Libcint: An efficient general integral library for Gaussian basis functions

An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian‐type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one‐electron overlap and nuclear attraction, two‐electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4‐component relativistic Hartree–Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth‐order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi‐CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20‐core node, we are able to achieve an average output 8.3 GB/s for C60 molecule with cc‐pVTZ basis. © 2015 Wiley Periodicals, Inc.

[1]  Georg Schreckenbach,et al.  Calculation of NMR shielding tensors based on density functional theory and a scalar relativistic Pauli-type Hamiltonian. The application to transition metal complexes , 1997 .

[2]  G. Schreckenbach.  Density functional calculations of 19F and 235U NMR chemical shifts in uranium (VI) chloride fluorides UF6−nCln: Influence of the relativistic approximation and role of the exchange‐correlation functional , 2005 .

[3]  Peter M. W. Gill,et al.  The prism algorithm for two-electron integrals , 1991 .

[4]  R. C. Binning,et al.  Dirac-Fock discrete-basis calculations on the beryllium atom , 1983 .

[5]  F. London,et al.  Théorie quantique des courants interatomiques dans les combinaisons aromatiques , 1937 .

[6]  Wenjian Liu,et al.  Four-component relativistic theory for nuclear magnetic shielding constants: the orbital decomposition approach. , 2007, The Journal of chemical physics.

[7]  Victor Lotrich,et al.  Efficient electronic integrals and their generalized derivatives for object oriented implementations of electronic structure calculations , 2008, J. Comput. Chem..

[8]  Shigeru Obara,et al.  General recurrence formulas for molecular integrals over Cartesian Gaussian functions , 1988 .

[9]  Wenjian Liu,et al.  Relativistic MCSCF by means of quasidegenerate direct perturbation theory. I. Theory , 2000 .

[10]  Roland Lindh,et al.  The reduced multiplication scheme of the Rys quadrature and new recurrence relations for auxiliary function based two‐electron integral evaluation , 1991 .

[11]  K. Dyall.  Relativistic and nonrelativistic finite nucleus optimized triple-zeta basis sets for the 4p, 5p and 6p elements , 1998 .

[12]  Kazuhiro Ishida.  ACE algorithm for the rapid evaluation of the electron‐repulsion integral over Gaussian‐type orbitals , 1996 .

[13]  Todd J. Martínez,et al.  Generating Efficient Quantum Chemistry Codes for Novel Architectures. , 2013, Journal of chemical theory and computation.

[14]  Shigeru Obara,et al.  Efficient recursive computation of molecular integrals over Cartesian Gaussian functions , 1986 .

[15]  T. Shiozaki,et al.  Large-scale Dirac-Fock-Breit method using density fitting and 2-spinor basis functions. , 2013, The Journal of chemical physics.

[16]  Warren J. Hehre,et al.  Computation of electron repulsion integrals involving contracted Gaussian basis functions , 1978 .

[17]  Wenjian Liu,et al.  Fully relativistic theories and methods for NMR parameters , 2012, Theoretical Chemistry Accounts.

[18]  Roland Lindh,et al.  The reduced multiplication scheme of the Rys-Gauss quadrature for 1st order integral derivatives , 1993 .

[19]  J. Gauss,et al.  Analytic energy derivatives in relativistic quantum chemistry , 2014 .

[20]  Martin Head-Gordon,et al.  Efficient computation of two-electron - repulsion integrals and their nth-order derivatives using contracted Gaussian basis sets , 1990 .

[21]  R. Lindh,et al.  Structure and energetics of Cr(CO)6 and Cr(CO)5 , 1993 .

[22]  B. Rees,et al.  Electronic structure of chromium hexacarbonyl at 78 K. I. Neutron diffraction study , 1975 .

[23]  K. Dyall,et al.  Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set , 1990 .

[24]  Trygve Helgaker,et al.  Recent advances in wave function-based methods of molecular-property calculations. , 2012, Chemical reviews.

[25]  Koji Yasuda,et al.  Two‐electron integral evaluation on the graphics processor unit , 2008, J. Comput. Chem..

[26]  Michel Dupuis,et al.  Computation of electron repulsion integrals using the rys quadrature method , 1983 .

[27]  Wenjian Liu,et al.  Four-component relativistic theory for nuclear magnetic shielding: magnetically balanced gauge-including atomic orbitals. , 2009, The Journal of chemical physics.

[28]  Lemin Li,et al.  Analytical energy gradient evaluation in relativistic and nonrelativistic density functional calculations , 2002, J. Comput. Chem..

[29]  H. Bernhard Schlegel,et al.  An efficient algorithm for calculating ab initio energy gradients using s, p Cartesian Gaussians , 1982 .

[30]  Michel Dupuis,et al.  The Rys quadrature revisited: A novel formulation for the efficient computation of electron repulsion integrals over Gaussian functions , 2001 .

[31]  G. Breit.  The Effect of Retardation on the Interaction of Two Electrons , 1929 .

[32]  Christof Hättig,et al.  Explicitly correlated electrons in molecules. , 2012, Chemical reviews.

[33]  Martin Head-Gordon,et al.  A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relations , 1988 .

[34]  E. Baerends,et al.  Dissociating energies, vibrational frequencies and 13C NMR chemical shifts of the 18 electron species [M(CO)6]n (M=Hf-Ir, Mo, Tc, Ru, Cr, Mn, Fe). A density functional study. , 1997 .

[35]  Martin Head-Gordon,et al.  An efficient algorithm for the generation of two‐electron repulsion integrals over gaussian basis functions , 2009 .

[36]  Wenjian Liu,et al.  Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations , 2011 .

[37]  S. F. Boys.  Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[38]  Pekka Pyykkö,et al.  Relativistic effects in structural chemistry , 1988 .

[39]  Theresa L. Windus,et al.  Components for integral evaluation in quantum chemistry , 2008, J. Comput. Chem..

[40]  Toru Shiozaki,et al.  Analytical Nuclear Gradients of Density-Fitted Dirac-Fock Theory with a 2-Spinor Basis. , 2013, Journal of chemical theory and computation.

[41]  Kimihiko Hirao,et al.  A highly efficient algorithm for electron repulsion integrals over relativistic four-component Gaussian-type spinors , 2002 .

[42]  Mark S. Gordon,et al.  New Multithreaded Hybrid CPU/GPU Approach to Hartree-Fock. , 2012, Journal of chemical theory and computation.

[43]  Kenneth Ruud,et al.  Arbitrary-Order Density Functional Response Theory from Automatic Differentiation. , 2010, Journal of chemical theory and computation.

[44]  M. Kaupp,et al.  A fully relativistic method for calculation of nuclear magnetic shielding tensors with a restricted magnetically balanced basis in the framework of the matrix Dirac-Kohn-Sham equation. , 2008, The Journal of chemical physics.

[45]  Ivan S. Ufimtsev,et al.  Dynamic Precision for Electron Repulsion Integral Evaluation on Graphical Processing Units (GPUs). , 2011, Journal of chemical theory and computation.

[46]  Wim Klopper,et al.  Computation of some new two-electron Gaussian integrals , 1992 .

[47]  E. Davidson,et al.  One- and two-electron integrals over cartesian gaussian functions , 1978 .

[48]  Michel Dupuis,et al.  Evaluation of molecular integrals over Gaussian basis functions , 1976 .

[49]  H. Schlegel,et al.  Dipole Moments, Polarizabilities, and Infrared Intensities Calculated with Electric Field Dependent Functions , 1994 .

[50]  Guy E. Blelloch,et al.  Implementation of a portable nested data-parallel language , 1993, PPOPP '93.

[51]  Han,et al.  Supersymmetrization of N=1 ten-dimensional supergravity with Lorentz Chern-Simons term. , 1986, Physical review. D, Particles and fields.

[52]  R. Ditchfield,et al.  Molecular Orbital Theory of Magnetic Shielding and Magnetic Susceptibility , 1972 .

[53]  Brett M. Bode,et al.  Uncontracted Rys Quadrature Implementation of up to G Functions on Graphical Processing Units. , 2010, Journal of chemical theory and computation.

[54]  Richard E. Stanton,et al.  Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations , 1984 .