automatic code generation for quantum chemistry applications · xc functional zoo i true functional...

Automatic code generation for

quantum chemistry applications

Grzegorz Mazur1,2, Marcin Makowski1,2, Roman Lazarski3,Rados law W lodarczyk4, Ewa Czajkowska1

1Faculty of Chemistry, Jagiellonian University2Academic Computer Centre CYFRONET AGH

3Otto-Schott-Institut fur Materialforschung, Friedrich-Schiller-Universitat Jena4Institute of Chemistry, Humboldt-Universitat zu Berlin

Quantum Chemistry Landscape

Exchange-Correlation Functional

I Theory: E xc [ρα, ρβ] – spin density functional

I Practice:I Local Density Approximation (LDA)

E xc =

∫R3

exc (ρα(r), ρβ(r)) dr3

I Generalized Gradient Approximation (GGA)

E xc =

∫R3

exc (ρα(r), ρβ(r), σαα(r), σαβ(r), σββ(r)) dr3

I meta-GGA – additional dependency on kinetic energydensity

XC Functional zoo

I True functional not known

I Plethora of approximationsI SlaterI SVWNI SLYPI BLYPI PBE/revPBE/mPBEI PW91/PW92I OPTXI B3LYPI PBE0I . . .

I and new ones showing up every so often. . .

XC Functional derivativesI Energy straight from xc functional

I Ground state determination requires xc functionalderivatives

I Excited state energies require xc functional secondderivatives

state number of derivativesLDA GGA

ground 1(∂Exc

∂ρσ

)3(∂Exc

∂ρσ, ∂Exc

∂γστ

)excited 2

(∂2Exc

∂ρσ∂ρτ

)9(

∂2Exc

∂ρσ∂ρτ, ∂2Exc

∂ρσ∂γστ, . . .

). . . . . . . . .

I Geometry optimization goes one order further

I Hessian requires another order of differentiation

XC Functional – example

E Slater(ρα, ρβ) := −3

4

(6

π

) 13 (ρ

43α + ρ

43β

)


EVWN (ρα, ρβ) := ρEc(x , ζ)

where

ρ := ρα + ρβ x :=

(3

4πρ

)1/6

ζ :=ρα − ρβ

ρ

Ec(x , ζ) := EPc (x) + EA

c (x)g(ζ)(1 + B2(x)ζ4)

g(ζ) :=9

8

((1 + ζ)4/3 + (1− ζ)4/3 − 2

)B2(x) :=

4

9(21/3 − 1)

EFc (x)− EP

c (x)

EAc (x)

− 1

EAc (x) := E

(− 1

6π2,−0.00475840, 1.13107, 13.0045, x

)EFc (x) := E (0.01554535000,−0.32500, 7.06042, 18.0578, x)

EPc (x) := E (0.03109070000,−0.10498, 3.72744, 12.9352, x)


E (A,X0,B ,C , x) :=

A

(ln

x2

X (x ,B ,C )+ 2B

arctan Q(B,C)2x+B

Q (B ,C )+

−BX0

ln (x−X0)2

X (x ,B,C)+ 2 (B + 2X0)

arctan Q(B,C)2x+B

Q(B,C)

X (X0,B ,C )

Q (b, c) :=

√4c − b2

X (x , b, c) := x2 + bx + c

We’ve got a problem

I calculating derivatives tedious and error prone

I transforming them to C or FORTRAN even more so

Computer Algebra System

I high-level, math oriented language

I built-in arithmetics, elementary functions, differentiation,integration, etc.

I automatic formula simplification

I usually ability to generate code in C and/or FORTRAN

yacas

I Yet Another Computer Algebra System

I term-rewriting based

I built-in basic math

I built-in C code generation

I easy to extend

I highly portable

I easily accessible (LGPL)


XC’ D e f i n e ( X S l a t e r ( rhoa , rhob ) ,[−3./4∗(6/ Pi ) ˆ ( 1 . / 3 . ) ∗

( rhoa ˆ ( 4 . / 3 . ) + rhob ˆ ( 4 . / 3 . ) ) ;] ) ;

XC Functional – exampleXC’ D e f i n e (CVWN( rhoa , rhob ) ,[

L o c a l ( rho , z , x , EcP , EcF , EcA , B2 , dEe , Ec , X , Q, E ) ;

X( x , b , c ) := x ˆ2 + b∗x + c ;Q( b , c ) := S q r t (4 ∗ c − b ˆ 2 ) ;

E(A , X0 , B , C , x ) := A∗( Ln ( x ˆ2 / X( x , B , C ) ) +2 ∗ B ∗ ArcTan (Q(B , C) / (2 ∗ x + B) ) / Q(B , C) −B ∗ X0 ∗ ( Ln ( ( x − X0)ˆ2/X( x , B , C ) )

+ 2∗(B + 2 ∗ X0 ) ∗ArcTan (Q(B , C) / ( 2 ∗ x + B) ) / Q(B , C ) ) / X(X0 , B , C ) ) ;

rho := rhoa + rhob ;z := ( rhoa − rhob ) / rho ;x := (3/(4∗ Pi ∗ rho ) ) ˆ ( 1 . / 6 . ) ;EcP := E ( 0 . 0 3 1 0 9 0 7 0 0 0 0 , −0 . 1 0 4 9 8 , 3 . 7 2 7 4 4 , 1 2 . 9 3 5 2 , x ) ;EcF := E ( 0 . 0 1 5 5 4 5 3 5 0 0 0 , −0 . 3 2 5 0 0 , 7 . 0 6 0 4 2 , 1 8 . 0 5 7 8 , x ) ;EcA := E (−1 ./(6 . ∗ Pi ˆ 2 ) , −0.00475840 ,1 .13107 ,13 .0045 , x ) ;B2 := 4 . / ( 9 .∗ ( 2 . ˆ ( 1 / 3 ) −1 ) ) ∗ ( EcF − EcP ) / EcA − 1 ;g := 9 . / 8 .∗ ( ( 1 . + z ) ˆ ( 4 . / 3 . ) + (1.− z ) ˆ ( 4 . / 3 . ) ) ;Ec := EcP + EcA∗g∗(1.+B2∗z ˆ 4 ) ;rho ∗ Ec ;

] ) ;

Code generation

I define functionals

I differentiate

I Common Subexpression Elimination

I generate code

I compile

Does it really work?

I functionality already present in:I niedoida, up to second derivatives (excited states

energies at the hybrid-GGA level)I DFT-B parameter-suite generation package

I new version of automateuszI supports derivatives up to third-orderI code generation in C, C++, FORTRAN95, CUDA C,

OpenCLI used in adorycja, GPGPU quantum chemistry program

(part of the niedoida package)

Electron Repulsion Integrals

gµνκλ =

∫R3

∫R3

χµ(r1)χν(r1)χκ(r2)χλ(r2)

r12d3r1d

3r2

I 6 dimensions

I N4 integrals

I analytically integrable for Gaussian Type Orbitals (GTO)

Obara-Saika scheme

I auxilliary integral

ΘN000000000000 = (ss|ss)N =

2π52

pq√p + q

K xyzab K xyz

cd FN(αR2PQ),

where for primitive Cartesian Gaussian functions centeredon A,B ,C ,D with exponents equal to a, b, c , d

p = a + b q = c + d α =pq

p + q

Px =aAx + bBx

a + bXPQ = Px−Qx K xyz

ab = exp

(− ab

a + bR2AB

)I Obara-Saika recurrences

ΘN000000000000 → Θ0

ix iy iz jx jy jzkxkykz lx ly lz = (aix iy izbjx jy jz |ckxkykzdlx ly lz )

Obara-Saika recurrencesI horizontal

ΘNijkl = ΘN

ij ,k+1,l−1 + XCDΘNijk,l−1

ΘNijkl = ΘN

i+1,j−1,kl + XABΘNi ,j−1,kl

I electron transfer

ΘNi0k0 = XQCΘN

i0,k−1,0 +α

qXPQΘN+1

i0,k−1,0+

+k − 1

2q

(ΘN

i0,k−2,0 −α

qΘN+1

i0,k−2,0

)+

+i

2(p + q)ΘN+1

i−1,0,k−1,0

I vertical

ΘNi000 = XPAΘN

i−1,000 −α

pXPQΘN+1

i−1,000+

+i − 1

2p

(ΘN

i−2,000 −α

pΘN+1

i−2,000

)

Implementation glimpse

// h o r i z o n t a l r e c u r s i o n i n t h e r i g h t p a i r

500 # ERI ’ OS1P ’ h r r 1 (ERI ’ OS1P ’ g ( ma , mb , p , b , xab , xp , kab ,

mc , md , q , d , xcd , xq , kcd ,N , b s ) ) (md [ 1 ] > 0) <−−

[ERI ’ OS1P ’ g (ma , mb, p , b , xab , xp , kab ,

mc + {1 , 0 , 0} , md − {1 , 0 , 0} ,q , d , xcd , xq , kcd , N, bs ) +

xcd [ 1 ] ∗ ERI ’ OS1P ’ g (ma , mb, p , b , xab , xp , kab ,mc , md − {1 , 0 , 0} ,q , d , xcd , xq , kcd , N, bs ) ;

] ;

Code generation

I define recurrences

I write down requested integral

I recurrences are applied automatically

I Common Subexpression Elimination

I apply platform-specific optimizations

I generate code

I compile

Does it really work?

I functionality present in adorycja, GPGPU quantumchemistry program (part of the niedoida package)

I part of niedoida HF energy gradient routine for CPU

ERI gradients and beyond

I gradientsI formulas derived by manual differentiation of

Obara-Saika recurrencesI implementation analogous to ERII used to implement HF energy gradientI DFT gradient being worked on

I HessianI Hermite Gaussians more involved but easier to

differentiate than Cartesian GaussiansI HF energy Hessian is being worked on

Summary

I Computational quantum chemistry involves intricatekernels

I Computer Algebra System significantly reduces thesemantic gap

I On-the-fly common subexpression elimination makes theapproach viable

I As a result, developing the kernels less tedious anderror-prone

Software

I http://www.yacas.org

I http://www.chemia.uj.edu.pl/~niedoida

http://www.yacas.org

http://www.chemia.uj.edu.pl/~niedoida

Acknowledgements

This research has been partially supported by the EuropeanUnion within the European Regional Development Fundprogram no. POIG.02.03.00-12-138/13 as part of the PLGridNG project.

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