Molecular Integral Evaluation

Molecular Integral Evaluation Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 13th Sostrup Summer School Quantum Chemistry and Molecular Properties July 6 18 2014 Trygve Helgaker (CTCC, University of Oslo) Molecular Integral Evaluation 13th Sostrup Summer School (2014) 1 / 34

Molecular integral evaluation one-electron integrals overlap, multipole-moment, and kinetic-energy integrals nuclear-attraction integrals O ab = χ a(r)ô(r)χ b(r) dr two-electron integrals Coulomb integrals χa(r1 )χ b (r 1 )χ c(r 2 )χ d (r 2 ) g abcd = dr 1 dr 2 r 12 Coulomb and exchange contributions to Fock/KS matrix F ab = ( ) 2gabcd D cd g acbd D cd cd basis functions primitive Cartesian GTOs integration schemes McMurchie Davidson, Obara Saika and Rys schemes screening, density fitting, fast multipole methods Trygve Helgaker (CTCC, University of Oslo) Introduction 13th Sostrup Summer School (2014) 2 / 34

Overview 1 Cartesian and Hermite Gaussians properties of Cartesian Gaussians properties of Hermite Gaussians 2 Simple one-electron integrals Gaussian product rule and overlap distributions overlap distributions expanded in Hermite Gaussians overlap and kinetic-energy integrals 3 Coulomb integrals Gaussian electrostatics and the Boys function one- and two-electron Coulomb integrals over Cartesian Gaussians 4 Sparsity and screening Cauchy Schwarz screening 5 Coulomb potential early density-matrix contraction density fitting Trygve Helgaker (CTCC, University of Oslo) Introduction 13th Sostrup Summer School (2014) 3 / 34

Cartesian Gaussian-type orbitals (GTOs) We shall consider integration over primitive Cartesian Gaussians centered at A: a > 0 G ijk (r, a, A) = xa i y j A zk A exp( ar A 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 Gaussians of a given l constitute a shell: 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 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( ax2 A ) note: this is not true for spherical-harmonic Gaussians nor for Slater-type orbitals Gaussians satisfy a simple recurrence relation: x A G i (a, x A ) = G i+1 (a, x A ) The differentiation of a Gaussian yields a linear combination of two Gaussians G i (a, x A ) = G i (a, x A ) = 2aG i+1 (a, x A ) ig i 1 (a, x A ) A x x Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 13th Sostrup Summer School (2014) 4 / 34

Hermite Gaussians The Hermite Gaussians are defined as ( ) 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 t(x P ) = ( / P x ) t exp ( px 2 P) a Gaussian times a polynomial of degree t Hermite Gaussians yield the same spherical-harmonic functions as do Cartesian Gaussians they may therefore be used as basis functions in place of Cartesian functions We shall only consider their use as intermediates in the evaluation of Gaussian integrals i+j Cartesian product G i (x A )G j (x B ) = E ij t Λt(x P) Hermite expansion t=0 such expansions are useful because of simple integration properties (next slide) McMurchie and Davidson (1978) Λ t(x) dx = δ t0 π p We shall make extensive use of the McMurchie Davidson scheme for integration Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 13th Sostrup Summer School (2014) 5 / 34

Integration over Hermite Gaussians From the definition of Hermite Gaussians, we have ( ) t Λ t(x) dx = exp ( px P P) 2 dx x We now change the order of differentiation and integration by Leibniz rule: ( ) t Λ t(x) dx = exp ( px P P) 2 dx x The basic Gaussian integral is given by exp ( pxp 2 ) π dx = p Since the integral is independent of P x, differentiation with respect to P x gives zero: Λ t(x) dx = δ t0 π p only integrals over Hermite s functions do not vanish Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 13th Sostrup Summer School (2014) 6 / 34

Hermite recurrence relation We recall the simple recurrence relation of Cartesian Gaussians: x A G i = G i+1 The corresponding Hermite recurrence relation is slightly more complicated: 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( pxp 2 P x P ) = 2p x P Λ 0 x P x inserting the identity ( we then obtain P x ) t x P = x P ( P x ) t ( ) t 1 t P x [ ( ) t ( ) ] t 1 Λ t+1 = 2p x P t Λ 0 = 2p (x P Λ t tλ t 1 ) P x P x Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 13th Sostrup Summer School (2014) 7 / 34

Comparison of Cartesian and Hermite Gaussians Cartesian Gaussians Hermite Gaussians definition G i = xa i exp ( axa 2 ) Λ t = t Px t exp ( pxp 2 ) recurrence x A G i = G i+1 x P Λ t = 1 2p Λ t+1 + tλ t 1 G differentiation i Λ = 2aG A x i+1 ig t i 1 = Λ P x t+1 2 2 1 1 3 2 1 1 2 3 1 3 2 1 1 2 3 1 2 2 Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 13th Sostrup Summer School (2014) 8 / 34

The Gaussian product rule An important property of Gaussians is the Gaussian product rule The product of two Gaussians is another Gaussian centered somewhere on the line connecting the original Gaussians; its exponent is the sum of the original exponents: where P x = exp ( axa 2 ) ( ) ( ) exp bx 2 B = exp µx 2 AB exp ( px 2 ) P }{{}}{{} exponential prefactor product Gaussian aax + bbx p p = a + b X AB = A x B x, µ = ab a + b center of mass total exponent relative separation reduced exponent 1 0.8 0.6 0.4 0.2-3 -2-1 1 2 3 The Gaussian product rule greatly simplifies integral evaluation two-center integrals are reduced to one-center integrals four-center integrals are reduced to two-center integrals Trygve Helgaker (CTCC, University of Oslo) Gaussian product rule and Hermite expansions 13th Sostrup Summer School (2014) 9 / 34

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 Ω ij (x) dx = KAB x xa i xj B exp ( pxp) 2 dx, K x AB = exp ( µxab 2 ) the Cartesian monomials still make the integration awkward we would like to utilize the simple integration properties of Hermite Gaussians We therefore expand Cartesian overlap distributions in Hermite Gaussians overlap distribution Ω ij (x) = i+j t=0 E ij t Λt(x P) Hermite Gaussians note: Ω ij (x) is a single 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 The expansion coefficients may be evaluated recursively from E 00 0 = K x AB : E i+1,j t = 1 2p E ij t 1 + X PAE ij t + (t + 1)E ij t+1 we shall now prove these recurrence relations Trygve Helgaker (CTCC, University of Oslo) Gaussian product rule and Hermite expansions 13th Sostrup Summer School (2014) 10 / 34

Recurrence relations for Hermite expansion coefficients A straighforward Hermite expansion of Ω i+1,j Ω i+1,j = KAB x xi+1 A xj B exp ( prp 2 An alternative Hermite expansion of Ω i+1,j Ω i+1,j = x A Ω ij = (x A x )Ω ij ) = t E i+1,j t Λ t = (x P x )Ω ij + (P x A x )Ω ij = x P Ω ij + X PA Ω ij = E ij t x PΛ t + X PA E ij t Λt t t = ( ) E ij 1 t 2p Λ t+1 + tλ t 1 + X PA Λ t t = [ ] 1 2p E ij ij t 1 + (t + 1)Et+1 + X PAE ij t Λ t t we have here used the recurrence x P Λ t = 1 2p Λ t+1 + tλ t 1 A comparison of the two expansions yields the recurrence relations E i+1,j t = 1 2p E ij t 1 + X PAE ij t + (t + 1)E ij t+1 Trygve Helgaker (CTCC, University of Oslo) Gaussian product rule and Hermite expansions 13th Sostrup Summer School (2014) 11 / 34

Overlap integrals We now have everything we need to evaluate 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 factorizes in the Cartesian directions: S ab = S ij S kl S mn, S ij = G i (x A ) Gj (x B ) Use the Gaussian product rule, the Hermite expansion, and the Hermite integration rule: S ij = i+j Ω ij (x) dx = only one term survives! t=0 E ij t The total overlap integral is therefore given by i+j Λ t(x P ) dx = S ab = E ij 0 E 0 kl E 0 mn t=0 ( ) π 3/2 p E ij t δ π t0 p = E ij π 0 p Trygve Helgaker (CTCC, University of Oslo) Simple one-electron integrals 13th Sostrup Summer School (2014) 12 / 34

Dipole-moment integrals Many other integrals may be evaluated in the same manner For example, dipole-moment integrals are obtained as D ij = G i (x A ) x C G j (x B ) = Ω ij (x P )x C dx i+j = E ij t Λ t(x P )x cdx (expand in Hermite Gaussians) t=0 i+j = E ij t [x P Λ t(x P ) + X PC Λ t(x P )] dx (use x C = x P + X PC ) t=0 i+j [ ] = E ij 1 t 2p Λ t+1(x P ) + X PC Λ t(x P ) + tλ t 1 (x P ) dx (use x P Λ t = 1 2p Λt+1 + tλt 1) t=0 = E ij π 0 X PC p + E ij π 1 p (s functions integration) The final dipole-moment integral is given by ( ) D ij = E ij 1 + X PC E ij π 0 p Trygve Helgaker (CTCC, University of Oslo) Simple one-electron integrals 13th Sostrup Summer School (2014) 13 / 34

Kinetic-energy integrals As a final example of non-coulomb integrals, we consider the kinetic-energy integrals: T ab = G 1 2 2 a x 2 + 2 y 2 + 2 z 2 Gb Differentiation of Cartesian Gaussians gives 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 ) Gj (x B ) T ij = 1 G 2 i (x A ) 2 x 2 Gj (x B ) d dx G j (x B ) = 2bG j+1 + jg j 1 d 2 dx 2 G j (x B ) = 4b 2 G j+2 2b(2j + 1)G j + j(j 1)G j 2 the first derivative is linear combination of two undifferentiated Gaussians the second derivative is linear combination of three undifferentiated Gaussians Kinetic-energy integrals are linear combinations of up to three overlap integrals: T ij = 2b 2 S i,j+2 + b(2j + 1)S i,j 1 2 j(j 1)S i,j 2 Trygve Helgaker (CTCC, University of Oslo) Simple one-electron integrals 13th Sostrup Summer School (2014) 14 / 34

Coulomb integrals We shall consider two types of Coulomb integrals: one-electron nuclear-attraction integrals G a(r A ) r 1 C G b (r B ) two-electron repulsion integrals G a(r 1A )G b (r 1B ) r 1 12 G c(r 2C )G d (r 2D ) Coulomb integrals are not separable in the Cartesian directions however, they may be reduced to a one-dimensional integral on a finite interval: 1 F n(x) = exp ( xt 2) t 2n dt the Boys function 0 this makes the evaluation of Gaussian Coulomb integrals relatively simple We begin by evaluating the one-electron spherical Coulomb integral: ( ) exp pr 2 V p = P dr = 2π r C p F ( ) 0 pr 2 PC we shall next go on to consider more general Coulomb integrals Trygve Helgaker (CTCC, University of Oslo) Coulomb integrals 13th Sostrup Summer School (2014) 15 / 34

Coulomb integral over a spherical Gaussian I We would like to evaluate the three-dimensional Coulomb-potential integral ( ) exp pr 2 V p = P dr r C 1 The presence of r 1 C is awkward and is avoided by the substitution 1 = 1 exp ( r r C π C 2 t2) dt Laplace transform to yield the four-dimensional integral V p = 1 exp ( pr 2 ) π P exp ( t 2 r 2 ) C dt dr 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 p + t 2 ) rs 2 ] to obtain (where the exact value of S in r S does not matter): V p = 1 exp [ (p + t 2 )r 2 ] π S dr exp ( pt2 ) p + t 2 R2 CP dt to be continued... Trygve Helgaker (CTCC, University of Oslo) Spherical Coulomb integrals 13th Sostrup Summer School (2014) 16 / 34

Coulomb integral over a spherical Gaussian II carried over from previous slide: V p = 1 π [ (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 ( ) π 3/2 exp( π 0 p + t 2 prcp 2 t 2 ) p + t 2 dt 4 To introduce a finite integration range, we perform the substitution u 2 = t2 p + t 2 and obtain ( t 1 + pt 2 = u 2 pt 3 dt = u 3 du dt = p 1 2 ) 3/2 u 2 du V p = 2π 1 exp ( prcp 2 p u2) du = 2π 0 p F ( ) 0 pr 2 PC We have here introduced the Boys function 1 F n(x) = exp ( xt 2) t 2n dt 0 a 3D integral over all space has been reduced to a 1D integral over [0, 1] Trygve Helgaker (CTCC, University of Oslo) Spherical Coulomb integrals 13th Sostrup Summer School (2014) 17 / 34

Gaussian electrostatics The Boys function is related to the error function erf(x) = 2 x exp ( t 2) dt = 2xF ( 0 x 2 ) π 0 π Introducing unit charge distributions and the reduced exponent as ( p ) 3/2 ( ) ρ p(r P ) = exp pr 2 pq π P, α = p + q we obtain formulas similar to those of point-charge electrostatics (but damped) ρp (r P ) dr = erf ( ) prpc r C R PC ρp (r 1P ) ρ q (r 2Q ) dr 1 dr 2 r 12 1.5 1.0 = erf ( αr PQ ) R PQ 0.5 Trygve Helgaker (CTCC, University of Oslo) Spherical Coulomb integrals 13th Sostrup Summer School (2014) 18 / 34

The Boys function F n (x) The Boys function is central to molecular integral evaluation it is evaluated by a combination of numerical techniques F n(x) > 0 since the integrand is positive: 1 F n(x) = exp ( xt 2) t 2n dt > 0 0 F n(x) is strictly decreasing and convex: 1.0 0.8 0.6 F n(x) = F n+1 (x) < 0 0.4 F n (x) = F n+2 (x) > 0 0.2 Evaluation for small and large values: F n(x) = 1 2n + 1 + ( x) k k!(2n + 2k + 1) k=1 F n(x) exp ( xt 2) t 2n (2n 1)!! dt = 0 2 n+1 (x small) π x 2n+1 Different n-values are related by downward recurrence relations: F n 1 (x) = 2xFn(x) + exp( x) 2n 1 (x large) Trygve Helgaker (CTCC, University of Oslo) Spherical Coulomb integrals 13th Sostrup Summer School (2014) 19 / 34

Cartesian Coulomb integrals We have obtained a simple result for one-center spherical Gaussians: ( ) exp pr 2 P dr = 2π r C p F ( ) 0 pr 2 PC We shall now consider general two-center one-electron Coulomb integrals: r 1 Ωab (r) V ab = G a C G b = dr r C The overlap distribution is expanded in Hermite Gaussians Ω ab (r) = Ω ij (x)ω kl (y)ω mn(z), Ω ij (x) = i+j t=0 E ij t Λt(x P) The total overlap distribution may therefore be written in the form Ω ab (r) = E ij t E u kl Ev mn Λ tuv (r P ) = tuv tuv E ab tuv Λ tuv (r P ) We now expand two-center Coulomb integrals in one-center Hermite Coulomb integrals: two-center integral V ab = Etuv ab Λtuv (r P ) dr one-center integrals r tuv C Our next task is therefore to evaluate Hermite Coulomb integrals Trygve Helgaker (CTCC, University of Oslo) Cartesian Coulomb integrals 13th Sostrup Summer School (2014) 20 / 34

Coulomb integrals over Hermite Gaussians Expansion of two-center Coulomb integrals in one-center Hermite integrals V ab = ( ) Etuv ab Λtuv (r P ) exp pr 2 dr, P dr = 2π r tuv C r C p F ( ) 0 pr 2 PC Changing the order of integration and differentiation by Leibniz rule, we obtain Λtuv (r P ) r C dr = t+u+v exp( pr 2 P ) Px t Py u Pz v dr r C = 2π p t+u+v F 0 (prpc 2 ) Px t Py u Pz v = 2π p Rtuv (p, R PC ) Expansion of the Coulomb integrals in one-center Hermite Coulomb integrals R tuv : V ab = 2π ( ) Etuv ab p Rtuv (p, R PC ), R tuv (p, R PC ) = t+u+v F 0 pr 2 PC P tuv x t Py u Pz v The Hermite Coulomb 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 Trygve Helgaker (CTCC, University of Oslo) Cartesian Coulomb integrals 13th Sostrup Summer School (2014) 21 / 34

Evaluation of Hermite Coulomb integrals To set up recursion for the Hermite integrals, we introduce the auxiliary Hermite integrals R n tuv (p, P) = ( 2p)n t+u+v F n ( pr 2 PC ) P t x Pu y Pv z which include as special cases the source and target integrals R n 000 = ( 2p)Fn R0 tuv = t P t x u v Py u Pz v F 0 The Hermite integrals can now be generated from the recurrence relations: Rt+1,u,v n = trn+1 t 1,u,v + X PC Rtuv n+1 Rt,u+1,v n = urn+1 t,u 1,v + Y PC Rtuv n+1 Rt,u,v+1 n = vrn+1 t,u,v 1 + Z PC Rtuv n+1 Trygve Helgaker (CTCC, University of Oslo) Cartesian Coulomb integrals 13th Sostrup Summer School (2014) 22 / 34

Summary one-electron Coulomb integral evaluation 1 Calculate Hermite expansion coefficients for overlap distributions by recurrence E0 00 = exp ( µxab 2 ) E i+1,j t = 1 2p E ij t 1 + X PAE ij t + (t + 1)E ij t+1 2 Calculate the Boys function by a variety of numerical techniques 1 F n(x) = exp ( xt 2) t 2n dt (x = prpc 2 ) 0 F n(x) = 2xF n+1(x) + exp( x) 2n + 1 3 Calculate Hermite Coulomb integrals by recurrence R n 000 = ( 2p)n F n R n t+1,u,v = X PC R n+1 tuv + tr n+1 t 1,u,v 4 Obtain the Cartesian Coulomb integrals by expansion in Hermite integrals r 1 G a C G b = 2π E ij t p E u kl E v mn Rtuv 0 tuv Trygve Helgaker (CTCC, University of Oslo) Cartesian Coulomb integrals 13th Sostrup Summer School (2014) 23 / 34

Two-electron Coulomb integrals Two-electron integrals are treated in the same way as one-electron integrals Evaluation of one-electron integrals: r 1 Ωab G a C G b = dr r C = Etuv ab Λtuv dr = 2π Etuv ab r tuv C p Rtuv (p, R PC ) tuv Evaluation of two-electron integrals: G a(a)g b (1) r 1 12 G c(2)g d (2) = tuv E ab tuv τνφ = 2π5/2 pq p + q Ωab (1)Ω cd (2) = r 12 dr 1 dr 2 Eτνφ cd Λtuv (1)Λ τνφ (2) dr 1 dr 2 r 12 tuv E ab tuv τνφ E cd τνφ ( 1)τ+ν+φ R t+τ,u+ν,v+φ (α, R PQ ) Hermit expansion for both electrons, around centers P and Q integrals depend on R PQ and α = pq/(p + q), with p = a + b and q = c + d translational invariance simplifies two-electron Hermite integrals Trygve Helgaker (CTCC, University of Oslo) Cartesian Coulomb integrals 13th Sostrup Summer School (2014) 24 / 34

Direct SCF and other techniques The general expression for two-electron integrals is given by g abcd = 2π5/2 pq p + q tuv E ab tuv τνφ E cd τνφ ( 1)τ+ν+φ R t+τ,u+ν,v+φ (α, R PQ ) In SCF theories, these integrals make contributions to the Fock/KS matrix: F ab = ( ) 2gabcd D cd g acbd D cd }{{}}{{} 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 screening of integrals early contraction with density matrices density fitting multipole methods Trygve Helgaker (CTCC, University of Oslo) Cartesian Coulomb integrals 13th Sostrup Summer School (2014) 25 / 34

Screening of overlap integrals The product of two s functions is by the Gaussian product rule exp ( ara 2 ) ( ) exp br 2 B = exp( ab a+b R2 AB ) exp ( (a + b)rp 2 ) This gives a simple expression for the overlap integral between two such orbitals: ( ) π 3/2 ( S ab = exp ab ) a + b a + b R2 AB the number of such integrals scales quadratically with system size however, S ab decreases rapidly with the separation R AB Let us assume that we may neglect all integrals smaller than 10 k : S ab < 10 k insignificant integrals We may then neglect integrals separated by more than [( R AB > a 1 π ) 310 ] 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 Trygve Helgaker (CTCC, University of Oslo) Screening of integrals 13th Sostrup Summer School (2014) 26 / 34

Screening of two-electron integrals A two-electron ssss integral may be written in the form: g abcd = erf ( αr PQ ) S ab S cd R PQ 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 PQ 0 very slowly Let us decompose the integral into classical and nonclassical parts: g abcd = S abs cd erfc ( ) S ab S cd αr PQ R PQ R PQ }{{}}{{} classical 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 PQ ) exp( αr2 PQ ) παrpo Trygve Helgaker (CTCC, University of Oslo) Screening of integrals 13th Sostrup Summer School (2014) 27 / 34

The scaling properties of molecular integrals linear system of up to 16 1s GTOs of unit exponent, separated by 1a 0 2 10 2 overlap n 2 2 10 2 2 10 2 kinetic energy n 2 1 10 2 n 1 1 10 2 n 1 3 5 7 9 11 13 15 3 5 7 9 11 13 15 3 10 3 1 10 3 nuclear attraction n 3 n 2 n 1 3 5 7 9 11 13 15 5 10 4 2 10 4 electron repulsion n 4 n 2 n 1 3 5 7 9 11 13 15 Trygve Helgaker (CTCC, University of Oslo) Screening of integrals 13th Sostrup Summer School (2014) 28 / 34

Integral prescreening Small integrals (< 10 10 ) are not needed and should be avoided by prescreening The two-electron integrals Ωab (1)Ω cd (2) g abcd = dr 1 dr 2 r 12 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 g ab,ab g ab,ab gcd,cd Precalculate and prescreen G ab = g ab,ab gab,cd Gab G cd Trygve Helgaker (CTCC, University of Oslo) Screening of integrals 13th Sostrup Summer School (2014) 29 / 34

Early contraction with density matrix In direct SCF, two-electron integrals are only used for Fock/KS-matrix construction J ab = 2π5/2 pq p + q tuv E ab tuv τνφ cd E cd τνφ ( 1)τ+ν+φ R t+τ,u+ν,v+φ (α, R PQ )D cd only integrals that contribute significantly are evaluated screening is made based on the size of the density matrix It is not necessary to evaluate significant integrals fully before contraction with D cd An early contraction of D cd with the coefficients Eτνφ cd is more efficient: K Q τνφ = ( 1)τ+ν+φ D cd Eτνφ cd cd Q Timings for benzene: J ab = 2π5/2 pq p + q tuv E ab tuv K Q τνφ R t+τ,u+ν,v+φ(α, R PQ ) Q τνφ cc-pvdz cc-pvtz cc-pvqz late contraction 27 374 3823 early contraction 10 90 687 Trygve Helgaker (CTCC, University of Oslo) Early density-matrix contraction 13th Sostrup Summer School (2014) 30 / 34

Density fitting Traditionally, the electron density is expanded in orbital products ρ(r) = 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 ) r 1 12 ρ(r 2 ) the formal cost of this evaluation is therefore quartic (n 4 ) Consider now an approximate density ρ(r), expanded in an auxiliary basis ω α(r): ρ(r) = α cαωα(r), ρ(r) ρ(r) the size of the auxiliary basis N increases linearly with system size We may now evaluate the Coulomb contribution approximately as Jab = Ω ab (r 1 ) r 1 12 ρ(r 2 ) the cost of the evaluation is therefore cubic (n 2 N) if ρ is sufficiently accurate and easy to obtain, this may give large savings Trygve Helgaker (CTCC, University of Oslo) Density fitting 13th Sostrup Summer School (2014) 31 / 34

Coulomb density fitting Consider the evaluation of the Coulomb potential J ab = cd (ab cd ) D cd We now introduce an auxiliary basis and invoke the resolution of identity: (ab cd ) 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 = cd (ab cd ) D cd = (ab ρ ) cd αβ (ab α ) (α β ) 1 (β cd ) D cd = α (ab α ) cα = (ab ρ ) = J ab where we have introduced the approximate density ρ(r) = α c αα(r) whose coefficients c α are obtained from the linear sets of equations α (β α ) cα = cd (β cd ) D cd (β ρ ) = (β ρ ) no four-center integrals, the formal cost is cubic or smaller note: the solution of the linear equations scales cubically This particular approach to density fitting is called Coulomb density fitting Trygve Helgaker (CTCC, University of Oslo) Density fitting 13th Sostrup Summer School (2014) 32 / 34

Robust density fitting In Coulomb density fitting, the exact and fitted Coulomb matrices are given by J ab = (ab ρ ), Jab = (ab ρ ), (α ρ ) = (α ρ ) We may then calculate the exact and density-fitted Coulomb energies (multiplied by two) as E cou = ab J abd ab = (ρ ρ ), Ẽ cou = ab J ab D ab = (ρ ρ ) The true Coulomb energy is an upper bound to the density-fitted energy: E cou Ẽcou = (ρ ρ ) (ρ ρ ) = (ρ ρ ) (ρ ρ ) ( ρ ρ ) + ( ρ ρ ) = (ρ ρ ρ ρ ) where we have used the relation ( ρ ρ) = ( ρ ρ) which follows from (α ρ ) = (α ρ ) the error in the density-fitted energy is quadratic in the error in the density Coulomb fitting is thus equivalent to minimization of the self-interaction error: E cou Ẽcou = (ρ ρ ρ ρ ) 0 It is possible to determine the approximate density in other ways (not by Coulomb fitting) quadratic energy error is always ensured by using the robust formula Ẽ cou = ( ρ ρ ) + (ρ ρ ) ( ρ ρ ) = E cou (ρ ρ ρ ρ ) however, the error in the energy is no longer minimized Trygve Helgaker (CTCC, University of Oslo) Density fitting 13th Sostrup Summer School (2014) 33 / 34

Density fitting: sample calculations Calculation on benzene cc-pvdz cc-pvtz cc-pvqz exact density late 27 374 3823 exact density early 10 90 687 fitted density 1 7 34 exact energy 230.09671 230.17927 230.19463 density-fitted energy 230.09733 230.17933 230.19466 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 Trygve Helgaker (CTCC, University of Oslo) Density fitting 13th Sostrup Summer School (2014) 34 / 34