theoretical potential energy surfaces for excited mercury...

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Chemical Physics xxx (2006) xxx–xxx

Theoretical potential energy surfaces for excited mercury trimers

Hikaru Kitamura *

Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

Received 21 June 2005; accepted 9 December 2005

Abstract

Potential energy surfaces are calculated for electronic excited states of mercury trimers (Hg3) correlating to Hg(61P,63P) +Hg(61S) + Hg(61S) asymptotes. The theory is based on the diatomics-in-molecules (DIM) method with the aid of the existing ab initiodiatomic potential energy curves and the atomic spin–orbit interaction parameters. Stability of the trimers with C2v symmetry is inves-tigated in detail. It is thus shown that the lowest excited level corresponds to the metastable A0

1 state in the equilateral triangle (D3h)configuration, which is located at 3.04 eV above the energy of infinitely separated ground-state atoms. The energy of dipole allowed tran-sition from the bottom of the A0þu state to the ground X0þg state in the symmetric linear (D1h) configuration is predicted to be 2.55 eV(4870 A), which accounts for the well known 4850 A continuum emission observed through many previous experiments. The origin of the2170 A fluorescence band is discussed in connection with higher excited states of Hg3.� 2005 Elsevier B.V. All rights reserved.

Keywords: Potential energy surface; Mercury; Trimer; Excited states; Diatomics-in-molecules

1. Introduction

Elucidation of interatomic interactions in mercury com-plexes is a fundamental issue for understanding how the bulkmetallic liquid evolves from nonmetallic gas through aggre-gation of constituent atoms [1,2]. Detailed experimentalstudies on Hg2 dimers have revealed that the interactionbetween two ground-state Hg(6s2 1S0) atoms is dominatedby short-range repulsion, while the van derWaals attractionis quite weak and the binding energy is as small as 0.045–0.05 eV [3,4]. In contrast, rather strong attractive forces arepredicted when one of the atoms takes the excited 6s6p 3Por 6s6p 1P configuration [3]. Turkevich and Cohen [5] raisedthe possibility of excimer formation in dense mercury vapor.Uchtmann et al. [6] showed that creation of excimers insupersaturated mercury vapor by photo-illumintation givesrise to enhancement of nucleation rates.

There have been experimental reports on absorption andfluorescence spectra in optically excited mercury vaporindicating the existence of Hg3. In particular, considerable

0301-0104/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2005.12.007

* Tel.: +81 75 753 3750; fax: +81 75 753 3751.E-mail address: [email protected].

attention has been paid to the origin of 4850 A continuumemission called the blue-green emission. Drullinger et al. [7]detected fluorescence band centered at 4850 A in opticallyexcited pure mercury vapor and assigned it as an emissionfrom an excited state of Hg3 which lies about 0.806 eVbelow the Hg2D1u state. The same band was studied laterby Skordoulis et al. [8] through laser-induced fluorescencespectroscopy of mercury clusters created via photodissoci-ation of HgBr2. Callear and Lai [9] reported absorptionspectra arising from Hg2 and Hg3 by using the flash photol-ysis of mercury vapor with added N2 gas. They ascribed the4850 A emission to a radiative decay from the Hg3A0þustate, which is in equilibrium with the lowest excited stateof the dimer as Hg2A0�g þHg¢Hg3A0�u . This conclusionwas obtained solely from experiments, because no theoret-ical data were available on Hg3 energy levels. On the otherhand, the vapor cell experiment using pump-and-probeexcitation by Niefer et al. [10] observed 2170 A fluorescenceband; they speculated that this emission might arise fromhigher state of Hg3 in the vicinity of 5.50–5.60 eV. Koper-ski et al. [11] observed both 4850 and 2125 A fluorescencebands in laser-excited supersonic expansion beam consist-ing of mercury and noble-atom carrier gas.

mailto:[email protected]

2 H. Kitamura / Chemical Physics xxx (2006) xxx–xxx

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Theoretical computation of interatomic potentials formercury complexes requires accurate treatments of electroncorrelation and relativistic effects inherent to heavy-ele-ment compounds. A number of authors investigated geo-metrical and electronic structures of ground-state Hgnclusters on the basis of the ab initio quantum chemicalmethods [12–14], whereas relatively few works exist onthe properties of excited-state clusters [13,14]. Bas�tuget al. [13] carried out relativistic all-electron Dirac–Fock–Slater SCF calculations of the potential energy surfacesfor the ground-state Hg3 and showed that the stable struc-ture has the D3h symmetry. Furthermore, they estimatedthe corresponding excitation energy as 4.6 eV; this valueturns out considerably larger than the energy of the blue-green emission. Recently, Omary et al. [14] performed cou-pled-cluster calculations of the ground and low-lyingexcited states (3Rþ

u ,3Pu) of Hg3 in the D1h geometry by

using various types of orbital basis sets and effective corepotentials including scalar relativistic effects. They treated5d electrons explicitly as valence states. They showed thatthe emission wavelength from the 3Pu state amounts to4800 A, thus providing the first theoretical assignment ofthe blue-green emission. In their calculations, however,the spin–orbit interactions were neglected.

While the ab initio studies mentioned above [12–14] arerestricted to particular energy levels or geometries, weexpect that a semi-empirical approach would be a suitableand efficient way of computing complete potential energysurfaces for Hg3 excimers over wider energy range. Thediatomics-in-molecules (DIM) method enables one to con-struct approximate potential energy surfaces for groundand excited states of a large cluster with the knowledgeof potential curves for the corresponding diatomic frag-ments [15–17]. The DIM duly takes into account the direc-tional nature of chemical bonding and has been appliedsuccessfully to a wide range of triatomic and larger clusterscontaining p and/or d electrons [15,17,18]. It can incorpo-rate spin–orbit interactions as well [16]. Recently, DIM wasapplied to the analysis of absorption spectra of Hg2 embed-ded in rare-gas matrices [21].

In order to apply the DIM theory to Hgn clusters, wemust utilize existing potential energy curves for the groundand excited states of Hg2. Czuchaj et al. [19] performedself-consistent-field (SCF) and multi-reference configura-tion-interaction (MRCI) calculations of potential energycurves for Hg2 that dissociate to Hg(1S0) + Hg(63P, 61P,71S, 73S, 71P, 73P) asymptotes, where the electrons apartfrom the two outermost electrons were treated with thepseudopotentials. Analogous calculations were carriedout by Balasubramanian et al. [20] on the basis of the rel-ativistic complete active space multi-configuration self-con-sistent-field calculations with the outer 12 valence electronsbeing treated explicitly. These theoretical results wereexamined by Koperski [3] who constructed experimentalpotential energy curves for excited D1u, E1u, F0þu , andG0þu states of Hg2 through combining ro-vibrational spec-tra over a broad range of internuclear separations. It was

thus concluded that the potential curves by Czuchaj et al.[19] are in closest agreement with the experimental data [3].

In this paper, we calculate potential energy surfaces forexcited states of Hg3 with the DIM formalism by using theab initio diatomic potential curves by Czuchaj et al. [19].Spin–orbit interactions are treated through superpositionof individual atomic contributions. Stable geometries ofHg3 excimers with the C2v configuration are thereby stud-ied for the energy range of 3.0–5.7 eV. For each excitedstate potential minimum, we compute an emission wave-length and an approximate transition moment of theFranck–Condon vertical transition to the ground state.Possible origins of the fluorescence bands centered at4850 and 2170 A are discussed.

2. The DIM formalism

We apply the DIM method to a Hg3 cluster that disso-ciates to Hg(63P, 61P) + Hg(61S) + Hg(61S) asymptote. Itis convenient to choose the laboratory frame so that atom1 is at the origin, atom 2 is placed on the z-axis, and atom 3lies in the xz-plane. We denote rij as the distance betweenatoms i and j. We also introduce hij which measures theangle between the molecular axis of the dimer ij and thez-axis. The total Hamiltonian of the trimer is decomposedinto diatomic fragments and expressed as [15,17]

H ¼ Hð12Þð0Þ þHð13Þðh13Þ þHð23Þðh23Þ �HðatomÞ þ Vso. ð1ÞHere, H(ij)(hij) is the part of the Hamiltonian containing thecontributions from atoms i, j, and their mutual interaction;H(atom) is the sum of three Hamiltonians of isolated atoms,and the last term, Vso, describes the spin–orbit interaction.

The prescription of the DIM theory is to express H in amatrix form in terms of the polyatomic basis functions andto calculate energy eigenvalues and eigenfunctions of acluster through its diagonalization. In this work, weapproximate the polyatomic basis functions by productsof localized atomic wavefunctions, which describe a set ofstates in which any one of the atoms is in the 63P or 61Pstate and the rest of the atoms are in the ground 1S state.Overlap of these wavefunctions is neglected for simplicity[15,17]. Because the atomic Hg(63P) and Hg(61P) statesare ninefold and threefold degenerate, respectively, in theabsence of spin–orbit interaction, we must consider 36basis functions, which we denote as |kSMLMSi(k = 1,2,3; S = 0,1; ML = 1,0,�1; MS = S,S-1, . . . ,�S).Here, the angular momentum and spin quantum numbersML and MS are those associated with the excited atomspecified by k. In addition, we consider the state |0iin whichall the atoms are in the ground 1S state. Each term on theright-hand side of Eq. (1) can therefore be expressed in a37 · 37 matrix form.

In order to evaluate the matrix elements of H(ij)(hij) withthe laboratory-frame basis functions |kSMLMSi, it is con-venient to calculate the matrix elements of H(ij)(0) withthe molecular-fixed basis functions |kSKMSi, where Kdenotes the component of the orbital angular momentum

H. Kitamura / Chemical Physics xxx (2006) xxx–xxx 3

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projected onto the molecular axis of the dimer ij. The trans-formation from molecular to laboratory frame can be facil-itated with the aid of the rotation matrix, as we shalldescribe later in this section. The matrix elements ofH(ij)(0) can be directly related to the potential energy curveE(2S+1K;rij) of the dimer ij in the 2S+1K state with inter-atomic distance rij. To establish such relations, we adoptsimple valence-bond forms of wavefunctions for the ij

dimer in terms of the atomic orbitals:

jX 1Rgi ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ n2ij

q j0i þ nijffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2ij

q j1Rð0Þg i; ð2aÞ

j1Rgi ¼�nijffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2ij

q j0i þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2ij

q j1Rð0Þg i; ð2bÞ

j3Rgi ¼1ffiffiffi2

p ðjj10MSi � ji10MSiÞ ð2cÞ

j2Sþ1Rui ¼1ffiffiffi2

p ðjjS0MSi þ jiS0MSiÞ; ð2dÞ

j2Sþ1Pgi ¼1ffiffiffi2

p ðjjS � 1MSi � jiS � 1MSiÞ; ð2eÞ

j2Sþ1Pui ¼1ffiffiffi2

p ðjjS � 1MSi þ jiS � 1MSiÞ ð2fÞ

with the unmixed wavefunction of the excited 1Rg state,

j1Rð0Þg i ¼ 1ffiffi

2p ðjj00MSi � ji00MSiÞ. ð2gÞ

The parameter nij in Eqs. (2a) and (2b) describes a mix-ing of wavefunctions for the ground X 1Rg state and theexcited 1Rg state. It depends on the interatomic distancerij, and additional information is required to establish itsfunctional form [15,18]. In this study, nij is determinedthrough an empirical valence bond method according to

nij ffi �h0jHðijÞð0Þj1Rð0Þ

g i � h0j1Rð0Þg iEðX 1Rg; rijÞ

Eð1Rg; rijÞ � EðX 1Rg; rijÞ. ð3Þ

The resonance and overlap integrals in the numerator maybe evaluated by expressing the molecular wavefunctions|0iand j1Rð0Þ

g i explicitly in terms of four-electron Slaterdeterminants constructed from 6s and 6pz atomic orbitals,/iðjÞ

6s and /iðjÞ6pz, localized on atom i (j). Retaining only the

lowest-order term of the overlap integrals Qz � h/i6pzj/j

6siand S � h/i

6sj/j6si for large rij, we arrive at an expression,

nij ffi �QzS U � ðe2=rijÞ

� �Eð1Rg; rijÞ � EðX 1Rg; rijÞ

; ð4Þ

where

U ¼Z

dr

Zdr0/i�

6sðrÞ/i�6sðr0Þ

e2

jr� r0j/i6sðrÞ/

i6sðr0Þ ð5Þ

refers to the energy of on-site Coulomb repulsion. A roughestimation of U may be given by the balance between thesecond and the first ionization potentials of a Hg atom,that is, U � Ip2 � Ip1 = 18.756 � 10.437 = 8.319 eV. Theoverlap integrals Qz and S are calculated numerically with

the aid of the published Hartree–Fock atomic wavefunc-tions by McLean and McLean [22] for /iðjÞ

6s and those byHyman [23] for /iðjÞ

6pz. Eq. (4) has thus been computed for

4 6 rij/aB 6 10, with aB denoting the Bohr radius; the re-sults are well reproduced by the fitting formula,

nij ¼ �1:1� 10�3ðrij=aBÞ4 exp½�0:08ðrij=aBÞ2�. ð6Þ

The maximum magnitude of nij takes on a relatively smallvalue of 0.093 (which is attained at rij/aB = 5.0), so that thelevel mixing does not drastically affect the potential energycurves.

With the help of the relations (2a)–(2g), non-zero matrixelements of the dimer fragment Hamiltonians may be writ-ten as follows:

h0jHðijÞð0Þj0i ¼ hkSKMS jHðijÞð0ÞjkSKMSi ¼ Sij

for k 6¼ i; j; ð7aÞh0jHðijÞð0ÞjiSKMSi ¼ �h0jHðijÞð0ÞjjSKMSi ¼ T ij; ð7bÞhiSKMS jHðijÞð0ÞjiSKMSi ¼ hjSKMS jHðijÞð0ÞjjSKMSi

¼2Sþ1Qij for K ¼ 0;2Sþ1 �Qij for K ¼ �1;

(ð7cÞ

hiSKMS jHðijÞð0ÞjjSKMSi ¼ hjSKMS jHðijÞð0ÞjiSKMSi

¼2Sþ1J ij for K ¼ 0;2Sþ1�J ij for K ¼ �1.

(ð7dÞ

The quantities on the right-hand side of Eqs. (7a) and (7b)are related to the potential energy curves of X 1Rg and

1Rg

states,

Sij ¼EðX 1Rg; rijÞ þ n2ijEð1Rg; rijÞ

1þ n2ij; ð8Þ

T ij ¼nijffiffiffi

2p

ð1þ n2ijÞ½Eð1Rg; rijÞ � EðX 1Rg; rijÞ�. ð9Þ

The functions,

1Qij �1

2Eð1Ru; rijÞ þ

n2ijEðX 1Rg; rijÞ þ Eð1Rg; rijÞ1þ n2ij

" #; ð10aÞ

3Qij �1

2½Eð3Ru; rijÞ þ Eð3Rg; rijÞ�; ð10bÞ

2Sþ1 �Qij �1

2½Eð2Sþ1Pu; rijÞ þ Eð2Sþ1Pg; rijÞ�; ð10cÞ

represent the energy band center of the excited 2S+1R and2S+1P states [5]; the excitation transfer integrals for thosestates are defined by [5],

1J ij �1

2Eð1Ru; rijÞ �

n2ijEðX 1Rg; rijÞ þ Eð1Rg; rijÞ1þ n2ij

" #; ð11aÞ

3J ij �1

2½Eð3Ru; rijÞ � Eð3Rg; rijÞ�; ð11bÞ

2Sþ1�J ij �1

2½Eð2Sþ1Pu; rijÞ � Eð2Sþ1Pg; rijÞ�. ð11cÞ


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Combining Eqs. (7)–(11), we obtain matrix representa-tions of the fragment Hamiltonians as

0 1 2 3

0

1

2

3

(13) (13)13

(13)† (13) (13)(13)

(13)

(13)† (13) (13)

(0)

k k k

k

k

k

S= = =

=

=

=

=

T T

T Q J

S

T J Q

H

,

,

,

, ð12Þ

, ð13aÞ

, ð13bÞ

with

, ð14aÞ

, ð14bÞ

TðijÞ ¼ 0 0 0 0 0 0 0 0 0 0 T ij 0½ �; ð15ÞSðijÞ ¼ SijI12. ð16Þ

Here, In denotes the n · n unit matrix. We adopt the non-relativistic dimer potential energy curves E(2S+1K;rij) forthe ground state (X 1Rg) and excited states (3Rg,

3Ru,3Pg,

3Pu,1Rg,

1Ru,1Pg, and

1Pu) computed with the SCF andMRCI schemes by Czuchaj et al. [19], which are availablein the tabulated forms for the range 3.50 6 rij/aB 6 14.00.

Once the dimer fragment matrices are evaluated in theirrespective molecular frames, they can be transformed tothe laboratory-frame representations in accordance with

Hð13Þðh13Þ ¼ Dð13Þðh13Þ Hð13Þð0Þ Dð13Þðh13Þ�1; ð17aÞ

Hð23Þðh23Þ ¼ Dð23Þðh23Þ Hð23Þð0Þ Dð23Þðh23Þ�1; ð17bÞ

Dð13Þðh13Þ ¼

1

Rðh13ÞI12

Rðh13Þ

26664

37775;

Dð23Þðh23Þ ¼

1

I12

Rðh23ÞRðh23Þ

26664

37775; ð18Þ

RðhÞ ¼

H1ðhÞH1ðhÞ

H1ðhÞH1ðhÞ

26664

37775; ð19Þ

where

ð20Þ

refers to the rotation matrix [24] that transforms the spher-ical basis functions of the p orbitals by angle h.

The atomic contribution H(atom) in Eq. (1) is describedas

HðatomÞ ¼

0

Eð0ÞT I9

Eð0ÞS I3

Eð0ÞT I9

Eð0ÞS I3

Eð0ÞT I9

Eð0ÞS I3

26666666666664

37777777777775.

ð21Þ

with Eð0ÞT ¼ 5:1962 eV and Eð0Þ

S ¼ 6:6584 eV denoting theenergy levels of 63P and 61P states, respectively, withoutspin–orbit coupling. Throughout this paper, the energyof infinitely separated ground-state atoms is chosen aszero.

The spin–orbit interaction matrix Vso for the trimer isapproximated by a superposition of the contributionsVatom

so from isolated atoms, assuming that the dependenceof spin–orbit interaction on the interatomic distance is neg-ligible. Czuchaj et al. [19] showed that such ‘‘atoms-in-mol-ecules’’ treatment works well for Hg2 in spite of the


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significantly large spin–orbit interaction. The atomic spin–orbit interaction matrix is expressed compactly in the(J,MJ) representation,

, ð22Þ

with k = 0.5288 eV and k0= 0.4007 eV [19]. The parame-

ter k characterizes the splitting of the 3P manifold into3P2,

3P1, and3P0 states, whereas k 0 accounts for the cou-

pling between 3P1 and 1P1 states. The resultant spin–orbitatomic energy levels are E(1P1) = 6.7036 eV, E(3P2) =5.4606 eV, E(3P1) = 4.8864 eV, and E(3P0) = 4.6673 eV.Eq. (22) can be transformed to the (ML,MS) representa-tion with the aid of the Clebsch–Gordon coefficients,leading to

.

ð23Þ

The spin–orbit interaction matrix which enters the trimerHamiltonian (1) is thus evaluated as

Vso ¼

0

Vatomso

Vatomso

Vatomso

26664

37775. ð24Þ

Numerical diagonalization of the 37 · 37 Hamiltonianmatrix (1) yields energy eigenvalues for the ground stateand a set of excited states of the trimer. The eigenvectorof each excited state is expressed in a form,

jWi ¼ c0j0i þX3i¼1

X1S¼0

X1ML¼�1

XSMS¼�S

ciSMLMS jiSMLMSi

¼ c0j0i þX3i¼1

X1S¼0

XSþ1

J¼jS�1j

XJMJ¼�J

c0iSJMJjiSJMJ i. ð25Þ

Though this is a crude wavefunction constructed from re-stricted basis functions without overlap, it may be helpfulin qualitative understanding of the electronic structuresand transition properties of a trimer. Specifically, thestrength of radiative transitions between excited andground states of Hg3 may be characterized by the transitiondipole moment,

l ¼ jhWjljX0þg ij

ffiX1

MJ¼�1

X3i¼1

c0i01MJ

!lð1S0 � 1P1Þ

"(

þX3i¼1

c0i11MJ

!lð1S0 � 3P1Þ

#29=;

1=2

; ð26Þ

with l(1S0 � 1P1) = 1.55eaB and l(1S0 � 3P1) = 0.269eaBbeing atomic transition moments for the singlet and tripletstates, respectively [19,25]. Since l(1S0 � 1P1) >l(1S0 � 3P1), the magnitude of l may reflect the degree ofmixing of the Hg(1P1) state in the trimer wavefunction.Spatial distribution of excitation within a trimer may bemeasured by the net population of p state for ith atom(i = 1,2,3) introduced as

P i ¼X1S¼0

X1ML¼�1

XSMS¼�S

jciSMLMS j2

¼X1S¼0

XSþ1

J¼jS�1j

XJMJ¼�J

jc0iSJMJj2. ð27Þ

3. Results and discussion

We have confirmed that the DIM scheme in Section 2correctly reproduces the spin–orbit energy levels and equi-librium bond lengths for low-lying bound excited states ofHg2 calculated earlier by Czuchaj et al. [19]; the resultsare summarized in Table 1. The full potential energycurves are given in Fig. 3 of Ref. [19] and are not repeatedhere. Numerical results for Hg3 will be presented belowseparately for the energy ranges of 3.0–4.1, 4.1–5.1, and5.1–5.7 eV.

3.1. Low-lying excited states (3.0–4.1 eV)

Fig. 1 exhibits the potential energy curves of Hg3 in lin-ear configuration (h13 = 0, h23 = 0) describing the collisionof Hg(1S0) with low-lying excited states of Hg2 (A0�g , B1g,C0�u , D1u, 2g, 1g). In each curve, the dimer bond length r12is kept constant at the corresponding equilibrium valuelisted in Table 1. In all the cases treated, we clearly observeattractive interactions between Hg2 and Hg to form stableHg3 bound states. The minima of the potential energycurves dissociating to Hg2ðA0�g Þ �Hgð1S0Þ and Hg2ðA0þg Þ�Hgð1S0Þ asymptotes constitute the lowest excited statesof Hg3, that lie at 3.18–3.20 eV with r23 = 5.3aB.

4 5 6 7 8 93

3.5

4

4.5

E(e

V)

r3−12/aB

C2v trimer

B2

A1

A2

B1

A2

Hg2 A0g−

Hg2 A0g+

Hg2 B1g

Hg2 C0u−

Hg2 D1u

Hg2 2g(3P2)

B2

A1

Hg2 1g(3P2)

B1

A1

A2

B2

2

1

3

Fig. 2. Potential energy curves for Hg3 in C2v configuration that correlateswith Hg2(6

3P0,1,2) + Hg(61S0) asymptotes; r3�12 denotes the distancebetween the Hg atom and the center of mass of Hg2. In each curve, r12 isfixed at the corresponding equilibrium bond length (listed in Table 1) ofthe asymptotic Hg2 state designated on the right edge.

4 5 6 7 8 93

3.5

4

4.5

E(e

V)

r23/aB

linear trimer

Hg2 A0g−

Hg2 A0g+

Hg2 B1g

Hg2 C0u−

Hg2 D1u

Hg2 2g(3P2)

Hg2 1g(3P2)

1 2 3

Fig. 1. Potential energy curves for low-lying excited states of Hg3 in linearconfiguration that correlates with Hg2(6

3P0,1,2) + Hg(61S0) asymptotes. Ineach curve, r12 is fixed at the corresponding equilibrium bond length (listedin Table 1) of the asymptotic Hg2 state designated on the right edge.

Table 1Optimized bond lengths r, potential energy minima E, and transition dipole moments l for the ground and excited states of Hg2

State X0þg ð1S0Þ A0�g ð3P0Þ A0þg ð3P1Þ B1g(3P1) C0�u ð3P0Þ D1u(

3P1) 2g(3P2)

r/aB 7.42 5.16 5.16 5.16 5.35 5.33 5.16E (eV) �0.0365 3.6163 3.6393 3.8421 4.0635 4.1114 4.1923l/eaB – 0 0 0 0 0.107 0

State 1g(3P2) F0þu ð3P1Þ 0�u ð3P2Þ E1u(

3P2) 2u(3P2) G0þu ð1P1Þ 1g(

1P1)

r/aB 5.14 7.6 6.21 6.61 8.27 5.58 6.76E (eV) 4.7007 4.8479 5.1668 5.2983 5.4367 5.7309 5.8294l/eaB 0 0.110 0 0.0139 0 2.15 0


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We have performed similar calculations for the casewhere the direction of the incident atom is perpendicularto the dimer axis and the triatomic system as a whole keepsthe C2v symmetry. The resultant potential curves are shownin Fig. 2 as a function of the distance r3�12 between theincident atom and the dimer center of mass. As the Hgatom approaches Hg2, the degeneracy of the dimer statewith K5 0 is broken and the state is split into two levels.It turns out that the three states (A1, B2, A2) correlating toHg2ðA0þg , A0�g , B1g) + Hg(1S0) asymptotes have potentialminima of 3.1 eV, which is lower than the correspondingvalue for the linear configuration shown in Fig. 1.

In order for detailed investigations of structural stabilityof excited Hg3 in the C2v geometry, we plot in Fig. 3 thepotential energy curves against the bond angle h(= p � 2h13), where the bond length r (= r13 = r23) is opti-mized for a given h so as to minimize the potential energy.The corresponding potential curve for the ground state isalso indicated. Numerical data on the energy levels forthe D1h (h = 180�) and D3h (h = 60�) configurations arelisted in Tables 2 and 3, respectively; the other local and

global potential minima in the C2v configuration are sum-marized in Table 4. The lowest excited level is identifiedas the A0

1 state with D3h configuration, with E = 3.035 eVand r = 5.38aB (see Table 3). The ground-state Hg3 hasthe same A0

1 symmetry, with substantially larger bondlength, r = 7.20aB. Since A0

1 ! A01 dipole transition is for-

bidden, the excited A01 state is expected to be metastable

and long-lived. Located slightly above the A01 state is the

doubly degenerate E00 state at 3.050 eV. This state is unsta-ble and undergoes Jahn–Teller distortion to nearby A2 andB2 states with C2v geometry with E = 3.049 eV, as listed inTable 4. Of these, the B2 state can make dipole transition tothe dissociative ground state (A1), but the magnitude of thetransition dipole moment is minuscule (on the order of10�4eaB).

The A0�u and B1u states in the D1h configuration cor-respond to local minima of the potential energy surfacesand are expected to be stable, except for the A1 state. Ash decreases, their potential energies first increase and thendecrease steeply until they reach deep potential minima

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

60 80 100 120 140 160 180−0.14

−0.1

−0.06

θ(deg)

A1Hg2 X0g

+

A1’

X0g+

E(e

V)

C2v trimer

A2

B1

B2

A1

A0u−

A0u+

B1u

2u

A1’E’’

C0u−

D1u

1u

A2’

E’

E’’

A1’

E’’

A2

A1 B2

B1

A2B2

A1B1

A2’

A2 A1

A1’’

Hg2 A0g−

Hg2 A0g+

Hg2 B1g

Hg2 C0u−

Hg2 D1u

Hg2 1g(3P2)

Hg2 2g(3P2)

1

3

2θ

Fig. 3. Potential energy curves of Hg3 for the ground state (lower panel)and excited states in the range 3.0–4.1 eV (upper panel) in C2v configurationas functions of the bond angle h. The D1h and D3h configurationscorrespond to the cases with h = 180� and 60�, respectively. For each state,the bond length r is optimized for a given h so that the potential energy isminimized; the asymptotic state to which the Hg3 in the D1h configurationis dissociated is designated on the right edge of each curve.


ARTICLE IN PRESS

(A01 and E00 states) at h = 60�. On the other hand, the A0þu

state corresponds to the global minimum of the potentialenergy curve and hence highly stable; it can decay to thedissociative ground state ðX0þg Þ by emitting 4865 A radi-

Table 2Properties of the ground and excited states of Hg3 in the symmetrical linear (Dtransition wavelengths k to the ground state, transition dipole moments l, an

State X0þg ð1S0Þ A0�u A0þu

r/aB 7.72 5.28 5.28E (eV) �0.06467 3.1630 3.1810k (A) – – 4865l/eaB – 0 0.0550P3 0 0.695 0.681

State 1u 0�g F0þg

r/aB 5.30 (6.14) 6.02E (eV) 3.9133 (4.4178) 4.5431k (A) 3749 – –l/eaB 0.257 (0) 0P3 0.607 (0) 0

State E1u 1g 0þg

r/aB 6.80 6.90 7.11E (eV) 5.2663 5.3139 5.3373k (A) 2336 – –l/eaB 0.0175 0 0P3 0.367 0 0

The numbers written in parentheses designate the states that lie on unstable p

ation, which is in good agreement with the observed peakwavelength (4850 A) of the blue-green emission.

Summarizing the foregoing arguments, the origin of the4850 A emission can be ascribed to the Hg3A0þu state in theD1h configuration, which is formed through collision ofHg2ðA0þg Þ with a ground-state atom as indicated inFig. 1. This conclusion is essentially the same as the theo-retical explanation by Omary et al. [14] (though the spin–orbit interaction was neglected there) and corroboratesthe earlier prediction by Callear and Lai [9], namely,

Hg2ðA0�g Þ þHgð61S0Þ ! Hg3ðA0�u Þ;Hg3ðA0þu Þ ! 3Hgð61S0Þ þ hmð4850 AÞ. ð28Þ

On the basis of the observed dimer and trimer absorptionintensities, Callear and Lai [9] also estimated the dissocia-tion energy of Hg3A0þu state to be 0.57 ± 0.03 eV. The cor-responding theoretical value, calculated directly throughthe balance between Hg2A0þg and Hg3A0þu energy levelslisted in Tables 1 and 2, amounts to 0.46 eV, which is some-what smaller than the prediction by Callear and Lai. Itshould be noted that the formation of trimer with theD3h configuration (h ffi 60�) is likewise suggested fromFig. 2 through the reactions,

Hg2ðA0þg Þ þHgð61S0Þ ! Hg3ðA01; 3:035 eVÞ;

Hg2ðA0�g Þ þHgð61S0Þ ! Hg3ðB2; 3:049 eVÞ. ð29Þ

These trimer configurations may not contribute to the4850 A emission, however, since dipole transition to theground state is forbidden.

Koperski et al. [11] also discovered fluorescence bandcentered at 4040 A when the carrier gas pressure was suffi-ciently high. Since the position and peak of this band wasindependent of the species of the carrier gas, these authors

1h) configuration: optimized bond lengths r, potential energy minima E,d population P3 of excitation in the central atom

B1u 2u C0�u D1u

5.27 5.28 (5.45) (5.40)3.3371 3.7328 (3.8327) (3.8575)4607 – – (3692)0.0580 0 (0) (0.108)0.709 0.677 (0.469) (0.511)

1g 0þu 2g 0�u

6.15 (7.40) (6.02) 6.674.6621 (4.8095) (5.1116) 5.1226– (2542) – –0 (0.304) (0) 00 (0.328) (0) 0.439

2u G0þu 0�u 1u

(8.10) 5.93 9.50 (7.22)(5.4112) 5.4216 5.4376 (5.7437)– 2349 – (2135)(0) 2.251 0 (0.0823)(0.342) 0.515 0.344 (0.535)

oints of the potential surfaces.

Table 3The same as Table 2 but for the equilateral triangular (D3h) configuration

State A01ð1S0Þ A0

1 E00 A02 E

0E00 A0

1 A02

r/aB 7.20 5.38 (5.38) 5.37 (5.48) (5.46) 5.45 5.43E (eV) �0.1275 3.0352 (3.0496) 3.3866 (3.6702) (3.7222) 3.8581 3.8748k (A) – – – – (3920) – – –l/eaB – 0 (0) 0 (0.102) (0) 0 0

State A001 E00 E

0E00 E

0E00 E

0

r/aB 5.92 (5.34) (5.86) (5.69) (5.60) (6.40) (7.40)E (eV) 3.9898 (4.0022) (4.2078) (4.4380) (4.9928) (5.2198) (5.3195)k (A) – – (3054) – (2684) – (2275)l/eaB 0 (0) (0.0546) (0) (1.39) (0) (0.0158)

State A001 A0

2 E00 E0

E00

r/aB 7.80 8.90 (9.10) (6.19) (7.23)E (eV) 5.3764 5.3978 (5.4026) (5.4604) (5.7366)k (A) – – – (2263) –l/eaB 0 0 (0) (1.032) (0)

Excitation is distributed equally among the three atoms (P1 = P2 = P3) except for E0and E00 states.

Table 4The same as Table 2 but for the C2v configuration; h denotes the optimized bond angle

State A2 B2 A2 A1 B2 B2 B1

r/aB 5.37 5.39 5.31 5.32 5.81 5.37 5.38h (deg) 60.4 59.7 73.0 72.6 53.4 96.7 64.4E (eV) 3.0493 3.0493 3.5653 3.5793 3.6173 3.6509 3.6577k (A) – 5136 – 3996 3897 3850 3936l/eaB 0 1.64 · 10�4 0 0.0946 0.0142 0.0396 0.110

State A2 B2 B2 A2 A1 B1 A2

r/aB 5.34 5.26 5.31 5.42 5.92 5.67 5.75h (deg) 108.0 114.6 62.6 58.0 55.4 64.2 58.6E (eV) 3.7674 3.9019 3.9717 3.9760 4.1534 4.1635 4.4360k (A) – 3748 3738 – 3162 3144 –l/eaB 0 0.269 0.0443 0 0.0470 0.0353 0

State B2 A1 B1 B1 B1 B2 A2

r/aB 5.67 5.41 6.32 5.95 5.80 6.12 6.81h (deg) 61.2 84.8 133.4 88.6 54.4 75.8 53.4E (eV) 4.4362 4.7642 4.7909 4.8670 4.9273 5.1027 5.1551k (A) 2980 2821 2593 2573 2728 2416 –l/eaB 7.32 · 10�4 0.766 0.485 0.321 1.343 6.40 · 10�3 0

State B1 A1 B2 A2 A1 B1 B2

r/aB 7.06 7.58 9.05 9.20 6.19 6.19 7.01h (deg) 67.6 55.4 60.8 57.0 60.2 59.9 91.4E (eV) 5.3080 5.3106 5.4024 5.4018 5.4588 5.4601 5.6124k (A) 2282 2280 2264 – 2262 2263 2175l/eaB 0.0393 0.0434 2.62 · 10�4 0 1.504 0.711 0.0557

State A2

r/aB 7.66h (deg) 54.0E (eV) 5.6975k (A) –l/eaB 0


ARTICLE IN PRESS

inferred that this band originates from Hg3. As possiblecandidate states responsible for this emission, the A1 state(3.579 eV), B1 state (3.658 eV), and B2 states (3.617 eV,3.651 eV) might be considered (see Table 4). Of these, theA1 and B2 states can be formed from the Hg2D1u state;

the presence of Hg2D1u has been confirmed in the experi-ment [11] through the well-known 3250 A emission dueto the Hg2D1u ! X0þg transition.

It can be found from Tables 2–4 that the excited statesbelow 4 eV exhibit the following general features: (i) The


ARTICLE IN PRESS

optimized bond length r takes on values appreciably smal-ler than those in the ground state (7.2aB), because thesestates correlate with tightly bound excited dimers whosebond lengths are as small as 5.16–5.33aB; (ii) The variationof r against bond bending is relatively small and is confinedwithin 0.1aB for the A0�u state, for instance, when h is var-ied from 60� to 180�; (iii) P3 > 1/3 for the D1h configura-tion, indicating that excitation is localized mainly on thecentral atom.

Concerning the properties of the ground-state Hg3, thepresent theory predicts D3h structure; its cohesive energyper bond (0.0425 eV) is larger than that of Hg2(0.0365 eV), and its bond length (7.20aB) is smaller thanthat of Hg2 (7.42aB). These trends are qualitatively consis-tent with the ab initio theories [12,13] and are attributedto the mixing effect described by Eq. (6). Quantitatively,however, the binding is still weaker than the ab initioresults (0.061 and 0.076 eV/atom in Refs. [12] and [14],respectively); furthermore, D1h structure was obtainedin Ref. [14] instead of D3h. These discrepancies may arisenot only from underestimation of binding energy in theground-state Hg2 potential curve [19] and our crude esti-mation of the mixing parameter, but also from our neglectof the mixing of Hg+–Hg� charge-transfer states in theground-state dimer [4,26]. More elaborate treatment ofthe ground-state clusters will not be pursued here, how-ever, because the excited-state property is the main subjectof this study.

3.2. Excited states in the range 4.1–5.1 eV

Fig. 4 shows the potential energy curves of the excitedstates in the range 4.1–5.1 eV. We find that the Hg3F0

þg

state dissociates to Hg2F0þu þHg1S0 asymptote; the

Hg3(1g,2g) states correlate with the ground-state dimerðHg2X0þg Þ and atomic Hg(3P1,

3P2) states. These asymp-

60 80 100 120 140 160 1804.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

5.1

E(e

V)

θ(deg)

C2v trimer

A2

B1

A1

F0g+

1g

0u+

B1

B2

A1

B1

E’

E’’

E’

Hg2 F0u+

Hg2 D1u

Hg 3P1

A2

A1

Hg 3P2B2 2g

Fig. 4. The same as Fig. 3, but for the energy range of 4.1–5.1 eV.

totic dimer states are much weakly bound compared tothose in the low-lying Hg2 states. Moreover, the energyof the Hg3 1g state (4.66 eV) turns out even higher than thatof the asymptotic Hg2D1u state (4.11 eV): As shown inFig. 5, when the trimer is stretched, its potential energy firstincreases from 4.66 to 4.71 eV as the distance r13 betweenthe outermost atoms is increased from 12.3aB to 14aB.The potential energy then decreases suddenly when r12exceeds about 17aB, where the potential energy curve inter-sects that of D1u state and the trimer spontaneously disso-ciates to Hg2D1u (4.11 eV) and Hg 61S0 states.

Though not shown in Fig. 4, there exists linear symmet-ric 0�g state at 4.418 eV, as listed in Table 2. This statebecomes unstable once h is slightly decreased from 180�.We observe in Fig. 6 that the potential minimum of the0�g state is located at r = 6.14aB whereas a crossing of 0�gand 1u levels is observed at r ffi 6.0aB. This level crossingno longer occurs for h < 180� when these two states acquirethe same B2 symmetry. As a result, the potential minimumat r = 6.14aB disappears at h = 170�, as indicated by thesolid curve.

Summarizing the numerical results presented in Tables2–4, we notice that typical wavelength of emissionexpected from Hg3 excited states in the energy range of4.1–5.1 eV extends from 2400 to 3200 A. So far, therehas been no report on the detection of fluorescence fromHg3 in this wavelength regime, however. In the analysisof spectroscopic data, it has usually been assumed thatthe formation of Hg3 might proceed through attachmentof a Hg atom to a metastable Hg2 reservoir state existingin a high-density vapor [8,9]. As we have mentionedabove, the seed dimers may not be stable enough to cre-ate Hg3 efficiently in the energy range under consider-ation, though definite conclusions should await for

5 6 7 8 94

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

E(e

V)

r12/aB

linear trimer

1 2 3

r13=14aB

r13=17aB

Hg2(D1u)+HgD1u

1g

1g

D1u

Fig. 5. Change in the energies of the 1g and D1u states as the position ofthe central atom deviates from the center of symmetry. The solid anddashed curves correspond to the cases with r13 = 17aB and 14aB,respectively.

C2v trimer

AE’’ Hg2 1g(

1P1)1u

A2 B1

5 5.5 6

3.9

4

4.1

4.2

4.3

4.4

4.5E

(eV

)

r/aB

C2v

1u

0g−

θ=180deg

θ=170deg

B2

B2

Fig. 6. Crossing of 0�g and 1u states at h = 180� (dashed curves) andrepulsion of the two B2 states at h = 170� (solid curves).


ARTICLE IN PRESS

further analysis on the kinetics of various chemical spe-cies [9] under the experimental conditions.

We mention other notable features associated with theHg3 states in this energy range: (i) In the F0þg , 1g and 2gtrimer states, excitation is localized at the outermostatoms instead of the central atom (i.e., P3 = 0). (ii) Thepotential energy curve of the A2 state changes its slopesuddenly at h ffi 82�, E ffi 5.1 eV. Such abrupt change ofthe potential curves may generally be expected whenthe potential curve has two minima as a function of r

and relative stability between the two states is reversedat a certain value of h. As shown in Fig. 7, this happensat h ffi 82�, giving rise to discontinuous change of theoptimized bond length from 6.1aB to 5.24aB. Similartransitions are predicted for the B1 state at E = 4.8 eV,h ffi 90� and for other higher excited states presented inthe next subsection.

5 5.5 6 6.54.9

5

5.1

5.2

5.3

r/aB

E (

eV)

80deg

A2 state

84deg

θ=88deg

Fig. 7. Potential energy curves for the A2 state at h = 80� (solid curve), 84�(dotted curve), and 88� (dashed curve).

3.3. Higher excited state (5.1–5.7 eV)

The potential energy curves for higher excited states ofHg3 (5.1–5.7 eV) are exhibited in Fig. 8. The states locatedbelow 5.4 eV correlate to the atomic 63P2 state at 5.46 eV.The Hg3 states correlating with the Hg2G0þu ð1P1Þ þHgð1S0Þand Hg21g(

1P1) + Hg(1S0) asymptotes lie above 5.4 eV;radiative decay from these Hg3 states mixed with Hg 61P1

state could be a dominant source of fluorescence in theUV range. Experimentally, Niefer et al.[10] reported obser-vation of 2170 A fluorescence band which was tentativelyassigned as an emission from Hg3 excited state in the vicin-ity of 5.5–5.6 eV; existence of this trimer state was inferredfrom the spectra indicating absorption from the Hg3A0�ustate (near 3.09 eV) induced by the probe laser. Theobserved fluorescence spectra exhibit broad distributions,extending up to about 2300 A. Supersonic expansion beamexperiment by Koperski et al. [11] also detected the 2125 Afluorescence band that is very sensitive to the carrier gaspressure and may be attributed to an emission from Hg3.

Numerical results presented in Fig. 8 reveal that,though no stable trimer levels exist near 5.5 eV, several tri-mer levels are predicted near 5.4 eV and also there is oneB2 state that has potential minimum of 5.6 eV. The wave-length of emission from the bottom of the B2 state at5.612 eV amounts to 2180 A, which is in good agreementwith observed wavelength [10]. To elucidate other possibleorigins of the 2170 A UV emission from Hg3, we show inFig. 9 the potential energy curves of relevant Hg3 excitedstates in the D1h configuration that lie in the range of5.4–5.7 eV. The corresponding curves for the lowestexcited states ðA0�u Þ are also indicated for comparison.

60 80 100 120 140 160 180

5.1

5.2

5.3

5.4

5.5

5.6

5.7

E(e

V)

θ(deg)

A2

B1

A1

2g

1g

B1

B2

E’’

E’

A2

E’’

1u

0u−

E’

Hg2 2u(3P2)

Hg2 0u−(3P2)

Hg2 E1u(3P2)

Hg 3P2

B2

A1

1

G0u+

Hg2 G0u+

B1

Hg 3P2

B2

0g+

B2

2u

A1’’A2’

A2

A2 0u−

Hg 3P2

Hg2 0g−(3P2)

B1

(to 5.4eV)

A2’

Fig. 8. The same as Fig. 3, but for the energy range of 5.1–5.7 eV.

4 5 6 7 8 9

0

1

2

3

4

5

6

E(e

V)

r/aB

linear symmetric trimer

X0g+

A0u−

A0u+

G0u+ 2u

1u

rr

Fig. 9. Potential energy curves for high-lying excited states of Hg3 in therange 5.4–5.7 eV, superimposed on those for the ground ðX0þg Þ state andthe first excited ðA0�u Þ states; the D1h configuration is assumed.

4 5 6 7 8 9

0

1

2

3

4

5

6

E(e

V)

r/aB

D3h trimer

A1’

E’’

A1’

E’

E’

A2’

E’’

E’’

r

r

r

Fig. 10. Potential energy curves for high-lying excited states of Hg3 in therange 5.4–5.7 eV, superimposed on those for the ground ðA0

1Þ state and thefirst excited (A0

1, E00) states; the D3h configuration is assumed.

6 7 85.4

5.45

5.5

5.55

E(e

V)

r/aB

C2v

G0u+

2u

θ=180deg

θ=160deg

B1

B1

Fig. 11. Potential energy curves for the two B1 states at h = 160� (solidcurves) and those for G0þu and 2u states at h = 180� (dashed curves).


ARTICLE IN PRESS

We find that the 1u state (5.744 eV) correlating toHg21g(

1P1) can decay to the ground state through emissionof k = 2140 A, though this state continues to the more sta-ble B2 state mentioned above. On the other hand, thewavelengths of dipole transitions from the other levels,such as the Hg3G0þu state (5.422 eV), B1 state (h = 68�,E = 5.308 eV), and the flat B1 band at 5.46 eV continuedfrom the 2u state, amount to 2260–2350 A, which aresomewhat larger than the peak wavelength (2170 A) ofthe band [10]. It should be noted here that all of theHg3 states described above possess odd (u) parity so thatthey cannot be populated directly through absorptionfrom the lower Hg3A0�u states, contrary to the scenariosuggested by Niefer et al. [10].

Similar potential curves for the D3h geometry are exhib-ited in Fig. 10. As shown in Tables 3 and 4, the E 0 state(5.460 eV, r = 6.19aB) as well as the nearby A1 and B1

states formed through Jahn–Teller distortion can makedipole transitions with k = 2260 A. The E00 state at5.74 eV is also important because it continues to the B2

state (5.612 eV) which can emit 2180 A radiation. Contraryto the case of D1h configuration, it is in principle possibleto populate E 0 and E00 states through excitation from thelower A0

1 and E00 states, respectively, at 3.0 eV. It shouldbe noted, however, that the bond length of the upper E00

state is as large as 7.2aB; it is questionable if this state couldbe populated efficiently from the lower E00 state that hassmaller bond length (5.4 a.u.).

Finally, we give the following remarks: (i) The E00 andA0

2 states at 5.4 eV have significantly large bond length ofabout 9aB and correspond to local minima of the potential

curves: the global minima exist at shorter bond lengths, asindicated by dotted curves in Fig. 10. (ii) The B1 state plot-ted by the thin solid curve in Fig. 8 is proven to be unstableat h = 180�, as illustrated in Fig. 11: When h is slightlysmaller than 180�, the B1 state has a potential minimumnear 5.5 eV and continues to the G0þu state at large bondlength, whereas we find another B1 state that has lowerenergy and continues to the 2u state. The potential mini-mum of the upper B1 state disappears at h = 180� due tothe onset of G0þu � 2u level crossing, as shown by the dot-ted curves. (iii) Abrupt changes of the optimum bond


ARTICLE IN PRESS

lengths similar to those illustrated in Fig. 7 are predictedfor the A1 state (E = 5.38 eV, h = 72�) and the A2 state(E = 5.34 eV, h = 90�).

4. Concluding remarks

In conclusion, we have presented theoretical potentialenergy surfaces for excited states of Hg3 calculated withthe DIM formalism by using the interatomic potentials ofHg2 excited states by Czuchaj et al [19]. Stable geometriesof Hg3 excimers have been investigated for the C2v config-uration in the energy range 3.0–5.7 eV. Implications of thepresent results for the observed fluorescence spectra inoptically excited mercury vapor have been discussed. Theoutcome of this work is summarized as follows:

(i) The Hg3 excited states below 4.1 eV are formedthrough attachment of the ground-state atom totightly bound low-lying Hg2 excimers. The structureof the potential energy surfaces of the low-lyingexcited states is consistent with the earlier conjectureby Callear and Lai [9] that the origin of the blue-green emission centered at 4850 A may be attributedto the Hg3A0þu state. The lowest excited state of Hg3is proven to be the A0

1 state at 3.035 eV, for whichdipole transition to the ground state is forbidden.The fluorescence band near 4040 A observed by Kop-erski et al. [11] may arise from the trimer states in therange 3.58–3.65 eV that correlate with the Hg2D1ustate.

(ii) The trimers in the range 4.1–5.1 eV dissociate eitherto weakly bound Hg2 excited states near the atomicHg(3P1) and Hg(3P2) levels or to the ground Hg2state. The optimized bond lengths and potential ener-gies depend sensitively on the bond angles.

(iii) The higher-lying Hg3 states correlating withHg2G0þu ð1P1Þ þHgð1S0Þ or Hg21g(

1P1) + Hg(1S0)asymptote lie in the range 5.4–5.7 eV, though no sta-ble levels exist near 5.5 eV. Of these, only the B2 stateat 5.612 eV (h = 91�) can account for the observed2170 A fluorescence band [10,11], but the kinetic pro-cesses of populating this state in the real experimentalsituation remains uncertain. The wavelength of dipo-lar transitions from the other states near 5.4 eV is pre-dicted as 2300 A, which is somewhat larger thanobservation.

A number of improvements on the theory should bemade in the future. In the present work, diatomic poten-tials for high-lying states dissociating to Hg(71S, 73S, 71P,73P) + Hg61S asymptotes have been neglected. The poten-tial curves for those Rydberg states are strongly attractiveand their minima lie close to the lower-lying states correlat-ing to Hg61P + Hg61S asymptotes [19]. Consideration ofHg(7S, 7P) configurations in DIM might therefore resultin some modifications of the high-lying excited states ofHg3 relevant to the 2170 A fluorescence band. It was shown

that ionic (Hg+–Hg�) configuration provides the mecha-nism of strong binding for those Rydberg states [19] andsoftening of short-range repulsion in the ground-statedimer [4,26]. Mixing of ionic states could be incorporatedthrough the diatomics-in-ionic-systems framework [18,28],which would enable one to analyze the influence of s � pmixing on the transition dipole moments [27] and polariz-abilities [4,26] in larger clusters and dense gas. Moredetailed analysis of experimental spectra would also requireevaluations of vibrational wavefunctions and Franck–Con-don factors. The principal achievement of this work hasbeen to produce theoretical potential energy surfaces ofHg3 excited states for a wide energy range within the sim-plest version of the DIM theory.

Acknowledgments

The author thank Dr. M. Yao for pertinent discussionson this and related subjects, and Dr. F. Hensel forintroducing Ref. [4]. This work was supported in partthrough Grant-in-Aid for Scientific Research provided bythe Japanese Ministry of Education, Science, Sports andCulture.

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[2] F. Hensel, W.W. Warren Jr., Fluid Metals, Princeton Univ. Press,Princeton, 1999, Chap. 4.

[3] J. Koperski, Phys. Rep. 369 (2002) 177.[4] J.N. Greif-Wustenbecker, Ph. D. Thesis, Phillips University, Marburg

(2000).[5] L.A. Turkevich, M.H. Cohen, J. Phys. Chem. 88 (1984) 3751;

L.A. Turkevich, M.H. Cohen, Phys. Rev. Lett. 53 (1984) 2323.[6] H. Uchtmann, R. Dettmer, S.D. Baranovskii, F. Hensel, J. Chem.

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theoretical potential energy surfaces for excited mercury...

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