Surface Physics: Lecture 3

9.4 Introduction to Bonding & Reactivity of Metal Surfaces

Surface Electronic Structure

rs = 2rs = 5

π/kF

N.D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970)

Ion cores approximated byuniform, positive backgroundcharge (spatial average),confined to semi-infinite space

Failures:- exponential instead of power

law image potential (LDA-problem) - breaks down for TMs

(localized d-electrons)

Achievements:- electron spill out

(dipole layer, hence finite work function)- Friedel oscillations- oscillatory multilayer relaxation- qualitative description of simple metals (Na, Rb): work function, surface energy…

Surface Electronic Structure• The work function is the energy cost to remove an electron from a solid

• Chemical potential of the electrons

µ=E(N+1)-E(N)=EF

• Work function

Φ=φ (∞)-µ = ∆φ−µ

Adapted from M. Gajdos, PhD Thesis, U. Wein, 2004

• Adsorbates alter the surface dipole and Φ and thus critical information about bonding at interfaces can be obtained

HOMO

LUMO

HOMO

LUMO

Surface Electronic Structure

Electropositive Adsorbates: ∆Φ is negative

Electronegative Adsorbates: ∆Φ is positive

-

+µµ

+

-

• Very successful approach. Widely used to make interpretations about bonding, charge transfer, reactivity. • Even used as a structural probe (LEEM), but…

“Conventional” example: Adsorbate induced work function change

∆Φ[e

V]

Tang et al., Surf. Sci. Lett. 255, L497 (1991).

Properties of Na/Cu(111)

“Conventional” example: Adsorbate induced work function change

charge depletion

+

-

charge accumulation

adsorbate induced dipole

WW NN

∆∆ρρ (z

)(z

)

∆∆ρρ(z) = (z) = ρρ(z)(z)N/W{100} N/W{100} -- ρρ(z)(z)W{100} W{100} -- ρρ(z)(z)NN

“Unconventional” example: N/W(100)

Must consider height of the adsorbate relative to the spillout electron density of the substrate

Phys. Rev. Lett. 90, 246103 (2003)

Elementary processes (in heterogeneous catalysis)

i) (dissociative) adsorption of reactants

ii) diffusion / ordering at the surface

iii) chemical reactions

iv) desorption of reaction products

bond making and breaking:sometimes fast, sometimes slow

Importance of intermediate bond strength: Sabatier Principle (1913)

Low bond strength: no incentive to form/bind intermediates (adsorption is rate limiting)High bond strength: no incentive to form products (reaction/desorption is rate limiting)

HDSH. Toulhoat et al., Catal. Today

50, 629 (1999)

NH3J.R. Jennings, Ammonia Synthesis, Plenum (1991)

N2 + 3H2 → 2NH3

Catalysis in Action: Dissociation Reactions

• In the absence of the surface (catalyst) the energy would rise continuously up to the bond dissociation energy (4-5 eV). • The surface provides ‘new’ bonds to compensate for the broken one.

JCP, 112 8120 (2000)

Transition States and Reaction Pathways

∆E

Reaction coordinate q

IS

TS

FS

• A chemical reaction in the Born-Oppenheimer picture is described as the movement of nuclei on a multidimensional potential energy (PES)• Chemical reactions are interesting and important ‘rare events’ – rare events on the timescale of atomic vibrations• Minima and transition states are stationary points on the PES, i.e. gradient of energy with regard to displacement is zero• Transition states are first order saddle points, i.e. energy rises in all directions but one

Transition States and Reaction Pathways

• Reaction coordinate, q, is not generally a simple geometrical coordinate (e.g. bond length, bond angle)•Given some configuration Ri and Hamiltonian H(Ri) get Hessian matrix:

•At a transition state there will be one (and only one) real and negative eigenvalue

RjRi

Hδδ

δ 2

• So how to determine transition states, minimum energy pathways and reaction rates?

• No general methods guaranteed to find saddle points. • Almost as many methods as there are chemical reactions! …new approaches published nearly every week

The minimum energy path follows the ‘intrinsic reaction coordinate’ – the steepest descent path in mass weighted coordinates

Rare event dynamics

• Direct approach to get rate constants: Molecular Dynamics – within DFT Ab initio MD “Car-Parinello MD (CPMD)

Run MD trajectory so long that it establishesequilibrium, crossing from IS to FS many, manytimes back and forth:

⇒ k =no. of crossings IS → FS per unit time

fraction of time system has spent in IS

⇒ require approximate theories to obtain process rates!

Yet:- Relevant time step in MD run is fs (vibrations)- Surface processes are often activated: ∆E ~ 1 eV ⇒ 10-2 - 102 events per second (TOF!)- Requires to run trajectory over about 1015 – 1020 time steps → unfeasable…

…and essentially 99,9999% of the time, the system willjust vibrate around IS basin (short time dynamics)

∆E

Reaction coordinate q

IS

TS

FS

Rare event dynamics

IS

TS

FS

Accelerated Molecular Dynamics: Apply some bias/boost potential to artificially raise the energy of the minimum whilst not altering the

transition states

Reaction coordinate q

⇒ require approximate theories to obtain process rates!

Many implementations, many people (Voter, Fichthorn et al., Wilkins et al., Parinello et el. ) and many names (hyperdynamics, metadynamics, …)

A more indirect approach: Transition state (activated complex) theory

Assumptions:

i) Reaction system passes the barrier only once (no re-crossings)

ii) Energy distribution of reactant DOF is Boltzmann-like (many collisions compared to reaction eventsyield equilibrium between activated complex and IS,except with respect to the reaction coordinate)

iii) Passage over barrier is the motion of only one DOF,the reaction coordinate, which is independent of allother motions of the activated complex (no concertedmotions)

iv) Passage over barrier is a classical event (no tunneling, but easy to include in an approximate way)

( Eyring, Evans, Polanyi, ~1935 )

IS FS

eq.

XX

Derivation: see e.g.R.J. Laidler, Chemical kinetics,

Harper & Row, New York (1987)

⇒ kTST = [ ( ) e ∆S/k ] e-∆E/kTkThIS→FS

~ 1013 sec-1 ~ 1 (or vibrations!)

I Grid method:- Compute PES on a (regular) grid- Scales like: (no. of points) dim

- e.g. 59 ~ 2 million grid points• Almost always unfeasible

II Interpolation Methods:

(i) Constrained minimization (Drag method):

- Choose appropriate reaction coordinate q- Constrain q and relax all other DOF- Move from IS to FS

• Can be efficient and simple to implement & use

• highly dependent on good reaction coordinate

• hysteresis!

IS

FS

TSx

x

xx

x

x

x

x

x

x

x

x

x

Some transition state search algorithms

Some transition state search algorithms

(ii) Nudged Elastic Band (NEB) Method

H. Jónsson, G. Mills, and K.W. Jacobsen, in Classical and Quantum Dynamics in Condensed Phase Simulations, edited by B.J. Berne, G. Ciccotti, and D.F. Coker (World Scientific, Singapore, 1998), p. 385

Initialize with several images {Ri} along astraight-line interpolation

- Problem:- elastic band cuts corners- images tend to slide down towards

low-energy IS/FS regions, leaving fewimages for relevant TS region

- Solution:- only spring force component parallel

to path (no corner cutting)- only true force component perpendicular

to path (no down-sliding)

→ widely applied workhorse→ has problems, if energy varies largely along path,

but very little perpendicular to it (kinky PES)

Some transition state search algorithms

(ii) Nudged Elastic Band (NEB) Method

Spring force between images of system:

Effective force on each image:

tangent along path

where the perpendicular force is

H. Jónsson, G. Mills, and K.W. Jacobsen, in Classical and Quantum Dynamics in Condensed Phase Simulations, edited by B.J. Berne, G. Ciccotti, and D.F. Coker (World Scientific, Singapore, 1998), p. 385

Some transition state search algorithms

Climbing Image NEB: The highest energy image in the band is pulled up to the saddle point, gives more rigorous estimate of saddle point

G. Henkelman, B.P. Uberuaga and H. Jónsson, J. Chem. Phys. 113, 9902 (2000); 113, 9978 (2000).

Some transition state search algorithms

(iii) …many, many other interpolation based methods

III ‘eigenvector following’ or ‘local’ methods:

At the transition state the negative eigenvector gives the reaction coordinate, close to the transition state points to TS

+ only works with good starting geometry and a ‘hard’ PES

- Away from transition state lowest eigenvector not directed to TS –wrong mode can be followed (especially problematic for metals)

So with some way to compute transition states what can we learn?

d-band center is important…

s-band

d-band

EF

clean surface

(anti-bonding)

(bonding)

d-band center is important…

•Adsorption energies and activation energies for dissociation often correlate with the depth of the “d band center” beneath the Fermi level prior to adsorption

Tailoring the reactivity of a surface

PRL, 81 2819 (1998)

Dissociation Reactions: General trends

NorskovNorskov et al. J. Catal. 209, 275 (2002).et al. J. Catal. 209, 275 (2002).

•• BrBr∅∅nsnsted, Evans, and Polanyi (BEP); Hammett and Marcusted, Evans, and Polanyi (BEP); Hammett and Marcus

Dissociation Reactions: General trends

Ea

∆H

Ener

gyEn

ergy

→→

Reaction Coordinate Reaction Coordinate →→

Ener

gy→

Reaction Coordinate →

Ener

gy→

Reaction Coordinate →

Ener

gy→

Reaction Coordinate →

AA BB CC

Overly simple yet clear picture for surface reactions!

-8

-6

-4

-2

0

2

4

-10 -7.5 -5 -2.5 0 2.5 5

∆H

E aga

s

-2

-1

0

1

2

3

4

-4 -1.5 1 3.5

∆H

E aga

s

-3

-2

-1

0

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3 4

∆H

E aga

s

-8

-6

-4

-2

0

2

4

-10 -7.5 -5 -2.5 0 2.5 5

∆H

E aga

s

Ea = 0.89∆H + 1.08 Ea = 0.92∆H + 0.87

Ea = 0.74∆H + 1.03 Ea = 0.97∆H + 1.69

All Dehydrogenation

Diatomics…Triatomics

Dissociation Reactions: General trends

CHx, NHx,

OHx…

CO2, N2O,

NO2…

CO, NO, N2..

DFT for many common dissociation reactions on transition metal surfaces indicate that BEP correlations generally exist in surface physics

Dissociation Reactions: General trendsEn

ergy

Ener

gy→→

δδ

δδ

Ea = γ∆H + Ea0

γ ~ 1

Association barriers change very little with ∆H!!

… and what we didn’t talk about …

Growth (epitaxy, CVD / MBE, facetting / roughening) Nanotechnology (self-assembly, atom manipulation)Surface magnetism (storage devices…)Optical properties/excited states (fs-laser…)

Carbon nanotubesSolid-liquid interface / electrochemistryBiomolecules at surfaces…

There’s lots more:surface physics is certainly neither

“superficial” nor “marginal”…

Frontiers in surface and interface scienceC.B. Duke and E.W. Plummer (Eds.),

North-Holland (Amsterdam, 2002); Surf. Sci. 500 (2002)