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Lecture 8-9: Multi-electron atoms

- Alkali atom spectra.
- Central field approximation.
- Shell model.
- Effective potentials and screening.
- Experimental evidence for shell model.

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Energy levels in alkali metals

- Alkali atoms: in ground state, contain a set of completely filled subshells with a single valence electron in the next s subshell.
- Electrons inpsubshells are not excited in any low-energy processes. selectron is the single optically active electron and core of filled subshells can be ignored.

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Energy levels in alkali metals

- In alkali atoms, the l degeneracy is lifted: states with the same principal quantum number n and different orbital quantum number l have different energies.
- Relative to H-atom terms, alkali terms lie at lower energies. This shift increases the smaller l is.
- For larger values of n, i.e., greater orbital radii, the terms are only slightly different from hydrogen.
- Also, electrons with small l are more strongly bound and their terms lie at lower energies.
- These effects become stronger with increasing Z.
- Non-Coulombic potential breaks degeneracy of levels with the same principal quantum number.

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Hartree theory

- For multi-electron atom, must consider Coulomb interactions between its Z electrons and its nucleus of charge +Ze. Largest effects due to large nuclear charge.
- Must also consider Coulomb interactions between each electron and all other electrons in atom. Effect is weak.
- Assume electrons are moving independently in a spherically symmetric net potential.
- The net potential is the sum of the spherically symmetric attractive Coulomb potential due to the nucleus and a spherically symmetric repulsive Coulomb potential which represents the average effect of the electrons and its Z - 1 colleagues.
- Hartree (1928) attempted to solve the time-independent Schrödinger equation for Z electrons in a net potential.
- Total potential of the atom can be written as the sum of a set of Z identical net potentials V( r), each depending on r of the electron only.

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Screening

- Hartree theory results in a shell model of atomic structure, which includes the concept of screening.
- For example, alkali atom can be modelled as having a valence electron at a large distance from nucleus.
- Moves in an electrostatic field of nucleus +Ze which is screened by the (Z-1)inner electrons. This is described by the effective potential Veff( r ).
- At r small, Veff(r ) ~ -Ze2/r
- Unscreened nuclear Coulomb potential.
- At r large, Veff(r ) ~ -e2/r
- Nuclear charge is screened to one unit of charge.

-e

r

+Ze

-(Z-1)e

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Central field approximation

- The Hamiltonian for an N-electron atom with nuclear charge +Ze can be written:

where N = Z for a neutral atom. First summation accounts for kinetic energy of electrons , second their Coulomb interaction with the nuclues, third accounts for electron-electron repulsion.

- Not possible to find exact solution to Schrodinger equation using this Hamiltonian.
- Must use the central field approximation in which we write the Hamiltonian as:

where Vcentralis the central field and Vresidual is the residual electrostatic interaction.

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Central field approximation

- The central field approximation work in the limit where
- In this case, Vresidual can be treated as a perturbation and solved later.
- By writing we end up with N separate Schrödinger equations:

with E = E1 + E2 + … + EN

- Normally solved numerically, but analytic solutions can be found using the separation of variables technique.

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Central field approximation

- As potentials only depend on radial coordinate, can use separation of variables:

where Ri(ri) are a set of radial wave functions and Yi(i, i ) are a set of spherical harmonic functions.

- Following the same procedure as Lectures 3-4, we end up with three equations, one for each polar coordinate.
- Each electron will therefore have four quantum numbers:
- l and ml: result from angular equations.
- n: arises from solving radial equation. n and l determine the radial wave function Rnl(r ) and the energy of the electron.
- ms: Electron can either have spin up (ms= +1/2)or down (ms= -1/2).
- State of multi-electron atom is then found by working out the wave functions of the individual electrons and then finding the total energy of the atom (E = E1 + E2 + … + EN).

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Shell model

- Hartree theory predicts shell model structure, which only considers gross structure:
- States are specified by four quantum numbers, n, l, ml, and ms.
- Gross structure of spectrum is determined by n and l.
- Each (n,l) term of the gross structure contains 2(2l + 1) degenerate levels.
- Shell model assumes that we can order energies of gross terms in a multielectron atom according to n and l. As electrons are added, electrons fill up the lowest available shell first.
- Experimental evidence for shell model proves that central approximation is appropriate.

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Shell model

- Periodic table can be built up using this shell-filling process. Electronic configuration of first 11 elements is listed below:
- Must apply
- Pauli exclusion principle: Only two electrons with opposite spin can occupy an atomic orbital. i.e., no two electrons have the same 4 quantum numbers.
- Hunds rule: Electrons fill each orbital in the subshell before pairing up with opposite spins.

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Shell model

- Below are atomic shells listed in order of increasing energy. Nshell = 2(2l + 1) is the number of electrons that can fill a shell due to the degeneracy of the mland mslevels. Naccumis the accumulated number of electrons that can be held by atom.
- Note, 19th electron occupies 4s shell rather than 3d shell. Same for 37th. Happens because energy of shell with large l may be higher than shell with higher n and lower l.

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Shell model

- 4s level has lower energy than 3d level due to penetration.
- Electron in 3s orbital has a probability of being found close to nucleus. Therefore experiences unshielded potential of nucleus and is more tightly bound.

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Shell model

Radial probabilities for 4s 3d

4s - red

3d - blue

Note: Movies from http://chemlinks.beloit.edu/Stars/pages/radial.htm

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Quantum defect

- Alkali are approximately one-electron atoms: filled inner shells and one valence electron.
- Consider sodium atom: 1s2 2s2 2p6 3s1.
- Optical spectra are determined by outermost 3s electron. The energy of each (n, l) term of the valence electron is

where (l) is the quantum defect - allows for penetration of the inner shells by the valence electron.

- Shaded region in figure near r = 0 represent the inner n = 1 and n = 2 shells. 3s and 3p penetrate the inner shells.
- Much larger penetration for 3s => electron sees large nuclear potential => lower energy.

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Quantum defect

- (l) depends mainly on l. Values for sodium are shown at right.
- Can therefore estimate wavelength of a transition via
- For sodium the D lines are 3p 3s transitions. Using values for (l) from table,

=> = 589 nm

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Shell model justification

- Consider sodium, which has 11 electrons.
- Nucleus has a charge of +11e with 11 electrons orbiting about it.
- From Bohr model, radii and energies of the electrons in their shells are
- First two electrons occupy n =1 shell. These electrons see full charge of +11e. => r1 = 12/11 a0 = 0.05 Å and E1 = -13.6 x 112/12 ~ -1650 eV.
- Next two electrons experience screened potential by two electrons in n = 1 shell. Zeff =+9e => r2 = 22/9 a0 = 0.24 Å and E2 = -13.6 x 92/22 = -275 eV.

and

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Experimental evidence for shell model

- Ionisation potentials and atomic radii:
- Ionisation potentials of noble gas elements are highest within a particular period of periodic table, while those of the alkali are lowest.
- Ionisation potential gradually increases until shell is filled and then drops.
- Filled shells are most stable and valence electrons occupy larger, less tightly bound orbits.
- Noble gas atoms require large amount of energy to liberate their outermost electrons, whereas outer shell electrons of alkali metals can be easily liberated.

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Experimental evidence for shell model

- X-ray spectra:
- Enables energies of inner shells to be determined.
- Accelerated electrons used to eject core electrons from inner shells. X-ray photon emitted by electrons from higher shell filling lower shell.
- K-shell (n = 1), L-shell (n = 2), etc.
- Emission lines are caused by radiative transitions after the electron beam ejects an inner shell electron.
- Higher electron energies excite inner shell transitions.

80 keV

40 keV

Wavelength (A)

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Experimental evidence for shell model

- Wavelength of various series of emission lines are found to obey Moseley’s law.
- For example, the K-shell lines are given by

where accounts for the screening effect of other electrons.

- Similarly, the L-shell spectra obey:
- Same wavelength as predicted by Bohr, but now have

and effective charge (Z - ) instead of Z.

- L ~ 10 and K ~ 3.

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Bohr model including screening

- Assume net charge is ( Z - 1 )e.
- Therefore, the potential energy is
- Total energy of orbit is
- Modified Bohr formula taking into account screening.
- Can therefore easily show that

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Shell model summary

- Electrons in orbitals with large principal quantum numbers (n) will be shielded from the nucleus by inner-shell electrons.

Zeff = Z - nl.

- nl increases with n => Zeffdecreases with n.
- nl increases with l => Zeffdecreases with l.

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Shell model summary

- In hydrogenic one-electron model, the energy levels of a given n are degenerate in l:
- Not the case in multi-electron atoms. Orbitals with the same n quantum number have different energies for differing values of l.
- As Zeff = Z - nl is a function of n and l, the l degeneracy is broken by modified potential.

3s

3p

3d

3s

3p

3d

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Shell model summary

- Wave functions of electrons with different l are found to have different amount of penetration into the region occupied by the 1s electrons.
- This penetration of the shielding 1s electrons exposes them to more of the influence of the nucleus and causes them to be more tightly bound, lowering their associated energy states.

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Shell model summary

- In the case of Li, the 2s electron shows more penetration inside the first Bohr radius and is therefore lower than the 2p.
- In the case of Na with two filled shells, the 3s electron penetrates the inner shielding shells more than the 3p and is significantly lower in energy.

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