Normality orthonormality

The Hermitian property means that the solutions can be chosen to be orthogonal and normalized orthonormal). 

The eigenvectors are normally orthonormalized with the overlap matrix S as metric... 

A function T is normalized if the product T integrated over all configuration space is unity. An orthonormal set contains functions that are normalized and orthogonal to each other. 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

A further simplification is made. The wave functions pi and p2, which are orthogonal and normalized in the hydrogen atom, are assumed to retain their orthonormality in the molecule. Orthonormality requires that... 

Here (1/2%) exp(ikx) is the normalized eigenfunction ofF =-ihd/dx corresponding to momentum eigenvalue hk. These momentum eigenfunctions are orthonormal ... 

Express the three resultant orthonormal orbitals as linear eombinations of these three normalized STO s. 

There are a few interesting points about the treatment. First of all, there is no variational HF-LCAO calculation (because every available x is doubly occupied) and so the energy evaluation is straightforward. For a wavefunction comprising m doubly occupied orthonormal x s the normalizing factor N is... 

This contribution considers systems which can be described with just the Hamiltonian, and do not need a dissipative term so that TZd = 0- This would be the case for an isolated system, or in phenomena where the dissipation effects can be represented by an additional operator to form a new effective non-Hermitian Hamiltonian. These will be called here Hamiltonian systems. For isolated systems with a Hermitian Hamiltonian, the normalization is constant over time and the density operator may be constructed in a simpler way. In effect, the initial operator may be expanded in its orthonormal eigenstates (density amplitudes) and eigenvalues Wn (positive populations), where n labels the states, in the form... 

In counting the number of orthonormalization conditions on C, CGM apparently did not assume the hermiticity of the scalar product in the subspace, but rather chose to impose it. Their calculation of K ran along the following lines a complex projector, which is hermitian and normalized, may be factored into [13]... 

For cn = 1, or An = j2]l the eigenfunctions, like the unit vectors of particle mechanics, are normalized. Thus, for Sn = y/2/lsinmrx/l, the functions are said to be orthonormal and the normalization condition reads... 

Because of these properties of Hermitian functions it is accepted as a basic postulate of wave mechanics that operators which represent physical quantities or observables must be Hermitian. The normalized eigenfunctions of a Hermitian operator constitute an orthonormal set, i.e. 

Now the expression Normal Equations starts to make sense. The residual vector r is normal to the grey plane and thus normal to both vectors f ,i and f , 2 As outlined earlier, in Chapter Orthogonal and Orthonormal Matrices (p.25), for orthogonal (normal) vectors the scalar product is zero. Thus, the scalar product between each column of F and vector r is zero. The system of equations corresponding to this statement is ... 

From Eq.(73) the Gram-Schmidt orthonormalization is applied to obtain the normalized values [Sxin, Sx2n, Sx2n, Sx2n]- The following integration step is carried out with the values ... 

Sturmian basis set obeys a potential-weighted orthogonality relationship analogous to equation (10). This still does not tell us how to normalize the functions, and in fact the choice is arbitrary. However, it will be convenient to choose the normalization in such a way that in momentum space the orthonormality relations become ... 

Associate the Lagrange multiplier ji (chemical potential) with the normalization condition in Eq. (6), the set of Hermitian-Lagrange multipliers X[ with orthonormality constraints in Eq. (4), and define the auxiliary functional Q, by the formula... 

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... 

This special normalization is convenient because it leads to momentum-space orthonormality relations of the form ... 

When dealing with spin functions it is normally convenient to arrange the bases to be orthonormal, and we obtain two functions,... 

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