Question

How do you calculate the ionization energy of lithium?

Answer 1

By using a computer... we obtained a value of `"5.39271223 eV"`, compared to the true value of `5.39171495 (pm 0.00000004) "eV"`. Not bad.

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Lithium clearly has more than one electron; that makes it so the ground-state energy is not readily able to be calculated by hand, since the electronic coordinates are mutually dependent due to the inherent electron-electron correlation.

Instead, we would have to supply input files to a computer software and calculate the ground-state energies that way, of `"Li"` and `"Li"^(+)`.

Using the so-called Feller-Peterson-Dixon method to get practically perfect accuracy, one would have to calculate (or consider):

`DeltaE_"IE" = DeltaE_("IE",0) + DeltaE_"corr" + DeltaE_"CBS" + DeltaE_"CV" + DeltaE_"QED" + DeltaE_"SR" + DeltaE_"SO" + DeltaE_"Gaunt"`

where:

• `DeltaE_("IE",0)` is the initial ionization energy calculated from Multi-Configurational Self-Consistent Field (MCSCF) theory.
• `DeltaE_"corr"` is the dynamic correlation energy contribution not accounted for in Multi-Configurational Self-Consistent Field theory, but recovered in Multi-Reference Configuration Interaction (MRCI).
• `DeltaE_"CBS"` is the energy contribution from extrapolating to the limit of an infinite set of basis functions that represent atomic orbitals.
• `DeltaE_"CV"` is the energy contribution from correlating the core electrons with the valence electron(s).
• `DeltaE_"QED"` is the energy contribution from the so-called Lamb Shift, a quantum electrodynamics interaction present primarily among `s` orbitals.
• `DeltaE_"SR"` is the energy contribution from relativistic effects. This is negligible in `"Li"` but is automatically accounted for using the 2nd order Douglas-Kroll-Hess (DKH) Hamiltonian for light atoms (3rd order DKH for heavy atoms).
• `DeltaE_"SO"` is the energy contribution from spin-orbit coupling.
• `DeltaE_"Gaunt"` is the energy contribution from high-order two-electron correlation in the relativistic scheme.

That WOULD be extremely involved for a heavier atom... Here are some things that save time:

• No electron correlation is present here since only one state is possible (spin up in a `2s` orbital!).

So we can get by from a simple Hartree-Fock calculation. There will be a tiny bit of core-valence (`1s"-"2s`) correlation, so `DeltaE_"CV" ne 0` and that can be taken care of with an MRCI using a weighted-core basis set.

• The model potentials for QED only are made for `Z >= 23` (quote: "Fails completely for `Z < 23`"), so there is no point in including the Lamb Shift at all.

• `DeltaE_"SR"` is included by default by the 2nd order DKH Hamiltonian.

• The `DeltaE_"SO"` contribution can be included but [it has been done before...](http://www7b.biglobe.ne.jp/~kcy05t/spincal.html) It is `"0.000041 eV"`, or `"0.000945 kcal/mol"`. Gaunt is unimportant here, based on how small the spin-orbit value is.

Here are the (not so interesting) results:

From this, we had gotten that:

`color(blue)(DeltaE_"IE") = "123.195465 kcal/mol" + "0.000000 kcal/mol" + "0.000000 kcal/mol" + "1.162450 kcal/mol" + "0.000000 kcal/mol" + "accounted for" + "0.000945 kcal/mol" + "0.000000 kcal/mol"`

`=` `color(blue)(ul"124.358860 kcal/mol")`

`=` `color(blue)(ul"5.39271223 eV")`

whereas the value on NIST is practically the same, at `5.39171495 (pm 0.00000004) "eV"`, a mere `0.01849%` error...