Shape resonances localization and analysis by means of the Single Center Expansion e-molecule scattering theory Andrea Grandi and N.Sanna and F.A.Gianturco Caspur Supercomputing Center and University of Rome ”La Sapienza” URLS node of the EPIC Network

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

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 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 The SCE method

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

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

The SCE method Where S stays for

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

The quadrature is carried out using Gauss- Legendre abscissas and weights for  and Gauss- Chebyshev abscissas and weights for , over a dicrete variable radial grid The radial coefficients are computed by integration 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

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

The SCE method The Static Potential And as usual:

The SCE method Where :

The SCE method The polarization potential: where r c  is the cut-off radius Short range interaction Long range interaction For r ≤ r c For r > r c

Short-range first model: Free-Electron Gas Correlation Potential with and  =0.1423,  1 =1.0529,  2 = The SCE method

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

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

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

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

Problems with the radial part: The SCE method 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)

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

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

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

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

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 V pol

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

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

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)

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.

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

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

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

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’

SCELib(API)-VOLLOC code

SCELIB-VOLLOC code

Serial / Parallel ( open MP / MPI )

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

SCELIB-VOLLOC code Test cases:

SCELIB-VOLLOC code Hardware tested:

Shape resonance analysis

 we fit the eigenphases sum with the Briet-Wigner formula and evaluate  and 

Uracil Uracil

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

Uracil E R =9.07 eV  R =0.38 eV  =0.1257* s

Uracil

Uracil

Uracil

Uracil

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

Thymine Exp J.Phys.Chem. A, Vol. 102, No.31, 1998 (E.T.S.)

Work in progress on Thymine Exp J.Phys.Chem. Vol. 114, No.13, 2001

Work in progress on ThymineExp Chem.Phys.Lett Vol. 377, (2003) (crossed beams)

Work in progress on ThymineExp Nature 231, 262, (1971) (E.E.L.)

Cubane Cubane

E r =9.24 eV  =3.7 eV  =1.8* s

Cubane Cubane

E r =14.35 eV  =4.2 eV  =1.5* s

Cubane Cubane

Conclusion and future perspectives  Shape resonance analysis (S-matrix poles)  Transient Negative Ion Orbitals analysis (post- SCF multi-det w/f)  Dissociative Attachment with charge migration seen through bond stretching (  (R)   (R) )  Study of the other DNA bases (thymine t.b.p., A,C,G planned)  Development of new codes (SCELib-API & parallel VOLLOC)