadened (in frequency) as the
harmonic order is increased, as is evident in Fig. 1 from the 1st to the 7th harmonics. As the
harmonic order is further increased, the time profiles (see particularly the 11th harmonic
in Fig. 1) develop extended fine structures. This can be attributed to the effect of excited
states and the onset of the ionization threshold. (c) Third, for those high harmonics in
the plateau regime well above the ionization threshold, the most prominent feature is the
development of fast burst time profiles. At a given time, we see that such bursts actually
form a continuous frequency profile in Fig. 1. This is a clear evidence of the existence of
the bremsstrahlung radiation emitted by each recollision of the electron wavepacket with
the parent ionic core(s). In contrast, we find that the (multiphoton-dominant) lowest-order
harmonics form a continuous time profile at a given frequency. In the intermediate energy
regime where both multiphoton and tunneling mechanisms contribute, the time-frequency
profiles show a net-like structure.
12
Telnov, Heslar & Chu
18
16
Time (Optical cycle)
14
12
10
8
6
4
2
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Frequency (Harmonic order)
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Log scale
Figure 1. The time-frequency spectra (modulus) of H2 (at R = 1.4a0 ) in (1064 nm, 20
o.c., sin2 pulse shape) laser fields with peak intensity 1014 W/cm2 . The colors shown are
in logarithmic scale (in the powers of 10).
3.3.
Orientation-Dependent Multiphoton Ionization and High-Order Harmonic Generation of Diatomic Molecules N2 and F2
The exact form of the exchange-correlation (xc) potential vxc,σ (r, t) is unknown. However,
high-quality approximations to the xc potential are becoming available. When these potentials, determined by time-independent ground-state DFT, are used along with TDDFT in the
electronic structure calculations, both inner shell and excited states can be calculated rather
accurately [50]. In the time-dependent calculations, we adopt the commonly used adiabatic
approximation, where the xc potential is calculated with the time-dependent density. The
adiabatic approximation had recently many successful applications to atomic and molecular processes in intense external fields [50, 51]. For the studies of the diatomic molecules
[43, 52, 53], we utilize the LBα (van Leeuwen – Baerends) xc potential [54]:
LBα
LSDA
LSDA
vxc,σ
(r, t) = αvx,σ
(r, t) + vc,σ
(r, t)
1/3
βx2σ (r, t)ρσ (r, t)
−
.
1 + 3βxσ (r, t) ln{xσ (r, t) + [x2σ (r, t) + 1]1/2 }
(39)
The LBα potential contains two parameters, α and β, which have been adjusted in timeindependent DFT calculations of several molecular systems and have the values α = 1.19
LSDA and v LSDA are the exchange and
and β = 0.01 [54]. The first two terms in Eq. (39), vx,σ
c,σ
TDDFT methods for molecular systems in laser fields
13
correlation potentials within the local spin density approximation (LSDA). The last term
4/3
in Eq. (39) is the gradient correction with xσ (r) = |∇ρσ (r)|/ρσ (r), which ensures the
LBα → −1/r as r → ∞. The potential (39) has
proper long-range asymptotic behavior vxc,σ
proved to be reliable in molecular TDDFT studies [53, 55]. The correct long-range asymptotic behavior of the LBα potential is crucial in photoionization problems since it allows to
reproduce accurate MO energies, and the proper treatment of the molecular continuum.
In our calculations of multiphoton processes in N2 and F2 , we used the laser wavelength
800 nm (ω0 = 0.056954 a.u.) and the sine-squared envelope with 20 optical cycles. The
propagation procedure based on Eq. (33) is applied sequentially starting at t = 0 and ending
at t = T . As a result, the spin orbitals ψnσ (ξ, η, ϕ, t) are obtained on a uniform time grid
within the interval [0, T ]. The space domain is finite with the linear dimension restricted
by the end point Rb . We choose Rb = 40 a.u.; the corresponding space volume contains
all relevant physics for the laser field parameters used in the calculations. Between 20 a.u.
and 40 a.u. we apply an absorber which smoothly brings down the wave function for each
spin orbital without spurious reflections. Absorbed parts of the wave packet localized beyond 20 a.u. describe unbound states populated during the ionization process. Because of
the absorber, the normalization integrals of the wave functions ψnσ (r, t) decrease in time.
(s)
Calculated after the pulse, they give the survival probabilities Pnσ for each spin orbital:
Z
(s)
Pnσ
= d3 r|ψnσ (r, T )|2 .
(40)
(i)
Then one can define the spin orbital ionization probabilities Pnσ as
(i)
(s)
Pnσ
= 1 − Pnσ
.
(41)
(s)
We note that the quantities Pnσ represent the survival probabilities for the electron occupying the unperturbed ψnσ (r, t = 0) spin orbital before the laser pulse. Accordingly, the
(i)
quantity Pnσ represents the ionization probability for the electron originally occupying the
unperturbed ψnσ (r, t = 0) spin orbital.
The total survival probability P (s) can be calculated as a product of the spin orbital
survival probabilities:
Y
Y
(s)
(i)
,
(42)
P (s) =
Pnσ
=
1 − Pnσ
nσ
nσ
and the total ionization probability can be written as
Y
(i)
P (i) = 1 − P (s) = 1 −
1 − Pnσ
.
(43)
nσ
The total ionization probability as defined by Eq. (43) reduces to the sum of the spin orbital
(i)
probabilities only in the limit of the weak laser field (small Pnσ ). In the calculations, we
used the experimental values of the equilibrium internuclear separations for the diatomic
molecules [59] (2.074 a.u. for N2 and 2.668 a.u. for F2 ). In Table 1, we summarize the
energies for the spin orbitals that have a significant contribution to MPI and HHG and the
corresponding experimental vertical ionization energies. Also presented are the data for
the companion Ar atom which has an ionization potential close to that of N2 and F2 and
14
Telnov, Heslar & Chu
Table 1. Absolute values of spin orbital energies of N2 , F2 , and Ar. (A) DFT calculations
[43] (eV). (B) Experimental ionization energies (eV).
Molecule
Spin-orbital
A
B
N2
2σu
1πu
3σg (HOMO)
18.5
16.9
15.5
18.7 (Ref. [56])
17.2 (Ref. [56])
15.6 (Ref. [56])
F2
3σg
1πu
1πg (HOMO)
21.9
19.2
16.0
21.0 (Ref. [57])
19.0 (Ref. [57])
15.7 (Ref. [57])
Ar
3s
3p
29.0
15.3
29.3
15.8
(Ref. [58])
(Ref. [58])
is expected to manifest close ionization probabilities as well. The agreement between the
calculated and experimental values is fairly good for all three systems indicating a good
quality of the LBα exchange-correlation potential.
3.3.1.
Multiphoton ionization
We present the orientation-dependent MPI probabilities for N2 molecule at the peak intensity 2 × 1014 W/cm2 (Fig. 2). The orientation dependence of the total MPI probability is in
a good accord with the experimental observations [60, 61] for this molecule and reflects the
symmetry of its HOMO: the maximum corresponds to the parallel orientation. However,
multielectron effects are quite important for N2 , particularly at intermediate orientation angles. In the angle range around 30◦ , the orbital probability of HOMO−1 (1πu ) is larger than
that of HOMO (3σg ). Despite the orbital probabilities have local minima and maxima, the
total probability shows monotonous dependence on the orientation angle. With increasing
the peak intensity of the laser field, the orientation angle distribution of the total ionization
probability becomes more isotropic. For comparison, we show also the ionization probability of the Ar atom. As one can see from Fig. 2, the absolute values of the ionization
probabilities of N2 and Ar are close to each other. However, the inner shell contributions
are less important for Ar: the total probability is dominated by the highest-occupied (3p)
shell contribution. An analysis of the spin orbital energies (Table 1) can help to understand
the relative importance of MPI from the inner shells in N2 compared to that in Ar. The
smaller the ionization potential of the electronic shell, the easier it can be ionized. That
is why HOMO is generally expected to give the main contribution to the MPI probability.
However, in N2 the ionization potential of HOMO−1 is quite close to that of HOMO (the
difference between the calculated values is 1.4 eV), and in the strong enough laser field both
shells show comparable ionization probabilities (a possible resonance between HOMO and
HOMO−1 in the 800 nm laser field also favors that; see discussion of HHG in Sec. 3.3.2.
TDDFT methods for molecular systems in laser fields
A
B
90°
0.120
e
a
45°
a
b
180°
0°
225°
Ionization probability
135°
15
e
0.100
f
b
0.080
c
0.060
0.040
0.020
315°
d
0.000
270°
0
30
90
60
Orientation angle (degrees)
Figure 2. MPI probabilities of N2 molecule and Ar atom for the peak intensity 2 ×
1014 W/cm2 in polar (panel A) and Cartesian (panel B) coordinates: (a) total probability for N2 , (b) 3σg (HOMO) probability for N2 , (c) 1πu (HOMO−1) probability for N2 , (d)
2σu probability for N2 , (e) total probability for Ar, (f) 3p0 probability for Ar.
A
B
90°
a
45°
a
b
180°
0°
225°
Ionization probability
135°
0.025
0.020
0.015
b
0.010
0.005
315°
270°
c
d
0.000
0
30
90
60
Orientation angle (degrees)
Figure 3. MPI probabilities of F2 molecule for the peak intensity 2 × 1014 W/cm2 in
polar (panel A) and Cartesian (panel B) coordinates: (a) total probability, (b) 1πg (HOMO)
probability, (c) 1πu (HOMO−1) probability, (d) 3σg probability.
16
Telnov, Heslar & Chu
below). At the same time, the gap between the 3p and 3s spin orbital energies in Ar is
much larger (our calculation gives the value 13.7 eV), and the 3p contribution to the MPI
probability remains dominant for all three laser intensities.
For F2 , the total ionization probability appears smaller than that of N2 (and Ar) at the
same laser intensity 2 × 1014 W/cm2 (Fig. 3). The ratio of the MPI probabilities of Ar and
F2 (at 40◦ ) is approximately equal to 4.2. The pattern for the orientation dependence of
MPI in F2 resembles that experimentally observed in O2 [60] since both molecules have
the HOMO of the same symmetry (1πg ), and the HOMO contribution is dominant at this
intensity. The maximum in the orientation angle distribution of the total MPI probability
points at 40◦ . The HOMO−1 contribution is less important than that in N2 , and this is well
explained by the larger gap between the HOMO and HOMO−1 energies (3.2 eV).
3.3.2.
High-order harmonic generation
For non-monochromatic fields, the spectral density of the radiation energy emitted for all
the time is given by the following expression [62]:
S(ω) =
4ω 4 e
|D(ω)|2 .
6πc3
(44)
e
Here ω is the frequency of radiation, c is the velocity of light, and D(ω)
is a Fourier
transform of the time-dependent dipole moment:
Z ∞
e
dtD(t) exp(iωt).
(45)
D(ω) =
−∞
The dipole moment is evaluated as an expectation value of the electron radius-vector with
the time-dependent total electron density ρ(r, t):
Z
D(t) = d3 r rρ(r, t).
(46)
The total energy E emitted in the harmonic radiation can be calculated by integration of
S(ω):
Z ∞
E=
dωS(ω).
(47)
0
For a long enough laser pulse, the radiation energy spectrum (44) contains peaks corresponding to odd harmonics of the carrier frequency ω0 . We define the energy E(Nh ) emitted
in the Nh th harmonic (Nh is an odd integer number) as follows:
Z
(Nh +1)ω0
E(Nh ) =
dωS(ω).
(48)
(Nh −1)ω0
In Figs. 4 and 5 we present the HHG data for N2 and F2 molecules, respectively, at the
peak intensity 2 × 1014 W/cm2 . The cutoff position in the HHG spectrum for this intensity
is expected at the harmonic order 35, in fair agreement with the computed data. To show the
orientation dependence of the HHG spectra, we choose three values of the orientation angle
γ: 0◦ , 40◦ , and 90◦ which represent the limiting cases of the parallel and perpendicular
TDDFT methods for molecular systems in laser fields
0°
40°
90°
-9
Harmonic energy (a.u.)
17
10
-10
10
-11
10
-12
10
3
7
11
15
19 23 27 31
Harmonic order
35
39
43
Figure 4. Energy emitted in harmonic radiation by N2 molecule for the peak intensity
2 × 1014 W/cm2 : left (blue) bar, orientation angle γ = 0◦ ; middle (green) bar, orientation
angle γ = 40◦ ; right (red) bar, orientation angle γ = 90◦ .
-9
10
0°
40°
90°
Harmonic energy (a.u.)
-10
10
-11
10
-12
10
-13
10
-14
10
3
7
11
15
19 23 27 31
Harmonic order
35
39
43
Figure 5. Energy emitted in harmonic radiation by F2 molecule for the peak intensity 2 ×
1014 W/cm2 : left (blue) bar, orientation angle γ = 0◦ ; middle (green) bar, orientation angle
γ = 40◦ ; right (red) bar, orientation angle γ = 90◦ .
18
Telnov, Heslar & Chu
orientation as well as the intermediate angle case. For all three orientations, the HHG
signal from N2 is about an order of magnitude stronger than that from F2 ; this observation
is consistent with the MPI results of Sec. 3.3.1.: at this intensity, the MPI signal from F2
is 4 to 10 times weaker than that from N2 , depending on the orientation. The orientation
dependence of HHG also resembles that of MPI: HHG is more intense for the orientations
where MPI reaches its maximum. It is clearly seen for F2 where the radiation energy at 40◦
exceeds that at other orientations for almost every harmonic. For N2 , the HHG signal at
0◦ is dominant in the low-order part of the spectrum whereas in the central part a stronger
signal is observed at 40◦ . One can also see that the emission of the harmonic radiation at
the perpendicular orientation (γ = 90◦ ) is suppressed for both N2 and F2 in the low-order
and central parts of the HHG spectra. The maximum in the harmonic energy distribution at
90◦ is shifted to higher orders. This result is in a good accord with the recent experimental
measurements on N2 [63].
3.4.
Multielectron Effects on the Orientation Dependence of Multiphoton
Ionization of CO2
In this section, we present all-electron TDDFT calculations of the orientation-dependent
MPI of the three-center CO2 molecule [64]. The electronic structure of CO2 is solved with
the help of the Voronoi-cell finite difference (VFD) method [65]. In contrast to the ordinary
finite difference method with regular uniform grids, the VFD method can accommodate any
type of grid distributions, so-called unstructured grids, with the help of geometrical flexibility of the Voronoi diagram. To attack multicenter Coulombic singularity in all-electron
calculations of polyatomic molecules, highly adaptive molecular grids are used [64] in this
study.
Table 2 compares experimental vertical ionization potentials [66] of CO2 and absolute values of orbital binding energies computed with the LBα potential. Molecular
grids are constructed by a combination of spherical atomic grids covering large distances
(rmax ∼20 Å). The C–O bond length is fixed at 1.162 Å [67]. As one can see from Table 2,
the calculated orbital binding energies are in fairly good agreement with the experimental
data, particularly those for HOMO (1πg ) and HOMO−1 (1πu ).
Fig. 6 shows the orientation dependence of the total ionization probability. The laser
parameters used are 20-optical-cycle sin2 -envelope laser pulses with two different sets of
the wavelength and the peak intensity: (a) 820 nm and 1.1 × 1014 W/cm2 , and (b) 800 nm
and 5 × 1013 W/cm2 . For comparison, Fig. 6 includes experimental measurements [60, 61]
and MO–ADK results [60, 68]. All data sets are normalized to their maximum value. In
Fig. 6(a), two dashed lines of experiment are due to uncertainty of the measured alignment distribution [60]. The total ionization probability computed by TDDFT manifests the
center-fat propeller shape with the peak at 40◦ . The position of the peak agrees well with
both experiments [60, 61] which give it at 45◦ . As for the broadness of the central pattern,
our results agree well with the data of Ref. [61] but are different from that of Ref. [60],
the latter showing a narrower pattern. This discrepancy may be related to the experimental
uncertainty in the molecular alignment processes.
We now examine contributions of the individual orbitals to the total ionization probability. Fig. 7 shows individual ionization probabilities of multiple orbitals with 820 nm and
TDDFT methods for molecular systems in laser fields
90°
19
90°
135°
45°
180°
135°
45°
0° 180°
315°
225°
0°
315°
270°
225°
270°
Present work
Experimenta
MO-ADKa
Present work
Experimentb
MO-ADKc
14
(a) 820 nm, 1.1×10
2
13
W/cm
(b) 800 nm, 5×10
2
W/cm
Figure 6. Orientation dependence of total ionization probability of CO2 .
b Ref. [61]; c Ref. [68]
1πg,x
1πg,y
3σu (×10)
1πu,x (×10)
1πu,y (×10)
4σg (×10)
a Ref.
[60];
Figure 7. Orientation dependence of individual ionization probability of multiple orbitals
with 820 nm and 1.1 × 1014 W/cm2 .
20
Telnov, Heslar & Chu
Table 2. Absolute values of spin-orbital energies of CO2 . (A) Present DFT calculations
(eV). (B) Experimental vertical ionization energies (eV).
Spin-orbital
A
1πg (HOMO)
1πu
3σu
4σg
13.9
17.5
17.2
18.5
B
13.8 (Ref.
17.6 (Ref.
18.1 (Ref.
19.4 (Ref.
[66])
[66])
[66])
[66])
1.1×1014 W/cm2 . As the calculations at 820 nm and 1.1×1014 W/cm2 reveal [64], HOMO
(1πg ) is dominant in the total ionization and others (1πu , 3σu , and 4σg ) are scaled by 10
times. Contributions of 2σu and 3σg are negligible. In fact, the unperturbed π orbitals are
degenerate: one lies on the xz-plane (1πg,x and 1πu,x ) and the other lies on the yz-plane
(1πg,y and 1πu,y ). As the field whose polarization vector varies in the xz-plane is applied to
the molecule, 1πg,x provides the most dominant contribution to the orientation dependence
of the total ionization probability, which has the maximum at 45◦ ; 1πg,y shows a dumbbell
shape with the same probability as 1πg,x at 0◦ . Thus the center-fat propeller shape of the
total ionization probability in Fig. 6 is mostly reflected by contributions of the two HOMOs (1πg,x and 1πg,y ). On the other side, the MO–ADK model predicts the butterfly shape
with the peak at 25◦ [60, 68]. In contrast with MO–ADK model, the self-interaction-free
TDDFT approach [64] incorporates multi-electron correlation and multiple orbital effects,
and the results are in excellent agreement with experimental observation. The TDDFT [64]
results show the significance of the electron correlations and suggest that all the valence
orbitals should be taken into account, even when HOMO dominates the ionization process.
4.
4.1.
HIGH-ORDER HARMONIC GENERATION OF HETERONUCLEAR AND HOMONUCLEAR DIATOMIC
MOLECULES IN INTENSE ULTRASHORT LASER
FIELDS: AN ALL-ELECTRON TDDFT STUDY
Multiphoton Ionization of Heteronuclear and Homonuclear Diatomic
Molecular Systems
In this section, we present all-electron TDDFT calculations of the MPI of N2
and CO diatomic molecules [53].
The ground-state electronic configurations is
1σg2 1σu2 2σg2 2σu2 1πu4 3σg2 for N2 and 1σ 2 2σ 2 3σ 2 4σ 2 1π 4 5σ 2 for CO, respectively. N2 and
CO are isoelectronic molecules, both having 14 electrons and triple bonds. Since the CO
molecule has unequal nuclear charges, its ground electronic state possesses a permanent
dipole moment, calculated here to be 0.149 Debye. The corresponding experimental value
is 0.112 Debye [69]. Furthermore, there is no concept of gerade and ungerade orbitals for
CO (or any other heteronuclear diatomic molecule) since the inversion symmetry of the
TDDFT methods for molecular systems in laser fields
21
Table 3. Comparison of the field-free molecular orbital energy levels of CO and N2 , calculated with the LBα potential, and the experimental ionization potentials (in a.u.).
Orbital
Expt. [70]
LBα
1σ
19.9367
19.6142
Orbital
Expt. [66, 71, 72]
LBα
1σg
15.0492
14.7962
CO
2σ
3σ
10.8742 1.3964
10.6556 1.2549
N2
1σu
2σg
15.0492 1.3708
14.7950 1.2162
4σ
0.7239
0.7071
1π
0.6247
0.6276
5σ
0.5144
0.5086
2σu
0.6883
0.6786
1πu
0.6233
0.6199
3σg
0.5726
0.5682
potential is broken. Table 3 lists the MO energies calculated with the LBα potential, using
50 grid points in ξ and 30 grid points in η. The agreement of the calculated valence MO
energies with the experimental data is well within 0.01 a.u.
Once the time-dependent wave functions and the time-dependent electron densities are
obtained, we can calculate the time-dependent (multiphoton) ionization probability of an
individual spin-orbital according to
Pi,σ = 1 − Ni,σ (t)
(49)
Ni,σ (t) = hψi,σ (t)|ψi,σ (t)i
(50)
where
is the time-dependent population (survival probability) of the iσ-th spin-orbital.
Figure 8 presents the time-dependent (multiphoton) ionization probability of individual
spin orbital, as defined in Eq. (49). The slope of the decay of the electron population
in time determines the ionization rate. The laser (electric) field is assumed to be parallel
to the internuclear axis, and the internuclear distance for the CO (Re = 2.132 a0 ) and
N2 (Re = 2.072 a0 ) molecules is fixed at its equilibrium distance Re . Results for two
laser intensities (5 × 1013 W/cm2 and 1 × 1014 W/cm2 ) and a wavelength of 800 nm,
20-optical-cycle laser pulse are shown for CO and N2 . The orbital structure and ionization
potentials of the two molecules under consideration are close to each other. That is why one
can expect similar behavior in the laser field with the same wavelength and intensity. The
multiphoton ionization in the laser field is dominated by HOMO, that is 3σg in N2 and 5σ in
CO. As one can see from Figs. 8(a) and 8(c), at lower intensity 5 × 1013 W/cm2 , the HOMO
survival probabilities of N2 and CO are close to each other. However, at higher intensities,
the difference becomes more pronounced, at the intensity 1 × 1014 W/cm2 , the ionization
probability of CO is much larger than that of N2 (Figs. 8(b) and 8(d)). The explanation of
the phenomenon can be as follows. In intense low-frequency laser fields, the multiphoton
ionization occurs mainly in the tunneling regime. In this picture, the ionization takes place
in the DC field with slowly varying amplitude from zero to its peak value. The width of
the potential barrier depends on the field strength; the stronger the field, the narrower the
barrier. Thus the ionization occurs mainly at the peak values of the field strength. The
22
Telnov, Heslar & Chu
0
0
2σu
1πu
12
a
20
0
5
3σg
10
Optical cycles
0
4σ
3
Pi,σ × 10
5
5
10
Optical cycles
20
15
4σ 1π
CO
8
12
d
c
5σ
5
3σg
4
40
80
0
b
0
CO
60
40
0
1π
20
20
30
20
15
N2
4
N2
16
Pi,σ × 10
10
2σu
Pi,σ × 10
Pi,σ × 10
5
4
8
1πu
10
Optical cycles
15
5σ
16
20
20
0
5
10
Optical cycles
15
20
Figure 8. The time-dependent population of electrons in different spin orbitals of CO and
N2 in 800 nm, sin2 pulse laser field, with 20 optical cycles in pulse duration. N2 molecule
(a) 5 × 1013 W/cm2 , (b) 1 × 1014 W/cm2 , CO molecule (c) 5 × 1013 W/cm2 , (d) 1 × 1014
W/cm2 .
probability of the tunneling ionization is very sensitive with respect to the HOMO energy.
However, in the external field this energy is changed due to the Stark shift. The nitrogen
molecule is symmetric with respect to inversion, that is why the Stark shift in the DC field
is quadratic in the field strength and its value is quite small. On the contrary, the carbon
monoxide molecule has a permanent dipole moment, and the DC Stark shift is linear in
the field strength; at the peak values of the field, the HOMO energy can differ significantly
from its unperturbed value. We have performed the self-consistent DFT calculations of
N2 and CO in the DC electric field parallel to the molecular axis to see how large the
Stark shift can change the ionization potential of the molecule. On Table 4 we show the
HOMO energies computed at the field strength 0.7549 × 10−2 a.u. which corresponds to
Table 4. HOMO energies of N2 and CO molecules in DC electric field (positive field direction is from C to O).
Electric field (a.u.)
0
0.7549 × 10−2
−0.7549 × 10−2
N2 HOMO energy (a.u.)
−0.5682
−0.5681
−0.5681
CO HOMO energy (a.u.)
−0.5086
−0.5149
−0.5026
TDDFT methods for molecular systems in laser fields
23
the intensity 2 × 1012 W/cm2 . As one can see, even in the field as weak as 2 × 1012 W/cm2 ,
the shift of the HOMO energy in CO molecule is large. The shift depends on the direction
of the external field with respect to the position of the carbon and oxygen nuclei. In one
direction the energy level becomes higher, and in the other direction it becomes lower than
the unperturbed level. Decrease of the binding energy will result in the enhanced ionization.
In intense low-frequency laser fields, this effect can be responsible for the enhancement of
ionization of CO molecule as compared with N2 .
4.2.
High-Order Harmonic Generation of Heteronuclear and Homonuclear
Diatomic Molecules in Intense Laser Fields
After the time-dependent single electron wave functions {ψiσ } are obtained, the total electron density ρ(r, t) can be determined. The time-dependent induced dipole moment can
now be calculated as
Z
X
d(t) = d3 r zρ(r, t) =
diσ (t),
(51)
iσ
where
diσ (t) = niσ hψiσ (r, t) |z|ψiσ (r, t)i,
(52)
is the induced dipole moment of the iσ-th spin orbital, and niσ is its electron occupation
number. The power spectrum of the HHG is then acquired by taking the Fourier transform
of the total time-dependent induced dipole moment d(t):
2
Z tf
4ω 4 1
−iωt S(ω) = 3 d(t)e
dt .
(53)
3c tf − ti ti
Here c is the speed of light, and S(ω) has the meaning of the energy emitted per unit time
at the particular photon frequency ω. In figures 9–10 we present the HHG power spectra
(Eq. (53)) for the laser field intensities 5 × 1013 W/cm2 , and 1 × 1014 W/cm2 . An important
difference between the N2 and CO spectra is that the latter contain even as well as odd
harmonics. Generation of even harmonics is forbidden in systems with inversion symmetry,
such as atoms and homonuclear diatomic molecules. This selection rule does not apply to
the heteronuclear molecules with no inversion center (CO). From Figs. 9–10, one can see
that in general HHG is more efficient in CO than in N2 . However, for higher harmonics
(17 and above) the N2 spectra become dominant at the same laser intensity. As the laser
intensity increases, the maximum in the power spectra is shifted towards higher harmonics.
To investigate the detailed spectral and temporal structure of HHG for homonuclear and
heteronuclear systems, we perform the time-frequency analysis by means of the wavelet
transform of the total induced dipole moment d(t) [36, 48],
r
Z
ω iω(t−t0 ) −(ω(t−t0 ))2 /2τ 2
dω (t) = d(t)
e
e
dt.
(54)
τ
The parameter τ = 15 is chosen to perform the wavelet transformation in the following
study.
The peak emission times, te , represent the instance when the maxima of the dipole
time profile occur, and semiclassically are interpreted as the electron-ion recollision times
24
Telnov, Heslar & Chu
1.8
CO
N2
1.6
1.2
1
4
2
15
ω |d(ω)| ×10 (a.u.)
1.4
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10 12 14 16
Harmonic order
18
20
22
24
26
Figure 9. Comparison of the HHG power spectra of CO and N2 , in 800 nm, 5×1013 W/cm2
sin2 pulse laser field.
3
2.7
CO
N2
14
2.1
1.8
2
ω |d(ω)| ×10 (a.u.)
2.4
1.5
4
1.2
0.9
0.6
0.3
0
0
2
4
6
8
10 12 14 16
Harmonic order
18
20
22
24
26
Figure 10. Comparison of the HHG power spectra of CO and N2 , in 800 nm, 1 × 1014
W/cm2 sin2 pulse laser field.
TDDFT methods for molecular systems in laser fields
25
20
(a)
15
N
2
Time Profile, |d (t)| (10
-5
a.u.)
19
21
10
23
25
5
0
12
(b)
CO
8
22
23
4
24
25
26
0
6
7
8
9
10
11
12
13
14
15
Time (o.c.)
Figure 11. Time profiles for (a) N2 and (b) CO. Laser intensity used is 5 × 1013 W/cm2 ,
wavelength used is 800 nm, with 20 optical cycles in pulse duration.
[48]. For the case of the N2 molecule, the time profiles of the 19th to 25th harmonic orders
are shown in Fig. 11(a). There are two emissions occurring at each optical cycle, and
the most prominent bursts take place at the center of the laser field envelope. The time
profiles of the superimposed harmonics are rather uniform among themselves implicating
that the harmonics are partially synchronized. More importantly for the CO molecule, a
distinct feature possibly characteristic of all heteronuclear diatomic systems is observed in
Fig. 11(b) for the harmonic orders 22th to 26th. The number of dominant emissions per
optical cycle is now limited to only one. This finding is in contrast with results normally
obtained in the HHG for atoms and homogeneous molecules in which two bursts per optical
cycle are observed. The spectral profiles are as uniform as those obtained for N2 , though the
CO harmonics appear to be more synchronized than those of N2 . In this research, we present
a detailed comparison of the very high-order nonlinear optical response of the homonuclear
N2 and heteronuclear CO diatomic molecules in intense ultrashort laser fields by means of
a TDDFT with correct asymptotic long-range (−1/r) potential to ensure individual spinorbital has the proper ionization potential. We consider only the case that the molecular axis
is aligned with the laser beam direction. This is justified based on the recent experimental
development of the laser molecular alignment techniques [73–76]. We found that although
CO has only a very small permanent dipole moment, qualitatively different nonlinear optical
26
Telnov, Heslar & Chu
responses are predicted for CO and N2 . First, the MPI rate for the heteronuclear diatomic
CO molecules is larger than that for the N2 homonuclear diatomic molecules. Second, while
the laser excitation of the N2 molecules can generate only odd harmonics, both even and odd
harmonics can be produced for the CO case. In this connection, we note that the even-order
harmonics were also predicted in an earlier study of the HHG of a one-dimensional model
HD with unequal nuclear mass [77]. In this model, even-order harmonics can be produced
only by means of the breakdown of the Born-Oppenheimer approximation. However, in
our ab initio 3D study of CO with unequal nuclear mass and charge, even-order harmonics
can still be produced when the internuclear separation is fixed. Third, from our wavelet
time-frequency analysis, we found that there are two dominant rescattering (and harmonic
emission) events within each optical cycle for the N2 molecules, while there is only one
dominant rescattering event for the CO molecules.
4.3.
HHG Spectra of Homonuclear and Heteronuclear Diatomic Molecules:
Exploration of Multiple Orbital Contributions
In this section, we explore the nonlinear response of individual molecular orbitals (MO) to
the laser field and their dynamic role in formation of the HHG spectra of multielectron heteronuclear and homonuclear diatomic molecules [78]. We analyze the effect of asymmetry
of the heteronuclear molecules on their HHG spectra, a subject of largely unexplored area of
intense field molecular physics. In contrast with the homonuclear molecules which generate only odd harmonics of the laser frequency, aligned heteronuclear molecules can produce
even harmonics as well [53]. In the high-energy part of the HHG spectra, the interference
of contributions from different MO is mostly constructive for the heteronuclear molecules
and destructive for the homonuclear molecules. We present an all-electron nonperturbative
investigation of the HHG mechanisms taking into account the detailed electronic structure
and the responses of individual electrons and using N2 and CO as examples (N2 and CO
are isoelectronic molecules with 14 electrons). The ground state electronic configurations
of N2 (homonuclear molecule) is 1σg2 1σu2 2σg2 2σu2 1πu4 3σg2 . The heteronuclear molecule CO
have the ground state electronic configuration of 1σ 2 2σ 2 3σ 2 4σ 2 1π 4 5σ 2 . We analyze the
HHG spectra of the diatomic molecules and investigate the role of the contributions from
different molecular orbitals. We identify the patterns of constructive and destructive interference of the orbital contributions to the total HHG spectrum. In the high-energy part
of the spectrum, the interference appears destructive for the homonuclear molecules and
constructive for the heteronuclear molecules.
For the CO molecule, the HOMO (5σ) is dominant for the whole HHG spectrum
(Fig. 12); other orbitals contribute much less. The distinct harmonic peaks are seen up
to the order 30. In CO, the 5σ (HOMO) permanent dipole is 1.57 a.u. which is much larger
than that of the other orbitals. The density for the 5σ (and 2σ) is mostly localized on the
carbon atom. For all other orbitals the density is localized on the oxygen atom. Looking at
the time-dependent orbital dipoles for the CO molecule (Fig. 13), we can see that all other
orbitals that are localized on the oxygen atom are in phase and 5σ (carbon) is out of phase
in time.
The N2 molecule has dipole amplitudes that follow the trend:
d2σg < d1πu < dtotal < d3σg < d2σu .
(55)
TDDFT methods for molecular systems in laser fields
27
-2
(a)
total
-6
log
10
( P( ) )
-4
-8
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
32
34
-2
(b)
1
5
-6
log
10
( P( ) )
-4
-8
-10
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Harmonic order
Figure 12. Total (a) and orbital (b) harmonic power spectra of the CO molecule in the sin2
laser pulse with a peak intensity of I0 = 1 × 1014 W/cm2 . The laser pulse has a wavelength
of 800 nm and a time duration of 20 optical cycles.
1.6
a
1.5
4
Orbital dipole
1.4
5
1
1.3
-0.3
total
c
-0.6
d
-0.9
-1.2
b
-1.5
0
2
4
6
8
10
12
14
16
18
20
Time (optical cycle)
Figure 13. Time-dependent orbital (a, 5σ; b, 4σ; c, 1π) and total (d) dipole moments of the
CO molecule in the sin2 laser pulse with a peak intensity of I0 = 1 × 1014 W/cm2 . The
laser pulse has a wavelength of 800 nm and a time duration of 20 optical cycles. All dipoles
are in a.u.
28
Telnov, Heslar & Chu
The orbital dipole moments 2σg , 1πu , and 2σu are oscillating with the same sign (in phase)
of the total dipole, the 3σg has opposite sign (out of phase). So when we look at the two
orbital dipole amplitudes 2σu and 3σg with similar intensities, they oscillate in time with
different sign, canceling each other out (Fig. 14), leading to a smaller total dipole.
The behavior of the HHG spectrum (Fig. 15) for N2 is quite different than that of a
heteronuclear diatomic molecule. The HOMO (3σg ) is dominant in the middle part of the
spectrum (again up to the order 31). However, starting the 33th harmonic, the 2σu orbital has a comparable contribution which interferes destructively with that of 3σg . Thus
the result of interference is much lower than both of the single orbital spectra. Then this
two-orbital spectrum becomes comparable in magnitude with the 2σg -only HHG. Again,
the destructive contributions takes place, and the result is much lower than any of the single orbital contributions. In summary, we can say that the long HHG spectrum of N2 is a
collective multielectron effect. Destructive interference between 2σu and 3σg orbital contributions makes the resulting total HHG spectrum lower in amplitude with distinct harmonic
peaks up to the order 49. Note that single orbital HHG do not have distinct peaks, just a
smooth background, so the peaks in the high harmonic part of the total HHG spectra are the
pure multielectron interference effect.
In contrast to the N2 molecule, the full destructive interference is not possible in CO
because of the broken g − u symmetry. The induced dipole moment of the HOMO 5σ
is so large that no other orbital can make a comparable contribution. This observation is
confirmed by the calculations of other heteronuclear molecules, BF and HF [78]. In BF
molecule, the 5σ HOMO (like that of CO, since they are isoelectronic) has the greatest
dipole moment. This is understandable, since most of the electronic density resides on the
fluorine atom. The fluorine atom is more electronegative than that of the boron atom, so
it will have the greater electron density. As previous seen in the CO molecule, BF has
the same physical characteristics of its orbital HHG spectrum. Another heteronuclear
molecule, hydrogen fluoride, has the greatest permanent dipole moment (1.820 Debye),
where most of the electron density resides on the fluorine atom. For HF molecule, the
HOMO is the 1π orbital, and it has the greatest negative value of the orbital dipole moment.
However, the greatest orbital contribution to the HHG spectrum comes from the 3σ orbital,
not the HOMO (1π). Since the field is linearly polarized to the z axis and the 1π orbital is
perpendicular to the z axis, the induced dipole moment is aligned not with the 1π orbital
but with the 3σ orbital. Between the harmonic order peaks (even and odd), the 1π orbital
has greater intensity than that of the 3σ orbital. However, at the peak values (even and
odd harmonic orders) the 3σ orbital has almost the same intensity as that of the total HHG
spectrum. Thus we conclude that HF manifests the same orbital characteristics as other
heteronuclear molecules studied (CO and BF).
A more detailed investigation and comparison of the very high-order nonlinear optical
response of the homonuclear (N2 and F2 ) and heteronuclear (CO, BF, and HF) diatomic
molecules in intense ultrashort laser fields can be found in Ref. [78]. Our analysis of the
HHG spectra for the parallel orientation of the molecular axis with respect to the polarization of the laser field reveals that homonuclear molecules have destructive interference
between the orbital contributions to the total harmonic signal. This happens because the
induced dipole moments of different orbitals oscillate in time with opposite phases, so their
contributions are canceled out in the total dipole moment. Therefore a close look at the in-
TDDFT methods for molecular systems in laser fields
29
total
1.0
3
2
g
u
Orbital dipole
0.5
0.0
-0.5
-1.0
0
2
4
6
8
10
12
14
16
18
20
Time (optical cylces)
Figure 14. Time-dependent orbital and total dipole moments of the N2 molecule in the sin2
laser pulse with a peak intensity of I0 = 1 × 1014 W/cm2 . The laser pulse has a wavelength
of 800 nm and a time duration of 20 optical cycles. All dipoles are in a.u.
-6
(a)
) )
-12
log
10
-10
( P(
-8
total
-14
-16
0
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
-6
(b)
) )
-12
log
10
-10
( P(
-8
2
3
u
g
-14
-16
15
18
21
24
27
30
33
36
39
42
45
48
Harmonic order
Figure 15. Total and orbital harmonic power spectra of the N2 molecule in the sin2 laser
pulse with a peak intensity of I0 = 1 × 1014 W/cm2 . The laser pulse has a wavelength of
800 nm and a time duration of 20 optical cycles.
30
Telnov, Heslar & Chu
duced dipole moment of the homonuclear diatomic molecules can give predictions of what
orbital contributions will interfere destructively in the total HHG spectrum. The destructive
interference in the HHG spectrum accounts for the unexpected observation that some of the
individual orbitals harmonic power spectra have greater intensity than that of the total HHG.
The HHG process by the heteronuclear diatomic molecules has a quite different characteristic. First, heteronuclear molecules can generate even and odd harmonics since they lack
the inversion symmetry. Second, for all the studied heteronuclear molecules, the HOMO
contribution to the total HHG spectrum is by far dominant, and the total harmonic signal has
almost the same intensity as that produced by HOMO only. The interference between the
different orbitals in the total HHG spectrum is mostly constructive. The same features are
observed in the time-dependent induced dipole moments, with the HOMO dipole moment
having the largest oscillation magnitude.
5.
CONCLUSION
In this chapter, we have presented TDDFT approaches recently developed for accurate and
efficient treatment of the time-dependent dynamics of many-electron quantum molecular
systems. They allow the construction of orbital-independent single-particle local exchangecorrelation potential which possesses the correct (−1/r) long-range asymptotic behavior.
With the asymptotically correct potential, the energy of the highest occupied spin-orbital
provides a good approximation to the ionization potential. The generalized pseudospectral
(GPS) technique allows the construction of non-uniform and optimal spatial grids, denser
mesh nearby each nucleus and sparser mesh at longer range, leading to high-precision solution of both electronic structure and time-dependent quantum dynamics with the use of
only a modest number of spatial grid points. The TDDFT formalism along with the use
of the time-dependent GPS numerical technique provides a powerful new nonperturbative
time-dependent approach for exploration of the electron correlation and multiple orbitals
effects on strong field multiphoton processes.
Like the steady-state case, the exact form of time-dependent xc energy functional is
unknown. Most of the strong-field calculations so far (including those discussed in this
chapter) have used the adiabatic approximation, neglecting the memory-effect terms in the
xc potential. As shown by the recent study [79], the adiabatic approximation is well justified
in the case of medium-strong low-frequency laser fields. However, its validity in very strong
fields still remains to be investigated. More rigorous nonadiabatic treatment of the timedependent xc energy functional can be facilitated if some information regarding the electron
density for N -electron systems can be determined by means of the ab initio wavefunction
approach. But this task is not feasible at the current time for N > 2. Since the exact timedependent xc energy functional form is supposed to be universal and independent of N , the
information of the strong-field behavior of the simplest but nontrivial two-electron systems
will be very valuable for the future construction of time-dependent xc energy functional.
Applicability of the modern TDDFT approaches for the treatment of multiple electron
ionization processes is another problem related to the quality of time-dependent xc energy
functionals. Most of approximate xc functionals lack the important property of the exact
functional, the discontinuity of its derivative with respect to the number of particles N ,
when N passes through integer values [80]. Several attempts to apply TDDFT with such
TDDFT methods for molecular systems in laser fields
31
approximate functionals for calculations of nonsequential double ionization were unsuccessful [81, 82]. Recently it was shown [83] that the derivative discontinuity is crucial for
correct description of double ionization.
At this time, the TDDFT is the primary approach available for the treatment of timedependent processes of many-electron quantum systems in strong fields. Further extension
of the self-interaction-free TDDFT approaches to larger molecular systems will be valuable and can lead to significant advancement in the understanding of strong-field chemical
physics and atomic and molecular physics in the future.
ACKNOWLEDGMENTS
This work was partially supported by the Chemical Sciences, Geosciences and Biosciences
Division of the Office of Basic Energy Sciences, Office of Sciences, U. S. Department of
Energy, and by the U. S. National Science Foundation. We also acknowledge the partial
support of National Science Council of Taiwan (Grant No. 97-2112-M-002-003-MY3) and
National Taiwan University (Grants No. 98R0045 and 99R80870).
References
[1] Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford
University Press: Oxford, 1989.
[2] Dreizler, R. M.; Gross, E. K. U. Density Functional Theory, An Approach to the Quantum Many-Body Problem; Springer: Berlin, 1990.
[3] Gross, E. K. U., Dreizler, R. M., Eds. Density Functional Theory; NATO Advanced
Studies Institute, Series B: Physics; Plenum: New York, 1995; Vol. 337.
[4] March, N. H. Electron Density Theory of Atoms and Molecules; Academic: San
Diego, 1992.
[5] Labanowski, J. K., Andzelm, J. W., Eds. Density Functional Methods in Chemistry;
Springer: Berlin, 1991.
[6] Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864.
[7] Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1113.
[8] Vosko, S. J.; Wilk, L.; Nusair, M. T. Can. J. Phys. 1980, 58, 1200.
[9] Becke, A. D. Phys. Rev. A 1988, 38, 3098.
[10] Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785.
[11] Perdew, J. P.; Wang, Y. Phys. Rev. B 1986, 33, 8800.
[12] Zhao, Q.; Parr, R. G. Phys. Rev. A 1992, 46, R5320.
[13] Tong, X. M.; Chu, S. I. Phys. Rev. A 1997, 55, 3406.
32
Telnov, Heslar & Chu
[14] Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, 5048.
[15] Runge, E.; Gross, E. K. U. Phys. Rev. Lett. 1984, 52, 997.
[16] Gross, E. K. U.; Kohn, W. Phys. Rev. Lett. 1985, 55, 2850.
[17] Mahan, G. D.; Subbaswamy, K. R. Local density theory of polarizability; Plenum
Press: New York, 1990.
[18] Petersilka, M.; Gossmann, U. J.; Gross, E. K. U. Phys. Rev. Lett. 1996, 76, 1212.
[19] Casida, M. E. In Recent Developments and Applications of Modern Density Functional Theory; Seminario, J. M., Ed.; Elsevier: Amsterdam, 1996; p 391.
[20] Hsu, C. P.; Hirata, S.; Head-Gordon, M. J. Phys. Chem. A 2001, 105, 451.
[21] Onida, G.; Reining, L.; Rubio, A. Rev. Mod. Phys. 2002, 74, 601.
[22] Osinga, V. P.; van Gisbergen, S. J. A.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys.
1997, 106, 5091.
[23] Hohm, U.; Goebel, D.; Grimme, S. Chem. Phys. Lett. 1997, 272, 1059.
[24] Chelikowsky, J. R.; Kronik, L.; Vasiliev, I. J. Phys. Condens. Matter 2003, 123,
062207.
[25] Yabana, K.; Bertsch, G. F. Phys. Rev. A 1999, 60, 1271.
[26] Burke, K.; Werschnik, J.; Gross, E. K. U. J. Chem. Phys. 2005, 123, 062206.
[27] Levy, M. Proc. Nat. Acad. Sci. U.S.A. 1979, 76, 6062.
[28] Lieb, E. H. Int. J. Quant. Chem. 1983, 24, 243.
[29] Marques, M. A. L.; Gross, E. K. U. In A Primer in Density-Functional Theory; Fiolhais, C., Nogueira, F., Marques, M. A. L., Eds.; Lecture Notes in Physics; Springer:
Berlin, 2003; Vol. 620; pp 144–184.
[30] Tong, X. M.; Chu, S. I. Phys. Rev. A 1998, 57, 452.
[31] Sharp, R. T.; Horton, G. K. Phys. Rev. 1953, 90, 317.
[32] Talman, J. D.; Shadwick, W. F. Phys. Rev. A 1976, 14, 36.
[33] Krieger, J. B.; Li, Y.; Iafrate, G. F. Phys. Lett. A 1990, 146, 256.
[34] Krieger, J. B.; Li, Y.; Iafrate, G. F. Phys. Rev. A 1992, 45, 101.
[35] Ullrich, C. A.; Gossmann, U. J.; Gross, E. K. U. Phys. Rev. Lett. 1995, 74, 872.
[36] Chu, X.; Chu, S. I. Phys. Rev. A 2001, 63, 023411.
[37] Telnov, D. A.; Chu, S. I. Phys. Rev. A 2007, 76, 043412.
TDDFT methods for molecular systems in laser fields
33
[38] Telnov, D. A.; Chu, S. I. Phys. Rev. A 2005, 71, 013408.
[39] Chu, X.; Chu, S. I. Phys. Rev. A 2001, 64, 063404.
[40] Abramowitz, M., Stegun, I., Eds. Handbook of Mathematical Functions; Dover: New
York, 1965.
[41] Tao, L.; McCurdy, C. W.; Rescigno, T. N. Phys. Rev. A 2009, 79, 012719.
[42] Marston, C. C.; Balint-Kurti, G. G. J. Chem. Phys. 1989, 91, 3571.
[43] Telnov, D. A.; Chu, S. I. Phys. Rev. A 2009, 80, 043412.
[44] Chu, X.; Chu, S. I. Phys. Rev. A 2001, 63, 013414.
[45] Hermann, M. R.; Fleck, J. A. Phys. Rev. A 1988, 38, 6000.
[46] Feit, M. D.; Fleck, Jr., J. A.; Steiger, A. J. Comput. Phys. 1982, 47, 412.
[47] Tong, X. M.; Chu, S. I. Chem. Phys. 1997, 217, 119.
[48] Tong, X. M.; Chu, S. I. Phys. Rev. A 2000, 61, 021802(R).
[49] Chui, C. K. An Introduction to Wavelets; Academic Press: New York, 1992.
[50] Marques, M. A. L., Ullrich, C. A., Nogueira, F., Rubio, A., Burke, K., Gross, E. K. U.,
Eds. Time-Dependent Density Functional Theory; Springer: Berlin, 2006.
[51] Chu, S. I. J. Chem. Phys. 2005, 15, R1517.
[52] Telnov, D. A.; Chu, S. I. Phys. Rev. A 2009, 79, 041401(R).
[53] Heslar, J.; Carrera, J. J.; Telnov, D. A.; Chu, S. I. Int. J. Quant. Chem. 2007, 107,
3159.
[54] Schipper, P. R. T.; Gritsenko, O. V.; van Gisbergen, S. J. A.; Baerends, E. J. J. J. Chem.
Phys. 2000, 112, 1344.
[55] Chu, X.; Chu, S. I. Phys. Rev. A 2004, 70, 061402(R).
[56] Lofthus, A.; Krupenie, P. H. J. Phys. Chem. Ref. Data 1977, 6, 113.
[57] Cornford, A. B.; Frost, D. C.; McDowell, C. A.; Ragle, J. L.; Stenhouse, I. A. J. Chem.
Phys. 1971, 54, 2651.
[58] Siegbahn, K.; Nordling, C.; Johansson, G.; Hedman, J.; Hedén, P. F.; Hamrin, K.;
Gelius, U.; Bergmark, T.; Werme, L. O.; Manne, R.; Baer, Y. ESCA Applied to Free
Molecules; North Holland: Amsterdam, 1969.
[59] Huber, K. P.; Herzberg, G. Molecular spectra and molecular structure. IV. Constants
of diatomic molecules; Van Nostrand Reinhold: New York, 1979.
34
Telnov, Heslar & Chu
[60] Pavičić, D.; Lee, K. F.; Rayner, D. M.; Corkum, P. B.; Villeneuve, D. M. Phys. Rev.
Lett. 2007, 98, 243001.
[61] Thomann, I.; Lock, R.; Sharma, V.; Gagnon, E.; Pratt, S. T.; Kapteyn, H. C.; Murnane, M. M.; Li, W. J. Phys. Chem. A 2007, 112, 9382.
[62] Landau, L. D.; Lifshitz, E. M. The classical theory of fields; Pergamon Press: Oxford,
1975.
[63] McFarland, B. K.; Farrell, J. P.; Bucksbaum, P. H.; Gühr, M. Science 2008, 322, 1232.
[64] Son, S. K.; Chu, S. I. Phys. Rev. A 2009, 80, 011403(R).
[65] Sukumar, N. Int. J. Numer. Meth. Eng. 2003, 57, 1.
[66] Turner, D. W.; Baker, C.; Baker, A. D.; Brundle, C. R. Molecular photoelectron spectroscopy; Wiley: London, 1970.
[67] Herzberg, G. Molecular spectra and molecular structure. III. Electronic spectra and
electronic structure of polyatomic molecules; Van Nostrand: New York, 1966.
[68] Le, A. T.; Tong, X. M.; Lin, C. D. J. Mod. Opt. 2007, 54, 967.
[69] Nelson, Jr., R. D.; Lide, D. R.; Maryott, A. A. A. A. National Standard Reference Data
Series Natl. Bur. Stand. (U.S.) Circ. No. 10; US GPO: Washington, DC, 1967.
[70] Siegbahn, K. J. Electron Spectrosc. Relat. Phenom. 1974, 5, 3.
[71] Johansson, G.; Hedman, J.; Berndtsson, A.; Klasson, M.; Nilsson, R. J. Electron Spectrosc. Relat. Phenom. 1973, 2, 295.
[72] Hamnett, H.; Stoll, W.; Brion, C. E. J. Electron Spectrosc. Relat. Phenom. 1976, 8,
367.
[73] Leibscher, M.; Averbukh, I. S.; Rabitz, H. Phys. Rev. A 2004, 69, 013402.
[74] Stapelfeldt, H.; Seideman, T. Rev. Mod. Phys. 2003, 75, 543.
[75] Pinkham, D.; Jones, R. R. Phys. Rev. A 2005, 72, 023418.
[76] Pinkham, D.; Mooney, K. E.; Jones, R. R. Phys. Rev. A 2007, 75, 013422.
[77] Kreibich, T.; Lein, M.; Engel, V.; Gross, E. K. U. Phys. Rev. Lett. 2001, 87, 103901.
[78] Heslar, J.; Telnov, D. A.; Chu, S. I. Phys. Rev. A 2011, accepted.
[79] Thiele, M.; Gross, E. K. U.; Kümmel, S. Phys. Rev. Lett. 2008, 100, 153004.
[80] Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, Jr., J. L. Phys. Rev. Lett. 1982, 49, 1691.
[81] Petersilka, M.; Gross, E. K. U. Laser Phys. 1999, 9, 105.
[82] Lappas, D. G.; van Leeuwen, R. J. Phys. B 1998, 31, L249.
[83] Lein, M.; Kümmel, S. Phys. Rev. Lett. 2005, 94, 143003.