Orthonormal

Wlien nomialized, the eigenfiinctions corresponding to a Hennitian operator are said to represent an orthonormal set. 

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. 

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... 

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... 

Some details of END using a multiconfigurational electronic wave function with a complete active space (CASMC) have been introduced in terms of an orthonormal basis and for a fixed nuclear framework [25], and were recently [26] discussed in some detail for a nonoithogonal basis with electron translation factors. 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, hamonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the bajectoiy, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

From the preceding analysis, it is seen that the coordinate space neai R can be usefully partitioned into the branching space described in tenns of intersection adapted coordinates (p, 9, ) or (x,y,z) and its orthogonal complement the seam space spanned by a set of mutually orthonormal set w, = 4 — M . From Eq. (27), spherical radius p is the parameter that lifts the degeneracy linearly in the branching space spanned by x, y, and z. 

We here describe the alternative of approximating <,c(S)b via Lanczos method. The Lanczos process [18, 22] recursively generates an orthonormal basis Qm = [qi,.., qm] of the mth Krylov subspace... 

Iris type of constrained minimisation problem can be tackled using the method of Lagrange nultipliers. In this approach (see Section 1.10.5 for a brief introduction to Lagrange nultipliers) the derivative of the function to be minimised is added to the derivatives of he constraint(s) multiplied by a constant called a Lagrange multiplier. The sum is then et equal to zero. If the Lagrange multiplier for each of the orthonormality conditions is... 

Kohn and Sham wrote the density p(r) of the system as the sum of the square moduli of a set of one-electron orthonormal orbitals ... 

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... 

We also know that exact atomic orbitals are orthonormal, that is, Sa = 1 and = 0 for an LCAO. If we assume that orthonormality is canied from an LCAO into the molecular orbital, then S = I and, from Eq. (7-17),... 

The first two integrals are simplified by the faet that orthonormal funetions yield (ri) (ri)dn = 1 (9-13a)... 

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

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

If the functions Oj are orthonormal, then the overlap matrix S reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form ... 

Projecting this equation against < hj (r R) (integrating only over the electronic coordinates because the hj are orthonormal only when so integrated) gives ... 

For the hermitian matrix in review exereise 3b show that the pair of degenerate eigenvalues ean be made to have orthonormal eigenfunetions. 

Note that this identity enables you to utilize the orthonormality of the spherieal harmonies. 

Yim V Yoo> vanishes (from orthonormality). In quantum ehemistry when using... 

Sets of orbitals that are not orthonormal ean be eombined to form new orthonormal funetions in many ways. One teehnique that is espeeially attraetive when the original funetions are orthonormal in the absenee of "interaetions" (e.g., at large interatomie... 

---