Transcript Document 7296400

ISSPI: Time-dependent DFT
Kieron Burke and friends
UC Irvine Physics and Chemistry Departments
http://dft.uci.edu
ISSP I
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Recent reviews of TDDFT
To appear in Reviews of Computational Chemistry
ISSP I
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Book: TDDFT from Springer
ISSP I
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TDDFT publications in recent years
Search ISI web of Science for topic ‘TDDFT’
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250
TDDFT pubs
200
150
100
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0
1997
1999
2001
2003
2005
• Warning! By 2300, entire mass of universe will be TTDFT papers
Road map
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TD quantum mechanics->TDDFT
Linear response
Overview of all TDDFT
Does TDDFT really work?
Complications for solids
Currents versus densities
Elastic scattering from TDDFT
ISSP I
5
•
TDDFT
Basic points
– is an addition to DFT, using a different theorem
– allows you to convert your KS orbitals into optical
excitations of the system
– for excitations usually uses ground-state approximations
that usually work OK
– has not been very useful for strong laser fields
– is in its expansion phase: Being extended to whole new
areas, not much known about functionals
– with present approximations has problems for solids
– with currents is more powerful, but harder to follow
– yields a new expensive way to get ground-state Exc.
ISSP I
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TD quantum mechanics
ISSP I
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Current and continuity
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Current operator:
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Acting on wavefunction:
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Continuity:
ISSP I
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Runge-Gross theorem (1984)
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Any given current density, j(r,t), and initial
wavefunction, statistics, and interaction, there’s
only one external potential, vext(r,t), that can
produce it.
Imposing a surface condition and using continuity,
find also true for n(r,t).
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Action in RG paper is WRONG
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von Leeuwen gave a constructive proof (PRL98?)
ISSP I
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TD Kohn-Sham equations
•Time-dependent KS equations:
di (rt )
 1 2

   vS (rt ) i (rt )  i
dt
 2

• Density:
N
 (rt )   i (rt )
2
i 1
• XC potential:
Depends on entire
history(MEMORY)
vS (rt )  vext (rt )   d r '
3
 (r' t )
r  r'
 vXC [n; (0), (0)](rt )
initial state(s)
dependence(MEMORY)
Road map
•
•
•
•
•
•
•
TD quantum mechanics->TDDFT
Linear response
Overview of all TDDFT
Does TDDFT really work?
Complications for solids
Currents versus densities
Elastic scattering from TDDFT
ISSP I
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Optical response in box
ISSP I
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Excitations from DFT
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Many approaches to excitations in DFT
There is no HK theorem from excited-state
density (PRL with Rene Gaudoin)
Would rather have variational approach
(ensembles, constrained search, etc.)
TDDFT yields a response approach, i.e, looks at TD
perturbations around ground-state
ISSP I
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TDDFT linear response
5
4
In time-dependent
external field
He atom density
3
2
1
0
-2
4
-1.5
-1
-0.5
0
0.5
potential
1.5
2
vext (rt )
0
-4
1
vS (r)
vext (r)
vS (rt )
-8
-12
-2
-1.5
-1
For a given interaction
and
statistics:
HS:
KS:
vext [  ](r)
vS[  ](r)
-0.5
0
0.5
RG:
KS:
1
1.5
2
vext [  (t ), (0)](rt )
vS[  (t ), (0), (0)](rt )
Density response
3
 0 (r )
2
(r )
1
0
vS (r )
-1
vS (r )
vext (r ' )
vext (r )
-2
-3
-2
-1.5
-1
-0.5
3
0
0.5
1
1.5
(r )   d r ' [  0 ](rr ' ) vext (r ' )
where
2
  d 3 r ' S [  0 ](rr ' ) vS (r ' )
 1

vS (r )  vext (r )   d r ' 
 f XC [  0 ](rr ' ) (r ' )
 r  r '

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Dyson-like equation
n(t )   (rt )
Key quantity is susceptibility
 (rr ' , t  t ' ) 
v(r ' t ' )
n(rt )
v(r' t ' )
Dyson-like equation for a susceptibility:
 1

 (rr ' ,  )   S (rr '  )   d r1  d r2  S (rr1 )
 f XC [  0](r1r2 )  (r2r '  )
 r1  r2

3
3
Two inputs: KS susceptibility
 j (r )k* (r ) *j (r' ) j (r' )
S (rr ' ,  )   ( f k  f j )
  ( j   k )  i 0
jk
and XC kernel
f XC (r, r' , t  t ' ) 
vXC (r, t )
(r' , t ' )
TDDFT linear response
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Probe system with AC field of freq 
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Ask at what  you find a self-sustaining response
That’s a transition frequency!
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Need a new functional, the XC kernel, fxc[0](r,r’,)
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Almost always ignore -dependence (called adiabatic
approximation)
Can view as corrections to KS response
ISSP I
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Eigenvalue equations
Casida’s matrix formulation (1996)
~
2

(

)
v


 qq' q' vq
q'
True transition
frequencies
 qq'   qq' q2  4  q q ' q f HXC ( ) q'
q f
KS transition
frequencies
HXC
( ) q     (r' ) f HXC (r, r ' ;  ) q (r )drdr '
*
q
q (r)  i* (r)a (r)
ISSP I
Unoccupied KS orbital
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Occupied KS orbital
Transitions in TDDFT
In this equation, fHXC is the Hartree-exchangecorrelation kernel, 1 r  r'  f XC (r, r' , ) , where fXC is
the unknown XC kernel
b
b
b
a
q   a  i
q
b
q'
i
KS susceptibility
ISSP I
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How good the KS response is
ISSP I
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Extracting Exc
ISSP I
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Adiabatic approximation
ISSP I
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Road map
•
•
•
•
•
•
•
TD quantum mechanics->TDDFT
Linear response
Overview of all TDDFT
Does TDDFT really work?
Complications for solids
Currents versus densities
Elastic scattering from TDDFT
ISSP I
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Overview of ALL TDDFT
1. General Time-dependent Density Functional Theory
• Any e- system subjected to any
• Only unknown:
vext (rt )
vXC [  ](rt )
• Treat atoms and molecules in INTENSE laser fields
2. TDDFT linear response to weak fields
• Linear response:
• Only unknown:
(rt )   d 3r '  dt '  (rr ' , t  t ' ) vext (r ' t ' )
vXC (rt ) near ground state
vXC [  0  ](r )  vXC [  0 ](r )   d 3 r ' f XC [  0 ](rr ' ) (r ' )
• Treat electronic excitations in atoms + molecules + solids
3. Ground-state Energy from TDDFT
•Fluctuation–dissipation theorem: Exc from susceptibility
•Van der Waals; seamless dissociation
ISSP I
Basic approximation: ALDA
unif
vXC
(  (rt ))
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Methodology for TDDFT
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In general: Propagate TDKS equations forward in
time, and then transform the dipole moment, eg.
Octopus code
Linear response: Convert problem of finding
transitions to eigenvalue problem (Casida, 1996).
ISSP I
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Green fluorescent Protein
TDDFT
approach for
Biological
Chromophores,
Marques et al,
Phys Rev Lett
90, 258101
(2003)
ISSP I
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Success of TDDFT for excited states
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Energies to within about 0.4 eV
Bonds to within about 1%
Dipoles good to about 5%
Vibrational frequencies good to 5%
Cost scales as N2, vs N5 for CCSD
Available now in your favorite quantum chemical
code
ISSP I
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Naphthalene
TDDFT results
for vertical
singlet
excitations in
Naphthalene
Elliot, Furche, KB,
Reviews Comp
Chem, sub. 07.
ISSP I
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Road map
•
•
•
•
•
•
•
TD quantum mechanics->TDDFT
Linear response
Overview of all TDDFT
Does TDDFT really work?
Complications for solids
Currents versus densities
Elastic scattering from TDDFT
ISSP I
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How good the KS response is
ISSP I
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Quantum defect of Rydberg series
nl  I  2( n 1
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nl )
2
I=ionization potential, n=principal, l=angular
quantum no.s
Due to long-ranged Coulomb potential
Effective one-electron potential decays as -1/r.
Absurdly precise test of excitation theory, and
very difficult to get right.
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Be s quantum defect: expt
Top:
triplet,
bottom:
singlet
ISSP I
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Be s quantum defect: KS
ISSP I
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Be s quantum defect: RPA
KS=triplet
fH
RPA
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Be s quantum defect: ALDAX
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Be s quantum defect: ALDA
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General notes
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Most papers are lin resp, looking at excitations:
need gs potential, plus kernel
Rydberg excitations can be bad due to poor
potentials (then use OEP, or be clever!).
Simple generalization to current TDDFT
Charge transfer fails, because little oscillator
strength in KS response.
Double excitations lost in adiabatic approximation
(but we can put them back in by hand)
Typically not useful in strong fields
Exc schemes still under development
ISSP I
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Road map
•
•
•
•
•
•
•
TD quantum mechanics->TDDFT
Linear response
Overview of all TDDFT
Does TDDFT really work?
Complications for solids
Currents versus densities
Elastic scattering from TDDFT
ISSP I
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Complications for solids and long-chain polymers
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Locality of XC approximations implies no
corrections to (g=0,g’=0) RPA matrix element in
thermodynamic limit!
fH (r-r’) =1/|r-r’|, but fxcALDA = (3)(r-r’) fxcunif(n(r))
As q->0, need q2 fxc -> constant to get effects.
Consequences for solids with periodic boundary
conditions:
– Polarization problem in static limit
– Optical response:
• Don’t get much correction to RPA, missing excitons
• To get optical gap right, because we expect fxc to shift all
lowest excitations upwards, it must have a branch cut in w
starting at EgKS
ISSP I
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Two ways to think of solids in E fields
•
A: Apply Esin(qx), and take
q->0
– Keeps everything static
– Needs great care to take
q->0 limit
•
B
B: Turn on TD vector
potential A(t)
– Retains period of unit cell
– Need TD current DFT,
take w->0.
Au
Au
Au
ISSP I
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Relationship between q->0 and ->0
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Find terms of type: C/((q+ng)2-2)
For n finite, no divergence; can interchange q->0
and ->0 limits
For n=0:
– if =0 (static), have to treat q->0 carefully to cancel
divergences
– if doing q=0 calculation, have to do t-dependent, and take
->0 at end
ISSP I
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Road map
•
•
•
•
•
•
•
TD quantum mechanics->TDDFT
Linear response
Overview of all TDDFT
Does TDDFT really work?
Complications for solids
Currents versus densities
Elastic scattering from TDDFT
ISSP I
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TD current DFT
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RG theorem I actually proves functional of j(r,t).
Easily generalized to magnetic fields
Naturally avoids Dobson’s dilemma: Gross-Kohn
approximation violates Kohn’s theorem.
Gradient expansion exists, called Vignale-Kohn
(VK).
TDDFT is a special case
Gives tensor fxc, simply related to scalar fxc (but
only for purely longitudinal case).
ISSP I
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Currents versus densities
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Origin of current formalism: Gross-Kohn
approximation violates Kohn’s theorem.
Equations much simpler with n(r,t).
But, j(r,t) more general, and can have B-fields.
No gradient expansion in n(rt).
n(r,t) has problems with periodic boundary
conditions – complications for solids, long-chain
conjugated polymers
ISSP I
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Beyond explicit density functionals
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Current-density functionals
– VK Vignale-Kohn (96): Gradient expansion in current
– Various attempts to generalize to strong fields
– But is just gradient expansion, so rarely quantitatively
accurate
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Orbital-dependent functionals
– Build in exact exchange, good potentials, no selfinteraction error, improved gaps(?),…
ISSP I
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Basic problem for thermo limit
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Uniform gas:
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Uniform gas moving with velocity v:
ISSP I
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Polarization problem
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Polarization from current:
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Decompose current:
where
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Continuity:
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First, longitudinal case:
– Since j0(t) not determined by n(r,t), P is not!
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What can happen in 3d case (Vanderbilt picture frame)?
– In TDDFT, jT (r,t) not correct in KS system
– So, Ps not same as P in general.
– Of course, TDCDFT gets right (Maitra, Souza, KB, PRB03).
ISSP I
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Improvements for solids: currents
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Current-dependence: Snijders, de Boeij, et al –
improved optical response (excitons) via ‘adjusted’
VK
Also yields improved polarizabilities of long chain
conjugated polymers.
But VK not good for finite systems
ISSP I
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Improvements for solids:
orbital-dependence
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Reining, Rubio, etc.
Find what terms needed in fxc to reproduce BetheSalpeter results.
Reproduces optical response accurately, especially
excitons, but not a general functional.
In practice, folks use GW susceptibility as
starting point, so don’t need effective fxc to have
branch cut
ISSP I
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Our recent work
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Floquet theory
Double excitations
Understanding how it works
– Single- and Double-pole approximations
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X-ray spectra
Rydberg series from LDA potential
Quantum defects
Errors in DFT for transport
TDDFT for open systems
Elastic electron-atom scattering
ISSP I
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Road map
•
•
•
•
•
•
•
TD quantum mechanics->TDDFT
Linear response
Overview of all TDDFT
Does TDDFT really work?
Complications for solids
Currents versus densities
Elastic scattering from TDDFT
ISSP I
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Elastic scattering from TDDFT
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Huge interest in low energy scattering from
biomolecules, since resonances can lead to
cleavage of DNA
Traditional methods cannot go beyond 13 atoms
Can we use TDDFT? Yes!
ISSP I
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Simple scheme for spherical case
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Eg e- scattering from H.
Put H- into spherical box, and consider E>0 states.
Old formula due to Fano (1935):
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Exact for any Rb beyond potential.
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ISSP I
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Is KS a good starting place?
ISSP I
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Is the LDA potential good enough?
ISSP I
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TDDFT corrections
ISSP I
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Summary
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TDDFT is different from DFT
Linear response TDDFT turns KS orbital
differences into single optical excitations
Value is in semi-quantitative spectra
– Can help determine geometry
– Identify significant excitations
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Troubles with strong fields
Troubles with solids
Current- or orbital-dependence are promising
alternatives for solids and long-chain polymers
ISSP I
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