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Simulation of Charge Transport in Oxides and Organic Simulation of Charge Transport in Oxides and Organic Semiconducting MaterialsSemiconducting Materials
Jochen Blumberger
University College London
Trieste, 23.05.2014
Workshop on Materials Challenges in Devices for Fuel Solar Production and Employment
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OverviewI. Theoretical background II. CT in oxide materials
III. CT in organic solar cell materials
IV. CT in bacterial nanowiresElectrode
e-
Food
Food
Microbe
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Photoelectrochemical cell
e-
oxide good charge transport
essential for high efficiency
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e-
Clarke and Durrant, Chem Rev. (2010)
Organic photovoltaic cell
good charge transport essential for high efficiency
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Ishii, S. et al. Appl. Environ. Microbiol. 71, 7838 (2006).
Electrode
Food
e-Microbe
Mediatorless microbial fuel cell
Summers, Z. M. et al. Science 330, 1413 (2010).
good charge transport essential for high efficiency
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Why computation ?
Nature of charge carrier (localised/delocalised)
Mechanism of charge transport (band/hopping/wavepacket)
Atomistic interpretation of measured charge mobilities, I-V curves
NanoStructure-property relationships
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Overview
• I. Theoretical Background
• II. Electron tunneling between O-vacancies in MgO
• III. Electron transport in fullerenes
• IV. Electron transport in bacterial nanowires
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Which theory is adequate?
Cha
rge
mob
ilit
y
electron-phonon couplingor reorganisation energy or trapping energy
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Which theory is adequate?
Cha
rge
mob
ilit
y
electron-phonon couplingor reorganisation energy or trapping energy
metals
band theory
redox in solutionredox proteins
Small polaron hoppingET theories (Marcus)
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Which theory is adequate?
Cha
rge
mob
ilit
y
electron-phonon couplingor reorganisation energy or trapping energy
metals
band theory
redox in solution redox proteins
Small polaron hoppingET theories (Marcus)
?
holes/e- in oxidesholes/e- in organicsemiconductors
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Electron transfer theory (Thermally activated polaron hopping)
initial diabatic state final diabatic state
e-e-
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Diabatic and adiabatic electronic states
Q reaction coordinate
λ reorganization energy
Hab electronic coupling matrix element
initial diabatic state
final diabatic statee-e-
adiabatic ground & 1st excited ET state
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Diabatic and adiabatic electronic states
e-e-
Transition state theory
Nuclear tunneling
Landau-Zener theory
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Diabatic states from constrained DFT (CDFT)
Idea: - Construct Hamiltonian in diabatic = charge localized basis
-Create charge localized states by adding an external
potential to Hamiltonian
=
Martin Karplus (1963) (?), Arieh Warshel (1993), Troy Van Voorhis (2005), John Tully
(2008),…
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Charge constrained DFT (CDFT)
1. Define donor, acceptor and external potential w(r)
Q. Wu and T. van Voorhis, Phys. Rev. A 72, 024502 (2005)
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Charge constrained DFT (CDFT)
1. Define donor, acceptor and external potential w(r)
2. Add Vw(r) to KS-equation
and vary V so that charge constraint
is fulfilled.
Q. Wu and T. van Voorhis, Phys. Rev. A 72, 024502 (2005)
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Charge constrained DFT (CDFT)
1. Define donor, acceptor and external potential w(r)
2. Add Vw(r) to KS-equation
and vary V so that charge constraint
is fulfilled.
Q. Wu and T. van Voorhis, Phys. Rev. A 72, 024502 (2005)
diabatic state B ψB, EB = FB-VBN
1 AD qq
e-
e-
1 AD qq
diabatic state AψA, EA = FA-VAN
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Charge constrained DFT (CDFT)
1. Define donor, acceptor and external potential w(r)
Q Wu and T Van Voorhis, Phys. Rev. A 72, 024502 (2005)
diabatic state B ψB, EB = FB-VBN
1 AD qq
e-
e-
1 AD qq
diabatic state AψA, EA = FA-VAN
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CDFT Implementation in the CPMD code
• CDFT weight function for charge constraint:
where ρi are promolecular atomic densities (pseudo AO, Slater, Gaussians)
• CDFT wavefunction optimisation, geometry optimisation and BOMD
• GGA, hybrid and range separated hybrid functionals (with new HFX parallelisation)
• Troullier-Martins or Goedecker-Hutter pseudo potentials for core electrons
• Available in CPMD Version 3.15.1 (thanks to M. Boero and T. Laino)
H. Oberhofer, JB, J. Chem. Phys. 131, 064101 (2009)H. Oberhofer, JB, J. Chem. Phys. 133, 244105 (2010)
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Benchmarking CDFT electronic couplings (Hab):The HAB11 database
A. Kubas, F. Hoffmann, A. Heck, H. Oberhofer, M. Elstner JB, J. Chem. Phys. 140, 104105 (2014).
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Hab CDFT error wrt MRCI+Q/NEVPT2
Relatively small dependence on %HFX25-50% HFX gives excellent accuracy
A. Kubas, F. Hoffmann, A. Heck, H. Oberhofer, M. Elstner JB, J. Chem. Phys. 140, 104105 (2014).
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Overview
• I. Theoretical Background
• II. Electron tunneling between O-vacancies in MgO
• III. Electron transport in fullerenes
• IV. Electron transport in bacterial nanowires
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Electron tunneling between F+-F0 centres in MgO
positively charged oxygen vacancy
neutral oxygenvacancy
F0F+
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Initial diabatic state from CDFT
Isosurface of spin density, PBE0 (CPMD code):
K. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).
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Initial diabatic state from CDFT
Isosurface of spin density, PBE0 (CPMD code):
K. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).
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Final diabatic state from CDFT
Isosurface of spin density, PBE0 (CPMD code):
K. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).
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Electronic coupling
• CDFT calculation with PBE0 for
• F-centres along 100, 110, 111, 211 and 310 crystallographic directions
• distance between F-centres ranging from 3 to 16 Angstroms
K. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).
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Hab CDFT for hole transfer in MgO
5 10 15defect separation (Angstroms)
PBE with different %ages of Hartree-Fock exchange (HFX):
JB, K. P. McKenna, Phys. Chem. Chem. Phys. 15, 2184 (2013).
Strong dependence on % HFX
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Coupling decay constant versus band gap Eg
Eg: band gap of MgO
0%
25%
50%75%
100%
PBE + x % HFX
JB, K. P. McKenna, Phys. Chem. Chem. Phys. 15, 2184 (2013).
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Square barrier tunneling model
Exact solution of Schroedinger equation:
Use functional that gets band gap right (PBE0 i.e. 25%HFX)
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Parallelisation of HFX in CPMD
PBE
PBE0 PBE0hfx parallelized
V. Weber, T. Laino, A. Curioni (IBM Zurich)http://cpmd.org/the-code/performance-and-scale-out
Mg80O78+
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Accounting for finite size effects on electronic couplingJB, K. P. McKenna, Phys. Chem. Chem. Phys. 15, 2184 (2013).
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Accounting for finite size effects on electronic couplingJB, K. P. McKenna, Phys. Chem. Chem. Phys. 15, 2184 (2013).
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Electronic coupling versus defect separation
Overall approximately exponential and isotropic decay
CDFT, PBE0, for MgO 100, 110, 111, 211 and 310 crystallographic directions
K. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).
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Reorganization energy versus defect separation
CDFT, PBE0, for MgO 100, 110, 111, 211 and 310 crystallographic directions
long range: Marcus like = const – 1/d short range: non-Marcus due to large distortions of e--Mg2+ distances
K. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).
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Hole transfer rates between F-center defects in MgO K. P. McKenna, JB Phys. Rev B. 86,45110 (2012).
5
6
7
8
9
10
11
12
13
442 6 8 10 12 14 16 18 20
log
(kE
T/s
-1)
distance between F centers (Angstroms) 0
hello hellohello
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Three ET regimes
Hab << λ Hab > 3/8 λ Hab < 3/8 λ
“non-adiabatic ET” “adiabatic ET” delocalized carrier
no ET rate defined
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5
6
7
8
9
10
11
12
13
442 6 8 10 12 14 16 18 20
log
(kE
T/s
-1)
distance between F centers (Angstroms) 0
crossover incoherent coherenttransport
Rates & Transport regimesK. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).
no polaron no rate
hole hopping
hole hopping
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Overview
• I. Theoretical Background
• II. Electron tunneling between O-vacancies in MgO
• III. Electron transport in fullerenes
• IV. Electron transport in bacterial nanowires
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Electron transport in fullerenes
e-
exciton dissociation efficiency
electron mobility nanoscale/mesoscale structure
Clarke and Durrant, Chem Rev. (2010)
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Dabirian et al. PCCP 12, 4473 (2010).
Kim et al. ACS Nano 3, 2557 (2009).
distorted bcc (model)
hexagonal (no X-ray)
monoclinic w. solvent
Rispens et al. Chem Commun 2116, (2003).
triclinic w. solvent
Phenyl-C61-Butyric acid Methyl ester (PCBM)
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First solvent-free PCBM single crystalG. Paterno, J. Spencer, JB, F. Cacialli and co-workers, J. Mater. Chem. C (2013)
monoclinic4 PCBM/unit cell
obtained by slow drying from chlorobenzene See also Casalegno et al.
Chem. Commun. 49, 4525 (2013)
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First solvent-free PCBM single crystalG. Paterno, J. Spencer, JB, F. Cacialli and co-workers, J. Mater. Chem. C (2013)
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First solvent-free PCBM single crystalG. Paterno, J. Spencer, JB, F. Cacialli and co-workers, J. Mater. Chem. C (2013)
MD exp
density (g/cm3) 1.653 1.631
RDF 1. peak (A) 10.05 10.2
coordination number 7 7
very good agreement with experimental X-ray structure
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Electronic couplings vs distance in PCBM crystalsF. Gajdos, H. Oberhofer, M. Dupuis, JB J. Phys. Chem. Lett. 4, 1012 (2013).
I
(Gajdos)
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Electronic couplings vs distance in PCBM crystalsF. Gajdos, H. Oberhofer, M. Dupuis, JB J. Phys. Chem. Lett. 4, 1012 (2013).
constructive orbital overlap
destructiveorbital overlap
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Polaron hopping
calculate Hab , for all possible hops
hopping rate kET for all possible hops
time
field
e-
is currently the state-of-the-art, used by many groups: Bredas, Nelson, Andrienko,…
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Polaron hopping
calculate Hab , for all possible hops
hopping rate kET for all possible hops
experiment
time
field
e-
mobility factor of 6 too large not bad….
is currently the state-of-the-art, used by many groups: Bredas, Nelson, Andrienko,…
Kinetic Monte Carlo
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Polaron hopping
calculate Hab , for all possible hops
hopping rate kET for all possible hops
experiment
time
field
e-
mobility factor of 6 too large not bad….
is currently the state-of-the-art, used by many groups: Bredas, Nelson, Andrienko,…
Kinetic Monte Carlo
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Polaron does not exist
|HAB| = up to 150 meV
λ = 150 meV
|HAB| > 3/8 λ
no barrier, ET rate not defined
(though one can still insert into the rate equation and get some number)
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Non-adiabatic MD for ET in organic materials
field
time
Don’t integrate electron dynamics out (as done in rate theory)
Solve coupled electron-nuclear dynamics
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Non-adiabatic MD for ET in organic materials
field
time
Don’t integrate electron dynamics out (as done in rate theory)
Solve coupled electron-nuclear dynamics
Fast Tully surface hopping molecular dynamics:
Wavefunction expansion:
work in progress
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“Message passing” parametrisation of non-adiabatic MD
off-diagonal Hamiltonian
elements:
can be calculated
ultrafast
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Speed-up of 6 orders of magnitude, little loss of accuracy
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Overview
• I. Theoretical Background
• II. Electron tunneling between O-vacancies in MgO
• III. Electron transport in fullerenes
• IV. Electron transport in bacterial nanowires
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Very long-ranged electron transport (ET) in biology
pili
e-extracellular ET via conductive pili
Summers et al. Science 330, 1413 (2010).
1 m
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extracellular ET via conductive pili
Very long-ranged electron transport (ET) in biology
I-V measurementson pili
nA over 0.6 m
1 Siemens/cm
El-Naggar et al., PNAS 107, 18127 (2010).
AFM tip
e-
Summers et al. Science 330, 1413 (2010).
Au
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m
What is mediating the ET?
10-100 nm
5-10 nm
1 nm
Protein thermal fluctuations
Electronic coupling
Protein-protein interactions
Cellular/environmental interactions
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MtrF: a deca-heme “nanowire” protein
• binds 10 hemes
• all hemes bis-his (low spin)
• staggered octaheme bisected by planar tetra-heme chain
4.5 nm
6.5 nm
Why a tri-furcated electron transfer path?
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Band structure of deca-heme protein
CB water
CB protein
VB water
VB protein
‘VB heme’
‘CB heme’
~ 0.3 eV
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Computation of heme reduction potentials
redox
oxred
AMBER03/TIP3P
exp: from proteinfilm voltammetry
assignment of heme reduction potentials
M Breuer, PP Zarzycki, JB, KM Rosso, J. Am. Chem. Soc. 134, 9868 (2012).
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Free energy landscape for electron transport
1
6
4
3
2
5
7
8
9
10
largest barrier for forward flow: 0.35 eV
largest barrier for reverse flow: 0.34 eV
M Breuer, PP Zarzycki, JB, KM Rosso, J. Am. Chem. Soc. 134, 9868 (2012).
How can protein cope with these barriers?
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Electronic coupling versus heme-heme distance
2.1±1.4 0.4±0.3 0.2±0.2
stacked T-shaped coplanar
in meV
FODFT(PBE), 100 ns MD for each heme-pair
M Breuer, KM Rosso, JB PNAS 111, 611 (2014)
1
stacked
T-shaped
coplanar
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Free energy (solid) vs electronic coupling Hab (circles)
(Breuer)
stacked
T-shapedco-planar
ET is a balancing act between electronic coupling and free energy
Strongest electronic couplings where free energy barriers are largest
Stacked hemes facilitate transport into protein interior
M Breuer, KM Rosso, JB PNAS 111, 611 (2014)
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Heme-to-heme ET rates in MtrF
(Breuer)
rates in the range (100 s)-1 to (100 ps) -1 range
rates ~ the same in and directions
full : e- shaded: e-
M Breuer, KM Rosso, JB PNAS 111, 611 (2014)
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MV+e-
Solution kineticsGF White et al., PNAS 110, 6346 (2013).
How can we make contact to experiment?
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MV+e-
Solution kinetics I-V measurements
a
GF White et al., PNAS 110, 6346 (2013). MY El-Naggar et al., PNAS 107, 18127 (2010).
How can we make contact to experiment?
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Electron flux via Master Equation
5
4
7
6
8
3
2
1
• Electron hopping: i = D, heme 1-8, A
(i)red + (i+1)ox (i)ox + (i+1)red
• Electron flux between two neighbour hemes:
P(i) = electron population on heme i (≤1)
• Steady state flux:
solve iterativelyD
A
k1D kD1
kA8 k8A
k9,10
k10,9J
M Breuer, KM Rosso, JB PNAS 111, 611 (2014)
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Electron flux J through MtrF
simulation
experimental lower limit
protein limited
5
4
7
6
8
3
2
1
MV+
Fe2O3
kin
kout
k9,10
k10,9J
kin >>kout
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Electron flux J through MtrF
simulation
experimental lower limit
protein limited
5
4
7
6
8
3
2
1
MV+
Fe2O3
kin
kout
k9,10
k10,9J
kin >>kout
experiment: 8x103 electrons/sec (lower limit) simulation: 104-105 electrons/sec “a first indication” that computed rates are OK
simulation
experimental lower limit
acceptor limited protein limited
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5
4
7
6
8
3
2
1
I-V characteristic of MtrF
kin, EETe-
e- kout, EET
J = -I
V
anode
cathode
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5
4
7
6
8
3
2
1
J
I-V characteristic of MtrF
kin, EETe-
e-
simulation: ~ pAexperiment: ~ nA !
kout, EET
J = -I
V
anode
cathode
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Conclusions: Three ET regimes
Hab << λ Hab > 3/8 λ Hab < 3/8 λ
“non-adiabatic ET” “adiabatic ET” delocalized carrier
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Acknowledgment
CDFT & FODFT implementationDr Harald Oberhofer (U Cambridge, TU Munich)
MgODr Keith McKenna (York)Prof Alex Shluger (UCL)
fullerenesFruzsina Gajdos (UCL)Jacob Spencer (UCL)
Dr. Michel Dupuis (PNNL)Prof Franco Cacialli (UCL)
£££:
Biological nanowireMarian Breuer (UCL)
Dr Kevin Rosso (PNNL)Prof Julea Butt (UEA)
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Sensitivity on definition of charge constraint
Charge constraint 1: 6 Mg coord. O vacancies
Charge constraint 2: all atoms in left/right half
JB, K. P. McKenna, Phys. Chem. Chem. Phys. 15, 2184 (2013).
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CPMD INPUT file: donor (D) atoms, acceptor (A) atoms
weight function w (r) = wA,D (pseudo AO, Slater, Gaussian)
charge difference (NC = +1 or -1 for transfer of 1 electron)
initial guess for Lagrange multiplier V=V0
wavefunction optimization
is charge constraint fulfilled ? i.e.
Yes, ψA, EA = FA-VANC
new Lagrange multiplier
No
(Oberhofer)