Hamiltonian motion equation

We now wish to analyse the potential part of the Hamiltonian in equation (1) in order to understand the combination of vibrational and (pseudo)rotational motion of the system. To do this, it is very convenient to follow O Brien [26,27] and use the angular parametrization... 

Stanton JF, Gauss J (1995) Perturbative treatment of the similarity transformed Hamiltonian in equation-of-motion coupled-cluster approximations. J Chem Phys 103 1064-1076. 

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... 

The n-electron wave function r t) describes the motion of the electrons in the field of the nuclei. Due to the electron-electron interaction term in the Hamiltonian, this equation cannot be solved without approximations. The HF approximation assumes that the n-particle wave function r) can be written as an antisymmetrized product of one-electron functions lF (ri) (so-called orbitals) ... 

Here, is the Hamiltonian in Equation 4.1 with the first term neglected I, 2,. .., N stand for the electronic coordinates 1 = (xi,yi,Zi, i), etc. and E IQ) forms a connected surface as a function of Q, called the potential energy surface (PES). By fitting with known mathematical functions, PES may be written with an explicit nuclear coordinate dependence. We will soon show that the PES, that is, Eg(Q), is a potential surface for the motion of the nuclei. 

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... 

Typically, however, the more channels there are that are strongly coupled in an inelastic collision the better it is to approximate the dynamics by classical mechanics i.e., there are more channels the heavier the particles, but this is also the limit in which classical mechanics is a better approximation. Thus there have been many classical trajectory simulations of inelastic collision processes [71]. These have the advantage that no approximations other than the use of classical mechanics need be made, and the number of classical equations of motion to be solved (Hamiltonian s equations) grows linearly with the number of particles, while the numbers of coupled channels in the coupled-channel Schrodinger equation grows exponentially with this number. 

Spin-orbit interaction implies a coupling of the two "motions" of spin and orbit. From the discussion in Chapter 8, we would expect that this coupling may mix the hydrogenic spin-orbital states with the resulting wavefunctions no longer assured to be eigenfimc-tions of the and operators. The interaction Hamiltonian in Equation 10.20 can be rewritten following Equation 8.65 ... 

This paper describes some new developments and applications based on the reaction path model. In section II the original form of the reaction path Hamiltonian [of. equation (1) below] is transformed to a new representation that has a more desirable structure for some applications. Section III shows how the reaction path model makes application of the unified statistical model for chemical reactions especially simple, and a generalized version of the unified statistical model is also developed there. Finally, in section IV the fact that the reaction path model consists of one special degree of freedom — i.e., the reaction coordinate — coupled to a number of harmonic oscillators is exploited to derive a generalized Langevin equation (GLE) for motion along the reaction path. This is a reduced equation of motion for only the reaction coordinate, but it experiences friction" and a "random force" because of coupling to the transverse vibrational modes. 

Here, ctq refers to thermal equilibrium and the starred quantities are to be taken in the interaction representation. Neglecting nonsecular terms introducing the narrowing limit (a small correlation time t compared to the inverse Larmor frequency, cUoT< 1) and using the rotational invariance of the Hamiltonian describing the molecular motions. Equation (3) simplifies as follows. 

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

The basic equation [8] is tlie equation of motion for the density matrix, p, given in equation (B2.4.25), in which H is the Hamiltonian. 

Hamiltonian in the second-quantization fomi, only one //appears in this fmal so-called equation of motion (EOM) f//, <7/]+ = AJr 7 p(i e. in the second-quantized fomi, // and //are one and the same). 

The total effective Hamiltonian H, in the presence of a vector potential for an A + B2 system is defined in Section II.B and the coupled first-order Hamilton equations of motion for all the coordinates are derived from the new effective Hamiltonian by the usual prescription [74], that is. 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

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