introduction to density functional theory · density functional theory - an advanced course...
TRANSCRIPT
the team
Introduction to DensityFunctional Theory
# electronsCO
ST Exponential
wall
the team
Goal: determine material properties directly from fundamental equations
Challenge: develop efficient andaccurate methods to achieve that goal
# electronsCO
ST Exponential
wall
any observable:
From
degrees of freedom: Note: only electronic degrees of freedom; Born-Oppeheimer approximation
Challenge of electronic structure problem
the team
Use simpler quantities than wavefunction
reduce # degree of freedom by averaging out information I do not need
e.g. density matrices
: all info's about the system: do I really need that?!?
The simplest of them all: density
electron density: just 3 degrees of freedom!
?Which information is contained in the density
?Can we use the density to calculate materials properties
the team
the team
Which information does the density contain?
Formal proof: H-K theorem (reductio ad absurdum)
one-to-one correspondence:
ground state unique, universal functional of the density:
any ground state observable is a density functional:
Hohenberg-Kohn (1964)
the team
Can we use the density for calculations?
one-to-one correspondence + Ritz variational principle:
Ground state energy is a density functional:
With this minimum principle we can develop a computationalmethod to calculate GS properties of a system
the team
What is density-functional theory?
based on the Hohenberg-Kohn theorem
Electronic structure approach whose key quantity is the density
to calculate GS properties of a system
(ensures that many-particle system in its GS is fully characterized by its GS density)
minimizes
the team
Note: HK theorem has clear limits
It considers: * nondegenerate* nonmagnetic (only scalar potential!), spinless (only spin unpolarized!)* ground-states
z
y
x
S
L
SL = +
SN
Extensions: spin-density, current-density, B-density, density-polarization von Barth-Hedin 1972, Vignale-Rasolt 1988, Grayce-Harris 1994, Gonze-Ghosez-Godby 1995....list is not complete
spin-orbit coupling external magnetic field
++ + + + +
systems with no ground-state(anti)ferromagneticity
Not described*:
* for B=0, m = m[n] but unknown!
the team
Generalization to spin-polarized systems
2x2 electron density:
spin-dependent 1-e potential: 2x2 Hermitian matrices
e.g.: scalar potential + magnetic field/magnetic moment coupling (zero if no B ext)scalar potential + spin-orbit coupling (relativistic effects)
the team
Can HK theorem be generalized to 2x2 case?
Similar proof to H-K theorem (reductio ad absurdum)
NO one-to-one correspondence:
ground state unique, universal functional of the 2x2 density:
any ground state observable is a spin-density functional:
von Barth-Hedin (1972)
X
+variational principle
the team
How do we put the HK theorem in practice?
Challenge:
Fit many-particle intricacies in such simple object as the density
We need:
We have:
the team
Get large part of T via non-interacting system
Kohn-Sham (1965)
Physical system (N-body problem)
Kohn-Sham system (N X 1-body problems)
the team
Kohn-Sham equations
Kohn-Sham (1965)
Defining the exchange-correlation energy functional
+ applying Hohenberg-Kohn II ( minimize E) for both systems:
{
Exchange-correlation energy functional defined in the same way
but dealing with a 2x2 external potential, thus xc potential
{the team
von Barth-Hedin (1972)
Spin-polarized Kohn-Sham equations
the team
How do we approximate the xc functional
Introduction to DFT - Myrta Grüning
How to solve the KS equations in practice
{Nonlinear, integro-differential equations
1. solution through self-consistency2. basis set expansion to get an algebraic problem
{ }
the team
Solution through self-consistency
GUESSDENSITY
CALCULATEKS POTENTIAL
SOLVE KSEQUATIONS
CALCULATEENERGY/NEW DENSITY
CHECKCRITERIA
the team
Basis set expansion to get algebraic problem
Expansion in a convenient basis set
Hamiltonian (overlap) matrix elements
Solve (generalized) eigenproblem
Possible choices:Localized basis sets
e.g.: Gaussians, Slater
Delocalized basis sets Plane-waves
the team
Periodic crystals are described in terms of:
Crystal
Unit cell
Primitive Lattice vectors
Basis
(for simplicity 2D example, trivially extended to 3D)
the team
Electrons in a crystal potential
Free-electron periodic potential
a
Ene
rgy
k
Ene
rgy
k
1/k
Plane waves
Bloch-states
the team
Periodic crystals are described in terms of:
Reciprocal lattice vectors:
1st Brillouin zone:
Primitive reciprocalLattice vectors
translation respresented by with
with
Wigner-Seitz primitive cell in k-space
Direct, real spaceFourier trasform
Reciprocal,momentum, k-space
the team
Planewave basis set
pseudopotential:
Expand:
Diagonalize:
Matrix elements:
the team
You need to "converge" wrt parameters
a. energy cutoff used to define the size of the planewave basis set
I need to evaluate integrals of the type(e.g. for the charge density)
numerically on a discrete (uniform) grid as:
b. number & density of k points used to sample k-space
with
the team
Choice of the Hamiltonian
Schrödinger Hamiltonian is not 'exact': relativistic effects (QED/Dirac)
Visible relativistic effects (splitting of order of tenths of eV) from third-row semiconductors: spin-orbit coupling
Example optical absorption in GaSb
the team
?
Hamiltonian with spin-orbit coupling
What changes to include spin-orbit coupling:
where Pauli matrices:
2x2 external potential : 4-density (noncollinear spin formalism) we considered earlier
Main differences with spinless approach:
(spinor)
spinless: non-collinear spin:
full-relativistic pseudopotentials
4-vxc (or 2x2 ) xc approximations e.g. spin-LDA
the team
DFT with PWs in practice:
System:
Hamiltonian: (physical approx.)
Numerical approx:
Physical quantities
1-particle quantities
GUESSDENSITY
CALCULATEKS POTENTIAL
SOLVE KSEQUATIONS
CALCULATEENERGY/NEW DENSITY
CHECKCRITERIA
Solve KS equations
IN: OUT:
RUN:
xc-approximationso-interaction(relativistic effects)
unit celllattice vectorsbasis
energy cut-offk-points gridpseudopotentialsSCF procedure/threshold
density and related quantities
total energy and components
any GS observable (in principle)
Kohn-Sham 1-p wavefunctions
Kohn-Sham 1-p energies
Energ
y (
eV
)
L LZ A D A
the team
Connection KS and physical quantities?
Can be related to the fundamental band gap?
Ek
e-
Ei
Ef
hv
e-Ei
Evac
Efhv
Ek
EF
e-
e-
with (charged excitation)
Connection KS and physical quantities?
Can be related to neutral excitations of the systems?
hv
Ei
Ef
EF
hv
the team
In conclusions:
the team
1-p model:independent
particles Hartree
independentfermions in
effective potential
independentfermions
Hartree-Fock
partially interactingparticles in
effective potential
generalizedKohn-Sham
cost:approach:
N3
N3
N4
N4
Kohn-Sham
The Kohn-Sham energies/wavefunctions do not have a precise physical meaning
However KS is a privileged systems of independent particles:
* Reproduce (in principle) GS density * with relatively low computational effort
That is why it is often used as starting point for many-body
perturbation theory calculations
the team
References&material&further reading
A Chemist’s Guide to Density Functional Theory. Second EditionWolfram Koch, Max C. Holthausen (2001) Wiley-VCH Verlag GmbH
A Primer in Density Functional Theory
Eds. C. Fiolhais F. Nogueira M. Marques (2003) Springer-Verlag Berlin Heidelberg Density Functional Theory - An Advanced Course
Eberhard Engel · Reiner M. Dreizler (2001) Springer-Verlag Berlin Heidelberg
Kohn-Sham potentials in density functional theoryPh.D thesis - Robert van Leeuwen
Electronic Structure of Matter – Wave Functions and Density FunctionalsW. Kohn - Nobel Lecture, January 28, 1999
the team
1. Many-body perturbation theory calculations using the yambo code Journal of Physics: Condensed Matter 31, 325902 (2019)2. Yambo: an ab initio tool for excited state calculations Comp. Phys. Comm. 144, 180 (2009)