Water: Ionization Energies and Dyson Orbitals (MRSF-EKT)

The extended Koopmans' theorem (EKT) gives ionization energies from a correlated wavefunction, together with the Dyson orbital of each ionization: the orbital from which the electron is removed. Dyson orbitals are what photoelectron spectroscopy measures. OpenQP computes them from the MRSF-TDDFT ground state.

Dyson orbital, first ionization (1b1)

Second ionization (3a1)

Third ionization (1b2)

Copy and run

water-ekt.oqp

mrsf(nstate=10)/bhhlyp/6-31g* ekt scf(save_molden=true)
geom="""
O      0.000000     0.000000     0.122745
H      0.000000     0.759541    -0.471922
H      0.000000    -0.759541    -0.471922
"""

Download water-ekt.oqp

openqp water-ekt.oqp

ekt computes ionization potentials (the default; ekt(ip=false,ea=true) gives electron affinities instead) from the relaxed MRSF S0 density. scf(save_molden=true) writes the Dyson orbitals to water-ekt_dyson_rohf_bhhlyp_6-31gs.molden. The run takes about 2 seconds.

What you get

  MRSF-EKT ionization potentials (Dyson roots)
  Rows index EKT Dyson roots/orbitals, not TDDFT excited states.
  dyson      eig(ha)        eBE(ha)        eBE(eV)       metric       strength
      1      -19.692297       19.692297      535.8547      1.000000      1.000000
      2       -1.136394        1.136394       30.9228      1.000000      1.000409
      3       -0.593666        0.593666       16.1545      1.000000      1.005962
      4       -0.454481        0.454481       12.3671      1.000000      1.003072
      5       -0.378120        0.378120       10.2892      1.000000      1.002174

eBE is the electron binding energy, that is, the ionization energy, and strength the pole strength of each Dyson orbital.

Ionization	MRSF-EKT (eV)	Experiment, vertical (eV)
1b1 (out-of-plane lone pair)	10.29	12.62
3a1	12.37	14.74
1b2	16.15	18.51
2a1	30.92	—
1a1 (O 1s)	535.85	—

Reading the result

• The order of the ionizations and the shape of each Dyson orbital match the photoelectron spectrum: the first electron is removed from the out-of-plane oxygen lone pair.
• With this small basis set and BHHLYP, the three outer-valence ionization energies are 2.3–2.4 eV below the experimental vertical values, so the spacing between the bands is reproduced much better than the absolute values. Larger, diffuse basis sets change the absolute values.
• The Dyson orbitals are in the Molden file with the labels Dyson-IP-state-1 to Dyson-IP-state-5; open the file in OQP Studio Analysis or OpenqpView and select them by label.

Electron affinities: ozone

ekt(ip=false,ea=true) gives electron affinities instead. Ozone binds an extra electron (experimental electron affinity 2.10 eV), and a basis set with diffuse functions is needed for the anion:

ozone-ekt-ea.oqp

mrsf(nstate=10)/bhhlyp/6-31+g* ekt(ip=false,ea=true)
geom="""
O      0.000000     0.000000     0.444340
O      0.000000     1.083397    -0.222170
O      0.000000    -1.083397    -0.222170
"""

Download ozone-ekt-ea.oqp (experimental geometry, r(O–O) = 1.272 Å, ∠OOO = 116.8°; about 6 seconds)

  MRSF-EKT electron affinities (Dyson roots)
  Rows index EKT Dyson roots/orbitals, not TDDFT excited states.
  dyson      eig(ha)        eBE(ha)        eBE(eV)       metric       strength
      1       -0.163309        0.163309        4.4439      1.000000      1.006303
      2        0.095599       -0.095599       -2.6014      1.000000      0.999194

Only the first root is bound (positive electron binding energy): MRSF-EKT gives an electron affinity of 4.44 eV, 2.3 eV above experiment. Together with the ionization energies of water, which are about 2.3 eV too low, this shows that the EKT ionization and attachment energies at this level are shifted by a similar amount in opposite directions; compare trends between molecules rather than absolute values.

Try next

• Compare 6-31g* with aug-cc-pvdz for the first ionization energy.

See also: MRSF-EKT, ekt.

Scientific references

Excited-state analysis and ionization; MRSF-TDDFT.