Transcript Models of Chemical Potential and the Concept of Atomistic Embedding Steve Valone (Structure-Property Relations, LANL) LA-UR-09- The concept of embedding some fragment in a.

Models of Chemical Potential and
the Concept of Atomistic Embedding
Steve Valone (Structure-Property Relations, LANL)
LA-UR-09-
The concept of embedding some fragment in a larger system or reservoir appears
everywhere in our scientific deliberations. Examples cover such diverse situations as
an atom in a crystal, a functional group in a molecule, a molecular wire between a
source and sink, or a protein in aqueous solution. Virtually every atomistic potential
energy surface is built on one embedding strategy or another. This situation presents
a difficulty for systems where charges are being transferred among fragments.
Governance of transfer of charge is embodied in the concept of the chemical potential.
When an embedded fragment and a reservoir can be readily identified, transfer of
electrons between them can be described by an open ensemble. However, when
interactions are strong as happens in crystals and molecules, the ensemble picture is
not strictly applicable. Nevertheless, we still think in terms of open systems.
Customary atomistic models use a linear model of chemical potential that is highly
problematic. Here one strategy is presented to accommodate the open system point
of view, while overcoming several limitations. That strategy focuses on decomposition
into fragments of the many-body electronic hamiltonian itself, rather than decomposing
the ground-state energy. The strategy produces a new, nonlinear model of chemical
potential that has numerous ramifications for designing atomic potentials that can
account for charge transfer.
• Mix covalent and ionic wavefunctions [1,4]
+q -q
0
0
• Variational energy transcribed to
charges and eigenenergies
0.00
0.25
0.50
0.75
1.00
(q,qgs)(dimensionless)
0.8
 (q, d) 
0.0
0.0
0.2
0.4
0.6
0.8
1.0
• When q0 is zero, dependence is
linear in
• When q0 not zero, dependence is
quadratic near q0
• Transcribing between
wavefunction/density parameters and
properties
– Nonlinearities
– Branches
Charge Transfer Regulation
Test problem: A and B neutral, well separated, start to interact,
different ionization potentials, hardnesses
Only small amount of charge should transfer between fragments
-q
Pritchard & Sumner [1]
E    tr  H 
E0  inf E  :  K 
  1    N   N 1
 1     N  N    N 1  N 1
 Physically we know
that  = 1 for atoms:
Keep adding electrons until
they don’t bind any more
Nonanalytical minimization
E nN     inf E  :   1   nN   nN 1 

 inf 1    EN   EN 1 

Fractional charge implies quantum-mechanical averaging over two or more charge states
of an atom. When quantum mechanical averaging is properly applied, new functional
forms for the charge dependence emerge. How do we get between these two limits?
 A    , A c i  i C A
i
i
2
A
   , A c
i
i
   0      
2
i
ˆ 0
H
A


E    EA  1 2  VAB 

A 
B° A
, '
VAB   C AB ' C ABVAB
'   tr VABAB 
, '
qA 
ˆ 
H
A
• Energy
– Collect contributions by charge state
– Density matrix form for convenience
– Energy matrices have other charge dependencies
E A   C A ' C A H A' 
 tr H AA 
ˆ 
H
A
H A'    A' Hˆ A'  Hˆ A  A 2
C A2   C A2 –
• Actual charges
– A “property”, not unique
(A)
(B)
q Aq B q A C i q B C i
– Yet another collect of variables

d AB
d AB
– Not essential, but convenient
– Could go q -> C, instead of C -> q
 EA

E
V
AB
 
 12 

– Quantum charge evolution for q, instead
q
q
q


 
A
B° A
of Carr-Parrinello
  
– Not all definitions of charge related to CP
 H A
E A
A 
 tr 
A  H A
– New contributions

C A2 0  C A2   C A2 –
 d, q  q 
 (arb. units)
1  2q0 d   1  2q 
0.0
-0.5
q0 d 1  q0 d 
q 1  q 
q0 regulates amount of charge
transferred
q0
0.005
0.05
0.2
0.5
coulombic
perturbation
I-M model
Iczkowski-Margrave: linear
2q d   A   B  q A  B 
-1.0
0.0
0.2
0.4
0.6
q (arb. units)
0.8
1.0
Atomic properties regulate
amount of charge transferred
Charge equilibration condition at crossing of black
line crosses the colored curves
Regulated by derivative discontinuity seen in DFT
Captured in occupation number
Strong similarity to exp. I-V curve
q  
Pseudo-inverses
Time evolution
Representing transformations (q <-> C <-> c)
E   A   B
0.5
 q 
Mathematical Issues
2 q d   d, q  q E
This model is the cornerstone in models such as QEq, ReaxFF, BSK, ES+, etc.
Eminently reasonable, simple, sound in many cases. If charges allowed to vary too much
with configuration, quantum mechanics has something else to say.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
• States [5]
– Nonadiabatic; no other restriction
– Integer numbers of electrons on each fragment

q 
Quantum
1.0
 q related to occupation
number of the two charge
states
EG, if adding a charge,
q=–
2
 A   C A  A
   
Hardness or coulomb self-repulsion
Iczkowski & Margrave [1]; Sanderson [2]
 N-particle density matrices

1 qq0 d 
• Occupation number as a fcn of q
– Get two branches, ground &
excited
• Hamiltonian in fragment form
– Numbers of electrons on each fragment
must be integer
– Numbers of electrons on each fragment undefined
i
C
q (dimensionless)
Classical
Electrostatic
PPLB [3] Chemical Potential
0
0.4
0.2
PPLB
 q1 q d  


ˆ
ˆ
1
ˆ
H    HA  2  VAB 

A 
B° A
   ci  i
E(q,d)  E0 d    q, d E1 d   E0 d 
0.6
EIM q, d   E0 d   q  A   B  1 2q2 A  B  q2 d
+q
-q
qgs
Classical electrostatic potentials have long been used to describe charged interactions
among atoms in a molecule. The usual structure of an electrostatic interaction between
a neutral pair of atoms/sites/fragments and charge q not an integer
d
+q
1.0
Iczkowski-Margrave Chemical Potential
Short-range
(Exponential,
Electronegativity,
Morse, LJ, Rose) Chemical Potential
General Charge Model
Two-State Model with Charged Pair
ABSTRACT
Conclusions
• Charge fluctuations on fragments yields quantum I-M model
• Map between wavefunction parameters and properties: branches
• No artificial spatial boundaries
• New term in chemical potential of fragments, new atomistic potentials
References
[1] RP Iczkowski and JL Margrave, J Am Chem Soc 83, 3547 (1961);
HO Pritchard and FH Sumner, Proc Roy Soc, London A 235, 136-143 (1956).
[2] RT Sanderson, Science 114, 670 (1951).
[3] JP Perdew, RG Parr, M Levy, JL Balduz, Jr, Phys Rev Lett 49, 1691 (1982).
[4] SM Valone and SR Atlas, J Chem Phys 120, 7262 (2004).
[5] J Morales and TJ Martinez, J Phys Chem A 105, 2842 (2001).
Acknowledgement
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
This work was performed at Los Alamos National Laboratory under the auspices of the
U. S. Department of Energy, under contract No. DE-AC52-06NA25396. This work was
supported in part by the Global Nuclear Energy Partnership program from U. S.
Department of Energy Office of Nuclear Energy, Los Alamos National Laboratory
Laboratory-Directed Research and Development Program, and National Science
Foundation, Institute for Mathematics and Its Applications, University of Minnesota.