Matrix, unitary

The D matrix is by definition a unitary matrix (it is product of two unitary mati ices) and since the adiabatic eigenvalues are uniquely defined in CS, we have, u(0) = m(P). Then, Eq. (57) can be written as... 

The B matrix is, by definition, a unitary matrix (it is a product of two unitary matrices) and at this stage except for being dependent on F and, eventually, on So, it is rather arbitrary. In what follows, we shall derive some features of B. 

The derivation of the D matrix for a given contour is based on first deriving the adiabatic-to-diabatic transformation matrix, A, as a function of s and then obtaining its value at the end of the arbitrary closed contours (when s becomes io). Since A is a real unitary matrix it can be expressed in terms of cosine and sine functions of given angles. First, we shall consider briefly the two special cases with M = 2 and 3. 

We start by proving that Am is a unitary matrix and as such it will have an inverse (the proof is given here again for the sake of completeness). Let us consider the... 

Thus Ap is a unitary matrix at any point in configuration space. 

The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

In this form, one sees why the array Dj m, m is viewed as a unitary matrix, with M and M as indices, that describes the effect of rotation on the set of functions Yl,m - This... 

The orbital rotation is given by a unitary matrix U, which can be written as an exponential transformation. 

Kinetic energy functional, calculated from a Slater determinant Internal energy Unitary matrix... 

In order to try to approach the HF scheme as much as possible, we will now introduce the basic orthonormal set fc which has maximum occupation numbers. Let U be the unitary matrix which brings the hermitean matrix (ylk) to diagonal form ... 

Now remove the first row and first column and repeat with the submatrix of order n — 1. After continuing the process until the reduced matrix is of third order, restore the rows and columns that had been dropped, and border the transforming matrices with ones on the diagonal and zeros elsewhere. There results, then, a unitary matrix W, the product of the Wt, such that... 

The Hermitian Hamiltonian matrix H, the diagonal matrix E, and the unitary matrix... 

For example, according to Hamermesh (see Ref. [11]), the number of real conditions to uniquely determine an (N x N) (complex) unitary matrix is N2, while the number of real conditions to uniquely fix a (real) orthogonal matrix of same dimensions is not N2/2 but N(N + l)/2. 

However, the decomposition of P into C+C is not unique, since, as Pecora wrote, any C = VC (where V is understood to be an (Ax A) unitary matrix) will generate the same P matrix, which" is just a basic fact of Quantum Mechanics or, more generally,... 

As V is a unitary matrix, Y = VTX is just an equivalent set of Cartesian coordinates, and = UTZ is just an equivalent set of internal coordinates, simply linear combinations of the Zn. The i, , N-6, change independently, in proportion to changes in linear combinations of the Cartesian coordinates. So, locally, we have defined 3N — 6 independent internal coordinates. Every different configuration of the molecule, X, will have a different B matrix, and hence a different definition of local internal coordinates, defined automatically. 

QR Factorization of a Matrix If A is an m x n matrix with m > n, there exists inmxji unitary matrix = [qij q2,..., q ] and an m X n right triangular matrix R such that A = QR. The OR factorization is frequently used in the actual computations when the other transformations are unstable. 

An mxm unitary matrix U is formed from the eigenvectors u. of the first matrix. 

An n x n unitary matrix V is formed from the eigenvectors v. of the second matrix. 

A unitary matrix is one whose inverse is equal to its hermitian conjugate, A"1 = At = A. ... 

Since the scalars are complex, this last condition is equivalent to two conditions. The number of parameters is therefore reduced from 8 to 4. The most general unitary matrix of order two involving four real parameters can be expressed in the form... 

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