Luis A. MonteroUniversidad de La Habana, Cuba, 2012

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MODELING MATTER AT NANOSCALES

1. Introduction and overview

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Sizes and scales

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Sizes and Scales in Nature

We humans develop knowledge and the perception of the surrounding universe from our standing point and our dimensions.

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Sizes and Scales in Nature

Science is essentially a systemic assembling of knowledge. Consequently, it was also founded on our perception of the world at our scale.

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Sizes and Scales in Nature

• We also created space measurements at our scale.• SI space unit is meter and a person could range

between 1 and 2 m tall.

Length Area Scale name

< 1 mm < 1 mm2 Nanoscopic

1 mm – 1 mm 1 mm2 – 1 mm2 Microscopic

1 mm – 1 m 1 mm2 – 1 m2 Personal

1 m - 1 km 1 m2 - 1 km2 Local

1 km - 100 km 1 km2 - 10 000 km2 Regional

100 km - 10 000 km 10 000 km2 - 100 000 000 km2 Continental

> 10 000 km >100 000 000 km2 Global

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SI PrefixesPrefix Symbol Factor

yotta Y 1024 

zetta Z 1021 

Exa E 1018 

Peta P 1015 

Tera T 1012 

giga G 109 

mega M 106 

miria ma 104 

kilo k 103 

hecto h 102 

deca da 101 

Prefix Symbol Factor

deci D 10-1 

centi c 10-2 

mili m 10-3 

micro µ 10-6 

nano n 10-9 

pico p 10-12 

femto f 10-15 

atto a 10-18 

zepto z 10-21 

yocto y 10-24 

Models and modeling in the nanoworld

What is a model?

•A model is a representation of any object, made or created for a given purpose.

•The Encyclopædia Britannica understands a model as a description or analogy used for aiding visualization of something (as could be an atom) that can not be directly observed.

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What is a model?

•A model is a representation of any object, made or created for a given purpose.

•The Encyclopædia Britannica understands a model as a description or analogy used for aiding visualization of something (as could be an atom) that can not be directly observed.

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SOLUTION STRUCTURE OF THE ALZHEIMER'S DISEASE AMYLOID BETA-PEPTIDE (1-42)

O.CRESCENZI,S.TOMASELLI,R.GUERRINI,S.SALVADORI, A.M.D'URSI,P.A.TEMUSSI,D.PICONE “SOLUTION STRUCTURE OF THE ALZHEIMER AMYLOID BETA-PEPTIDE (1-42) IN AN APOLAR MICROENVIRONMENT. SIMILARITY WITH A VIRUS FUSION DOMAIN” EUR.J.BIOCHEM. 269 5642 (2002)

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Modeling at Nanoscales

The fundamental purpose of “molecular” or “nanoscale modeling” is building virtual models on structures and processes occurring mostly at dimensions around 10-9 m that were both perceptible and reliable.

Nanoscale modeling works with tools developed by mathematics, chemistry, physics and computer sciences.

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Modeling at Nanoscales

The fundamental purpose of “molecular” or “nanoscale modeling” is building virtual models on structures and processes occurring mostly at dimensions around 10-9 m that were both perceptible and reliable.

Nanoscale modeling works with tools developed by mathematics, chemistry, physics and computer sciences.

Computer representation of “nanoscopic” systems

NO

N

N

OON

NO

Adenosine

What we represent?

• Molecular modeling holds by representing molecular and crystal structures in computer output devices or any other information supporting material.

• Such structures are defined in terms of the relative positions of centers or typical reference points of the system components.

What we represent?

• Molecular modeling holds by representing molecular and crystal structures in computer output devices or any other information supporting material.

• Such structures are defined in terms of the relative positions of centers or typical reference points of the system components.

Conventional standardsThe most widely accepted standards can be summarized as follows:

• Reference centers fixed in the tridimensional space are atomic nuclei of simple molecules, or aggregates, or crystals constituting the nanoscopic system.

• The position of each nuclei is established according their spatial coordinates in any orthogonal base system (ex. Cartesian, aspheric, cylindrical, internal, etc.).

• Proportions of sums of tabulated covalent radii of each element are used to establish the presence of bonding between two different centers.

Conventional standardsThe most widely accepted standards can be summarized as follows:

• Reference centers fixed in the tridimensional space are atomic nuclei of simple molecules, or aggregates, or crystals constituting the nanoscopic system.

• The position of each nuclei is established according their spatial coordinates on any orthogonal base system (ex. Cartesian, aspheric, cylindrical, internal, etc.).

• Proportions of sums of tabulated covalent radii of each element are used to establish the presence of bonding between two different centers.

Conventional standardsThe most widely accepted standards can be summarized as follows:

• Reference centers fixed in the tridimensional space are atomic nuclei of simple molecules, or aggregates, or crystals constituting the nanoscopic system.

• The position of each nuclei is established according their spatial coordinates on any orthogonal base system (ex. Cartesian, aspheric, cylindrical, internal, etc.).

• Proportions of sums of tabulated covalent radii of each element are used to establish the presence of bonding between two different centers.

Methyl amine and water

Most extended forms of representation

Traditional models of ethane

• C2H6, is a model of empirical formula based on stoichiometry.

• H3C-CH3 is a model of structural formula.• The nearest approach to an structural

representation is the “stick model”:

Traditional models of ethane

• C2H6, is a model of empirical formula based on stoichiometry.

• H3C-CH3 is a model of structural formula.• The nearest approach to an structural

representation is the “stick model”:

Traditional models of ethane

• C2H6, is a model of empirical formula based on stoichiometry.

• H3C-CH3 is a model of structural formula.• The nearest approach to an structural

representation is the “stick model”:

Sticks linking bonded centers of ethane

Balls linked by sticks for ethane

Space filling balls for ethane

Ammonia (NH3) in Cartesian coordinates would be:

Each center i is defined by xi, yi, zi values referred to an arbitrary center of coordinates in x0, y0, z0.

Internal coordinates of ethane would be:

Each center i is designed by coordinates:

ri (the distance vector of separation from any other center j),

qi (the angle of centers i, j, k) and

ji (the spatial angle of centers i, j, k, l).

Reference centers for any atom must have been previously defined.

Typical geometries

Simple bonds

J. A. Pople and M. Gordon, J. Am. Chem. Soc. 89 (17), 4253 (1967)

H-H 0.74 C4-C3 1.52 C3-N2 1.40 N3-N2 1.45

C4-H 1.09 C4-C2 1.46 C3-O2 1.36 N3-O2 1.36

C3-H 1.08 C4-N3 1.47 C3-F1 1.33 N3-F1 1.36

C2-H 1.06 C4-N2 1.47 C2-C2 1.38 N2-N2 1.45

N3-H 1.01 C4-O2 1.43 C2-N3 1.33 N2-O2 1.41

N2-H 0.99 C4-F1 1.36 C2-N2 1.33 N2-F1 1.36

O2-H 0.96 C3-C3 1.46 C2-O2 1.36 O2-O2 1.48

F1-H O.92 C3-C2 1.45 C2-F1 1.30 O2-F1 1.42

C4-C4 1.54 C3-N3 1.40b N3-N3 1.45 F1-F1 1.42

Typical geometriesDouble bonds

J. A. Pople and M. Gordon, J. Am. Chem. Soc. 89 (17), 4253 (1967)

C3-C3 1.34 C3-O1 1.22 C2-O1 1.16 N2-O1 1.22a

C3-C2 1.31 C2-C2 1.28 N3-O1 1.24b O1-O1 1.21

C3-N2 1.32 C2-N2 1.32 N2-N2 1.25    

Triple bondsC2-C2 1.20 C2-N1 1.16 N1-N1 1.10

Aromatic bondsC3-C3 1.40 C3-N2 1.34 N2-N2 1.35

C3-Cl 1.77 C4-Br 1.93 C4-I 2.14

C3-Cl 1.73 C3-Br 1.87(arom) C3-I 2.09

C2-Cl 1.63 C2-Br 1.79 C2-I 1.99

C3-Cl 1.71(arom) C3-Br 1.87    

Carbon - halogen bonds (from other source)

a.- For partial double bonds in NO2 and NO3 groups.

Some common input formats of computational chemistry

programs

There are computer programs and program packages that became reference input data format for molecular structures like Gaussian, Mopac, etc.

Gaussian%mem=3700000 #KEYWORDS GO HERE

Ethane

Ch Mu C C 1 r2 H 2 r3 1 a3 H 1 r4 2 a4 3 d4 H 1 r5 2 a5 3 d5 H 1 r6 2 a6 3 d6 H 2 r7 1 a7 3 d7 H 2 r8 1 a8 3 d8Variables:r2= 1.5424r3= 1.0841a3= 110.79r4= 1.0841a4= 110.79d4= 180.00r5= 1.0841a5= 110.79d5= 60.00r6= 1.0841a6= 110.79d6= 300.00r7= 1.0841a7= 110.79d7= 240.00r8= 1.0841a8= 110.79d8= 120.00

Gaussian considering symmetry :%mem=3700000#KEYWORDS GO HERE

Ethane

Ch Mu C C 1 rCC H 2 rCH 1 aCCH H 1 rCH 2 aCCH 3 d4 H 1 rCH 2 aCCH 3 d5 H 1 rCH 2 aCCH 3 d6 H 2 rCH 1 aCCH 3 d7 H 2 rCH 1 aCCH 3 d8Variables:rCC= 1.5424rCH= 1.0841aCCH= 110.79d4= 180.00d5= 60.00d6= 300.00d7= 240.00d8= 120.00

Mopac internal coordinates:

comandos mopac Ethane

C 00000.0000 0 00000.0000 0 00000.0000 0 0 0 0C 00001.5424 1 00000.0000 0 00000.0000 0 1 0 0H 00001.0841 1 00110.7955 1 00000.0000 0 2 1 0H 00001.0841 1 00110.7949 1 00180.0590 1 1 2 3H 00001.0841 1 00110.7950 1 00060.0570 1 1 2 3H 00001.0841 1 00110.7945 1 00300.0595 1 1 2 3H 00001.0841 1 00110.7956 1 00300.0532 1 2 1 5H 00001.0841 1 00110.7932 1 00060.0535 1 2 1 60

Other well known program, GAMESS, uses mostly Cartesian coordinates

$CONTROL COORD=CART UNITS=ANGS $END$DATAethanePut symmetry info hereC 6.0 0.00000 0.00000 0.00000 C 6.0 1.54240 0.00000 0.00000 H 1.0 1.92729 1.01347 0.00000 H 1.0 -0.38488 -1.01348 0.00104 H 1.0 -0.38488 0.50587 -0.87820 H 1.0 -0.38487 0.50765 0.87717 H 1.0 1.92729 -0.50680 -0.87766 H 1.0 1.92725 -0.50665 0.87776 $END

“Protein data bank” (PDB) of Brookhaven format

HEADER PROTEINCOMPND ETHANEAUTHOR GENERATED BY BABEL 1.05dATOM 1 C1 UNK0 1 0.000 0.000 0.000 1.00 0.00 ATOM 2 C2 UNK0 1 1.542 0.000 0.000 1.00 0.00 ATOM 3 H1 UNK0 1 1.084 110.796 0.000 1.00 0.00 ATOM 4 H2 UNK0 1 1.084 110.795 180.059 1.00 0.00 ATOM 5 H3 UNK0 1 1.084 110.795 60.057 1.00 0.00 ATOM 6 H4 UNK0 1 1.084 110.794 300.060 1.00 0.00 ATOM 7 H5 UNK0 1 1.084 110.796 300.053 1.00 0.00 ATOM 8 H6 UNK0 1 1.084 110.793 60.053 1.00 0.00 CONECT 1 2CONECT 2 1CONECT 3 0CONECT 4 0CONECT 5 8CONECT 6 7CONECT 7 6CONECT 8 5MASTER 0 0 0 0 0 0 0 0 8 0 8 0END

H.M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T.N. Bhat, H. Weissig, I.N. Shindyalov, P.E. Bourne: The Protein Data Bank. Nucleic Acids Research, 28 pp. 235-242 (2000).

Potential energy surfaces in the nanoworld

Second Newton's law and modeling nanoscales

Confident models at nanoscales are grounded on the consideration that any system is more stable in conditions of minimal potential energy (or internal energy):

where Fi is a force that could change the position of a body at a point .

0

i

ii r

rEF

ir

ir

Potential energy surfaces

In order to find the most probable molecular structures is necessary a function that expresses the total potential energy, or simply the total energy of the system, in terms of the number and kind of nuclei (Z) and their respective spatial coordinates given by a matrix R, as well as those of electrons:

This function is known as the potential energy surface (PES) of the system or hypersurface.

),( RZEE

The main problem of methods for modeling

nanoscale objects is finding the appropriate analytical or numerical function of such

hypersurfaces:

),( RZEE

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Quantum mechanics as a physical theory for nanoscopic

systems

Quantum mechanics is the only known theory, until now, providing valid a priori results for

modeling and describing nanoscopic phenomena, as is the case of molecular

interactions and chemical bonding.

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Quantum Hypersurfaces for Quantum Models

Quantum models are those where the hypersurface is calculated from wave functions associated to the involved particles. It uses to be the most reliable approach to such purpose.

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Quantum Hypersurfaces for Quantum Models

tr

trHE

tr

,

,ˆ

,

Quantum models are those where the hypersurface is calculated from wave functions associated to the involved particles. It uses to be the most reliable approach to such purpose.

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Only quantum mechanics?

Classical models are those where hypersurfaces are built with known functions of classical mechanics that are adjusted for reproducing experimental results o previous confident quantum calculations.

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Only quantum mechanics?

Classical models are those where hypersurfaces are built with known functions of classical mechanics that are adjusted for reproducing experimental results o previous confident quantum calculations.

tf

tr

trHE

,

,

,ˆ

R

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“Optimized”molecular geometries

A certain coordinate set Req that provides a minimal energy Eeq for the whole system is known as the optimized geometry and it means a global minimum of the hypersurface.

0

i i

eq

r

E R

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Hypersurface of H2 according the Morse's potential

-1 0 1 2 3 4 5 60

1

2

3

4

5

6

r (10-10 m)

E (

ev)

E=DH

2

(1-e-1.1(0.74-r))2

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Real world = Multiple minima of hypersurfaces

Hypersurfaces of nanosystems can also contain one or much other local or secondary minima, that represent alternative system geometries, less stable, although being able to show significant populations in the statistical configuration space.

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Real world = Multiple minima of hypersurfaces

Statistical relationships could help to model populations by the partition function:

1i

kTEi

eq

Being the relative population of each state:

qe

NkT

E

i

i

50

Furan + Ethyne

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Entropy in molecular models

Modeling a molecular system is usually performed with the global or absolute minimum of the hypersurface. In such cases, the entropy of the corresponding macroscopic system is absent.

However, the great majority of molecular systems behaves macroscopically as presenting multiple local minima with significant probabilities (and populations) each, in addition to the global. This is equivalent to take into account the system entropy when is desired to project modeling to reality.

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Entropy in molecular models

Modeling a molecular system is usually performed with the global or absolute minimum of the hypersurface. In such cases, the entropy of the corresponding macroscopic system is absent.

However, the great majority of molecular systems behaves macroscopically as presenting multiple local minima with significant probabilities (and populations) each, in addition to the global. This is equivalent to take into account the system entropy when is desired to project modeling to reality.

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Searching multiple minima

Identifying multiple minima in a hypersurface can only be performed as starting from a series of initial molecular arrangements, preferably generated by a random procedure and furtherer optimized.The configurational entropy of a polyatomic system can be evaluated by this means.

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Searching multiple minima

Identifying multiple minima in a hypersurface can only be performed as starting from a series of initial molecular arrangements, preferably generated by a random procedure and furtherer optimized.The configurational entropy of a polyatomic system can be evaluated by this means.

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Multiple minima in mathematical language

The potential energy surface is given by E = E(Z,R) where R is the nuclei position vector matrix• A stable stationary structure (a conformation,

reactant, product, aggregate etc.) is a minimum of E = E(Z,R)

• A search for Req means that:

RRMRRR

0M

,,, ZEZE eqeq

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Multiple minima in mathematical language

There are two kinds of minimal geometries of nanoscopic systems:• Local minimum: Req is a minimum in its

neighborhood (M)• Global minimum: Req is a minimum for all R

RRR ,,, ZEZE eq