Electron-molecule scattering from time-dependent density functional theory
Adam Wasserman,1 Neepa T. Maitra,2 and Kieron Burke1
1
Department of Chemistry and Chemical Biology,
Rutgers University, 610 Taylor Rd., Piscataway, NJ 08854, USA
2
Department of Physics and Astronomy, Hunter College of the City
University of New York, 695 Park Av., New York, NY 10021, USA
(Dated: December 23, 2003)
An approach to low-energy electron-molecule scattering based on the linear response formalism
of time-dependent density functional theory is given. Exact formulas to extract the transmission
amplitude from the susceptibility in one-dimension are tested on a simple model. Scattering from
the ground-state Kohn-Sham potential of the neutral atom can be a good approximation at low
energies, as shown in electron-He+ scattering.
PACS numbers:
The study of electron-atom and electron-molecule collisions is of pivotal importance in a variety of fields, including semiconductor plasma etching, astrochemistry,
and atmospheric science [1, 2]. When the collision occurs at low energies, the calculations become especially
difficult due to correlation effects between the projectile
electron and those of the target [3, 4]. Several methods exist for the accurate calculation of scattering cross
sections for electrons colliding with atoms or molecules
[2, 5], but they share an intrinsic limitation: their computational cost grows rapidly with the number of electrons
in the target, rendering them useless in practice when the
systems of interest are large. The field of quantum chemistry faced similar difficulties until recently, when only
wavefunction-based methods were available for the calculation of bound-state properties of molecular systems.
With the advent of density functional theory (DFT) [6],
computational studies of the ground-state properties of
many systems became feasible for the first time.
Although correlation effects are the speciality of DFT,
no rigorous theory based on DFT presently exists to
study electronic scattering. Time-dependent density
functional theory (TDDFT)[7] is being applied to calculate optical response, response to very intense laser fields,
and even as a method for approximating non-local effects
in ground-state calculations [8]. Here, we extend the
application of linear response TDDFT [9] to electronatom scattering. A generalization to the problem of
electron-molecule scattering is straightforward whenever
the Born-Oppenheimer approximation is valid (otherwise
multicomponent DFT [10] is required). The formalism we present yields exact scattering amplitudes if two
quantities, the ground-state exchange-correlation potential (vXC ) and the exchange-correlation kernel of TDDFT
(fXC ), are known exactly.
The basic idea is simple. We want to calculate lowenergy electron scattering from an N -electron target. In
a classical picture, for very low energy, the projectile electron spends a long time in the neighborhood of the target,
and can become fully correlated with the target electrons.
We calculate the ground-state KS potential for N + 1
electrons, e.g., the negative ion, if the target is neutral.
We then apply linear-response TDDFT in a novel way to
extract scattering information from the asymptotic behavior of the continuum states of the (N + 1)-electron
system, analogous to the use of R-Matrix theory[11].
Our starting point is the Dyson-like response equation
that relates the susceptibility χ(r, r0 ; ω) of an interacting system of ground-state density n(r), with that of its
ground-state Kohn-Sham analog, χs (r, r0 ; ω) [9]. In operator form (∗ indicates spatial convolution):
χ = χs + χs ∗ fHXC∗χ
(1)
We treat only systems that bind N + 1 electrons,
so our formalism does not apply to electron scattering from many negative ions. In Eq.(1), fHXC is the
Hartree-exchange-correlation kernel (we use atomic units
throughout):
δ(t − t0 )
δvXC (r, t) 0
0
fHXC[n](r, r ; t − t ) ≡
, (2)
+
|r − r0 |
δn(r0 , t0 ) n
which is a functional of the (N + 1)-ground-state density
n(r). In Eq.(2), vXC (r, t) is the time-dependent exchangecorrelation potential when a time-dependent perturbation is applied to the (N + 1)-ground state. Our prescription, proven below, is: (1) Solve the ground-state
Kohn-Sham equations for the N + 1 system; (2) construct χs ; (3) solve Eq.(1) to obtain χ; and (4) extract the
scattering information at energy ε from the susceptibility
evaluated at a frequency ε + I, where I is the ionization
potential of the N + 1 system. Only the asymptotic behavior of the susceptibility is required, so step (3) doesn’t
require a full solution of Eq.(1).
At this point, we limit the derivation to onedimensional systems with finite-range interactions.
Later, we present results for a real system that leave
little doubt that the derivation can be generalized. Begin by writing the spin-decomposed susceptibility in the
Lehman representation:
"
#
X Fnσ (x)F ∗ 0 (x0 )
0
nσ
+ cc(ω → −ω) ,
χσσ0 (x, x ; ω) =
ω − Ωn + i0+
n
(3)
2
with
Fnσ (x) = hΨ0 |n̂σ (x)|Ψn i ; n̂σ (x) =
N
+1
X
i=1
δ(x − x̂i )δσσ̂i
(4)
where Ψ0 is the ground-state of the (N + 1)-electron system, Ψn its nth excited state, and n̂σ (x) is the σ-spin
density operator. In Eq.(3) Ωn is the Ψ0 → Ψn transition frequency, and “cc” indicates complex conjugate.
Consider now large x and x0 , where all the (N +1)-bound
states are exponentially small and the density is dominated by the decay of the highest occupied orbital. The
ground-state wavefunction behaves as [12]:
r
n(x)
N
Ψ0 → ψ0 (x2 , ...xN +1 )
S0 (σ, σ2 , ...σN +1 ) (5)
x→∞
N +1
where ψ0N is the ground-state wavefunction of the N electron system (the target), S0 the spin function of the
ground state and n(x) the (N + 1)-ground-state density.
Similarly,
φk (x)
Ψn → ψnNt (x2 , ...xN +1 ) √ n
x→∞
N +1
Sn (σ, σ2 , ...σN +1 ) (6)
where ψnNt is an eigenstate of the target (labeled by nt ),
Sn is the spin function of the nth excited state, and
φkn (x) is a one-electron orbital.
We restrict our attention to elastic scattering, so the
contribution to Fnσ (x) from channels where the target is
excited vanishes as x → ∞ due to orthogonality. Inserting Eqs.(5) and (6) into Eq.(4), and taking into account
the antisymmetry of both Ψ0 and Ψn , we find
p
n(x)φkn (x)δ0,nt
Fnσ (x) →
x→∞
X
×
S0∗ (σ...σN +1 )Sn (σ...σN +1 )
(7)
σ2 ...σN +1
The susceptibility at large distances is then obtained
from using Eq.(7) in Eq.(3):
X
p
n(x)n(x0 )
χ(x, x0 ; ω) =
χσσ0 (x, x0 ; ω) 0→
σσ 0
×
X
n
φkn (x)φ∗kn (x0 )
ω − Ωn + iη
x,x →±∞
δ0,nt δS0 ,Sn + cc(ω → −ω) (8)
Since only scattering states of the (N + 1)-optical potential contribute to the sum in Eq.(8) at large distances,
it
√
becomes an integral over wavenumbers k = 2ε, where
ε is the energy of the projectile electron:
Z ∞
X φkn (x)φ∗k (x0 )
1
φk (x)φ∗k (x0 )
n
→
dk
ω − Ωn + iη x,x0 →±∞ 2π 0[R],[L] ω − Ωk + iη
n
(9)
In this notation, the √
functions φkn are box-normalized,
and φkn (x) = φk (x)/ L, where L → ∞ is the length of
the box. The transition frequency Ωn = EnN +1 − E0N +1
is now simply Ωk = E0N + k 2 /2 − E0N +1 = k 2 /2 + I,
where I is the first ionization potential of the (N + 1)system, and E0M and EnM are the ground and nth excited
state energies of the M -electron system. The subscript
“[R],[L]” implies that the integral is over both orbitals
satisfying right and left boundary conditions:
±ikx
[R]
e
+ rk e∓ikx , x → ∓∞
(10)
φk[L] (x) →
±ikx
tk e
, x → ±∞
When x → −∞ and x0 = −x the integral of Eq.(9)
is dominated by
p a term that oscillates in space with
wavenumber 2 2(ε − I) and amplitude given by the
transmission amplitude for spin-conserving collisions tk
at that wavenumber. Denoting this oscillatory part of χ
by χosc , we obtain:
t(ε) = lim
x→−∞
"
#
√
i 2ε
p
χosc (x, −x; ε + I) . (11)
n(x)n(−x)
While this formula also applies to the KS system, its
transmission ts (ε) can be easily obtained by solving a potential scattering problem (i.e. scattering off the (N + 1)ground-state Kohn-Sham potential). The exact amplitudes t(ε) of the many-body problem are related to the
ts (ε) through Eqs.(11) and (1). The main result of this
work is that the time-dependent response of the (N + 1)electron ground-state contains the scattering information, and this is accesible via TDDFT.
We illustrate our method on a simple 1-d model of
an electron scattering off a one-electron atom of nuclear
charge Z [13]:
1 d2
1 d2
−
− Zδ(x1 ) − Zδ(x2 ) + λδ(x1 − x2 ) ,
2 dx21 2 dx22
(12)
Electrons interact via a delta-function repulsion, scaled
by λ. With λ = 0 the ground state density is a simple
exponential, analogous to hydrogenic atoms in 3d.
(i) Exact solution in the weak interaction limit: First,
we solve for the exact transmission amplitudes to first
order in λ using the static exchange method [14]. The
total energy must be stationary with respect to variations of both the bound (φb ) and scattering (φs ) orbitals that form the spatial part of the
√ Slater determinant: (φb (x1 )φs (x2 ) ± φb (x2 )φs (x1 )) / 2, where the upper sign corresponds to the singlet, and the lower sign to
the triplet case. The static-exchange equations are:
1 d2
2
+ γ|φs,b (x)| − Zδ(x) φb,s (x) = µb,s φb,s (x) ,
−
2 dx2
(13)
where γ = 2λ for the singlet, and 0 for the triplet. Thus
the triplet transmission amplitude is that of a simple δfunction:
Ĥ = −
ttrip = t0 =
ik
,
Z + ik
(14)
3
√
with k = 2µs . For the singlet, even though Eqs.(13)
are coupled, the effect of the scattered electron on the
bound electron can be neglected to first order in λ due
to the delocalized nature of φs . The resulting scattering
state has transmission amplitude:
−ik 2
t1 =
(k − iZ)2 (k + iZ)
1
Re(t)
0.8
0.4
0.3
0.2
0.1
0
1
Ψ0 (x1 σ1 , x2 σ2 ) = √ φ0 (x1 )φ0 (x2 ) [δσ1 ↑ δσ2 ↓ − δσ1 ↓ δσ2 ↑ ] ,
2
(16)
where the orbital φ0 (x) satisfies [15, 16]:
1 d2
2
−
Zδ(x)
+
λ|φ
(x)|
−
φ0 (x) = µφ0 (x) (17)
0
2 dx2
To first order in λ,
√
λ −3Z|x|
Ze−Z|x| + √
2e
+ e−Z|x| (4Z|x| − 3)
8 Z
(18)
The bare Kohn-Sham transmission amplitudes ts (ε) characterize the asymptotic behavior of the positive-energy
solutions of vs (x) = −Zδ(x) + λ|φ0 (x)|2 , and can be
obtained exactly to first order in λ by a distorted-wave
Born approximation (see e.g. Ref.[17]):
ts = t0 + λt1
(19)
The result is plotted in Fig. 1, along with the singlet and
triplet transmission amplitudes of the interacting system.
We note that ts averages the interacting triplet and singlet t’s at any energy, and approaches either at both low
and high energies.
We now apply Eq.(11) to show that the fHXC -term of
Eq.(1) corrects the ts values to their exact singlet and
triplet amplitudes. We need fHXC only to first order in
λ:
σσ 0
0
0
fHX (x, x ; ω) = λδ(x − x )(1 − δσσ0 ) ,
(20)
where the fHXC of Eq.(1) is given to first order in λ by
P
σσ 0
fHX = fH + fX = 41 σσ0 fHX
. For two electrons fHX =
1
f
.
Eq.(20)
yields:
2 H
χ(x, x0 ; ω) = χs (x, x0 ; ω)+
λ
2
Z
dx00 χs (x, x00 ; ω)χ(x00 , x0 ; ω)
(21)
Since the ground-state of the (N + 1)-system is a
spin-singlet, the Kronecker delta δS0 ,Sn in Eq.(8) implies that only singlet scattering information may be
extracted from χ, whereas information about triplet
scattering requires the magnetic susceptibility M =
Im(t)
(ii) Our DFT solution: We now test our DFT approach
to see if it can reproduce these exact results. The groundstate of the N + 1 system (N = 1) is given to first order
in λ by:
φ0 (x) =
interacting
KS
0.2
0
,
triplet
0.4
(15)
tsing = t0 + 2λt1
singlet
0.6
triplet
singlet
0
1
2
3
4
k
5
6
7
8
FIG. 1: Real and imaginary parts of the Kohn-Sham transmission amplitude ts , and of the interacting singlet and triplet
amplitudes. Z = 2 and λ=0.5 in this plot.
P
σσ 0 (σσ
0
)χσσ0 , related to the Kohn-Sham susceptibility
by spin-TDDFT [18]:
Z
λ
M(x, x0 ; ω) = χs (x, x0 ; ω)−
dx00 χs (x, x00 ; ω)M(x00 , x0 ; ω)
2
(22)
For either singlet or triplet case, since the correction to χs is already multiplied by λ, the leading correction to ts (ε) is determined by the same quantity,
(0)
(0)
(0)
χ̂s ∗ χ̂s , where χ̂s is the 0th order approximation
to the Kohn-Sham susceptibility (i.e. with vs (x) =
(0)
vs (x) = −Zδ(x)), given by [19]:
p
0
χ(0)
n(x)n(x0 ) g > (x, x0 ; ε)
(23)
s (x, x ; ε + I) =
>∗
0
+ g (x, x ; −ε − 2I) .
The outgoing Green’s function g > (x, x0 ; ε) is constructed
according to the prescription [20]:
g > (x, x0 ; ε) = −
2 [L]
[R]
φ (x< )φk (x> ) ,
W k
(24)
with x< and x> denoting, respectively, the smaller and
greater of (x, x0 ), and W being the wronskian between
[L]
[R]
(0)
φk and φk (the scattering states of vs (x)). The result
0
when x → −∞ and x → −x is:
Z
λ
00
(0) 00
lim
dx00 χ(0)
s (x, x ; ε + I)χs (x , −x; ε + I) =
x→−∞ 2
p
√
n(x)n(−x)
√
(25)
(λt1 ) e−2i 2εx + n.o.
i 2ε
where “n.o” stands for non-oscillatory terms.
Eqs.(11) and (21) we find
tsing = ts + λt1 , ttrip = ts − λt1 ,
in complete agreement with Eqs.(14) and (15).
From
(26)
4
triplet
0.8
0.6
0.2
0
singlet
KS
interacting
TDDFT
0
0.5
*
0.4
*
s−phase shifts
found by inverting the KS equation using the groundstate density of an extremely accurate wavefunction calculation of the He atom [21]. We calculate the low-energy
Kohn-Sham s-phase shifts from this potential, and compare with the accurate results in Fig. 2. The results
demonstrate that the N + 1-electron ground-state KS
potential is an excellent starting point for approximating bound-free correlation, and are similar to those of
our one-dimensional model.
We go further, and find the TDDFT corrections to
these phase-shifts. The quantum defect µnl of the boundbound transitions (angular momentum l) is defined by
*
1
1
E
1.5
2
FIG. 2: He atom: s-phase shifts as a function of energy (in
Hartrees), for the exact KS potential, and accurate wavefunction calculations of the interacting system from Ref.[22].
TDDFT results at zero energy are indicated by stars.
The method tested in the preceeding example is
applicable to any one-dimensional scattering problem.
Eqs.(11) and (1) provide a way to obtain scattering information for an electron that collides with an N -electron
target entirely from the (N +1)-ground-state Kohn-Sham
susceptibilty (and a given approximation to fXC ). A potential scattering problem is solved first from the (N +1)ground-state KS potential, and the scattering amplitudes
thus obtained are further corrected by fXC to account for
e.g. polarization effects.
We conclude our results with a real case, electron scattering from the He+ ion. In Fig. 2, we plot the results of
a recent highly accurate wavefunction calculation[22] of
the singlet and triplet s-phase shifts (which immediately
yield the corresponding elastic scattering cross-sections).
For this system, an essentially exact ground-state potential for the N + 1 electron system is known. This was
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KS scattering cross sections is ongoing.
We thank Michael Morrison for inspiring discussions.
This work was supported by the Petroleum Research
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