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Molecular Integral Evaluation Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 11th Sostrup Summer School Quantum Chemistry and Molecular Properties July 4–16 2010 Trygve Helgaker (CTCC, University of Oslo) Molecular Integral Evaluation 11th Sostrup Summer School (2010) 1 / 34

Molecular integral evaluation ◮ one-electron interactions ◮ overlap, multipole-moment, and kinetic-energy integrals ◮ Coulomb attraction integrals � χ a ( r ) ˆ O ab = O ( r ) χ b ( r ) d r ◮ two-electron interactions ◮ Coulomb integrals �� χ a ( r 1 ) χ b ( r 1 ) χ c ( r 2 ) χ d ( r 2 ) g abcd = d r 1 d r 2 r 12 ◮ Coulomb and exchange contributions to Fock/KS matrix � � � F ab = 2 g abcd D cd − g acbd D cd cd ◮ basis functions ◮ primitive Cartesian GTOs ◮ integration schemes ◮ McMurchie–Davidson, Obara–Saika and Rys schemes Trygve Helgaker (CTCC, University of Oslo) Introduction 11th Sostrup Summer School (2010) 2 / 34

Overview Cartesian and Hermite Gaussians 1 ◮ properties of Cartesian Gaussians ◮ properties of Hermite Gaussians Simple one-electron integrals 2 ◮ Gaussian product rule and overlap distributions ◮ overlap distributions expanded in Hermite Gaussians ◮ overlap and kinetic-energy integrals Coulomb integrals 3 ◮ Gaussian electrostatics and the Boys function ◮ one- and two-electron Coulomb integrals over Cartesian Gaussians Sparsity and screening 4 ◮ Cauchy–Schwarz screening Coulomb potential 5 ◮ early density matrix contraction ◮ density fitting Trygve Helgaker (CTCC, University of Oslo) Introduction 11th Sostrup Summer School (2010) 3 / 34

Cartesian Gaussian-type orbitals (GTOs) ◮ We shall consider integration over primitive Cartesian Gaussians centered at A :   a > 0 orbital exponent  G ijk ( r , a , A ) = x i A y j A z k A exp( − ar 2 A ) , r A = r − A electronic coordinates   i ≥ 0 , j ≥ 0 , k ≥ 0 quantum numbers ◮ total angular-momentum quantum number ℓ = i + j + k ≥ 0 ◮ Gaussians of a given ℓ constitutes 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 i ( a , x A ) = x i A exp( − ax 2 G ijk ( a , r A ) = G i ( a , x A ) G j ( a , y A ) G k ( a , z A ) , 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 ) = 2 aG i +1 ( a , x A ) − iG i − 1 ( a , x A ) ∂ A x ∂ x Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 11th Sostrup Summer School (2010) 4 / 34

Hermite Gaussians ◮ The Hermite Gaussians are defined as � ∂ � t � ∂ � u � ∂ � v � � − pr 2 Λ tuv ( r , p , P ) = exp , r P = r − P P ∂ P x ∂ P y ∂ P z ◮ Like Cartesian Gaussians, they also factorize in the Cartesian directions: � � Λ t ( x P ) = ( ∂/∂ P x ) t exp − px 2 ← a Gaussian times a polynomial of degree t P ◮ Hermite Gaussians yields 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 � E ij Cartesian product → G i ( x A ) G j ( x B ) = t Λ t ( x P ) ← Hermite expansion t =0 ◮ such expansions are useful because of simple integration properties such as � ∞ � π Λ t ( x ) d x = δ t 0 p −∞ ◮ McMurchie and Davidson (1978) ◮ We shall make extensive use of the McMurchie–Davidson scheme for integration Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 11th Sostrup Summer School (2010) 5 / 34

Integration over Hermite Gaussians ◮ From the definition of Hermite Gaussians, we have � ∂ � t � ∞ � ∞ � � − px 2 Λ t ( x ) d x = exp d x P ∂ P x −∞ −∞ ◮ We now change the order of differentiation and integration by Leibniz’ rule: � ∞ � ∂ � t � ∞ � � − px 2 Λ t ( x ) d x = exp d x P ∂ P x −∞ −∞ ◮ The basic Gaussian integral is given by � ∞ � π � � − px 2 exp d x = P p −∞ ◮ Since the integral is independent of P , differentiation with respect to P gives zero: � ∞ � π Λ t ( x ) d x = δ t 0 p −∞ ◮ only integrals over Hermite s functions do not vanish Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 11th Sostrup Summer School (2010) 6 / 34

Hermite recurrence relation ◮ Cartesian Gaussians satisfy the simple recurrence relation x A G i = G i +1 ◮ The corresponding Hermite recurrence relation is slightly more complicated: x P Λ t = 1 2 p Λ t +1 + t Λ t − 1 ◮ The proof is simple: ◮ from the definition of Hermite Gaussians, we have � ∂ � t � ∂ � t ∂ exp( − px 2 Λ t +1 = P ) = 2 p 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 � � ∂ � t � ∂ � t − 1 � Λ t +1 = 2 p x P − t Λ 0 = 2 p ( x P Λ t − t Λ t − 1 ) ∂ P x ∂ P x Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 11th Sostrup Summer School (2010) 7 / 34

Comparison of Cartesian and Hermite Gaussians Cartesian Gaussians Hermite Gaussians � � � � ∂ t G i = x i − ax 2 − px 2 definition A exp Λ t = x exp ∂ P t A P 1 recurrence x A G i = G i +1 x P Λ t = 2 p Λ t +1 + t Λ t − 1 ∂ G i ∂ Λ t differentiation ∂ A x = 2 aG i +1 − iG i − 1 ∂ P x = Λ t +1 2 2 1 1 � 3 � 2 � 1 1 2 3 � 3 � 2 � 1 1 2 3 � 1 � 1 � 2 � 2 Trygve Helgaker (CTCC, University of Oslo) Cartesian and Hermite Gaussians 11th Sostrup Summer School (2010) 8 / 34

The Gaussian product rule ◮ The most 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: � � � � � � � � − ax 2 − bx 2 − µ X 2 − px 2 exp exp = exp exp A B AB P � �� � � �� � exponential prefactor product Gaussian where 1 P x = aA x + bB x ← “center of mass” 0.8 p p = a + b ← total exponent 0.6 X AB = A x − B x , ← relative separation 0.4 ab 0.2 µ = ← reduced exponent a + b -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 11th Sostrup Summer School (2010) 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 � � � � x i A x j − px 2 Ω ij ( x ) d x = K AB B exp d x P ◮ 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 Λ t ( 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: E i +1 , j 2 p E ij 1 t − 1 + X PA E ij t + ( t + 1) E ij = t t +1 ◮ we shall now prove these recurrence relations Trygve Helgaker (CTCC, University of Oslo) Gaussian product rule and Hermite expansions 11th Sostrup Summer School (2010) 10 / 34

Recurrence relations for Hermite expansion coefficients ◮ A straighforward Hermite expansion of Ω i +1 , j � � � E i +1 , j Ω i +1 , j = K AB x i +1 x j − pr 2 B exp = Λ t A P t t ◮ An alternative Hermite expansion of Ω i +1 , j 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 PA Ω ij � � E ij E ij = t x P Λ t + X PA t Λ t t t � � � E ij 1 = 2 p Λ t +1 + t Λ t − 1 + X PA Λ t t t � � � 2 p E ij 1 t − 1 + ( t + 1) E ij t +1 + X PA E ij = Λ t t t 1 ◮ we have here used the recurrence x P Λ t = 2 p Λ t +1 + t Λ t − 1 ◮ A comparison of the two expansions yields the recurrence relations E i +1 , j 2 p E ij 1 t − 1 + X PA E ij t + ( t + 1) E ij = t t +1 Trygve Helgaker (CTCC, University of Oslo) Gaussian product rule and Hermite expansions 11th Sostrup Summer School (2010) 11 / 34