short course on density functional theory and applications ix....
TRANSCRIPT
Short Course on Density Functional Theory and Applications
IX. Brief Mentions: GW Method, Kinetic Energy Density Functionals
Quantum Theory ProjectDept. of Physics and Dept. of Chemistry
Samuel B. Trickey©Sept. 2008
GW Method for ExcitationsStrictly, the GW method is not a part of DFT. As generally formulated and practiced, however, GW is a practical realization of the relatively little-noticed third Hohenberg-Kohn-Sham theorem. It is this. The one-particle QM Green’s function is a unique functional of the density alone which attains the correct physical one-particle spectrum at the ground state density n0 . [L.J. Sham and W. Kohn, Phys. Rev. 145, 561 (1966)]
†ˆ ˆ( , ; , ) 0 | { ( , ) ( , )} | 0G t t t t′ ′ ′ ′= ⟨ Ψ Ψ ⟩x x x xT
Brief reminder: For a system of a finite number of electrons, the electronionization energies and electron affinities are found most generally from the one-electron QM Green's function or propagator. The basic structure is
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A sum of ground state expectation values.
†
ˆ ˆ
† † †
ˆ ˆ( , ; , ) 0 | { ( , ) ( , )} | 0
ˆtime ordering operator; ( , ) electron creation operator
ˆ ˆ ˆ ˆ( , ) ( ) ( ) ( )
ˆ ˆ ˆ ˆ ˆ ˆ, , , 0
{ ( )} comple
;
;
iHt iHtj j
j
i j ij i j i j
j
G t t t t
t
t e e a
a a a a a a
ϕ
δ
ϕ
−
++ +
′ ′ ′ ′= ⟨ Ψ Ψ ⟩
= Ψ =
Ψ = Ψ Ψ =
= = =
=
∑
x x x x
x
x x x x
x
T
T
†
te 1-electron basis set
ˆ ˆ( , ; , ) ( , ) 0 | | 0ij i jij
G t t g t t a a′ ′ ′⇒ = ⟨ ⟩∑x x
GW (cont’d.) - Photoelectron Excitation and Quasi-particle statesConsider
( )
( )
: particle ground state with total energy
1, : 1particle excited state "j"
1, total energy for excited state "j"
e e e
e e
e
N N E N
N j N
E N j
=
− = −
− =
Electron removal Dyson orbitals and energies (photoemission):
( ) ( )ˆ[ ] 1, ( )
1,
j e eN j N
E N E N j
χ
ε
= − Ψ
= − −
x x
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Credit: Patrick Rinke.
( ) ( )ˆ[ ] , ( ) 1
1 ,
j e e
j e e
N j N
E N E N j
χ
ε
= Ψ +
= + −
x x
Electron addition Dyson orbitals and energies (inverse photoemission):
( ) ( )1,j e eE N E N jε = − −
GW (cont’d.) -
which requires self-consistency in the energy E=εεεεj . The self-energy operator ΣΣΣΣ is defined in terms of the Green’s function
212
[ ( ) ( )] [ , ] [ , , ] [ , ] [ ] ,H ext j j j jV V dχ χ χε′ ′ ′− ∇ + + + Σ =∫r r x x x x x xE E E E
The self-energy operator takes account of the energy response of the electron population that the quasi-particle experiences due to its own presence in that population.
212
[ ( ) ( )] [ , , ] [ , , ] [ , , ] ( ) H extV V G d G δ′ ′ ′+ ∇ − − − Σ = −∫r r x x z x z z x x xE E E E
The Dyson orbitals obey the one-body equation
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population.
Casida [Phys. Rev. A 51, 2005-13 (1995)] has proved that, in a certain well-defined sense (Klein’s energy minimization principle), the KS potential is the best local approximation to the non-local self-energy operator ΣΣΣΣ.
High precision numerical studies of exact KS eigenvalue differences for simple systems are consistent with this interpretation. [C.J. Umrigar, A. Savin, and X. Gonze in Electronic Density Functional Theory: Recent Progress and New Directions J.F. Dobson, G. Vignale, and M.P. Das eds. (Plenum, 1998), pp 167-76].
GW (cont’d.) -
1( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ; )
( )( ; ) ( ) ( ) ( ) ( ) ( ; )
( )
( ) ( ) ( ) ( ; )
W d g
d g
i d d G G
d d d d G GG
i d d W G
δ
δδ δδ
−=
= −
= − Γ
ΣΓ = + Γ
Σ = Γ
∫∫
∫
∫
∫
12 3 13 32
12 12 3 13 32
12 3 4 23 42 34 1
1212 3 12 13 4 5 6 7 46 75 67 3
45
12 3 4 13 14 42 3
ε
ε P
P
Self-energy operator
Vertex function
Irreducible polarization
Microscopic dielectric function
Screened Coulomb interaction
Solution of the Dyson equation involves a nest of equations, written here in a common short hand x1t1 →→→→1, and with some carelessness about time-ordering
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( ) ( ) ( ) ( ; )i d d W GΣ = Γ∫12 3 4 13 14 42 3 Self-energy operator
which simplifies the equations to
The GW approximation is ( ; ) ( ) ( )δ δΓ ≈34 1 34 31
1
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
iG W
iG G
W d g
i d g G Gδ
−
Σ ≈≈ −
=
≈ +
∫∫
12 12 12
12 21 12
12 3 13 32
12 12 3 13 32 23
P
ε
ε
Even with the GW simplification, solution of the Dyson equation is a demanding problem. Often models of the dielectric function are used, non-iterative solutions are done, etc. The connection with DFT is that the orbitals and eigenvalues of a DFT calculation often (almost always) are used as the starting approximations. Especially when combined with OPE X, the results are very good for semiconductor band gaps [P. Rinke et al. New J. Phys. 7, 126 (2005)]:
GW (cont’d.) -
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GW (cont’d.) -Basic references in GW:1. L. Hedin and H. Lundqvist, Solid State Phys. 23, 1 (1969) 2. F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998)3. W.G. Aulbur, L. Jönsson, and J.W. Wilkins, Solid State Phys. 54, 1 (2000)4. G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002)
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1 2( , , , )I I I inter particle Nm V −= −∇R R R R�
ɺɺ …
is the potential energy for all “particle” coordinates. { }( )inter- particleV R
As remarked at the end of Lect. VII, Molecular Dynamics(MD) is at the heart of much of materials physics and biomolecular simulation. At base, it is justNewton’s 2nd Principle.
A “particle” is a fixed, rigid object that can be identified with a point location. For simplicity, just think of point nuclei.
In principle, the inter-particle potential is theBorn-Oppenheimerenergysurface
Molecular Dynamics and the Born-Oppenheimer Potential Energy Surface
{ }( ) { }( ) { }( )0inter particle NNV E E− = +R R R
In principle, the inter-particle potential is theBorn-Oppenheimerenergysurface
where E0 ({R}) is the ground state electronictotal energy and ENN is the nuclear-nuclear repulsion.
In practice Vinter-particle({R}) often is represented by parameterization of a classical potential function to some set QM calculations or by an empirically calibrated “force field”. For bond-breaking, fracture, etc. this is not good enough. The real B-O surface is needed.
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• The KS eigenvalue form of the DFT ground state energy illustrates the computational barrier to using ordinary KS DFT to provide the B-O forces (“gradients” in quantum chemistry jargon) at every MD step:
{ }0
2
2
1 ( ) ( )( ) [ ] [ ( ) ( )]
2 | |
( ) | ( ) |
1ˆ ˆ( ) ( ); ( ); ( ) ( ) ( ) ( )2
k xc xck
k kk
KSk k k KS KS ext H xc
n nE d d E n d n v
n n
h h v v v v v
ε
φ
ϕ ε ϕ
′′= − + −′−
=
= = − ∇ + = + +
∑ ∫ ∫
Σ
r rR r r r r r
r r
r r
r r r r r r r
B-O Forces from DFT
( ) ( ); ( ); ( ) ( ) ( ) ( )2k k k KS KS ext H xch h v v v v vϕ ε ϕ= = − ∇ + = + +r r r r r r r
This eigenvalue approach is slow. There are “order-N” approximate methods but they introduce additional assumptions (e.g. about basis locality, etc.).
Challenge: to get the content of KS DFT without doing the KS eigenvalueproblem, yet accurately enough to do physics and chemistry studies on realistically available computer clusters.
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The KS construction does a very good thing. It separates the KE into a tractable part TS and a remainder which goes into Exc , recall II-6:
◊ Most DFT methods, even so-called order-N, are focused on determining the Kohn-Sham orbitals in a basis.
◊ But in DFT, E[n] is fundamental, the eigenvaluesεεεεj[φK-S[n]] are not.
[ ] [ ] [ ] [ ] [ ][ ] ( ) ( )21
2:
extv ee xc ext
S j j jj
STE n E n E n
n d
n
T n
E n
ϕ ϕ= ∇
= + + +
− ∑ ∫ r r r
Unfortunately, KS also introduces an orbital eigenvalueproblem to calculate T .
Orbital-free DFT: Motivation
Unfortunately, KS also introduces an orbital eigenvalueproblem to calculate TS.This is a computational bottleneck. What we need instead is an explicit density functional for the KS KE:
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* 2i
1 ( ) ( )
2[ ] [ ( )]i is Sn dT n d t nϕ ϕ−= ∇ ≡ ∫∑ ∫ r r rr r
Were we to have the K-S KE density kernel, and a good density-dependent (NOT orbitally dependent) EXC, then DFT B-O forces would be simple:
{ }{ }
[ ] [ ] [ ] [ ]
[ ] ( ) 0 [ ( )]
[ ( )] [ ] [ ] [ ]
[ ]
[ ] [ ( )]
ee xc Ne
sKS
KS ee xc
s
Ne
s S
E n E n E n E n
TE n d n V n
n n
V n E n E n E nn
T n
T n d t n
δδ µ µδ δ
δδ
= + + +
=
− = ⇒ + =
= + +
∫
∫ r r r
r
r r
← “Hydrodynamic” Form of DFT
Orbital-free DFT: General Structure
[ ] ( ) ( ) [ ( )] (r)OF DFT sI I I NN I Ne KS I
nT
E n E d n v d V n nn
δδδ
− = −∇ = −∇ − ∇ − + ∇ ∫ ∫F r r r r r
Electronic contributions to forces become easy
• Is this goal realistic?
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Remark: The QM Coulomb virial theorem reinforces our focus on TS [n] rather than the full T[n]:
0
0 0
0 0 0
0 0
0 0 0
2 [ ] / [ ] 1
[ ] [ ] [ ]
[ ] 2 [ ]
[ ] [ ]
total potential
total potential
total potential
T n E n
n T n E n
E n T n
n T n
− =
= +
⇒ = −
⇒ = −
E
E
Orbital-free DFT – Considerations that shape approaches
← ← ← ← Coulomb Virial Theorem
• Implication: Seeking the full T[n] would be roughly equivalent to seeking the exact, universal HKS functional
• Clearly this is far too challenging a goal!
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Search for OF-KE has a long history of big names and big difficulties:
�Thomas-Fermi-Dirac:
Teller’s non-binding theorem (1962)
[E. Lieb, Rev. Mod. Phys. 53, 603-41 (1981)]
�Thomas-Fermi-von Weizäcker:
( )2 2 / 3
0
5/30
3(3 )
10
[ ] r rTF
c
T n c d n
π=
= ∫
21 n∇
Orbital-free DFT: History as Guidance
�Thomas-Fermi-von Weizäcker:
When combined with ENe[n] + Eee[n] + ENN: binds neutrals, has negative ions.
( )
21
[ ] r8 rTFvW TF vW TF
nT n T T T d
n
∇= + = + ∫
But not accurately.
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�Thomas-Fermi-von Weizäcker used in OFDT:
Paul Ayers (Sanibel 2007): “An ab initio quantum chemist will wonder- “Is an N-representability constraint missing?”
[ ] [ ] [ ] [[ ] ]TFvW ee xcT evW nFE n E n E n En nT= + + +
0 0[ ] [ ]TFvW KSE n E n<
Orbital-free KE: More Problems (for guidance)
Variational treatment – or evaluation with theactual KS n0 for a given XC – can lead to disaster
“If so, should we surrender? N-representability problems are very difficult ….”“If so, should we surrender? N-representability problems are very difficult ….”
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Orbital-free KE: Fermion N-representability (constraint)
Defn: A KS kinetic energy functional TKS[n] is N-representable iff for eachproper density n, there exists a proper N-fermion state (or ensemble) which has the same KS kinetic energy for that density. Remark – there are infinitely many N-fermion states associated with each proper n.
Consider the constrainedsearchdefinition of the K-S KE:
[ ] ( ) ( )2
*1 1 1 1 1 12min , , , ,
ej
N
s N N N N N Nn
j
T n d d d dσ σ σ σ σ σΨ→
−∇ = Ψ Ψ
∑∫ r r r r r r… … …
Implications: 1. An approximate KE functional which delivers a value below the KS
KE Ts for any system is NOTN-representable.2. Any non-N-representable KE functional will give a value below the KS
KE for at leastone system.
Credit: Paul Ayers
Observation – the risks of non-N-representability are serious.
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Nevertheless, calculations with Thomas-Fermi-von Weizäcker in OFDFT continue to appear. Some very clever finite element techniques for NON-periodic systems and some interesting results for electronic structure of defects in Al are in, for example, V. Gavini et al. [J. Mech. Phys. Solids 55, 6987-718 (2007)]
Orbital-free KE: TF-vW anyway
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Several groups, e.g. Madden et al., Carter et al., Teeter, and others, have focused on Ts[n] with a response function strategy for developing OF-KE approximations. Use Carter et al.,as an example of much of the literature. [See Y.A. Wang and E.A. Carter in Theoretical Methods in Condensed Phase Chemistry, S.D. Schwartz ed. (Kluwer NY 2000) p. 117]
They calibrate their KE density models to linear response on grounds that 1. In atoms and molecules, “shell structure is the barometer” of a good OF-KE2. “short range [density] oscillations and Friedel oscillations” are the “corresponding physical standard” in solid state physics, and 3. “correct linear response behavior is the key to predicting such oscillations.”
Orbital-free KE Approaches – Response Function
3. “correct linear response behavior is the key to predicting such oscillations.”
Result is a set of non-local or two-point approximations – work moderately well for metals, not insulators.
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Here, in somewhat sloppy translationally invariant notation, is the basic theme of the response function approach
2
(r) r (r - r ) (r ) (q) (q) (q)
(r )(r)(r - r ) r (q - q ) (q) (q,q )
(r ) (r )
[ (r)] (q - q ) (q) (q,q )
KS KS
KS KS
KS
KSs s
n d V n V
V Vnd
V n n
T TV n
n n n
δ χ δ δ χ δ
δ δδδ δ χδ δ δ
δ δµ δ χδ δ δ
′ ′ ′= ⇒ =
′′′ ′′ ′ ′= ⇒ =′′ ′
′ ′+ = ⇒ = −
∫
∫
Orbital-free KE Approaches – Response Function
[ (r)] (q - q ) (q) (q,q )V nn n n
µ δ χδ δ δ
+ = ⇒ = −
• Outcome of efforts by Carter et al.: models based on Average Density Approximation, Weighted Density Approximation, and Nonlocal Approximations.• Related work by García-Aldea, Alvarellos, Stott, García-Cervera, Ghringhelli, Delle Sitte, ….
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Orbital-free KE Approaches – Response Function
An older example of the Response Function Approach[Phys. Rev. B 60, 16350 (1999),Erratum, Phys. Rev. B 64, 089903(2001)]
The “surface” actually is a slaband the axis should be “atomic units” not “arb units”.
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OFDFT generally gives the “smooth” part of the density, so why not use the OFDFT density as an input to a Harris functional approach? [J. Chem. Phys. 124, 081107 (2006)]
Orbital-free KE Approaches – Orbital Corrections
{ }
{ } { }
(1)
(1)
(1) (1) (1)
( ) ( )1( ) [ ( )] ( ) [ ( )]
2 | |
( ) ( )1( ) [ ( )] ( ) [ ( )] [ ( )]
2 | |
[ ( )]
OF OFHarris k xc OF OF xc OF
k
OO OOHK k xc OO OO Hi OF xc OF
k
KS OF k k k
OO k
n nE d d E n d n V n
n nE d d E n d n V n V n
h n r
n n
ε
ε
φ ε φ
′′= − + −′−
′′= + + − +′−
=
=
∑ ∫ ∫
∑ ∫ ∫
r rR r r r r r r
r r
r rR r r r r r r r
r r�
(1) (1)k k
k
φ φ∑The third line is a “one-shot” diagonalization, not self-consistent. Zhou and Wang propose an interpolated functional “ZW λ” as being superior. Here’s one functional “ZW λ” as being superior. Here’s one example of their results.
The approach is closely related to “Strutinsky” shell corrections[Yannouleas and Landman, Chem. Phys. Lett. 210, 437 (1993); Phys. Rev. B 48, 8376 (1993); Ullmo, Nagano, Tomsovic, and Baranger, Phys. Rev. B 63, 125339 (2001)]
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• Models based on Average Density Approximation, Weighted Density Approximation, and Nonlocal Approximations are all two-point.
• Too complicated for our (Karasiev, Harris, Jones, and SBT) purposes!
• OBJECTIVE: torb[n(r)] be no more complicated than GGA EXC (depends ongradient of the density) or perhaps those meta-GGAs which depend on Laplacianof the density also.
• Assumption: continued progress on pure EXC approximations, i.e.,NOT hybrids or OEP. • Goal: a workable recipe for ts[n] primarily for driving MD. Note the
Orbital-free KE: Constraint-based Approach
• Goal: a workable recipe for ts[n] primarily for driving MD. Note the wording: we do NOT seek a KE density kernel that will do everything that is in the basic DFT theorems
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2) Use Pauli KE and Pauli potential positivity [M. Levy and H. Ou-Yang, Phys. Rev. B 38,
( ) { }
( )( ) ( )
4/3 2 1/3 4/3
5/3
[ ] r (r) (r) ; (r) = 2(3 )
exchange enhancement factor
Let [ ] r (r) (r)
(r) (r) conjoint KE enhancement factor
GGAx x x
x
GGAs t
t x
E n c d n F s s n n
F
T n d n F s
F s F s
π= − ∇
=
=
∝
∫
∫
Orbital-free KE: Constraint-based Approach (cont’d.) 1) Use “conjointness” [H. Lee, C. Lee, and R.G. Parr, Phys. Rev. A 44, 768 (1991)]to build the initial model of the functional but not for the parameters. (Note: the conjoint-ness hypothesis itself is false.) Conjointness is:
2) Use Pauli KE and Pauli potential positivity [M. Levy and H. Ou-Yang, Phys. Rev. B 38, 625 (1988); A. Holas and N.H. March, Phys. Rev. A 44, 5521 (1991); E.V. Ludeña, V.V. Karasiev, R.
López-Boada, E. Valderama, and J. Maldonado, J. Comp. Chem. 20, 155 (1999) and refs. therein]. An exactdecomposition is
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( )212
[ ] [ ] [ ], [ ] 0
(r) (r) (r) (r)
(r) : 0 r
s W
KS
T n T n T n T n
v v n n
v T n
θ θ
θ
θ θ
µ
δ δ
= + ≥
− ∇ + + =
= ≥ ∀
3) Use the gradient expansion for Tθθθθ[n], but not the expansion coefficients. Instead, choose the coefficients to remove nuclear-site singularities from vθθθθ[n] to a specified order in the expansion.
4) Test existing conjoint (or nearly) models of Ts[n] by inputting the density from a conventional KS calculation with a selected Exc approximation.
5) Upon step 4) success, use ts[n] to generate n from Euler equation (in a basis) and compare with n0 from conventional KS calc. Continue or loop back as required.
6) Apply to MD calculation on target system(s) related to training set.
Orbital-free KE: Constraint-based Approach (cont’d.)
6) Apply to MD calculation on target system(s) related to training set.
7) Replace basis calculation of density with parameterized density model (parameters become dynamical variables in MD).
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SiO Stretch; Total energy vs. bond length.Exact is KS (LDA-VWN).
OF-DFT: Tests of Existing KE Functionals• Tests of 6 existing functionals, three conjoint; two others close.
PW91: Lacks and Gordon, J.Chem. Phys. 100, 4446 (1994) [conjoint]PBE-TW: Tran and Wesolowski, Internat. J. Quantum Chem. 89, 441 (2002) [conjoint]GGA-Perdew: Perdew, Phys. Lett. A 165, 79 (1992) [conjoint]DPK: DePristo and Kress, Phys. Rev. A 35, 438 (1987)Thakkar: Thakkar, Phys. Rev. A 46, 6920 (1992)SGA: Second order Gradient Approx. TS = TTF + (1/9) TW
−359
−358
−357
−356
exactSGAGGA−PerdewPW91DPKThakkarExact is KS (LDA-VWN).
Input to KE functionals is the KS density.
All six Ts approximations fail to bind! Vθθθθ (not shown) violates positivity.
J. Comput. Aided Matl. Design 13, 111 (2006)0.626 1.126 1.626 2.126 2.626
R, angstrom
−366
−365
−364
−363
−362
−361
−360
E, h
artr
ee
ThakkarPBE−TW
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5/3 2 20
5 5[ ] r (r) (r) ; (r)
3 3W t tT n c d n s F F s = ≡ − ∫ ɶ
Write TW in GGA-like form and define a shifted enhancement factor
Small s (nominally weakly inhomogeneous) limit: 2
0
40( (r)) 1
27ts
F s s→
−ɶ ∼
• Conjoint Pauli KE and potential for GGA-level models:
5/30
22/3 5/3 5/3
[ ] [ ] r [ (r)] r (r) ( (r))
5 5
GGA GGAt
t t
T n T n d t n c d n F s
F Fs n s s sv c n F c n c n s
θ θ θ
δ δδ δ δ δ
≈ = ≡
∇ = + − − ∇ − ∇
∫ ∫ ɶ
ɶ ɶɶ i i i
Modified Conjoint GGA OFKE Functionals
2/3 5/3 5/30 0 0 2
5 5
3 3t t
t
F Fs n s s sv c n F c n c n s
s n n n n s nθδ δδ δ δ δδ δ δ δ δ δ
∇ = + − − ∇ − ∇ ∇ ∇ ∇
ɶ ɶɶ i i i
Ft comes fromFX Parameterize to enforce positivity of vθθθθ
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Si-O bond stretching gradient in H6Si2O7.→→→→OF-KE parameters from 3-member training set (SiO, H4SiO4, and H6Si2O7) except PBE2 model. Exact is KS (LDA-VWN)
0.81 1.31 1.81 2.31 2.81R, angstrom
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
dE
/dR
, har
tree
/ang
stro
m
exactPBE2PBE3PBE4exp4
Modified Conjoint GGA OFKE Functionals
0
H O – gradient for single-bond stretching
J. Comput. Aided Matl. Design 13, 111 (2006)
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0.57 1.07 1.57 2.07R, angstrom
−5
−4
−3
−2
−1
dE
/dR
, har
tree
/ang
stro
m
exactPBE2PBE3PBE4exp4
H2O – gradient for single-bond stretching←←←←NO information about H2O in the training set.
But all the OFKE models give bond lengths that are too long. Problem – positive singularities at nuclei.
Orbital-free KE: Constraint-based Approach (cont’d.) Solution to “too positive” a Pauli potential (nuclear site singularities): Analyze the gradient expansion for Tθθθθ[n] using known (Kato cusp condition) near-nucleus behavior of the density. Determine expansion coefficients through given order that suppress the singularities. [V.V. Karasiev, R.S. Jones, S.B.T. and F.E. Harris, Phys. Rev. B (submitted)]
Comparison of Ts errorsat KS equilibrium geometries for the Thakkar, Perdew-Constantin (MGGA)and Karasiev et al.’s new “RDA”
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and Karasiev et al.’s new “RDA” OFKE functionals. RDA is generally better than the others -but not always.
GW and OFKE - Some Commentary
• GW is still an active area of research• There has been considerable controversy about the role of the multiple self-consistency requirements in GW and the impact of basis sets.• A simplified, computationally fast, yet basically accurate approximate version of GW would be highly desirable.• OFKE is also an area of active research, though for a smaller community. • There is a strong divergence of approaches between the “response function”(two-point schemes) and the newer, less-well-developed “constraint-based” schemes.• Note that, in all fairness, the response function approach is also “constraint-
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• Note that, in all fairness, the response function approach is also “constraint-based”.