Let us next consider how one can calculate one- electron elements of the long- range nuclear- electron potential:

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1 Hatee- Fock Long Range In the following assume the patition of 1/ into shot ange and long ance components 1 efc( / a) ef ( / a) + Hee a is a paamete detemining the patitioning between shot ange and long ange behaviou. In the following it is impotant that the Fouie tansfom of the long ange pat, fo a peiodic potential is known in a convenient fom: V() 1 ef ( / a) ei. d 1 ef ( / a) e i.( ) d 1 ef ( / a) all space e i. d 1 4πe 2 a 2 2 f a () It depends on the paamete a. Moeove the Fouie seies is apidly conveging due to the exponential in the Fouie coefficients. Let us also conside V(, R i ) 1 ef ( R i / a) ei. d R i 1 e i.(+r i ) ef ( / a) d 1 ef ( / a) e i.r i all space e i. d 1 4πe 2 a 2 2 f a ()e i.r i e i.r i Let us next conside how one can calculate one- electon elements of the long- ange nuclea- electon potential:

2 V qb * p0 dϕ p ( ( B) Z i f ( R i + ) i, * Z i dϕ p ( ( B)e i. e i.' f (' R i + ) i, Z i p0qb f a ()e i.r i i, p0qb ( Z i e i.r i f a ()) p0qb * ( Z i e i.r i f a ()) i i The Fouie Coefficients fo a poduct of aussian Basis functions play a cucial ole. We will assume that the unique integals listed above can be calculated. Let us now conside the so- called Coulomb pat of the ee contibution to Fock matix: F p s q p qs D F qb p0 F qb p0,s p0,c qb,sd D C,s,C,D p0,c qb,sd C D 0,s,C,D D d C d 1 2 )ϕ * C) f 2 B C, D 0 C, C d 1 d 2 ϕ p * )ϕ * ) f 2 + C B (D C)) D 0 d 1 d 2 )ϕ * )( f 2 + C) B D 0 d 1 d 2 B) f 2 )ϕ * C D 0 d 1 d 2 B)e i(2 1) f a ()ϕ *, D 0, p0qb D 0 p0qb [ p0qb f a (0) 0] f a () 0 * f a () 0 D 0

3 Hee the integals have been teated using a Fouie tansfom, simila to the nuclea potetnial tem. The quantity is squae backets is defined using the line above. Let us now conside the exchange tem to the long ange Fock matix. F p s q p sq D F qb p0 F qb p0,s p0,c SD,qB D C,s,C,D p0,c SD,qB C D 0,s,C,D D d C d 1 2 )ϕ * C) f a 2 ϕ q B) C, C, d 1 d 2 f a 2 )ϕ * C B)D C g,,c, d 1 e i(g+)1 f a (g + ) d 2 ei(g+) 2 ϕ * C B)D C g,,c, g,,c, g,c, (g + ) p0 f a (g + ) d 2 ei(g+)+c ) ϕ * (B C))D C (g + ) p0 * f a (g + ) g + 0q(B C) e ig.c D C ( (g + ) p0 * f a (g + ) g + 0q(B C) g,c, [ p0 f a (g) 0q(B C) ]e ig.c D C )e ig.c D C The final fomulas fo the full long ange pat of the fock matix take a faily simple fom: F p0 qb V p0 qb + [ p0qb f a (0) 0]D 0 [ ]e ig.c C D 0 w g p0 f a (g) 0q(B C) g,c, V qb p0 + g,c, w g [ p0qb f a (0) 0]D 0 [ p0 f a (g) CqB]D C whee the quantities in squae backets ae defined as [ p0 f a (g) 0qC) ] ( (g + ) p0 * f a (g + ) g + 0qC ) This gives us a good indication of how to tuncate the lattice. The integals involve the poduct of two O basis funtions, ϕ p ( ( C), one is centeed in the cental unit cell, the othe is located at the lattice vecto C. Wheneve this poduct vanishes

4 suffiently well (below a theshold) the integals vanish, and can be neglected. lso the sum ove the lattice sum is well behaved. f a (g + ) 1 4πe (g+)2 a 2 (g + ) 2 In the lage limit the sum apidly conveges. This detemines how many lattice vectos to include. The point 0 is poblematic. I think it is most consistent to emove the lattice vecto 0 fom all ecipocal lattice ssummations. The agument is a bit involved, and thee might have been discussion in the liteatue. To follow the agument let us conside the following quantity (fo a closed shell system): Q pqb 0 + g 0 + g C,B,C,D,0,g,,s,p,q p0qb g e ig.( C ) g 0 C,B,C,D,0,g,,s,p,q p0qb g e ig.( C ) g 0,B,C,D,0,g,,s,p,q L(S qb p0 (g)d qb p0 )(S 0 (g)d 0 ),g e ig. L(S qb p0 (0)D qb p0 )(S 0 (0)D 0 ) B,C,,p,q L *n 2 p0 el (SDSD) p0 L *n 2 el n el 1 L (Ln el (Ln el 1) (D qb p D C D C p D qb ) (D qb p0 D C 0 D C p D qb ) D qb p0 D 0 pqb g,b,c,d,0,g,,s,p,q S qb p0 (g)d qb S C p0 (g)d C S qb p0 (g)d qb S,g g C D C p D qb ) e ig. C p0 (g)d C e ig. This quantity is pecisely the numbe of electon pais pe unit cell, fo a Bon von Kaman cell with L units, and n el. We have used the idempotency condition of the density matix ove this full peiod: SDSD SD. The scaling of this quantity is weid as it is not a quantity that scales linealy. It deives fom the 0 pat of the sum. We take this as evidence that the 0 component of the long ange integals should be eliminated completely. It would be instuctive to calculate the 0 component to the long- ange HF enegy, including all tems: nuclea- nuclea, nuclea electon and electon electon. One would expect this to yield a constant, pehaps zeo. The above calculation can be epeated to obtain the enegy, inseting eveywhee an exta facto f a (g). The sum ove, still yields δ (g), and this means we get an exta facto of f a (0), which has a divegence. Let us also investigate the nuclea- electon attaction tem which would yield L *n el * * f a (0). The final tem aises fom the nuclea nuclea i Z i

5 attaction tem, which has the value 1 2 ( Z I ) 2 f 0 a 1 2 Z I. The main issue is I α π I that thee may be a bet divegence in the enegy fom the 0 tem, but in any case it is independent of the wave function. The tem is a tue constant, pehaps infinity. It does not depend on any details of the calculation. Theefoe we can safely ignoe it. In fact we should ignoe it. We should discad the divegent 0 tem fom the Fouie tansfom of the long ange pat. Long ange second quantized pat of the Hamiltonian: ˆV p,qb pqb p0qb p,qb p0qb p,qb p0qb p,qb p0 V Ne qc p,qc V p p,q,,s,,b,c,d V p p,q,,s,,b,c,d,g, V p p,q,,s,,b,c,d,g, V p p Ê qc+ p0 V Ne qc p,qb Ê p qc p,q,,s,,b,c,d,g, p,q,,s,c,d,g p,q,,s,,b,c,d,g pqb C ÊC pqb pc g + f a (g + ) g + qb ÊC pqb p0c g + f a (g + )e ig( B) * g + q0 B ÊC pqb [ p0c f (g) q0 B] e ig( B) Ê C [ p0c f (g) q0] e ig( B) pqb Ê C+,+B pqb [ p0c f (g) q0 B] e ig( B) pqb Ê C, This is the usual second quantized hamiltonian, including both lattice and site indices, but the matix elements ae expessed in an ieducible fashion, and indicate the minimum numbe of quantities that have to be calculated. Both of the final foms of the opeatos ae potentially useful. In the penultimate expession the lattice summations ove C,D ae shot ange. They ae detemined by the Fouie tansfoms. The lattice sums ove, B can be long ange. The many- body theoy has to tuncate these tems. If we have connected tems only in equations and the extenal labels ae limited to the shot- ange lattice, then, since the ieducible H matix elements ae shot- ange, I think eveything educes to shot ange labels.

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