3.320: Lecture 9 (Mar 3 2005) SUCCESS AND FAILURE
FAILURE
Photos of soccer players removed for copyright reasons.
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Pseudopotentials
Occupation, Eigenvalue
Al
Z =13
3p
3s
1 -2.7 eV
2 -7.8 eV
2p
2s
6 -69.8 eV
2 -108 eV
1s
2 -1512 eV
(-
1 ∆2
+ υeff) ψj = εjψj
2
Valence
2p
1s
1 -2.7 eV
2 -7.8 eV
Pseudo- Al
Z =3
Core-states
1 ∆2
(- 2
(ps)
(ps)
+ υeff ) ψj
Figure by MIT OCW.
After Pehlke, Eckhard. Lecture on "The Plane-Wave Pseudopotential Method."
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
(ps)
= εjψj
Bloch Theorem
& &
&
&
\ nk& (r ) unk& (r ) exp ik ˜ r
periodic u is expanded in planewaves, labeled
according to the reciprocal lattice vectors
&
unk& (r )
& &
¦& c exp(i G ˜r )
&
G
&
nk
G
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Other possibilities - many
• Gaussian basis sets (Hartree-Fock codes)
• Real space representations
• LCAO (Linear combination of atomic
orbitals)
• LMTO (Linear muffin-tin orbitals), LAPW
(Linearized augmented plane waves), PAW
(Projector-augmented wave)
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Brillouin Zone integrations
• Sampling at one point (the best – Baldereschi point, or the
simplest – Gamma point)
• Sampling at regular meshes (Monkhorst-Pack grids)
• For metallic systems, integration of the discontinuity is
improved introducing a fictitious electronic temperature
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Iterations to Selfconsistency
• Construct the external potential (array of non-local psp)
• Choose the plane-wave basis set cutoff, k-point sampling
• Pick a trial electronic density
• Construct the Hamiltonian operator: Hartree and
exchange-correlation
• Solve Kohn-Sham equations for the given Hamiltonian (e.g. by diagonalization)
• Calculate the new charge density
• Iterate
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Self-consistent ground state
• Iterative diagonalizations (Davidson,
Lancsoz, non-scf conjugate gradients +
mixing strategy)
Hˆ
&
\ (r)
&
ªn
r
º¼
¬
&
E\ (r ) • Direct minimization (Car-Parrinello,
conjugate gradients)
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Matrix Diagonalization
Hˆ
&
\ (r )
&
ªn
r
º¼
¬
&
E\ (r )
\
¦c
n
Mn
n 1,k
¦c
n
M
m Ĥ M n
Ecm
n 1, k
§ H11
¨
¨ .
¨ .
¨
¨ .
¨H
© k1
Mar 3 2005
......
......
H1k · § c1 ·
¸ ¨ ¸
. ¸ ¨ .¸
. ¸ ˜ ¨ .
¸
¸ ¨ ¸
. ¸ ¨ .¸
H kk ¸¹ ¨© ck ¸¹
§ c1 ·
¨ ¸
¨ .
¸
E ¨ . ¸
¨ ¸
¨ . ¸
¨ c ¸
© k¹
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Constrained non-linear minimization
E >^\ i `@
N
&
1 * & 2 & &
³\ i (r ) ’ \ i (r ) dr E H [n(r )] ¦
2
i 1
&
& & &
E xc [n(r )] ³ Vext (r ) n(r ) dr
&
EH [n(r )]
Mar 3 2005
& &
1 n(r1 )n(r2 ) & &
& & d
r1 dr2
³³
2
| r1 r2
|
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Constrained non-linear minimization
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Practice and Outlook
• Techniques and applications (from k-points
to DNA and superconductivity)
• Beyond GGA: WDA, TDDFT, QMC
• Connection to approximate methods
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Phonons
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Phonon Dispersions (Linear Response Theory)
600
Frequency (cm-1)
Si
400
200
0
Γ
K
X
Γ
L
X
W
Figure by MIT OCW.
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
L
DOS
B0(T)/B0(298 K)
Thermodynamical Properties
(Vibrational Free Energy)
1.002
1
0.998
0
50
100
150
200
250
300
350
400
TEMPERATURE (K)
Our ab-initio results
Experimental data (McSkimin, H. J., et. al.
J Appl Phys 43 (1972): 2944.
Figure by MIT OCW.
Ab-initio results compared to experimental data of H.J. McSkimin et al., J. Appl. Phys. 43, 2944 -1972.
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Dynamics in Al(110)
<110>
<001>
Surface
2nd layer
“Bulk”
“Bulk”
2nd layer
Surface
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Al(110) Mean Square Displacements
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Computational LEED
Graph removed for copyright reasons.
Interlayer relaxation vs. Temperature (K).
Source: Marzari, N., et al. “Thermal contraction and disordering of the Al(110) surface.” Physical Review Letters 82, no.16 (1999): pp.3296-9.
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
References (theory)
• W. Koch, M. C. Holthausen, A Chemist’s Guide to Density
Functional Theory
• R. G. Parr, W. Yang, Density-Functional Theory of Atoms and
Molecules
• W. Kohn, Nobel lecture
• F. Jensen, Introduction to Computational Chemistry
• J. M. Thijssen, Computational Physics
• B. H. Bransden and C. J. Joachim, Physics of Atoms and
Molecules
• K. Burke: The ABC of DFT, http://dft.rutgers.edu/kieron/beta/
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
References (practice)
• Payne, Teter, Allan, Arias, Joannopoulos,
Review of Modern Physics 64, 1045 (1992).
• Lecture notes from
http://www.FHI-Berlin.MPG.DE/th/Meetings/FHImd2001/program.html ,
(L3 Pehlke, L2 Kratzer, L4 Fuchs)
• Hartree-Fock for solids, Dovesi et al.,
Physica Status Solidi (b) 217, 63 (2000).
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Software
•
Gaussian (http://www.gaussian.com) ($$) (chemistry, Hartree-Fock, DFT,
correlated approaches)
• Gamess-UK (http://www.cse.clrc.ac.uk/qcg/gamess-uk/ ) ($) (chemistry,
Hartree-Fock, DFT, correlated approaches)
• Materials Studio/Cerius (http://www.accelrys.com) ($$) (DFT, planewave,
ultrasoft)
• Crystal (http://www.chimifm.unito.it/teorica/crystal) ($) (Hartree-Fock)
• VASP (http://cms.mpi.univie.ac.at/vasp) ($) (DFT, planewave, ultrasoft,
PAW)
• ESPRESSO (http://www.pwscf.org) (free) (DFT, planewave, ultrasoft, linearresponse theory, Car-Parrinello)
• ABINIT (http://www.abinit.org) (free) (DFT, planewave, linear-response
theory, GW)
• CPMD (http://www.cpmd.org) (free) (DFT, planewave, Car-Parrinello, timedependent DFT)
• CASINO (http://www.tcm.phy.cam.ac.uk/~mdt26/cqmc.html ) (free)
(Quantum Monte Carlo)
Mar 3 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
3.320/SMA5107: Lecture 9b (3/3/05)
Applications and Performance of DFT
methods
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Spin Polarization: The Local Spin Density
Approximation (LSDA)
• Electrons have spin +(-) 1/2 µB
• Spin is treated as a scalar quantity (this is
approximate, as relation to angular momentum makes
it a vector quantity)
• Two spin states often referred to as “up”
and “down”
• Up-Up interaction is different from Up-Down
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Exchange Effects: Refresher
Hund’s rule: A consequence of Pauling exclusion principle
e.g. for atomic d-levels
Even in solids where energy levels are split, can get
parallel filling if splitting is not too large
Why is this important ?
Filling of different orbitals
may give the atom different
chemical properties
Materials can carry a magnetic moment
ρ↑ ≠ ρ↓
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Spins in Density Functional Theory
In principle Exc[ρ] “knows” about this effect, but in practice it is
poorly approximated since only total charge density is
variable
But in practice, need to help LDA along …
Solution: Treat up and down densities separately
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Local Spin Density (LSD) = LDA with
different charge density for up and down
electrons
↑
veff
= v(r) +
↓
veff = v(r) +
ρ (r' )
∫ r − r'
ρ (r' )
∫ r − r'
dr' +
dr' +
[
∂Exc ρ ↑ , ρ ↓
]
∂ρ ↑
[
∂Exc ρ ↑ , ρ ↓
∂ρ ↓
]
Up and down charge density can be different
ρ↑ ≠ ρ b
Similar to restricted and unrestricted Hartree Fock
Regular LDA can not capture exchange effect well since
it is non-local
A spin-polarized version of GGA exists as well
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Charge Density: LiCoO2
O
Co
Li
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Spin Density:
filters out unpolarized ions (e.g. O2- )
Ni2+
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
How good is the output from LDA/GGA ?
Computational quantum mechanics provides very basic
information: Energy, charge density, band structure, optimized
atomic positions, etc.
How do I make a faster car from this ?
First evaluate accuracy of basic information
-> then understands how that propagates
into higher order models
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Photo of sports car removed
for copyright reasons.
Energies of Atoms
energy in Ry
Li
Be
GGA: PW91
LDA
Expt.
214.928
214.668
214.958
229.296
228.892
229.334
energy in Ry
Na
Mg
GGA: PW91
LDA
Expt.
energy in Ry
GGA: PW91
LDA
2324.514
2322.867
2324.49
2400.12
2398.265
2400.086
B
249.24
248.686
249.308
Al
2484.686
2482.618
2484.672
C
275.561
274.849
275.688
Si
2578.669
2576.384
2578.696
N
2108.926
2108.045
2109.174
O
2149.997
2148.939
2150.126
P
2682.386
2679.88
2682.764
S
2796.152
2793.419
2796.2
2199.433
2198.189
2199.45
Cl
2920.278
2917.313
2920.298
Ne
2257.893
2256.455
2257.856
Ar
21055.077
21051.876
21055.098
K
Ca
Ga
Ge
As
Se
Br
Kr
21199.825
21196.383
21355.144
21351.466
23850.018
23843.66
24154.2
24147.583
24471.917
24465.038
24803.334
24796.191
25148.619
25141.209
25507.943
25500.263
LDA underestimates stability of atom, GGA is closer
3/3/05
F
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Small Molecules
Binding Energy
exp (ev)
H2
LiH
O2
H20
F2
-4.753
-2.509
-5.230
-10.078
-1.66
LDA
-4.913
-2.648
-7.595
-11.567
-3.32
GGA
-4.540
-2.322
-6.237
-10.165
Binding energy too high in LDA, GGA is closer but
sometimes bound to weak. Pure Hartree Fock without
corrections is terrible.
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
HF
-3.64
-1.28
1.37
Lattice Parameters in Solids
exp
Si
Ge
GaAs
Al
Cu
Ag
Ta
W
Pt
Au
LDA
∆
GGA
∆
5.427
5.65
5.65
5.4
5.62
5.62
-0.50%
-0.53%
-0.53%
5.49
5.74
5.73
1.16%
1.59%
1.42%
4.03
3.60
4.07
3.30
3.16
3.91
4.06
3.98
3.52
4.00
3.26
3.14
3.90
4.05
-1.31%
-2.35%
-1.69%
-1.12%
-0.67%
-0.41%
-0.13%
4.09
3.62
4.17
3.32
3.18
3.97
4.16
1.57%
0.44%
2.47%
0.80%
0.67%
1.49%
2.48%
LDA tends to “overbind”, GGA “underbinds” GGA error more
variable
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Bulk Modulus in Solids (in GPA)
exp
Si
Ge
GaAs
Al
Cu
Ag
Ta
W
Pt
Au
∆
LDA
99
77
76
96
78
74
-3.03%
1.30%
-2.63%
83
61
65
77
138
102
193
310
283
172
84
192
139
224
337
307
198
9.09%
39.13%
36.27%
16.06%
8.71%
8.48%
15.12%
73
151
85
197
307
246
142
LDA tends to be too stiff. GGA too soft
3/3/05
∆
GGA
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
-16.16%
-20.78%
-14.47%
-5.19%
9.42%
-16.67%
2.07%
-0.97%
-13.07%
-17.44%
Oxides
exp
MgO
TiO2 (a)
TiO2 (c)
Al2O2
BaTiO3
PbTiO3
SnO2
β-MnO2 (a)
b-MnO2 (c)
3/3/05
4.21
4.59
2.958
5.128
4
3.9
4.737
4.404
2.876
LDA
4.17
4.548
2.944
5.091
3.94
3.833
4.637
4.346
2.81
∆
GGA
-0.95%
-0.92%
-0.47%
-0.72%
-1.50%
-1.72%
-2.11%
-1.32%
-2.29%
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
∆
4.623
2.987
5.185
0.72%
0.98%
1.11%
3.891
-0.23%
4.444
2.891
0.91%
0.52%
Summary of Geometry Prediction
LDA under-predicts bond lengths (always ?)
GGA error is less systematic though over-prediction is
common.
errors are in many cases < 1%, for transition metal
oxides < 5%
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Predicting Structure: The Energy Scales
Accuracy required to predict physical behavior is astonishing
V
Atomic energy:
-1894.074 Ry
Fcc V
:
-1894.7325 Ry
Bcc V
:
-1894.7125 Ry
Cohesive energy is 0.638 Ry (0.03% of total E)
Fcc/bcc difference is 0.02 Ry (0.001% of total E)
Mixing energies are also order 10-6 fraction of total E
How can we ever get physical behavior correct ?
Large cancellation of errors !
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Elemental Crystal Structures: GGA pseudopotential
experimentally found to be fcc
method
H
experimentally found to be bcc
-0.12
Ebcc - Efcc
Li
Be
0.13
0.11
0.11
2.19
0.04
0.50
Na
Mg
0.12
0.05
0.05
1.37
0.50
0.50
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
0.04
-0.05
-0.05
1.64
1.41
0.93
5.80
-3.02
4.79
0.48
-23.95
-7.50
-15.30
-36.76
-6.13
-9.19
7.41
0.78
1.80
-8.45
-7.97
8.36
1.71
4.20
9.23
7.99
7.49
2.84
4.02
4.02
(kJ/mole)
B
C
N
O
F
34.77
-19.71
-21.12
10.24
-4.53
-6.00
Al
Si
P
S
Cl
9.21
10.08
10.08
-1.89
-4.00
-4.00
-16.04
7.95
-17.65
-4.46
Zn
Ga
Ge
As
Se
Br
5.94
-0.08
6.03
1.48
0.70
0.70
0.70
-1.90
-1.90
-10.71
-14.67
-2.85
VASP-PAW
SGTE data
Saunders et al.
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
0.08
-0.20
-0.20
0.43
1.33
0.75
10.02
1.19
3.61
-0.29
-31.20
-13.50
-22.00
-38.74
-15.20
-28.00
19.04
8.00
8.00
48.93
9.00
14.00
32.39
19.00
19.00
3.74
10.50
10.50
2.27
3.40
3.40
4.90
1.02
0.64
0.65
0.99
-1.11
0.25
-8.96
-11.19
-1.26
Po
At
Cs
Ba
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
0.10
-0.50
-0.50
-1.62
-1.80
-1.80
10.14
2.38
-4.14
-23.75
-16.00
-26.50
-45.03
-19.30
-33.00
24.87
6.00
18.20
70.92
14.50
30.50
59.39
32.00
32.00
7.85
15.00
15.00
1.90
4.25
4.25
-1.02
-1.40
-0.09
0.07
4.06
2.40
2.40
-4.53
1.40
Fr
Ra
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
12.22
22.40
11.55
11.99
12.55
12.88
-1.61
13.11
12.97
12.73
12.36
11.86
Tm
Yb
Am
Cm
Bk
Cf
Es
Fm
Md
No
Ac
Th
Pa
U
Np
Pu
12.56
13.95
17.09
-10.36
-23.17
11.73
Lu
9.91
Lr
data taken from:
Y. Wang,*a S. Curtarolo, et al. Ab Initio Lattice Stability in Comparison with CALPHAD Lattice Stability, Computer Coupling of Phase Diagrams
and Thermochemistry (Calphad) Vol. 28, Issue 1, March 2004, Pages 79-90.
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Elemental Crystal Structures: GGA pseudopotential
experimentally found to be hcp
method
H
-0.01
experimentally found to be fcc
Ehcp - Efcc
Li
Be
0.19
-0.05
-0.05
-7.91
-6.35
-6.35
Na
Mg
0.06
-0.05
-0.05
-1.22
-2.60
-2.60
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
0.26
0.31
0.50
0.50
-4.48
-5.00
-5.51
-6.00
-6.00
0.53
-3.50
--4.80
0.91
-2.85
-1.82
-3.01
-1.00
-1.00
-7.76
-2.24
-1.95
-0.43
-0.43
2.22
2.89
1.50
0.52
0.60
0.60
-0.79
-2.97
0.69
0.70
0.70
-0.28
-1.00
-1.00
-4.83
-35.43
3.00
0.00
(kJ/mole)
B
C
N
O
F
-78.73
-6.18
-34.15
1.00
-14.64
-3.00
VASP-PAW
SGTE data
Saunders et al.
Al
Si
P
S
Cl
2.85
5.48
5.48
-3.26
-1.80
-1.80
-3.77
-43.63
-16.81
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
-0.01
0.38
0.25
0.25
-2.13
-6.00
-3.69
-7.60
-7.60
-3.08
-3.50
-5.00
1.14
-3.65
-5.00
-6.53
-10.00
-10.00
-10.79
-12.50
-12.50
3.26
3.00
3.00
2.50
2.00
2.00
0.29
0.30
0.30
-1.00
-0.89
0.35
0.37
0.65
-0.50
-1.61
-0.25
-3.94
23.40
0.99
0.00
Cs
Ba
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
-0.06
-0.40
0.20
0.20
-6.82
-10.00
-10.00
3.06
-4.00
-6.50
-1.79
-4.55
-6.00
-6.26
-11.00
-11.00
-13.26
-13.00
-13.00
6.55
4.00
4.00
5.02
2.50
2.50
0.08
0.24
0.55
-1.51
-2.07
-1.80
-0.31
-0.31
1.80
0.30
0.30
-4.03
Po
At
0.00
Fr
Ra
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
2.63
8.50
2.07
1.94
1.77
1.53
0.24
0.77
0.24
-0.41
-1.18
-1.97
Tm
Yb
-3.85
Ac
Th
Pa
U
Np
Pu
0.93
4.00
0.49
-15.79
-14.01
0.69
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
Lu
data taken from:
Y. Wang,*a S. Curtarolo, et al. Ab Initio Lattice Stability in Comparison with CALPHAD Lattice Stability, Computer Coupling of Phase Diagram
and Thermochemistry (Calphad) Vol. 28, Issue 1, March 2004, Pages 79-90.
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Summary: Comparing Energy of Structures
For most elements, both LDA and GGA predict the correct structure for a
material (as far as we know)
Notable exceptions: Fe in LDA; materials with substantial electron
correlation effects (e.g. Pu)
High Throughput studies are now possible.
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Metal Hydrides
Hydride formation energy: M + H2 -> MH2
LDA Calculation
Experiment
Correct structure !
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Spin Polarization can be Important (in
particular for transition metal compounds
Pmmn: structure of LiMnO2
300
250
C2/m
200
Pmmn
150
C2/m: other common structure
meV
100
50
0
-50
-100
-150
-200
NSP
Ferro
A-Ferro
Spin polarized calculations
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Orbital filling depends on spin polarization
5d orbitals
3/3/05
Mn3+ (high spin)
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Mn3+ (no spin)
Reaction Energies
Reaction
Exp (eV)
Li (bcc) + Al (fcc) -> LiAl (compound)
0.5 Cu(fcc) + 0.5 Au(fcc) -> CuAu
Li(bcc) + CoO2 -> LiCoO2
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-0.2457
-0.053
-4.25
LDA (eV)
-0.2234
-0.0193
-3.75
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
∆
9.08%
63.58%
11.76%
Redox Reactions can be more Problematic
Exp
3.5 eV
FePO4 + Li
-> LiFePO4
GGA
2.8 eV
MnO2 + Li
-> LiMn2O4
3.6 eV
4.1 eV
3.3 eV
4.6 eV
V2(PO4)3 + Li
-> LiV2(PO4)3
All these reactions involve the transfer of an electron from a
delocalized state in Li metal to a localized state in the transition
metal oxide (phosphate)
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Structure of LiV2(PO4)3
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
In standard DFT an electron interacts with the effective
potential generated by all the electrons (including itself)
H=
∑H
i
i
ρ(r)
Metal
=
Ne
Ne
∑∇ + ∑V
i =1
2
i
i =1
nuclear
Ne
(ri ) + ∑ Veffective (ri )
i =1
Small self-interaction error
Self interaction in DFT
is key problem in transition
metal oxides
r
ρ(r)
TM Oxide
3/3/05
Large self-interaction
error
r
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
Li insertion into cathode transfers electron from a delocalized to a
localized state
ρ(r)
Metal
Li metal
Less selfinteraction
r
Li+
e-
electron is transferred from
delocalized state to localized state
ρ(r)
M4+(O2-)2
TM Oxide
More selfinteraction
Li+ M3+(O2-)2
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3.320 Atomistic Modeling of Materials G. Ceder and N Marzari
r
Summary (LDA)
Lattice constants: 1-3% too small
Cohesive Energies: 5-20% too strongly bound
Bulk Modulus: 5-20% (largest errors for late TM)
Bandgaps: too small
GGA gives better cohesive energies. Effect on lattice
parameters is more random. GGA important for magnetic
systems.
3/3/05
3.320 Atomistic Modeling of Materials G. Ceder and N Marzari