Computational chemistry from the perspective of numerical analysis

We present the field of computational chemistry from the standpoint of numerical analysis. We introduce the most commonly used models and comment on their applicability. We briefly outline the results of mathematical analysis and then mostly concentrate on the main issues raised by numerical simulations. A special emphasis is laid on recent results in numerical analysis, recent developments of new methods and challenging open issues.

[1]  R. Hill,et al.  Rates of convergence and error estimation formulas for the Rayleigh–Ritz variational method , 1985 .

[2]  G. Henkelman,et al.  A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives , 1999 .

[3]  Hill Dependence of the rate of convergence of the Rayleigh-Ritz method on a nonlinear parameter. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[4]  D. Brenner,et al.  Convergence Acceleration Scheme for Self-consistent Orthogonal-basis-set Electronic Structure Methods , 2003 .

[5]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[6]  E. Cancès,et al.  On the convergence of SCF algorithms for the Hartree-Fock equations , 2000 .

[7]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[8]  D.R.Bowler,et al.  Density matrices in O(N) electronic structure calculations: theory and applications , 1998 .

[9]  J. Lions,et al.  Résolution d'EDP par un schéma en temps « pararéel » , 2001 .

[10]  E. B. Starikov,et al.  On the convergence of the hartree-fock selfconsistency procedure , 1993 .

[11]  David Chandler,et al.  Barrier crossings:. classical theory of rare but important events , 1998 .

[12]  David R. Bowler,et al.  A Comparison of Linear Scaling Tight Binding Methods , 1997 .

[13]  Gustavo E. Scuseria,et al.  A fast multipole algorithm for the efficient treatment of the Coulomb problem in electronic structure calculations of periodic systems with Gaussian orbitals , 1998 .

[14]  Pierre-Louis Lions,et al.  Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 4: Binding of neutral systems for the Hartree model , 1993 .

[15]  J. D. Morgan,et al.  Convergence properties of Fock's expansion for S-state eigenfunctions of the helium atom , 1986 .

[16]  H. Schlegel,et al.  A combined method for determining reaction paths, minima, and transition state geometries , 1997 .

[17]  Eric Darve,et al.  Calculating Free Energies Using a Scaled-Force Molecular Dynamics Algorithm , 2002 .

[18]  M. Griebel,et al.  On the computation of the eigenproblems of hydrogen helium in strong magnetic and electric fields with the sparse grid combination technique , 2000 .

[19]  Berend Smit,et al.  Understanding Molecular Simulation , 2001 .

[20]  X. Blanc A mathematical insight into ab initio simulations of the solid phase , 2000 .

[21]  T. Schlick Molecular modeling and simulation , 2002 .

[22]  M. Tuckerman,et al.  Understanding Modern Molecular Dynamics: Techniques and Applications , 2000 .

[23]  A. Voter Hyperdynamics: Accelerated Molecular Dynamics of Infrequent Events , 1997 .

[24]  A. Voter Parallel replica method for dynamics of infrequent events , 1998 .

[25]  M. Tuckerman Ab initio molecular dynamics: basic concepts, current trends and novel applications , 2002 .

[26]  Michael Griebel,et al.  Sparse grids for boundary integral equations , 1999, Numerische Mathematik.

[27]  Volker Bach,et al.  Error bound for the Hartree-Fock energy of atoms and molecules , 1992 .

[28]  B. Lévy,et al.  Exponential transformation of molecular orbitals: A quadratically convergent SCF procedure. I. General formulation and application to closed-shell ground states , 1980 .

[29]  M. Ortiz,et al.  Quasicontinuum analysis of defects in solids , 1996 .

[30]  Frédéric Legoll,et al.  High-order averaging schemes with error bounds for thermodynamical properties calculations by MD simulations , 2003 .

[31]  Klaus Ruedenberg,et al.  Systematic approach to extended even-tempered orbital bases for atomic and molecular calculations , 1979 .

[32]  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.

[33]  X Blanc,et al.  Nonlinear instability of density-independent orbital-free kinetic-energy functionals. , 2005, The Journal of chemical physics.

[34]  Martin,et al.  Linear system-size scaling methods for electronic-structure calculations. , 1995, Physical review. B, Condensed matter.

[35]  Harry Yserentant,et al.  Sparse grid spaces for the numerical solution of the electronic Schrödinger equation , 2005, Numerische Mathematik.

[36]  Jan Almlöf,et al.  General methods for geometry and wave function optimization , 1992 .

[37]  Roger Smith,et al.  Numerical calculations using the hyper-molecular dynamics simulation method , 2001 .

[38]  Arnold Neumaier,et al.  Molecular Modeling of Proteins and Mathematical Prediction of Protein Structure , 1997, SIAM Rev..

[39]  David A Mazziotti,et al.  Realization of quantum chemistry without wave functions through first-order semidefinite programming. , 2004, Physical review letters.

[40]  Eric Schwegler,et al.  Linear scaling computation of the Fock matrix , 1997 .

[41]  Peter M. W. Gill,et al.  Molecular integrals Over Gaussian Basis Functions , 1994 .

[42]  M. Ortiz,et al.  An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method , 1997, cond-mat/9710027.

[43]  Egil A. Hylleraas ON THE GROUND STATE OF THE HELIUM ATOM , 2000 .

[44]  J. Koutecký,et al.  On Convergence Difficulties in the Iterative Hartree—Fock Procedure , 1971 .

[45]  Y. Saad,et al.  Electronic structure calculations for plane-wave codes without diagonalization , 1999 .

[46]  Barry Simon,et al.  The Hartree-Fock theory for Coulomb systems , 1977 .

[47]  Bruno Klahn,et al.  The convergence of the Rayleigh-Ritz Method in quantum chemistry , 1977 .

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

[49]  M. Overton,et al.  The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions. , 2004, The Journal of chemical physics.

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

[51]  Giulia Galli,et al.  Large‐Scale Electronic Structure Calculations Using Linear Scaling Methods , 2000 .

[52]  Michael Ortiz,et al.  Mixed Atomistic and Continuum Models of Deformation in Solids , 1996 .

[53]  A. Laio,et al.  Escaping free-energy minima , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[54]  Walter Kohn,et al.  Nobel Lecture: Electronic structure of matter-wave functions and density functionals , 1999 .

[55]  David A. Mazziotti,et al.  3,5-CONTRACTED SCHRODINGER EQUATION : DETERMINING QUANTUM ENERGIES AND REDUCED DENSITY MATRICES WITHOUT WAVE FUNCTIONS , 1998 .

[56]  Gabriel Turinici,et al.  Quadratically convergent algorithm for fractional occupation numbers in density functional theory , 2003 .

[57]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.

[58]  Tobias Jahnke,et al.  Long-Time-Step Integrators for Almost-Adiabatic Quantum Dynamics , 2004, SIAM J. Sci. Comput..

[59]  Benjamin Jourdain,et al.  QUANTUM MONTE CARLO SIMULATIONS OF FERMIONS: A MATHEMATICAL ANALYSIS OF THE FIXED-NODE APPROXIMATION , 2006 .

[60]  Werner Kutzelnigg,et al.  Theory of the expansion of wave functions in a gaussian basis , 1994 .

[61]  Mario A. Natiello,et al.  Convergence properties of Hartree-Fock SCF molecular calculations , 1984 .

[62]  Normand Mousseau,et al.  Efficient sampling in complex materials at finite temperature: the thermodynamically-weighted activation-relaxation technique , 2003 .

[63]  C. Schütte,et al.  Quantum‐classical molecular dynamics as an approximation to full quantum dynamics , 1996 .

[64]  H. C. Andersen Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations , 1983 .

[65]  J. D. Morgan,et al.  Rates of convergence of variational calculations and of expectation values , 1984 .

[66]  Pierre-Louis Lions,et al.  On some periodic Hartree-type models for crystals , 2002 .

[67]  Claude Le Bris,et al.  ON THE PERTURBATION METHODS FOR SOME NONLINEAR QUANTUM CHEMISTRY MODELS , 1998 .

[68]  Li,et al.  Density-matrix electronic-structure method with linear system-size scaling. , 1993, Physical review. B, Condensed matter.

[69]  C. Brooks Computer simulation of liquids , 1989 .

[70]  Pierre-Louis Lions,et al.  Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 2 : Stability is equivalent to the binding of neutral subsystems , 1993 .

[71]  Stephen D. Bond,et al.  The Nosé-Poincaré Method for Constant Temperature Molecular Dynamics , 1999 .

[72]  S. Nosé A molecular dynamics method for simulations in the canonical ensemble , 2002 .

[73]  Werner Kutzelnigg,et al.  Hund's rules, the alternating rule, and symmetry holes , 1993 .

[74]  Pierre-Louis Lions,et al.  Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 3 : Binding of neutral subsystems , 1993 .

[75]  H. Bungartz,et al.  Sparse grids , 2004, Acta Numerica.

[76]  E. Hairer,et al.  Geometric numerical integration illustrated by the Störmer–Verlet method , 2003, Acta Numerica.

[77]  Paul Indelicato,et al.  Computational approaches of relativistic models in quantum chemistry , 2003 .

[78]  C. Schwab,et al.  NUMERICAL SOLUTION OF PARABOLIC EQUATIONS IN HIGH DIMENSIONS , 2004 .

[79]  R. Glassey,et al.  Global existence of solutions to the Cauchy problem for time‐dependent Hartree equations , 1975 .

[80]  P. Deuflhard,et al.  Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains , 2000 .

[81]  G. V. Chester,et al.  Solid State Physics , 2000 .

[82]  Mohammed Lemou,et al.  Fast multipole method for multidimensional integrals , 1998 .

[83]  T. Hoffmann-Ostenhof,et al.  Electron Wavefunctions and Densities for Atoms , 2000, math/0005018.

[84]  Yihan Shao,et al.  Curvy steps for density matrix-based energy minimization: Application to large-scale self-consistent-field calculations , 2003 .

[85]  Pierre-Louis Lions,et al.  The Mathematical Theory of Thermodynamic Limits: Thomas--Fermi Type Models , 1998 .

[86]  Pierre-Louis Lions,et al.  On the thermodynamic limit for Hartree–Fock type models , 2001 .

[87]  C. Bris,et al.  Can we outperform the DIIS approach for electronic structure calculations , 2000 .

[88]  P. Deuflhard,et al.  A Direct Approach to Conformational Dynamics Based on Hybrid Monte Carlo , 1999 .

[89]  T. Beck Real-space mesh techniques in density-functional theory , 2000, cond-mat/0006239.

[90]  Yihan Shao,et al.  Improved Fermi operator expansion methods for fast electronic structure calculations , 2003 .

[91]  Søren Fournais,et al.  The Electron Density is Smooth Away from the Nuclei , 2002 .

[92]  Richard E. Stanton,et al.  Intrinsic convergence in closed‐shell SCF calculations. A general criterion , 1981 .

[93]  Frédéric Legoll,et al.  Long-time averaging for integrable Hamiltonian dynamics , 2005, Numerische Mathematik.

[94]  P. Stern,et al.  Automatic choice of global shape functions in structural analysis , 1978 .

[95]  D. Mazziotti Contracted Schrödinger equation: Determining quantum energies and two-particle density matrices without wave functions , 1998 .

[96]  Matt Challacombe,et al.  Linear scaling computation of the Fock matrix. V. Hierarchical Cubature for numerical integration of the exchange-correlation matrix , 2000 .

[97]  C. Valdemoro,et al.  N-representability problem within the framework of the contracted Schrodinger equation , 2000 .

[98]  G. Henkelman,et al.  Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points , 2000 .

[99]  Noam Bernstein,et al.  Mixed finite element and atomistic formulation for complex crystals , 1999 .

[100]  Ron Shepard,et al.  Elimination of the diagonalization bottleneck in parallel Direct-SCF methods , 1993 .

[101]  Y Maday,et al.  Parallel-in-time molecular-dynamics simulations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[102]  Michael Hehenberger,et al.  A dynamical damping scheme for converging molecular scf calculations , 1979 .

[103]  Yoshiyuki Kawazoe,et al.  Mesoscopic Dynamics of Fracture , 1998 .

[104]  Bertha Swirles,et al.  The Relativistic Self-Consistent Field , 1935 .

[105]  Arthur F. Voter,et al.  Accelerating Atomistic Simulations of Defect Dynamics: Hyperdynamics, Parallel Replica Dynamics, and Temperature-Accelerated Dynamics , 1998 .

[106]  E Weinan,et al.  Minimum action method for the study of rare events , 2004 .

[107]  E. Lieb,et al.  The Thomas-Fermi theory of atoms, molecules and solids , 1977 .

[108]  Richard E. Stanton,et al.  The existence and cure of intrinsic divergence in closed shell SCF calculations , 1981 .

[109]  Eric Canccs Scf Algorithms for Hartree-fock Electronic Calculations , 2022 .

[110]  Clemens C. J. Roothaan,et al.  New Developments in Molecular Orbital Theory , 1951 .

[111]  C. Lubich A variational splitting integrator for quantum molecular dynamics , 2004 .

[112]  J. C. Slater Atomic Shielding Constants , 1930 .

[113]  Wolfgang Hackbusch,et al.  The Efficient Computation of Certain Determinants Arising in the Treatment of Schrödinger's Equations , 2001, Computing.

[114]  Maria J. Esteban,et al.  An overview on linear and nonlinear Dirac equations , 2002 .

[115]  Michael Ortiz,et al.  Quasicontinuum simulation of fracture at the atomic scale , 1998 .

[116]  Eric Cancès,et al.  Computing electronic structures: A new multiconfiguration approach for excited states , 2006, J. Comput. Phys..

[117]  Valdemoro Approximating the second-order reduced density matrix in terms of the first-order one. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[118]  Patrick Fischer,et al.  Numerical Solution of the Schrödinger Equation in a Wavelet Basis for Hydrogen-like Atoms , 1998 .

[119]  Alfredo Bellen,et al.  Parallel algorithms for initial-value problems for difference and differential equations , 1989 .

[120]  E. Clementi,et al.  Electronic structure of large molecular systems , 1966 .

[121]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1979 .

[122]  E. Carter,et al.  Orbital-Free Kinetic-Energy Density Functional Theory , 2002 .

[123]  C. Pekeris,et al.  Ground State of Two-Electron Atoms , 1958 .

[124]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[125]  R. O. Jones,et al.  The density functional formalism, its applications and prospects , 1989 .

[126]  Jean Dolbeault,et al.  A variational method for relativistic computations in atomic and molecular physics , 2003 .

[127]  Christof Schütte,et al.  A mathematical investigation of the Car-Parrinello method , 1998 .

[128]  Kenneth R. Meyer,et al.  Generic Hamiltonian dynamical systems are neither integrable nor ergodic , 1974 .

[129]  Tobias Jahnke,et al.  Numerische Verfahren für fast adiabatische Quantendynamik , 2003 .

[130]  Paul F. Tupper,et al.  Ergodicity and the Numerical Simulation of Hamiltonian Systems , 2005, SIAM J. Appl. Dyn. Syst..

[131]  D. Vanderbilt,et al.  Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. , 1990, Physical review. B, Condensed matter.

[132]  J. Dolbeault,et al.  Minimization methods for the one-particle dirac equation. , 2000, Physical review letters.

[133]  L. Verlet Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules , 1967 .

[134]  Claude Le Bris,et al.  A general approach for multiconfiguration methods in quantum molecular chemistry , 1994 .

[135]  T. Hoffmann-Ostenhof,et al.  On the regularity of the density of electronic wavefunctions , 2002 .

[136]  Roberto Dovesi,et al.  The Periodic Hartree‐Fock Method and Its Implementation in the CRYSTAL Code , 2000 .

[137]  John A. Pople,et al.  Self-consistent molecular orbital methods. XVI. Numerically stable direct energy minimization procedures for solution of Hartree-Fock equations , 1976 .

[138]  Bertha Swirles,et al.  The Relativistic Interaction of Two Electrons in the Self-Consistent Field Method , 1936 .

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

[140]  Werner Kutzelnigg,et al.  Rates of convergence of the partial‐wave expansions of atomic correlation energies , 1992 .

[141]  Jacopo Tomasi,et al.  INTEGRAL EQUATION METHODS FOR MOLECULAR SCALE CALCULATIONS IN THE LIQUID PHASE , 1999 .

[142]  J. Martins,et al.  A straightforward method for generating soft transferable pseudopotentials , 1990 .

[143]  H. James,et al.  The Ground State of the Hydrogen Molecule , 1933 .

[144]  David R. Bowler,et al.  Recent progress in linear scaling ab initio electronic structure techniques , 2002 .

[145]  Harry Yserentant,et al.  On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives , 2004, Numerische Mathematik.

[146]  George A. Hagedorn,et al.  Crossing the Interface between Chemistry and Mathematics , 1997 .

[147]  F. London,et al.  Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik , 1927 .

[148]  Angela K. Wilson,et al.  Benchmark calculations with correlated molecular wave functions XII. Core correlation effects on the homonuclear diatomic molecules B2-F2 , 1997 .

[149]  David Chandler,et al.  Transition path sampling: throwing ropes over rough mountain passes, in the dark. , 2002, Annual review of physical chemistry.

[150]  Larry Spruch,et al.  Pedagogic notes on Thomas-Fermi theory (and on some improvements): atoms, stars, and the stability of bulk matter , 1991 .

[151]  Joseph Delhalle,et al.  Numerical determination of the electronic structure of atoms, diatomic and polyatomic molecules , 1989 .

[152]  Blöchl,et al.  Projector augmented-wave method. , 1994, Physical review. B, Condensed matter.

[153]  Benedict J. Leimkuhler,et al.  Generating generalized distributions from dynamical simulation , 2003 .

[154]  C. Schwartz,et al.  Importance of Angular Correlations between Atomic Electrons , 1962 .

[155]  Gustavo E. Scuseria,et al.  What is the Best Alternative to Diagonalization of the Hamiltonian in Large Scale Semiempirical Calculations , 1999 .

[156]  Eric Vanden Eijnden Numerical techniques for multi-scale dynamical systems with stochastic effects , 2003 .

[157]  Michael W. Schmidt,et al.  Effective convergence to complete orbital bases and to the atomic Hartree–Fock limit through systematic sequences of Gaussian primitives , 1979 .

[158]  Wim Klopper,et al.  Gaussian basis sets and the nuclear cusp problem , 1986 .

[159]  T. Hoffmann-Ostenhof,et al.  Analyticity of the density of electronic wavefunctions , 2002, math-ph/0211075.

[160]  Yvon Maday,et al.  Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations , 2003, Numerische Mathematik.

[161]  T. Woolf,et al.  DYNAMIC REACTION PATHS AND RATES THROUGH IMPORTANCE-SAMPLED STOCHASTIC DYNAMICS , 1999 .

[162]  George B. Bacskay,et al.  A quadratically convergent Hartree—Fock (QC-SCF) method. Application to closed shell systems , 1981 .

[163]  Kohn,et al.  Density functional and density matrix method scaling linearly with the number of atoms. , 1996, Physical review letters.

[164]  Pierre-Louis Lions,et al.  From atoms to crystals: a mathematical journey , 2005 .

[165]  E. Vanden-Eijnden,et al.  String method for the study of rare events , 2002, cond-mat/0205527.

[166]  Folkmar Bornemann,et al.  Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems , 1998 .

[167]  Car,et al.  Unified approach for molecular dynamics and density-functional theory. , 1985, Physical review letters.

[168]  G. Ciccotti,et al.  Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of n-Alkanes , 1977 .

[169]  Bach,et al.  There are no unfilled shells in unrestricted Hartree-Fock theory. , 1994, Physical review letters.

[170]  Warren J. Hehre,et al.  AB INITIO Molecular Orbital Theory , 1986 .

[171]  W. E,et al.  Finite temperature string method for the study of rare events. , 2002, Journal of Physical Chemistry B.

[172]  Wl,et al.  Improved theoretical dissociation energy and ionization potential for the ground state of the hydrogen molecule , 1993 .

[173]  Gero Friesecke,et al.  The Multiconfiguration Equations for Atoms and Molecules: Charge Quantization and Existence of Solutions , 2003 .

[174]  Arne Lüchow,et al.  Accurate upper and lower bounds to the 2S states of the lithium atom , 1994 .

[175]  G. A. Worth,et al.  Applying Direct Molecular Dynamics to Non‐Adiabatic Systems , 2003 .

[176]  Werner Kutzelnigg,et al.  Convergence of Expansions in a Gaussian Basis , 1996 .

[177]  E. Hylleraas,et al.  Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium , 1929 .

[178]  S. Goedecker Linear scaling electronic structure methods , 1999 .

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

[180]  David A. Mazziotti,et al.  Comparison of contracted Schrödinger and coupled-cluster theories , 1999 .

[181]  G. Scuseria,et al.  A black-box self-consistent field convergence algorithm: One step closer , 2002 .

[182]  Bernard Philippe,et al.  A parallel shooting technique for solving dissipative ODE's , 1993, Computing.

[183]  Wilhelm Huisinga,et al.  An Averaging Principle for Fast Degrees of Freedom Exhibiting Long-Term Correlations , 2004, Multiscale Model. Simul..

[184]  Elliott H. Lieb Density functionals for coulomb systems , 1983 .

[185]  E. Lieb Thomas-fermi and related theories of atoms and molecules , 1981 .

[186]  E. B. Tadmor,et al.  Quasicontinuum models of interfacial structure and deformation , 1998 .

[187]  Tobias Jahnke,et al.  Numerical integrators for quantum dynamics close to the adiabatic limit , 2003, Numerische Mathematik.

[188]  A R Plummer,et al.  Introduction to Solid State Physics , 1967 .

[189]  Mark S. Gordon,et al.  Approximate second order method for orbital optimization of SCF and MCSCF wavefunctions , 1997 .

[190]  David Chandler,et al.  Geometrical explanation and scaling of dynamical heterogeneities in glass forming systems. , 2002, Physical review letters.

[191]  L. Jay Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems , 1996 .

[192]  M.G.B. Drew,et al.  The art of molecular dynamics simulation , 1996 .

[193]  Jean Dolbeault,et al.  Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues , 2000 .

[194]  Claude Le Bris,et al.  Computing a molecule in its environment: A mathematical viewpoint , 1999 .

[195]  Wilhelm Huisinga,et al.  Biomolecular Conformations can be Identified as Metastable Sets of Molecular Dynamics , 2003 .

[196]  K. Merz,et al.  Combined Quantum Mechanical/Molecular Mechanical Methodologies Applied to Biomolecular Systems , 1999 .

[197]  Hoover,et al.  Canonical dynamics: Equilibrium phase-space distributions. , 1985, Physical review. A, General physics.

[198]  C. Withers,et al.  Schrödinger equation for the helium atom , 1984 .

[199]  C. Bris,et al.  Computing a molecule: A mathematical viewpoint , 1997 .

[200]  Mathieu Lewin,et al.  Solutions of the Multiconfiguration Equations in Quantum Chemistry , 2004 .

[201]  V. R. Saunders,et al.  A “Level–Shifting” method for converging closed shell Hartree–Fock wave functions , 1973 .

[202]  Pierre-Louis Lions,et al.  Solutions of Hartree-Fock equations for Coulomb systems , 1987 .

[203]  Arthur F. Voter,et al.  Exploiting past visits or minimum-barrier knowledge to gain further boost in the temperature-accelerated dynamics method , 2002 .

[204]  Jorge V. José,et al.  Chaos in classical and quantum mechanics , 1990 .

[205]  Anthony T. Patera,et al.  A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem* , 1999 .

[206]  Anthony T. Patera,et al.  NUMERICAL ANALYSIS OF A POSTERIORI FINITE ELEMENT BOUNDS FOR LINEAR FUNCTIONAL OUTPUTS , 2000 .

[207]  B. Forrest,et al.  Accelerated equilibration of polymer melts by time‐coarse‐graining , 1995 .

[208]  Gustavo E. Scuseria,et al.  Linear Scaling Density Functional Calculations with Gaussian Orbitals , 1999 .

[209]  Jonathan C. Mattingly,et al.  Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise , 2002 .

[210]  S. Nosé,et al.  An extension of the canonical ensemble molecular dynamics method , 1986 .

[211]  Giles Auchmuty,et al.  Convergent iterative methods for the Hartree eigenproblem , 1994 .