shape resonances localization and analysis by means of the single center expansion e-molecule...

Shape resonances localization and Shape resonances localization and analysis by means of the Single Center analysis by means of the Single Center Expansion e-molecule scattering theoryExpansion e-molecule scattering theory

Andrea GrandiAndrea Grandiandand

N.Sanna and F.A.GianturcoN.Sanna and F.A.Gianturco

[email protected]

Caspur Supercomputing Centerand

University of Rome ”La Sapienza” URLS node of the EPIC Network

IntroductionIntroductionThe 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 SCE method

The Single Center Expansion method

( , , ) ( ) ( , )lm lmlm

F r F r Y

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



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

1( ) ( ) ( , )k pk kk i hl hl

hl

x r u r X


),(Υb),(Χ lmpkhlmm

pkhl

),(Sb),(Χ lmpkhlm

pkhl

l

lm

Symmetry adapted generalized harmonicsSymmetry adapted generalized harmonics

Symmetry adapted real spherical harmonicsSymmetry adapted real spherical harmonics


)(cosP4π

1φ),(S l0

2

1

l0

)cosmφ(cosPm)!(l

m)!(l

2π

12l1)(φ),(S lm

2

1

mlm

Where S stays forWhere S stays for

)sinmφ(cosPm)!(l

m)!(l

2π

12l1)(φ),(S lm

2

1

mm-l

The bound orbitals are computed in a multicentre description using GTO basis functions of near-HF-limit quality - gk(a,rk)


2krcba

jkjkj ezyx);c,b,a(N)r,(g

Where N is the normalization coefficient

2

1

cba2

3

j !!1c2!!1b2!!1a2

42N

2

0 0

( ) sin( ) ( ) ( , )k i i i kj kjhl lm lm kj v v j k

k j v lm

u r d b S C R d g r d

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 The SCE method Once evaluated the radial coefficients each

bound one-electron M.O. is expanded as:

,Xrurru phl

hl

ihl

1i

So the one-electron density for a closed shell may beexpressed as

2

iiN2

2N21 |ru|2dx...dx|x,...,x,x|r


and so we have the electron density as:

11 ( ) ( , )Ahlm hlmhlm

r r r X

then, from all of the relevantquantities are computed.

r

Where

drurudsin2r i

2

0

ii0

lm


The Static Potential

N

i i

iSt

|Rr|

Zds|sr|

1srV

And as usual:

,XrVrV 1Alm

lm

Stlm

St


Where :

dsr

rs

1l2

4rV

1l

lStlm

a

a

r

0

0

r1llm

lllm1l

drr

1rrdrrr

r

1

1l2

4


The polarization potential:

rVrV corrcp

where rc is the cut-off radius

rVrV polcp

Short range interaction

Long range interaction

For r ≤ rc

For r > rc

Short-range first model:Free-Electron Gas Correlation Potential

1

0.0311ln 0.0584 0.00133 ln 0.0084 for 1.0

7 41/ 2

(1 )1 26 3

1/ 2 2

(1 )1 2

( ) ( , )

( )

r r r r rs s s s s

r rs s

r rs s

FEG AFEGcorr hlmhlmhlm

FEG

hlm

V r r X

r

V

V

for r 1.0s

with and =0.1423,1=1.0529,2=0.3334.

3 ( )4sr r


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

where is the Correlation EnergycE

1( ) ( , )

( )

DFT ADFTcorr hlmhlmhlm

C

V r r X

E r

V


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

21

22'1

2"1

21

22'1

2"1

1'1

3

5

F1'

1DFTcorr

G465GG372

ab

G423GG4

ab

G3

8GabCFFarV

32

2F

3

1

3

5

11

3

1

1

310

3C

cexprFrG

d1rF

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


( , , ) ( ) ( , )lm lmlm

F r F r Y

lm

lmθlmlm

rlmlm

lmlmlm

eXsinθ

me

θ

X

r

FeX

dr

dF

θ,XrFθ,r,F

ˆˆˆ

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


lm2

22

Xr

1ll

r

2

lm

lm2lm

2

lmlmlm

dr

dF

dr

Fd

θ,XrFθ,r,F

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

The polarization model potential is then corrected to take into account the long range behaviour

1l1l2

l

r

Apol r2

RlimR/rV 1


zzyzxz

yzyyxy

xzxyxx

2

1

2

12

1

2

12

1

2

1

In the simple case of dipole-polarizability

av

av

av

00

00

00


0

3

1

yzxzxy

zzyyxxavzzyyxxav

Where

cosPr2r2

V 242

40

pol

Usually in the case of a linear molecule one has


Where

20zz20yyxx

22

2

1

1cos32

1cosP


In a more general case

3,2,1j,iz,y,xq

qqr2

1z,y,xV

j.i

ij

3

1i

3

1jji6pol

Once evaluated the long range polarization potentialwe generate a matching function to link the short / long range part of Vpol

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

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

1/32( ) 3 ( )FK r r

1

2

( ) ( , )

2 1 1 1( ) ln

2 4 1

FEGE AFEGEhlmhlmhlm

F

V r r X

K r

V

/ Fk K

212Fpcoll rKIE2rk


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)

2

1

s

2

s

2AST0

AST0

2SCE 8rVE

2

1rVE

2

1k,rV 11


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.

2

12

3

22A

ST03

22A

ST02

MSCE r4r310

3rVE

2

1r3

10

3rVE

2

1k,rV 11

The solution of the SCE coupled radial equations

Once the potentials are computed, one has to solve the integro-differential equation

2 2

*

1 1( ) ( )

2 2

1( ) ( ) ( )

k V r F r

u s F s dsu rr s




The quantum scattering equation single center expanded generate a set of coupling integro-differential equation

'h'l

p'h'l,lh'h'l,lh

,p'h'l,lh2

22

2

rfR/rVrfr

1llk

dr

d



Where the potential coupling elements are given as:

r̂XR/rVr̂Xr̂d

r̂XR/rVr̂XR/rV

p'h'l

plh

p'h'l

plh

p'h'l,lh

The SCE method The SCE method The solution of the SCE coupled radial equations

The standard Green’s function technique allows us to rewritethe previous differential equations in an integral form:

n

r

0

pnjl

l'h'l,lhp

'h'l,lh

R/'rfR/'rV'r,rg'dr

krjR,rf

This equation is recognised as Volterra-type equation



In terms of the S matrix one has:

jijj l

2

1kri

pij

l2

1krir

,pij eSerf

i,j identify the angular channel lh,l’h’

SCELib(API)-VOLLOC codeSCELib(API)-VOLLOC code

SCELIB-VOLLOC codeSCELIB-VOLLOC code

SCELIB-VOLLOC codeSCELIB-VOLLOC codeSerial / Parallel ( open MP / MPI )


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


Test cases:


Hardware tested:

Shape resonance analysis Shape resonance analysis

Shape resonance analysis Shape resonance analysis

2 1( ) ( ) ( ) tan2( )R R

R

E a b E E c E EE E

we fit the eigenphases sum with we fit the eigenphases sum with the the Briet-WignerBriet-Wigner formula and formula and evaluate evaluate and and

UracilUracil

UracilUracil

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

UracilUracil

• ER=9.07 eV R=0.38 eV =0.1257*10-15 s

UracilUracil

Thymine

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

CubaneCubane

CubaneCubane

EErr=9.24 eV =9.24 eV =3.7 eV =3.7 eV =1.8*10=1.8*10-16-16ss

CubaneCubane

CubaneCubane

EErr=14.35 eV =14.35 eV =4.2 eV =4.2 eV =1.5*10=1.5*10-16-16ss

CubaneCubane

Conclusion and future Conclusion and future perspectivesperspectives

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)

shape resonances localization and analysis by means of the single center expansion e-molecule...

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