Internal excitation operators

Table Vlll lists the norms of internal, semi-internal, and external excitation amplitudes obtained in the CASSCF(6e, 5o, NO)/L-CTSD calculations with the 6-3IG basis set. The percentage of retained amplitudes for internal and semi-internal excitations are also shown in the table. The norm of the external amplitudes, which primarily contribute to dynamic correlation, does not fluctuate much across the potential curve. The maximums of the internal and semi-internal amplitudes are found at the intermediate bond region. More internal and semi-internal excitation operators are retained as the bond length ron is increased.

The second approximate scheme we will discuss here is the internally contracted Cl (ICCI) method. In this method correlating configurations are formed by applying excitation operators (the generators of the unitary group) directly on the full reference Cl vector. The four types of configurations thus formed can be written as,... 

An alternative contraction scheme which has received more attention is internally contracted multireference CISD (usually denoted simply CMRCI), which was first discussed by Meyer99 and Siegbahn.100 This method applies the single and double excitation operators to a single multiconfigurational reference wavefunction as a whole, including the reference coefficients. Thus, if the reference wavefunction is... 

The second possible contraction scheme was first proposed by Meyer , and discussed in the context of the direct Cl method by Siegbahn ". In this case all configurations which have the same external but different internal parts are contracted, and the scheme is therefore called the internal contraction. The internally contracted configurations are generated by applying pair excitation operators to the complete MCSCF reference function. Therefore, the number of contracted configurations and variational parameters is independent of the number of reference configurations. It only depends on the number of correlated internal orbitals and the size of the basis set. ... 

We define an operator as closed , if its action on any model function G P produces only internal excitations within the IMS. An operator is quasi-open , if there exists at least one model function which gets excited to the complementary model space R by its action. Obviously, both closed and quasi-open operators are all labeled by only active orbitals. An operator is open , if it involves at least one hole or particle excitation, leading to excitations to the g-space by acting on any P-space function. It was shown by Mukheijee [28] that a size-extensive formulation within the effective Hamiltonians is possible for an IMS, if the cluster operators are chosen as all possible quasi-open and open excitations, and demand that the effective Hamiltonian is a closed operator. Mukhopadhyay et al. [61] developed an analogous Hilbert-space approach using the same idea. We note that the definition of the quasi-open and closed operators depends only on the IMS chosen by us, and not on any individual model function. 

An important insight in the development of size-extensive formulations in a IMS was the realization that the intermediate normalization convention for the wave operator, viz. PflP = P, should be abandoned in favor of a more appropriate normalization [28,61]. For the IMS, in general, products of quasi-open operators may lead to internal excitations, or may even be closed, so that if we choose il = 2 exp(7 )l< X< l, with = Top-f Tq-op, then powers of Tq. op coming from the exponential might lead from 4>p to internal excitations to some other model function or it may contain closed operators. We would have to bear this in mind while developing our formalism, and would not force POP = P in our developments. 

In the first application of the CASCC method, the external part of the CC operator, T in the CASCC wave function contained only doubles excitations with respect to Fermi s vacuum. We should note that generates external and semi-internal excitations with respect to 0). In the semi-internal excitations electrons are distributed among active (occupied in 0)) and inactive (unoccupied in 0)) orbitals or between inactive (occupied) and active (unoccupied) orbitals. In general, has the following form ... 

There are eight types of double external and semi-internal excitations included in operator f4. Their amplitudes are ... 

If we are only interested in elastic scattering, a useful procedure [Wa53] is to absorb the contributions from virtual excitation of the target nucleus (and internal excitation of the projectile) into an effective operator (the optical potential operator) and to solve a scattering equation in which only elastic channel intermediate states appear. (Analogous procedures can be used for inelastic scattering.) Projecting onto the elastic channel in eq. (2.18) yields... 

ABSTRACT. The principle of operation of drift tubes and their application to the determination of ion-neutral reaction rate coefficients, k, as a function of the ion/reactant molecule (E ) and the ion/buffer gas (Ej,) centre-of-mass energies are discussed. It is shovm that drift tube data of k versus Ej., for atomic ion/neutral reactions can be used with confidence in modelling the ion chemistry of shocked interstellar gas. However, it is stressed that drift tube data relating to molecular ion reactions must be used with caution since internal excitation of the ions can occur in collisions with the buffer gas. Some consideration is given to the variation with Ej, and Ej. of the rate coefficients, k3, for ternary association reactions and to the relevance of the data in estimating radiative association rate coefficients appropriate to shocked interstellar gas. 

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