# QTPIE and water (Part 1)

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## Technology

#### Description

Slides for group meeting in Fall 2007.

#### Related documents

- 1. QTPIE and water Jiahao Chen October 23, 2007

- How to pickk ij ?

- Most na ïve choice:k ij= 1

- Recall for QEq:

- Comparing with QTPIE (rightmost):

- Want agreement at some geometry:

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

- A better choice ofk ij :

- 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 }

- The problem is numerically unstable

- The matrixAis singular & rank deficient

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

- The usual solution for numerically awkward problems is SVD, but can we do better?

- QR decomposition factorizes an arbitrary full-rank (complex) square matrix into an orthogonal matrixQand an upper triangular matrixR

- Rank-revealing QR decomposition uses column pivoting to delay processing of zeroes

- From the RRQR factorization, we can construct a projection ofAonto the nonzero subspace

- Only the rows ofQspanning span( P ) contribute, so can omit the other rows:

- 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

- O( N 6 ) computational complexity!

- Not practical

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

- Can we write down equations withN -1 unknowns?

- Write the relation as a matrixT :

- The inverse relation is given byT -1 :

- Tis (usually)notsquare, soT -1is a pseudoinverse, not a regular inverse

- It turns out that it can be shown that

- Therefore,

- We getNsimultaneous equations

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

- STO-1G basis set *

- Maximize overlap integral

- After some algebra, want to solve

- Dipole moment of water increasesfrom 1.854 Debye 1in gas phaseto2.95±0.20 Debye 2at r.t.p. liquid phase

- Polarization enhances dipole moments

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

- Reproduce ab initio electrostatics

- Dipole moments, polarizabilities

- Water monomer only

- For the point charges, the dipole is

- And the polarizability is

- 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,

- Identical to TIP3P/QEq

- No out of plane polarizability

- In-plane components underestimated

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

- Reproduce ab initio electrostatics

- Dipole moments, polarizabilities

- Water monomer and dimer

- Weak bias toward initial guess (gradually relaxed)

- 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