Recent developments of xc functionals for large systems
Weitao Yang
Duke University
Peking University
South China Normal University
2H
Funding
NSF
NIH
DOE
Theory
Biological
Nano
Material
FHI-Beida
July 31 20181
Outline
● Chemical Potentials, Derivative Discontinuity, Band
Gaps, and LUMO Energy
● SCAN, a meta GGA
● Localized Orbital Scaling Correction to
Approximate Functionals
Paula Mori-Sanchez
(Duke, now Univ. Autonoma Madrid)
Aron J. Cohen
(Duke, now Cambridge)
1. “Fractional charge perspective on the band-gap in density-functional
theory,” PRB, 2008.
2. “Derivative discontinuity, band gap and LUMO orbital in density-
functional theory”, JCP 2012
Story of the chemical potential:Derivative discontinuity, band gap and
LUMO orbital in DFT
DFT for fractional number of electronsfrom grand ensembles,
Perdew, Parr, Levy, and Balduz, PRL. 1982
How to do calculations for fractional charges
1. Ensemble calculations are not useful at all– it always give the
correct linear behavior
2. Fractional occupation calculations --YES
E(N)
Molecule Fractional-Charge Atom
Localization Error
Delocalization Error
Delocalization and Localization Error
Mori-Sanchez, Cohen and WY, PRL 2008
Band Gap
Definition of fundamental gap
In terms of derivative information
Only if is linear.
Band Gap
Definition of fundamental gap
derivative
Only if is linear.
Background on DFT
Ev[½] = Ts[½] + J[½] +Exc[½] +Rv(r)½(r)dr
Density functional
Ground state energy
Ev(N) = minR½(r)dr=N
Ev[½]
¹ =±Ev[½]
±½(r)=±Ts[½]
±½(r)+±Exc[½]
±½(r)+ vJ(r) + v(r)
The 1st approach for solution (formal, OFDFT)
¹ =
µ@Ev(N)
@N
¶
v
Chemical potential, - electronegativity
Spin is implied, in all the discussion
Gap: Conventional Perdew-Levy-Sham-Schluter Analysis
A formal Euler-Lagrange approach
--in terms of the discontinuity of the functional derivatives
Gap: Perdew-Levy-Sham-Schluter Analysis
•Important in revealing the role of the functional
derivative discontinuity
•even the exact KS gap is not enough
Issues
•Defined at every point r in the 3-dimensional space
•Difficult/impossible to evaluate the functional derivative
discontinuity
•Can DFT even be used to predict fundamental band
gaps?
¹ =±Ev[½]
±½(r)
What is the Chemical potential ?
Ts[½] =X
i
ni
ZÁ¤i (r)
µ¡1
2r2
¶Ái(r)dr
½(r) = ½s(r) = ½s(r; r) =X
i
niÁi(r)Á¤i (r)
Two choices for carrying out practical DFT calculations:
Kohn-Sham (KS) (eg, LDA, GGA, OEP)
Generalized Kohn-Sham (GKS) (eg. HF, B3LYP, PBE0, HSE)
The 2nd approach for DFT solution (uses orbitals,
practical)
¹ =
µ@Ev(N)
@N
¶
v
Cohen, Mori-Sanchez and Yang (PRB 2008), based on potential
functional theory (PFT) (Yang, Ayers and Wu, PRL 2005)
Computing the chemical potential based on PFT
Cohen, Mori-Sanchez and WY, PRB, 2008
GaGap as the discontinuity of energy derivatives- chemical potentials
Comparison
1. -Perdew-Levy-Sham-Schluter view
The KS potential discontinuity
2. -new view based potential functional
theory
The constant in potential is irrelevant
Ev[vs;N] =Ev[vs+¢;N]
¹ =
µ@Ev(N)
@N
¶
v
¹ =±Ev[½]
±½(r)
Convex curve (LDA, GGA):
derivative underestimates I, overestimates A, I-A is too small
Concave curve (HF):
derivative overestimates I, underestimate A, I-A is too large
1,E
N EN
For Linear E(N)
How can fundamental gap be predicted in DFT
• LUMO energy is the chemical potential for electron addition
• HOMO is the chemical potential for electron removal
• Fundamental gaps predicted from DFT with KS, or GKS
calculations, as the KS gap or the GKS gap
• For orbital functionals, the LUMO of the KS (OEP) eigenvalue
is NOT the chemical potential of electron addition.
Thus the KS gap is not the fundamental gap predicted by the
functional. @Ev(N)
@N= hÁf jHe® jÁfi
How well can fundamental gap be predicted in DFT
• Fundamental gaps predicted from DFT with KS, or GKS
calculations, as the KS gap or the GKS gap
• Only works well if functionals have minimal
delocalization/localization error.
--HOMO energy is I, describing electron removal.
--Any meaning for LUMO in DFT?
Observation (Savin, Umrigar, Baerends…)
Accurate KS gaps, obtained from accurate density,
DO NOT correspond to I-A, but approximate
excitation energies (good for atoms, not so good for
molecules, Wu, Cohen, WY, MP 2005).
The meaning of LUMO in DFT
The meaning of LUMO in DFT: chemical potential
The HOMO in describes electron removal:
PRB 2008, JCP, 2012
KS eigenvalues if is a continuous functional of density
GKS eigenvalues if is a continuous functional of density matrix
The LUMO describes electron addition:
The meaning of LUMO in DFT
• The LUMO orbital energy in LDA, GGA, Hartree-Fock,
B3LYP, PBE0, HSE … is the chemical potential of
adding an electron, as described by its functional.
• Different from Koopmans’s theorem on HF LUMO
(frozen orbitals)
Yang, Mori-Sanchez and Cohen, JCP, 2012
What is the right form of the XC functional
1. Theory: Fundamental gaps, I-A, predicted from DFT as
the KS gap, if XC is an explicit and differentiable
functional of the density.
2. Observation (Savin, Umrigar, Baerends…) Accurate KS
gaps, obtained from accurate density, DO NOT
correspond to I-A, but approximate excitation energies
(good for atoms, not so good for molecules, Wu, Cohen,
WY, MP 2005).
Conclusion:
The exact functional is NOT an differentiable, explicit and
differentiable functional of the electron density.
Consequence of the Flat Plane Condition
Conclusion: Exact functional is NOT a continuous functional of density/orbs.
Exact conditions on DFT—all coming from QM
for degeneracy
, ,[ ]i N i N i NiE c E E
1 1(1 ) (1 )N N N NE E E
, 1, 1(1 ) (1 )i N i j N j N Ni jE c d E E
•!! The exact XC functional cannot be an explicit and differentiable functional of the electron density/density matrix, either local or nonlocal.
•Valid for density functionals, and also for 1-body density matrix functionals, 2-RDM theory, and other many-body theories.
Fractional Charge: 1982: Perdew, Levy, Parr and Baldus
Fractional Spins: 2000, PRL, WY, Zhang and Ayers;2008, JCP, Cohen, Moris-Sanchez, and WY
Fractional Charges and Spins: 2009: PRL, Moris-Sanchez, Cohen and WY
Main points
• Two views of chemical potentials: and
• Chemical potentials equal to the KS/GKS eigenvalues of HOMO
and LUMO
• LUMO has as much meaning in describing electron addition as
HOMO in describing electron removal.
• The exact XC functional cannot be an explicit and differentiable
functional of the electron density, either local or nonlocal.
• For Mott insulators, the exact exchange-correlation functional
cannot be an explicit and differentiable functional of the density
matrix.
¹ =
µ@Ev(N)
@N
¶
v
¹ =±Ev[½]
±½(r)
Recent Meta GGA development
SCAN: Jianwei Sun, Adrienn Ruzsinszky, and John P.
Perdew Phys. Rev. Lett. 115, 036402, 2015
Improving band gap prediction in density functional theory from molecules to solids
PRL, 2011, Xiao Zheng, Aron J. Cohen, Paula Mori-Sanchez, Xiangqian Hu, and Weitao Yang
Paula Mori-Sanchez(Univ. Autonoma Madrid)
Xiao Zheng(USTC)
Aron J. Cohen(Cambridge) Xiangqiang Hu
Use the exact conditions to improve approximations
Local Scaling Correction, PRL, 2015
Paula Mori-Sanchez
(Univ. Autonoma Madrid)
Xiao Zheng
(USTC)
Aron J. Cohen
(Cambridge))
Chen Li
(Duke)
Delocalization Error—Size dependent manifestation
Delocalization Error—Size dependent manifestation
Deviations between the calculated 𝜀𝐻𝑂𝑀𝑂 and
−𝐼𝑣𝑒 and between 𝐼𝑣𝑒 and 𝐼𝑒𝑥𝑝 for a series of 𝐻𝑒𝑀clusters (non-interacting).
𝐼𝑣𝑒= E(N-1)-E(N) 𝜀𝐻𝑂𝑀𝑂= (𝜕𝐸(𝑁)
𝜕𝑁)𝑣
Mori-Sancehz, Cohen and Yang PRL 2008, National Science Review 2018
Localized Orbital Scaling Correction (LOSC)
Chen Li, Xiao Zheng, Neil Qiang Su and WY (arXiv:1707.00856v1)
National Science Review, 2018
• Orbitalets: Novel localized orbitals to represent density
matrix.
• Size-consistent, functional of the GKS density matrix for
corrections to common DFA.
• Accurately characterization of the distributions of global and
local fractional electrons.
• Systematic improvements: the dissociation of cationic
species, the band gaps of molecules and polymers, the
energy and density changes upon electron addition and
removal, and photoemission spectra.
Chen Li Xiao Zheng Neil Qiang Su
Orbitalets: Novel Localized Orbitals
E-Constrained Optimization
Novel Localized Orbitals
-- Span both occupied and virtual space
-- Localization both in the physical
space and in the energy space.
Traditional
-- localized orbitals -- localization in the physical space
-- canonical orbitals -- localization only in the energy space
(energy eigenstates of an one-particle Hamiltonian)
Orbitalets
Delocalization Error—Size dependent manifestation
Distribution of LO densities i𝑛 𝐻2+
At small R, R=1A
• Large energy gap between
HOMO and LUMO
• Little mixing, LO ~ Canonical
Orb, integer occupations
At large R, R=5A
• Small energy gap between
HOMO and LUMO
• Much mixing, LO localized,
fractional occupations
Previous Global and Local Scaling Approach
PRL 2011 and PRL 2015
New LOSC, as correction to DFA
Non-empirical parameter to
get correct limit for 𝐻2+
Orbital energy corrections
Linear E(N) and Size-Consistent: 𝐻𝑒2(𝑅)
LOSC: Linear E(N) and Size-Consistent
LOSC: Size-Consistent Corrections
LOSC: HOMO, LUMO and Energy Gaps
LOSC: HOMO, LUMO and Energy Gaps
Method IP EA
scGW 0.47 0.34
G0W0-PBE 0.51 0.37
LOSC-BLYP 0.47 0.32
LOSC-PBE 0.37 0.32
LOSC-B3LYP 0.26 0.27
LOSC-LDA 0.34 0.48
BLYP 2.98 1.99
PBE 2.81 2.17
B3LYP 2.00 1.58
LDA 2.58 2.44
Mean absolute error (eV) of ionization potential and electron affinity results on 40 test molecules.
Photoemission spectrum of nitrobenzene
Photoemission spectrum of anthracene
Photoemission spectrum of C60
Photoemission spectrum of H2TPP
LOSC: photoemission spectra
LOSC: corrections to electron density
2 6( )Cl H O
LOSC: Summary
--Very different from conventional density
functionals
--Novel localized Orbitals with energy and space
localization – Orbitalets
--Functional of the Generalized Kohn-Sham
density matrix
--Size-consistent
Prospective of DFT Approximations—bright future
Semilocal functionals + Nonlocal corrections
• LOSC: Eliminating delocalization error • Band gaps
• Energy alignment
• Charge transfer
• ….
• Describing strong correlation (static correlation) —using fractional spins
Strategy of nonlocal corrections
---Imposing the exact constraints of fractional charges and fractional spins
Reducing delocalization and strong correlation error
Chen LiNeil Su
HF molecular dissociation, restricted
Neil Su, C. Li, W. Yang, submitted 2018
Reducing delocalization and strong correlation error
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