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QTPIE and water
Jiahao ChenOctober 23, 2007
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“To include polarization [in force fields] is to model not only the forces or energetics
but also the electronic structure.”
Clifford E. DykstraChem. Rev. 93 (1993), 2339-53 QuickTime™ and a
TIFF (Uncompressed) decompressorare needed to see this picture.
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I. Tying up some loose ends
Choosing a better definition of fij
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The QTPIE model
Coulomb integral
Slater-type orbitals
Charge-transfer variables Attenuatedelectronegativity
Overlap integral
“Variationally solved”: Minimize E to solve for charge distribution
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Scaling the Slater exponent
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Normalizing the attenuator fij
How to pick kij?
Most naïve choice: kij = 1
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Planar water chains
Charge on first oxygen
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0 2 4 6 8 10 12 14 16 18 20Number of water molecules, N
q/e
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A better choice of kij
• Recall for QEq:
• Comparing with QTPIE (rightmost):
• Want agreement at some geometry:
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A better choice of kij (cont’d)
• Within QTPIE, there is a natural choice of length scale for each pair of atoms:
• A better choice of kij:
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Result of new fij
Charge on first oxygen-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0 2 4 6 8 10 12 14 16 18 20Number of water molecules, N
q/e
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II. Practical QTPIE
Summary: QTPIE doesn’t have to be more expensive than
Hartree-Fock
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“It is a wondrous human characteristic to be
able to slip into and out of idiocy many times
a day without noticing the change or
accidentally killing innocent bystanders in the
process.”
Scott Adams, The Dilbert Principle
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How we first solved QTPIE
1. Solve for charge-transfer variables {pji}
(standard linear algebra problem: Ax+b=0)
2. Sum to get atomic partial charges {qi}
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Numerical issues
• The problem is numerically unstable– The matrix A is singular & rank deficient
– The unknowns {pij} are redundant: for N atoms, have N(N-1)/2 unknowns but only N-1 linearly independent {pij}
• The usual solution for numerically awkward problems is SVD, but can we do better?
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Rank-revealing QR decomposition
• QR decomposition factorizes an arbitrary full-rank (complex) square matrix into an orthogonal matrix Q and an upper triangular matrix R
• Rank-revealing QR decomposition uses column pivoting to delay processing of zeroes
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Rank-revealing QR decomposition
• From the RRQR factorization, we can construct a projection of A onto the nonzero subspace
• Only the rows of Q spanning span(P) contribute, so can omit the other rows:
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The projected equations
• We can then rewrite the equations as
• Since this full-rank, symmetric and real, we can solve this with Cholesky decomposition
• Use DGELSY in LAPACK
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Performance issues
• O(N6) computational complexity!– Not practical
Why bother? Naïve HF has only O(N3) complexity!
• Can we write down equations with N-1 unknowns?
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Relating {pji} and {qi}
• Write the relation as a matrix T:
• The inverse relation is given by T-1:
• T is (usually) not square, so T-1 is a pseudoinverse, not a regular inverse
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The solution
• It turns out that it can be shown that
• Therefore,
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The equations in terms of {qi}
• We get N simultaneous equations
with 1 constraint on the total charge (enforce either with a Lagrange multiplier or by substitution)
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Computer timey = 0.0003x1.7918
R2 = 0.9998
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 20 40 60 80 100 120No. of atoms
time/s
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III. Interlude
How to construct the STO-1G basis set
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Constructing a Gaussian basis
• STO-1G basis set*
• Maximize overlap integral
• After some algebra, want to solve
*A. Szabo, N. S. Ostlund, Modern Quantum Chemistry, Dover, 1982, Table 3.1, p.157.
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The STO-1G basis setn 1 0.27094980
89
2 0.2527430925
3 0.2097635701
4 0.1760307725
5 0.1507985107
6 0.1315902101
7 0.1165917484
Integrals being coded… results soon!
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IV. Electrostatics of QTPIE-water
Image credit: J. Phys. D: Appl. Phys. 40 (2007) 6112–6114
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“Water is a very fundamental
substance[3].”E. V. Tsiper, Phys. Rev. Lett.
94 (2005), 013204
[3] Genesis 1:1-2
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Cooperative polarization
• Dipole moment of water increases from 1.854 Debye1 in gas phase to 2.95±0.20 Debye2 at r.t.p. liquid phase
• Polarization enhances dipole moments
• Water models with implicit or no polarization can’t describe local electrical fluctuations
1D. R. Lide, CRC Handbook of Chemistry and Physics, 73rd ed., 1992.2A. V. Gubskaya and P. G. Kusalik, J. Chem. Phys. 117 (2002) 5290-5302.
+
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Choosing parameters
• Reproduce ab initio electrostatics– Dipole moments, polarizabilities– Water monomer only
eV H H O O
QEq 4.528 13.890 8.741 13.364
new 4.960 8.285 10.125 20.680
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Dipole moment of planar chains of water
0
5
10
15
20
25
0 5 10 15 20 25
No. of water molecules,
Dipole per molecule/D
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Eigenvalues of the polarizability tensor of planar chains of water
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 5 10 15 20 25
No. of water molecules,
Polarizability (xx, yy)/Å
3
0
200
400
600
800
1000
1200
1400
1600
Polarizability (zz)/Å
3
Polxx/N
Polyy/N
Polzz/N
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Calculating dipoles and polarizabilities
• For the point charges, the dipole is
• And the polarizability is
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“Distributed” properties
• Instead of calculating properties of the whole system directly, calculate them as a sum of molecular properties
• Define sum centered on molecular centers of mass; e.g. for dipole,
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Mean dipole moment per water
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( /N)/Debye
TIP3P
AMOEBA
DF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
planar
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Mean dipole moment per water
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( /N)/Debye
TIP3P
AMOEBADF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
twisted
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TIP3P/QTPIE doesn’t predict polarizabilities well• Identical to TIP3P/QEq• No out of plane polarizability• In-plane components
underestimated
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( zz
TIP3P
AMOEBADF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)twisted
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( zz
TIP3P
AMOEBADF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
planar
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( yy/N)/Å
TIP3P
AMOEBA
DF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( yy/N)/Å
TIP3P
AMOEBA
DF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( xx/N)/Å
TIP3P
AMOEBADF-LMP2/aug-cc-pVDZ
TIP3P/QTPIETIP3P/QEq
gas phase (experimental)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( xx/N)/Å
TIP3P
AMOEBA
DF-LMP2/aug-cc-pVDZ
TIP3P/QTPIETIP3P/QEq
gas phase (experimental)
out of plane in plane dipole axis
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Out-of-plane polarizability per water
planar
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( xx/N)/Å
TIP3P
AMOEBADF-LMP2/aug-cc-pVDZ
TIP3P/QTPIETIP3P/QEq
gas phase (experimental)
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Out-of-plane polarizability per water
twisted
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( xx/N)/Å
TIP3P
AMOEBA
DF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
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In-plane polarizability per water
planar
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( yy/N)/Å
TIP3P
AMOEBA
DF-LMP2/aug-cc-pVDZ TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
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In-plane polarizability per water
twisted
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( yy/N)/Å
TIP3P
AMOEBA
DF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
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Dipole-axis polarizability per water
planar
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( zz
TIP3P
AMOEBA
DF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
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Dipole-axis polarizability per water
twisted
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 5 10 15 20 25 30 35 40
Number of water molecules, N
( zz
TIP3P
AMOEBADF-LMP2/aug-cc-pVDZ
TIP3P/QTPIE
TIP3P/QEq
gas phase (experimental)
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Lack of translational invariance
• Polarizabilities are supposed to be translationally invariant, but ours aren’t!
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Waterd/D xx/Å3 yy/Å3 zz/Å3
C 1.864
1.419 1.474 1.363
D 1.864
1.419 1.474 1.363
Using analytic point charges
C 1.684
0.058 0.326 0.000
D 1.684
23.660
0.326 0.000
Using numerical finite field
C 3.369
1.176 14.994
0.000
D 3.369
1.176 14.994
0.000
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Choosing parameters
• Reproduce ab initio electrostatics– Dipole moments, polarizabilities– Water monomer and dimer– Weak bias toward initial guess
(gradually relaxed)
eV H H O O
QEq 4.528 13.890 8.741 13.364
new 2.213 17.841 4.386 11.274
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Dipole moment of planar chains of water
0
5
10
15
20
25
0 5 10 15 20 25
No. of water molecules,
Dipole per molecule/D
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Eigenvalues of the polarizability tensor of planar chains of water
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 5 10 15 20 25
No. of water molecules,
Polarizability (xx, yy)/Å
3
0
200
400
600
800
1000
1200
1400
1600
Polarizability (zz)/Å
3
Polxx/N
Polyy/N
Polzz/N
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Conclusions
• There is most likely an error in the polarizability formula (missing terms?)
• Using the method of finite fields solves the translational invariance problem but not the “distribution” problem