We could writeEMP1¼EtotalHF ¼EMP0þEð^1 Þ, whereEMP0is the sum of one-elec-
tron energies and internuclear repulsions and E(1) is the J, K correction
(corresponding respectively to the two terms in Eqs.5.85and5.90), regarding the
second term as a kind of perturbational correction to the sum of one-electron
energies.
MP2 is the first MP level to go beyond the HF treatment: it is the first “real”
Møller–Plesset level. The MP2 energy is the HF energy plus a correction term (a
perturbational adjustment) that represents a lowering of energy brought about by
allowing the electrons to avoid one another better than in the HF treatment:
EMP2¼EtotalHF þEð^2 Þ ð 5 : 161 ÞThe HF term includes internuclear repulsions, and the perturbation correction
E(2)is a purely electronic term.E(2)is a sum of terms each of which models the
promotion of pairs of electrons. So-calleddouble excitationsfrom occupied to
formally unoccupied MOs (virtual MOs) are required by Brillouin’s theorem
[ 89 ], which says, essentially, that a wavefunction based on the HF determinant
D 1 plus a determinant corresponding to exciting just one electron fromD 1 cannot
improve the energy.
Let’s do an MP2 energy calculation on HHe+, the molecule for which a
Hartree–Fock (i.e. an SCF) calculation was shown in detail in Section 5.2.3.6.5.
As we did for the HF calculation, we will take the internuclear distance as 0.800 A ̊
and use the STO-1G basis set; we can then use for our MP2 calculation these HF
results that we obtained in Section 5.2.3.6.5:
The MO coefficients
For the occupied MOc 1 ,c 11 ¼0.3178,c 21 ¼0.8020
Recall that these are respectively the coefficient of basis function 1,f 1 , in MO1
and the coefficient of basis function 2,f 2 , in MO1. In this simple case there is one
function on each atom:f 1 andf 2 on atoms 1 and 2 (H and He).
For the unoccupied (virtual) MOc 2 ,c 12 ¼1.1114,c 22 ¼#0.8325
The two-electron repulsion integrals:
ðÞ¼ 11 j 11 0 : 7283 ðÞ¼ 21 j 21 0 : 2192ðÞ¼ 21 j 11 0 : 3418 ðÞ¼ 22 j 21 0 : 4368ðÞ¼ 22 j 11 0 : 5850 ðÞ¼ 22 j 22 0 : 9927The energy levels
Occupied MO,e 1 ¼#1.4470, virtual MO,e 2 ¼#0.1051
The HF energy:EtotalHF ¼#2.4438
The MP2 energy correction for a closed-shell two-electron/two-MO system
is [ 90 ]
262 5 Ab initio Calculations