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Go Large and Go Long?. Todd J. Martinez. Extending Quantum Chemistry. Extend accuracy and/or size range of quantum chemistry? Remember the canon!. “Right Answer”. Minimal Basis Set Full CI. Minimal Basis Set/Hartree-Fock. Electron Correlation. Complete Basis Set/Hartree-Fock. Basis set.

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Go large and go long l.jpg

Go Large and Go Long?

Todd J. Martinez


Extending quantum chemistry l.jpg
Extending Quantum Chemistry

  • Extend accuracy and/or size range of quantum chemistry?

  • Remember the canon!

“Right Answer”

Minimal Basis Set Full CI

Minimal Basis Set/Hartree-Fock

Electron Correlation

Complete Basis Set/Hartree-Fock

Basis set


Taking the canon seriously l.jpg
Taking the Canon Seriously

Can we estimate the exact answer?

Hypothesis: One- and Many-particle basis set contributions

to energy are additive

Implies that electron correlation and the flexibility of the

electronic wfn are independent – cannot be true…

Examples: Gaussian-2 (G2); Complete Basis Set (CBS)

These methods only work well

when the SBS is big enough to

qualitatively describe correlation,

i.e. polarized double-zeta or preferably

better

Extrapolated

Corr/LBS

Corr/SBS

G2/G3 – Curtiss, et al. J. Phys. Chem. 105 227 (2001)

CBS – Montgomery, et al. J. Chem. Phys. 112 6532 (2000)

HF/SBS

HF/LBS


Beyond the canon l.jpg
Beyond the Canon…

Can consider a 3-dimensional version of the canon –

the new dimension is model size/faithfulness

For example, consider the following sequence of models:

Should not ask about total energy, but rather about energy differences, e.g. De(OH) in the above examples.

Always looking for E anyway – total energies are not experimentally observable for molecules.


Extending the canon imomo l.jpg

Chemical Realism of Model

Correlation

Basis Set

Extending the Canon - IMOMO

Canon is now a cube…

  • Again, assume additivity:

Ereal/LBS/CorrEsmall/SBS/HF+

(Esmall/LBS/HF-Esmal/SBS/HF)+

(Esmall/SBS/Corr-Esmall/SBS/HF)+

(Ereal/SBS/HF-Esmall/SBS/HF)

  • Can be very sensitive to

  • choice of small model…

  • Test thoroughly for your problem!

Vreven, et al. J. Comp. Chem. 21 1419 (2000)


Multi level for transition states l.jpg
Multi-Level for Transition States?

  • Simple variant of previous ideas

    • Optimize w/low-level method (e.g. HF/3-21G)

    • Energies w/high-level method (e.g. CCSD/cc-pvtz)

    • Predict heat of reaction by difference of high-level E

  • Why not do the same for TS?


Why do rxns have barriers l.jpg
Why do Rxns have Barriers?

It’s the electrons, …

Simple example: H2+HH+H2

“diabats,” often

reasonably approximated

as harmonic

VH H-H

VH-H H

Adiabatic PES – w/barrier

Crudely approx’d as constant


Shift and distort l.jpg

High-Level

Low-Level

TS moves!

Shift and Distort…

To see the point, we need to complicate things…

Consider XCH3 + Y- X- + H3CY

  • Correlation and basis set

  • affect frequency and

  • relative energy of

  • diabatic states


Hope springs eternal l.jpg
Hope springs eternal…

  • It turns out that the MEP does not change much…

  • Determine MEP at low-level first

  • Search along low-level MEP for maximum to get

    • estimate for high-level barrier height – “IRCMax”

High-Level

Low-Level

Malick, Petersson, and Montgomery, J. Chem. Phys. 108 5704 (1998)


Empirical valence bond evb l.jpg
Empirical Valence Bond (EVB)

  • Parameterize diabats and couplings

  • One potential energy surface per bonding topology

  • More potential energy surfaces, but advantage is that

    • they are simpler than adiabatic surfaces

  • Possible to incorporate solvent effects

  • Disadvantages

    • Diagonalize a matrix to get PES

    • Number of diabats quickly gets large unless few

      • reactions are allowed…

  • Proposed by Warshel and Weiss

  • Recent applications – Voth, Hammes-Schiffer, others

Warshel, et al. – J. Amer. Chem. Soc. 102 6218 (1980)

Cuma, et al. J. Phys. Chem. 105 2814 (2001)


Large molecules directly l.jpg
Large Molecules Directly…

  • Is there any way to solve electronic SE for

  • large molecules w/o additivity approximations?

  • O(N) Methods

    • Richard Martin will cover for DFT

    • Same ideas are applicable in ALL e- structure methods

    • Generally harder to implement for correlated methods

    • Available in commercial code (e.g. Qchem)

  • Pseudospectral Methods

    • Closely related to FFT methods in DFT


Pseudospectral methods intro l.jpg
Pseudospectral Methods-Intro

Integral Contractions are major bottleneck in

Gaussian-based methods

N4 work!

Try a numerical grid…

2N3 work!


Pseudospectral methods l.jpg
Pseudospectral Methods

Problem: # grid pts scales w/molecular size, but

prefactor is usually very large

Pseudospectral Idea – Don’t think of numerical integration,

but of transform between spaces

Rspectral=physical

Qphysical=spectral

Q must be R-1…

Least-squares fitting matrix


Pseudospectral performance l.jpg
Pseudospectral Performance

  • PS advantage depends on Ng/N – smaller is better

  • Not useful for MBS/small molecules

  • HF and Hybrid DFT, 10x faster/100 atoms

  • Advantage partly additive w/locality –

    • local MP230x faster/100 atoms

  • Only available in commercial

    • code – Jaguar (Schrödinger)

    • (accessible at NCSA)

  • Eg where PS-B3LYP optimization and PS-LMP2 energy calculations are possible – active site of cytochrome c oxidase

    Moore and Martínez, J. Phys. Chem. 104, 2367 (2000)


    Quantum effects l.jpg
    Quantum Effects

    • Is there any need for quantum mechanics of nuclei in

      • large molecules?

    • Answer not completely known, but certainly yes for:

      • Tunneling – H+ transfer

      • Electronic Excited States – Photo-chemistry/biology

    • Classical mechanics only works with one PES?!

    What should happen


    Traditional methods l.jpg
    Traditional Methods

    Need to solve TDSE for nuclear wavefunction:

    • Grid methods (Kosloff and Kosloff, J. Comp. Phys. 52 35 1983)

      • Solve TDSE exactly

      • Require entire PES at every time step

      • Only feasible for < 10 degrees of freedom

    • Mean-Field (Meyer and Miller, J. Chem. Phys. 70 3214 1979)

      • Classical Mechanics on Averaged PES

      • Problematic if PES’s are very different

    E


    Spawning methods l.jpg
    Spawning Methods

    • Classical mechanics guides basis set

    • Adaptively increase basis set when quantum effects occur

    • Best for t-localized quantum effects

    • Effort  N Classical Trajectories,

      • size of N controls accuracy

    Nuclear wavefunction

    Electronic state

    M. Ben-Nun and T. J. Martínez, to appear in Adv. Chem. Phys.


    Spawning application l.jpg
    Spawning Application

    • Transmembrane protein

    • 248 AA/7 helices

    • Chromophore: all-trans retinal

    • 3762 atoms = 11,286 DOF

    • Light-driven proton pump

    Light-induced isomerization:


    Br photocycle l.jpg
    bR Photocycle

    Can simulate first

    steps directly

    200 fs

    < 1ps

    bR568

    bR*

    J625

    5 ms

    3-5 ps

    O640

    K590

    2 ms

    ms

    H+

    ms

    70 ms

    N520

    M412

    L550

    H+

    Initial Geometry of RPSB



    Time scale problem l.jpg
    Time Scale Problem

    • Reality check…

      • We can model the first picosecond… (10-12s)

      • But the photocycle takes 5ms… (10-3s)

      • Only 9 orders of magnitude to go?!

    • Abandon QM for the nuclei and microsecond simulation

      • is feasible, but still leaves 3 orders of magnitude

    • What do we do?

    • Back to the MEP…


    Minimum energy path mep l.jpg
    Minimum Energy Path (MEP)

    E†

    Don’t be deceived! – MEP is usually highly curved

    What happens if we assume it is straight?

    MEP


    Transition state theory l.jpg
    Transition State Theory

    The canon of rate theories!

    • Need enough energy to surmount barrier

    partition

    functions

    • For “other” degrees of freedom

    • For MEP coordinate

    • TS has finite lifetime


    Tst ii l.jpg
    TST II

    • What is ?

    Assume it corresponds to oscillator with -1 = TSLifetime

    • Put it all together…


    Tst iii l.jpg
    TST III

    All we need are:

    Barrier Height

    Frequencies at reactant and TS

    From Quantum Chemistry!

    What did we assume?

    Classical mechanics

    V separable along MEP Not good approx…

    • > 1 implicates

      tunneling

    True if TSLifetime

    is short

    Can always write:


    Polanyi rules l.jpg
    Polanyi Rules

    Vibrational excitation promotes rxn

    Translational excitation promotes rxn

    Dynamics Matters!


    Corrections to tst l.jpg
    Corrections to TST

    • TST almost always within 2 orders of magnitude of

      • correct rate

    • Often within factor of 5….

    • To do better, need dynamics and tunneling…

    • Correct TST by doing dynamics around TS?

      • Avoid rarity of events

      • Issue is how to sample initial conditions…

    • Is TS a good idea in the first place?

      • Hard to find the “right” TS in large molecules

      • Condensed phase rxns do not have a unique TS

      • Better to think of ensemble of TS’s

        • Chandler’s Transition Path Sampling

    Dellago, et al. J. Chem. Phys. 110 6617 (1999)


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