Electronic Structure Calculations of Structure and Reactivity: Opportunities and Challenges

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Electronic Structure Calculations of Structure
and Reactivity Opportunities and Challenges

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• MARTIN HEAD-GORDON,
• Department of Chemistry, University of
  California, and
• Chemical Sciences Division,
• Lawrence Berkeley National Laboratory
• Berkeley CA 94720, USA

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Calculations of Structure and Reactivity Outline

• (1) Current Status of electronic structure
  calculations
• A (very) brief overview of modern electronic
  structure theory.
• (2) Expanding the scope of electronic structure
  calculations
• Fast methods for DFT and MP2
• New methods for bond-breaking
• A new theorem

3
3 engines now drive the development of chemistry
experiments
electronic structure theory
chemical concepts
simulations
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Time is on our side computers get exponentially
faster.
supercomputers
workstations
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Branches of the family tree

• Wavefunction-based electronic structure theory
• Minimize the energy by varying the wavefunction
• Tremendously complicated unknown function
• Modeling the wavefunction yields model
  chemistries
• Density functional theory
• The unknown is very simple
• Hohenberg-Kohn theorem guarantees that
• True functional is unknown and probably
  unknowable
• Modeling the functional gives DFT model
  chemistries.

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Wave function approaches to electronic structure

• Hartree-Fock (MO) theory (mean field) A3-A4
  cost.
• HF 99! Correlation energy is roughly 1 eV
  per pair.
• Perturbative treatment of the electron
  correlations A5
• MP2 80 of the last 1. Pair correlations
  (doubles).
• Treat single double substitutions
  self-consistently A6
• CCSD 95 of the last 1.
• Correct for the triple substitutions
  perturbatively A7
• CCSD(T) gt 99 of the last 1. Chemical
  accuracy.

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State of the art thermochemistry

• Up to 4 first row atoms ? 0.3 kcal/mol
• Martin, Taylor, Dunning, Helgaker, Klopper
• include relativistic effects, core correlation,
  anharmonicity
• CCSD(T) with extrapolation to the complete basis
  set limit
• limited by (?) extrapolation, sometimes CCSD(T)
  itself.
• Up to 10 or 15 first row atoms ? 1 kcal/mol
• Curtiss, Raghavachari, Pople G3 methods
• CBS methods of Petersson et al
• based on methods like CCSD(T), with either
  additivity corrections for basis set, or
  extrapolation.

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A brief overview of density functional theory

• Hohenberg-Kohn theorem a universal functional
  exists describing kinetic, exchange, and
  correlation energy.
• Largest energy contribution is the kinetic
  energy.
• No satisfactory functional yet exists.
• Kohn-Sham framework begin with the kinetic
  energy of a non-interacting system with the same
  electron density.
• This leaves exchange and electron correlation to
  specify.
• Kohn-Sham computational cost similar to
  Hartree-Fock. Cheap enough to apply to large
  systems.

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Modern density functional theory methods

• Local density approximation (LDA) 1960s, 1970s
• Functional depends only on the density at each
  point, r
• LDA overbinds almost as much as HF underbinds!
• Generalized gradient approximations (GGAs) 1988
• Functional depends on density and its gradients
  at each r
• Improved results! 5-8 kcal/mol error for BLYP,
  PW91
• Beckes exact exchange mixing 1992
• Best yet! 3-5 kcal/mol error for B3LYP
• Still no dispersion energy, though, and poorer
  for barriers

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Calculations of Structure and Reactivity Outline

• (1) Current Status of electronic structure
  calculations
• A (very) brief overview of modern electronic
  structure theory.
• (2) Expanding the scope of electronic structure
  calculations
• Fast methods for DFT and MP2
• New methods for bond-breaking
• A new theorem

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The importance of linear scaling algorithms

• (Naïve) exact solution of the Schrodinger
  equation involves exponentially increasing cost
  with size.
• Theoretical model chemistries scale algebraically
• DFT methods scale as N3-N4
• Wavefunction methods scale as N3-N7
• Nonlinear scaling means that improvements in
  chemistry will be sublinear against improvements
  in computing.
• N3 scaling ? 8 ? faster computer for 2? larger
  molecule
• Development of reduced scaling methods is
  important.

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Computational advances in DFT

• First main step is formation of the effective
  Hamiltonian.
• Traditionally scales between O(N2) and O(N4)
• Accurate O(N) algorithms are now
  well-established.
• cross-overs between 15 and 50 first row atoms
• See Shao and Head-Gordon, JCP 114 (April 22,
  2001)
• Second main step is diagonalization of the
  Hamiltonian.
• Traditionally scales as O(N3)
• O(N) replacements are not yet established as
  practical for large basis sets with high
  accuracy, but will be important in the 1000 atom
  regime.

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Density functional methods typical timings

• BLYP/6-31G (25 functions per water), timings in
  minutes.
• The hundred atom regime is becoming feasible.

Calculations performed by Yihan Shao (Berkeley)
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Fast wavefunction-based methods

• Electron correlation is primarily a local
  phenomenon
• Much interest in exploiting this to develop
  fast methods
• Pioneering work by Saebo and Pulay
• Atoms in molecules local correlation models
• Reduces scaling while preserving all features of
  a good model chemistry.
• Efficiently implemented for the MP2 method.
• Lee, Maslen and MHG, JCP 112, 3592 (2000).

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A hierarchy of atomic truncations for doubles
Exact (tetra-atomics in molecules)
Tri-atomics in molecules (TRIM)
Diatomics in molecules (DIM)
Atoms in molecules (AIM)
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Fractional correlation recovery in linear alkanes

• 6-31G basis.
• TRIM yields 99.7 of correlation energy
• DIM yields 95 of correlation energy

TRIM
DIM
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Local MP2 timings (Mike Lee)
600 MHz Alpha 21164, cc-pVDZ basis
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Calculations of Structure and Reactivity Outline

• (1) Current Status of electronic structure
  calculations
• A (very) brief overview of modern electronic
  structure theory.
• (2) Expanding the scope of electronic structure
  calculations
• Fast methods for DFT and MP2
• New methods for bond-breaking
• A new theorem

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STO-3G N2 dissociation (Troy Van Voorhis)
VCCD
FCI
How important are quadruples and hextuples?
CCD
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Less is more? Coupled cluster perfect
pairingTroy Van Voorhis

• Whats the simplest useful form of CCD
  wavefunction?
• It is to keep only alpha-beta correlation within
  a bond.
• Perfect pairing is a simplified coupled cluster
  doubles (CCD) wavefunction, as first recognized
  by Cullen.
• The CCD equations simplify drastically each
  correlated pair is uncoupled from all others.
• Leads to improved numerical results for
  bond-breaking!

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N2 dissociation revisited (Troy Van Voorhis)
PP-CC
NOPP-CC
CASSCF(10,10)
CCD
6-31G
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Imperfect pairing (Troy Van Voorhis)

• Imperfect pairing (IP) introduces pair-pair
  coupling.
• Looks very promising
• The number of amplitudes is small (quadratic)
• Pairs are recoupled so that same spin correlation
  is obtained, as well as additional mixed spin
  correlation.
• Perhaps like a coupled cluster analog of GVB-RCI?
• But, it turns out to behave just as poorly as CCD
  at dissociation.

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Coupled cluster wavefunctions for
bond-dissociation (Troy Van Voorhis)

• The success of PP relative to IP (or full CCD)
  hints that coupling between double excitations is
  not properly treated at dissociation.
• IP for 4 electrons in 4 orbitals homolytically
  dissociating into 2 two-electron triplets is not
  exact at dissociation.
• Evidently the 4 electron terms contain spurious
  ionic character (and singlet character).
• A careful analysis shows that product terms with
  a repeated index (e.g. T12T12 in the 4-electron
  case) yield ionic contributions at dissociation.
• This is a basic flaw in limited cluster
  wavefunctions in general.

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Restricted pairing (Troy Van Voorhis)

• Restricted pairing (RP) is a modification of
  imperfect pairing to eliminate the 4-electron
  terms (and higher analogs) responsible for
  incorrect dissociation.
• To eliminate these terms, one could
• Define a modified doubles operator such that
  products vanish when there is a repeated index
  or
• Simply eliminate the terms that involve a
  repeated index in the cluster equations
• Test in 2 stages
• 1) variationally 2) in cluster
  equations

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Coupled cluster solution for Imperfect and
Restricted Pairing for N2 (Troy Van Voorhis)
Perfect pairing
Hartree-Fock
Restricted pairing
Exact (FCI)
Imperfect pairing
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Computationally efficient implementation

• Quadratic number of amplitudes (iterations are
  free)
• Hence solve for amplitudes in inner iterations,
  while optimizing the orbitals in outer
  iterations.
• Cubic computational effort in transforming
  integrals
• Linear number of Coulomb and exchange matrices
  needed
• Fast Coulomb (CFMM) and exchange (LinK)
  algorithms can be applied to accelerate this step
  (in progress).
• Systems with hundreds of active electrons are
  feasible
• So far the largest we have done is 91 active
  orbital pairs (182 active orbitals) (C30H62).
• Conventional CASSCF is limited to 14 active
  orbitals

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Calculations of Structure and Reactivity Outline

• (1) Current Status of electronic structure
  calculations
• A (very) brief overview of modern electronic
  structure theory.
• (2) Expanding the scope of electronic structure
  calculations
• Fast methods for DFT and MP2
• New methods for bond-breaking
• A new theorem

28
General coupled cluster theory

• The exact wavefunction can be expressed in terms
  of a generalized coupled cluster doubles (GCCD)
  operator
• M. Nooijen, Phys. Rev. Lett. 84, 2108 (2000).
• H. Nakatsuji, J. Chem. Phys. 113, 2949 (2000).
• T. Van Voorhis and M. Head-Gordon, J. Chem. Phys.
  (in press)
• The GCCD operator contains terms which give zero
  acting on the quasiparticle reference
• But acting on higher substitutions, they permit a
  description of all beyond doubles correlation
  effects.

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Why is GCCD exact? (Troy Van Voorhis)

• Long-time limit of imaginary time propagation (T
  i t)
• 
  (1)
• gives the exact ground state as can be seen
  from
• But the Hamiltonian has the form of a GCCD
  operator, so one solution for the GCCD amplitudes
  is Eq. (1)!
• Because the GCCD wavefunction is highly
  nonlinear, we expect that other solutions with
  normed amplitudes exist.

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Truncated GCCD (Troy Van Voorhis)

• Consider methods that augment conventional
  doubles by just one of the additional types of
  generalized excitation
• GCC2,1
• GCC2,0
• GCC2,-1
• GCC2,-2
• Separate the problem of assessing these
  truncations from the problem of obtaining a
  practical way of solving
• Perform variational solution for benchmark
  purposes
• A constrained full configuration interaction
  (again!).

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Converging GCC (Troy Van Voorhis)

• We observe extreme ill-conditioning in the
  problem of minimizing a GCC functional with
  respect to amplitudes
• Eigenvalues of the second derivative matrix
  typically span roughly 16 orders of magnitude
  (!).
• Hence minimization to high precision is
  impossible.
• Obtain microHartree convergence for neon
• Obtain converge to roughly 0.1 milliHartree for
  nitrogen
• Challenging implications for nonvariational
  schemes!

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Effect of generalized doubles on N2 dissociation
Errors relative to FCI
STO-3G
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Summary and outlook

• Electronic structure calculations can already
  complement experimental studies of catalytic
  processes.
• I have summarized some research that ranges from
  immediately applicable to highly speculative
• Fast methods for DFT, MP2
• Improved bond-breaking methods for large systems
• An extremely compact form of the exact
  wavefunction
• Compared to other simulation areas like fluid
  mechanics, electronic structure theory is in a
  state of flux.

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Q-Chem 2.0 A review of features and capabilities

• J.Kong, C.A.White, A.I.Krylov, C.D.Sherrill,
  R.D.Adamson, T.R.Furlani, M.S.Lee, A.M.Lee,
  S.R.Gwaltney, T.R.Adams, C.Ochsenfeld,
  A.T.B.Gilbert, G.S.Kedziora, V.A.Rassolov,
  D.R.Maurice, N.Nair, Y.Shao, N.A.Besley,
  P.E.Maslen, J.P.Dombroski, H.Daschel, W.Zhang,
  P.P.Korambath, J.Baker, E.F.C.Byrd, T.Van
  Voorhis, M.Oumi, S.Hirata,
• C.-P.Hsu, N.Ishikawa, J.Florian, A.Warshel,
  B.G.Johnson, P.M.W.Gill, M.Head-Gordon,
  J.A.Pople.
• Journal of Computational Chemistry (2000) 21,
  1532
• (Special issue on ab initio methods for large
  molecules)

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(Very Grateful) Acknowledgements.

• Past group members (amongst others)
• Prof. R. Baer (Hebrew) Dr. D. Maurice (APAM)
• Prof. A. I. Krylov (USC) Dr. C. Ochsenfeld
  (Mainz)
• Dr. M. S. Lee (Scripps) Dr. C. A. White
  (Agere/Lucent)
• Prof. C. D. Sherrill (GA Tech) Prof. P. E.
  Maslen (Rutgers)
• Present group members
• Ed Byrd Dr. Troy Van Voorhis
• Yihan Shao Jenni Weisman
• Dr. Steve Gwaltney Dr. Cherri Hsu
• Dr. WanZhen Liang Dr. Chandra Saravanan
• Dr. Garnet Chan Dr. Barry Dunietz
• Greg Beran Tony Dutoi Alex Sodt
• Funding ACS-PRF, DOE, NSF, NIH, Packard
  Foundation.