Electrons occupying orbitals

Section 1 1 A review of some fundamental knowledge about atoms and electrons leads to a discussion of wave functions, orbitals, and the electron con figurations of atoms Neutral atoms have as many electrons as the num ber of protons m the nucleus These electrons occupy orbitals m order of increasing energy with no more than two electrons m any one orbital The most frequently encountered atomic orbitals m this text are s orbitals (spherically symmetrical) and p orbitals ( dumbbell shaped)... 

Electrons occupy orbitals in such a way as to minimize the total energy of an atom by maximizing attractions and minimizing repulsions in accord with the Pauli exclusion principle and Hund s rule. 

In molecular orbital theory, electrons occupy orbitals called molecular orbitals that spread throughout the entire molecule. In other words, whereas in the Lewis and valence-bond models of molecular structure the electrons are localized on atoms or between pairs of atoms, in molecular orbital theory all valence electrons are delocalized over the whole molecule, not confined to individual bonds. 

In Fig. 1 there is indicated the division of the nine outer orbitals into these two classes. It is assumed that electrons occupying orbitals of the first class (weak interatomic interactions) in an atom tend to remain unpaired (Hund s rule of maximum multiplicity), and that electrons occupying orbitals of the second class pair with similar electrons of adjacent atoms. Let us call these orbitals atomic orbitals and bond orbitals, respectively. In copper all of the atomic orbitals are occupied by pairs. In nickel, with ou = 0.61, there are 0.61 unpaired electrons in atomic orbitals, and in cobalt 1.71. (The deviation from unity of the difference between the values for cobalt and nickel may be the result of experimental error in the cobalt value, which is uncertain because of the magnetic hardness of this element.) This indicates that the energy diagram of Fig. 1 does not change very much from metal to metal. Substantiation of this is provided by the values of cra for copper-nickel alloys,12 which decrease linearly with mole fraction of copper from mole fraction 0.6 of copper, and by the related values for zinc-nickel and other alloys.13 The value a a = 2.61 would accordingly be expected for iron, if there were 2.61 or more d orbitals in the atomic orbital class. We conclude from the observed value [Pg.347]

Arrangements 2 and 3 look spatially equivalent, but experiments show that a configuration that gives unpaired electrons the same spin orientation is always more stable than one that gives them opposite orientations. Hund s rule summarizes the way in which electrons occupy orbitals of equal energies. 

If the electrons occupy orbitals different from the most stable (ground) electronic state, the bonding between the atoms also changes. Therefore, an entirely different potential energy surface is produced for each new electronic configuration. This is illustrated in Figure 6.6 for a diatomic molecule. 

In the SCF approach, each electron is assumed to move in an average field due to nuclei and the remaining electrons. Consequently, electronic repulsions are formally considered in the procedure. For example, consider a hypothetical atom (nuclear charge Z) containing two electrons as shown in Fig. 5. The electrons occupy orbitals (f>i and (j>2, with corresponding electron densities of —e and —e4>l, respectively. According to the SCF approach, the potential seen by electron 2 would be... 

The Aufbau or build-up principle states that electrons occupy orbitals of progressively increasing energies. 

In an excited configuration one or more electrons occupy orbitals other than the lowest ones available. In a singly excited configuration only one electron... 

Diene n,w triplets have much n bonding character between carbons 2 and 3 and, in acyclic compounds, are probably twisted at the ends, so that their two unpaired electrons occupy orbitals that are either orthogonal to each other or close to it. 

The second derivative (92E/dnidtij)0 is the energy of the Coulomb interaction between two electrons occupying orbitals i and and j, when taking the screening from all other electrons into account we designate this quantity as Uq. Choosing... 

An atom, then, is in a valence state when its electrons occupy orbitals of energies and shapes that they would occupy if they were subject to the interactions that they would experience in some molecule thus one could speak of the valence state of carbon in CH4 (above). Clearly a valence state is an abstract concept. 

Each quantum mechanical operator is related to one physical property. The Hamiltonian operator is associated with energy and allows the energy of an electron occupying orbital cp to be calculated [Equation (2.3)]. We will never need to perform such a calculation. In fact, in perturbation theory and the Hiickel method, the mathematical expressions of the various operators are never given and calculations cannot be done. Any expression containing an operator is treated merely as an empirical parameter. 

As the temperature rises, the probability of the sixth electron occupying orbital energy levels above the lowest level increases, and so too does the Sel of Fe2+ located in a distorted site, eq. (2.27). However, when Fe2+ is present in a regular octahedron, its sixth 3d electron remains delocalized over the three equivalent orbitals so that Sel remains constant at 9.13 J/(deg. g.ion). Therefore,... 

Orbital diagrams assign electrons to individual orbitals so the energy state of individual electrons may be found. This requires knowledge of how electrons occupy orbitals within a subshell. Hund s rule states that before any two electrons occupy the same orbital, other orbitals in that subshell must first contain one electron each with parallel spins. Electrons with up and down spins are shown by half-arrows, and these are placed in lines of orbitals (represented as boxes or dashes) according to Hund s rule, the Aufbau principle, and the Pauli exclusion principle. Below is the orbital diagram for vanadium ... 

Closely related to the Pauli exclusion principle is the third rule, Hund s rule, which states that when electrons occupy orbitals of equal energy (e.g., the five 3d orbitals), one electron enters each orbital until all the orbitals contain one electron. In this configuration, all electrons will have parallel spin (same direction). Second electrons then add to each orbital so that their spins are opposite to the first electrons in the orbital. Atoms with all outer orbitals half-filled are very stable. 

Fig. 5. The feasibility erf PET is dictated by the relative energies of donor and acceptor electrons. In this simplified orbital picture, the electrons occupying orbitals can move to other orbitals depending on their ionization potentials (IP) and electron affinities (EA). IP and EA are related to orbital energies by Koopman s theorem...

For the first term of (7.95) the sum rule (3.89) gives 1. The second term is nonzero only for 7Z = v, in which case we rearrange the left-hand Clebsch—Gordan coefficient and again use the sum rule to obtain (—l) jiv if J is even. Odd J is forbidden for electrons occupying orbitals with identical radial parts, i.e. in the same shell. The normalisation factor is... 

These are known as the Roothaan equations. They represent an algebraic equivalent to the Hartree-Fock equations. The approximate eigenvalues represent orbital energies. By Koopmans theorem, — approximates the ionization energy for an electron occupying orbital a. The orbital energies can be determined directly Ifom the n roots of the secular equation... 

In molecules, as in isolated atoms, electrons occupy orbitals, and in accordance with much the same rules. These molecular orbitals are considered to be centered about many nuclei, perhaps covering the entire molecule the distribution of nuclei and electrons is simply the one thkt results in the most stable molecule. 

Fig. 5.20 Energy level diagram for the molecular orbitals of carbon monoxide. Note that upon bond formation electrons occupy orbitals that are more oxygen-like than carbon-like. Note carefully the bond order The Icr and la MOs are essentially nonbonding. The bond order, as in the N2 molecule is three.

Atoms are formed from protons, neutrons and electrons. The protons and neutrons together constitute the nucleus while the electrons occupy orbitals,... 

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