Integral formulae for the phase shifts. Convergence of the partial-wave expansion. Hard sphere scattering. Absolute phase shifts. Levinson Theorem. Scattering from a square-well. Low energy scattering. Coulomb scattering. Classical mechanics description of Coulomb scattering. Quantum mechanical description. Partial wave expansion. Coulomb plus short-range potentials.

Green's functions, T- and S-matrices. Lippmann-Schwinger equations. The transition and the scattering operators. The time-dependent picture. Scattering from non-local separable potentials. Scattering from the sum of two potentials. Partial-wave expansions. Long range potentials. Evaluation of partial-wave Green's functions. Approximate methods in potential scattering. Perturbative approximations. Semiclassical approximations. Spin and identical particles. Collisions of particles with spin. Identical particles. For small oscillations, we may again neglect all but the leading term and arrive at the potential Eqn.

A system of harmonic oscillators can be formed by simple addition of the individual Hamiltonians and decoupling them through a coordinate transformation. In general, a system of N coupled harmonic oscillators can be decoupled into N separate harmonic oscillators. Two important relations can be immediately obtained: [a,f] a [at,II] -at.

However, we must remember that the quadratic terms in H make it a positive operator with positive eigenvalues. Since there are no degeneracies in one dimension these constitute all of the eigenstates of H. The energy eigenstates can be succinctly written in terms of the creation operator at at n In - n - n! A rigid body is a system of mass points subject to the constraints that the distances between all pairs of points remain constant throughout the motion. This is something of an idealization, especially in the case of molecular motion, but it allows us to discuss the important aspects of rotational kinematics and dynamics.

In the case of a rigid body with N particles it can at most have 3N degrees of freedom. Although, as we see by definition there exists a set of constraints. These constraints serve to reduce the number of degrees of freedom greatly. We only need to establish the position of just three non-collinear particles of the body to define its location.

This gives us nine degrees of freedom, except we are not free to alter the distances between the particles which reduces the total number to just 6 degrees of freedom.

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Three of which describe the location of the rigid body and the other three describe the rotations. Rotations of rigid bodies are thought of as orthogonal transformations where the coordinates of the position of the particles in the rigid body are transformed into another set of coordinates. This matrix, which will be the representation of an operator A, is orthogonal and therefore must satisfy Saijaik -ik 4. Here we describe how these transformations are constructed and practically implemented.

The simplest rotation is one about a particular axis through and angle 0. For such a rotation of the transformation matrix can be easily calculated using basic trigonometry, for example a rotation about the x-axis in three dimensions: 1 0 A 0 cos0 sin0 4. The choice of these parameters is one made of convenience and appropriateness for particular problems, here we will describe two of the more popular formulations. The most popular set of parameters are the Euler Angles.

They are defined as three successive rotations performed in a specific sequence through three angles. Within limits, the choice of rotation angles is arbitrary but the most common is to rotate the system about the coordinate axis as in Eqn. One such sequence is started by rotation the initial system of axes, xyz, by an angle k counterclockwise about the z-axis, giving the new axis as x'y'z. In the second stage the intermediate axes are rotated about the x'-axis counterclockwise by an angle 0 to produce another intermediate set of axes x'y"z '.

Finally these axes are rotated again about the x'-axis in a counterclockwise direction by and angle V. The three angles 0, 0 and b constitute the three Euler angles and they completely specify the orientation, labeled XYZ, of the new system relative to the original coordinates. The elements of the complete transformation A can be obtained by writing the matrix as the triple product of the separate rotations, each of which has a relatively simple matrix form similar to Eqn.

The Euler angles are difficult to use in numerical computation because of the large number of trigonometric functions involved, and the four-parameter representations are much better adapted for use on computers. We can write the matrix A in terms of four real parameters eo, e1, e2, e This formula represents a single finite rotation about an axis to transform the body to its new orientation. It is given by 4. As a result we have 4.

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Consider some arbitrary vector F' involved in a mechanical problem, such as the position vector of a point in the body, or the total angular momentum. Usually such a vector will vary in time as the body moves, but the change will often depend on the coordinate system to which the observations are referred. For example, if the vector happens to be the radius vector from the origin of the body set of axes to a point in the rigid body then, clearly, such a vector appears constant when measured in body set of axes.

For an observer fixed in the space set of axes finds that the components of the vector, vary in time, when the body is in motion. The change dt in a time of the components of a general vector? We can derive a relation between the two differential changes in F based on physical arguments. As the body rotates there is of course no change in the components of this vector as observed from the body frame. The only contribution to drI' space is then the effect of the rotation of the body.

But since the vector is fixed in the body system, it rotates with it counterclockwise, and the change in the vector as observed in space is that given infinitesimal version of Eqn. The magnitude of w- measures the instantaneous rate of rotation of the body. The angular velocity can be expressed in terms of the Euler angles and their derivatives in both the space and body frames. The eigenvalues-I1, 12, and are commonly referred to as the principal axes of inertia.

We can transform the coordinate description of the orientation of the rigid body so that the I is diagonal. We consider either an inertial frame whose origin is at the fixed point of the rigid body, or a system of space axes with origin at the center of mass.

Additionally, Eijk is the Levi-Civita tensor also known as the antisymmetric tensor of rank 3. Now we see that we can find a basis common to H, L2, and one of the components of L, since the components do not commute see Eqn. The complete solution of the eigenvalue problem actually includes half integer angular momentum but these are not of interest at this time. The parallel construction between harmonic oscillators and angular momentum begins to break down here where the eigenvalue is bounded from above and below.

Expanding the operators in the coordinate basis allows us to determine the eigenvectors for the rotationally invariant problem. Hence, the closed linear span of Coherent States is the Hilbert space, H. Coherent States have proven very useful to many areas of the physical sciences, and occur in great variety. A state, I 7p , is said to be quasiclassical when the expectation values of the position, momenta, and energy satisfy the classical Hamiltonian equations of motion.

OHqc q Hqc 5. It is important to note that this is not the same demand as the semi classical limit, when h -- 0, is invoked.

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There is no guarantee that a quasiclasscial state even exists for a given Hamiltionian. A method to investigate if a Hamiltonian allows quasiclassical states is to apply Ehrenfest's theorem and show the resulting equations reduce to those of the classical case. In this manner, we can show that the canonical CS are quasiclassical. The harmonic oscillator creation operators can be expressed in terms of the self-adjoint position i3 and momentum P operators a i - 5. So the distribution of the vibrational hw levels are 5.

A similar derivation can be made for CS associated with the Hamiltonian of a force free rotor, and these also have the quasiclassical property. Additionally, CS have been constructed which combine rotational and vibrational modes. Here we have a powerful tool to analyze molecular dynamics, and motivation to develop a method of obtaining the relevant quantities from the classical evolution of a rovibrational molecule.

It is our intention to generate a general method of analyzing the classical vibration and rotations of molecular systems in order to evaluate their wave functions in terms of quasiclassical CS. Morales, et al. Although these derived CS do no exhibit an exact quasi-classical behavior their departure from the classical equations can be easily remedied. Morales was able to show that the exclusion of the half-integer rotational states from his construction leads to exact quasi-classical rotational CS.

The formulation of half-integer only rotational coherent states is an open research problem. Rotational Hamiltonian and Related Operators. For molecular systems the values of the orbital angular momentum are restricted to integer quantum numbers appropriate for bosons. Observe the asymmetric commutation relationship for t components of Li.

Since we can immediately observe that the the Hamiltonian Eqn. Example: Spherical Rotor. In order to establish among the generators of the elements of the irreducible representation Morales, et al. Moreover, it has been shown that the IMK do in fact span the irreducible representations of the product group. Janssen[36] altered this construction to generate a set of quasi-classical rotational CS.

Morales modified Janssen's construction even further to generate CS for the bosonic systems. I01 5. All of these results are still valid for integer and half-integer angular momentum eigenstates. I- 21! The second condition requires a positive measure that is a resolution of unity, Morales, et. Evaluating the operator averages and examining their dynamical properties they were able derive[38] relationships between the parameters and physical quantities. Connecting Operator Averages and Physical Parameters.

Connecting the coherent state parameters with physical ones was accomplished by employing the following relationships x -. The angle -y determines the relative orientation of the body-fixed frame with respect to the space-fixed one. The probability is the product of binomial distributions in p and q, and a Poisson distribution in r. Evolution of the Coherent State. They demonstrate that the derived CS are almost quasi-classical. It is highly regarded for its accuracy and stability. It seeks to fit a deterministic exponential model to the data. The standard PM is used only to analyze one degree of freedom, but we will require an analysis of many more degrees of freedom.

We will build an analogous method to analyze our simultaneously rotating and vibrating system, but first we present the modem version of Prony's technique. Evidently, this is a difficult nonlinear problem, even if the value of p is known. Iterative algorithms, such as steepest descent procedures or Newton's method, have been devised to minimize this error with respect to all the parameters. Unfortunately, these algorithms have proven to be computationally expensive.

The Prony method embeds the nonlinear aspects of the exponential model into a polynomial factoring, for which reasonably fast solution algorithms are available. Prony's keen insight allowed him to develop just such a method. In order to fit a p exponential model using PM, at least 2p data samples are required. The method has been extended to optimize the 2p parameters when more data is available using least squares fitting.

The key to the separation is to recognize that the Eqn. In summary, PM consists of three distinct steps.

Finally, the matrix equation Eqn. A computer algorithm is implemented to utilize this method in order to extract normal coordinates and rotational motion from ENDyne molecular simulations. In the regime of low energies and small vibrations, it is possible to separate the rotational and vibrational motion. The time step between data points is given by At, p represents the number of vibrational modes present. The Qj and mOj are the jth normal frequencies and phase respectively. This will interfere with the separation vibrational and rotational motion. So we choose as our fundamental variable: p t 1t?

The Xt,q variable is precisely the form prescribed in the PM, which allows the Prony procedure to extract the desired information for this model of several degrees of freedom. For our case of real valued data there is a modified version of Prony's method. Instead of determining the smoothing coefficients by solving a matrix equation similar to Eqn.

Z hq, Xl,q zI Az1 hq,2 X2,q 1 2 6. Fundamental to utilizing the GPM scheme is the minimization of Eqn. The minimization method we have currently implemented is the Fletcher-Reeves-Polak-Ribiere [47] conjugate gradient method. This requires calculation of the gradient of the function to be minimized. The angular momentum is given by N? Determination of the angular velocity allows us to calculate the rotation matrix. Since we are interested in the inverse of the rotation matrix we redefine the rotation matrix so that it is orthogonal, to ensure a well defined process.

We can reformulate Eqn. With further manipulation, Eqn. Using ENDyne[8] we calculated the dynamical evolution of water for various magnitudes of vibrational excitation. Our study of water was motivated chiefly by the large amount of experimental data available for this chemically and biologically imporNormal Modes of Water v3 Figure 7. It's three atoms and bent geometry provide a theoretical model which is not simple, but also not very complex. The bent shape of the H20 molecule is a result of the character of its molecular orbitals[48].

Normal Freq. Source: Bartlett[50] 7. The current experimental values are given in Table 7. It is important to realize that both theory and experiment do not represent the situation we are presented with in the dynamics of nuclei in molecules. These calculations rely on the assumption that the nuclei experience a truly harmonic potential, specifically that the potential they encounter is proportional their displacement from equilibrium.

In real molecules this can only be considered an approximation especially as the vibrations lead to large displacements from equilibrium. The effect of this breakdown of the harmonic assumption is commonly called Anharmonicity and is visually represented in Fig. Firstly, near the equilibrium distance the differences with the harmonic potential are small breaking down at greater displacements.

Importantly, unlike the harmonic model V,. Also the potential V exhibits an asymmetry about the equilibrium distance. All of these features represent a breakdown of the harmonic potential. The deficiencies of the harmonic approximation posed a problem for our analysis. This means that realistic intramolecular potentials deviate from the harmonic potential more for higher vibrational excitations, and we are faced with the fact the molecular motion will not consist strictly of harmonic vibrations, especially for highly excited molecules. Often realistic potentials are examined by using empirical and semi-empirical approximations such as the Lennard-Jones potential or the Morse potential.

Here we again see the effects of the realistic poten- Morse Curve 0 1 2 3 4 5 Internuclear distance Req Figure 7. Since the potential allows for bound states the energy levels are discrete but the energy levels are not evenly spaced as they are for the harmonic approximation. In fact, the spacing between sequential energy levels becomes increasingly smaller for larger energy states. Additionally, the disassociation energy limits the number of energy levels as opposed the infinite number of levels available to for the harmonic oscillator.

We used one method of obtaining an expression for the energy levels by employing the Dunham[52] expansion of the morse potential. Immediately we can identify the frequency of vibration given by Morse potential by making an analogy with the structure of the harmonic energy levels. This is precisely the type of signal required to perform the Prony analysis. Since we are only concerned with the low lying vibrational states we will leave the discussion of higher order corrections for a later time. But they are also expressible as sums of sinusoids. Examining Eqn.

The second term g cos 2wet oscillates with twice the fundamental frequency of the main component of the motion. As a result the motion will appear assymetric with the peaks of the main component being more spread out and the troughs being more narrow as in Fig. We did this by performing a series of Electron Nuclear Dynamics trajectories for only H20 with a single vibrational mode excited to varying degrees.

The results of the trajectories of the symmetric bending mode of water from equilibrium are given in Fig. Here the effects described previously are plainly seen. The frequency of vibration decreases with increasing excitation in energy and there is a noticeable difference between the peaks and troughs of the motion. Vibration of Water 2. Applying the GPM in this way to these resulted in the data in Table 7.

The additional sinusoidal terms were clearly measured by our GPM and found to have magnitudes as predicted by Eqn. Moreover, the frequencies decline with increasing energy as predicted by Eqn. Performing a linear regression using the data from Table 7. It should be kept in mind that these results are the product of a first order correction and that the disassociation energy corresponds to a physical parameter far away from the equilibrium.

We observed that these values demonstrate that our morse potential approximation is consistent with physical reality and provides a useful tool in analyzing the effects of anharmonicity on our GPM. Importantly the analysis demonstrates that the anharmonicities do not cripple our ability to use the methods we developed. Whenever computers and numerical methods are used we must remain conscious of numerical instability and loss of precision.

This matrix Z is a function of the roots of 0 z 0o 0 0 z1 z An ill conditioned matrix is one whose condition number is greater than or of the order of the reciprocal of the machine precision, i. We will consider a matrix to be poorly conditioned if the square of the condition number is greater than the reciprocal of the machine precision.

While a general result relating the condition number to the parameters of our particular implementation is desirable, we first only investigate our application of GPM to the H20 molecule. This molecule has three normal mode frequencies. For the Prony analysis this indicates that the number of exponential parameters is six, consisting of three complex conjugate pairs. The free parameters in this analysis are the sampling period, At, and the number of data points n.

We will evaluate the condition number for several reasonable values of these parameters. Additionally, we will investigate the effect of small errors in the frequency Q. The location of the roots in the complex plane can be observed in Fig. In the presence of noise in the prony signal the location of the roots no longer lie on the unit circle in the complex plane[39]. Magnus Hedstrom for his advice and love of film; to Dr. Along with them 1 would like to thank Minky and Mozu for their constant affection and for providing pleasant and fanciful distractions.

Time-Dependent Scattering 12 2. Utilizing the nearly classical behavior of nuclei in chemical processes we were able to devise a system to map quasi-classical coherent states to trajectories calculated v PAGE 6 from the semi-classical approximation of the Electron Nuclear Dynamics theory. Finally we will have some concluding remarks in Chapter 1 1. Our goal in this chapter is to present the essence of molecular collision theory and its use as an introduction to our own theoretical developments in studying state resolved reaction dynamics and scatter4 PAGE 11 5 ing.

The collisions of molecules leads to scattering in all directions and the resulting intensity distribution in the CM frame gives the differential cross section DCS defined by the ratio da ij Number of scattered particles per unit time per unit solid angle Number of incident particles per unit time per unit area 2. The first derived relevant quantity is the integral cross section r 2n P7T sin 9 d9d J. Classical mechanics provides the roots for modem physics. The trajectory must lie in a plane since L is conserved. The equations of motion may be written in polar coordinates: 2.

The complete classical trajectory may now be determined by integrating Eqn. The purely formal solutions are usually obtained from an integral equation reformulation of the original Schrodinger equation in both the time-dependent and the time-independent cases. In the usual case of a state to-state scattering problem, the initial wave function is a stationary eigenfunction of the whole Hamiltonian with a trivial time dependence through a phase.

Here we will discuss the Bom Approximation and the method of partial waves which comprise the main quantum mechanical approximations, then we will address the semiclassical JWKB approximation, and Eikonal approximation. Clearly T is diagonal in both l and m. The second iteration gives 2. This condition allows us to replace the exact wave function ipW by a semiclassical wave function where S x can be interpreted as the phase of the wave function. Time-dependent theories were not studied in earnest until the advent of modem computers in the s.

However, this calculation is very intensive if PES are not PAGE 28 22 used so practical methods have so far been mostly restricted to classical, semiclassical, or quasiclassical descriptions of the nuclear degrees of freedom. The trajectory surface hopping method TSH uses a probabilistic description [12, 13] to make the solution of the time-dependent equations easier.

Direct calculation of the dynamics of the electronic states and phase is ignored in the TSH method[14]. Sometimes the full ab initio Flamiltonian is considered, in other cases model Hamiltonians are set up to drive the dynamics. This novel approach generates a theoretical framework to calculate the full dynamics of molecular collisions and dynamics without the use of the Bom-Oppenheimer approximation thus allowing the study of nonadiabatic interactions.

Here we will introduce the quantum mechanical action A and demonstrate its connection to the Schrodingers equation by introducing a general set of complex variational parameters. If we choose to specify a constrained form for the wave functions 'k then we would restrict ourselves to particular regions of the Hilbert space resulting in a dynamical evolution that would be an approximate solution to the Schodinger equation.

Even before making any assumptions about the form of we can derive a set of general equations of motion. PAGE 35 29 3. The constant a k can be related to standard deviation or width of the Gaussian functions. PAGE 36 30 The physical meaning of the previous parameters and constants imply that each nuclear wave packet can be seen as parametric plane wave function Furthermore, since the constants a k are time-independent and the evolution is through the TDVP, each nuclear Gaussian will remain a Gaussian during the evolution without changing shape. This type of function has been long established in the TDHF theory of nuclear many-body theory [28, 29, 30].

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Both the atomic and molecular 0 depend parametrically on the Gaussian parameter R t. The parameters z pq are complex numbers. We naturally cast our system in terms of deviations r] l from equilibrium: 4. The Lagrangian is given now given by Using the Vi as the generalized coordinates, the Lagrangian leads to the following n equations of the motion: where explicit use has been made of the symmetry property of the V tJ and coefficients.

We then must solve this set of simultaneous differential equations involving all of the coordinates Vi to obtain the motion near the equilibrium. Substituting our ansatz into the equations of motion leads to 4. It is possible to transform the t? All of the particles in each mode vibrate with the same frequency and with the same phase; the relative amplitudes being determined by the matrix elements a ik.

The complete motion is then built up out the the sum of the normal modes weighted with appropriate amplitude and phase factors contained in the C k s. PAGE 45 39 Harmonics of the fundamental frequencies are absent in the complete motion essentially because of the stipulation that the amplitude of oscillation be small. The normal coordinate transformation emphasizes this point, for the Lagrangian in the normal coordinates is seen to be the sum of Lagrangians for harmonic oscillators of frequencies uj k. The dynamics of a collection of noninteracting oscillators is no more complicated than that of a single oscillator aside from the.

A particle moving in a potential V x is placed at one of its minima x 0 , it will remain there in a state of stable, static equilibrium. From the basis independent commutation relation 4. Two important relations can be immediately obtained: a, H a f , H 4. PAGE 49 43 With some manipulation it can be shown that the coefficients are given by 4.

In the case of a rigid body with N particles it can at most have 3 N degrees of freedom. We only need to establish the position of just three non-collinear 4. PAGE 51 45 4. The simplest rotation is one about a particular axis through and angle 6. One such sequence is started by rotation the initial system of axes, xyz, by an angle p counterclockwise about the z-axis, giving the new axis as x'y'z.

## Semiclassical Theory of Elastic Scattering

In the second stage the intermediate axes are rotated about the x'-axis counterclockwise by an angle 9 to produce another intermediate set of axes x'y" z'. Finally these axes are rotated again about the x'-axis in a counterclockwise direction by and angle ip. The three angles 9, p and ip constitute the three Euler angles and they completely specify the orientation, labeled XYZ, of the new system relative to the original coordinates. PAGE 52 46 The elements of the complete transformation A can be obtained by writing the matrix as the triple product of the separate rotations, each of which has a relatively simple matrix form similar to Eqn.

The rotation formula can be rewritten Figure 4. Consider some arbitrary vector r involved in a mechanical problem, such as the position vector of a point in the body, or the total angular momentum. The change d t in a time of the components of a general vector f as seen by an observer in the body system of axes will differ from the corresponding change as seen by an observer in the space system. We can derive a relation between the two differential changes in f based on physical arguments. The only contribution to d r space is then the effect of the rotation of the body.

But since the vector is fixed in the body system , it rotates with it counterclockwise, and the change in the vector as observed in space is that given infinitesimal version of Eqn. The magnitude of u measures the instantaneous rate of rotation of the body. In the CM we may express the rotational inertia tensor as a matrix with elements It is possible to diagonalize I by solving the appropriate eigenvalue problem yielding a transformation matrix. We consider either an inertial frame whose origin is at the fixed point of the rigid body, or a system of space axes with PAGE 58 52 origin at the center of mass.

## Semiclassical theories of molecular scattering / B.C. Eu | National Library of Australia

In these situations we have space 4. This equation can also be expressed in terms of the body frame derivitives by space LU X L. Additionally, is the Levi-Civita tensor also known as the antisymmetric tensor of rank 3. Now we see that we can find a basis common to H, L 2 , and one of the components of L, since the components do not commute see Eqn. A state, ip , is said to be quasiclassical when the expectation values of the position, momenta, and energy satisfy the classical Hamiltonian equations of motion. It is important to note that this is not the same demand as the semi classical limit, when h 0, is invoked.

The Glauber states are considered the canonical Coherent States and are quasiclassical. The harmonic oscillator creation operators can be expressed in terms of the self-adjoint position x and momentum p operators where m is the oscillator mass. Figure 5. It has been demonstrated that an a posteriori vibrational analysis in terms of these harmonic oscillator CS can be utilized to obtain vibrationally resolved differential cross PAGE 65 59 sections when the vibrational energy levels of a product wave function are approximately equidistant.

For molecular systems the values of the orbital PAGE 66 60 angular momentum are restricted to integer quantum numbers appropriate for bosons. The irreducible representation of the rotational eigenstates IMK is the semidirect product of the 50 3 x 50 3 with an Abelian group. The generators of the 50 3 Lie groups are the L t and the J,, respectively. It is important to note the action of one particular generator of the Abelian group on one particular angular momentum eigenstates: 5.

In this case his construction would require three complex parameters x , y, and z, one for each quantum number, and the set of CS will be generated by Unfortunately, this set of CS does not exhibit the quasi-classical properties for a rotor. By construction the rotational CS, xyz , satisfy the first condition of the definition of a CS see Eqn. In the limit of high angular momentum r oo these equations would match the classical result.

In order to fit a p exponential model using PM, at least 2 p data samples are required. The method has been extended to optimize the 2 p parameters when more data is available using least squares fitting. In the regime of low energies and small vibrations, it is possible to separate the rotational and vibrational PAGE 79 73 motion. Cooperative Agreement Conditions. Special Conditions. Federal Demonstration Partnership. Policy Office Website. Eric Heller heller physics.

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