Modelings

Modelings for a better imagination of matter  

Electron side - Density Functional Theroy

For atoms and molecules, a wavefunction, derived from the Schrodinger equation, is used to describe all electrons and nuclei (N particles) in a system has to consider 3N coordinates in Cartesian basis. It is quite heavy in terms of resources of calculations and still not accessible in today’s technology.

The energy operator, H, is composed of two kinds of terms, (i) kinetic for electrons and nuclei, (ii) potential for all possible interactions between nuclei and between electrons and nuclei and (iii) exchange correlation term describing the interactions between electrons. The exchange-correlation is a complex term designing the electron-electron interactions, this is due to the wave-particle duality of electrons, many approximations exist for this term e.g. Hartree-Fock, local density approximation (LDA), Generalized Gradient approximation (GGA) e.g. PBE and some hybrid forms like B3LYP. Other approximations are often used to help solving the equation for systems constituted by many particles. Born-Oppenheimer is one of the first approximations stating that nuclei are much heavier than electrons and are supposed fixed in the space compared to moving electrons. DFT allows to overcome the astronomical number of particles for real systems with many atoms by considering the electronic density instead of the total wavefunction of the system departing from the assumption that the total energy of the system is a unique functional of the electron density, Kohn and Shan theorem. DFT is the most used theory nowadays, it describes the three-dimensional electronic density of atoms and molecules. The ground-state wave function that derives and its corresponding energy permits the calculation of spectroscopic observables, adsorption energies, and reactivity. Among these observables, NMR parameters are calculated using specific operators, they include the chemical shielding, the quadrupolar coupling, the indirect dipolar coupling and obviously the direct one. Some programs deal with crystals while in others it is possible to do the calculations on a cluster of atoms, dynamics may be added to simulate the effect of temperature. Some programs are very popular in the field: CASTEP, VASP, GAUSSIAN, CRYSTAL etc. Some of them describe the electronic shells using pseudo-potential plane waves with a certain cut-off energy, for NMR calculation, Gauge Including Projector Augmented Waves (GIPAW) methods is used to consider the inner electron shell and represent the atoms well close to the nuclei prior to calculations. Other programs describe the electrons with localized orbital Gaussian functions. Dispersion corrections including London, Keesom, Debye and van Der Waals terms may also be added to correct the electronic density considering short range interactions. The combination of NMR, spin dynamics and DFT on structures issued from XRD opened the door for a new domain called NMR crystallography that allowed the resolution of some XRD structures based on the NMR parameters. Note that DFT deos not consider temperature. Molecular dynamics are considered by adding the Newton equations of motion for atoms allowing them to distribute on Boltzmann energy levels depending on the temperature set.... Reactivity modelling is still under development.

Nuclear spin side - Spin Dynamics

The dynamics of nuclear spins are described by the Liouville-von Neuman equation of motion. This equation describes the evolution of the so-called spin density operator,  under the action of a group of Hamiltonians. The time evolution of the spin density operator can be written in terms of propagators.

 The effective Hamiltonian approximation considers time independent Hamiltonians using Average Hamiltonian theory-Magnus series or Floquet theory-Fourier series. Among the most popular programs in the field, SIMPSON and SPINACH. The aim of these simulations is to give insights on the dynamics of spins under the effect of internal and external Hamiltonians and simulate their answer to understand and optimize experimental methods.

It helps in both (i) finding the optimal conditions for an NMR sequence, thus decreasing the time consumed on the spectrometer for optimization and (ii) extracting precise information on the size of the internal Hamiltonians, a widely spread example is the extraction of interatomic distances, when dipolar couplings are involved in a system of spins. Nowadays, optimal control approaches are applied to find optimal NMR sequences for systems of interest when looking for specific information.

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