Operator manifolds

The RPA method may be improved either by choosing an MCSCF reference wave function, leading to the MCRPA method, or by extending the operator manifold beyond... 

Partitioning the operator manifold can lead to efficient strategies for finding poles and residues that are based on solutions of one-electron equations with energy-dependent effective operators [16]. In equation 15, only the upper left block of the inverse matrix is relevant. After a few elementary matrix manipulations, a convenient form of the inverse-propagator matrix emerges, where... 

Extensive numerical investigations of this formalism were undertaken by Nakatsuji/52/ and Hirao/53/ for IP and EE computations. For IP calculations, the operator manifold taken by them were W and W2, and product excitations of the form W T2 were also included. A similar approximation scheme for EE was also used, although all the spin-adapted W operators for the triplet EE calculations were not included. This should be contrasted with the scheme of... 

Here h is a complete excitation operator manifold arranged as a column vector and li is the transposed row vector. Eq. (58) can also be derived from Eq. (56) using the identity (Simons, 1976)... 

Using successively larger and larger operator manifolds we may define a hierarchy of approximate propagator methods based on Eq. (58). Also the choice of reference state influences Eq. (58) through the definition of the superoperator binary products in Eqs (52) and (60). It has been our experience that it is important to maintain a balance between the level of sophistication of the operator manifold and of the reference state. Better results are generally obtained this way, as we will see examples of in the subsequent sections. 

As shown in Section II, we wish to calculate the poles and residues of P Q . However, even using moderately large operator manifolds, the inverse matrix becomes so large that we cannot evaluate all elements of it equally well. We therefore wish to treat one part of it better than the rest. Which part we choose will be directed by the physics of the problem. In order to do so it is convenient to partition (Lowdin, 1963) the inverse matrix, for instance in the following way (Nielsen et al, 1980), letting hf, = h —... 

Not all electron propagator calculations are based upon an order-by-order evaluation of the propagator. Redmon et al. (1975) included the hs operator manifold, without computing all terms in in the next order of perturbation... 

For closed-shell reference states, spin adaptation of the operator manifolds reduces the rank of H. Amplitudes for doublet final states have the following... 

This choice of operator manifold was shown to be capable of producing EAs and IPs that were precise through third order [17] in the MP perturbation, which is why this choice was made. 

The resultant variant of Eq. (11) was not solved by finding the eigenvalues of this matrix eigenvalue equation whose dimension is the sum of the dimensions of the and a, a b p operator manifolds. Rather, that large matrix eigenvalue... 

A possible way to construct a bctsis for the Hilbert spaw e Y may also be the definition of operator sets A and B, such that the set of states A,B) A A,B B span the Hilbert space Y. In the context of traditional one- and two-particle propagators of many-body theory, such complete operator manifolds have been found [23] and used for deriving approximation schemes (see e. g. [24] and references cited therein). In the present context, however, such a construction seems difficult because of the complicated nature of the extended states A,B). It is certainly an interesting open question whether a convenient construction can be found. 

The above results having to do with completeness of operator manifolds permit us to write a resolution of the identity as... 

Because of the high research activity level on how to use an MCSCF reference in the GFs (EP and PP), it is not presently clear how to optimally choose truncated sets of 1V operators. It is likely that many workers will carry out test calculations involving many choices of the pertinent operator manifolds before this situation is improved. Moreover, questions concerning when and how to partition the resulting (1V / T/) matrix so as to reduce the dimension of the matrix whose poles are to be found remain unanswered... 

This choice of A and is made because we are interested in studying primary ionization events [ionization potentials (Cederbaum, 1973 Pickup and Goscinski, 1973 Doll and Reinhardt, 1972 Purvis and Ohrn, 1974) and electron affinities (Simons and Smith, 1973 Jorgensen and Simons, 1975)], which may be reasonably described through acting with a singleelectron operator r or r) on the reference state 0>, To obtain computationally useful expressions for G (E) specific choices must be made for the reference state 0> and for the operator manifold T in Eq. (6.32). We describe a few of the most commonly employed choices of these quantities and the resulting GF. 

All transition amplitudes corresponding to primary ionization events thus become equal to unity at this level of approximation. The above result expresses the EP analog of Koopmans theorem. To go beyond Koopmans theorem, better choices must be made for the reference state and operator manifold. 

In a simple and very commonly used approximation to the PP, the reference state 0> is chosen to be a single-configuration (but not necessarily single determinant) HF wavefunction. The operator manifold T then is taken as the set of particle-hole excitation and deexcitation operators used for optimizing the reference state ... 

Two major approximations must be made to obtain the response functions a choice of a reference function and a choice of an operator manifold. For the linear response function, the choices of a Hartree-Fock reference state and simple particle-hole excitation operators (in the second quantization sense) lead to an approximation known as the random phase approximation (RPA) and is equivalent to the TDFiF method discussed earlier. 

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