Characterization of molecular shape resonances using different
decouplings of the dilated electron propagator with application to 2ΠCO−
and 2B2gC2H4− shape resonances
Milan N. Medikeri and Manoj K. Mishra
Citation: J. Chem. Phys. 103, 676 (1995); doi: 10.1063/1.470101
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Characterization of molecular shape resonances using different
decouplings of the dilated electron propagator with application
to 2 P CO2 and 2 B 2 g C2H2
4 shape resonances
Milan N. Medikeria) and Manoj K. Mishra
Department of Chemistry, Indian Institute of Technology, Powai, Bombay 400 076, India
~Received 11 October 1994; accepted 29 March 1995!
The zeroth ~S0!, second order ~S2!, quasiparticle second order (S 2q ), diagonal two-particle one-hole
Tamm Dancoff approximation ~S2ph-TDA! and the quasiparticle diagonal 2ph-TDA (S 2ph-TDA
)
q
decouplings have been applied to investigate the 2 P CO2 and 2 B 2g C2H2
4 shape resonances. An
examination of the resonant roots and the corresponding Feynman Dyson amplitudes ~FDAs!
reveals that the most economic and effective description is offered by the second order decoupling.
The more demanding diagonal two-particle one-hole Tamm Dancoff approximation ~2ph-TDA! is
shown to be less effective and the quasiparticle decouplings are shown to be no better than the
zeroth order ~bivariational self-consistent field! approximation in the description of molecular shape
resonances. The correlation and relaxation effects incorporated by the S2 and S2ph-TDA decouplings
are shown to assist resonance formation by lowering the antibonding nature of the lowest
unoccupied molecular orbitals ~LUMOs! on the real line and by turning these into anionic diffuse
orbitals suitable for metastable electron attachment for the optimal value of the complex scaling
parameter. The use of complex resonance energies calculated here to construct a nonempirical
optical potential for the investigation of vibrational dynamics of these resonances is
suggested. © 1995 American Institute of Physics.
I. INTRODUCTION
The effectiveness of the one electron propagator
theory1,2 in the treatment of electronic structure and dynamics is well established. The methodology and applications of
the real one electron propagator theory have been reviewed
by many authors3–10 and newer developments11–16 continue
to extend the scope of utility of this extensively applied technique. The method of complex scaled17 electron
propagator18 –23 where all the electronic coordinates in the
Hamiltonian have been scaled by a complex scale factor
( h 5 a e i u ) has been quite effective in the treatment of
atomic19,22,24 –31 and molecular shape resonances.32–36 The
decouplings utilizing biorthogonal orbital bases generated by
a complex, bivariational self-consistent field ~SCF!
procedure37–39 take full cognizance of the non-Hermiticity of
the complex scaled Hamiltonians and also preserve the
simple formal structure of the well established real electron
propagator method.1–9,18,23,28 In these schemes based on the
bivariational SCF, the orbital energies and amplitudes are the
zeroth order poles and Feynman–Dyson amplitudes ~FDAs!
of the biorthogonal dilated electron propagator18,23,28 and for
many systems, shape resonances have been uncovered at the
level of the bivariational SCF itself,26,28,29,35,36 permitting a
detailed analysis of the correlation effects in the formation
and decay of resonances.
The biorthogonal dilated electron propagator has, however, been applied mostly to atomic resonances.25–31 It has
been extended to permit its use in the investigation of molecular resonances only recently and a preliminary investigation of the 2 P g shape resonance in e-N2 scattering using the
a!
CSIR Senior Research Fellow.
676
J. Chem. Phys. 103 (2), 8 July 1995
zeroth and the second order decouplings has provided interesting mechanistic insights.35 These promising results, however, need to be substantiated through further applications of
the biorthogonal dilated electron propagator method to other
well characterized molecular resonances. Also, the preliminary application has utilized only the zeroth and the second
order decouplings.35 Attachment of an additional electron is
bound to relax all orbitals of the target molecule and also
effect changes in their correlated motion. The renormalized
decouplings such as the diagonal 2ph-TDA ~Refs. 5, 7, 9, 40!
are expected to incorporate greater extent of correlation and
relaxation and the quasiparticle approximations30,41– 43 offer
substantial computational economy. Investigation of these
decouplings in the treatment of molecular resonances is
therefore useful in extending the scope of utility of the molecular biorthogonal dilated electron propagator. Additional
applications should also serve as a probe for the crosssystemic validity of the mechanistic picture of shape resonance formation and decay and a comparison of the effectiveness of the different decouplings in the characterization
of molecular resonances is an automatic adjunct of such an
investigation.
In this paper we apply the zeroth, second order, diagonal
2ph-TDA and quasiparticle decouplings to investigate the
prototypical 2 P CO2 and 2 B 2g C2H2
4 molecular shape resonances. The poles and FDAs from different decouplings are
examined individually and contrasted with each other to
elicit the role of correlation and relaxation in the formation
and decay of molecular resonances. The formal and computational methodology for the construction and pole search of
the biorthogonal dilated electron propagator and the framework for its interpretation is well documented18 –20,27 and we
only offer some final formulas in Sec. II. The resonant poles
0021-9606/95/103(2)/676/7/$6.00
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M. N. Medikeri and M. K. Mishra: Molecular shape resonances
and FDAs for CO and C2H4 are presented and discussed in
Sec. III and a summary of the main results concludes this
paper.
S 2i j ~ h ,E ! 5
1
2
677
^ ik i lm &^ lm i jk &
( N klm ~ E1 e k 2 e l 2 e m ! ,
k,l,m
~8!
where
N klm 5 ^ n k & 2 ^ n k &^ n l & 2 ^ n k &^ n m & 1 ^ n l &^ n m &
II. METHOD
The Dyson equation for the biorthogonal matrix electron
propagator G( h ,E) may be expressed as23
G21 ~ h ,E ! 5G21
0 ~ h ,E ! 2S ~ h ,E ! ,
~1!
where G0( h ,E) is the zeroth order propagator for the uncorrelated electron motion, here chosen as given by the bivariational SCF approximation.37–39 The self-energy matrix
S( h ,E) contains the relaxation and correlation effects.1,2 Solution of the bivariational SCF equations for the N-electron
ground state yields a set of occupied and unoccupied spin
orbitals. In terms of these spin orbitals the matrix elements of
G21
0 ( h ,E) are
@ G21
0 ~ h ,E !# i j 5 ~ E2 e i ! d i j ,
G21 ~ h ,E ! 5E12L~ h ,E ! ; L5 e 1S
~3!
and the poles of the dilated electron propagator are the energy dependent eigenvalues of L( h ,E) given by
L~ h ,E ! x n ~ h ,E ! 5E n ~ h ,E ! x n ~ h ,E ! ,
~4!
which satisfy the relation
E n ~ h ,E ! 5E
~5!
and may be obtained by iterative diagonalization19 of L.
The Feynman–Dyson amplitude x n corresponding to the
nth N11 electron state is given by
x n5
E
* ~ 1,2,3,...,N,N11,n !
C N11
n
3C N0 ~ 1,2,3,...,N ! d ~ 1 ! d ~ 2 ! d ~ 3 ! •••d ~ N ! ,
~6!
where C N0 is the optimal single determinantal description of
the N-electron target based on the bivariationally determined
is the nth N11 state generated
SCF orbitals $ c i % and C N11
n
by the creation operator manifold $a i `, a i `a j `a k z;
j.i%.23,28 In the bivariationally obtained biorthogonal orbital
basis $ c i % , x n is a linear combination44
x n ~ r! 5
(i C ni c i~ r! ,
with ^ n k & being the occupation number for the kth spin orbital and the antisymmetric two-electron integral,
^ i j i kl & 5 h 21
~7!
where the mixing of the canonical orbitals allows for the
incorporation of correlation and relaxation effects. In the zeroth ~S50! and quasiparticle approximations ~diagonal S!,
there is no mixing. The difference between perturbative second order ~S2! or renormalized diagonal 2ph-TDA ~S2ph-TDA!
decouplings manifests itself through differences between the
mixing coefficients C ni from these approximations.
Through the second order in electron interaction, the elements of the self-energy matrix are
Ec
i~ 1 ! c j~ 2 !
12 P 12
r 12
3 c k ~ 1 ! c l ~ 2 ! dx 1 dx 2 ; h 5 a e i u .
~10!
The lack of complex conjugation stems from the biorthogonal set of orbitals resulting from bivariational SCF being the
complex conjugate of each other.37 For the diagonal 2phTDA decoupling40 of the dilated electron propagator28
S 2ph-TDA
~ h ,E ! 5
ij
1
2
^ ik i lm &^ lm i jk &
( N klm ~ E1 e k 2 e l 2 e m ! 2D ,
k,l,m
~11!
~2!
where e i , the orbital energy of the ith orbital, is obviously a
zeroth-order pole of the dilated electron propagator. The corresponding orbital c i is the zeroth order FDA. The propagator Eq. ~1! may be recast as
~9!
where
D5 21 ^ ml i ml & ~ 12 ^ n m & 2 ^ n l & ! 2 ^ km i km & ~ ^ n k &
2 ^ n m & ! 2 ^ kl i kl & ~ ^ n k & 2 ^ n l & ! .
~12!
The quasiparticle (S q ) approximation30 for dilated electron propagator results from a diagonal approximation to the
self-energy matrix S( h ,E) with poles of the dilated electron
propagator being given by
E ~ h ! 5 e i 1S ii ~ h ,E !
~13!
which is solved iteratively beginning with E5 e i and S ii may
correspond to any perturbative or renormalized decoupling
such as the second order and the diagonal 2ph-TDA approximations mentioned earlier.
The basic computational difference between the atomic
and the molecular dilated electron propagator stems from the
nondilatation-analyticity of the Born–Oppenheimer Hamiltonian when only the electronic coordinates are subjected to
complex scaling. A computationally tractable solution is offered by the Moiseyev–Corcoran method of treating the
nuclear attraction integrals for the dilated Hamiltonian.45
This approach has been employed profitably32–36 and its
equivalence with other methods for the treatment of molecular resonances has been established earlier.45– 48 We follow
the same modified procedure for the calculation of the complex three center nuclear attraction integrals in the construction of the molecular biorthogonal dilated electron propagator.
III. RESULTS AND DISCUSSION
Characterization of resonances is particularly sensitive to
the choice of the underlying basis set. In all such applications, the basis sets are chosen such that there is at least one
virtual orbital energy of appropriate symmetry in the vicinity
of the experimental resonance energy. The propagator root
corresponding to this positive orbital energy under complex
scaling by appropriate amount ~uopt! is expected to unmask
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678
M. N. Medikeri and M. K. Mishra: Molecular shape resonances
the discrete complex pole corresponding to the shape resonance. u and a trajectories ( h 5 a e i u ) for these roots are next
plotted by studying their response to variations in a and u.
The roots which meet the stability criterion ( ]e i / ]h )50 are
classified as resonances.19,24 –36
Even though the energies and the widths calculated by us
will be shown to be reasonable, it is not our purpose to
discuss the energetics of resonance formation in any detail.
We employ the same primitive bases as that of Donnelly33,34
but different decouplings of the complex scaled electron
propagator. The results obtained by us will be seen to be
close to those of Donnelly who has compared these values to
the energies and widths obtained by other methods. The underlying zeroth order decoupling being real, the Donnelly–
Simons approach19 is not suitable for isolation of resonances
at the level of SCF itself and the correlation and relaxation
effects in resonance formation therefore cannot be apportioned conveniently. A comparative investigation of the different decouplings allows a probe of the role of correlation
and relaxation and an examination of the chemical content of
the resonant FDAs is useful for eliciting mechanistic clues. It
is these two concerns which will be the focus of our efforts
and in the following subsections we examine the resonant
roots and the corresponding FDAs to investigate the mechanism of shape resonance formation in e-CO and e-C2H4 scattering.
A. CO
For CO we have utilized the C(4s,5p) and O(4s,5 p)
CGTO basis used in an earlier study of the 2 P CO2
resonance.33 The resonant u-trajectories from different decouplings are displayed in Figs. 1~a! and 1~b! and we note
that optimal value of the radial scale factor a is different for
the methods with ~S2 and S2ph-TDA! and without
(S 0 ,S 2q ,S 2ph-TDA
) orbital relaxation. Table I has the values
q
for the resonant energy and the width for this resonance obtained by the different decouplings along with those from
other methods as well as experiments. The radial scale factors required to uncover the 2 P CO2 resonance using the
methods without orbital relaxation (S 0 ,S 2q ,S 2ph-TDA
) and
q
those with orbital relaxation and correlation ~S2 and
S2ph-TDA! are different but the energies and the widths computed by all these decouplings are close to each other ~Table
I!. The extreme sensitivity to a-values seen in Figs. 1~a! and
1~b! denotes the delicate nature of the resonance characterization and its acute dependence on both the primitive and
the orbital bases. It is somewhat reassuring that though the
resonant pole is approached from opposite sides by the S0,
S 2q , S 2ph-TDA
decouplings @Fig. 1~a!# and the S2 and S2ph-TDA
q
decouplings @Fig. 1~b!#, these become resonant at almost the
same point in the complex E-plane. The width of the resonance calculated is however much narrower than the values
obtained by other methods.
The resonant FDA from the second order and other decouplings on the real line once again are the familiar p*
LUMO for CO with greater amplitude on the C atom and the
resonant FDA from the S2 decoupling is displayed in Fig.
2~a!. In Fig. 2~b! we have plotted the difference between
FIG. 1. Resonant trajectories from different decouplings for CO. The radial
scale factor required to unmask the resonance using the S0, S 2q , and
S 2ph-TDA
decouplings ~a! is different from that at the S2 and S2ph-TDA levels,
q
~b!. All values in the trajectories of ~a! are offset by 1.71 eV, while those in
~b! are offset by 1.68 eV—the resonance energies from these calculations.
FDAs from S2 and S0 decouplings which shows that the
major effect of the S2 correction to the SCF p* LUMO of
CO is the diminution in its antibonding nature through shifting of electron amplitude to the more electronegative O
atom. This lowers the LUMO energy level bringing it closer
to the HOMO orbital energy. The greater effectiveness of S2
vis-à-vis S2ph-TDA decoupling is indicated by Fig. 2~c!, where
TABLE I. Energy and width of the 2 P CO2 shape resonance.
Method/reference
a
Experiment
Boomerang modelb
T matrix methodc
Close coupling methodd
Second order dilated electron propagator
~real SCF!e
Zeroth order, quasiparticle second order,
and quasiparticle diagonal 2ph-TDA
biorthogonal dilated electron propagatorf
Second order biorthogonal dilated
electron propagatorf
Diagonal 2ph-TDA biorthogonal dilated
electron propagatorf
Energy ~eV!
Width ~eV!
1.50
1.52
3.40
1.75
1.71
0.40
0.8
1.65
0.28
0.08
1.71
0.10
1.68
0.09
1.69
0.08
a
Reference 59.
Reference 60.
c
Reference 61.
d
Reference 51.
e
Reference 33.
f
This work.
b
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M. N. Medikeri and M. K. Mishra: Molecular shape resonances
679
FIG. 2. ~a! Resonant FDA from the S2 decoupling for the CO on the real line ~u50.0!. ~b! Difference between the resonant FDAs from the S2 and the S0
decouplings and ~c! between S2 and S2ph-TDA decouplings at u50.0. The diminution in the antibonding nature of the p* LUMO through shifting of the orbital
amplitude towards the O atom by the correlated decouplings is seen. The difference between the real part of the resonant FDA from the S2 decoupling at
u5uopt and the same FDA on the real line is displayed in ~d!. The synergy between the role of complex scaling and correlation/relaxation effects in diminishing
the HOMO-LUMO gap by shifting the amplitude towards the O atom is made obvious by ~b!, ~c!, and ~d!.
the second order approximation is seen to pull an additional
amount ~albeit small in magnitude! of electron amplitude on
both C and O where the extra buildup on the O atom is
somewhat larger and more compact.
The role of complex scaling is explored in Fig. 2~d!,
where we have plotted the difference between the real part of
the resonant FDA at u5uopt and the same FDA on the real
line ~u50.0!. It is clear from Fig. 2~d! that the role of complex scaling is to assist in diminishing the HOMO-LUMO
gap by transferring of probability amplitude towards the O
atom. Though the imaginary part of the FDAs from both the
S2 and S2ph-TDA decouplings is much smaller and is therefore
not depicted here, the imaginary part of the resonant FDAs
from both these decouplings at optimal u has larger amplitude on the O atom. This coupled with the difference amplitude profiles of Figs. 2~b!–2~d! seem to indicate a synergy
between the effects of complex scaling and corrections induced by the correlated decouplings ~S2 and S2ph-TDA!, with
both reinforcing each other to suitably modify the antibonding SCF LUMO for metastable attachment of an additional
electron.
B. C2H4
The calculations on C2H4 were done using a (5s,7p)
CGTO basis centered on the C atoms and 2 s-type Gaussians
centered on the H-atoms.34 The resonant u-trajectories from
different decouplings are plotted in Fig. 3. The energy and
the width for this resonance from different decouplings and
those obtained by experiment and other methods are collected in Table II. That there is a cross-systemic validity to
our explanations is made clear by Fig. 3 where just like in
the case of CO, the decouplings devoid of orbital relaxation
(S 0 ,S 2q ,S 2ph-TDA
) clump together and once again maximum
q
lowering of the HOMO-LUMO gap is offered by the second
order self-energy approximant.
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680
M. N. Medikeri and M. K. Mishra: Molecular shape resonances
here it is responsible for accumulation of electron amplitude
in the internuclear region. The complex scaling and correlation effects again seem to act in tandem to turn an antibonding LUMO for u50.0 into a diffuse anionic orbital for
u5uopt .
IV. CONCLUDING REMARKS
FIG. 3. Resonant trajectories from different decouplings for C2H4 . The lowering of LUMO energy by the decouplings incorporating orbital relaxation
~S2 and S2ph-TDA! is seen for this system as well.
In Fig. 4~a! we have plotted the resonant FDA from the
S2 decoupling on the real line. This displays the familiar
nodal pattern of the p* LUMO of the ethylene molecule.
Those from the S0, and S2ph-TDA have identical features and
we explore the role of correlation by plotting the difference
between resonant FDAs from the S2 and S0 decouplings in
Fig. 4~b!. In the case of C2H4 the reduction of antibonding
nature of the p* LUMO through depletion of small amounts
of probability amplitude away from the C–H s bond region
and its accumulation near the C–C bond seems to be the
major contribution from the correlated decouplings.
That the major role perhaps is that of relaxation is indicated by Fig. 4~c!, where we have plotted the real part of the
resonant FDA from the S2 decoupling for u5uopt . The most
striking feature is that the optimal value of the complex scaling parameter has turned it into a diffuse anionic orbital preparing it for the metastable electron attachment.
Differences in the description of the 2 B 2g C2H2
4 shape
resonance by the S2 and the S0 decouplings may be probed
by plotting the difference between the values of the resonant
FDAs from these decouplings. These results show that the
major effect of correlation and relaxation incorporated by the
S2 and S2ph-TDA decouplings is through greater diffusion of
both the real and imaginary parts of the resonant amplitude.
Though the imaginary part of the resonant FDA is two orders
of magnitude smaller than the real part and is not depicted
TABLE II. Energy and width of the 2 B 2g C2H2
4 shape resonance.
Method/reference
Experimenta
Complex Kohnb
Second order dilated electron propagator
~real SCF!c
Zeroth order, quasiparticle second order,
and quasiparticle diagonal 2ph-TDA
biorthogonal dilated electron propagatord
Second order biorthogonal dilated
electron propagatord
Diagonal 2ph-TDA biorthogonal dilated
electron propagatord
a
Reference 62.
Reference 63.
c
Reference 34.
d
This work.
b
Energy ~eV!
Width ~eV!
1.78
1.84
1.94
0.70
0.46
0.11
1.93
0.19
1.86
0.18
1.89
0.18
In this paper we have employed the zeroth, second order,
second order quasiparticle, diagonal 2ph-TDA and quasiparticle diagonal 2ph-TDA decouplings to investigate the relaxation and correlation effects in the formation of molecular
shape resonances. The quasiparticle decouplings seemed to
have resonant trajectories identical to that of their much
more demanding nondiagonal counterparts for atomic shape
resonances30 but this promise is not fulfilled for molecular
resonances. The orbital relaxation seems to be critical to
metastable molecular anion formation and the quasiparticle
approximations are no better than the uncorrelated zeroth
order ~bivariational SCF! decoupling. Maximum lowering of
the HOMO-LUMO gap through the lowering of the antibonding nature of the LUMO is offered by the second order
decoupling and the computationally more demanding diagonal 2ph-TDA decoupling does not seem to be worth the extra
effort involved in the computation of the denominator shift D
in Eq. ~11!. Optimal complex scaling is seen to turn the
compact antibonding LUMOs on the real line into anionic
diffuse orbitals preparing them for metastable electron attachment. That these trends persist for diverse systems like
the CO and C2H4 molecules generates faith in the ability of
the biorthogonal dilated electron propagator to unmask molecular shape resonances and to unfold descriptive insights
with cross-systematic validity.
Some limitations of this study need sharper focus and
the first and foremost is that while the calculated energies are
plausible, the calculated widths are much narrower than the
widths calculated by other methods for both the systems investigated here. The larger widths obtained from other methods have been contested50 as being due to the inadequacy of
the empirical optical potential. The widths calculated by us
like those from Donnelly’s alternative S2 decoupling however seem to be much narrower. This may be due to insufficiency of the primitive basis sets and/or the amount of correlation and relaxation incorporated by the decouplings
employed here. That the different trajectories coalesce or
come together from different directions near the same value
in the complex energy plane inspires some confidence in the
ability of the propagator decouplings to correctly characterize the energetics of resonance formation. The need for a
comprehensive study of the basis set effects and incorporation of the higher order decouplings like the third order43
~S3!, quasifourth order43,44 or balanced renormalized decouplings such as the algebraic diagrammatic construction
@ADC~3!# ~Refs. 9 and 13! to extend the utility of the molecular biorthogonal propagator is obvious. One of the
strengths of the biorthogonal dilated electron propagator is
that it preserves the formal structure of the real electron
propagator decouplings. The biorthogonal orbital basis preserves the symmetries of the two-electron integrals and different decouplings of the dilated electron propagator can be
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M. N. Medikeri and M. K. Mishra: Molecular shape resonances
681
FIG. 4. ~a! Resonant FDA from the S2 decoupling for the C2H4 molecule on
the real line ~u50.0!. ~b! Difference between the resonant FDAs from the S2
and the S0 decouplings for C2H4 on the real line. The real part of the diffuse
resonant FDA at u5uopt is plotted in ~c!.
implemented without too many modifications of the corresponding real electron propagator code. Incorporation of the
S3, S 4q or ADC~3! decouplings therefore should not be very
difficult and is a planned extension of this work.
Furthermore, our calculations are for the equilibrium internuclear distance of both systems. The single bond length
calculations presented here may be extended by allowing
bond-stretching and calculating E res5k 2res as a function of the
bond length R. One could then employ the Chandra
N
2
approximation51,52 E N11
res (R) . E 0 (R) 1 k res(R) in conjunction with semiclassical/quantal wave packet dynamics53
on this complex E N11
res (R) to unravel the vibrational dynamics attending electron attachment resonances without having
to employ empirical optical potentials. An effort along these
lines is underway in our group.
ACKNOWLEDGMENTS
This investigation has been sponsored by the Department of Science and Technology, India, through Grant No.
SP/S1/F60/88. Their support is gratefully acknowledged.
M.N.M. is grateful to the CSIR, India for a predoctoral fellowship @No. 9/87~160!/93 EMR-I#.
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1
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682
18
M. N. Medikeri and M. K. Mishra: Molecular shape resonances
A review of the different approaches to the construction of complex scaled
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J. Chem. Phys., Vol. 103, No. 2, 8 July 1995
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