Computational Chemistry

Eð^2 Þ¼

RR

c 1 ð 1 Þc 1 ð 2 Þr^112



c 2 ð 1 Þc 2 ð 2 Þdv 1 dv 2

hi 2

2 ðe 1 #e 2 Þ

ð 5 : 162 Þ

Applying this formula “by hand” is straightforward, although the arithmetic is
tedious. Nevertheless it is worth doing (as was true for theHartree–Fockcalcula-
tion in Section 5.2.3.6.5) in order to appreciate how much arithmetical work is
involved in even this simplest molecular MP2 job. Consider the integral in the
numerator of Eq.5.162; substituting forc 1 andc 2 :

ZZ

c 1 ð 1 Þc 1 ð 2 Þ

1

r 12



c 2 ð 1 Þc 2 ð 2 Þdv 1 dv 2

¼

ZZ

ðc 11 f 1 ð 1 Þþc 21 f 2 ð 1 ÞÞðc 11 f 1 ð 2 Þþc 21 f 2 ð 2 ÞÞ

1

r 12



'ðc 12 f 1 ð 1 Þþc 22 f 2 ð 1 ÞÞðc 12 f 1 ð 2 ÞÞ



Multiplying out the integrand gives a total of 16 terms (from four terms to the
left of 1/r 12 and four terms to the right), and leads to a sum of 16 integrals:

ZZ

c 1 ð 1 Þc 1 ð 2 Þ

1

r 12



c 2 ð 1 Þc 2 ð 2 Þdv 1 dv 2

¼c^211 c^212

Z

f 1 ð 1 Þf 1 ð 2 Þ

1

r 12



f 1 ð 1 Þf 1 ð 2 Þdv 1 dv 2 þ(((þc^221 c^222

Z

f 2 ð 1 Þf 2 ð 2 Þ

1

r 12



¼c^211 c^212 ð 11 j 11 Þþ(((þc^221 c^222 ð 22 j 22 Þ;

recalling the notational degeneracy in the two-electron integrals (Section 5.2.3.6.5
“Step 2– Calculating the integrals”). Substituting the values of the coefficients and
the two-electron integrals:

ZZ

c 1 ð 1 Þc 1 ð 2 Þ

1

r 12



c 2 ð 1 Þc 2 ð 2 Þdv 1 dv 2

¼ 0 : 12475 ð 0 : 7283 Þþ(((þ 0 : 44577 ð 0 : 9927 Þh¼ 0 :12932 h

So from Eq.5.162

Eð^2 Þ¼

0 : 129322

2 ðe 1 #e 2 Þ

h¼

0 : 129322

2 ð# 1 : 4470 þ 0 : 1051 Þ

h¼# 0 :00623 h

The MP2 energy is the Hartree–Fock energy plus the MP2 correction
(Eq.5.162):

EMP2¼EtotalHF þEð^2 Þ¼# 2 :4438 h# 0 :00623 h¼# 2 :4500 h

5.4 Post-Hartree–Fock Calculations: Electron Correlation 263