Density matrix reduced scalar

Abstract. The elements of the second-order reduced density matrix are pointed out to be written exactly as scalar products of specially defined vectors. Our considerations work in an arbitrarily large, but finite orthonormal basis, and the underlying wave function is a full-CI type wave function. Using basic rules of vector operations, inequalities are formulated without the use of wave function, including only elements of density matrix. 

We can word the results up to this point for an N-particle fermion system, using M-dimensional one-particle function basis, the elements of the second-order reduced density matrix in geminal basis are scalar products of ( ) piece of ( 2) dimensional vectors. 

Elements of second order reduced density matrix of a fermion system are written in geminal basis. Matrix elements are pointed out to be scalar product of special vectors. Based on elementary vector operations inequalities are formulated relating the density matrix elements. While the inequalities are based only on the features of scalar product, not the full information is exploited carried by the vectors D. Recently there are two object of research. The first is theoretical investigation of inequalities, reducibility of the large system of them. Further work may have the chance for reaching deeper insight of the so-called N-representability problem. The second object is a practical one examine the possibility of computational applications, associate conditions above with known methods and conditions for calculating density matrices. 

It can be seen that, in the average density matrix formalism which is based on spin system model, the scalar couplings and the exchange processes are handled simultaneously. Thus they cannot be separated and a larger atomic basis (spin system) is required for their description. Meanwhile, the Monte Carlo method based on spin sets separates the two interactions, and thus spin systems can be reduced to smaller spin sets. 

Before going on to consider more complicated systems, we review here some of the basic behavior of a two-state quantum system in the presence of a fast stochastic bath. This highly simplified bath model is useful because it allows qualitatively meaningful results to be obtained from a density matrix calculation when bath correlation functions are not available in fact, the bath coupling to any given system operator is reduced to a scalar. In the case of the two-level system, analytic results for the density matrix dynamics are easily obtained, and these provide an important reference point for discussing more complicated systems, both because it is often possible to isolate important parts of more complicated systems as effective two-level systems and because many aspects of the dynamics of multilevel systems appear already at this level. An earlier discussion of the two-level system can be found in Ref. 80. The more... 

Each of these columns of this symmetrical matrix may be seen as representing a molecule in the subspace formed by the density functions of the N molecules that constitute the set. Such a vector may also be seen as a molecular descriptor, where the infinite dimensionality of the electron density has been reduced to just N scalars that are real and positive definite. Furthermore, once chosen a certain operator in the MQSM, the descriptor is unbiased. A different way of looking at Z is to consider it as an iV-dimensional representation of the operator within a set of density functions. Every molecule then corresponds to a point in this /V-dimensional space. For the collection of all points, one can construct the so-called point clouds, which allow one to graphically represent the similarity between molecules and to investigate possible relations between molecules and their properties [23-28]. 

Read carefully the discussion of the electron-nuclear contact coupling (pp 387-389) and make a similar analysis of the electron-nuclear dipolar coupling that arises from the term (11.7.9). Hence verify that the coupling with any nucleus is described by a coupling tensor whose components depend on the form of the spin density in the vicinity of the nucleus. [Hint Express the scalar products in terms of spherical components as in (11.7.3) and (11.7.4), and reduce the matrix elements (within the degenerate manifold of electron-nuclear product functions) in a parallel fashion, focusing attention on the coefficient of S l . The cartesian form can be obtained at the end.]... 

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