1. Comparison of Born-Oppenheimer and Atom-Centered Density Matrix Propagation Methods for Ab Initio Molecular Dynamics H. Bernhard Schlegel Dept. of Chemistry, Wayne State U. Detroit, Michigan 48202, USA

2. Born-Oppenheimer Dynamics –to move on an accurate PES, converge the electronic structure calculation at each point Extended Lagrangian Dynamics –propagate the electronic structure as well as nuclei using an extended Lagrangian (e.g. Car-Parrinello, ADMP, etc.) Approaches for Ab Initio Molecular Dynamics

3. Born-Oppenheimer approach –to get an accurate PES at each step, converge wavefunction rather than propagate –gradient-based integrators standard numerical methods small step sizes –Hessian-based integrators local quadratic surface - larger steps –Helgaker et al. CPL 173, 145 (1990) predictor – corrector based on a local 5 th order surface –CPL 228, 436 (1994), JCP 111, 3800 and 8773 (1999)

4. Ab Initio Classical Trajectory on the Born-Oppenheimer Surface Using a Hessian-based Predictor-Corrector Method Calculate the energy, gradient and Hessian Solve the classical equations of motion on a local 5 th order polynomial surface

5. Energy Conservation as a Function of Step Size for Hessian Based Integrators -12 -10 -8 -6 -4 -2 0 -2.5-2-1.5-0.50 Logarithm of mass-weighted step size (amu bohr) 1/2 Logarithm of the average error in the conservation of total energy (hartree) 5th order fit (3.00) Rational fit (2.47) 2nd order fit (2.76)

6. Updating methods for Hessian-based trajectory integration Significant savings in CPU time if the Hessian can be updated for a few steps before being recalculated BFGS and BSP updating formulas used in geometry optimization are not suitable here Murtaugh-Sargent (Symmetric Rank 1) update is satisfactory

7. Relative CPU Time as a Function of the number of Hessian Updates 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0369 Number of Hessian updates Relative CPU usage C3N3H3C3N3H3 H 2 CO + CH 3 F NCCHO + CH 3 Cl

8. Dynamics with Atom-centered Basis Functions and Density Matrix Propagation (ADMP) Schlegel, Millam, Iyengar, Voth, Daniels, Scuseria, Frisch, JCP 114, 9758 (2001), JCP 115, 10291 (2001), JCP 117, 8694 (2002) Lagrangian in an orthonormal basis L = ½ Tr(V T MV) + ½ Tr((   W   ) 2 ) - E(R,P)  - Tr[  (P 2 -P)] R, V, M – nuclear coordinates, velocities and masses P, W – density matrix and density matrix velocity in an orthonormal basis (Löwdin or Cholesky)  – fictitious electronic ‘mass’ matrix E(R,P) – energy (electronic + V NN ) Tr[  (P 2 -P)] – constraint for idempotency and N-representability

9. Euler-Lagrange Equations of Motion equations of motion for the nuclei M d  R/dt 2 = -  E/  R| P equations of motion for the density matrix d  P/dt 2 = -  -½ [  E/  P| R +  P+P  -  ]  -½ integrate using velocity Verlet iteratively solve for  to impose the idempotency constraints on P and W

10. Basis functions for representing the electronic structure Method of orthonormalization Calculation of the energy derivatives Integration of the equations of motion Satisfying the idempotency constraints Mass weighting Behind the Scenes: Choices and Challenges

11. Car-Parrinello method (CP) PRL 55, 2471 (1985) –expand in plane waves (appropriate for condensed phase) –most integrals easy to calculate with fast Fourier transforms –only Hellmann-Feynman terms required for gradients Atom-centered Density Matrix Propagation (ADMP) –expand in atom centered functions (e.g. gaussians) –use methods from standard molecular orbital calculations –far fewer basis functions needed than plane waves –fast routines for multi-centered integrals –gradients require Hellmann-Feynman and Pulay terms Basis functions for representing the electronic structure: plane waves vs atom centered functions

12. molecular orbitals vs density matrix –arbitrary rotations among occupied molecular orbitals do not change the energy or the density –density matrices become sparse for large systems and calculations scale linearly atomic orbitals vs orthonormal orbitals –in an AO basis, changes in the density matrix reflect changes in bonding and changes in overlap –in an AO basis, effect of changes in the overlap must be handled by the propagation –in an orthonormal basis, changes in the overlap are handled separately and only changes in the bonding are handled by propagation –equations of motion simpler in an orthonormal basis Representing the electronic structure

13. Derivative of the energy wrt the density energy calculated with the purified density in an orthonormal basis (like CG-DMS method) E = Tr[h P + ½ G( P ) P ] + V NN P = 3 P 2 – 2 P 3 – McWeeny purified density h, G( P ) – one and two electron integral matrices derivative of the energy with respect to the density in an orthonormal basis  E/  P| R = 3 F P + 3 P F – 2 F P 2 – 2 P F P – 2 P 2 F F = h + G( P ) – Fock matrix, Kohn-Sham matrix derivative contains only occupied-virtual blocks (occupied- occupied and virtual-virtual blocks are zero) equations of motion satisfy idempotency condition to first order

14. Derivative of the energy wrt the nuclei  E/  R| P = Tr[dh/dR P + 1 / 2  G( P )/  R| P P ] + dV NN /dR where h, G and P are matrices in the orthonormal basis transformation to the orthonormal basis S’ = U T U, P = U P’ U T, h = U -T h’ U -1, etc. where S’, h’, etc. are matrices in the atomic orbital basis  E/  R| P = Tr[dh’/dR P’ + 1 / 2  G’(P’)/  R| P’ P’] + dV NN /dR - Tr[F’ U -1 dU/dR P’ + P’ dU T /dR U -T F’] (obtained using UU -1 = I, UdU -1 /dR = dU/dR U -1 and P 2 =P)  E/  R| P = Tr[dh’/dR P’ + 1 / 2  G’(P’)/  R| P’ P’] + dV NN /dR - Tr[F’ P’ dS’/dR P’] + Tr{[F, P] (Q dU/dRU -1 -P U -T dU T dR)} for a converged SCF calculation, [F, P]=0

15. Derivatives of the transformation matrix for the Löwdin orthonormalization, U= S’ 1/2  U/  R =  s i (  i 1/2 +  j 1/2 ) -1 (s i T  S’/  R s j ) s j T where  and s are the eigenvalues and eigenvectors of S’ for the Cholesky basis, S’=U T U and U is upper triangular (  U/  R U -1 )  = (U -T  S’/  R U -1 )  for  = ½ (U -T  S’/  R U -1 )  for  = 0 for  for large, sparse systems, Cholesky is O(N) and the transformed F and P are sparse

16. Velocity Verlet step for the density –symplectic integrators provide excellent energy conservation over long periods P i+1 =P i +W i  t -  -½ [  E(R i,P i )/  P| R +  i P i +P i  i -  i ]  -½  t 2 /2 W i+½ =W i -  -½ [  E(R i,P i )/  P| R +  i P i +P i  i -  i ]  - ½  t/2 = (P i+1 -P i )/  t W i+1 =W i+½ -  -½ [  E(R i+1,P i+1 )/  P| R +  i+1 P i+1 +P i+1  i+1 -  i+1 ]  -½  t/2 –Lagrangian multipliers chosen so that the idempotency condition and its time derivative are satisfied, but must not affect the conservation of energy, etc.

17. Constraint for Idempotency: scalar mass   P = P-P 0 = W 0  t-  -1 [  E(R 0,P 0 )/  P| R +  P 0 +P 0  -  ]  t 2 /2 P 0 is idempotent, choose  so that P 2 = P W 0 and  E(R 0,P 0 )/  P| R contain only occ-virt blocks but  P 0 +P 0  -  contains only occ-occ and virt-virt blocks closed form solution, P 0  P P 0 = -(I-(I-4AA T ) 1/2 )/2, Q 0  P Q 0 = (I-(I-4A T A) 1/2 )/2, where A=P 0 {W 0  t -  -1 [  E(R 0,P 0 )/  P| R ]  t 2 /2}Q 0 iterative solution, B  (A+A T ) 2 + B 2, P 0  P P 0 = -P 0 B P 0, Q 0  P Q 0 = Q 0 B Q 0 W must satisfy WP + P W = W W = W* - PW*P - QW*Q where W*=(P-P 0 )/  t -  -1 [  E(R 0,P 0 )/  P| R ]  t/2

18. Mass-weighting in the course of a trajectory, core orbitals change more slowly than valence orbitals second derivatives wrt density matrix element involving core orbitals are much larger than for valence orbitals core functions can be assigned heavier fictitious masses so that their dynamics are similar to valence functions in initial tests,  chosen to be a diagonal matrix  =  for valence orbitals  ½ =  ½ (2 (F ii + 2) 1/2 +1) for core orbitals with F ii < -2 hartree more general choice may be desirable, depending on the elements and the basis set

19. Iterative solution for the Lagrangian multipliers for mass weighted case,   i is chosen so that P i+1 2 = P i+1 (1) P i+1 = P i +W i  t -  -½  E(R i,P i )/  P| R  -½  t 2 /2 (2) P i+1 = 3 P i+1 2 – 2 P i+1 3 (3)  i =  ½ ( P i+1 - P i+1 )  ½ (4) P i+1 = P i+1 +  -½ (P i  i P i + Q i  i Q i )  -½ go to (2) if not converged  i+1 is chosen so that W i+1 P i+1 + P i+1 W i+1 = W i+1 (1) W i+1 = (P i+1 -P i )/  t -  -½  E(R i+1,P i+1 )/  P| R  -½  t/2 (2) W i+1 = W i+1 - P i+1 W i+1 P i+1 - Q i+1 W i+1 Q i+1 (3)  i+1 =  ½ ( W i+1 - W i+1 )  ½ (4) W i+1 = W i+1 +  -½ (P i+1  i+1 P i+1 + Q i+1  i+1 Q i+1 )  -½ go to (2) if not converged

20. Comparison of Resource Requirements relative timings can be estimated from the times for computing the Fock matrix, total energy, gradient and Hessian –BO with Hessian based predictor-corrector t = (t(energy) + t(Hessian)) /  t –BO with Hessian based predictor-corrector with 5 updates t = (t(energy) + 1 / 6 t(Hessian) + 5 / 6 t(gradient)) /  t –BO with gradient based integrators with one gradient eval per  t t = (t(energy) + t(gradient)) /  t –ADMP with velocity Verlet integrator t = (t(Fock) + t(gradient)) /  t time step,  t, chosen so that energy is conserved to ca 10 -5 Hartree (  t = ca 0.7 fs for Hessian methods, ca. 0.1 for gradient methods)

21. Relative timings for HF/6-31G(d) calculations on linear hydrocarbons

22. Relative timings for B3LYP/6-31G(d) calculations on linear hydrocarbons

23. Chloride – Water Cluster (0.10 amu bohr 2 (182 a.u.) fictitious ‘mass’ for the density)

24. Energy Conservation for Cl - (H 2 O) 25

25. Comparison of the Fourier transform of the velocity-velocity autocorrelation function for Cl - (H 2 O) 25 at PBE/3-21G*

26. Fourier transform of the velocity-velocity autocorrelation function for Cl - (H 2 O) 25 at B3LYP/6-31G* (~1 ps) (compare with 3620-3833 cm -1 for OH str in (H 2 O) 2 and 3379 cm -1 in Cl - -H 2 O)

27. Effect of Fictitious Mass on Vibrational Frequencies for CP calculations of ionic systems, effective mass of ion is the nuclear mass plus the fictitious mass of the electrons vibrational frequencies depend on the fictitious mass, but can be rescaled by a factor depending on the electron-nuclear coupling (a) (b) Phonon density of states for (a) crystalline silicon and (b) crystalline MgO P. Tangney and S. Scandolo, JCP 116, 14 (2002)

28. Effect of Fictitious Mass on Vibrational Frequencies for ADMP calculations, basis functions move with the nuclei fictitious mass affects only the response of the density, not the dynamics of the nuclei vibrational frequencies do not depend on the fictitious mass Vibrational motion of diatomic NaCl (fs)

29. Ab initio classical trajectory study of H 2 CO  H 2 + CO dissociation. Important test case, since studied intensively, both experimentally and theoretically. Excitation to S 1 followed by internal conversion to S 0, with sufficient energy to dissociate. Products are rotationally and vibrationally excited. Born-Oppenheimer ab initio classical trajectory studies: W. Chen, W. L. Hase, H. B. Schlegel, CPL 228, 436 (1994) X. Li, J. M. Millam, H. B. Schlegel, JCP 113, 10062 (2000)

30. Effect of fictitious mass on energy conservation and adiabaticity  valence = 0.40 amu bohr 2  valence = 0.20 amu bohr 2  valence = 0.10 amu bohr 2..... kinetic energy of the density --- total energy __ real energy (total – kinetic)

31. Effect of the Fictitious Mass on the Vibrational Distributions for CO and H 2

32. Effect of the Fictitious Mass on the Rotational Distributions for CO and H 2

33. Three Body Photofragmentation of Glyoxal Xiaosong Li, John M. Millam, H. B. Schlegel JCP 114, 8 (2001), JCP 114, 8897 (2001), JCP 115 6907 (2001) Under collision free conditions, excitation to S 1 followed by internal conversion to S 0, with 63 kcal/mol excess energy. High pressure limit for rate constant: E a = 55 kcal/mol. CO formed with J max = 42 and a broad distribution, but vibrationally cold H 2 has significant population in v=1, but low J

34. Transition States for Glyoxal Unimolecular Dissociation Barrier enthalpies (CBS-APNO) glyoxal  H 2 + 2 CO59.2 kcal/mol H 2 CO + CO54.4 kcal/mol HCOH + CO59.7 kcal/mol 2 HCO70.7 kcal/mol

35. Vibrational energy distribution ADMP HF/3-21G

36. Acknowledgments Group members –Xiaosong Li, Smriti Anand, Hrant Hratchian, Jie Li, Jason Cross, Stanley Smith, John Knox Collaborators –S. S. Iyengar, G. E. Scuseria, G. A. Voth, J. M. Millam, M. J. Frisch, W. L. Hase, Ö. Farkas, C. H. Winter, M. A. Robb, S. Shaik, T. Helgaker, Funding –National Science Foundation, DOE, Gaussian Inc., Wayne State University

37. Xiaosong Li, John Knox, Hrant Hratchian, Stan Smith, Smriti Anand, HBS, Jie Li