The wave operator

Equation (2.1), the original Schrodinger equation, can then be written 

This equation hold true for any model function, Thus we can write 

The wave operator, i is seen to be characterized by three relations  

The wave operator also satisfies the following relations  

Operating on the effective eigenequation (2.23) from the left by the wave operator, 17, gives 

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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.

These equations (14) and (15) determine the scalar and vector potentials in terms of p and J. When p and J are zero, these equations become wave equations with wave velocity c = y/l/pe. That is, A and are solutions of decoupled equations, where they are related by the wave operator... 

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... 

In order to obtain the wave operator A in a form suitable for practical calculations, we project Eq. (15) onto configurations from the Q and P subspaces... 

So far, we have specified the wave operator H in the BW form (15). If we adopt an exponential ansatz for the wave operator Cl, we can speak about the single-root multireference Brillouin-Wigner coupled-cluster (MR BWCC) theory. The simplest way how to accomplish the idea of an exponential expansion is to exploit the so-called state universal or Hilbert space exponential ansatz of Jeziorski and Monkhorst [23]... 

If we substitute the Hilbert space exponential ansatz (26) for the wave operator fl, we obtain the system of equations... 

For the sake of completeness, we recall that the idea of the single-root formalism exploiting the Hilbert space approach was also proposed by Banerjee and Simons [31] and Laidig and Bartlett [34,35]. In both approaches they start from the complete active space MC SCF wave function, however, in order to eliminate redundant cluster amplitudes they approximate the wave operator by... 

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,... 

We shall now employ the SU Ansatz for the wave operator U, Eq. (4), with the cluster operator T i) having the same general form as in the SR case, namely... 

Notice that in RS theory the wave operator is the same for all states under investigation. It is convenient to use also another (correlation) operator x, defined by... 

Here a complete model space (P-space), defined on eigenfunctions of Ho, representing all possible distributions of electrons between open shells, is utilized. In our case the model space consists of the ls22s22p4 and ls22p6 configurations for the even parity states and of the ls22s2p5 configuration for odd parity states. The wave-operator may be written as... 

D.A. Micha, E. Brandas, Variational Methods in the Wave Operator Formalism. A Unified Treatment for Bound and Quasi-Bound Electronic and Molecular States, J. Chem. Phys. 55 (1971) 4792. 

Van der Avoird A (1967) Perturbation theory for intermolecular interactions in the wave-operator formalism. J Chem Phys 47 3649-3653... 

The form of the wave operators need not be defined, but, in principle, they can describe any type of wavefunction, for example, Hartree-Fock or coupled-cluster wavefunctions. However, at their core, they always consist of strings of creation operators. We define the supermolecular wavefunction as... 

For the core-extensive theories (i.e., with the feature (al)), there must be an explicit cluster expansion structure with respect to the core electrons, and no such cluster expansion maintained for the valence electrons. The wave-operator ft should have then either of the following forms... 

It is interesting to note that ft in eq.(7.2.4) can be regarded as a valence universal wave-operator, i.e.,ft is also the wave-operator for the core-problem/94(a)/. This assertion follows from the simple observation that all T operators have destruction operators and hence they annihilate S. ... 

Mukhopadhyay e t a 1 /122/ generalized the ansatz (7.2.4) still further. They suggested the use of a recursive procedure whereby the wave-operator ft n at the n-valence lavel is recursively generated from the lower valence ft s. Thus, they define... 

The above formulation is quite general and applies equally well to quasi-complete model spaces having m holes and n partic 1es.When there are several p-h valence ranks in the parent model space, the situation is fairly complicated. The subduced model spaces in this case may belong to the parent model space itself. The valence-universality of ft in such a situation implies that ft is the wave—operator for all the subduced model spaces, in addition to those which have same number of electrons as in the parent model space. It appears that a more convenient route to solve this problem is to redefine the core in such a way that holes for the problem become particles and treat it as an IMS involving valence particles only. 

Now we can define the wave operator, with the following properties... 

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