Constrained Density Functional Theory: A Potential-Based Self-Consistency Approach

Chemical reactions, charge transfer reactions, and magnetic materials are notoriously difficult to describe within Kohn-Sham density functional theory, which is strictly a ground-state technique. However, over the last few decades, an approximate method known as constrained density functional theory (cDFT) has been developed to model low-lying excitations linked to charge transfer or spin fluctuations. Nevertheless, despite becoming very popular due to its versatility, low computational cost, and availability in numerous software applications, none of the previous cDFT implementations is strictly similar to the corresponding ground-state self-consistent density functional theory: the target value of constraints (e.g., local magnetization) is not treated equivalently with atomic positions or lattice parameters. In the present work, by considering a potential-based formulation of the self-consistency problem, the cDFT is recast in the same framework as Kohn-Sham DFT: a new functional of the potential that includes the constraints is proposed, where the constraints, the atomic positions, or the lattice parameters are treated all alike, while all other ingredients of the usual potential-based DFT algorithms are unchanged, thanks to the formulation of the adequate residual. Tests of this approach for the case of spin constraints (collinear and noncollinear) and charge constraints are performed. Expressions for the derivatives with respect to constraints (e.g., the spin torque) for the atomic forces and the stress tensor in cDFT are provided. The latter allows one to study striction effects as a function of the angle between spins. We apply this formalism to body-centered cubic iron and first reproduce the well-known magnetization amplitude as a function of the angle between local magnetizations. We also study stress as a function of such an angle. Then, the local collinear magnetization and the local atomic charge are varied together. Since the atomic spin magnetizations, local atomic charges, atomic positions, and lattice parameters are treated on an equal footing, this formalism is an ideal starting point for the generation of model Hamiltonians and machine-learning potentials, computation of second or third derivatives of the energy as delivered from density-functional perturbation theory, or for second-principles approaches.