Hamiltonian wave operators

From Adiabatic to Effective Hamiltonian Matrices Through the Wave Operator Procedure... 

Let us look at the standard Hamiltonian (13). Its representation restricted to the ground state and the first excited state of the fast mode may be written according to the wave operator procedure [62] by aid of the four equations... 

Figure 5. Illustration of the equivalence between the spectral densities obtained within the adiabatic approximation and those resulting from the effective Hamiltonian procedure, using the wave operator. Common parameters a0 = 0.4, co0 = 3000cm-1, C0Oo = 150cm-1, y = 30cm-1, and T = 300 K.

Durand P (1983) Direct determination of effective Hamiltonians by wave-operator methods. I. General formalism. Phys Rev A 28 3184... 

As concerns cluster amplitudes, if we employ the exact Hamiltonian in the normal-ordered-product form (31) with the /i-th configuration as a Fermi vacuum, the basic equation for the single-root wave operator (25) takes the form... 

Note that all the above expressions characterize the effective Hamiltonian formalism per se, and are independent of a particular form of the wave operator U. Indeed, this formalism can be exploited directly, without any cluster Ansatz for the wave operator U (see Ref. [75]). We also see that by relying on the intermediate normalization, we can easily carry out the SU-Ansatz-based cluster analysis We only have to transform the relevant set of states into the form given by Eq. (16) and employ the SU CC Ansatz,... 

D. Maynau, P. Durand, J. P. Duadey, and J. P. Malrieu, Phys. Rep. A, 28, 3193 (1983). Direct Determination of Effective-Hamiltonians by Wave-Operator Methods. 2. Application to Effective-Spin Interactions in -Electron Systems. P. Durand and J. P. Malrieu, in Advances in Chemical Physics (Ah Initio Methods in Quantum Chemistry—I), K. P. Lawley, Ed., Wiley, New York, 1987, Vol. 67, pp. 321-412. Effective Hamiltonians and Pseudo-Operators as Tools for Rigorous Modelling. 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

If we now consider the generalized Silverstone-Sinanoglu strategy discussed above, then the most natural way to proceed is by way of an effective hamiltonian formalism. We introduce a single (state-universal) wave-operator O, whose action on 4 s produce the functions 4>k, defined by eq. (6.1.1) or (6.1.7), and write Schrodinger equations for 4[Pg.327]

It should be emphasized that the absence of terms in the wave operators in the preceding equations does not. reflect further truncation [e.g., with respect to exp(7 + T2 + T3)] rather it is a consequence of the triangle inequalities involving the (irreducible) ranks of the Hamiltonian, the external space operators, and the cluster operators. More specifically, a matrix element vanishes identically unless it has an overall rank of 0 from... 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

Hamiltonian An operator which when operating on the wave function of a quantum chemical system returns the energy of that system. The classical Hamiltonian function H = T + Fis the sum of the kinetic energy function Land the potential energy function V representing the total energy E of a system. 

This relation defines a p-fold fimction Zj = /(z), and one then looks for the "crossing" points with the straight line Zj = z, which gives the eigenvalues z = z = E. Substitution into the relation (4.3) gives then the exact wave functions. In comparison to the previous sections, this approach deals also with a secular equation of order p, but the wave operator now contains the energy E explicitly, and further all the degeneracies of the Hamiltonian H are removed. This means that the connection with the idea of the existence of a "model Hamiltonian" and a set of "model functions" is definitely lost. However, from the point-of-view of ab-initio applications this approach may offer other... 

Another consequence is that n0 now does determine not only the GS wave function but the complete Hamiltonian (the operators T and U are fixed), and thus all excited states, too ... 

Once we know the wave operator through Eqs. (13) and (14), we are ready to evaluate the matrix elements of the effective Hamiltonian in the basis of reference configurations because the effective Hamiltonian is given by the following relationship (see, for example. Ref. [55])... 

Before considering approximations for the wave operator, we will establish an important equation. We will show that the operator HQ, can be expressed in terms of the effective Hamiltonian. First, we note that... 

This original approach, first proposed by Bloch (22) in 1958, follows a pedagogical approach to obtain both the wave operator and the effective Hamiltonian. However, from a computational point of view, the perturbative expansion [Eq. (41)] frequently diverges and the first few terms give only an approximation to the exact solution. In vibrational... 

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