Electron in the SET

The idea of stable three membered rings for systems with less than a half-filled collection of orbitals, as indicated by the third moment curve of Fig. 6 has bearing on the structures of molecules too. Figure 32a shows examples of known, isoelectronic molecules which contain three membered XHH rings. (The first two have been structurally characterized, the last has only been seen in a mass spectrometer). In each case there are a total of two electrons in the set of molecular orbitals generated by the ring. So x = 0.33. We can see this quite clearly in 49. The frontier orbitals of H, Cr(CO)5 and CH( are empty but two electrons are provided by the Hj moiety. 

For a C atom (Z = 6) there are a number of possible configurations for the second electron in the set of three 2p orbitals. We use Hund s rule to determine where the... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

A I lai lcee-Fock calculation provides a set of orbital energies, e,. What is the significance oi these The energy of an electron in a spin orbital is calculated by adding the core inleraclion to the Coulomb and exchange interactions with the other electrons in the svstein ... 

There are several issues to consider when using ECP basis sets. The core potential may represent all but the outermost electrons. In other ECP sets, the outermost electrons and the last filled shell will be in the valence orbital space. Having more electrons in the core will speed the calculation, but results are more accurate if the —1 shell is outside of the core potential. Some ECP sets are designated as shape-consistent sets, which means that the shape of the atomic orbitals in the valence region matches that for all electron basis sets. ECP sets are usually named with an acronym that stands for the authors names or the location where it was developed. Some common core potential basis sets are listed below. The number of primitives given are those describing the valence region. 

Iron carries half the charge of a whole electron. The calculation produces a set of molecular orbitals appropriate for this pseudowave function. HyperChem then assigns the unpaired electron its proper spin (alpha), substitutes this electron in the orbital formerly occupied by the half electrons, and calculates energy and other properties. 

The semi-empirical methods of HyperChem are quantum mechanical methods that can describe the breaking and formation of chemical bonds, as well as provide information about the distribution of electrons in the system. HyperChem s molecular mechanics techniques, on the other hand, do not explicitly treat the electrons, but instead describe the energetics only as interactions among the nuclei. Since these approximations result in substantial computational savings, the molecular mechanics methods can be applied to much larger systems than the quantum mechanical methods. There are many molecular properties, however, which are not accurately described by these methods. For instance, molecular bonds are neither formed nor broken during HyperChem s molecular mechanics computations the set of fixed bonds is provided as input to the computation. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2, but in somewhat different ways. Both assume that electron waves behave like more familiar waves, such as sound and light waves. One important property of waves is called interference in physics. Constructive interference occurs when two waves combine so as to reinforce each other (in phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2.2). Recall from Section 1.1 that electron waves in atoms are characterized by then- wave function, which is the same as an orbital. For an electron in the most stable state of a hydrogen atom, for example, this state is defined by the I5 wave function and is often called the I5 orbital. The valence bond model bases the connection between two atoms on the overlap between half-filled orbitals of the two atoms. The molecular orbital model assembles a set of molecular- orbitals by combining the atomic orbitals of all of the atoms in the molecule. 

You should remember the basic physical idea behind the HF model each electron experiences an average potential due to the other electrons (and of course the nuclei), so that the HF Hamiltonian operator contains within itself the averaged electron density due to the other electrons. In the LCAO version, we seek to expand the HF orbitals i/ in terms of a set of fixed basis functions X X2 > and write... 

The first step in reducing the computational problem is to consider only the valence electrons explicitly, the core electrons are accounted for by reducing the nuclear charge or introducing functions to model the combined repulsion due to the nuclei and core electrons. Furthermore, only a minimum basis set (the minimum number of functions necessary for accommodating the electrons in the neutral atom) is used for the valence electrons. Hydrogen thus has one basis function, and all atoms in the second and third rows of the periodic table have four basis functions (one s- and one set of p-orbitals, pj, , Pj, and Pj). The large majority of semi-empirical methods to date use only s- and p-functions, and the basis functions are taken to be Slater type orbitals (see Chapter 5), i.e. exponential functions. 

Table 11.4 H2O geometry as a function of basis set at the MP2 level of theory including all electrons in the correlation ...

No currently known elements contain electrons in g (< = 4) orbitals in the ground state. If an element is discovered that has electrons in the g orbital, what is the lowest value for n in which these g orbitals could exist What are the possible values of mi How many electrons could a set of g orbitals hold ... 

Soon after Bohr developed his initial configuration Arnold Sommerfeld in Munich realized the need to characterize the stationary states of the electron in the hydrogen atom by. means of a second quantum number—the so-called angular-momentum quantum number, Bohr immediately applied this discovery to many-electron atoms and in 1922 produced a set of more detailed electronic configurations. In turn, Sommerfeld went on to discover the third or inner, quantum number, thus enabling the British physicist Edmund Stoner to come up with an even more refined set of electronic configurations in 1924. 

Now consider the alkynes, hydrocarbons with carbon-carbon triple bonds. The Lewis structure of the linear molecule ethyne (acetylene) is H—O C- H. To describe the bonding in a linear molecule, we need a hybridization scheme that produces two equivalent orbitals at 180° from each other this is sp hybridization. Each C atom has one electron in each of its two sp hybrid orbitals and one electron in each of its two perpendicular unhybridized 2p-orbitals (43). The electrons in the sp hybrid orbitals on the two carbon atoms pair and form a carbon—carbon tr-bond. The electrons in the remaining sp hybrid orbitals pair with hydrogen Ls-elec-trons to form two carbon—hydrogen o-bonds. The electrons in the two perpendicular sets of 2/z-orbitals pair with a side-by-side overlap, forming two ir-honds at 90° to each other. As in the N2 molecule, the electron density in the o-bonds forms a cylinder about the C—C bond axis. The resulting bonding pattern is shown in Fig. 3.23. 

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