228 research outputs found
Enlarging Maurer-Cartan form via Kronecker product and construction of Coupled Integrable systems by Nilpotent, Hadamard, Idempotent and K-idempotent matrix
Coupled nonlinear integrable systems are generated from usual zero curvature
equation. The relevant Maurer-Cartan forms are constructed by combining
suitably chosen matrices (nilpotent, Hadamard, idempotent and k-idempotent) and
Lie algebraic elements via Kronecker product. In each case a closure type
property among the matrices chosen is found to be playing a key role to produce
both the coupling and nonlinearity present in the system of equations obtained.
The method is highly flexible and can be used to construct general systems
containing 'p' number of equations. It is also shown that these new equations
can be written in the Hamiltonian form (with a preassigned symplectic operator)
with the trace identity introduced by Tu. Since the Lax operator is known one
can obtain the hereditary operators signifying the complete integrability.
Various properties of Kronecker product are found to be useful in our
construction.Comment: 16 page
Eigenvalues of the Anti-periodic Calogero - Sutherland Model
The U(1) Calogero Sutherland Model (CSM) with anti-periodic boundary
condition is studied. The Hamiltonian is reduced to a convenient form by
similarity transformation. The matrix representation of the Hamiltonian acting
on a partially ordered state space is obtained in an upper triangular form.
Consequently the diagonal elements become the energy eigenvalues.Comment: 1 figur
Derivation of time-dependent transition probability for generation from state in the presence of external electromagnetic field
In this work, we investigate the effect of electromagnetic (EM) field on the
generation of 2e-2h states from 1e-2h states. One of the fundamental ways by
which electromagnetic (EM) waves interact with matter is by the generation of
excited electronic states. The interaction of EM field with atoms and molecules
is given by the field-dependent Hamiltonian. Excited states are intrinsically
transient in nature because they are not stationary states of the
field-dependent Hamiltonian. Consequently, the time-dependent dynamics of
excited states depend strongly on the external electromagnetic field. Starting
with the 1e-1h excitation in a general many-electron system, the system was
propagated in time using time-dependent perturbation theory (TDPT). The
expression for time-dependent transition probability of
was evaluated
for a given time up to second-order in TDPT using diagrammatic techniques.
The derivation does not assume any a priori approximations to the
electron-electron correlation operator and presents the derivation of a
complete set of contributing diagrams associated with the full configuration
interaction wave function. The result from this work show that the calculation
of time-dependent transition probability can be factored into a
time-independent and time-dependent components. This is a significant outcome
for efficient computation of the time-dependent transition probability because
it allows for pre-computation of time-independent components before the start
of the calculations.Comment: 18 pages, 5 figures, 106 equations, diagrammatic representatio
Effect of dot size on exciton binding energy and electron-hole recombination probability in CdSe quantum dots
Exciton binding energy and electron-hole recombination probability are
presented as the two important metrics for investigating effect of dot size on
electron-hole interaction in CdSe quantum dots. Direct computation of
electron-hole recombination probability is challenging because it requires an
accurate mathematical description of electron-hole wavefunction in the
neighborhood of the electron-hole coalescence point. In this work, we address
this challenge by solving the electron-hole Schrodinger equation using the
electron-hole explicitly correlated Hartree-Fock (eh-XCHF) method. The
calculations were performed for a series of CdSe clusters ranging from
to
that correspond to dot diameter range of 1-20 nm. The calculated exciton
binding energies and electron-hole recombination probabilities were found to
decrease with increasing dot size. Both of these quantities were found to scale
as with respect to the dot diameter D. One of the key
insights from this study is that the electron-hole recombination probability
decreases at a much faster rate than the exciton binding energy as a function
of dot size. It was found that an increase in the dot size by a factor of 16.1,
resulted in a decrease in the exciton binding energy and electron-hole
recombination probability by a factor of 14.4 and ,
respectively.Comment: 10 pages, 4 figure
A New form of Kr\"onecker product-Lax pair and coupled systems
A new method is proposed to generate nonlinear integrable systems by starting
with existing Lax pair and a new form of Kr\"onecker product. It is observed
that such equation can be generated with the help of a Hamiltonian structure.
Actually they are all bi-Hamiltonian. A few explicit form of such equations are
given alsoComment: 12 pages no figure
On A New Form of Darboux-B\"acklund Transformation for DNLS Equation-Mixed and Rational Type Solutions
A new form of Darboux-B\"acklund transformation and its higher order form is
derived for Derivative Nonlinear Schrodinger Equation(DNLS). The new form
arises due to the different form of Lax pair. It is observed that by a special
choice of the eigenvalue of DB transformation one can generate a mixed form of
solution(containing both algebraic and exponential dependence on (x, t) can be
generated. On the other hand by adopting a new methodology due to Neugebauer
et. al. it is found that purely rational solution can be constructed. The two
different approach yields different class of solution and are compared.Comment: 12 pages 8 figure
Development of composite control-variate stratified sampling approach for efficient stochastic calculation of molecular integrals
In this work, the composite control-variate stratified sampling (CCSS) method
is presented for calculation of MO integrals without transformation of AO
integrals. The central idea of this approach is to obtain the 2-electron MO
integrals by direct integration of 2-electron coordinates. This method does not
require or use pre-computed AO integrals and the value of the MOs at any point
in space is obtained directly from the linear combination of AOs. The
integration over the electronic coordinates was performed using stratified
sampling Monte Carlo method. This approach was implemented by dividing the
integration region into a set of non-overlapping segments and performing Monte
Carlo calculations on each segment. The Monte Carlo sampling points for each
segment were optimized to minimize the total variance of the sample mean.
Additional variance reduction of the overall calculations was achieved by
introducing control-variate in the stratified sampling scheme. The composite
aspect of the CCSS allows for simultaneous computation of multiple MO integrals
during the stratified sampling evaluation. The main advantage of the CCSS
method is that unlike rejection sampling Monte Carlo methods such as Metropolis
algorithm, the stratified sampling uses all instances of the calculated
functions for the evaluation of the sample mean. The CCSS method is designed to
be used for large systems where AO-to-MO transformation is computationally
prohibitive. Because it is based on numerical integration, the CCSS method can
be applied to a wide variety of integration kernels and does not require
\textit{a priori} knowledge of analytical integrals. In this work, the
developed CCSS method was applied for calculation of excitonic properties in
CdSe quantum dots using electron-hole explicitly correlated Hartree-Fock
(eh-XCHF) and geminal-screened electron-hole interaction kernel (GSIK) methods.Comment: 13 page
On The Darboux B\"acklund Transformation of Optical Solitons with Resonant and Nonresonant Nonlinearity
Solitons in nonlinear optics holds a special role both in theoretical and
experimental studies. Several types of evolution equations are seen to govern
different situation of physical relevance. One such is the existence of both
resonant and nonresonant situation in optical fibre. The corresponding
evolution equation was devised by Doktorov et. al., which consists of a forced
NLS equation along with two other equations for population difference and
polarization. Here, we have followed an earlier formulation of Neugebauer for
Darboux-B\"acklund transformation for this coupled systems. This formalism has
the advantage that one can write the N-soliton solution altogether.An important
difference with the usual non-linear system is that all the field variables are
not present in both part of Lax operator. So we are to apply the DT to both
part separately.Comment: 6 pages no figure
Determination of electron-hole correlation length in CdSe quantum dots using explicitly correlated two-particle cumulant
The electron-hole correlation length serves as an intrinsic length scale for
analyzing excitonic interactions in semiconductor nanoparticles. In this work,
the derivation of electron-hole correlation length using the two-particle
reduced density is presented. The correlation length was obtained by first
calculating the electron-hole cumulant from the pair density,and then
transforming the cumulant into intracular coordinates, and finally then
imposing exact sum-rule conditions on the radial integral of the cumulant. The
excitonic wave function for the calculation was obtained variationally using
the electron-hole explicitly correlated Hartree-Fock method. As a consequence,
both the pair density and the cumulant were explicit functions of the
electron-hole separation distance. The use of explicitly correlated wave
function and the integral sum-rule condition are the two key features of this
derivation. The method was applied to a series of CdSe quantum dots with
diameters 1-20 nm and the effect of dot size on the correlation length was
analyzed.Comment: keywords: explicitly correlated, Gaussian-type geminal, electron-hole
correlation, reduced density matrix, cumulant, transition density matri
Calogero-Sutherland Model with Anti-periodic Boundary Conditions: Eigenvalues and Eigenstates
The U(1) Calogero Sutherland Model with anti-periodic boundary condition is
studied. The Hamiltonian is reduced to a convenient form by similarity
transformation. The matrix representation of the Hamiltonian acting on a
partially ordered state space is obtained in an upper triangular form.
Consequently the diagonal elements become the energy eigenvalues. The
eigenstates are constructed using Young diagram and represented in terms of
Jack symmetric polynomials. The eigenstates so obtained are orthonormalized.Comment: 9 pages, 4 figure
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