Education

Partition Spin-Density Functional Theory

Martín MosqueraChem. PhD student

Martín Mosquera
Chem. PhD student

Molecules are made of atoms, yet quantum mechanics, the theory that describes molecules, treats a molecule as a single object. However, when molecules collide at high energies, atoms or molecular fragments are scattered, revealing that there are atoms in a molecule. Thus one may wonder: "What is an atom inside a molecule?" or "How many electrons belong to an atom in a molecule?". A unique answer to this question may be unexistent, but certain criteria can be established to give an answer. In our group we believe the electronic density is all needed to define an atom in a molecule. Density-functional theory (DFT) does, of course, employ such variable to calculate the state of lowest energy of a molecule. DFT despite is exact in principle, needs approximations to the exchange-correlation energy, the missing piece of the puzzle. Most approximations fail to describe molecular dissociation, a process in which an atom (or a fragment) is pulled away from a molecule. For example, we know molecular hydrogen (H_2) dissociates into two hydrogen atoms. Thus, the energy of the dissociated molecule is just twice the energy of a hydrogen atom; many approximations within DFT fails this simple test.

Elliot et al. [1] proposed partition DFT (PDFT), a variation of DFT that, among other features, determines the electronic density of an atom (or fragment) in a molecule and introduces a new way to calculate the ground-state energy of a molecule. In PDFT, an atom, or a fragment, is defined as a Hamiltonian that describes the interaction between the electrons and nucleus of the atom (or nuclei of the fragment), and their kinetic and repulsion energies. The atoms are assumed to be isolated from the rest of the world, but they all "feel" a potential, termed partition potential, whose purpose is to ensure that the electronic densities of the atoms add up to the exact total density of the real system.  The partition potential is shown to be related to the residual energy, called partition energy, that the isolated atoms need to match the true energy of the system. Inspired by Elliot et al. [1], we extended PDFT to include electronic spin-densities.  An electron spin-density is roughly speaking the number of spin-up or spin-down electrons  per volume. We show that the density-functionals of PDFT, i.e., the sum of isolated-fragment energies and the partition energy, are easily extended to the spin-polarized regime. The total spin-density of a molecule is thus partitioned into spin-densities that are localized around their respective fragment nuclei. A variation of any of these densities is equivalent to a variation in the total density. However, a special set of densities that are functionals of the total density exists; this set is employed to suggest a new way to estimate the partition potential. We illustrated that the estimation of the response of the fragment density to perturbations in the total density is important to estimate the partition potential.

Partition spin-DFT (PSDFT) [2], our extension of PDFT, includes a technique to introduce external potentials like magnetic or electric fields. Every fragment in the molecule must be subject to the same external field, allowing us, for example, to employ the partition potential of the molecule in absence of fields as a zero-order approximation to estimate properties like polarizabilities when the external field is weak. The future challenge for our work is to find approximations to the partition potential as a function of the spin-densities, and implement it to simulate 3D molecules. For this sake, the many approximations for the exchange-correlation energy can be used. However, approximations to the kinetic energy are also required, which are usually harder to find. The kinetic effects are not only important to us, but for traditional DFT as well.

References
[1] Partition Density Functional Theory, P. Elliott, K. Burke, M.H. Cohen, and A. Wasserman, Phys. Rev. A 82, 024501 (2010).

[2] Partition density functional theory and its extension to the spin-polarized case, M. Mosquera, and A. Wasserman, Mol. Phys. 111, 505-515 (2013).

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