1. Abinit : Excited states within TDLDA & Langevin molecular dynamics
Jean-Yves Raty and Xavier Gonze
Universite de Liege Universite Catholique de Louvain

2. Outlook Langevin dynamics Time dependent LDA Background Implementation
Test case
Examples
a. Structural optimization via simulated annealing
b. Structure and dynamics of liquids
Time dependent LDA
Example
a. TDLDA spectra of carbon and silicon nanoclusters
b. Discussion TDLDA vs other methods

3. Finite T Molecular dynamics
2 thermostats included
Ionmov 8 Nose-Hoover thermostat
Variables : dtion, mditemp, mdftemp, noseinert
Ionmov 9 Langevin thermostat
Variables : dtion, mditemp, mdftemp, friction
mdwall, sigperm,delayperm
8=> Molecular dynamics with Nose-Hoover thermostat, using the Verlet algorithm. Although partly coded, optcell/=0 is not available. Related parameters : the time step (dtion), the initial temperature (mditemp), the final temperature (mdftemp), and the thermostat mass (noseinert).
noseinert Mnemonics: NOSE INERTia factor Characteristic: Variable type: real noseinert Default is 1.0d5 Give the inertia factor WT of the Nose-Hoover thermostat (when ionmov=8), in atomic units of weight*length2, that is (electron mass)*(bohr)2. The equations of motion are : MI d2RI/dt2= FI - dX/dt MI dRI/dt and WT d2X/dt2= Sum(I) MI (dRI/dt)2 - 3NkBT where I represent each nucleus, MI is the mass of each nucleus (see amu), RI is the coordinate of each nucleus (see xcart), dX/dt is a dynamical friction coefficient, and T is the temperature of the thermostat (see mditemp and mdftemp). mditemp Mnemonics: Molecular Dynamics Initial Temperature Characteristic: Variable type: real mditemp Default is Give the initial temperature (for itime=1) of the Nose-Hoover thermostat (ionmov=8) and Langevin dynamics (ionmov=9), in Kelvin. This temperature will change linearly to reach the temperature mdftemp at the end of the ntime timesteps. ######################################################################
9=> Langevin molecular dynamics. Although partly coded, optcell/=0 is not available. Related parameters : the time step (dtion), the initial temperature (mditemp), the final temperature (mdftemp), and the friction coefficient (friction).
mdwall Mnemonics: Molecular Dynamics WALL location Characteristic: Variable type: real parameter Default is Bohr (the walls are extremely far away). Gives the location (atomic units) of walls on which the atoms will bounce back. when ionmov=6, 7, 8 or 9. For each cartesian direction idir=1, 2 or 3, there is a pair of walls with coordinates xcart(idir)=-wall and xcart(idir)=rprimd(idir,idir)+wall . Supposing the particle will cross the wall, its velocity normal to the wall is reversed, so that it bounces back. By default, given in bohr atomic units (1 bohr= Angstroms), although Angstrom can be specified, if preferred, since mdwall has the 'LENGTH' characteristics. friction Mnemonics: internal FRICTION coefficient Characteristic: Variable type: real parameter Default is Gives the internal friction coefficient (atomic units) for Langevin dynamics (when ionmov=9): fixed temperature simulations with random forces. The equation of motion is : MI d2RI/dt2= FI - friction MI dRI/dt - F_randomI where F_randomI is a Gaussian random force with average zero, and variance 2 friction MI kT. The atomic unit of friction is hartrees*electronic mass*(atomic time units)/bohr2. See J. Chelikowsky, J. Phys. D : Appl Phys. 33(2000)R33.

4. Langevin dynamics Movement equations :
NOTE SI PAS G et gamma bien choisi : th de type Nose
friction Mnemonics: internal FRICTION coefficient Characteristic: Variable type: real parameter Default is Gives the internal friction coefficient (atomic units) for Langevin dynamics (when ionmov=9): fixed temperature simulations with random forces. The equation of motion is : MI d2RI/dt2= FI - friction MI dRI/dt - F_randomI where F_randomI is a Gaussian random force with average zero, and variance 2 friction MI kT. The atomic unit of friction is hartrees*electronic mass*(atomic time units)/bohr2. See J. Chelikowsky, J. Phys. D : Appl Phys. 33(2000)R33.
See J. Chelikowsky, J. Phys. D : Appl Phys. 33(2000)R33.

5. Implementation Tests Src9drive/moldyn.f Test_v2/
87. Ge liquid. Test of Nose dynamics. 2 atoms in a cell.
Allows 4 time steps. .
88. Ge liquid. Test of Langevin dynamics. 2 atoms in a cell.
Tests

6. Simulated annealing : Clusters optimization
Variables :
Ionmov 9
dtion ~100
Mditemp
Mdftemp 300
Ntime
Optional variables :
mdwall
getvel
signperm
delayperm
signperm Mnemonics: SIGN of PERMutation potential Characteristic: Variable type: integer Default is 1. In development. See the routine moldyn.f.
See also delayperm.
+1 favors alternation of species
-1 favors segregation
delayperm Mnemonics: DELAY between trials to PERMUTE atoms Characteristic: Variable type: integer Default is 0. Delay (number of time steps) between trials to permute two atoms, in view of accelerated search of minima. Still in development. See the routine moldyn.f. See also signperm. When delayperm is zero, there is not permutation trials.

7. Si_n, Ge_n n<11

8. Sample Input Adapt tsmear to T_ion Getvel -1 ! At startup :
#Cluster optimization via simulated annealing
#*****************************************************
acell 3*25
xcart ………
occopt 4
ionmov #Langevin dynamics
ntime 100
dtion 100
friction 0.001
mdwall # optional
mditemp1 3000
mdftemp1 2500
tsmear1 0.05
tolvrs1 1.0d-2
mditemp
mdftemp
tsmear
getxcart
getvel
tolvrs d-2
………..
# Then end with a Broyden minimization
Adapt tsmear to T_ion
Getvel -1 !
At startup :
If E_kin = 0, atoms are given random velocities (mditemp)
Otherwise, velocities are rescaled to the requested mditemp.
Mnemonics: internal FRICTION coefficient Characteristic: Variable type: real parameter Default is Gives the internal friction coefficient (atomic units) for Langevin dynamics (when ionmov=9): fixed temperature simulations with random forces. The equation of motion is : MI d2RI/dt2= FI - friction MI dRI/dt - F_randomI where F_randomI is a Gaussian random force with average zero, and variance 2 friction MI kT. The atomic unit of friction is hartrees*electronic mass*(atomic time units)/bohr2. See J. Chelikowsky, J. Phys. D : Appl Phys. 33(2000)R33.
mdwall (a.u.)

9. ?
Ge10
1. Langevin
2.Broyden

10. Optional variables For binary systems :
delayperm : nr of time steps between attemps for exchanging 2 atoms from different type
signperm favors alternation
segregation
Under development, not optimized

11. Liquids simulation System 56 atoms GS stuff LDA & GGA
Molecular dynamics strategy
1) Random cfg, 5ps at 3000K
2) Temperature ramp mditemp 3000, mdftemp 700, ntime 1000, dtion 200 (5ps)
3) Thermalization at mdftemp (typical 5ps)
Check : diffusivity + potential energy
4) Data acquisition
Keep friction as low as possible

12. Eutectic Ge-Te alloy Why ? Glass forming,
strong anomalous behavior pseudo Liquid-Liquid transition
LDA vs GGA
LDA ok GeTe, not that ok for Ge1Te6…
Limitation : fixed volume
Importance of the density

13. GeTe6 : comparison with experiments
Agreement could be better but general trends OK
Problems : 85 % Te ! Large temperature fluctuations

14. GeTe6 : partial structure factors
SGeTe(q)
STeTe(q)
More changes on SGeTe(q) than on STeTe(q)

15. GeTe6 : partial pair correlation functions
gGeTe(r)
gTeTe(r)
first peak of gGeTe(r) becomes broader changes localized around Germanium atoms

16. What kind atomic arrangements ?
3+1+2
Significant changes around Ge 6 Ge Ge
volume
4+2 Te Ge Te
Few changes around Te Te
temperature

17. What kind atomic arrangements ?
Average distances of nearest neighbors around
Ge atoms
Te atoms
Temperature
Run E
Run D
Run C
Run B
Run A

18. A simple picture …. Ge Te Low temperature Broken symmetry
Energetically favored (Peierls like mechanism)
High temperature
More symmetry
Entropically favored vibrational entropy

19. Future developments ?
Constant pressure dynamics !

20. Excited States : Time dependent LDA
Need for improved description of the excited states
Methods available : computationnally demanding
GW + BSE, DFT + QMC
TDLDA : ‘cheap’ calculation on top of standart LDA calculation

21. Time-Dependent DFT 2 ways
Linear response method (TD-DFRT)
“Time-Dependent Density Functional Response Theory of Molecular systems: Theory, Computational Methods, and Functionals", by M.E. Casida, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier, Amsterdam, 1996).)
Real time method

22. TDLDA calculation Simple eigenvalue problem on top of LDA : Where
are the Kohn-Sham transition energies
are the diff. in occupation numbers
are the TDDFT excitations energies
are related to the oscillator strengths

23. The coupling matrix K Adiabatic approximation : Linear response :
xc term
What is treated is w_eff = w(t) + dv_SCF(r,t)
If LDA : 1 center integral that’s what is in abinit
Coulomb term
Oscillator strength :

24. Src_5common/tddft.f Double loop + Mpi coming (-> src_9seqpar)
NxN Matrix to diagonalize with
coupling the excitation i-j with the excitation l-k
The Matrix to be diagonalized in the Casida framework (see "Time-Dependent Density Functional Response Theory of Molecular systems: Theory, Computational Methods, and Functionals", by M.E. Casida, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier, Amsterdam, 1996).) is a NxN matrix, where, by default, N is the product of the number of occupied states by the number of unoccupied states. The input variable td_maxene allows to diminish N : it selects only the pairs of occupied and unoccupied states for which the Kohn-Sham energy difference is less than td_maxene. See td_mexcit for an alternative way to decrease N.
Default is 0. The Matrix to be diagonalized in the Casida framework (see "Time-Dependent Density Functional Response Theory of Molecular systems: Theory, Computational Methods, and Functionals", by M.E. Casida, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier, Amsterdam, 1996).) is a NxN matrix, where, by default, N is the product of the number of occupied states by the number of unoccupied states. The input variable td_mexcit allows to diminish N : it selects the first td_mexcit pairs of occupied and unoccupied states, ordered with respect to increasing Kohn-Sham energy difference. However, when td_mexcit is zero, all pairs are allowed. See td_maxene for an alternative way to decrease N.
Double loop + Mpi coming (-> src_9seqpar)
Restart – disabled
N Excitations window (by default, N= nval x ncond)
Optional variables : td_maxene, td_mexcit

25. Test_v1/t69 Iscf -1 Mkmem 0 (opt) : td_matrix is stored on disk
# Magnesium atom. acell much too small.
# Excited states computation
ndtset 3
iscf1 5
nband1 1
prtden1 1
iscf #non-scf followed by tdlda
nband2 10
getden2 1
getwfk2 1
iscf #same, but with mkmem 0
nband3 10
mkmem3 0
getden3 1
getwfk3 1
#Common
acell
boxcenter 3*0
ecut 3.5
natom 1
xangst 0 0 0
Test_v1/t69
Iscf -1
Mkmem 0 (opt) :
td_matrix is stored on disk
(for eventual restart – to debug)
are stored on disk !
Take center of molecule as boxcenter
+Test_v2/t42.in : ixc=20,21,22

26. Test_v3/t55.in Test : Excitation window : Default: N = nval * ncond
# N2 system.
# Excited state computation, using LDA/TDLDA
# with different XC kernels
ndtset 4
#DATASET 1 SCF
#DATASET 2 TDDFT
#Common to all except GS calculations
getden 1
tolwfr d-9
iscf -1
getwfk 1
nband 12
#DATASET 3 SCF with another ixc
iscf3 5
nband3 5
prtden3 1
getwfk3 1
tolwfr d-15
ixc3 7
#DATASET 4 TDDFT
getden4 3
getwfk4 3
ixc4 7
#Common
acell 6 2*5 Angstrom
boxcenter 3*0.0d0
diemac 1.0d0 diemix 0.5d0
ecut 25
ixc 1
Test :
TDDFT excitation energies
oscillator strengths,
Polarisability
Cauchy coefficients.
Excitation window :
Default:
N = nval * ncond
Optional :
Td_maxene xxx :
Keep excitation i-j such
E_j-E_i < xxx
Td_mexcit yyy :
Keep yyy lowest KS excitations

27. *** TDDFT : computation of excited states ***
Splitting of 12 bands in 5 occupied bands, and 7 unoccupied bands, giving excitations.
Kohn-Sham energy differences, corresponding total energies and oscillator strengths (X,Y,Z and average)-
(oscillator strengths smaller than 1.e-6 are set to zero)
Transition (Ha) and (eV) Tot. Ene. (Ha) Aver XX YY ZZ
5-> E E E E E E E+00 …
1-> E E E E E E E-04
Sum of osc. strength : E+00
TDDFT singlet excitation energies (at most 20 of them are printed), and corresponding total energies.
Excit# (Ha) and (eV) total energy (Ha) major contributions
E E E ( 5-> 6) 0.25( 5-> 7)
E E E ( 4-> 11) 0.17( 2-> 7)
Oscillator strengths : (elements smaller than 1.e-6 are set to zero)
Excit# (Ha) Average XX YY ZZ XY XZ YZ
E E E E E E E E+00
E E E E E E E E-03
E E E E E E E E-06
Cauchy coeffs (au) : ( -2)-> E+00, ( -4)-> E+01, ( -6)-> E+02
(-8)-> E+02, (-10)-> E+03, (-12)-> E+04, (-14)-> E+04
TDDFT triplet excitation energies (at most 20 of them are printed),
and corresponding total energies.
…………………….
Test_v3/t55.out

28. Application : Nanoclusters
Carbon & Silicon clusters
Description
Convergence vs excitation window
Nanodiamonds : quantum confinement
Comparison with experiment Si29H24
Silicon : comparison with other methods

29. Nanodiamonds & TDLDA They do exist ! 2-6 nanometers Crystalline
Huge quantum confinement
(still ~1 eV at 35 nm) PRL xxxxxx
But
LLNL x-ray exp. show NO QC for 3nm particles
What is the gap of 3 nm particles ? GGA vs TDLDA

30. Simulation C211H140 1.4 nm Surface effects are dominant Needs for
At this size, surface effect are DOMINANT -> detailed, reliable information on the structure
Needs to include - many-body potentials
– quantum electronic description
Dynamics
Accurate description of the excited states
Needs for
-structure optimization
-quantum description
-better excited states
Surface effects are dominant

31. Convergence with excitations window
Default : N = 52*148 = 7696
Td_mexcit = 600 : converged on ~2 eV
Td_mexcit = 1600 : ~3eV

32. C59H60 GGA 0.8 nm C-C : 1.61-1.64 Å Bulk : 1.54 Å HOMO 3x deg.
Egap = 4.3 eV
Bulk : 4.23 eV
Redo the figure…
HOMO in the center, lumo on the surface (as in SI case, usually gap is large)
Simulation

33. TDLDA Optical Spectra Arbitrary units E (eV) CH C H 0.91 nm 12 10 8 64 2
E (eV) CH C H 16 5 29 36 66 64 35 87 76
0.91 nm

34. Optical gap vs size Simulation GGA TDLDA onset TDLDA first peak (eV)9
GGA
TDLDA onset 8
TDLDA first peak 7
(eV) 6
gap E 5
TDLDA onset is close to GGA, but very low osc.strength – should be detected in fluorescence measurements
The first large TDLDA abs. Peak would correspond to the edge NOOOOO FAUX
Below the bulk value due to surface stress…
Bulk diamond 4 3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Diameter (nm)
Simulation

35. Surface reconstruction
Explain the atoms colors
Remove pairs of hydrogens
Reconstructs at ~0 K
HOMO-LUMO changed
C29H36 – 6 H2 -> C29H24
Simulation

36. Surface reconstruction effect9 8 7 6 5 4 3 E
gap
(eV)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Diameter (nm)
C29H24
GGA
C29H36
TDLDA
TDLDA Absorption spectra (Arbitrary units)
Bulk diamond 4 5 6 7 8
E (eV)
Gap is reduced by ~0.5 eV
No quantum confinement effect for D > 1-2 nm
Simulation

37. Optical gaps of Si clusters5
Experiments 4
Size dependence of model clusters
Surface effects
Preparation conditions 3
Optical Gap (eV) 2 1 2 4 6 8
Diameter (nm)
Predictive simulation techniques are needed
to model and characterize silicon nanostructures

38. State of the simulations (QMC)
Si148H120
2 nm diameter
268 atoms
712 electrons
>1000 CPU hours
2 nm
A.J. Williamson et al. Physical Review Letters, (2001)

39. QMC Calculations explain the largest gaps5
QMC calculations
Experiments 4 3
Optical Gap (eV) 2 1 2 4 6 8
Diameter (nm)
Too expensive ! Is TDLDA a way to go ?

40. TDLDA vs Experiment Not normalized (sum rule not verified)
L. Mitas et al.
Not normalized
(sum rule not verified)

41. Optical Gap of Silicon Dots
Number of silicon atoms 1 2 5 10 17 29 35 66 87
148 10
QMC (LLNL)
EPM (NREL)
TD-LDA
LDA HOMO-LUMO
GW+BSE (Berkeley) 8
Optical gap (eV) 6 4 2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Diameter (nm)

42. TDLDA vs BSE
Lorin Benedict et al.

44. Kcoul, Kxc contributions

46. Future developments MPI version Other functionals
(TDLDA/LB94, TDLDA/ACLDA)
M. Casida et al. J. Chem. Phys. 108 (1998) 4439

47. Thanks to : The FNRS Christophe Bichara (CRMC2-Univ. Marseille)
Igor Vasiliev (Univ. Illinois)
Jim Chelikowsky (Univ. Minnesota)
Giulia Galli
Lorin Benedict
Andrew Williamson
& Jeff Grossman (Lawrence Livermore Natl Lab.)