Hamiltonian potential energies associated

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. 

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)... 

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... 

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 ... 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

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... 

The potential energy surface associated with the 3-DOF Hamiltonian, Eq. (31), resembles the usual 2-DOF one. In particular, a threshold energy may be defined, Et = co /6e, below which the motion is bound and above which the motion becomes unbound. At = configuration space is an equilateral triangle for the 2-DOF version and a cone for the 3-DOF case (see Fig. 14). 

Every solution of equation 1 defines a different electronic state of the molecular system. Associated with each state is its energy, Ei, which, because it is a function of the nuclear coordinates, defines a many-dimensional surface called the potential energy surface (PES) for that state. Physically, it represents the effective potential energy of interaction between the nuclei. Equation 1 may be rewritten to define the state energy as an expectation value of the state wavefunction over the electronic Hamiltonian ... 

The equation is written in velocity gauge. Atomic units are used. The particle has charge unity and mass m in units of the free electron mass. V is the constant potential energy appropriate for the interval under consideration. The vector potential is supposed to be spatially constant at the length scale of the structure. With such a vector potential, the A2 term contributes an irrelevant phase factor which can be omitted. For a one-mode field A(t) is written as Ao cos(ut). The associated electric field is 0 sin(ut), with 0 = uAq. px is the linear momentum i ld/dx. For such a time-periodic Hamiltonian, a scattering approach can be developped, with a well-defined initial energy, and time-independent transition probabilities for reflection and transmission. 

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