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 We use the exact solutions 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 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 We introduce the following approximation

(the orbital approximation):

For the helium atom, this amounts to

1 2 1 1 2 2, r r r r

Hydrogenic orbital

The orbital approximation The approximation is equivalent to neglecting

interaction between electrons 1 and 2,

… so that,Hydrogenic electron 1 Hydrogenic electron 2 Interaction

The orbital approximation

exact hydrogenic problem

Eigenfunction

The orbital approximation We construct a helium wave function as the

product of hydrogenic orbitals 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).

1 2 1 1 2 2, r r r r

Spins Let us first append spin factors

None of these is antisymmetric yet

1 2 1 1 2 2

1 2 1 1 2 2

1 2 1 1 2 2

1 2 1 1 2 2

, (1) (2)

, (1) (2)

, (1) (2)

, (1) (2)

r r r r

r r r r

r r r r

r r r r

(Anti)symmetrization Symmetrization:

Antisymmetrization:

Sym.

Antisym.

Antisymmetric function

Sym. Antisym.Antisym.

Helium wave functionsAlready symmetric and cannot be made antisymmetric

Neither sym. or antisym.

Neither sym. or antisym.

Sym.Antisym.

Sym.Antisym.

Antisym.Sym.

Triplet states These three have the same spatial shape –

the same probability density and energy – triply degenerate (triplet states)

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)

Sym.Antisym.

Singlet state There is another state which is non-

degenerate (singlet state):

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.

Opposite spins

Antisym.Sym.

Energy ordering For the helium atom, depicting α- and β-spin

electrons by upward and downward arrows, we can specify its electron configurations.

1s

2s

Triplet states

1s

2s

Singlet state B

1s

2s

Singlet stateA

The orbital approximation

1s

2s

Triplet states 1s

2s

Singlet B

1s

2s

Singlet A

159856 cm-1166277 cm-1

0 cm-1

Hydrogenic electron 1 Hydrogenic electron 2 Interaction

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 hydrogenic orbitals 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.

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.

Shielding

Shielding Electrons in outer, more

diffuse orbitals experience Coulomb potential of nuclear charge less than Z because inner electrons shield it.

effZ Z Effective nuclear charge

Shielding The s functions 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

3s

3p3d

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

Aufbau principle

1s2s 2p3s 3p 3d4s 4p 4d 4f5s 5p 5d 5f 5g6s 6p 6d 6f 6g 6g

Hund’s rule An atom in its ground state adopts a

configuration with the greatest number of unpaired electrons (exceptions exist) – why?

Oxygen

2p

1s

2s

Hund’s rule Spin correlation or Pauli exclusion rule

explains Hund’s rule.

2p

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

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

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’s rule.

Shielding explains the aufbau principle.