Extended free energy Lagrangians are proposed for first principles molecular dynamics simulations at finite electronic temperatures for plane-wave pseudopotential and local orbital density matrix based calculations. Thanks to the extended Lagrangian description the electronic degrees of freedom can be integrated by stable geometric schemes that conserve the free energy. For the local orbital representations both the nuclear and electronic forces have simple and numerically efficient expressions that are well suited for reduced complexity calculations. A rapidly converging recursive Fermi operator expansion method that does not require the calculation of eigenvalues and eigenfunctions for the construction of the fractionally occupied density matrix is discussed. An efficient expression for the Pulay force that is valid also for density matrices with fractional occupation occurring at finite electronic temperatures is also demonstrated.
Molecular dynamics simulations are widely used in materials science, chemistry and molecular biology. Unfortunately, the most accurate molecular dynamics methods that are based on self-consistent field (SCF) theory [1] are often limited by some fundamental shortcomings such as a very high computational cost or unbalanced phase space trajectories with unphysical hysteresis effects, numerical instabilities and a systematic long-term energy drift [2,3]. Recently an extended Lagrangian framework for first principles molecular dynamics simulations was proposed that overcomes most of the previous problems [4]. The new framework was originally designed for ground state Born-Oppenheimer molecular dynamics simulations based on Hartree-Fock [5] or density functional theory [6,7] at zero electronic temperatures. In this paper we supplement previous work by proposing extended free energy Lagrangians for first principles molecular dynamics simulations also at finite electronic temperatures [8,9].
Our extended Lagrangian free energy dynamics is formulated and demonstrated both for plane-wave pseudopotential and local orbital density matrix based calculations. For the density matrix formulation our approach is given by a generalization of the original extended Lagrangian framework [4] using a density matrix representation of the electronic degrees of freedom with fractional occupation of the states. The plane-wave formulation is based on the wave function extended Lagrangian molecular dynamics method that recently was proposed by Steneteg et al. [10].
First we formulate the extended Lagrangian approach to first principles molecular dynamics at finite electronic * Email: amn@lanl.gov temperatures for local orbital density matrix representations and thereafter for plane wave pseudopotential calculations. A numerically efficient expression for a generalized Pulay force that does not require idempotent density matrices is demonstrated and a recursive Fermi operator expansion algorithm for the density matrix is discussed. Thereafter a Verlet integration scheme for the equations of motion is presented. Finally, we illustrate the extended Lagrangian free energy dynamics by some examples before giving a brief summary.
The first principles molecular dynamics scheme presented in this paper is a finite electronic temperature generalization of the extended Lagrangian framework of time-reversible Born-Oppenheimer molecular dynamics [3,4]. In many ways the generalization is straightforward, except for an additional entropy term, problems arising in the construction of the density matrix, the calculation of the Pulay force for fractional occupations and the phase alignment problem in the integration of the wave functions [10]. Our two different formulations of extended Lagrangian free energy molecular dynamics are based on either an underlying local orbital representation or a plane wave pseudo potential description. Alternative free energy formulations suitable for extended Lagrangian self-consistent tight-binding molecular dynamics simulations [11,12] will be presented elsewhere.
an electronic temperature, T e , by a density matrix (DM) extended free energy (XFE) Lagrangian, which we define as
The extended dynamical variables P and the velocity Ṗ in the Lagrangian L XFE DM are auxiliary electronic degrees of freedom evolving in a harmonic potential centered around the self-consistent free energy ground-state solution D [4]. Here P , Ṗ , and the self-consistent free energy ground state D, are orthogonal density matrix representations of the electronic degrees of freedom. The relation between D and the non-orthogonal (atomic orbital) representation, D, is given by the congruence transformation
where Z is a congruence factor determined by the overlap matrix S (e.g. see Ref. [13]) from the condition that
The constants µ and ω are fictitious electron mass and frequency parameters. The potential U [R; D] is defined at the self-consistent field (SCF) electron ground state D, at which the free energy functional
has its minimum for a given nuclear configuration, R = {R i }, [8,9] under the constraints of correct electron occupation, Tr[D] = N occ . The electronic temperature T e can be different from the ionic temperature T ion . We assume that U [R; D] is the total electronic energy, including nuclear ion-ion repulsions, in self-consistent density functional theory, Hartree-Fock theory or some of their extensions that are based on an underlying SCF description. The mean field entropy term in L XFE ,
is included to provide the variationally correct energetics and dynamics [14,15]. The factor 2 above is used for restricted calculations (not spin polarized) when each state is doubly occupied, which is assumed throughout the paper.
The time evolution of the system described by L XFE DM is determined by the Euler-Lagrange equations,
which give the equations of motion for the nuclear and electronic degrees of freedom,
µ P = µω 2 (D -P )
This content is AI-processed based on open access ArXiv data.
