Current versus density in TDDFT

1
Current versus density in TDDFT
Kieron Burke and friends Departments of Physics
and of Chemistry UC Irvine
http//dft.uci.edu
2
Recent reviews of TDDFT
To appear in Reviews of Computational Chemistry
3
Road map

• Quick review of TDDFT
• Currents, densities, orbitals
• Transport weak bias and strong

4
Road map

• Quick review of TDDFT
• Currents, densities, orbitals
• Transport weak bias and strong

5
Runge-Gross theorem (1984)

• Any given current density, j(r,t), and initial
  wavefunction, statistics, and interaction,
  theres 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).
• von Leeuwen gave a constructive proof (PRL98)

6
Overview of ALL TDDFT
1. General Time-dependent Density Functional
Theory
2. TDDFT linear response to weak fields
3. Ground-state Energy from TDDFT

• Fluctuationdissipation theorem Exc from
  susceptibility
• Van der Waals seamless dissociation

7
Comment on recent paper

• Critique of the foundations of time-dependent
  density-functional theory, by J. Schirmer and A.
  Dreuw (PRA 75,022513(2007))

• criticizes TDDFT on several grounds
• RG action -- but now long-recognized that there
  are problems with the action in the original
  formulation (van Leeuwen, need Keldysh action),
  and no ominous consequences. NO problem.
• problems with non-local perturbations but
  these are explicitly excluded from the start! NO
  problem
• TDDFT is not predictive this is sole true
  criticism of the theory, and it turns out this is
  also wrong! Again, NO problem.

Comment by Maitra, KB, van Leeuwen, to be
finished this week
8
Quantum defect of Rydberg series Meta van
Faassen

• Iionization potential, nprincipal, langular
  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.

9
Be s quantum defect expt
Top triplet, bottom singlet
10
Be s quantum defect KS
11
Be s quantum defect RPA
KStriplet
fH
RPA
12
Be s quantum defect ALDAX
13
Be s quantum defect ALDA
14
Road map

• Quick review of TDDFT
• Currents, densities, orbitals
• Transport weak bias and strong

15
So, whats the problem?

• If TDDFT is working so well, and stands up to
  scrutiny, why worry about it?

16
Basic problem with using density

• Uniform gas
• Uniform gas moving with velocity v

17
Current and continuity

• Current operator
• Acting on wavefunction
• Continuity

18
TD current DFT

• RG theorem I actually proves functional of
  j(r,t).
• Easily generalized to magnetic fields
• Naturally avoids Dobsons dilemma Gross-Kohn
  approximation violates Kohns 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).

19
Currents versus densities

• Origin of current formalism Gross-Kohn
  approximation violates Kohns theorem.
• Equations much simpler with n(r,t).
• But, j(r,t) more general, and can have B-fields.
• No gradient expansion in n(r,t).
• n(r,t) has problems with periodic boundary
  conditions complications for solids, long-chain
  conjugated polymers

20
Complications for solids and long-chain polymers

• Locality of XC approximations implies no
  corrections to (g0,g0) RPA matrix element in
  thermodynamic limit!
• fH (r-r) 1/r-r, but fxcALDA d(3)(r-r)
  fxcunif(n(r))
• As q-gt0, need q2 fxc -gt constant to get effects.
• Consequences for solids with periodic boundary
  conditions
• Polarization problem in static limit
• Optical response
• Dont 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

21
Two ways to think of solids in E fields

• A Apply Esin(qx), and take q-gt0
• Keeps everything static
• Needs great care to take q-gt0 limit
• B Turn on TD vector potential A(t)
• Retains period of unit cell
• Need TD current DFT, take w-gt0.

22
Relationship between q-gt0 and w-gt0

• Find terms of type C/((qng)2-w2)
• For n finite, no divergence can interchange
  q-gt0 and w-gt0 limits
• For n0
• if w0 (static), have to treat q-gt0 carefully to
  cancel divergences
• if doing q0 calculation, have to do t-dependent,
  and take w-gt0 at end

23
Polarization problem

• Polarization from current
• Decompose current
• where
• Continuity
• First, purely longitudinal case
• Since j0(t) not determined by n(r,t), P is not!
• What can happen in 3d case (Vanderbilt picture
  frame)?
• In TDDFT, jT (r,t) not correct in KS system
  (Vignale..)
• So, Ps not same as P in general.
• Of course, TDCDFT gets right (Maitra, Souza, KB,
  PRB03).

24
Dynamical contributions in VK not dynamical

• Careful definition of adiabatic contribution
• Define fxc(r,r,?-gt0)fxcadia(r,r), should
  recover static DFT
• Define fxcdyn(r,r,?) fxc (r,r,?)-fxcadia(r,r)
• For conditions of derivation, agrees with VK
  definition.
• But for finite systems, dyn VK yields finite
  contribution as ?-gt0!, ie is NOT dynamical.

Definition of adiabatic DAmico and Vignale, PRB
99.
25
Beyond explicit density functionals

• 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
• Orbital-dependent functionals
• Build in exact exchange, good potentials, no
  self-interaction error, improved gaps(?),

26
Improvements for solids currents

• Current-dependence Snijders, de Boeij, et al
  improved optical response (but not excitons) via
  adjusted VK
• Also yields improved polarizabilities of long
  chain conjugated polymers.
• But VK not good for finite systems
• Van Faassen and de Boeij, JCP 2004
• Ullrich and KB, JCP, 2004..

27
Improvements for solids orbital-dependence

• Reining, Rubio, etc.
• Find what terms needed in fxc to reproduce
  Bethe-Salpeter results.
• Reproduces optical response accurately,
  especially excitons, but not a general
  functional.
• In practice, folks use GW susceptibility as
  starting point, so dont need effective fxc to
  have branch cut

28
Road map

• Quick review of TDDFT
• Currents, densities, orbitals
• Transport weak bias and strong

29
Recent review
Topical review, submitted to J Phys C, available
on cond-mat Summary of work so far Warning Only
contains about ½ of background
30
Break junction expts
31
Standard approach
32
Three different questions

• 1. Within standard model
• Are present calculations good enough? No,
  possible origin of overestimate of conductance
  (with Sanvito et al.).
• 2. Test standard model
• A. Weak bias allows linear response.
• Find missing XC contributions (with Koentopp
  and Evers).
• B. What to do for finite bias?
• Change gauge, put on ring, and add dissipation
  (with Car and Gebauer).

33
Three different questions

• 1. Within standard model
• Are present calculations good enough? No,
  possible origin of overestimate of conductance
  (with Sanvito et al.).
• 2. Test standard model
• A. Weak bias allows linear response.
• Find missing XC contributions (with Koentopp
  and Evers).
• B. What to do for finite bias?
• Change gauge, put on ring, and add dissipation
  (with Car and Gebauer).

34
Effect on resonant tunneling
(Koentopp, Evers, and KB. PRB 04).

• double barrier resonance shape and position
• compare smooth functional with exact result

• conductance of benzenedithiolHF instead of
  DFT/GGA

Peaks too broad, wrong postion
-2
T reduced by 100
35
Missing derivative discontinuity

• Local functionals miss derivative discontinuity
• Resonances smeared in LDA, yielding overestimated
  current

36
Molecule weakly coupled to leads
Tohar, Filipetti, Sanvito, and KB (PRL, 2005).

• For weak coupling, see much lower conductance
  when SIC turned on.
• No effect for normal (chemical) bonding.

weak
normal
37
Recent developments

• Ke, Baranger, and Yang (JCP 07) find these
  effects in OEP-EXX calculations of transmission.
• Tohar and Sanvito (PRL last month) have
  implemented poormans SIC into full transport
  code, with similar results.
• Prodan and Car (arXiv) get good values of ß from
  LDA calculations, suggesting no misalignment.
• Capelle et al (PRL07) get derivative
  discontinuity from LDA orbital energy differences
  for harmonic confinement.

38
Three different questions

• 1. Within standard model
• Are present calculations good enough? No,
  possible origin of overestimate of conductance
  (with Sanvito et al.).
• 2. Test standard model
• A. Weak bias allows linear response.
• Find missing XC contributions (with Koentopp
  and Evers).
• B. What to do for finite bias?
• Change gauge, put on ring, and add dissipation
  (with Car and Gebauer).

39
Static density response eqns

• Three different ways to calculate dr
• Full non-local susceptibility in response to
  external field
• Proper susceptibility in response to total
  potential
• KS susceptibility in response to KS pot

40
TDCDFT response eqns

• Three different ways to calculate dj
• Full non-local conductivity in response to
  external E-field
• Proper cond. in response to total field
• KS conductivity in response to KS pot

41
Treatment of length scales as w-gt0

• L length of leads
• lF Fermi wavelength
• lb width of barrier
• lel elastic scattering length
• lper vF/w distance traveled by a Fermi
  electron during one period of external field, if
  free
• lTF Thomas-Fermi screening length
• vF/wp, where wp is the plasmon
  frequency.
• Long clean leads
• lb, lTF, lF ltlt lper ltlt L, lel.

42
Extreme simplicity at w0

• For one dimensional case (complications in 3D)
• And inserting in Rs yields ss yields independent
  of positions, and depending only on transmission
  thru barrier at EF
• Generalization to 3d by Prodan and Car (arXiv).

43
Low frequency limit

• As w-gt0, ss indep of r,r and equals Ts(eF)/p.
• Becomes
• But integral of field is just potential drop
• Compare with Landauer

44
References

• Basic derivation for non-interacting electrons
  first done nicely by Baranger and Stone (PRB88)
• Careful derivation in 1d for Hartree interacting
  particles by Kamenev Kohn (PRB03)
• 1d result by Koentopp,Evers, KB (PRB05)
• Generalization to 3d by Prodan and Car, arXiv, to
  appear in PRB.

45
Consequences good

• If Vxc?0, there are XC corrections to Landauer!
• Two types
• Adiabatic (show up in static DFT calculation)
• Dynamic (show up as ?-gt0 limit of TDCDFT).
• Adiabatic No contribution from LDA or GGA
• Thus, present calculations with standard
  functionals, dont need to go looking for this.
• Even in TD(C)DFT within eg ALDA, get no
  corrections.

46
Likely corrections

• Adiabatic
• Do EXX static orbital-dependent calculation
• No reason why there wont be an overall drop in
  Vx across molecule
• Dynamic
• Use VK to estimate (Na Sai et al, PRL 05)
• Find small but finite corrections
• But VK is for high ?, might not apply here.
• Missed some other terms
• (see comment by Bokes et al in PRL month ago).

47
Adiabatic XC field from orbital dependenec

• Well-known problem for LDA/GGA
• Overestimate of static polarizabilities of
  long-chain polymers
• Cured by OEP-EXX or now by LDA-SIC
• Exact OEP (Körzdöffer, Mundt, Kümmel, see arXiv)
• KLI-SIC (Das, Sanvito, KB, see arXiv.)
• LDA-SIC inexpensive alternative to full EXX

48
Counteracting XC field in H2 chains
49
Alkane-dithiol results

• Clean experiments by Xu and Tau on
  polyalkane-dithiols.
• Strongly coupled to leads, not conjugated.
• KaunSeideman (preprint) get excellent
  quantitative agreement using standard approach
• Can explain other expts by changing contacts

-Au-S-(CH2)N-S-Au-
Conductance/G0
50
Three different questions

• 1. Within standard model
• Are present calculations good enough? No,
  possible origin of overestimate of conductance
  (with Sanvito et al.).
• 2. Test standard model
• A. Weak bias allows linear response.
• Find missing XC contributions (with Koentopp
  and Evers).
• B. What to do for finite bias?
• Change gauge, put on ring, and add dissipation
  (with Car and Gebauer).

51
Formal DFT for electron-nuclear system

• Can prove 1-1 correspondence for r(r,t) and
  G(R1RM). - dont use single-particle nuclear
  density, despite textbooks.
• Basic papers by Kreibich and Gross about 10 years
  ago.
• But only applied to H2, and need good
  approximations as it dissociates.

52
Chapter in TDDFT book
53
Alternative approach

• Coupled dynamics is approximated in many
  different ways for different purposes, eg
  friction vs branching
• Take known approximate treatments of QM, and
  generalize DFT to those approximations

54
Beyond standard approach

• Many groups now working on microscopic derivation
• Charging big electrodes Todorov, Di Ventra,
  Vignale.
• Real Non-equilibrium Greens functions Kurth,
  Rubio, Gross, Almbladh, Stefanucci, van Leeuwen
• Dissipation Car, Gebauer, Burke
• Complex Hamiltonians Ernzerhof
• All time-dependent (involve TD(C)DFT), watching
  steady current evolve.
• Each has own advantages

55
TDDFT for Open Systems

• Put electrons on finite ring in solenoidal field
• Prove TDDFT theorems about Lindblad form of
  Master equation for N electrons coupled to a bath
  of phonons.
• In principle, can get coupling from phonons, in
  practice, must be much stronger to dissipate
  energy, but can then extrapolate to zero
  coupling.
• Two contributions to current Hamiltonian and
  dissipative

Burke, Car, Gebauer, PRL 05
56
Results for finite bias

• Evolve QM in master eqn, not Schrödinger
  equation.
• Generalize TDCDFT to include dissipation
• Produces a TD KS master equation
• KB, Roberto Car, Ralph Gebauer, PRL 2005
• Steady-state solutions look like Landauer.
• Predicts optical bistability, heating, etc.

57
Recent realistic calculations

• Three atom gold chain
• Conductance depends on wire length

(Piccinin, Gebauer, Car, KB, in prep
58
Dithiolated benzene

• Supercell approach with 12 layers Au
• Get conductance very similar to standard approach

59
Master equation for dissipation

• HHelHphKel-ph
• Assume relaxation time much longer than time for
  transitions or phonon periods
• Coarse-grain over electronic transitions and
  average over bath fluctuations
• Master equation for system density matrix

60
Master equation continued

• Operator C is from Fermis golden rule applied to
  Kel-ph
• Transition probabilities satisfy detailed balance
• Builds in irreversibility to evolution
• Allows off-diagonal density matrix elements, so
  not a pure state evolution
• Prototype lifetime of two-level atom coupled to
  quantized photon field

61
Restoring continuity in Master equation(Gebauer
and Car, 2005)

• For A, define BiH,A, so ltBgtdltAgt/dt
• Eg Bgrad j if Ar
• In Master equation, find
• dltAgt/dt ltBgt Tr CA
• Or dltAgt/dtltBgt Tr DB

62
1-1 correspond. for Master eqn

• Assume potential is Taylor-expandable about t0.
• Consider two potentials that differ by more than
  a time-dep constant
• Show that current densities must then differ
• Use (restored) continuity to prove densities
  differ.

63
Kohn-Sham Master equation

• Define a Kohn-Sham Master equation yielding same
  r(r,t) from vs (r,t), but choose Cs to
  equilibrate to the Mermin-Kohn-Sham Ss(0)

64
Return to weak bias

• Usual Kubo calculation yields adiabatic
  conductivity
• Our approach produces true isothermal
  conductivity
• Can show, as Cs-gt0, it becomes ih in Kubo formula

65
Comparison of Electronic vs Open
US
Euro

• dissipation.
• periodic boundary conditions
• current density as basic variable
• allows non-steady processes
• shows Joule heating to phonons
• slightly new functionals needed

• steady state via continuum purely electronic
• finite system with sinks and sources
• density as basic variable
• allows non steady processes
• no dissipation at present.
• no new functionals needed

66
Questions about Euro-TDDFT approach

• RG theorem only true for finite systems, but need
  continuum for steady state?
• Proof of steady state only if KS Hamiltonian
  steady as t-gt8
• Requires smooth DOS on molecule, but what happens
  when molecule only weakly coupled to leads?

67
Limitations of present DFT calculations for
single molecule transport?

• Several significant open questions about
  standard DFT transport
• Missing derivative discontinuity how big?
• Missing non-local XC corrections to weak bias
  how big?
• Microscopic derivations for finite bias do they
  agree with each other, and with standard?
• Thanks to friends and funders (DOE).

transport
Quantum defect
68
Recent realistic calculations
(Picinnin thesis with Car, in prep)
69
More about realistic calculations