Basis sets symmetry orbitals

We have studied the FCl results for the ground, Xg, state of the linear, D i,BeH2 molecule with a minimal basis set. The orbitals 1, 2, and 3 are of the a type 4 and 5 area and 6 and 7 are degenerate The HF corresponds to a double occupancy of orbitals 1, 2 and 4. A bar over the orbital label indicates that its spin is p. It is important to note that 2 and 4 may be considered the homo of the Og and a symmetry-shells respectively. Similarly, 3 and 5 are the respective lumo for the same symmetry-shells. In the following analysis we will denote genetically the homos by hi and the lumos by U. 

An LCAO description of the electronic structure requires at least the minimal basis set (all orbitals that may be occupied in the ground state of the atom) of five d states per atom and the s state. Consideration of the bands from F ig. 20-1 indicates that in fact the highest-cnergy slates shown (for examples, H,s and X4) have p-like symmetry, and we shall not reproduce this with our minimal set, but the bands at this energy arc unoccupied in any case and it will be of little consequence. For constructing bands in solids, the angular forms for the d states in terms of cartesian coordinates, shown in Eq. (1-21), arc most convenient. Here we shall carry out the calculation explicitly for chromium, in the body-centered cubic structure it is carried out for the face-centered cubic structure in Problem... 

An alternative analysis of sigmatropic reactions involves drawing the basis set atomic orbitals and classifying the resulting system as Htickel or Mobius in character. When this classification has been done, the electrons involved in the process are counted to determine if the TS is aromatic or antiaromatic. The conclusions reached are the same as for the frontier orbital approach. The suprafacial 1,3-shift of hydrogen is forbidden but the suprafacial 1,5-shift is allowed. Analysis of a 1,7-shift of hydrogen shows that the antarafacial shift is allowed. This analysis is illustrated in Figure 10.31. These conclusions based on orbital symmetry considerations are supported by HF/6-31G calculations, which conclude that 1,5-shifts should be suprafacial, whereas... 

Table 3. Symmetries of Rh + d-orbitals in the bpy x,y, z reference frame (C2v) expressed in the basis set d-orbitals in the x, /, /-frame (Fig. 9)...

There are two ways to correct our treatment of methane by inclusion of symmetry-correct MOs. The first approach is to consider the descriptions of C-H bonds to be localized molecular orbitals (LMOs)—that is, molecular orbitals that have electron density on only a portion of a molecule. We can then consider these LMOs to be the basis set of orbitals for a new MO calculation to determine the symmetry-correct, delocalized MOs for the... 

We call these new basis functions symmetry orbitals or symmetry-adapted basis functions. The labels on these linear combinations are explained in Appendix 1. The new basis set consists of the oxygen orbitals r/ris, fis, ipy, fip, and the symmetry orbitals and. The canonical orbitals will be simpler linear combinations of these basis orbitals than if we used the original basis set. 

What will become obvious, in the next chapters, is that the primary valence orbitals that we will use are derived from t2g, 2Cg, and sometimes 2o g in Figure 15.1. Therefore, our basis set of orbitals will never be larger than five or six. Those orbitals are concentrated on the metal and they lie at intermediate energies. They will be the HOMOs and LUMOs in any transition metal complex. When the octahedral symmetry is perturbed by removing a ligand or distorting the geometry, the 2a g and 2t orbitals may be utilized. For example, metal s or p may mix into the members of Cg or t2g. In other words, 2t u and 2o g provide a mechanism for the hybridization of the valence orbitals. 

As mentioned above, HMO theory is not used much any more except to illustrate the principles involved in MO theory. However, a variation of HMO theory, extended Huckel theory (EHT), was introduced by Roald Hof nann in 1963 [10]. EHT is a one-electron theory just Hke HMO theory. It is, however, three-dimensional. The AOs used now correspond to a minimal basis set (the minimum number of AOs necessary to accommodate the electrons of the neutral atom and retain spherical symmetry) for the valence shell of the element. This means, for instance, for carbon a 2s-, and three 2p-orbitals (2p, 2p, 2p ). Because EHT deals with three-dimensional structures, we need better approximations for the Huckel matrix than... 

ADF uses a STO basis set along with STO fit functions to improve the efficiency of calculating multicenter integrals. It uses a fragment orbital approach. This is, in essence, a set of localized orbitals that have been symmetry-adapted. This approach is designed to make it possible to analyze molecular properties in terms of functional groups. Frozen core calculations can also be performed. 

The researchers established that the potential energy surface is dependent on the basis set (the description of individual atomic orbitals). Using an ab initio method (6-3IG ), they found eight Cg stationary points for the conformational potential energy surface, including four minima. They also found four minima of Cg symmetry. Both the AMI and PM3 semi-empirical methods found three minima. Only one of these minima corresponded to the 6-3IG conformational potential energy surface. 

Carbon atoms in free space have spherical symmetry, but a carbon atom in a molecule is a quite different entity because its charge density may well distort from spherical symmetry. To take account of the finer points of this distortion, we very often need to include d, f,. .. atomic orbitals in the basis set. Such atomic orbitals are referred to as polarization functions because their inclusion would allow a free atom to take account of the polarization induced by an external electric field or by molecule formation. 1 mentioned polarization functions briefly in Section 9.3.1. 

In this method, the orbital symmetry rules are related to the Hiickel aromaticity rule discussed in Chapter 2. Huckel s mle, which states that a cyclic system of electrons is aromatic (hence, stable) when it consists of 4n + 2 electrons, applies of course to molecules in their ground states. In applying the orbital symmetry principle, we are not concerned with ground states, but with transition states. In the present method, we do not examine the molecular orbitals themselves but rather the p orbitals before they overlap to form the MO. Such a set of p orbitals is called a basis set (Fig. 15.5). In investigating the possibility of a concerted reaction, we put the basis sets into the position they would occupy in the transition state. Figure 15.6 shows this for both the... 

In practice, unfortunately, not even the HF equations can be solved precisely due to the complicated shapes that the orbitals assume for general low-symmetry molecular environments. Hence, one introduces a fixed basis set ip that is used to expand the HF orbitals ... 

The second approach typically involves expanding the wave functions in terms of atomic or atomic-like orbitals. Frequently s- and p-symmetry functions suffice for silicon and impurities up through the third period. A minimal basis set for silicon would consist of four basis functions, one 5-function and three p-functions on each atom. Some approaches supplement the minimal basis either with more atomic-like functions or with additional types of functions, such as plane waves. Some calculations use only plane waves for the basis. 

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