A Valence Bond Study of the Dioxygen Molecule
PEIFENG SU1, LINGCHUN SONG,1 WEI WU,1 PHILIPPE C. HIBERTY,2 SASON SHAIK3
1
Department of Chemistry, State Key Laboratory of Physical Chemistry of Solid Surfaces,
Center for Theoretical Chemistry, Xiamen University, Xiamen 361005, People’s Republic of China
2
Laboratoire de Chimie Physique, Groupe de Chimie The´orique, Universite´ de Paris-Sud,
91405 Orsay Ce´dex, France
3
Department of Organic Chemistry and Lise Meitner-Minerva Center for Computational Quantum
Chemistry, The Hebrew University, Jerusalem 91904, Israel
Received 31 March 2006; Accepted 22 April 2006
DOI 10.1002/jcc.20490
Published online 23 October 2006 in Wiley InterScience (www.interscience.wiley.com).
Abstract: The dioxygen molecule has been the subject of valence bond (VB) studies since 1930s, as it was considered as the first ‘‘failure’’ of VB theory. The object of this article is to provide an unambiguous VB interpretation for
the nature of chemical bonding of the molecule by means of modern VB computational methods, VBSCF, BOVB, and
VBCI. It is shown that though the VBSCF method can not provide quantitative accuracy for the strongly electronegative and electron-delocalized molecule because of the lack of dynamic correlation, it still gives a correct qualitative
analysis for wave function of the molecule and provides intuitive insights into chemical bonding. An accurate quantitative description for the molecule requires higher levels of VB methods that incorporate dynamic correlation. The
potential energy curves of the molecule are computed at the various VB levels. It is shown that there exists a small
hump in the PECs of VBSCF for the ground state, as found in previous studies. However, higher levels of VB methods
dissolve the hump. The BOVB and VBCI methods reproduce the dissociation energies and other physical properties of
the ground state and the two lowest excited states in very good agreement with experiment and with sophisticated MO
based methods, such as the MRCI method.
q 2006 Wiley Periodicals, Inc.
J Comput Chem 28: 185–197, 2007
Key words: oxygen molecule; valence bond theory; VBSCF; BOVB; VBCI
Introduction
Owing to the rapid progresses in computer science and technology,
computational chemistry is becoming a powerful tool for studying
chemical problems, ranging from the various properties of small
molecules to the simulation of biochemical systems. However,
there are still many small molecules that even high levels of theory
do not tackle very well and do not describe a simple bonding picture compatible with the chemist’s view. The dioxygen molecule
is one of these small molecules which require very high levels of
theoretical methods to be properly described throughout the intermolecular distance. This molecule is also one of the molecular
icons in chemistry, connected with the rivalry of the two theories
of quantum chemistry, molecular orbital (MO) and valence bond
(VB) theories, and its electronic structure description is often used
as the reason why MO theory should be favored over VB theory.
Allegedly, the latter theory provides a wrong description of the
ground state of this molecule. And even though a simple Hückeltype VB theory shows that this is not true, the ‘‘failure’’ has somehow stuck to VB theory.1,2 Our article addresses the VB description of the O2 molecule, its bonding and features, from the equilib-
rium distance to the dissociation limit. Such a study seems to
match the general theme of the volume that celebrates 90 years for
the concept of the ‘‘chemical bond.’’
The O2 molecule has a triplet ground state and it appears in the
atmosphere as a persistent diradical; oxidation of molecules by oxygen is thermodynamically favored but kinetically slow.3 The first
theoretical description of O2 was given by Lennard-Jones4 who
used MO theory to predict a triplet ground state in accord with
experiment. Early VB theory gave the same physical description,
and in his landmark paper,5 Pauling was careful to state that the
molecule does not possess a ‘‘normal perfectly paired’’ state, but
rather a diradical one with two three-electron bonds, and so did
Wheland in his 1937 paper,6 as well as on page 39 of his book.7
There is also a 1934 Nature paper by Heitler and Pöschl8 who
treated the O2 molecule with VB principles and concluded that
‘‘the 3Sg term . . . giving the fundamental state of the molecule.’’
Correspondence to: W. Wu; e-mail: [email protected]
Contract grant sponsors: Natural Science Foundation of China; Israel
Science Foundation (ISF)
q 2006 Wiley Periodicals, Inc.
186
Su et al. • Vol. 28, No. 1 • Journal of Computational Chemistry
Clearly, therefore, early VB theory gave a correct description of
the O2 molecule. So the real cause of the myth that is propagated
even today via textbooks,1,2 about the failure of VB theory,
remains somewhat of a mystery.9 It is certainly true that the simple
Lewis picture of O2 fails, and that the MO picture4 was exceedingly simpler than the VB picture, but the VB description of O2 is
not the simple Lewis picture, and hence none of these hand waving
arguments really justifies the statement that has accompanied VB
theory like a shadow throughout the years. What has happened in
the interim time between those early treatments and the more modern days of quantum chemistry? Well, both MO and VB theories
have found that O2 requires more than the simple MO or VB treatment; MO requires extensive CI,1–4,10–12 and VB required extensive post-Pauling/Wheland treatments.13–16 Since this article
focuses on VB theory, the following discussions address only some
landmark VB studies of O2.
In 1975, Goddard and coworkers used their generalized valence
bond (GVB) method to carry out the quantitative VB studies on O2
molecule.13 Their study showed that GVB with the perfect pairing
approximation (GVB-PP) accounts for the bonding in O2 quite
simply as a resonance between two three-electron VB structures;
thus, O2 possesses in addition to the bond also two three-electron
hemi-bonds. However, the GVB-PP by itself was unable to properly describe the dissociation, and good dissociation energies required a GVB-CI calculations; thus sacrificing somewhat the simplicity advantage of VB theory. McWeeny14 calculated the ground
state of O2 at the experimental equilibrium distance using a minimal basis with eight VB structures. He calculated the ground state
energies of wave functions containing two, four, and eight structures, and concluded that the double bond arises from resonance
involving two dominant ionic structures. He also presented the
potential energy curves (PECs) of the lower triplet and singlet
states using a double-zeta basis. Subsequently, based on McWeeny’s
results, Harcourt15 showed that the one-electron transfer resonance
between each covalent structure and a pair of ionic structures contributes most to the bonding energy, using four of the eight VB
structures given by McWeeny. In 1995, a VBSCF study was carried out by Byrman and van Lenthe.16 The three lowest states of
the oxygen molecule (3Sg, 1Dg, and 1Sgþ) were studied by means
of two models, one being called proper dissociation model (PD),
the other called proper reference model (PR). The value of dissociation energy for the ground state was 2.832 and 3.672 eV, respectively for the two models, thus covering 54 and 70% of experimental values. The authors observed a small barrier on the dissociation
potential curve of the ground state and stated that the hump originates from a ‘‘spin-recoupling.’’ However, they did not reach a definitive conclusion about the precise origins of the small barrier,
whether the hump is ‘‘real’’ or an artifact of the calculation. For
the two excited states, they also presented the PEC and the dissociation energies of 2.799 eV for 1Dg and 2.036 eV for 1Sgþ state, 66
and 57% of experimental values respectively, for the PR model.
It is obvious that even though the previous VB methods provided a qualitatively correct prediction of the ground state of oxygen molecule, the quantitative performance was still unsatisfactory, unless the wave function lost its simplicity by extensive CI.
One of the deficiencies of past VB applications, using minimal sets
of VB structures, was that the numerical results, such as bond energies and reaction barriers and so on, were lacking quantitative
accuracy. However, thanks to the rapid progresses in computer science, the VB method has been enjoying renaissance in the last
two-three decades. The BOVB and VBCI methods enable us to
carry out quite accurate VB calculations for small molecules, while
keeping the wave function simple and compact.17–20 As such, it is
worthwhile to revisit the dioxygen molecule by means of ab initio
VB methods using high computational levels. The aim of the present article is to perform a VB study of the dioxygen molecule and
provide not only a lucid interpretation of the nature of the bonding,
but also to achieve this lucidity along with considerable accuracy
of numerical results. The ground state and the two lowest excited
states are being both considered in this article.
The article is organized as follows: It starts with a brief review
of the necessary theory of the used VB methods. The computational details are reported in the next section, including the qualitative description for the wave functions for the ground state and
the excited states, the choice of required VB for calculations,
computational results, and discussions. Finally, a brief conclusion
is given.
Theoretical Methods
In VB theory, a many-electron wave function is expressed in
terms of VB functions FK,
¼
X
CK K
(1)
K
where FK corresponds to the traditional VB structure, which may
be a spin-coupled function, or a spin-free form of VB function.21,22
The coefficients CK in eq. (1) are subsequently determined by solving the usual secular equation HC ¼ EMC.
Since VB structures are not mutually orthogonal, normalized
structural weights are defined as:23
WK ¼ C2K þ
X
CK CL hK jL i:
(2)
L6¼K
The modern VB computational methods, which will be used here,
are VBSCF, BOVB, and VBCI. In the VBSCF method,24 both the
VB orbitals and structural coefficients CK are optimized simultaneously to minimize the total energy. The VBSCF method takes
care of static electron correlation, but lacks dynamic correlation,25 an absolutely essential ingredient for attaining quantitative
accuracy. As such, the VBSCF results are only qualitatively correct, and this is re-validated in the present article.
A VB method that incorporates dynamic correlation is the
breathing orbital VB (BOVB) method due to Hiberty et al.25,26
BOVB improves the description of the VB structures by allowing
different orbitals for different structures. In this manner, the orbitals can fluctuate in size and shape so as to fit the instantaneous
charges of the atoms on which these orbitals are located, as well as
adapting to the interaction with the other VB structures.
Recently, another VB method,27 called VBCI, was introduced;
it starts from a VBSCF wave function, followed by a subsequent
VBCI calculation involving the entire set of fundamental and
excited VB structures. Similar to MO-based CI methods, the excited
Journal of Computational Chemistry
DOI 10.1002/jcc
A VB Study of the O2 Molecule
VB structures are generated by replacing occupied orbitals with
virtual orbitals. To keep the lucidity of the VB wave function in
the VBCI expansion, the virtual orbitals should be strictly localized on precisely the same atom as the corresponding occupied
orbitals. Furthermore, the occupied orbitals are allowed to be
replaced by only those virtual orbitals that are localized on the
same atoms. In this manner, the entire VBCI wave function can be
compacted into a linear combination of the same minimal number
of VB structures as in the VBSCF and BOVB methods. In the
present article, VBCISD that involve single and double excitations
is applied to the ground state, while the VBCIS that involves only
single excitations is used for the two excited states of O2. There is
no doubt that VBCISD is definitely more accurate than VBCIS,
but it is also much more expensive.27
The Symmetries of States of the Dioxygen
Molecule O2
The dioxygen molecule consists of 12 valence electrons, which
form three kinds of VB orbitals: , , and lone electronic pairs.
All these 12 electrons are involved in VB calculations. As shown
in Scheme 1, the two atoms lie on z-axis, and the 2pz and 2s orbitals of the oxygen atom are hybridized to form two p-type bonding orbitals, labeled as p1z and p2z, and two lone pair orbitals of
2s character (held doubly occupied in all configurations, and not
shown in Schemes 1 and 2). The six electrons occupy four orbitals: p1x, p1y, p2x, and p2y; the px AOs are in the plane of the paper, while the py AOs are out of plane and are drawn as circles
with one lobe pointing at the observer.
Various VB methods, including VBSCF, BOVB, VBCIS, and
VBCISD, and three basis sets, 6-311þG*, cc-pVDZ, and ccpVTZ, are employed in this article. All orbitals are strictly localized to prevent any obscure interpretations. The bond lengths of
O2 in the ground state 3Sg and excited states 1Dg and 1Sgþ are
optimized at different levels of VB and MO methods. The VB
wave functions are kept at D2h symmetry during the computations.
In the CASSCF and MRCI calculations for the ground state, the
active space involves all the valence electrons, 12 electrons, in
eight orbitals. The VB calculations are carried out with the Xiamen
Valence Bond (XMVB) package of programs.28 To obtain basis
set integral and nuclear repulsion energy, the ROHF calculations
are carried out using GAUSSIAN 98.29 CASSCF and MRCI calculations are performed using MOLPRO 2000.30,31
A VB description of a molecule is usually based on atomic
orbitals. We begin with the separated oxygen atoms. The atomic
Scheme 1. The VB orbitals representation in a coordinate axis. The
py orbitals are drawn with one lobe pointing at the observer. The lone
pair orbitals along the z-axis are not drawn.
187
term of oxygen is 3P, having electronic configuration 1s2 2s2 2p4;
the doubly occupied 1s and 2s orbitals do not affect the state symmetries which are determined by the electron distribution in the
2p4 subshell. If we focus on those cases in which the two atoms
are neutral and linked by a covalent bond, there are four configurations which differ in the occupancy of the p-type orbitals, as
shown in Scheme 2. These four configurations are divided into
two degenerate pairs, (A1, A2) and (B1, B2), as done initially by
Goddard.13 In the pair (A1, A2), the two unpaired electrons are
located on the two mutually orthogonal orbitals px and py, such
that each plane (xz and yz) has three p electrons. On the other
hand, in the pair (B1, B2), there are two electrons in one plane
(xz or yz), while the other four electrons are in the other plane.
Different spin couplings for these four configurations lead to
different VB structures; our notation is Ai and Bi for configurations with no specific spin coupling, while the terms Ti and Si are
used for structures with definite spin quantum numbers. The total
eight structures generated from configurations Ai and Bi are
shown in Scheme 2, where the bond is drawn by a line. It is
commonly accepted14 that the two unpaired electrons in configurations A1 and A2 prefer the triplet spin coupling, which leads to
structure T1 and T2; while configurations B1 and B2, which have
two overlapping singly occupied orbitals, prefer the singlet coupling that leads to structures S1 and S2, which are the perfectly
paired structures. In addition to these four energy-preferred structures, there are four structures, S10 , S20 , T10 , and T20 , generated
from A1, A2, B1, and B2 by unfavorable spin coupling, namely
singlet (A1, A2) and triplet (B1, B2) couplings.
It can be seen that the ground state has a dilemma of choice
between being a diradical corresponding to a linear combination of
the T1 and T2 structures, with one bond and resonating threeelectron bonds in the xz and yz planes, or being perfectly paired
as a linear combination of structures S1 and S2, with a double
bond. As was shown already in Wheland’s work,6 the repulsion
between the two doubly occupied orbitals in B1 and B2 raises significantly the energy of the doubly bonded structure made from B1
and B2. Goddard13 argued that the repulsion overrides bonding.
It was further demonstrated by Shaik and Hiberty1,2 using a simple
Hückel-type VB theory that this is indeed true; the diradical
structure with resonating three-electron bonds does not suffer
from overlap repulsion and is inherently more stable than the doubly bonded structure. To assess these qualitative ideas, we performed energy calculations of the structures as well as configuration pairs by higher levels of VB methods. As mentioned
before, strictly localized orbitals are adopted to prevent any obscure interpretations.
Table 1 collects the energies of individual structures T1(T2),
S1(S2), T10 (T20 ), and S10 (S20 ) and the structure pairs by different
VB methods with three different basis sets. According to the rules
of qualitative VB theory,1,32 the reduced Hamiltonian matrix elements hT1|H|T2i and hS10 |H|S20 i are both negative, while hS1|H|S2i
and hT10 |H|T20 i are positive. Therefore, the most stable combinations of these individual configurations are (T1 þ T2), (S10 þ S20 ),
(S1 S2), and (T10 T20 ). Moreover, T2 is obtained from T1 by
replacement of two singly occupied orbitals (p1x, p2y) by two new
orbitals that overlap strongly with the former (p2x, p1y). Therefore,
the reduced matrix element hT1|H|T2i is expected to be large,1,32
and the (T1 þ T2) combination is expected to be much lower in
Journal of Computational Chemistry
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Su et al. • Vol. 28, No. 1 • Journal of Computational Chemistry
188
Scheme 2. The four important configurations (Ai and Bi; i ¼ 1, 2) and the eight VB structures generated from different spin couplings of these key configurations.
energy than T1(T2). On the other hand, T20 is obtained from T10
by substituting (p1y, p2y) by (p1x, p2x), and since the new orbitals
are orthogonal to the former, (T10 T20 ) is predicted to be only
marginally lower than the individual structures, T10 (T20 ). The same
reasoning explains why (S1 S2) is weakly stabilized relative to
its constituent configurations, while (S10 þ S20 ) is strongly stabilized. Scheme 3 provides a pictorial representation of the states
made from these structures, using VB mixing diagrams.1,32 Based
on these preliminary results, in the following discussions we use
A1 and A2 as the most important configurations for the ground
Table 1. The Energies of Two Pairs and Individual Structures with the Equilibrium Geometries (in Hartree).
6-311þG*
T1(T2)
S10 (S20 )
S1(S2)
T10 (T20 )
(T1 þ T2)
(S1 S2)
(S10 þ S20 )
(T10 T20 )
cc-pVDZ
cc-pVTZ
VBSCF
BOVB
VBCISD
VBSCF
BOVB
VBCISD
VBSCF
BOVB
VBCISD
149.3400
149.3360
149.3777
149.2719
149.3935
149.3807
149.3807
149.2732
149.3400
149.3360
149.3777
149.2719
149.4001
149.3842
149.3867
149.2745
149.5140
149.5096
149.5528
149.4444
149.5768
149.5604
149.5626
149.4470
149.4167
149.4126
149.4561
149.3476
149.4722
149.4594
149.4594
149.3492
149.4167
149.4126
149.4561
149.3476
149.4773
149.4625
149.4639
149.3502
149.5671
149.5626
149.6070
149.4936
149.6297
149.6142
149.6154
149.4995
149.4988
149.4940
149.5409
149.4251
149.5558
149.5414
149.5414
149.4264
149.4988
149.4940
149.5409
149.4251
149.5637
149.5483
149.5483
149.4278
149.7042
149.6991
149.7484
149.6292
149.7733
149.7586
149.7567
149.6318
Journal of Computational Chemistry
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A VB Study of the O2 Molecule
189
Scheme 3. (a) The ground state, 3Sg, and the triplet excited state, 3Du, generated from the mixture
of structures T1 and T2. (b) The first excited state, 1Dg, and the second excited state, 1Sgþ, generated
from the mixture of structures S1 and S2, S10 , and S20 .
state, while the excited states will be formed from the singlet coupling of the A1 and A2 forms, as well as from the two singlet combinations of the B1 and B2 forms.
Let us now find the symmetries of the states. Though one can
use simple MO theory (using complex * orbitals) to show that the
ground state of oxygen molecule is 3Sg and the lowest excited
states are 1Dg and 1Sgþ, here we prefer to derive the symmetries
directly from VB wave functions. These assignments are already
included in Scheme 3, and the reader should refer to the scheme.
As shown in Table 1 and in Scheme 3a, the ground state of
the molecule is nascent from triplet coupling of configurations A1
and A2, and the wave function of the ground state is the positive
combination of triplet structures2 T1 and T2:
1 ¼ T1 þ T2 :
(3)
Neglecting all doubly occupied orbitals and orbitals, which do
not make contributions to the symmetries of the states, eq. (3)
may be written as,
Journal of Computational Chemistry
1 ¼ jp1x p2y j þ jp2x p1y j
DOI 10.1002/jcc
(4)
Su et al. • Vol. 28, No. 1 • Journal of Computational Chemistry
190
where all the spin orbitals are associated with spin and coupled to
a triplet state. Applying symmetry operations immediately shows
that the triplet ground state C1 [eq. (4)] is antisymmetrical with
respect to reflection in a vertical plane, symmetrical with respect to
the center of inversion, and antisymmetrical with respect to C2 axes,
which characterize a 3Sg state in the D?h point group. Similarly, it
appears by inspection that the singlet state (S1 S2) transforms like
(x2 y2), which characterizes a 1Dg state, and that (S10 þ S20 ) transforms like xy, which characterizes the companion component of the
1
Dg irreducible representation, degenerate with the former (Scheme
3b). Finally, the (S1 þ S2) state, which transforms like (x2 þ y2),
must be of 1Sgþ symmetry. Owing to the weak hS1|H|S2i matrix
element, this latter state can be expected to be only slightly higher
in energy than the individual structures, S1/S2, and way below the
other excited states built on anti-resonating combinations of Ti, Ti 0 ,
Si, or Si 0 . Thus, simple considerations lead to the ordering 3Sg,
two degenerate 1Dg, and 1Sgþ for the low-lying states of dioxygen.
Let us now improve this simple scheme by performing quantitative VB calculations, including all relevant configurations that
can be predicted to participate to the lowest-lying states.
For the ground state, one begins with the ‘‘parent’’ configurations,
A1 and A2 (Scheme 2). To cover all the important structures, one
considers configurations in which three electrons are in the two px orbitals, three electrons are in the two py, and two in the two pz orbitals.
Thus, in total there are 12 configurations, as shown in Scheme 4. It is
clear that configurations A1–A4 correspond to covalent structures
(two neutral atoms, each possessing four electrons), A5–A10 are for
mono-ionic structures, and the last two, A11 and A12, are for diionic structures, which can be expected to be very high-lying and
will be negligible.
Methodology of VB Structure Selection
To have a balance between accuracy of the numerical results and
a compact form of the VB wave function, one of the most important steps is to choose VB structures that will be used in VB calculations. In the previous VB studies,14–16 the choice of eight
structures were based on Mcweeny’s work.14 In this article, we
begin with a full VB structure set and then condense it to the final
VB structure set step by step.
For a system of spin S with N electrons and m orbitals, the number of independent VB structures is given by the Weyl formula:33
Dðm; N; SÞ ¼
2S þ 1
mþ1
mþ1
1
2N þ S þ 1
mþ1
:
1
2N S
(5)
For the triplet ground state of oxygen molecule, if we take all six
p-type orbitals and eight electrons into account, there are totally
105 VB canonical structures. Let us recall that the number of canonical structures is independent of the basis set, and that, in basis sets that
are larger than minimal, the AOs that are used to represent a canonical
structure are made of combinations of basis functions of the same
symmetry. For the two excited states, there are also 105 VB structures
with singlet spin coupling. But it is easy to show that most of VB
structures make minor or zero contributions to the states because of
unfavorable bonding patterns or because of mismatch of orbital symmetry. In this article, the choice of VB structures that are involved in
the calculations is made in two different ways. One is to start with the
most important configurations, which are A1 and A2 for the triplet
ground state and B1 and B2 for the singlet excited states, and then to
derive all the important structures from them by the uses of mathematical consideration and chemical reasoning. The other way is to
perform a full VBSCF calculation of 105 structures, and select the
structures that have the largest coefficients CK in the wave function,
for subsequent calculations. It is obvious that the former choice is
more physical, while the latter is more mathematical. As can be seen
later, both of them lead to the same selection of VB structures.
Scheme 4. The 12 configurations necessary for producing a consist-
ent and accurate VB structure set for the ground state of O2.
Journal of Computational Chemistry
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A VB Study of the O2 Molecule
191
Let us now couple the electrons of the above configurations so
as to generate triplet VB structures, which ensure a balanced
description of the molecule at both short and large interatomic distances. At long distances, only configurations A1 and A2 are important, since they correlate with the ground states of the separate
oxygen atoms. For a four-electron four-orbital system with triplet
coupling, there are three structures with independent coupling
modes. The choice of these three structures is arbitrary. For configuration A1, McWeeny14 and van Lenthe16 argued that in order to
ensure correct dissociation, one has to take along with structure
T1, two other structures, labeled as W1 and W2, which are shown
in Scheme 5 (both are derived from the D atomic states). The symmetry adaptation of the W1 and W2 structures to give a triplet state
leads to the following combination:14
aT1 þ bðW1 W2 Þ:
(6)
Expressed explicitly, these VB structures are linear combinations
of two determinants, i.e:
T1 ¼ jp1z p2z p1x p2y j jp1z p2z p1x p2y j
(7)
W1 ¼ jp1z p1x p2z p2y j jp1z p1x p2z p2y j
¼ jp1x p2y p1z p2z j þ jp1z p2z p1x p2y j
ð8Þ
W2 ¼ jp2z p2y p1z p1x j jp2z p2y p1z p1x j
¼ jp1x p2y p1z p2z j þ jp1z p2z p1x p2y j:
ð9Þ
Using eqs. (7)–(9), the McWeeny–van Lenthe wave function in
eq. (6) becomes:
aT1 þ bðW1 W2 Þ ¼ ða bÞ jp1z p2z p1x p2y j jp1z p2z p1x p2y j
þ b jp1x p2y p1z p2z j jp1x p2y p1z p2z j ¼ ða bÞT1 þ bT3 : ð10Þ
Here, T3 is a new structure, shown at the bottom of Scheme 5.
This is often the case in VB theory that the combination of two
non-orthogonal VB structures leads to a third structure.32 Thus,
Scheme 6. The selected 12 structures that lead to a consistent and
accurate calculation of the ground state of O2.
Scheme 5. The two structures (W1, W2) required for the work of
McWeeny14 and van Lenthe16 and the new structure (T3) generated
from them; see eq. (10) in the text.
eq. (10) shows that instead of using three structures, one can economize and use the more compact form made from T1 and T3 only.
Based on this result, we can use only four structures that are nascent from the diradical forms, A1 and A2, and are denoted as
T1–T4 in Scheme 6.
At short distances, the two electrons that are in orbitals in
A3–A10 must be singlet-coupled to form the bond, and there
remains to couple the additional two unpaired electrons in a
triplet manner, thus leading to T5–T12. As such, with the above
simple analysis, we remain with the 12 structures T1–T12 that
Journal of Computational Chemistry
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Su et al. • Vol. 28, No. 1 • Journal of Computational Chemistry
192
Table 2. The Weights (WK) and Coefficients(CK) of the 12 Selected
Structures in the 105-Structure VBSCF/cc-pVTZ Calculation for the
Ground State.
R ¼ 10 Å
R ¼ R0
Structure
T1
T2
T3
T4
T7
T8
T5
T6
T9
T10
T11
T12
WK
CK
WK
CK
0.2292
0.2292
0.0004
0.0004
0.0856
0.0856
0.0754
0.0754
0.0540
0.0540
0.0540
0.0540
0.3572
0.3572
0.0206
0.0206
0.1684
0.1684
0.1727
0.1727
0.1173
0.1173
0.1173
0.1173
0.2500
0.2500
0.2500
0.2500
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.5000
0.5000
0.5000
0.5000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
are supposed to describe the dioxygen ground state in a balanced
way at any interatomic distance.
As mentioned earlier, a ‘‘computational’’ way to choose the
structures is by performing a full VBSCF calculation of all the 105
structures and selecting the most important ones for subsequent calculations. The corresponding VBSCF calculation shows that among
these 105 structures, at the equilibrium geometry, only 10 structures, T1, T2, and T5–T12, have weights larger than 0.001 at the
equilibrium geometry, while only T1–T4 possess nonzero weights
at large distances. Table 2 lists the weights and coefficients of these
12 structures in the full VBSCF calculation of 105 structures. As
can be seen, the combined weights for T1–T12 in the full VBSCF
reaches 99.9% of the total wave function, thus confirming the validity of the selection of VB structures based on physical principles.
Therefore, based on the analysis and the test calculations
above, 12 structures, T1–T12, are adopted for all the levels of VB
calculations employed in the study, including VBSCF, BOVB,
and VBCI, for the ground state. To double-check, a full VBSCF
calculation of 105 structures is also performed to validate the
compact wave function of T1–T12 for the entire dissociation
curve. By comparison, the van Lenthe’s study used six structures
in the ‘‘proper dissociation model’’ and eight structures in ‘‘proper
reference model.’’ In addition, the VB orbitals, in the latter study,
were allowed to be semi-delocalized so that the contributions
from ionic structures were implicitly included.16
The excited states, 1Dg and 1Sgþ, can be generated from the
negative and positive combinations of the S1 and S2 structures,
the classical doubly bonded O¼
¼O structures (see Scheme 3b). At
large distances, the alternative coupling of this four-electron fourorbital system is important, so the S3 and S4 structures must be
added (Scheme 7). At short distances, the ionic components of
the and bonds must be added to S1 and S2 to attain quantitative accuracy. Further removal of the very high-lying di-ionic
Scheme 7. The selected 16 structures that lead to a consistent and
accurate calculation of the excited states of O2.
Journal of Computational Chemistry
DOI 10.1002/jcc
A VB Study of the O2 Molecule
Table 3. The Weights (WK) and Coefficients (CK) of 16 Selected
Structures in the 105-Structure VBSCF Calculation for the 1Dg State.
R ¼ 10 Å
R ¼ R0
Structure
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
WK
CK
WK
CK
0.2720
0.2710
0.0158
0.0158
0.0588
0.0587
0.0588
0.0587
0.0331
0.0330
0.0331
0.0330
0.0302
0.0302
0.0302
0.0302
0.4236
0.4229
0.0449
0.0448
0.1294
0.1292
0.1294
0.1292
0.0997
0.0995
0.0997
0.0995
0.1070
0.1068
0.1070
0.1068
0.5000
0.5000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.8165
0.8165
0.4083
0.4083
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
structures of the type O2þO2– leads to the 16 VB structures
S1–S16 displayed in Scheme 7. In a similar strategy as employed
for the ground state, here too a full 105-structure VBSCF calculation is performed. Table 3 shows the contributions from the 16
structures of Scheme 7 to the total wave function of the first
excited state, 1Dg. As can be seen, the sum of the weights of these
16 structures reaches 99.99% for the equilibrium geometry. At
the dissociation limit, only four structures possess nonzero contributions to the wave function, and only two of these have also
nonzero weights. It should be pointed out that though the weights
Table 4. The Spectroscopic Constants of the O2 Ground State with
Various Methods.
Method
Basis set
Re (Å)
De (eV)
!e (cm1)
VBSCF
6-311þg*
cc-pVDZ
cc-pVTZ
1.242
1.253
1.242
3.19
3.05
3.33
1508.2
1576.6
1521.4
BOVB
6-311þg*
cc-pVDZ
cc-pVTZ
1.242
1.252
1.242
4.28
4.12
4.48
1516.6
1619.2
1536.7
VBCISD
6-311þg*
cc-pVDZ
cc-pVTZ
1.242
1.252
1.236
4.37
4.22
4.77
1500.1
1571.7
1544.6
VBSCF(105)a
Goddard13
Byrman16
Guberman11
Schaefer III12
Pittner10
CASSCF
MRCI
Expt34
cc-pVTZ
Dzd
EZPP
[3s/2p/1d]
[4s/2p]
cc-pVTZ
cc-pVTZ
cc-pVTZ
1.242
1.238
1.218
1.227
1.220
1.201
1.218
1.214
1.208
3.37
4.88
3.67
3.72
4.72
1521.4
1693.0
1549.0
1539.2
1614.0
1661.4
1574.6
1579.9
1580.0
4.07
4.86
5.21
193
of structures S3 and S4 are zero, their coefficients are nonzero
and they are important for the dissociation limit. Hence, the two
ways of selecting the configuration lead to the same 16 structures.
Therefore, these 16 structures are used for VBSCF and VBCIS
calculations for the excited states. By comparison, the van Lenthe’s
study16 used four structures in the PD model and eight structures
in the PR model (still with semi-delocalized orbitals).16
Computational Results
Spectroscopic Constants and the Potential Energy Curves
Table 4 shows the VB calculated spectroscopic constants at the
various levels and various basis sets, alongside the computational
results obtained by use of sophisticated MO-based methods. As
can be seen, the VB optimized equilibrium bond lengths range in
between 1.236 and 1.253 Å, which is somewhat longer than the
experimental value,34 1.208 Å, by 0.03–0.04 Å. The value of
VBCISD, 1.236 Å, is virtually identical to the GVB-CI value of
Goddard,13 1.238 Å.
It can be seen from Table 4 that as expected, the VBSCF values
of dissociation energy cover only 59–64% of the experimental
value for various basis sets. The BOVB and VBCISD values get
significant improvements from the VBSCF method. The most accurate VB dissociation energy is the 4.77 eV result of the VBCISD/
cc-pVTZ calculation which is in very good agreement with the
value of MRCI/cc-pVTZ, 4.86 eV, reaching 92% value of the experimental value,34 5.21 eV. The results show that VB theory with
good computational levels, such as BOVB and VBCI, is able to provide not only intuitive insights into chemical problems but also
accurate quantitative results, which match sophisticated MO-based
methods.
Table 4 also collects the vibrational frequencies obtained with
the various methods. As can be seen, the VB methods give also
reasonably good values of !e, compared to those of MO-based
methods and experiment.34
Table 5 shows the VB calculated spectroscopic constants for the
two excited states. Only VBSCF and VBCIS are performed, as
BOVB and VBCISD calculations of 16 structures are too demanding for the available computational resources. It is obvious that here
Table 5. The Spectroscopic Constants of the First and Second Excited
States of O2.
Dg
1
Sgþ
1
Method
Basis set
Re (Å)
De (eV)
!e (cm1)
VBSCF
VBCIS
Byrman16
Goddard13
Pittner10
Expt34
VBSCF
VBCIS
Byrman16
Pittner10
Expt34
cc-pVTZ
cc-pVTZ
EZPP
Dzd
cc-pVTZ
1.261
1.250
1.233
1.249
1.210
1.216
1.290
1.275
1.252
1.222
1.227
2.450
3.880
2.799
3.790
1408.3
1492.5
1445.0
1595.0
1583.9
1509.0
1481.7
1260.8
1320.0
1491.0
1433.0
cc-pVTZ
cc-pVTZ
EZPP
cc-pVTZ
a
A full 105-structure VBSCF calculation.
Journal of Computational Chemistry
DOI 10.1002/jcc
4.232
1.670
2.980
2.036
3.578
Su et al. • Vol. 28, No. 1 • Journal of Computational Chemistry
194
Table 6. The Excitation Energy Computed at the Optimized Equilibrium
Distance (eV).
Method
Basis set
Te(3Sg ? 1Dg)
Te(3Sg ? 1Sgþ)
VBSCF
VBCIS
Goddard13
Pittner10
Expt34
cc-pVTZ
cc-pVTZ
Dzd
cc-pVTZ
0.885
0.954
1.089
1.026
0.982
1.661
1.862
1.777
1.636
too, the VBSCF method is not accurate enough in reproducing the
dissociation energies and the optimized bond lengths of the two
excited states. The values of dissociation energy and bond length
computed by VBCIS method are closer to the experimental data34
because of partial accounting of dynamic correlation by this
method. However, the value of the vibrational frequency for the
1
Sgþ state is still underestimated by *170 cm1.
Table 6 shows the excitation energy from the ground state to
the two excited states computed by VBSCF and VBCIS with the
cc-pVTZ basis set. The energies of the three states are computed
at the optimized bond length. It can be seen that for the excitation
energy from the ground state to the first excited state, Te(3Sg ?
1
Dg), the VBCIS method gives a very good value with a deviation
of 0.038 eV from the experiment value,34 but the VBSCF method
underestimates the values. For the excitation energy from ground
state to the second excited state, Te(3Sg ? 1Sgþ), the VBSCF
method gives good results, and its deviation is 0.025 eV from that
experimental data. However, the value of VBCIS method is overestimated by 0.226 eV relative to experiment.
Figure 1 shows the PECs of the ground state with various VB
methods. Infact, the PEC of VBSCF with the 105 structures,
denoted as VBSCF(105) coincides exactly with the VBSCF curve
based on the 12-structure. This confirms that the selected 12
structures account for virtually all the contribution to the total
energy not only for equilibrium geometry but also for the entire
energy curve. Both the VBSCF and BOVB curves dissociate to
Figure 1. The dissociation energy curve of the ground state of O2,
computed by various VB methods with cc-pVTZ basis set.
the same values, which is precisely the sum of the energies of
two oxygen atoms of 3P electronic states. The VBCISD curve in
Figure 1 is lower in energy than others throughout, but it runs
almost parallel to the VBSCF curves. The computed energy
curves show at all levels, that the ground state, 3Sg, dissociates
to two oxygen atoms of triplet state 3P. This will be discussed in
the analysis of the wave function later.
One of the interesting features that come up in the studies of
O2 is the existence of a small barrier in the PEC, at about 2.1 Å, in
many MO and VB studies.14,16 The curves between 2.0 and 4.0 Å
in the ground state potential surface with VBSCF, CASSCF, and
BOVB calculations are shown in Figure 2, where the total energy
at dissociation limit for all methods is set to zero by shifting PECs.
Such a barrier of 1.0 kcal/mol is also observed in the VBSCF calculation. By contrast, both the BOVB and VBCISD curves have
no humps. The same phenomenon occurs in the MO-based calculations: a small barrier is observed in CASSCF calculation but there
is none on the MRCI curve. It is reasonable to assume that the
small barrier in the curve is due to the lack of dynamic correlation
in VBSCF and CASSCF calculation. The VBCISD, BOVB, and
MRCI methods all take dynamic correlation into account so that
the small barrier en route to dissociation disappears.
Figure 3 shows the dissociation energy curves of the two excited states, computed with the VBSCF and VBCIS methods. It
can be seen that the dissociations of 1Dg and 1Sgþ lead to the same
dissociation limit as the 3Sg state, as expected from the lineage of
these states to the A1 and A2 configurations (Scheme 2).
The VB Wave Functions of O2 at Equilibrium Geometry
One of the advantages of VB theory is its ability to provide intuitive insights through its compact wave function. Table 7 collects
the structural weights for the ground state’s wave function at equilibrium geometry. The values of weights for the three VB methods
are in good mutual agreement. Particularly, the 12-structure
VBSCF weights (Table 7) are virtually identical to those of the
Figure 2. The VB computed curves for the ground state of O2 in
the range of 2.0–4.0 Å, with various VB methods with cc-pVTZ
basis set.
Journal of Computational Chemistry
DOI 10.1002/jcc
A VB Study of the O2 Molecule
Figure 3. The dissociation energy curves of the two excited states
computed by VBSCF and VBCIS methods with cc-pVTZ basis set.
105-structure VBSCF calculation (see Table 2). As we reasoned already, structures T1 and T2 dominate the wave function of the
ground state; however, the weights of T1 and T2 only cover 36–
46% of total wave function in the three VB methods. This suggests
that the resonance including ionic structures makes a very important contribution to the total energy of O2, as deduced already by
Galbraith et al. for the bonding in the 1Dg state.35 The resonance
energy arising from covalent–ionic mixing in the bond can be
accurately estimated by comparing the energy of a state displaying
a pure covalent bond (made of an optimized combination of T1,
T2, T7, T8) to the energy of the full ground state (T1–T12). The
difference, which accounts for the covalent–ionic resonance
energy of the bond, amounts to 1.87 eV (43 kcal/mol) at the
VBCISD level in cc-pVTZ basis set. This can be compared to resonance energy of 51 kcal/mol that has been calculated for the
O
O single bond in HO
OH,36 which accounts for the totality
of the bonding energy of this latter molecule. Therefore, the bond in dioxygen, just as its homolog in hydrogen peroxide, is
195
characterized by charge-shift bonding, which should be reflected
in other properties of the molecule such as depleted electron density in the bonding region, etc.36,37 One must remember that the
optimized structure of T1, T2, T7, and T8 involves resonance
energy in the system. The -resonance energy can be quantified
relative to the structure pair T1 þ T2, and is 2.57 eV. As such,
both the and bonds in dioxygen are charge-shift bonds.
Similarly, Table 8 collects the structure weights and coefficients for one of the 1Dg excited states at its equilibrium geometry
(the second sate, which is degenerate with the former, has not
been calculated). The wave function of the 1Sgþ state only differs
from the previous one in the relative signs of the structure coefficients. As such, the results for the 1Sgþ state are not collected. It
is shown that structures S1 and S2 dominate the wave function,
and their weights cover 43–54% of the total wave function in the
1
Dg and 1Sgþ excited states.
S1 and S2 are parent structures for the excited states, with
covalent bonds in the and senses. The structures S5, S7, S9,
S11, S13, and S15 are from the corresponding and ionic structures of S1. Thus, the combination of these latter structures, together with S1 and S3, form a classical doubly bonded structure,
K1, which displays a bond and a bond in the yz plane, both
bonds being of the Lewis type, involving their covalent and ionic
components. Similarly, the combination of S2, S4, S6, S8, S10,
S12, S14, and S16 forms an analogous structure K2, in which the
bond is now in the xz plane. The negative combination of K1
and K2 corresponds to the 1Dg state, and the positive combination
to the 1Sgþ state.
The Dissociation to Oxygen Atoms
From the computed PECs, all three states dissociate to two oxygen atoms of triplet state 3P. However, this feature has to be confirmed by the analysis that demonstrates that indeed, the wave
Table 8. The Weights (WK) and Coefficients (CK) of Structures with
cc-pVTZ Basis Set for the 1Dg State in its Equilibrium Geometry.
VBSCF
BOVB
VBCISD
Table 7. The Weights (WK) and Coefficients (CK) of Structures with
cc-pVTZ Basis Set for the Ground State in its Equilibrium Geometry.
VBSCF
Structure
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
BOVB
VBCISD
WK
CK
WK
CK
WK
CK
0.2302
0.2302
0.0004
0.0004
0.0859
0.0859
0.0746
0.0746
0.0545
0.0545
0.0545
0.0545
0.3580
0.3580
0.0206
0.0206
0.1694
0.1694
0.1711
0.1711
0.1182
0.1182
0.1182
0.1182
0.1749
0.1819
0.0004
0.0004
0.1218
0.1218
0.0675
0.0675
0.0646
0.0673
0.0673
0.0646
0.2845
0.2936
0.0207
0.0207
0.2211
0.2211
0.1566
0.1566
0.1405
0.1448
0.1448
0.1405
0.2273
0.2273
0.0004
0.0004
0.1184
0.1184
0.0660
0.0660
0.0439
0.0439
0.0439
0.0439
0.3448
0.3448
0.0180
0.0180
0.1996
0.1996
0.1486
0.1486
0.0978
0.0978
0.0978
0.0978
Structure
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
Journal of Computational Chemistry
WK
0.2725
0.2717
0.0158
0.0157
0.0587
0.0586
0.0587
0.0586
0.0332
0.0332
0.0331
0.0331
0.0301
0.0300
0.0301
0.0300
CK
WK
CK
WK
CK
0.4243
0.2158
0.3446
0.2449
0.3801
0.4237
0.2153 0.3442
0.2443 0.3796
0.0447 0.0270
0.0742 0.0125
0.0358
0.0446 0.0269 0.0741 0.0125 0.0358
0.1293
0.0992
0.1986
0.0614
0.1289
0.1291
0.0988 0.1982
0.0612 0.1288
0.1293
0.0622
0.1387
0.0614
0.1288
0.1291
0.0620 0.1385
0.0612 0.1287
0.0997
0.0518
0.1402
0.0441
0.1185
0.0997
0.0517 0.1401
0.0440 0.1183
0.0996
0.0482
0.1331
0.0441
0.1185
0.0996
0.0481 0.1330
0.0440 0.1183
0.1065
0.0250
0.0893
0.0287
0.0945
0.1064
0.0250 0.0892
0.0286 0.0944
0.1065
0.0254
0.0906
0.0287
0.0945
0.1064
0.0253 0.0905
0.0286 0.0944
DOI 10.1002/jcc
Su et al. • Vol. 28, No. 1 • Journal of Computational Chemistry
196
function evolves into the wave functions of the two oxygen atoms
each with a 3P state.
Tables 2 and 3 show the coefficients and weights of structures
at infinity for the ground state and the first excited state. As can
be seen, for the ground state, the wave function is expressed in
linear combinations of four structures,
3Sg ðR ¼ 1Þ ¼ 0:5ðT1 þ T3 Þ þ 0:5ðT2 T4 Þ:
(11)
The first parentheses can be written explicitly in terms of VB
determinants,
T1 þ T3 ¼ jðp1z p2z p1z p2z Þp1x p2y j þ jðp1x p2y p1x p2y Þp1z p2z j
p2z p2y Þj:
¼ jðp1z p1x þ p1z p1x Þp2z p2y j jp1z p1x ðp2z p2y þ (12)
At infinity, the electrons between the two oxygen atoms do not
interact. Thus, eq. (12) is equivalent to the form of simple products of the wave functions for the two atoms,
T1 þ T3 ¼ jðp1z p1x þ p1z p1x Þkp2z p2y j jp1z p1x kðp2z p2y
þ p2z p2y Þj:
ð13Þ
In the first term of the first determinant eq. (13) contains displays
a 3P triplet coupling (S ¼ 1, Ms ¼ 0) between the orbitals p1z and
p1x of the first oxygen atom, and the other determinant displays
another 3P triplet coupling (S ¼ 1, Ms ¼ 1) in the second atom.
Similarly, the second product of determinants in eq. (13) corresponds to two independent oxygen atoms, both in a 3P triplet
state. Thus, the combination of T1 and T3 describes two triplet
oxygen atoms coupled to a triplet molecular state. In a similar
fashion, it would be shown that the combination of T2 and T4
also describes the coupling of the two triplet oxygen atoms.
Based on the derivations, it is clear that the VB wave function for
the triplet ground state of dioxygen dissociates to the two triplet
oxygen atoms.
For the 1Dg and 1Sgþ excited states, the nonzero contributions
to the wave function are from structures S1–S4, i.e.,
1g ðR ¼ 1Þ ¼ ðS1 þ 0:5S3 Þ ðS2 þ 0:5S4 Þ
(14)
1Sþ ðR ¼ 1Þ ¼ ðS1 þ 0:5S3 Þ þ ðS2 þ 0:5S4 Þ
(15)
g
where the normalization factor is neglected. Similarly to the
ground state, we begin here with the combination of S1 and S3.
S1 þ 0:5S3 ¼ jp1z p1y k
p2z p1z p2y j j
p1y kp2z p2y j
þ 0:5jp1z p1z p1y kp2z p1y þ p2y þ p2z p2y j:
(17)
In eq. (17), the first term corresponds to a coupling of the two triplet oxygen atoms, one is for MS ¼ 1, the other is for MS ¼ 1,
and so is the second term. The last term also describes the coupling of the two triplet oxygen atoms, but it is for MS ¼ 0 for both
atoms. The equation corresponds to the structure K1 mentioned
earlier for dissociation limit. Similar to S1 and S3, the combination of S2 and S4 also describes the triplet oxygen atoms coupled
to a singlet state of the dioxygen molecule and corresponding to
the structure K2 for the dissociation limit. The negative combination of the two at the dissociation limit describes the first excited
state 1Dg; then the positive combination of them describes the
second excited state 1Sgþ. The two excited states have the same
energy in the dissociation limit, both converging to the two 3P
oxygen atoms limit.
Conclusion
In this article, the dioxygen molecule is studied by ab initio VB
methods, including the spectroscopic data, PECs, and the analysis
of the wave functions. VB structures are carefully selected to avoid
missing any important structures. The 12 structures are used for
VBSCF, BOVB, and VBCISD calculations for the ground state.
The computed spectroscopic properties of VB methods are in good
agreement with the previous studies and experimental values. Particularly, the high levels VB methods, BOVB and VBCISD, provide very accurate values of dissociation energy. The VBCISD/
cc-pVTZ value covers 92% of experimental data, which matches
the MRCI result very well. For the excited states, 16 structures also
provide quantitatively correct description. In addition, a full set of
105 structures are employed for VBSCF calculations. The computation results show the validity of the choice of structures.
Like in other previous VB studies,14,16 a small barrier exists in
the VBSCF dissociation energy curve. However, higher level VB
methods, BOVB and VBCISD, dissolve the barrier. This means
that the origin of the barrier is due to an artifact of calculations that
lack dynamic correlation. The study of this paper shows that the
‘‘mythical failure’’ of VB theory in the early VB period may have
originated in the lack of quantitative studies. Modern VB methods
are able to provide a very clear description for the nature of bonding for oxygen molecule, not only for qualitative interpretation,
but also for quantitative purpose. Furthermore, recalling that the
three wave functions are dominated by a few structures, e.g.,
T1 and T2 for the ground state, one can qualitatively understand
the states of the O2 molecule with ease and facility, comparable to
the MO method.1,2
S1 þ 0:5S3 ¼ jðp1z p2z p1z p2z Þðp1y p2y p1y p2y Þj
References
þ 0:5jðp1z p1y p1z p1y Þðp2z p2y p2z p2y Þj
¼ jp1z p1y p2z p2y j jp1z p1y p2z p2y j
þ 0:5jp1z p1y þ p1z p1y kp2z p2y þ p2z p2y j:
Then we have,
(16)
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A VB Study of the O2 Molecule
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Journal of Computational Chemistry
DOI 10.1002/jcc