Orthogonality projector

We assume that the space on which H operates is a direct sum of two subspaces characterized by orthogonal projectors P and Q (P + Q = l, PQ = QP = 0). If we are interested in propagations in the subspace P alone, it is possible to eliminate quite formally propagations in the subspace Q, for each energy z at which the system is interrogated. Putting... 

The criterion outlined previously clearly expresses statisticality as a property of continuum states, and as such has little to do with bound-state dynamics. However, if the process is dominated by a resonance, or a number of resonances, then contact can be made with chaotic behavior in bound systems.75 Denoting Q as a projection operator onto a bound manifold and P = I - Q as its orthogonal projector (which includes the unbound manifold), it is always possible to express E,n"> in terms of the scattering solutions in the P space, as [with G = (E - ie - PHP)- ] ... 

Therefore, can be automatically defined the second sub-representation, H 2,7 2), by the action of specific orthogonal projectors... 

These equations are expressed in the spin-orbital formalism and the products of orbitals are assumed to be antisymmetrized. The coefficients are the explicitly correlated analogues of the conventional amplitudes. The xy indices refer to the space of geminal replacements which is usually spanned by the occupied orbitals. The operator Q12 in Eq. (21) is the strong orthogonality projector and /12 is the correlation factor. In Eq. (18) the /12 correlation factor was chosen as linear ri2 term. It is not necessary to use it in such form. Recent advances in R12 theory have shown that Slater-type correlation factors, referred here as /12, are advantageous. Depending on the choice of the Ansatz of the wave function, the formula for the projector varies, but the detailed discussion of these issues is postponed until Subsection 4.2. The minimization of the Hylleraas functional... 

Similar to the MP2-E12 formalism, the strong orthogonality projector in the geminal basis leads to many-electron integrals in the amplitude equations. Explicit evaluation of these integrals severely restricts the range of application and the successful approaches are thc e that involve two-electron integration at most. The implementation of the CCSD(F12) in Turbomole fulfills this requirement. 

The current Section is organized as follows in Subsection 4.2 the general information about the excitation operators, strong orthogonality projectors and correlation factors is... 

Excitation operators, strong orthogonality projectors and correlation factors... 

The most successful Ansatz in F12 theory uses the following form of the strong orthogonality projector [19, 20]... 

It is convenient to begin with the simplest case, i.e. the V intermediate derived within Anl and the RHF reference, without the CABS contribution (Al-noCABS). If we take the general formula of the V intermediate [Eq. (87)] and substitute the expression of the strong orthogonality projector that defines Ansatz 1 [Eq. (61)] we will get the following equation... 

It is also useful to consider an Af -dimensional subspace S spanned by N exact solutions of the exact Hamiltonian H. These solutions will correspond to the part of the spectrum in which we are interested. The orthogonal complement of S is denoted S- -. The orthogonal projectors associated with S and are P and Q ... 

Thus Q is the orthogonal projector onto its orthogonal complement is ... 

The states [ ) and their duals (model space. The imaginary part of the complex energy Si — Ei irj/2 is negative. The non-orthogonal projectors (f>i) i enable to turn the... 

The operator (1 - P) is usually called the orthogonal projector since P(1 - P) annihilates a phase-space function. Equation (35) is useful because it expresses the memory function explicitly as a projected correlation function of the phase-space variables. 

I P is an orthogonal projector with respect to the scalar product... 

This term contains three-electron integrals due to the strong-orthogonality projector l2occ(12), for example... 

Fig. 2 Left Orbital spaces considered in F12 theory and index conventions used in this chapter. Right Representation of the two-particle function space (geminal space), the grey shaded subspaces are projected out by the strong- orthogonality projector Q 2-...

The constituents of this term are (from right to left) A pair of orbitals from the generating space, the correlation factor, the strong-orthogonality projector, and the corresponding coefficients. These elements will be detailed in the following. 

We note that the correlation factor is often given as/(r ) = —In fact, this form is equivalent with Eq.(12) when the strong orthogonality projector Qu (to be discussed in the following) is applied. In most present work, the Slater type correlation factor is approximated by a contracted set of Gaussian geminals (typically n = 6), " ... 

At this point the question might occur, why we did not use a direct expansion of the strong orthogonality projector in terms of orbitals... 

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