Matrix transformation orthogonal

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... 

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

Backtransform the resultant function using the orthogonal transformation matrix of Eq. (4.63). 

A complementary approach to the parabolic barrier problem is obtained by considering the Hamiltonian equivalent representation of the GLE. If the potential is parabolic, then the Hamiltonian may be diagonalized" using a normal mode transformation. One rewrites the Hamiltonian using mass weighted coordinates q Vmd. An orthogonal transformation matrix... 

An orthogonal transformation matrix U may be found to diagonalize the symmetric matrix C of Eq. (220). The matrix may be faetored, as U = Us, into the product of a rotation matrix U and a diagonal sign matrix s where = 1. The matrix U may further be written as U = exp(K) where K is antisymmetric as discussed in Section II. Eq. (18) shows Det(U) = exp(Tr(K)) = -I-1 as required for rotation matrices. An arbitrary two-electron wavefunction may then be written in the form... 

Tridiagonaliae Given a symmetric matrix this operation returns a similar symmetric tridiagonal matrix and the corresponding orthogonal transformation matrix. 

A similar calculation of an element of the orthogonal transformation matrix between irreducible quadruple products yields... 

The local coordinate systems always can be reoriented using a simple similarity transformation of the fragment density matrix P ((p(ATt)), based on a suitable orthogonal transformation matrix of the AO sets. The nuclear families f for the... 

Two central identities have been derived for the orthogonal transformation matrix elements and associated frequencies. The first identity (64) uses the determinant of the force constant matrix K to show that for any value of the parameter e ... 

Here VXj denotes partial differentiation with respect to the variable This matrix is symmetric and so may be diagonalized with an orthogonal transformation matrix O leading to a new set of bath coordinates Zj, j = 1,. .., N, such that... 

A change in coordinate system is represented by an orthogonal transformation matrix U (unitary coordinate system),... 

Changing the coordinate system thus changes a matrix by pre- and post-multiplication of a unitary matrix and its inverse, a procedure called a similarity transformation. Since the U matrix describes a rotation of the coordinate system in an arbitrary direction, one person s U may be another person s U . There is thus no significance whether the transformation is written as U AU or UAU and for an orthogonal transformation matrix (U = U ), the transformation may also be written as IPAU or UAU . 

By comparison with (Al), equation (A5) implies the definition of an orthogonal transformation matrix T(p)... 

Apart from the condition 8, which ensures that the new MOs Xj form an orthonormal set, and that the observables remain invariant, we are completely free in the choice of the orthogonal transformation matrix U. This means that for a given molecule, there is in principle a great ambiguity of equally acceptable MO sets. 

Here S defines the orthogonal transformation matrix. For a 2 x 2 Hamiltonian S is given by... 

The index runs over the eigenvalues k ic of Z, which are coimected to the zeros zm,kji of Am(z) by the relation k k — - m,k - Here is the square of the matrix element C k of the complex orthogonal transformation matrix that would be obtained in the diagonalization of by means of a complex orthogonal transforma-... 

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