Row-orthonormal

Kuppermann A 1997 Reactive scattering with row-orthonormal hyperspherical coordinates. 2. 

Kuppermann A 1996 Reactive scattering with row-orthonormal hyperspherical coordinates. I. Transformation properties and Hamiltonian for triatomic systems J. Phys. Chem. 100 2621... 

D. Wang, A. Kuppermann, Analytical derivation of row-orthonormal hyperspherical harmonics for triatomic systems, J. Phys. Chem. A 113 (2009) 15384. 

K. Museth and A. Kuppermann, Asymptotic analysis of state-to-state tetraatomic reactions using row-orthonormal hyperspherical coordinates. J. Chem. Phys., 115 8285-8297, 2001. 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

The 3x3 matrix U shown below is both row- and column-orthonormal ... 

We have seen above that the r columns of U represent r orthonormal vectors in row-space 5". Hence, the r columns of U can be regarded as a basis of an r-dimensional subspace 5 of 5". Similarly, the r columns of V can be regarded as a basis of an r-dimensional subspace S of column-space 5. We will refer to S as the factor space which is embedded in the dual spaces S" and SP. Note that r

factor-spaces will be more fully developed in the next section. 

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

Once we have obtained the projections S and L of X upon the latent vectors V and U, we can do away with the original data spaces S and 5". Since V and U are orthonormal vectors that span the space of latent vectors each row i and each column j of X is now represented as a point in as shown in Figs. 31.2c and d. The... 

In this way, an nxpxq table X is decomposed into an rxsxt core matrix Z and the nxr, pxs, qxt loading matrices A, B, C for the row-, column- and layer-items of X. The loading matrices are column-wise orthonormal, which means that ... 

Expanding the Z matrix in terms of Ax, Ay and Az and the components of the -matrix, and simplifying making use of the orthonormality of rows of the -matrix, we have ... 

If columns (or rows) of X are normalised to the square root of the sum of their squared elements (i.e. to unity length), the matrix is called orthonormal. Recall that earlier this kind of normalisation was solved most elegantly by right (left) multiplication with a diagonal matrix comprising the appropriate normalisation coefficients. See the section introducing diagonal matrices for more details. 

Orthonormal matrices have very special properties. If a matrix X is comprised of orthonormal rows then... 

If matrix X is square and has orthonormal rows, its columns are also orthonormal. The inverse is then equal to its transpose... 

The elements of D represent the sum over all unit cells of the interaction between a pair of atoms. D has 3n x 3n elements for a specific q and j, though the numerical value of the elements will rapidly decrease as pairs of atoms at greater distances are considered. Its eigenvectors, labeled e ( fcq), where k is the branch index, represent the directions and relative size of the displacements of the atoms for each of the normal modes of the crystal. Eigenvector ejj Icq) is a column matrix with three rows for each of the n atoms in the unit cell. Because the dynamical matrix is Hermitian, the eigenvectors obey the orthonormality condition... 

The first equation in (2.24) states that column vectors / and j of a unitary matrix are orthonormal the second equation states that the row vectors of a unitary matrix form an orthonormal set. For a real orthogonal matrix, the row vectors are orthonormal, and so are the column vectors. 

The t x t matrix of coefficients /J on the left is non-singular and there are (s - t ) = r independent ways of assigning arbitrary values to the a s on the right hand side. The simplest way would be to set aj +l = <5, which would give the Hooyman form (Rem. 9). Alternatively the resulting row vectors may be orthonormalized by the Gramm-Schmidt process and this will give the required stoicheiometric matrix a. Clearly this can only be defined to within an arbitrary rotation that does not confound the subspaces of reaction and invariance. ... 

In the unlikely event that none of the basis functions overlap, then S is a unit matrix. We usually require the LCAO orbitals ipA, Pb, , V m to be orthonormal and this fact can be summarized in a single matrix statement. A little manipulation will show that UTSU is then a unit matrix (with m rows and m columns), and also that... 

Example 4.4-3 Using the partial character table for C3v in Table 4.3, show that the character systems ixi and xf satisfy the orthonormality condition for the rows. 

Equation (7) describes the transformation of the set of basis vectors ei e2 e31 that are firmly embedded in configuration space and were originally coincident with fixed orthonormal axes x y z prior to the application of the symmetry operator R(n (3 7). In eq. (8) the column matrix x y z) contains the variables x y z, which are the components of the vector r = OP and the coordinates of the point P. In eq. (9) the row matrix (x y z contains the functions x y z (for example, the angle-dependent factors in the three atomic p functions px, py, pz). 

Equations (21) and (22) state that the rows or columns of an orthogonal matrix are orthonormal. A is a unitary matrix if... 

Equations (25) and (26) show that the rows or the columns of a unitary matrix are orthonormal when the scalar product is defined to be the Hermitian scalar product. 

Suppose that h(n i is diagonalized in a basis of dimension n — 1, and this basis is extended by adding an orthonormalized function q . The diagonalized matrix is augmented by a final row and column, with elements h i,hi respectively, for i < n. The added diagonal element is hnn. Modified eigenvalues are determined by the condition that the bordered determinant of the augmented matrix h(n) — e should vanish. This is expressed by... 

General linear relations between the elements of the HOs residing on a heavy atom as taken in the quaternion form represent some interest. The orthonormality condition for the HOs written in the quaternion form allows us to establish the shape of the hybridization tetrahedra through eq. (3.61). On the other hand, the 4 x 4 matrix formed by HOs expansion coefficients is orthogonal not only with respect to rows, each representing one HO, but also with respect to columns, so that ... 

The eigenvectors in Vt can be used to form a set of orthonormal row basis vectors for A. The eigenvectors are called loadings or sometimes abstract factors or eigenspectra, indicating that while the vectors form a basis set for the row space of A, physical interpretation of the vectors is not always possible (see Figure 4.2). 

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