Topological matrix

A. The Necessary Conditions for Obtaining Single-Valued Diabatic Potentials and the Introduction of the Topological Matrix... 

It is known that multivalued adiabatic electronic manifolds create topological effects [23,25,45]. Since the newly introduced D matrix contains the information relevant for this manifold (the number of functions that flip sign and their identification) we shall define it as the Topological Matrix. Accordingly, K will be defined as the Topological Number. Since D is dependent on the contour F the same applies to K thus K = f(F),... 

One of the main outcomes of the analysis so far is that the topological matrix D, presented in Eq. (38), is identical to an adiabatic-to-diabatic transformation matrix calculated at the end point of a closed contour. From Eq. (38), it is noticed that D does not depend on any particular point along the contour but on the contour itself. Since the integration is carried out over the non-adiabatic coupling matrix, x, and since D has to be a diagonal matrix with numbers of norm 1 for any contour in configuration space, these two facts impose severe restrictions on the non-adiabatic coupling terms. 

The fact that there is a one-to-one relation between the (—1) terms in the diagonal of the topological matrix and the fact that the eigenfunctions flip sign along closed contours (see discussion at the end of Section IV.A) hints at the possibility that these sign flips are related to a kind of a spin quantum number and in particular to its magnetic components. 

In Section IV, we introduced the topological matrix D [see Eq. (38)] and showed that for a sub-Hilbert space this matrix is diagonal with (-1-1) and (—1) terms a feature that was defined as quantization of the non-adiabatic coupling matrix. If the present three-state system forms a sub-Hilbert space the resulting D matrix has to be a diagonal matrix as just mentioned. From Eq. (38) it is noticed that the D matrix is calculated along contours, F, that surround conical intersections. Our task in this section is to calculate the D matrix and we do this, again, for circular contours. 

Single-valued potential, adiabatic-to-diabatic transformation matrix, non-adiabatic coupling, 49-50 topological matrix, 50-53 Skew symmetric matrix, electronic states adiabatic representation, 290-291 adiabatic-to-diabatic transformation, two-state system, 302-309 Slater determinants ... 

If all nearest-neighbor exchange parameters are equal to J and all the others vanish, then Jij/ J is the N x N topological matrix of the molecule (its elements are 1 for neighboring atoms and 0 otherwise). In this case the summation over all exchange parameter combinations in Eq.(69) can be explicitly performed leading to... 

An =0, i.e. a vertex is not connected with itself and the graph has no loops. Such a matrix is customarily called an adjacency or a topological matrix. As an example, a topological matrix of a styrene molecule is illustrated in Fig. 11. 

In search of invariants. Are there possibly any other characteristics of a graph (or its topological matrix) that are independent of the vertex enumeration mode Yes, such invariant characteristics do exist. However, they can be obtained only after certain refinements of the theory. 

The column matrix is called an i-th eigenvector of the topological matrix A, and Xi an i-th eigenvalue of this matrix. The totality of eigenvalues of the matrix A is called a spectrum of the graph corresponding to this matrix. 

Corollary two the sum of the roots of a characteristic polynomial, that is, the sum of the eigenvalues of the topological matrix of a molecule, is equal to zerort ... 

Topological orbitals... Probably the smartest of our readers have already appreciated the close relationship between the graph theory and the Hiickel method. Actually in quantum chemistry there is a great variety of problems in which the Hamiltonian of a molecule can be written in a matrix form as a one-valued function of the topological matrix of that molecule ... 

First all it should be noted that for some types of the matrix S (see below) the eigenvectors c, of the topological matrix of the molecule coincide precisely with those of the Htickel method, i.e. with the series of coefficients defining the molecular orbitals calculated by this method. 

Thus, we have expressed the orbital energies of the simple MO method in terms of the eigenvalues of the topological matrix. If we now pass over to the simplest ver-... 

We have thereby clarified the meaning of eigenvjalues of a topological matrix of a molecule these values, when expressed in units of p, coincide with the-orbital energies of the HMO method. In other words, the Hiickel [method determines the MO energy values specified only by the molecular topology. 

Topological matrix of a bipartite graph. Bipartite graphs, just as the corresponding alternant systems, possess a number of remarkable properties. In particular, their vertices can always be enumerated so that the topological matrix is simplified and reduced to the block form... 

The maUix composed of the eigenvalues of topological matrix of a bipartite graph whose vertices are properly enumerated also takes the sufficiently simple block form ... 

As has already been said, the system representing an even alternant hydrocarbon has a topological matrix of the type (23). Then, in accordance with the Hall theorem the matrix of orders of bonds between marked and unmarked atoms (P °) is expressed by the following topological formula ... 

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