Automatic code generation in density functional theory

Transcription

1 Computer Physics Communications 136 (2001) Automatic code generation in density functional theory R. Strange F.R. Manby P.J. Knowles School of Chemistry University of Birmingham Edgbaston Birmingham B15 2TT UK Received 18 December 2000 Abstract We present a program dfauto that uses automatic code generation to produce Fortran code and LATEX documentation for implementing density functionals in a Kohn Sham program. The user provides the formulae that define the density functional and dfauto produces Fortran to evaluate the exchange-correlation kernel on an integration grid along with the gradients necessary for Kohn Sham calculations. The program is implemented in Bourne shell and Maple Elsevier Science B.V. All rights reserved. PACS: Ew Keywords: DFT; Density functional theory; Kohn Sham theory; Automatic code generation PROGRAM SUMMARY Title of program: dfauto Catalogue identifier: ADNY Program Summary URL: Program obtainable from: CPC Program Library Queen s University of Belfast N. Ireland Licensing provision: Those pertaining to Maple Computers: UNIX platforms Operating systems under which program has been tested: Linux OSF1 Programming languages used: Maple V (version 5) Bourne shell (sh) Fortran LATEX No. of bits in a word: 32 No. of bytes in distributed program including test data: Distribution format: gzip file Keywords: DFT density functional theory Kohn Sham theory automatic code generation Nature of physical problem Density functional theory of electronic structure. Method of solution Automatic code generation. Restrictions on the complexity of the problem Functionals only of the density its first and second derivatives and the kinetic energy density are treated. Typical running time Functional dependent but typically of the order of one minute. * Corresponding author. address: P.J.Knowles@bham.ac.uk (P.J. Knowles) /01/$ see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S (01)

2 R. Strange et al. / Computer Physics Communications 136 (2001) Introduction The Kohn Sham (KS) [1] construction of density functional theory (DFT) [23] has proven to be extremely useful in treating the electronic structure of large systems. DFT uses the electronic density a simple function in three-dimensional space as the primary variational quantity in place of the complicated N-electron wavefunction. In KS theory the density ρ is generated from the KS determinant which also provides the non-interacting kinetic energy. The non-local effects of exchange and correlation (including the correction to the kinetic energy) are treated by a local effective potential v xc related to the elusive exchange-correlation functional E xc by v xc = δe xc δρ. (1) The usual Fock equations of Hartree Fock (HF) theory are then replaced by the KS equations [ ] + v ext + v J + v xc ψi = ɛ i ψ i (2) where v ext and v J are the external and Coulomb potentials. The orbitals ψ i are typically expanded in an atomic orbital (AO) basis ψ i = a c ia η a (3) and the coefficients c ia are determined by solving the matrix form of Eq. (2) to self-consistency. The general exchange-correlation functional has the form E xc [ρτ]= d rk ( ρ( r)σ( r)υ( r)τ( r) ) (4) where σ = ρ 2 υ = 2 ρ and τ is the kinetic energy density of the KS determinant. The electronic and kinetic energy densities can be represented in canonical or atomic orbitals: ρ = ψ i 2 = γ ab ηa η b (5) i occ ab and τ = ψ i 2 = γ ab [ η a ] [ η b ] (6) i occ ab where γ is the AO density matrix. Higher gradients of the density may be used in forming the exchange-correlation functional but we do not consider them here. The exchange-correlation contribution f xc to the Fock matrix should apparently involve an integration over the exchange correlation potential v xc. However one can and should avoid the evaluation of the functional derivative in Eq. (1) by considering the derivatives of E xc with respect to the elements of the density matrix: fab xc = E xc[ρτ] = d r K. (7) γ ab γ ab These and not the integrals involving the functional derivative are exactly what are required for the variational minimization of the energy with respect to the orbital coefficients. Using the chain rule we have f xc ab = ξ {ρσυτ} d r ξ( r) γ ab v ξ ( r) where v ξ K/ ξ. The partial derivatives with respect to γ ab have simple forms in terms of the AOs and their gradients for instance ρ/ γ ab = η a η b. The contributions to the Fock matrix arising from exact exchange (present in so-called hybrid functionals [4]) have the usual HF form. (8)

3 312 R. Strange et al. / Computer Physics Communications 136 (2001) The integral in Eq. (8) is usually performed on a numerical quadrature grid [5 9] by evaluating the orbitals their gradients K and its partial derivatives v ξ at each grid point. Writing the subroutines for complicated functionals is a time-consuming and error-prone process: in the current work we use automatic code generation to produce the subroutine and its documentation directly from the formulae that define the density functional. 2. Implementation For a general treatment of an N-electron system the spin densities ρ α and ρ β have to be treated independently. K consists of a spin-summed term (as for exchange) and a coupling term between the spins (present in correlation functionals): K = g(ρ s ρ s σ ss σ s s σ s s υ s υ s τ s τ s ) s {αβ} + f(ρ α ρ β σ αα σ αβ σ ββ υ α υ β τ α τ β ) where s is the conjugate spin to s. Functionals are usually derived in terms of these spin indexed densities (and related objects) but for computational convenience functionals are implemented using the total density ρ ρ c = ρ α + ρ β and the spin (or open-shell) density ρ o = ρ α ρ β. Similar total- and open-shell quantities are used for the gradient terms with υ c υ o and τ c τ o being obvious generalizations of ρ c ρ o ; σ cc σ co and σ oo are constructed from ρ c and ρ o to give e.g. σ co σ oc =[ ρ c ] [ ρ o ]. The exchange-correlationkernel that appears in Eq. (4) is then a function of the nine co-indexed quantities listed above and we require in addition to K at each grid point the values of the partial derivatives v ξ with respect to each of these objects. For closed-shell systems all quantities with subscripts containing an o can be neglected increasing the efficiency of the computation. The derivatives with respect to quantities absent from the density functional can of course be neglected. For example to evaluate the Dirac exchange for a closed-shell system a possible fragment of Fortran code is do i=1n zk(i)=-cf*rhoc(i)**(4d0/3d0) vrhoc(i)=-4d0/3d0*cf*rhoc(i)**(1d0/3d0) enddo where the array zk stores the exchange-correlation kernel on the n grid points labeled i rhoc ρ c vrhoc v ρc and cf is the closed-shell Fermi constant. The program presented here dfauto converts a Maple [10] expression for a density functional given in α β form to Fortran code like that above and produces a manual page documenting the functional. The program takes care of the transition from the usual α β form of functionals to the computationally convenient co form. dfauto is implemented in Bourne shell (sh) under the UNIX operating system and the mathematical manipulations and translations to Fortran and LATEX are performed by Maple. To generate the body of the loop in the above code the Maple input k:=-cf*rhoc[i]^(4/3): fortran([zk[i]=kvrhoc[i]=diff(krhoc[i])]precision=double); returns zk(i) = -cf*rhoc(i)**(4.d0/3.d0) vrhoc(i) = -4.D0/3.D0*cf*rhoc(i)**(1.D0/3.D0) and the command latex(k=k):

4 R. Strange et al. / Computer Physics Communications 136 (2001) produces the LATEX K=-{\it cf}\{{\it rhoc}_{{i}}}^{4/3} for documentation. The Fortran generated by dfauto is optimized by Maple but for technical reasons Maple s simplifying procedure is not invoked. 3. Using dfauto In order to use dfauto aunix system with Maple a Fortran compiler and LATEX with the standard packages breqn [11] and inlinebib [12] are needed. Although dfauto was originally written to introduce new density functionals into the quantum chemistry package MOLPRO [13] it can be used with any suitably designed Fortran file. To run dfauto a Maple input file must be created with the suffix.df e.g. fn.df. The file should contain the definitions of f and/or g along with any auxiliary definitions such as constants that appear in the formulae. Although not all features of Maple have been tested with dfauto most should be acceptable in the.df file so that for example user defined procedures arrays and summations present no difficulty. Given input fn.df dfauto appends a subroutine dftacg_fn to a specified Fortran file and creates a manual page in the file fn.tex. The density functional program must test a character string key which specifies the density functional to be used. By default the files dftfun.f dftuser.f and base/src/dft/dftfun.f are scanned to prevent repetition of a key; the defaults can be overridden by setting the environment variable DFTFILES. In order to insert the call for a new subroutine dfauto requires the presence of a marker in the Fortran file. The tag c:dfauto needs to appear directly above an elseif in or the endif of the key test and the call to the new functional is inserted directly above it. The relevant part of such a file might be if(key.eq. FUNC1 ) then call func1(...) elseif(key.eq. FUNC2 ) then call func2(...) c:dfauto endif and after running the command dfauto fn.df would be if(key.eq. FUNC1 ) then call func1(...)... c:fncallstart elseif(key.eq. FN ) then call dftacg_fn(namefderivopenigradnptrhocrhoo > sigmaccsigmacosigmaootauctauo > upsiloncupsilono > zkvrhocvrhoo > vsigmaccvsigmacovsigmaoovtaucvtauo > vupsiloncvupsilono) c:fncallend c:dfauto endif The c:fncallstart and c:fncallend tags are introduced to allow dfauto to remove the functional FN.

5 314 R. Strange et al. / Computer Physics Communications 136 (2001) The subroutine dftacg_fn takes as input the number of grid points npt; arrays rhoc rhoo sigmacc sigmaco sigmaoo tauc tauo upsilonc upsilono defining the corresponding functions on the grid; the logicals fderiv and open indicating whether gradients should be computed and if the system is open-shell; the value of the functional is returnedin zk and the derivatives with respect to ξ are accumulated in the arrays vξ; name is the string Automatically generated FN. igrad is set to an integer between 0 and 2 by the subroutine specifying the highest order gradient present in the functional; thus the subroutine can be called with zero grid points to ascertain which of ρστ and υ must be computed. Everynewequationinfn.df must occur on a new line but very long equations may be split over many lines using a backslash for continuation. The variables ρ s σ s s υ s τ s and the reduced (or dimensionless) spin density gradients χ s = ρ s ρ 4/3 s = σss ρ 4/3 s are entered rho sigma upsilon tau and chi with the subscripts α β s and s appended a b s and s_ in parentheses. For example ρ α and σ s s are entered rho(a) and sigma(ss_) respectively. The objects in parentheses are protected names and cannot be assigned in an input file e.g. the line s:=rho(s): will cause an error. Spin summed quantities are entered unindexed so that rho ρ α + ρ β ; there is no spin-summed χ. Special attention has to be paid to functionals that contain χ s or other objects that diverge when ρ s 0. Consider Becke s 1986 exchange functional [14] K = s [ 3 2 ( 3 4π ) 1/3 ρs 4/3 + β ρ4/3 s χs λχs 2 ] where β and λ are constants. The second term vanishes as ρ s 0 despite the presence of χ s.ifthiskindof singularity occurs dfauto prompts the user to insert an equation in the input file defining the limit G when one of the spin densities goes to zero. G is inputed in the same way as g without terms involving s_; seesample input and output for an example. The value of the functional for the hydrogen atom (ρ α = e 2r /π) is evaluated on a 90 point grid to test the generated Fortran. For compilation the environment variable MFILES lists the make dependencies of the Fortran file to which the subroutine is appended. For the manual generation a title reference and description of the functional may be given in the variables title ref and blurb protected by double quotes. References are entered in Bibtex format and dfauto places them in the file acg.bib multiple entries in acg.bib are not checked for. The file fn.tex does not contain any preamble information dfauto wraps the output in suitable LATEX and produces a DVI and/or postscript file acg.dvi or.ps. The usage of dfauto is: dfauto [-lpdz] [-x ptsize] [-t paper] [-f font] [-mras] [-o ffile] <.df files > -l produce LATEX only no Fortran -p postscript output no DVI -d DVI output -z suppress G in manual -x ptsize font size in pts -t paper paper size -f font font option

6 R. Strange et al. / Computer Physics Communications 136 (2001) m produce Fortran only no LATEX -r remove subroutine -a append subroutine -s replace subroutine -o ffile Fortran file The arguments <.df files > are the input files. By default ffile is dftuser.f and the DVI file is generated. With the flag -p only postscript output is produced;with -dp both files are produced.the printing of G may be suppressed using the -z flag. Acknowledgements The authors acknowledge support from the EPSRC (grant number: GR/N07912) and FRM is grateful for financial support from the Royal Society. RS would like to thank Dr S.J. McNicholas for helpful discussions. References [1] W. Kohn L.J. Sham Phys. Rev. A 140 (1965) [2] P. Hohenberg W. Kohn Phys. Rev. B 136 (1964) 864. [3] R. Parr W. Yang Density-Functional Theory of Atoms and Molecules Oxford University Press New York [4] A.D. Becke J. Chem. Phys. 98 (1993) [5] S.F. Boys P. Rajagopal Adv. Quantum Chem. 2 (1965) 1. [6] A.D. Becke J. Chem. Phys. 88 (1988) [7] C.W. Murray N.C. Handy G.J. Laming Mol. Phys. 78 (1993) 997. [8] O. Treutler R. Ahlrichs Chem. Phys. Lett. 102 (1995) 346. [9] M.E. Mura P.J. Knowles J. Chem. Phys. 104 (1996) [10] B.W. Char K.O. Geddes G.H. Gonnet B.L. Leong M.B. Monagan S.M. Watt Maple V Language Reference Manual Springer New York [11] ftp://ftp.tex.ac.uk/tex-archive/help/catalogue/entries/breqn.html. [12] ftp://ftp.tex.ac.uk/tex-archive/help/catalogue/entries/inlinebib.html. [13] MOLPRO is a package of ab initio programs written by H.-J. Werner and P.J. Knowles with contributions from R.D. Amos A. Bernhardsson A. Berning P. Celani D.L. Cooper M.J.O. Deegan A.J. Dobbyn F. Eckert C. Hampel G. Hetzer T. Korona R. Lindh A.W. Lloyd S.J. McNicholas F.R. Manby W. Meyer M.E. Mura A. Nicklass P. Palmieri R. Pitzer G. Rauhut M. Schütz H. Stoll A.J. Stone R. Tarroni and T. Thorsteinsson. [14] A.D. Becke J. Chem. Phys. 84 (1986) [15] D.J. Tozer N.C. Handy J. Chem. Phys. 108 (1998) 2545.

7 316 R. Strange et al. / Computer Physics Communications 136 (2001) Sample input The TH1 functional of Tozer and Handy [15] involves a summation over a range of optimized parameters which may be conveniently expressed in an array. Unlike other variables arrays must be defined before they are used. The following file th1.df illustrates the use of arrays to input the TH1 functional. Note the use of backslashes to protect the quotes in the reference. t:=array([7/68/69/610/68/69/610/611/69/610/6\ 11/612/69/610/611/612/67/68/69/610/61]): u:=array([ ]): v:=array([ ]): w:=array([ ]): omega:=array([ \ \ \ \ ]): n:=21: R[i]:=rho(a)^t[i]+rho(b)^t[i]: S[i]:=((rho(a)-rho(b))/rho)^(2*u[i]): X[i]:=(sqrt(sigma(aa))^v[i] + sqrt(sigma(bb))^v[i]) / (2*rho^(4*v[i]/3)): Y[i]:=((sigma(aa)+sigma(bb)-2*(sigma(aa)^(1/2)*sigma(bb)^(1/2)))/\ rho^(8/3))^w[i]: f:=sum(omega[i]*r[i]*s[i]*x[i]*y[i]i=1..n): ref:="@article{th1 author=\"d. J. Tozer and N. C. Handy\"\ journal=\"j. Chem.\\ Phys.\" volume=108 number=6 pages=2545 year=1998}": blurb:="density and gradient dependent first row exchange-correlation \ functional": title:="tozer and Handy 1998": G:=sum(omega[i]*rho(s)^t[i]*sqrt(sigma(ss))^v[i]/(2*rho(s)^(4*v[i]/3))*\ (sigma(ss)/rho(s)^(8/3))^w[i]i=1..n):

8 R. Strange et al. / Computer Physics Communications 136 (2001) Sample output On issuing the command dfauto -o mydft.f th1.df the following standard output is generated: ACG: th1.df --> th1.tex ACG: th1.df --> subroutine dftacg_th1 in mydft.f Testing for singularities No singularities found in TH1 Functional value for the Hydrogen atom = au and 500 lines of Fortran are placed in mydft.f. A manual page is generated and forms the remainder of this section: TH1: Tozer and Handy 1998 D.J. Tozer and N.C. Handy J. Chem. Phys. 108 (6) (1998) Density and gradient dependent first row exchange-correlation functional where n K = ω i R i S i X i Y i i=1 n = 21 R i = (ρ α ) t i + (ρ β ) t i t = [ 7/6 4/3 3/2 5/3 4/3 3/2 5/ ( ) ρα ρ 2 ui β S i = ρ X i = 1/2 ( σ αα ) v i + ( σ ββ ) v i ρ 4/3 v i ( σαα + σ ββ 2 ) σ αα σββ wi Y i = ρ 8/ /2 5/ /2 5/ /6 4/3 3/2 5/3 1] u =[ ] v =[ ] w =[ ] and ω =[ ]. To avoid singularities in the limit ρ s 0 G = n i=1 1/2 ω i (ρ s ) t ( ( ) ) wi i vi σss ( σss (ρs (ρ s ) 8/3 ) 4/3 v ) i 1.

9 318 R. Strange et al. / Computer Physics Communications 136 (2001) Bibliography D.J. Tozer and N.C. Handy J. Chem. Phys. 108 (6):