a.grandi@caspur.it

1. Shape resonances localization and analysis by means of the Single Center Expansion e-molecule scattering theoryAndrea GrandiandN.Sanna and F.A.Gianturco a.grandi@caspur.it Caspur Supercomputing CenterandUniversity of Rome ”La Sapienza” URLS node of the EPIC Network

2. Introduction The talk will be organized as follows: • Introduction to the e-molecule scattering theory based on the S.C.E. approach • SCELib(API)-VOLLOC code • Shape resonances analysis • Examples and possible applications • Conclusions and future perspectives

3. The SCE method The Single Center Expansion method • Central field model : • Factorization of the wave-function in radial and angular components • Bound and continuum electronic states of atoms • Extension to bound molecular systems • Electron molecule dynamics, molecular dynamics, surface science, biomodelling

4. The SCE method The Single Center Expansion method In the S.C.E. method we have a representation of the physical world based on a single point of reference so that any quantity involved can be written as

5. The SCE method In the SCE method the bound state wavefunction of the target molecule is written as

6. The SCE method Symmetry adapted generalized harmonics Symmetry adapted real spherical harmonics

7. The SCE method Where S stays for

8. The SCE method The bound orbitals are computed in a multicentre description using GTO basis functions of near-HF-limit quality - gk(a,rk) Where N is the normalization coefficient

9. The SCE method The radial coefficients are computed by integration The quadrature is carried out using Gauss-Legendreabscissas and weights for  and Gauss-Chebyshevabscissas and weightsfor , over a dicrete variable radial grid

10. The SCE method Once evaluated the radial coefficients each bound one-electron M.O. is expanded as: So the one-electron density for a closed shell may be expressed as

11. The SCE method and so we have the electron density as: Where then, from all of the relevant quantities are computed.

12. The SCE method The Static Potential And as usual:

13. The SCE method Where :

14. The SCE method The polarization potential: Short range interaction For r ≤ rc Long range interaction For r > rc whererc is the cut-off radius

15. The SCE method Short-range first model: Free-Electron Gas Correlation Potential with and g=0.1423,b1=1.0529,b2=0.3334.

16. The SCE method Short-range second model: Ab-Initio Density Functional (DFT) Correlation Potential where is the Correlation Energy

17. Short-range second model: Ab-Initio Density Functional (DFT) Correlation Potential

18. The SCE method We need to evaluate the first and second derivative of (r) In a general case we have:

19. The SCE method We need to evaluate the first and second derivative of (r) In a general case we have:

20. The SCE method Problems with the radial part: Single center expansion of F,F’, and F” are time consuming We performe a cubic spline of F to simplify the evaluation of the first and second derivative Problems with the angular part: For large values of the angular momentum L it is possible to reach the limit of the double precision floating point arithmetic To overcame this problem it is possible to use a quadrupole precision floating point arithmetic (64 bits computers)

21. The SCE method Long-range : The asymptotic polarization potential The polarization model potential is then corrected to take into account the long range behaviour

22. The SCE method Long-range : The asymptotic polarization potential In the simple case of dipole-polarizability

23. The SCE method Long-range : The asymptotic polarization potential Where Usually in the case of a linear molecule one has

24. The SCE method Long-range : The asymptotic polarization potential Where

25. The SCE method Long-range : The asymptotic polarization potential In a more general case Once evaluated the long range polarization potential we generate a matching function to link the short / long range part of Vpol

26. The SCE method The exchange potential: first model The Free Electron Gas Exchange (FEGE) Potential Two great approximations: • Molecular electrons are treated as in a free electron gas, with a charge density determined by the ground electronic state • The impinging projectile is considered a plane wave

27. The SCE method The exchange potential: first model The Free Electron Gas Exchange (FEGE) Potential

28. The SCE method The exchange potential: second model The SemiClassical Exchange (SCE) Potential and modified SCE (MSCE) SCE: • The local momentum of bound electrons can be disregarded with respect to that of the impinging projectile (good at high energy collisions)

29. The SCE method The exchange potential: second model The SemiClassical Exchange (SCE) Potential and modified SCE (MSCE) MSCE: • The local velocity of continuum particles is modified by both the static potential and the local velocity of the bound electrons.

30. The SCE method The solution of the SCE coupled radial equations Once the potentials are computed, one has to solve the integro-differential equation

31. The SCE method The solution of the SCE coupled radial equations The quantum scattering equation single center expanded generate a set of coupling integro-differential equation

32. The SCE method The solution of the SCE coupled radial equations Where the potential coupling elements are given as:

33. The SCE method The solution of the SCE coupled radial equations The standard Green’s function technique allows us to rewrite the previous differential equations in an integral form: This equation is recognised as Volterra-type equation

34. The SCE method The solution of the SCE coupled radial equations In terms of the S matrix one has: i,j identify the angular channel lh,l’h’

35. SCELib(API)-VOLLOC code

36. SCELIB-VOLLOC code

37. SCELIB-VOLLOC code

38. SCELIB-VOLLOC code Serial / Parallel ( open MP / MPI )

39. SCELIB-VOLLOC code Typical running time depends on: • Hardware / O.S. chosen • Number of G.T.O. functions • Radial / Angular grid size • Number of atoms / electrons • Maximum L value

40. SCELIB-VOLLOC code Test cases:

41. SCELIB-VOLLOC code Hardware tested:

42. Shape resonance analysis

43. Shape resonance analysis • we fit the eigenphases sum with the Briet-Wigner formula and evaluate G and t

44. Uracil

45. Uracil J.Chem.Phys., Vol.114, No.13, 2001

46. Uracil • ER=9.07 eV R=0.38 eV =0.1257*10-15 s

47. Uracil

48. Thymine J.Phys.Chem. A, Vol. 102, No.31, 1998

49. Cubane

50. Cubane