Operator, wave

We next introduce the Meller wave operator l(+> which transforms the state vectors Qn> into T >+... 

Mintzer, David, 1 Mitropolsky, Y. A361,362 Mixed groups, 727 Modality of distribution, 123 Models in operations research, 251 Modification, method of, 67 Molecular chaos, assumption of, 17 Miller wave operator, 600 Moment generating function, 269 Moment, 119 nth central, 120... 

Now let us consider the plane wave operator O = , i.e. that which... 

In order to determine the relations among orbital exponents in a basis which will follow these guidelines, we look at the matrix elements contributing to on(q). To that end we consider eq. 17 for the plane wave operator (eq. 21) which involves evaluation of terms of the sort (p e i /i). We wish to determine how these matrix elements behave as a function of orbital exponent and momentum transfer, and we then propose a scheme for choice of orbital exponents that will keep the BSR satisfied to as high momentum transfer as possible. 

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... 

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... 

All chemical laser systems discussed so far were operated in a pulsed mode and they need either flashlamp excitation or a gas discharge to initiate the chemical reaction by producing free radicals. Recently these limitations have been overcome and pure chemical lasers with continuous-wave operation have been developed. 

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]... 

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... 

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... 

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