Orthogonality transformation

It is important to note that the two surface calculations will be carried out in the diabatic representation. One can get the initial diabatic wave function matrix for the two surface calculations using the above adiabatic initial wave function by the following orthogonal transformation,... 

We consider a 2D diabatic framework that is characterized by an angle, P(i), associated with the orthogonal transformation that diagonalizes the diabatic potential matrix. Thus, if V is the diabatic potential matrix and if u is the adiabatic one, the two are related by the orthogonal transformation matrix A [34] ... 

The standard analytic procedure involves calculating the orthogonal transformation matrix T that diagonalizes the mass weighted Hessian approximation H = M 2HM 2, namely... 

Orthogonal transformations preserve the lengths of vectors. If the same orthogonal transformation is applied to two vectors, the angle between them is preserved as well. Because of these restrictions, we can think of orthogonal transfomiations as rotations in a plane (although the formal definition is a little more complicated). 

Given the modified Huekel matrix and the orthogonal transform in Exereise 7-1, earry/ out the multiplieation... 

As F is a symmetric matrix, there exists an orthogonal transformation that diagonalizes F ... 

Normally the orbitals are real, and the unitary transformation becomes an orthogonal transformation. In the case of only two orbitals, the X matrix contains the rotation angle a, and the U matrix describes a 2 by 2 rotation. The connection between X and U is illustrated in Chapter 13 (Figure 13.2) and involves diagonalization of X (to give eigenvalues of ia), exponentiation (to give complex exponentials which may be witten as cos a i sin a), follow by backtransformation. 

A Hessenberg form H (the same form but not the same matrix) can also be obtained by a sequence of orthogonal transformations, either by plane rotations (the method of Givens), each rotation annihilating an individual element, or by using unitary hermitians, I — 2wiwf, wfwt = 1 (the method of Householder), each of which annihilates ill possible elements in a column. Thus, at the first step, if A = A, and... 

The major problem is to find the rotation/reflection which gives the best match between the two centered configurations. Mathematically, rotations and reflections are both described by orthogonal transformations (see Section 29.8). These are linear transformations with an orthonormal matrix (see Section 29.4), i.e. a square matrix R satisfying = RR = I, or R = R" . When its determinant is positive R represents a pure rotation, when the determinant is negative R also involves a reflection. 

The first term on the right-hand side represents the total sum of squares of Y, that obviously does not depend on R. Likewise, the last term represents the total sum of squares of the transformed X-configuration, viz. XR. Since the rotation/reflection given by R does not affect the distance of an object from the origin, the total sum of squares is invariant under the orthogonal transformation R. (This also follows from tr(R" X XR) = tr(X rXRR T) = tr(X XI) = tr(X" X).) The only term then in eq. (35.2) that depends on R is tr(Y XR), which we must seek to maximize. 

Operator definitions, phase properties, 206-207 Optical phases, properties, 206-207 Orbital overlap mechanism, phase-change rule, chemical reactions, 450-453 Orthogonal transformation matrix ... 

The set of all orthogonal transformations in a three-dimensional real vector space (i.e. a space defined over the field of real numbers) constitutes a group denoted by 0/(3). Alternatively it may be defined as the group of all 3 x 3 orthogonal matrices. The two groups are isomorphic. 

The Lorentz transformation is an orthogonal transformation in the four dimensions of Minkowski space. The condition of constant c is equivalent to the requirement that the magnitude of the 4-vector s be held invariant under the transformation. In matrix notation... 

The new orthogonal hybrids have 60% d character (sdL5 hybridization) as a result of the Lowdin procedure. However, because these expressions involve three atomic orbitals, there must be one other hybrid (in addition to the two bonding hybrids) affected by the orthogonalization transformation. This hybrid, denoted n(h), belongs... 

Due to the invariance of the free energy (3.4) — and also (2.12) — to an orthogonal transformation of its constituent matrices and vectors, we are allowed to carry out this analysis in a more convenient solvent coordinates... 

The three state orthogonal transformation to the space of natural solvent coordinates is the 6 x 6 matrix (see BH-I)... 

An equivalent decomposition can be performed using the Q-R orthogonal transformation (Sanchez and Romagnoli, 1996). Orthogonal factorizations were first used by Swartz (1989), in the context of successive linearization techniques, to eliminate the unmeasured variables from the constraint equations. 

Thus, we have identified the subset of redundant equations containing only the redundant process variables fa, fa, fu, fH, and fa5. Furthermore, the rank of Ru is equal to 6, which means that at least one of the unmeasured variables is indeterminable. The remaining ones can be written in terms of it, as indicated by Eq. (4.15). In this case, from the orthogonal transformation, the subsets of u are defined as... 

A method for decomposing unmeasured process variables from the measured ones, using the Q-R orthogonal transformation, was discussed before for the linear case. A similar procedure is applied twice in order to resolve the nonlinear reconciliation problem. 

Sanchez, M., and Romagnoli, J. (1996). Use of orthogonal transformations in data classification— reconciliation. Comput. Chem. Eng. 20, 483-493. 

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