Slides for group meeting presentation, Spring 2008.

1. QTPIE and water
Jiahao Chen
2008-05-20
quot;We can't solve problems by using the same kind of thinking we used when we created them.quot;
- attributed to Albert Einstein

2. Since last time...
• QTPIE has become much faster
• We now know why dipole moments and
polarizabilities previously weren’t
translationally invariant, and why they aren’t
size extensive.
• We have (some) parameters for a new
water model
• We’ve shown that QTPIE gets the correct
direction of intermolecular charge transfer

3. What is QTPIE: a
scientific POV
• A way to model polarization and
intermolecular charge transfer in molecular
mechanics
• One of the simplest electronic structure
methods, except without electrons
• Give me a geometry, and I will give you a
charge distribution

4. What is QTPIE? A
numerical POV
• Give me a geometry, and I will give you a
charge distribution
f : R = R1 , . . . , RN → (q1 , . . . , qN )
• Minimize the quadratic form1
E(q1 , . . . , qN ; R) = qi vi (R) + qi qj Jij (R)
i
2 ij
• subject to the constraint
qi = 0
i

5. How to make QTPIE faster
• Using GTOs in place of STOs
• Integral prescreening
• Sparse matrix data structure for overlap
matrix
• Conjugate gradients to solve linear problem
(vs GMRES)
• Initial guess from previous solution
• Fast multipole methods (future work)

6. Optimizing GTOs for
STOs

7. n
n!e−α αν
An (α) =
αn+1 ν=0
ν!
n ν
n!e−α (−α) − αν
Bn (α) =
αn+1 ν=0
ν!
k m n
mn
Dp = (−1)
p−k k
k
2m
1 (2α) R2m−1
K2 (α, β, m, n, R) = + (2αR − 2m) A2m−1 (2αR) − e−2αR
R (2m)!
2n 2m−1+ν
α2m+1 R2m 2n − ν ν
− (βR) 2m−1,ν
Dp Bp (R (α − β)) A2m+ν−1−p (R (α + β))
2n (2m)! ν=0
ν! p=0
2m
dK2 2m + 1 2m (2αR)
= − + K2 + 2m (1 + 2m − 2αR) A2m−1 (2αR) + (1 + 2m) e−2αR
dR R 2 R (2m)!
2n−1 2m+ν
α2m+1 R2m 2n − ν − 1 ν
−β (βR) 2m−1,ν+1
Dp Bp (R (α − β)) A2m+ν−p (R (α + β))
2n (2m)! ν=0
ν! p=0
2m 2n 2m−1+ν
α2m+1 R 2n − ν ν
+ (βR) 2m−1,ν
Dp ×
2n (2m)! ν=0
ν! p=0
[(α − β) Bp+1 (R (α − β)) A2m+ν−1−p (R (α + β)) + (α + β) Bp (R (α − β)) A2m+ν−p (R (α + β))]
Unlike GTOs, STOs are really, really nasty to work with.
ab
p =
a+b
dK2 2p −p2 R2 erf (pR)
= √ e −
dR R π R2

8. STO-nG orbitals were defined by fitting an orbital of n
contracted Gaussian primitives to an STO to reproduce
the orbital in a least-squares sense. The conventional
wisdom is n > 2 for some kind of useful, reasonable fit.
STO-1Gs suck when used in QEq/QTPIE.
However, there is a way to get accurate results with just
primitive Gaussians...

9. Instead of maximizing overlap in a least-squares sense,
minimize the deviation in the Coulomb integrals.
min J ST O − J GT O (α)
α
This is equivalent to minimizing
J GT O − J ST O 2
2 = J GT O , J GT O − 2 J GT O , J ST O + J ST O , J ST O
= J GT O , J GT O − 2J ST O + J ST O , J ST O
which is minimized by the exponent such that
∂
0 = J GT O (α) , J GT O (α) − 2J ST O
∂α
∂ GT O ∂ GT O
= J (α) , J GT O (α) − 2J ST O + J GT O (α) , J (α)
∂α ∂α
∂ GT O
= 2J GT O (α) − 2J ST O , J (α)
∂α
i.e.
∞
1 −αR2
J GT O
(R; α) − J ST O
(R) √ e =0
0 απ

10. Element Best Coulomb Least-squares
H 0.5343 0.3101
Li 0.1668 0.0440
C 0.2069 0.1853
N 0.2214 0.2088
O 0.2240 0.2400
F 0.2312 0.2142
Na 0.0959 0.0399
Si 0.1032 0.1256
P 0.1085 0.1430
S 0.1156 0.1584
Cl 0.1137 0.1758
K 0.0602 0.0361
Br 0.0701 0.1850
Rb 0.0420 0.0402
I 0.0469 0.1735
Cs 0.0307 0.0374

12. Summary
• STOs are really, really nasty to work with
• GTOs fit to STOs by reproducing the
Coulomb self-repulsion integrals yield
excellent approximations
• Use in QEq and QTPIE result in very little
error (< 0.00001e)
• There is very little basis to the claim that
STOs can be used “for extra accuracy”

13. Sparse matrices in
QTPIE

14. Matrix-vector multiplication
in computer memory
 
1.0 0.4 0.0 do i = 1, N
do j = 1, N
 0.4 1.0 0.0  u(i) = M(i, j) * v(j)
end do
0.0 0.0 1.0 end do
In (linear) memory, matrix data structure looks like this:
M(:) 1.0 0.4 0.0 0.4 1.0 0.0 0.0 0.0 1.0
k = 0
do i = 1, N
do j = 1, N
k = k + 1
u(i) = M(k) * v(j)
end do
end do

15. Conventional matrix data structure
1.0 0.4 0.0 0.4 1.0 0.0 0.0 0.0 1.0
  k = 0
1.0 0.4 0.0 do i = 1, N
do j = 1, N
 0.4 1.0 0.0  k = k + 1
u(i) = M(k) * v(j)
0.0 0.0 1.0 end do
end do
Compressed sparse row (CSR) data structure
row start 1 3 5 6 do i = 1, N
do k=M%r(i),M%r(i+1)-1
j = M%c(k)
column 1 2 1 2 3 u(i) = M%d(k) * v(j)
end do
data 1.0 0.4 0.4 1.0 1.0 end do
Fewer operations and lower memory latency, so faster!

16. Calculating the linear
coefficients in QTPIE
• In QTPIE, minimize 1
E(q1 , . . . , qN ; R) = qi vi (R) + qi qj Jij (R)
i
2 ij
• The linear coefficients are given by
j (χi − χj )Sij
vi =
Sij
• The main costs are matrix-vector
j
multiplication and memory latency

17. Using conjugate
gradients for
constrained problems

18. QTPIE is a saddle-point
problem
• In matrix notation, 1 T
min q v + q Jq
T
q: q·1=0 2
• Solving QTPIE with a Lagrange multiplier,
J 1 q −v
=
1T 0 µ 0
• J is positive definite but this constrained problem
is not (the constraint introduces a negative
eigenvalue). Conjugate gradients can (and does)
fail by going uphill, thinking it’s going downhill.

19. However...

20. Solving the saddle-point
problem
• There exists a block inversion formula1 for
2 x 2 structured matrices
−1
J 1 J −1 + J −1 1S1T J −1 −J −1 1S
=
1T 0 −S1T J −1 S
1
S = − T −1
1 J 1
• The analytic solution is given by
−1
q J 1 −v −J −1 (v − µ1)
= =
µ 1T 0 0 1T J −1 v S
1. e.g. P.-O. Löwdin, Linear algebra for quantum mechanics

21. Solving the saddle-point
problem with CG
• Analytic solution can be solved numerically
with two symmetric, positive definite
problems (CG is guaranteed to work)
q −J −1 (v − µ1) Jw = 1
= w·v
µ 1T J −1 v S µ=−
w·1
1 Jy = −v
S = − T −1
1 J 1
q = y − µw

22. II. Fixing the
translational invariance
of dipole moments and
polarizabilities

23. Dipoles and
polarizabilities in QEq
• The regular story: the charge model is
solved by q = −J −1 v
• The energy is therefore minimized by
1
E = − v T J −1 v
2
• Want to calculate dipole moments and
polarizabilities
∂E ∂dν
dν = ανλ =
∂Riν j
∂Rjλ
i

24. How to calculate dipoles
and polarizabilities
• Dipole coupling prescription
1
E(q1 , . . . , qN ; R, ) = qi vi (R) + qi qj Jij (R) − q i Ri ·
i
2 ij i
• The external field shifts the voltages on each atom
by an external potential
1
E(q1 , . . . , qN ; R, ) = qi vi (R) − Ri · + qi qj Jij (R)
i
2 ij
• Now calculate dipole moments and polarizabilities
∂E ∂2E
dν = ανλ =
∂ ν ∂ ν∂ λ

25. Physically, the universe is translationally invariant.
Therefore, electrostatic properties (mostly) don’t
depend on the choice of origin.
x→x+δ
⇒ d → d + δQ
ανλ → ανλ

26. This is not the case in QEq!
d→d+δ 1 J T −1
v
ανλ → ανλ − 1 JT −1
· (δν Rλ + Rν δλ ) − δν δλ 1 J
T −1
1
The solution in the literature is to fix an origin
arbitrarily, even though nobody has a good physical
reason why that should be the “correct” origin

27. As it turns out, QEq and other fluctuating charge
models do obey the correct translational
properties, as long as one works with the correct
solution of the constrained minimization
q = −J −1 v
J −1 1
q = −J −1 (v − µ1) = −J −1 v − T −1
1 J 1
The second term actually generates counterterms
under translation that kill all the pathological terms.

28. The analytic solutions for the dipole
moment and polarizability in QTPIE
are given by
µ T −1 1T J−1 Rµ
dµ = − (R ) J v − 1T J−1 v + Q
1T J−1 1
µ T −1 ν 1T J−1 Rµ 1T J−1 Rν
αµν = − (R ) J R −
1T J−1 1

29. T
dµ → − (R + δ 1) J−1 v
µ µ
1T J−1
− 1T J−1 v + Q T −1 (Rµ + δ µ 1)
1 J 1
T −1 1T J−1 Rµ
= − (Rµ ) J v + 1T J−1 v + Q
1T J−1 1
1T J−1 1
−δ µ 1T J−1 v + δ µ 1T J−1 v + Q T −1
1 J 1
= dµ − δ µ 1T J−1 v + δ µ 1T J−1 v + Q
= dµ + δ µ Q

30. T
αµν → − (Rµ + δ µ 1) J−1 (Rν + δ ν 1)
1T J−1 (Rµ + δ µ 1) 1T J−1 (Rν + δ ν 1)
−
1T J−1 1
T
= − (Rµ ) J−1 Rν − δ µ δ ν 1T J−1 1
T
−δ µ 1T J−1 Rν − δ ν (Rµ ) J−1 1
1T J−1 Rµ 1T J−1 Rν
−
1T J−1 1
1T J−1 1 1T J−1 1
−δ µ δ ν
1T J−1 1
1T J−1 1 1T J−1 Rν
−δ µ
1T J−1 1
1T J−1 Rµ 1T J−1 1
−δ ν
1T J−1 1
= αµν

31. III. Fixing the size
extensivity of dipole
moments and
polarizabilities

32. Not size extensive!
2n
lim = 2
n→∞
4
n
Why?

33. Work it out analytically for N identical, noninteracting
subsystems...
     
¯
Rµ ¯
Rµ 0
 ¯
Rµ + ∆µ 1   Rµ ¯   1 
    µ 
R =
µ
. = . +∆  . 
 .
.   . .   .
. 
¯
Rµ + (N − 1) ∆µ 1 ¯
Rµ (N − 1) 1
 ¯ 0 ··· 0   
J ¯
v
 .. . 
 0 J ¯ . . 
. 

 ¯
v 

J=
 . .. .  v= .
. 
. ..  
 . . . . 
. .
0 ··· ··· J ¯ ¯
v

34. T 1
dµ = − (Rµ ) J−1 v − T −1 1T J−1 v + Q
1 J 1
   −1   
¯µ
R ¯ v
J ¯ ¯
J−1 1
 ¯
Rµ + ∆µ 1 ¯ ¯
  J−1 v  N 1T J−1 v + Q 
¯ ¯ ¯
J−1 1 
     
= − .  ·  . − ¯  . 
 .
.   .
.  N 1T J−1 1  .
. 
¯
Rµ + (N − 1) ∆µ 1 ¯ ¯
J−1 v ¯
J−1 1
T −1 ¯
= −N ¯
Rµ
T ¯ −1 v − 1 J v + Q/N Rµ
J ¯ ¯ T ¯
J−1 1
1T J−1 1
N −1
T ¯ −1 ¯
1T J−1 v + Q/N T ¯ −1
−∆ µ
n 1 J ¯
v− 1 J 1
n=0
1T J−1 1
¯ (N − 1) (N − 2) µ T ¯ −1
= N dµ − ¯ ¯
∆ 1 J v − 1T J−1 v − Q/N
2
¯ (N − 1) (N − 2) Q µ
= N dµ − ∆
2N

35. µ T 1T J−1 Rµ 1T J−1 Rν
αµν = − (R ) J R + −1 ν
1T J−1 1
 T  J−1
¯ 0 0
 
R¯µ ··· ¯
Rν
 ¯
Rµ + ∆µ 1   .. . 
.  ¯
Rν + ∆ν 1 
   0 ¯
J−1 . .  
= − . 
  . .
 .
.   . .. .. . 
.  .
.


. . . .
R¯ µ + (N − 1) ∆µ 1 ¯ ¯ ν + (N − 1) ∆ν 1
R
0 · · · · · · J−1
  
¯ ¯
1T J−1 Rµ ¯ ¯
1T J−1 Rν
 ¯ ¯
1T J−1 Rµ + ∆µ 1  ¯ ¯
1T J−1 Rν + ∆ν 1 
  
 .
.  .
. 
 .  . 
¯ ¯
1T J−1 Rµ + (N − 1) ∆µ 1 ¯ ¯
1T J−1 Rν + (N − 1) ∆ν 1
+ ¯
N 1T J−1 1
N −1
= − ¯
Rµ
T ¯ ¯ ¯
J−1 Rν + n∆ν Rµ
T ¯ ¯ ¯ ¯
J−1 1 + n∆µ 1J−1 Rν + n2 ∆µ ∆ν 1T J−1 1
n=0
N −1 ¯
Rµ
T ¯ ¯
J−1 1 + n∆µ 1T J−1 1
N −1 ¯
Rν
T ¯ ¯
J−1 1 + n ∆ν 1T J−1 1
n=0 n =0
+ ¯
N 1T J−1 1
¯ T¯ ¯ (N − 1) (N − 2) ν ¯ µ T ¯ −1
= −N Rµ J−1 Rν − ∆ R J 1
2
(N − 1) (N − 2) µ ¯ −1 ¯ ν (N − 1) (N − 2) (2N − 3) µ ν T ¯ −1
− ∆ 1J R − ∆ ∆ 1 J 1
2 6
¯ T¯
Rµ J−1 1 ¯ T¯
Rν J−1 1 2 2
(N − 1) (N − 2) µ ν T ¯ −1
+N ¯ + ∆ ∆ 1 J 1
1T J−1 1 N
(N − 1) (N − 2) ¯ T¯ ¯ T¯
+ Rµ J−1 1∆ν + Rν J−1 1∆µ
2
(N − 1) (N − 2) ¯
= N αµν −
¯ N 2 − 3N − 6 ∆µ ∆ν 1T J−1 1
6N

36. Dipole moments have the correct translational
properties because the terms and counterterms
cancel perfectly.
The terms in the polarizabilities cancel only to
second order; the cubic terms do not cancel
perfectly, giving rise to anomalous cubic scaling.

37. Modified dipole coupling
In QEq, the external field shifts the electronegativities on
each atom by an external potential
1
E(q1 , . . . , qN ; R) = qi vi (R) + qi qj Jij (R)
i
2 ij
vi (R) = χi → χi − Ri ·
1
E(q1 , . . . , qN ; R, ) = qi vi (R) − Ri · + qi qj Jij (R)
i
2 ij

38. Modified dipole coupling
We propose to apply the same coupling in QTPIE, which
shifts the atomic voltages in a less trivial manner
1
E(q1 , . . . , qN ; R) = qi vi (R) + qi qj Jij (R)
i
2 ij
j (χi − χj )Sij
vi =
j Sij
χi → χi − Ri ·
(χi − χj ) Sij j Ri − Rj Sij
j
vi (R) → vi (R, ) = − ·
j Sij j Sij

39. With this coupling, the dipole moments and polarizabilities
become
µ µ
Sik (Ri − Rk ) −1
dµ = k
J v i
i l Sil
P “ ”
µ µ
Si k R −R
1 J
T −1
v i 1 J
T −1
i
k
P
Si
i k
l l
−
1T J−1 1
µ µ
k Sik (Ri − Rk ) J−1 ij l
ν ν
Sjl Rj − Rl
αµν = −
ij k Sik l Sjl
P P
k Sik (Ri −Rk ) (Ri −Rk )
µ µ ν ν
Si k
i 1 J T −1 P
i 1 J T −1 k P
i l Sil i l Si l
+
1T J−1 1

40. More importantly, these expressions are still translationally
invariant, but are now size-extensive
dµ → dµ α→α
dµ = N dµ α = Nα
¯
The key difference is that the overlap matrix attenuates any
contributions across subsystems to terms of this form
 ¯ −1   ¯ 
J 0 ··· 0 S 0 ··· 0
 .. .
.   .. .
. 
 0 ¯
J−1 . .   0 S¯ . . 
J−1 ν ν
Sjl Rj − Rl =  .
 .



 . . .


ij
 . .. .. .  . .. .. .
l . . . .  l . . . 
0 ··· ··· ¯
J −1
0 ··· ··· ¯
S
ij jl
   
¯
Rν ¯
Rν
 ¯
Rν + ∆ν 1   ¯
Rν + ∆ν 1  
    
×  .
.  − .
.  
 .   .  
¯
Rν + (N − 1) ∆ν 1 ¯
Rν + (N − 1) ∆ν 1
j l
= ¯
J−1 ¯ ¯ν ¯ν
Sjl Rj − Rl
ij
l

41. Planar water chains with z-spacing 10.00 Angstroms
0
zz-polarizability
linear fit (gradient = -0.325919, intercept = 0.007088)
-100
-200
-300
Polarizability (bohr^3)
-400
-500
-600
-700
-800
-900
0 500 1000 1500 2000 2500 3000
Number of atoms
Planar water chains with z-spacing 10.00 Angstroms
5e+09
zz-polarizability
linear fit (gradient = -5773165.409344, intercept = 1434581215.053739)
0
-5e+09
Polarizability (bohr^3)
-1e+10
-1.5e+10
-2e+10
-2.5e+10
0 500 1000 1500 2000 2500 3000
Number of atoms

42. IV. Parameters for
water

43. Strategy
5
2
• Aim: reproduce ab initio 1
data (LCCSD(T)/aug-cc- 4
pVTZ) with three-site 3
6
water model
E = EB (R12 ) + EB (R13 ) + EB (R45 ) + EB (R46 )
OH OH OH OH
• +EA (R12 , R13 , R23 ) + EA (R45 , R46 , R56 )
HOH HOH
First determine QTPIE +EvdW (R14 )
OO
parameters (4) by fitting +EvdW (R15 ) + EvdW (R16 ) + EvdW (R42 ) + EvdW (R43 )
OH OH OH OH
+EvdW (R25 ) + EvdW (R26 ) + EvdW (R35 ) + EvdW (R36 )
HH HH HH HH
to dipole moments +EQT P IE (R)
•
2
Then fit other parameters EB (r) = k OH r − r0
OH OH
2
(10) to energies (up to HOH
EA (θ) = κOH θ − θ0
HOH
σAB 12 σAB 6
constant) EvdW (r) = AB
AB
r
−
r

44. Geometries
• 1,230 water monomer
geometries generated
by directly varying
internal coordinates
• 890 water dimer geometries generated by classical MD
sampling (TINKER, flexible SPC water model, 1000K)
• O-O distance constrained by varying vdW radius of O
• 10 ps equilibration then 10 geometries at 1ps intervals

45. Fitting method
• Weighted least-squares fitting with
Boltzmann weight (β = 100)
f (ζ) = e−βEi (dab initio − dQTPIE (ζ))2
iν iν
i ν∈{x,y,z}
• Fitting done with Neader-Mead downhill
simplex algorithm in scipy
import scipy.optimize
z0 = (8.741, 13.364, 4.528, 13.890)
zopt = scipy.optimize.fmin(f, z0)
print quot;Optimal parameters = quot;, zopt

46. Best-fit parameters
Par./eV QEq New
χ(H) 4.5280 5.6841
η(H) 13.8904 12.4166
χ(O) 8.741 7.9173
η(O) 13.364 13.1643

47. QTPIE
Dipoles correlation
ab initio

48. Dipole moment per water / D
Number of water molecules

49. yy-polarizability per water / ų
Number of water molecules

50. zz-polarizability per water / ų
Number of water molecules

51. That’s all, folks!