Functionals of exchange and correlation (Ch 5,8)
Pseudopotentials (Ch 10,11)
Choosing basis (Ch 12-15)
KKR approach (Ch 16.3)
no generic exact solution
Minimizing the constrained energy functional:
Exchange-correlation potential:
HK theorems tell us that we can progress based just on the density \(n(\mathbf{r})\) !
use an exact numerical method:
QMC
Ceperley and Alder Phys. Rev. Lett. 45, 566 (1980)
Perdew and Zunger; Phys. Rev. B 23, 5048 (1981)
Vosko, Wilkes, and Nussair; Can. J. Phys. 58, 1200 (1980)
No fitting to experiments, based only on general rules of quantum mechanics!
A mixture of the exactly calculable HF exchange and other \(E_{xc}\) parametrizations
A. D. Becke, J. Chem. Phys. 98, 5648 (1993)
Addresses some of the spectral problems of LDA and GGA.
The prefered choice among chemists.
Fitted to experimental data!!
Rappoport et al. Encyclopedia of Inorganic Chemistry (2009)
Motivation:
Approximations:
Potential is not local, each angular momentum channel feels different potential:
\( |p_{\mathbf {R} ,i}\rangle \) are duals of \(|\tilde{\phi }_{\mathbf{R},j}\rangle \) inside the cutoff
$$ {\displaystyle \langle p_{\mathbf {R} ,i}|{\tilde {\phi }}_{\mathbf {R} ,j}\rangle _{r<r_{c}}=\delta _{i,j}}$$
Vanderbilt, Phys. Rev. B 41, 7892 (1990).
$$ {\displaystyle q_{\mathbf {R} ,ij}=\langle \phi _{\mathbf {R} ,i}|\phi _{\mathbf {R} ,j}\rangle -\langle {\tilde {\phi }}_{\mathbf {R} ,i}|{\tilde {\phi }}_{\mathbf {R} ,j}\rangle }, $$
$${\displaystyle {\hat {H}}|\Psi _{i}\rangle =\epsilon _{i}{\hat {S}}|\Psi _{i}\rangle }$$
$$ {\displaystyle {\hat {S}}=1+\sum _{\mathbf {R} ,i,j}|p_{\mathbf {R} ,i}\rangle q_{\mathbf {R} ,ij}\langle p_{\mathbf {R} ,j}|}, $$
\(r_c\)=?
Softness:
Transferability:
Large \(r_c\) makes more soft potentials, but throwing out a large chunk of the core we loose physics
Comp. Mat. Sci. 98, 372 (2015)
PRO | CON | |
---|---|---|
Plane waves |
-simple to implement -one convergence parameter |
- resource hungry - big basis set needed |
Atomic like orbitals | - small basis - sparse matrix methods - scales to BIG systems |
-harder to implement -many things influence convergence |
KKR method baseless method.. relies on scattering approach and Green's functons |
-relativistic effects are easy -standard for magnetic systems -most accurate for some problems |
-harder to implement -smaller community -some features not yet implemented.. |
Single convergence parameter:
size of basis, #\(p\), usually BIG
P. E. Blöchl Phys. Rev. B 50, 17953 (1994)
energy shift
single-\(\zeta\)
Energy shift enforces a node in the wavefunction! It is cut after the node!
double-\(\zeta\):
more radial freedom
split orbital at radius \(r^s_l\)
polarization:
more angular freedom
J. Phys.: Condens. Matt. 14, 2745 (2002).
Instead of Schrödinger, we have Green!
$$\displaystyle \left[{-{\frac {\hbar ^{2}}{2m}}{\nabla ^{2}}+V({\bf {r}})-E}\right]G(E,{\bf {r}},{\bf {r'}})=-\delta ({\bf {r}}-{\bf {r'}})$$
$$({G}^{-1}_0-{V}){G}={I}$$
We know \({G}_0\) analytically even in the case of fully relativistic (Dirac) physics!
$$\displaystyle \left[{-{\frac {\hbar ^{2}}{2m}}{\nabla ^{2}}+V({\bf {r}})}\right]\psi({\bf {r}})=E\psi({\bf {r}})$$
\(G=G_0+G_0VG_0+\dots\)
\(G=G_0+G_0TG\)
\(T=V+VG_0V\dots\)
\(T=V+VG_0T\)
\(T=(V^{-1}-G_0)^{-1} \)
LDA+U:
J. Phys.: Cond. Matt. 9,767 (1997)
DMFT:
Phys. Rev. Lett. 62, 324 (1989)
QMC:
Rev. Mod. Phys. 73, 33 (2001)
GW:
Phys. Rev. 139, A796 (1965)
BSE:
Phys. Rev. Lett. 75, 818 (1995)
6000 atom
3000 proc
120s/scf
Massively
parallel
Phonon spectrum
J. Chem. Phys. 143, 064710 (2015)
Excitations with
GW method
Phys. Rev. B 75, 235102 (2007)
J. Phys.: Cond. Matt. 27, 054004 (2015)
J. Phys.: Cond. Matt. 26, 305503 (2014)
6000 atom
2000 proc
10 sec/scf
SIESTA-PEXSI parallelization
New J. Phys., 16, 093029 (2014)
Phys. Rev. B 65, 165401 (2002)
Local quantities (e.g. STM)
Ideal as an input for transport calculations
https://launchpad.net/siesta
KKRNano @ http://www.judft.de/ massive parallelization
PRB 94, 104511 (2016)
abinitio superconductivity
PRB 89, 224401 (2014)
finite temperature magnetism
PRB 82, 024411 (2010)
ARPES+DMFT
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