How do you build a good Hamiltonian for CEID?
Andrew Horsfield, Lorenzo Stella, Andrew Fisher
Simplification of Chinese Sounds
“A prominent and peculiar phenomenon in Chinese is that
during last 3,000 years or so, the language has experienced
repeated and massive losses of important phonological
distinctions, and become increasingly homophonous.
http://www.pinyinology.com/diaoHao2a/group/simple2m.html
Something similar is happening to Correlated Electron-Ion
Dynamics, except for the Hamiltonian.
Dynamics:
adiabatic and nonadiabatic
Adiabatic dynamics
• For molecular dynamics we assume atoms move
classically on a fixed energy surface.
• Electrons take a known configuration defined by
instantaneous nuclear coordinates
• Do not respond to the nuclear velocity
• Can be computed accurately with ground state DFT
• Job done.
Non-adiabatic dynamics
• Heat can flow between
electrons and nuclei
• From hot electrons to nuclei
• Heating of light bulbs
• Energy transfer in
photoexcited conjugated
polymers
• From hot nuclei to electrons
• Drag on atoms in radiation
damage cascades
Finnis et al, PRB 44, 567 (1991)
Ehrenfest
Dynamics
Ehrenfest dynamics: the idea
• Simplest non-adiabatic method
d 2R
H ( Rr )
2
M 2  F ( Rt )   
(rt ) dr Newton
dt
R
 (rt )
H ( Rr ) (rt )  i
t
Schrödinger
• Relatively easy to implement efficiently
• Fast ions can excite electrons
• But hot electrons cannot heat ions
Ehrenfest dynamics: the problem
H ( Rr )
2
F ( Rt )   
 (rt ) dr
R
Number of electrons
Force per electron
•Ions see electrons as cold fluid (no fluctuations)
•Electrons see ionic fluctuations (could be hot)
Ehrenfest dynamics: the problem
Correlated ElectronIon Dynamics
Correlated Electron-Ion Dynamics (CEID)
• Current flows past
one dynamic atom
• Classical kinetic
energy
2
Tclassical
P
=
2M
• Quantum kinetic
energy
DP
Tquantum =
2M
2
J. Phys.: Condens. Matter 16 (2004)
8251–8266
Basis set formulation of CEID
• Original moment formulation does not converge
systematically
• Now use harmonic oscillator basis for ions
• L. Stella, M. Meister, A. J. Fisher, and A. P. Horsfield, J.
Chem. Phys. 127, 214104 (2007).
• L. Stella. R. P. Miranda, A. P. Horsfield, and A. J.
Fisher, J. Chem. Phys. 134, 194105 (2011)
Correlated Electron-Ion Dynamics (CEID)
• Retain form of molecular dynamics
dR P

dt M
dP
 H 
M
 F  Tr 

dt
 R 
• Note: these are exact, but insoluble without
approximation.
© Imperial College
London
Correlated Electron-Ion Dynamics (CEID)
•To develop approximate
scheme, use narrowness of
nuclear wavefunction
•Expand Hamiltonian in
Taylor series about average
position of nucleus
2
dH ( R ) 1
d
H (R )
2
H ( R)  H ( R )  ( R  R )
 (R  R )

2
dR
2
dR
Basis set formulation of CEID
• For the electrons we use the Ehrenfest states

ˆ
H e R    i 
t
• For the nuclei we use moving harmonic oscillator
states
R n  exp( iP R)n (R)
• Note: the formalism of Stella (2011) is more general
than this.
Basis set formulation of CEID
• The density matrix n , 'n ' then has the following
equation of motion
i
n , 'n '
t
  H n , ''n ''  ''n '', 'n '  n , ''n '' H  ''n '', 'n ' 
 '' n ''
• where the fluctuation Hamiltonian is given by
Makes nuclei quantum
2 Couples electron and nuclear fluctuations
2

P 
H 
1
 FR  K R   
2m
2
Constrains spread of nuclear packets
Correlated electron ion dynamics (CEID)
Basis set formulation of CEID
• Matrix elements of the Hamiltonian become easy to
evaluate if use ladder operators


R 
a  a
2 M

M

P  i
aa
2


• Can select just few modes to undergo quantum
fluctuations: great reduction in computational cost
Basis set formulation of CEID
L. Stella, M. Meister, A. J. Fisher, and
A. P. Horsfield, J. Chem. Phys. 127,
214104 (2007).
Basis set formulation of CEID
L. Stella. R. P. Miranda, A. P.
Horsfield, and A. J. Fisher, J. Chem.
Phys. 134, 194105 (2011)
Basis set formulation of CEID
L. Stella. R. P. Miranda, A. P.
Horsfield, and A. J. Fisher, J. Chem.
Phys. 134, 194105 (2011)
The Hamiltonian
What is the problem with the Hamiltonian?
dH e (R) 1
2 d H e (R)
H e (R) = H e (R) + (R - R)
+ (R - R)
+…
2
dR
2
dR
2
How do we compute these terms?
What is the problem with the Hamiltonian?
What basis set do we use for the electrons?
• Planewaves
• Removes many problems associated with atom centered
orbitals
• Often very large number of functions
• Inefficient for molecules
• Atom centered (e.g. gaussians)
• Efficient for many static problems
• But being atom centered creates problems in dynamic
simulations
• Even for Ehrenfest, overlap matrix is problematic
• For CEID also need to decide where to put the orbitals
What is the problem with the Hamiltonian?
Need to take matrix elements with basis before differentiating
What is the problem with the Hamiltonian?
Interacting electrons
• In past tried to reduce interacting electron problem to
effective independent electron problem
• Based on Hartree-Fock
• Coupling between electrons and nuclear fluctuations
results in proliferation of matrices
• Not well controlled
What is the problem with the Hamiltonian?
Solution?
• Could construct energy surfaces
• But opposes philosophy of CEID
• How do we extend to metals?
• Very labour intensive