Introduction to the very basic computational aspects of the modern Quantum Chemistry for Software Engineers

1
Introduction to the very basic computational
aspects of the modern Quantum Chemistry for
Software Engineers

• Alexander A. Granovsky
• The PC GAMESS/Firefly Project

July 23, 2009 MSU, Moscow, Russia
2
Outline

• Quantum Chemistry purpose and methods
• Typical tasks, their parameters and computational
  complexity
• Conventional, direct, and semi-direct methods
• Standard and fast methods
• Typical parallel algorithms key features and
  open problems
• Canonical example four index integral
  transformation step

3
Quantum Chemistry purpose and methods

• Quantum Chemistry (QC) is the science based on
  the applications of the first principles of
  Quantum Mechanics to the modeling of chemical
  systems and processes.
• All chemical systems are treated as the sets of
  electrons and nuclei described by the molecular
  Hamiltonian operator. Solutions of the molecular
  Schrödinger Equation contain information on all
  the molecular properties.
• The molecular Schrödinger Equation has to be
  solved approximately to obtain information on the
  properties of the molecular system of interest.

4
Quantum Chemistry standard model

• Non-relativistic or weakly relativistic theory
  mainly based on the standard Quantum Mechanics
• Most widely used approach
• Note, spins of electrons are still very important
  variables!
• More or less quasi-relativistic and purely
  relativistic approaches are primary used to
  describe systems with heavy nuclei
• Adiabatic or Born-Oppenheimer approximations
• Nuclei are fixed or moving slowly.
• Molecular Hamiltonian now acts on electronic
  variables and depends parametrically on nuclear
  variables
• Algebraic approach
• Use of finite basis sets to solve
  eigenvalue/eigenvector problem
• Modern QC is the highly algebraic science!

5
Quantum Chemistry algebraic approach

• Hamiltonian is a two-particle operator acting on
  the functions of 3n variables (electronic
  degrees of freedom)
• One needs a suitable basis to deal with
• Electrons are fermions
• Basis functions are thus the antisymmetrized
  direct products (Slater determinants) of the
  (orthogonal) single-electron basis functions
  (Molecular Orbitals or MOs)
• The set of single-electron basis functions can be
  obtained e.g. from the mean-field SCF
  calculations
• Finally, single-electron basis functions are
  expressed as the linear combinations (MO LCAO) of
  the nuclei-centered properly chosen
  (non-orthogonal) atomic basis set functions
  (Atomic Orbitals or AOs).

6
Some important facts

• One needs the rules to compute matrix elements of
  Hamiltonian and other operators
• These are so-called Slater rules
• Most important consequences of the two-body
  nature of electronic Hamiltonian
• Matrix elements can be expressed as the
  combinations of four-index quantities (ijkl) -
  so called two-electron integrals
• Called atomic integrals in the original AO
  basis set
• (????)
• Called molecular integrals being transformed to
  the MO basis
• (ijkl)
• Simple consequence use of four-index quantities
  (tensors) are more or less unavoidable in QC!

7
Some important collisions

• Let N be the number of atomic basis functions
  (AOs) the main parameter controlling complexity
• The native size of dense matrices typical to QC
  methods is about of N by N, e.g. 1000x1000
• Relatively small matrices
• Has nothing common with HPL
• The native size of sparse matrices typical to QC
  methods varies but is usually very large (e.g. up
  to ca. N!)
• No any regular structure usually
• The native size of intermediate quantities to be
  computed and reused can be up to N4 (two-electron
  integrals in MO basis) and more.
• 10004 double precision numbers would require 8
  TBytes of RAM or storage

8
Typical tasks, their parameters and computational
complexity

• QC myriads of theoretical approximations
• To name just a few
• Hartree-Fock (Self-Consistent Field) and Density
  Functional Theory
• Simplest Mean Field Theories
• Perturbative approaches
• Single-reference RS-type perturbation theories
• MP2, MP3, MP4 etc
• Various Multi-Reference and/or Quasi-Degenerate
  perturbation theories
• Configuration Interaction (CI)
• Linear variational principle
• Lots of different types of CI
• Coupled Clusters
• Truncated exponential Ansatz
• Lots of different approximations/variants
• Lots of multi-reference methods
• Green functions, propagators and similar
  approaches
• Time-dependent approaches

9
Quantum Chemistry computation complexity

• Hartree-Fock (Self-Consistent Field) and Density
  Functional Theory
• From N2 to N4
• Perturbative approaches
• N5 at the second order, N6 at the third, N7 at
  the fourth order of PT
• Configuration Interaction
• Lots of different CI types
• E.g., N6 for CISD
• Up to N! for Full CI
• Coupled Clusters
• Lots of different approximations/variants
• Most widely used approaches - N6 and worse

10
Conventional, direct and semidirect methods

• Basically, the question is whether to store
  intermediates on disk or recompute them as needed
• Conventional
• store almost all, never recompute
• More advanced variants use real-time data
  compression and may store some metadata instead
  of raw intermediates
• Direct
• recompute as much as computationally feasible,
  store minimal amount of data
• Semidirect
• Reasonable compromise between fully Conventional
  and fully Direct limits

11
Standard (canonical) and fast methods

• Fast methods
• An attempt to improve algorithmic complexity for
  large problems
• Some examples
• Use of Quantum Fast Multipole Method (QFMM)
• Based on FMM ideas but much more involved
• Use of Laplace transform or other tricks to avoid
  so-called energy denominators (e.g. Laplace
  transform MP2)
• Use of spatially-localized intermediate basis
  functions
• (Density) fitting and related approximations
• Two classes of methods
• Allowing to get exact answer within given
  theoretical model
• Resulting only in approximate answers

12
Typical large-scale QC calculation requirements

• Petaflops of operations
• Terabytes of data
• Gigabytes of memory

Efficient highly-scalable parallel algorithms are
mandatory
13
Typical parallel algorithms key features and
open problems

• Key features and open problems
• Efficient I/O is very important
• Use of advanced I/O features of OS directly
• On the fly data compression/decompression
• Efficient memory management is very important
• Efficient multithreading is very important
• Typically, OpenMP is just not enough flexible to
  be used.
• Direct use of OS-level API
• Efficient communications are very important
• In particular, MPI-1 and MPI-2 are just not
  enough flexible to use in all situations.
• Use of proprietary communication interfaces.
• Main problem myriads of very different
  theoretical and hence computational methods
• each has a set of different combinations of
  controlling parameters with their own optimal
  computational strategy
• For optimal efficiency, each theoretical model
  has to be coded multiple times as a set of
  several separate, very complex algorithms.
• The degree of code reuse is not too high
  unfortunately

14
Canonical problem Integral transformation step

• (pqrs) ?? ?? ?? ?? Cp?Cq?Cr?Cs? (????)
• Formally N8 step
• Usually considered as a sequence of four
  sequential quarter-transformations
• (p???) ?? Cp?(????)
• (pq??) ?? Cq?(p???),
• etc
• Computation complexity N5 or below!
• Lots of different strategies
• complete integral transformation vs. partial
  transformation specific to particular
  approximation
• different requirements to the size of RAM and
  intermediate files to be used
• different parallelization strategies
• different requirements to the way of distribution
  of computed quantities across nodes
• Etc
• Hundreds of publications so far

15
Medium size moleculeFullerene dimer C120
16
MP2 calculation (PC GAMESS, Spring 2004) for
Fullerene dimer
Pentium 4C 2.4 GHz / 1024MB / 120GB / Gigabit
Ethernet
17
Thank you for your attention!