Orbitals symmetrically orthogonalized

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... 

The subscripts x, y, and z indicate the angular dependencies. As already mentioned, the three p orbitals are orthogonal to each other, and it is obvious that they are not spherically symmetrical about the nucleus. A boundary surface for each of the three p orbitals is given in Figure 1-3. The radial function is of course the same for all three p orbitals, and the first radial function is... 

Fig. 14. Linear combination of the atomic valence orbitals s,px,py to form sp and sp2 orbitals in the (x,y)-plane. The resulting orbitals are symmetric perpendicular to the (x,y)-plane and build the structure framework in the (x,y)-plane with the orbitals of the neighboring atoms. The remaining antisymmetric pz orbital is orthogonal to the orbitals in the (x,y)-plane and is used for the delocalized electron orbitals...

Symmetrically Orthogonalized INDO (Method) Minimum Basis Set of Slater-type Orbitals, each Represented by 3 Gaussian Orbitals... 

Firstly, the function (70) is invariant under a linear transformation of the m doubly occupied orbitals amongst themselves. A proof of this statement seems hardly necessary as, in the case m = N, equation (70) is equivalent to a Slater determinant, and this property of a determinant is well-known. The m orbitals m may therefore be orthogonalized amongst themselves by a linear transformation, without altering the total wavefunction. This, of course, may be done in several ways, by transforming to MOs for example, but perhaps the most convenient method is to employ the Lowdin symmetric orthogonalization method 73... 

Further properties of the model arise from interpretations based on mutually orthogonal and normalized atomic orbitals. The common Lowdin symmetrically orthogonalized basis gives... 

Symmetrically orthogonalize the L frozen orbitals and compute the corresponding density matrix. 

There is another very important property of the symmetrically orthogonalized orbitals that leads to the observations in Table 1. These orbitals can be shown to be the orthogonalized set that most resembles the original basis orbitals in the sense that... 

Let us begin by transforming the atomic orbitals into an orthogonal set, via a symmetrical orthogonalization. 

The constituent atomic orbitals are orthogonal and if they are also normalized, these hybrid orbitals can be seen to be normal and orthogonal by forming their squares and products. Since the s function is spherically symmetrical, and since the angular dependence of px, py, and pz is proportional to x/r, y/r, and z/r, respectively, the four sp3 hybrids can be seen to have their maxima directed respectively toward the four corners of a tetrahedron centred at the origin of coordinates. 

Exercise 3.7. Application of the canonical and symmetric orthogonalization routines to the construction of sto-3g) approximations to the hydrogen Is and 2s radial orbitals. 

The new orbitals (p. as lineariy independent combinations of the occupied canonical orbitals cpj, span the space of flie canonical occupied HForbitals symmetric orthogonalization see Appendix I available at booksite.elsevier.com/978-0-444-59436-5). 

Orbital energies from other ab initio calculations pertain to assumed bond lengths [9] or to the experimental geometry [13, 14]. Orbital energies were also calculated semiempirically by the extended Hiickel method [11], by CNDO [15], INDO [15, 16], MNDO [17], and SINDO [18] (symmetrically orthogonalized INDO, see [19]). 

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