Atomic number degenerate

The number of electrons in an atom affects the properties of the atom. The hydrogen atom, with one electron, has no electron-electron repulsions therefore, all the orbitals of a given shell in the hydrogen atom are degenerate. For instance, the 2s-orbital and all three 2p-orbitaIs have the same energy. In many-electron atoms, however, the results of spectroscopic experiments and calculations show... 

A symmetry-based molecular orbital description of the unusual four-coordinate C3v W(RC=CR)3(CO) series of molecules was presented by King in 1968 (32). The tt orbitals of the three alkynes yield linear combinations of A2 and E symmetry. Since there is no metal orbital of A2 symmetry only the degenerate E combination of n orbitals finds metal orbital mates for bonding and antibonding combinations. The three alkyne 7T orbitals serve as er donors [(Ax + E) symmetry] as does the fourth ligand (Al symmetry). Thus the total metal electron count adheres to the effective atomic number rule [W(0)(6) + 37T j(6) + 2ir (4) + lo-(2) = 18 electrons]. 

Because only n and values affect the energies of electrons, the electrons with the same n value and the same value all have the same energy. In other words, all the electrons in a given subshell are degenerate. Each subshell is denoted by its principal quantum number and the letter designation for . For example, for neon, with atomic number 10, the sets of quantum numbers for the 10 electrons are listed in Table 4.2. We can group them as follows ... 

This is the simplest measure with regard to molecular size, defined as the total number of atoms in a molecule. It is a global, zero-dimensional, highly degenerate descriptor. In several applications for the calculation of - molecular descriptors, the atom number A refers only to non-hydrogen atoms. 

The spherical jellium model has been applied to alkali metal clusters by many authors (see Ref. [6]). Fig. (1) shows the self-consistent effective potential for a sodium cluster with twenty atoms. The degenerate levels are filled up to electron number = 20. In a spherical cluster with 21 electrons, the last electron will have to occupy the If level above (dashed line). This electron is less... 

Analyze and Plan Because oxygen has an atomic number of 8, each oxygen atom has 8 electrons. Figure 6.24 shows the ordering of orbitals. The electrons (represented as arrows) are placed in the orbitals (represented as boxes) beginning with the lowest-enei orbital, the Is. Each orbital can hold a maximum of two electrons (the Pauli exclusion principle). Because the 2p orbitals are degenerate, we place one electron in each of these orbitals (spin-up) before pairing any electrons (Hund s rule). 

The energy levels of the hydrogen-like atom are degenerate because more than one state corresponds to a specific value of n (the principal quantum number). These degenerate states are characterized by the quantum numbers I and m, which characterize the spherical harmonics of the wave function. For each value of n there are n values of l(l = 0,.,., n — 1) and for each value of / there are 2/ -h 1 values of m(m = — / -h 1,..., 0,..., / — 1, /) giving... 

The degenerate MOs 02 and 03 should have one nodal plane each, and these should be perpendicular to each other.If we take one plane as shown in (VIII), we can immediately write down 02. The node for 03 is given in (IX). It is obvious that X6, XI, X2 have coefficients of the same sign, and that C2 = ce = —cs = —cs, and also c = —C4. However, c need not equal C2 as these atoms are differently placed with respect to the nodal plane. To determine these coefficients, we will use the fact that the neutral ground-state tt densities are all unity in this system. We consider first atom number 1. Its electron density due to two electrons in 0i and two electrons in 02 is... 

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