Pratical course – Theoretical Chemistry
CO on Palladium adsorption energies vibrations diffusion path
Dr. Carine MICHEL
[email protected]
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Introduction This hands-on session aims at providing you a first experience in simulating a metallic catalyst (here Pd) and its interaction with a molecule (here CO). The calculations will be done at the Density Functional Theory (DFT) level using the VASP package (http://www.vasp.at). In this code, the wave function is expanded on a plane wave basis set leading to 3D-periodic simulations. The visualization will be performed using VMD (http://www.ks.uiuc.edu/Research/vmd/). Editing and modifying a text file can be done using GEDIT. The operating system of the computers is linux-based. This text assumes that the participants are beginners and relies 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 as much as possible the computational cost, not to provide accurate results.
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1 1.1
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Bulk Atomic positions – The POSCAR file
Pd crystalizes in the face centered cubic (fcc) lattice. The corresponding conventional 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 in the bulk directory. You can open it using GEDIT clicking on the file. You can also visualize the corresponding structure using VMD (see the Survivor Guide) clicking on poscar.vmd.
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POSCAR Pd bulk 3.89 1.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 4 Direct 0.0 0.0 0.0 0.5 0.5 0.0 0.5 0.0 0.5 0.0 0.5 0.5
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Comments Title line ˚ Cell parameter in A(scaling factor applied to the cell vectors) A vector B vector C vector Number of atoms in the cell Type of coordinates (direct or cartesian) Pd1 Pd2 Pd3 Pd4
Atom kind – The POTCAR file
With a plane wave basis set, one has to use a pseudo-potential to replace the core electrons. The related information is in the POTCAR file. Open it and report here the following information: Atom name Atomic number Electronic configuration Minimal energy cutoff ENMIN Maximum energy cutoff ENMAX The DFT functional
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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 Pd bulk PREC = Normal ALGO = Fast ENCUT = 200 IBRION = -1 ISMEAR = -5
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Comments Title line Global keyword that controls the numerical precision Type of algorithm to converge the electronic energy Energy cut off on the plane wave basis set Compute the energy of the system Smearing method
K-points grid – The KPOINTS file
The mesh of K-points is used to integrate over the 3D-Brillouin zone. It is defined by the KPOINTS file. KPOINTS Automatic Mesh 0 Gamma Monkhorst Pack 11 11 11 000
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Comments Title line Automatic generation scheme Generate a Γ-centered mesh Scheme used to generate the mesh Number of K-points in each direction Optional shift of the mesh
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 end of the job. We want to know the energy of the system that can be found in several files. Open the OSZICAR file. Report here the energy: E0 =
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Optimizing the cell parameter
We want to find the cell parameter that minimizes the energy. Redo the previous calculation changing the cell parameter in the POSCAR file. You can share the calculations with other participants. ˚ Cell parameter a (A) 3.89 ... ... ... ... ... ... ...
Energy (E0 = ) (eV)
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 corresponding energy? How does this compare with the experimental cell parameter?
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Influence of the energy cutoff
Now, we want to see the importance of an appropriate energy cutoff on the plane wave basis set. We need to redo the same set of calculations for various values of the energy cut off ENCUT. The successive steps (changing the lattice parameter, click on vasp.j, report the energy found in the OSZICAR file) have been automatized in a script called bulk.j. Clicking on this file will generate a table in the file bulk.dat that collects the cell parameter and the corresponding energy. Modify the INCAR file to have the appropriate ENCUT value, run bulk.j, open bulk.dat and report the 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 converged to 10 meV/atom: ENCUT = . . .
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Influence of the k-point mesh
In addition to the energy cutoff, an other important parameter is the quality of the K-points mesh. It is defined in the KPOINTS file. You should test this with a sufficient energy cutoff ENCUT (see the previous paragraph). What is the minimal K-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 energy cutoff
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The best set up
To conclude this section, what is the best set up? and what is the cell parameter and the corresponding energy using it?
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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 slab of 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 adsorption of CO on each surface (see Paragraph 4 page 16). Then, we need larger cells to screen 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 this cell 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 surface model consists in a slab of a given thickness, 2D periodic but repeated also periodically in the third direction (perpendicularly to the surface plane). The C vector has to be chosen large enough to avoid interactions between the periodic images of the slab in this direction. See Figure 3.
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Figure 3: Side view of the Pd(111) slab. The three layers are represented in red/blue/white. The p(2x2) cell is repeated in U,V,z directions.
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The (001) surface Energy calculation
To start, we will compute the energy of a (001) surface as cut from the bulk. Here again, we need to use four input files. The POSCAR has been modified to provide the corresponding vectors and coordinates. We use the optimal Pd-Pd distance corresponding to the optimal cell parameter obtained in the previous section on bulk calculations. In addition, we will freeze the two bottom layers and optimize the positions of the two up layers using VASP. The POTCAR is unchanged. The INCAR has been adapted to slab calculation, changing the smearing method. Extra keywords are also added in this input file to perform a geometry optimization. More details can be found on the VASP website (https://www.vasp.at/). Last, in the direction perpendicular to the slab, we don’t need k-points. Thus, the KPOINTS file is modified to reduce the k-points mesh at the Gamma point in this direction. It is also adapted to the size of the supercell p(2x2). See next page.
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POSCAR Pd(001) 2.80 1.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 7.071067811 4 Selective dynamics Direct 0.00 0.00 0.0 F F F 0.50 0.00 0.0 F F F 0.00 0.50 0.0 F F F 0.50 0.50 0.0 F F F 0.25 0.25 0.1 F F F 0.75 0.25 0.1 F F F 0.25 0.75 0.1 F F F 0.75 0.75 0.1 F F F 0.00 0.00 0.2 T T T 0.50 0.00 0.2 T T T 0.00 0.50 0.2 T T T 0.50 0.50 0.2 T T T INCAR Geometry Optimisation Electronic minimization PREC = Normal ALGO = Fast ENCUT = 400 LREAL = Auto EDIFF = 1e-6 Ionic relaxation NSW = 50 IBRION = 2 POTIM = 0.5 EDIFFG = -0.01 DOS related values ISMEAR = 2 KPOINTS Automatic Mesh 0 Monkhorst Pack 551 000
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˚ (scaling factor) Pd-Pd distance in A U vector V vector C vector, 10 times the interlayer distance Only some coordinates will be optimized (T flags) Layer 1, x frozen, y frozen, z frozen Layer 1, x frozen, y frozen, z frozen Layer 1, x frozen, y frozen, z frozen Layer 1, x frozen, y frozen, z frozen Layer 2, x frozen, y frozen, z frozen Layer 2, x frozen, y frozen, z frozen Layer 2, x frozen, y frozen, z frozen Layer 2, x frozen, y frozen, z frozen Layer 3, x free, y free, z free Layer 3, x free, y free, z free Layer 3, x free, y free, z free Layer 3, x free, y free, z free
Recommended for large systems Precision on the wavefunction Maximum number of optimization steps Geometry optimisation algorithm Step size Convergence criterium on the gradient Smearing method, different from the bulk one
12 At the Gamma-point in the third direction
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Run VASP clicking on vasp.j. Open the OSZICAR file. You can observe that you have performed several steps of optimization. Report the corresponding energies. E0 = E0 = E0 = ... The first one corresponds to the initial structure, without any relaxation. The last one 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 possible one. The final geometry is reported in the CONTCAR file. You can visualize them with VMD and look at the Pd-Pd distance between the upper layers. You can also easily compute the interlayer distance using the CONTCAR file. How does it evolve during the geometry optimization?
2.2
The (111) surface
In the slab111 directory, you will find the input files for VASP. Redo the same analysis than for the Pd(001) surface and compare the two facets.
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CO molecule
3.1
Optimal distance and energy
First, we need to know the equilibrium CO distance and the corresponding energy of 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 periodic ˚ 3 and do the computation code. Thus, we have to insert CO in a large box of 10A at the Γ-point. The POTCAR concatenates the information concerning the C atom and then the O atom. Files are provided in the CO_gasphase directory. POSCAR CO molecule alone 1.0 10.0 0.0 0.0 0.0 10.0 0.0 0.0 0.0 10.0 11 Cartesian 5.0 5.0 5.0 6.2 5.0 5.0 KPOINTS Automatic Mesh 0 Monkhorst Pack 111 000
scaling factor A vector B vector C vector 1 carbon, 1 oxygen (the order is defined by the POTCAR) Cartesian coordinates C atom at the center of the box O atom, CO along the x axis
At the Gamma-point in the three directions
Report here the energy and the CO distance in the optimal geometry. The energy will be used later to compute adsorption energies and the CO distance will be a good indicator of the back donation from the Pd d orbitals to the CO vacant π ∗.
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Vibration frequency
We want to follow the evolution of the CO vibration frequency depending on the facet and the adsorption site. To start, we compute here the vibration frequency of CO vibration in vacuum in the CO_freq directory. We have to restart from the optimal geometry, provided in the POSCAR file. The INCAR is modified to do a frequency calculation. INCAR Freq PREC = Normal ALGO = Fast ENCUT = 400 IBRION = 5 POTIM = 0.02 NFREE = 2 NSW = 50 ISMEAR = 2
Frequency calculation Step used for the frequency calculations Number of points used for the frequency calculations
Click on vasp.j. The frequency is provided in the freq.dat file: f
=
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CO@Pd CO adsorption
For each fact (Pd(001) and Pd(111)), several sites have to be considered (see Figure 4) : • 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 possible site. 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 between the surface and the molecule: Eads = ECO@slab − ECO − Eslab
(5)
What is the most stable configuration on each facet? Can you comment the CO distance evolution?
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ECO@slab
ECO
Eslab
Pd(001) top bridge hollow
Pd(111) top bridge hcp fcc
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Eads (eV)
˚ CO distance (A)
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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 minimum of the potential energy surface (PES), all the frequencies are real. The highest wave number corresponds to the CO vibration. If the structure is a maximum of the PES, all the frequencies are imaginary. If the structure is a transition state, one and 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 PROCAR file in the DOS directory. At each K-point, each band (=molecular orbital) is decomposed in term of atomic orbitals. C is the first atom, O the second one. Try
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to complete the CO molecular orbitals diagram with the energies and the sketch of the MOs.
E
To know how those orbitals are modified upon adsorption, we consider now the density 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 is the energy in eV. The second one is the PDOS on the MO3 of CO. The third one os the 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
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CO diffusion on Pd(001)
We will focus here on the CO diffusion on the Pd(001) surface as example of reactions paths. Three paths are possible (see Figure 5). You will find the results in the corresponding directories slab001_CO_PATH. Path 3
Path 1 Path 2
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 images) using the Nudge Elastic Band method (NEB). The results are in the NEB subdirectory. The structures are reported in the XDATCAR and the energies in the energy profile.dat file. Plot the energy in function of geometrical parameters such as a Pd-C distance or the C-O distance. What is the structure closest to the transition state?
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TS
Starting from the structure the closest to the TS, we have optimize it. Looking at the 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 frequencies have been computed (see TS_FREQ). Report the imaginary frequencies. Conclude.
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4.4.3 Path
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Comparison Energy Profile
TS Energy
Activation Barrier
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Imaginary Frequencies
