This energy, which includes internuclear repulsion, sinceEtotalHF includes this
(Eq.5.93), is the MP2 energy normally printed out at the end of the calculation.
To get an intuitive feel for the physical significance of the calculation just per-
formed look again at Eq.5.162, which applies to any two-electron/two-basis
function species. The equation shows that the absolute value (the correction is
negative sincee 1 is smaller thane 2 – the occupied MO has a lower energy than the
virtual one) of the correlation correction increases, i.e. the energy decreases, with
the magnitude of the integral (which is positive). This integral represents the
decrease in energy arising from allowing an electron pair in the occupied MO
(c 1 ) to spill over into the virtual MO (c 2 ):
c 1 (1) represents electron 1 inc 1 andc 1 (2) represents electron 2 inc 1.
c 2 (1) represents electron 1 inc 2 andc 2 (2) represents electron 2 inc 2.
The operator 1/r 12 brings in coulombic interaction: the coulombic repulsion
energy between infinitesimal volume elementsc 1 (1)c 1 (2)dv 1 andc 2 (1)c 2 (2)dv 2
separated by a distancer 12 is (c 1 (1)c 1 (2)dv 1 )(c 2 (1)c 2 (2)dv 2 )/r 12 , and the integral is
simply the sum over all such volume elements (cf. the discussion in connection with
Fig.5.3and the average-field integralsJandKinSection 5.2.3.2). Physically, the
decrease in energy makes sense: allowing the electrons to be partly in the formally
unoccupied virtual MO rather than confining them strictly to the formally occupied
MO enables them to avoid one another better than in the HF treatment, which is
based on a Slater determinant consisting only of occupied MOs (Section 5.2.3.1).
The essence of the Møller–Plesset method(MP2, MP3, etc.) is that the correction
term handles electron correlation by promoting electrons from occupied to unoccu-
pied (virtual) MOs, giving electrons, in some sense, more room to move and thus
making it easier for them to avoid one another; the decreased interelectronic
repulsion results in a lower electronic energy. The contribution of the “c 1 /c 2 inter-
action” toE(2)decreases as the occupied/virtual MO gape 1 #e 2 , increases, since this
is in the denominator. Physically, this makes sense: the bigger the gap between the
occupied and higher-energy virtual MO, the harder it is to promote electrons from
the one into the other, so the less can such promotion contribute to electronic
stabilization. So in the expression forE(2)(Eq.5.162), the numerator represents
the promotion of electrons from the occupied to the virtual orbital, and the denomi-
nator represents a check on how hard it is to do this.
As we just saw, MP2 calculations utilize the Hartree–Fock MOs (their coeffi-
cientscand energiese). The HF method gives the bestoccupiedMOs obtainable
from a given basis set and a one-determinant total wavefunctionc, but it does not
optimize thevirtualorbitals (after all, in the HF procedure we start with a determi-
nant consisting of only theoccupiedMOs –Sections 5.2.3.1–5.2.3.4). To get a
reasonable description of the virtual orbitals and to obtain a reasonable number of
them into which to promote electrons, we need a basis set that is not too small. The
use of the STO-1G basis in the above example was purely illustrative; the smallest
basis set generally considered acceptable for correlated calculations is the 6–31G*,
and in fact this is perhaps the one most frequently used for MP2 calculations. The
6–311G** basis set is also widely used for MP2 and MP4 calculations. Both bases
264 5 Ab initio Calculations