arXiv subjects
Publications and source records attributed to Aniruddha Chakraborty.
Understanding electron correlation requires solving inseparable Schrodinger equation. In general, inseparable Schrödinger equations cannot be solved analytically. So their solutions are obtained numerically. In this paper we investigate electron correlation problem using the Dirac delta function repulsion between two electrons, where each electron is bound in their respective Dirac delta potential well.
We study the passage of proton (H$^{+}$) through different five membered aromatic ring systems by considering one dimensional motion of the proton along a line perpendicular to the plane of the ring. The potential for the motion of (H$^{+}$) from one side of the ring to the other through the center of the ring is found to be a symmetric double well and such complexes are suggested as molecular rattles which can exhibit a ring umbrella like inversion. Our study reveals C$_{2}$H$_{2}$P$_{3}^{-}$ - H$^{+}$ to be a molecular rattle with a barrier height of 44.277 kcal mol$^{-1}$ and exciting it to the third vibrational level makes easy for the proton to go through the ring.
Can a $C_{60}$ molecule chemisorbed on an aluminium surface, exhibit rolling motion and even move on the surface? Here we presents the results of our calculation which shows that $C_{60}$ can move on that surface utilizing its own rolling motion. This rolling motion involves breaking and making of several $Al-C$ bonds. We have analyzed three possible mechanisms, in one, only carbon atoms of hexagonal rings are involved in the process, in another case only carbon atoms of pentagonal rings are involved in the process and in the last case carbon atoms of both the hexagonal and pentagonal rings are involved in the process. We found that the activation energy for rolling motion involving, only hexagonal rings is $1.69$ kcal mol$^{-1}$, only pentagonal rings is $5.97$ kcal mol$^{-1}$ and both hexagonal and pentagonal rings is $121.38$ kcal mol$^{-1}$. Rate constants are estimated using classical version of transition state theory for rolling motion involving, only hexagonal rings is $3.45 \times 10^{13} sec^{-1}$, only pentagonal rings is $17.01 \times 10^{-79} sec^{-1}$ and both hexagonal and pentagonal rings is $28.92 \times 10^{2} sec^{-1}$.
We present a method to calculate exact dynamics of a wave-packet in a quantum two-state problem with Dirac delta coupling. The advantage of our method is that the calculations are done in the time domain. Hence inverting the solutions from other domains (Laplace or Fourier domain) do not intervene us from presenting the solution in the time domain. The initial wave packet is considered to be a Gaussian function. The wave propagation in the quantum state is governed by time-dependent Schrödinger equation and the coupling is given by non-diagonal element in the matrix representation. We present the exact analytic forms of the wave function for both the states in the time domain.
We give a general method for finding the exact solution for the problem of electronic relaxation in solution, modeled by a particle undergoing diffusive motion under a potential in the presence of a sink of finite width. The solution requires the knowledge of the Green's function in Laplace domain in the absence of any sink. We find the exact solution for the case of parabolic potential. This model has considerable improvement over the existing models for understanding non-radiative electonic relaxation of a molecule in solution, in fact this is the first model where a simple analytical solution is possible in the case of a sink of finite width.
We give a method for finding the exact analytical solution for the problem of a particle undergoing diffusive motion in a flat potential in the presence of a new localized sink. The Diffusive motion is described using the Smoluchowski equation. Our method requires the knowledge of Laplace transform of Green's function for the motion in absence of the sink. The same model for sink can be used to deal with other potentials.
A simple analytical method for solving intra-molecular reactions of polymer chain in dilute solution is formulated. The physical problem of looping can be modeled mathematically with the use of a Smoluchowski-like equation with a Dirac delta function sink of finite strength. Here we have proposed a very simple method of solution. The solution is expressed in terms of Laplace Transform of the Green's function for end-to-end motion of the polymer in absence of the sink. We have used two different rate constants, the long term rate constant and the average rate constant. The average rate constant and long term rate constant varies with several parameters such as length of the polymer (N), bond length (b) and the relaxation time ${τ_R}$. Our results are in agreement with that produced by more general and detailed method.
We investigate theoretically the end-to-end looping time of a long chain polymer molecule immersed in a solvent. The dynamics of the end-to-end distance is governed by a Smoluchowski-like equation of a particle moving under the influence of a parabolic potential in presence of a Dirac delta sink of arbitrary strength and location. Using Wilemski-Fixman [ G. Wilemski and M. Fixman, J. Chem. Phys. {\bf 60}, {\it 866} (1974)] approach we calculate the looping time for a long chain molecule immersed in a solvent. We find that looping time varies with several parameters such as length of the polymer (N), bond length (b) and the relaxation time ${τ_R}$.
A diffusion-like theory for real time end-to-end distance of a long polymer chain in dilute solution is formulated. We give a detailed analytical expression for the end-to-end distance auto-correlation function of a long chain polymer in solution. The physical problem of dynamics of end-to-end distance can be modeled mathematically with the use of a Smoluchowski-like equation. Using this equation analytical expression for end-to-end distance auto-correlation function is derived. We find that this auto-correlation function varies with several parameters such as length of the polymer (N), bond length (b) and the relaxation time ${τ_R}$.
In this paper, the looping kinetics of a long polymer chain in solution has been investigated by using two state model, where one state represents open chain polymer molecule and the other represents closed chain polymer molecule. The dynamics of end-to end distance of both open chain and closed chain polymer is represented by a Smoluchowski-like equation for a single particle under two different harmonic potentials. The coupling between these two potentials are assumed to be a Dirac Delta function in our model. We evaluate two rate constants, the long term and the average rate constant. We have also incorporated the effect of all other chemical reactions involving at least one end of the open chain polymer molecule - on rate of end-to-end loop formation. The closed chain polymer molecule can be converted to a open chain molecule by breaking of any bond - the effect of this reaction on the rate of end-to-end loop formation is also considered in our model.
We model quasi-two-dimensional two-electron Quantum Dots in a parabolic confinement potential with rovibrational and purely vibrational effective Hamiltonian operators. These are optimized by non-linear least-square fits to the exact energy levels. We find, that the vibrational Hamiltonian describes the energy levels well and reveals how relative contributions change on varying the confinement strength. The rovibrational model suggests the formation of a rigid two-electron molecule in weak confinement and we further present a modified model, that allows a very accurate transition from weak to strong confinement regimes.
Assume that the eigenvalues of a finite hermitian linear operator have been deduced accurately but the linear operator itself could not be determined with precision. Given a set of eigenvalues $λ$ and a hermitian matrix $M$, this paper will explain, with proofs, how to find a hermitian matrix $A$ with the desired eigenvalues $λ$ that is as close as possible to the given operator $M$ according to the operator 2-norm metric. Furthermore the effects of this solution are put to a test using random matrices and grayscale images which evidently show the smoothing property of eigenvalue corrections.
The Boltzmann distribution (the most probable distribution) is one of the most important concepts used in physics, chemistry and biology. Suppose we put the system initially in one of the less probable state then the system will find the most probable state by a random search among all possible energy distributions and thus can take long time depending on the size of the system. In the following, simple analysis using our simple model shows that a small and physically reasonable energy bias against locally unfavorable energy distribution, of the order of a few $kT$, can reduce the time-scale of the process by a significant size.
A theory of diffusion-controlled intramolecular reactions of polymer chain in dilute solution is formulated. Our model is based on the widely used diffusion-reaction formalism of Wilemski and Fixman (J. Chem. Phys. 60, 866 (1974). Our model is more general than the model proposed by Wilemski and Fixman, in the sense that our model considers the motion of both close and open chain polymer explicitely. It is quite unlikely that the motion of close chain polymer do not play any role in the loop formation dynamics, but unfortunately this fact was not considered in any of the earlier studies.
The present work focuses on the calculation of a non-adiabatic transition probability between two states which may or may not cross with each other and are coupled to each other by a moving $δ$ function potential. Here, the time dependent Schrodinger equation is converted to time independent one by using a scaling factor which is function of time. This time independent Schrodinger equation is then considered for two potentials coupled by a moving $δ$ potential and an expression for non-adiabatic transition probability has been derived.
The Smoluchowski equation with a time dependent delta function sink is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation. It is shown that by knowing the probability distribution at the position of sink, one can derive analytical expression for probability distribution everywhere. Thus the problem is reduced from a PDE in two variables to an integral equation of one. As far as the authors knowledge, we are the first one to provide an exact analytical solution of Smoluchowski equation for a parabolic potential with time dependent sink.
The effective thermal diffusivity is evaluated for a two dimensional real solid and the real solid is modeled with periodic surface. The result contain the scale factor for the conformal transformation that flattens the surface. We find that the effective thermal diffusivity is always less than the local thermal diffusivity and this conclusion is independent of the model.
We propose an analytical model based on diffusion-reaction equation approach for electrochemical electron transfer reaction, where the rate is limited by the electron transfer process. The electron transfer from an ion in solution to the metal electrode would occur as soon as the energy of the orbital on the ion matches the Fermi energy of the metal and a new ion with more positive charge is formed. Obviously the ion before electron transfer and the new ion, which is formed after electron has transferred, moves under the influence of different potentials. The coupling between these two potentials is assumed to be represented by a Dirac Delta function. The diffusive motion in this paper is described by the Smoluchowski equation. Our solution requires only the knowledge of the Laplace transform of the Green's function for the motion in both the uncoupled potentials. Our model is more general than all the earlier models, because we are the first one to consider the potential of both the ions explicitly.