Effective Hamiltonian diagonalizing

It is appropriate to refer these quantities to the molecular-principal-axis system (o, b, c). In this way, the tensor components Dj., g -, and uj,. become uniquely defined molecular properties (g, g = , b, c). However, only the diagonal tensor components are useful in analysing hfs spectra as non-diagonal terms, g+g, do not enter matrix elements diagonal in J, and higher-than-first-order rotational perturbations of JT are barely observable. Thus it is sufficient to consider an effective Hamiltonian diagonal in J. This Hamiltonian is customarily written in terms of rotationally dependent interaction strengths g(A-, ... 

The ground-state effective Hamiltonian is diagonal with eigenvalues ha n + 5], whereas the excited state one is that of a driven quantum harmonic oscillator that must lead to coherent states. 

The eigenvalue equations of the two diagonal blocks of the effective Hamiltonian matrix is characterized by the equations... 

The Hiickel method attempts to solve these equations without explicitly writing down an expression for the effective Hamiltonian. It firstly assumes that the integrals Ha along the diagonal... 

The next step is to introduce temperature by averaging out the bath operators appearing in the time dependent terms of Eq.(51) [137] over an adequate ensemble. To this end, the partial trace (or sum of the diagonal elements) over the surrounding subsystem has to be taken. For the system in interaction, the effective Hamiltonian of the solvent Hmeff must be defined in such a way that the sum of HSeff+Hmeff leads to the... 

In this case, X is to be determined by requiring that the off-diagonal blocks of the resulting transformed Dirac Hamiltonian vanishes. It can be shown that the equation for X is identical to the one we obtained in the case of a unitary transformation as given in equation (38). In this case, the effective Hamiltonian hn and wave function xp, can be written as... 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

Obviously, we gain precisely the same expressions as in the multi-root theory since we postulated the same form of the effective Hamiltonian. We recall that all matrix elements of the effective Hamiltonian are expressed by means of connected diagrams only in the case of diagonal elements just connected vacuum diagrams may come into consideration and in the case of off-diagonal elements at least one part of a disconnected diagram would correspond to an internal excitation. 

If we take into account the expression for diagonal elements of the effective Hamiltonian (33), we get a system of nonlinear equations... 

The desired exact states j l i), given as linear combinations of jfl i) (see Sec. 2), are then obtained by diagonalizing the effective Hamiltonian = PHUP, defined on Mo, as guaranteed by the projector P onto Mo, i.e.. 

Once all the relevant cluster amplitudes have been determined by solving the SU CCSD equations, Eq. (21), while taking into account the C-conditions, we obtain the desired energies and eigenstates by diagonalizing the effective Hamiltonian. For details, we refer the reader to [62]. 

A conical intersection is expected above 10,000 cm 1 that has not yet been rigorously identified. This conical intersection also creates important anharmonicities in the fully diagonalized effective Hamiltonians of Weigert... 

The effective Hamiltonian by Abbouti Temsamani and Herman [123] is composed of a diagonal part given by a Dunham expansion with all the x s and g s as well as y244, and of a nondiagonal part including the following resonances [123] ... 

Figure 8. Time evolution, obtained via a numerical diagonalization of the effective Hamiltonian M (adapted from Ref. 45) when N = K and N > K for two different initial states. Shown is the survival probability of the initial state (dashed line) and the probability to remain bound (full line) vs. time. The reason for the difference between these two is due to the system sampling the rest of the bound phase space. At higher number N of bound states, for a given value of K, the delayed decay would be shifted to even longer times while the survival probability will remain essentially unchanged, showing that the delay is due to sampling of the bound phase space.

As in (3.13) we take the matrices in this equation to be defined in the basis that diagonalizes T so that, in the absence of Hi, the effective Hamiltonian is diagonal and the dynamics manifests trapping. 

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