1. IMPACT OF COMPLEX-VALUED ENERGY FUNCTION SINGULARITIES ON THE BEHAVIOUR OF RAYLEIGH-SCHRODINGER PERTURBATION SERIES. H 2 CO MOLECULE VIBRATIONAL ENERGY SPECTRUM. Authors: A.N. Duchko, A.D. Bykov

2. 2 Problem statement [A. V. Nikitin et. al, JCCP, 15, 10071, 2013]

3. “Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever“ Niels Hendrik Abel, 1828 3

4. 4 Application to vibrational energy calculations z=1 corresponds to physical case. Series coefficients (even the divergent one) contain information about inital function. Thus, by applying more complicated techniques than direct summation one can find its values.

5. 5 Summation technique N-th order Pade-Hermite method PH(z) — complex multivalued function (N-valued) PH(1) — approximation to the physical case (E V (1))

6. 6 Structure of series and branch points Fig.1. Structure of series and branch points for 2v 2 and v 3 +2v 4 vibrational states of H 2 CO. Low energy state E=3464,03 Low energy state E=3824,36

7. 7 Structure of series and branch points Fig.2. Structure of series and branch points for 4v 4 and 4v 4 +2v 6 vibrational states of H 2 CO. E=4616,40 Highly excited state E=7112,45

8. 8 RSPT coefficients modelling Fig.3. RSPT coefficients modelling: RSPT/estimated coefficient ratios for v 3 vibrational state of H 2 CO. [A.V. Sergeev, D. Z. Goodson, J. Chem. Phys, 124, 094111, 2006]

9. 9 Convergence of Pade-Hermite approximants Fig.4. Convergence of Pade-Hermite approximants for the ground (a); v 3 +v 6 (b); v 3 +2v 4 (c), and v 3 +3v 6 (d) vibrational states of H 2 CO. Green line – 1-st order Pade- Hermite approximant, red line – 2 order, blue line – 3-rd order, black line – 4-th order. (a)(b) (c) (d)

10. 10 Hamiltonian repartitioning (shifting frequencies) where

11. 11 Effect of repartitioning Fig.5. RSPT series and convergence of Pade-Hermite approximants for 3v 2 vibrational state of H 2 CO without shift (a,b) and with shift of the second frequency equal to 20 cm -1. (a) (b) (c) (d)

12. Carrier’s Rule: “Divergent series converge faster than convergent series because they don’t have to converge.”

13. 13 Effect of repartioning: shifting singularities Fig.6. Singularities of 3v 2 vibrational state of H 2 CO. 2 states are in resonance if they have the same Katz's branch point inside the unit circle

14. 14 Impact of singularities Fig.7. Singularities of v 2 +2v 6 vibrational state of H 2 CO (symmetry type A1) v 5 +v 6 Katz's points

15. 15 Impact of singularities Fig.8. Number of same digits of high order RSPT coefficients for v 2 +2v 6 and v 5 +v 6 vibrational states of H 2 CO. According to Katz theorem high order RSPT series coefficients of resonating states coincide in magnitude but differ in sign

16. 16 Branch identification State v 2 +2v 6. Energy: 4275.06 Pade-Hermite approximation (5-th order) BranchEnergy, cm -1 Error of resummation, cm -1 Branch identification 14275.06 3*10 -31 v 2 +2v 6 24175.36 1.9*10 -12 v 5 +v 6 34316.66 7.5*10 -5 v 1 +2v 3 43939.42 0.6v 3 +2v 6 5not converged Table 1. Convergence of all branches of 5-th order Pade-Hermite approximant for v 2 +2v 6 vibrational energy state of H 2 CO molecule.

17. 17 Convergence of all branches of Pade-Hermite approximants corresponding to RSPT series of the state v 2 +2v 6 Fig.9. Convergence of all branches of 1-st (a), 2-nd (b), 3-rd (c), 4- th (d), 5-th (e), 6-th (f) Pade-Hermite approximants for v 2 +2v 6 vibrational state of H 2 CO. v 2 +2v 6 v 5 +v 6 v 1 +v 3 v 3 +2v 6 (a)(b)(c) (f) (e) (d) 1-st order 1 branch 2-nd order 2 branches 3-rd order 3 branches 6-th order 6 branches 5-th order 5 branches 4-th order 4 branches

18. 18 Program complex 1. Big data processing 2. Long real arythmetics 3. High accuracy calculations 4. Flexibility of algorithms 5. User-friendly interface 6. Cross-platform project 7. Ready for parallel calculations

19. 19 Program screenshot

20. 20 Conclusions Extremely high order perturbation theory without resonances (up to 200 lowest states) High-order algebraic approximants (up to 10) Technique of repartitioning Calculations of energy of highly excited vibrational states (up to 200) Multiple branches of the same algebraic approximant allow to get energy of different states from one RSPT series

21. Thank you for your attention!