2,752 research outputs found
On an "interaction by moments" property of four center integrals
The four center integrals needed in the Hartree Fock approximation and in
TDDFT linear response are known to be difficult to calculate for orbitals of
the Slater type or of finite range. We show that the interaction of pairs of
products that do not mutually intersect may be replaced by the interaction of
their moments, of which there are O(N). Only quadruplets of orbitals 'close' to
one another need an explicit calculation and the total calculational effort
therefore scales as O(N). We provide a new and concise proof of this
"interaction by moments" property.Comment: The context of this note is the implementation of TDDFT linear
response for extended molecular system
Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals
An integral scheme for the efficient evaluation of two-center integrals over
contracted solid harmonic Gaussian functions is presented. Integral expressions
are derived for local operators that depend on the position vector of one of
the two Gaussian centers. These expressions are then used to derive the formula
for three-index overlap integrals where two of the three Gaussians are located
at the same center. The efficient evaluation of the latter is essential for
local resolution-of-the-identity techniques that employ an overlap metric. We
compare the performance of our integral scheme to the widely used Cartesian
Gaussian-based method of Obara and Saika (OS). Non-local interaction potentials
such as standard Coulomb, modified Coulomb and Gaussian-type operators, that
occur in range-separated hybrid functionals, are also included in the
performance tests. The speed-up with respect to the OS scheme is up to three
orders of magnitude for both, integrals and their derivatives. In particular,
our method is increasingly efficient for large angular momenta and highly
contracted basis sets.Comment: 18 pages, 2 figures; accepted manuscript. v2: supplementary material
include
On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance
A new numerical method is introduced to study the problem of time evolution
of generic non-linear dynamical systems in four-dimensional spacetimes. It is
assumed that the time level surfaces are foliated by a one-parameter family of
codimension two compact surfaces with no boundary and which are conformal to a
Riemannian manifold C. The method is based on the use of a multipole expansion
determined uniquely by the induced metric structure on C. The approach is fully
spectral in the angular directions. The dynamics in the complementary 1+1
Lorentzian spacetime is followed by making use of a fourth order finite
differencing scheme with adaptive mesh refinement.
In checking the reliability of the introduced new method the evolution of a
massless scalar field on a fixed Kerr spacetime is investigated. In particular,
the angular distribution of the evolving field in to be superradiant scattering
is studied. The primary aim was to check the validity of some of the recent
arguments claiming that the Penrose process, or its field theoretical
correspondence---superradiance---does play crucial role in jet formation in
black hole spacetimes while matter accretes onto the central object. Our
findings appear to be on contrary to these claims as the angular dependence of
a to be superradiant scattering of a massless scalar field does not show any
preference of the axis of rotation. In addition, the process of superradiance,
in case of a massless scalar field, was also investigated. On contrary to the
general expectations no energy extraction from black hole was found even though
the incident wave packets was fine tuned to be maximally superradiant. Instead
of energy extraction the to be superradiant part of the incident wave packet
fails to reach the ergoregion rather it suffers a total reflection which
appears to be a new phenomenon.Comment: 49 pages, 11 figure
Quantum mechanical calculation of Rydberg-Rydberg autoionization rates
We present quantum mechanical calculations of Auger decay rates for two
Rubidium Rydberg atoms with weakly overlapping electron clouds. We neglect
exchange effects and consider tensor products of independent atom states
forming an approximate basis of the two-electron state space. We consider large
sets of two-atom states with randomly chosen quantum numbers and find that the
charge overlap between the two Rydberg electrons allows one to characterise the
magnitude of the Auger decay rates. If the electron clouds overlap by more than
one percent, the Auger decay rates increase approximately exponentially with
the charge overlap. This finding is independent of the energy of the initial
state.Comment: 8 pages, 5 figure
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