Projection operators, properties

The subset 0kv 0k2> 0k3,. . . formed from the complete set by means of the projection operator 0k is called /l-adapted or symmetry-adapted in the case when A is a symmetry operator. From Eqs. III.81 and III.86 it follows that the projection operators 0k commute with H and, using this property, the quantum-mechanical turn-over rule/ and Eq. III.91, we obtain... 

The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

Symmetry properties which have so far been successfully treated by the projection operator method, include translational symmetry in crystals, cyclic systems, spin, orbital and total angular momenta, and further applications are in progress. ... 

If the wave function W has a symmetry property characterized by a projection operator 0, the expansion IV. 1 may be replaced by the series... 

The von Neumann Projection Operators.—Consider the eigenstates n > in the Hilbert space of N particles with the properties ... 

To construct a chirality function belonging to a particular representation JW of S, one proceeds, in principle, as follows Starting with an arbitrary function ip(l,2,..., n), one applies one of the Young operators (arbitrarily chosen) which project onto. T. If the result it not zero, it will be a function belonging to though not necessarily a chirality function. One then applies the projection operator onto the chiral representation of . If the result is still not zero, it will be a chirality function having the desired properties. In mathematical form,... 

The projection operator ( ) of Eq. (5) is not normalized. Since normalization plays no particular role in the theory, we shall ignore it in the remainder of this section, with the result that some equations are true only up to a multiplicative constant. Such a constant does not affect transformation properties, so the chirality functions obtained retain their validity. 

It is obvious that the projection operators for the different species have different numbers of terms in them. The HON species have 12 terms (3 x 2 ) while the A2B-type species have four terms, and the HDT+ isotopomer has only two terms. This results in different sizes of the spin-projected basis sets, and for this reason the properties obtained in this work are not precisely comparable between the A3, A2B, and ABC systems, although a very good idea of the trends may be obtained from the data in Table XVI. While all of the above are given in terms of the original particles, it should be noted that the permutations used in the internal particle basis functions are pseudo -permutations induced by the permutations on real particles. 

The separation can be formally realized by means of two projection operators V and C, with the following properties ... 

One may then try to generalize this operation, by defining another decomposition, different from the previous one. Two new projection operators n and n, were first timidly defined by C. George in 1967 and were later introduced definitively in the toolbox of statistical mechanics in 1969 by I. Prigogine, C. George, and F. Henin (MSN.60). The new objects possess the usual properties of projectors ... 

Let us look more closely at the analogy between vector projection operators and character projection operators, and list some of the more important properties that they share. 

The following important properties of projection operators are most easily visualized in terms of vectors, but are true for the character projection operators as well ... 

A description in terms of local spins, however, raises the question of how to define these localized surrogate spins, where we have to move from one-electron spin operators Sj to multi-electron spin operators Sa that are located at one center and summarize all electronic contributions attributed to this center. This can be solved by the introduction of projection operators Pa (33,113,114,122-124). The projection operators do not alter any property of the molecule but divide it into local basins A. They add up to the identity operator,... 

Form SALCs from the set of equivalent orbitals on the pendent atoms. As noted above and emphasized in Figure 8.9, we could use either the hybrid orbitals on atom A or the o orbitals on the B atoms, since their symmetry properties are the same. We choose the orbitals on the pendent atoms because the application of projection operators to these is exactly as previously explained in Chapter 6. The results obtained are... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

Zwanzig showed how a powerful but simple technique, known as the projection operator technique, can be used to derive the relaxation equations [12]. Let us consider a vector A(t), which represents an arbitrary property of the system. Since the time evolution of the system is given by the Liouville operator, the time evolution of the vector can be written as /4(q, t) = e,uA q), where A is the initial value. A projection operator P is defined such that it projects an arbitrary vector on A(q). P can be written as... 

The quantum mechanical forms of the correlation function expressions for transport coefficients are well known and may be derived by invoking linear response theory [64] or the Mori-Zwanzig projection operator formalism [66,67], However, we would like to evaluate transport properties for quantum-classical systems. We thus take the quantum mechanical expression for a transport coefficient as a starting point and then consider a limit where the dynamics is approximated by quantum-classical dynamics [68-70], The advantage of this approach is that the full quantum equilibrium structure can be retained. 

The aim of this work is to demonstrate that the above-mentioned unusual properties of cuprates can be interpreted in the framework of the t-J model of a Cu-O plane which is a common structure element of these crystals. The model was shown to describe correctly the low-energy part of the spectrum of the realistic extended Hubbard model [4], To take proper account of strong electron correlations inherent in moderately doped cuprate perovskites the description in terms of Hubbard operators and Mori s projection operator technique [5] are used. The self-energy equations for hole and spin Green s functions obtained in this approach are self-consistently solved for the ranges of hole concentrations 0 < x < 0.16 and temperatures 2 K< T <1200 K. Lattices with 20x20 sites and larger are used. 

These formulas update the rank-m projection of F° or G°, using the nonhermitian projection operator Vm = Ap(Aqf Ap) 1Aqf, such that V nAq = Aq, VmAp = Ap, and VmVm = Vm. This operator projects onto the m-dimensional vector space spanned by the specified set of gradient vectors. The Rm update has the undesirable property of altering the complementary projection of the updated matrix. 

For systems with high symmetry, in particular for atoms, symmetry properties can be used to reduce the matrix of the //-electron Hamiltonian to separate noninteracting blocks characterized by global symmetry quantum numbers. A particular method will be outlined here [263], to complete the discussion of basis-set expansions. A symmetry-adapted function is defined by 0 = 04>, where O is an Hermitian projection operator (O2 = O) that characterizes a particular irreducible representation of the symmetry group of the electronic Hamiltonian. Thus H commutes with O. This implies the turnover rule (0 > II 0 >) = (), which removes the projection operator from one side of the matrix element. Since the expansion of OT may run to many individual terms, this can greatly simplify formulas and computing algorithms. Matrix elements (0/x H ) simplify to (4 H v) or... 

This holds even if p is not a projection operator, since the property to be a projection operator is not generally conserved by linear operations, but just a matrix). Thus can be considered a linear superoperator transforming one 2M x 2M matrix to another one of the same dimension. 

Usually, an individual VB structure assembled from the localized bonding components does not share the point group symmetry of the molecule anymore. However, the overall VB wavefunction, PVB, should retain the same symmetry properties as the MO wavefunction (in the sense of full Cl, they are in fact identical). Therefore, TVs can be classified by an irreducible representation associated with a given point group. In order to sort vFra by symmetry, a project operator can be introduced as follows ... 

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