Lecture 20 helium and heavier atoms
This presentation is the property of its rightful owner.
Sponsored Links
1 / 23

Lecture 20 Helium and heavier atoms PowerPoint PPT Presentation


  • 32 Views
  • Uploaded on
  • Presentation posted in: General

Lecture 20 Helium and heavier atoms.

Download Presentation

Lecture 20 Helium and heavier atoms

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Lecture 20 helium and heavier atoms

Lecture 20Helium and heavier atoms

(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.


Helium and heavier atoms

Helium and heavier atoms

  • We use the exactsolutions of hydrogenic Schrödinger equation or orbitals to construct an approximate wave function of a many-electron atom, the helium and heavier atoms.

  • Unlike the hydrogenic atom, the discussion here is approximate and some rules introduced can have exceptions.

  • Spins and antisymmetry of fermion wave functions start to play a critical role.


Helium and heavier atoms1

Helium and heavier atoms

  • The Schrödinger equation for hydrogenic atoms can be solved exactly, analytically. Those for many-electron atoms and molecules cannot be solved analytically.

  • The wave function is a coupled function of many variables:

Coordinates of electron 1


The orbital approximation

The orbital approximation

  • We introduce the following approximation (the orbital approximation):

  • For the helium atom, this amounts to

Hydrogenic orbital


The orbital approximation1

The orbital approximation

  • The approximation is equivalent to neglecting interaction between electrons 1 and 2,

  • … so that,

Interaction

Hydrogenic electron 1

Hydrogenic electron 2


The orbital approximation2

The orbital approximation

exact hydrogenic problem

Eigenfunction


The orbital approximation3

The orbital approximation

  • We construct a helium wave function as the product of hydrogenicorbitals with Z = 2.

  • Issue #1: an electron is fermion and fermions’ wave function must be antisymmetric with respect to interchange (the above isn’t):

  • Issue #2: each electron must be either spin α or β(the above neglects spins).


Spins

Spins

  • Let us first append spin factors

  • None of these is antisymmetric yet


Anti symmetrization

(Anti)symmetrization

  • Symmetrization:

  • Antisymmetrization:

Sym.

Antisym.


Antisymmetric function

Antisymmetric function

Antisym.

Sym.

Antisym.


Helium wave functions

Helium wave functions

Already symmetric and cannot be made antisymmetric

Neither sym. or antisym.

Antisym.

Sym.

Antisym.

Sym.

Neither sym. or antisym.

Sym.

Antisym.


Triplet states

Triplet states

  • These three have the same spatial shape – the same probability density and energy – triply degenerate (triplet states)

Antisym.

Sym.

φ1 and φ2 cannot have the same spatial form (otherwise this part becomes zero). Electrons 1 and 2 cannot be in the same orbital or same spatial position in triplet states (cf. Pauli exclusion principle)


Singlet state

Singlet state

  • There is another state which is non-degenerate (singlet state):

Sym.

Antisym.

Opposite spins

φ1 and φ2 can have the same spatial form because the anti-symmetry is ensured by the spin part. Electrons 1 and 2 can be found at the same spatial position.


Energy ordering

Energy ordering

  • For the helium atom, depicting α-and β-spin electrons by upward and downward arrows, we can specify its electron configurations.

2s

2s

2s

1s

1s

1s

Singlet state B

Singlet state

A

Triplet states


The orbital approximation4

The orbital approximation

Interaction

Hydrogenic electron 1

Hydrogenic electron 2

2s

2s

1s

1s

2s

Singlet A

Triplet states

166277 cm-1

1s

159856 cm-1

0 cm-1

Singlet B


Beyond helium

Beyond helium …

  • A many-electron atom’s ground-state configuration can be obtained by filling two electrons (α and β spin) in each of the corresponding hydrogenicorbitals from below.

  • When a shell (K, L, M, etc.) is completely filled, the atom becomes a closed shell – a chemically inert species like rare gas species.

  • Electrons partially filling the outermost shell are chemically active valence electrons.


Shielding

Shielding

  • In a hydrogenic atom (with only one electron), s, p, d orbitals in the same shell are degenerate.

  • However, for more than one electrons, this will no longer be true.

  • Nuclear charge is partially shielded by other electrons making the outer orbitals energies higher.


Shielding1

Shielding

  • Electrons in outer, more diffuse orbitals experience Coulomb potential of nuclear charge less than Z because inner electrons shield it.

Effective nuclear charge


Shielding2

Shielding

  • The sfunctions have greater probability density near the nucleus than p or d in the same shell and experience less shielding.

  • Consequently, the energy ordering in a shell is

Lower energy

3p

3d

3s


Aufbau principle

Aufbau principle

  • This explains the well-known building-up (aufbau) principle of atomic configuration based on the order (exceptions exist).

6s

6p

6d

6f

6g

6g

5s

5p

5d

5f

5g

4s

4p

4d

4f

3s

3p

3d

2s

2p

1s


Hund s rule

Hund’s rule

  • An atom in its ground state adopts a configuration with the greatest number of unpaired electrons (exceptions exist) – why?

2p

2s

1s

Oxygen


Hund s rule1

Hund’s rule

  • Spin correlation or Pauli exclusion rule explains Hund’s rule.

2p

Two electrons can be in the same spatial orbitals and the same position

Spatial part is antisymmetricand the two electron cannot occupy the same spatial orbitals or the same position – energetically more favorable


Summary

Summary

  • We have learned the orbital approximation,an approximate wave function of a many-electron atom that is an antisymmetric product of hydrogenic orbitals.

  • We have learned how the (anti)symmetry of spin part affects the spatial part and hence energies and the singlet & triplet helium atom and explains Hund’srule.

  • Shieldingexplains the aufbau principle.


  • Login