Ordering matrix

The full bond order matrix is a symmetric tridiagonal matrix (Chapter 2). It is symmetric because the bond order Cji Y Na,ack is the same as tiie bond order Na,i, a,j. Elements off the tridiagonal (-.4472 in the butadiene example) are artifacts of the minimization atid should be disi egarded. The full bond order matrix for butadietie is... 

Internally, molecules can be represented several different ways. One possibility is to use a bond-order matrix representation. A second possibility is to use a list of bonds. Matrices are convenient for carrying out mathematical operations, but they waste memory due to many zero entries corresponding to pairs of atoms that are not bonded. For this reason, bond lists are the more widely used technique. 

Individual chains are planar with alternating cis—trans conformation along the chain, and the interaction with neighboring chains leads to a highly ordered matrix. 

The antisymmetrization of the trial function has a definite effect on the generalized density matrices P(x x 2.. . x x1x2. . . xv) defined by Eq. II.9 since, except for the first-order matrix, they will now all be antisymmetric in each set of the indices. For p = 2, we have in particular ... 

For k = J and for degenerate eigenvalues the elements of f are taken equal to zero. Let us next consider the variation of the population-bond-order matrix, which, in the orthogonal case of eqn 1, is just ... 

D is the charge and bond order matrix 4>v(r) and r) are the basis functions... 

One of the characteristic features of mesogenic molecules in uniaxial mesophases is their orientational order specified by a set of order parameters, which forms a symmetric and traceless order matrix S. For rigid molecules, the elements of order matrix Sa/Y defined by Saupe,52 are given by... 

The potential of mean torque, parameterized by the local interaction tensors Xa and Xc (assume to be cylindrical) for the aromatic core and the C C segment, respectively, can be mapped out at each temperature by fitting the observed quadrupolar splittings in the mesophase. Furthermore, the order matrix of an average conformer of the molecule can also be calculated.17 Now pn is needed to describe the internal dynamics. 

Another method to determine the magnitude and rhombicity of the alignment tensor is based on the determination of the Saupe order matrix. The anisotropic parameter of motional averaging is represented by this order matrix, which is diagonalized by a transformation matrix that relates the principal frame, in which the order matrix is diagonal,... 

The approximate polarizabilities calculated by the steepest descent method are useful mainly because they are easily calculated from the charge-and-bond-order matrix of the imperturbed molecule but their... 

A recent re-examination of the perturbation problem (McWeeny, 1961 Diercksen and McWeeny, 1965), leads to a method of determining, to any prescribed order, the perturbation of an electronic system due to any applied field. This method involves the direct calculation of the first-, second-, third-order corrections to the charge-and-bond-order matrix... 

The new charge-and-bond-order matrix may be used as a basis for discussion of the reactivity of the heterocyclic system. 

Note 3 Molecules which constitute nematogens are not strictly cylindrically symmetric and have their orientational order given by the Saupe ordering matrix which has elements 5 aP = (3< a P> - 5ap)/2, where la and ip are the direction cosines between the director and the molecular axes a and P, 5ap is the Kronecker delta and a, p denote the molecular axes X, Y, Z. 

Note 4 The constituent molecules of a nematogen are rarely rigid and their orientational order is strictly defined, at the second-rank level, by a Saupe ordering matrix for each rigid sub-unit. 

For density matrices generated from a determinantal wave function IF (which occurs, e.g., in the HF method or the KS approach), the p-th order matrix can be obtained from the 1-st order one... 

In 1974 Coleman [73] proposed to decompose any Hermitian antisymmetric second-order matrix as... 

The three parts of this decomposition reveal the structure of the matrix with respect to the contraction operations. These parts have been called the 0-, 1- and 2-body part of the second-order matrix A, respectively. Following the notation introduced in Ref. [73], each of these parts have been identified by a left-lower index. 

As has been mentioned, the MZ purification procedure is based on Coleman s unitary decomposition of an antisymmetric Hermitian second-order matrix described earlier. When applied to singlet states of atoms and molecules, the computational cost of this purification procedure is reduced, since the 2-RDM (and thus the 1-RDM obtained by contraction) presents only two different spin-blocks, the aa- and a/i-blocks (and only one spin-block for the 1-RDM). For the remaining part of this section only this type of state will be treated. 

It must be noted that, due to the arbitrary symmetry under permutation of indices of this second-order matrix, a larger set of contractions into the 0- and 1 -body space must be taken into account. 

---