Anders M. N. Niklasson

Expansion algorithm for the density matrix

A purification algorithm for expanding the single-particle density matrix in terms of the Hamiltonian operator is proposed. The scheme works with a predefined occupation and requires less than half the number of matrix-matrix multiplications compared to existing methods at low (10%) and high (g90%) occupancies. The expansion can be used with a fixed chemical potential, in which case it is an asymmetric generalization of and a substantial improvement over grand canonical McWeeny purification. It is shown that the computational complexity, measured as the number of matrix multiplications, essentially is independent of system size even for metallic materials with a vanishing band gap.

Time-reversible Born-Oppenheimer molecular dynamics.

We present a time-reversible Born-Oppenheimer molecular dynamics scheme, based on self-consistent Hartree-Fock or density functional theory, where both the nuclear and the electronic degrees of freedom are propagated in time. We show how a time-reversible adiabatic propagation of the electronic degrees of freedom is possible despite the nonlinearity and incompleteness of the self-consistent field procedure. With a time-reversible lossless propagation the simulated dynamics is stabilized with respect to a systematic long-term energy drift and the number of self-consistency cycles can be kept low thanks to a good initial guess given from the electronic propagation. The proposed molecular dynamics scheme therefore combines a low computational cost with a physically correct time-reversible representation, which preserves a detailed balance between propagation forwards and backwards in time.

Trace resetting density matrix purification in O(N) self-consistent-field theory

A new approach to linear scaling construction of the density matrix is proposed, based on trace resetting purification of an effective Hamiltonian. Trace resetting is related to the trace preserving canonical purification scheme of Palser and Manolopoulos [Phys. Rev. B 58, 12704 (1999)] in that they both work with a predefined occupation number and do not require adjustment or prior knowledge of the chemical potential. In the trace resetting approach, trace conservation is not strictly enforced, allowing greater flexibility in the choice of purification polynomial and improved performance for Hamiltonian systems with high or low filling. However, optimal polynomials may in some cases admit unstable solutions, requiring a resetting mechanism to bring the solution back into the domain of convergent purification. A quartic trace resetting method is developed, along with analysis of stability and error accumulation due to incomplete sparse-matrix methods that employ a threshold τ to achieve sparsity. It is ar...

Representation independent algorithms for molecular response calculations in time-dependent self-consistent field theories

Four different numerical algorithms suitable for a linear scaling implementation of time-dependent Hartree-Fock and Kohn-Sham self-consistent field theories are examined. We compare the performance of modified Lanczos, Arooldi, Davidson, and Rayleigh quotient iterative procedures to solve the random-phase approximation (RPA) (non-Hermitian) and Tamm-Dancoff approximation (TDA) (Hermitian) eigenvalue equations in the molecular orbital-free framework. Semiempirical Hamiltonian models are used to numerically benchmark algorithms for the computation of excited states of realistic molecular systems (conjugated polymers and carbon nanotubes). Convergence behavior and stability are tested with respect to a numerical noise imposed to simulate linear scaling conditions. The results single out the most suitable procedures for linear scaling large-scale time-dependent perturbation theory calculations of electronic excitations.

Density Matrix Perturbation Theory

An orbital-free quantum perturbation theory is proposed. It gives the response of the density matrix upon variation of the Hamiltonian by quadratically convergent recursions based on perturbed projections. The technique allows treatment of embedded quantum subsystems with a computational cost scaling linearly with the size of the perturbed region, O(N(pert.)), and as O(1) with the total system size. The method allows efficient high order perturbation expansions, as demonstrated with an example involving a 10th order expansion. Density matrix analogs of Wigners 2n+1 rule are also presented.

Modeling the actinides with disordered local moments

A first-principles disordered local moment (DLM) picture within the local-spin-density and coherent potential approximations of the actinides is presented. The parameter-free theory gives an accurate description of bond lengths and bulk modulus. The case of

Energy conserving, linear scaling Born-Oppenheimer molecular dynamics

\ensuremath{\delta}

Lagrangian formulation with dissipation of Born-Oppenheimer molecular dynamics using the density-functional tight-binding method

-Pu is studied in particular, and the calculated density of states is compared to data from photoelectron spectroscopy. The relation between the DLM description, the dynamical mean-field approach, and spin-polarized magnetically ordered modeling is discussed.

Ab Initio Linear Scaling Response Theory: Electric Polarizability by Perturbed Projection

Born-Oppenheimer molecular dynamics simulations with long-term conservation of the total energy and a computational cost that scales linearly with system size have been obtained simultaneously. Linear scaling with a low pre-factor is achieved using density matrix purification with sparse matrix algebra and a numerical threshold on matrix elements. The extended Lagrangian Born-Oppenheimer molecular dynamics formalism [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008)] yields microcanonical trajectories with the approximate forces obtained from the linear scaling method that exhibit no systematic drift over hundreds of picoseconds and which are indistinguishable from trajectories computed using exact forces.

Direct energy functional minimization under orthogonality constraints

An important element determining the time requirements of Born-Oppenheimer molecular dynamics (BOMD) is the convergence rate of the self-consistent solution of Roothaan equations (SCF). We show here that improved convergence and dynamics stability can be achieved by use of a Lagrangian formalism of BOMD with dissipation (DXL-BOMD). In the DXL-BOMD algorithm, an auxiliary electronic variable (e.g., the electron density or Fock matrix) is propagated and a dissipative force is added in the propagation to maintain the stability of the dynamics. Implementation of the approach in the self-consistent charge density functional tight-binding method makes possible simulations that are several hundred picoseconds in lengths, in contrast to earlier DFT-based BOMD calculations, which have been limited to tens of picoseconds or less. The increase in the simulation time results in a more meaningful evaluation of the DXL-BOMD method. A comparison is made of the number of iterations (and time) required for convergence of the SCF with DXL-BOMD and a standard method (starting with a zero charge guess for all atoms at each step), which gives accurate propagation with reasonable SCF convergence criteria. From tests using NVE simulations of C(2)F(4) and 20 neutral amino acid molecules in the gas phase, it is found that DXL-BOMD can improve SCF convergence by up to a factor of two over the standard method. Corresponding results are obtained in simulations of 32 water molecules in a periodic box. Linear response theory is used to analyze the relationship between the energy drift and the correlation of geometry propagation errors.