Quantum Chemistry of Solids / 14-Hartree-Fock LCAO Method for Periodic Systems

4.3 Density Matrix of Crystals in the Hartree–Fock Method	145
ρ0(Rm0 )ρ0(Rn0 − Rm0 ) = 2ρ0(Rn0 )	(4.163)
R0
m

In the strict sense, (4.162) is not an idempotency relation, because summation is carried out only over the vectors R0m, lying within the LUC, whereas the vector di erence (R0n − R0m) can lie outside the LUC. If we perform summation over all Bravais lattice vectors, the right-hand side of (4.162) will diverge, because ρ0(R0n) does not vanish at infinity.

For the approximate DM to have the proper point symmetry, the LUC should be taken to be the Wigner–Seitz (WS) cell. In this case, however, the symmetry can be broken if on the boundary of the WS cell, there are atoms of the crystal. Indeed, if an atom lies on the WS cell boundary, then there is one or several equivalent atoms that also lie on the boundary of the cell and their position vectors di er from that of the former atom by a superlattice vector A. When constructing the approximate density matrix ρ˜ we assigned only one of several equivalent atoms to the WS cell. In other words, in the set {R0n}, there are no two vectors that di er from each other by a superlattice vector A. In this case, if a point-symmetry operation takes one boundary atom into another atom assigned to another WS cell, then the point symmetry of the density matrix ρ˜ is broken, because the symmetry of the weighting function ωµν (Rn) of (4.157) is broken.

Since it is desirable to preserve the point symmetry when calculating the electronic structure, the approximate DM can be replaced by an averaged density matrix (see also [99, 100]):

		Ns
	1
ρs =	Ns	ρ˜α(Rn) = ωs(Rn)ρ0(Rn)	(4.164)
		α=1

Here, the index α = 1, 2, . . . , Ns labels all Ns possible ways in which one of the equivalent boundary atoms can be assigned to a given WS cell and the symmetrical weighting function is defined as

			Ns			1
										−
		1				nµνs , if			Rn + dν
s				α						dµ VA
ωµν	(Rn) =	Ns α=1		ωµν	(Rn) =	0, if			Rn + dν	− dµ / VA	(4.165)

where nsµν is the number of atoms in the WS cell (including its boundary) that are translationally equivalent to atom B (ν B) in the case where the WS cell is centered at the atom-A site (µ A). In other words, nsµν is the number of WS cells that have atom B in common. If atom B is strictly inside the WS cell, then nsµν = 1 and (4.165) is identical to (4.157). We note that the density matrix ρs(Rn) does not satisfy idempotency relation (4.162) and corresponds to a mixed state of the system, However, as the LUC enlarges, the e ect of the boundary atoms decreases and the density matrix ρs(Rn) approaches the idempotent density matrix ρ(Rn).

The above consideration can be interpreted as deduction of the cyclic cluster model of the infinite crystal when the Hartree–Fock LCAO method (or its semiempirical version with nonlocal exchange) is applied.

The study of the approximate density matrix properties allowed the implementation of the cyclic cluster model in the Hartree–Fock LCAO calculations of crystalline systems [100] based on the idempotency relations of the density matrix. The results

146 4 Hartree–Fock LCAO Method for Periodic Systems

obtained for cyclic-cluster modeling the rutile TiO2 structure are discussed in Chap. 9.

The consideration presented allows one to better understand the features of Hartree–Fock self-consistent calculations of the electronic structure of an infinite crystal with a nonlocal exchange potential determined by the o diagonal elements of the one-electron density matrix. It has been shown that the number of k points chosen in the BZ for calculations should be in accord with the size of the interaction range in the coordinate space. In other words, there should be a correlation between summations over the Bravais lattice in the coordinate space and over the BZ in the reciprocal space. It was also shown that when the Fock operator (or the Kohn–Sham operator in hybrid DFT methods, see Chap. 7) contains o diagonal DM elements, the BZ special-point technique should be modified: weighting-function introduction in the direct-lattice sums entering the exchange term establishes the necessary balance and removes the artificial divergences. The HF LCAO method is a well-defined starting point for more sophisticated techniques allowing inclusion of the electron correlation. We consider this problem in the next chapter.