Orthonormal basis functions

The definitions (22) and (23) don t require ortho-normalized basis functions. But calculated results fairly depends on basis functions. The absolute value of partitioned energy with orthonormalized basis functions is usually larger and seems more reasonable than the counterpart with non-orthonormalized basis functions, like the case of Okada s bond order. The basis functions orthonormalized by Eq. (19) and Eq. (20) are used for the energy partition calculations in the following sections. But systematic and theoretical studies are necessary to clarify the dependence of bond parameters on the basis functions and the optimization of them. 

Armstrong, Perkins and Stewart s bond index and valency, which have been used in MOP AC, were extended to general MO methods including DV-Xa, by the way of basis function orthonormalization. 

In the unlikely event that none of the basis functions overlap, then S is a unit matrix. We usually require the LCAO orbitals . to be orthonormal... 

Looking back, 1 seem to have made two contradictory statements about the basis fiinctions Xt used in the PPP model. On the one hand, I appealed to your chemical Intuition and prior knowledge by suggesting that the basis functions should be j garded as ordinary atomic orbitals of the correct symmetry (i.e. 2p orbitals). On the other hand, 1 told you that the basis functions used in such calculations are taken to be orthonormal and so... 

These new basis functions can easily be shown to be orthonormal. It also turns out that two-electron integrals calculated using these orthogonalized basis functions do indeed satisfy the ZDO approximation much more closely than the ordinart basis functions. 

The variational problem is to minimize the energy of a single Slater determinant by choosing suitable values for the MO coefficients, under the constraint that the MOs remain orthonormal. With cj) being an MO written as a linear combination of the basis functions (atomic orbitals) /, this leads to a set of secular equations, F being the Fock matrix, S the overlap matrix and C containing the MO coefficients (Section 3.5). 

The basis functions constructed in this manner automatically satisfy the necessary boundary conditions for a magnetic cell. They are orthonormal in virtue of being eigenfunctions of the Hermitian operator Ho, therefore the overlapping integrals(6) take on the form... 

A second way of resolving this question is provided by examining the constraint itself. Indeed, the condition CC+ = N is equivalent to requiring the orthonormalization of the basis functions 33(0 of the occupied subspace. That is to say, 33(/) s must satisfy... 

In this section the symbols Hilbert space L, which may be finite or infinite, and x stands for the variables on which the functions of L may depend. An operator defined on L has the action Tf(x) = g(x) where g L. The action of T on a basis function 4>n x) is described by... 

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

Let us consider the simple case of the H atom and its variational approximation at the standard HF/3-21G level, for which we can follow a few of the steps in terms of corresponding density-matrix manipulations. After symmetrically orthogonalizing the two basis orbitals of the 3-21G set to obtain orthonormal basis functions A s and dA, we obtain the corresponding AO form of the density operator (i.e., the 2 x 2 matrix representation of y in the... 

Assuming that the basis functions used in discretization ( < ) )) are complete and orthonormal, the wave function in Eq. [1] can be expressed in the following expansion ... 

It is well established that the eigenvalues of an Hermitian matrix are all real, and their corresponding eigenvectors can be made orthonormal. A special case arises when the elements of the Hermitian matrix A are real, which can be achieved by using real basis functions. Under such circumstances, the Hermitian matrix is reduced to a real-symmetric matrix ... 

Obviously, we can examine the effect of the Oh symmetry operations over a different set of orthonormal basis functions, so that another set of 48 matrices (another representation) can be constructed. It is then clear that each set of orthonormal basis functions transformation equation as follows ... 

If the solution 4>(t) to the time-dependent Schrodinger equation is expanded as a linear combination of time-independent orthonormal many-electron basis functions i.e. 

L6wdin[25] suggested that one find the orthonormal set of functions that most closely approximates the original nonorthogonal set in the least squares sense and use these to determine the weights of various basis functions. An analysis shows that the appropriate transformation in the notation of Eq. (1.18) is... 

Note that the orthonormality of basis functions of different angular momentum both residing on the same atom k causes many terms in the second sum of Eq. (9.14) to be zero. The... 

To alleviate a number of these problems, Lowdin proposed that population analysis not be carried out until the AO basis functions tp were transformed into an orthonormal set of basis functions / using a symmetric orthogonalization scheme (Lowdin 1970 Cusachs and Politzer 1968)... 

The Teller proof (33) assumes that the eigenfunctions of the two states of concern and V/j say) may be written as a linear combination of two orthonormal basis functions and o, . The energies of the two states are then identical to the eigenvalues of the 2 x 2 Hamiltonian matrix... 

An n-dimensional function space is defined by specifying n mutually orthogonal, normalized, linearly-independent functions, [et, e j and es define physical space] they are called orthonormal basis functions. 

If we choose our basis functions for a particular function space to be orthonormal (orthogonal and normalized) i.e. (/ /,) = J/, /, dr then, since the transformation operators are unitary ( 5-7), the representation created will consist of unitary matrices. This is proved in Appendix A.6-1. It should be stated that it is always possible to find an orthonormal basis and one way, the Schmidt orthogonalization process, is given in Appendix A.6-2. 

If an n dimensional space is characterized by the n orthonormal basis functions /i, /s, / , then, by definition, the scalar product is... 

Suppose that we have found Jc different function spaces for a given point group, where k is the number of classes or irreducible representations for the point group, and suppose that each function space provides the basis functions for one of the Jc irreducible representations. If the dimension of the rth irreducible representation is nv, there will be nv orthonormal basis functions describing the rth function space. We will write these sets of basis functions as... 

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