Electronic Hamiltonian mapping

Analytic, exact solutions cannot be obtained except for the simplest systems, i.e. hydrogen-like atoms with just one electron and one nucleus. Good approximate solutions can be found by means of the self-consistent field (SCF) method, the details of which need not concern us. If all the electrons have been explicitly considered in the Hamiltonian, the wave functions V, will be many-electron functions V, will contain the coordinates of all the electrons, and a complete electron density map can be obtained by plotting Vf. The associated energies E, are the energy states of the molecule (see Section 2.6) the lowest will be the ground state , and the calculated energy differences En — El should match the spectroscopic transitions in the electronic spectrum. 

Because of the off-diagonal nature of the electronic Hamiltonian in the di-abatic basis, transitions among different electronic states can, and in general will, occur during the forward and backward time evolutions. As a first step towards a computationally convenient expression for the non-adiabatic correlation function, we shall make use of the mapping Hamiltonian formalism to account for these transitions [28-31]. 

The second mapping relation acts on the electronic Hamiltonian operator. This quantity can be rewritten in the diabatic basis as... 

The /V-electron Hamiltonian is H = f + U + V, where T is kinetic energy, U, the interelectronic Coulomb interaction, and V. the external potential term. A system of N interacting electrons is modeled by a system of N noninteracting electrons moving in a self-consistent mean field. This requires a postulated mapping rule by... 

The PCM method is one of the best known of such models. In essence, it involves the generation of a solvent cavity from spheres centered at each atom in the solute the polarization of the solvent is represented by means of virtual point charges mapped onto the cavity surface and proportional to the derivative of the solute electrostatic potential at each point, calculated from the molecular wavefunction. The point charges are then included into the one-electron Hamiltonian, and therefore they induce a polarization of the solute. An iterative procedure is performed until the wavefunction and the point charges are self-consistent. 

From the above discussion, we can see that the purpose of this paper is to present a microscopic model that can analyze the absorption spectra, describe internal conversion, photoinduced ET, and energy transfer in the ps and sub-ps range, and construct the fs time-resolved profiles or spectra, as well as other fs time-resolved experiments. We shall show that in the sub-ps range, the system is best described by the Hamiltonian with various electronic interactions, because when the timescale is ultrashort, all the rate constants lose their meaning. Needless to say, the microscopic approach presented in this paper can be used for other ultrafast phenomena of complicated systems. In particular, we will show how one can prepare a vibronic model based on the adiabatic approximation and show how the spectroscopic properties are mapped onto the resulting model Hamiltonian. We will also show how the resulting model Hamiltonian can be used, with time-resolved spectroscopic data, to obtain internal... 

The problem of T[p] is cleverly dealt with by mapping the interacting many-electron system on to a system of noninteracting electrons. For a determinantal wave function of a system of N noninteracting electrons, each electron occupying a normalized orbital >p, (r), the Hamiltonian is given by... 

Here, for notational convenience, we have assumed that Vnm = We would like to emphasize that the mapping to the continuous Hamiltonian (88) does not involve any approximation, but merely represents the discrete Hamiltonian (1) in an extended Hilbert space. The quantum dynamics generated by both Hamilton operators is thus equivalent. The Hamiltonian (88) describes a general vibronically coupled molecular system, whereby both electronic and nuclear DoF are represented by continuous variables. Contrary to Eq. (1), the quantum-mechanical system described by Eq. (88) therefore has a well-defined classical analog. 

Let us first consider the relation to the mean-field trajectory method discussed in Section III. To make contact to the classical limit of the mapping formalism, we express the complex electronic variables imaginary parts, that is, mean-field Hamiltonian function which may be defined as... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

The Hamiltonian of valence electrons (39), in the so-called orthogonal representation (or in the most localized representation, neglecting orbital overlap) can be mapped on a tight-binding form Hamiltonian... 

S. Bonella and D.F. Coker. A semi-classical limit for the mapping Hamiltonian approach to electronically non-adiabatic dynamics. J. Chem. Phys., 114 7778, 2001. 

If we disregard electronic phenomena, ignore the residual coupling terms between atomic quasi-particles, and specialize to the case of a single kind of composite quasi-particle, i.e. a pure chemical substance, at low density, the dynamical map P = W(X) allows us to write the Hamiltonian iC in the form,... 

We have described a mixed MOVB model for describing the potential energy surface of reactive systems, and presented results from applications to SN2 reactions in aqueous solution. The MOVB model is based on a BLW method to define diabatic electronic state functions. Then, a configuration interaction Hamiltonian is constructed using these diabatic VB states as basis functions. The computed geometrical and energetic results for these systems are in accord with previous experimental and theoretical studies. These studies show that the MOVB model can be adequately used as a mapping potential to derive solvent reaction coordinates for... 

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