Correlated maximum probability domains in
crystals
Mauro Causa’ Andreas Savin
Universita’ di Napoli “Federico II”
Universitè “Pierre et Marie Curie”, LCT-CNRS, Paris
Vallico, july 2013
In so far quantum mechanics is correct, chemical questions are
problems in “applied” mathematics (Eyring, Walter, Kimball, 1944)
O
C
Chemical “observable”
Observable ρ (r)
Chemists like a local topology
Typical questions:
Atomic charges ?
Oxidation numbers ?
Bonds (order)?
Is more covalent or ionic?
AIM
Atoms in
Molecule
(Bader)
ELF
Electron
Localization
Function
(Becke)
MPD
Maximum
Probability
Domains
(Savin)
Atoms in Molecules R.Bader, 1960->
ρ(r) , number electronic density

ρ
r
dr
=

N

Ω

Ω
Bader, R. F. W. “Atoms in molecules: a quantum theory” Oxford
University Press: Oxford,1990
Analysis of the ρ(r) scalar field
Electron Localization Function (ELF)
1.
(Local (Ek) cost for respecting Pauli principle)normalized
Elf region definition
Topological analysis and bifurcation
TiO2: ionic(0.6) and covalent(0.3) elf
Probability functions
p2(bond)=0.3-0.5
Pn(core)=0.7-1.0
Probability and number density

ρ
r
dr
=

N

Ω

Ω
Maximum Probability Domains (MPD)
unique definition of a domain
molecules
MPD for solids
Crystalline solids only shows extreme and
peculiar chemical bonds
(ionic, metallic, covalent)
Based ol Localized Orbitals (for insulators)
For delocalized system->QMC
Ionic crystals
MC, AS JPC 115 2011 13139
Anions
Covalent crystals
Voronoi
ELF
MC, AS ZAAC 637 2011 882
MPD
ELF-MPD
Semi-ionic crystals: α-SiO2
P(cov)=0.38;
p(O-)=0.37,p(O2-)=0.58
MPD-QMC(CASINO)
25 crystals in test
•Covalent
•Ionic
•Semi-ionic
•Metals
•Rare gases
•Easy work on configuration file for
VMC and DMC
•Configurations are distributed like
•|Ψvmc|2 |Ψdmc Ψvmc |
•Try to reducing numerical noise of
Shape Derivative
MPD-QMC preliminary results (VMC)
•For covalent and ionic crystal the shape of the domain
• doesn’t change (accuracy 0.1%)
•In covalent systems the core and bond probability increases:
•P2(bond,Si): 0.42 ---> 0.44
•P10(core,Si): 0.69 ---> 0.73
•In ionic crystals the Ionic probability seems unaffected by
electronic correlation
Next steps
Complete the data collection on all the 25 test
systems
Metals !!!!!! (QMC only)
DMC
Better accuracy (number of configurations)
Multiple domain probabilities (QMC only)