Algorithms for Predicting the Physical Properties of Nanocrystals and Large Clusters

JAMES R. CHELIKOWSKY

Center for Computational Materials, Institute for Computational

Engineering and Sciences, University of Texas at Austin, Austin, Texas 78712, USA; Departments of Physics and Chemical Engineering, University of Texas at Austin, Austin, Texas 78712, USA

1.1 Introduction

Quantum confinement offers one the opportunity to alter physical properties of matter without changing the chemical composition. For example, quantum confinement can be used to tune the optical gap across the visible spectrum in CdSe nanocrystals or it can be used to modify the magnetic moment of ferromagnetic clusters.

However, constructing an efficacious algorithm for predicting the role of quantum confinement and its role in determining the properties of nanocrystals is a difficult task owing to the complexity of nanocrystals, which often contain thousands of atoms. Here, I will illustrate new algorithms for nanoscale systems. These algorithms are especially designed for highly parallel platforms. The nature of these algorithms will be discussed and applied to complex systems such as magnetic clusters, and doped semiconductor nanocrystals.

1.2 The Electronic Structure Problem

Many properties of materials can be predicted by a solution of the Kohn–Sham equation:

[MATHEMATICAL EXPRESSION OMITTED], (1.1)

where Vpion is an ionic pseudopotential, VH is the Hartree or Coulomb potential, and Vxc is the exchange-correlation potential. The Hartree and exchange-correlation potentials can be determined from the electronic charge density. The density is given by

[MATHEMATICAL EXPRESSION OMITTED]. (1.2)

The summation is over all occupied states. The Hartree potential is determined by

[MATHEMATICAL EXPRESSION OMITTED]. (1.3)

This term can be interpreted as the electrostatic interaction of a point charge with the charge density of the system.

A physical interpretation of the exchange-correlation potential is more difficult as it has no classical analogue. This potential can be evaluated using a local density approximation. The central tenet of this approximation is that the total exchange-correlation energy may be written as a universal functional of the density:

[MATHEMATICAL EXPRESSION OMITTED], (1.4)

where εxc is the exchange-correlation energy density. Exc and εxc are to be interpreted as depending solely on the charge density. The exchangecorrelation potential, Vxc, is then obtained as [MATHEMATICAL EXPRESSION OMITTED].

It is not difficult to solve the Kohn–Sham equation (eqn (1.1)) for an atom as the atomic charge density can be taken to be spherically symmetric. Thus, the Kohn–Sham problem reduces to solving a one-dimensional problem. The Hartree and exchange-correlation potentials can be iterated to form a self-consistent field. This atomic solution provides the input to construct a pseudo-potential representing the effect of the core electrons and nucleus. The "ion-core" pseudopotential, Vpion, can be transferred to other systems such as molecules and nanocrystals.

The Kohn–Sham equations represent a nonlinear, self-consistent eigenvalue problem. Typically, a solution is obtained by first approximating the Hartree and exchange-correlation potentials using a superposition of atomic charge densities. The Kohn–Sham equation is then solved using these approximate potentials. From the solution, new wavefunctions and charge densities are obtained and used to construct updated Hartree and exchange-correlation potentials. The process is repeated until the "input" and "output" potentials agree and a self-consistent solution is realised. At this point, the total electronic energy can be computed along with a variety of other electronic properties. Once the Kohn–Sham equation is solved, the total electronic energy, ET, of the system can be evaluated from

[MATHEMATICAL EXPRESSION OMITTED]. (1.5)

The structural energy can be obtained by adding the ion-core electrostatic terms. The forces can be obtained by taking the derivative of the energy with respect to position as from the Hellmann–Feynman theorem.

1.3 Algorithms for Solving the Kohn–Sham Equation

The Kohn–Sham equation as cast in eqn (1.1) can be solved using a variety of techniques. Often, the wavefunctions can be expanded in a basis such as plane waves or gaussians and the resulting secular equations can be solved using standard diagonalisation packages such as those found in VASP.

Here, we focus on a different approach. We solve the Kohn–Sham equation without resort to an explicit basis. We solve for the wavefunctions on a grid with a fixed domain, which encompasses the physical system of interest. The grid need not be uniform, but it greatly simplifies the problem if it is. The wavefunctions outside of the domain are required to vanish for confined sys- tems or assume periodic boundary conditions for systems with translational symmetry. In contrast to methods employing an explicit basis, such boundary conditions are easily incorporated. In particular, real-space methods do not require the use of supercells for localised systems. As such, charged systems can easily be examined without considering any electrostatic divergences. The Hartree potential is also solved on the grid from eqn (1.3) using conjugate-gradient minimisation. For confined systems, the boundary condition for this Poisson equation can be obtained by first performing a multipole expansion of the electronic charge density, the Hartree potential on the boundary of the domain can then be evaluated using the multipoles. A few...