QTPIE and water

Jiahao ChenOctober 23, 2007

“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.

I. Tying up some loose ends

Choosing a better definition of fij

The QTPIE model

Coulomb integral

Slater-type orbitals

Charge-transfer variables Attenuatedelectronegativity

Overlap integral

“Variationally solved”: Minimize E to solve for charge distribution

Scaling the Slater exponent

Normalizing the attenuator fij

How to pick kij?

Most naïve choice: kij = 1

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

A better choice of kij

• Recall for QEq:

• Comparing with QTPIE (rightmost):

• Want agreement at some geometry:

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:

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

II. Practical QTPIE

Summary: QTPIE doesn’t have to be more expensive than

Hartree-Fock

“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

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}

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?

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

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:

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

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?

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

The solution

• It turns out that it can be shown that

• Therefore,

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)

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

III. Interlude

How to construct the STO-1G basis set

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.

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!

IV. Electrostatics of QTPIE-water

Image credit: J. Phys. D: Appl. Phys. 40 (2007) 6112–6114

“Water is a very fundamental

substance[3].”E. V. Tsiper, Phys. Rev. Lett.

94 (2005), 013204

[3] Genesis 1:1-2

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.

+

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

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

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

Calculating dipoles and polarizabilities

• For the point charges, the dipole is

• And the polarizability is

“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,

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

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

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

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)

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)

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)

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)

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)

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)

Lack of translational invariance

• Polarizabilities are supposed to be translationally invariant, but ours aren’t!

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

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

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

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

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