TDCDFT beyond the adiabatic approximation: from multiple excitations to dissipation

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TDCDFT beyond the adiabatic approximation from
multiple excitations to dissipation
Carsten A. Ullrich University of Missouri-Columbia
Thanks to H.O. Wijewardane, I.V. Tokatly, G.
Vignale
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Overview
? Beyond the adiabatic approximation why and
how? ? Lagrangian formulation of
TDDFT ? TDCDFT and the VK functional ? TDKS
equation with memory ? Dissipation where does
the energy go?
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TDSE versus TDKS
Full many-body TDSE linear equation,
instantaneous interactions.
TDKS equation nonlinear (Hxc), memory-dependent
(xc) Hamiltonian.
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Two kinds of xc memory in TDDFT
Dependence on initial states, except when
starting from the ground state
E. Runge and E.K.U. Gross, PRL 52, 997 (1984) R.
van Leeuwen, PRL 82, 3863 (1999)
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The adiabatic approximation
Adiabatic means no history dependence, no
memory.
ALDA depends only on the density at the same
space-time point
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The adiabatic approximation
? In general, the adiabatic approximation works
well for excitations which have an analogue in
the KS system (single excitations) ? formally
justified only for infinitely slow electron
dynamics. But why is it that the frequency
dependence seems less important?
The frequency scale of vxc is set by correlated
multiple excitations, which are absent in the KS
spectrum.
? Adiabatic approximation fails for more
complicated excitations (multiple,
charge-transfer) (see N. Maitra and K. Burke) ?
misses dissipation of long-wavelength plasmon
excitations
Fundamental question what is the proper
extension of the LDA into the dynamical regime?
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Ultranonlocality and the density
?
x
x0
An xc functional that depends only on the local
density (or its gradients) cannot see the motion
of the entire slab. A density functional needs
to have a long range to see the motion through
the changes at the edges.
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Upgrading TDDFT TDCDFT
nonlocal
nonlocal
nonlocal
local
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TDDFT in the Lagrangian frame
I.V. Tokatly, PRB 71, 165104 and 165105
(2005) C.A. Ullrich and I.V. Tokatly, PRB 73,
235102 (2006)
? use a local reference frame that moves with
the fluid. ? basic variables positions of
fluid elements and their deformations ?
nonlinear coordinate transformation
Lagrangian coordinate
Cauchys deformation tensor in the laboratory
frame
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TDDFT in the Lagrangian frame stress tensor
? xc stress tensor is a spatially local
functional of (which, in turn, is a functional
of the velocity) ? in general, it contains both
elastic and dissipative effects. ? This gives
the exact extension of LDA into the dynamical
regime ? In the limit of small deformations,
this reduces to the nonlinear VK expression
of TDCDFT
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Nonlinear elastic approximation
If we neglect dissipation, a nonlinear local
approximation for the stress tensor can be
rigorously derived
where
and is a known function.
? Exact in the high-frequency limit, for any
deformation ? For small deformations, this
reduces to the purely elastic high-frequency
limit of TDCDFT. ? deviations of the deformation
tensor g from dij can be viewed as a measure
of nonadiabaticity.
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Simple 1D models
C.A.Ullrich and I.V. Tokatly, PRB 73, 235102
(2006)
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Breakdown of the ALDA
L-TDDFT in the high-frequency, purely elastic
limit (?gtgt?p)
(exact)
Sloshing mode small deformation,
minor corrections to ALDA
Breathing mode large deformation,
ALDA breaks down
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The VK functional in TDCDFT
G. Vignale and W. Kohn, PRL 77, 2037 (1996)
G. Vignale, C.A.Ullrich, S. Conti, PRL 79, 4878
(1997)
? Valid up to second order in the spatial
derivatives ? The gradients need to be small, but
the velocities themselves can be large
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Nonlinear TDCDFT xc stress tensor
time-dependent velocity field
where the viscosity coefficients are defined as
Fourier transforms
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xc viscosity coefficients
The xc complex xc viscosities describe
dissipative and elastic behavior
shear modulus
reflect the stiffness of Fermi surface against
defor- mations
dynamical bulk modulus
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xc kernels of the homogeneous electron gas
GK E.K.U. Gross and W. Kohn, PRL 55, 2850
(1985) NCT R. Nifosi, S. Conti, and M.P. Tosi,
PRB 58, 12758 (1998) QV X. Qian and G. Vignale,
PRB 65, 235121 (2002)
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Nonlinear C-TDDFT 1D systems
z
Consider a 3D system which is uniform along two
directions can transform xc vector
potential into scalar potential
with the memory-dependent xc potential
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The xc memory kernel
where the memory kernel is given by
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The xc memory kernel
H.O. Wijewardane and C.A.U., PRL 95, 086401 (2005)
rs3

• Y(0) has zero slope
• purely elastic in
• high-frequency limit

0.25 Tplasma
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xc potential with memory simple model
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xc potential with memory full TDKS calculation
40 nm GaAs/AlGaAs
Weak excitation (initial field 0.01)
ALDA ALDAM
Strong excitation (initial field 0.5)
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...but where does the energy go?
? The system is not driven by external fields, so
the energy should be conserved. ? In linear
response calculations of atomic excitation
energies, the VK functional gives a finite
linewidth, which is unphysical.
R. DAgosta and G. Vignale, PRL 96, 016405 (2006)
? collective motion along z is coupled to the
in-plane degrees of freedom ? the x-y degrees of
freedom act like a reservoir ? decay into
multiple particle-hole excitations
This is the situation for infinite systems. But
what about finite systems?
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Example two electrons on a 2D quantum strip
C.A. Ullrich, J. Chem. Phys. 125, 234108 (2006)
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Example two electrons on a 2D quantum strip
initial-state density
?10 L50 F0.02
exact
LDA
? Compare exact calculation (time-dependent CI)
with TDKS ? Initial state constant electric
field, which is suddenly switched off ? After
switch-off, free propagation of the
charge-density oscillations
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Construction of the exact xc potential
Step 1 solve full 2-electron Schrödinger equation
Step 2 calculate the exact time-dependent density
Step 3 find that TDKS system which reproduces
the density
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Construction of the exact xc potential
Ansatz
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2D quantum strip charge-density oscillations
z
density
adiabatic Vxc
exact Vxc
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2D quantum strip time-dependent dipole moment
?10, L50, F0.02
?10, L100, F0.02
exact
ALDA
? Exact calculations give a beating pattern of
d(t), due to a superposition of dipole
oscillations involving single and double
excitations ? Recurrence time increases with
length of the strip ? ALDA misses the beating
pattern since it has no multiple excitations
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2D quantum strip TDCDFT
? d(t) is exponentially damped ? The VK
functional cannot tell that the system is
finite. It treats the system locally like
a homogeneous electron gas. ? infinite
recurrence time emerges in the thermo-
dynamic limit of the system ? damping of d(t) is
due to decoherence, involving many
excitations with a continuous spectrum
L50
L100
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Energy of the TDKS system (no TD external force)
without TD external force, is constant
defines
Rate of change of adiabatic energy work done by
dynamic xc forces
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2-electron system adiabatic energy
? ALDA gives constant energy, VK dissipates
energy exponentially ? The exact adiabatic
energy is not constant but oscillates ? In
order to reproduce the modulations of the
dipole amplitude, the exact xc potential
acts like an external force which
alternatingly damps and drives the system.
L50
VK
L100
VK
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Summary
? density-based nonadiabatic xc functionals in
TDDFT are plagued by ultranonlocality ?
upgrading to TDCDFT makes a local approximation
possible ? nonlinear VK functional emerges from
Lagrangian-TDDFT ? Time-dependent VK unphysical
dissipation in finite systems, but OK in the
thermodynamic limit ? Multiple excitations in
TDDFT require nonadiabatic xc potential ?
Breakdown of ALDA for large, rapid deformations