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### 固体表面化学的理论研究方法、模型和应用

2005.5.19

State Key Laboratory for Physical Chemistry of Solid Surfaces

（实验研究）

• 分类(理论方法、模型方法、物理体系）
• 层板模型方法与应用（Slab Model and Its Applications)

3. 簇模型方法及其应用（Cluster Model and Its Applications)

• 理论方法分类：

• 模型分类：

3. 应用体系分类：

2 Slab Model2.1 First-Principle Method
• 量子化学问题均在于求解Schrödinger方程，对于大块固体，其Schrödinger方程表示为：

H(Rm,rn) = E (Rm,rn) (1.1)

Problem: H将是无限维的，上式很难求解。

Solutions: Introducing some approximations.

A. Born-Oppenheimer Approximation:

H({Rm}) '(rn)= E({Rm}) '(rn) (1.2)

(核运动和电子运动分离)

B. Single-Particle Approximation for Solving The Wavefunctions of Electrons

(电子波函数的单粒子近似)

C. Energy Band Theory (DFT) and Crystal Orbital Theory (HF)

(see A. Gross, Surf. Sci. Rep. 1998, 32, 291)

2.2 Density functional Theory, Kohn-Sham Equation

Eec(n) exchange-correlation functional

ec(n) exchange-correlation energy per particle

2.3 PSPW and Super Cell
• Pseudopotentials for inner shells
• Plane-wave functions for valence shells
• Periodic Boundary Conditions and Super Cell Method for Solid
• Slab Model for Solid Surface
• Car-Parrilleno Molecular Dynamics Method
Example 1: SO2 on MgO(100) and CuMgO(100)

Slab model:

4 atomic layers

QM Method:

DFT-GGA, PSPW

(see J. A. Rodriguez et al, J. Phys. Chem. B 2000, 104, 7439.)

Bonding Modes of SO2 on MgO(100) Surface

Cu-Free

2-O,O on Mg 8

1-S on O 11

3-S,O,O 21

Cu-dopping

2-O,O 28

1-S on O 25

3. Cluster Model 3.1 Concept

• 1)      FC: 小簇C的Fock算符，包括簇C内的动能与各种相互作用能；
• 2)     VSlr :环境S对簇C的长程作用势，包括簇C与环境S间的电子—电子、电子—核、核—电子、核—核等四种库仑势；
• 3)      VSsr : 环境S对簇C的短程作用势，包括簇C与环境S间的电子交换势，反映出簇C与环境S间的轨道相互作用。
• 4)      ρ VSsr ρ: 定域化势（亦称屏蔽势）
3.2 How to reach a successful cluster modeling?

• 怎样选择簇模型，使之与环境的短程作用尽可能小（必须注意，这并不意味着簇与环境的相互作用能很小）?
• 怎样合理地考虑环境对簇的长程作用?
3.3 Schemes of Cluster Modeling
• Simple Cluster Model
• Embedded Cluster Model (for ionic solids)
• Saturated Cluster Model (for covalent solids)
• ONIOM Model (hybrid QM/QM or QM/MM method, readily for covalent solids)
3.4 Simple Cluster Model
• Simple cut-out !!!!!
• Capacity? (may give qualitatively reasonable simulation results for VDW, HB, metal and ionic solids)
• How to make a reasonable cut-out?
• How to determine the electronic state of the cluster?
3.4.1 Cluster Model for Metal Surface
• Dilemma: The larger, the more reasonable, but more expensive; the smaller, the more economical with higher accuracy, but less reasonable.
• What’s the way out?

“Surface molecule”

Convergence problem

*H/Ni(111):

Ni19 (2.75kcal/mol)

Ni22(15 kcal/mol)

Ni40(46.5kcal/mol)

Ni4(55kcal/mol)

Expt(63kcal/mol)

• Examples: 1) P.S. Bagus, et al , J. Chem. Phys., 78 (1983) 1390; 2) C.W. Bauschlicher Jr., Chem. Phys. Lett., 129(1986) 586; 3) P.E.M. Siegbahn et al., Chem. Phys. Lett., 149(1988) 265.*
Concept of “Metallic Atom”
• Two kind of motions of electrons in bulk metal: 1) Localized ; 2) Delocalized--Free electrons.
• The atom in a bulk metal should be quite different from a simple atom, e.g.

a) R(Cr-Cr):1.68 Å ( Cr2 ), 2.49 Å (bulk Cr)

b) Pd atom: 4d10// bulk Pd:(4d9.635sp0.37)

Metallic Basis Functions
• The attractive potential of a metallic atom is:
• m(r) = -(Z*/r)exp(-kSr) vs a(r) = -Z*/r
• 1/kS --- Thomas-Fermi Screening Length.
• Slater exponents: m = a +  (1)
• With the help of Free Electron Theory, we have:
•  = - (a(n))/n(inner shell) (2)
•  = (a(n-1))/n(outermost valence-shell) (3)

( N. Wang et al., J. Mol. Struct. (Theochem), 262(1992) 105.)

1s

2sp

3sp

3d

4sp

a

26.47

11.09

4.55

3.94

1.40

m

26.46

10.96

4.01

3.35

1.84

Metallic m and Atomic a of Co Atom.
CO-like

Co-CO

CO/Co

Ni-CO

CO/Ni

MO’s

a

m

UPS

a

m

UPS

4

21.99

16.68

16.8

20.75

16.43

16.6

1

16.46

13.10

13.2

15.52

13.35

13.6

5

18.27

12.70

13.8

16.14

12.33

12.3

4-1

5.53

3.58

3.5

5.23

3.08

3.0

5-1

1.81

0.4

0.6

0.62

1.02

1.3

UHF/STO-3G Calculations M-CO cluster

X. Xu et al., Surf Sci., 274 (1992) 378

Choice of Multiplicity
• Metallic Cr: 3d5.244s0.76 (3d64s0 -- 3d54s1)
• 3d64s0: 5, 3, 1; 3d54s1: 7, 5, 3,1
• Note: UHF wavefunctions of a quintet are mixtures of wavefunctions from quintet and septet, rather than a pure quintet.
Multipl.

1

(3)

(5)

7

CO/Cr

4

18.82

16.80

16.74

17.48

16.6

1

14.66

12.66

12.60

13.23

12.6

5

13.91

11.67

12.03

12.68

Multiplicity Dependency in the UHF Calculations of Cr-CO
• *Fe: 3d7.344s0.61// (3d84s0 - 3d74s1)//(3),1 - 5,3,1
• *Co: 3d8.374s0.63//(3d94s0 - 3d84s1)//(2) - 4,2
Metallic State Principle

M Mn M Mn M

Ground State Bulk Metallic State

Some relative methods
• Bond-Prepared State Principle

(P.E. M. Siegbahn et al., Stockholm, 1988)

(H. Nakatsuji, Kyoto, 1991)

• Many-Electron Embedding Theory

(J.L. Whitten, 1980; 1987)

Example 2:NO2/Au(111)

X. Lu, J.Phys.Chem. A, 103 (1999) 10969.

B3LYP calculations of NO2Au2
• NO2 (2A1) + Au2 (1g)  NO2Au2 (2A1)
• NO2 (2A1) + Au2 (3u)  NO2Au2 (2B2)
More Cluster Models: Au7 and Au12
• Results omitted from here
3.4.2 Simple cluster model for ionic solids

How to cut out a cluster?

• Three Principles: Neutrality, Stoichiometry and Coordination Principles.
• Coordination number principle: 1) fewest dangling bonds at the edge of a cut-out; 2) maintain the stronger dative bonds within the cluster.

X. Lu et al., 1) Chem. Phys. Lett. 291(1998) 457; 2) Int. J. Quant. Chem. 73 (1999) 377; 3) Theor. Chem. Acc. 102(1999) 179.

CO/MgO

X. Lu et al., J. Phys. Chem. B, 105(2001) 10024.

3.5 Embedded Cluster Model for Ionic Solid

For ionic solid, VSsrcan be replaced byVIsr:

For ideally ionic solid, VIsrwould be negligible:

i.e. Simple embedded cluster model

3.5.1 Simple embedded cluster model
• A cut-out cluster is embedded into an array of point charges (always in formal charge) to represent the Madelung Potential of the ionic surroundings.
Example 4: CO/MgO(100) and NiO(100)
• See in G. Pacchioni et al. Surf. Sci. 255 (1991) 344.

Simple embedded cluster model for MgO(100) and NiO(100) ( Mg(Ni) +2; O: -2 )

Demerits of simple embedded cluster model

Most of the ionic solids are not ideally ionic. Hence,

• the ionic charges are always fractional;
• the short range interaction between the cut-out cluster and its surrounding is seldom negligible.

Way-out:

• Charge consistency
• Minimize the short range interaction.

Different embedding charge Q gives different C with different charges at the in-cluster atoms. Hence charge consistence between the embedding charges and the equivalent in-cluster atoms is essential and can be readily reached.

SPC Embedded Cluster Model

Cutout Cluster

SPC Embedding

Nuetrality Principle

Coordination Principle

Spherical

Point Charges

Self-consistency of Charge Density

Stoichiometry Principle

X. Lu et al, J. Phys. Chem. B 103(1999) 2689.

Example: SPC Cluster Models for MgO

X. Lu et al., J. Phys. Chem. B, 103(1999) 3373.

NxOx+12- (X=1,2) Species Formed on MgO

X. Lu et al., J. Phys. Chem. B, 103(1999) 5657.

3.6 Saturated Cluster Model for Covalent Solids
• Saturating the radical-like dangling bonds at the edge of the cut-outs by using suitable saturators (e.g. H or other pseudoatoms).
• Widely employed in the study of covalent solid surfaces, e.g., Silicon, Diamond, Zeolite and so on.
• Examples shown below include Chemical Reactions on Silicon Surfaces.
Atomic arrangements of a) X(100)-21 (X= Si, Ge) and b) Si(111)-77 reconstructed surfaces.

buckling

Reconstruction of X(100) X= C, Si, Ge

In the solid state, each atom adopts sp3 hybridization and tetrahedral coordination.

Controversy on the Mechanism
• p-complex mechanism:
• FTIR spectra of dideuterioethylene/Si(100) suggested that the adsorption is stereospecific and stereoselective. (Liu et al., J. Am. Chem. Soc., 1997, 119, 7593.)
• STM images of 2-butene/Si(100) indicates the adsorption is not stereospecific, thought with a high stereoselectivity of 98%. (Lopinski et al., J. Am. Chem. Soc., 2000, 122, 3548.)
C4H4X(X=S,O) on Si(100)-2x1 surface

X. Lu et al, J. Phys. Chem. B, 105(2001) 10069.

Example: HN3 reaction with C(100)-2x1

X. Lu et al., Chem. Phys. Lett. 343(2001) 212.

X. Lu et al., 1) J. Org. Chem. 67(2002) 515; 2) J. Phys. Chem. B, 106(2002) in press.

Example: NH3 on Si(111)-7x7

X. Lu et al, Chem. Phys. Lett. 355(2002) 365.

Organic functionalization of Si(111)

(X. Lu et al, J. Am. Chem. Soc. 2003, 125, 7923)

Is the direct [4+2] pathway realistic? No!!!!

The key point P4 on this pathway is indeed diradicaloid! Its UB3LYP wavefunction is 3.4 kcal/mol more stable than the RB3LYP one!!!

Is the direct [4+2] pathway realistic? No!!!!

The key point P4b on this pathway is indeed diradicaloid! Its UB3LYP wavefunction is more stable than the RB3LYP one!!!

Model System = A + H

Real System = A + B

outer layer

X (set 4)

inner layer

A(set 1)

H (set 2)

B (set 3)

3.7 ONIOM Model

EONIOM= Ehow(A+H) – Elow(A+H) + Elow(A+B)

(K. Morokuma et al., J. Mol. Struct. (Theochem) 461-462(1999) 1.)

Adsorption of Methanol, Formaldehyde and Formic Acid on Si(100)-21 Surface

( see X. Lu et al., Phys. Chem. Chem. Phys. 3(2001) 2156.)

Methanol

[email protected]

CCSD(T):B3LYP

Sidewall functionalization by F and H (Bauschilicher, Chem. Phys. Lett. 322(2000) 237.)

a) SWNT(10,0)片断

b) ONIOM中最内层的C24簇

ONIOM(B3LYP:UFF)

Results:

• F atoms appear to favor bonding next to existing F atoms.
• Hydrogenation of the sidewall of SWNT is probably endothermic.
Sidewall Functionalization of SWNT by1,3-Dipolar Cycloadditions

ONIOM(B3LYP/6-31G*:AM1)

1,3-DC of nitrile ylide with an olefin

Predicted Reaction Energies (kcal/mol)

1

3

2

3

SWNT(5,5) 片断

X. Lu et al., 1) J. Phys. Chem. B, 106(2002), 2136; 2) J. Am. Chem. Soc., 2003, 125, 10459-10464.

3.8 Cluster modeling of electrodes
• Charged cluster: [Cluster]
• Cluster in electric field.
• More realistic models are required.

Liao, M. et al. Int. J. Quant. Chem., 67(1998), 175.

Concluding Remarks
• Methods of simulation vary with and depend largely on the solids to be concerned.
• A simulation process is meaningless itself, unless certain physical criteria have been introduced to guarantee the consistence between the physical model and the real physical system.
• More significant is the scientific problem to be concerned.
• Simulation can be found everywhere nowadays.