Diabatic states Hamiltonian

SCF) techniques (11). In a similar fashion, t jg is determined variationally by minimizing Hgg. Once SCF solutions for the diabatic states and i ig are available, H g is straightforwardly calculated using the same system Hamiltonian. Note that in... 

X + P )/4, which by construction varies between 0 (system is in /i)) and 1 (system is in /2)). Describing, as usual, the nuclear motion through the position X, the vibronic PO can then be drawn in the (A dia,v) plane. Here, the subscript dia emphasizes that we refer to the population of the diabatic states which are used to define the molecular Hamiltonian H. For inteipretational purposes, on the other hand, it is often advantageous to change to the adiabatic electronic representation. Introducing the adiabatic population A ad. where Nad = 0 corresponds to the lower and A ad = 1 to the upper adiabatic electronic state, the vibronic PO can be viewed in the (Nad,x) plane. Alternatively, one may represent the vibronic PO as a curve N d i + (1 — A ad)IFi between the... 

Suppose Va and V/, are diabatic surfaces which cross at some configuration and let cj/2 be the matrix element of the electronic Hamiltonian between the two diabatic states. In other words, the adiabatic curves are the eigenvalues of the potential energy matrix. 

Resonances in the absorption spectrum are the fingerprints of the eigenenergies of the binding part of the zeroth-order Hamiltonian (i.e., the Hamiltonian in the absence of coupling between the two diabatic states). 

Pacher, T., Cederbaum, L.S., and Koppel, H. (1988). Approximately diabatic states from block diagonalization of the electronic Hamiltonian, J. Chem. Phys. 89, 7367-7381. 

The free energies of the initial (i) and final (f) states, the so-called diabatic states in the ET process (discussed in more detail in Section 3.54, Reaction Field Hamiltonian, Electronic Structure models), are given by [28]... 

As a result of its dependence on the density (pa), the one-electron operator H is a pseudo-Hamiltonian, and the corresponding Schrodinger equation is nonlinear, so that its solution (for a fixed pin) must be self consistently adjusted to (e.g., by iteration) [3,53], In the case of full equilibrium, when pm = pa, both optical and inertial potentials (4>RF) depend on pa. As discussed below, the eigenstates of H (i.e., the electronically adiabatic states) are distinct from the diabatic states used to characterize the ET process (see footnote 5). 

One of the most valuable features of theoretical methods based on classical VB structures is their ability to calculate the energy of a diabatic state. Contrary to adiabatic states, a diabatic state is not an eigenfunction of the Hamiltonian. Such a state can be a single VB structure, separate VB curves of covalent and... 

While the definition of an adiabatic state is straightforward, as an eigenfunction of the Hamiltonian within the complete set of VB structures, the concept of diabatic state is less clear-cut and accepts different definitions. Strictly speaking, a basis of diabatic states (fi, ...) should be such that eq 21 is satisfied for any variation <5Q of the geometrical coordinates. 

Having defined a diabatic state as a unique VB structure, or more generally as a linear combination of a subset of the full VB structure set that describes the adiabatic state, in the next step one has to specify the orbitals needed to construct the VB structure(s) of this diabatic state. One first possibility is to keep for the diabatic state the same orbitals that optimize the adiabatic state. This has the advantage of simplicity. Practically, once the orbitals have been determined at the end of the BOVB orbital optimization process, the hamiltonian matrix is constructed in the space of the VB structures and the... 

A linear combination of the three diabatic states in equation (16) provides a good description of the ground state potential surface in all regions along the reaction coordinate, and the potential energy of the system is obtained by solving the secular equation (15) by diagonalizing the Hamiltonian matrix to yield... 

For the two-state Hamiltonian the adiabatic functions can be easily transformed into diabatic functions. In the latter representation the Hamiltonian is not diagonal, and the model is completely specified by the matrix Hik as a function of R. If, besides, the semiclassical approximation is adopted there will be only two essential functions AH(t) = Hn(t) — and... 

The energetics needed to characterize ET between weakly coupled diabatic states, ij/j and [pf, can be expressed in terms of free energy quantities based on the Hamiltonians for nuclear motion (//, and ///) at the level of the BOA [47],... 

In this appendix we generalise the expressions of the diabatic quantities first introduced in Sec. 2 for the ideal case of an exact two-level problem to a more realistic description. In a normal situation, the Hamiltonian has an infinite number of eigenstates, and there is no finite number of strictly diabatic states [76] that can describe a given pair of adiabatic states [77-80]. Instead, one can define a unitary transformation of the adiabatic states generating two quasidiabatic states characterised by a residual non-adiabatic coupling, as small as possible, but never zero (see, e.g., [5,24,32-35]). In practice, the electronic Hilbert space is always truncated to a finite number of configurations. In what follows, we consider the case of MCSCF wavefunctions and make use of generalised crude adiabatic states adapted to this. 

It is clear from (A.8) and (A.9) that the gradient difference and derivative coupling in the adiabatic representation can be related to Hamiltonian derivatives in a quasidiabatic representation. In the two-level approximation used in Section 2, the crude adiabatic states are trivial diabatic states. In practice (see (A.9)), the fully frozen states at Qo are not convenient because the CSF basis set l Q) is not complete and the states may not be expanded in a CSF basis set evaluated at another value of Q (this would require an infinite number of states). However, generalized crude adiabatic states are introduced for multiconfiguration methods by freezing the expansion coefficients but letting the CSFs relax as in the adiabatic states ... 

Notice that the electronic Hamiltonian is in general non-diagonal in the coordinate-diabatic state representation, and its matrix elements are given by... 

Diabatic states are obtained from a similar approach, except that additional term (or terms) in the Hamiltonian are disregarded in order to adopt a specific physical picture. For example, suppose we want to describe a process where an electron e is transferred between two centers of attraction, A and B, of a molecular systems. We may choose to work in a basis of vibronic states obtained for the e-A system in the absence of e-B attraction, and for the e-B system in the absence of the e-A attraction. To get these vibronic states we again use a Born-Oppenheimer procedure as described above. The potential surfaces for the nuclear motion obtained in this approximation are the corresponding diabatic potentials. By the nature of the approximation made, these potentials will correspond to electronic states that describe an electron localized on A or on B, and electron transfer between centers A and B implies that the system has crossed from one diabatic potential surface to the other. 

The classical two-state Hamiltonian has been widely used by scientists. It is, for example, the same one that was classically used for the motion of planets (with elliptic diabatic orbitals instead of parabolic ones). One could easily imagine that there might be problems applying classical theory to the motion of electrons within... 

Before now discussing the solution of the coupled equations (4) we introduce another important approximation viz. the two-state approximation. Anticipating that in the inelastic processes which we will discuss here only two diabatic states j and 2 play a role, we limit the expansion of the complete wave function to these two wave functions. This approximation seems justified if transitions are confined to the crossing region and if all other states remain far from these two states for all R. Now one can show that these wave functions can be chosen real if magnetic interactions are neglected. In that case the relation Htj = holds. The adiabatic states are easily found by diagonalizing the electronic Hamiltonian Hel, so one obtains... 

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