Dr. Aniruddha Chakraborty
School of Basic Sciences
Indian Institute of Technology Mandi
Kamand, Himachal Pradesh 175005
India
ph: +(91)1905-237930
achakrab
Here we will investigate the algebraic structure of few Smoluchowski equation. Using the Lie-algebraic approach we plan to arrive at an exact form of the time evolution operator which, in turn, will enables us to derive the propagator readily. We will apply our method to few very interesting problems.
Understanding the dynamics of barrierless reactions in solution is an interesting problem for theoretical as well as experimental research. A reasonably good model for such reaction would be a particle executing one dimensional random walk on two potential wells in presence of point-like coupling between them. Exact solution of this model was proposed by us. This is an interesting & important work in this field. Our method of solution is used further to solve few related problems. Also we have generalized our model to the case where the coupling has a finite width. Very recently we have proposed time domain solution, which was so far considered to be intractable. Even we are able to derive closed form analytical solution in time domain for delocalized sink case as well.
Nonadiabatic transition due to potential curve crossing is one of the most important mechanism to effectively induce electronic transitions in collisions. We consider two curves crossing each other & there is a point-like coupling between two curves, which causes transition from one curve to another. Exact solution of this model was proposed by us! This is an interesting & important work in this field. Our method of solution is used extensively to solve few related problems.
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Two-state nano-mechanical systems (mechanical equivalent of 'bit') are very interesting examples for the possibility of observing quantum effects in them. The compression of a nano-rod would cause it to buckle. There are now two possible states, and the system is interesting, as a potential nano-sized device. We have developed quantum mechanical methods for the calculation of the rate of transitions between the two states, due to thermal fluctuations and tunneling. Our method goes beyond the standard method (Transition State Theory). In future our plan is to analyze the same problem in detail using dynamical transition state theory.
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Proton tunneling thorugh ring compound with pi electrons is an interesting problem to study. It may be helpful in understanding electrophilic substitution reaction in solution as well as proton tunneling in bio-systems. Here we will consider two types of planer ring compunds, one is aromatic and the other is antiaromatic. Our aim is to study te effect of electron transfer from the ring compound to proton. So we will assume one dimensional model for motion of proton along a line perpendicular to the plane of benzene ring as well as through the middle of the benzene ring and constrcut a potential energy curve for the corresponding motion. We will repeat the same procedure for motion of neutral H-atom through positively charges benzene ring. We will use computational chemistry software for computation.
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Understanding the rattling motion of proton through the benzene ring, Somesh Chamoli* & A. Chakraborty (in progress).
Is it possible to have C60 molecules chemisorbed on a surface, but at the same time exhibit some kind of rotational motion and even move on the surface? C60 is a molecule that consists of 60 carbon atoms, arranged as 12 pentagons and 20 hexagons. The shape is the same as that of a soccer ball. It can move on the surface by utilizing 'pentagon' - path or by utilizing 'hexagon' - path. We will use computational chemistry software for computation.
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Cyanide (CN−) and carbon monoxide (CO) have long been of interest as classic inhibitors of respiration. CN− is iso-electronic and isosteric with CO. We will investigate both CO & CN− in presence of a unit positive charge placed at different distances along the bond axis. We plan to use the force approach, to look into the nature of the individual molecular orbitals.
Understanding the kinetics of loop formation of long chain polymer molecules has been an interesting research field both, to experimentalists and theoreticians. Loop formation is believed to be an initial step in understanding several protein events. The theories of loop formation dynamics are in general approximate. The looping dynamics of a single polymer chain having reactive end-groups hav been modeled by simplest possible option. In our model, dynamics of end-to end distance is mathematically represented by an equation for a random walking under harmonic potential. Looping process is ensured by adding a sink of arbitrary strength in that equation. We have also incorporated the effect of all other chemical reactions involving either one or both of the end-group on rate of end-to-end loop formation - to the best of our knowledge this is the first time this has been done in any analytical model. An important future direction of this research is to understand the dynamics of contact formation between different parts of a long chain molecule.
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A reliable potential energy surface is required for studying the dynamics of a molecule in a particular electronic state. For polyatomic molecules, a global potential energy surface is out of reach of current computational capabilities. So one possible way to make progress is to use analytical functions with several parameters to fit the experimental data for constructing an effective Hamiltonian. One can use these effective Hamiltonians to extract new information about dynamics from information encoded in experimental spectra and to use these informations to understand internal molecular energy flow and reaction dynamics. Even when the potential energy surface is available, effective Hamiltonians are constructed from the exact Hamiltonian in order to simplify the subsequent analysis. We have recently developed the first genrealized polyad-breaking effective spectroscopic Hamiltonian for understanding bond dissociation process.
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One of our research goal is to develop effective spectroscopic Hamiltonian for systems undergoing chemical reactions e.g., dissociation and isomerization reaction. Once we will be able to construct an effective spectroscopic Hamiltonians for energies above the activation energy, then only these Hamiltonians can be used to understands reaction mechanism in more details than ever before. Then by using dynamical transition state theory we will be able to analyze importance of each term of the effective spectroscopic Hamiltonian.
Two electron atom is the most simplest system to study electron correlation. The doubly excited states of atoms with two outer electrons, exhibit molecule-like collective motion. There are even frozen planet state observed in two electron systems, where the electrons become locked into place on the same side of the nucleus. There are evidence for molecule like collective effects in atoms with three outer electrons. A simple effective spectroscopic Hamiltonian is proposed for double excited two electron atom for understanding independent particle, shielding and correlation effect. The Hamiltonian is constructed by nonlinear least square fit of spectra of two electron systems. An open question in this area is the applicability of this type of correlation to motion of electrons in molecule.
The motion of electrons in quantum dots have many properties in common with ro-vibrational motion of molecules. In fact for highly excited two electron quantum dot, the motion of electrons is much like the ro-vibrational motion of triatomic molecule. Naturally all the effects of anharmonicity are expected to be present in case of motion of electrons in a quantum dot. We have very recently constructed the effective spectroscopic Hamiltonian for motion of electrons in a quantum dot by non linear least square fit of the exact spectrum & we attemped to assign quantum numbers to different electronic states using our effective Hamiltonian.
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Temperature and heat flow are two important quantities in understanding the heat conduction. When temperure distribution is not uniform at all points of a system, then heat flows in the direction of decreasing temperature. In general Fourier equation is used in understanding heat conduction, which is valid only for a homogeneous isotropic solid. But in reality we have solids of different shapes, e.g., we can have a cylinder with nonuniform cylindricity. We solve two dimensional version of this problem to understand the effect of this type of nonuniformity on thermal diffusivity. Our conclusion is valid for systems with higher dimension & is independent of shape of the system.
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Now the question we would be addressing here is the following, what fraction of pixel data is necessary to regenerate the whole pixel data. The pixel data set can be easily converted to a Hermitian data matrix. For a large enough matrix it is difficult, if not impossible to regenerate the whole pixel data. Our method is based on how to express the large matrix in terms of suitable but smaller effective matrices, so it is computationally less expensive & fast. Instead of storing the big data matrix one may store all eigenevalues and a smaller part of the whole matrix. The same method is used for correcting 'defects' in image.
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The Boltzmann distribution (the most probable distribution of energy - in thermal equilibrium) is one of the most important concepts used in physics, chemistry and biology. Often in different experimetal situations, system may not be in thermal equilibrium to start with, then the system will find the most probable state by a random search among all possible energy distributions and thus in principle, can take long time depending on the size of the system. An 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.
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An electron near a metal surface feels the charge of its image in the metal. So near a metal surface the electron moves under the influence of that attractive potential that supports quantized energy levels whose wave function lie outside the metal surface & whose energy lie in the band gap. Harris et. al., have carried out interesting studies of this image potential states using two-photon photo-emission proces in the case of metal with polar adsorbates. As there does not seem to be any theoretical investigation on this problem, we investigated this problem. In our model, we account for the interaction of the dipole moment of the adsorbate with the lectric field exerted by the lectron perpendicular to the surface.
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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. Recently we have extended our model to the case where the coupling between two potentials is represented by a function of finite width.
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Intermolecular electron transfer reaction often occurs over long range distances (i.e. up to several tens of angstroms) and plays a key role in various physical, chemical and biological processes. In these reaction the rate constant of long range electron transfer depends upon electronic coupling between the donor and acceptor. The coupling between donor and acceptor may increases by the atoms located between them which form a kind of bridge for electron tunneling. By using exac analytical method we calculated the value of electronic coupling for the above mentioned processes in which the interaction of an electron with the donor, acceptor are represented as Dirac delta functions and conjugated bridge is represented by finite square well.
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The Algebraic Resonance Quantization (ARQ) method is used for finding a simplified approximate matrix Hamiltonian in place of impossibly large exact Hamiltonian matrix in full basis. This is generally done by two ways. First a sparse matrix approximation is made by considering the fact that most of the resonance coupling matrix elements can have negligible contribution. So the Hamiltonian matrix with few resonance coupling terms would be good enough to represent coupling between different modes. This is called Sparse Multi Resonance Approximation (SMRA). The second approximation is made by considering the fact that most basis elements of the full basis are relatively unimportant. So inclusion of few appropriately chosen basis elements for constructing the Hamiltonian matrix would be enough to represent coupling between different modes. This is called Truncated Multi Resonance Approximation (TMRA). In TMRA, all resonance terms are included in the Hamiltonian matrix, so it is not a sparse matrix approximation. When sparse matrix approximation is combined with a truncated basis, that is called Sparse Truncated Multi Resonance Approximation (STMRA). As the number of dimensions of the system increase, the TMRA basis size increases immensely less quickly, in comparison with the complete many dimensional basis. The rapidly increasing sparseness and basis set truncation shows optimism that the ARQ method would be very useful for numerical calculations involving highly excited vibrational states of polyatomic molecules. This method has been applied earlier to kinetically coupled Morse Oscillators with moderate to strong coupling below dissociation energy of a single bond and sparse matrix multi-resonance approximation (SMRA) was found to produce good results even when there is a global chaos, while truncated basis multi-resonance approximation (TMRA) was found to be successful for moderate coupling, but it breaks down for strong coupling. Now the plan is to to apply the ARQ method to coupled Morse oscillators systems above dissociation energy of a single bond and multiple wells systems with above barrier motion. For the last case, examples include isomerizing systems (e.g., coupled stretch-bend model of acetylene-vinylidene isomerization and systems with torsional motion (e.g., torsion- vibration model of methanol. Results of this analysis will be important to learn about the possibility of nonlinear least square fitting of energy levels using standard effective spectroscopic Hamiltonian in these systems.
In nonlinear least square fit of an energy level spectrum extracted from experimental data, the unknown parameters, defining the spectroscopic Hamiltonian are varied continuously until, the energy levels produced by this trial Hamiltonian show the best possible match to the energy levels generated from experimental data. At each step of variation, it is necessary to compare the energy of each ‘experimental’ level with that calculated from the trial Hamiltonian. The experimental levels are simply numbered in order of ascending energy and the n-th experimental level is matched with the n-th calculated level. For fitting the ‘experimental’ levels, it is conventional to assign to the ‘experimental’ levels a set of normal mode or local mode quantum numbers. The assigned quantum numbers in conventional fits are only used for the purpose of labelling the experimental levels. All that is really necessary is correctly labelled number of experimental levels according to the energy, so that each experimental level can be compared with the right predicted level at each stage of iteration of the optimization. So the key idea of the feasibility of fitting a spectrum lies in the fact, whether the levels can be labelled and not whether they can be assigned. Bootstrap method was used earlier for the purpose of fitting the unassigned spectra and in this works by starting with the fit of the lowest energy range, say R of the spectrum to calculate predicted levels in the next slightly higher energy range R'. The experimental levels in the range R' were then labelled by matching them with these predicted levels. This newly labelled levels were next included in a new fit of the expanded range R+R'. The information from this fit was used to label the next range R'' and so on. The process continued until one was able gradually to bootstrap up through the entire spectrum. Now, the plan is to construct an ab initio type Hamiltonian which will work even above the dissociation energy of a single bond by Bootstrap method of fitting the spectrum generated from coupled Morse oscillator system. Another interesting future research would be to construct an ab initio type Hamiltonian by Bootstrap method for systems with multiple wells and above barrier motion. For the last case, examples include isomerizing systems (e.g., coupled stretch-bend model of acetylene-vinylidene isomerization and systems with torsional motion (e.g. torsion- vibration model of methanol.
The dressed basis method is a method in which a many-mode, multi resonance effective Hamiltonianis approximated by a Hamiltonian with a new zero order basis within which there acts a residual effective resonance coupling operators. For this to work, it is required to label the new zero ordersequences with a set of 'n' numbers that work like zero order quantum numbers. This labelling method is based on the reliability of diabatic correlation diagram assignments. This dressed basis is used for simplifying the analysis of a very complicated system, with multiple resonance couplings between many interacting modes. So the system can be viewed from several simplified perspectives, each involving effectively a single resonance coupling. Numerical test shows that this procedure works remarkably well. However until now nobody has actually examined the possibility of fitting the spectra using dressed basis method. Now the plan is to construct an effective spectroscopic Hamiltonian for weakly coupled Morse Oscillators systems in dressed basis for testing the feasibility of this method. In future I plan to apply this method to more complex, i.e., more realistic systems.
For molecules such as H2O, D2O and CH2 large volumes of classical vibrational phase space are found to be non-dissociating even well above dissociation energy of a single bond. Classical dynamical study suggests the existence of dynamical constant of motion (conserved polyad number), which will have the effect of creating a classical dynamical barrier, preventing dissociation and hence the resulting non-RRKM behavior. Fully quantum calculations show families of exceptionally long-lived quantum states corresponding to these trapped volumes. These long lived states corresponds to simultaneous high overtone excitations in both modes i.e., doubly vibrationally excited states. The standard approach to build effective spectroscopic Hamiltonian is well understood for systems below the dissociation energy of a single bond. But here I need to incorporate the effect of continuum into the effective spectroscopic Hamiltonian, which is one of the key challenges in this area.
Recent high resolution ion imaging studies for HCHO unimolecular decomposition, in combination with quasi-classical trajectory calculation have shown evidences for novel pathways in unimolecular decomposition that does not proceed via the conventional transition state geometry. The dynamics are dominated by the long range abstraction of H from HCHO by the ``roaming" hydrogen atom. Recent work from several groups has identified analogous behaviour in other systems e.g. CH3CHO. I plan to construct an effective spectroscopic Hamiltonian for HCHO system to understand the origin of this roaming atom mechanism in more details.
It is known from literature that HOCl vibrational dynamics, for energies at or below HO+Cl dissociation energy can still be well described by a model in which the H--O bond is held fixed. The H--O bond only starts to play an important role in HOCl dynamics at energies well above the HO+Cl dissociation energy where the HClO isomer can be formed. The work of M. Joyeux et. al., focused on the bound energy regime below dissociation. Below the dissociation energy HOCl dynamics is mostly integrable. It is only slightly below and above dissociation that large scale chaos becomes mixed with regions of regular dynamics in the classical phase space. Quantum Mechanically, the HO+Cl dissociation process involves decay rate with values that range over several orders of magnitude, indicating that there likely is complex collection of quasi-bound states that determine the dissociation of the molecule. It has been shown that quasi bound states can find support on periodic orbits above dissociation. Recently Barr et. al., studied the classical dynamics of bound state and scattering trajectories of the chlorine atom interacting with the HO molecule using a two dimensional model in which the H--O bond length is held fixed. The plan of this project is to construct an effective spectroscopic Hamiltonian for this system to understand the important dynamical effects in detail.
Recently, the quantum monodromy concept is introduced and shown to be an important qualitative feature of many different realistic models e.g., vibrational structure of molecules, electronic states of hydrogen atom in presence external fields, coupling of angular momenta. Starting from these examples new qualitative features of molecular systems leading naturally to generalized monodromy notions is used. Going finally to really complex systems the relationship between phyllotaxis and monodromy is alos possible. Recent proof of the persistence of monodromy under small perturbations spoiling the integrability strengthens the interest of applications to non-integrable systems.
In 1930s, Eyring and Polanyi studied a chemical reaction, providing the first calculation of the potential energy surface of chemical reaction. This surface contains one minimum associated with the reactant and another minimum for the product. They are separated by a barrier that needs to be crossed for the chemical reaction to occur. Eyring and Polanyi defined the surface's transition state as the path of steepest ascent from the barriers saddle point in coordinate space. Once crossed, this transition state can never be recrossed. The notion of a transition state as a surface of 'no return' defined in coordinate space was soon recognized as fundamentally flawed, because recrossing is possible from dynamical effects originated from coupling terms in kinetic energy. So transition state theory provides upper bound of the exact classical reaction rate. The exact classical reaction rate in general will depend on the choice of dividing surface. This is the key idea behind the variational transition state theory, in which one searches the space between reactants and products for a dividing surface that minimizes the reaction rate. E. Wigner recognized very early that in order to develop a rigorous theory of reaction rates, one must extend the notions above from configuration space to phase space.
The breaking of a polymer chain that is under strain: The breaking of a polymer chain that is under strain is relevant for understanding material failure under stress, polymer rupture, adhesion, friction, mechanochemistry and biological applications of dynamical force microscopy. Theoretical studies suggest that, simple approaches e.g., Transition State Theory, are likely to fail for this problem. Here we plan to use phase space method for calculating this very interesting problem of breaking of a polymer chain under strain.
Mechanical unzipping of DNA: The replication of the DNA molecule is a very interesting problem of current interest. The first step in the replication is the unzipping of the two strands. Usually, this is done by enzymes, and mechanical force at the molecular level is involved in the action of these enzymes. There have been interesting investigations of the theory of denaturation of DNA under a torque. Motivated by all these research we plan to study the problem of pulling a polymer out of a potential well by a force applied to one of its ends using phase space method.
Polymer looping and unlooping problem: The dynamics of contact formation between different parts of a long chain molecule is a very interesting topic in biology. The related processes of opening of a loop or closing to form a loop also are of considerable interest. Here we plan to apply our phase space method for opening of a polymer ring by breaking a weak bond between the two ends, also the reverese problem of closing of a loop by forming a new bond.
Barrier Crossing of a long chain polymer molecule:. The barrier crossing problem for polymer is important in Biology as many biological processes involve the translocation of a chain molecule from one side of the membrane to the other through a pore in the membrane. We plan to apply the phase space method for calculating the rate of translocation from one side of the barrier to the other side.
An extreme challenge in the complexity of internal dynamics comes with long chain biological molecules. One of the major classes of this are proteins, with large number of amino acid residues. A protein with a particular sequence of amino acid residues is biologically active and folds to a native structure. An important question here is the connection between the sequence of amino acid residues and the native structure. Also it is not known how a particular sequence of amino acid residues makes protein a good folder. An understanding of sequence-structure connection would be very useful. Also the understanding of structure-reactivity connection would be another interesting aspect of research. A related question is the dynamical process by which a good protein folds to its native structure. However both the dynamic and static aspects of the protein folding problems are complicated by the fact that the folding takes place within water. It is presently unknown which type of dynamical analysis would be useful for protein folding problems. In a protein it is likely that, there is a small number of relevant large amplitude, low frequency motions that are crucial to the folding process and also one has to include the effect of bath of water molecules in which the folding takes place. It will be of great interest to discover if techniques of dynamical systems can be used to reveal the folding modes of proteins.
Larger molecules are formed by linking together several smaller components. So understanding dynamical informations from small molecules may be useful for studying the dynamical nature of internal motion of larger molecule. For understanding dynamics of a hydrocarbon chain CH3CH2.......CH2-CH3, one would probably start working with CH3-CH3, in which there is hindered rotational motion of the two -CH3 groups. The construction of effective spectroscopic Hamiltonian and the bifurcation analysis of this system can be a very useful starting point for dynamical analysis of hydrocarbon chain larger than CH3-CH3, i.e., n > 1. The dynamics become qualitatively very different with the incorporation of more -CH2- group in the hydrocarbon chain due to large amplitude, flexible twisting motions generally seen in short chain molecule. Also there are faster high frequency vibrations, such as those of the individual bonds. So for constructing effective spectroscopic Hamiltonian for larger molecules we plan to use all the required informations obtained from the dynamical studies of smaller molecules
The vibrational dynamics of molecules in gas phase has been studied extensively by effective spectroscopic Hamiltonian method. One of our research plan is to construct effective spectroscopic Hamiltonian for studying the vibrational dynamics in condensed phase. We plan to use kinetically coupled Morse Oscillators system coupled with a reservoir as our first model system to construct an effective spectroscopic Hamiltonian. The result using this simple model system will be a good starting point for the future exploration of the real systems. The key question in this area are the role of polyad numbers in energy transfer in condensed phase.
Another class of problems in large systems are the dynamical analysis of large amplitude motions in clusters of atoms and molecules. Clusters are simple systems that exhibit complex behavior such as freezing or melting or even has the possibility of glassy behavior. The phenomena of internal rearrangement and seeking minima on highly rugged potential energy surfaces of high dimensionality has been extensively investigated. The dynamical nature of large amplitude motions in these systems has yet to be explored. Here we plan to identify the essential features of the potential energy surface that has the strongest influence on dynamical behavior, so that we can construct a reduced dimensional model.
Nonadiabetic transitions due to potential energy curve or surface is one of the most important mechanism to efficiently induce electronic transition in collisions. This is very interesting concept and appears in various areas of physics, chemistry and biology. The most typical examples are, of course, a variety of atomic and molecular processes such as atomic and molecular collisions, chemical reactions and molecular spectroscopic processes. Other examples are dynamic processes on solid surfaces, energy relaxation and phase transitions in condensed phase physics, and electron and proton transfer processes in biology. In real systems, shape of potential energy curves & nature of coupling both are not very simple in general and so the problem is not in general analytically tractable. So it would be very interesting to construct effective spectroscopic Hamiltonians from experimental or simulated spectra for curve or surface crossing problems.
Dr. Aniruddha Chakraborty
School of Basic Sciences
Indian Institute of Technology Mandi
Kamand, Himachal Pradesh 175005
India
ph: +(91)1905-237930
achakrab