One particle irreducibility

The object of investigation are one particle irreducible vertex functions defined... 

However, as it has been noticed in Ref. [62], in the double limit m, n —t 0 the terms with o and Vo become of the same symmetry, joining these terms one passes to an effective Hamiltonian with only one coupling constant of the 0 mn = 0)-symmetry. This can be proved for the expressions of the one-particle irreducible vertex functions... 

Figure 5. The one-loop contributions to the one-particle irreducible vertex function r J k) in the diagrammatical representation.

There exist such important properties of graphs as their one-particle reducibility (IPR) and, correspondingly, one-particle irreducibility (IPI). 

This property of graphs due to the absence of integration over one-line-connectcd variables is called a one-particle reduction (IPR). In the contrary version, a graph is called one-particle irreducible (IPl). 

For instance, a one-light particle irreducible amplitude in QCD of two gluons and two quarks is matched as... 

Formulating conditions for the energy to be stationary with respect to variations of the wavefunction P in this generalized normal ordering, one is led to the irreducible Brillouin conditions and irreducible contracted Schrodinger equations, which are conditions on the one-particle density matrix and the fe-particle cumulants k, and which differ from their traditional counterparts (even after reconstruction [4]) in being strictly separable (size consistent) and describable in terms of connected diagrams only. 

In terms of these conditions, a fc-particle hierarchy of approximations can be defined, with Hartree-Fock as the one-particle approximation for closed-shell states. Unfortunately, the stationarity conditions do not determine the fully, and for their constmction additional information is required, which essentially guarantees -representability. Nevertheless, the fe-particle hierarchy based on the irreducible stationarity conditions opens a promising way for the solution of the -electron problem. 

Examples for non-totally-symmetric components in the decomposition of density matrix into irreducible tensor components are the one-particle spin density matrices ... 

Figure 1. Diagrammatic representation of the terms that contribute to the irreducible one-particle BriUouin conditions, Eq. (167), with their algebraic equivalents. The last line represents the commutators (i.e., the difference of the above values).

In Chapter 14 we have shown how an expansion in terms of irreducible tensors in the spaces of orbital and spin angular momenta for one shell can be obtained for the operators corresponding to physical quantities. The tensors introduced above enable the terms of a similar expansion to be also defined in the space of a two-shell configuration. So, for the one-particle operator of the most general tensorial structure (14.51) we find, instead of (14.52),... 

In the catalysis community, it is generally accepted that there are two types of support materials for heterogeneous oxidation catalysts [84]. One variety is the reducible supports such as iron, titanium, and nickel oxide. These materials have the capacity to adsorb and store large quantities of molecules. The adsorbed molecules diffuse across the surface of the support to the catalyst particle where they are activated to a superoxide or atomically bound state. The catalytic reaction then takes place between the reactant molecules and the activated on the catalyst particle. Irreducible supports, in contrast, have a very low ability to adsorb O. Therefore, can only become available for reaction through direct adsorption onto the catalyst particle. For this reason, catalysts deposited on irreducible supports generally exhibit turnover frequencies that are much lower than those deposited on reducible supports [84]. More recent efforts in our laboratory are focused on characterizing catalyst support materials that are commonly used in industry. These studies are aimed at deciphering how specific catalyst and support material combinations result in superior catalytic activity and selectivity. 

The propagator [6-9] or Green s function method (GFM) [10-14] provides another approach to calculate the quasi-particle energy bands. The Dyson equation provides the exact E(N 1) energies in a formally one-particle picture, but the equation can only be solved approximately in real applications [57], With the irreducible self-energy part in the diagonal approximation being correct to second-order, the inverse Dyson equation can be written as [26]... 

All CS manifolds in based on the one-particle group U 2s) are families of AGP states, some of these manifolds are irreducible Riemannian manifolds and correspond to cosets formed by the maximal compact subgroups U 2M) XU 2s - 2M) and USp(25), while others are reducible and correspond to non-maximal compact subgroups USp(2basic physical properties, e.g., U 2M) X U 2s - 2M) invariant manifold describes uncorrelated IPSs, the USp(2s) invariant manifold describes highly correlated extreme AGP states that are superconducting, while the USp(2a>i) X X USp(2wp) X SU(2) X SU(2 ) invariant manifold for general (Oj,..., cr, describe intermediate types of correlation and linear response properties, see, for a particular example. Ref. [35], most of which have not been explored in any depth. 

Tensor representations, synonymous for product representations and their decomposition into irreducible constituents, are useful concepts for the treatment of several problems in spectroscopy. Important examples are the classification of the electronic states in atoms and the derivation of selection rules for infrared absorption or the vibrational Raman or hyper-Raman effect in crystals. In the first case the goal is to reduce tensors which are defined as products of one-particle wave functions, while in the second case tensors for the dipole moment, the electric susceptibility or the susceptibilities of higher orders have to be reduced according to the irreducible representations of the relevant point groups. 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

Its irreducibility the prime means that the only transitions allowed in Eq. (429) are such that one starts with a single excited particle wave number (k 0) and arrives at the final state kf without any such intermediate state. As discussed previously in other contexts (see Section II-C), this condition implies that describes a dynamical process that is localized both in space and time ... 

We study next the dynamical irreducibility condition which appeared in the definition of the transport operator. It eliminates from this quantity the reducible collision processes where the particles coming from infinity interact, recede to an infinite distance from one another, and then interact again. We define an extended transport operator from which the irreducibility condition is eliminated and which involves this time the reducible collisions. The relation between the transport operator and the extended transport operator is made explicit by means of a correspondence between the dynamical processes and the Mayer graphs for equilibrium. In this respect, we demonstrate, in these graphs, the importance of the role of the articulation points. 

Let us now specify the nature of the dynamical irreducibility condition in Eq. (56). The conservation rides of the wave vectors (Eq. 63) impose the condition that the k of certain particles is zero in certain intermediate states. For example, in Fig. 2a particles 2 and 3 have their k zero in the second state of propagation. It may be that the structure of the diagram is such that for one or many intermediate states the k of every particle is identically zero. The diagram is then reducible (see Fig. 2c) and is not contained in Eq. (56). This leads us to extend the definition of jijru. ) so as to include in it the reducible contributions. We shall define... 

One can formulate a two-particle analogue of the Brillouin condition, the IBC2 ( I stands for irreducible, which essentially means connected. For details see Refs. [20, 29].)... 

If one formulates the conditions for stationarity of the energy expectation value in terms of generalized normal ordering, one is led to either the irreducible fc-particle Brillouin conditions IBCj or the irreducible A -particle contracted Schrodinger equations (IBC ), which are conditions to be satisfied by y = yj and the k. One gets a hierarchy of k-particle approximations that can be truncated at any desired order, without any need for a reconstruction, as is required for the reducible counterparts. 

The external field approximation is clearly inadequate for calculation of the recoil corrections and, in principle, one needs the machinery of the relativistic two-particle equations to deal with such contributions to the energy levels. The first nontrivial recoil corrections are generated by kernels with two-photon exchanges. Naively one might expect that all corrections of order Za) m/M)m are generated only by the two-photon exchanges in Fig. 4.1. However, the situation is more complicated. More detailed consideration shows that the two-photon kernels are not sufficient and irreducible kernels in Fig. 4.2 with arbitrary number of the exchanged Coulomb pho-... 

We have established above that one-electron operators are expressible in terms of tensors W(kK) related to triple tensors W KkK by (15.59). Therefore, we shall find here the expansion in terms of irreducible tensors in quasispin space only for the two-particle operator that is a scalar in the total momentum. 

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