Molecular Models and Potential Energy Surfaces

CNDOL Hamiltonians and extended configuration interaction to approach nanoscopic systems Luis A. Montero Cabrera Ana L. Montero Alejo Universidad de La Habana 1 Nanoscopic systems are usually calculated and modeled as perfect crystals to take advantage of periodicity The problem Nanoscopic systems are real challenges for computational modeling when related with life sciences and materials, even with the most crude quantum procedures. These cases are those where the environment or a non-periodic arrangement disallow a reduction to simple molecules to be understood. Particularly, simulations of the collective capacity of absorption and redistribution of charge and energy in those systems appear now as essential to explain such phenomena and

facilitate their applications. The problem Nanoscopic systems are real challenges for computational modeling when related with life sciences and materials, even with the most crude quantum procedures. These cases are those where the environment or a non-periodic arrangement disallow a reduction to simple molecules to be understood. Particularly, simulations of the collective capacity of absorption and redistribution of charge and energy in those systems appear now as essential to explain such phenomena and facilitate their applications. The problem Nanoscopic systems are real challenges for computational modeling when related with life sciences and materials, even with the most crude quantum procedures. These cases are those where the environment or a non-periodic arrangement disallow a reduction to simple molecules to be understood. Particularly, simulations of the collective capacity of absorption and redistribution of

charge and energy in those systems appear now as essential to explain such phenomena and facilitate their applications. Molecular dynamics, stochastic methods and QM/MM The choices of today, and perhaps the future, remain on molecular dynamics and/ or stochastic procedures to represent the atomic structure of large non periodic systems. However, Hamiltonians of molecular electronic states are unavoidably required to model relations between nanoscopic systems and electromagnetic fields. The old Hartree Focks treatment The Hartree Focks Hamiltonian considers electrons as navigating in the field of nuclei and uses antisymmetrized wave functions to allow electron exchange interactions, together with pure Coulombic potentials. Operators of matrix elements are:

Hartree Focks treatment The Hartree Focks Hamiltonian considers electrons as navigating in the field of nuclei and uses antisymmetrized wave functions to allow electron exchange interactions, together with pure Coulombic potentials. Operators of matrix elements are: f (rm ) h(rm ) 2 J j (rn ) K j (rn ) j The old Hartree Focks treatment The main advantage as a tool for understanding behaviors of nanoscopic systems is that electronic states are treated explicitly. The main disadvantage is that electron correlation is partially missing. The old Hartree Focks

treatment The main advantage as a tool for understanding behaviors of nanoscopic systems is that electronic states are treated explicitly. The main disadvantage is that electron correlation is partially missing. The NDOL Formalism Translated to an atomic basis, the NDOL formalism is an approximate Hartree-Focks procedure for calculations of energy and composition of valence molecular orbitals: F H P l B

ll AB k PB lk AB A B F H 1 2 p

lk AB All terms are a priori evaluated and provide consistent and reliable results The NDOL Formalism Translated to an atomic basis, the NDOL formalism is an approximate Hartree-Focks procedure for calculations of energy and composition of valence molecular orbitals: F H P l B

ll AB k PB lk AB A B F H 1 2 p

lk AB All terms are a priori evaluated and provide consistent and reliable results The NDOL Formalism The method is based on the CNDO and CNDO/S procedures as formulated by Poplea, Jaffb and others in the 60s and Chens Hamiltonianc of 1975. a.- Pople, J. A.; Santry, D. P.; Segal, G. A., Approximate self-consistent molecular orbital theory. I. Invariant procedures. J. Chem. Phys. 1965, 43 (10;Pt. 2), S129-S135. b.- Del Bene, J.; Jaffe, H. H., Use of the complete neglect of differential overlap method in spectroscopy. I. Benzene, pyridine, and the diazines. J. Chem. Phys. 1968, 48 (4), 1807-13. c.- Chen, C.; Chen, S. C.; Leu, S. W., Approximate Self-Consistent Molecular Orbital Theory. I. S-P Separation Model of CNDO-MO. J. Chinese Chem. Soc. (Taiwan) 1975, 22, 205-213. The NDOL Formalism Description and proposed nomenclature for the CNDOL/1 diagonal one electron matrix elements (H)a.

H U V l AB B A V l AB B A CNDO origin U Z l

B ll lk AB Z Bk AB B A ll ll lk I AA Z Al AA Z Ak AA Z B l AB

B A Split core charge: S Complete core charge: C CNDOL/1SS CNDOL/1SC CNDOL/1CS CNDOL/1CC Split core charge: 1S CNDO/1 ll ll I AA Z A AA Complete core charge: 1C a. I and A are the valence state ionization potential and electron affinity, respectively, of the AO ; Z is the effective nuclear charge of the atomic core; lAB is the interaction integral between an electron with azimutal quantum number l on atom A with the core of atom B. CNDOL/1SS and CNDOL/1CC correspond with the

previous names CNDOL/11 and CNDOL/12, respectively. The NDOL Formalism Description and proposed nomenclature for the CNDOL/2 diagonal one electron matrix elements (H)a. H U V l AB B A V l AB B A CNDO origin U

Z l B ll lk AB Z Bk AB B A 1 2 I ll

ll lk A 1 2 AA Z Al AA Z Ak AA Z B l AB B A Split core charge: S Complete core charge: C CNDOL/2SS CNDOL/2SC CNDOL/2CS

CNDOL/2CC Split core charge: 1S CNDO/2 1 2 I ll ll A 1 2 AA Z A AA Complete core charge: 1C a. I and A are the valence state ionization potential and electron affinity, respectively, of the AO ; Z is the effective nuclear charge of the atomic core; lAB is the interaction integral between an electron with azimutal quantum number l on atom A with the core of atom B. CNDOL/2SS and CNDOL/2CC correspond with the

previous names CNDOL/21 and CNDOL/22, respectively. The NDOL Formalism While one center off diagonal elements are neglected, two center terms appear for all options as: H 1 I I f r , l , l , q 2 being the projection term: f r , l , l , q S f ' q The NDOL Formalism: term evaluation Two electron integrals on a single center A are evaluated by means of the Parisers relation : ll AA I A The NDOL Formalism: term evaluation A general formula has been used for two center two electron integrals:

lk AB a lk 2 AB cRAB a lk AB RAB 2 1 2

that becomes the Mataga Nishimotos when c = 2, Ohnos when c = 0, and the so called modified Ohnos when c = 1. The energy term is: lk ll kk 1 a AB 2 AA BB The NDOL Formalism: term evaluation Core-electron potential terms for one center are evaluated by: ll A AA I A The NDOL Formalism: term evaluation When the electron in the orbital on atom A interacts with core B, the two center coreelectron potentials are calculated as: B

ll 2 AA cRAB ll 1 AA RAB 2 1 2 The NDOL Formalism: term evaluation The total energy is: E p H F Z A Z B R

A B 1 AB A priori character confirmed Only I, A and Slater exponents for calculating S are basis parameters, and they are always taken from data published elsewhere. How reliable are these molecular wave functions? CNDOL HOMOs vs. molecular ionization potentials 12.0 CNDOL/2SS [-E HOMO = IP exp. + 0.73] r = 0.92 11.5 CNDOL/1CS [-E HOMO = IP exp. + 0.01] r = 0.94 11.0

-E HOMO (eV) 10.5 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 6.0 6.5 7.0 7.5 8.0 8.5 9.0

9.5 10.0 10.5 11.0 11.5 12.0 IP exp. (eV) Linear regression between the experimental ionization potentials (IPexp) of a molecule set and the corresponding module of the HOMO energies obtained by the CNDOL/2SS (filled square) and CNDOL/1CS (open circle) SCF procedure, imposing a unitary slope. Excited states from singly excited configuration interaction Hartree-Fock self consistency is unable to optimize j* virtual molecular orbitals. Therefore, the SCF transition energies: 1

H j* i J ij* 2 K ij* 3 H j* i J ij* must be used as diagonal elements* in the single excitation CI matrix (i and j* are molecular orbital eigenvalues of occupied and unoccupied levels, respectively) * Non diagonal elements are made by four center two electron integrals, alone. Each M excited state is obtained from a variational procedure well known as configuration interaction: CIS M aM H Ea 0 where E is a diagonal matrix of eigenvalues

where each term corresponds to an optimized excited state on the basis of the ground state configuration and a selected set of monoexcited determinants. One electron MOs vs. CIS for electronic transitions MO view Based on one electron states (MOs) resulting from SCF or other iterative energy optimizations. Virtual states are not optimized because density elements of matrix have no value for them. Believable OMOs, uncertain UMOs. One electron MOs vs. CIS for electronic transitions MO view CIS view Based on one electron states (MOs) resulting from SCF or

other iterative energy optimizations. Virtual states are not optimized because density elements of matrix have no value for them. Believable OMOs, uncertain UMOs. Resulting from a variational optimization of every state above the fundamental, and accounting both Coulombic and exchange interactions between every electron pair in the molecule. One electron MOs vs. CIS for electronic transitions E S3 eLUMO S2 S1

eHOMO S0 The variational procedure provides us that as much reference functions (Slater determinants) are taken for defining a state in CIS, the result will bring a lower overall energy and therefore will better approach the exact solution of quantum mechanics. In our procedure all possible SCF single transition energies are evaluated and ordered according to their increasing values. CIS matrix is then built with the lowest energy terms of such singly excited determinants, where is chosen by a procedure based in the appearance of bands, and then diagonalized. In our procedure all possible SCF single transition energies are evaluated and ordered according to their increasing values. CIS matrix is then built with the lowest energy terms of such singly excited

determinants, where is chosen by a procedure based in the appearance of bands, and then diagonalized. Benchmarking Full CIS CNDOL/2SS (Dashed line) CNDOL/1CS (Dotted line) CNDOL - Condensed Aromatic Polycyclic Excitation energies eV 8.00 7.00 6.00 n 6.76 6.66 FCIS 5.00

4.84 4.51 4.00 2.00 4.55 4.1 3.98 FCIS 3.00 5.88 5.62 5.61 n 6.62

3.54 3.48 n FCIS n 2.63 FCIS 2.29 n FCIS CNDOL/2SS (solid line) and CNDOL/1CS (dotted line) excitation energies for a set of aromatic condensed-polycyclic hydrocarbons. The experimental maxima of the absorption spectra in hydrocarbon solution (marked as filled) were used as reference data. Cut-off selected CIS Electron density terms of each singly excited configuration comes from SCF

matrices: N p ni ci ci i 1 where ni is the occupation number of each i MO in the state (being ni = 0, 1, 2 according to the number of electrons in each MO corresponding to the determinant) and ci are the MO coefficients. Then, density elements of each CI state is given by the squares of the corresponding coefficients and densities of SCF excited states: p M 2

a p 1 3 Charge displacements upon excitation of acrolein (CNDOL/2CC full CIS) 4 1 2 The calculated photochemical behavior points to the approach and reaction of a C1 that becomes negative charged upon excitation to a non excited monomer showing a positive charge at the equivalent site QO4

QO4(p) QO4 QC1 QC1(p) QC1 QC2 S1(n) -0.511 0.296 0.296 -0.226 -0.363 -0.364

0.109 S0 -0.807 0.138 0.102 Excitons and configuration interaction Electron excitations in atomic and molecular aggregates can be considered as excitons. Typically, excitons describe quasi-particles related with the pairing of an electron that has been kicked into a higher energy state by a photon with a hole left within the shell or orbit around the nuclei. We prefer to understand excitons as quasi-particles expressing charge bounding between the excited and the ground state originated in Coulomb and exchange effects, They are mostly depending on charge migrations occurring upon excitation. Excitons and configuration

interaction Electron excitations in atomic and molecular aggregates can be considered as excitons. Typically, excitons describe quasi-particles related with the pairing of an electron, which has been kicked into a higher energy state by a photon, with a hole left within the shell or orbit around the nuclei. We prefer to understand excitons as quasi-particles expressing charge bounding between the excited and the ground state originated in Coulomb and exchange effects. They are mostly depending on charge migrations occurring upon excitation. Excitons and configuration interaction Electron excitations in atomic and molecular aggregates can be considered as excitons. Typically, excitons describe quasi-particles related with the pairing of an electron, which has been kicked into a higher energy state by a photon, with a hole left within the shell or orbit around the nuclei. We prefer to understand excitons as quasi-particles expressing charge bounding between the excited and the ground state originated in Coulomb and exchange effects. They are mostly depending on charge migrations occurring upon excitation.

An illuminating recent review describes that Related electronhole exchange interactions mix the HartreeFock (single particle) configurations, so that the exciton states are finally obtained through a configuration interaction (CI-singles) calculation. Correct spin eigenstates can only be obtained by antisymmetrizing wavefunctions and mixing configurations through exchange interactions. Scholes, G. D.; Rumbles, G., Excitons in nanoscale systems. Nature Materials 2006, 5, 683-696. Therefore, in the above described HF excited matrix elements: 1 H j* i J ij* 2 K ij* 3 H j* i J ij* the MO Coulombic (Js) and exchange (Ks) integrals could be considered as related with

the degree of bounding between electrons and their holes in the case of single determinant excitations. Therefore Coulomb-Exchange terms for each excited state: 1 M 2 J ij* 2K ij* ECE a 1 3 CE E M 2

a 1 J ij * could be considered as related with the excitons, or charge bounding energy after CIS. Coulomb exchange terms in benzaldehyde Coulomb exchange energies (1ECE) in eV obtained by tested CNDOL theoretical approaches. Molecules Benzaldehyde CHO ECE (eV) 1

State s CNDOL/2SS CNDOL/1CS n* 4.98 5.17 * 5.34 5.18 * 3.66 3.67 *

3.83 4.01 Coulomb exchange terms in benzaldehyde (CNDOL/1CS) ECE (ev) 5.17 5.18 3.67 4.01 Atom Q Q Q Q

C1 0.0525 0.0518 0.0140 -0.0159 C2 -0.0249 0.0793 0.0078 0.0694 C3 0.0097 -0.0027

-0.0161 0.0539 C4 -0.0035 -0.1104 -0.0476 -0.0539 C5 -0.0047 -0.0011 -0.0023 0.0526 C6

-0.0226 0.0455 0.0091 0.0642 C7 -0.1149 -0.1744 -0.1349 -0.1917 O8 -0.0858 0.0061 0.0603

-0.0191 H9 0.0515 0.0044 0.0145 0.0068 H10 0.0418 0.0189 0.0136 0.0086 H11 0.0034

0.0277 0.0199 0.0068 H12 0.0552 0.0074 0.0155 0.0073 H13 0.0361 0.0114 0.0173 0.0070

H14 0.0060 0.0362 0.0289 0.0039 Coulomb exchange terms of polyacenes Molecule Benzene Naphthalene Anthracene Naphthacene Pentacene Structural formula

Origin CNDOL/2SS CNDOL/1CS 3.91 3.96 3.33 3.34 3.63 3.64

3.69 3.73 2.91 2.92 3.33 3.36 3.51 3.51 3.52

3.49 2.62 2.63 3.09 3.10 3.08 3.08 3.13 3.14

2.40 2.41 2.90 2.90 2.76 2.76 Timing and sustainability Calculation times of the methods, using porphine and pentacene as reference molecules for the comparison. Calculation times (seg.) Methods Porphine

(n = 110; FCIS = 3021) Pentacene (n = 102; FCIS = 2601) 1.38 0.95 1977 (~ 33 min) 1132 (~19 min) 1.35 0.96 1551 (~ 26 min) 948 (~16 min) TD-DFRT (3-21+G*) (5 h, 50 min) (4 h, 24 min)

TD-DFRT (6-31+G*) (16 h, 20 min) CNDOL/2SS (SECs = n) CNDOL/2SS (SECs = FCIS) CNDOL/1CS (SECs = n) CNDOL/1CS (SECs = FCIS) ZINDO/S (SECs = 100) * 12 (11 h, 14 min) 30 The number of SECs used in CNDOL and ZINDO calculations has been specified in parenthesis. In the same way have been added the basis used in each TD-DFRT calculations. The TD-DFRT and ZINDO calculations have been performed with Gaussian03 package, while the CNDOL with the NDOL2009 computer program. n means the number basis functions in each case i.e. valence atomic orbitals. The consumed calculation times have been specified in hours (h.) and minutes (min.) when required.

Computers Calculations have been performed with either: Linux version 2.6.21.5-smp, Memory: 1020736 KB, Intel(R) Pentium(R) 4 CPU, 3.20 GHz. Total of 2 processors. Linux version 2.6.29.6 #2 SMP, Memory: 1921140 KB, AMD Athlon(tm) 64 X2 Dual Core Processor 5200+, 2712.126 MHz. Total of 2 processors. Some applications to excited states of polyatomic systems Nanotubes Models of Single-Walled Carbon Nanotubes (SWCNT) Zig-zag type SWCNT, within (n-m=3M) rule L 2 Length (nm) 0.71

4 1.56 6 2.42 8 3.27 10 4.12 12 4.98 14 5.83 16

6.68 18 7.54 20 8.39 SWCNT(13,0,16) Three visions of the same phenomenon MO-DOS CIS-DOS Calculated transition energies Experiment vs. theory for (5,0,20) SWCNT Middle, experimental optical

absorption spectra of SWCNT with diameter near 4 located in the pores of a zeolite. Top, calculated imaginary part of the dielectric function, obtained with the BetheSalpeter equation. Salpeter equation. Bottom, CNDOL absorption spectrum of (5,0,20) SWCNT. The electric field of light is polarized along the nanotube axis. The case of (9,0) SWCNT The case of (13,0) SWCNT How they evolve when projected to infinite lengths? SWCNT (13,0) E 2.0 1.5 2.0

1.5 1.0 1.0 0.5 0.5 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

-1 (1/Length) nm When length tend to infinite the lowest excited state of (9,0) SWCNT converges to about 0.4 ev A (f) B (w) C (m) D (m) E (s) F (w) G (s) H (s) 2.5 E CNDOL-CIS (eV)

2.5 (eV) A (w) B (s) C (m) D (w) E (w) F (m) G (w) H (w) 3.0 3.0 CNDOL-CIS SWCNT (9,0) 0.0 0.00 0.05

0.10 0.15 0.20 0.25 0.30 0.35 -1 (1/Length) nm When length tend to infinite the lowest excited state of (13,0) SWCNT remains unchanged and forbidden for direct light absorption around 0.6 ev Montero-Alejo, A. L.; Fuentes, M. E.; Menndez-Proupin, E.; Orellana, W.; Bunge, C. F.; Montero, L. A.; Garca de la Vega, J. M., Approximate quantum mechanical method for describing excitations and related properties of finite single-walled carbon nanotubes. Phys. Rev. B 2010, 81 (23), 235409.

The rhodopsin adventure Retinal, the chromophore Retinal, both at the 11-cis form and the all trans are among the very scarce natural chromophores selected in nature H O H O How rhodopsin sees? The upper figure shows retinal Schiffs base in green with 11-12 cis bond in blue inside the crystal structure of rhodopsin. The lower is the same image, although showing the van der Waals surface of the protein. RBPs lowest excitations CNDOL/1CS of RBP (SECs = 10 x n)

6 CNDOL/1CS of RBP (SECs = 10 x n) (f ) +5 6 5 5 4 4 3 2 2 1 1 0 0 2.0

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6 5 5 4 4 3 3

CE (eV) 3 CE (eV) Theo. Abs.=log 7 7 Theo. Abs.=log (f ) +5 6 2 2 1 1 Excitation energies (eV)

0 0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 Excitation energies (eV)

CNDOL/1CS vertical electron transitions bellow 6.0 eV of the RBP model (left). A zoom of the low excitation energies (between 2.0 and 4.0 eV) is on the right. SECs = 10 x n, where n is 622 basis functions. Montero-Cabrera, L. A.; Rhrig, U.; Padron-Garca, J. A.; Crespo-Otero, R.; Montero-Alejo, A. L.; Garca de la Vega, J. M.; Chergui, M.; Rthlisberger, U., CNDOL: A fast and reliable method for the calculation of electronic properties of very large systems. Applications to retinal binding pocket in rhodopsin and gas phase porphine. J. Chem. Phys. 2007, 127 (14), 145102. Origin of RBPs excitations CNDOL/1CS description of the first electron transitions of the RBP model, by the molecular orbitals involved. Transitions E f CE Molecular Orbitals involved (qualitative) 1 2.53

0.01 2.12 82% HOMO (Glu113) LUMO (ret. polyene) 2 2.91 0.15 2.28 70% HOMO-1(Glu113) LUMO (ret. polyene) 3 2.92 0.70 2.42 70% HOMO-2, HOMO-3 (Glu113) LUMO (ret. polyene)

4 3.46 0.01 2.17 60 % A LUMO (ret. polyene) 5 3.55 1*10-3 2.45 90% HOMO-2, HOMO-3 (Glu113) LUMO (ret. polyene) 6 3.58 1*10-4

6.29 77% (Glu122) (Glu122) 7 3.61 0.07 2.12 50 % B LUMO (ret. polyene) 8 3.66 0 5.84 75% HOMO (Glu113) (Glu113) 9

3.77 1*10-3 5.91 80% (His211) (His211) 10 3.92 0.07 2.31 42% (ret. cyclohexadiene) LUMO (ret. polyene) * E and CE are excitation energy and their Coulomb and Exchange contribution energy, respectively. Both are in eV. The oscillator strength (f) are in arbitrary units. A and B represent a set of orbitals delocalized over different residues of the systems. A (Tyr268, Phe261, Glu113) and B (Trp265, Phe261, Tyr191). Calculation as published in 2007 with a CIS active space reduced to the n lower energy excited determinants.

Membrane rhodopsin and intermediates Bovine rhodopsin photolysis intermediates were characterized since the early 90s and this scheme continues alive as a description of what happens in the retinal binding pocket once a photon excites the photo or dark state. [Lewis, J. W.; Kliger, D. S., Photointermediates of visual pigments. Journal of Bioenergetics and Biomembranes 1992, 24 (2), 201-210.] Dimerization as a pathway for signals [Neri, M.; Vanni, S.; Tavernelli, I.; Rothlisberger, U., Role of Aggregation in Rhodopsin Signal Transduction. Biochemistry 2010, 49 (23), 4827-4832.] The first transition of dark state is here only 20 % from GLU113-. 55 % is a local transition on retinal and appears 8 % from the other GLU- above retinal.

Therefore, what we have and what we can do? We have A collection of a priori approximate Hamiltonians that can afford large (nanoscopic) polyatomic systems, giving reliable results, fair modeling and predictive possibilities because using well known procedures for improvements. A set of calculable values that can characterize electronic states of such polyatomic systems and the corresponding charge displacements and bounding. We have A collection of a priori approximate Hamiltonians that can afford large (nanoscopic) polyatomic systems, giving reliable results, fair modeling and predictive possibilities because using well known procedures for improvements. A set of calculable values that can characterize electronic states of such polyatomic systems and the corresponding

charge displacements and bounding. We foresee A systematic increase of computer power at sustainable costs that could allow to model greater and more complex polyatomic systems in every lab. A development of tools for understanding quantum properties as modeled for very large polyatomic systems. We foresee A systematic increase of computer power at sustainable costs that could allow to model greater and more complex polyatomic systems in every lab. A development of tools for understanding quantum properties as modeled for very large polyatomic systems. Can us get useful information from quantum calculations of a complete protein or aggregate? 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) What about an amorphous photosensitive material? Scanning tunneling microscopy of poly[2-methyloxy-5-(3=,7=dimethyloctyloxy)-p-phenylene vinylene]: 1-(3-methoxycarbonyl) propyl-1phenyl[6,6]C61 thin films spincast from toluene: (a) 1 m 1 m topography taken at positive sample bias in the dark Maturov, K.; Janssen, R. A. J.; Kemerink, M., Connecting Scanning Tunneling Spectroscopy to Device Performance for Polymer:Fullerene Organic Solar Cells. ACS Nano 2010, 4 (3), 1385-1392. ACKNOWLEDGEMENTS The authors are greatly indebted to the University of Havana, in Havana, Cuba, and the Deutscher Akademischer Austauschdienst (DAAD) in Bonn, Germany, for their support to this work. The Universidad Autnoma de Madrid, and the Ministry of Education and Science, and the Spanish Agency for Foreign Cooperation for Development has provided the latest essential support.

Collaborators Laboratorio de Qumica Computacional y Terica, Universidad de La Habana Ana L. Montero-Alejo Rachel Crespo-Otero Susana Gonzlez-Santana Luis A. Montero-Cabrera Nelaine Mora-Dez Juan A. Padrn-Garca Instituto de Ciberntica, Matemtica y Fsica, La Habana Augusto Gonzlez Alan Delgado-Gran Centro de Investigaciones Biomdicas, Instituto Superior de Ciencias Mdicas de La Habana Erix Wiliam Hernndez-Rodrguez Jos Carlos Garca-Pieiro cole Polytechnique Fderale de Lausanne Ute Rhring Ursula Rthlisberger Majed Chergui Marilisa Neri Universidad Autnoma de Madrid Jos Manuel Garca de la Vega Ral H. Gonzlez Jontes Universidad Nacional Autnoma de Mxico Carlos F. Bunge

Universidad Autnoma de Chihuahua Mara E. Fuentes Universidad de Chile, Santiago de Chile Eduardo Menndez And Wilfredo Lam, for this paints

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