Orthonormal set

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. 

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. 

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 ... 

Hol0ien, E., Phys. Rev. 104, 1301, Radial configurational interaction in He and similar atomic systems." An orthonormal set of associated Laguerre functions is used. 

We now introduce a second complete orthonormal set of functions v, which we shall take to be functions of t, vt(t). First, however, we merely consider the set (0). Since they are by definition constant in time, they behave like the u, and all preceding equations can be written in the v(0)-representation. A transformation of the functions has the form... 

If the hamiltonian is truly stationary, then the wx are the space-parts of the state function but if H is a function of t, the wx are not strictly state functions at all. Still, Eq. (7-65) defines a complete orthonormal set, each wx being time-dependent, and the quasi-eigenvalues Et will also be functions of t. It is clear on physical grounds, however, that to, will be an approximation to the true states if H varies sufficiently slowly. Hence the name, adiabatic representation. 

Although the eigenvalues 0 and 1 are universal, there are many possible eigenvectors r that depend on the kind of states p is to designate. To each one of an orthonormal set of functions , for instance,... 

If a basis is found that does not satisfy this condition, an orthonormal set can be constructed from it by the Schmidt process analogous to the familiar device in three-dimensional vector analysis.8... 

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. 

If we expand the state function ja ) in terms of the complete, orthonormal set... 

Thus, the operator Hi is hermitian and the radial functions REi(r) constitute an orthonormal set with a weighting function w(r) equal to r ... 

To prove the variation theorem, we assume that the eigenfunctions 0 form a complete, orthonormal set and expand the trial function 0 in terms of that set... 

We assume in this section and in Section 10.2 that equation (10.6) has been solved and that the eigenfunctions Q) and eigenvalues k(Q) are known for any arbitrary set of values for the parameters Q. Further, we assume that the eigenfunctions form a complete orthonormal set, so that... 

According to equation (F.22), the functions Xigip) constitute an orthonormal set. We now show that this set is complete. 

Here it is most convenient to introduce an orthonormal set of unit vectors ... 

It should be noted that the functions Xn need not necessarily form an orthonormal set The linearly independent coefficients c can be considered to be variable parameters that are determined by minimization of the variational energy, W. If the functions Xn are not orthonormal, Eq. (105) can be rewritten in the form... 

According to the postulates of QM, any if/ representing a physical state of the system can be expressed as a linear combination of energy eigenfunctions forming an infinite orthonormal set ... 

Transformations that take one orthonormal set of basis vectors into another orthonormal set are called unitary transformations, the operators associated with them are called unitary operators. This definition preserves the norms and scalar products of vectors in Ln. The transformation (4) is in fact a set of linear equations... 

Together these unit vectors define an n-dimensional vector space and they are said to constitute an orthonormal set. State vectors of this type are... 

A complete orthonormal set of n-component vectors is a set vi,v2,---, vn, satisfying the orthonormality relations... 

The eigenvalue-eigenvector decomposition of a Hermitian matrix with the complete orthonormal set of eigenvectors Vi and eigenvalues A, is written as... 

To summarize, suppose that all possible states for any given observable (spin, polarization, energy, momentum, etc.) are known and that each can be formulated in terms of a column vector a = a,i, <22, an. These vectors form an orthonormal set and are represented by an n x n matrix... 

Since H is Hermitian, the eigenvectors Vj of H form a complete orthonormal set and the vector representing a general state at t = 0 may be expressed as a linear superposition of these eigenvectors, (0) = CjVj, ... 

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. 

When choosing combinations of basis functions which constitute an orthonormal set, S becomes the identity matrix, i.e. Sij = SlJ. 

Let us begin by sketching the general polyatomic formulation of hybridization from the NBO viewpoint. A general hybrid h/A) on atom A can be expanded in the complete orthonormal set of NAOs, (A) on this atom ... 

Except for the initial AO —NAO transformation, which starts from non-orthogonal AOs, each step in (3.38) is a unitary transformation from one complete orthonormal set to another. Each localized set gives an exact matrix representation of any property or function that can be described by the original AO basis. 

Solution Let us first choose a reference orthonormal set (j>, jp) to be used consistently in displaying the various matrices and vectors under discussion. For simplicity, we choose (pi, (p2 to be the ( Lowdin-orthogonalized ) functions that are closest to xa and Xb >n the mean-squared-deviation sense. The non-orthogonal functions xa and Xb (with(xJXb) = S ) can then be expressed in terms of reference orthonormal functions as... 

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