Hamiltonian configuration space

Next, consider an ensemble defined in configuration space, so that the density matrix has the form of Eq. (8-190). We assume that the eigenvectors X> are not eigenvectors of the hamiltonian. We have... 

In concluding this section we note that the hamiltonian describing the system of noninteracting charged spin 0 particles, Eq. (9-199), can be expressed in terms of the configuration space operators <(>(x) and (x) as follows ... 

The Hamiltonian again has the basic form of Eq. (63). The system is described by the nuclear coordinates, Q, which are relative to a suitable nuclear configuration Q0. In contrast to Section III.C, this may be any point in configuration space. As a diabatic representation has been assumed, the kinetic energy operator matrix, T, is diagonal with elements... 

Let us take a simple example, namely a generic Sn2 reaction mechanism and construct the state functions for the active precursor and successor complexes. To accomplish this task, it is useful to introduce a coordinate set where an interconversion coordinate (%-) can again be defined. This is sketched in Figure 2. The reactant and product channels are labelled as Hc(i) and Hc(j), and the chemical interconversion step can usually be related to a stationary Hamiltonian Hc(ij) whose characterization, at the adiabatic level, corresponds to a saddle point of index one [89, 175]. The stationarity required for the interconversion Hamiltonian Hc(ij) defines a point (geometry) on the configurational space. We assume that the quantum states of the active precursor and successor complexes that have non zero transition matrix elements, if they exist, will be found in the neighborhood of this point. 

We may transform this to the quantum-mechanical Hamiltonian operator by substitution of the configuration space operators... 

The definition of the pair part or, equivalently the no-pair part of Hmat is not unique. The precise meaning of no-pair implicitly depends on the choice of external potential, so that the operator Hm t depends implicitly on the external potential, whereas the sum Hmat Hm t + Hm t is independent of the choice of external potential. Since the no-pair part conserves the number of particles (electrons, positrons and photons) we can look for eigenstates of Hm j in the sector of Fock space with N fermions and no photons or positrons. Following Sucher [18,26,28], the resulting no-pair Hamiltonian in configuration space can be written as... 

The Fock-space Hamiltonian H is equivalent to the configuration-space Hamiltonian H insofar as both have the same matrix elements between n-electron Slater determinants. The main difference is that H has eigenstates of arbitrary particle number n it is, in a way, the direct sum of aU // . Another difference, of course, is that H is defined independently of a basis and hence does not depend on the dimension of the latter. One can also define a basis-independent Fock-space Hamiltonian H, in terms of field operators [11], but this is not very convenient for our purposes. 

Let us focus on molecular systems for which we know molecular Hamiltonian models, H(q,Q). Electronic and nuclear configuration coordinates are designated with the vectors q and Q, respectively x = (q,Q) = (qi,..., qn, Qi,---, Qm- For an n-electron system, q has dimension 3n Q has dimension 3m for an m-nuclei system. The wave function is the projection in configuration space of a particular abstract quantum state, namely P(x) P(q,Q), and base state func-... 

The caveat is that integration over electronic configuration space is performed first. Any physical base quantum state with respect to the Coulomb Hamiltonian is a robust species. Observe that no structural features are implied yet. Thus, separability via electronic quantum numbers is achieved although the general quantum states E(q,Q,t) are not separable. 

Here r represents the coordinates for the system (x, y, z for each electron and each nucleus) and H is the Hamiltonian operator correspond ing to the total energy of the system. The integral is taken over the whole of configuration space for the system. if is the complex conjugate of if. 

The form of the new energy function ( Hamiltonian ) Sa( u, c) reflects the representation chosen for the region of configuration space relevant to the phase because this energy function carries a phase label, it can also be used to absorb the constraint [Eq. (6)] that restricts the integral to that region [60]. 

When these matrices are applied in an N-particle configuration space (N k) the moments of the hamiltonian behave on ensemble averaging in the limit of large dimensionality as in the case of noninteracting particles without averaging. This reflects the dominance of binary correlations in the operator products HP when the ensemble averaging is performed in the dilute system (k N). 

Note that the amplitudes are invariant elements when compared with that in Eq. (1) the configuration space is the support for the complex functions (x I, f) and the set of base functions (x) relates to eigenfunctions of the time-independent Schrodinger equation. The Hamiltonian incorporates constitutive parameters of the material system. 

It is a phase space rather than configuration space theory, so it can treat Hamiltonian systems containing unconserved angular momenta like Coriolis interactions which prevent the Hamiltonian from being written as a sum of the kinetic and potential energies [6,18]. The resulting hypersurfaces are dynamical in that they involve momenta as well as coordinates. 

In order to understand the problem of finding TS with three or more DOFs, it is useful to address the question of dimensionalities, in configuration and phase space. In classical, Hamiltonian dynamics, transition states are grounded on the idea that certain surfaces (more precisely, certain manifolds) act as barriers in phase space. It is possible to devise barriers in phase space, since in phase space, in contrast to configuration space, two trajectories never cross [uniqueness of solutions of ODEs, see Eq. (4)]. In order to construct a barrier in phase space, the first step is to construct a manifold if that is made of a set of trajectories [8]. 

The potential energy surface associated with the 3-DOF Hamiltonian, Eq. (31), resembles the usual 2-DOF one. In particular, a threshold energy may be defined, Et = co /6e, below which the motion is bound and above which the motion becomes unbound. At = configuration space is an equilateral triangle for the 2-DOF version and a cone for the 3-DOF case (see Fig. 14). 

Since spin-orbit coupling is normally not included in the Born-Oppenheimer Hamiltonian, singlet and triplet states can be distinguished. In a discussion of photochemical processes, large areas of the nuclear configuration space are of interest, and it is useful to label the energy surfaces in a way that differs from the one conventionally used by spectmscopists. Ai inv... 

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