KI 2141 - Struktur dan Ikatan Kimia

Achmad Rochliadi, Ph.D.Program Studi Kimia

Institut Teknologi Bandung The Structure of Many-electron

Atoms. Personal Used Only .

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The orbital approximation

Interaction in many-electron atoms : (eg. 2 electron atoms)– electron-1 vs nuclei– electron-2 vs nuclei– electron-1 vs electron-2

No analytical expression of many-electron atom for the orbital wavefunction and energy can be given.

Schrodinger equation CANNOT be solved exactly formany-electron atoms.

Approximation and numerical computation is use to solved this.

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Justification of orbital approximation

The wavefunction of many-electron atom is a complicated function of the coordinate of all the electron

An orbital approximation is made by assuming that each electron is occupying its own orbital

Each individual orbital resemble the hydrogenic orbital. The corresponding nuclear charge is modified due to the presence of other electrons.

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Justification of orbital approximation

The orbital approximation will be exact if there are no interaction between electrons.

The complete Hamiltonian, with electron interaction.

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The helium atom

The orbital approximation express the electronic structure of an atom by configuration, a statement of its occupied orbitals. A ground state of a hydrogenic atom consists of the single electron in a 1s orbital, and its report as 1s1

A He atom has two electrons. The forming the atom by adding the electrons in succession to the orbitals of the bare nucleus (of charge 2e-). The first electron occupies a 1s hydrogenic orbital.

He have Z=2 so the orbital is more compact than in H itself. The second electron joins the first in the 1s orbital, so the electron configuration of the ground state of He is 1s2

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The lithium atom

Lithium have Z = 3 and 3 electron. The first 2 electron occupied 1s orbital.

Pauli exclusion principleNo more than two electrons may occupy any given orbital, and if two do occupy one orbital, then their spins must be paired.

Electrons with paired spins, denoted ↑↓, have zero net spin angular momentum.

By ‘total wavefunction’ including the spin of the particles for two electrons ψ(1,2).

The Pauli principle implies that it is a fact of nature that for fermion the wavefunction must change sign if we interchange the labels 1 and 2 wherever they occur in the function:

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The lithium atom

Possibilites of spin of the electron α = spin up, β = spin down, (1) = electron 1, (2) = electron 2

Spin states as normalized linear combination.

The total wavefunction of the system is therefore the product of the orbital part and one of the four spin states:

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pau

The Pauli principle : A wavefunction to be acceptable (for electrons), it must change sign when the electrons are exchanged. factor ψ(1)ψ(2) into ψ(2)ψ(1), which is the same, because the order of multiplying the functions does not change the value of the product. The same is true of α(1)α(2) and β(1)β(2). Therefore, the first two overall products are not allowed, because they do not change sign.

The acceptable product wavefunction ψ(1)ψ(2)σ−(1,2) can be expressed as a determinant:

Survive Pauli

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Any acceptable wavefunction for a closed-shell species can be expressed as a Slater determinant, as such determinants are known. In general, for N electrons in orbitals ψa, ψb

Writing simplified Slater :

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For lithium. In Li (Z=3), the third electron cannot enter the 1s orbital because that orbital is already full: we say the K shell is complete and that the two electrons form a closed shell. Because a similar closed shell is characteristic of the He atom, we denote it [He]. The third electron is excluded from the K shell and must occupy the next available orbital, which is one with n=2 and hence belonging to the L shell.

Question is which available orbital (2s orbital or a 2p orbital) is going to be used for the 3rd electron [He]2s1or [He]2p1

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Penetrating and shielding

Due to coulombic repultion between electron the 2s and 2p orbitals in many-electron atom are not degenerate.

The effect coulombic, when averaged over all the locations of the electron, is to reduce the full charge of the nucleus from Ze to Zeff e, the effective nuclear charge.

We say that the electron experiences a shielded nuclear charge, and the difference between Zand Zeff is called the shielding constant,σ:

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Energies of subshells : s < p < d < f(for same l)

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The building up principles

The Aufbau principles, (German for building up), the rules of occupation for electron in the orbital of atoms. The order of occupation is

Each orbital may accomodate up to 2 electrons.

For carbon :

Electrons occupy different orbitals of a given subshell before doubly occupying any one of them

Hund’s maximum multiplicity rule:An atom in its ground state adopts a configuration with the greatest number of unpaired electrons

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Strong electron-electron repulsion in 3dOrbital of Sc

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Self-consistent field orbitals

The Schrödinger equation for many-electron atoms is solved numerically and iteratively until the solutions are self-consistent.

The interaction of electron-electron repultion make the Shrodinger equation hard to be solved. The potential energy for the electrons is. Personal Used Only .

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HF-SCF

Hartree-Fock Self-consistent fieldFor example to solve 1s2 2s2 2p6. For 1 electron in p

orbital. The schrodinger equation has the form of

● The first term on the left is the contribution of the kinetic energy and the attraction of the electron to the nucleus, just as in a hydrogenic atom.

● The second term takes into account the potential energy of the electron of interest due to the electrons in the other occupied orbitals.

● The third term is an exchange correctionthat takes into account the spin correlation effects discussed earlier.

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The radial distribution functions for the orbitals of Na based on SCF calculations. Note the shell-like structure, with the 3s orbital outside the inner K and L shells.

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