Wavefunction Stability

Example 1 Molecular Oxygen

(i) RHF Calculation

%chk=o2.chk
#RHF/sto-5g scf=qc

O2 singlet

0,1
O
O 1 1.1695
 

Orbital Symmetries:

Occupied (SGG) (SGU) (SGG) (SGU) (PIU) (SGG) (PIU) (PIG)

Virtual (PIG) (SGU)

E(RHF) =-148.8788617 a.u.
 

(ii) Test of the wave function stability

#RHF/sto-5g stable guess=read

The wavefunction has an RHF -> UHF instability.
 

(iii) UHF Calculation of triplet oxygen

%chk=o2.chk
#UHF/sto-5g scf=qc

O2 triplet

0,3
O
O 1 1.1695

Orbital Symmetries:

Alpha Orbitals:

Occupied (SGU) (SGG) (SGG) (SGU) (PIU) (PIU) (SGG) (PIG)

(PIG)

Virtual (SGU)

Beta Orbitals:

Occupied (SGG) (SGU) (SGG) (SGU) (SGG) (PIU) (PIU)

Virtual (PIG) (PIG) (SGU)

The electronic state is 3-SGG.
 

E(UHF,triplet) = -148.9626395a.u. (lower than the singlet)
 

The wavefunction is stable under the perturbations considered.
 

Lewis representation (Pauling three-electron bond and Linnett's non-paired spatial orbital structure)

Example 2 Ozone
 

(i) RHF calculation

%chk=o3.chk
#RHF/6-31G(d) scf=qc

ozone

0,1
O
O 1 1.2227
O 1 1.2227 2 114.0451

Orbital Symmetries:

Occupied (A1) (B2) (A1) (A1) (B2) (A1) (A1) (B1) (B2) (B2)

(A1) (A2)

Virtual (B1) (A1) (B2) (A1) (B2) (A1) (B1) (A1) (B2) (A1)

(A2) (B1) (B2) (A1) (B2) (A2) (A1) (B1) (B2) (A2)

(B1) (A1) (B2) (A1) (B1) (B2) (A2) (A1) (B2) (A1)

(A1) (B2) (A1)

The electronic state is 1-A1.

E(RHF)=-224.2585378
 

The wavefunction has an RHF -> UHF instability.
 

(ii) UHF calculation (triplet).

%chk=o3.chk
#UHF/6-31G(d) scf=qc

ozone

0,3
O
O 1 1.2227
O 1 1.2227 2 114.0451

Orbital Symmetries:

Alpha Orbitals:

Occupied (A1) (B2) (A1) (A1) (B2) (A1) (B1) (A1) (B2) (A2)

(A1) (B2) (B1)

Virtual (A1) (B2) (A1) (A1) (B2) (B1) (A1) (B2) (A2) (A1)

(B1) (B2) (A1) (B2) (A2) (A1) (B1) (B2) (B1) (A2)

(A1) (B2) (A1) (B1) (B2) (A2) (A1) (B2) (A1) (A1)

(B2) (A1)

Beta Orbitals:

Occupied (A1) (B2) (A1) (A1) (B2) (A1) (A1) (B2) (B1) (A2)

(B2)

Virtual (A1) (B1) (A1) (B2) (A1) (B2) (A1) (B1) (A1) (B2)

(A1) (A2) (B1) (B2) (B2) (A1) (A2) (A1) (B1) (B2)

(A2) (B1) (A1) (B2) (A1) (B1) (B2) (A1) (A2) (B2)

(A1) (A1) (B2) (A1)

The electronic state is 3-B1.
 
 

E(UHF, trip) =-224.2266113\S2=2.062463\S2-1=0.\S2A=2.001812
 

The wavefunction has an internal instability.

Energy is higher than for the singlet state!

We have to consider an open shell singlet (biradical)!
 
 
 
 

(iii) UHF calculation (singlet)

%chk=o3.chk
#UHF/6-31G(d) scf=qc

ozone

0,1
O
O 1 1.2227
O 1 1.2227 2 114.0451

Orbital Symmetries:

Alpha Orbitals:

Occupied (A1) (B2) (A1) (A1) (B2) (A1) (A1) (B1) (B2) (B2)

(A1) (A2)

Virtual (B1) (A1) (B2) (A1) (B2) (A1) (B1) (A1) (B2) (A1)

(A2) (B1) (B2) (A1) (B2) (A2) (A1) (B1) (B2) (A2)

(B1) (A1) (B2) (A1) (B1) (B2) (A2) (A1) (B2) (A1)

(A1) (B2) (A1)

Beta Orbitals:

Occupied (A1) (B2) (A1) (A1) (B2) (A1) (A1) (B1) (B2) (B2)

(A1) (A2)

Virtual (B1) (A1) (B2) (A1) (B2) (A1) (B1) (A1) (B2) (A1)

(A2) (B1) (B2) (A1) (B2) (A2) (A1) (B1) (B2) (A2)

(B1) (A1) (B2) (A1) (B1) (B2) (A2) (A1) (B2) (A1)

(A1) (B2) (A1)

The electronic state is 1-A1.

E(UHF)=-224.2585378

The wavefunction has an internal instability.

We find the same energy and electronic state as in the RHF calculation.
 

(iv) We have to consider an open shell singlet (biradical) and we have to destroy a and b spin symmetry.

%chk=o3.chk
#UHF/6-31G(d) scf=qc guess=mix

guess=read stable

ozone

0,1
O
O 1 1.2227
O 1 1.2227 2 114.0451

E(UHF, LUMO HOMO mix)=-224.3272072 a.u. / S2=0.88567
 

The wavefunction is stable under the perturbations considered.

The wavefunction is contaminated!
 

(I) VB Bild

a) Lewis

b) Increased Valence structures can be obtained from delocalizing 2 nonbonding electrons into the bonding O-O p LMO.

(II) MO: 3-Zentren-4-Elektronen-p-Bindung

This MO diagram is not correct as the Yⁿ is only singly occupied! The biradical character (see VB picture) results from the coupling singly occupied p orbitals
 
 

Summary:
 

Instabilities can be of many types:

· The lowest energy wavefunction is a singlet, but not a closed shell singlet (e.g. biradical)

RHF Þ UHF instability
 

· There is a lower-lying triplet state than the singlet (e.g. O2)

RHF Þ UHF instability

RHF Þ UHF instability means E(UHF) < E(RHF)

1O2 : E(RHF) = -149.532997 a.u.

3O2 : E(UHF) = -149.617908 a.u. D E = 53.3 kcal/mol
 

· There is more than one solution to the SCF equations for the system, and the calculation converges to a solution which is not the minimum.

RHF Þ RHF instability or UHF Þ UHF instability
 
 

Example Ozone:
 

· Ozone is a singlet ground state

· Calculation: RHF/6-31G(d) stable=opt

finds: RHF Þ UHF instability

· Calculation: UHF/6-31G(d) stable=opt

finds: UHF Þ UHF instability (internal instability)

· solution: modify the default electronic configuration

(biradical character resulting from the coupling singly occupied p orbitals, a and b symmetry and spatial symmetries needs to be destroyed)

Calculation: UHF/6-31G(d) guess=mix stable

finds: stable wavefunction

last changes: 01.04.2008, AS
questions & comments to: axel.schulz@uni-rostock.de