1 MODELING MATTER AT NANOSCALES 2. Energy of intermolecular processes

Understanding molecular interactions

Rough classification Strong interactions: r  2 Å (r  0.2 nm) geometry and electronic states of interacting molecules are noticeably perturbed chemical bonds could be formed or destroyed

Rough classification Strong interactions: r  2 Å (r  0.2 nm) geometry and electronic states of interacting molecules are noticeably perturbed chemical bonds could be formed or destroyed Weak interactions: 2 Å < r < 6 Å (0.2 nm < r < 0.6 nm) geometry and electronic states of interacting molecules are very slightly perturbed appearance or destruction of chemical bonds is quite impossible

Modeling the approach between two molecules Hypersurfaces Hypersurfaces are multidimensional functions where the potential energy of a system is depending on coordinates and mass of each and every particle.

Modeling the approach between two molecules Hypersurfaces Hypersurfaces are multidimensional functions where the potential energy of a system is depending on coordinates and mass of each and every particle. They must contain information to describe the behavior of interactions that can occur between the involved molecular bodies:

Modeling the approach between two molecules Hypersurfaces Hypersurfaces are multidimensional functions where the potential energy of a system is depending on coordinates and mass of each and every particle. They must contain information to describe the behavior of interactions that can occur between the involved molecular bodies: Finding wells and valleys in hypersurfaces means owning the most probable pathways and structures involved in molecular interactions.

Modeling the approach between two molecules A process of approaching a body A to B, interacting, and giving an associated AB in the nanospace can be represented as: A + B  AB

Modeling the approach between two molecules A process of approaching a body A to B, interacting, and giving an associated AB in the nanospace can be represented as: A + B  AB The association energy hypersurface can then be expressed, generally, by:

Quantum variational modeling of molecular associations

11 Quasi exact solution of molecular associations A “quasi exact” solution consists in the direct quantum calculation of the many body system by means of an adequate wave function  0.

12 Quasi exact solution of molecular associations A “quasi exact” solution consists in the direct quantum calculation of the many body system by means of an adequate wave function  0. This kind of calculation uses to be limited to consider other than very small systems because the huge computational requirements.

13 The quantum expression The wave equation of the associated species could be written as: and the corresponding to the isolated bodies:

14 Variational solution The calculation of energy differences between the tied species and those corresponding to the original isolated components provides a reliable and “a priori” value of the association energy:

15 Variational solution It must be observed that searching variables for a variational energy optimization will strongly depend on the character and size of AB.

16 Variational solution It must be observed that searching variables for a variational energy optimization will strongly depend on the character and size of AB. The calculation of E AB must be determined with “size consistency” respect to the reference of the isolated species A and B.

17 Variational solution A general overview, still valid, can be found in:

18 Variational solution A general overview with more recent tools:

19 Variational solution A variational optimization of E AB has been proposed several times in literature, ranging from first principle ab initio calculations of the associated species to the selection of relevant geometrical parameters during the interaction process to monitor the evolution of relevant quantum properties. Ayers, P. W.; Parr, R. G., Variational Principles for Describing Chemical Reactions. Reactivity Indices Based on the External Potential. J. Am. Chem. Soc. 2001, 123 (9), Specchio, R.; Famulari, A.; Sironi, M.; Raimondi, M., A new variational coupled-electron pair approach to the intermolecular interaction calculation in the framework of the valence bond theory: The case of the water dimer system. J. Chem. Phys. 1999, 111 (14),

Quantum perturbational understanding of molecular associations

23 The quantum expression in terms of a perturbation Departing from the wave equation: the Hamiltonian can be considered as: And the energy eigenvalue can be decomposed as:

24 The quantum expression in terms of a perturbation Departing from the wave equation: the Hamiltonian can be considered as: And the energy eigenvalue can be decomposed as: We must take into account that this approach allows a direct identification of the Hamiltonian and the energy originated from molecular associations and, consequently, facilitates the understanding of the process.

The quantum expression in terms of a perturbation The Hamiltonian of association according the theory of perturbations applied to non reactive molecular interactions can be developed following the “electrostatic approach” as: where i,j are sub-indexes for electronic wave functions and k,l for nuclei in the A and B systems, respectively. 25

26 Case A. Wave functions of non overlapping bodies (long distances): According to the Rayleigh - Schrödinger perturbation theory (RSPT), the wave function of the system is: being the sums on ground and excited vibronic states a, b of systems A and B, respectively.

27 In this case the energy of the system can be expressed in terms of the previous general expression: as the eigenvalue developed in series from the non perturbed energy of the system E 0 : Case A. Wave functions of non overlapping bodies (long distances):

28 In this case the energy of the system can be expressed in terms of the previous general expression: as the eigenvalue developed in series from the non perturbed energy of the system E 0 : Then, it can be considered that: Case A. Wave functions of non overlapping bodies (long distances):

29 The zero th order energy is then: Case A. Wave functions of non overlapping bodies (long distances):

30 The zero th order energy is then: and the association energy is then: where E 1 expresses the first order perturbation, E 2 the second and so on. Case A. Wave functions of non overlapping bodies (long distances):

The first order: electrostatic energy First order perturbation energy is given by purely electrostatic interactions (Coulombic) only depending on charge distributions ( ) of the ground state:

The first order: electrostatic energy First order perturbation energy is given by purely electrostatic interactions (Coulombic) only depending on charge distributions ( ) of the ground state: It can be expressed after some algebra as: where  X is the charge distribution of the ground state for each pair of particles placed according the position vectors.

33 where is the A molecule dipole moment in the a state, R is the distance between their charge centers, and is a non-dimensional unitary vector in the direction of R. The first order: electrostatic energy R B A BB AA It can be shown that the E 1 energy corresponding to interactions between permanent dipole moments of two interacting bodies with a neglected net charge is:

34 Accurate expressions for electrostatic energy are only useful in cases and distances where other perturbing effects are absent. The first order: electrostatic energy

35 Accurate expressions for electrostatic energy are only useful in cases and distances where other perturbing effects are absent. The simplest representation is to find or assign a constant fractional charge to each atom in the system, in the framework of the position of nuclei: The first order: electrostatic energy

36 Accurate expressions for electrostatic energy are only useful in cases and distances where other perturbing effects are absent. The simplest representation is to find or assign a constant fractional charge to each atom in the system, in the framework of the position of nuclei: More accurate representations must take into account anisotropy of orbitals in molecules and therefore, include a multipole analysis. The first order: electrostatic energy

37 The second order: induction and dispersion Second order perturbation energy is intrinsic negative (stabilizing) and it is given by:

38 The second order: induction and dispersion Second order perturbation energy is intrinsic negative (stabilizing) and it is given by: This term can be decomposed as: where: is the polarization or induction energy because a redistribution of charge in A owing to the interaction with the charge distribution in system B. is the “dispersive energy”.

39 Inductive effect The expressions for inductive terms are:

40 Inductive effect The expressions for inductive terms are: They are related with the effects of non – perturbed charge distributions in a molecule with a associated wave function on the other, as developed in terms of their excited vibronic states a, b,… to model distortions in their charge distribution.

41 This term becomes very important if ions are present. Inductive effect

42 This term becomes very important if ions are present. For an ion of net charge q B and a molecule with  A polarizability: Inductive effect

43 This term becomes very important if ions are present. For an ion of net charge q B and a molecule with  A polarizability: In the case of neutral molecules this term is not important. Inductive effect

44 A molecule with   dipole moment acting together with a non polar molecule A develops and inductive energy given by: where   is the angle between   and the internuclear axis. Inductive effect R AB B A BB BB

45 Dispersive effect Dispersive effects can be modeled in the framework of perturbation theory as originated in the overlapping space of excited ( a, b,…) and fundamental ( 0 ) vibronic states unperturbed molecules.

46 Dispersive effect Dispersive effects can be modeled in the framework of perturbation theory as originated in the overlapping space of excited ( a, b,…) and fundamental ( 0 ) vibronic states unperturbed molecules. This is a very theoretical expression of dynamic correlation between electrons in both interacting systems on the grounds of their local electronic configurations:

47 The energy of dispersive term can be expanded as: Dispersive effect

48 The energy of dispersive term can be expanded as: The leading term of this series results to be: where is the modular component in z coordinate of the electric moment of transition a  0. Dispersive effect

49 The energy of dispersive term can be expanded as: The leading term of this series results to be: where is the modular component in z coordinate of the electric moment of transition a  0. I t arises from the correlation of the instantaneous dipole fluctuations in the charge density of the two nanoscopic bodies. Dispersive effect

50 This term brings up the London dispersion energy as: Dispersive effect

51 This term brings up the London dispersion energy as: For most effects, the London dispersive term is approached as the total dispersive energy Dispersive effect

52 Higher terms in the expansion: are related dipole - quadrupole interactions ( R -8 ), both quadrupole - quadrupole and dipole - octopole ( R -10 ), etc. Dispersive effect

53 Higher terms in the expansion: are related dipole - quadrupole interactions ( R -8 ), both quadrupole - quadrupole and dipole - octopole ( R -10 ), etc. They are not frequently taken into account. Dispersive effect

54 Case B. Overlapping bodies (intermediate distances): It must be observed that exchange interaction E ex is absent as either a first or second order term because bonding is a negligible contribution to energy at large separations.

55 Case B. Overlapping bodies (intermediate distances): It must be observed that exchange interaction E ex is absent as either a first or second order term because bonding is a negligible contribution to energy at large separations. Overlapping of electronic clouds during the approach of two nanoscopic bodies means the increase of electronic repulsion (because both electrostatic effects and the Pauli principle) and attraction (because electron exchange effects arise).

56 Case B. Overlapping bodies (intermediate distances): It must be observed that exchange interaction E ex is absent as either a first or second order term because bonding is a negligible contribution to energy at large separations. Overlapping of electronic clouds during the approach of two nanoscopic bodies means the increase of electronic repulsion (because both electrostatic effects and the Pauli principle) and attraction (because electron exchange effects arise). At intermediate distances, repulsion is usually predominant, although the true behavior is strongly dependent on the character of the system.

57 Case B. Overlapping bodies (intermediate distances): It must be observed that exchange interaction E ex is absent as either a first or second order term because bonding is a negligible contribution to energy at large separations. Overlapping of electronic clouds during the approach of two nanoscopic bodies means the increase of electronic repulsion (because both electrostatic effects and the Pauli principle) and attraction (because electron exchange effects arise). At intermediate distances, repulsion is usually predominant, although the true behavior is strongly dependent on the character of the system. The use of perturbative procedures for accounting such effects must necessarily take into account isotropy features of the interacting electron clouds.

58 Case B. Overlapping bodies (intermediate distances): These cases are treated nowadays by variational procedures as implemented in current program packages.

59 Case C. Prereactive interactions: At very small separations between different nanoscopic bodies the unperturbed Hamiltonian is very similar to the one corresponding to a united molecule. Therefore, effects are mainly originated in Coulomb – exchange interactions among electrons.

60 Case C. Prereactive interactions: At very small separations between different nanoscopic bodies the unperturbed Hamiltonian is very similar to the one corresponding to a united molecule. Therefore, effects are mainly originated in Coulomb – exchange interactions among electrons. A comprehensive treatment is more appropriate than the perturbative solution because pre-reactive effects are usually present and they are hardly considered as a factor in any kind of pairwise development.

61 Case C. Prereactive interactions: A detailed study of the method and the quality of the hypesurface is usually very important.

Approximate calculations of molecular associations

63 Resumé on the physical origin of the main interactions E assoc = E Coul + E ex + E ind + E disp

64 Resumé on the physical origin of the main interactions E assoc = E Coul + E ex + E ind + E disp The important terms for strong interactions at short distances are: E coul is the Coulombic or purely electrostatic interaction energy between polarized zones of molecules

65 Resumé on the physical origin of the main interactions E assoc = E Coul + E ex + E ind + E disp The important terms for strong interactions at short distances are: E coul is the Coulombic or purely electrostatic interaction energy between polarized zones of molecules E ex is the orbital exchange energy if electronic overlap is present

66 E assoc = E Coul + E ex + E ind + E disp The active terms in the case of weak interactions at intermediate distances are: E ind means inductive effect energy of charges in one molecule on the electronic cloud of the other Resumé on the physical origin of the main interactions

67 E assoc = E Coul + E ex + E ind + E disp The active terms in the case of weak interactions at intermediate distances are: E ind means inductive effect energy of charges in one molecule on the electronic cloud of the other E disp means dispersive or London energy, characterizing the so called van der Waals forces Resumé on the physical origin of the main interactions

68 Aproximation by a power series Finding E assoc must be solved in a practical way and is usually developed in a power series of distances among involved nanoscopic bodies: where V n values will depend on the kind of interactions to consider in each case.

69 Practical solutions for neutral spherical or quasi-spherical systems: Lennard - Jones potential: where  is the deepness of the energy well between the two bodies, and  is the distance where E assoc is zero.

70 Buckingham potential: where a, b, c 6 y c 8 are constants. Practical solutions for neutral spherical or quasi-spherical systems:

Practical solutions for neutral isotropic systems: Stockmayer potential: where  A is the angle between R AB joining the centers of mass and the direction of the dipole moment  A, and  is the dihedral angle between planes and AA R AB B A BB AA BB

72 Practical solutions for general cases: If the energy of association is expressed as a series in terms of interactions among atoms i, j,… : it will converge over N atoms, being among them aggregate entities that could qualify as molecules or clusters.

73 Practical solutions for general cases: The “pairwise additive approximation” also allows to write: where potentials are different to those of the series with interactions of more than two bodies.

74 Practical solutions for general cases: The “pairwise additive approximation” also allows to write: where potentials are different to those of the series with interactions of more than two bodies. Nevertheless, this pairwise additive approximation is expected to prevent convergence.

75 Clementi potentials: where, and are adjusted constants for each pair of atoms of types A and B with a net electrostatic charge q. Practical solutions for general cases:

76 Clementi potentials: where, and are adjusted constants for each pair of atoms of types A and B with a net electrostatic charge q. Constants are adjusted against accurate quantum mechanical ab initio calculations. Practical solutions for general cases:

77 Fraga potentials: where A, B, C, and D are adjusted to give the appropriate units,  i is the atomic polarizability, q i the net charge and n i = Z i - q i the number of electrons on atom i. Practical solutions for general cases:

78 Fraga potentials: where A, B, C, and D are adjusted to give the appropriate units,  i is the atomic polarizability, q i the net charge and n i = Z i - q i the number of electrons on atom i. Parameters c i and f i are adjusted to fit previous accurate ab initio values. Practical solutions for general cases:

79 Atom types Either the Clementi’s and Fraga’s potentials, as well as many other approaches for approximate non bonding and bonding energies, are not parameterized by elements but by atom types that mean elements according their particular bonding conditions in the considered nanoscopic system.

80 Hydrogen bond treatments “Hydrogen bond” is an ubiquitous case of molecular interactions behaving sometimes as electron exchange driven (quasi covalent bonding) as well as other times being described as electrostatic associations.

81 Hydrogen bond treatments There are some potential formulas fitting hydrogen bond properties in the case of different classes of atoms involved. Solutions are: 1.A separate hydrogen bond potential, i.e Lennard - Jones, or a Buckingham potential with different parameters.

82 Hydrogen bond treatments There are some potential formulas fitting hydrogen bond properties in the case of different classes of atoms involved. Solutions are: 1.A separate hydrogen bond potential, i.e Lennard - Jones, or a Buckingham potential with different parameters. 2. The use of normal dispersive potentials with parameters dedicated to hydrogen bonding classes of atoms.

Main conclusions

84 Covalent vs. non – covalent associations The border line for the separation between molecules in covalent and non-covalent interactions could be considered as 2 Å for most organic systems.

85 Associative factors The main interacting forces at larger distances could be classified as those between: permanent multipoles (electrostatic)

86 Associative factors The main interacting forces at larger distances could be classified as those between: permanent multipoles (electrostatic) permanent multipoles and induced multipoles (inductive)

87 Associative factors The main interacting forces at larger distances could be classified as those between: permanent multipoles (electrostatic) permanent multipoles and induced multipoles (inductive) instantaneous variable multipoles (dispersive)

88 Energy scales for molecular associations E assoc ranges between 1 and 20 kcal mol -1 (4 and 85 kJ mol -1 or 350 and 7000 cm -1 ) in non covalent molecular aggregates

89 Energy scales for molecular associations E assoc ranges between 1 and 20 kcal mol -1 (4 and 85 kJ mol -1 or 350 and 7000 cm -1 ) in non covalent molecular aggregates van der Waals interactions reach values between 0.1 and 0.2 kcal mol -1 (0.4 and 0.8 kJ mol -1 or 35 and 70 cm -1 )

90 Energy scales for molecular associations E assoc ranges between 1 and 20 kcal mol -1 (4 and 85 kJ mol -1 or 350 and 7000 cm -1 ) in non covalent molecular aggregates van der Waals interactions reach values between 0.1 and 0.2 kcal mol -1 (0.4 and 0.8 kJ mol -1 or 35 and 70 cm -1 ) Hydrogen bonding behaves between 2 and 5 kcal mol -1 (8 and 21 kJ mol -1 or 700 and 1750 cm -1 )

91 Energy scales for molecular associations E assoc ranges between 1 and 20 kcal mol -1 (4 and 85 kJ mol -1 or 350 and 7000 cm -1 ) in non covalent molecular aggregates van der Waals interactions reach values between 0.1 and 0.2 kcal mol -1 (0.4 and 0.8 kJ mol -1 or 35 and 70 cm -1 ) Hydrogen bonding behaves between 2 and 5 kcal mol -1 (8 and 21 kJ mol -1 or 700 and 1750 cm -1 ) Covalent bonds range around 100 kcal mol -1 (420 kJ mol -1 or cm -1 ).