265 research outputs found
An efficient and accurate decomposition of the Fermi operator
We present a method to compute the Fermi function of the Hamiltonian for a
system of independent fermions, based on an exact decomposition of the
grand-canonical potential. This scheme does not rely on the localization of the
orbitals and is insensitive to ill-conditioned Hamiltonians. It lends itself
naturally to linear scaling, as soon as the sparsity of the system's density
matrix is exploited. By using a combination of polynomial expansion and
Newton-like iterative techniques, an arbitrarily large number of terms can be
employed in the expansion, overcoming some of the difficulties encountered in
previous papers. Moreover, this hybrid approach allows us to obtain a very
favorable scaling of the computational cost with increasing inverse
temperature, which makes the method competitive with other Fermi operator
expansion techniques. After performing an in-depth theoretical analysis of
computational cost and accuracy, we test our approach on the DFT Hamiltonian
for the metallic phase of the LiAl alloy.Comment: 8 pages, 7 figure
Computing the Absolute Gibbs Free Energy in Atomistic Simulations: Applications to Defects in Solids
The Gibbs free energy is the fundamental thermodynamic potential underlying
the relative stability of different states of matter under constant-pressure
conditions. However, computing this quantity from atomic-scale simulations is
far from trivial. As a consequence, all too often the potential energy of the
system is used as a proxy, overlooking entropic and anharmonic effects. Here we
discuss a combination of different thermodynamic integration routes to obtain
the absolute Gibbs free energy of a solid system starting from a harmonic
reference state. This approach enables the direct comparison between the free
energy of different structures, circumventing the need to sample the transition
paths between them. We showcase this thermodynamic integration scheme by
computing the Gibbs free energy associated with a vacancy in BCC iron, and the
intrinsic stacking fault free energy of nickel. These examples highlight the
pitfalls of estimating the free energy of crystallographic defects only using
the minimum potential energy, which overestimates the vacancy free energy by
60% and the stacking-fault energy by almost 300% at temperatures close to the
melting point
A hybrid approach to Fermi operator expansion
In a recent paper we have suggested that the finite temperature density
matrix can be computed efficiently by a combination of polynomial expansion and
iterative inversion techniques. We present here significant improvements over
this scheme. The original complex-valued formalism is turned into a purely real
one. In addition, we use Chebyshev polynomials expansion and fast summation
techniques. This drastically reduces the scaling of the algorithm with the
width of the Hamiltonian spectrum, which is now of the order of the cubic root
of such parameter. This makes our method very competitive for applications to
ab-initio simulations, when high energy resolution is required.Comment: preprint of ICCMSE08 proceeding
Efficient methods and practical guidelines for simulating isotope effects
The shift in chemical equilibria due to isotope substitution is often
exploited to gain insight into a wide variety of chemical and physical
processes. It is a purely quantum mechanical effect, which can be computed
exactly using simulations based on the path integral formalism. Here we discuss
how these techniques can be made dramatically more efficient, and how they
ultimately outperform quasi-harmonic approximations to treat quantum liquids
not only in terms of accuracy, but also in terms of computational efficiency.
To achieve this goal we introduce path integral quantum mechanics estimators
based on free energy perturbation, which enable the evaluation of isotope
effects using only a single path integral molecular dynamics trajectory of the
naturally abundant isotope. We use as an example the calculation of the free
energy change associated with H/D and 16O/18O substitutions in liquid water,
and of the fractionation of those isotopes between the liquid and the vapor
phase. In doing so, we demonstrate and discuss quantitatively the relative
benefits of each approach, thereby providing a set of guidelines that should
facilitate the choice of the most appropriate method in different, commonly
encountered scenarios. The efficiency of the estimators we introduce and the
analysis that we perform should in particular facilitate accurate ab initio
calculation of isotope effects in condensed phase systems
Evaluating functions of positive-definite matrices using colored noise thermostats
Many applications in computational science require computing the elements of
a function of a large matrix. A commonly used approach is based on the the
evaluation of the eigenvalue decomposition, a task that, in general, involves a
computing time that scales with the cube of the size of the matrix. We present
here a method that can be used to evaluate the elements of a function of a
positive-definite matrix with a scaling that is linear for sparse matrices and
quadratic in the general case. This methodology is based on the properties of
the dynamics of a multidimensional harmonic potential coupled with colored
noise generalized Langevin equation (GLE) thermostats. This "thermostat"
(FTH) approach allows us to calculate directly elements of functions of a
positive-definite matrix by carefully tailoring the properties of the
stochastic dynamics. We demonstrate the scaling and the accuracy of this
approach for both dense and sparse problems and compare the results with other
established methodologies.Comment: 8 pages, 4 figure
Efficient first-principles calculation of the quantum kinetic energy and momentum distribution of nuclei
Light nuclei at room temperature and below exhibit a kinetic energy which
significantly deviates from the predictions of classical statistical mechanics.
This quantum kinetic energy is responsible for a wide variety of isotope
effects of interest in fields ranging from chemistry to climatology. It also
furnishes the second moment of the nuclear momentum distribution, which
contains subtle information about the chemical environment and has recently
become accessible to deep inelastic neutron scattering experiments. Here we
show how, by combining imaginary time path integral dynamics with a carefully
designed generalized Langevin equation, it is possible to dramatically reduce
the expense of computing the quantum kinetic energy. We also introduce a
transient anisotropic Gaussian approximation to the nuclear momentum
distribution which can be calculated with negligible additional effort. As an
example, we evaluate the structural properties, the quantum kinetic energy, and
the nuclear momentum distribution for a first-principles simulation of liquid
water
Theoretical prediction of the homogeneous ice nucleation rate: disentangling thermodynamics and kinetics
Estimating the homogeneous ice nucleation rate from undercooled liquid water
is at the same time crucial for understanding many important physical phenomena
and technological applications, and challenging for both experiments and
theory. From a theoretical point of view, difficulties arise due to the long
time scales required, as well as the numerous nucleation pathways involved to
form ice nuclei with different stacking disorders. We computed the homogeneous
ice nucleation rate at a physically relevant undercooling for a single-site
water model, taking into account the diffuse nature of ice-water interfaces,
stacking disorders in ice nuclei, and the addition rate of particles to the
critical nucleus.We disentangled and investigated the relative importance of
all the terms, including interfacial free energy, entropic contributions and
the kinetic prefactor, that contribute to the overall nucleation rate.There has
been a long-standing discrepancy for the predicted homogeneous ice nucleation
rates, and our estimate is faster by 9 orders of magnitude compared with
previous literature values. Breaking down the problem into segments and
considering each term carefully can help us understand where the discrepancy
may come from and how to systematically improve the existing computational
methods
Fine Tuning Classical and Quantum Molecular Dynamics using a Generalized Langevin Equation
Generalized Langevin Equation (GLE) thermostats have been used very
effectively as a tool to manipulate and optimize the sampling of thermodynamic
ensembles and the associated static properties. Here we show that a similar,
exquisite level of control can be achieved for the dynamical properties
computed from thermostatted trajectories. By developing quantitative measures
of the disturbance induced by the GLE to the Hamiltonian dynamics of a harmonic
oscillator, we show that these analytical results accurately predict the
behavior of strongly anharmonic systems. We also show that it is possible to
correct, to a significant extent, the effects of the GLE term onto the
corresponding microcanonical dynamics, which puts on more solid grounds the use
of non-equilibrium Langevin dynamics to approximate quantum nuclear effects and
could help improve the prediction of dynamical quantities from techniques that
use a Langevin term to stabilize dynamics. Finally we address the use of
thermostats in the context of approximate path-integral-based models of quantum
nuclear dynamics. We demonstrate that a custom-tailored GLE can alleviate some
of the artifacts associated with these techniques, improving the quality of
results for the modelling of vibrational dynamics of molecules, liquids and
solids
Atom-Density Representations for Machine Learning
The applications of machine learning techniques to chemistry and materials
science become more numerous by the day. The main challenge is to devise
representations of atomic systems that are at the same time complete and
concise, so as to reduce the number of reference calculations that are needed
to predict the properties of different types of materials reliably. This has
led to a proliferation of alternative ways to convert an atomic structure into
an input for a machine-learning model. We introduce an abstract definition of
chemical environments that is based on a smoothed atomic density, using a
bra-ket notation to emphasize basis set independence and to highlight the
connections with some popular choices of representations for describing atomic
systems. The correlations between the spatial distribution of atoms and their
chemical identities are computed as inner products between these feature kets,
which can be given an explicit representation in terms of the expansion of the
atom density on orthogonal basis functions, that is equivalent to the smooth
overlap of atomic positions (SOAP) power spectrum, but also in real space,
corresponding to -body correlations of the atom density. This formalism lays
the foundations for a more systematic tuning of the behavior of the
representations, by introducing operators that represent the correlations
between structure, composition, and the target properties. It provides a
unifying picture of recent developments in the field and indicates a way
forward towards more effective and computationally affordable machine-learning
schemes for molecules and materials
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