143 research outputs found
On the evaluation of derivatives of Gaussian integrals
We show that by a suitable change of variables, the derivatives of molecular integrals over Gaussian-type functions required for analytic energy derivatives can be evaluated with significantly less computational effort than current formulations. The reduction in effort increases with the order of differentiation
Fermion -representability for prescribed density and paramagnetic current density
The -representability problem is the problem of determining whether or not
there exists -particle states with some prescribed property. Here we report
an affirmative solution to the fermion -representability problem when both
the density and paramagnetic current density are prescribed. This problem
arises in current-density functional theory and is a generalization of the
well-studied corresponding problem (only the density prescribed) in density
functional theory. Given any density and paramagnetic current density
satisfying a minimal regularity condition (essentially that a von
Weiz\"acker-like the canonical kinetic energy density is locally integrable),
we prove that there exist a corresponding -particle state. We prove this by
constructing an explicit one-particle reduced density matrix in the form of a
position-space kernel, i.e.\ a function of two continuous position variables.
In order to make minimal assumptions, we also address mathematical subtleties
regarding the diagonal of, and how to rigorously extract paramagnetic current
densities from, one-particle reduced density matrices in kernel form
Alternative separation of exchange and correlation energies in multi-configuration range-separated density-functional theory
The alternative separation of exchange and correlation energies proposed by
Toulouse et al. [Theor. Chem. Acc. 114, 305 (2005)] is explored in the context
of multi-configuration range-separated density-functional theory. The new
decomposition of the short-range exchange-correlation energy relies on the
auxiliary long-range interacting wavefunction rather than the Kohn-Sham (KS)
determinant. The advantage, relative to the traditional KS decomposition, is
that the wavefunction part of the energy is now computed with the regular
(fully-interacting) Hamiltonian. One potential drawback is that, because of
double counting, the wavefunction used to compute the energy cannot be obtained
by minimizing the energy expression with respect to the wavefunction
parameters. The problem is overcome by using short-range optimized effective
potentials (OEPs). The resulting combination of OEP techniques with
wavefunction theory has been investigated in this work, at the Hartree-Fock
(HF) and multi-configuration self-consistent-field (MCSCF) levels. In the HF
case, an analytical expression for the energy gradient has been derived and
implemented. Calculations have been performed within the short-range local
density approximation on H2, N2, Li2 and H2O. Significant improvements in
binding energies are obtained with the new decomposition of the short-range
energy. The importance of optimizing the short-range OEP at the MCSCF level
when static correlation becomes significant has also been demonstrated for H2,
using a finite-difference gradient. The implementation of the analytical
gradient for MCSCF wavefunctions is currently in progress.Comment: 5 figure
Kohn-Sham theory with paramagnetic currents: compatibility and functional differentiability
Recent work has established Moreau-Yosida regularization as a mathematical
tool to achieve rigorous functional differentiability in density-functional
theory. In this article, we extend this tool to paramagnetic
current-density-functional theory, the most common density-functional framework
for magnetic field effects. The extension includes a well-defined Kohn-Sham
iteration scheme with a partial convergence result. To this end, we rely on a
formulation of Moreau-Yosida regularization for reflexive and strictly convex
function spaces. The optimal -characterization of the paramagnetic current
density is derived from the -representability conditions.
A crucial prerequisite for the convex formulation of paramagnetic
current-density-functional theory, termed compatibility between function spaces
for the particle density and the current density, is pointed out and analyzed.
Several results about compatible function spaces are given, including their
recursive construction. The regularized, exact functionals are calculated
numerically for a Kohn-Sham iteration on a quantum ring, illustrating their
performance for different regularization parameters
The choice of basic variables in current-density functional theory
The selection of basic variables in current-density functional theory and
formal properties of the resulting formulations are critically examined. Focus
is placed on the extent to which the Hohenberg--Kohn theorem,
constrained-search approach and Lieb's formulation (in terms of convex and
concave conjugation) of standard density-functional theory can be generalized
to provide foundations for current-density functional theory. For the
well-known case with the gauge-dependent paramagnetic current density as a
basic variable, we find that the resulting total energy functional is not
concave. It is shown that a simple redefinition of the scalar potential
restores concavity and enables the application of convex analysis and
convex/concave conjugation. As a result, the solution sets arising in
potential-optimization problems can be given a simple characterization. We also
review attempts to establish theories with the physical current density as a
basic variable. Despite the appealing physical motivation behind this choice of
basic variables, we find that the mathematical foundations of the theories
proposed to date are unsatisfactory. Moreover, the analogy to standard
density-functional theory is substantially weaker as neither the
constrained-search approach nor the convex analysis framework carry over to a
theory making use of the physical current density
Molecular vibrations in the presence of velocity-dependent forces
A semiclassical theory of small oscillations is developed for nuclei that are
subject to velocity-dependent forces in addition to the usual interatomic
forces. When the velocity-dependent forces are due to a strong magnetic field,
novel effects arise -- for example, the coupling of vibrational, rotational,
and translational modes. The theory is first developed using Newtonian
mechanics and we provide a simple quantification of the coupling between these
types of modes. We also discuss the mathematical structure of the problem,
which turns out to be a quadratic eigenvalue problem rather than a standard
eigenvalue problem. The theory is then re-derived using the Hamiltonian
formalism, which brings additional insight, including a close analogy to the
quantum-mechanical treatment of the problem. Finally, we provide numerical
examples for the H, HT, and HCN molecules in a strong magnetic field
The integral‐direct coupled cluster singles and doubles model
An efficient and highly vectorized implementation of the coupled cluster singles and doubles (CCSD) model using a direct atomic integral technique is presented. The minimal number of n6 processes has been implemented for the most time consuming terms and point group symmetry is used to further reduce operation counts and memory requirements. The significantly increased application range of the CCSD method is illustrated with sample calculations on several systems with more than 500 basis functions. Furthermore, we present the basic trends of an open ended algorithm and discuss the use of integral [email protected]
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