Hamiltonian-Jacobi equation

We now substitute the series (5) for S in the Hamiltonian-Jacobi equation for the perturbed motion... 

Our first problem will therefore be to find the proper variables wp°, Jp° in place of the wp° s, Jp° s, to serve as the limiting values in an approximation to wp, Jp. For this purpose we make use of the method of secular perturbations already discussed (cf. 18). It consists in finding a transformation w°J°->-Mj0J0 such that the first term of the perturbation function, when averaged over the unperturbed motion, depends only on the J° s. We assume at the start that Hj is not identically zero we shall return later to the case where it vanishes identically. Wc have now, as before, to solve a Hamiltonian-Jacobi equation... 

These equations show that it is the classical action Sn that satisfies the Hamiltonian-Jacoby equation (3.11) with coordinate x, momentum p = dSn (x)/dx, and Hamiltonian equal to zero (stationary condition). The Hamiltonian equations of motion for the system are... 

S, Chapman, B. C. Garrett, and W. H. Miller, Semiclassical eigenvalues for nonseparable systems Nonperturbative solution of the Hamiltonian-Jacobi equation in action-angle variables,... 

It may happen that the Hamiltonian function does not consist o a sum of terms depending on only one pair of variables qkpk, but that the Hamilton-Jacobi equation may be solved by separation of the variables, i.e. on the assumption that... 

After calculating the unperturbed motion of the inner electron, we can find the secular motions of the remaining variables by introducing a new Hamiltonian function, the mean value of Hx taken over the unperturbed motion of the inner electron. The integration of the corresponding Hamilton-Jacobi equation is again performed by the methods of the theory of perturbations. 

For systems of n degrees of freedom, take a normally hyperbolic invariant manifold with 2r normal directions in phase space. Note that for Hamiltonian systems, the dimension of the normal directions in phase space is alway even, because the eigenvalues of the variational equation (Jacobi equation) is symmetric around the value 0. Thus, the dimension of the normally hyperbolic invariant manifold is 2n — 2r and, for its stable and unstable manifolds, their dimensions are 2n — r, respectively. The dimension of their homoclinic intersection, if it exists, is 2n — 2r in the 2n-dimensional phase space. When we consider the intersection manifold on the equi-energy surface, its dimension on the surface is 2n — 2r — 1. Thus, the dimension d of the intersection on the Poincare section is d = 2n — 2r — 2. 

The instanton theory of tunneling splittings in hydrogen-bonded systems and decay of metastable states in polyatomic molecules was studied by Nakamura et al. [182, 192, 195, 201-204, 216] They formulated a rigorous solution of the multidimensional Hamiltonian-Jacobi and transport equations, developed numerical methods to construct a multidimensional tunneling instanton path, and applied this method to HO [201], malonaldehyde [192, 195], vinyl radical [203], and formic acid dimer [202]. Coupled electron and proton transfer reactions were recently reviewed by Hammes-Schiffer and Stuchebrukhov [209]. 

Equation (5.2) is a combination of the two two-dimensional Hamiltonians (2.39) and (3.15) which describe the vibrational and rotational excitations of BC separately. The Jacobi coordinates R, r, and 7 are defined in Figures 2.1 and 3.1 and P and p denote the linear momenta corresponding to R and r, respectively, j is the classical angular momentum vector of BC and 1 stands for the classical orbital angular momentum vector describing the rotation of A with respect to BC. For zero total angular momentum J=j+l = 0we have 1 = — j and the Hamilton function reduces to... 

The extension to more than one dimension is rather straightforward within the time-dependent approach (Heller 1978a, 1981a,b). For simplicity we restrict the discussion to two degrees of freedom and consider the dissociation of the linear triatomic molecule ABC into A and BC(n) as outlined in Section 2.5 where n is the vibrational quantum number of the free oscillator. The Jacobi coordinates R and r are defined in Figure 2.1, Equation (2.39) gives the Hamiltonian, and the transition dipole function is assumed to be constant. The parent molecule in the ground electronic state is represented by two uncoupled harmonic oscillators with frequencies ur and ur, respectively. 

In order to illustrate electronic transitions we discuss the simple two-dimensional model of a linear triatomic molecule ABC as depicted in Figure 2.1. R and r are the appropriate Jacobi coordinates to describe the nuclear motion and the vector q comprises all electronic coordinates. The total molecular Hamiltonian Hmoi, including all nuclear and electronic degrees of freedom, is given by Equation (2.28) with Hei and Tnu being the electronic Hamiltonian and the kinetic energy of the nuclei, respectively. 

In order to obtain a more compact formulation of the mixed quantum-classical equations we use a Hamilton-Jacobi-like formalism for the propagation of the quantum degree of freedom as in earlier studies [23], A similar approach has been introduced by Nettesheim, Schiitte and coworkers [54, 55, 56], TTie formalism presented here is based on recent investigations of the present authors [23], This formalism can be summarized as follows. Starting from the Hamiltonian Eqn. (2.2) and averaging over the x- and y-mode, respectively, gives... 

We here derive the coupled channel equations in Jacobi and hyperspherical coordinates. By using Eqs. (1) and (2). we can obtain the Hamiltonian in Jacobi coordinates for either channel as... 

It is the Hamiltonian function of a system in which all co-ordinates but one are cyclic. The motion may be found in the usual way by solving a Hamilton-Jacobi differential equation for one degree of freedom. Since and ij (like ° and rj°) must vanish with A, we need only consider small motions, that is, those belonging to a system whose Hamiltonian function is... 

In Fig. lb we show an alternative coordinate system for the same four atoms. In the absence of an external force, the center of mass remains stationary. There is therefore no need to include the coordinate in the motion. The dynamics reduces to 18 equations of motion in three coordinates, r12, r34, and R. These are called the Jacobi coordinates. Instead of the masses of the individual atoms appearing in the Hamiltonian, we now have the reduced masses, x12, jx34, and x ... 

We have assumed logistic population growth, with K being the carrying capacity of the environment. The probability of a newborn being a disperser is /u. = rj/(rj +r2>. Note that if rj 0, the whole population disperses and (7.17) becomes the standard RD equation. Since the kinetics satisfy the KPP criteria, we can calculate the front velocity from the Hamilton-Jacobi formalism. The corresponding Hamiltonian is... 

The FSCC equation (2.7) is solved iteratively, usually by the Jacobi algorithm. As in other CC approaches, denominators of the form Eq —E ) appear, originating in the left-hand side of the equation. The well-known intruder state problem, appearing when some Q states are close to and strongly interacting with P states, may lead to divergence of the CC iterations. The intermediate Hamiltonian method avoids this problem in many cases and allows much larger and more flexible P spaces. 

---