Download User Manual

50
CHAPTER 4. THEORY
4.1.4
Dimer–es approximation
For the dimers constructed with largely separated fragments, the E ′ HFIJ is approximately calculated by considering only the electrostatic interaction between the two fragments. This treatment
is called ”dimer–es approximation”. In this approximation, the E ′ HFIJ is evaluated as the following equation:
′ HF
E ′ HF
+ E ′ HF
IJ = E I
J +
∫
∑
−ZA ρI (r1 )
A∈J
∫
∑ ∫ −ZA ρJ (r1 )
ρI (r1 )ρJ (r2 )
dr1 +
dr1 +
dr1 dr2 .
| RA − r1 |
| RA − r1 |
| r1 − r2 |
(4.1.29)
A∈I
This equation can be written with the density matrices as
′ HF
E ′ HF
+ E ′ HF
IJ = E I
J +
∑
∑
∑ ∑
HF
DIHF
DJHFµν uJ(I)µν +
DIHF
µν uI(J)µν +
µν DJ λσ ( µν | λσ ).
µν∈I
µν∈J
(4.1.30)
µν∈I λσ∈J
Additionally, for the dimer–es pairs, the following equation is satisfied:
(
)
( HF
)
( HF
)
( HF
)
T r ∆DHF
IJ VIJ = T r DIJ VIJ − T r DI(J) VIJ − T r DJ(I) VIJ = 0.
(4.1.31)
HF
Thus, by combining this equation with Eq. (4.1.27) , the ∆EIJ
is approximately written as
′ HF
HF
− E ′ HF
= E ′ HF
∆EIJ
J .
IJ − E I
(4.1.32)
Consequently, by substituting Eq. (4.1.30) to Eq. (4.1.32), we obtain the following equation:
∑
∑
HF
DJHFµν uJ(I)µν
=
DIHF
∆EIJ
µν uI(J)µν +
µν∈J
µν∈I
+
∑ ∑
HF
DIHF
µν DJ λσ ( µν | λσ ).
(4.1.33)
µν∈I λσ∈J
This is the IFIE for the dimer–es pairs.
4.1.5
Environmental electrostatic potential
The matrix of the environmental electrostatic potential (VX ) is written as sum of the attractive
potential and repulsive potential form the other fragments, i.e.,
∑
∑
VX =
uX(K) +
vX(K) ,
(4.1.34)
K̸=X
K̸=X
where the first term of this equation is calculated as
∑
K̸=X
uX(K)µν = ⟨ µ |
∑ ∑
K̸=X A∈K
−ZA
| ν ⟩.
| RA − r1 |
(4.1.35)
On the other hands, the second term of this equation is evaluated using three types of manner:
1. evaluated without any approximation, i.e., 4-center electron repulsion integrals are used.
2. approximately evaluated with 3-center integrals, which is called ”esp-aoc” approximation.
3. approximately evaluated with point charges, which is called ”esp-ptc” approximation.