Block-adaptive quantum mechanics: An adaptive divide-and-conquer approach to interactive quantum chemistry

Block-adaptive quantum mechanics: An adaptive divide-and-conquer approach to interactive quantum chemistry
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  Block-Adaptive Quantum Mechanics: an adaptive divide-and-conquerapproach to interactive quantum chemistry Mäel Bosson, Sergei Grudinin, Stephane RedonNANO-D - INRIA Grenoble - Rhone-Alpes655, avenue de l’Europe, 38335 Saint-Ismier Cedex, FranceFebruary 28, 2014 Abstract We present a novel  Block-Adaptive Quantum Mechanics   (BAQM) approach to interactive quantum chemistry. Al-though quantum chemistry models are known to be computationally demanding, we achieve interactive rates by focusingcomputational resources on the most active parts of the system. BAQM is based on a divide-and-conquer technique, andconstrains some nucleus positions and some electronic degrees of freedom on the fly to simplify the simulation. As a result,each time step may be performed significantly faster, which in turn may accelerate attraction to the neighboring localminima. By applying our approach to the non-self-consistent ASED-MO (Atom Superposition and Electron DelocalizationMolecular Orbital) theory, we demonstrate interactive rates and efficient virtual prototyping for systems containing morethan a thousand of atoms on a standard desktop computer. Keywords:  Interactive Quantum Chemistry, Reduced Basis, Adaptive, Divide-And-Conquer, ASED-MO.1  Block-Adaptive Quantum Mechanics (BAQM) is a new approach to interactive quantum chemistry. BAQM is based on adivide-and-conquer technique, and constrains some nucleus positions and some electronic degrees of freedom on the fly tosimplify the simulation. By applying our approach to the non-self-consistent ASED-MO theory, we demonstrate interactiverates and efficient virtual prototyping for systems containing more than a thousand of atoms on a standard desktop computer.2  1  Introduction The fundamental Schrödinger equation for nuclei and electrons is a fascinating problem that has been attracting a lot of attention in the computational chemistry community. In theory, solving this equation makes it possible to accurately describethe behavior of particles at the atomic scale. Thus, it seems natural that software applications for computer-aided design(CAD) of nanosystems should simulate quantum physics. In particular, CAD applications should interactively provide theuser with physically-based feedback when editing the structure of a nanosystem.Because of the high computational cost of underlying numerical methods, though, interactively solving the Schrödingerequation is a challenging problem. Fortunately, many efficient computational methods have been deduced from approximatetheories 15 . In general, these methods solve the one-electron Schrödinger equation after it has been projected to a finite basisset. For instance, employing a basis set composed of atomic orbitals (denoted by  φ µ ) 30 leads to the following generalizedeigenvalue problem: HC   =  SCD,  (1)where H  µν   =   φ µ | H  | φ ν    and  S  µν   =   φ µ | φ ν   .  (2)The diagonal matrix  D  contains the sorted eigenvalues and the matrix  C   contains the corresponding eigenvectors ( e i  denotesthe  i th lowest eigenvalue and  C  i  the corresponding eigenvector). The potential energy of the system is the sum: E   = N/ 2  i =1 2 e i ,  (3)where  N   is the number of electrons in the system. The gradient of the potential energy is: ∇ x E   =  µ  ν  P  µν  ∇ x H  µν   −  µ  ν  W  µν  ∇ x S  µν  ,  (4)where  P   is the density matrix and  W   is the energy-weighted density matrix: P   = N/ 2  i =1 2 C  i C  T i  , W   = N/ 2  i =1 2 e i C  i C  T i  .  (5)One approach to efficiently evaluate the Hamiltonian matrix  H   is to use a semi-empirical model such as the ASED-MO(Atom Superposition and Electron Delocalization Molecular Orbital) theory 3 . In this theory, we have recently presented aninteractive quantum chemistry approach red 7 based on the Divide-And-Conquer (D&C) method 16 . In particular, we havedemonstrated that interactively solving the one-electron Schrödinger equation is possible on current desktop computers forsystems composed of a few hundreds of atoms. By subdividing the system into many overlapping subsystems, this approachhas a linear time complexity in the number of atoms, as well as a good parallel scaling 32 , which should thus allow forcontinued improvements with current hardware trends in personal computers.Despite this, it will still be difficult to achieve interactive rates in two situations: •  Large number of subsystems : since the number of subsystems increases linearly with the number of atoms, somesystems will simply be too large to allow for interactive rates. •  Large subsystems : to reach high accuracy, the D&C approach needs to employ sufficiently large overlapping sub-systems 7 . In this case, solving even a single subsystem’s eigendecomposition problem may be too costly to achieveinteractive rates. Furthermore, it may be difficult to expect important speed-ups in the near future because diagonal-ization algorithms typically have poor parallel scaling 5,8 and the serial speed of processing cores is reaching a physicallimit 41 . One approach to speed-up electronic structure calculations consists in incrementally updating eigenvectors,as in the Residual Minimization – Direct Inversion of the Iterative Subspace” (RM-DIIS) approach 33 . Unfortunately,this may be as slow as the direct approach when too many eigenvectors have to be updated. Another approach could3  be to directly freeze the density matrix while letting atomic nuclei move 17 . However, when a non-orthogonal basisset is used, this may produce non-orthogonal molecular orbitals which might attract the system in configurations withactually higher potential energy.To address both issues, we propose a novel  Block-Adaptive Quantum Mechanics   (BAQM) approach, based on the Divide-And-Conquer method and two new components.First, in order to decouple the computational complexity from the system’s size, we propose to  adaptively simulate   thenucleus degrees of freedom. In general, the nearsightedness principle 23 makes it possible to perform a fast incremental updateof the electronic structure when only some atoms have moved 17,26,39,40 . In the Divide-And-Conquer approach 16 , the systemis divided into nearly independent overlapping subsystems. In the context of a non self-consistent theory, when all atomsof a subsystem are frozen in space, both the Hamiltonian and its eigendecomposition are constant. To take advantage of this fact, we extend the approach we previously introduced for adaptive Cartesian mechanics coordinates 6 . Precisely, wefreeze and unfreeze  groups   of atoms, according to the applied atomic forces and the system’s decomposition into overlappingsubsystems. We call this first component  Block-Adaptive Cartesian Mechanics  .Second, to be able to deal with large subsystems for which diagonalization is the bottleneck, we propose to use an  adap-tively updated   reduced basis which takes advantage of temporal coherence between successive eigendecomposition problems.For some methods, evaluating the Hamiltonian and overlap matrices may be computationally demanding. However, thesecomputations are intrinsically parallel and can benefit from modern hardware architectures such as Graphics ProcessingUnits (GPUs) 43 . Similarly, the computation of the density matrix has a cubic complexity in the number of basis functions,but dense matrix multiplications are memory-friendly 42,46 and can be efficiently handled on modern hierarchical-memorymulticore architectures 13,45 . As a result, we have focused our efforts on the computation of molecular orbitals. A natural wayto accelerate the resolution of many similar differential equations is to use a reduced basis approach 31 . This methodologyhas been applied in specific contexts for electronic structure calculation 12,29 . In this paper, we propose to use an adaptivereduced basis which is automatically updated during the simulation. We call this second component  Adaptive Reduced-Basis Quantum Mechanics  .We demonstrate that the BAQM approach may significantly speed-up energy minimization, as well as enable  interactive  quantum chemistry for large molecular systems. Figure 1 illustrates interactive virtual prototyping of a polyfluorene chainmolecule.Figure 1:  Block-Adaptive Quantum Mechanics (BAQM) in SAMSON (Software for Adaptive Modeling andSimulation Of Nanosystems) 1 . In this example the system is divided into four subsystems. The energy is minimizedcontinuously as the user edits the molecular system. At each time step, both the geometry and the electronic structureare incrementally and adaptively updated. Because the user pulls one atom (red arrow) in the left part of the system,the electronic structure is updated with the full basis for the leftmost subsystem (all atoms are red). In the neighboringsubsystem, the electronic structure is updated according to a reduced-basis approximation (some carbons are black and somehydrogens are white). In the right part of the molecule, the user force does not have a sufficiently large impact, and atomspositions are frozen (all atoms are blue).4  2  Overview In general, adaptive approaches automatically focus computational resources on the most relevant parts of a problem. We usesuch an approach to maintain interactive rates while modeling chemical structures based on quantum chemistry principles.In this section, we provide an overview of our approach, and introduce its two main components: block-adaptive Cartesianmechanics, and adaptive reduced-basis quantum mechanics. For completeness, we first briefly recall the ASED-MO theoryand the Divide-And-Conquer (D&C) technique. red We refer the reader to our previous publication 7 for more details aboutour ASED-MO D&C method. 2.1  The ASED-MO theory In this paper, we used the ASED-MO theory 3 to test and validate our BAQM approach. In this theory, the electronicdensity function is split into two terms: a  perfectly-following term   (the electron density when atoms do not interact), anda  non-perfectly-following term   (corresponding to the bonds formation). This last term is computed based on the ExtendedHückel Molecular Orbital theory (EHMO) 20 , a simple semi-empirical quantum chemistry method which approximates theHamiltonian matrix terms as: H  µν   = K I  µ  + I  ν  2  S  µν  ,  (6)where  I  µ  is the ionization energy of the atomic orbital  φ µ , and  K   is the Wolfsberg-Helmholtz constant. 2.2  The Divide-And-Conquer (D&C) technique redThe D&C approach is attractive because of its efficiency (nearly perfect parallel scaling 32,35 ), simplicity for non-orthogonalbasis sets, and accuracy 16,22,25,44,47,48 . There are three main steps in it: •  Dividing the system The srcinal system  S   is first divided into  M   non-overlapping subsystems  S  1 ,  ... ,  S  M  . Then, for each subsystem  S  i ,an  extended   subsystem  S  ∗ i  is defined as the one containing all atoms from  S  i  and those closer to these atoms than acertain distance cutoff. •  Computing each subsystem electronic structure independently red A basis set is associated to each extended subsystem  S  ∗ i  ( 1  i  M  ). The projection of the one-electron Schrödingerequation in redthis basis leads to the generalized eigenvalue problems: H  i C  i  = S  i C  i D i ,  1  i  M.  (7)red Each local generalized eigenvalue problem (7) provides a set of molecular orbitals, which are then globally rankedaccording to their corresponding energies. We then populate these molecular orbitals until there are exactly N   electronsin the system, as detailed in 7 . •  Summing up the various contributions red The occupied molecular orbitals determine the local density matrices  P  i  and energy-weighted density matrices  W  i ,from which the density matrix  P   and the energy-weighted density matrix  W   are obtained  via   a superposition scheme 7 .Once  P   and  W   have been obtained, the potential energy is expressed as E   = Tr( HP  )  (8)and the gradient of the potential energy is approximated as: ∇ x E   =  µ  ν  P  µν  ∇ x H  µν   −  µ  ν  W  µν  ∇ x S  µν  .  (9)5
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