Atoms with More Than Three Electrons

Atoms with More Than Three Electrons 

The treatment of other atoms in zero order is analogous to the helium and lithium treatments. For an atom with atomic number Z (Z protons in the nucleus and Z electrons), the stationary-nucleus Hamiltonian operator is 

The first two sums in Eq. (18.6-9) are a sum of hydrogen-like one-electron Hamiltonian operators. In zero order, we ignore the double sum representing the electron-electron repulsions and obtain the Hamiltonian operator  

The time-independent Schrodinger equation corresponding to the zero-order Hamiltonian can be solved by separation of variables, using the trial function 

We must antisymmetrize the orbital wave function of Eq. (18.6-11). This can be done by writing a Slater determinant with one row for each spin orbital and one column for each electron  

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Atoms with More Than Three Electrons... 

In many crystalline salts of these elements values of p are observed that are close to those for the aqueous ions some of these are given in Table 5-2. For central atoms with more than three 3d electrons... 

This is the simplest case. It is usually sufficient to have the valence orbitals active, perhaps with added Rydberg type orbitals for studies of excited states. One can normally leave the ns orbital inactive for main group atoms with more than three np electrons. First row transition metals are, however, more demanding. It has been shown that in order to be able to accurately describe the relative correlation effects in atomic states, which differ in the number of 3d electrons, one needs to use two sets of d-orbitals, 3d and 3d where the second set describes the strong radial correlation effects in the 3d shell [29]. Adding the 4s and 4p orbital one is faced with an active space of 14 orbitals. The importance of the second 3d orbital decreases for second and, in particular, for third row transition metals. 

Calculate the third ionization potential of the lithium atom. Solution. The lithium atom is composed of a nucleus of charge +3(Z = 3) and three electrons. The first ionization potential IPi of an atom with more than one electron is the energy required to remove one electron for lithium,... 

There are three new characteristics required to explain the behavior of atoms with more than one electron (all elements except hydrogen) a fourth quantum number (mj specifies electron spin an orbital holds no more than two electrons (exclusion principle) and interactions between electrons and between nucleus and electrons (shielding and penetration) cause energy levels to split into sublevels of different energy. (Section 8.1)... 

We have so far considered valence shells containing four pairs of electrons, but we can extend the same arguments to other numbers of valence shell electron pairs. The most probable arrangements of pairs of opposite spin electrons in the valence shell of an atom in a molecule are two pairs, collinear three pairs, equilateral triangular four pairs, tetrahedral five pairs, trigonal bipyramidal six pairs, octahedral. This is because, as we will now see, these are the arrangements that keep the electron pairs as far apart as possible. We discuss valence shells with more than six electron pairs in Chapter 9. 

Double and Triple Bonds Sometimes an atom shares more than one electron with another atom. In the molecule carbon dioxide, shown in Figure 17, each of the oxygen atoms shares two electrons with the carbon atom. The carbon atom shares two of its electrons with each oxygen atom. When two pairs of electrons are involved in a covalent bond, the bond is called a double bond. Figure 17 also shows the sharing of three pairs of electrons between two nitrogen atoms in the nitrogen molecule. When three pairs of electrons are shared by two atoms, the bond is called a triple bond. 

However, although the Bohr theory, involving a single quantum number n, was adequate to explain the line spectrum of the hydrogen atom with a single valence electron (Figures 2.4 and 2.5, respectively), it was inadequate to explain, in detail, the line spectrum of elements with more than one electron. To do this, it was found necessary to introduce the idea of three further quantum numbers, in addition to the principal quantum number, n. These arise from the wave nature of the electron. 

Ion cyclotron resonance (ICR) and flowing afterglow experiments can also be used to derive relative affinities. Neutral beam experiments, where a beam of alkali atoms such as Cs is crossed with a beam of molecules such as PCI3 or (012)2 have been used to derive thermochemistry for anions such as PCli" and Cli", but proper analysis of this type of data is difficult. High-resolution negative ion photoelectron spectroscopy (NIPES) experiments can provide otherwise unobtainable information on hypervalent anions, including precise electron affinities and vibrational frequencies.This technique has limited applicability to hypervalent species with more than three atoms because of vibrational congestion from low-frequency modes. 

Rigorous descriptions of nonadiabatic transitions in molecules with more than three atoms are, for several reasons, much more complicated than for triatoms. First, the dimensionality is considerably higher and therefore more energy points are required to construct global PESs. As an unavoidable consequence, the accuracy of the electronic structure calculations has to be lower. Second, the dynamical calculations become much more demanding. Exact quantum mechanical calculations are possible with more than three degrees of freedom, as has been demonstrated by Hammerich... 

The Schrodinger equation (introduced in Chapter 7) does not give exact solutions for the energy levels of many-electron atoms, those with more than one electron—that is, all atoms except hydrogen. However, unlike the Bohr model, it gives excellent approximate solutions. Three additional features become important in many-electron atoms (1) a fourth quantum number, (2) the number of electrons that can occupy an orbital, and (3) a splitting of energy levels into sublevels. 

Evidence has been advanced8 that the neutral helium molecule which gives rise to the helium bands is formed from one normal and one excited helium atom. Excitation of one atom leaves an unpaired Is electron which can then interact with the pair of Is electrons of the other atom to form a three-electron bond. The outer electron will not contribute very much to the bond forces, and will occupy any one of a large number of approximately hydrogen-like states, giving rise to a roughly hydrogenlike spectrum. The small influence of the outer electron is shown by the variation of the equilibrium intemuclear distance within only the narrow limits 1.05-1.13 A. for all of the more than 25 known states of the helium molecule. 

The concept of an octet of electrons is one of the foundations of chemical bonding. In fact, C, N, and O, the three elements that occur most frequently in organic and biological molecules, rarely stray from the pattern of octets. Nevertheless, an octet of electrons does not guarantee that an inner atom is in its most stable configuration. In particular, elements that occupy the third and higher rows of the periodic table and have more than four valence electrons may be most stable with more than an octet of electrons. Atoms of these elements have valence d orbitals, which allow them to accommodate more than eight electrons. In the third row, phosphoms, with five valence electrons, can form as many as five bonds. Sulfur, with six valence electrons, can form six bonds, and chlorine, with seven valence electrons, can form as many as seven bonds. 

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