Multipole Models Fit to 6-31G** Electric Potentials
Molecule
Points
Methane
Water
Ammonia
Methanol
Acetone
Acetylene
Formamide
Me Acetate
DiMe Amine
NMA
# Data
455
363
404
483
614
438
492
559
583
685
Relative RMS Error
M only
M+D
M+D+Q
13.53
8.44
9.90
8.35
2.31
1.34
3.68
6.03
16.27
3.26
1.04
0.88
2.31
1.31
0.98
0.06
0.65
0.67
1.48
0.30
0.39
0.01
0.02
0.02
0.02
0.02
0.03
0.02
0.03
0.01
from D. E. Williams, J. Comput. Chem., 9, 745-763 (1988)
Polytensor Formulation of Multipole Interactions
Two Atoms A and B
with Atom Centered
Multipole Moments
r
A
B
Atom A
t
∆
∆
∆
3
1
()
●
●
r
4
3
∆
2
∆
=
2
2
∆
Multipole
Energy
∆
1
∆
Atom B
∆
● Monopole (1 Component)
● Dipole (3 Components)
● Quadrupole (9 Components)
( n)
i
j
k
∂ ∂ ∂ , i+j+k=n
=
∂xi ∂yj ∂z k
Dipole Polarizability
All matter is polarized in direct proportion to the strength of an external field,
where the proportionality constant is α , the polarizability:
µinduced = α E (i.e., µ is linear, provided E is not too big!)
Imagine a one-electron (e) atom with a radius of R placed in an electric field E.
The electron’s orbit will be shifted away from the nucleus by a distance d. Then
the induced dipole is given by:
µinduced = αE = de
At the equilibrium value of d , the external force on the electron due to the field
must exactly counterbalance the internal force of displacement between the
nucleus and the electron. These forces are:
Fext = eE
e2
sinθ
, and Fint =
4πε0R2
e2d
≈ 4πε R3
0
=
e
4πε0R3
µinduced
Since Fext is equal to Fint , we obtain for the polarizability: α = 4π ε0 R3. Thus,
neglecting the permittivity term, the polarizability should be roughly equal to
the volume of the atom or molecule. For water, the experimental value of α =
1.48 Å3 suggests a radius of 1.14 Å, about 20% less than the standard water
radius of 1.4 Å used in surface area calculations.
The Importance of Polarization
● Inter-molecular polarization is
necessary to describe gas-phase
and condensed-phase properties
within a single model
● Intra-molecular polarization is
needed to treat the conformational
dependence of electrostatics
Water Dimer Structure and Energy
O-O Distance (Ång)
O-O Bisector Angle (°)
Dimerization Energy
● MP2/CBS
Expt
QM
AMOEBA
2.98
57 ± 10
5.4 ± 0.7
2.907
56.9
4.98
2.892
57.1
4.95
● TIP3P
● AMOEBA
π-Cation Interactions
Catastrophic Failure of the Standard Model
OPLS-AA
CHARMM27
Amber ff94
Amber ff02
AMOEBA
MP2/6-311+G(2d,2p)
MP2/aVQZ
CCSD(T)/CBS
Expt (HPMS)
Expt (CID)
+
∆E0
K -Centroid
-9.32
-11.06
-12.55
-15.87
-19.27
-18.4
-19.9
-20.6
2.90
2.81
2.74
2.63
2.81
2.81
2.79
2.79
∆H298
-18.15
-20.1
-18.3
-17.7
Ion Selectivity by Benzene-Water
+
Na
K
+
Molecular Dipole Polarizability
E = [0.1, 0.1, 0.1]
●
Based on Thole's modified dipole
interaction model
●
Isotropic atomic dipole
polarizabilities are sufficient to
reproduce experimental molecular
polarizability tensors
●
Induced dipoles further
interactively induce each other
within the molecule
●
The field and interaction involved
in induction are modified (damped)
at short range
Intermolecular Polarization for NMA
#
Dimer
Monomer
Monomer
DMA/
QM Dipoles
DMA
DMA
+ Polarization
ESP RRMS
Total Dipole
Dx
Dy
Dz
6.9%
7.88
7.73
0.09
1.51
16.6%
6.64
6.46
0.01
1.52
6.5%
7.83
7.69
0.03
1.48
ESP RRMS
Total Dipole
Dx
Dy
Dz
6.4%
8.85
-8.82
0.76
0.02
15.9%
7.49
-7.44
0.75
0.00
5.6%
8.85
-8.81
0.82
0.00
All calculations performed at MP2/6-311G++(2d,2p)
interaction energy (kcal/mol)
12
H 2 - He
10
8
8
6
6
4
4
2
2
0
0
1.5 2.0 2.5 3.0 3.5 4.0
H 2 - Ne
1.5 2.0 2.5 3.0 3.5 4.0
distance (Å)
distance (Å)
rf = b / a
a
b
12
12
interaction energy (kcal/mol)
Probing
vdW
Anisotropy
10
12
10
H 2 - He
10
8
8
6
6
4
4
2
2
0
0
H 2 - Ne
1.5 2.0 2.5 3.0 3.5 4.0
1.5 2.0 2.5 3.0 3.5 4.0
distance (Å)
distance (Å)
Small Molecule Database for Parametrization
Neon
Argon
Krypton
Graphite
Dinitrogen
Dioxygen
Difluorine
Methane
Ethane
Propane
Butane
Isobutane
Pentane
Cyclohexane
Benzene
Toluene/Xylenes
Phenol/Cresols
Ammonia
Water
Hydrogen Fluoride
Hydrogen Sulfide
Hydrogen Chloride
Methanol
Ethanol
Isopropanol
Methyl Amine
Ethyl Amine
n-Propyl Amine
Dimethyl Amine
Trimethyl Amine
Formic Acid
Acetic Acid
Carboxylates
Ureas
Methyl Sulfide
Ethyl Sulfide
Dimethyl Sulfide
Dimethyl Disulfide
Formamide
Acetamide
N-Methyl Formamide
Dimethyl Formamide
N-Methyl Acetamide
Dimethyl Acetamide
Imidazoles
Guanidine
Indoles
Tryptamine
+ various strained
hydrocarbons, ions
and monofunctionals
Noble Gases and Homodiatomic Molecules
H2
3.50 0.010 0.80
N2
3.72 0.076 1.04
O2
3.39 0.106 0.99
F2
3.22 0.109 0.96
Ne
3.15 0.073
Cl 2
3.925 0.340 0.935
Ar
3.82 0.260
Kr
4.09 0.359
T
Ne
Ar
Kr
Xe
H2
N2
O2
F2
Cl 2
Density (g/cm3)
(K)
P
(atm)
expt
27.1
87.5
120.3
166.1
20.3
77.3
90.2
85.0
243.2
-4
-34
8
6
-3
-52
-5
-23
7
1.200
1.390
2.400
3.100
0.070
0.804
1.142
1.512
1.552
calc %error
1.203
1.394
2.405
3.100
0.069
0.830
1.145
1.511
1.564
0.3
0.3
0.2
0.0
-1.4
3.2
0.3
-0.1
0.8
H vap (kcal/mol)
expt
calc %error
0.420
1.554
2.161
3.018
0.218
1.332
1.628
1.554
4.857
0.421
1.554
2.163
3.018
0.216
1.346
1.617
1.565
4.851
0.2
0.0
0.1
0.0
-0.9
1.0
-0.6
0.7
0.1
Xe
4.37 0.498
AMOEBA Parameters for Water
O-H Bond
H-O-H Angle
Urey-Bradley
van der Waals
O
H
Hreduction
Polarizability
O
H
b0 (Å)
0.9572
θ0 (deg)
108.50
l0 (Å)
1.5537
0
R (Å)
3.405
2.655
91%
α (Å )
0.837
0.496
3
Kb
(kcal/Å2/mol)
529.6
Kθ (kcal/deg /mol)
34.05
Kl (kcal/Å /mol )
38.25
2
2
ε (kcal/mol)
0.110
0.0135
O Multipoles
Q
dz
Qxx
Qyy
Qzz
(a.u.)
-0.51966
0.14279
0.37928
-0.41809
0.03881
H Multipoles
Q
dx
dy
Qxx
Qyy
Qxz
Qzz
(a.u.)
0.25983
-0.03859
-0.05818
-0.03673
-0.10739
-0.00203
0.14412
Water Dimer Equilibrium Properties
aAMOEBA
De
rO-O
4.96
2.892
4.18
57.2
2.02
2.54
α
θ
<µmol>
µtot
exp
ab initio
E tttt
a
5.44 ± 0.7
f
2.976
f
-1 ± 10
f
57 ± 10
b
4.98
a
2.907
a
4.2
a
56.9
5.02
b
2.912
b
5.5
b
55.6
e
c
2.1
d
2.76
2.643
a
f
Based on calculations at CCSD(T)/TZ2P(f,d)+dif corrected for BSSE.
Complete basis set estimate from correction of CCSD(T) calculations.
c
Derived from DMA calculation directly on water dimer minimum.
d
From MP2/TZ2P++ calculations.
e
Estimate after vibrational correction of experimental ∆H at 373 K.
f
Microwave spectra from molecular beam resonance experiments at 20 K.
b
H
O
α
H
O
H
θ
H
Liquid Water Properties
3.0
Soper ’00
Expt
2.0
AMOEBA
Soper ’86
Heat Vaporization (kcal)
298K
10.51
10.48
Density (g/cc)
298K
0.997
323K
0.988
363K
0.962
1.0
0.0
AMOEBA
0
1
2
2.0
3
4
5
O - O Distance
6
7
8
1.000
0.992
0.964
Dielectric Constant
273K
87.7
86.8
298K
78.3
80.7
323K
69.9
66.5
-5
2
1.5
Diffusion (10 cm/s )
298K
2.3
2.0
1.0
Cp (cal/mol K)
298K
18.0
0.5
0.0
20.9 / 27.6
Avg Mol Dipole (Debye)
2.78
2.6-3.0
0
1
2
3
4
5
O - H Distance
6
7
8
Epol / (Epol+Eperm)
30%
Ammonia
Monomer, Dimer
and Liquid
MONOMER
Dipole
Quadrupole
Expt
AMOEBA (unscaled)
AMOEBA (60% Q)
1.471
1.528
1.528
-2.42, -2.45
-3.093
-2.491
DIMER
ab Initio *
AMOEBA
Energy
N..H
3.09
3.19
2.226
2.248
N..N
3.224
3.265
<HN..H
135
120
* aug-cc-pVQZ energy at 6-31+G* minimum
LIQUID
Hvap Pressure
Expt
AMOEBA
5.58
5.54
1
99
5
Dx10
T(K)
5.8
5.0
240
240
Methanol Dimer
QM
a
#
#
AMOEBA CHARMM
E
5.44
5.38
6.99
RO..O
(Å)
2.87
2.91
2.80
a
(deg)
44
44
23
b
(deg)
179
174
178
(kcal/mol)
MP2 Calculations from Mooij, et al., 1999
Self-Diffusion Coefficient (10 -5 cm2/s)
6
AMOEBA
5
Ammonia
4
3
NMA
Methanol
2
Water
1
DMF
0
0
1
2
3
Expt
4
5
6
Heat of Vaporization (kcal/mol)
16
Alcohols
Amines
Amides
Sulfides
Aromatics
14
Expt
12
10
8
Pressure (NVT)
74±139 Atm
6
4
4
6
8
10
12
AMOEBA
14
16
Dielectric Constants: AMOEBA vs. Expt
100
AMOEBA
Expt
Dieletric Constant
Formamide
80
Water
60
Acetamide
40
NH3
20
Methanol
Ethanol
0
MeNH2
Formamide Dimer Energy Minima
B
C
A
E = -16.0 kcal/mol
rms = 0.02 Å
D
E = -7.5 kcal/mol
rms = 0.28 Å
E = -10.3 kcal/mol
rms = 0.04 Å
E
E = -7.3 kcal/mol
rms = 0.03 Å
E = -9.0 kcal/mol
rms = 0.09 Å
F
E = -5.5 kcal/mol
rms = 0.05 Å
Formamide Dimer Association Energies
A
cyclic
B
side
C
D
nonplan nonplan
E
side
F
head-tail
MP2/6-31G**
B3LYP/6-31G(d)
-13.4
-13.4
-8.5
-8.3
-6.9
-6.7
-5.8
-6.0
-6.1
-6.7
-3.6
-3.2
MP2/aug-cc-pVTZ
-16.1
-10.6
-8.2
-7.2
-6.9
-5.4
AMOEBA
RMS
-16.0
0.02
-10.3
0.04
-9.0
0.09
-7.5
0.28
-7.3
0.03
-5.5
0.05
OPLS-AA
RMS
-14.2
0.06
-7.8
0.24
-8.2
0.82
-8.2
1.03
-8.0
0.63
-2.7
0.16
AMBER
RMS
-16.8
0.06
-9.5
0.09
-9.6
0.22
-9.0
1.03
-8.9
0.67
-3.8
0.12
CHARMM
RMS
-13.0
0.05
-8.2
0.13
-8.0
0.21
-7.7
1.06
-7.6
0.74
-4.3
0.10
MM3
RMS
-12.0
0.06
-6.5
0.24
-6.8
0.38
-6.8
1.37
-6.4
0.24
-1.5
0.25
Comparison of DMF Dimers: AMOEBA vs Dixon/Hay
A
C
E = -4.97 kcal/mol
rms = 0.08Å
E = -7.80 kcal/mol
rms = 0.24Å
B
D
E = -5.37 kcal/mol
rms = 0.09Å
E = -8.79 kcal/mol
rms = 0.15Å
Dimethylformamide Dimer Structure and Energy
A
B
C
D
BSSE Corrected
-6.61
-4.07
-6.95
-5.35
-5.51
-3.31
-5.82
-4.14
-10.32
-5.86
-11.41
-8.34
-10.98
-6.36
-12.11
-8.90
AMOEBA (single)
AMOEBA (opt)
RMS
-4.94
-4.97
0.08
-5.03
-5.37
0.09
-7.37
-7.80
0.24
-8.60
-8.79
0.15
OPLS-AA (single)
OPLS-AA (opt)
RMS
-1.68
-3.45
0.25
-0.60
-3.54
0.64
-3.41
-5.42
0.41
-2.48
-5.18
0.69
MP2/DZP+diffuse #
BSSE Corrected
MP2/aug-cc-pVTZ #
# QM Results from Vargas, et al., JACS, 122, 4750-4755 (2000)
Parameterization for Polypeptides
● vdW parameters and atomic polarizabilities
transfered from small molecules
● Atomic multipole parameters
-> from small molecule fragments (?)
-> from capped amino acids (?)
conformational dependence via intramolecular polarization
● Torsional parameters obtained by fitting to
conformational energy surfaces
MP2/6-311+G(2d,2p)
LMP2/cc-pVTZ(-f)
180
120
Psi
60
0
-60
-120
-180
-180
-120
-60
0
60
120
180
-180
-120
-60
1
2
3
60
Phi
Phi
4
5
6
7
0
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
120
180
AMBER ff99
CHARMM27
180
180
120
120
3.2
2.8
2.4
2
1.6
1.2
0.8
0.4
60
0
-60
-120
-180
-180
-120
-60
0
60
120
60
0
-60
-120
-180
-180
180
-120
-60
0
60
120
OPLS-AA
180
120
Free
Energy
Surfaces
60
Psi
Solvated
Alanine
Dipeptide
0
-60
-120
-180
-180
-120
-60
0
Phi
60
120
180
180
Torsional Energy Functional Forms
●
Fourier series
Etors = k1 [1+cos(f)] + k2 [1-cos(2f)] + k3 [1+cos(3f)] + ....
• Bicubic spline
Input:
●❷z
●
●
●
Output:
●
●
●
¶z/¶x
¶z/¶y
¶z2/¶x¶y
z(xi ,yi)
Smooth first derivative
Continuous second
derivatives
Comparison of QM/MM, PDB and AMOEBA Results
AMOEBA
QM/MM vs. PDB
(Fixed Charge Water)
-180
QM/MM, Jan Hermans, UNC
PDB, Jane Richardson, Duke
-120
-60
3.6
3.2
2.8
0
2.4
2
1.6
60
120
1.2
0.8
0.4
180
Conformational Populations
Alpha
Pass
Beta
Other
Amber ff94
Amber ff99
CHARMM27
OPLS-AA
OPLS-AA/L
68
77
46
13
23
5
10
2
9
8
26
13
52
75
65
1
1
0
3
4
SCCDFTB (Amber)
SCCDFTB (CHARMM)
SCCDFTB (CEDAR)
27
33
27
16
14
12
48
48
61
9
4
0
AMOEBA (Polar Water)
AMOEBA (Fixed Water)
29
32
16
13
54
54
1
1
Valine Sidechain Energetics
12
LMP2/cc-pVTZ(-f)
Energy (kcal/mol)
10
φ=-78 ψ=-32
AMOEBA
8
6
4
2
φ=-86 ψ=89
0
0
50
100
150
200
χ (degrees)
250
300
350
cis-N-Methylacetamide vs β-Sheet Model
H
CH3
CH3
C
O
N
N
CH3
CH3
C
CH3
-20.5
-16.2
-18.7
-17.3
-11.3
-11.6
-11.5
CH3
N
O
cis-NMA
MP2/(CEP)4-31G+(2d)
BP/DZVP (BSSE)
SIBFA
TINKER
AMBER94
CHARMM27
OPLS-AA
O
H
O
CH3
H
O
H
H
#
N
CH3
N
O
CH3
N
CH3
H
β -Sheet
∆E
-17.5
-8.4
-17.1
-11.5
-14.8
-16.9
-16.9
+3.0
+7.8
+1.6
+5.8
-3.5
-5.3
-5.4
QM and SIBFA data from Gresh, et al., JACS, 121, 7885-7894 (1999)