Robert
Monckton
II. József
1782
Multilevel Methods
Toward Accurate Estimation of
Thermochemical Properties
The world’s first ice-calorimeter
Milán Szőri
2014
2014
„Virtual calorimeter”
http://en.wikipedia.org/wiki/Calorimetry
http://gizmodo.com/298029/worlds-biggest-supercomputer-is-a-virus
Viktor Orbán
1
Importance of thermochemical
calculations
• Predictive power
– Check the experiment and its evaluation are done properly
„The recommended values of this paper affect a large number of
other thermochemical quantities which directly or indirectly rely
on or refer to D0(H−OH), D0(OH) and ΔHf(OH).”
J. Phys. Chem. A, 2002, 106, 2727–2747.
– Check these values consistency
• Provide missing thermochemical properties
–
–
–
–
For unknown species
Unable to measure
Estimation of rate constants
Input in atmospheric and combustion chemistry
2
Important Properties
• Classical Thermochemical Properties
G=H-TS
H=U+pV=U+RT
IE, IP, BDE, EA etc.
• Needed
– Geometry (rotational contribution and repulsion
energy)
– Frequencies (vibrational contribution)
– Energy
Reward as little as possible!!!
3
Intermezzo
The energy is a function of position and of electrons and nuclei and the time
Very high dimension!!!!
Decoupling (Separation) of coordinates is highly needed!
Separation in practise
Exact nonrelativistic Hamiltonian in field-free space:
Nuclei
Small
Electron
Constant
Nuclei
Nuclei Electron
+
Electron
R is only parameter
if the coordinates of nuclei are fixed (PHYSICALLY!!!)
Clamped-nuclei Schrödinger equation:
Model:
Classical treatment for nuclei
QM for electron
Clamped-nuclei Schrödinger equation
Electronic Schrödinger equation:
Nuclear Schrödinger equation:
Electronic energy
the nuclei move in a potential set up by the electrons
At fixed geometry (PHYSICALLY!!!):
Total energy
neglected
Etot≈ Eel +ENN
Nuclear-nuclear repulsion
Total energy ≈ (Electronic energy)+ (Nuclear-Nuclear repulsion)
Model
• Molecular SE
– No general solution (high dimension), so instead:
• Clamped-nuclei SE: Classical nuclei + QM electrons
– It is possible to solve it with some approximations
(see Molecular Methods)
– It gives solution for the electronic problem
– Correction for the nuclei SE (+QM electrons)
• Frequency calculation (Vibrational analysis)
Approximation of the molecular SE
7
One consequence of BO
• Definition of a molecule:
– An electrically neutral entity consisting of more
than one atom which must correspond to a
depression on the potential energy surface that is
deep enough to confine at least one vibrational
state. (IUPAC)
Molecular methods
• In practise accuracy, robustness and system size
are coupled
• Neat ab initio or ideal DFT
• Parametrized methods:
– Scaling correlation energies (SAC-x, PCI-80, SCS-MPx)
– Extrapolated methods (CBS)
– Composite methods/Multilevel methods/Model
chemistry
– Applied DFT functionals (B3LYP)
– Semiempirical methods (PM6)
– Force fields (CHARMM-AA)
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Molecular methods
• In practise accuracy, robustness and system size
are coupled
• Neat ab initio or ideal DFT
• Parametrized methods:
– Scaling correlation energies (SAC-x, PCI-80, SCS-MPx)
– Extrapolated methods (CBS)
– Composite methods/Multilevel methods/Model
chemistry
– Applied DFT functionals (B3LYP)
– Semiempirical methods (PM6)
– Force fields (CHARMM-AA)
10
Definition
• Quantum chemistry multilevel methods
/composite methods/model chemistries:
are computational chemistry methods that aim for high
accuracy by combining the results of several (individual)
calculations.
components
c jk  j  k

Emultilevel   ci i 
i
 cl l
ci and cjk can be: dependent
independent (!)
from experimental data
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Multilevel Methods
Extrapolative/additive
protocols:
Purely additive protocols:
CBS-4, CBS-q, CBS-Q, CBS-APNO,
W1, W1U, W1BD, W2, W3, W4, …
G2, G3, G2MP2, G3MP2, G3B3,
G3MP2B3, G3-RAD, …
Scaled/additive protocols:
SAC, MCQCISD, MCG3, G3S,
G3S(MP2), G3X, …
Bond-correcting protocols:
BAC-MP4, PDDG/MNDO,
PDDG/PM3
Multilevel Methods
• Single-Point – trivial method
– X1 –Xiamen
• Additivity/Extrapolation/Scaled
>1
NO!
–
–
–
–
–
–
MCCM - Multicoefficient (correlation) models
Gn – Gaussian
ccCA-x – Correlation Consistent Composite Approach
CBS-n – Complete Basis Set
Wn – Weizmann
HEAT – High accuracy Extrapolated Ab initio Thermochemistry
empirical parameter(s)
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Measuring the calcs performance
•
•
In the case of a single molecule:
Deviation/Error (D):
D=experimental value – calculated value
Chemical accuracy: AD< 1 kcal/mol
Absolute deviation/absolute error (AD):
Spectroscopic accuracy: AD< 1 kJ/mol
AD=abs(D)
Relative deviation (RD): RD=AD/experimental value
For set of molecules (usually standard test sets): – Mean Deviation (MD):
– Mean Unsigned Error (MUE):
MD=mean(Di)
MUE=mean(ADi)
– Largest deviation (LD):
– Maximum absolute deviation (MAD):
LD=max(Di)
MAD=max(ADi)
– Root-Mean-Squared Error (RMSE):
RMSE
– Standard Deviation (SD):
SD
– Error distribution (Histogram):
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Measuring the calcs performance
Set of molecules (standard test sets):
Sets
Δf H0
IP
EA
PA
H-bond
G2/97
148
88
58
8
-
G3/99
223
88
58
8
-
G3/05
270
105
63
10
6
Database/3
109
Atomization energy
13
13
44
Barrier height
X1/07
Extended G3/99
S22
Often transition metals are not well-represented
(the largest experimental error > 40 kJ/mol)
http://www.cmt.anl.gov/OldCHMwebsiteContent/compmat/g3-05.htm
http://www.cmt.anl.gov/OldCHMwebsiteContent/compmat/g2geoma.htm
http://www.cmt.anl.gov/OldCHMwebsiteContent/compmat/g3-99.htm
http://www.begdb.com/
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Multilevel Methods
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Single-Point
John A. Pople
• Experience: geometry is not that sensitive
to the level of theory as energy.
• Example:ozone (1O3) a non-trivial case
O3
d(O-O) in Å
experiment
1.278
HF
1.204
B3LYP
1.264
MP2
1.3
MP3
1.256
MP4
1.306
B2PLYP
1.288
CID
1.241
CISD
1.244
QCISD
1.275
QCISD(T)
1.298
CCD
1.259
CCSD
1.271
CCSD(T)
1.296
• Notation: QCISD(T)/6-31G(d)//B3LYP/6-31G(d)
Properties//Geometry
It is not always a simple energy calculation! (NMR)
http://cccbdb.nist.gov/
RD
in %
6%
1%
2%
2%
2%
1%
3%
3%
0%
2%
1%
1%
1%
AD
in Å
0.058
0.011
0.017
0.017
0.022
0.008
0.029
0.027
0.002
0.016
0.015
0.005
0.014
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Single-Point
Length of a line between two points does not always give you the smallest
distance in computational chemistry.
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Larry A. Curtiss John A. Pople
Johann Carl Friedrich Gauss
G3MP2B3
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G3MP2B3
model
chemistry
Its elements:
(1) Geometry and frequencies: B3LYP/6-31G*
ΔE(ZPE) =0.96*ZPE
(2) Additional higher polarization:
ΔE(G3Large)=E(MP2/G3Large))-E(MP2/6-31G(d))
(3) Correction for MP2 truncation:
ΔE(QCI)=E(QCISD(T)/6-31G*)-E(MP2/6-31G*)
(4) Remaining deficiencies:
ΔE(HLC)=-Anβ -B (nα -nβ )
ΔE(SO)
From table
nα ≥nβ
A(Hartree) B(Hartree)
Molecules
0.010041
0.004995
Atoms
ΔE(SO) are the same as used in G3
0.010188
0.002323
E0 (G3MP2B3)=E(MP2/6-31G*)+ΔE(G3Large)+ΔE(QCI)+E(HLC)+E(ZPE) +E(SO)
E0(G3MP2B3)=E(QCISD(T)/6-31G*)+ΔE(G3Large)+E(HLC)+E(ZPE) +E(SO)
A. G. Baboul, L. A. Curtiss, P. C. Redfern, and K. Raghavachari, J. Chem. Phys., 1999, 110, 7650-7657.
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G3MP2B3 model chemistry
E
E(MP2/6-31G(d)//B3LYP/6-31G(d))
ΔE(G3Large)
ΔE(QCI)
E(MP2/G3MP2Large//B3LYP/6-31G(d))
‘Vector sum’
E(QCISD(T)/6-31G(d)//B3LYP/6-31G(d))
QCISD(T) limit
G3MP2B3 energy-(ΔE(HLC)+ΔE(ZPE))
6-31G(d)
6-311+G(3df,2p)
An estimation of QCISD(T)/6-311++G(3df,2p) level of theory
Curtiss, L. A.; Raghavachari, K.; Pople, J. A. J. Chem. Phys. 1999, 110, 7650.
21
ΔE(HLC) correction
• E.g. OH radical
O 1s1 2s2 2p4
H 1s1
Ms=2
valence: 6evalence: 1enα = 4
≥ nβ = 3
ΔE(HLC)=-Anβ -B (nα -nβ )
A(Hartree) B(Hartree)
Molecules
0.010041
0.004995
Atoms
0.010188
0.002323
ΔE(HLC)= E(empiric)=
-0.010041×3-0.004995×(4-3)=-0.035118 Hartree
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ΔE(SO) correction for atoms
Atom
SO in milliHartree
Atom
SO in milliHartree
H
0.0
Ne
0.0
He
0.0
Na
0.0
Li
0.0
Mg
0.0
Be
0.0
Al
-0.34
B
-0.05
Si
-0.68
C
-0.14
P
0.0
N
0.0
S
-0.89
O
-0.36
Cl
-1.34
F
-0.61
Ar
0.0
Unit conversion!!!
It is not included in the G3MP2 energy in the output file !!!
This correction needs to be added manually!!!
J. Chem. Phys., 1998, 109, 7764-7776.
23
E(QCISD(T)/6-31G(d))
Practise
T = 298.15 K
Ecorr
Temperature=
E(ZPE)=
E(QCISD(T))=
DE(MP2)=
G3MP2(0 K)=
G3MP2 Enthalpy=
298.150000 Pressure=
0.007972 E(Thermal)=
-75.537195 E(Empiric)=
ΔE(HLC)
-0.093259
-75.657600 G3MP2 Energy=
-75.654295 G3MP2 Free Energy=
1.000000
0.010332
-0.035118
-75.655240
-75.674542
ΔE(G3Large)
E(G3MP2B3)=Etot(G3MP2B3) + Ecorr
H(G3MP2B3)=Etot(G3MP2B3) + Hcorr
G(G3MP2B3)=Etot(G3MP2B3) + Gcorr
T = 298.15 K and P = 1 atm
24
MUE from experiment
Type (No of species)
G3MP2B3 (kJ/mol)
Enthalpies of formation
(148)
4.94
Nonhydrogen (35)
8.87
Hydrocarbons (22)
2.93
Substituted hydrocarbons (47)
3.10
Inorganic hydrides (15)
4.31
Radical (29)
5.15
Ionization enrgies (85)
5.90
Electron affinities (58)
6.11
Proton affinities (8)
4.27
All (299)
5.44
1 kcal/mol (=4.184 kJ/mol) is the chemical accuracy
25
Multilevel Methods
Performance
Measured by Publications
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G2
G3
G3MP2B3
Thank for your
attention!
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