Mutual orthonormality

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. 

Clearly, while the configurations i) are mutually orthonormal, and similarly for l i), their tilded counterparts are not, and none of the target states Ti) is orthogonal to Mo- Thus, relating the two basis sets of Mo as follows... 

Table 1.1 The three sets of mutually-orthonormal polynomial functions required to provide basis functions for the three components of Tju symmetry in the Ij, regular representation.

For example, the 5Hu command button on the keypad leads to the display. Figure 1.26, of the 5 copy of five polynomial functions, which, on the unit sphere, are mutually orthonormal to one another and to the 20 other polynomial functions of this irreducible symmetry forming the F , 2", and 4 sets of functions also of this symmetry, displayed when the other buttons on the keypad labelled with this symmetry are selected by a mouse click. The third function of 5hu irreducible symmetry in Figure 1.26 has leading polynomial terms... 

The general spherical harmonics are familiar, in low order, as the mutually orthonormal angular components of valence atomic orbitals. Now, the sufficient number of these functions to provide basis functions for the regular representations of the molecular point groups, in... 

Basis functions of a, tt and 5-types, for the irreducible components of the reducible characters of each kind, then, can be formed by the taking of the linear combinations generated by the actions of the structure point group symmetry operations on the sets [a]j, [ne, and [See, 5e ]j about each vertex, Pj, of a structure orbit with the rendering of these functions to be mutually orthonormal, where this is necessary as an additional step. 

These functions have the appropriate transformation properties for applications in calculation, as illustrated on the CDROM in the BonusPack , where sets of Hiickel theory calculations can be found for the energy spectra of regular orbits of I and Ih point symmetry decorated with His atomic orbitals. It is to be noted that, while the continuous functions are mutually orthonormal on the unit sphere, this property is not maintained in their discrete samplings on the 60 and 120-vertex orbit cages, and so a further orthogonalization transformation is required to restore orthogonality. 

The aim of these exercises, in the BonusPack, is to demonstrate that the polynomial functions of Table A 1.1, which are mutually orthonormal on the unit sphere, can also be made to transform appropriate as basis functions for the irreducible spaces of a Hamiltonian with icosahedral point symmetry. 

Note, carefully, the entries for the normalization constants, the squares of which, render the integrals to be of unit value. Distinct normalization constants have been included for radial and angular parts. To get the overall constant, it is necessary to multiply the two partial constants together. The table has been constructed, in this manner, to draw attention to a possible confusion in basis set theory. Often, the normalization condition is not clear for particular basis sets. Moreover, only rarely are basis sets mutually orthogonal, one with respect to another. Thus, it will be important to check the normalization data in Table 1.1 as an exercise in using the numerical integration techniques developed in Chapter 2. Orthonormalization is the subject of Chapter 3 because, in the end, all calculations in quantum chemistry require the rendering of approximate wave functions mutually orthonormal. 

Mutual orthonormality is also a key requirement in our interpretation of the physical information about the electron distribution implicit in one-electron atomic or molecular... 

Subject to the normality condition of equation 3.13, the doubie-zeta dementi 2s function, equation 3.12 is rendered mutually orthonoimal to the doubie-zeta Clementi Is function using the spreadsheet of two worksheets set out in Figure 3.8. Thus, with the identifications, [1) and T) — [Pg.92]

It is appropriate therefore to attempt calculations based on the use of mutually orthonormal linear combinations of the simple Slater functions. In particular, let us make a 2s Slater function orthonormal to the Is function and calculate the 2s orbital energy again. The calculation on a spreadsheet involves two stages, the construction of the 2s function orthogonal to the 1 s function and then the calculation of the energy terms using our numerical procedure. 

The forerunner of Cl is the self-consistent field (SCF) method [1, 2]. A version that properly accounts for the antisymmetry of the electronic wave function was developed independently by Fock [3] and Slater [4] shortly after Schrodinger s papers. It is characterized by an approximate wave function that is a single determinant whose elements are one-electron functions (spin orbitals). The latter orbitals are optimized under two conditions minimization of the energy expectation value and mutual orthonormality. The method produces both the occupied orbitals appearing in the determinant but also a potentially infinite number of unoccupied functions that prove to be the basis for the Cl method. One can look upon a Slater determinant formed by substituting unoccupied for occupied one-electron functions as a representation of an excited state of the molecular system. The possible applications to spectroscopy were obvious. 

The pair (x, y) define the branching plane or g-h plane. The remainder of the intersection adapted coordinate system, w , i = l-(Ar " — 2), spans the seam space. These — 2 mutually orthonormal vectors need only be orthogonal to the branching space. It is also convenient to define... 

Corollary ENV2-1 (spectral decomposition of SR ) Let A be a real, symmetric matrix, A = A, and thus Hermitian. From Aw = Xjw, as both A and Xj are real, it is always possible to find a real set of mutually orthonormal eigenvectors for A. Therefore, we may write any vector v e 91 as the eigenvector expansion... 

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