State weighting matrix

State weighting matrix %Control weighting matrix... 

Collect the statistical weights into statistical weight matrices (of dimension Vj.j X v , where v,- is the number of rotational isomeric states of bond i), one per skeletal bond subject to conformational change, indexing rows and columns of the matrices with the RIS of the bond. The generic statistical weight matrix for bond i is termed U,-. 

In order to calculate the partition function and molecular averages, a 4x4 statistical weight matrix for the fth residue is formulated to correlate the states of residues i-1, i, and / + 1 of the polymer chain ... 

Matrix schemes which assign end effects to each end of a helical segment are also proposed. A 3 x3 statistical weight matrix B using this weighting scheme is used here. Row index states of amino acid residues i-1 and /, while columns index states of amino acid residues i and /+ 1. 

Correction phase Correct the prior estimates of state at k — 1 and update the weighting matrix... 

Computing the partition function for an /V-statc chain requires enumerating all possible states. The clever trick associated with the helix-coil transition theory is to generalize this calculation using the statistical-weight matrix ... 

Flory has adopted the use of a matrix to represent the dependency between rotational states of neighboring bonds. For the typical case of a polymer with three rotational states such as trans (t), gauche plus (g+), and gauche minus (g ), the statistical weight matrix is... 

In Eq. 6, is the electronic matrix element involving the promoting mode p F j is the FC factor for mode j and p( nj ) is the density of states weighted FC factor for the set n, where Ft,- represents the most probable (i.e., optimal) distribution of the vibrational quanta in the final state. The function p( fij ) accounts for the contributions to the FC factor of modes other than). For the radiationless transition originating from the zero-point level of mode j, the FC factor is given by... 

We now address the issue of computing the matrix elements X%,v), assuming that the stationary state ug is known explicitly and that the weight matrix has finite elements. We shall discuss the following increasingly complex possibilities (1) [X, G] = 0 and X is near-diagonal in the S... 

The calculation of vibrational frequencies (called phonons) is important to the study of the solid state. Indeed, the calculation of and study of phonons is often given a special name, lattice dynamics. To calculate the vibrational frequencies for a solid one follows a very similar approach to that described earlier for molecules, with the exception that when a shell model is being used then their effect must be incorporated into the mass-weighted matrix of second derivatives (though not directly as they have no mass) ... 

Character-state weight matrices have usually been estimated more or less by eye, but they can also be derived from a rate matrix. For example, if it is presumed that each of the two transitions occurs at double the frequency of each transversion, a weight matrix can simply specify, for example, that the cost of A-G is 1 and the cost of A-T is 2 (Fig. 14.5). (The parsimony method dictates that the diagonal elements of the matrix, or the cost of having the same base in different sequences, be zero. This proves to be a shortcoming of parsimony this will be discussed further below.) In the subsequent tree-building step, this set of assumptions will minimize the overall number of transversions and tend to cluster sequences differing mainly by transitions. 

In the Monte Carlo method, the bonds are assigned to the different rotational states according to their conditional probabilities. These are calculated, using the eigenvalue method outlined by Flory (1969), from the corresponding statistical weight matrix ... 

Let the polymer chain be capable of assuming two different conformational states, coil and helix, the latter capable of binding iodine as I or I. Thus an amylose molecule may be described as a sequence of coil and helix sites, which for present purposes can each be taken to comprise approximately six glucose residues. Coil states can be denoted by c, unbound helix states by h, and helix states bound either to I or I by b. We define "complex" as any uninterrupted sequence of b states, regardless of length. Then in the usual fashion (3 ) we define the statistical weight matrix U indexed for sites i-1 and i as... 

Calculation of the Chains Unperturbed Dimensions. Inspection of the conformational energy maps (Figures 2 and 3) reveals that adoption of the familiar three-state (i.e., T, G", G") scheme is sufficient for application of the RIS theory to these chains.The corresponding statistical weight matrix U, inclusive of both first-order and second-order interactions (those interactions depending on, respectively, one and two skeletal-bond rotations), is represented by ... 

As an example, the statistical weight matrix for polyethylene with its three-fold rotational states (trans, gauche plus, gauche minus) is... 

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