Hamiltonian equation normal form theory

Until now, we have discussed NHIMs in general dynamical systems. In this section, we limit our argument to Hamiltonian systems and show how singular perturbation theory works. In particular, we discuss NHIMs in the context of reaction dynamics. First, we explain how NHIMs appear in conventional reaction theory. Then, we will show that Lie permrbation theory applied to the Hamiltonian near a saddle with index 1 acmally transforms the equation of motion near the saddle to the Fenichel normal form. This normal form can be considered as an extension of the Birkhoff normal form from stable fixed points to saddles with index 1 [2]. Finally, we discuss the transformation near saddles with index larger than 1. 

The theory is based on an optimized reference state that is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions (r). This is the simplest form of the more general orbital functional theory (OFT) for an iV-electron system. The energy functional E = (4> // < >)is required to be stationary, subject to the orbital orthonormality constraint (i j) = Sij, imposed by introducing a matrix of Lagrange multipliers kj,. The general OEL equations derived above reduce to the UHF equations if correlation energy Ec and the implied correlation potential vc are omitted. The effective Hamiltonian operator is... 

A deeper argument is that local density functional derivatives appear to be implied by functional analysis [2,21,22]. The KS density function has an orbital structure, p = Y.i niPi = X fa- For a density functional Fs, strictly defined only for normalized ground states, functional analysis implies the existence of functional derivatives of the form SFj/ Sp, = e, — v(r), where the constants e, are undetermined. On extending the strict ground-state theory to an OFT in which OEL equations can be derived, these constants are determined and are just the eigenvalues of the one-electron effective Hamiltonian. Since they differ for each different orbital energy level, the implied functional derivative depends on a direction in the function-space of densities. Such a Gateaux derivative [1,2] is equivalent in the DFT context to a linear operator that acts on orbital functions [23]. 

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