how do you build a good hamiltonian for ceid? andrew horsfield, lorenzo stella, andrew fisher

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 non-

adiabatic

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)

EhrenfestDynamics

Ehrenfest dynamics: the idea

• Simplest non-adiabatic method

• Relatively easy to implement efficiently• Fast ions can excite electrons• But hot electrons cannot heat ions

drrt

RRrHRtF

dtRdM 2

2

2

)()()(

trtirtRrH

)()()(

Newton

Schrödinger

Ehrenfest dynamics: the problem

•Ions see electrons as cold fluid (no fluctuations)

•Electrons see ionic fluctuations (could be hot)

drrt

RRrHRtF 2)()()(

Number of electronsForce per electron

Ehrenfest dynamics: the problem

Correlated Electron-Ion Dynamics

Correlated Electron-Ion Dynamics (CEID)

• Current flows past one dynamic atom

• Classical kinetic energy

• Quantum kinetic energy

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)

© Imperial College London

Correlated Electron-Ion Dynamics (CEID)

• Retain form of molecular dynamics

• Note: these are exact, but insoluble without approximation.

RHTrF

dtPdM

MP

dtRd

•To develop approximate scheme, use narrowness of nuclear wavefunction•Expand Hamiltonian in Taylor series about average position of nucleus

2

22 )()(

21)()()()(

dRRHdRR

dRRdHRRRHRH

Correlated Electron-Ion Dynamics (CEID)

Basis set formulation of CEID

• For the electrons we use the Ehrenfest states

• For the nuclei we use moving harmonic oscillator states

• Note: the formalism of Stella (2011) is more general than this.

t

iRH e

ˆ

)()exp( RRPinR n

Basis set formulation of CEID

• The density matrix then has the following equation of motion

• where the fluctuation Hamiltonian is given by

'', nn

'''''','''''''','','''''''',

'',

nnnnnnnnn

nn HHt

i

22

21

2RKRF

mPH

Makes nuclei quantumCouples electron and nuclear fluctuations

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

• Can select just few modes to undergo quantum fluctuations: great reduction in computational cost

aaMiP

aaM

R

2

2

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?

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