Potential energies associated with Hamiltonian

Typical potential energies associated with such a Hamiltonian are shown in Figure 4 as a function of the parameter 0 = x/2J. The coordinate is the antisymmetric combination. The symmetric mode clearly adds a term to the total energy independent of coupling. 

When expressed in terms of normal coordinates, the classical kinetic and potential energies associated with vibration in a polyatomic molecule are both diagonalized (cf. Eqs. 6.47). The classical vibrational Hamiltonian becomes... 

Suppose that there are two different potentials, v(r) and v (r) with ground states T(r) and IGr) respectively, which yield the same density. Then, unless f (r) — w(r) = const., (r) is different from T(r) since they solve different Schrodinger equations. So, if the Hamiltonians and energies associated with T fr)... 

A unimolecular reaction can be viewed as a reaction flux in phase space. It is best to have in mind a potential energy surface with a real barrier in the product channel, that is, a saddle point. Figure 6.4 shows both the reaction coordinate and a picture of the phase space associated with the molecule and the transition state. Recall, that a molecule of several atoms having a total of m internal degrees of freedom can be fully described by the motion of m positions (q) and m momenta (p). At any instant in time, the system is thus fully described by 2m coordinates. A constant energy molecule (a microcanonical system) has its phase space limited to a surface in which the Hamiltonian H = E. Thus, the dimensionality of this hypersurface is reduced to 2m — 1. 

Tlierc are two major sources of error associated with the calculation of free energies fi computer simulations. Errors may arise from inaccuracies in the Hamiltonian, be it potential model chosen or its implementation (the treatment of long-range forces, e j lie second source of error arises from an insufficient sampling of phase space. 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

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 order to obtain the potential energy surfaces associated with chemical reactions we, typically, need the lowest eigenvalue of the electronic Hamiltonian. Unlike systems such as a harmonic oscillator and the hydrogen atom, most problems in quantum mechanics cannot be solved exactly. There are, however, approximate methods that can be used to obtain solutions to almost any degree of accuracy. One such method is the variational method. This method is based on the variational principle, which says... 

The motion in the reaction coordinate Q is, like in gas-phase transition-state theory, described as a free translational motion in a very narrow range of the reaction coordinate at the transition state, that is, for Q = 0 hence the subscript trans on the Hamiltonian. The potential may be considered to be constant and with zero slope in the direction of the reaction coordinate (that is, zero force in that direction) at the transition state. The central assumption in the theory is now that the flow about the transition state is given solely by the free motion at the transition state with no recrossings. So when we associate a free translational motion with that coordinate, it does not mean that the interaction potential energy is independent of the reaction coordinate, but rather that it has been set to its value at the transition state, Q j = 0, because we only consider the motion at that point. The Hamiltonian HXlans accordingly only depends on Px, as for a free translational motion, so... 

Here, Vf is the kinetic energy operator for particle i, (with h = 1, 2m, = 1, for all i), Vi represents the interaction of particle i with an external potential, such as that associated with nuclei in the system, and Vij represents the mutual interaction between particles i and j. The definition of the singleparticle Hamiltonian, hi, is evident. We are interested in the solutions of the many-particle time-independent Schrodinger equation,... 

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