MODELING MATTER AT NANOSCALES. 1. Introduction and overview

Transcription

1 MODELING MATTER AT NANOSCALES 1. Introduction and overview 1 Luis A. Mont ero Firmado digitalmente por Luis A. Montero Nombre de reconocimiento (DN): cn=luis A. Montero, o=universidad de La Habana, ou=dep. de Química Física, =luis.montero@qui mica.uh.cu, c=cu Fecha: :44:49-04'00'

2 Sizes and scales 2

3 Sizes and Scales in Nature We humans develop knowledge and the perception of the surrounding universe from our standing point and our dimensions. 3

4 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. 4

5 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 µm < 1 µm 2 Nanoscopic 1 µm 1 mm 1 µm 2 1 mm 2 Microscopic 1 mm 1 m 1 mm 2 1 m 2 Personal 1 m - 1 km 1 m 2-1 km 2 Local 1 km km 1 km km 2 Regional 100 km km km km 2 Continental > km > km 2 Global 5

6 SI Prefixes Prefix Symbol Factor yotta Y zetta Z Exa E Peta P Tera T giga G 10 9 mega M 10 6 miria ma 10 4 kilo k 10 3 hecto h 10 2 deca da 10 1 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 femto f atto a zepto z yocto y

7 Models and modeling in the nanoworld

8 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.

9 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. 9

10 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 (2002) 10

11 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. 11

12 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. 12

13 Computer representation of nanoscopic systems O O N N N O O N N Adenosine

14 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.

15 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.

16 Conventional standards The 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.

17 Conventional standards The 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.

18 Conventional standards The 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.

19 Methyl amine and water

20 Most extended forms of representation

21 Traditional models of ethane C 2 H 6, is a model of empirical formula based on stoichiometry. H 3 C-CH 3 is a model of structural formula. The nearest approach to an structural representation is the stick model :

22 Traditional models of ethane C 2 H 6, is a model of empirical formula based on stoichiometry. H 3 C-CH 3 is a model of structural formula. The nearest approach to an structural representation is the stick model :

23 Traditional models of ethane C 2 H 6, is a model of empirical formula based on stoichiometry. H 3 C-CH 3 is a model of structural formula. The nearest approach to an structural representation is the stick model :

24 Sticks linking bonded centers of ethane

25 Balls linked by sticks for ethane

26 Space filling balls for ethane

27 Ammonia (NH 3 ) in Cartesian coordinates would be: Each center i is defined by x i, y i, z i values referred to an arbitrary center of coordinates in x 0, y 0, z 0.

28 Internal coordinates of ethane Each center i is designed by coordinates: r i (the distance vector of separation from any other center j), θ i (the angle of centers i, j, k) and ϕ i (the spatial angle of centers i, j, k, l). Reference centers for any atom must have been previously defined. would be:

29 Typical geometries Simple bonds H-H 0.74 C4-C C3-N N3-N C4-H 1.09 C4-C C3-O N3-O C3-H 1.08 C4-N C3-F N3-F C2-H 1.06 C4-N C2-C N2-N N3-H 1.01 C4-O C2-N N2-O N2-H 0.99 C4-F C2-N N2-F O2-H 0.96 C3-C C2-O O2-O F1-H O.92 C3-C C2-F O2-F C4-C C3-N b N3-N F1-F J. A. Pople and M. Gordon, J. Am. Chem. Soc. 89 (17), 4253 (1967)

30 Double bonds Typical geometries C3-C C3-O C2-O N2-O a C3-C C2-C N3-O b O1-O C3-N C2-N N2-N a.- For partial double bonds in NO 2 and NO 3 groups. Triple bonds C2-C C2-N N1-N Aromatic bonds C3-C C3-N N2-N Carbon - halogen bonds (from other source) 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 J. A. Pople and M. Gordon, J. Am. Chem. Soc. 89 (17), 4253 (1967)

31 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.

32 %mem= #KEYWORDS GO HERE Ethane Gaussian 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 d8 Variables: r2= r3= a3= r4= a4= d4= r5= a5= d5= r6= a6= d6= r7= a7= d7= r8= a8= d8=

33 Gaussian considering symmetry : %mem= #KEYWORDS GO HERE Ethane Ch Mu C C H H H H H H Variables: rcc= rch= acch= d4= d5= d6= d7= d8= rcc 2 rch 1 acch 1 rch 2 acch 3 d4 1 rch 2 acch 3 d5 1 rch 2 acch 3 d6 2 rch 1 acch 3 d7 2 rch 1 acch 3 d8

34 Mopac internal coordinates: comandos mopac Ethane C C H H H H H H

35 Other well known program, GAMESS, uses mostly Cartesian coordinates $CONTROL COORD=CART UNITS=ANGS $END $DATA ethane Put symmetry info here C C H H H H H H $END

36 Protein data bank (PDB) of Brookhaven format HEADER PROTEIN COMPND ETHANE AUTHOR GENERATED BY BABEL 1.05d ATOM 1 C1 UNK ATOM 2 C2 UNK ATOM 3 H1 UNK ATOM 4 H2 UNK ATOM 5 H3 UNK ATOM 6 H4 UNK ATOM 7 H5 UNK ATOM 8 H6 UNK CONECT 1 2 CONECT 2 1 CONECT 3 0 CONECT 4 0 CONECT 5 8 CONECT 6 7 CONECT 7 6 CONECT 8 5 MASTER END 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 (2000).

37 Potential energy surfaces in the nanoworld

38 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): r i F i E = i r ( r ) = 0 where F i is a force that could change the position of a body at a point. i r i

39 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: E = E( Z, R This function is known as the potential energy surface (PES) of the system or hypersurface. )

40 The main problem of methods for modeling nanoscale objects is finding the appropriate analytical or numerical function of such hypersurfaces: E = E( Z, R )

41 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. 41

42 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. 42

43 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. Ψ = Ψ Hˆ Ψ E = Ψ ( r, t) ( r, t) ( r, t) 43

44 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. 44

45 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. E Hˆ Ψ = Ψ f ( r, t) ( r, t) ( R, t) 45

46 Optimized molecular geometries A certain coordinate set R eq that provides a minimal energy E eq for the whole system is known as the optimized geometry and it means a global minimum of the hypersurface. i ( ) E R r i eq = 0 46

47 Hypersurface of H 2 according the Morse's potential 6 5 E=D H2 (1-e -1.1(0.74-r) ) 2 4 E (ev) r (10-10 m) 47

48 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. 48

49 Real world = Multiple minima of hypersurfaces Statistical relationships could help to model populations by the partition function: q = i= 1 e E i kt Being the relative population of each state: N i = e Ei kt q 49

50 Furan + Ethyne 50

51 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. 51

52 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. 52

53 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. 53

54 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. 54

55 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 R eq means that: M (, R ) (, R) R, R R M E Z E Z eq 0 eq 55

56 Multiple minima in mathematical language There are two kinds of minimal geometries of nanoscopic systems: Local minimum: R eq is a minimum in its neighborhood ( M) Global minimum: R eq is a minimum for all R ( R ) ( R) R, E Z, E Z, eq 56