Get Complete Project Material File(s) Now! »

## Post-Hartree-Fock methods

There are three main post-Hartree-Fock methods, Møller-Plesset perturbative methods (MPn), Coupled Cluster approaches and configuration interaction (CI) methods. We present here, briefly, the second-order Møller-Plesset perturbation method (MP2) [52], which we have preferably used in this thesis. This method estimates the correlation of electrons as a perturbative correction to the HF problem. The MP theory is based on the Rayleigh-Schrödinger perturbation theory (RS-PT), considering an undisturbed Hamiltonian (H0) to which a small (weak) perturbation (V ) is added:

H =H0+λV (2.17)

Here λ is an arbitrary real parameter that controls the magnitude of the perturbation. In MP theory, the zeroth-order wave function is the exact eigenfunction of the Fock operator. The perturbation therefore shall recover the correlation energy. If one writes the total wave function |Yi and the total energy E as Taylor series of the λ perturbative factor:

|Yi = Y(0)E + λ Y(1)E + λ(2) Y(2)E + …

E = E(0) + λE(1) + λ(2)E(2) + …,

### Density functional theory DFT

Density Functional Theory offers a different approach to deal with electronic correlation. It uses the electron density ρ(r) instead of the n−electron wave function as the system variable. Electron density is, in principle, an observable physical property of molecules, unlike the wave function which is a mathematical tool without any physical meaning. The electronic energy, E[ρ], is a functional (a function of a function) of electron density, with ρ the electron density, a function of three variables (x, y, z). The Density Functional Theory is based on the two theorems of Hohenberg and Kohn [54]. In the first theorem, they showed that there is an exact correspondence between the electron density and the external potential of a given physical system; the external potential (Vext) is determined with a single density to within a constant. E[ρ] = Vext(r).ρ(r).dr + F (ρ(r)), (2.26)

where ρ(r) is the electron density, F (ρ(r)) is a universal density function which contains the kinetic and Coulomb contribution to the energy.

#### Relativistic effects and Effective core potentials

The regions near the nuclei, the « core regions », are composed primarily of tightly bound core electrons which respond very little to the presence of neighboring atoms, while the remaining volume contains the valence electron density which is involved in the interatomic bonding. Hence, in order to reduce the computational cost of all-electron calculations for elements containing a lot of electrons, namely heavy elements, the core electrons and the potential due to the charge of the nuclei can be replaced by an effective potential. This approach consists in assuming that there is no significant overlap between the core and valence wave functions. The side advantage is that it is possible to get rid of the atomic basis functions describing the core orbitals, thus reducing the computational cost. In the effective core potential approach, only the chemically active valence electrons are dealt with explicitly, because they are the only ones involved in the establishment of chemical bonds, while the core electrons are ’frozen’. There are several pseudopotential formalisms which differ in whether or not conserving the charge in the core region, for further details see [75–78] When dealing with heavy atoms, relativistic effects become quite relevant [79]. There are namely two types of relativistic effects: the scalar relativistic effects and the spin-orbit coupling. In a simplified view, the scalar relativistic effects result from the variation of mass with the velocity of the electrons, which is roughly proportional to the nuclear charge Z. Thus in heavy elements, the scalar relativistic effects lead to a contraction of the core s and p orbitals and in the expansion of the d and f orbitals. Accordingly, a mass-velocity correction is applied to the kinetic energy of the electrons.3. The spin-orbit coupling originates from the interaction of the spin of the electron with this magnetic field due to the relative motion of the charges.

For a heavy atom (as heavy we refer to all atoms beyond the last two rows of the periodic table), the pseudopotential that replaces the core electrons must account for these relativistic effects. These are the so-called Relativistic Effective Core Potentials [77, 78], the parameters of which are adjusted to reproduce all-electron relativistic calculations. The RECPs differ by the choice of the all-electron reference method and the number of core electrons modeled. Namely in the case of Pu, one can choose a « small-core » RECP, ECP60MWB, that was adjusted to multiconfigurational Wood-Boring relativistic calculations and replaces 60 core electrons, leaving the valence 5s, 5p, 5d, 5f, 6s, 6p, 7s electrons active. Alternatively, the « large 5f-in-core » pseudopotential have been optimized for the tetravalent oxidation state of Pu, that incorporates the four unpaired 5f4 electrons into the core, thus turning Pu(IV) into a fictive closed-shell atom. The latter ECP82MWB was used in Chapter 3, knowing it yields very good accuracy for structural parameters and energetics [80].

**Inner-sphere complexation with monoamides**

The optimized structure of [Pu(NO3)4L2] is represented in Figure 3.2 for PEE ligand and structural parameters are listed in Table 3.2 for all the ligands. Additional structural parameters are given in annex A section. In the most stable geometry of these systems, the coordination number of plutonium is 10, with the nitrate groups acting as bidentate ligands. Amide ligands act as monodentate ligands, which interact with plutonium through the carbonyl oxygen. Changing the ligand alkyl chains does not significantly alter the structure. In the gas phase, the P u − OL distance varies from 2.34 to 2.35 Å while the mean value of P u − Onit varies from 2.47 to 2.49 Å. Ligand interatomic distances, such as C-O or C-N distances, also remain similar for all the ligands. In the polar DMA solvent P u − OL distances are shortened by 0.03 – 0.04 Å compared to the gas phase while P u − Onit distances are lengthened by 0.02 – 0.03 Å. These calculated distances are all longer than the solid phase distances reported from XRD of 2.26 Å and 2.44 Å respectively for P u − OL and P u − ON O3 distances in the Pu(IV)-tetranitrate complex with N,N-dibutyl-butanamide (DBBA) [29].

The variation of the complexation energies between the ligands and the tetra-nitrate plutonium complex are reported in Table 3.3. As detailed in computational methods section, energies are given as ligand exchange complexation energies. Energy differences were also calculated by adding empirical dispersion with Grimme’s D3 method to account for dispersion interactions. On the carbonyl side, substituting the methyl by an ethyl group in R1 position (from MMM to EMM or from MEE to EEE) has a small destabilizing.

**Table of contents :**

Abstract

Résumé

Acknowledgments

List of Figures

List of Tables

**1 Introduction **

1.1 Nuclear fuel reprocessing

1.2 Liquid-Liquid extraction technique

1.3 Extraction properties of monoamides

1.4 Why molecular modeling?

1.5 Thesis organization

**2 Quantum chemistry methods **

2.1 The Schrödinger equation

2.2 Wave Function Theory methods

2.2.1 Hartree-Fock Method

2.2.2 Electronic correlation

2.2.3 Post-Hartree-Fock methods

2.3 Density functional theory DFT

2.4 Atomic basis sets

2.5 Relativistic effects and Effective core potentials

2.6 Solvent effects

**3 Quantum chemical study of plutonium nitrates complexes with amides derivatives **

3.1 Computational Methods

3.2 Results and Discussion

3.2.1 Inner-sphere complexation with monoamides

3.2.2 Outer-sphere complexation with monoamides

3.2.3 Complexation with carbamides

3.3 Conclusion

**4 Molecular dynamics simulations **

4.1 Molecular Dynamics simulations

4.2 Molecular mechanics: the Force-Field Models

4.2.1 Bonded interactions

4.2.1.1 Bond Stretching

4.2.1.2 Angle bending

4.2.1.3 Proper dihedral torsion

4.2.1.4 Improper dihedral torsion

4.2.1.5 Cross terms

4.2.2 Non-bonded interactions

4.2.2.1 Electrostatic interactions

4.2.2.2 Van der Waals interactions

4.2.2.3 Polarization

4.2.2.4 Specific terms

4.2.2.5 1 – 4 interactions

4.2.3 Parameterization of a Force Field

4.2.3.1 Overview of the parameterization approach

4.2.3.2 The followed parameterization approach

4.3 Molecular Dynamics in Various Statistical ensembles

4.3.1 Constant temperature: canonical ensemble (NVT)

4.3.2 Constant pressure and temperature: NPT ensemble

4.4 Periodic Boundary Conditions (PBC)

4.5 Physical properties derived from MD simulations

4.5.1 The bulk density

4.5.2 Heat of Vaporization

4.5.3 Radial distribution function (RDF)

4.5.4 Diffusion coefficient

**5 Force field for alkanes and amides derivatives **

5.1 Parametrization methodology

5.2 Results of parametrization: bonded and non bonded parameters

5.3 Force-field evaluation: physical properties

5.3.1 Density and heat of vaporization

5.3.2 Structural properties

5.4 Molecular properties of dodecane/monoamides mixtures

5.5 Conclusions

**References**