Atoms many electron

3 Many-electron atoms electronic configuration and spectroscopic terms 

For an atom with n electrons, the Schrodinger equation has the form (in a.u.) 

This equation cannot be solved exactly. The most often used approximation model to solve this equation is called the self-consistent field (SCF) method, first introduced by D. R. Hartree and V. A. Fock. The physical picture of this method is very similar to our treatment of helium each electron sees an effective nuclear charge contributed by the nuclear charge and the remaining electrons. 

The radial distribution functions of 3s, 3p, and 3d orbitals together with that of the sodium core (ls22s22p6). 

In an atom with more than one electron, each electron (or orbital) is characterized by the following quantum numbers  

In an atom containing more than one electron, each electron can be considered as having its own set of values of n, l, mi and ms- An extremely important rule, known 

The first and fourth of these combinations have a definite symmetry when electrons 1 

The first combination is symmetric, and the second is antisymmetric with respect to permutation. In the ground state of the helium atom both electrons occupy the same spatial orbital (Is), so that they must have the antisymmetric spin function for the total wave function to be antisymmetric they therefore form a singlet spin state. In order to find a helium atom with a triplet spin state (so-called spins parallel), the spatial part of the wave function must be antisymmetric with respect to interchange. 

Whilst we are discussing the Pauli principle, it is worthwhile to introduce a further method of expressing the antisymmetric nature of the electronic wave function, namely, the Slater determinant [1], Since both electrons in the ground state of the helium atom occupy the same space orbital, the wave function may be written in the form 

This may be expressed in a more abbreviated form by introducing the spin-orbital //((.( I ), with a = a or 0, so that (6.18) can be written in the form 

Write down a set of quantum numbers that describes an electron in a 5 atomic orbital. How does this set of quantum numbers differ if you are describing the second electron in the same orbital  

The notation for the gromd state electronic configuration of an H atom is, signifying that one electron occupies the Is atomic 

Worked example 1.4 Quantum numbers an electron in an atomic orbital 

Write down two possible sets of quantum numbers that describe an electron in a 2s atomic orbital. What is the physical significance of these unique sets  

An electron in a 2s atomic orbital may have one of two sets of four quantum numbers  

The set of quantum numbers in many-electron atoms has to be completed by the spin quantum number 

Then the one-electron wave function characterised by the four quantum numbers is termed the atomic spin-orbital 

Because of the vector nature of the angular momenta, addition of angular momenta in atoms is feasible. The total orbital angular momentum and its projection are expressed through the sums 

The same algebra holds true for the total spin angular momentum S, its projection Sz and the corresponding quantum numbers S and Ms. 

On the set of atomic spin-orbitals various configuration functions 

Like the Bohr model, the Schrodinger equation does not give exact solutions for many-electron atoms. However, unlike the Bohr model, the Schrodinger equation gives very good approximate solutions. These solutions show that the atomic orbitals of many-electron atoms resemble those of the H atom, which means we can use the same quantum numbers that we used for the H atom to describe the orbitals of other atoms. 

Nevertheless, the existence of more than one electron in an atom requires us to consider three features that were not relevant in the case of hydrogen (1) the need for a fourth quantum number, (2) a limit on the number of electrons allowed in a given orbital, and (3) a more complex set of orbital energy levels. Let s examine these new features and then go on to determine the electron configuration for each element. 

One of our goals in this chapter has been to determine the electronic structures of atoms. So far, we have seen that quantum mechanics leads to an elegant description of the hydrogen atom. This atom, however, has only one electron. How does our description change when we consider an atom with two or more electrons (a many-electron atom) To describe such an atom, we must consider the nature of orbitals and their relative energies as well as how the electrons populate the available orbitals. 

Not all of the orbitals in the n = shell are shown in this figure. Which subshells are missing  

Nr Orbitals in a subshdl are degenerate (have same energy) 

A FIGURE 6.24 General energy ordering of orbitals for a many-electron atom. 

Relativistic Quantum Chemistry. Markus Reiher and Alexander Wolf 

In the context of numerical solution methods, we should also mention the special case of linear molecules for which fully numerical methods are feasible 

After this short (and unavoidably selective) overview and introduction we shall now work out the theory of relativistic atomic structure calculations. 

Transformation of the Many-Electron Hamiltonian to Poiar Coordinates 

We have seen in chapter 8 that the many-electron Dirac-Coulomb(-Breit) Hamiltonian can be written as 

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The true value of tk for a many-electron atom or a molecule is unknown. If we could set it equal ( expand it) to a linear combination of an infinite number of basis functions, each defined in a space of infinite dimensions, we could carry out an exact calculation of (k. Such a set of basis functions would be a complete set. 

Petersson and coworkers have extended this two-electron formulation of asymptotic convergence to many-electron atoms. They note that the second-order MoUer-Plesset correlation energy for a many-electron system may be written as a sum of pair energies, each describing the energetic effect of the electron correlation between that pair of electrons ... 

You are probably used to this idea from descriptive chemistry, where we build up the configurations for many-electron atoms in terms of atomic wavefunctions, and where we would write an electronic configuration for Ne as... 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

For the following pairs of orbitals, indicate which is lower in energy in a many-electron atom. 

Analysis of the spectra of many-electron atoms shows the following similarities to the hydrogen atom case. 

Figure 15-11 shows a schematic energy level diagram of a many-electron atom. Blue patterns... 

It is possible to remove two or more electrons from a many-electron atom. Of course it is always harder to remove the second electron than the first because the second electron to come off leaves an ion with a double positive charge instead of a single positive charge. This gives an additional electrical attraction. Even so, the values of successive ionization energies have great interest to the chemist. 

Turn back to Figure 15-11, the energy level diagram of a many-electron atom, and consider the occupied orbitals of the element potassium. With 19 electrons placed, two at a time, in the orbitals of lowest energy, the electron configuration is... 

These days students are presented with the four quantum number description of electrons in many-electron atoms as though these quantum numbers somehow drop out of quantum mechanics in a seamless manner. In fact, they do not and furthermore they emerged, one at a time, beginning with Bohr s use of just one quantum number and culminating with Pauli s introduction of the fourth quantum number and his associated Exclusion Principle. 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

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. 

This problem clearly did not worry Stoner, who just went ahead and assumed that three quantum numbers could be specified in many-electron atoms. In any case, Stoner s scheme solved certain problems present in Bohr s configurations. For example, Bohr had assigned phosphorus the configuration 2,4,4,41, but this failed to explain the fact that phosphorus shows valencies of three and five. Stoner s configuration for phosphorus was 2,2,2,4,2,2,1, which easily explains the valencies, since it becomes plausible that either the two or the three outermost subshells of electrons form bonds. 

In many electron atoms the maximum contributions to the polarizability and to London forces arise from configurations with more than one electron contributing to the net dipole moment of the atom. But in such configurations the electronic repulsion is especially high. The physical meaning to be attributed to the Qkl terms is just the additional electron repulsive energy which these configurations require. 

Boys, S. F., Proc. Roy. Soc. London) A207, 181, Electronic wave functions. IV. Some general theorems for the calculation of Schrodinger integrals between complicated vector-coupled functions for many-electron atoms."... 

A many-electron atom is also called a polyelectron atom. 

The next step in our journey takes us from hydrogen and its single electron to the atoms of all the other elements in the periodic table. A neutral atom other than a hydrogen atom has more than one electron and is known as a many-electron atom. In the next three sections, we build on what we have learned about the hydrogen atom to see how the presence of more than one electron affects the energies of... 

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... 

FIGURE 1.41 The relative energies of the shells, subshells, and orbitals in a many-electron atom. Each of the boxes can hold at most two electrons. Note the change in the order of energies of the 3d- and 4s-orbitals after Z = 20. 

As well as being attracted to the nucleus, each electron in a many-electron atom is repelled by the other electrons present. As a result, it is less tightly bound to the nucleus than it would be if those other electrons were absent. We say that each electron is shielded from the full attraction of the nucleus by the other electrons in the atom. The shielding effectively reduces the pull of the nucleus on an electron. The effective nuclear charge, Z lle, experienced by the electron is always less than the actual nuclear charge, Ze, because the electron-electron repulsions work against the pull of the nucleus. A very approximate form of the energy of an electron in a many-electron atom is a version of Eq. 14b in which the true atomic number is replaced by the effective atomic number ... 

In a many-electron atom, because of the effects of penetration and shielding, the order of energies of orbitals in a given shell is s < p < d < f. 

Describe the factors affecting the energy of an electron in a many-electron atom (Section l.f2). 

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