Compact expressions for charge distributions

As will be discussed in the following sections, the essential problem in HF studies o f periodic systems is how to deal with infinite sums o f Coulomb integrals, necessary to calculate the electrostatic interaction between an infinite number o f charge distributions. To do so, it is useful to introduce some formalisms for the expressions o f the charge distributions and their Coulomb interactions in the crystal, that was originally used by the authors o f CRYSTAL.

p(r) = {type; location}

(3.3.5)

indicates a generic charge distribution, where “type” indicates the set o f indices that describe the characteristics o f the charge distribution with respect to the crystal cell(s) specified by “location”.

({ty l; lo c l) I {ty2;loc2)) =

J

(3.3.6)

indicates the interaction between two distributions Pj and p2- In particular, the following notations will be used:

i) AO overlap distributions, that are the product of two A O ’s, are written as:

{12g ; 0 } h {12g) =

x° (r) x f (r)

(3-3.7)

{12g ; h) s { 12g) = Xj (r) Xh2g «

(3-3-8)

The interaction between two AO overlap distributions can be written as:

(10 2gl 3h 4n+h) = ({12g)l {34n;h)) =

= I dr dr’ %° (r)

Zf

(r)

(r)

(3-3.9)

ii) A capital letter B used for “location” corresponds to a sub-set o f lattice cells over which the local distribution {type;h) is summed. For example, if we consider a sub-set B o f lattice cells where all the two-electron integrals are evaluated exactly , we can write:

U 2 g ; B } = I hEB { 12g;h)

(3.3.10)

If the sum is extended to all lattice cells, a translationally invariant distribution is obtained, whose location will be denoted by the letter T.

{ 12g ; T} = I h { 12g;h) (3.3.11)

iii) A Greek letter identifies the charge distribution o f ‘shells’, that is , the electronic charge attributed to shell X according to a Mulliken analysis:

{A.;h} = Z l e X I 2 I n P12n ( 12n;h) (3.3.12)

where P is the density matrix element referring to the AOs ^ %2 *n zero and n-th cell.

iv) Unit nuclear charge distributions are defined as:

{ Z ; h ) =8 (r-h-sz) (3.3.13)

The nuclear charge Z can formally be split into integer nuclear shell charges Z^, each one approximately compensating the electron charge attributed to a given shell. Nuclear attraction integrals are expressed as:

( ( 12g)l{Z;h)) = / dr z? (r) (r )/ Ir-h-sJ (3.3.14)

3 .3 .2 B asic equations

In this section we first sketch the equations and the general procedures. A more detailed discussion then follows, in which we analyze the form o f the HF Hamiltonian and the computational parameters involved. We also comment on the procedure adopted for the integration in reciprocal space. More details o f CRYSTAL can be found in the literature cited above . We shall emphasize those computational features that will be used in the embedding method discussed in Chapter 4.

A periodical HF Hamiltonian can be defined:

F f2 = H f

+ B

(3.3.15)

where H8 12 and B8 1 2 are respectively the one- and two-electron contributions

in the basis set o f the AO pq8, g being a direct lattice vector (%£ = (r - Sj - g)) and si the position in the reference cell of the atom to which jq belongs.

The Fock matrix and the overlap matrix S8 are then Fourier transformed into

the reciprocal space, in the Bloch function (BF) basis; eigenvalues (E) and eigenvectors (A) are obtained after solving for each k vector, in the first Brillouin zone (BZ), the matrix equation:

F(k) A(k) = S(k) A(k) E(k)

(3.3.16)

Once the eigenvalues are known, the new density matrix, in the AO basis, is calculated, by integration over the BZ volume:

P 12g = 2 X n Jbz dk exp(i k .g) a*ln(k) (k) 0 [eF - en (k)]

(3.3.17)

where the ajn (k) element o f A(k) is the coefficient o f the j-th BF in the n-th crystalline orbital at point k; en(k) is the corresponding eigenvalue and £p is the Fermi level (note that here, and in the following, only closed shell systems are considered).

The knowledge o f P allows the Fock matrix, in direct space, to be re­ evaluated. The process continues, iteratively, until convergence in the total energy is reached.

The implementation o f these equations requires the calculation o f infinite summations over direct lattice vectors (Coulomb and exchange series) and a technique for integration in reciprocal space. The scheme adopted in CRYSTAL will be discussed with particular reference to these problems.

3.3.2. 1 Coulomb terms

In eq. 3.3. I f , the 'one-electronic* term includes the kinetic contribution

^ 1 2)

TjS2 = < 10 I - V2 / 2 1 2g> (3.3.18)

and the nuclear attraction term (Zjg2), that corresponds to the Coulomb interaction between the AO overlap distribution {12 g) with the nuclear charges in the crystal:

Z 182 = - ? X ( Z X ( ( 1 2 8 ) 1 ( Z X ; T ) ) =

= - S x I h ( M g i z ^ h ) (3.3.19)

B f2 is the sum o f the Coulomb and exchange terms:

C12

= (( 12 g) I {X; T )) (3.3.20)

X f2 = -1 / 2 £ 34n P34n [ 2 h (10 3h I 2g 4n+h) ] (3.3.21)

where P is the density matrix; and the n and h summations are in principle extended to all direct lattice vectors. Because o f translational invariance the first vector can always be centred in the origin cell, identified by the null vector 0.

We first rearrange the Z and C contributions:

C 12

+ z i

=

J

* * ’ X°i (r ) Zg2(r ) 1 r -r ’ d P x (r’-h) =

= S h [ Z 34n (10 2g I 3h 4n+h) - Z x (12g I h )]

(3.3.22) defining Mulliken shell net charge distributions:

p x (rM i) = (X;h} - Zx {Z^h} (3.3.23)

In this last expression we have two infinite summations, over h and n vectors.

We first note that, due to the localized nature o f the basis set, the total amount o f charge q x and q2 associated with the two overlap distributions {GjG2 g) and

{G3G4 n;h) (where Gj is the adjoined Gaussian o f the shell to which the AO

belongs) decays exponentially to zero with increasing Igl and Ini. A Coulomb overlap threshold Sc (or sc = -log1 0 S£) can be defined, such that when either qj

and q2 are smaller than Sc the two-electron integral is disregarded

The problem o f the h summation is more delicate. Each shell X can be classified in two different sets:

i) a long-range set o f h vectors that includes the interactions between the charge distribution X°j(r) %g2(r) and the A,-th shell distributions p x within the

cell h, when their penetration is smaller than a threshold SM (or

These interactions are evaluated by an Ewald type technique, after a multipole expansion o f p^

ii) a short-range set (B) o f h vectors, complementary to the long-range one, whose integrals are evaluated exactly.