Molecular Integral Evaluation. The 9th Sostrup Summer School. Quantum Chemistry and Molecular Properties June 25 July 7, 2006.

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1 1 Molecular Integral Evaluation The 9th Sostrup Summer School Quantum Chemistry and Molecular Properties June 25 July 7, 2006 Trygve Helgaker Department of Chemistry, University of Oslo, Norway Overview one-electron integrals two-electron integrals Coulomb evaluation in large systems

2 2 Molecular integral evaluation one-electron interactions overlap, multipole-moment, momentum, and kinetic-energy integrals Coulomb attraction integrals Z O µν = χ µ (r)ô(r)χ ν(r)dr two-electron interactions Coulomb integrals g µνλσ = Coulomb potential ZZ χµ (r 1 )χ ν (r 1 )χ λ (r 2 )χ σ (r 2 ) r 12 dr 1 dr 2 J µν = X λσ g µνλσ D λσ one-electron functions primitive Cartesian GTOs integration schemes McMurchie Davidson, Obara Saika and Rys schemes

3 3 Contracted spherical-harmonic GTOs We shall discuss integration over primitive Cartesian GTOs: G ijk (r A, a) = x i Ay j A zk A exp ` ara 2 Final integrals are needed over contracted spherical-harmonic GTOs: χ lm (r A ) = S lm (r A ) X d µ exp ` a µ r 2 A µ This may be accomplished by a simple linear transformation χ lm (r A ) = X X Cijk lm d µ G ijk (r A, a µ ) ijk µ The number of contracted spherical-harmonic GTOs is usually much smaller than the number of primitive Cartesian GTOs Properly implemented, the transformation to a contracted spherical-harmonic basis reduces CPU time (relative to that required for the corresponding primitive Cartesian basis).

4 4 Overview Cartesian and Hermite Gaussians properties of Cartesian Gaussians properties of Hermite Gaussians expansion of products of Cartesians in Hermite Gaussians Simple one-electron integrals overlap and multipole kinetic energy Coulomb integrals electrostatics of Gaussians the Boys function one- and two-electron Coulomb integrals over Cartesian Gaussians Coulomb potential for large systems density fitting multipole techniques

5 5 Cartesian Gaussians Primitive Cartesian Gaussians centered at A: 8 >< a > 0 G ijk (r, a,a) = x i Ay j A zk A exp( ara), 2 r A = r A >: i 0, j 0, k 0 orbital exponent electronic coordinates quantum numbers total angular-momentum quantum number l = i + j + k 0 simple nodal structure, with a single node at the origin The full set of Gaussians of a given a and l constitutes a shell of GTOs: each shell contains (l + 1)(l + 2)/2 GTOs s shell: G 000 p shell: G 100, G 010, G 001 d shell: G 200, G 110, G 101, G 020, G 011, G 002

6 6 Properties of Cartesian Gaussians Definition of a primitive Cartesian Gaussian: G ijk (a,r A ) = x i Ay j A zk A exp( ar 2 A), r A = r A Each Gaussian factorizes in the Cartesian directions: G ijk (a,r A ) = G i (a, x A )G j (a, y A )G k (a, z A ) G i (a, x A ) = x i A exp( ax 2 A) note: this is not true for the solid-harmonic Gaussians nor for STOs Differentiation of Gaussians: G i (a, x A ) = G i(a, x A ) A x x = 2aG i+1 (a, x A ) ig i 1 (a, x A ) a linear combination two undifferentiated Gaussians Gaussian recurrence relation: x A G i (a, x A ) = G i+1 (a, x A )

7 7 The Hermite Gaussians are defined as Hermite Gaussians Λ tuv (r, p,p) = «t «u P x P y P z «v exp ` pr 2 P, rp = r P The Hermite Gaussians are not used as basis functions by themselves rather, they are used as intermediates in the evaluation of Gaussian integrals Cartesian product G i (x A )G j (x B ) = i+j X t=0 E ij t Λ t (x P ) Hermite expansion useful for integration Z π t (x) dx = δ t0r Λ p McMurchie and Davidson (1978) We shall make extensive use of the McMurchie Davidson scheme for integration other methods: Obara Saika, Rys quadrature

8 8 Properties of Hermite Gaussians Definition of Hermite Gaussians: «t «u Λ tuv (r, p,p) = P x P y P z «v exp ` pr 2 P, rp = r P Like Cartesian Gaussians, they also factorize in the Cartesian directions: Λ tuv (r P ) = Λ t (x P )Λ u (y P )Λ v (z P ) Λ t (x P ) = ( / P x ) t exp ` px 2 P Different nodal structure: 1 Λ t has t nodes Λ t+1 = Λ t x P symmetric and antisymmetric what is the relationship to Hermite polynomials?

9 9 Relationship to harmonic-oscillator (HO) functions Rodrigues expression for Hermite polynomials slight rearrangement H t (x) = ( 1) t exp `x 2 d t H t ( px P ) exp ` px 2 P = exp ` x 2 dxt d t d ` exp ` px 2 P ppx t Hermite Gaussians are therefore given by Λ t = p t/2 H t ( px P ) exp ` px 2 P {z } {z } Hermite polynomial Gaussian the standard harmonic-oscillator (HO) functions are given by Ψ t = H t ( «px P ) exp px2 P 2 differ from Hermite Gaussians by factor of one half in the Gaussian

10 10 Hermite recurrence relation An important property of the Hermite Gaussians is the recurrence relation The proof is simple: x P Λ t = 1 2p Λ t+1 + tλ t 1 from the definition of Hermite Gaussians, we have «t «t Λ t+1 = exp( px 2 P) = 2p x P Λ 0 P x P x P x inserting the identity «t «t «t 1 x P = x P t P x P x P x we then obtain h t t 1i Λ t+1 = 2p x P t Λ 0 = 2p (x P Λ t tλ t 1 ) P x P x

11 11 definition Comparison of Cartesian and Hermite Gaussians Cartesian Gaussians G i = x i A exp( ) ax 2 A Hermite Gaussians Λ t = t P t x exp ( px 2 P recurrence x A G i = G i+1 x P Λ t = 1 2p Λ t+1 + tλ t 1 differentiation G i A x = 2aG i+1 ig i 1 Λ t P x = Λ t+1 )

12 12 Integration over Hermite Gaussians From the definition of Hermite Gaussians, we have Z Λ t (x) dx = Z P x «t exp ` px 2 P dx We now change the order of differentiation and integration (Leibniz rule): Z Λ t (x) dx = «t Z exp ` px 2 P dx P x The Gaussian integral is given by Z exp ` px r 2 π P dx = p Since the integral is independent of P, differentiation with respect to P gives zero: Z π t (x)dx = δ t0r Λ p only integrals over s functions do not vanish

13 13 What we have done so far Properties of Cartesian Gaussians G i (x A ) = x i A exp ` ax 2 A Properties of Hermite Gaussians Λ t (x P ) = t P t x exp ` px 2 P What we are going to do The Gaussian product rule exp ` ax 2 2 A exp ` bx B = KAB exp ` px 2 P Overlap distributions expanded in Hermite Gaussians: G i (x A ) G j (x B ) = X t E ij t Λ t (x P ) Calculation of integrals

14 14 The Gaussian product rule The product of two Gaussians is another Gaussian centered somewhere on the line segment connecting the original Gaussians; its exponent is the sum of the original exponents: exp 2 ` bx B = exp 2 ` µxab where exp ` ax 2 A {z } exponential prefactor exp ` px 2 P {z } product Gaussian P x = aa x + bb x p p = a + b X AB = A x B x, µ = ab a + b center of mass total exponent relative separation reduced exponent The Gaussian product rule greatly simplifies the calculation of integrals two-center integrals are reduced to one-center integrals four-center integrals are reduced to two-center integrals

15 15 Overlap distributions The product of two Cartesian Gaussians is known as an overlap distribution: Ω ij (x) = G i (x, a, A x )G j (x, b, B x ) The Gaussian product rule reduces two-center integrals to one-center integrals Z Ω ij (x) dx = K AB Z x i Ax j B exp ` px 2 P dx the Cartesian monomials still make the integration awkward To simplify integration, we expand overlap distributions in Hermite Gaussians overlap distribution Ω ij (x) = X i+j E ij t=0 t Λ t (x P ) in Hermite Gaussians note: Ω ij (x) is a Gaussian times a polynomial in x of degree i + j it may be exactly represented as a linear combination of Λ t with 0 t i + j recall: integration over Hermite Gaussians Λ t is simple: R Λ t(x)dx = δ 0t q π p Our task now is to determine the expansion coefficients E ij t

16 16 Hermite expansion of Ω i+1,j : Recurrence relations for Hermite coefficients Ω i+1,j = K AB x i+1 A x j B exp ` prp 2 = X t E i+1,j t Λ t Alternative Hermite expansion of Ω i+1,j (with use of x P Λ t = 1 2p Λ t+1 + tλ t 1 ) Ω i+1,j = x A Ω ij = (x A x )Ω ij = (x P x )Ω ij + (P x A x )Ω ij = x P Ω ij + X P A Ω ij X E ij t x P Λ t + X P A E ij t Λ t = X t = X t = X t E ij t h t 1 Λ 2p t+1 + tλ t 1 + X P A Λ t 1 2p Eij t 1 + X P AE ij t i + (t + 1)E ij t+1 Λ t A comparison of the two expansions yields the recurrence relations E i+1,j t = 1 2p Eij t 1 + X P A E ij t + (t + 1)E ij t+1

17 17 Recurrence relations: example Recurrence relations E 00 0 = K AB E i+1,j t = 1 2p Eij t 1 + X P A E ij t E i,j+1 t = 1 2p Eij t 1 + X P B E ij t + (t + 1)E ij t+1 + (t + 1)E ij t+1 note: E ij t = 0 for t < 0 and for t > i + j Simple example: Ω 10 (x) = E 10 1 Λ 1 (x) + E 10 0 Λ 0 (x) E 00 0 = K AB E1 10 = 1 2p E00 0 = 1 K 2p AB E 10 0 = X P A E 00 0 = X P A K AB Final expansion Ω 10 (x) = 1 2p K ABΛ 1 (x) + X AB K AB Λ 0 (x)

18 18 Overlap integrals Overlap integrals S ab = G a G b G a = G ikm (r,a,a) = G i (x A )G k (y A )G m (z A ) G b = G jln (r,b,b) = G j (x B )G l (y B )G n (z B ) The overlap integral separates in the Cartesian directions: S ab = S ij S kl S mn, S ij = G i (x A ) G j (x B ) Use the Gaussian product rule and Hermite expansion: Z i+j X Z i+j X S ij = Ω ij (x)dx = Λ t (x P )dx = only one term survives t=0 E ij t The total overlap integral is therefore given by t=0 «3/2 π S ab = E ij 0 Ekl 0 E0 mn p E i+j t δ t0 q π p = Eij 0 q π p

19 19 Dipole integrals Multipole integrals follow the same scheme as overlap integrals Z D ij = G i (x A ) x C G j (x B ) = Ω ij (x P )x C dx = = = i+j X t=0 i+j X t=0 i+j X t=0 E ij t E ij t E ij t Z Z Λ t (x P )x c dx [x P Λ t (x P ) + X P C Λ t (x P )] dx Z h i 1 Λ 2p t+1(x P ) + X P C Λ t (x P ) + tλ t 1 (x P ) dx r r π π = E ij 0 X P C p + Eij 1 p Final dipole integral D ij = r π E ij 1 + X P C E ij 0 p

20 20 Kinetic-energy integrals Kinetic-energy integral T ab = 1 2 D G a 2 x y z 2 Gb E T ab = T ij S kl S mn + S ij T kl S mn + S ij S kl T mn S ij = G i (x A ) G j (x B ) D E T ij = 1 G 2 i (x A ) 2 x 2 Gj (x B ) Differentiation of Cartesian Gaussians x G j(x B ) = 2bG j+1 + jg j 1 2 x 2 G j(x B ) = 4b 2 G j+2 2b(2j + 1)G j + j(j 1)G j 2 the second derivative is linear combination of three undifferentiated Gaussians Kinetic-energy integrals are linear combinations of overlap integrals T ij = 4b 2 S i,j+2 2b(2j + 1)S i,j + j(j 1)S i,j 2

21 21 Summary Gaussian product rule exp ` ax 2 2 A exp ` bx B = KAB exp ` px 2 P Expansion of overlap distributions in Hermite Gaussians Ω ij = G i (x A )G j (x B ) = i+j X t=0 E ij t Λ t (x P ) recurrence relations for E ij t coefficients Simple integrals overlap integrals «3/2 π S ab = E ij 0 E0 kl E0 mn p kinetic-energy integrals as linear combinations of overlap integrals

22 22 Coulomb integrals One-electron nuclear-attraction integrals Two-electron repulsion integrals Ga (r A ) r 1 C Ga (r 1A )G b (r 1B ) r 1 12 Coulomb integrals are not separable Gb (r B ) Gc (r 2C )G d (r 2D ) The Gaussian Coulomb integrals may be reduced to a one-dimensional finite integral: the Boys function a special case of the Kummer confluent hypergeometric function makes the evaluation of Gaussian Coulomb integrals relatively simple this is not true for STOs The relative ease of evaluating Coulomb integrals over Gaussians was the main motivation for their introduction

23 23 Cartesian Coulomb integrals I The Coulomb potential integrals are given by V ab = G r 1 a C Gb = Z Ωab (r) r C dr The overlap distribution factorizes in the Cartesian directions Ω ab (r) = Ω ij (x)ω kl (y)ω mn (z) The overlap distributions are expanded in Hermite Gaussians Ω ij (x) = P i+j t=0 Eij t Λ t (x P ) The total overlap distribution may therefore be written in the form Ω ab (r) = X tuv E ij t E kl u Ev mn Λ tuv (r P ) = X tuv E ab tuvλ tuv (r P ) Expansion of two-center Coulomb integrals in one-center Hermite integrals: two-center integral V ab = X Z Etuv ab Λtuv (r P ) dr one-center integrals r C tuv

24 24 Cartesian Coulomb integrals II We have expanded two-center one-electron integrals in one-center integrals: V ab = X tuv E ab tuv Z Λtuv (r P ) r C dr We may likewise expand four-center two-electron integrals in two-center integrals: Z Z Ωab (r 1 )Ω cd (r 2 ) g abcd = dr 1 dr 2 r 12 = X X Z Z Etuv ab Eτνφ cd Λtuv (r 1P )Λ τνφ (r 2Q ) dr 1 dr 2 r 12 tuv τνφ Therefore, we need only consider integrals over Hermite Gaussians first, we consider integrals over s functions (Λ 000 (r p ) = exp( pr 2 P)) V p (C) = Z exp( pr 2 P ) r C dr, V pq = Z exp( pr 2 1P ) exp( qr 2 2Q) r 12 dr 1 dr 2 next, we consider integrals over general Hermite Gaussians

25 25 1. The presence of r 1 C Coulomb integrals over spherical charge distributions I V p = Z r 1 C exp ` pr 2 P dr is awkward and is avoided by the substitution 1 r C = 1 π Z exp ` r 2 Ct 2 dt to yield the four-dimensional integral V p = 1 Z exp ` pr 2 Z P exp ` t 2 rc 2 dt dr π 2. To prepare for integration over r, we invoke the Gaussian product rule exp ` prp 2 exp ` t 2 rc 2 pt 2 = exp` p + t 2 R2 CP exp ˆ `p + t 2 rs 2 to obtain (where the exact value of S in r S does not matter): V p = 1 Z Z expˆ (p + t 2 )rs 2 drexp pt2 π p + t 2 R2 CP dt

26 26 Coulomb integrals over spherical charge distributions II V p = 1 π Z Z ˆ (p + t 2 )rs 2 dr exp pt2 p + t 2 R2 CP dt 3. Integration over all space r now yields a one-dimensional integral V p = 2 Z «3/2 π exp pr 2 t 2 π p + t 2 CP dt p + t To introduce a finite integration range, we perform the substitution u 2 = «t2 t p + t pt 2 = u 2 pt 3 dt = u 3 du dt = p 1 2 3/2 du u 2 and obtain V p = 2π p Z 1 0 exp ` pr 2 CPu 2 du we have thus reduced a three-dimensional integral over all space to a one-dimensional integral over [0, 1]

27 27 Coulomb integrals over spherical charge distributions III For the two-electron integrals, we proceed in the same manner: Z Z exp( pr 2 1P )exp( qr 2 Z 2Q) V pq = dr 1 dr 2 = V p (r 2 )exp( qr2q)dr 2 2 r 12 insert derived expression for V p (r 2 ) and use the Gaussian product rule Introducing the Boys function F 0 (x) = Z 1 0 exp ` xt 2 dt the final results for the Coulomb potential and interaction integrals are now: Z exp 2 ` prp dr 1 = 2π r C p F 2 0 `prp C ZZ exp 2 2 «` pr1p exp ` qr2q dr 1 dr 2 = 2π5/2 pq r 12 pq p + q F 0 p + q R2 P Q we must now learn to evaluate the Boys function

28 28 Gaussian electrostatics The previous results may be written compactly in terms of the error function erf(x) = 2 π Z x 0 exp ` t 2 dt = 2 π xf 0 `x2 Introducing unit charge distributions and the reduced exponent as p 3/2 ρ p (r P ) = exp 2 pq ` prp, α = π p + q we obtain formulas similar to those of electrostatics Z ρp (r P ) r C dr 1 = erf ` prp C R P C ZZ ρp (r 1P ) ρ q (r 2Q ) r 12 dr 1 dr 2 = erf ( αr P Q ) R P Q

29 29 The Boys function F n (x) F n (x) > 0 since the integrand is positive: F n (x) = R 1 0 exp ` xt 2 t 2n dt > 0 F n (x) is concave and decreasing: F n(x) = F n+1 (x) < 0 F n(x) = F n+2 (x) > 0 F n (x) decreases with increasing n: F n (x) F n+1 (x) = Z 1 Special value and asymptotic form: F n (0) = F n (x) Z 1 0 Z 0 0 t 2n dt = exp ` xt 2 t 2n `1 t 2 dt > 0 F n (x) > F n+1 (x) 1 2n + 1 exp ` xt 2 t 2n dt = (2n 1)!! 2 n+1 r π x 2n+1 (x large)

30 30 Evaluation of the Boys function F n (x) = R 1 0 exp ` xt 2 t 2n dt For large x, we use the asymptotic form F n (x) = (2n 1)!! 2 n+1 r π x 2n+1 For small x, we Taylor expand about x = 0: X ( x) k F n (x) = k!(2n + 2k + 1) k=0 For intermediate x, we Taylor expand about pretabulated values of F n (x) We use recurrence relations: F n+1 (x) = (2n + 1)F n(x) exp( x) 2x F n 1 (x) = 2xF n(x) + exp( x) 2n 1 it is sufficient to calculate F n (x) for one value of n then use recursion upward recursion is numerically unstable

31 31 We are almost there... We know how to expand Cartesian integrals in Hermite integrals V ab = X Z Etuv ab Λtuv (r P ) dr r C tuv where we know how to calculate the E coefficients E i+1,j t = 1 2p Eij t 1 + X P AE ij t + (t + 1)E ij t+1 We know how to do Hermite integrals for s functions Z Λ000 (r) dr = 2π r C p F 2 0 `prp C where we know how to calculate the Boys function F n (x) = Z 1 0 exp( xt 2 )t 2n dt But we do not know how to calculate general Hermite integrals Z Λtuv (r) r C dr =?

32 32 Coulomb integrals over Hermite Gaussians The Coulomb integral over a one-center spherical Gaussian is given by Z exp( pr 2 P ) dτ = 2p π F 2 0 `prp C r C We now return to the more general Coulomb integral over a Hermite Gaussian Λ tuv = t+u+v exp ` pr P 2 Px P t y u Pz v Changing the order of integration and differentiation by Leibniz rule, we obtain Z Λtuv dτ = 2p t+u+v Z exp( pr 2 P ) dτ = 2π r C π Px P t y u Pz v r C p R tuv where the one-center Hermite Coulomb integral is given by R tuv (p,r P C ) = t+u+v F 0 `pr 2 P C P t x P u y P v z the Hermite integrals are derivatives of the Boys function the final two-center Coulomb integrals are to be expanded in Hermite integrals

33 33 Two-electron Coulomb integrals over Hermite Gaussians The two-electron integrals may be done in a similar manner Z Z Λtuv (r 1P )Λ τνφ (r 2Q ) dr 1 dr 2 r 12 t+u+v τ+ν+φ Z Z exp( pr 2 1P ) exp( qr2q) 2 = dr Px P t y u Pz v Q τ x Q ν y Q φ 1 dr 2 z r 12 = t+u+v P t x P u y P v z = 2π5/2 pq p + q τ+ν+φ Q τ x Q ν y Q φ z t+u+v P t x P u y P v z 2π 5/2 pq p + q F 2 0 `αrp Q ( 1) τ+ν+φ τ+ν+φ 2 F Px τ Py ν Pz φ 0 `αrp Q = ( 1) τ+ν+φ 2π 5/2 pq p + q R t+τ,u+ν,v+φ(α,r P Q ) where we have used Leibniz rule and previous results for spherical charges one- and two-electron Coulomb integrals both calculated from R tuv (a,a) We must learn to calculated R tuv (a,a).

34 34 Evaluation of Hermite Coulomb integrals The Hermite integrals are derivatives of the Boys function can be obtained by repeated differentiation, using F n(x) = F n+1 (x) recursion is simpler and more efficient To set up recursion, we introduce the auxiliary Hermite integrals R n tuv(a,a) = ( 2a) n t+u+v F n `aa 2 A t x A u y A v z which include as special cases the source and target integrals R n 000 = ( 2a)F n R 0 tuv = t A t x u A u y v A v z F 0 The Hermite integrals can now be generated from the recurrence relations Rt+1,u,v n = tr n+1 t 1,u,v + A x R n+1 tuv Rt,u+1,v n = ur n+1 t,u 1,v + A y R n+1 tuv Rt,u,v+1 n = vr n+1 t,u,v 1 + A z R n+1 tuv

35 35 Proof of the Hermite recurrence relations The auxiliary Hermite integrals are given by R n tuv(a,a) = ( 2a) n t x u y v zf n (aa 2 ) where for example t x indicates t times differentiation with respect to A x. Incrementing v now gives Rt,u,v+1 n = ( 2a) n x t y u z v+1 F n = ( 2a) n t x u y v z( 2aA z ) F n+1 = ( 2a) n+1 x t y u za v z F n+1 = ( 2a) n+1 x t y u v 1 `v z + A z z v Fn+1 = vr n+1 t,u,v 1 + A z R n+1 tuv where in the second line we have used F n(x) = F n+1 (x) and the chain rule. The recurrence relations for t and u are demonstrated in the same manner: Rt+1,u,v n = tr n+1 t 1,u,v + A x R n+1 tuv Rt,u+1,v n = ur n+1 t,u 1,v + A y R n+1 tuv

36 36 Example: R tuv when t + u + v 2 1. Calculate F 2 and then F 1 and F 0 by downward recursion. 2. Generate R000 n R000 0 = F 0, R000 1 = 2aF 1, R000 2 = ( 2a) 2 F 2 3. Generate Rtuv 0 with t + u + v = 1: R100 0 = A x R000, 1 R010 0 = A y R000, 1 R001 0 = A z R Generate Rtuv 1 with t + u + v = 1: R100 1 = A x R000, 2 R010 1 = A y R000, 2 R001 1 = A z R Generate Rtuv 0 with t + u + v = 2: R200 0 = A x R R000, 1 R020 0 = A y R R000, 1 R002 0 = A z R R000 1 R110 0 = A x R010, 1 R101 0 = A x R001, 1 R011 0 = A y R001 1

37 37 Final Coulomb integrals One-electron integrals Z Ga r 1 Ωab C Gb = dr r C = X tuv E ab tuv Z Λtuv r C dr = 2π p X tuv E ab tuvr tuv (p,r P C ) Two-electron integrals Ga (a)g b (1) r 1 12 Gc (2)G d (2) = = X tuv E ab tuv X τνφ = 2π5/2 pq p + q E cd τνφ X tuv Z Z Ωab (1)Ω cd (2) dr 1 dr 2 r 12 Z Z Λtuv (1)Λ τνφ (2) dr 1 dr 2 r 12 E ab tuv X τνφ E cd τνφ( 1) τ+ν+φ R t+τ,u+ν,v+φ (α,r P Q )

38 38 Summary one-electron Coulomb integral evaluation 1. Calculate expansion coefficients E0 00 = exp ` µxab 2 2. Calculate the Boys function E i+1,j t = 1 2p Eij t 1 + X P AE ij t + (t + 1)E ij t+1 F n (x) = Z 1 0 exp ` xt 2 t 2n dt (with x = pr 2 P C) F n (x) = 2xF n+1(x) + exp( x) 2n Calculated Hermite integrals 4. Calculate Cartesian integrals R n 000 = ( 2p) n F n R n t+1,u,v = X P C R n+1 tuv + tr n+1 t 1,u,v Ga r 1 C X Gb = tuv E ij t E kl u Ev mn Rtuv 0

39 39 Evaluation of two-electron contributions Fock/Kohn Sham matrices We have derived a general expression for two-electron integrals g abcd = 2π5/2 pq p + q X tuv E ab tuv X τνφ E cd τνφ( 1) τ+ν+φ R t+τ,u+ν,v+φ (α,r P Q ) In SCF theories, we these integrals make contribution to the Fock/KS matrix: F ab = X ` 2gabcd D cd g acbd D cd {z } {z } cd Coulomb exchange in early days, one would write all integrals to disk and read back as required in direct SCF theories, integrals are calculated as needed this development (1980) made much larger calculations possible Many developments have since improved the efficiency of direct SCF efficient screening of integrals early contraction with density matrices (Coulomb) density fitting (Coulomb) multipole methods (Coulomb)

40 40 Scaling properties Product of two s functions from Gaussian product rule exp ` ara 2 2 exp ` brb = exp ` ab/(a + b)r 2 AB exp ` (a + b)r 2 P Overlap integral between two s orbitals: π S ab = a + b «3/2 exp ab «a + b R2 AB the number of such integrals scales quadratically with system size however, S ab decreases rapidly with R AB Let us assume that we may neglect all integrals S ab < 10 k insignificant integrals We may neglect all integrals separated by more than r h π 310 i R AB > a 1 min ln 2k 2a min in a large system, most integrals becomes small and may be neglected the number of significant integrals increases linearly with system size

41 41 The sparsity of overlap and electron-repulsion integrals linear system of 16 Gaussian 1s functions of unit exponent, separated by 1a 0

42 42 Two-electron ssss integrals: Two-electron integral scaling g abcd = erf ` αr P Q S ab S cd R P Q the total number of integrals scales quartically with system size the number of significant integrals scales quadratically S ab, S cd 0 rapidly R 1 P Q 0 very slowly Decompose integral into classical and nonclassical parts: g abcd = S abs cd R P Q {z } classical erfc ` αr P Q S ab S cd R P Q {z } nonclassical quadratic scaling of classical part, can be treated by multipole methods linear scaling of nonclassical part since S ab 0, S cd 0, erfc 0 rapidly erfc( αr P Q ) exp( αr2 P Q) παrp O

43 43 The scaling properties of molecular integrals linear system of up to 16 1s GTOs of unit exponent, separated by 1a overlap n kinetic energy n n n nuclear attraction n 3 n 2 n electron repulsion n 4 n 2 n

44 44 The sparsity of overlap and electron-repulsion integrals linear system of 16 Gaussian 1s functions of unit exponent, separated by 1a 0

45 45 Sparsity in linear alkanes and alkenes Percentage of matrix elements greater than 10 6 in alkane and alkene chains overlap matrix: - sparse 100 LDA alkanes 100 HF alkanes density matrix: - nonsparse for alkenes - sparse for alkanes Fock/KS matrices: KS matrix like overlap - Fock matrix intermediate between overlap and LDA alkenes HF alkenes density matrices

46 46 Integral prescreening Small integrals (< ) are not needed and should be avoided by some prescreening technique The two-electron integrals g abcd = Z Z Ωab (1)Ω cd (2) r 12 dr 1 dr 2 are the elements of a positive definite matrix with diagonal elements g ab,ab 0 The conditions for an inner product are thus satisfied The Cauchy Schwarz inequality yields Precalculated g ab,ab g ab,ab gcd,cd and prescreen G ab = g ab,ab g ab,cd G ab G cd

47 47 Calculation of Coulomb potential The traditional Coulomb-potential evaluation is a two-step procedure 1. evaluate significant two-electron integrals (in batches) g abcd = 2π5/2 pq p + q X tuv X Etuv ab τνφ E cd τνφ( 1) τ+ν+φ R t+τ,u+ν,v+φ (α,r P Q ) 2. add their contribution to the Fock or Kohn Sham matrix: J ab = P cd g abcdd cd The cost of the different steps (L is ang. mom., p number of primitives): Boys E coef R int τνφ cont S lm tuv cont S lm J ab Lp 4 L 3 p 2 L 4 p 4 L 10 p 4 L 7 p 4 L 8 p 2 L 9 p 2 L 6 p 2 L 7 L 4 for low L, evaluation of the Boys function dominates for large L, the τνφ contraction dominates It is not necessary to assemble the integrals fully before contraction with D cd

48 48 Early contraction with density matrix for Coulomb potential Two-electron Coulomb contribution to Fock/Kohn Sham matrix: J ab = 2π5/2 pq p + q X tuv E ab tuv X X τνφ Early contraction of density matrix with E coefficients: K Q τνφ = X ( 1)τ+ν+φ D cd Eτνφ cd J ab = 2π5/2 pq p + q cd Q X tuv cd E ab tuv D cd E cd τνφ( 1) τ+ν+φ R t+τ,u+ν,v+φ (α,r P Q ) X X K Q τνφ R t+τ,u+ν,v+φ(α,r P Q ) the τνφ (tuv) cost reduced from L 10 p 4 to L 6 p 4 (from L 9 p 2 to L 7 p 2 ) Timings (seconds) for benzene: Q τνφ cc-pvdz cc-pvtz cc-pvqz late contraction early contraction

49 49 Density fitting Traditionally, the electron density is expanded in overlap distributions ρ(r) = P ab D ab Ω ab (r) the number of terms is n 2, where n is the number of AOs The Coulomb contribution to the Fock/KS matrix is evaluated as J ab = Ω ab (r 1 ) ρ(r 2 ) r 1 12 the formal cost of this evaluation is therefore quartic (n 4 ) Consider now an approximate density eρ(r), expanded in an auxiliary basis ω α (r): eρ(r) = P α c αω α (r), eρ(r) ρ(r) the size of the auxiliary basis N increases linearly with system size We may now evaluate the Coulomb contribution approximately as ej ab = Ω ab (r 1 ) r 1 12 eρ(r2 ) the cost of the evaluation is therefore cubic (n 2 N) if the eρ is sufficiently accurate and easy to obtain, this may give huge savings

50 50 Coulomb density fitting Consider the evaluation of the Coulomb potential J ab = P cd (ab cd) D cd We now introduce an auxiliary basis and invoke the resolution of identity: (ab cd) P αβ (ab α)(α β ) 1 (β cd) in a complete auxiliary basis, the two expressions are identical We may now calculate the Coulomb potential in two different ways: J ab = P cd (ab cd) D cd = (ab ρ) P P cd αβ (ab α) (α β ) 1 (β cd) D cd = P α (ab α) c α = (ab eρ) = J e ab where the coefficients c α are obtained from the linear sets of equations P α (β α) c α = P cd (β cd) D cd (β eρ) = (β ρ) no four-center integrals, the formal cost cubic or smaller note: the solution of the linear equations scales cubically This particular approach to density fitting is called Coulomb density fitting

51 51 Robust density fitting In Coulomb density fitting, the exact and fitted Coulomb matrices are given by J ab = (ab ρ), Jab e = (ab eρ), (α eρ) = (α ρ) We may then calculate the exact and density-fitted Coulomb energies as E = P ab J abd ab = (ρ ρ), E e = Pab J e ab D ab = (ρ eρ) The true Coulomb energy is an upper bound to the density-fitted energy: E E e = (ρ ρ) (ρ eρ) = (ρ ρ) (ρ eρ) (eρ ρ) + (eρ eρ) = (ρ eρ ρ eρ) the error in the density-fitted energy is quadratic in the error in the density Coulomb fitting is thus equivalent to minimization of E E e = (ρ eρ ρ eρ) It is possible to determine the approximate density in other ways quadratic energy error is ensured by using the robust formula ee = (eρ ρ) + (ρ eρ) (eρ eρ) = E (ρ eρ ρ eρ) however, the error in the energy is no longer minimized

52 52 Density fitting: sample calculations no symmetry used; exact energy: E h cc-pvdz cc-pvtz cc-pvqz cc-pv5z exact density late na exact density early fitted density exact energy na density-fitted energy Clearly, very large gains can be achieved with density fitting Formal scaling is cubical for large systems, screening yields quadratic scaling for integral evaluation the cubic cost of solving linear equations remains for large systems Linear scaling is achieved by boxed density-fitting and fast multipole methods the density is partitioned into boxes, which are fitted one at a time fast multipole methods for matrix elements

53 53 Coulomb integral Multipole expansion g abcd = Z Z Ωab (1)Ω cd (2) r 12 dr 1 dr 2 If Ω ab and Ω cd do not overlap significantly, we can use multipole expansion: g abcd = X qlmt ab lm,jk qjk cd a b R lmjk multipole vector and interaction matrix Z qlm ab = Ω ab (r)r lm (r P )dr regular and irregular solid harmonics T lm,jk = ( i) j I l+j,m+k(r QP ) R lm r l Y lm I lm r l 1 Y lm Can be applied to individual contracted integrals but also to more complicated electron distributions

54 54 Multipole method FF FF FF FF FF FF FF FF Divide molecule into boxes Create multipole expansions in each box Evaluate interactions between nonneighbouring boxes by multipole expansion FF FF FF FF FF FF FF FF FF FF FF NN NN NN FF FF FF FF FF NN C NN FF FF FF FF FF NN NN NN FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF FF Multipole expansions are expansions in r R = size of boxes their separation we can use larger boxes for more distant interactions at no loss of accuracy Taking advantage of this, we can calculate classical Coulomb interactions at costs N log N tree code N fast multipole method (FMM) rather than N 2, where N is the number of boxes

55 55 Illustration: alanine residue peptides Features of the code diagonalization-free trust-region Roothaan Hall (TRRH) energy minimization trust-region density-subspace minimization (TRDSM) for density averaging boxed density-fitting with FMM for Coulomb evaluation (Simen Reine) LinK for exact exchange, linear-scaling exchange-correlation evaluation compressed sparse-row (CSR) representation of few-atom blocks alanine residue peptides CPU time against atoms HF/6-31G 5th SCF iteration dominated by exchange RH step least expensive full lines: sparse algebra dashed lines: dens algebra exchange Coulomb DSM RH

56 56 Exchange in large systems In Hartree Fock theory, we also need the exchange contribution F ab = X ` 2gabcd D cd g acbd D cd {z } {z } cd Coulomb exchange The treatement of exchange differs from that of the Coulomb contribution density fitting and multipole expansions cannot (easily) be used for exchange Still, it is possible to reduce cost and achieve linear scaling by screening K ac = X bd erf ( αr P Q ) R P Q S ab D bd S cd the factor erf ( αr P Q ) R 1 P Q decays slowly with R P Q the overlap factors S ab and S cd decay rapidly, giving quadratic scaling the density-matrix elements D bd decay fairly rapidly in nonmetals Since techniques such as density fitting and early density matrix contraction cannot be applied, the evaluation of exchange is expensive