Adiabatic separation

Yet another new method for calculating PMFs has recently appeared, which appears promising in initial tests.53 In this method, an adiabatic separation between the reaction coordinate and the remaining degrees of freedom is imposed. This allows improved sampling while alleviating the need for (often difficult) post processing. 

The technique can be used either to perform geometry optimization, by simultaneously annealing the wavefunction and the geometry, or to simulate real dynamics, if the temperature of the fictitious (electronic) parameters is kept close to zero. A drawback of the method is that small masses must be chosen for the electronic parameters in order to achieve an adiabatic separation of the nuclear and the fictitious parameter motions. As a consequence, time steps smaller than MD simulations involving only nuclear motion, are required. 

Here, the diabatic PES for the motion of the skeleton modes are Ea(Q) = Ea — faaQ + 1/2QKqqQ and the coupling between the diabatic states is given by V Q) = — fa0Q + 1/2QKq/3Q. The fo0 and Ko0 are the diabatic state matrix elements of the forces and the Hessian for the skeleton modes, respectively. Exemplary PES are shown in Fig. 1. Note that Eq. (1) is exact within the reaction surface approximation, i.e., no assumption concerning an adiabatic separation of H-atom and skeleton motions has been made. 

In the quantum mechanical description of molecules (atoms and clusters) one problem has been the identification and validity of adiabatic separations of electronic ip) md nuclear (R) coordinates [30] This problem has been with us ever since the Bom-Oppenheimer (BO) theory was published in 1927 [1,2]. But this approach, as implemented in quantum chemistry, has serious deceiving aspects. 

Goscinski, 0. and Mujica, V. (1987) Adiabatic separation, broken symmetries and geometry optimization, in Erdahl, R. and Smith Jr, V. H. (eds) Density matrices and density functionals, Reidel, Dordrecht. 

O.I. Tolstikhin, S. Watanabe, M. Matsuzawa, Slow variable discretization a novel approach for Hamiltonians allowing adiabatic separation of variables, J. Phys. B At. Mol. Opt. Phys. 29 (1996) L389. 

His early works on adiabatic separation of valence electrons from fast moving core electrons in atoms attracted much attention a few years later, when the time came for the theory of pseudopotentials. These papers are still being referred to, more than forty years after their publication. 

When conventional spectroscopic pictures and means, like normal and local modes or adiabatic separation, are useless for the interpretation of absorption spectra because either the displacements from equilibrium are too large or the coupling between the degrees of freedom is too strong, unstable periodic orbits might provide the desired link between molecular... 

Now, we may observe that for weak H-bonds, it is possible to perform an adiabatic separation between the motions of the high and low frequency stretching modes that will allow us to treat many important situations that cannot be handled beyond this approximation, within the straightforward approach of Section II.B. 

The idea of a classical treatment of the nuclear motion within the molecular dynamics (MD) scheme with ab initio determined, quantum-mechanical forces acing on nuclei is as old as quantum mechanics.11,12 The commonly used Born-Oppenheimer approximation12 introduces the concept of potential energy surface (PES). Different time-scales for nuclear and electronic motion allows for the adiabatic separation of the nuclear and electronic wave-function. In the Born-Oppenheimer molecular dynamics (BO-MD) the nuclei move according to Newton laws, while the quantum mechanics is required to determine the potential for this motion ... 

We could return to the exact equation (2.131) and examine the matrix elements of the Cn, n terms in the same manner as we dealt with the diagonal f,, terms. It is, however, easier to turn to the form of the exact Hamiltonian given in equations (2.113) and (2.120). The terms in this Hamiltonian which cause a breakdown of the adiabatic separation of nuclear and electronic motion are... 

The purpose of this chapter is to review some properties of isomerizing (ABC BCA) and dissociating (ABC AB + C) prototype triatomic molecules, which are revealed by the analysis of their dynamics on precise ab initio potential energy surfaces (PESs). The systems investigated will be considered from all possible viewpoints—quanmm, classical, and semiclassical mechanics—and several techniques will be applied to extract information from the PES, such as Canonical Perturbation Theory, adiabatic separation of motions, and Periodic Orbit Theory. 

So far, we have only separated out the HX vibrational motion. Generally, such a solution of the stationary Schiodinger equation is not computationally feasible for clusters with more atoms. Therefore, other approximations have to be employed. At the same time, all phenoniena important for the cluster structure have to be properly included. We have performed an adiabatic separation of the HX libra-tional motion from the motion of the heavy particles, i.e., from the cage modes. Moreover, the cage modes have been calculated within the harmonic approach, i.e., by a diagonalization of the Hessian matrix. Formally, the wavefunction is expressed as... 

In the AFED method, an adiabatic separation is imposed between the degrees of freedom of interest, i = 1... n, and the rest of the system by choosing a set of masses, mj, where rrii > mjij=n..3N- This separation gives rise to an effective AFED Hamiltonian,... 

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