Essentials of Computational Chemistry

8.4 EXCHANGE-CORRELATION FUNCTIONALS 265
show that the exchange-correlation energy can then be computed as
1
Exc =
0
〈(λ)|Vxc(λ)|(λ)〉dλ (8.31)
where λ describes the extent of interelectronic interaction, ranging from 0 (none) to 1 (exact).
To evaluate this integral, it is helpful to adopt a geometric picture, as illustrated in Figure 8.2.
We seek the area under the curve defined by the expectation value of Vxc. While we know
very little about V and as functions of λ in general, we can evaluate the left endpoint
of the curve. In the non-interacting limit, the only component of V is exchange (deriving
from antisymmetry of the wave function). Moreover, as discussed in Section 8.3, the Slater
determinant of KS orbitals is the exact wave function for the non-interacting Hamiltonian
operator. Thus, the expectation value is the exact exchange for the non-interacting system,
which can be computed just as it is in HF calculations except that the KS orbitals are used.
The total area under the expectation value curve thus contains the rectangle having the
curve’s left endpoint as its upper left corner, which has area EHF x . The remaining area is
−E
(0, 〈Ψ(0)|K|Ψ(0)〉)
A
0 1
l
B
(1, 〈Ψ(1)|V xc |Ψ(1)〉)
(1, 〈Ψ(0)|K|Ψ(0)〉)
Figure 8.2 Geometrical schematic for the evaluation of the integral in Eq. (8.31). The area under
the curve is the sum of the areas of regions A and B. AsregionA is a rectangle, its area is trivially
computed to be one times the expectation value of the HF exchange operator acting on the Slater
determinantal wave function for the non-interacting system, (0). The area of region B is less easily
determined. One simplification is to assume (i) that 〈(1)|Vxc(1)|(1)〉 is equal to the corresponding
computed value from an approximate DFT calculation and (ii) that area B will be some characteristic
fraction of the area of the rectangle having dotted lines for 2 sides (e.g., if the curve is well approximated
by a line, clearly the characteristic fraction would be 0.5). Note that if the curve rises very
steeply from its left endpoint, i.e., the value of z in Eq. (8.32) is very close to 1, then the adiabatic
connection method is of limited value