MOLECULAR MODELING OF MATTER :

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MOLECULAR MODELING OF MATTER FROM REALISTIC
HAMILTONIANS TO SIMPLE MODELS AND THEIR
APPLICATIONS Ivo NEZBEDA E. Hala Lab. of
Thermodynamics, Acad. Sci., Prague, Czech
Rep. and Dept. of Theoret. Physics, Charles
University, Prague, Czech Rep. COLLABORATORS J
. Kolafa, M. Lisal, M. Predota, L. Vlcek Acad.
Sci., Prague A. A. Chialvo Oak Ridge Natl. Lab.,
Oak Ridge P. T. Cummings Vanderbilt Univ.,
Nashville M. Kettler Univ. of Leipzig,
Leipzig SUPPORT Grant Agency of the Czech
Republic Grant Agency of the Academy of Sciences
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Roughly speaking, there are TWO WAYS OF MOLECULAR
MODELING OF MATTER SPECULATIVE
MATHEMATICAL
to model INTUITIVELY certain features (e.g. to
capture the essence of intermolecular interactions
) or a specific property (e.g. free energy models)
starting from a certain mathematical description
of the problem (e.g. from a realistic
Hamiltonian), well-defined approximations are
introduced in order to facilitate finding a
solution.
Although both methods may end up with the same
model, there is a substantial difference between
them.
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WAYS OF MODELINGcontd. SPECULATIVE
MATHEMATICAL
EXAMPLE 1 van der Waals EOS Molecules are not
immaterial points but OBJECTS with their own
impenetrable volume ? molecules may be viewed as
HARD SPHERES ?
EXAMPLE 1 Theory of simple fluids The simple
fluid is defined by a REALISTIC potential, u (r).
Too complex for theory. 1. The structure is
determined by short-range REPULSIVE
interactions. ? u(r)u(r)rep ?u(r)
and use a perturbation expansion 2. To solve the
problem for the reference, u(r)rep, is
still too difficult ? properties of the
u(r)rep fluid are mapped onto those of hard
spheres Xrep ? Xhard sphere ?
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WAYS OF MODELINGcontd. SPECULATIVE
MATHEMATICAL
PROBLEMS (1) How to determine ?z? (2) How to
refine/improve the approach? (3) How this
is related to reality?
Hard sphere model is a part of the well-defined
scheme The correction term is well-defined ? There
are evident ways to improve performance of the
method
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ULTIMATE GOAL OF THE PROJECT Using a
molecular-based theory, to develop workable (and
reliable) expressions for the thermodynamic
properties of fluids
HOW TO REACH THE GOAL? Start from the best
realistic potential models and use a perturbation
expansion. SIMPLE (THEORETICALLY FOOTED) MODELS
ARE AN INDISPENSABLE PART OF THIS SCHEME.
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PERTURBATION EXPANSION general
considerations   Given an intermolecular pair
potential u, the perturbation expansion
method proceeds as follows (1) u is first
decomposed into a reference part, uref, and a
perturbation part, upert u uref upert
The decomposition is not unique and is
dictated by both physical and mathematical
considerations. This is the crucial step
of the method that determines convergence
(physical considerations) and feasibility
(mathematical considerations) of the
expansion. (2) The properties of the reference
system must be estimated accurately and
relatively simply so that the evaluation
of the perturbation terms is feasible. (3)
Finally, property X of the original system is
then estimated as X Xref ?X where
?X denotes the contribution that has its origin
in the perturbation potential upert.
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STEP 1 Separation of the total u into a
reference part and a perturbation part, u
uref upert THIS PROBLEM SEEMS
TO HAVE BEEN SOLVED DURING THE LAST
DECADE AND THE RESULTS MAY BE SUMMARIZED AS
FOLLOWS  Regardless of temperature and
density, the effect of the long-range
forces on the spatial arrangement of the
molecules of PURE fluids is very
small. Specifically
(1) The structure of both polar and associating
realistic fluids and their short-
range counterparts, described by the set of
the site-site correlation functions,
is very similar (nearly identical).
(2) The thermodynamic properties of
realistic fluids are very well estimated by
those of suitable short-range
models (3) The long-range forces affect
only details of the orientational correlations.
? THE REFERENCE MODEL IS A SHORT-RANGE
FLUID uref ushort-range model
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STEP 2 Estimate the properties of the
short-range reference accurately
(and relatively simply) in a CLOSED form
HOW TO ACCOMPLISH THIS STEP ?
HINT Recall theories of simple
fluids uLJ usoft spheres ?u
(decomposition into ref and pert
parts) XLJ Xsoft spheres ?X

XHARD SPHERES ?X
SOLUTION Find a simple model (called primitive
model) that (i) approximates
reasonably well the STRUCTURE of the
short-range reference, and (ii) is
amenable to theoretical treatment
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• SUBSTEPS OF STEP 2
• construct a primitive model
• apply (develop) theory to get its properties

Re SUBSTEP (1) Early (intuitive/empirical)
attempts (for associating fluids) Ben-Naim, 1971
M-B model of water (2D) Dahl, Andersen, 1983
double SW model of water Bol, 1982 4-site model
of water Smith, Nezbeda, 1984 2-site model of
associated fluids Nezbeda, et al., 1987, 1991,
1997 models of water,
methanol, ammonia Kolafa, Nezbeda, 1995 hard
tetrahedron model of water Nezbeda, Slovak, 1997
extended primitive models of water
PROBLEM These models capture QUALITATIVELY the
main features of real associating fluids,
BUT they are not linked to any realistic
interaction potential model.
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GOAL 1 Given a short-range REALISTIC (parent)
site-site potential model,
develop a methodology to construct from FIRST
PRINCIPLES a simple
(primitive) model which reproduces the structural
properties of the parent model. IDEA Use
the geometry (arrangement of the interaction
sites) of the parent model,
and mimic short-range REPULSIONS by a HARD-SPHERE
interaction,
,
Example carbon dioxide
?
and short-range ATTRACTIONS by a SQUARE-WELL
interaction.
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PROBLEM We need to specify the parameters of
interaction 1. HARD CORES (size of the
molecule) 2. STRENGTH AND RANGE OF
ATTRACTION
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1-1. HOW (to set hard cores) ??? FACTS
Because of strong cooperativity, site-site
interactions cannot be treated
independently. HINT Recall successful
perturbation theories of molecular fluids (e.g.
RAM) that use sphericalized
effective site-site potentials and which are
known to produce quite accurate
site-site correlation functions.
SOLUTION Use the reference molecular fluid
defined by the average site-site Boltzmann
factors, and apply then the hybrid
Barker-Henderson theory (i.e. WCAHB) to get
effective HARD CORES (diameters dij)
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EXAMPLES
?
SPC water
OPLS methanol
?
carbon dioxide
?
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1-2. HOW (to set the strength and range of
attractive interaction) ??? HINT Make use of
(i) various constraints, e.g. that
no hydrogen site can form no more than
one hydrogen bond. This is purely
geometrical problem. For instance, for OPLS
methanol we get for the upper
limit of the range, ?, the relation
The upper limit is used for
all models. (ii) the known facts
on dimer, e.g. that for carbon dioxide the stable
configuration is T-shaped.
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SELECTED RESULTS (OPLS methanol)
filled circles OPLS methanol solid line
primitive model
Average bonding angles ? and f ?
f prim. model 147 114 OPLS
model 156 113
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AMMONIA circles .. reality lines . prim.
model
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ETHANOL
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SPC/E WATER circles .. reality lines . prim.
model
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POLAR FLUIDS Although they do not form hydrogen
bonds, the same methodology can be applied also
to them.
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ACETONE site-site correlation functions
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ACETONE site-site correlation functions
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APPLICATIONS (of primitive models) 1. As a
reference in perturbed equations for the
thermodynamic properties of REAL fluids STEP
3 of the above scheme X Xref ?X .
Example equation of state for water Nezbeda
Weingerl, 2001 Projects under way
equations of state for METHANOL, ETHANOL,
AMMONIA, CARBON DIOXIDE
2. Used in molecular simulations to understand
basic mechanism governing the behavior of
fluids. Examples (i) Hydration of
inerts and lower alkanes entropy/enthalpy driven
changes Predota Nezbeda, 1999,
2002 Vlcek Nezbeda, 2002 (ii)
Solvation of the interaction sites of water
Predota, Ben-Naim Nezbeda, 2003
(iii) Mixtures water-alcohols, water-carbon
dioxide, (iv) Preferential solvation in
mixed (e.g. water-methanol) solvents (v)
Aqueous solutions of polymers (with EXPLICIT
solvent) (vi) Clustering/Condensation (i.e.
nucleation) (vii) Water at interface
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Excess thermo-properties WATER-METHANOL at
ambient conditions circles.. exptl.
data squares prim. model
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Excess thermo-properties CO_2 - WATER at
supercritical CO_2 conditions circles.. exptl.
data squares prim. model
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THANK YOU for your attention