Basis set, correlation and relaxation effects in the characterisation of electron-molecule scattering resonances using the dilated electron propagator method with application to the ²II_g N₂^- shape resonances

Basis set, correlation and relaxation effects in the characterisation of electron-molecule scattering resonances using the dilated electron propagator method with application to the _{2TIg N2-}shape resonances

S Mahalakshmi & Manoj K Mishra*

Department of Chemistry, Indian Institute of Technology, Bombay Powai, Mumbai 400 076 , India

Received 4 October /999; accepted ²⁶ November /999

The effect of basis set variations on molecular resonance attributes is investigated using systematically augmented basis functions and correlating the resulting changes in resonance energy and width with the alterations induced in the resonant Feynman Dyson amplitudes. Application to the prototypical 2ng N2' shape resonances reveals that basis set effects can be quite large and that basis sets effecting large electron density accumulation near the target nucleus to facilitate resonance formation and sufficiently large electron density away from the target nucleus to provide for its decay offer optimal characterisation of these resonances.

A comparison of results from the zeroth order, second order, diagonal 2ph-TDA and quasi-particle decouplings shows that the basis set effects can be larger than the correlation and relaxation effects and systematic basis set saturation is required for dependable characterisation of molecular resonances.

I. Introduction

The electron propagator theory

I.

tron detachment and attachment energies and correlated treatment of electronic structure and dynamics3-R• The spectral representation of the matrix-dilated electron propagator9- 1 1,

� [(

^{la, 1 �+I )(�}

^l

1 n (�13;1 �-I )(�- l la ,1 n ]

G,, (E) =

!��. "",

EN+I - E N ) + iE + E _ (EN - EN-I ) - iE

S s (I (

)

... ( 1 ) where all electronic coordinates have been scaled by a complex scale factor II = aei^S has been a convenient method for the direct calculation of energies and widths of shape resonances in electron-atom and electron-mol­

ecule scattering 12-20. The dilated Hamiltonian is non­

Hermitian and only the dilated biorthogonal electron propagatorl l based on a bi-variational SCF21,22 preserves the formal simplicity of the real (undilated) electron propagator decouplings. As can be seen from Eq. ^{( I ),} the poles of the dilated electron propagator provide for the direct calculation of the energy (real part) and width (twice the imaginary part) of electron attachment shape and electron detachment Auger resonances from the poles (EsN+1 - EoN ) and (EoN - EsN-I) and the corre­

sponding Feynman-Dyson amplitudes <oN I as I sN+1 > and

< sN+1 la/ I ON > provide a correlated picture of struc­

tural changes accompanying electron attachment/de­

tachment respectively. nhe resonant eigenvalues E,

-

(rt2) have a negative imaginary part to account for the finite life time23 of the metastable resonances. The tar­

get ground state energy EoN is completely real. There­

fore, the poles corresponding to the shape resonances (EsN+1 - EoN ) will have a negative imaginary part and their trajectory as a function of variations in the com­

plex scaling parameter II will move in the fourth quad­

rant of the complex energy plane24. As per the complex scaling theorems25-27, the complex poles in the fourth quadrant displaying quasi-stability with respect to variations in II are associated with shape reso­

nances.

As will be seen later in this paper, these complex poles are searched through an iterative diagonalisation proce­

dure by making multiple passes through self energy lists with approximately N5 elements where N is the basis set size. Furthermore, the resonances are identified through a search for regions of quasi-stability requiring the construction of the dilated electron propagator for a large number ofll values with 4-6a and approximately 30 e values per a being fairly representative. This makes the dilated electron propagator calculations at least 1 00 times more demanding in comparison to their real coun­

terparts.

MAHALAKSHMI et ^{al. :} CHARACTERISATION OF ELECTRON-MOLECULE SCATTERING RESONANCES 23

The decouplings of the bi-orthogonal dilated electron propagator are based on an underlying bivariational self­

consistent field (SCF) to take into account the non­

Hermiticity of the dilated (complex scaled) Hamilto­

nian21. In these schemes based on the bi-variational SCF, the orbital energies and amplitudes are the zeroth order poles and Feynman-Dyson Amplitudes (FDAs) of the bi-orthogonal dilated electron propagatorll.12 and for many systems, shape resonances have been uncovered at the level of the bi-variational SCF, permitting a de­

tailed analysis of the correlation and relaxation effects in the formation and decay of resonances. Second order, diagonal 2ph-TDA (two-particle one hole-Tamm­

Dancoff approximation) and quasi-particle approxima­

tions have also been applied to many atomic and mo­

lecular systemsI2.

The renormalised decouplings such as the diagonal 2ph-TDA 3,4 are expected to incorporate greater extent of correlation and relaxation effects by summing the more important diagonal ring and ladder diagrams to all order and thereby improve the systemic description. However, in dilated electron propagator calculations, the diago­

nal 2ph-TDA has not provided the expected improve­

ment to the results obtained using the second order decouplings 12. It has, therefore, become imperative that improvement in the description of resonances be sought through the incorporation of much more demanding third and the partial fourth order decouplings which have be­

come the preferred decouplings in the application of the real (undilated) electron propagator schemes2R.3o.

Due to the computational demands discussed above, earlier demonstrative applications of the dilated elec­

tron propagator utilized modest primitive bases to dem­

onstrate its promise and utility. The more recent appli­

cations31.32 using comparatively larger bases to treat e­

C2 H2 and e-HCHO resonances have shown that the reso­

nance attributes are quite sensitive to the nature of the primitive bases utilized in these applications. The third and the partial fourth order decouplings scale roughly as N7 and it is, therefore, imperative that an attempt be made to understand the basis set attributes which can provide an economic and balanced treatment of molecular reso­

nances.

This paper is an initial attempt to study the basis set dependence of resonance energies and widths of the very extensively studied prototypical 2IT N?' shape reso-_{g -} nance in e_N27.15.33.43 scattering using the bi-orthogonal dilated electron propagator method. The bi-orthogonal dilated electron propagator has earlier been used to

characterise the 2ITg shape resonance in e-N2 scattering using the zeroth order (1:0), second order (1:2), the di­

agonal two-particle one hole Tamm Dancoff approxi­

mation (1:2ph.TDA) and the quasi-particle decouplings (1:2 and 1: 2pH·IDA )15 . The 1:2 self energy approximant has _q q

consistently provided more dependable and balanced resultsl6 and therefore we confine this initial investiga­

tion of the basis set effects to the use of the 1:2 decoupling alone and proceed by systematically augmenting some standard basis sets and correlating the resulting changes in resonance energy and width with alterations induced in the resonant Feynman Dyson Amplitudes (FDAs).

For the saturated primitive basis, in addition to the 1:2 decoupling the 1:0 1:2pH.TDA 1:2 and 1: 2pH·IDA decouplings , , ' q q

are utilized to assess the role of correlation and relax- ation in the formation and decay of molecular resonances.

The biorthogonal dilated electron propagator has been reviewed recently 12 and only a skeletal outline of final theoretical expressions is offered in Section 2. The ef­

fect of basis set variations on the energies and widths of shape resonances in e-N2 scattering are discussed in Section 3 and some concluding remarks are collected in Section 4.

II. Method

Just as in the case of the real, undilated one electron propagatof, the bi-orthogonal dilated matrix electron propagator G(l1,E) may also be expressed asl2

G -I

G�I

where Go(l1 ,E) is the zeroth order propagator for the uncorrelated motion, here chosen as given by the bi-varia­

tional SCF approximation21.22• The self-energy matrix 1:(l1,E) contains the relaxation and correlation effects.

Solution of the bi-variational SCF equations21 for the N-electron ground state yields a set of occupied and unoccupied orbitals. In terms of these spin orbitals

{ 'l'j } , the matrix elements are

...(3) where tj is the orbital energy corresponding to the i-th spin orbital and the poles of the zeroth order (1:0 ⁼ 0) electron propagator Go are the orbital energies them­

selves.

Through the second order in electron interaction, the elements of the self-energy matrix arell

where

2 ⁼ � ^� (ik Ilfm)(em IUk)

ki/TJ, E) 2 L. N k,l.m (E ) ... (4)

k,l.1II + E k - E , - Em

with (nk) being the occupation number for the k-th spin orbital and the antisymmetric two-electron integral

(ijllkl)

f

... (6) where Pl2 is the permutation operator and r'2 is the distance between electrons I and 2.

The lack of complex conjugation stems from the bi­

orthogonal set of orbitals, resulting from bi-variational SCF, being the complex conjugate of each other21 . For the diagonal 2ph-TDA decoupling of the dilated elec­

tron propagatorl4

... (7) where

� = � (m ₂ ^e 11 m e)(l ^- (11 ^on)^-(11 , » - (km Il km )« n , ) - .^.. (8)

(^{11 "}. ) -(k ^e II

k

The usual dilated electron propagator calculations proceed by iterative solution of the eigenvalue prob­

lem

L (11 , E) X i (11 , E ) = E ⁱ (11 , E ) X i (11 , E ) ... (9)

with

L (TJ , E) = £ + :E ( TJ , E)

. . . ( 1 0) where £ is the diagonal matrix of orbital energies and I.j is the self energy matrix. The propagator pole E;

(ll,E) is obtained by iterative diagonalisation such that one of the eigenvalues E/ll, E) fulfills the condi­

tion Ej(ll, E) = E (ref. 1 2). These Ej(ll, E) represent the poles of the dilated electron propagator G (ll, E). From among these poles, the resonant pole Er (ll, E) and the

corresponding eigenvector (FDA) X, (ll, E) are selected as per the prescription of the complex scaling theo­

rems44,45 wherein those roots in the continua which are invariant to changes in the complex scaling param­

eter II are to be associated with resonances. In a lim­

ited basis set calculation, instead of absolute stability one finds quasi-stability where the 8 trajectory dis­

plays kinks, cusps, loops or inflections which indicate the proximity of a stationary point46 and in our case the resonance attributes have been extracted from the value at the kink in 8 trajectories (aE

/

The 8 value at the kink is taken as being the optimal 8 (8 _� ). The real part of the resonant pole at 8 _� fur- nishes the energy and the imaginary part the half width of the resonance.

The quasiparticle approximation47 for the dilated electron propagatorl7 results from a diagonal approxi­

mation to the self energy I.(ll, E) with poles of the di­

lated electron propagator given by

. . . ( I I )

which are determined iteratively beginning with E ⁼ £j

and I.;; may correspol,d to any perturbative (I.2) or renormalised decoupling such as the diagonal 2ph-TDA (I.2ph'TDA) .

In the bi-variationally obtained bi-orthogonal or­

bital basis { "," ) , Xn is a linear combination

x,.cf) ⁼

L

lows for the incorporation of correlation and relaxation effects. In the zeroth (I. ⁼ 0) and quasi-particle approxi­

mations (diagonal I.), there is no mixing. The differ­

ence between perturbati ve second order (I.2) or

Table 1 - Energy and width o f the 2ng N2' shape resonance using different primitive basis and :E2 decoupling of

the bi-orthogonal dilated electron propagator Basis Set Energy (eV) Width (eV)

I . 4s9p 2. 1 1 0. 1 8

2. 4591' I d 2.06 0.22

3. 5s7p 2.56 0.24

4. 5s7p l d 2.49 0. 1 8

5. 5s9p 1 .98 0. 1 5

6. 5s l 0p 2.08 0. 1 7

7. 551 11' 2. 1 3 0. 1 8 8. 5s l lp l d 2. 1 0 0. 1 5

MAHALAKSHMI ^et al. : CHARACTERISATION OF ELECTRON-MOLECULE SCATTERING RESONANCES 25

-0.01 b)

-0.02 5s_{1 1}p

-0.03 ^\

> -0.04

@ Q) ^-0.05

§ _-0.06

5s9p 5s1 Op

�

-0.07

-0.08 _\-.-- L.----I---

a .s O.995

� �

-0.09 6, .. - 0.001 rads.

"rIg

-0.1

1 .95 1 .975 2 2.025 2.05 2.075 2.1 2.125 2.15 2.1 75 2.2

Re (E) eV

Fig. I -Theta trajectories from second order dilated (11 _{= a} e;9 ) electron propagator (l:2) calculations on 20g N2- shape resonance using different CGTO bases. The trajectories start on the real line (9 ⁼ 0.0) and 9 increments are in steps (9;,,) of 0.00 1 radians

renormalised diagonal 2ph-TDA (l:2ph.TDA) decouplings manifests itselfthrough the differences between the mix­

ing coefficients Cni from these approximations. For any chosen decoupling, the Cni vary with the change in the primitive CGTO basis and, therefore, differ­

ences between Xn( t' ) from different bases may be employed to gauge the impact of these variations.

III . Results and Discussion

The 2ng N2' shape resonance in scattering of elec­

trons off N2 has been investigated using many differ­

ent theoretical and experimental methods7,12,15,18,3.1.43 and is perhaps the most studied molecular resonance, The 2ng N2' shape resonance is, therefore, a natural choice for our investigation of basis set effects in

"-

M -0

&

�

til "-

N &

>=: ^e;)

x

�

<{ oS>

0 ^,

LL "-

.-.-

&

& _,

�

& _,

>=: ffi _"-

. oS>

X .&

<{ ^.-".

0 ^N

LL ^& 0

g: N r- oS>

<Sl X _M ^, M ^{... m}

<{ &

0 0

LL ^,

"- m M

�

0 _I

L2 (4s9p) FDA at 9 = 0.0 _L2 (5s7p) FDA at 9 = 0.0

a) b)

Xl ..-

<Si

>=: N � _oS>

X . ^&

M ^;::..

<{ ^..-

0 ^&

11- e;) _,

M M

�

oS>

-0 ,

a-N

- . = = - 7 - .... . =-- <Sl ... _ � &iia

>< , ^- E!!Oo. --

Amplitude difference at e ⁼ 0.0 Amplitude difference at 9 ⁼ 90pt

><

Fig. 2 -

... '"

>=: _. ^{<Sl �} X ^.,;

<{ ^... M ^'"

0 ^oS>

LL 0

m _'"

>=: ^&

. oS>

x ^,

M <{ ^<Sl �

0 ^oS>

LL 0

Q) M 0::: ^... �

.,; - . = -

, - 7 . - �

><

Resonant Feynman Dyson Amplitude (FDA) from the . second order dilated electron propagator calculations on the 2n. N2- shape resonance using (a) 45917 and (b) 55717 atom centred CGTO bases. The difference between the resonant FDAs from these two bases at e = 0.0 and between the real parts at e = e"PI are plotted in 2c and 2d respectively.

characterisation ofmolecuiar resonances. This resonance has been studied earlier 'Jsing a 4s9p atom centred CGTO basis and second order complex scaled electron propa­

gator based on real SCF by Donnelly34 and using a Ss7p CGTO basis with different self energy approximants as optical potential in scattering calculations by Meyer7.

We too had utilized the 4s9p CGTO basis in an earlier investigationl5 of this resonance.

For a comparative assessment of basis set effects, we have, therefore, utilized both the 4s9p and the Ss7p primi­

tive CGTO bases to begin with and theta trajectories fer the resonant pole from the second order biorthogonal

}:: -

x LO _<X:

0 LL

MAHALAKSHMI ^et aL. : CHARACTERISATION OF ELECTRON-MOLECULE SCATTERING RESONANCES 27

r? (5s9p) ^{FDA at 9 = 0.0} Amplitude difference at 9 = 0.0

a)

}:: ^lJ)

'" ...

N M

Ln . M

M X <Sl

G M

<X: $

N M 0 -

� LL csi

csi ^"-

}:: � -

<Sl

.

CS> , X ^M

� ^{ill <X: <D M}

M 1:1 0 ^'"

G ""'" LL ,

...,..

'" ,

v '"

1Il

<S> , ^>-<

Amplitude difference at 9 = 9opt Amplitude difference at S ⁼ Sept

c) �

;:n }:: ^(S)

}:: v ..- ^csi

csi ^x

x - M _<X: ^{M '"}

0 � 0

LL I 1Il '"

V

}:: ^<Sl csi ,

x

LO ^{M N}

<X: �

0 <S>

LL ,

N Q) �

n::: t.n

M ^<X: ₀

�

LL

I "-

>-

�

x

LO .,. _"-

'"

<X: '"

0 '"

LL ,

� � ...::f.rTT7lrrnTTTTT7lrrnTTTrTT7"TTTrrr.rTT7"TTTTT7lrrnTTTr

!!{!l

� --

csi .,

Fig. 3 ^_ Resonant Feynman Dyson Amplitude (FDA) from the second order dilated electron propagator calculation using the 5s9p basis (al and difference between the resonant FDAs from the 5s9p and 5s7p bases at ^{B =} 0.0 (b) and at ^{B = B"r'} between the real parts

dilated electron propagator calculations using these two bases are plotted in Fig 1 a. The resonance energy (real part) and the width (twice the imaginary part) from the quasi-stable portion of these theta trajectories are col­

lected in Table ^{1 .} The resonance energy (2. 1 1 eV) cal­

culated using 4s9p basis is quite close to the experimen­

tal value (2.20 e V) but that obtained using the 5s7 p ba­

sis (2.56 e V) needs improvement. Both these bases are devoid of any d-type functions and we began our im­

provement by first checking for the impact of

d

functions by redoing the ^1.2 calculations using 4s9p I

d

and 5s7p I d bases with an uncontracted d function (ex­

ponent ⁼ 1 .5) added to both the 4s9p and the 5s7p bases.

As can be seen from the results in theta trajectories in Fig. I a and the corresponding values in Table I , the ad­

ditional d function reduced the resonance energy by 0.05 eV and 0.07 eV respectively for the two bases and we conclude that the additional d-function does not have a major impact in these calculations. Also, while addition of

d

bination will not give LeAO-MOs of the n' symmetry required to describe the 2ng N2- shape resonance. The

0.00 -0.01 -0.02 -0.03

-0.04

>

@ Q) ^-0.05

..5

-0.06 -0.07 -0.08 -0.09 -0. 1 0

r,2 r,2ph-TDA r,q

� N�

r,0

�2ph-TDA

« = 0.995 91ne = 0.001 rads.

2. 1 2 2.1 3 2.1 4 2.1 5 2. 1 6 2.1 7 2.18

Re (E)eV

Fig. 4 ^- Theta trajectories from different decouplings of the dilated electron propagator using the 55 l ip basis and a = 0.995 (11 = a ei9 ).

additional d-function will, however, add considerably to the computational effort and, therefore, we did not experiment any further with additional d-functions.

To understand the large difference in resonance en­

ergy obtained from the 4s9p (2. 1 1 e V) and the 5s7 p bases (2.56 e V) we have plotted the resonant Feynman-Dyson amplitudes from these two bases and the difference be­

tween the corresponding amplitudes both for 9 ⁼ 0.0 (on the real line) and at 9 ⁼ 90PI (9 value corresponding to the inflection in 9 trajectories of Fig. I a) in Fig 2. As can be seen from Figs 2a and 2b the resonant FDA from both the bases are indeed of n* symmetry and the elec­

tron attachment leading to the 2ITg N2- shape resonance is into the N2 LUMO. The reason for the higher energy of the resonant pole from the 5s7p basis becomes clear from Figs 2c and 2d where the amplitude differences show that the FDA from 5s7p basis accumulates elec­

tron density close to the nuclei resulting in an increase in interelectronic repulsion and, therefore, to a higher value for the resonance energy . .

These initial results show that the 4s9p CGTO basis delivers better agreement with the experimental reso-

nance energy compared to the Ss7 p basis which has two less p functions. The Ss7p results may therefore be improved by adding the two uncontracted p-type GTOs with smallest exponents i.e. the two most diffuse p functions in the 4s9p basis to the 5s7p basis to obtain a 5s9p basis. The theta trajectories from the r,2 calcula­

tions using this 5s9p basis is presented in Fig. 1 b. The resonance energy (Table 1 ) obtained using the 5s9p ba­

sis reduced drastically to 1 .98 e V and underscores the importance of having diffuse p functions in the primi­

tive basis.

The resonant FDA obtained using the 5s9p basis is plotted in Fig 3a and the diffuse anionic character of this FDA clearly establishes this as portraying a meta­

stable anionic complex for a centrosymmetric system like N2 with its marked diffuseness being instrumental in the lowering of interelectronic repulsion and thereby the resonance energy. The nodal topology remains that of a n* orbital and, therefore, the electron attachment in e-N2 scattering as portrayed by this FDA too is clearly to a n* LUMO of N2 and the dilated electron propagator may, therefore, be effectively employed for consistent

MAHALAKSHMI et al. : CHARACTERISATION OF ELECTRON-MOLECULE SCATTERING RESONANCES 29

5s1 1p basis

:5 o L M • 0-

N �

K < IS> . o

u. , i€ M

:c � , ^� i

� ,,;

K <

G: _{, ,,;}

�

, �

" · L ISl

0 9

"

'"

0 ₀ "

�

· ·

K ^ISl

<

G: � _c;;

':'

Amplitude difference at e ⁼ 0.0

Amplitude difference at 9 ⁼ 0.0

- . -- ^{- �} - - .

Amplitude difference at 9 ⁼ 90pt

Amplitude difference at 9 ⁼ 90pt

c) :( 8 ,

I

� " ..,

� i ₀₎

,

I

" . ₀ , .. ISl

0

I

.

" 0)

� � ^�

� .=;'f

r: c;;

-- 0)

Fig. 5 -Difference between resonant FDAs from the second order and the zeroth order decoupling at e = 0.0 (a), between the second order and the zeroth order at e = _e"p' (b), second order and 2ph-TDA at e ⁼ 0.0 (c) and second order and 2ph-TDA at e ⁼ e"Pt (d) and unequivocal identification of LUMOs as argued

earlierl? The amplitude differences between the reso­

nant FDAs from the 5s9p basis (FDA5) and the 5s7p basis (FDA3) on the real line (8 ⁼ 0.0) and at 8 ⁼ 80PI are plotted in Figs 3b-3d and demonstrate that the reso­

nance energy value of 2.56 eV from 5s7p basis is low­

ered to 1 .96 e V for the 5s9p basis by providing for more electron amplitude both near to and away from the nuclei by the FDA5. The 5s9p basis, therefore, of­

fers a more balanced description of both the formation and decay of the 2ITgN2- resonance.

Basis set saturation was attempted by generating 5s1 0p and 5s1 1p CGTO bases from the 5s9p basis by adding additional uncontracted GTOs with exponents

(0.0032 and 0.00 1 28 respectively) maintaining the same ratio as that between the two most diffuse p functions from the 4s9p basis added to the 5s7p basis initially. The theta trajectories of the resonant roots from the 1:2 cal­

culations using these bases are plotted in Fig. I b and the corresponding resonance energy and width is again collected in Table 1 . These results give a value of 2.08 e V for the resonance energy from the 5s 1 Op basis and 2. 1 3 e V from the 5s I Ip basis which are in close proximity to the experimental value (2.20 e V). The width of the 2ITg N2-resonance from these two bases are 0. 1 7 e V and 0. 1 8 e V respectively. The energies and widths from 5sl Op and 5s1 1p bases being indistinguishable within the limits of experimental accuracy, no further

Table 2-Energy and width of the 2n N -_g 2 shape resonance from experiments and other theoretical approaches.

Basis Set Energy (eV) Width (eV)

Experiment15•36 2.20 0.S7

Static exchange17 _{3.70 1.1 6}

Static exchange R-matrix1R 2. IS 0.34

Stabilization method39 2^.44 0.32

R-matrix40 _3.26 0.80

Stietljes imaging technique41 4.13 1. 1 4

Many-body optical potentiaF·42 3.80 1 .23

Complex SCF 43 3.19 0.44

Second order dilated electron34 2.14 0.26 Propagator (real SCF)

Present calculations (Ssl lp basis)

Zeroth order 2.IS 0. 1 8

Second order 2^.13 0.18

Quasi-particle second order 2. I S 0. 1 8

Diagonal 2ph-TDA 2. 1 4 0. 1 7

Quasi-particle diagonal 2ph-TDA 2. I S 0.18

attempt at basis set saturation was attempted except that the lone d-function used earlier to check the effect of p ­ function polarization was added to obtain the Ss I Ip l d primitive basis and that this leads to only a marginal change of 0.03 eV in energy and width led us to believe that further basis set changes are unlikely to provide more meaningful results.

To understand the role of correlation and relax­

ation in the formation and decay of resonances, we used the final Ssl lp basis to calculate resonance attributes using the LD, L2, L2ph-TDA , L� and Lq2Ph-TDA decouplings as well. The theta trajectories for the resonant pole from all these decouplings are plotted in Fig. 4 and dif­

ferences between FDAs from these decouplings are por­

trayed in Fig. S. The results from Fig. 4 show the same trend seen in dilated electron propagator calculations on other systems l2 as wel l , i . e . , the quasi-particle decouplings devoid of orbital relaxation provide results which are almost identical to the uncorrelated zeroth order (LO) results. The L2ph-TDA results are closer to the LO results and both L2ph-TDA and L2 resonance energies are lower than those from the quasi-particle (L 2 q and Lq2rb-IDA ) and zeroth order decouplings. The orbital re- laxation even in case of this saturated basis does lower resonance energy as expected, with the L2 decoupling once again providing the lowest resonance energy. The width from all the decouplings is almost identical and much lower compared to the experimental value indi-

cating that the broad 20g N2- resonance may require the presence of other resonant roots to describe other parts of the resonant wavepacket48• The resonant root determined by our calculation perhaps describes only a narrow part of the resonant wavepacket and presence of other roots within the range 2.S0 (energy) ±0.S7* 112 (halfwidths) eV will offer superior characterization of the resonant wavepacket. Instead of mere basis set satu­

ration, a different approach to basis set design may there­

fore be required for proper description of wide reso­

nances like the 20g N2- investigated here. This view seems to receive some affirmation from Figs Sb and Sc where the L2 decoupling in comparison to the L2ph-TDA decoupling is seen to build greater electron amplitude both nearer to and away from the nitrogen nuclei. The balanced description of resonances requires amplitude close to the nucleus for electron attachment and away from the nuclei to facilitate decay and our results reaf­

firm the need to approach basis set design for resonances so that the contrary demands for both the formation and decay are incorporated adequately.

Also, as can be seen from the trajectories in Fig .4 and the corresponding values for the width and reso­

nance energy in Table 2, with a sufficiently saturated basis which provides an apt description of the diffuse metastable anionic resonant complex, there is only a marginal difference between the results from different decouplings and all of these provide excellent agreement with the experimental result for the energy but as seen in other calculations earlierl2 the widths are much narrower. It has been argued elsewherel2 that this may be due to underestimation of interelectronic re­

pulsion in the second and pseudo-second order decouplings employed in the dilated electron propaga­

tor calculation so far and its rectification may require third and fourth order decouplings which remain to be implemented. Our results indicate that presence of other roots within the 2.S0 ± 1 /2(0.S7)eV band may also be required for a more accurate description.

IV. Concluding Remarks

The dilated electron propagator calculation is in­

herently demanding and from the modest investigation of basis set and correlation/relaxation effects undertaken in this paper only a few definitive conclusions are pos­

sible. First and foremost is of course the need to saturate the basis sets. The basis set effects can be considerably larger than the correlation/relaxation effects permitted by the lower order decouplings employed in this inves­

tigation. The FDAs establish that the resonance forma-

MAHALAKSHMI et af. : CHARACTERISATION OF ELECTRON-MOLECULE SCATTERING RESONANCES 3 1

tion i s clearly due to electron attachment into the N2 1t*

LUMO leading to a diffuse anionic complex and only the primitive bases with enough flexibility to describe electron amplitude both close to and away from the nucleus can offer appropriate description of these meta­

stable resonances. For the final 5s1 1p basis, the results for resonance energy from all decouplings are very close to the experimental value; however, the widths are much narrower than the experimental results. Though the prox­

imity of the calculated results to the experimental value for the resonance energy has been taken as the criterion for basis set adequacy, we should mention that not only electron propagator method is perturbation theoretic and there is no variational bound, the states in question are those in the continuum and apart from energetics, an appropriate description of the asymptotic form of these metastable anionic states embedded in the continuum is a desirable requirement for the integrity of resonance calculations. Only the resonant FDAs from saturated bases seem to satisfy both these demands.The dominant role of interelectronic correlation is clearly shown by the FDAs plotted here since diffuse FDAs lower the resonance energy by as much as 0.6 e V. The unreason­

ably narrow widths underscore the need for investigat­

ing the role of third and fourth order decouplings. An effort along these lines is underway in our group.

Acknowledgement

We are grateful to the Department of Science and Technology, India (grant No. SP/S IIH26/96) for finan­

cial support.

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