a valence bond study of the dioxygen...

A Valence Bond Study of the Dioxygen Molecule

PEIFENG SU1, LINGCHUN SONG,1 WEI WU,1 PHILIPPE C. HIBERTY,2 SASON SHAIK3

1Department of Chemistry, State Key Laboratory of Physical Chemistry of Solid Surfaces,Center for Theoretical Chemistry, Xiamen University, Xiamen 361005, People’s Republic of China

2Laboratoire de Chimie Physique, Groupe de Chimie Theorique, Universite de Paris-Sud,91405 Orsay Cedex, France

3Department of Organic Chemistry and Lise Meitner-Minerva Center for Computational QuantumChemistry, The Hebrew University, Jerusalem 91904, Israel

Received 31 March 2006; Accepted 22 April 2006DOI 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 consid-

ered 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 electronega-

tive 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 quantita-

tive 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 pic-

ture 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 inter-

molecular 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 Huckel-

type VB theory shows that this is not true, the ‘‘failure’’ has some-

how stuck to VB theory.1,2 Our article addresses the VB descrip-

tion 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 ox-

ygen 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 Poschl8 who

treated the O2 molecule with VB principles and concluded that

‘‘the 3Sg� term . . . giving the fundamental state of the molecule.’’

Contract grant sponsors: Natural Science Foundation of China; Israel

Science Foundation (ISF)

Correspondence to: W. Wu; e-mail: [email protected]

q 2006 Wiley Periodicals, Inc.

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 exceed-

ingly 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 mod-

ern days of quantum chemistry? Well, both MO and VB theories

have found that O2 requires more than the simple MO or VB treat-

ment; MO requires extensive CI,1–4,10–12 and VB required exten-

sive 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 prop-

erly describe the dissociation, and good dissociation energies re-

quired a GVB-CI calculations; thus sacrificing somewhat the sim-

plicity advantage of VB theory. McWeeny14 calculated the ground

state of O2 at the experimental equilibrium distance using a mini-

mal basis with eight VB structures. He calculated the ground state

energies of wave functions containing two, four, and eight struc-

tures, 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 con-

tributes most to the bonding energy, using four of the eight VB

structures given by McWeeny. In 1995, a VBSCF study was car-

ried 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 dissoci-

ation energy for the ground state was 2.832 and 3.672 eV, respec-

tively for the two models, thus covering 54 and 70% of experimen-

tal values. The authors observed a small barrier on the dissociation

potential curve of the ground state and stated that the hump origi-

nates from a ‘‘spin-recoupling.’’ However, they did not reach a de-

finitive 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 dissoci-

ation 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 pro-

vided a qualitatively correct prediction of the ground state of oxy-

gen molecule, the quantitative performance was still unsatisfac-

tory, 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 ener-

gies and reaction barriers and so on, were lacking quantitative

accuracy. However, thanks to the rapid progresses in computer sci-

ence, 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 initioVB methods using high computational levels. The aim of the pres-

ent 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 computa-

tional details are reported in the next section, including the quali-

tative 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,

� ¼XK

CK�K (1)

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 solv-

ing 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

XL6¼K

CKCLh�K j�Li: (2)

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 simulta-

neously to minimize the total energy. The VBSCF method takes

care of static electron correlation, but lacks dynamic correla-

tion,25 an absolutely essential ingredient for attaining quantitative

accuracy. As such, the VBSCF results are only qualitatively cor-

rect, 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 orbi-

tals 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

186 Su et al. • Vol. 28, No. 1 • Journal of Computational Chemistry

Journal of Computational Chemistry DOI 10.1002/jcc

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 local-

ized 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 orbi-

tals of the oxygen atom are hybridized to form two p�-type bond-

ing 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 orbi-

tals: p1x, p1y, p2x, and p2y; the px AOs are in the plane of the pa-

per, 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 cc-

pVTZ, are employed in this article. All orbitals are strictly local-

ized to prevent any obscure interpretations. The bond lengths of

O2 in the ground state 3Sg� and excited states 1Dg and 1Sg

þ 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 calcu-

lations 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

term of oxygen is 3P, having electronic configuration 1s2 2s2 2p4;

the doubly occupied 1s and 2s orbitals do not affect the state sym-

metries 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 config-

urations 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 configura-tions 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 configu-

rations 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 cou-

pling that leads to structures S1 and S2, which are the perfectly

paired structures. In addition to these four energy-preferred struc-

tures, there are four structures, S10, S20, T10, and T20, generatedfrom 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 three-

electron � 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 sig-

nificantly 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

Huckel-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 dou-

bly bonded structure. To assess these qualitative ideas, we per-

formed energy calculations of the structures as well as configu-

ration pairs by higher levels of VB methods. As mentioned

before, strictly localized orbitals are adopted to prevent any ob-

scure 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 ele-

ments hT1|H|T2i and hS10|H|S20i are both negative, while hS1|H|S2iand hT10|H|T20i are positive. Therefore, the most stable combina-

tions 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

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.

187A VB Study of the O2 Molecule


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 stabi-

lized. 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

Scheme 2. The four important configurations (Ai and Bi; i ¼ 1, 2) and the eight VB structures gener-

ated from different spin couplings of these key configurations.

Table 1. The Energies of Two Pairs and Individual Structures with the Equilibrium Geometries (in Hartree).

6-311þG* cc-pVDZ cc-pVTZ

VBSCF BOVB VBCISD VBSCF BOVB VBCISD VBSCF BOVB VBCISD

T1(T2) �149.3400 �149.3400 �149.5140 �149.4167 �149.4167 �149.5671 �149.4988 �149.4988 �149.7042

S10(S20) �149.3360 �149.3360 �149.5096 �149.4126 �149.4126 �149.5626 �149.4940 �149.4940 �149.6991

S1(S2) �149.3777 �149.3777 �149.5528 �149.4561 �149.4561 �149.6070 �149.5409 �149.5409 �149.7484

T10(T20) �149.2719 �149.2719 �149.4444 �149.3476 �149.3476 �149.4936 �149.4251 �149.4251 �149.6292

(T1 þ T2) �149.3935 �149.4001 �149.5768 �149.4722 �149.4773 �149.6297 �149.5558 �149.5637 �149.7733

(S1 � S2) �149.3807 �149.3842 �149.5604 �149.4594 �149.4625 �149.6142 �149.5414 �149.5483 �149.7586

(S10 þ S20) �149.3807 �149.3867 �149.5626 �149.4594 �149.4639 �149.6154 �149.5414 �149.5483 �149.7567

(T10 � T20) �149.2732 �149.2745 �149.4470 �149.3492 �149.3502 �149.4995 �149.4264 �149.4278 �149.6318



state, while the excited states will be formed from the singlet cou-

pling of the A1 and A2 forms, as well as from the two singlet com-

binations 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,

�1 ¼ jp1xp2yj þ jp2xp1yj (4)

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.



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) trans-

forms like xy, which characterizes the companion component of the1Dg irreducible representation, degenerate with the former (Scheme

3b). Finally, the (S1 þ S2) state, which transforms like (x2 þ y2),must be of 1Sg

þ 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, Ti0,Si, or Si0. 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 quanti-

tative VB calculations, including all relevant configurations that

can be predicted to participate to the lowest-lying states.

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 impor-

tant steps is to choose VB structures that will be used in VB cal-

culations. 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 num-

ber of independent VB structures is given by the Weyl formula:33

Dðm;N; SÞ ¼ 2Sþ 1

mþ 1

mþ 112N þ Sþ 1

� �mþ 112N � 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 canoni-

cal 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 sym-

metry. 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 mathe-

matical 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.

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 or-

bitals, 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 di-

ionic structures, which can be expected to be very high-lying and

will be negligible.

Scheme 4. The 12 configurations necessary for producing a consist-

ent and accurate VB structure set for the ground state of O2.



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 dis-

tances. At long distances, only configurations A1 and A2 are im-

portant, 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 config-

uration 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 sym-

metry adaptation of the W1 and W2 structures to give a triplet state

leads to the following combination:14

a�T1 þ bð�W1 � �W2Þ: (6)

Expressed explicitly, these VB structures are linear combinations

of two determinants, i.e:

�T1 ¼ jp1z�p2zp1xp2yj � j�p1zp2zp1xp2yj (7)

�W1 ¼ jp1z�p1xp2zp2yj � j�p1zp1xp2zp2yj¼ �j�p1xp2yp1zp2zj þ j�p1zp2zp1xp2yj ð8Þ

�W2 ¼ jp2z�p2yp1zp1xj � j�p2zp2yp1zp1xj¼ �jp1x�p2yp1zp2zj þ jp1z�p2zp1xp2yj: ð9Þ

Using eqs. (7)–(9), the McWeeny–van Lenthe wave function in

eq. (6) becomes:

a�T1 þ bð�W1 � �W2Þ ¼ ða� bÞ jp1z�p2zp1xp2yj � j�p1zp2zp1xp2yj� �

þ b jp1x�p2yp1zp2zj � j�p1xp2yp1zp2zj� �

¼ ða� bÞ�T1 þ b�T3: 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,

eq. (10) shows that instead of using three structures, one can econ-

omize and use the more compact form made from T1 and T3 only.

Based on this result, we can use only four structures that are nas-

cent 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

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.

Scheme 6. The selected 12 structures that lead to a consistent and

accurate calculation of the ground state of O2.



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 cal-

culations. The corresponding VBSCF calculation shows that among

these 105 structures, at the equilibrium geometry, only 10 struc-

tures, 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 valid-

ity 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 O¼¼O structures (see Scheme 3b). At

large distances, the alternative coupling of this four-electron four-

orbital 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 quantita-

tive accuracy. Further removal of the very high-lying di-ionic

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.

Structure

R ¼ R0 R ¼ 10 A

WK CK WK CK

T1 0.2292 0.3572 0.2500 0.5000

T2 0.2292 0.3572 0.2500 0.5000

T3 0.0004 0.0206 0.2500 0.5000

T4 0.0004 �0.0206 0.2500 �0.5000

T7 0.0856 �0.1684 0.0000 0.0000

T8 0.0856 �0.1684 0.0000 0.0000

T5 0.0754 �0.1727 0.0000 0.0000

T6 0.0754 �0.1727 0.0000 0.0000

T9 0.0540 0.1173 0.0000 0.0000

T10 0.0540 0.1173 0.0000 0.0000

T11 0.0540 0.1173 0.0000 0.0000

T12 0.0540 0.1173 0.0000 0.0000

Scheme 7. The selected 16 structures that lead to a consistent and

accurate calculation of the excited states of O2.



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 calcula-

tion 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 contri-

butions to the wave function, and only two of these have also

nonzero weights. It should be pointed out that though the weights

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 A, which is somewhat longer than the

experimental value,34 1.208 A, by 0.03–0.04 A. The value of

VBCISD, 1.236 A, is virtually identical to the GVB-CI value of

Goddard,13 1.238 A.

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 accu-

rate 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 ex-

perimental value,34 5.21 eV. The results show that VB theory with

good computational levels, such as BOVB and VBCI, is able to pro-

vide 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 demand-

ing for the available computational resources. It is obvious that here

Table 4. The Spectroscopic Constants of the O2 Ground State with

Various Methods.

Method Basis set Re (A) De (eV) !e (cm�1)

VBSCF 6-311þg* 1.242 3.19 1508.2

cc-pVDZ 1.253 3.05 1576.6

cc-pVTZ 1.242 3.33 1521.4

BOVB 6-311þg* 1.242 4.28 1516.6

cc-pVDZ 1.252 4.12 1619.2

cc-pVTZ 1.242 4.48 1536.7

VBCISD 6-311þg* 1.242 4.37 1500.1

cc-pVDZ 1.252 4.22 1571.7

cc-pVTZ 1.236 4.77 1544.6

VBSCF(105)a cc-pVTZ 1.242 3.37 1521.4

Goddard13 Dzd 1.238 4.88 1693.0

Byrman16 EZPP 1.218 3.67 1549.0

Guberman11 [3s/2p/1d] 1.227 3.72 1539.2

Schaefer III12 [4s/2p] 1.220 4.72 1614.0

Pittner10 cc-pVTZ 1.201 1661.4

CASSCF cc-pVTZ 1.218 4.07 1574.6

MRCI cc-pVTZ 1.214 4.86 1579.9

Expt34 1.208 5.21 1580.0

aA full 105-structure VBSCF calculation.

Table 3. The Weights (WK) and Coefficients (CK) of 16 Selected

Structures in the 105-Structure VBSCF Calculation for the 1Dg State.

Structure

R ¼ R0 R ¼ 10 A

WK CK WK CK

S1 0.2720 0.4236 0.5000 0.8165

S2 0.2710 �0.4229 0.5000 �0.8165

S3 �0.0158 0.0449 0.0000 0.4083

S4 �0.0158 �0.0448 0.0000 �0.4083

S5 0.0588 0.1294 0.0000 0.0000

S6 0.0587 �0.1292 0.0000 0.0000

S7 0.0588 0.1294 0.0000 0.0000

S8 0.0587 �0.1292 0.0000 0.0000

S9 0.0331 0.0997 0.0000 0.0000

S10 0.0330 �0.0995 0.0000 0.0000

S11 0.0331 0.0997 0.0000 0.0000

S12 0.0330 �0.0995 0.0000 0.0000

S13 0.0302 0.1070 0.0000 0.0000

S14 0.0302 �0.1068 0.0000 0.0000

S15 0.0302 0.1070 0.0000 0.0000

S16 0.0302 �0.1068 0.0000 0.0000

Table 5. The Spectroscopic Constants of the First and Second Excited

States of O2.

Method Basis set Re (A) De (eV) !e (cm�1)

1Dg VBSCF cc-pVTZ 1.261 2.450 1408.3

VBCIS cc-pVTZ 1.250 3.880 1492.5

Byrman16 EZPP 1.233 2.799 1445.0

Goddard13 Dzd 1.249 3.790 1595.0


Expt34 1.216 4.232 1509.01Sg

þ VBSCF cc-pVTZ 1.290 1.670 1481.7

VBCIS cc-pVTZ 1.275 2.980 1260.8

Byrman16 EZPP 1.252 2.036 1320.0


Expt34 1.227 3.578 1433.0



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 the1Sg

þ state is still underestimated by*170 cm�1.

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

� ?1Dg), 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 over-

estimated 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

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 A, in

many MO and VB studies.14,16 The curves between 2.0 and 4.0 A

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 cal-

culation. By contrast, both the BOVB and VBCISD curves have

no humps. The same phenomenon occurs in the MO-based calcula-

tions: 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 ex-

cited states, computed with the VBSCF and VBCIS methods. It

can be seen that the dissociations of 1Dg and1Sg

þ 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 intui-

tive insights through its compact wave function. Table 7 collects

the structural weights for the ground state’s wave function at equi-

librium 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 1. The dissociation energy curve of the ground state of O2,

computed by various VB methods with cc-pVTZ basis set.

Figure 2. The VB computed curves for the ground state of O2 in

the range of 2.0–4.0 A, with various VB methods with cc-pVTZ

basis set.

Table 6. The Excitation Energy Computed at the Optimized Equilibrium

Distance (eV).

Method Basis set Te(3Sg

� ? 1Dg) Te(3Sg

� ? 1Sgþ)

VBSCF cc-pVTZ 0.885 1.661

VBCIS cc-pVTZ 0.954 1.862

Goddard13 Dzd 1.089


Expt34 0.982 1.636



105-structure VBSCF calculation (see Table 2). As we reasoned al-

ready, 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 impor-

tant 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 res-

onance 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

characterized by charge-shift bonding, which should be reflected

in other properties of the molecule such as depleted electron den-

sity 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 coeffi-

cients 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 coeffi-

cients. 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 the1Dg 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 struc-

tures of S1. Thus, the combination of these latter structures, to-

gether 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 oxy-

gen atoms of triplet state 3P. However, this feature has to be con-

firmed by the analysis that demonstrates that indeed, the wave

Table 7. The Weights (WK) and Coefficients (CK) of Structures with

cc-pVTZ Basis Set for the Ground State in its Equilibrium Geometry.

Structure

VBSCF BOVB VBCISD

WK CK WK CK WK CK

T1 0.2302 0.3580 0.1749 0.2845 0.2273 0.3448

T2 0.2302 0.3580 0.1819 0.2936 0.2273 0.3448

T3 0.0004 0.0206 0.0004 0.0207 0.0004 0.0180

T4 0.0004 �0.0206 0.0004 �0.0207 0.0004 �0.0180

T5 0.0859 �0.1694 0.1218 �0.2211 0.1184 �0.1996

T6 0.0859 �0.1694 0.1218 �0.2211 0.1184 �0.1996

T7 0.0746 �0.1711 0.0675 �0.1566 0.0660 �0.1486

T8 0.0746 �0.1711 0.0675 �0.1566 0.0660 �0.1486

T9 0.0545 0.1182 0.0646 0.1405 0.0439 0.0978

T10 0.0545 0.1182 0.0673 0.1448 0.0439 0.0978

T11 0.0545 0.1182 0.0673 0.1448 0.0439 0.0978

T12 0.0545 0.1182 0.0646 0.1405 0.0439 0.0978

Table 8. The Weights (WK) and Coefficients (CK) of Structures with

cc-pVTZ Basis Set for the 1Dg State in its Equilibrium Geometry.

Structure

VBSCF BOVB VBCISD

WK CK WK CK WK CK

S1 0.2725 0.4243 0.2158 0.3446 0.2449 0.3801

S2 0.2717 �0.4237 0.2153 �0.3442 0.2443 �0.3796

S3 �0.0158 0.0447 �0.0270 0.0742 �0.0125 0.0358

S4 �0.0157 �0.0446 �0.0269 �0.0741 �0.0125 �0.0358

S5 0.0587 0.1293 0.0992 0.1986 0.0614 0.1289

S6 0.0586 �0.1291 0.0988 �0.1982 0.0612 �0.1288

S7 0.0587 0.1293 0.0622 0.1387 0.0614 0.1288

S8 0.0586 �0.1291 0.0620 �0.1385 0.0612 �0.1287

S9 0.0332 0.0997 0.0518 0.1402 0.0441 0.1185

S10 0.0332 0.0997 0.0517 �0.1401 0.0440 �0.1183

S11 0.0331 �0.0996 0.0482 0.1331 0.0441 0.1185

S12 0.0331 �0.0996 0.0481 �0.1330 0.0440 �0.1183

S13 0.0301 0.1065 0.0250 0.0893 0.0287 0.0945

S14 0.0300 �0.1064 0.0250 �0.0892 0.0286 �0.0944

S15 0.0301 0.1065 0.0254 0.0906 0.0287 0.0945

S16 0.0300 �0.1064 0.0253 �0.0905 0.0286 �0.0944

Figure 3. The dissociation energy curves of the two excited states

computed by VBSCF and VBCIS methods with cc-pVTZ basis set.



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,

�3S�gðR ¼ 1Þ ¼ 0:5ð�T1 þ �T3Þ þ 0:5ð�T2 � �T4Þ: (11)

The first parentheses can be written explicitly in terms of VB

determinants,

�T1 þ �T3 ¼ jðp1z�p2z � �p1zp2zÞp1xp2yj þ jðp1x�p2y � �p1xp2yÞp1zp2zj¼ jðp1z�p1x þ �p1zp1xÞp2zp2yj � jp1zp1xðp2z�p2y þ �p2zp2yÞj:

(12)

At infinity, the electrons between the two oxygen atoms do not

interact. Thus, eq. (12) is equivalent to the form of simple prod-

ucts of the wave functions for the two atoms,

�T1 þ �T3 ¼ jðp1z�p1x þ �p1zp1xÞkp2zp2yj � jp1zp1xkðp2z�p2yþ �p2zp2yÞj: 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) corre-

sponds 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 and1Sg

þ excited states, the nonzero contributions

to the wave function are from structures S1–S4, i.e.,

�1�gðR ¼ 1Þ ¼ ð�S1 þ 0:5�S3Þ � ð�S2 þ 0:5�S4Þ (14)

�1SþgðR ¼ 1Þ ¼ ð�S1 þ 0:5�S3Þ þ ð�S2 þ 0:5�S4Þ (15)

where the normalization factor is neglected. Similarly to the

ground state, we begin here with the combination of S1 and S3.

�S1 þ 0:5�S3 ¼ jðp1z�p2z � �p1zp2zÞðp1y�p2y � �p1yp2yÞjþ 0:5jðp1z�p1y � �p1zp1yÞðp2z�p2y � �p2zp2yÞj

¼ �jp1zp1y�p2z�p2yj � j�p1z�p1yp2zp2yjþ 0:5jp1z�p1y þ �p1zp1ykp2z�p2y þ p2zp2yj: (16)

Then we have,

�S1 þ 0:5�S3 ¼� jp1zp1yk�p2z�p2yj � j�p1z�p1ykp2zp2yjþ 0:5jp1z�p1y þ �p1zp1ykp2z�p2y þ p2zp2yj: (17)

In eq. (17), the first term corresponds to a coupling of the two tri-

plet 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 cou-

pling 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 combina-

tion 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 combina-

tion 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. Par-

ticularly, the high levels VB methods, BOVB and VBCISD, pro-

vide 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 compu-

tation 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 bond-

ing 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

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a valence bond study of the dioxygen...

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