Lecture 9: Molecular integral. Integrals of the Hamiltonian matrix over Gaussian-type orbitals

Transcription

1 Lecture 9: Molecular integral evaluation Integrals of the Hamiltonian matrix over Gaussian-type orbitals

2 Gaussian-type orbitals The de-facto standard for electronic-structure calculations is to use Gaussian-type orbitals with variable exponents This is because they lead to much more efficient evaluation of two-electron integrals

3 Gaussian-type orbitals The usefulness of the GTOs is based on the Gaussian product rule

4 GTO basis sets Primitive set Contraction Valence & core polarization Further augmentation Various established generation philosophies Correlation-consistent & polarization-consistent sets ANO sets Pople-style segmented contraction Completeness-optimized

5 Molecular integrals We will now consider two types of integrals: One-electron integrals Two electron integrals Compare with the integrals related to the molecular Hamiltonian

6 Cartesian Gaussians The orbitals β will be realvalued Cartesian GTOs Here i+j+k=l ( Cartesian quantum numbers ) From the integrals over the CGTOs we will obtain the integrals in (contracted) spherical-harmonic GTO basis as linear combinations

7 Gaussian overlap distributions The Cartesian GTOs can be factored in Cartesian directions Consequently, the Gaussian overlap distribution will factorize in Cartesian directions

8 Overlap integrals Let s begin with overlap integrals S <ÚG ( r, a, R ) G ( r, b, R ) dr ab ikm a jln b that also factorize as Employing the Gaussian product rule, we obtain where ab m < a b p < a b ax bx A x < p a b X < X, X AB A B B

9 Overlap integrals We can similarly write the direction S 00 in each Cartesian Then, by invoking the following (Obara-Saika) recurrence relations we can obtain the overlap integrals up to arbitrary Cartesian quantum number in each Cartesian direction The final overlap integral is obtained as a product of different Cartesian components

10 Multipole-moment integrals Integrals of the form S < G xyz G < SSS efg e f g e f g ab a C C C b ij kl mn are obtained through relations e e 1 1 1, ( e e e, S < X S is js es i j PA ij i, 1, j i, j, 1 ij ) 2p e e 1 1, 1 ( e e e, S < X S is js es i j PB ij i, 1, j i, j, 1 ij ) 2p e 1 e 1 ( e e e, 1 S < X S is js es ij PC ij i, 1, j i, j, 1 ij ) 2p Special cases: overlap, dipole and quadrupole integrals

11 One-electron integrals with differential operators We will need e.g. in evaluation of the kinetic energy operator in the one-electron Hamiltonian integrals of kind These will again factorize as The Obara-Saika relations are D < DDD efg e f g ab ij kl mn 0 with D ij < S ij

12 One-electron integrals with differential Now we can obtain for operators r P <, i G G ab a b r L <, i G r G ab a b 1 T <, G G 2 2 ab a b the following expressions (taking z-component only for the momentum integrals) P <, issd z ab ij kl mn z <,, ab ij kl mn ij kl mn L isds ( DSS ) 1 T <, D S S S D S S S D ( ab ij kl mn ij kl mn ij kl mn

13 One-electron Coulombic integrals Let us set up an analogous scheme for one-electron Coulombic integrals The Obara-Saika relations are written for auxillary integrals Π as

14 One-electron Coulombic integrals The auxillary integrals have the special cases where and F is the Boys function

15 Boys function For integrals featuring the 1/r singularity, i.e. Coulombic integrals, we will need a special function called the Boys function

16 Two-electron Coulombic integrals Let us finally set up the Obara-Saika (like) scheme for two-electron Coulombic integrals Employ again the auxillary integrals that feature the special cases

17 Two-electron Coulombic integrals 1. Generate Boys functions 2. Vertical recursion 3. Electron-transfer recursion 4. Horizontal recursion

18 Step Computational requirements of twoelectron integrals Floating point operations Boys functions Lp 4 Lp 4 Vertical recursion L 4 p 4 L 3 p 4 Transfer recursion L 6 p 4 L 6 p 4 Primitive contraction L 6 p 4 L 6 Horizontal recursion I L 9 L 7 Harmonics conversion I L 8 L 5 Horizontal recursion II L 8 L 6 Harmonics conversion II L 7 L 4 Memory requirement L is the angular momentum (1-4) p is the number of primitives in the shell (~1-20)

19 The multipole method Let us consider a 2D system decomposed three times Level-2 Level-3

20 The multipole method The fast multipole method (FMM): Only the NN interactions are treated explicitly; while the rest are obtained as multipole expansions of each box The number of NN contributions scales linearly with the size of the system With continuous charge distributions (Gaussians) the FMM has to be generalized (CFMM) Linearly scaling number of nonclassical integrals The rest can be computed from multipole expansions