density functional theory · 2018. 7. 31. · density functional theory (dft) (1964, 1965) 1. 2....
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Density Functional Theory
Weitao Yang
Duke University
Peking University
South China Normal University
2H
Funding
NSF
NIH
DOE
Theory
Biological
Nano
Material
FHI-Beida
July 30 20181
Outline of Three Topics
1. Electronic Structure Theory Overview and
Introduction to DFT
HK Theorems, Kohn-Sham Equations and Density Functional
Approximations
2. Insight into DFT
Adiabatic Connections, Density Functional Approximations, Fractional
Perspectives of DFT, Challenges in DFT—Delocalization and Static
Correlation Error, Band Gaps, Chemical potentials, Derivative Discontinuity,
Physical Meaning of Frontier Orbital Energies (HOMO and LUMO).
3. Recent Developments of xc Functionals for Large
Systems
SCAN meta GGA
Localized Orbital Scaling Corrections
2
Key references
R. G. Parr and W. Yang, “Density functional theory of atoms and
molecules”, Oxford Univ. Press, 1989
RM Dreizler, EKU Gross, “Density Functional Theory: An
Approximation to the Quantum Many-Body Problems”
Cohen, A. J., Mori-Sánchez, P. & Yang, W. “Challenges for
Density Functional Theory”. Chemical Reviews 112, 289–320,
2012.
SCAN: Jianwei Sun, Adrienn Ruzsinszky, and John P. Perdew
Phys. Rev. Lett. 115, 036402, 2015
LOSC: C. Li, X. Zheng, N. Q. Su, and W. Yang, National Science
Review, vol. 5, no. 2, pp. 203–215, 2018.3
Roles of Electrons
Matter
Molecules, Atoms
Nuclei, Electrons
C1
H
HH
H
Electrons are the glue that bind the
repulsive nuclei and determine the
structure of matter, from molecules
to macromolecules, to bulk materials. 6
Quantum Mechanics
The wave-like behavior of electrons
Electron Beam
7
Quantum Mechanics
Erwin Schrödinger
8
Adiabatic Approximation:
Born-Oppenheimer (BO) Approximation
9
The time-dependent Schrodinger equation for the entire system
(containing both electrons and nuclei)
The Quantum Adiabatic Theorem: A physical system remains in its
instantaneous eigenstate if a given perturbation is acting on it slowly
enough (and if there is a gap between the eigenvalue and the rest of the
Hamiltonian's spectrum).
The nuclei move slowly, compared to electrons, because of the difference in
mass.
Considering the nuclei motion as perturbation to the electrons, and
assuming that the nuclei move slowly enough, then the electrons are in its
instantaneous eigenstate of a given nuclei configuration. This is the BO
approximation.
The Schrodinger Equation for N electrons
Hª(x1;x2;:::xN) = Eª(x1;x2;:::xN)
x= r; s
H = T + Vee +
NX
i
vext(ri)
T =
NX
i
¡1
2r2
Vee =
NX
i<j
1
rij
vext(r) = ¡X
A
¡ZA
rAi
10
Methods for Solving the Schrodinger Equation
for N electrons
Hª(x1;x2;:::xN) = Eª(x1;x2;:::xN)
11
1. Hartree-Fock mean-field theory
2. Perturbation Theory
3. Configuration interaction
4. Coupled-cluster theory
5. Quantum Monte Carlo
6. Green function theory (for excitation spectrum)
Wavefunction--the curse of dimensionality
Hª=Eª# of electrons wavefunction amount of data
1
2
… … …
N
ª(r1)
ª(r1; r2) 106103
•The amount of data contained in wavefunction grows exponentially
with the number of electrons!
•The problem is caused by the exponential increase in volume
associated with adding extra dimensions to a (mathematical) space.
ª(r1;r2; :::rN) 103N
N » 1¡10000
12
The curse of dimensionality in QM
The problem is caused by the exponential increase in
volume associated with adding extra dimensions to a
(mathematical) space.
13
Exponential growth of information
N, positions and types of atoms
H
103N in ª(r1;r2; :::rN)
+
+, N, v(r)
v(r) is the electrostatic potential from the nuclei
m
v(r) =
AtomsX
A
¡ ZA
jr¡RAj
1.
2.
?
14
Electron density , 3-dimentional only
15
•Electron density is the measure of the probability of an
electron being present at a specific location.
•Experimental observables in X-ray diffraction for structural
determination of small and large molecules
½(r)
Electron density for anilineElectron density for the hydrogen atom
Distance from the nucleus
½(r) =1
¼exp(¡2r)
½(r)
Density Functional Theory
Density functional theory (DFT) (1964, 1965)
1.
2. Variational principle for ground state energy over all densities.
3.16
½(r)()N;v(r)()ª
The Nobel Prize in Chemistry 1998
Walter Kohn Pierre C. Hohenberg Lu J. Sham
½(r) =X
i
jÁi(r)j2 Just the sum of molecular orbital densities
DFT: Exchange-correlation energy
17
E = E[½(r)]
= Kinetic energy + potential energy
+ Coulomb interaction energy
+ Exc[½(r)]
E
Exc[½(r)]
•Exchange-correlation energy
•The only unknown piece in the energy
•About 10% of E
Approximations in exchange-correlation energy
18
Exc[½(r)]
• 1965: Kohn and Sham, Local Density Approximation (LDA)
• 1980s-1990s: Axel Becke, Robert Parr, John Perdew
•Generalized Gradient Approximation (GGA)
•Hybrid Functionals (B3LYP, PBE0)
Density Functional Theory
• Structure of matter: atom, molecule, nano,
condensed matter
• Chemical and biological functions
• Electronic
• Vibrational
• Magnetic
• Optical (TD-DFT)
19
ISI Web of Science search for articles with topic “density functional theory”
20
0
10000
20000
30000
40000
50000
60000
103573 DFT records, March 2014
records
A deeper look at the Hamiltonian
PN
i vext(ri) =Rdrvext(r)
PN
i ±(r¡ ri)
H = T + Vee +
NX
i
vext(ri)
PN
i vext(ri)Only depends on atoms and molecules
Rdrf(r)±(r¡ r0) = f(r0)
21
Introducing the electron density
hªjNX
i
vext(ri) jªi =
ZdxN jª(x1;x2;:::xN )j2
NX
i
vext(ri)
=
ZdxN jª(x1;x2;:::xN )j2
Zdrvext(r)
NX
i
±(r¡ ri)
=
Zdrvext(r)
ZdxN jª(x1;x2;:::xN )j2
NX
i
±(r¡ ri)
=
Zdrvext(r)½(r)
22
½(r) =
ZdxN jª(x1;x2;:::xN )j2
NX
i
±(r¡ ri)
= hªjNX
i
±(r¡ ri) jªi
= N hªj ±(r¡ r1) jªi
= N
ZdxN jª(x1;x2;:::xN )j2 ±(r¡ r1)
= N
Zds1dx2:::dxN jª(rs1;x2;:::xN )j2
Introducing the electron density
23
½(r)the electron density
½(r)
1. The electron density at r in physical space
2. The probability of finding an electron at r
3. A function of three variables
½(r) ¸ 0
Zdr½(r) = N
Zdr jr½(r)j2 <1
24
For a determinant wave function
j©0i = jÂ1Â2:::ÂiÂj:::ÂNi½(r) = h©0j
NX
i
±(r¡ ri) j©0i
=
NX
i
h©0j ±(r¡ ri) j©0i
=
NX
i
Zdxi jÂi(xi)j2 ±(r¡ ri)
=
NX
i
Zdxi jÂi(xi)j2 ±(r¡ ri)
=
NX
i
ZdriX
si
jÁi(ri)¾(si)j2 ±(r¡ ri)
=
NX
i
jÁi(r)j2 25
The variational principle for ground states
Most QM methods use the wavefunction as the
computational variable and work on its optimization.
26
An alternative route
Goal: Looking for the max height, and the tallest kid in a school.
1) We can first look for the tallest kid in each class
2) then compare the result from each class
27
An alternative route
Goal: Looking for the max height, and the tallest kid in a school.
1) We can first look for the tallest kid in each class
2) then compare the result from each class
h(Ki) is the height of kid Ki
H(J) = maxKi2class(J) fh(Ki)g, the max height of class J28
An alternative route
is the original variable
H(J) = maxKi2class(J) fh(Ki)g is a constrained search
i
is the reduced variableJ
29
The constrained-search formulation of DFT
Levy, 197930
The constrained search
F [½(r)] = minª!½(r)
hªjT + Vee jªi1. A functional of density; given a density, it gives a
number.
2. A universal functional of density, independent of
atoms, or molecules.
B. The ground state energy is the minimum
E0 = min½(r)
Ev[½(r)]
= min½(r)
½F [½(r)] +
Zdrvext(r)½(r)
¾
A. We have a new
C. is the new reduced variable ½(r)31
The condition on the density
For what density the energy functional is defined?
-- It has to come from some wavefunciton, N-representable
-- the N-representability is insured if
F [½(r)] = minª!½(r)
hªjT + Vee jªi
½(r) ¸ 0
Zdr½(r) = N
Zdr jr½(r)j2 <1 32
The Thomas Fermi Approximation for T
1. Divide the space into many small cubes
2. Approximate the electrons in each cubes as independent
particles in a box (Fermions)
3. T= Sum of the contributions from all the cubes
² =h2
8ml2(n2x + n2y + n2z) =
h2
8ml2R2²
Eigenstate energies for particle in a cubic box
With quantum numbers nx; ny; nz = 1;2;3; ::::
Each state can have two electrons (spin up and spin down).
33
The number of states (with energy < ²) ©(²)
©(²) =1
8
4¼R3²
3=
¼
6(8ml2
h2²)
32
Density of states
g(²) =d©(²)
d²=
¼
4(8ml2
h2)32 ²
Two electrons occupy each state. The kinetic energy is
t = 2
Z ²F
0
²g(²)d² =8¼
5(2m
h2)32 l3²
52
F
N = 2
Z ²F
0
g(²)d² =8¼
3(2m
h2)32 l3²
32
F
The Thomas Fermi Approximation for T
34
The Thomas Fermi Approximation for T
To express t as a function of density
t =3
10(3¼2)
23 l3(
N
l3)53
N
l3! ½
l3 ! dr
TTF [½] =X
t = CF
Zdr½(r)
53
CF =3
10(3¼2)
23
---Local density approximation for T 35
The Thomas Fermi energy functional
J[½] = 12
Rdrdr0 ½(r)½(r
0)jr¡r0j
For the Vee functional, use the classical electron-electron interaction
energy
ETF [½] = CF
Zdr½(r)
53 +
1
2
Zdrdr0
½(r)½(r0)
jr¡ r0j +
Zdrvext(r)½(r)
The Thomas-Fermi total energy functional is
The minimization of ETF [½] leads to the Euler-Lagrange equation, with ¹
for the constraint ofRdr½(r) = N .
¹ =±Ev[½]
±½(r)=
±TTF [½]
±½(r)+ vJ(r) + vext(r)
or
5
3CF½(r)
23 +
Zdr0
½(r0)
jr¡ r0j + vext(r) = ¹
36
The Thomas Fermi Dirac energy functional
Dirac exchange energy functional
ELDAx [½] = ¡CD
Z½(r)
43 dr
Slater, Gaspar, the X® approximation
EX®x [½] = ¡3
2®CD
Z½(r)
43 dr
CD = 34( 3¼)13
Adding the local approximation to the exchange energy
37
1 Question to the students
Question: What is the Thomas –Fermi kinetic energy functional in
2 dimensional space?
In 3dTTF [½] = CF
Zdr½(r)
53
CF =3
10(3¼2)
23
38
Comments on TF theory
•Beautiful and simple theory, with density as
the basic variable
BUT,
•the energy is NOT a bound to the exact
energy
•no shell structure for atoms, no quantum
oscillation
•no bounding for molecules
•Main problem is in the local
approximation for T
39
Functions and Functionals
I Ã f(x) I =R badxf(x)
E Ã ª(x) E[ª] = hªjH jªiE Ã ½(r) E[½] =?
• A function of one variable is a mapping from a number to a number
• A function of multi-variables is a mapping from muny numbers to a
number
• A functional is a mapping from a function to a number.
Functional is the continuous version of function of
infinitively many variables.
40
Functional derivatives
Functional Derivatives: Continuous extension of multivariable derivatives
41
Functional derivatives
dy(t) = lim²!0
[y(t+ ²dt)¡ y(t)]
²
= y0(t)dt
= lim²!0
dy(t+ ²t)
d²
±F [f(x)] = lim²!0
[F [f(x) + ²±f(x)]]¡ F [f(x)]
²
=
Zdx
±F
±f [x]±f [x]
= lim²!0
dF [f(x) + ²±f(x)]
d²
dy(t1; t2; :::) =X
i
@y(t1; t2:::)
@tidti
42
Functional derivatives
±F [f(x)] =
Zdx
±F
±f(x)±f(x)
The meaning of functional derivative: let ±f(x) = ±(x¡ x0), then
±F [f(x)] =
Zdx
±F
±f(x)±(x¡ x0) =
±F
±f(x0)
43
How to take functional derivatives
±F [f(x)] =
Zdx
±F
±f(x)±f(x)
F [f(x)] =
Zdxg(f(x))
±F [f(x)] = lim²!0
d
d²
Zdxg(f(x) + ²±f(x))
= lim²!0
d
d²
Zdx
½g(f(x)) +
dg(f(x))
df²±f(x)
¾
=
Zdx
½dg(f(x))
df±f(x)
¾
±F
±f(x)=
dg(f(x))
df 44
Functional derivatives
F [f(x)] =
Zdxg(f(x))
±F
±f(x)=dg(f(x))
df
F [f(x)] =
Zdxf(x) ±F
±f(x)= 1
TTF [½] = CF
Zdr½(r)
53
±TTF [½]
±½(r)=
5
3CF½(r)
23
45
Functional derivatives
ETF [½] = CF
Zdr½(r)
53 +
1
2
Zdrdr0
½(r)½(r0)
jr¡ r0j +
Zdrvext(r)½(r)
use ¹ for the constraint ofRdr½(r) =N
±
±½
½ETF [½]¡ ¹(
Zdr½(r)¡N)
¾= 0
The condition for the minimization of the energy functional is
5
3CF½(r)
23 +
Zdr0
½(r0)
jr¡ r0j + vext(r) = ¹
46
How to take functional derivatives
±F [f(x)] =
Zdx
±F
±f(x)±f(x)F [f(x)] =
Zdxg(f(x); f 0(x))
±F [f(x)] = lim²!0
d
d²
Zdxg(f(x) + ²±f(x); f 0(x) + ²±f 0(x))
= lim²!0
d
d²
Zdx
½g(f(x)) +
@g
@f²±f(x) +
@g
@f 0²±f 0(x)
¾
=
Zdx
½@g
@f±f(x)
¾+
Zdx
@g
@f 0±f 0(x)
=
Zdx
½@g
@f±f(x)
¾+
Z@g
@f 0d(±f(x))
=
Zdx
½@g
@f±f(x)
¾¡Z
dx
µd
dx
@g
@f 0
¶±f(x) +
@g
@f 0±f(x) jboundary
±F
±f(x)=dg(f(x))
df¡ d
dx
@g
@f 047
Kohn-Sham theory, going beyond the Thomas-Fermi
Approximation
Use a non-interacting electron system, —Kohn-Sham reference system,
to calculate the electron density and kinetic energy
j©si = jÂ1Â2:::ÂiÂj:::ÂNi
½(r) =
NX
i
jÁi(r)j2
Ts[½] = h©sjNX
i
¡1
2r2i j©si
=
NX
i
ZdriÁ
¤i (ri)
µ¡1
2r2i
¶Ái(ri)
=
NX
i
hÁij ¡1
2r2 jÁii
49
Kohn-Sham theory
½(r) =
NX
i
jÁi(r)j2 Ts[½] =
NX
i
hÁij ¡1
2r2 jÁii
² ½(r) =P
i jÁi(r)j2 is the true electron den-
sity, but Ts[½] is not T [½]: However, Ts[½] is
a very good approximation to Ts[½]
² The essence of KS theory is to solve Ts[½]
exactly in terms of orbitals
50
Kohn-Sham theory
½(r) =
NX
i
jÁi(r)j2 Ts[½] =
NX
i
hÁij ¡1
2r2 jÁii
Introduce the exchange correlation energy functional
F [½] = T [½] + Vee[½]
= Ts[½] + J [½] +Exc[½]
Exc[½] = T [½]¡ Ts[½] + Vee[½]¡ J [½]
J[½] = 12
Rdrdr0 ½(r)½(r
0)jr¡r0j
51
Kohn-Sham Equations
½(r) =
NX
i
jÁi(r)j2 Ts[½] =
NX
i
hÁij ¡1
2r2 jÁii
In terms of the orbitals, the KS total energy functional is now
Ev[½] =
NX
i
hÁij ¡1
2r2 jÁii+ J[½] +Exc[½] +
Zdrvext(r)½(r)
The g.s. energy is the minimum of the functional w.r.t. all
possible densities. The minimum can be attained by
searching all possible orbitals
W [Ái] = Ev[Ái]¡NX
i
"i f< ÁijÁi > ¡1g
±W [Ái]
±Ái(r)= 0 52
W [Ái] = Ev[Ái]¡NX
i
"i f< ÁijÁi > ¡1g ±W [Ái]
±Ái(r)= 0
The orbitals fjÁiig are the eigenstates of an
one-electron local potential vs(r) if we have
explicit density functional for Exc[½] (KS equa-
tions)
µ¡12r2+ vs(r)
¶jÁii = "i jÁii ;
or a nonlocal potential vNLs (r,r0), if we have
orbital functionals for Exc[Ái] (generalized KS
equations)
µ¡12r2+ vNLs (r,r0)
¶jÁii = "GKSi jÁii : 53
vs(r) =±Exc[½]
±½(r)+ vJ(r) + vext(r)
vNLs (r,r0) =±Exc[±½s(r
0; r)]
±½s(r0; r)+ [vJ(r) + vext(r)] ±(r
0 ¡ r)
µ¡12r2+ vs(r)
¶jÁii= "i jÁii ;
µ¡12r2+ vNLs (r,r0)
¶jÁii= "GKSi jÁii :
Kohn-Sham (KS)
Generalized Kohn-Sham (GKS)
One electron Equations
54
Feastures of Kohn-Sham theory
1. With N orbitals, the kinetic energy is treated rigorously. In
comparison with the Thomas-Fermi theory in terms of
density, it is a trade of computational difficulty for
accuracy.
2. The KS or GKS equations are in the similar form as the
Hartree-Fock equations and can be solved with similar
efforts.
3. KS or GKS changes a N interacting electron problem into
an N non-interacting electrons in an effective potential.
4. The E_xc is not known, explicitly. It is about 10% of the
energy for atoms. 55
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