Quantum Chemical Molecular Dynamics Simulations of Graphene Hydrogenation

Stephan IrleDepartment of Chemistry, Graduate School of Science

Nagoya University

第 30 回 量子物理化学セミナーTachikawa & Kita Group

Yokohama City University, Yokohama, February 13, 2012

CFC: Carbon Fiber CompositeCourtesy of A. M. Ito

What do these processes have in common?

2

Introduction

Chemical reduction byhydrogenases

Buckminsterfullerene self-assembly

H2

2H+ + 2e-

Chemical reaction mechanisms are almost entirely unknown!

Chemical sputtering

3HCN CNH

+15

+30+45

+60

+90

+105

+120

+45

+30

+15

+60+90

+75

R

q

TS

R

P

Example: HCN CNH isomerization

Introduction

Experimental study of complex chemical reaction mechanism nearly impossible

Chemical reactions are theoretically studied mostly based on

•Born-Oppenheimer (BO) potential energy surfaces (PESs)

•Minimum energy reaction pathways (MERPs, result of “intrinsic reaction coordinate” (IRC) calculations) Kenichi Fukui

Acc. Chem. Res. 1981

Born & Oppenheimer

Problems of the MERP approach:•BO approximation and adiabatic wavefunctions may be unsuitable, for example due to

• conical intersections/state crossings

• mixed quantum states

• high nuclear velocities (tight minima may be missed)

• entropic effects (0 Kelvin)

• quantum tunneling (hydrogen!)

4

Molecule: HCN moleculeNumber of atoms N = 3NDOF = 3N-6 = 3

•Theoretical study of chemical reactions difficult due to dimensionality problem

Number of degrees of freedom(NDOF):

nuclear coordinates

Brute force scan:10 pts/DOF: 103 = 1000energy calculations

OK!

Introduction

Hase et al. JACS 129, 9976 (2007)

5

nuclear coordinates

Brute force scan:10 pts/DOF: 10174 grid points

Molecule: C60

Number of atoms N = 60NDOF = 3N-6 = 174

Not OK!

IRC of C60 formation?

IntroductionSelf-assembly mechanism of C60

Automatized MERP SearchIf we can find all TSs, then we can find all reaction pathwaysK. Ohno, S. MaedaChem. Phys. Lett. 384,277 (2004)

Starting from an EQ, reaction channels can be found by following Anharmonic Downward Distortions (ADD): Compass on the PES

6

Introduction

Automatized MERP Search

Global Reaction Route Mapping (GRRM)

K. Ohno, S. Maeda, Chem. Phys. Lett. 384,277 (2004)

BUT: Number of reaction pathways presents a combinatorial explosion problem!

Newton’s equations of motion for the N-particle system:

Fi can be calculated as . There are several approximate methods

to solve this system of equations. Some commonly used methods are:

Verlet’s algorithm

Beeman’s algorithm

Velocity Verlet algorithm:

7

RTS

P

i

iiii m

ttttttt

2)( 2 F

vrr

i

iiii m

ttttttt

2

FFvv

MD for Chemical Reactions

Introduction

Practical implementation requires discrete Dt

E can be either classical potential or Born-Oppenheimer total electronic energy

Density-Functional Tight-Binding: Method using atomic parameters from DFT (PBE, GGA-type), diatomic repulsive potentials from B3LYP

• Seifert, Eschrig (1980-86): minimum STO-LCAO; 2-center approximation, Slater-Koster parameter files, NO integrals!•Porezag, Frauenheim, et al. (1995): efficient parameterization scheme: NCC-DFTB

• Elstner et al. (1998): charge self-consistency: SCC-DFTB• Köhler et al. (2001): spin-polarized DFTB: SDFTB

Marcus Elstner

Christof Köhler

HelmutEschrig

GotthardSeifert

Thomas Frauenheim

8

DFTBAlternative to DFT-based MD:Semiclassical MD based on approximate DFT potential

Self-consistent-charge density-functional tight-binding (SCC-DFTB)

M. Elstner et al., Phys. Rev. B 58 7260 (1998)

Approximate density functional theory (DFT) method!

Second order Taylor expansion of DFT energy in terms of reference density r0 and charge fluctuation r1 ( ≅ r r0 + r1) yields:

Density-functional tight-binding (DFTB) method is derived from terms 1-6 Self-consistent-charge density-functional tight-binding (SCC-DFTB) method is derived from terms 1-8

o(3)

DFTB

9

DFTB and SCC-DFTB methods

where ni and i — occupation and orbital energy ot the ith Kohn-Sham

eigenstate Erep — distance-dependent diatomic repulsive potentials

qA — induced Mulliken charge on atom A

AB — distance-dependent charge-charge interaction functional; AB = AB (UA, UB,RAB) for RAB : Coulomb potential 1/RAB

AA = AA (UA, UA,RAA) for RAA 0: Hubbard UA = ½(IPA – EAA)

DFTB

10

DFTB method are tabulated for ~40 intervals as splines and have a

cutoff radius shorter than 2nd-neighbor distances; empirically fitted

Reference density 0 is constructed from atomic densities

Kohn-Sham eigenstates i are expanded via LCAO-MO scheme in

Slater basis of valence pseudoatomic orbitals i

The DFTB energy is obtained by solving a generalized DFTB eigenvalue problem with H0 computed by atomic and diatomic DFT

DFTB

Eigensolver: LAPACK 3.0Divide and Conquer: DSYGVD()Standard: DSYGV()Intel MKL SMP-threaded parallel up to ~8 CPU cores 11

DFTB repulsive potential Erep

Which molecular systems to include?

DFTB

Development of (semi-)automatic fitting:•Knaup, J. et al., JPCA, 111, 5637, (2007)•Gaus, M. et al., JPCA, 113, 11866, (2009)•Bodrog Z. et al., JCTC, 7, 2654, (2011)

repEab

repEab

12

Typical number of SCC iterations: ~10-20

Therefore: SCC-DFTB is ~10-20 times more expensive than DFTB

Additional induced-charges term allows for a proper description of polarization, charge-transfer

Induced charge qA on atom A is determined from Mulliken

population analysis, or equivalent

Kohn-Sham eigenenergies are obtained from a generalized, self-consistent SCC-DFTB eigenvalue problem

SCC-DFTB method

DFTB

13

Gradient for the (SCC)DFTB methods

The DFTB force formula

The SCC-DFTB force formula

computational effort: energy calculation 90%

gradient calculation 10%

DFTB

14

Spin-polarized SCC-DFTB (SDFTB, sSCC-DFTB)

for systems with different and spin densities, we have total density = + magnetization density S = -

2nd-order expansion of DFT energy at (0,0) yields

The Spin-Polarized SCC-DFTB method is derived from terms 1-9

o(3)

C. Köhler et al., Phys. Chem. Chem. Phys. 3 5109 (2001)

DFTB

15

where pA l — spin population of shell l on atom A

WA ll’ — spin-population interaction functional

Spin populations pA l and induced charges qA are obtained from Mulliken population analysis

Spin-polarized SCC-DFTB (II)

DFTB

16

Kohn-Sham energies are obtained by solving generalized, self-consistent SDFTB eigenvalue problems

where

M,N,K: indexing specific atoms

Spin-polarized SCC-DFTB (III)

DFTB

17

Performance for small organic molecules (mean absolut deviations)

• Reaction energies: ~ 5 kcal/mol

• Bond lenghts: ~ 0.014 Å

• Bond angles: ~ 2°

• Vibrational Frequencies: ~6-7 %

SCC-DFTB: general comparison with experiment

DFTB

18

SCC-DFTB: Transition metalsDFTB

G. Zheng et al.J. Chem. Theor. Comput. 3 1349 (2007)

Bond lengths: ~0.1 ÅBond angles: ~10°Relative energies: ~20 kcal/mol

19

20/25

New Confining Potentials

Wa

Conventional potential

r0

Woods-Saxon potential

k

R

rrV

0

)(

R0 = 2.7, k=2

)}(exp{1)(

0rra

WrV

r0 = 3.0, a = 3.0, W = 3.0

Typically, electron density contracts during covalent bond formation.

In standard ab initio methods, this is easily handled by n-z basis sets.

DFTB uses minimal valence basis set: the confining potential is adopted to mimic contraction

• •+

• •

1s

σ1s

H H

H2

e.g.

Δρ = ρ – Σa ρa

H2 difference density1s

DFTB Parameterization

Prof. Henryk Witek, National Chiao Tung

University, Taiwan20

Each particle has randomly generated

parameter sets (r0, a, W)within some region

Generating one-center quantities (atomic

orbitals, densities, etc.)

“onecent”

Computing two-center overlap and Hamiltonian integrals for wide range of interatomic distances

“twocent”

“DFTB+”

Calculating DFTB band structure

Update the parameter sets of each particle

Memorizing the best fitness value and parameter sets

*a [2, 4]W [0.1, 5]r0 [1, 10]

Evaluating “fitness value”(Difference DFTB – DFT band

structure using specified fitness points) “VASP”

“Particle Swarm Optimization”DFTB Parameterization

21

Chou, Nishimura, Irle, Witek, In preparation

Error in DH for linear alkanes CnH2n+2

Automatization of Erep

Parameterization

22

DFTB ParameterizationElectronic parameters now available for Z=1-83!Yoshifumi Nishimura, D2

Future: GA-based Erep parameterizationOr on-the-fly parameterization 22

DFTB ParameterizationTransferability of optimum parameter sets for different structures

Artificial crystal structures can be reproduced well

e.g. : Si, parameters were optimized with bcc only

W (orb) 3.33938

a (orb) 4.52314

r (orb) 4.22512

W (dens) 1.68162

a (dens) 2.55174

r (dens) 9.96376

εs -0.39735

εp -0.14998

εd 0.21210

3s23p23d0

bcc 3.081

fcc 3.868

scl 2.532

diamond 5.431

Parameter sets:

Lattice constants:bcc fcc

scl diamond

Expt.

DFTB ParameterizationTransferability of optimum parameter sets for different structures C, diamond + graphite, 2s22p2

DFTDFTB

Orbital energy:2s = -0.505332p = -0.19423

diamond graphite

Band gap:5.35 eV (DFT)7.23 eV (DFTB)7.3 eV (expt.)

Rocksalt (space group No. 225)

•NaCl•MgO•MoC•AgCl…

•CsCl•FeAl…

B2 (space group No. 221)

Zincblende (space group No. 216)

•SiC•CuCl•ZnS•GaAs…

Others

•Wurtzite (BeO, AlO, ZnO, GaN, …)•Hexagonal (BN, WC)•Rhombohedral (ABCABC stacking sequence, BN)

No further optimization of parametersmore than 100 pairs tested

DFTB ParameterizationBinary compounds

25

•d7s1 is used in POTCAR (DFT)

Further improvement can be performed for specific purpose but this preliminary sets will work as good starting points

NaCl (rocksalt) FeAl (b2)

CsF (rocksalt) BN (wurtzite)

•matsci-0-2 for previous work

DFTB ParameterizationBinary compounds: Selected examples

26

ExperimentalChemisorption ofatomic hydrogen •Fully saturated

graphene with sp3 hybridization (diamond-like)•Band gap of ~3eV

Band insulator!

“Graphane”, J. O. Sofo et al., Phys. Rev. B, 77 153401 (2007).

DFT calculations for partially hydrogenated graphene show:

1. Band gap at K-opening2. Dispersionless hydrogen

acceptor level at EF

3. Spin splittingE. J. Duplock et al., Phys. Rev Lett., 92, 225502 (2004). 27

Hydrogen plasma - wall interactions (PWI) in nuclear fusion reactors

LHD (Large Herical Device)

A. Sagara et al, (LHD Experimental Group, National Institute for Fusion Science, Gifu), J. Nucl. Mater. 1, 313 (2003)

Divertor plate(Graphite)

Experimental

28

Hydrogen-wall interaction

⇒H2, CHX, C2HX formation

Observable on graphite divertor and in plasma-beam experiments

CyHX formation mechanism unknown

Atomic-scale simulation of CyHX formation

CFC: Carbon Fiber Composite

Experimental

Hydrogen plasma - wall interactions (PWI) in nuclear fusion reactors

29

Reactive Empirical Bond Order (REBO) force field MD simulations of atomic hydrogen reactions with graphite

(0001)

H incident energy: 5 eVInjection rate: 1 H/0.1 ps“Graphite peeling”

A. Ito, Y. Wang, SI, K. Morokuma, H. Nakamura, J. Nuclear Mater. 300, 157 (2009).

REBO vs DFTB

30

-Two-body potential

-No effects of -p conjugation or aromaticity included

-Typically too high sp3 carbon fraction (Marks et al. Phys. Rev. B 65, 075411 (2002))

-Typically too low fraction of sp carbons (SI, G. Zheng, Z. Wang, K. Morokuma, J. Phys. Chem. B 110, 14531 (2006))

How trustworthy is REBO in this case?

Parameterize cheap QM method for MD!

Drawbacks of REBO

REBO vs DFTB

31

Fitting of Density-Functional Tight-Binding:Adjusting Erep for H-graphene chemisorption

Extended Hückel type method using atomic parameters from DFT (PBE, GGA-type), diatomic repulsive potentials from B3LYP

• Seifert, Eschrig (1980-86): STO-LCAO; 2-center approximation• Porezag et al. (1995): efficient parameterization scheme: NCC-DFTB• Elstner et al. (1998): charge self-consistency: SCC-DFTB• Köhler et al. (2001): spin-polarized DFTB: SDFTB

Adjust Erep for C-H!

REBO vs DFTB

Self-consistent charge-charge interactions

Self-consistent spin-spin interactions

Zeroth-order Hamiltonian: no e-e interactions

32

PyreneC16H10

X//B3LYP/pVDZ relaxed energy profiles for PAH models

Barrier:+6.9 kcal/mol (B3LYP)+9.2 kcal/mol (G2MS)

Well depth:-9.1 kcal/mol (B3LYP)-9.0 kcal/mol (G2MS)

0.5 1 1.5 2 2.5 3 3.5

-0.6

-0.4

-0.2

-1.11022302462516E-16

0.2

0.4

0.6

UB3LYP

ROMP2/DZ//UB3LYP

ROCCSD//UB3LYP

C-H distance (in Angstrom)

Bind

ing

ener

gies

(in

eV)

REBO vs DFTB

CoroneneC24H12

Barrier:+6.8 kcal/mol (B3LYP)

+10.1 kcal/mol (G2MS)

Well depth:-10.2 kcal/mol (B3LYP)-10.1 kcal/mol (G2MS)

0.5 1 1.5 2 2.5 3 3.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

UB3LYPROMP2/DZ//UB3LYPROMP2/TZ//UB3LYPRCCSD//UB3LYPRCCSD(T)//UB3LYPG2MS

C-H distance (in Angstrom)

Bin

din

g e

ne

rgy

(in e

V)

~1.5 CPU years

33

0.5 1 1.5 2 2.5 3 3.5

-40.00

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

Erep

Erep-wish

Erep-sdftb

C-H distance (in A)

energ

y (in k

cal/m

ol)

0.5 1 1.5 2 2.5 3 3.5

-40.00

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

Erep

C-H distance (in A)

energ

y (in k

cal/m

ol)

Well depth nearly same, but barrier height should be enhanced by ~5 kcal/mol

Blue: DE(UB3LYP/pVDZ)//B3LYP/pVDZRed curve: original DE(SDFTB)//B3LYP/pVDZ

REBO vs DFTB

X//B3LYP/pVDZ relaxed energy profiles for PAH models

34

0.5 1.0 1.5 2.0 2.5 3.0 3.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

B3LYP

RCCSD(T)

G2MS

SCCDFTB

SDFTB-original

SDFTB*

C-H distance [Å]

Rel

ativ

e E

nerg

y [e

V]

REBO vs DFTB

X//B3LYP/pVDZ relaxed energy profiles for PAH models

SDFTB* now reproduces G2MS curve exactly at tiny amount of computer time

35

36

3 qualitatively distinct types of reaction outcomes:

Reflection: EI < 1eV

Adsorption: 1eV < EI < 7eV

Reflection: 7eV < EI < 30eV

Penetration: EI > 30eVREBO Simulations by Ito et al. Contrib. Plasma Phys. 48, 265 (2008)

REBO vs DFTB

37

REBO vs DFTB

0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

SDFTB*Absorption (forward)Absorption (backward)

Incident Energy （ in eV)

Ra

tioREBO

REBO: Barrier 0.5 eV, height OK, but too thin

Well -4.8 eV, much too low

REBO

38

REBO vs DFTB

0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

SDFTB*Absorption (forward)Absorption (backward)

Incident Energy （ in eV)

Ra

tioREBO

0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

DeuteriumAbsorption (forward)

Absorption (backward)

Reflection

Penetration

Incident Energy (in eV)

Ra

tio

0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

TritiumAbsorption (forward)Absorption (backward)ReflectionPenetration

Incident Energy(in eV)

Ra

tio

39

2D potential of hydrogen atom in the hexagon plane

REBO SDFTB*

REBO vs DFTB

40

1 1.1 1.3 1.5 1.7 2 2.5 3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

DFTB+

1H-M=2-Te=0

2H-M=1-Te=0

2H-M=3-Te=0-initial-spin

2H-M=1-Te=0-initial spin

C-H distance (in Angstrom)R

ela

tiv

e e

ne

rgy

(in

eV

)

2 Hydrogen atoms on graphene: Singlet or triplet?

Top view

Side view

1H: SDFTB* Doublet

2H: SCC-DFTB closed-shellsinglet

2H: SDFTB* triplet

2H: SDFTB* open-shell singlet

Evidence for long-distance spin correlation via -p conjugation!

Cannot be captured classically!

REBO vs DFTB

41

42

C4H Polymer

Experiment: chemical environment of hydrogenated graphene

Quasi-free-standing grapheneHydrogenated

quasi-free standing graphene

D. Haberer et al. Nano Lett. 10, 3360 (2010) 42

H coverage from High-resolution XPS

C4H Polymer

D. Haberer et al. Adv. Mater. 23, 4497 (2011) 43

H coverage as function of time

Why does it stop at 25%??

C4H Polymer

D. Haberer et al. Adv. Mater. 23, 4497 (2011) 44

Simulation details• Ten trajectories for 1 eV and 0.4 eV incident energies

• NVT (Tn=300 K, Nose-Hoover chain thermostat), 4*4 unit cell (32 carbon atoms)

• H were “shot” at perpendicular angular from 3 Å distance, random x and y coordinates, random spin

• Totally 100/400 H were “shot”

• 1H/0.5 ps, Dt = 0.2 fs (ensure energy conservation in NVE)

• New G2MS-derived C-H Erep

• SDFTB with Te=300 K

C4H Polymer

45

46

1 2 3 4 5

6 7 8 9 10

Boukhvalov. et.al JPCC 113,14176 (2009)

C4H Polymer

All H-frustratedFlores et.al, Nanotechnology, 20 465704 (2009)

Incident energy: 1 eV

46

Average H Coverage Reaction processes

0 10 20 30 40 500

0.1

0.2

0.3

0.4

Time (ps)

Ra

tio

of

H/C

0 10 20 30 40 500

20

40

60

80

100reflectionadsorptionh2 formation

Time (ps)

Nu

mb

er

of

H

C4H PolymerIncident energy: 1 eV

47

2 and 3 show perfect “para-structure”, others are mixed para/H-frustrated

C4H PolymerIncident energy: 0.4 eV

Much less H-frustration48

490 50 100 150 200

0

100

200

300

400

reflection

adsorption

h2 formation

Time (ps)

Num

ber

of H

Reaction processes(average over 10 trajectories)

C4H PolymerIncident energy: 0.4 eV

0 50 100 150 2000

5

10

15

Nu

mb

er o

f H

12

4

8

D. Haberer et al. Adv. Mater. 23, 4497 (2011)

49

Why 25%? C4H possesses an “all-para” structure with aromatic superlattice!

D. Haberer et al. Adv. Mater. 43, 4497 (2011)

C4H Polymer

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8C4H+H

C+H

C-H distance (Angstrom)

Re

lati

ve

en

erg

y (

eV

)• C4H has higher hydrogenation

barrier of about 0.63 eV, higher than incident energy of 0.4 eV.

• Potential energy profile of C4H+H is shallower than graphene+H.

graphene+H

50

51

C4H Polymer

How does the surface really look like?

• Maximum H coverage depends on the incident energy, higher incident energy gives higher coverage

• Higher incident energy (1 eV) yields H-frustrated structure, while lower incident energy (0.4 eV) can lead to self-assembled para-hydrogenated structure similarity to crystallization

• Stability of C4H para-hydrogenated structure caused by:

1. local aromaticity

2. High barrier for attack on aromatic hexagons

3. Low reverse barriers for hydrogen loss from aromatic hexagons

C4H Polymer

52

• Recently, Grüneis found substantial isotope effects (unpublished):

- Deuteration has higher adsorption maximum than H

- Deuterium can completely replace H on graphene, but not vice versa

D/H Isotope Effect

53

D/H Isotope Effect

54

Averaged coverage Reflecion-Adsorption-H2

H

R:A:H2=479.6:14.0:6.4

D

R:A:H2=479.7:14.7:5.6

D has more adsorption (14.7 VS. 14.0) and less D2/H2 leaving (5.6 VS. 6.4) than H---- larger coverage

RA H2

Incident energy: 0.4 eV

AcknowledgementsThe Group:

Dr. Ying WangDr. Hu-Jun QianDr. Matt Addicoat (JSPS)Dr. Cristopher CamachoMr. Yoshifumi Nishimura (D1)Mr. Yoshio Nishimoto (M2)UndergraduatesMs. Yae Imai (Administrative Assistant)

Collaborators: Keiji Morokuma (Kyoto U, Emory U)

CREST “Multiscale Physics” (2006-2011)CREST “Soft -p materials: (2011-2015)

SRPR tenure track program (2006-2011) JSPS KAKENHI

Funding:

July 8, 2011