Pratical course – Theoretical Chemistry

CO on Palladium

adsorption energiesvibrationsdiffusion path

Dr. Carine MICHELcarine.michel@ens-lyon.fr

C. MICHEL PRATICAL COURSE

Introduction

This hands-on session aims at providing you a first experience in simulating ametallic catalyst (here Pd) and its interaction with a molecule (here CO). The calcu-lations will be done at the Density Functional Theory (DFT) level using the VASPpackage (http://www.vasp.at). In this code, the wave function is expanded on aplane wave basis set leading to 3D-periodic simulations. The visualization will beperformed using VMD (http://www.ks.uiuc.edu/Research/vmd/). Editing andmodifying a text file can be done using GEDIT. The operating system of the com-puters is linux-based. This text assumes that the participants are beginners and re-lies on a short manual, ’The Survivor Guide’, to connect, get started and use VMD,GEDIT, etc.

WARNING: This is a practical course. Parameters have been chosen in order to reduce asmuch as possible the computational cost, not to provide accurate results.

2

C. MICHEL PRATICAL COURSE

1 Bulk

1.1 Atomic positions – The POSCAR file

Pd crystalizes in the face centered cubic (fcc) lattice. The corresponding conven-tional cell contains 4 atoms and is represented in Figure 1.

Figure 1: Conventional cell of the fcc lattice. This cube of side a contains 4 atoms:one at the corner, and three face-centered.

The cell vectors can be written:

A = (a 0 0)B = (0 a 0)C = (0 0 a)

And using those three vectors as a basis, one can write the coordinates of the atoms:

Pd1 = (0.0 0.0 0.0) (1)Pd2 = (0.5 0.5 0.0) (2)Pd3 = (0.5 0.0 0.5) (3)Pd4 = (0.0 0.5 0.5) (4)

Those structural informations are collected in the POSCAR file. It is provided inthe bulk directory. You can open it using GEDIT clicking on the file. You can alsovisualize the corresponding structure using VMD (see the Survivor Guide) clickingon poscar.vmd.

3

C. MICHEL PRATICAL COURSE

POSCAR CommentsPd bulk Title line3.89 Cell parameter in A(scaling factor applied to the cell vectors)1.0 0.0 0.0 A vector0.0 1.0 0.0 B vector0.0 0.0 1.0 C vector4 Number of atoms in the cellDirect Type of coordinates (direct or cartesian)0.0 0.0 0.0 Pd10.5 0.5 0.0 Pd20.5 0.0 0.5 Pd30.0 0.5 0.5 Pd4

1.2 Atom kind – The POTCAR file

With a plane wave basis set, one has to use a pseudo-potential to replace the coreelectrons. The related information is in the POTCAR file. Open it and report herethe following information:

Atom name

Atomic number

Electronic configuration

Minimal energy cutoff ENMIN

Maximum energy cutoff ENMAX

The DFT functional

4

C. MICHEL PRATICAL COURSE

1.3 Calculation parameters – The INCAR file

This file contains a list of keywords that controls the way the calculation is done(numerical precision, choice of the DFT functional, type of calculation, etc.)

INCAR CommentsPd bulk Title linePREC = Normal Global keyword that controls the numerical precisionALGO = Fast Type of algorithm to converge the electronic energyENCUT = 200 Energy cut off on the plane wave basis setIBRION = -1 Compute the energy of the systemISMEAR = -5 Smearing method

1.4 K-points grid – The KPOINTS file

The mesh of K-points is used to integrate over the 3D-Brillouin zone. It is definedby the KPOINTS file.

KPOINTS CommentsAutomatic Mesh Title line0 Automatic generation schemeGamma Generate a Γ-centered meshMonkhorst Pack Scheme used to generate the mesh11 11 11 Number of K-points in each direction0 0 0 Optional shift of the mesh

1.5 First calculation

We are now ready to perform our first calculation. Click on vasp.j to run VASP.VASP generates many files by default. Some are deleted automatically at the endof the job. We want to know the energy of the system that can be found in severalfiles. Open the OSZICAR file. Report here the energy:

E0 =

5

C. MICHEL PRATICAL COURSE

1.6 Optimizing the cell parameter

We want to find the cell parameter that minimizes the energy. Redo the previouscalculation changing the cell parameter in the POSCAR file. You can share thecalculations with other participants.

Cell parameter a (A) Energy (E0 = ) (eV)3.89. . .. . .. . .. . .. . .. . .. . .

Table 1: Energy in function on the cell parameter for standard set up

What is the cell parameter that minimizes the energy? What is the correspondingenergy? How does this compare with the experimental cell parameter?

6

C. MICHEL PRATICAL COURSE

1.7 Influence of the energy cutoff

Now, we want to see the importance of an appropriate energy cutoff on the planewave basis set. We need to redo the same set of calculations for various values of theenergy cut off ENCUT. The successive steps (changing the lattice parameter, clickon vasp.j, report the energy found in the OSZICAR file) have been automatized ina script called bulk.j. Clicking on this file will generate a table in the file bulk.datthat collects the cell parameter and the corresponding energy. Modify the INCARfile to have the appropriate ENCUT value, run bulk.j, open bulk.dat and reportthe optimal cell parameter and the corresponding energy in Table 2.

ENCUT (eV) Optimal cell parameter Energy (E0 =) (eV)

ENMIN

200

ENMAX

400

500

Table 2: Influence of the plane wave basis set (ENCUT) on the bulk calculation

For the following, we will choose the ENCUT value to ensure an energy convergedto 10 meV/atom: ENCUT = . . .

7

C. MICHEL PRATICAL COURSE

1.8 Influence of the k-point mesh

In addition to the energy cutoff, an other important parameter is the quality of theK-points mesh. It is defined in the KPOINTS file. You should test this with asufficient energy cutoff ENCUT (see the previous paragraph). What is the minimalK-point mesh to ensure an energy converged to 10 meV/atom ?

KPOINTS Optimal cell parameter Energy (E0 =) (eV)

7x7x7

9x9x9

11x11x11

13x13x13

Table 3: Influence of the K-points mesh on the bulk calculation at a given energycutoff

1.9 The best set up

To conclude this section, what is the best set up? and what is the cell parameter andthe corresponding energy using it?

8

C. MICHEL PRATICAL COURSE

2 Surface

Let’s now get closer to a catalyst, cleaving the bulk along the (001) direction and the(111) and compare the two surfaces we obtain. The surfaces are modeled by a slabof finite thickness. Those two surfaces can be described using a small primitive cell(u, v) containing only atom in each layer. However, we want to study the adsorp-tion of CO on each surface (see Paragraph 4 page 16). Then, we need larger cells toscreen coverage lower than 1ML. For instance, a p(2x2) corresponds to a supercell(U, V) that is twice the primitive cell in the two directions (U = 2 × u, V = 2 × v).This supercell contains 4 Pd atoms per layer and the introduction of one CO in thiscell leads to a coverage of 1/4ML. See Figure 2.

(a) (b)

Figure 2: Top view of the (a) Pd(001), p(2x2) cell (b) Pd(111), p(2x2) cell.

Slabs are intrinsically 2D-system but VASP is a 3D-periodic code. So, our surfacemodel consists in a slab of a given thickness, 2D periodic but repeated also period-ically in the third direction (perpendicularly to the surface plane). The C vector hasto be chosen large enough to avoid interactions between the periodic images of theslab in this direction. See Figure 3.

9

C. MICHEL PRATICAL COURSE

Figure 3: Side view of the Pd(111) slab. The three layers are represented inred/blue/white. The p(2x2) cell is repeated in U,V,z directions.

10

C. MICHEL PRATICAL COURSE

2.1 The (001) surface

2.1.1 Energy calculation

To start, we will compute the energy of a (001) surface as cut from the bulk. Hereagain, we need to use four input files. The POSCAR has been modified to providethe corresponding vectors and coordinates. We use the optimal Pd-Pd distance cor-responding to the optimal cell parameter obtained in the previous section on bulkcalculations. In addition, we will freeze the two bottom layers and optimize the po-sitions of the two up layers using VASP. The POTCAR is unchanged. The INCARhas been adapted to slab calculation, changing the smearing method. Extra key-words are also added in this input file to perform a geometry optimization. Moredetails can be found on the VASP website (https://www.vasp.at/). Last, in thedirection perpendicular to the slab, we don’t need k-points. Thus, the KPOINTSfile is modified to reduce the k-points mesh at the Gamma point in this direction. Itis also adapted to the size of the supercell p(2x2). See next page.

11

C. MICHEL PRATICAL COURSE

POSCARPd(001)2.80 Pd-Pd distance in A (scaling factor)1.0 0.0 0.0 U vector0.0 1.0 0.0 V vector0.0 0.0 7.071067811 C vector, 10 times the interlayer distance4Selective dynamics Only some coordinates will be optimized (T flags)Direct0.00 0.00 0.0 F F F Layer 1, x frozen, y frozen, z frozen0.50 0.00 0.0 F F F Layer 1, x frozen, y frozen, z frozen0.00 0.50 0.0 F F F Layer 1, x frozen, y frozen, z frozen0.50 0.50 0.0 F F F Layer 1, x frozen, y frozen, z frozen0.25 0.25 0.1 F F F Layer 2, x frozen, y frozen, z frozen0.75 0.25 0.1 F F F Layer 2, x frozen, y frozen, z frozen0.25 0.75 0.1 F F F Layer 2, x frozen, y frozen, z frozen0.75 0.75 0.1 F F F Layer 2, x frozen, y frozen, z frozen0.00 0.00 0.2 T T T Layer 3, x free, y free, z free0.50 0.00 0.2 T T T Layer 3, x free, y free, z free0.00 0.50 0.2 T T T Layer 3, x free, y free, z free0.50 0.50 0.2 T T T Layer 3, x free, y free, z free

INCARGeometry OptimisationElectronic minimizationPREC = NormalALGO = FastENCUT = 400LREAL = Auto Recommended for large systemsEDIFF = 1e-6 Precision on the wavefunctionIonic relaxationNSW = 50 Maximum number of optimization stepsIBRION = 2 Geometry optimisation algorithmPOTIM = 0.5 Step sizeEDIFFG = -0.01 Convergence criterium on the gradientDOS related valuesISMEAR = 2 Smearing method, different from the bulk one

KPOINTSAutomatic Mesh0Monkhorst Pack5 5 1 At the Gamma-point in the third direction0 0 0

12

C. MICHEL PRATICAL COURSE

Run VASP clicking on vasp.j. Open the OSZICAR file. You can observe that youhave performed several steps of optimization. Report the corresponding energies.

E0 =

E0 =

E0 =

...

The first one corresponds to the initial structure, without any relaxation. The lastone corresponds to the optimized structure. Other output files are of great interest.The XDATCAR contains all the geometries that VASP has tried to find the best pos-sible one. The final geometry is reported in the CONTCAR file. You can visualizethem with VMD and look at the Pd-Pd distance between the upper layers. You canalso easily compute the interlayer distance using the CONTCAR file. How does itevolve during the geometry optimization?

2.2 The (111) surface

In the slab111 directory, you will find the input files for VASP. Redo the sameanalysis than for the Pd(001) surface and compare the two facets.

13

C. MICHEL PRATICAL COURSE

3 CO molecule

3.1 Optimal distance and energy

First, we need to know the equilibrium CO distance and the corresponding energyof CO in gas phase. We have to adapt the POSCAR, POTCAR and KPOINTS files.The INCAR is unchanged. Even if a CO molecule is aperiodic, VASP is a periodiccode. Thus, we have to insert CO in a large box of 10A3 and do the computationat the Γ-point. The POTCAR concatenates the information concerning the C atomand then the O atom. Files are provided in the CO_gasphase directory.

POSCARCO molecule alone1.0 scaling factor10.0 0.0 0.0 A vector0.0 10.0 0.0 B vector0.0 0.0 10.0 C vector1 1 1 carbon, 1 oxygen (the order is defined by the POTCAR)Cartesian Cartesian coordinates5.0 5.0 5.0 C atom at the center of the box6.2 5.0 5.0 O atom, CO along the x axisKPOINTSAutomatic Mesh0Monkhorst Pack1 1 1 At the Gamma-point in the three directions0 0 0

Report here the energy and the CO distance in the optimal geometry. The energywill be used later to compute adsorption energies and the CO distance will be agood indicator of the back donation from the Pd d orbitals to the CO vacant π∗.

14

C. MICHEL PRATICAL COURSE

3.2 Vibration frequency

We want to follow the evolution of the CO vibration frequency depending on thefacet and the adsorption site. To start, we compute here the vibration frequencyof CO vibration in vacuum in the CO_freq directory. We have to restart from theoptimal geometry, provided in the POSCAR file. The INCAR is modified to do afrequency calculation.

INCARFreqPREC = NormalALGO = FastENCUT = 400IBRION = 5 Frequency calculationPOTIM = 0.02 Step used for the frequency calculationsNFREE = 2 Number of points used for the frequency calculationsNSW = 50ISMEAR = 2

Click on vasp.j. The frequency is provided in the freq.dat file:

f =

15

C. MICHEL PRATICAL COURSE

4 CO@Pd

4.1 CO adsorption

For each fact (Pd(001) and Pd(111)), several sites have to be considered (see Figure4) :

• top: the CO is sitting on a Pd atom

• bridge: the CO is bridging two Pd atoms

• hollow: the CO is coordinated to three (Pd(111)) or four (Pd(001)) Pd atoms.

(a) (b)

Figure 4: Top view of the (a) Pd(001): top (T), bridge (B) and hollow (h) sites (b)Pd(111): top (T), bridge (B) and hollow (fcc and hcp) sites

We have to optimize the geometry of the CO adsorbed on each facet in each possiblesite. It is a bit long. Thus, the corresponding results are provided in the slab111_

CO_OPT and slab001_CO_OPT.The adsorption energy Eads is defined as the energy gained by the interaction be-tween the surface and the molecule:

Eads = ECO@slab − ECO − Eslab (5)

What is the most stable configuration on each facet? Can you comment the COdistance evolution?

16

C. MICHEL PRATICAL COURSE

ECO@slab ECO Eslab Eads (eV) CO distance (A)

Pd(001)

top

bridge

hollow

Pd(111)

top

bridge

hcp

fcc

17

C. MICHEL PRATICAL COURSE

4.2 CO vibration

The frequencies corresponding to the optimal structures of CO adsorbed on Pd(111)and Pd(001) have been computed and collected in the slab111_CO_FREQ and slab001_

CO_FREQ directories. They are listed in the freq.dat file. If the structure is a mini-mum of the potential energy surface (PES), all the frequencies are real. The highestwave number corresponds to the CO vibration. If the structure is a maximum ofthe PES, all the frequencies are imaginary. If the structure is a transition state, oneand only one frequency is imaginary.

Eads (eV) Minimum? CO distance (A ) σ(CO) (cm−1)

Pd(001)

top

bridge

hollow

Pd(111)

top

bridge

hcp

fcc

Could we distinguish the adsorption sites using infrared spectroscopies?

4.3 PDOS

In this part, we will give a quick look at where are the electrons in CO@Pd(001),adsorbed in bridge position. The molecular orbitals of CO are given in the PRO-CAR file in the DOS directory. At each K-point, each band (=molecular orbital) isdecomposed in term of atomic orbitals. C is the first atom, O the second one. Try

18

C. MICHEL PRATICAL COURSE

to complete the CO molecular orbitals diagram with the energies and the sketch ofthe MOs.

E

To know how those orbitals are modified upon adsorption, we consider now thedensity of states (DOS) of the CO@Pd(001) projected on those molecular orbitals.The projected DOS (PDOS) are given in the file PDOS-3 7.dat. The first column isthe energy in eV. The second one is the PDOS on the MO3 of CO. The third one osthe PDOS on the MO4 of CO, etc. The fermi level of the CO@Pd(001) is -3.7965 eV.You can plot the PDOS using Gnumeric clicking on the file PDOS-3 7.gnumeric.You could pay attention to:

• the dispersion of each band (the ’energy width’)

• the split of some bands

• the shift in energy

• the occupation of molecular orbitals that were previously unoccupied

19

C. MICHEL PRATICAL COURSE

4.4 CO diffusion on Pd(001)

We will focus here on the CO diffusion on the Pd(001) surface as example of reac-tions paths. Three paths are possible (see Figure 5). You will find the results in thecorresponding directories slab001_CO_PATH.

Path 2

Path 1

Path 3

Figure 5: The three possible paths for CO diffusion on Pd(001)

4.4.1 NEB

For each reaction path, we have optimized structures along the path (called im-ages) using the Nudge Elastic Band method (NEB). The results are in the NEB sub-directory. The structures are reported in the XDATCAR and the energies in theenergy profile.dat file. Plot the energy in function of geometrical parameters suchas a Pd-C distance or the C-O distance. What is the structure closest to the transitionstate?

20

C. MICHEL PRATICAL COURSE

4.4.2 TS

Starting from the structure the closest to the TS, we have optimize it. Looking atthe results in TS_OPT sub-directory, report here the energy of this transition state.What is the energy barrier of this diffusion path?

E0 =

Activation energy =

To check if the structure corresponds at a saddle point of order one, the frequencieshave been computed (see TS_FREQ). Report the imaginary frequencies. Conclude.

21

C. MICHEL PRATICAL COURSE

4.4.3 Comparison

Path Energy Profile TS Energy Activation Barrier Imaginary Frequencies

22