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Reduced Density Matrix Methods for Quantum Chemistry and Physics David A. Mazziotti Department of Chemistry RDM Workshop, Oxford James Franck Institute 13 April 2016 The University of Chicago Quantum Chemistry What is Quantum Chemistry?

  1. Be Revisited: A Variational 2-RDM Calculation Method E (a.u.) % Correlation E HF -14.5034 0 MP2 -14.5273 45.5 MP3 -14.5417 72.7 MP4 -14.5496 87.8 D -17.8973 6437.0 DQ -14.5573 102.4 DQG -14.5561 100.0 FCI -14.5561 100.0 D. A. Mazziotti, Phys. Rev. A 65 , 062511 (2002).

  2. Variational 2-RDM Method Characteristics: 1. variational lower bound 2. systematic hierarchy of 2-RDM constraints 3. independence from a reference wavefunction 4. size consistent and size extensive 5. random selection of initial 2-RDM 6. global minimum in semidefinite programming D. A. Mazziotti, Acc. Chem. Res. 39 , 207 (2006).

  3. 2-RDM Set The 2-RDM set contains all wave functions including the most strongly correlated wave functions that have an exponentially scaling number of large probability amplitudes: All Wave Functions

  4. Two Advances Two Advances: (1) Boundary-point SDP algorithm: 10-100x faster (2) Systematic hierarchy of N -representability conditions D. A. Mazziotti, Phys. Rev. Lett. 106 , 083001 (2011). D. A. Mazziotti, Phys. Rev. Lett. 108 , 263002 (2012).

  5. N -representability Conditions: Positivity Conditions 1-Positivity Conditions (Pauli principle):   1 1 D 0 0 Q 2- Positivity Conditions (Coleman ’63; Garrod & Percus ’64):    2 2 2 D 0 G 0 Q 0 3-Positivity Conditions (Mazziotti & Erdahl ’01):     3 3 3 3 Q 0 F 0 0 0 D E T 2 Condition (Erdahl ’78; Zhao et al. ’04; Mazziotti ’05):    3 3 T E F 0 2 D. A. Mazziotti, Phys. Rev. Lett. 108 , 263002 (2012). D. A. Mazziotti, Phys. Rev. A 74 , 032501 (2006).

  6. N 2 Molecule

  7. N 2 Molecule

  8. N 2 Molecule

  9. N 2 Molecule

  10. Strong Electron Correlation: Hydrogen Chains and Lattices

  11. Bonding in a Hydrogen Chain: Stretched geometries D. A. Mazziotti, Phys. Rev. Lett. 93 , 213001 (2004).

  12. Bonding in a Hydrogen Chain: Stretched geometries D. A. Mazziotti, Phys. Rev. Lett. 93 , 213001 (2004).

  13. Bonding in a Hydrogen Chain: Metal-to-Insulator Transition where    2   1 D γ ij  i j R A. Sinitskiy, L. Greenman, and D. A. Mazziotti, J. Chem. Phys. 133 , 014104 (2010).

  14. Bonding in a Hydrogen Lattice: Computational Cost of 4 x 4 x 4 Lattice Wave Functions: Number of Important Configurations: 10 18 determinants! Probability of Each Configuration: 10 -18 Can this calculation be done? No. A. Sinitskiy, L. Greenman, and D. A. Mazziotti, J. Chem. Phys. 133 , 014104 (2010).

  15. Bonding in a Hydrogen Lattice: Stretched geometries -0.30 HF MP2 -0.35 CCSD(T) DQG -0.40 E -0.45 -0.50 -0.55 0.5 1 1.5 2 2.5 3 3.5 R A. Sinitskiy, L. Greenman, and D. A. Mazziotti, J. Chem. Phys. 133 , 014104 (2010).

  16. Bonding in a Hydrogen Lattice: Metal-to-Insulator Transition    2   1 D ij  i j γ R A. Sinitskiy, L. Greenman, and D. A. Mazziotti, J. Chem. Phys. 133 , 014104 (2010).

  17. Strong Electron Correlation: Polyaromatic Hydrocarbons

  18. n-Acenes: Polyaromatic Hydrocarbons Napthalene (2-acene): Anthracene (3-acene): Tetracene (4-acene): Pentacene (5-acene): Hexacene (6-acene): Heptacene (7-acene): G. Gidofalvi and D. A. Mazziotti, J. Chem. Phys. 129 , 134108 (2008).

  19. n-Acenes: Molecule Number of Variables in CI 2-acene 4936 3-acene 69116 4-acene 112298248 5-acene 19870984112 6-acene 3725330089248 7-acene 728422684135920 8-acene 147068001734374624 5- and higher-acenes cannot be treated by traditional CAS-SCF. 8-acene has 1.5 x 10 17 configuration state functions (CSFs). G. Gidofalvi and D. A. Mazziotti, J. Chem. Phys. 129 , 134108 (2008).

  20. Acenes : Memory and Timings CI- CI- 2-RDM 2-RDM CASSCF CASSCF CASSCF CASSCF molecule Memory Time Memory Time 2-acene 0.2 MB 0.02 min 12.6 MB 1.1 min 3-acene 44.9 MB 2.2 min 15.8 MB 4.0 min 4-acene 9.0 GB 25.4 hr 23.3 MB 30 min G. Gidofalvi and D. A. Mazziotti, J. Chem. Phys. 129 , 134108 (2008).

  21. Active-space Variational 2-RDM Method The occupation numbers of the HOMO and LUMO spatial orbitals approaches one as the length of the acene increases. G. Gidofalvi and D. A. Mazziotti, J. Chem. Phys. 129 , 134108 (2008).

  22. n-Arynes : L. Greenman and D. A. Mazziotti, J. Chem. Phys. 130 , 184101 (2009) .

  23. n-Arynes: (12,12) 2-RDM Calculation L. Greenman and D. A. Mazziotti, J. Chem. Phys. 130 , 184101 (2009) .

  24. n-Arynes: (n C +2,n C +2) 2-RDM Calculation L. Greenman and D. A. Mazziotti, J. Chem. Phys. 130 , 184101 (2009) .

  25. Planar Acenes: Size We also observe the emergence of polyradical character with system size in planar acenes. K. Pelzer, L. Greenman, G. Gidofalvi and D. A. Mazziotti, JPC A 114 , 583 (2011).

  26. Planar Acenes: Geometry Which of these molecules is most strongly correlated? linear triangular superbenzene K. Pelzer, L. Greenman, G. Gidofalvi and D. A. Mazziotti, JPC A 114 , 583 (2011).

  27. Planar Acenes: Geometry Which of these molecules is most strongly correlated? linear triangular superbenzene Answer: triangular > linear > superbenzene K. Pelzer, L. Greenman, G. Gidofalvi and D. A. Mazziotti, JPC A 114 , 583 (2011).

  28. Strongly Correlated Periodic Systems

  29. Basis for Polymers and Molecular Crystals Bloch orbitals composed of atomic orbitals: Use non-orthogonal Bloch functions instead of plane waves for a basis representing the crystal. Allows us to use the quantum chemical basis set technologies — correlation consistent, polarizability, etc... Payne, et. Al. Rev. Mod. Phys. 64 1045 (1992); Pisani, Lec. Notes. Chem. Springer, (1996)

  30. Crystalline-Orbital Hartree-Fock The momentum space representation of an operator is related to its position space representation by a Fourier transform. -3 -2 -1 0 1 2 3 4 Fourier transform formally involves an infinite number of cells. We need to employ a cut off to discretize k-space. Pisani, Lec. Notes. Chem. Springer, (1996); J. M. André et. al . J. Com. Chem. (1984)

  31. k-space advantage for Hartree-Fock equations The Fock operator is diagonalized in each irreducible representation of the translational group: CO-HF gives us a set of orbitals (a representation) of the crystal that obeys the correct symmetry.

  32. Electron correlation in an infinite Hydrogen chain calculated by variational 2-RDM (DQG) • Infinite chain of Hydrogen atoms • 2 Hydrogen atoms/cell, 10 neighboring cells > 10 24 determinants in active • space if traditional electronic structure is used. • 1-RDM • 2-RDM But RDM has failed? Why are we below the ground state by 50 mhartrees?

  33. Symmetries in quantum mechanics and time-reversal symmetry operator in a spin-orbital basis Time-reversal symmetry can be Time-reversal also rotates even or odd after operation spin-momenta

  34. Time reversal symmetry on one-body operators dictates symmetry between (k,-k) Kramers pairs Position space constraints for TR symmetry: Momentum space space constraints for TR symmetry: Equality constraints on the 1-particle and 2-particle density matrices Constraints are explicitly included in the SDP

  35. Variational 2-RDM with time-reversal equalities included in the constraints on the 2-RDM Infinite Hydrogen chain revisited • Time-reversal constraints are added to the SDP as equality constraints • Still lower bound! Time-reversal symmetry constraints restore accuracy of DQG constraints! N. C. Rubin, D. A. Mazziotti, in preparation (2016).

  36. Time-reversal fixes occupation number symmetry Occupation numbers of an infinite H-chain indexed by k-point: Symmetry broken solution is fixed automatically by constraining D(k) = D(-k)* N. C. Rubin, D. A. Mazziotti, in preparation (2016).

  37. Active space treatment of LiH crystal • LiH crystal with 5 unit cells in CO-HF summation • 10 7 determinants on active space • Core treated at the mean-field level by creating new effective one-electron operators • RDM without TR fails N. C. Rubin, D. A. Mazziotti, in preparation (2016).

  38. To Reduce or Not to Reduce: A Story of a Transition Metal Complex

  39. Main Character A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  40. The Facts • The synthesis of a vanadium oxo complex with low-valent vanadium (III) has been elusive. • Both ligand-field theory and computationally feasible wave function calculations predict a metal-centered reduction of V (IV) to V (III) in the complex through the addition of an electron to the d xy molecular orbital. A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  41. The Experiment The recent reduction of vanadium (IV) oxo 2,6- bis[1,1-bis(2-pyridyl)ethyl]pyridine to a dark blue substance suggested the potential first synthesis of low-valent vanadium (III) in a vanadium oxo complex. King, A . et al. Inorg. Chem. 53, 11388-11395 (2014).

  42. What’s Been Done Before [12,10] CASSCF Calculations: • active space = 12 electrons and 10 orbitals • active orbitals on V and O • 10,000 quantum degrees of freedom! King, A . et al. Inorg. Chem. 53, 11388-11395 (2014).

  43. What Was Found Before [12,10] CASSCF Calculations: Metal-centered reduction of the vanadium from V (IV) to V (III) in the vanadium oxo complex in agreement with ligand-field theory. King, A . et al. Inorg. Chem. 53, 11388-11395 (2014).

  44. A 2-RDM Calculation [42,40] 2-RDM Calculations: • active space = 42 electrons and 40 orbitals • active orbitals on V and O and pyridine ligands • 10 21 quantum degrees of freedom! A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  45. Natural-orbital Occupations CASSCF[12,10]: HOMO = 1.97 LUMO = 0.03 A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  46. Natural-orbital Occupations CASSCF[12,10]: HOMO = 1.97 LUMO = 0.03 2-RDM[12,10]: HOMO = 1.97 LUMO = 0.03 A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  47. Natural-orbital Occupations CASSCF[12,10]: HOMO = 1.97 LUMO = 0.03 2-RDM[12,10]: HOMO = 1.97 LUMO = 0.03 2-RDM[42,40]: HOMO = 1.37 LUMO = 0.26 A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  48. Fractional Occupation Numbers A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  49. CASSCF [12,10] HOMO Orbital A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  50. 2-RDM [42,40] HOMO Orbital A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  51. 2-RDM [42,40] Mulliken Populations and Charges A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  52. Pyridine Reduction But pyridine is NOT a great reducing agent! A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  53. Entangled Electrons! • 5 pyridine ligands • electrons become entangled among the 5 pyridine ligands! A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

  54. Some Conclusions • The elusive V (III) oxo complex has NOT yet been formed . • Ligand-centered reduction is stabilized by strong electron correlation . • Significant difference between the [12,10] and [42,40] active spaces with the latter space having 10 21 quantum variables. A. Schlimgen, C. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 2016 , 7, 627−631 .

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