Anomalous Electron Correlation Due To Near Degeneracy Effects: Low-lying
Ionic States of Ne and Ar
Paul S. Bagus1), Ria Broer2), and Fulvio Parmigiani3)
1
Department of Chemistry, University of North Texas, Denton, TX 76203-5070, USA
Department of Chemical Physics and Materials Science Centre, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
3)
Department of Physics, Università degli Studi di Trieste, Via Valerio, 2 I-34127, Trieste, Italy
and Sincrotrone Trieste, I-34012 Basovizza, Trieste, Italy
2
ABSTRACT:
This letter addresses a long-standing problem related to non-dynamical electron correlation
effects. The origin of the large differential electronic correlation energy among the neutral 1S
ground state, the lowest, 2P, ionic state and the first excited, 2S, ionic state of the Ne and Ar
atoms is explained in terms of the near degeneracy of low-lying excited configurations. There is
an anomalous correlation for the 2S state that is shown to be due to non-dynamical correlation
involving a low-lying excited configuration. The conceptual framework used here is also
appropriate to be used for other atomic and molecular systems.
In the Hartree-Fock, HF, ΔSCF method for calculating ionization potentials, IP’s, the IP
is determined by subtracting the HF self-consistent field, SCF, energies calculated for the initial,
ground state, of the system and for an appropriate state of the ion.[1] Specifically,
IP(ΔSCF) = ESCF(Ion) − ESCF(Ground State).
(1)
Especially for the IP to low-lying states of the ion, it is expected that IP(ΔSCF) will be smaller
than the observed IP; this was first pointed out by Mulliken. [2] The explanation for this
bounding of the IP arises from the decomposition of the electron correlation energy, Ecorr, into
contributions from pairs of electrons as being the dominant terms for Ecorr. [3] Since the ion has
N−1 electrons when the initial, neutral, system has N electrons; the ion has N-1 fewer pairs and
hence a smaller Ecorr than the initial state. However, Bagus [4] pointed out that this was not the
case for the ionization of Ne and Ar to the lowest 2S state of the ions. This state is the first
excited state of the ionic species. [5] For these states, the IP(ΔSCF) is actually larger than the
experimental value; for the 3s ion of Ar, the IP(ΔSCF) is nearly 4 eV larger than experiment.
This means that there must be unusual correlation effects for the 2S states of Ne+ and Ar+ and
related systems that lead to a larger Ecorr for the system with N−1 electrons than for the system
with N electrons. The situation is summarized in Table I where experimental and theoretical
values for the lowest 2P(np−1) and 2S(ns−1) ions of Ne and Ar are presented. The IP(ΔSCF) for np
ionization are smaller than experiment as expected by the general rules for the magnitude of the
correlation energy. [2] On the other hand, the IP(ΔSCF) for ns ionization are larger than the
experimental values. The principle object of this work is to investigate and identify the origin of
the unexpectedly large electron correlation effects for the 2S(ns−1) ions. It is shown that by taking
account of a near degeneracy electron correlation effect one returns, with a suitably extended
definition of IP(ΔSCF), to the expected situation that the calculated IP’s are smaller than those
2
measured. The near degeneracy effect that is included can arise only for the 2S state of the ion
but not for the 2P state of the ion or for the 1S ground state of the neutral atom. For representative
studies of atomic correlation effects within the theoretical framework of the mixing of nearly
degenerate configurations see Refs. 6-10. A classification scheme normally used to distinguish
near degeneracy from other types of electron correlation effects is to include the near degeneracy
contributions in the category of non-dynamical, or static, correlation as opposed to dynamical
correlation effects. [11-14] While, as is usually the case with non-dynamical correlation, the
definition of near degeneracy is qualitative, [12, 13, 15] this paper uses a simple relationship
from perturbation theory to estimate when such near degeneracy may be important. Thus, this
work may have general value, beyond the specific results for the low-lying ions of Ne and Ar.
The conceptual framework used for these systems is also appropriate to use to estimate, for other
atomic and molecular systems, when there may be anomalous electron correlation effects.
The analysis is based on the properties of wavefunctions, WF’s, for the initial and ionic
states of Ne and Ar. The WF’s are determined by numerical integration of the HF and multiconfiguration HF, MCHF, equations for the isolated atoms. The calculations were performed
with programs developed by C. F. Fischer; [16] the WF’s are non-relativistic and the usual spin
and space symmetry restrictions are used. For the neutral atoms and for the ions, HF WF’s are
determined. In addition for the 2S ions a two configuration MCHF WF was determined. The first
configuration for the MCSCF WF, Φ1=(ns)1(np)6, is just the HF configuration; the second
configuration, Φ2=(ns)2(np)4(nd)1, involves the excitation np2→ns3d. Only the coupling of the
open (np)4 to the 1D multiplet can lead to the required total coupling of 2S for Φ2. This
configuration is a low-lying excitation since one np electron is lowered in energy by being
3
placed into the lower ns sub-shell while the other np electron is raised in energy by being placed
into the higher 3d sub-shell. The net energetic cost involves a cancellation of these two terms;
specific numerical values will be discussed below. Such a low-lying excitation is not possible for
the 2P (ns)2(np)5 configuration. While the excitation of np→3d may not involve a large excitation
energy, especially for Ar, the configuration (ns)2(np)4(3d)1 has a different parity from (ns)2(np)5
and these two configurations cannot mix. The configurations for the double excitations
(np)2→(3d)2 will mix with the HF 2P configuration for the ion, but it is expected that this
configuration will involve a large excitation energy and will not be nearly degenerate. Similar
arguments apply to the 1S (ns)2(np)6 ground state of the neutral atoms and show that there will
not be near degeneracy in this case either.
The type of excitation involved in the second configuration has been used previously in
the treatment of correlation effects for the core-level, 3s, ionization of 3d series transition metal
ions. [8,17] The inclusion of such near degeneracy configurations in the WF is necessary in order
to correctly describe the complex structure observed with X-ray photemission spectroscopy,
XPS, for these ions. [18] However, this is the first time that such an excitation has been used to
describe low lying excited states. As was noted above, this configuration is expected to be nearly
degenerate since there is a gain of energy when an np electron is moved to fill the partly
occupied ns shell while there is an energetic cost when the second np electron is moved into the
empty 3d shell. Furthermore, the off-diagonal Hamiltonian matrix element between the first and
second configurations, H12, is proportional to a Slater R integral, R1(ns,3d;np,np). [19] This
exchange-like R integral could easily have a large value and, when H12 is sufficiently large, Φ2
4
will represent important static correlation effects. These qualitative considerations about near
degeneracy will be placed in a quantitative context in the discussion that follows.
A clear indication of the importance of the MCHF WF that includes Φ2 is seen from the
IP(ΔMCHF) values given in Table I. The Ne 2s IP is reduced by >2.5 eV from the ΔSCF value.
The IP(ΔMCHF) is smaller than the experimental value for the IP as is to be expected when the
differences of dynamical electron correlation effects for the neutal and the ionic states dominate
the error of the theoretical IP. In fact, the error for Ne of the 2s IP(ΔMCHF) is about the same as
the error of the 2p IP(ΔSCF). This suggests that the remaining electron correlation effects for
these IP’s may be due to dynamical correlation for the neutral and ionic states. The Ar 3s
IP(ΔMCHF) is reduced from IP(ΔSCF) by much more than was the case for Ne; the MCHF IP is
~6.25 eV smaller indicating a greater importance for the configuration mixing of Φ2 than in the
case of Ne. The reason for the greater importance in the case of Ar will be discussed below. For
the case of Ar, as well as for Ne, the IP(ΔMCHF) is smaller than experiment suggesting that
dynamic correlation may be dominant for the remaining errors. However, for Ar, the error of the
3s IP(ΔMCHF) is over 1 eV larger than the error of the 3p IP(ΔSCF). This suggests that there
may be additional static, or near degeneracy, correlation not yet accounted for; see discussion
below.
Tables II and III give information about the MCHF H matrix, describe the energetics of
the H matrix eigenvalues, and characterize the configuration interaction, CI, WF for the lower 2S
state. The information in these tables makes it possible to examine and quantify the concepts of
near degeneracy for the np2→ns3d excited configuration, Φ2. In particular, the ratio, R, of the
5
off-diagonal matrix element, H12, to the excitation energy, ΔH=H22−H11, provides a direct
measure of whether the second configuration is nearly degenerate and whether perturbation
theory can be used. When the ratio is small, the configuration Φ2 makes a small contribution to
the WF and perturbation theory can be used; the relevant formulas for the contributions of Φ2 to
the energy of the first root of the two configuration CI, ΔE, and to the CI WF, C2, are: [20]
ΔE=−H122/(H22−H11) and C2= −H12/(H22−H11).
(2)
As the ratio, R=H12/ΔH, increases, the relations of Eq.(2) become less accurate and, for
sufficiently large ratios, they cannot be used at all. The results for Ne are considered first, then
those for Ar.
From Table II, the ratio for the Ne MCHF, is R=−0.27. Although the diagonal excitation
energy, ΔH=38.0 eV, is fairly large, the off-diagonal matrix element, H12=−10.4 eV, is also large
giving a value for R that is not small. The reasonably large value of R indicates that Φ2 will have
modest importance in the WF although the perturbation formulas of Eq.(2) may still be
approximately valid. The perturbation theory value for the correlation energy lowering is 2.8 eV
or about 7% larger than the value directly calculated. Similarly, perturbation theory predicts a
slightly larger magnitude for the CI mixing coefficient, C2, than obtained by direct
diagonalization. This modest mixing leads to changes in the natural orbital, NO, occupation
numbers [21] that are slightly different from the HF occupations. The 2s occupation increases
from 1 to 1.06 and the 2p occupation decreases from 6 to 5.88. Thus, the static correlation due to
the 2p2→2s3d excitation is significant and the remaining ~2 eV errors in the 2s and 2p IP’s of Ne
are probably due to differential dynamical correlation effects between the 10 electron WF for Ne
6
and the 9 electron WF’s for Ne+. Overall, there is a modest contribution to the WF from Φ2 and
perturbation theory is approaching the limits of its accuracy.
The situation is quite different for Ar; see Table III. The the diagonal excitation energy,
ΔH=4.3 eV, is reasonably small, especially compared to the off-diagonal matrix element,
H12=−8.2 eV. This is hardly surprising since the 3d shell will begin to fill only a few elements in
the periodic table past Ar and one should expect that Φ2 will be nearly degenerate with the HF
configuration, Φ1. It is interesting that, although the absolute value of the excitation energy has
decreased for Ar from that in Ne by almost an order of magnitude, from 38 to 4 eV, the offdiagonal matrix elements for Ar and Ne are rather similar. The consequence of the near
degeneracy is that the ratio of the off-diagonal matrix element to the diagonal excitation energy
is quite large in magnitude, R=−1.9; for this magnitude, perturbation theory fails and direct
diagonalization of the Hamiltoinian matrix is necessary. This diagonalization shows that a large
correlation energy of Ecorr=6.3 eV is recovered and that the second configuration has a weight of
almost 40% in the MCHF WF; clearly, it is appropriate to consider this as a non-dynamical
correlation effect. Given the very low energy of the 3p2→3s3d excitation for the 2S state of Ar+,
it is to not unreasonable to expect that the 3p2→3d2 excitation may also be important for the
neutral Ar and for both the 3p−1 and 3s−1 ionic Ar+ states. Thus the 2C MCHF WF may not
include all of the static correlation effects for the low-lying ionizations of Ar+ and 3p2→3d2
excitations may be required to fully treat the static correlation, especially the differential static
correlation among these states. However, it is also clear that a very large part of the static
correlation has been included for the 3s−1 2S state of Ar+ and that this inclusion is sufficient to
correct the anomalous differential correlation energy found for the HF WF’s for Ar and Ar+(2S)
7
It is worth noting that near degeneracy, or non-dynamical, correlation effects arise
frequently. They are a common occurrence in molecules where they normally must be treated to
permit correct dissociation to ground state open shell atoms; see, for example Refs. 14 and 22.
They are also important for many atomic systems. They need to be taken into account to
correctly describe the energy splittings of different multiplets of the ground state electronic
configuration of open shell atoms; see, for example, Refs. 6 and 7. For the excitations from
atomic ground to excited states, near degeneracy effects lead to major changes in the transition
probabilities. [9] For the ionization of core levels in transition metal systems, [8, 10, 17, 23, 24]
dramatic changes, due to atomic near degeneracy effects, are found in relative energies and in the
number and intensities of the states observed in XPS. In the present letter, a new consequence of
near degeneracy has been shown to be the large differential correlation energy for the lowest 2P
and 2S states of Ne+ and Ar+. Given the pervasive importance of such effects, it is useful and
important to understand the role that they have for various properties of atomic and molecular
systems. An important advance in this understanding has been presented here.
In summary, this letter has discussed the anomalous differential correlation energy of the
lowest 2S states of Ne+ and Ar+. The anomaly is that the N−1 electron ionic state has a larger
correlation energy than the N electron ground state of the neutral atom. This anomaly leads to a
ΔSCF IP that is larger than experiment; this sign of the error of the IP is contrary to general
experience and expectation. The origin of this anomalous correlation energy has been shown to
be a non-dynamical effect that involves a nearly degenerate configuration, Φ2=(ns)2(np)4(3d)1,
with an excitation of np2→ns3d with respect to the HF configuration, Φ1=(ns)1(np)6. The
8
analysis for the importance of Φ2 uses first order perturbation theory and is based on the
magnitude of the ratio R=H12/(H22−H11); in effect, this ratio is used as a quantitative measure of
the near degeneracy of Φ2. The type of analysis used here is not restricted to the particular atoms
treated but may be applied in general. The considerations presented here can be used
qualitatively to identify potential anomalous correlation effects for ions of other atoms and
molecules. Thus, one may be able to identify a priori limitations and irregular behavior in the
IP’s predicted by one electron models. Indeed, the knowledge that such qualitative estimates can
be made may be one of the most important consequences of the results presented in this letter.
This research was supported, in part, by the Geosciences Research Program, Office of Basic
Energy Sciences, U. S. Department of Energy (DOE). One of us (P.S.B.) is pleased to
acknowledge partial computer support from the National Center for Supercomputing
Applications, Urbana-Champaign, Illinois.
References:
1.
2.
3.
4.
See for example, P. S. Bagus et al. J. Elec. Spec. and Related Phen. 100, 215 (1999).
R. S. Mulliken, J. Chim. Phys. 46, 497 (1949).
P.-O. Löwdin, Adv. Chem. Phys. 2, 207 (1959).
P. S. Bagus, Phys. Rev. 139, A619 (1965).
5.
C. E. Moore, Atomic Energy Levels, Vol. 1 Natl. Bur. Stand. No., 467, U. S. GPO,
Washington, D. C., 1952 and http://physics.nist.gov/cgi-bin/AtData/main_asd.
6.
D. R. Hartree, W. Hartree, and B. Swirles, Phil. Trans. Roy. Soc. London A238, 233
(1939).
7.
P. S. Bagus and C. M. Moser, Phys. Rev. 167, 13 (1968); P. S. Bagus, N. Bessis, and C.
M. Moser, ibid. 179, 39 (1969); P. S. Bagus, A. Hibbert, and C. Moser, J. Phys. B 4, 1611
(1971).
8.
P. S. Bagus, A. J. Freeman, and F. Sasaki, Phys. Rev. Lett. 30, 850 (1973).
9.
Yong-Ki Kim and P. S. Bagus, Phys. Rev. A 8, 1739 (1973).
9
10.
P. S. Bagus et al., Phys. Rev. Lett. 84, 2259 (2000).
11.
O. Sinano_lu, Adv. Chem. Phys. 6, 315 (1969).
12.
P. E. M. Siegbahn in Methods in Computational Physics ed by G. H. F. Diercksen and S.
Wilson (D. Reidel, Dordrecht, 1983) p. 189.
13.
J. Cioslowski, Phys. Rev. A 43, 1223 (1991).
14.
P. Knowles, M. Schütz, and H.-J. Werner in Modern Methods and Algorithms of
Quantum Chemistry Vol. 3 ed. by J. Grotendorst (j. von Neumann Inst. Jülich, 2000) p. 97; also
http://www.fz-juelich.de/nic-series/Volume3/knowles.pdf
15.
E. Valderrama, E. V. Lude_a, and J. Hinze. J. Chem. Phys. 110, 2343 (1999).
16.
C. F. Fischer, Comput. Phys. Commun. 64, 431 (1991) and Refs. therein.
17.
E.K. Viinikka and Y. Ohrn, Phys. Rev.B 11 (1975) 4168.
18.
B. Hermsmeier et al., Phys. Rev. Lett. 61, 2592 (1988).
19.
J. C. Slater, Quantum Theory of Atomic Structure, Vol. II, (McGraw-Hill, New York,
1960).
20.
H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,
(Academic Press, New York, 1957).
21.
P.-O. Löwdin, Phys. Rev. 97, 1474 (1955) and B. O. Roos, P. R. Taylor, and P.E. M.
Siegbahn, Chem. Phys. 48, 157 (1980).
22.
H. F. Schaefer III, The Electronic Structure of Atoms and Molecules (Addison-Wesley,
Reading, 1972)
23.
A. J. Freeman, P. S. Bagus, and J. V. Mallow, Int. J. Magnetism, 4, 35 (1973).
24.
R. P. Gupta and S. K. Sen, Phys. Rev. B, 10, 71 (1974); ibid. 12, 15 (1975).
10
Table I. Theoretical and experimental IP’s for ionization to the lowest 2P and 2S states of Ne+ and
Ar+. In the one configuration description, these states have the configurations ns2np5(2P) and
ns1np6(2S) where n=2 for Ne and n=3 for Ar. They are, respectively, the lowest and the first
excited states of the cation. Theoretical IP’s are obtained with SCF HF WF’s, for the neutral and
ionized states and with two configuration MCHF WF’s for the 2S ionic state. The theoretical IP’s
are labeled IP(ΔSCF) when the SCF WF is used for the ionic state and IP(ΔMCHF) when the
MCHF WF is used for the ionic state. The difference of the theoretical and experimental IP’s,
ΔIP, is defined so that ΔIP<0 means that the theoretical IP is less than experiment. All IP’s are in
eV.
Experimenta
IP(ΔSCF)/ΔIP IP(ΔMCHF)/ΔIP
Ne Ions
2
P
21.56
19.85/−1.72
2
S
48.47
49.32/+0.84
---46.67/−1.80
Ar Ions
a
2
P
15.76
14.78/−0.98
2
S
29.24
33.19/+3.95
---26.93/−2.31
See Ref. 5.
11
Table II. Energies and other properties of the two configuration MCHF WF for the lowest 2S
state of Ne+. The diagonal and off-diagonal Hamiltonian matrix elements, Hij, and the CI
eigenvalues, E1 and E2, are given. The difference ΔH=H22−H11 and the ratio ΔH/H12 are given to
show the near degeneracy of the two configurations. The correlation energy recovered with the
2C MCHF WF, Ecorr, and the difference of the MCHF energy eigenvalues, ΔE=E2−E1, are given
as additional measures of the importance of the near degeneracy CI. Finally the CI coefficients,
C1 and C2, and the natural orbital, NO, occupation numbers, ω, for the lower state are given.
Total energies and diagonal energies are given in hartrees; all other energies are given in eV.
Matrix Elements
Eigenvalues
Lowest
ω
Eigenvector
H11=−126.7345
E1=−126.8319
C1=+0.969
1s – 2.000
H22=−126.3374
E2=−125.2400
C2=+0.247
2s – 1.062
ΔH=+38.02
ΔE=43.32
2p – 5.876
H12=−10.38
Ecorr=2.64
3d – 0.062
H12/ΔH=−0.27
12
Table III. Energies and other properties of the two configuration MCHF WF for Ar+(2S). See the
caption to Table II.
Matrix Elements
Eigenvalues
Lowest
ω
Eigenvector
H11=−525.5966
E1=−525.8280
C1=+0.792
1s& 2s – 2.000
H22=−525.4393
E2=−525.2079
C2=+0.611
2p – 6.000
ΔH=+4.28
ΔE=16.87
3s – 1.375
H12=−8.17
Ecorr=6.26
3p – 5.250
H12/ΔH=−1.19
3d – 0.375
13