Quantum mechanical study of the vibrational–rotational structure of[O

2(1D

g) ]

2Part II¤

Vincent Veyret,a Be� atrice Bussery-Honvaultb and Stanislas Ya. Umanskiic

a L aboratoire de Ionique et UMR CNRS L yon I,Spectrome� trie Mole� culaire, 5579ÈUniversite�campus de la Doua, 69622 V illeurbanne Cedex, France

b L aboratoire de Physique des Atomes, L asers, et Surfaces, UMR CNRSMole� culesde Rennes I, campus de Beaulieu, 35042 Rennes Cedex, France.6627ÈUniversite�

E-mail : Beatrice.Honvault=univ-rennes1.frc Institute of Chemical Physics, Russian Academy of Sciences, 117977 Moscow, Russia

Received 13th April 1999, Accepted 25th May 1999

The methodology used in the calculation of the rovibrational levels of the singlet ground state of [O2(3&g~)]2has been extended to the treatment of the rovibrational levels of the four singlet states of Two[O2(1*g)]2 .cases have been considered. The Ðrst one is when the two monomers are in their ground vibrational level(v\ 0) and the second case is when one monomer is in the Ðrst excited vibrational level (v\ 1) while the otherremains in its ground vibrational level (v\ 0). Due to di†erent symmetry relations in thepermutation-inversion group, more levels are populated in the second case. The dissociation energy (D0@ \ 23cm~1) is lower than the predicted value of Long and Ewing cm~1) while the di†erence between the(D0@ \ 41dissociation energy of the ground and excited state, is in good agreement with the recent estimationD0A [ D0@ ,of Campargue et al. Using semiempirical potential energy surfaces, we found 27 bound vibrational levels( J \ 0) for and 37 bound levels forO2(1*g)(v/0) ] O2(1*g)(v/0) O2(1*g)(v/0)] O2(1*g)(v/1) .

I. IntroductionWhile triplet oxygen plays an important role in magneticcouplings and in phase transitions of solid oxygen, metastablesinglet oxygen is involved in photolysis, peroxide decomposi-tions, chemiluminescent phenomena, ozonolysis reactions andin other photochemical and photophysical processes.1 Thetwo Ðrst excited states and of are metastableb 1&g` a 1*g O2since transitions to the ground triplet state are forbidden byselection rules for electric dipole radiation. In the case,a 1*gthe transition is three times forbidden (by spin, by angularmomentum and by parity) and has been less investigated. Itsradiative probability and lifetime have been theoretically esti-mated by Minaev3 and numerous papers have been publishedon the role of solvent e†ects on its radiative deactivation.4Amiot and Verges5 have reported a high-resolution spectra ofthis transition emitted in the afterglow of the oxygen gas.

Conversely, the dimer composed of two singlet oxygen mol-ecules leads to a dipole allowed transition with the grounddimer state at a frequency close to two times the monomer IRforbidden transition. This mechanism has been investigatedfor a long time as it is responsible for the blue color of con-densed oxygen6 and numerous articles can be found on thisbimolecular radiative process, which has also been observed inthe chemical iodine oxygen laser (COIL.) by Yoshida et al.7 in1989.

To further analyse the spectra recently reported by(O2)2Campargue et al.,8 we present here the rovibrational analysisof the dimer involved in the visible transitions[O2(1*g)]2observed. The method of Tennyson and van der Avoird9 pre-viously adapted to the dimer case has been[O2(3&g~)]2extended to the case involving di†erent spin and[O2(1*g)]2angular momentum couplings, the close-coupling equationsare again solved using the LC-RAMP approach.

In Section II, we described the potential generated for the

¤ Part I : ref. 1.

four singlet states of the dimer dissociating into O2(1*g)and in Section III, the Hamiltonian, the eigen-] O2(1*g)function basis and the evaluation of the matrix elements arepresented together with the results of this quantum dynamicstreatment.

II. Potentials from perturbative calculationsThe interaction of gives rise to four dimerO2(1*g) ] O2(1*g)singlet states. The construction of these four potential energysurfaces has been previously described in ref. 10. As for

a perturbative treatment has been handled up to[O2(3&g~)]2 ,the second order of the perturbation, including electrostaticmultipoleÈmultipole and exchange (Ðrst order) interactions aswell as an empirical treatment of the polarization(dispersion ] induction) energy in R~6, where R is the inter-molecular distance between the two monomers. The two coor-dinate systems used in present work [the body-Ðxed (BF) andthe space-Ðxed (SF) frames] have been described in Fig. 1 ofthe preceding paper.1 In the case of theO2(1*g)] O2(1*g),potential is not spin-dependent but the projection of themonomer angular momentum on the monomer axis is non-zero The intermolecular potential between the(KA , KB \ ^2).two monomers is represented in the form of a spherical expan-sion involving Wigner rotation matrix elements such as

V (R, hA , hB , /) \ ;KK L

vLA, LB, LKA, KB‰ KA{ , KB{ (R)A

LA, LB, LKA, KB, KA{ , KB{ (hA , hB , /)

(2.1)

where

ALA, LB, LKA, KB‰ KA{ , KB{ (hA , hB , /) \ ;

MA,MB

A L AMA

L BMB

L0

B

] DMA, KA~KA{LA (/A , hA , 0)

] DMB, KB~KB{LB (/B , hB , 0) (2.2)

with L , andMKN\ ML A , L B , KA , KB , KA@ , KB@ N /\ /B [ /A .

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Table 1 Coefficients aKK L, bKK L and cKK L of and (R) in the multipolar expansion of V (R, /)g0KK L, vLA, LB, L`2,`2‰`2,`2(R) v

LA, LB, L~2, ~2‰~2, ~2 hA, hB,

L A L B L g0(K) a b c

0 0 0 1652.62469343 12.51342343 1.14022484 È2 0 2 1016.40002727 12.70635530 1.20317673 È2 2 0 167.49271756 12.2212797 2.48422624 È2 2 2 [259.63473562 12.52654573 1.95982752 È2 2 4 654.70768759 13.14454777 0.65927058 È4 0 4 95.05287458 15.71333458 6.42120702 È4 2 2 16.50166814 14.30077171 7.83052820 È4 2 4 [27.72702329 14.79788007 5.91802361 È4 2 6 95.30323960 15.83414837 4.59491742 È4 4 0 [1.21764906 11.99521543 [1.04767355 È4 4 4 1.94894217 15.81524466 10.28777447 È4 4 6 [4.16713394 15.90059474 5.75876752 È4 4 8 37.86298647 14.17909802 [4.21213705 È6 0 6 2.86852562 12.48721712 È [17.878278176 2 4 4.69220250 13.54021939 5.74989941 È6 2 6 [2.68212890 19.26816774 21.87969785 È6 2 8 2.16805780 13.63615430 È [25.237531056 4 2 1.69191569 11.82811013 2.49255714 È6 4 4 [0.70473021 11.01442329 5.99599716 È6 4 6 0.27694693 10.51991220 3.07642195 È6 4 8 [0.92088451 15.01066527 1.48587557 È6 4 10 3.77463162 15.81058484 [8.97822415 È6 6 0 [0.20530433 12.61018050 1.11721641 È6 6 2 0.21437069 12.58905261 0.51499362 È6 6 4 0.08040471 11.88284261 1.06725287 È6 6 6 [0.07929305 17.10110214 5.45774473 È6 6 8 0.05922102 12.93166805 È È7.263239356 6 10 0.09300250 12.28653091 [5.46095535 È6 6 12 1.66723658 14.70620266 [7.55456316 È

The coefficients of eqn. (2.1) have expres-vLA, LB, LKA, KB‰ KA{ , KB{ (R)

sions similar to those of the case except that the[O2(3&g~)]2indices L N are replaced byML A , L B , MKN\ ML A , L B , KA , KB ,KA@ , KB@ N :

vKK L

(R)\ vKK Lexp(R)] v

KK L(1)(R)] v

KK L(2)(R) (2.3)

with

vKK Lexp(R)\ g0KK L(1] cKK Lx)exp([aKK Lx [ bKK Lx2) (2.4)

vKK L(1)(R)\ d

LA`LB, LC

LA`LB`1RLA`LB`1

(2.5)

with

CLA`LB`1\ ([1)LA

C (2L A ] 2L B) !(2L A ] 1) !(2L B ] 1) !

D1@2Q

LAQ

LB(2.6)

and

x \R[ R0

R0

vKK L(2)(R)\ ;

n/3

`= C2nKK L

R2n(2.7)

is taken to be the nearest neighbor distance in solidR0 aIf the sign of changes with R,[ O2 (R0\ 6.05 a0). v

LA, LB, Lexp (R)then otherwise These terms rep-bLA, LB, L\ 0, cLA, LB, L\ 0.resent, respectively, the repulsive exchange, the electrostaticmultipole interaction and the polarization energy terms.

To evaluate the coefficients aKK L and bKK L or cKK L relativeg0KK L,to the Ðrst order interaction potential of the four singlet statesdissociating into the matrix elements of theO2(1*g)] O2(1*g),potential calculated in the adiabatic 9, 11, 15)11 basisU

i(i\ 6,

are transformed in the diabatic representation MoKA , KBTN\Mo[2, [2T, o[2, 2T, o2, [2T, o2, 2TN.

Following the demonstration given in the Appendix, we

have

U6 [12 12 12 [12 o[2, [2T

U9 0 [i

J2

i

J20 o[2, ]2T

\U11

i

J20 0 [

i

J2o]2, [2T

U15 12 12 12 12 o]2, ]2T

R S R SR St t t tt tt t t tt t

t t t tt t

t t t tt t

t t t tt t

t t t tt t

t t t tt t

t t t tt t

t t t tt t

T U T UT U(2.8)

We note U the unitary matrix

[12 12 12 [12

0 [i

J2

i

J20

U\ (2.9)i

J20 0 [

i

J2

12 12 12 12

R St t

t t

t t

t t

t t

t t

t t

t t

t t

T Uwith UU`\ U`U \ I where I is the unity matrix ; so thatU`\ U~1.

The matrix elements of V in the basis are evalu-MoKA , KBTNated from those in the MoUTN basis using

SKA@ , KB@ o V oKA , KBT \ SUjo UVU~1 oU

iT (2.10)

Each matrix element is then Ðtted ontoSKA@ , KB@ o V oKA , KBTthe analytical multipolar expansion given by eqn. (2.1).The angular functions verify

P0

psin hA dhA

P0

psin hB dhB

]P0

2pd/A

LA, LB, LKA, KB‰ KA{ , KB{ (hA , hB , /)ALA{ , LB{ , L{KA, KB‰ KA{ , KB{ (hA , hB , /)

\8p

(2L A ] 1)(2L B] 1)(2L ] 1)dLA, LA{

dLB, LB{

dL, L{ (2.11)

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We then have

vLA, LB, LKA, KB‰ KA{ , KB{ (R)\

(2L A ] 1)(2L B ] 1)(2L ] 1)

8p

]P0

psin hA dhA

P0

psin hB dhB

P0

2pd/

] SKA , KB o V (R, hA , hB , /) oKA@ , KB@ T

] ALA, LB, LKA, KB‰ KA{ , KB{ (hA , hB , /) (2.12)

The values have been obtained by an angular integra-vK"L

(R)tion of in theSKA , KB o V (R, hA , hB , /)oKA@ , KB@ T hA ½ [0, n],

and / ½ [0, 2n] ranges using a GaussÈLegendrehB ½ [0, n]quadrature over 2000 points of the hypersurfaces. Due tosymmetry rules, the intervals of integration can be reduced to

n/2], n/2] and / ½ [0, n]. For the integra-hA ½ [0, hB ½ [hA ,tion, we have used 5 points in the [0, n/2] range and 10points in the [0, n] range, giving rise to a total of 150 geome-tries at each value of R. The three parameters a, b or c)(g0 ,are then obtained by solving exactly a non-linear system ofequations on three values of R (5, 6 and 7 a0).This procedure has to be handled for the 16 matrix ele-ments Nevertheless, the nondiagonalSKA , KB o V oKA@ , KB@ T.elements are negligible and the following relations hold for thediagonal ones :

S[2, [2 o V o[2, [2T \ S2, 2 o V o 2, 2T

and

S[2, 2 o V o[2, 2T \ S2, [2 o V o 2, [2T

So that the Ðt has been done only for two matrix elements andhas been realized for and L O 12. With theseL A , L BO 6limits, the relative errors obtained between the initial ab initioand Ðtted values are less than 0.3%.

The values of the three Ðtted parameters are presented inTable 1 for and and in Table 2v

LA, LB, L2, 2‰ 2, 2(R) vLA, LB, L~2, ~2‰~2, ~2(R)

for andvLA, LB, L~2, 2‰~2, 2(R) v

LA, LB, L2, ~2‰ 2, ~2(R).

III. Dynamics of [O2(1D

g)v/0

+ O2(1D

g)v/0, 1

]

A. The Hamiltonian

For the nuclear Hamiltonian can be written in the[O2(1*g)]2 ,BF coordinates frame as

HΠ\ [1

2k1

RL2

LR2R]

1

2kR2[ JŒ2] j ü2[ 2j ü É JŒ ]

] b0 jüA2 ] b0 j üB2 ] VŒAB (3.1)

where is the total angular momentum,JŒ \ jü ] l ü j ü\ j üA ] j üB ,where and are the angular momentum operators associ-jüA jü Bated with the rotation of the monomers and is the angularl ümomentum associated with the rotation of the dimer axis.

The interaction potential is independent of the spin inVABthat case and is given by eqns. (2.1) and (2.2).

B. Basis functions and their symmetry

Following the LC-RAMP treatment proposed by van derAvoird and co-workers,9 the basis eigenfunctions are writtenas a product of radial and angular wavefunctions. Due to thenon-zero projections of and on their monomer axis (*j üA j üBelectronic state), rotation matrix elements are used instead ofspherical harmonics. In the BF frame, the basis functions thenread as

WKA, KB, K,MJn, ( jA, jB)j, J \

1

Rsn(R)D

jA, KA‰ jB, KBj, K (hA , hB , /)

]S2J ] 1

4pD

MJ, KJ* (U, h, 0) (3.2)

where are radial functions taken as developed by Tenny-sn(R)

son and Sutcli†e,12 based on three parameters A, b)(Re ,related to and are coupled(Re , De , ue), D

jA, KA‰ jB, KBj, K (hA , hB , /)

Table 2 Coefficients aKK L, bKK L and cKK L of and in the multipolar expansion of V (R, /)g0KK L, vLA, LB, L~2, `2‰~2,`2(R) v

LA , LB, L`2, ~2‰`2, ~2(R) hA , hB ,

L A L B L g0(K) a b c

0 0 0 1652.32053977 12.51393770 1.13584842 È2 0 2 1018.10453985 12.70731869 1.19184093 È2 2 0 180.37308633 12.11977070 2.13616857 È2 2 2 [271.88179468 12.47800718 1.77501145 È2 2 4 608.88938668 13.24108769 0.75348412 È4 0 4 96.54512863 15.66341441 6.18327955 È4 2 2 22.09688100 12.94815666 4.03371481 È4 2 4 [31.37143349 14.33427619 4.27375543 È4 2 6 62.36591589 18.79789858 15.23625009 È4 4 0 3.00687615 12.06984863 0.84426297 È4 4 2 [2.59312309 11.44563721 0.90663585 È4 4 4 2.16105625 12.24378908 1.18193228 È4 4 6 [5.13388564 15.91538692 4.75603648 È4 4 8 9.41800267 17.86853317 [4.00392265 È6 0 6 3.12914697 12.56088450 È [16.469273046 2 6 [2.27854929 12.06233383 È [6.450293036 2 8 0.64236937 13.68307955 È [73.372698596 4 2 [0.88601602 11.47382456 0.91257536 È6 4 4 [0.04421691 12.89009534 È [23.782181246 4 6 0.33209979 17.60043573 14.06433897 È6 4 8 [0.06153495 14.89981747 È [25.844512456 4 10 1.97260913 14.70052965 [15.77887402 È6 6 0 0.06702244 23.41104514 37.6378901 È6 6 2 0.14176412 11.65587161 1.85163745 È6 6 6 0.17825893 9.20624817 [8.43251974 È6 6 8 [0.02051968 12.57289405 È [24.587876936 6 10 [0.19338644 13.80271857 [8.24316437 È6 6 12 0.45384670 12.86688805 [18.84509397 È

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rotation matrix elements such as

DjA, KA‰ jB, KBj, K (hA , hB , /)\

S2jA ] 1

4p

S2jB ] 1

4p;

mA, mB] S jA , mA , jB , mB o j, KT

] DmA, KAj AR (/A , hA , 0)

] DmB, KBj BR (/B , hB , 0) (3.3)

As for the permutation-inversion group is[O2(3&g~]2 , G16again well adapted to the symmetries of and the[O2(1*g)]2permutations that leave invariant the dimer are P12 , P34 ,E* (the spacial inversion) and combinations of them.P13P24 ,

We will adapt the wavefunctions to these symmetries.Using the transformation properties of the BF coordinates

presented in Table 1 of ref. 13 and using the properties of theWigner matrix elements,14 we can easily deduce the trans-formations of the functions (3.2). These transformations aresummarized in Table 3. The basis functions are not yet sym-metry adapted to and For that, we need to build upP12 P34 .the following linear combinations

WmA, KAjA, pA \

1

J2

] [DmA, KAjAR (/A , hA , 0) ] ([1)pAD

mA, ~KAjAR (/A , hA , 0)]

(3.4)

and

WmB, KBjB, pB \

1

J2[D

mB, KBj BR (/B , 0) ] ([1)pBD

mB, ~KBj BR (/B , hB , 0)]

(3.5)

where 1.pA , pB \ 0,Then,

P12WmA, KAjA, pA \ ([1) jA`pAW

mA, KAjA, pA (3.6)

P34WmB, KBjB, pB \ ([1) jB`pBW

mB, KBjB, pB (3.7)

The oxygen atoms with no nuclear spin are bosons. Then, thetotal wavefunction must be symmetric under the permutations

and that is satisÐed when and areP12 P34 jA ] pA jB] pBeven, respectively.We will now use and instead of andW

mA, KAjA, pA W

mB, KBj B, pB D

mA, KAjAR

in eqn. (3.3). E* and acts onDmB, KBj BR P13P24 W

KA, KB, K,MJ( jA, pA, jB, pB)j, J

(ref. 15) as

E*WKA, KB, K,MJ( jA, pA, jB, pB)j, J\ ([1) j`JW

KA, KB, ~K,MJ( jA, pA, jB, pB)j, J (3.8)

and

P13P24WKA, KB, K,MJ( jA, pA, jB, pB)j, J\ ([1) jA`jB`JW

KB, KA, K,MJ( jB, pB, jA, pA)j, J (3.9)

Symmetry adapted basis functions are built on by linear com-binations of previous functions :

WKA, KB, K,MJ( jA, pA, jB, pB)j, J, i \

1

J2[W

KA, KB, K,MJ( jA, pA, jB, pB)j, J

] ([1)iWKA, KB, ~K,MJ( jA, pA, jB, pB)j, J]

if K D 0 (i \ 0, 1) (3.10)

\ WKA, KB, K,MJ( jA, pA, jB, pB)j, J if K \ 0 (3.11)

So,

E*WKA, KB, K,MJ( jA, pA, jB, pB)j, J, i \ ([1) jA`jB`J`iW

KA, KB, K,MJ( jA, pA, jB, pB)j, J, i

(3.12)

and

WKA, KB, K,MJ( jA, pA, jB, pB)j, J, i, i\

1

J2[W

KA, KB, K,MJ( jA, pA, jB, pB)j, J, i

] ([1)iWKA, KB, K,MJ( jB, pB, jA, pA)j, J, i] ;

if jA D jB (i \ 0, 1) (3.13)

\ WKA, KB, K,MJ( jA, pA, jB, pB)j, J, i, if jA \ jB (3.14)

So,

P13P24 WKA, KB, K,MJ( jA, pA, jB, pB)j, J, i, i\ ([1) jA`jB`J`i`i

] WKA, KB, K,MJ( jA, pA, jB, pB)j, J, i, i

(3.15)

From eqns. (3.6), (3.7), (3.12) and (3.15), the symmetries of thewavefunctions are presented in Table 4 andW

KA, KB, K,MJ( jA, pA, jB, pB)j, J, i, i

depend on the parity of i, j and J ] i.jA , jB ,Introducing the vibrational wavefunctions associ-o vA vBTated with the stretching of the monomers, the total nuclear

wavefunctions can be written as K, !T where o J, K,o vA vBTo J,!T is a rovibrational state, K is the projection of J on thez-axis of the BF frame and ! is the symmetry label of therovibrational state.

In the case of the two mono-[O2(1*g)v/0 ] O2(1*g)v/0],mers are in their vibrational ground state, then the vibrationalwavefunction o 0, 0T is totally symmetric and has the sym-metry in As the total wavefunction must be sym-A1` G16 .metric under the permutations of the nuclei (P12 , P34 , P13, 24),only the states of symmetry and are populated.A1` B1~In the case of one monomer is[O2(1*g)v/0 ] O2(1*g)v/1],in the Ðrst excited vibrational state while the other is in theground vibrational state, the vibrational wavefunctions of thedimer are built up by linear combinations of o 0, 1T and o 1, 0Tsuch as

o vA , vBBT \1

J2[ o vA , vBT ^ o vB , vAT]

\1

J2[ o 0, 1T ^ o 1, 0T] (3.16)

These vibrational wavefunctions are invariant under P12 , P34and E* while and are exchanged under sovA vB P13, 24 ,o 1, 0`T has the symmetry and o 1, 0~T has the symmetryA1`inB2` G16 .

Again, the total wavefunction must be totally symmetricunder the permutations of the nuclei. The electronic and1*gvibrational o 1, 0`T wavefunctions being symmetric under

and only the van der Waals rovibrationalP12 , P34 P13, 24 ,states o J, K, !T of symmetry and will be populated.A1` B1~In turn, the vibrational wavefunction o 1, 0~T is even under

but odd under thus, the rovibrational statesP12 , P34 P13, 24 ;o J, K, CT must be even under and odd underP12 , P34 P13, 24 ;corresponding to the symmetry or In conclusion, inA2~ B2` .the case of the rovibrational[O2(1*g)v/0] O2(1*g)v/1],states populated are of symmetry andA1`, B1~, A2~ B2` .

C. Matrix elements. We now perform angular integrationsover the Euler angles of the dimer (U and h) and over all elec-tronic spatial coordinates in the BF frame and summationover all electronic variables on the basis of the monomerelectronicÈrotational functions deÐned by eqns. (3.4) and (3.5),and obtain

SWKA, KB, K,MJ( jA, pA, jB, pB)j, J o VΠ(R, hA , hB , /) oW

KA{ , KB{ , K{,MJ{( jA{ , pA{ , jB{ , pB{ )j{, J{T

\ 14 ;KA, KB, KA{ , KB{

([1)pSWKA, KB, K,MJ( jA, jB)j, J o

] VΠ(R, hA , hB , /) oKA{ , KB{ , K{,MJ{( jA{ , jB{ )j{, J{ T (3.17)

with

p \ 14[pA(2 [ KA) ] pB(2 [ KB) ] pA@ (2 [ KA@ ) ] pB@ (2 [ KB@ )]

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Table 3 Transformation of the Wigner matrix elements by the symmetry elements of the permutation-inversion group G16D

mA, KAjAR (/A , hA , 0) D

mB, KBj BR D

MJ, KJR (U, h, 0)

P12 ([1) jADmA, ~KAjAR (/A , hA , 0) D

mB, KBj BR (/B , hB , 0) D

MJ, KJR (U, h, 0)

P34 DmA, KAjAR (/A , hA , 0) ([1) jBD

mB, ~KBj BR (/B , hB , 0) D

MJ, KJR (U, h, 0)

P13P24 ([1) jAD~mA, KBjAR (/B , hB , 0) ([1) jBD~mB, KB

j BR (/A , hA , 0) ([1)JDMJ, ~KJR (U, h, 0)

E* ([1)KAD~mA, ~KAj AR (/A , hA , 0) ([1)KBD~mB, ~KB

j BR (/B , hB , 0) ([1)JDMJ, ~KJR (U, h, 0)

and

SWKA, KB, K,MJ( jA, jB)j, J o VΠ(R, hA , hB , /) oW

KA{ , KB{ , K{,MJ{( jA{ , jB{ )j{, J{ T

\ ;LA, LB, L

VLA, LB, LKA, KB , KA{ , KB{ (R)SW

KA, KB, K,MJ( jA, jB)j, J o

ALA, LB, LKA , KB‰ KA{ , KB{ (hA , hB , /) oW

KA{ , KB{ , K{,MJ{( jA{ , jB{ )j{, J{ T (3.18)

with

SWKA, KB, K,MJ( jA, jB)j, J oA

LA, LB, LKA, KB‰ KA{ ~KB{ (hA , hB , /) oWKA{ , KB{ , K{,MJ{( jA{ , jB{ )j{, J{ T

\ ([1) jA{`jB{`j{~L~K~KA{ KB{ [(2jA ] 1)(2jA@ ] 1)

] (2jB ] 1)(2jB@ ] 1)(2j] 1)(2j@] 1)

] (2L A ] 1)(2L B ] 1)(2L ] 1)]1@2

]A jA[KA

L A0

jA@KA@BA jB

[KB

L B0

jB@KB@BA j

[KL0

j@KB

]7jAjBj

L AL BL

jA@jB@j@

8dJ, J{ dK, K{ dMJ,MJ{

(3.19)

is the angular part of the wavefunction (3.2).oWKA, KB, K,MJ( jA, jB)j, J T

The expression given by eqn. (3.19) is a generalization of theGaunt coefficient.

Angular integration over the kinetic energy terms isstraightforward and can be calculated analytically. Toseparate the diagonal terms from the extradiagonal ones, wenote

HΠrot\ [1

2k1

Rd2

dR2R]

1

2kR2(JŒ2 ] Òü2[ 2Òü

zÉ JŒ

z)

] b0 ÒüA2 ] b0 ÒüB2 (3.20)

and

HΠrotvib\ [1

2kR2( Òü`

É JŒ`

] Òü~ É JŒ~) (3.21)

where and are the ladder operators.ÒüB

\ Òüx^ Òü

yJŒB

\ JŒx^ JŒ

yThe resulting set of equations has the form

SWKA, KB, K,MJ( jA, pA, jB, pB)j, J oHΠrot oWKA{ , KB{ , K{,MJ{

( jA{ , pA{ , jB{ , pB{ )j{, J{T

\1

2kR2[ J( J ] 1)] j( j] 1)[ 2K2] b0 jA( jA ] 1)

Table 4 Symmetry of the rovibrational wavefunctions of [O2(1*g)]2jA jB i j J] i P12 P34 P13, 24 E* G16e/o e/o e e e ] ] ] ] A1`e/o e/o e e o ] ] [ [ A2~e/o e/o e o e ] ] ] [ B1~e/o e/o e o o ] ] [ ] B2`e/o e/o o e e ] ] [ ] B2`e/o e/o o e o ] ] ] [ B1~e/o e/o o o e ] ] [ [ A2~e/o e/o o o o ] ] ] ] A1`e/o o/e e e e ] ] [ ] B2`e/o o/e e e o ] ] ] [ B1~e/o o/e e o e ] ] [ [ A2~e/o o/e e o o ] ] ] ] A1`e/o o/e o e e ] ] ] ] A1`e/o o/e o e o ] ] [ [ A2~e/o o/e o o e ] ] ] [ B1~e/o o/e o o o ] ] [ ] B2`

] bo jB( jB ] 1)]dK, K{ df, f { (3.22)

(3.23)

and

SWKA, KB, K,MJ( jA, pA, jB, pB)j, J oHΠrotvib oW

KA{ , KB{ , K{,MJ{( jA{ , pA{ , jB{ , pB{ )j{, J{T

\ [1

2kR2[C

j, K` CJ, K` d

K, K{`1] Cj, K~ C

J, K~ dK, K{~1]df, f {

(3.24)

where f stands for j, J, This lastjA , pA , KA , jB , pB , KB , MJ.

term which couples vibration and rotation is often calledCoriolis coupling.

As for the radial integration over the radial[O2(3&g~)v/0]2 ,

kinetic energy is conducted analytically12 while the radialintegrations over the rotation terms and the potential areevaluated numerically in the same way as in the precedingpaper.1 Diagonalization of the Hamiltonian matrix for eachsymmetry of yields the rovibration states of theG16dimer.[O2(1*g)]2

D. Calculations and results

The internuclear distance of the monomer is Ðxed at R0 \(ref. 5) for and at (ref. 16)2.3043 a0 O2(1*g)v/0 R0 \ 2.3184 a0for These two values correspond to rotationalO2(1*g)v/1.constants cm~1 for andb0 \ 1.418 O2(1*g)v/0 b1 \ 1.401

cm~1 for O2(1*g)v/1.Again, the wavefunction basis includes all radial functionswith n O 5 and all angular functions with The opti-jA , jB O 9.mized parameters and of the radial basis functionsRe , ue Deare cm~1 and cm~1.sn(R) Re \ 7.68 a0 , De \ 97.0 ue \ 24.0

This leads to an a value of 16.The variation of the intramonomer distance between theR0vibrational levels v\ 0 and v\ 1 being small (D 0.01 wea0),consider as a Ðrst approximation that the di†erences in the

intermolecular potential energy surfaces between [O2(1*g)v/0and due to the] O2(1*g)v/0] [O2(1*g)v/0 ] O2(1*g)v/1]stretch of one intramonomer distance is negligible in regard ofthe accuracy of the potential. Thus we have used same poten-tial energy surfaces in both cases. The coefficients aKK L,g0KK L,bKK L, cKK L L , are taken from(MKN\ ML A , L B , KA , KB , KA@ , KB@ N)Tables 1 and 2 and the electrostatic and polariza-C5 , C7 , C9tion coefficients are taken from Table 1 of the precedingC6paper.1 The UU` basis transformation leaves themunchanged.

To keep the problem tractable, the extradiagonal elementsof the Coriolis coupling have been neglected as in the

case. Even with that approximation, the Hamilto-[O2(3&g~)]2nian matrix size is three to four times larger than in the pre-ceding case. The rovibrational levels are calculated fordi†erent J and K values, where K, the projection of the totalangular momentum J on the z-axis of the BF frame, is treatedas a good quantum number.

Fig. 1 shows the Ðrst rovibrational levels of [O2(1*g)v/0and Fig. 2 those of] O2(1*g)v/0] [O2(1*g)v/0] O2(1*g)v/1].We present in Tables 5 and 6 the energies of the Ðrst Ðvebound rovibrational levels of symmetry and respec-A1` B1~tively for K \ 0, 1, 2 and increasing sequences of J relative to

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Fig. 1 First rovibrational levels of O2(1*g)v/0] O2(1*g)v/0 .

Fig. 2 First rovibrational levels of O2(1*g)v/0] O2(1*g)v/1.

Full data on these rovibrational[O2(1*g)v/0 ] O2(1*g)v/0].levels are available as Supplementary Information¤ togetherwith the energies of the bound levels of symmetry A1`, B1~, A2~and relative to In this lastB2` [O2(1*g)v/0 ] O2(1*g)v/1].case, the levels of symmetry and or and areA1` A2~ B1~ B2`degenerate for K P 1 and are presented in the same table. Weget 27 vibrational levels for the J \ 0 bound states of

(see Table 7) and 37 vibrationalO2(1*g)v/0] O2(1*g)v/0levels for (see Table 8).O2(1*g)v/0 ] O2(1*g)v/1As for the transitions J@^ JA with *K \ 0[O2(3&g~)v/0]2 ,

correspond to pure rotational transitions and obey the rela-tion

*EJ{HJ_\

AB] C2

B[ J@( J@] 1) [ JA( JA ] 1)] (3.26)

Each sequence of energies is so characterized by the rotationalconstant (B] C)/2 mentioned in the last column of each table.The rotational constant (around 0.06 cm~1) is slightly smallerthan the one for It gives a rotationally-[O2(3&g~)

v/0]2 .averaged value of the intermolecular distance SRT of around7.9 a0 .

The vibrational ground state energy cm~1) of(D0 \ [23.7symmetry is smaller than the predicted value of Long andA1`Ewing17 cm~1) while the di†erence(D0@ \ 41 ^ 2 D0A [ D0@ \

49.5 cm~1 where is the vibrational ground state energy ofD0Asymmetry) is within the range estimated by[O2(3&g~)]2 (A1`Campargue et al., cm~1 [ref. (8)]. Never-32 O D0A [ Do@ O 47theless, these comparisons seem to indicate that the potentialenergy surfaces of the singlet dimer states under-[O2(1*g)]2estimate the binding energy.

Furthermore, we note that the rovibrational energies inTables 5 and 6 for J \ K can be grouped into three blocks,ranging around [[24, [20.9], [[14.5, [12.1] and [5cm~1. Similar behavior is reproduced in the O2(1*g)v/0case with slightly di†erent range values (see] O2(1*g)v/1Supplementary Information).

IV. ConclusionThe behavior of the rovibrational levels of is[O2(1*g)]2similar to the case of and similar conclusions can[O2(3&g~)]2be drawn: a nearly rigid-rotor behavior with J and an irregu-lar behavior with K, characterizing a van der Waals dimer.

¤ Available as supplementary material (SUP 57562, 16 pp.) depos-ited with the British Library. Details are available from the EditorialOffice. For direct electronic access see http ://www.rsc.org/suppdata/cp/1999/3395.

Table 5 First rovibrational energies (in cm~1) of symmetry for K \ 0, 1, 2 relative toA1` O2(1*g)v/0] O2(1*g)v/0

K J \ 0 J \ 1 J \ 2 J \ 3 J \ 4 J \ 5 J \ 6Bp \

B] C2

0 [23.8998 [23.5393 [22.6988 [21.3797 0.0600[22.7724 [22.4139 [21.5781 [20.2664 0.0597[21.2466 [20.8980 [20.0852 [18.8097 0.0581[13.8509 [13.4967 [12.6709 [11.3748 0.0590[12.8636 [12.5132 [11.6962 [10.4138 0.0584

1 [23.0346 [22.7952 [22.4362 [21.9578 [21.3603 [20.6438 0.0598[22.0763 [21.8404 [21.4866 [21.0152 [20.4262 [19.7201 0.0589[14.6212 [14.3837 [14.0277 [13.5531 [12.9603 [12.2494 0.0593[13.9156 [13.6790 [13.3241 [12.8513 [12.2606 [11.5523 0.0591[13.4633 [13.2279 [12.8749 [12.4045 [11.8168 [11.1122 0.0588

2 [23.2158 [22.8583 [22.3819 [21.7868 [21.0733 0.0595[22.1587 [21.8034 [21.3298 [20.7383 [20.0290 0.0592[14.6278 [14.2714 [13.7965 [13.2033 [12.4919 0.0594[14.1319 [13.7760 [13.3016 [12.7089 [11.9983 0.0593[13.4376 [13.0826 [12.6096 [12.0188 [11.3102 0.0591

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Table 6 First rovibrational energies (in cm~1) of symmetry for K \ 0, 1, 2 relative toB1~ O2(1*g)v/0] O2(1*g)v/0

K J \ 0 J \ 1 J \ 2 J \ 3 J \ 4 J \ 5 J \ 6Bp \

B] C2

0 [23.1393 [22.7808 [21.9450 [20.6332 0.0597[21.2503 [20.9018 [20.0890 [18.8135 0.0581[14.5252 [14.1692 [13.3390 [12.0360 0.0593[13.4282 [13.0736 [12.2469 [10.9496 0.0591[12.8295 [12.4784 [11.6596 [10.3740 0.0585

1 [23.7009 [23.4611 [23.1017 [22.6226 [22.0241 [21.3065 0.0599[22.1463 [21.9106 [21.5571 [21.0861 [20.4976 [19.7920 0.0589[14.6849 [14.4473 [14.0911 [13.6164 [13.0234 [12.3123 0.0594[14.0115 [13.7749 [13.4201 [12.9474 [12.3568 [11.6486 0.0591[13.4220 [13.1854 [12.8305 [12.3576 [11.7669 [11.0587 0.0591

2 [22.4772 [22.1221 [21.6488 [21.0577 [20.3488 0.0592[14.7423 [14.3855 [13.9101 [13.3161 [12.6039 0.0594[14.0185 [13.6626 [13.1882 [12.5956 [11.8850 0.0593[13.5292 [13.1738 [12.7001 [12.1083 [11.3988 0.0592[12.5666 [12.2156 [11.7478 [11.1635 [10.4628 0.0585

Table 7 Vibrational energies (in cm~1) of the J \ 0 bound states relative to O2(1*g)v/0] O2(1*g)v/0A1` [23.8998, [22.7724, [21.2466, [13.8509, [12.8636, [5.8321, [5.4503, [4.9749,

[4.6750, [3.7248, [3.2803, [2.2715, [1.7117, [0.7008, [0.1263B1~ [23.1393, [21.2503, [14.5252, [13.4282, [12.8295, [5.7460, [5.1482, [4.0647,

[4.0222, [1.9500, [1.2182, [0.3259

Table 8 Vibrational energies (in cm~1) of the J \ 0 bound states relative to O2(1*g)v/0] O2(1*g)v/1A1` [24.024, [22.8750, [21.3492, [13.9558, [12.9700, [6.0332, [5.6543, [5.0836,

[4.8702, [3.8317, [3.4801, [2.3755, [1.8154, [0.8045, [0.2306B1~ [23.2418, [21.3529, [14.6288, [13.5327, [12.9356, [5.9488, [5.3495, [4.2632,

[4.1279, [2.0541, [1.3217, [0.4300A2~ [14.8760, [13.5602, [12.9499, [2.2002, [1.0396B2` [14.1564, [13.1794, [3.0018, [1.5716, [0.8419

The average intermolecular distance SRT is slightly larger dueto smaller (B] C)/2 constants. The dissociation energy isD0@found to be smaller than the predicted value of Long andEwing17 while the di†erence between the ground and excitedstate dissociation energies, is in good agreementD0A [ D0@ ,with the value proposed by Campargue et al.8 Based on thecalculation of these rovibrational energy levels, the visiblespectrum of can be constructed through the evaluation(O2)2of the electric dipole moment surfaces. Such work is inprogress and will be presented in a forthcoming paper.

AppendixIn ref. 10, the four singlet dimer states dissociating into

have been constructed by linear com-O2(1*g) ] O2(1*g)binations of product of A and B monomer wavefunctions ofXY , X2 or Y 2 type. After factorisation, we show that

U6\ (XA` Y A~] Y A`XA~)(Y B`XB~] XB` Y B~) (4.1)

U9\ (XA`XA~[ Y A` Y A~)(XB` Y B~] Y B`XB~)

[ (XA` Y A~] Y A`XA~)(XB`XB~[ Y B` Y B~) (4.2)

U11\ (XA`XA~[ Y A` Y A~)(XB` Y B~] Y B`XB~)

] (XA` Y A~] Y A`XA~)(XB`XB~[ Y B` Y B~) (4.3)

U15\ (XA`XA~[ Y A` Y A~)(XB`XB~[ Y B` Y B~) (4.4)

The Cartesian and orbitals are expressed in termsX(px) Y (p

y)

of spherical orbitals18 aspB

X \1

J2[p~1[ p

`1] (4.5)

Y \i

J2[p~1] p

`1] (4.6)

We then have

X`Y ~] Y `X~\ i[p~1` p~1~ [ p`1` p

`1` ]

\ i[ o[2T [ o]2T] (4.7)

and

X`X~[ Y `Y ~\1

J2[p~1` [ p

`1` ]1

J2[p~1~ [ p

`1~ ][i

J2

] [p~1` ] p`1` ]

i

J2[p~1~ ] p

`1~ ]

\ [p~1` ] p~1~ ] p`1` p

`1~ ]

\ [ o[2T ] o]2T] (4.8)

Substituting eqns. (4.7), (4.8) into eqns. (4.1), (4.2), (4.3), (4.4),the wavefunctions are expressed in theU6 , U9 , U11, U15M oKA , KBN\ M o[2, [2T, o[2, ]2T, o]2, [2T, o]2, ]2TN

basis :

U6 \ i[ o[2TA [ o]2TA]i[ o[2TB[ o]2TB]\ [ o[2, [2T ] o[2, ]2T ] o]2, [2T [ o]2, ]2T

(4.9)

U9\ [ o[2TA ] o]2TA]i[ o[2TB[ o]2TB][ i[ o[2TA [ o]2TA]

] [ o[2TB] o]2TB]\ 2i[[ o[2, ]2T ] o]2, [2T] (4.10)

U11\ [ o[2TA ] o]2TA]i[ o[2TB[ o]2TB]] i[ o[2TA [ o]2TA]

] [ o[2TB] o]2TB]\ 2i[ o[2, [2T [ o]2, ]2T] (4.11)

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U15 \ [o[2TA ] o]2TA][ o[2TB] o]2TB]

\ o[2, [2T ] o[2, ]2T ] o]2, [2T ] o]2, ]2T

(4.12)

After normalisation of these wavefunctions with pU6p \ 2,and we getpU9p \ 2J2, pU11p \ 2J2 pU15p \ 2,

U6 \ 12[[ o[2, [2T ] o[2, ]2T ] o]2, [2T

[ o]2, ] 2T]

(4.13)

U9 \i

J2[[ o[2, ]2T ] o]2, [2T] (4.14)

U11\i

J2[ o[2, [2T [ o]2, ]2T] (4.15)

U15 \ 12[ o[2, [2T ] o[2, ]2T ] o]2, [2T ] o]2, ]2T]

(4.16)

Or in matrix form,

U6 [12 12 12 [12 o[2, [2T

U9 0 [i

J2

i

J20 o[2, ]2

\U11

i

J20 0 [

i

J2o]2, [2T

U15 12 12 12 12 o]2, ]2T

R S R SR St t t tt tt t t tt t

t t t tt t

t t t tt t

t t t tt t

t t t tt t

t t t tt t

t t t tt t

t t t tt t

T U T UT U(4.17)

Acknowledgementscomputations presented in this paper were performedThe

partly on computers Ðnanced by the ““Conseil Re� gional deBretagneÏÏ, the ““Conseil Ge� ne� ral dÏIlle et Vilaine ÏÏ and theDistrict of Rennes and partly at the ““Centre National deCalcul ÏÏ of the IN2P3 (France). We thank all of them for pro-viding us with computing power.

References1 B. Bussery-Honvault and V. Veyret, Phys. Chem. Chem. Phys.,

1999, 1, 000.2 D. R. Kears, Chem. Rev., 1971, 71, 395.3 B. F. Minaev, Int. J. Quantum Chem., 1980, 17, 367.4 B. F. Minaev, Opt. Spectrosc., 1985, 58, 761 ; B. F. Minaev, J.

Mol. Struct., 1989, 183, 207 ; K. V. Mikkelsen and H. Agren, J.Phys. Chem., 1990, 94, 6220.

5 C. Amiot and J. Verges, Can. J. Phys., 1981, 59, 1393.6 S. C. Tsai and G. W. Robinson, J. Chem. Phys., 1969, 51, 3559.7 S. Yoshida, M. Taniwaki, T. Sawano, K. Shimizu and T. Fujioka,

Japan. J. Appl. Phys., 1989, 28, 2500.8 A. Campargue, L. Biennier, A. Kachanov, R. Jost, B. Bussery-

Honvault, V. Veyret, S. Churassy and R. Bacis, Chem. Phys. L ett.,1998, 288, 734.

9 J. Tennyson and A. van der Avoird, J. Chem. Phys., 1982, 77,5664.

10 B. Bussery, Chem. Phys., 1994, 184, 29.11 B. Bussery and P. E. S. Wormer, J. Chem. Phys., 1993, 99, 1230.12 J. Tennyson and B. T. Sutcli†e, J. Chem. Phys., 1982, 77, 4061.13 A. van der Avoird and G. Brocks, J. Chem. Phys., 1987, 87, 5346.14 A. R. Edmonds, in Angular Momentum in Quantum Mechanics,

Princeton Press, New Jersey, 1974.15 V. Veyret, These de Doctorat, Universite� Lyon 1, France, 1998.16 L. Herzberg and G. Herzberg, Astrophys. J., 1947, 105, 353.17 C. A. Long and G. E. Ewing, J. Chem. Phys., 1973, 58, 4824.18 M. Weissbluth, in Atoms and Molecules, Academic Press, New

York, 1978.

Paper 9/02945A

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