Handbook of Solvents - George Wypych - ChemTech - Ventech!

438 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
The MP2 wave functions we have introduced in a preceding subsection are just the application
of this method to another problem, that of the electronic correlation. In this case,
the simpler unperturbed system is the HF approximation, the corrections are limited to the
second order, and the corresponding contributions to the energy are expressed as a simple
summation of elements. The MP2 method is currently used in the PT description of the
intermolecular potential: 13 in such cases, two different applications of PT are used at the
same time.
We pass now to apply PT to the calculation of ΔE AB(R). The most reasonable choice
consists of defining the unperturbed system as the sum of the two non-interacting monomers.
We thus have:
0
H = HA + HB
[8.36]
A B
Φ = | Ψ Ψ >
0 0 0
[8.37]
A B
E = E + E
[8.38]
0 0 0
Φ 0 is the simple product of the two monomers’ wave functions. The perturbation operator
is the difference of the two Hamiltonians, that of the dimer and H 0 . The perturbation parameter
λ may be set equal to 1:
0
V = H − H = VAB [8.39]
The set Φ1, Φ2, Φ3, �, ΦK
of the solutions of the unperturbed system can be obtained
by replacing within each monomer wave function, one, two or more occupied MO with vacant
MOs belonging to the same monomer. The perturbation operator V only contains oneand
two-body interactions, and so, being the MOs orthonormal, the only VLK integrals different
from zero are those in which L and K differ at the maximum by two MOs.
The formulation is quite appealing: there is no need for repeated calculations, the decomposition
of the interaction energy can be immediately obtained by separately collecting
contributions corresponding to different ways of replacing occupied with virtual orbitals.
Interrupting the expansion at the second order, one obtains the following result:
(1) (2)
ΔE≈ E + E =
−
−
−
ΨΨ A B V| ΨΨ A B + I order: electrostatic term
∑
K
∑
K
A B A B 2
| Ψ0 Ψ0 | VΨKΨ0
|
+ II order: polarization of A
A A
E − E
K
0
A B A B 2
| Ψ0 Ψ0 | VΨ0
ΨK
|
+ II order: polarization of B [8.40]
B B
E − E
K
0 0 K L
∑∑
A B A B
K L ( E + E ) − ( E + E )
0
A B A B 2
| Ψ Ψ | VΨ
Ψ |
K
L
0 0
+ II order: dispersion A-B + higher order terms