Larger Molecules / Longer Time Scales
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Larger Molecules / Longer Time Scales
Todd J. Martinez
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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!
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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)
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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
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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!
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IMOMO Simplified
If we lump basis set and correlation method together, we can just write:
, , ,mod ,modhigh real low real high el low elE E E E where high and low are the “high-level” and “low-level”methods and “real” and “model” are the target and truncated molecules
Example: Proton Affinity – See Lab this afternoon
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IMOMO ExampleSN2 Reaction Energy
Re, et al. J. Phys. Chem. 105A, 7185 (2001)
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IMOMO Example
Errors of approx. 2 kcal/mol per solvent molecule in absolute energies; and 1-2 kcal/mol in reaction energetics
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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?
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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
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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
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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
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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)
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Large Molecules Directly…• Is there any way to solve electronic SE for
large molecules w/o additivity approximations?• O(N) Methods
• Divide and conquer• 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 and wavepacket dynamics
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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!
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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…
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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)
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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
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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
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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, Adv. Chem. Phys. 121, 439 (2002)
; ;I Ij j
I j
t C t t I R R
Nuclear wavefunctionElectronic state
;Ij t R
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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:
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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
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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