Hamiltonian collision

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

The Q-projected Green s function (7.111) in the polarisation potential (7.115) contains the collision Hamiltonian H, but practical implementation of it involves the weak-coupling approximation (7.132). A second-Born calculation is equivalent to a calculation of the polarisation potential with Q=1 and Vopul being the reduced matrix element of U. This has... 

If both T and R are linear molecules, the collision Hamiltonian of eqn (14.17) becomes identical to the diatom-diatom Hamiltonian in ref. 11 with the replacement of mr by the moment of inertia 7. Thus the current SVRT model for polyatom-polyatom reaction can be viewed as a generalization of the exact treatment for diatom-diatom reaction. 

Consider collisions between two molecules A and B. For the moment, ignore the structure of the molecules, so that each is represented as a particle. After separating out the centre of mass motion, the classical Hamiltonian that describes tliis problem is... 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

A dynamical study of molecular collisions requires construction of the Hamiltonian H=T+V, where T and V are kinetic energy (KE) and potential energy (PE), respectively. A knowledge of the interaction potential, therefore, is essential for the study of molecular dynamics. 

The potential energy functions are expressed in terms of q, 0 = 1,2,. 6), which explicitly exhibits its independence of the coordinates of the center of mass. Again, since the momenta conjugate to coordinates 0/7. q, qg), i.e. Pi,p andp9 remains constants of motion during the entire collision, the term containing them in the Hamiltonian has been subtracted. 

Tully has discussed how the classical-path method, used originally for gas-phase collisions, can be applied to the study of atom-surface collisions. It is assumed that the motion of the atomic nucleus is associated with an effective potential energy surface and can be treated classically, thus leading to a classical trajectory R(t). The total Hamiltonian for the system can then be reduced to one for electronic motion only, associated with an electronic Hamiltonian Jf(R) = Jf t) which, as indicated, depends parametrically on the nuclear position and through that on time. Therefore, the problem becomes one of solving a time-dependent Schrodinger equation ... 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

The total Hamiltonian of the collision system can be most generally written as the sum of three terms the kinetic energy of the relative motion, the interaction potential between the colliding particles, and the asymptotic Hamiltonian describing the colliding particles at infinite separation. We make the following approximations ... 

Figure 8.1 Schematic structure of the Hamiltonian matrix for a collision system in an external field expressed in the total angular momentum representation.

A time-dependent process, such as radiative absorption, internal conversion, intersystem crossing, unimolecular isomerization, or collision, may be treated in terms of a zero-order Hamiltonian H0 and a perturbation T. An unperturbed eigenstate of H0 evolves in time, since it is not an eigenstate to the perturbed Hamiltonian... 

We note that the separation into the three types of transitions (7), (2), and (2) is somewhat artificial. In fact, molecular collisions and transitions due to external fields are special examples of prepared states. Time evolution of a system described by a time-independent Hamiltonian does occur in general, unless the initial state of the system is described by a ket which is an eigenket to the complete Hamiltonian. 

Here MA, MB, MA , and MB are the z-components of the spins of A and of B. Such collisions are usually treated7,34,142,177 in an adiabatic approximation using spin-free Hamiltonian and spin-free potential curves. Thus, MA and Mb are only approximate local quantum numbers during a collision and so may change. The total M quantum number is, however, conserved... 

We assume again that the atoms follow straight line trajectories, and we calculate the transition probability, P(b), from the initial to the final state in a collision with a given impact parameter, b. We then compute the cross section by integrating over impact parameter, and, if necessary, angle of v relative to E to obtain the cross section. The central problem is the calculation of the transition probability P(b). The Schroedinger equation for this problem has the Hamiltonian... 

To describe the shifts and intensities of the m-photon assisted collisional resonances with the microwave field Pillet et al. developed a picture based on dressed molecular states,3 and we follow that development here. As in the previous chapter, we break the Hamiltonian into an unperturbed Hamiltonian H(h and a perturbation V. The difference from our previous treatment of resonant collisions is that now H0 describes the isolated, noninteracting, atoms in both static and microwave fields. Each of the two atoms is described by a dressed atomic state, and we construct the dressed molecular state as a direct product of the two atomic states. The dipole-dipole interaction Vis still given by Eq. (14.12), and using it we can calculate the transition probabilities and cross sections for the radiatively assisted collisions. 

D A Micha. Effective hamiltonian methods for molecular collisions. Adv. Quantum Chem., 8 231, 1974. 

The events related to the suspace PaP > would correspond to direct collisions between reactants that may either leave the system as it is or it may interconvert into products via the interaction hamiltonian W. The component Q LP > contains events involving states in the orthogonal complement that includes supermolecule states. To populate these states require physical excitation processes these states may have finite lifetimes. The Q-component is called the time-delaying component by Feshbach in the context of nuclear reactions [29]. The Q T > component is given by... 

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