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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 structur...
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 integ...
Scaling the Slater exponent
Normalizing the attenuator fij <ul><li>How to pick  k ij ? </li></ul><ul><li>Most na ïve choice:  k ij  = 1 </li></ul>
Planar water chains
A better choice of  k ij <ul><li>Recall for QEq: </li></ul><ul><li>Comparing with QTPIE (rightmost): </li></ul><ul><li>Wan...
A better choice of  k ij  (cont’d) <ul><li>Within QTPIE, there is a natural choice of length scale for each pair of atoms:...
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 ch...
How we first solved QTPIE <ul><li>1. Solve for charge-transfer variables { p ji } </li></ul><ul><li>(standard linear algeb...
Numerical issues <ul><li>The problem is numerically unstable </li></ul><ul><ul><li>The matrix  A  is singular & rank defic...
Rank-revealing QR decomposition <ul><li>QR decomposition factorizes an arbitrary full-rank (complex) square matrix into an...
Rank-revealing QR decomposition <ul><li>From the RRQR factorization, we can construct a projection of  A  onto the nonzero...
The projected equations  <ul><li>We can then rewrite the equations as </li></ul><ul><li>Since this full-rank, symmetric an...
Performance issues <ul><li>O( N 6 ) computational complexity! </li></ul><ul><ul><li>Not practical </li></ul></ul><ul><ul><...
Relating { p ji } and { q i } <ul><li>Write the relation as a matrix  T : </li></ul><ul><li>The inverse relation is given ...
The solution <ul><li>It turns out that it can be shown that </li></ul><ul><li>Therefore,  </li></ul>
The equations in terms of { q i } <ul><li>We get  N  simultaneous equations </li></ul><ul><li>with 1 constraint on the tot...
Computer time
III. Interlude How to construct the STO-1G basis set
Constructing a Gaussian basis <ul><li>STO-1G basis set * </li></ul><ul><li>Maximize overlap integral </li></ul><ul><li>Aft...
The STO-1G basis set Integrals being coded… results soon! 0.1165917484 7 0.1315902101 6 0.1507985107 5 0.1760307725 4 0.20...
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 <ul><li>Dipole moment of water increases  from 1.854 Debye 1  in gas phase  to  2.95±0.20 Debye 2...
Choosing parameters <ul><li>Reproduce ab initio electrostatics </li></ul><ul><ul><li>Dipole moments, polarizabilities </li...
 
 
Calculating dipoles and polarizabilities <ul><li>For the point charges, the dipole is </li></ul><ul><li>And the polarizabi...
“Distributed” properties <ul><li>Instead of calculating properties of the whole system directly, calculate them as a sum o...
Mean dipole moment per water planar
Mean dipole moment per water twisted
TIP3P/QTPIE doesn’t predict polarizabilities well <ul><li>Identical to TIP3P/QEq </li></ul><ul><li>No out of plane polariz...
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 <ul><li>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....
Choosing parameters <ul><li>Reproduce ab initio electrostatics </li></ul><ul><ul><li>Dipole moments, polarizabilities </li...
 
 
Conclusions <ul><li>There is most likely an error in the polarizability formula (missing terms?) </li></ul><ul><li>Using t...
Water trimers DF-LMP2/aug-cc-pVTZ 1,716 cm -1 1,623 cm -1 984 cm -1 687 cm -1 dissociation limit
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QTPIE and water (Part 1)

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Transcript of "QTPIE and water (Part 1)"

  1. 1. QTPIE and water Jiahao Chen October 23, 2007
  2. 2. “ 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
  3. 3. I. Tying up some loose ends Choosing a better definition of f ij
  4. 4. The QTPIE model Coulomb integral Slater-type orbitals Charge-transfer variables Attenuated electronegativity Overlap integral “ Variationally solved”: Minimize E to solve for charge distribution
  5. 5. Scaling the Slater exponent
  6. 6. Normalizing the attenuator fij <ul><li>How to pick k ij ? </li></ul><ul><li>Most na ïve choice: k ij = 1 </li></ul>
  7. 7. Planar water chains
  8. 8. A better choice of k ij <ul><li>Recall for QEq: </li></ul><ul><li>Comparing with QTPIE (rightmost): </li></ul><ul><li>Want agreement at some geometry: </li></ul>
  9. 9. A better choice of k ij (cont’d) <ul><li>Within QTPIE, there is a natural choice of length scale for each pair of atoms: </li></ul><ul><li>A better choice of k ij : </li></ul>
  10. 10. Result of new f ij
  11. 11. II. Practical QTPIE Summary: QTPIE doesn’t have to be more expensive than Hartree-Fock
  12. 12. “ 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
  13. 13. How we first solved QTPIE <ul><li>1. Solve for charge-transfer variables { p ji } </li></ul><ul><li>(standard linear algebra problem: A x + b =0) </li></ul><ul><li>2. Sum to get atomic partial charges { q i } </li></ul>
  14. 14. Numerical issues <ul><li>The problem is numerically unstable </li></ul><ul><ul><li>The matrix A is singular & rank deficient </li></ul></ul><ul><ul><li>The unknowns { p ij } are redundant: for N atoms, have N(N-1)/2 unknowns but only N -1 linearly independent { p ij } </li></ul></ul><ul><li>The usual solution for numerically awkward problems is SVD, but can we do better? </li></ul>
  15. 15. Rank-revealing QR decomposition <ul><li>QR decomposition factorizes an arbitrary full-rank (complex) square matrix into an orthogonal matrix Q and an upper triangular matrix R </li></ul><ul><li>Rank-revealing QR decomposition uses column pivoting to delay processing of zeroes </li></ul>
  16. 16. Rank-revealing QR decomposition <ul><li>From the RRQR factorization, we can construct a projection of A onto the nonzero subspace </li></ul><ul><li>Only the rows of Q spanning span( P ) contribute, so can omit the other rows: </li></ul>
  17. 17. The projected equations <ul><li>We can then rewrite the equations as </li></ul><ul><li>Since this full-rank, symmetric and real, we can solve this with Cholesky decomposition </li></ul><ul><li>Use DGELSY in LAPACK </li></ul>
  18. 18. Performance issues <ul><li>O( N 6 ) computational complexity! </li></ul><ul><ul><li>Not practical </li></ul></ul><ul><ul><li>Why bother? Na ïve HF has only O( N 3 ) complexity! </li></ul></ul><ul><li>Can we write down equations with N -1 unknowns? </li></ul>
  19. 19. Relating { p ji } and { q i } <ul><li>Write the relation as a matrix T : </li></ul><ul><li>The inverse relation is given by T -1 : </li></ul><ul><li>T is (usually) not square, so T -1 is a pseudoinverse, not a regular inverse </li></ul>
  20. 20. The solution <ul><li>It turns out that it can be shown that </li></ul><ul><li>Therefore, </li></ul>
  21. 21. The equations in terms of { q i } <ul><li>We get N simultaneous equations </li></ul><ul><li>with 1 constraint on the total charge (enforce either with a Lagrange multiplier or by substitution) </li></ul>
  22. 22. Computer time
  23. 23. III. Interlude How to construct the STO-1G basis set
  24. 24. Constructing a Gaussian basis <ul><li>STO-1G basis set * </li></ul><ul><li>Maximize overlap integral </li></ul><ul><li>After some algebra, want to solve </li></ul>*A. Szabo, N. S. Ostlund, Modern Quantum Chemistry , Dover, 1982 , Table 3.1, p.157.
  25. 25. 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
  26. 26. IV. Electrostatics of QTPIE-water Image credit: J. Phys. D: Appl. Phys. 40 (2007) 6112–6114
  27. 27. “ Water is a very fundamental substance[3].” E. V. Tsiper, Phys. Rev. Lett. 94 (2005), 013204 [3] Genesis 1:1-2
  28. 28. Cooperative polarization <ul><li>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 </li></ul><ul><li>Polarization enhances dipole moments </li></ul><ul><li>Water models with implicit or no polarization can’t describe local electrical fluctuations </li></ul>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. +
  29. 29. Choosing parameters <ul><li>Reproduce ab initio electrostatics </li></ul><ul><ul><li>Dipole moments, polarizabilities </li></ul></ul><ul><ul><li>Water monomer only </li></ul></ul>20.680 10.125 8.285 4.960 new 13.364 8.741 13.890 4.528 QEq  O  O  H  H eV
  30. 32. Calculating dipoles and polarizabilities <ul><li>For the point charges, the dipole is </li></ul><ul><li>And the polarizability is </li></ul>
  31. 33. “Distributed” properties <ul><li>Instead of calculating properties of the whole system directly, calculate them as a sum of molecular properties </li></ul><ul><li>Define sum centered on molecular centers of mass; e.g. for dipole, </li></ul>
  32. 34. Mean dipole moment per water planar
  33. 35. Mean dipole moment per water twisted
  34. 36. TIP3P/QTPIE doesn’t predict polarizabilities well <ul><li>Identical to TIP3P/QEq </li></ul><ul><li>No out of plane polarizability </li></ul><ul><li>In-plane components underestimated </li></ul>twisted planar out of plane in plane dipole axis
  35. 37. Out-of-plane polarizability per water planar
  36. 38. Out-of-plane polarizability per water twisted
  37. 39. In-plane polarizability per water planar
  38. 40. In-plane polarizability per water twisted
  39. 41. Dipole-axis polarizability per water planar
  40. 42. Dipole-axis polarizability per water twisted
  41. 43. Lack of translational invariance <ul><li>Polarizabilities are supposed to be translationally invariant, but ours aren’t! </li></ul>
  42. 44. 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
  43. 45. Choosing parameters <ul><li>Reproduce ab initio electrostatics </li></ul><ul><ul><li>Dipole moments, polarizabilities </li></ul></ul><ul><ul><li>Water monomer and dimer </li></ul></ul><ul><ul><li>Weak bias toward initial guess (gradually relaxed) </li></ul></ul>11.274 4.386 17.841 2.213 new 13.364 8.741 13.890 4.528 QEq  O  O  H  H eV
  44. 48. Conclusions <ul><li>There is most likely an error in the polarizability formula (missing terms?) </li></ul><ul><li>Using the method of finite fields solves the translational invariance problem but not the “distribution” problem </li></ul>
  45. 49. Water trimers DF-LMP2/aug-cc-pVTZ 1,716 cm -1 1,623 cm -1 984 cm -1 687 cm -1 dissociation limit
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