How do you calculate the ionization energy of lithium?
By using a computer... we obtained a value of
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
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"#
#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
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.
#DeltaE_"SO"#contribution can be included but it has been done before... 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