By using SCF and stabilization‐like procedures, we have located a (π, π*) singlet resonance‐like state in ethylene at 10.21 eV. This state is embedded in the ionization continuum and carries an oscillator strength of 0.46 and is probably the analog to the spectroscopic V state in Hartree–Fock theory. Implications of these results for other systems are discussed

Bender, C. F.

Davidson, Ernest R.

McKoy, Vincent

English

Caltech Authors - Main

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An SCF -stabilization approach to excited states embedded 
in the continuuma> 
C. F. Bender 
Th~oretical Atomic and Molecular Physics Group, Lawrence Livermore Laboratory, University of 
Califorma, Livermore, California 94550 
Vincent McKoy 
Arthur Amos Noyes Laboratory of Chemical Physics, b) California Institute of Technology Pasadena 
California 91125 ' ' 
Ernest R. Davidson 
Department of Chemistry, University of Washington, Seattle, Washington, D. C. 98195 
By using SCF and stabilization-like procedures, we have located a (1T, 1r*) singlet resonance-like state in 
ethylene at !0.21 eV. This state is embedded in the ionization continuum and carries an oscillator 
stren~th. of 0.46 and is probably the analog to the spectroscopic V state in Hartree-Fock theory. 
Imphcatwns of these results for other systems are discussed. 
INTRODUCTION 
Recently there have been many calculations designed 
to characterize the (7T, 1r*) singlet state of ethylene. 1•2 
Much of this interest stems from the SCF calculations 
of Dunning et al. 3 which showed that the inclusion of dif-
fuse orbitals along with valence types gave a 1( 1T, 1r*) 
state which was essentially Rydberg-like, e. g., with a 
second moment of the 1r* orbital, ( 1r* I x 2 l1r*), of 42 tfo 
where x is perpendicular to the plane of the molecule. 
This result is surprising and does seem inconsistent with 
experimental evidence such as the absence of any signif-
icant shift in this transition when the spectrum is ob-
served in solid or liquid krypton. 4 Several conventional 
calculations designed to include the effects of electron 
correlation on the location and nature of this state also 
concluded that the V state of ethylene had considerable 
diffuse or Rydberg character. 1 •2 Recently, Davidson, 5 
using traditional large basis set configuration interac-
tion methods, has shown that the V state of ethylene is 
in fact relatively valence-like. These results along with 
an adequate review of the background and other relevant 
aspects of this problem are discussed in Ref. 5. Their 
conclusions agree with the earlier work of Shibuya and 
McKoy, 6 Tanaka, 7 and Iwata and Freed. 8 
The purpose of the present work is not to obtain a de-
scription of the true 1( 1r, 7T*) state of ethylene but rather 
to try to understand the nature of the Hartree-Fock 
(7T, 7T*) spectrum in terms of a simple physical model. 
Such a model can display the essential physics of the 
problem and can be used to interpret and predict the ex-
istence of similar effects in the Hartree-Fock descrip-
tion of other molecules, e. g., N2 and C02 • Our model 
of the Hartree-Fock spectrum of the 1(7T, n1r*) states in 
ethylene is based on the observation that, in the SCF ap-
proximation, the lowest state of the same symmetry as 
the spectroscopic V state e B1u) is in fact the first mem-
ber of a Rydberg series followed by the other members 
a>This work was performed under the auspices of the u. s. En-
ergy Research and Development Administration under contract 
No. W-7405-Eng-48. 
b>contribution No. 5455. 
of this series leading up to the ion at 8. 96 eV. Hence 
SCF calculations with basis sets successively augmented 
with diffuse functions should yield additional members of 
this Rydberg series. The present calculations actually 
give up to five members of the series with energies 
which can be fitted to the Rydberg formula and with os-
cillator strengths ranging down from the largest value 
of 0. 09. None of these states shows the expected char-
acteristics of an intravalence transition such as theN 
- V transition. The question that now arises is what and 
where is the analog to the spectroscopic V state in Har-
tree-Fock theory. We have previously suggested that 
what would be the V state in Hartree-Fock theory has 
been pushed into the Hartree- Fock ionization continuum 
and is heavily contaminated with the lower-energy Ryd-
berg states due to orthogonality requirements. 9 In this 
paper, we carry out a series of SCF calculations on the 
singlet (7T, n1r*) states of ethylene which locates a state 
embedded in the continuum at 10. 21 eV which is very 
probably the SCF analog of the spectroscopic V state. 
Moreover, the location and nature of this state is stable 
with respect to major changes in the composition of the 
basis set. 
In the next section, we will discuss the SCF and sta-
bilization procedure which we use to carry out these cal-
culations and then present the results for the ( 7T, n1r*) 
states in ethylene. Finally, we outline the implications 
of these results for electronic transitions in other sys-
tems, e. g., N2, CO, and C02 and show that these states 
embedded in the continuum with large oscillator 
strengths can lead to definite problems in using such 
discrete spectral distributions to derive photoionization 
cross sections. 
THEORETICAL ASPECTS AND RESULTS 
The basis of our model of the (7T, n1r*) or 1b3u(7T) 
- nb2g (n7T*) spectrum of ethylene in the SCF approxima-
tion is that those states with this electron configuration 
which are below the first ionization potential are all 
members of a Rydberg series. This model seems to 
imply that the analog of the valence-like (7T, JT*) state in 
2178 The Journal of Chemical Physics, Vol. 67, No. 5, 1 September 1977 Copyright© 1977 American Institute of Physics 
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Bender, McKoy, and Davidson: Excited states in the continuum 2179 
TABLE I. Summary of SCF calculations locating a (7T, 1r*) 
resonance-like state. a,b 
N Rt R2 R3 R4 R5 R~ 
8.91 7.43 7.35 7.35 7.35 7.35 7.33 
19.84 9.93 8.09 8.06 8.06 8.06 8.04 
20.95"'-..10.17 8.38 8.38 8.38 8.36 
21.11 "'-.. 10. 21 8.60 8.60 8.58 
21.13"'-..10.21 8.77 8.75 
21.14 '-......10. 21 10.21 
21.14 19.81 
aEnergies are in electron volts relative to a ground state energy 
of - 78. 04171 a. u. See text for a discussion of the basis sets 
N, R 1, • • ·, R5, R~. We list the energies of all the states ex-
cept the one at 2. 55 a. u. above the ground state. 
~he arrows follow the state which we identify as a (7T, 7T*) res-
onance. The SC F ionization potential in these calculations is 
0.96eV. 
SCF theory has been pushed beyond the ionization poten-
tial and hence should appear as a. resonance embedded in 
the continuum. What would some characteristics of such 
a state be? We can expect that it would be heavily con-
taminated with the lower-energy Rydberg states (because 
of orthogonality) and hence would have more diffuse 
character, e. g., a larger second moment, than the true 
spectroscopic V state. On the other hand the SCF V 
state should have a large transition moment and oscilla-
tor strength for the 1A,.- B1u absorption due to its valence 
character. 
To locate this SCF state embedded in the continuum, 
we carry out a series of SCF calculations on the (1b3", 
nb2,.) states and look for a state in the continuum which 
is stable with respect to extensive changes in the basis 
set. As the number of diffuse orbitals in the basis is 
increased, we should obtain additional members of the 
Rydberg series and observe very little effect on the lo-
cation and character of the resonance-like state or on the 
previously converged Rydberg states. This approach to 
locating such a state in the continuum is essentially a 
stabilization method which has been widely used in de-
scribing resonant scattering. 10 The necessary spin-re-
stricted excited state SCF calculations were carried out 
on these states following the excited state method de.-
scribed by Stenkamp and Davidson. 11 This method allows 
excited states having the same orbital symmetry as low-
er states to be obtained by a variationally stable proced-
ure. 
For these calculations, we used six different basis 
sets and obtained all the (11', n1r*) states given by each 
basis set. These basis sets are referred to as N, R1, 
R2, R3, ••• , R5 where the basis set N is a [3s2p1d/2s] 
valence-like Gaussian orbit"al basis with Dunning's sand 
p contractions12 and ad-orbital exponent of 0. 75. All 
six d functions were included. The basis sets R 1 through 
R5 contain the basis set N successively augmented by p,. 
basis functions on the carbon atoms (the molecule is in 
the yz plane) with exponents of 0. 0251, 0. 0063, 0. 0016, 
0. 0004, and 0. 0001. For example, the basis sets R1 and 
R5 contain three and seven p,.-type functions, respective-
ly. 
The energies of the (11', n1r*) states obtained from these 
size calculations are listed in the corresponding columns 
of Table I. A clear trend emerges from these calcula-
tions. In the N basis the only relevant state is the one 
at 8. 91 eV. Adding a single diffuse function on each C 
atom to the N basis gives a state at 7. 43 eV and another 
at 9. 93 eV which is hence in the continuum (Hartree-
Fock IP= 8. 96 eV). In going to the R2 basis the state at 
7. 43 eV in the R1 calculation moves slightly down to 
7. 35 eV, another (Rydberg) state appears at 8. 09 eV, 
and the state at 9. 93 eV is pushed up to 10.17 eV. This 
trend continues on through the calculations with basis 
sets R3 and R4 until we finally have five Rydberg states 
at energies 7. 35, 8. 06, 8. 38, 8. 60, and 8. 77 eV and a 
stabilized root at 10. 21 eV on the R5 calculation. Two 
important observations can be made. First, the early 
members of the manifold of calculated (71', mr*) states, 
e. g., the first five states on the column R5, form a 
series with energies which can be fitted to the Rydberg 
formula with effective quantum numbers n* = 2. 907, 
3. 907, .•. , 6. 907. Next, the state at 10. 21 eV is ob-
viously stable with respect to the addition of diffuse func-
tions to the basis. We will see later that this state is 
also stable under changes in the valence part of the basis. 
To characterize this (11', 1r*) state at 10.21 eV, we look 
at the second moment of the 11'* orbitals and dipole tran-
sition moments from the ground state for the states of 
the R5 calculation listed in Table I. The second mo-
ments are shown in Table II and those of the h*, 211'*, 
..• , 5II* orbitals are characteristic of successive 
members of a Rydberg series. The 61r* orbital has a 
large second moment of 35 ~. This large value for the 
second moment reflects considerable Rydberg character 
which must be present due to orthogonality requirements 
to the lower-energy Rydberg series h* to 511'*. In Ta-
ble III, we list the oscillator strengths for the transi-
tions into the ( 1r, n1r*) states of the calculation with the 
R5 basis. 
13 We now see the important result that the 
state at 10.21 eV carries a large oscillator strength even 
though the large value of the second moment of the 1r* 
orbital reflects the presence of considerable Rydberg 
character. The large value of the transition moment, 
and hence the oscillator strength, indicates that the "sta-
bilized" state at 10. 21 eV also has significant valence-
like components. 
The results of Table I show that the state at 10. 21 eV 
TABLE II. Second moment of the 7T* 
orbitals. a 
Orbital frz7T* I x2 1 n7T*)b 
17T* 42.4 
27T* 175.1 
37T* 474.8 
47T* 2081.9 
57T* 8595 
67T* 35.3 
77T* 7.1 
aFor those states listed in column R5 of 
Table I. 
~ a. u. The molecule is in the yz plane. 
J. Chern. Phys., Vol. 67, No. 5, 1 September 1977 
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2180 Bender, McKoy, and Davidson: Excited states in the continuum 
TABLE ID. Oscillator strengths for the 
"l!t(mr*). a 
Transition b 
"llt-"l!t(h*) 
"l!t(2rr*) 
"l!t(3rr*) 
"l!t(4rr*) 
"l!t(5rr*) 
"l!t(6rr*) 
"l!t(7rr*) 
0.112 
0.077 
0.038 
0,003 
0,0001 
0.456 
0.139 
aFor those states listed in column R5 of 
Table I. See Text. 
b"llt(nrr*) denotes the (rr, nrr*) state. 
'1=~ t:.E I (0 ll:;Z; I "l!t(nrr*)) 12• 
has been stabilized with respect to extensive changes in 
the Rydberg character of the orbital basis. To study the 
stability of this state under changes in the valence part 
of the basis, we carried out additional calculations using 
a [4s3p1d/2s] basis14 augmented with the same five dif-
fuse Ps basis functions on the C atom as in the R5 calcu-
lation of Table I. These calculations listed under Rt in 
Table I gave five Rydberg states at almost identical en-
ergies as those of Table I and,a stabilized state at 10.21 
eV. The oscillator strength of the transition to this 
state is again 0. 46. Notice also the apparent stabilized 
state at - 21. 1 eV of the Rydberg calculations is not sta-
ble with respect to variation of the valence basis compo-
nents. We, therefore, conclude the state embedded in 
the continuum at 10.21 eV is stable with respect to 
changes in both the Rydberg and valence character of 
the basis, and is the only such state within 10 eV of the 
Hartree-Fock continuum. 
We now ask, what is this state at 10. 21 eV? It is 
much more valence-like than the lower energy states 
and is embedded in the continuum, We believe that this 
is the analog to the spectroscopic V state in Hartree-
Fock theory. The excitation energy of 10.21 eV is about 
2. 6 eV larger than the observed value of 7. 66 eV. The 
incorrect description arising in the Hartree-Fock calcu-
lation is probably due to the larger correlation effects 
in the ionic V state than in the Rydberg states. A pro-
cedure for obtaining an accurate description of the V 
state would be to carry out a configuration interaction on 
this SCF (7T, 61r*) state. The major effect will be the 
correlation between the I:-II system and the "unmixing" 
of Rydberg and valence characters which would appear 
as extensive mixing of (7T, 67T*) with the lower "Rydberg" 
SCF states. The mixing of ( 1r, 61r*) and ( 1r, rr*) is so ex-
tensive, in fact, that the true V state is nearly a 50-50 
mix of these SCF states. The observed f value for the 
N- V system is 0. 34. In these calculations we can ex-
pect an increase from the SCF value of 0. 46 for the os-
cillator strength due to the removal of Rydberg charac-
ter from the states and a decrease due to the effects of 
electron correlation. 5•6 -
DISCUSSION AND CONCLUSIONS 
The purpose of the present work was to locate the 
analog to the spectroscopic V state in Hartree-Fock the-
ory and to understand its character and location in terms 
of a simple model. We believe that we have located such 
a state at 10. 21 eV which is in fact a resonance em-
bedded in the ionization continuum with an approximate 
oscillator strength of 0. 46. However, the state has con-
siderable diffuse character due to contamination with the 
lower-energy Rydberg states (due to orthogonality re-
quirements). This resonance state could be a convenient 
starting point for a configuration interaction calculation. 
The existence of such V or ionic states in the Hartree-
Fock spectrum will be a warning in studying other sys-
tems. For example, preliminary calculations on the 
[core]rr!3a~-[core]rr~3a~nrrKe:E:) transitions in N2 indi-
cate the same behavior of the intravalence rru- 1Tc transi-
tion as in C2H4• Here again, all the rr~n7TK(
1 I::) states ly-
ing below the 2IIu ion of N2 are members of a Rydberg 
series while a valence-like rr!rrK state lies in the contin-
uum. The presence of such "resonances" which are 
spuriously pushed into the continuum in SCF theory must 
be dealt with carefully in methods which use discrete 
spectral distributions to derive photoionization cross 
sections, e. g., by a Stieltjes imaging procedure. 15 
tR. J. Buenker and S. D. Peyerimhoff, Chern. Phys. 9, 75 
(1975). 
2C, F. Bender, T, H. Dunning, Jr., H. F. Schefferiii, W. 
A. Goddard III, and W. J. Hunt, Chern. Phys. Lett. 15, 171 
(1972). 
3T, H. Dunning, Jr., W. J. Hunt, and W. A. Goddard III, 
Chern. Phys. Lett. 4, 147 (1969). 
4E. Miron, B. Raz, and J. Jortner, Chern, Phys. Lett. 6, 563 
(1970). 
5Larry E. McMurchie and Ernest R. Davidson, J. Chern. Phys. 
(accepted for publication). 
sT. Shibuya and V. McKoy, J, Chern. Phys. 54, 1738 (1971). 
7K, Tanaka, Int. J. Quantum Chern. 6, 1087 (1972). 
8s, Iwata and K. F. Freed, J. Chern. Phys. 61, 1500 (1974). 
9Harold Basch and Vincent McKoy, J. Chern. Phys. 53, 1628 
(1970). 
10H. s. Taylor, Adv. Chern. Phys. 18, 91 (1970). 
11E. R. Davidson and L. z. Stenkamp, Int. J. Quantum Chern. 
810, 21 (1976). 
12T, H. Dunning, Jr. and P. Jeffrey Hay, Modern Theoretical 
Chemistry edited by H. F. Schaeffer ill, (Plenum, New York, 
1977) Vol. 2. The [3s) basis is the contracted segmented 
basis set obtained from a (9s5p) primitive basis. 
13The ground state and excited state SCF orbitals are not or-
thogonal. To evaluate the transition moment, we take vectors 
of a particular excited state calculation. These vectors also 
give good approximations to the other states including the 
ground state. The transition moments are calculated with 
these eigenvectors and are sufficiently accurate for the pres-
ent purposes. 
14T, H. Dunning, Jr., J. Chern. Phys. 53, 2823 (1970), 
15p, w. Langhoff, Int. J. Quantum Chern. S8, 347 (1974). 
J. Chern. Phys., Vol. 67, No. 5, 1 September 1977 


An SCF-stabilization approach to excited states embedded

in the continuum

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An SCF-stabilization approach to excited states embedded in the continuum

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