Go Large and Go Long?

Todd J. Martinez

Extending Quantum Chemistry

Basis set

Ele

ctro

n C

or r

ela

t ion

Minimal Basis Set/Hartree-Fock

Minimal Basis Set Full CI

Complete Basis Set/Hartree-Fock

“Right Answer”

•Extend accuracy and/or size range of quantum chemistry?•Remember the canon!

Taking the Canon SeriouslyCan we estimate the exact answer?Hypothesis: One- and Many-particle basis set contributions

to energy are additiveImplies that electron correlation and the flexibility of the

electronic wfn are independent – cannot be true…Examples: Gaussian-2 (G2); Complete Basis Set (CBS)

/ / / / /

/ / /

extrapolated HF SBS Corr SBS HF SBS HF LBS HF SBS

Corr SBS HF LBS HF SBS

E E E E E E

E E E

HF/SBS HF/LBS

Corr/SBSExtrapolatedCorr/LBS

These methods only work well when the SBS is big enough toqualitatively describe correlation,i.e. polarized double-zeta or preferably better

G2/G3 – Curtiss, et al. J. Phys. Chem. 105 227 (2001)CBS – Montgomery, et al. J. Chem. Phys. 112 6532 (2000)

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.

H3CO

HH

O

H

CH3

CH2

O

H

Extending the Canon - IMOMO

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

Basis Set

Corr

ela

tion

Chemical R

ealism of M

odelCanon is now a cube…

• Again, assume additivity:

Ereal/LBS/CorrEsmall/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!

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?

It’s the electrons, …Simple example: H2+HH+H2

VH-H HVH H-H

12

12

( ) ( )( )

( ) ( )H H H

H H H

V R V RV R

V R V R

“diabats,” often reasonably approximated as harmonic

ˆ( )H H H H H H el H H HV R H

12ˆ( ) H H H el H H HV R H

Adiabatic PES – w/barrier

Crudely approx’d as constant

H H H

Shift and Distort…To see the point, we need to complicate things…

Consider XCH3 + Y- X- + H3CY

C

H

HH

YX

TS moves!

High-Level

Low-Level• Correlation and basis set affect frequency and relative energy of diabatic states

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”

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

High-Level

Low-Level

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 PESNumber of diabats quickly gets large unless few

reactions are allowed…

Proposed by Warshel and WeissRecent 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…• 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

Integral Contractions are major bottleneck in Gaussian-based methods

2ˆ ˆ ˆ2ei i

i

F J K 2 , ( | ) ( | )e J MO MOpq ir is rs

irs rs

F c c pq rs P pq rs

1 1 2 2

1 2

1 2

( | )p q r sr r r r

pq rs drdrr r

N4 work!Try a numerical grid…2 ,e Jpq rs rsg pg qg

grs

pg qg rs rsgg rs

F P A R R

R R P A

1 11

1

r srsg

g

r rA dr

r r

pg p gR r

2N3 work!

Pseudospectral MethodsProblem: # grid pts scales w/molecular size, but

prefactor is usually very largePseudospectral Idea – Don’t think of numerical integration,

but of transform between spaces

1( )t tQ S R wR R w Least-squares fitting matrix

Rspectral=physical Qphysical=spectral

pg p gR r1MO

spectralMON

c

c

1g

physical

gM

r

r

Q must be R-1…

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 MP230x 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 oxidaseMoore and Martínez, J. Phys. Chem. 104, 2367 (2000)

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

• 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

ˆ, ,i

R t H R tt

Need to solve TDSE for nuclear wavefunction:

E( , ) ( ) ( )ave i ii electronic states

V R t n t V R

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

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

; ;I Ij j

I j

t C t t I R R

Nuclear wavefunctionElectronic state

;Ij t R

Spawning Application

• Transmembrane protein• 248 AA/7 helices• Chromophore: all-trans retinal• 3762 atoms = 11,286 DOF•Light-driven proton pump

C9

C10

C11C12

C13C14

C15N

H+

Light-induced isomerization:

bR Photocycle

M412

bR568 bR* J625

K590

L550N520

O640

H+

H+

200 fs < 1ps

3-5 ps

2 s

5 ms

70 sms

ms

Can simulate first steps directly

Initial Geometry of RPSB

Sample Results

0

10

20

30

40

50

60

0 30 60 90 120 150 180

Ene

rgy

/ kca

l mol

-1

Dihedral angle / degrees

hn

13-cisall-trans

singly excited

C9

C10

C11C12

C13C14

C15N

H+

0

45

90

135

180

0 300 600 900 1200

Time / fs

62%

9%

Asp-85

Asp-212

f / d

egre

es

singly excited 13-cis

N

H+

Asp212 Asp85

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)

MEP

0

0.5

1

1.5

2

2.5

-1.5 -1 -0.5 0 0.5 1 1.5E

nerg

y / k

cal m

ol-1

Reaction coordinate / amu 1/2 A

CH

CH

CH

O O

H

CHCH

CH

OH

O1.22

1.32

1.42

Transition state (C2V

)

1.01(0.969)

1.36(1.32)

1.38(1.348)

1.45(1.454)

1.27(1.234)

1.72(1.68)

Don’t be deceived! – MEP is usually highly curvedWhat happens if we assume it is straight?

( , ) ( ) ( )MEP other MEP MEP other otherV R R V R V R

E†

Transition State TheoryThe canon of rate theories!• Need enough energy to surmount barrier

† /E kTRate e• For “other” degrees of freedom

†other

reactant,other

# states at TS

# states in reactant

qRate

q

• For MEP coordinate†MEP

reactant,MEP

# states at TS along MEP

# states of reactant along MEP

qRate

q

•TS has finite lifetime1

lifetimeRate TS

partitionfunctions

TST II

•What is ?†MEPq

Assume it corresponds to oscillator with n-1 = TSLifetime †

MEPLifetimeTS kT

qh

•Put it all together…†

†1 /Lifetime other

LifetimeReactant,MEP Reactant,other

TSTS E kTkT q

Rate ehq q

††

/other

ReactantqE kT

TST

qkTRate e

h

TST III

All we need are:Barrier HeightFrequencies at reactant and TSFrom Quantum Chemistry!

What did we assume?Classical mechanicsV separable along MEP Not good

approx…reactant totalq q

Can always write: TSTRate Rate

> 1 implicates tunneling

True if TSLifetime

is short

Polanyi Rules

Vibrational excitation promotes rxnTranslational excitation promotes rxn

Dynamics Matters!

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