![Page 1: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/1.jpg)
ab initio ThermochemistryRalf Stoffel
IAC, RWTH Aachen
First principles simulations and standard thermochemical data
Application to the High Temperature Superconductor YBa2Cu3O6+x
![Page 2: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/2.jpg)
Starting from the beginning: ab initio
Aim of Quantum Chemistry: Solution of Schrödinger´s Equation
for molecular and crystalline systemsCannot be solved exactly for systems with more than one electron
approximations (e.g. Hartree‐Fock, DFT)
Solid state quantum chemistry:
Density functional theory (DFT)electronic band structures and energies, crystal structures,
forces etc.
Ψ=Ψ EH
![Page 3: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/3.jpg)
Theoretical Calculation of Enthalpies
Standard density‐functional total‐energy calculations give access to the total electronic ground state energy at 0 K, Etot ≈ UE–V diagrams ΔH(p)
Example:Platinum Nitride
stability of solid compounds / different structure typesprediction of high‐pressure phasesup to this point: Temperature neglected
⇒⋅+=⇒∂∂
−=⇒ VpEHVEp
![Page 4: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/4.jpg)
How to deal with temperature?
Main effect of temperature:VibrationsOther contributions: electronic excitations, configurational entropy
ab initiomethods involving Temperature:• Molecular Dynamics (MD), Monte Carlo (MC),• Simulation of lattice vibrations (Phonons), assuming harmonic
oscillator scenario‐ Mostly used: Direct Method ‐ Programs: PHONON, FROPHO (in combination with
electronic structure programs, e.g. VASP)
![Page 5: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/5.jpg)
Gibbs free energy from lattice vibrations
Calculations are carried out at const. Volume (relaxed structure)
Fph (T,V): phonon free energy (harmonic vibrational energy)
we only need to know the phonon frequencies ω
),(),(),()(),(
),(),(),(
ahvibidel
oncontributimain
phtot VTFFVTFVTFVEVTF
VVFVTFpVVTFTpG
T
++++=
⋅⎟⎠⎞
⎜⎝⎛
∂∂
−=+=
444 3444 21
![Page 6: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/6.jpg)
Phonon frequencies / Direct Method
supercell method (> 50 atoms, a > 8 Å)
Calculation of the Hellmann‐Feynman‐forces Fα acting on each atom when the other atoms are slightly dislocated (harmonic oscillator)
Vibrational frequencies ω (k) => Fph(T,V)
![Page 7: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/7.jpg)
Obtaining the Gibbs Free Energy(Example: Barium Oxide)
Our calculations result in F (T) at different V
⇒ F (V) at different T ⇒G (p,T)
![Page 8: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/8.jpg)
Handling Pressure & Temperature Example:
Tantalum oxide nitride:
β → γ Phase transition at
higher temperature and
ambient pressure
Vm(γ) > Vm(β)
⇒ No pressure‐induced
transition
Temperaturemust be
explicitly included!
![Page 9: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/9.jpg)
Standardized thermochemical data
Numerical calculation of H, S and Cp from the ab initio Gibbs Function by the well known relations:
Fitting to parametrized Equation (Cp => S => H => G)
Gibbs function is not related to the elements but to „absolute“ zero energy ⇒ pure ab initio treatment: no problem with ΔG
⇒ mixing with exp. data: values must be related to the elements
pp
p THCSTGH
TGS ⎟
⎠⎞
⎜⎝⎛
∂∂
=⋅+=⎟⎠⎞
⎜⎝⎛
∂∂
−=
TTTTTTG 63
52
4321CCClnCCC +++++=
![Page 10: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/10.jpg)
Cooperation with
Project ELSA: Computational modelling and preparation of thehigh temperature superconductor YBa2Cu3O6+x
Our „job“: Prediction of thermochemical potentials wherethere is no experimental data available
First step (done): Comparing properties of simple compounds with known thermochemical data (binary systems, e.g. CuO, BaO)
Recent work: System Y‐Ba‐Cu‐O,Handling of gaseous compounds (oxygen),ΔG (p,T), Phase transitions
![Page 11: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/11.jpg)
System Ba‐Y‐O
Binary oxides BaO and Y2O3: well known compounds, reliable thermochemical data available
Ternary Oxides BaY2O4 and Ba3Y4O9: wide range of different experimental data
Example: Heat of formation (ΔHox) of BaY2O4 from the oxides
Lit: −113 to +0.4 kJ/molGTT: −70.788 kJ/molAb initio: −10.1736 kJ/mol
![Page 12: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/12.jpg)
System Ba‐Y‐O
Heat Capacity and Entropy of BaY2O4 compared with the data base:
Heat Capacity not linear, Entropy slightly overestimated
![Page 13: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/13.jpg)
System Ba‐Y‐O
Heat Capacity and Entropy of Ba3Y4O9 compared with the data base:
Heat Capacity underestimated,
Entropy also slightly underestimated at higher Temperatures
![Page 14: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/14.jpg)
Gibbs Free Energy of Formation (ΔGR)Free energy of formation of YBa2Cu3O6.5 from the Oxides
½ Y2O3 + 2 BaO +3 CuO → YBa2Cu3O6.5
• ΔGR shows correct tendency
•Wide range in exp. function
![Page 15: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/15.jpg)
T‐x phase diagram of YBa2Cu3O6+x
3 possible crystal structures depending on oxygen site fractions in the CuOx‐planes
tetragonal: oxygen on every site (statistically)
orthorhombic I: oxygen on two opposite sites (e.g. 1 and 3)
orthorhombic II: oxygen only on one site
Site fraction depends on Temperature T and oxygen content x.
![Page 16: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/16.jpg)
T‐x phase diagram of YBa2Cu3O6+x
Calculation of the T‐x‐phase diagram using a sublattice model with four different sites in the CuOx‐planes
All other sites (Y, Ba, Cu, O) are fixed
⇒ 16 different functions, from which some are degenerated
⇒ finally 6 functions
corresponding to x = 0, 0.5, 1 (2 different), 1.5, 2
![Page 17: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/17.jpg)
Oxygen site fraction YBa2Cu3O6.5
Site fraction of oxygen atoms in YBa2Cu3O6.5 calculated by Klaus Hack (GTT) from the data base
![Page 18: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/18.jpg)
Phase transition of YBa2Cu3O6.5
Handling the phase transition with ab initio data:
Simple model: 1x2x1 unit cell
![Page 19: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/19.jpg)
Phase transition of YBa2Cu3O6.5
Configurational entropy Sconf has to be taken into account
⇒ Good estimation of the phase transition temperature
tetr
ortho II
ortho I
![Page 20: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/20.jpg)
Summary
‐ Standard ab initio calculations: many properties available from Density Functional total energy calculations
‐ Involving Temperature by the simulation of lattice vibrations
‐ Good reproduction of temperature dependent potentials like the Heat Capacity and the Entropy
‐ Prediction of Temperature‐ and Pressure‐induced phase transitions possible via ΔG
![Page 21: ab initio Thermochemistry - GTT-Technologies · ‐Standard ab initio calculations: many properties available from Density Functional total energy calculations ‐Involving Temperature](https://reader031.vdocument.in/reader031/viewer/2022020922/5ea3b1bba0d15c5a031b315b/html5/thumbnails/21.jpg)
www.ssc.rwth‐aachen.de
Further information: