Wave operators definition

As one can see, the operator has a property of the wave operator (it transforms the projection of the exact wave function into the exact wave function), however, it should be stressed that the operator converts just one projected wave function into the corresponding exact wave function so we will denote it as a state-specific wave operator in contrast to the so-called Bloch wave operator [46] that transforms all d projections into corresponding exact states. From definition (11) it is iimnediately seen that the state-specific wave operators obey the following system of equations for a = 1,..., d... 

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. 

For an example, see [8] where definition A is used with K as Bloch s wave operator [6]. 

That the wave function for an orbital theoretically has no outer limit as one moves outward from the nucleus raises interesting philosophical questions regarding the sizes of atoms. Chemists have agreed on an operational definition of atomic size, as we will see in later chapters. 

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... 

Owing to the definition of fhe wave operator in intermediate normalization, i.e. P2P = and P2Q = 0, then only those matrix elements will be non-zero for which we have q,) e M and p) M. For this reason, the energy difference is always (E° - E") 0 and the matrix elements of the wave operator become... 

Obviously, any excitation of a core orbitals or excitation into a virtual orbital always leads out of the model space owing to our definitions above. The other terms of the wave operator, in contrast, which include only creation and annihilation operators of the (core-)valence orbitals, may result in either internal or external excitations, in dependence also of the particular basis function [Pg.195]

With these definitions of the perturbation and the model space, we can now solve successively the Bloch equation (16) for the energy corrections (e ) and wave operator (w ). As discussed above, we shall need the wave operator up to order (n — 1) if we wish to determine the energy correction to order n ... 

HUkiro = EkUkiro, and from the definition of the CC wave operator, we get ... 

In the first subsection, we define the model function and various projection operators. For the case of a single-reference function, the term model function is synonymous with zero-order function . This is followed by a derivative of the effective Hamiltonian operator, and then definitions of the wave operator and the reaction operator. 

From their definition the DI operators are easily implemented. Nevertheless, this implementation work is unnecessary if IDL or pv-wave are used, where the respective operators are simply picked from the rich library. 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

The relationship between different components of orbital angular momentum such as Lz and Lx can be investigated by multiple SG experiments as discussed for electron spin and photon polarization before. The results are in fact no different. This is a consequence of the noncommutativity of the operators Lx and Lz. The two observables cannot be measured simultaneously. While total angular momentum is conserved, the components vary as the applied analyzing field changes. As in the case of spin or polarization, measurement of Lx, for instance, disturbs any previously known value of Lz. The structure of the wave function does not allow Lx to be made definite when Lz has an eigenvalue, and vice versa. 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

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