How do you calculate the ionization energy of lithium?
1 Answer
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"# 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... 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
