The twoparticle one holeTammDancoff approximation (2phTDA)
decoupling of the dilated electron propagator with application to 2P shape
resonances in eBe, eMg, and eCa scattering
Milan N. Medikeri, Jayraman Nair, and Manoj K. Mishra
Citation: J. Chem. Phys. 99, 1869 (1993); doi: 10.1063/1.465304
View online: http://dx.doi.org/10.1063/1.465304
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The two-particle one hole-Tamm-Dancoff approximation {2ph-TDAJ
decoupling of the dilated electron propagator with application to P
shape resonances in e-Be, e-Mg, and e-Ca scattering
Milan N. Medikeri, Jayraman Nair, and Manoj K. Mishra
Department oj Chemistry, Indian Institute oj Technology, Powai, Bombay 400 076, India
(Received 10 February 1993; accepted 30 March 1993)
Formulas for the renormalized full two-particle one hole-Tamm-Dancoff approximation
(2ph-TDA) decoupling of the dilated biorthogonal electron propagator based on complex
scaled bivariational self-consistent-field (SCF) procedure are derived and the diagonal
2ph-TDA approximation is implemented for the first time. The 2p shape resonances in e-Be,
e-Mg, and e-Ca scattering are characterized using the diagonal 2ph-TDA, the second-order and
the zeroth-order (bivariational SCF) approximations to the dilated electron propagator. A
comparative investigation of these different decouplings reveals that although the resonance
energies and widths depend on the level of correlation employed, greater correlation need not
lead to sharper resonances.
I. INTRODUCTION
The electron propagator method 1 has been quite Successful in the calculation of electron detachment2 and attachment energies3 and has emerged as a powerful tool for
the correlated treatment of electronic structure. 4 The complex scaled electron propagator, S where electronic coordinates have been scaled by a complex scale factor has offered a convenient method· for the direct calculation of
energies and widths of shape resonances in electronatom5- 12 and electron-molecule 13 scattering. All the approaches to the construction of the dilated electron propagator have, however, been limited to a second~order
approximation to the self-energy. The contribution of
higher-order self-energy terms has been shown to be quite
important in the calculation of ionization energies, 14 and in
the calculation of resonance energies and widths using
complex scaled cr, and the importance of the level of correlation employed in the characterization of these resonances has been clearly demonstrated. 15 It is, therefore,
desirable to formulate and implement higher-order decouplings of the dilated electron propagator as well, since the
extent of improvement in the calculation of valence ionization energies offered by the higher self-energy
approximations 2a is generally of the same magnitude as the
resonance energies themselves.
Our own biorthogonal approach to the dilated electron
propagatorS,8 is based on the underlying bivariational selfconsistent field (SCF) 16-19 to take account of the nonHermiticity of the dilated Hamiltonian. One of the advantages of our approach is the ease with which traditional
decouplings of the real unscaled electron propagator can be
formulated and implemented for the dilated electron propagator as well. Even our own implementation, however,
has so far been limited to only a second-order approximation of the self-energy. The renormalized decouplings like
the 2ph-TDA20,21 sum the more important ring and ladder
diagrams to all orders and in this paper we present the
formulas for the full 2ph-TDA decoupling of the dilated
J. Chern. Phys. 99 (3), 1 August 1993
electron propagator. Due to the extremely large dimension
of the 2ph operator manifold, in this first attempt, the
energies and the widths of the 2p shape resonances in e-Be,
e-Mg, and e-Ca scattering are calculated using only the
diagonal terms in the full 2ph-TDA decoupIing.
The formal considerations have been detailed
elsewhere2a.8 and only a skeletal outline and the final'theoretical expressions are offered in Sec. II. In Sec. III, we
discuss the numerical results, and some concluding remarks are collected in Sec. IV.
II. THEORY
The dilated atomic Hamiltonian
is non-Hermitian for complex values of 1] and the variational theorem does not apply. There exists a bivariational
theorem22 for non-Hermitian operators. The bivariational
SCF equations for the dilated Hamiltonians are derived by
using the extreme of the generalized functional
(2)
under the constraint that the solutions <1>0 and '1'0 be single
determinants and the constituent one-electron orbitals be
biorthonormal as follows:
<1>0= (N!) -1/2 det[tPl (XI)tP2(X2)" ·tPN(XN)],
'1'0= (N!) -1/2 det[ lPr (XI)lP2 (X2)" '1{;N(XN)],
(3)
<tPiltPj)=8ij .
The extreme of the functional in Eq. (2) results in the
following SCF equations:
ntPi=€itPj,
nttPi=€rtPi'
(4)
where
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1869
,.
Medikeri, Nair, and Mishra: 2p shape resonances in e-8e (Mg, Ca) scattering
1870
(5)
where the inner projection manifold h consists of elements
from the linear space of electron field operators. The
zeroth-order propagator results by choosing the inner projection manifold h as
(16)
with
in Eq. (15). Similarly, if h is chosen to be
oce
p=
L 1f,<pf .
h=hl E9 h 3 and h=hl E9 h 3
(6)
(17)
j
In temlS of the Ferolion-type creation and annihilation
operator manifolds h and Ii defined on orbital manifolds
{1fJ and {4>J, respectively, the operator spaces are
h={a i L,aj!\a j La k L, ... },
(7)
j<k,
with
and
(18)
and
(8)
Generally, the Grassmann algebra used to take care of
non-Heroliticity, 8 the aj!\ and a j L denote the exterior and
interior multiplications, being counterparts of the creation
and annihilation operators on the direct {1fi} and adjoint
{4>i} spaces. Using the scalar product
(9)
and the super-operator fOrolalism,23,24 the Fourier transform of the equation of motion of the dilated electron propagator may be written as 8
( a < (3 labeling occupied, p < q labeling unoccupied orbitals, and i,j labeling unspecified orbitals).
We can write the dilated electron propagator equation
compactly as
G- 1 ( 'T/,E) =EI-E( 'T/) -l:( 'T/,E) ,
where E is the diagonal matrix of the eigenvalues of the
one-electron Hamiltonian in Eq. (4). The self-energy matrix ~ is
l:( 'T/,E) =(aIH( 'T/)h3)(h31(Ei -H( 'T/»h3)-1
X (h31H( 'T/ )a).
(10)
where the super-operator HC'T/) is defined as HoC'T/)
+ V('T/) with
HOX= [X,Ho('T/)]_,
VX= [X,V('T/)]_,
(19)
(20)
Approximation of H by Ho in Eq. (20) results in the
second-order approximation to the dilated electron propagator 10,11 with
(11)
(21)
ix=x, \fXEh,
and
(12)
V('T/) =
L Lj Lk LI (4)'<Pjl\
N k1m = (nk) - (nk) (nl) - (nk) (n m) + (nl) (n m).
(22)
If H in Eq. (20) is retained in entirety, we obtain the
full 2ph-TDA decoupling with
1fk1fl)
j
X (~aj!\aj!\al Lak L -Djl(n/)aj!\a k L),
(13)
X {(h I(Ei -Ho( 'T/) )h)
(nl) being the occupation number for the lth orbital and
-(hi Vc'T/)h)}ki~,k'I'm,(l'm'l\
jk'), (23)
hklm=Nid;,;2[alaPm+Dkm(nk)az-Dk/(nk)a m ],
(24)
where
where g( 'T/) is the electron-electron interaction terol
['l]( l/r12)]' Equation (10) provides the starting point for
various approximations and different schemes for decoupiing of the dilated biorthogonal electron propagator. Using inner projection,25,26 the expression for the dilated electron propagator becomes8
(15)
(hk'/'m' I(Ei-Ho('l]»h kZm )
= (E+Ek-EI-Em)Dkk,DWDmm"
and
J. Chern. Phys., Vol. 99, No.3, 1 August 1993
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Medikeri. Nair. and Mishra: 2p shape resonances in e-Se (Mg. Cal scattering
1871
(hk'I'm' I(V( 1/) )h k1m )
=NJJ~;':!~,H(ml" m'I'){)kk'( 1- (n m) - (nl»
-0.1
Se(10sSpl
<1=0.75
Binc:: 0.01 rods
-(k'ml/ km'){)w«nk)-(nm»-(k'/I/ ki')
-0.2
(25)
X {)mm' ( (nk) - (nl»)}·
As a result of the large dimension of the {h k1m } operator
manifold, the fu1l2ph-TDA technique20,21 is computationally not very attractive and its diagonal approximation
which restricts the summation over the spin orbitals to
k=k', 1=1', and m=m' affords considerable computational savings. 2a Applying this approximation to Eq. (22),
the diagona12ph-TDA self-energy expression may be written as
_~
2ph-TDA
~ij
'"
(1/,E)-2 £."
~~m
(ikll Im){lmll jk)
Nk1m(E+€ _€_€ )-1:::..'
k
,
m
1:;
Q
E
~
-0.3
-0.
-O.s.+:r~,-r-;""""-;-.--f..,..,.~';':"-~c-ro-"-':""""-i--r-T-r-"-,-,,,,,,-;,::,,:,,,,-rr':,...j
0.&3
0.64
0.65
0.66
0.67
0.S8
Re(E)eV
0.69
0.70
0.71
0.72
0.73
FIG. 1. Theta trajectories for the e-Be 2p shape resonance from the
zeroth-order (bivariational SCF), second-order (l;2), and diagonal 2phTn:A (~2ph:TDA) decouplings of the dilated electron propagator.
(26)
III. RESULTS AND DISCUSSION
where
I:::..=Hmll/ ml) (1- (n m )
-
(nl» - (kml/ km)
X ((nk) - (n m » - (kill kl) ((nk) - (n,».
(27)
The results obtained from diagonal and full 2ph-TDA
decouplings for the valence electrons are almost indistinguishable27 and this is our preferred decoupling for the
present treatment of shape resonances. The propagator Eq.
(19) may be written as
(28)
and the poles of the dilated electron propagator are the
energy dependent eigenvalues of the equation
L( 1/,E)Xn( 1/,E) = ~ n( 1/,E)Xn( 1/,E) ,
(29)
which satisfy the relation
~n(1/,E)=E.
(30)
In terms of the eigenfunctions and eigenvalues of L, the
spectral representation of the dilated electron propagator is
G(1/,E) =
L
IXn><Xnl
n E- ~ n( TJ,E)
,
f
0.0
(31)
and the requisite poles of G are searched by iterative diagonalization of L as detailed elsewhere. 6
The fact that lfl (TJ) =H* ( TJ) suggests the assumption
cPo='l'(!' and the consequent association {l/IJ={tPH
whereby
(ikll Im)=TJ
The basis sets employed for calculations on Be, Mg,
and Ca 2p shape resonances are those from earlier work on
these systems (Refs. 16,9, and 28, respectively). The theta
trajectories for the zeroth- and second-order self-energy
approximants for the 2p shape resonances in e-Be and
e-Mg scattering have also been studied in detail elsewhere lO,12 and selection of resonant roots to be investigated
and the optimal a-values to be used for the diagonal 2phTDA calculations on these systems were made accordingly. Theta trajectories of the resonant root for these two
shape resonances using the diagonal 2ph-TDA self-energy
approximant, along with those from the lower-order approximations, are plotted in Figs. I and 2, respectively.
The resonance energy and the width from these plots and
other experimental and theoretical values are collected in
Tables I and II. The resonant root for the Ca 2p shape
resonance using all the three self-energy approximants has
been calculated for the first time. The theta trajectories
from the various self-energy approximants for Ca 2p shape
Mg (4s9pl
-0.02
einC;
0.01 rods.
-0:04
>~
§
.§
-0.06
I-P12
tPi(l)tPk(2)-r12
-0.08
(32)
and lack of complex conjugation should be noted. This
approximation preserves the usual symmetries of the twoelectron integrals Ukll 1m} and just as in the case of
second-order decoupling l 0- 12 all equations were solved
with this assumption.
-0.10
0.10
0.12
0.14
0.16
0.18
0.20
Re(EleV
FIG. 2. Theta trajectories for the e-Mg 2p shape resonance from the
second-order Cl;2) and diagonal 2ph-TDA C~2ph-TDA) decoupIings of the
dilated electron propagator.
J. Chern. Phys., Vol. 99, No.3, 1 August 1993
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Medikeri, Nair, and Mishra: 2p shape resonances in a-Be (Mg, Cal scattering
1872
0.0001r---~--------""'_-------'-"c--:_.:-lr------,
TABLE I. Energy and width ofthe 2p shape resonance in e-Be scattering.
Energy (eV)
Method/reference
Width (eV)
0.025
Static exchange phase shift"
Static exchange plus polarizability"
phase shift
Static exchange phase shiftb
Static exchange plus polarizabilityb
phase shift
Static exchange cross sectionb
Static exchange plus polarizabilityb
cross section
Complex ~SCF"
Singles and doubles complex Cld
Singles, doubles, and triples
complex Cld
Second -order dilated electron
propagator based on real SCF
Zeroth-order biorthogonal dilated
electron propagator (this work)
Second-order biorthogonal dilated
electron propagator (this work)
Diagonal 2ph-TDA biorthogonal
dilated electron propagator( this work)
"Reference 30.
bReference 33.
"Reference 31.
0.77
0.20
1.61
0.28
0.75
0.14
1.64
0.13
1.20
0.16
2.6
0.14
0.70
0.58
0.32
0.51
0.38
0.30
0.57
0.99
0.67
0.88
0.64
0.60
0.67
0.66
Call1s19p)
<1 =1.0
= 0.01 rods
9nc
'i:; 0.050
~2
.:
§
!3. 0.Q75
0.100
0.168
dReference 15.
"Reference 6.
resonance are plotted in Fig. 3. The results for resonance
energy and width of the 2p shape resonance in e-Ca scattering are collected in Table III.
Our results for energies and widths of the three systems are in good agreement with experimental and other
theoretical results. Even though the diagonal 2ph-TDA
sums diagonal ring and ladder diagrams to all orders, from
Table I it is obvious that unlike in another study15 where
increased correlation leads to lower values for resonance
energy and width, the main effect of correlation in our
0.170
0.172
0.171;
0.176
Re(E) in.V
FIG. 3. Theta trajectories for the e-Ca 2p shape resonance from the
zeroth-order (inset), second-order C~;2), and diagonal 2ph-TDA
C~:.2ph-TDA) decouplings of the dilated electron propagator.
study seems not so much on the resonance energies as on
the width. Greater correlation provides for higher level of
polarization and relaxation effects to be incorporated leading to a longer lifetime for the resonance but the resonance
energies remain close to values obtained from the lowerorder approximations. For all three systems, we find that
the results obtained from the second order and the diagonal2ph-TDA are more or less indistinguishable from each
other. There is, however, considerable difference in results
obtained from the uncorrelated zeroth-order (bivariational
SCF), and the correlated second-order and diagonal 2phTDA approximants. At the zeroth order there is no resonant root for Mg and the resonance energy and width for
the 2p shape resonance in e-Ca scattering at the zerothorder level is twice as large as that for second-order and
diagonal 2ph-TDA approximants.
TABLE II. Energy and width of the 2p shape resonance in e-Mg scattering.
Method/reference
Energy (eV)
Width (eV)
0.15
0.46
0.16
0.13
1.37
0.24
0.46
0.14
1.53
0.24
0.91
0.19
2.30
0.30
0.20
0.51
0.14
0.23
0.54
0.13
0.15
0.13
0.15
0.13
Experiment"
Static exchange phase shiftb
Static exchange plus polarizabilityb
phase shift
Static exchange phase shiftC
Static exchange plus polarizabilityc
phase shift
Static exchange cross sectionC
Static exchange plus polarizabiIity"
cross section
Cld
Complex ~SCF"
Dilated electron propagator based
on real SCF
Second-order biorthogonal dilated
electron propagator (this work)
Diagonal 2ph-TDA biorthogonal
dilated electron propagator (this work)
"Reference 32.
bReference 30.
"Reference 33.
dReference 36.
"Reference 28.
fReference 9.
TABLE III. Energy and width of the 2p shape resonance in e-Ca scattering.
Method/reference
Energy (eV)
Width (eV)
0.24
0.06
0.54
0.10
0.40
0.08
0.70
0.10
0.45
0.10
0.65
0.14
0.23
0.34
0.16
0.32
0.16
0.14
0.17
0.19
Static exchange phase shift"
Static exchange plus polarlzability"
phase shift
Static exchange cross section"
Static exchange plus polarizability"
Cross section
Static exchange total cross sectionb
Static exchange plus polarizabilityb
total cross section
Complex ~SCF"
Zeroth-order biorthogonal dilated
electron propagator (this work)
Second-order biorthogonal dilated
electron propagator (this work)
Diagonal 2ph-TDA biorthogonal
dilated electron propagator (this work)
"Reference 33.
bReference 34.
cReference 28.
J. Chern. Phys., Vol. 99, No.3, 1 August 1993
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Medikeri, Nair, and Mishra: 2p shape resonances in e-Be (Mg, Ca) scattering
1873
cay. An effort along these lines is underway in our group.
IV. CONCLUDING REMARKS
The basic purpose of this investigation is an extension
of the formal apparatus of the biorthogonal dilated electron propagator to enable calculations that may employ
the diagonal and full 2ph-TDA decouplings. This has allowed us to undertake a comparative investigation of the
effectiveness of the three self-energy approximants in the
characterization of shape resonances. Our results indicate
that the biorthogonal approach to the construction of the
dilated electron propagator does lead to the same formulas
as in the case of the unsealed real propagator and all the
approximations from the real electron propagator formalism may be implemented using the formal and computational strategies adopted earlier. 2,4 Relaxation effects seem
to be the most important in the formation and decay of
shape resonances, since there is no resonant root for Mg at
the bivariational SCF level and the values obtained for Be
and Ca from the zeroth-order approximation (bivariational SCF) are much larger than that from second-order
and diagonal2ph-TDA self-energy approximations. In the
complex aSCF calculations28 for the 2p shape resonance in
e-Ca scattering, the resonance energy and width have been
calculated from E(ea->*(Oopt) - Eea(O = 0), where the
asterisk denotes the resonant state. Our propagator calculations evaluate Ecea-)*(Oopt) - Eea(Oopt). Since the basis
set used for both the calculations is same and exhaustive,
the e dependence of Eea is expected to be negligibly small
as assumed in the aSCF calculation. 28 We have utilized the
same basis, and since aSCF provides full relaxation permitted by the basis set, the. difference in calculated resonance energy and width between aSCF values and those
from our own ~2 and ~2ph-TDA calculations must be due to
polarization effects. We may thus surmise that the polarization has considerable effect on the resonance energy but
has little impact on the width of the resonance. The results
from the more demanding diagonal 2ph-TDA approximation however, offer only a marginal improvement over
those obtained with the second-order approximation.
The use of a complex scaled electron propagator for
the treatment of molecular resonances13 has shown extreme sensitivity to even minor variations in the scaling
parameter, making the search for the resonant root much
more demanding. We have speculated earlier5 that this
may be due to the second-order self-energy approximant
being employed in the investigation. 13 Our results here,
however, show that not much improvement may be had by
employing the more demanding higher-order self-energy
approximants. This opens tempting avenues of economy by
adopting the quasiparticle approximants of the type shown
to provide excellent results for the calculation of ionization
energies and electron affinities. 29 Also, clear isolation of a
single virtual orbital of the p-type from the whole manifold
of all unoccupied orbital from the bivariational SCF calculations for the 2p shape resonances for both e- Be and
e-Ca scattering lends credence to the unoccupied orbital
based mechanistic picture of shape resonances 32,35 and an
examination of the corresponding orbital and FeynmanDyson amplitUdes for these roots should be useful in establishing the mechanism of resonance formation and de-
ACKNOWLEDGMENTS
This investigation has been sponsored by the Department of Science and Technology, India, through their
Grant No. SP/SI/F60/88 to MKM. Their support is
gratefully acknowledged.
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1874
Medikeri, Nair, and Mishra: 2p shape resonances in e-Be (Mg, Cal scattering
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