QTPIE and water Jiahao Chen October 23, 2007

“ To include polarization [in force fields] is to model not only the forces or energetics but also the electronic structure.” Clifford E. Dykstra Chem. Rev. 93 (1993), 2339-53

I. Tying up some loose ends Choosing a better definition of f ij

The QTPIE model Coulomb integral Slater-type orbitals Charge-transfer variables Attenuated electronegativity Overlap integral “ Variationally solved”: Minimize E to solve for charge distribution

Scaling the Slater exponent

Normalizing the attenuator fij How to pick k ij ? Most na ïve choice: k ij = 1

How to pick k ij ?

Most na ïve choice: k ij = 1

Planar water chains

A better choice of k ij Recall for QEq: Comparing with QTPIE (rightmost): Want agreement at some geometry:

Recall for QEq:

Comparing with QTPIE (rightmost):

Want agreement at some geometry:

A better choice of k ij (cont’d) Within QTPIE, there is a natural choice of length scale for each pair of atoms: A better choice of k ij :

Within QTPIE, there is a natural choice of length scale for each pair of atoms:

A better choice of k ij :

Result of new f ij

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 { p ji } (standard linear algebra problem: A x + b =0) 2. Sum to get atomic partial charges { q i }

1. Solve for charge-transfer variables { p ji }

(standard linear algebra problem: A x + b =0)

2. Sum to get atomic partial charges { q i }

Numerical issues The problem is numerically unstable The matrix A is singular & rank deficient The unknowns { p ij } are redundant: for N atoms, have N(N-1)/2 unknowns but only N -1 linearly independent { p ij } The usual solution for numerically awkward problems is SVD, but can we do better?

The problem is numerically unstable

The matrix A is singular & rank deficient

The unknowns { p ij } are redundant: for N atoms, have N(N-1)/2 unknowns but only N -1 linearly independent { p ij }

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

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:

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

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( N 6 ) computational complexity! Not practical Why bother? Na ïve HF has only O( N 3 ) complexity! Can we write down equations with N -1 unknowns?

O( N 6 ) computational complexity!

Not practical

Why bother? Na ïve HF has only O( N 3 ) complexity!

Can we write down equations with N -1 unknowns?

Relating { p ji } and { q i } 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

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,

It turns out that it can be shown that

Therefore,

The equations in terms of { q i } We get N simultaneous equations with 1 constraint on the total charge (enforce either with a Lagrange multiplier or by substitution)

We get N simultaneous equations

with 1 constraint on the total charge (enforce either with a Lagrange multiplier or by substitution)

Computer time

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.

STO-1G basis set *

Maximize overlap integral

After some algebra, want to solve

The STO-1G basis set Integrals being coded… results soon! 0.1165917484 7 0.1315902101 6 0.1507985107 5 0.1760307725 4 0.2097635701 3 0.2527430925 2 0.2709498089 1  n

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 Debye 1 in gas phase to 2.95±0.20 Debye 2 at r.t.p. liquid phase Polarization enhances dipole moments Water models with implicit or no polarization can’t describe local electrical fluctuations 1 D. R. Lide, CRC Handbook of Chemistry and Physics , 73rd ed., 1992 . 2 A. V. Gubskaya and P. G. Kusalik, J. Chem. Phys. 117 (2002) 5290-5302. +

Dipole moment of water increases from 1.854 Debye 1 in gas phase to 2.95±0.20 Debye 2 at r.t.p. liquid phase

Polarization enhances dipole moments

Water models with implicit or no polarization can’t describe local electrical fluctuations

Choosing parameters Reproduce ab initio electrostatics Dipole moments, polarizabilities Water monomer only 20.680 10.125 8.285 4.960 new 13.364 8.741 13.890 4.528 QEq  O  O  H  H eV

Reproduce ab initio electrostatics

Dipole moments, polarizabilities

Water monomer only

Calculating dipoles and polarizabilities For the point charges, the dipole is And the polarizability is

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,

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 planar

Mean dipole moment per water twisted

TIP3P/QTPIE doesn’t predict polarizabilities well Identical to TIP3P/QEq No out of plane polarizability In-plane components underestimated twisted planar out of plane in plane dipole axis

Identical to TIP3P/QEq

No out of plane polarizability

In-plane components underestimated

Out-of-plane polarizability per water planar

Out-of-plane polarizability per water twisted

In-plane polarizability per water planar

In-plane polarizability per water twisted

Dipole-axis polarizability per water planar

Dipole-axis polarizability per water twisted

Lack of translational invariance Polarizabilities are supposed to be translationally invariant, but ours aren’t!

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

Water   0.000 14.994 1.176 3.369 D 0.000 14.994 1.176 3.369 C Using numerical finite field 0.000 0.326 23.660 1.684 D 0.000 0.326 0.058 1.684 C Using analytic point charges 1.363 1.474 1.419 1.864 D 1.363 1.474 1.419 1.864 C  zz /Å 3  yy /Å 3  xx /Å 3 d/D

Choosing parameters Reproduce ab initio electrostatics Dipole moments, polarizabilities Water monomer and dimer Weak bias toward initial guess (gradually relaxed) 11.274 4.386 17.841 2.213 new 13.364 8.741 13.890 4.528 QEq  O  O  H  H eV

Reproduce ab initio electrostatics

Dipole moments, polarizabilities

Water monomer and dimer

Weak bias toward initial guess (gradually relaxed)

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

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

Water trimers DF-LMP2/aug-cc-pVTZ 1,716 cm -1 1,623 cm -1 984 cm -1 687 cm -1 dissociation limit