Projection-Based Embedding Theory for Solving Kohn--Sham Density Functional Theory

Abstract

Quantum embedding theories are playing an increasingly important role in bridging different levels of approximation to the many-body Schrödinger equation in physics, chemistry and materials science. In this paper, we present a linear algebra perspective of the recently developed projection based embedding theory (PET) [Manby et al, J. Chem. Theory Comput. 8, 2564, 2012, restricted to the context of Kohn-Sham density functional theory. By partitioning the global degrees of freedom into a "system" part and a "bath" part, and by choosing a proper projector from the bath, PET is an in-principle exact formulation to confine the calculation to the system part only, and hence can be performed with reduced computational cost. Viewed from the perspective of domain decomposition methods, one particularly interesting feature of PET is that it does not enforce a boundary condition explicitly, and remains applicable even when the discretized Hamiltonian matrix is dense, such as in the context of the planewave discretization. In practice, the accuracy of PET depends on the accuracy of the bath projector. Based on the linear algebra reformulation, we develop a first order perturbation correction to the projector from the bath to improve its accuracy. Numerical results for real chemical systems indicate that with a proper choice of reference system used to compute the bath projector, the perturbatively corrected PET can be sufficiently accurate even when strong perturbation is applied to very small systems, such as the computation of the ground state energy of a SiH 3 F molecule, using a SiH 4 molecule as the reference system. Since the dimension of Bis a diagonal matrix containing all the eigenvalues of H| B ⊥ 0 ordered non-decreasingly, and Ψ ∈ C N ×(N −N b ) is given by orthogonal columns of an unitary matrix in the subspace 6 7

Projection-Based Wavefunction-in-DFT Embedding

“…59 The approach is implemented in the widely used Molpro software package, 60 allowing straightforward application of projection-based WF-in-DFT embedding for quantum chemical studies. 58,65,[67][68][69][70][71][72][73][74][75][76][77][78][79][80] Looking forward, we anticipate continued refinement of the projection-based embedding methodology, as well as increasingly widespread application in diverse areas of chemistry, biology, and materials science.…”

“…Among the most important aspects of projection-based embedding is that it enables robust and efficient WF-in-DFT calculations on complex chemical systems. To date, projection-based embedding has been used in application studies by the groups of the authors ,,− and others, − including applications to periodic systems. , We now summarize applications of the method to transition-metal catalysis, enzyme catalysis, and electrochemistry.…”

“…Continued technical advances, including atomic orbital basis set truncation and even-handed subsystem partitioning, have improved the efficiency and robustness of the method, and the development of the gradient theory further broadens opportunities for chemical applications . The approach is implemented in the widely used Molpro software package, allowing straightforward application of projection-based WF-in-DFT embedding for quantum-chemical studies. ,,− …”

KSSOLV 2.0: An efficient MATLAB toolbox for solving the Kohn-Sham equations with plane-wave basis set

“…Over the last decade, KSSOLV has be-come a useful research and teaching tool for studying electronic structures of molecules and solids and developing new methods for solving the Kohn-Sham problem, as evidenced by an increasing number of publications that use KSSOLV to perform numerical experiments required to demonstrate improved convergence or better accuracy of new theoretical methods and numerical algorithms. Examples of such developments include tensor hypercontraction algorithm [4], improved generalized Davidson algorithm [5], elliptic preconditioner in self-consistent field iteration [6], low rank approximation in G 0 W 0 calculations [7], perturbation theory [8], quantum embedding theory [9], quantum computation [10], linear-response time-dependent density functional theory [11,12], phonon calculations [13], proximal gradient method [14], trace-penalty minimization method [15].…”

Spectral methods for nonlinear functionals and functional differential equations

“…Such map, which seems a rather abstract mathematical concept, plays a fundamental role in many areas of mathematical physics and applied sciences. In fact, nonlinear functionals were used, for example, by Wiener to describe Brownian motion mathematically [100], by Hohenberg and Kohn [47] to reduce the dimensionality of the Schrödinger equation in many-body quantum systems [56,67], by Hopf to describe the statistical properties of turbulence [2,48,63], and by Bogoliubov to model systems of interacting bosons in superfluid liquid helium [12,84]. Applications of nonlinear functionals to other areas of mathematical physics can be found in [3,37,52,54,59,92].…”