Linear-scaling method for calculating nuclear magnetic resonance chemical shifts using gauge-including atomic orbitals within Hartree-Fock and density-functional theory.

Details of a new density matrix-based formulation for calculating nuclear magnetic resonance chemical shifts at both Hartree-Fock and density functional theory levels are presented. For systems with a nonvanishing highest occupied molecular orbital-lowest unoccupied molecular orbital gap, the method allows us to reduce the asymptotic scaling order of the computational effort from cubic to linear, so that molecular systems with 1000 and more atoms can be tackled with today's computers. The key feature is a reformulation of the coupled-perturbed self-consistent field (CPSCF) theory in terms of the one-particle density matrix (D-CPSCF), which avoids entirely the use of canonical MOs. By means of a direct solution for the required perturbed density matrices and the adaptation of linear-scaling integral contraction schemes, the overall scaling of the computational effort is reduced to linear. A particular focus of our formulation is to ensure numerical stability when sparse-algebra routines are used to obtain an overall linear-scaling behavior.

[1]  W. Kutzelnigg,et al.  Theory of magnetic susceptibilities and NMR chemical shifts in terms of localized quantities. II. Application to some simple molecules , 1982 .

[2]  H. F. Hameka.  Relationship between proton shielding constants and electric dipole moments in the hydrogen halides , 1959 .

[3]  Matt Challacombe,et al.  Density matrix perturbation theory. , 2003, Physical review letters.

[4]  Trygve Helgaker,et al.  Direct optimization of the AO density matrix in Hartree-Fock and Kohn-Sham theories , 2000 .

[5]  Andreas Bohne,et al.  W3-SWEET: Carbohydrate Modeling By Internet , 1998 .

[6]  Eric Schwegler,et al.  Fast assembly of the Coulomb matrix: A quantum chemical tree code , 1996 .

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

[8]  C. Ochsenfeld.  An ab initio study of the relation between NMR chemical shifts and solid-state structures: hexabenzocoronene derivatives , 2000 .

[9]  Jürgen Gauss,et al.  Helical packing of discotic hexaphenyl hexa-peri-hexabenzocoronenes: theory and experiment. , 2007, The journal of physical chemistry. B.

[10]  M. Challacombe,et al.  Higher-order response in O(N) by perturbed projection. , 2004, The Journal of chemical physics.

[11]  Nonorthogonal density-matrix perturbation theory. , 2005, The Journal of chemical physics.

[12]  Trygve Helgaker,et al.  Ab Initio Methods for the Calculation of NMR Shielding and Indirect Spin-Spin Coupling Constants , 1999 .

[13]  J. Gauss.  Accurate Calculation of NMR Chemical Shifts , 1995 .

[14]  E. Cuthill,et al.  Reducing the bandwidth of sparse symmetric matrices , 1969, ACM '69.

[15]  M. Bühl,et al.  Calculation of NMR and EPR parameters : theory and applications , 2004 .

[16]  P. Pulay.  Improved SCF convergence acceleration , 1982 .

[17]  M. Head‐Gordon,et al.  Curvy steps for density matrix based energy minimization: tensor formulation and toy applications , 2003 .

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

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

[20]  Matt Challacombe,et al.  A simplified density matrix minimization for linear scaling self-consistent field theory , 1999 .

[21]  R. Mcweeny.  Some Recent Advances in Density Matrix Theory , 1960 .

[22]  Yihan Shao,et al.  Efficient evaluation of the Coulomb force in density-functional theory calculations , 2001 .

[23]  Christian Ochsenfeld,et al.  Linear and sublinear scaling formation of Hartree-Fock-type exchange matrices , 1998 .

[24]  Nicholas C. Handy,et al.  The density functional calculation of nuclear shielding constants using London atomic orbitals , 1995 .

[25]  Dennis R. Salahub,et al.  NUCLEAR MAGNETIC RESONANCE SHIELDING TENSORS CALCULATED WITH A SUM-OVER-STATES DENSITY FUNCTIONAL PERTURBATION THEORY , 1994 .

[26]  M. Challacombe.  A general parallel sparse-blocked matrix multiply for linear scaling SCF theory , 2000 .

[27]  Gustavo E. Scuseria,et al.  Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations , 1997 .

[28]  Eric Schwegler,et al.  Linear scaling computation of the Fock matrix. II. Rigorous bounds on exchange integrals and incremental Fock build , 1997 .

[29]  Trygve Helgaker,et al.  Nuclear shielding constants by density functional theory with gauge including atomic orbitals , 2000 .

[30]  Emanuel H. Rubensson,et al.  Systematic sparse matrix error control for linear scaling electronic structure calculations , 2005, J. Comput. Chem..

[31]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[32]  Trygve Helgaker,et al.  Geometrical derivatives and magnetic properties in atomic-orbital density-based Hartree-Fock theory , 2001 .

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

[34]  P. Hohenberg,et al.  Inhomogeneous Electron Gas , 1964 .

[35]  N. Ramsey.  Electron Coupled Interactions between Nuclear Spins in Molecules , 1953 .

[36]  H. F. Hameka.  On the nuclear magnetic shielding in the hydrogen molecule , 1958 .

[37]  Andreas Bohne,et al.  SWEET - WWW-based rapid 3D construction of oligo- and polysaccharides , 1999, Bioinform..

[38]  Paul Adrien Maurice Dirac,et al.  Note on the Interpretation of the Density Matrix in the Many-Electron Problem , 1931, Mathematical Proceedings of the Cambridge Philosophical Society.

[39]  William H. Press,et al.  Numerical recipes in C , 2002 .

[40]  R. Mcweeny,et al.  Methods Of Molecular Quantum Mechanics , 1969 .

[41]  Hans W. Horn,et al.  Direct computation of second-order SCF properties of large molecules on workstation computers with an application to large carbon clusters , 1992 .

[42]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[43]  William H. Press,et al.  Numerical Recipes in C, 2nd Edition , 1992 .

[44]  Martin Head-Gordon,et al.  A tensor formulation of many-electron theory in a nonorthogonal single-particle basis , 1998 .

[45]  T. Keith,et al.  A comparison of models for calculating nuclear magnetic resonance shielding tensors , 1996 .

[46]  Michael J. Frisch,et al.  A linear scaling method for Hartree–Fock exchange calculations of large molecules , 1996 .

[47]  J. Gauss,et al.  Structure assignment in the solid state by the coupling of quantum chemical calculations with NMR experiments: a columnar hexabenzocoronene derivative. , 2001, Journal of the American Chemical Society.

[48]  Jörg Kussmann,et al.  Density matrix-based variational quantum Monte Carlo providing an asymptotically linear scaling behavior for the local energy , 2007 .

[49]  R. Parr.  Density-functional theory of atoms and molecules , 1989 .

[50]  Christian Ochsenfeld,et al.  Linear scaling exchange gradients for Hartree–Fock and hybrid density functional theory , 2000 .

[51]  Michael J. Frisch,et al.  Direct analytic SCF second derivatives and electric field properties , 1990 .

[52]  Dennis R. Salahub,et al.  Calculations of NMR shielding constants by uncoupled density functional theory , 1993 .

[53]  Michael J. Frisch,et al.  Achieving Linear Scaling for the Electronic Quantum Coulomb Problem , 1996, Science.

[54]  Johannes Grotendorst,et al.  Modern methods and algorithms of quantum chemistry , 2000 .

[55]  R. Ditchfield,et al.  Self-consistent perturbation theory of diamagnetism , 1974 .

[56]  Benny G. Johnson,et al.  THE CONTINUOUS FAST MULTIPOLE METHOD , 1994 .

[57]  G. Schreckenbach,et al.  Calculation of NMR Shielding Tensors Using Gauge-Including Atomic Orbitals and Modern Density Functional Theory , 1995 .

[58]  Aage E. Hansen,et al.  Localized orbital/local origin method for calculation and analysis of NMR shieldings. Applications to 13C shielding tensors , 1985 .

[59]  Solomon L. Pollack,et al.  Proceedings of the 1969 24th national conference , 1969 .

[60]  C. Ochsenfeld,et al.  A study of a moleculartweezer host-guest system by a combination of quantum-chemical calculations and solid-state NMR experiments. , 2002, Solid state nuclear magnetic resonance.

[61]  Valéry Weber,et al.  Ab initio linear scaling response theory: electric polarizability by perturbed projection. , 2004, Physical review letters.

[62]  T. Thirunamachandran,et al.  Molecular Quantum Electrodynamics , 1984 .

[63]  Thomas Schrader,et al.  Molecular tweezer and clip in aqueous solution: unexpected self-assembly, powerful host-guest complex formation, quantum chemical 1H NMR shift calculation. , 2006, Journal of the American Chemical Society.

[64]  Jörg Kussmann,et al.  Ab initio NMR spectra for molecular systems with a thousand and more atoms: a linear-scaling method. , 2004, Angewandte Chemie.

[65]  Jörg Kussmann,et al.  Structure of molecular tweezer complexes in the solid state: NMR experiments, X-ray investigations, and quantum chemical calculations. , 2007, Journal of the American Chemical Society.

[66]  Norman Ramsey.  Magnetic Shielding of Nuclei in Molecules , 1950 .

[67]  Fred G. Gustavson,et al.  Two Fast Algorithms for Sparse Matrices: Multiplication and Permuted Transposition , 1978, TOMS.

[68]  A. Szabó,et al.  Modern quantum chemistry : introduction to advanced electronic structure theory , 1982 .

[69]  Yihan Shao,et al.  Sparse matrix multiplications for linear scaling electronic structure calculations in an atom‐centered basis set using multiatom blocks , 2003, J. Comput. Chem..

[70]  Christian Ochsenfeld,et al.  A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme , 1997 .

[71]  C. Ochsenfeld,et al.  Structure and Dynamics of the Host–Guest Complex of a Molecular Tweezer: Coupling Synthesis, Solid-State NMR, and Quantum-Chemical Calculations , 2001 .

[72]  Guntram Rauhut,et al.  Comparison of NMR Shieldings Calculated from Hartree−Fock and Density Functional Wave Functions Using Gauge-Including Atomic Orbitals , 1996 .

[73]  Peter Pulay,et al.  Efficient implementation of the gauge-independent atomic orbital method for NMR chemical shift calculations , 1990 .