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Spectrochimica Acta Part A 62 (2005) 790–799
Ab initio study of low-lying triplet states of the lithium dimer
Boris Minaeva,b,∗
b
a State University of Technology, Cherkassy 18006, Ukraine
Department of Biotechnology, SCFAB, The Royal Institute of Technology, SE-10691 Stockholm, Sweden
Received 3 January 2005; received in revised form 2 March 2005; accepted 2 March 2005
Abstract
Observation of Bose–Einstein condensation in 7 Li2 initiated the interest in the scattering length of two ground state lithium atoms when
they approach each other as a radical pair triplet a3 +
u state. But some properties of this state are still unknown. In present work, a number
of low-lying triplet states of lithium molecule are calculated by multi-configuration self-consistent field (MCSCF) and response techniques
with account of spin–orbit coupling, spin–spin coupling and some other magnetic perturbations. The singlet–triplet transition probabilities
to the ground state are also presented. Most results are connected with the weakly bound lowest triplet a3 +
u state, whose radiative lifetime
and spin-splitting are unknown so far in spite of its great importance in Bose–Einstein condensation. Calculations indicate that this state
has a very small spin-splitting, λss = −0.01 cm−1 , which is negligible in comparison with the line-width in experimental Fourier transform
3 +
spectra published so far. Similar splitting is obtained for the upper state of the 13 +
g –a u transition. This is in agreement with experimental
3 +
3 +
rovibronic analysis of the 1 g –a u band system in which the triplet structure was not resolved. The radiative lifetime of the a3 +
u state is
predicted to exceed 10 h.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Lithium dimer; Triplet states; Zero-field splitting; Spin–spin coupling
1. Introduction
The lithium dimer is the second simplest stable homonuclear diatomic molecule (after H2 molecule), therefore its
spectra received a great attention from both theoretical and
experimental studies [1–5]. Refs. [6,7] have, for example,
presented an extensive bibliography and collection of high
quality ab initio results for potential energy curves and transition dipole moments. One should note that until recent time
[8,9], only singlet state spectroscopy of the Li2 molecule were
extensively investigated [6,7,10]. Two main transitions between low-lying singlet states were studied very carefully,
1 +
1
1 +
namely 1(A)1 +
u –X g and 1(B) u –X g [4,6]. A number of singlet Rydberg states, including states very close to
ionization limit, were identified [2]. Compared with the empirical knowledge of the excited singlet states, much less
was known about triplet states of Li2 , because the transitions
from the electronic ground state X1 +
g to excited triplets are
∗
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doi:10.1016/j.saa.2005.03.006
strongly forbidden. Spin–orbit coupling (SOC) which is responsible for the singlet–triplet (S–T) transitions probability
is very weak in lithium [11] and the S–T mixing need to be
enhanced by collisions [12], or by special optical methods
like the perturbation-facilitated double-resonance technique
[13].
Triplet excited states of usually stable diamagnetic
molecules with closed-shell ground state are very important
in many respects; they are connected with catalytic reactions
[11,14], photochemistry, luminescence [15] and many optical
pumping experiments [3,16,17]. Finally, the triplet states provide an important test to the electronic structure theory. From
the valence bond method [16], it follows that the electron pair
3 +
in the singlet state, X1 +
g , and its triplet counterpart, a u
state, dissociate to the same limit. In hydrogen molecule, the
singlet state is bound and the triplet state is repulsive [16,18].
In Li2 , the triplet a3 +
u state is slightly bound with dissociation energy equal to 0.04 eV [3]. In spite of its instability,
the 13 +
u state of H2 molecule has been known more than
one century [16]; for all that the stable Li2 analogous state is
experimentally discovered just 20 years ago [13,19].
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
791
Fig. 1. Potential energy curves for a number of excited states of Li2 molecule
obtained by linear response calculation from the ground state MCSCF wave
function.
Fig. 2. MCSCF potential energy curve for the three lowest states of Li2
molecule.
Potential energy curves for a number of low-lying states of
the Li2 molecule calculated in present work are given in Fig.1,
which is presented just to recapitulate the results of previous
works and illustrate the main spectral features [6,7,20]. Three
lowest states are presented in larger scale in Fig. 2. Though
the triplet states of H2 molecule are well known from electric
discharge emission spectra since the beginning of molecular
spectroscopy [16], the first observation of the triplet state,
1(b)3 u , in Li2 molecule was reported only in 1983 [12].
Diffuse bands in emission spectra of dense alkali vapors have
also been known over 20 years [21], but triplet states of the
lithium dimer have not been observed directly until the work
of Engleke and Haage [12]. These authors have detected the
3 → b3 transition near 507 nm in collision-induced flug
u
orescence and obtained the lower state molecular constants
from the partly resolved rotational structure. The 1(b)3 u
state spectroscopic constants were also derived from examination of the accidental predissociation of the 1(A)1 +
u rovibronic levels [22]. The 1(b)3 u state of Li2 has received
a great attention not only because of its important role in
the perturbations and accidental predissociation of the 11 +
u
state (A state) [3,13], but also because of its fine structure
[2,23].
3
The crossing between 1(A)1 +
u and 1(b) u states in
Fig. 1, is one of the main features of the lithium dimer spectroscopy [6,8,9,13]. This A–b crossing is responsible for the
accidental predissociation of the 1(A)1 +
u state [23]. The
SOC perturbation between these singlet and triplet states provided a mixed “window” levels, through which the system
can penetrate from singlet to triplet manifolds. Using this
A–b window, a number of excited triplet states of lithium
dimer have been reached by perturbation-facilitated optical–
optical double-resonance (PFOODR) spectroscopy [8,9,13].
The 1(b)3 u state in turn predissociates via rotational–
electronic interaction with the 1(a)3 +
u continuum (Fig. 1)
[23]. This weakly bound lowest triplet state of Li2 molecule
is getting increasing particular interest in recent time [3–
5,24,25].
The first rotationally resolved observation of the 1(a)3 +
u
state has been reported by Xie and Field [13,19]. Using the
3
selected (11 +
u –1 u ) mixed levels of the A–b gateway,
the lowest triplet state in Li2 molecule has been studied by
double-resonance methods and by the 3 g → 13 +
u fluorescence detection [3,13]. All triplet states of Li2 belong to
the Hund’s case (b) coupling scheme [2,3,13,17], since spin–
orbit coupling is very weak even for 3 states. The emission
which should correspond to electronic transition from the
1 +
weakly bound triplet state, a3 +
u , to the ground state, X g ,
is rather strictly forbidden as electric dipole radiation, even
when spin–orbit coupling is accounted; a magnetic dipole
transition moment for such emission has been calculated in
the present work and found to be completely negligible.
Recent developments in atom trapping and cold-atom
spectroscopy have led to new possibilities in the triplet states
study of the Li2 molecule through combinations of measurements of cold collisions, photo-association spectroscopy and
magnetic-induced Feshbach resonances [1,26,27]. Collisions
of alkali metal atoms at ultra low temperatures (10−4 K) are
very sensitive to the details of the interatomic potentials. Observation of Bose–Einstein condensation in 7 Li2 [1] initiated
the interest in the scattering length of two ground state lithium
atoms when they approach each other as a triplet radical pair
[5]. This requires knowledge of the accurate potential energy
curve of the lowest triplet a3 +
u state of the Li2 molecule
[3,5].
The sign of the scattering length of two ground state
lithium atoms (the triplet radical pair) determines the stability of the Bose–Einstein condensate; it can be obtained
792
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
from the highest bound vibrational level wave function of the
3 +
a 3 +
u state [3]. Spectroscopic constants of the a u state
have been studied by indirect spectroscopy methods [3,13].
The a3 +
u state is not of easy access by direct excitation
1 + ), since spin–orbit coupling is very weak in
(a3 +
←X
u
g
lithium and the singlet–triplet transition is rather strongly forbidden.
The weakly bound a3 +
u state in Li2 molecule, produced
by the single electron excitation 2␴g → 2␴u has a shallow
minimum at re = 4.2 Å [3]. The lowest triplet state minimum
in lithium dimer is not deep (De = 333.7 cm−1 ) but still contains 10 vibrational quanta [3]. The properties of this weakly
bound triplet lithium dimer have been the subject of great
interest in recent time [1,3,5]. The depth of the minimum,
equilibrium distance, the RKR potential and the energy gap
with respect to the ground state, Te = 8183 cm−1 , have been
accurately determined [3]. A possible source of error lies in
the fact that all experimental data were combined into a single fit for the three spin components of the 13 +
u state, which
was treated as if it were a singlet 1 state [3]. In order to solve
this puzzle, one has to calculate the spin-splitting in the lowest triplet state of lithium dimer and this is the main purpose
of the present work.
Absorption triplet spectra of the lithium dimer at visible
wavelengths can be considered as an extension of collisional
broadening of the atomic lithium resonance lines. In all these
connections, the natural radiative lifetime of the lowest triplet
state of Li2 molecule is of great interest. The singlet–triplet
3 +
X 1 +
g –a u transition probability is also important for estimation of the photodissociation cross-section in the range of
880–910 nm (the vertical excitation from the ground state). In
the same forbidden region of the spectrum, the most intense
3
0,0 band of the X1 +
g –1 u transition is expected.
The radiative lifetime of the a3 +
u state is unknown since
1 +
neither the phosphorescent emission a3 +
u → X g nor the
a←X absorption have been observed so far. It is one of
the goals of this study to estimate the a–X and b–X transitions probability in Li2 molecule. In this paper, we are going
to present spin-property calculation for the three low-lying
triplet states of the lithium dimer: their spin-splitting and
singlet–triplet transition probabilities for each spin sublevels
in a non-rotating molecule.
2. Method of calculation
Multi-configuration self-consistent field (MCSCF) theory
has been used to solve the Schrödinger equation for electronic
wave function in the adiabatic approximation:
H0 κ ΨnMS = κ En κ ΨnMS .
(1)
In Eq. (1), κ = 2S + 1 is the multiplicity of the state and S is
the total spin quantum number:
2
Sˆ κ ΨnMS = S(S + 1)2 κ ΨnMS ,
(2)
and MS determines the projection of the total spin:
Sz κ ΨnMS = MS κ ΨnMS .
(3)
The response method [28–30] have been applied for the fine
3
3 +
structure calculations of the (a)13 +
u , (b)1 u and 1 g
triplet states of the Li2 dimer and for the a–X and b–X transition dipole moments prediction at different internuclear
distances. To this end, the relativistic terms due to the intramolecular magnetic interaction have been added to the
non-relativistic Hamiltonian. The spin–orbit coupling operator was used in a complete Breit–Pauli form [28]. Expectation
values of the spin–spin coupling (SSC) operator:
si · sj
3(si · rij )(sj · rij )
2 2
HSSC = µB ge
,
(4)
−
rij3
rij5
i,j
have been calculated by the recently developed code within
the MCSCF technique [30]. Here, si is a spin angular momentum of electron i, µB is the Bohr magneton and ge is a
g-factor of the free electron.
There is also magnetic interaction energy created by spins
and rotational motion of the electrically charge nuclei [17]:
· S,
HSR = γv N
(5)
where γv is the electronic spin–rotation constant in the v
vibrational quantum
state, N is the rotational angular mo
mentum and S = i si . The orbital motion of electrons also
contributes to this spin–rotation interaction. All these perturbations have been included here: the calculated electronic
g-factor [31] has been used in order to estimate the electronic spin–rotation coupling constant by the Kurl’s formula [17]: γv = −2Bv g. Here, Bv is a rotational constant,
g is the anisotropic correction to free-electron g-value:
ge = 2.002319. The g-tensor components perpendicular to
the molecular axis, gxx = gyy = ge + g, are calculated by
the MCSCF response method, which is described in detail
elsewhere [32,33].
The orbital angular momentum Λ about the internuclear
axis z is equal to zero in the 3 states and the electron spin vector is very weakly coupled to the rotation of the molecule. This
type of coupling between rotation and electronic motion is
known as the Hund’s case (b) [16]. When spin–spin and spin–
orbit coupling are taken into account the weak interaction of
spin angular momentum S with the internuclear axis occurs;
spin S is quantized in zero external magnetic field with respect to the molecular frame [16]. Spin interaction with the
is also present, but in weaker
rotational angular momentum N
extent. In that case, the spin-splittings of the rotational levels
of the 3 states are given by the following equations [17]:
F1 (N) = Bv N(N + 1) + (2N + 3)Bv − λv
− (2N + 3)2 Bv2 + λv 2 − 2λv Bv + γv (N + 1),
(6)
F2 (N) = Bv N(N + 1),
(7)
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
F3 (N) = Bv N(N + 1) − (2N − 1)Bv − λv
+ (2N − 1)2 Bv2 + λv 2 − 2λv Bv − γv N,
(8)
where F1 , F2 and F3 refer to the J = N + 1, N and N − 1
sublevels, respectively. γv is the electronic spin–rotation
coupling constant in the vibrational state v.
The spin-splitting parameter λ in Eqs. (6)–(8) is determined by the electron spin–spin coupling in the first order
of perturbation theory and by SOC in the second order [17].
A conventional sum-over-state expansion provides the SOC
contribution as a sum over singlet, triplet and quintet states:
2λ = Hz,z = 3 Ψ0z |HSSC |3 Ψ0z −
n,S MS
M
3 Ψ0z |HSOC |2S+1 ΨnMS 2S+1 Ψnk |HSOC |3 Ψ0 S ×
. (9)
2S+1 E − 3 E
n
0
In this expression, 2S+1 ΨnMS is an eigenfunction of the nonrelativistic Schrödinger equation, Eq. (1), which is the zerothorder wave function for perturbation treatment. The indices
z and MS determine the projection of the total spin and z
corresponds to MS = 0 in case of diatomic molecule with a
3 state. Accounting for rovibrational states, it leads to Eqs.
(6)–(8). Most of the observed a3 +
u levels of the Li2 molecule
lie higher than the 2s + 2s dissociation limit (Fig. 1) and are
quasi-bound by the centrifugal barrier to dissociation [13].
The spin-splitting can be represented by the general zerofield splitting Hamiltonian which is widely applied for polyatomic triplet molecules [15]:
1
HS = D Sz2 − S 2 + E(Sx2 − Sy2 ).
(10)
3
This is a compact form of more general spin Hamiltonian:
HS = −XSx2 − YSy2 − ZSz2 [15], where D = −3/2Z, E =
1/2(Y − X) and |Z| is assumed to be the largest value. For
the 3 state, both D and E parameters are not equal to zero,
whereas for the 3 states E = 0 and D = 2λ. In this case,
Eq. (9) just determines the λ parameter. In some experimental
work on Li2 spectra [34], the C parameter is used, which
is connected with our D value by the simple relation, D =
−3C .
For the 3 state in Hund’s case (b), the splitting of the rotational levels is more complicated; especially for the 1(b)3 u
state in Li2 where a strong SOC mixing between A and b excited rovibrational levels occurs [19,23]. The 1(b)3 u state of
Li2 is of great interest because of its role in predissociation
of states correlating with the 2s + 2p atomic limit (Fig. 1)
[19,23]. Theory predicts [35] the accidental predissociation
of the 1(A)+
u state through SOC mixing with excited rovibrational levels of the 1(b)3 u state which in turn predissociates via interaction with the 1(a)3 +
u continuum. The
accidental predissociation of the Li2 1(A)+
u state results in
anomalously short radiative lifetime at the J levels near the
A–b crossings [19,23]. As follows from our calculations (see
further), the SOC matrix element between two triplet states
793
1(b)3 u –1(a)3 +
u near their crossing at 2.57 Å (Fig. 1) is
negligibly small and does not contribute to the spin-splitting
of the type given by Eq. (9), in the vicinity of the equilibrium.
The states are doubly degenerate in orbital space (besides spin). For one component of the 3 state, one gets
D = −3/2X and E = 1/2(Y − Z); for the other component
X and Y should be interchanged, if the real wave functions
are used [36]. Besides such zero-field splitting, the 3 states
in the Hund’s case (b) are splitted by first-order SOC contribution, which is described by the following spin Hamiltonian
[16]:
Ee = E0 + AΛΣ
(11)
where E0 is the energy of the state when the SOC is neglected,
A is the SOC constant for a given multiplet state and Σ = MS
is a spin projection on the molecular z-axis. Both results, Eqs.
(10) and (11), have to be combined. This analysis concerns
a non-rotating molecule. From our calculations, A 1 cm−1
for the 1(b)3 u state of Li2 , therefore the state really belongs
to the Hund’s case (b). In the case of Λ = 0, the first-order
SOC contribution, Eq. (11), is zero. The spin–orbit contribution to the splitting in second-order perturbation theory, Eq.
(9), is obtained in the present work by expansion including
n = 10.
Two types of basis sets have been used: 6-311G* and 6311++G(3df,3pd) [37]. The first basis is rather small (11s 5p
1d → [4s 3p 1d]) and includes 36 contracted functions for
Li2 molecule; the second one is big enough (12s 6p 3d 1f
→ [5s 4p 3d 1f]) and includes 78 contracted Gaussians. As
the first step, full configuration interaction (FCI) calculations
for the two valence electrons have been performed with the
6-311G* basis, while four the core electrons (1␴g )2 (1␴u )2
have been kept doubly occupied. The results are practically
the same for complete active space (CAS) which includes
five ␴g , five ␴u , two ␲u , two ␲g , one ␦g , one ␦u molecular orbitals (MO). Since each ␲ and ␦ MO are doubly degenerate, this CAS includes 22 MOs for two electrons. This
CAS is denoted as [6, 2, 2, 1, 6, 2, 2, 1] showing the number of active orbitals in each symmetry species of the D2h
point group [Ag , B3u , B2u , B1g , B1u , B2g , B3g , Au ], respectively (one MO of the ag symmetry represents the ␦g orbital
together with one MO of the b1g type). Such complete active space is denoted as CAS-2. It includes only 45 determinants for the 3 +
u state. Even a smaller valence active space
[2, 1, 1, 0, 2, 1, 1, 0], which is called CAS-1, provides practically the same results as CAS-2. The next type of CAS
includes all six electrons calculation of Li2 . The number of
determinants increases dramatically in this case and a full
CI is not feasible. For the CAS [7, 2, 2, 1, 7, 2, 2, 1], which
corresponds to CAS-2 in the case of frozen core, the number
of determinants is about 300,000. This active space is called
CAS-2a.
To be compatible with the 6-311G* basis results the calculations in the big basis set have been done with much
larger CAS. The analogous to CAS-1 in small basis set is
794
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
called CAS-1b, which is [4, 2, 2, 0, 4, 2, 2, 0]. The analogous to CAS-2a in the big basis set is [11, 4, 4, 2, 11, 4, 4, 2]
named CAS-2ab/6-311++G(3df,3pd) with all six electrons
being correlated. These are the most accurate MCSCF response calculations in our study and they are compared with
the results obtained in smaller active spaces and basis sets
at a number of internuclear distances. Calculated properties
for a wide range of internuclear distances (2–6 Å) are presented on the ground of this comparison. The singlet–triplet
transition probabilities are calculated by quadratic response
method [28] with account of the Breit–Pauli SOC operator.
Static and dynamic polarizability are calculated by linear response (LR) method [28].
3. Results and discussion
3.1. Potential energy curves for the Li2 molecule
A number of low-lying excited singlet and triplet states
calculated by linear response method with the 6-311G* basis set and simple valence CAS are shown in Fig. 1. They
are in a reasonable agreement with the results of previous
calculations [6,7]. Only the 13 g potential deviates slightly
from a proper behavior near dissociation limit, but this cannot
influence main results of the S–T transition intensity calculations by quadratic response method. The LR potential energy curves (PEC) are obtained from the MCSCF optimized
3 +
ground state X1 +
g . Three PECs for excited states a u ,
3
3
+
b u and 1 g are also calculated by CASSCF optimization (Fig. 2) during the study of spin-splitting parameters at
different internuclear distances.
The minimum of the lowest triplet a3 +
u state in the CAS1 calculation with the 6-311G* basis set is found to be at
re = 4.31 Å; the depth is equal to De = 294 cm−1 in reasonable agreement with experimental data (4.2 Å and 333 cm−1 ,
respectively) [3]. The energy gap from the ground state minimum (Te = 7966 cm−1 ) is also in fair agreement with spectroscopic result (Te = 8183 cm−1 ) [3]. Thus, even the simplest method employed provides acceptable accuracy and all
internuclear distance dependent properties (Figs. 3 and 4)
are calculated in this approach. Many points were recalculated with much higher accuracy; the results are presented in
Table 1. An increase of the CAS and basis set at the MCSCF
level do not improve very much the agreement with experimental data obtained at the simple CAS-1 level for valence
electrons (Table 1). Our best MCSCF calculation with all six
electrons correlated and half million determinants in CAS2ab do not achieve the same perfect agreement with experimental re and ωe constants like the valence CI method with effective core polarization potential [7]. Our CAS-1 results for
re and ωe constants practically coincide with previous MCSCF calculations of Konowalow and Olsen [38]. The aim of
this work is not to improve potential energy curves calculated
in previous studies [7,38], but to predict new spin-dependent
properties which have not been studied before. To check the
Fig. 3. Spin-splitting parameter D = 2λ for the lowest triplet (a)13 +
u state
(dashed line) and for the 13 +
g state (solid line) of Li2 molecule obtained
by MCSCF method.
accuracy of the present methods, one can also compare some
other properties of the excited states which are available from
very accurate calculations [5,7,39].
3.2. Electric quadrupole moment and dipole
polarizability
The parallel component Qzz of the electric quadrupole tensor for calculated equilibrium distances of a number of states
is presented in Table 1. The perpendicular components are
Fig. 4. Electric dipole transition moments between the excited triplet states
and the ground state of Li2 molecule obtained by quadratic response MCSCF
method. MT–S,x (T z ) means the x projection of the T–S transition moment
1 +
from the T z spin sublevel (MS = 0). The 13 +
u –X g (a–X) transition mo1
+
3
ment is shown by solid line, the 1 u –X g (b–X) transition moments are
shown by dotted (perpendicular polarization) and dashed (parallel polarization) lines.
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
795
Table 1
Spectroscopic constants of low-lying states of 7 Li2 molecule
State
Method
Ee
X 1 +
g
CAS-1a
−14.9001
−14.9323
−14.90682
–
–
−14.99427
2.693
2.688
2.675
2.673f
2.659
2.673
347.2
341.5
351.0
CAS-2abc
Ab initiod
Ab initiog
Ab initioi
Experimentalh
−14.86500
−14.89608
−14.86974
–
–
−14.95698
4.309
4.202
4.182
4.166
4.2f
4.18
56.3
67.8
63.7
69
0.2587
0.2721
0.2743
66
0.27
1(b)3 u
CAS-1
CAS-2aa
Ab initiod
Experimentalh
−14.85553
−14.85084
−14.88109
−14.94307
2.626
2.624
2.595
2.592
336.1
339.5
345.9
345.6
0.6967
0.6971
0.7137
0.716
1(A)1 +
u
CAS-1
CAS-2a
Ab initiod
Ab initiog
Experimentalh
−14.83769
−14.86784
−14.84281
–
−14.93017
3.2
3.131
3.108
3.099
3.108
253.5
256.1
266
255.5
0.4900
0.4977
CAS-1a
CAS-2ba
Ab initiod
Ab initiog
Experimentalh
−14.82530
−14.85566
−14.83240
–
−14.91987
3.139
3.130
3.067
3.064
3.068
239.8
240.2
252.2
261
251.5
0.4876
0.4904
0.5108
1(a)3 +
u
13 +
g
CAS-2abc
Ab initiod
Ab initioe
Ab initiog
Experimentalh
CAS-1a
re
ωe
369
351.4
Be
0.6628
0.6647
0.6713
0.6726
Qzz
10.34b
αzz
αxx
αav
315
310
156
169
209
216
310
303
169
160
216
208
221
659
701
297
257
418
405
698
697
252
254
401
402
13.40
13.87
28.2
311
310
136
139
194
205
−8.27
−19
358
198
251
462
224
303
−523
−523
−646
−42
−605
−202
−511
69
−124
10.87
20.0
∼13
−2.04
−2.06
−2.98
0.4974
−13.0
−12.85
−20.8
0.5106
Ee : total energy (a.u.) at the equilibrium distance re (Å); ωe : vibrational frequency (cm−1 ); Be : rotational constant (cm−1 ); Qzz and αzz are the parallel
components of the quadrupole and polarizability tensors (a.u.). The parallel (αzz ) and perpendicular (αxx ) components of the static dipole polarizability tensor
provide an average value αav = 13 (αzz + 2αxx ).
a This work; 6-311G*.
b Q = −2Q .
zz
xx
c This work; 6-311++G(3df,3pd).
d Ref. [7].
e Ref. [47].
f Calculation at the fixed point.
g Ref. [39].
h Ref. [3,7,24,40,47].
i Ref. [5]. Calculation at the fixed point.
equal to: Qxx = Qyy = − 21 Qzz . Our MCSCF calculated Qzz
values are in qualitative agreement with results of SchmidtMink et al. [7], though they are almost twice as small. There
is a great reduction of the electric quadrupole moment upon
excitation from the ground state to the a3 +
u state. This is important for the S–T energy difference at long interatomic distances where electrostatic interactions prevail. At the dissociation limit, both states have no quadrupole moments, but at
r = 10a0 , the difference in the electric quadrupole moments
is equal to 4.6ea02 and it is still observable at r < 15a0 . Recombination of two Li atoms in an external electric field will
also be influenced by the difference in the electric quadrupole
tensor for the S and T states.
The sign of quadrupole moment is changed upon ex3 +
3 +
citation from the ground to the A1 +
u , 1 g and a u
states. The former two states dissociate to the Li(2s) + Li(2p)
dissociation limit with non-zero quadrupole moment [7].
The b3 u state also dissociates to the same limit, but its
quadrupole moment is of opposite sign [7]. These trends are
consistent with the molecular quadrupole tensors for the studied states at their equilibrium distances. It should be noted that
the b3 u state has the largest |Qzz | value (Table 1).
The electric dipole polarizability tensor is also changed
upon excitation from the ground state to the a3 +
u state
(Table 1). The calculated αii tensors for both states are in
a good agreement with results of recent studies of the excited triplet [5,39] and singlet ground state [40], respectively.
The reasons for the great increase of the αzz value upon
1 +
the a3 +
u ← X g excitation together with the polarizability changes at the dissociation limit have been discussed in
details recently [39] and our results agree with this analysis. In the triplet 13 +
g state, polarizability is getting negative because the largest contribution to αzz comes from the
3 +
a 3 +
u → 1 g transition with negative energy (Table 1).
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
796
While αzz (13 +
g ) is negative and large by absolute value,
the αxx (13 +
)
g component stongly depends on the CAS. In
CAS-2b, the αxx (13 +
g ) is nearly equal to zero (Table 1) in
agreement with results of [39], where the perpendicular αxx
component is found to be small and positive.
The frequency-dependent dynamic polarizability is also
studied in this work in the course of the linear response function construction during QR calculations of the S–T transitions probability. For example, at the frequency of the b–
X transition (0.0480 hartree) at the point r = 2.2 Å, we get
αxx = 187a03 and αzz = 368a03 . At the frequency of the vertical b3 u ← X1 +
g transition (0.0519 hartree), the calculated
dynamic polarizability is equal to αxx = 213a03 and αzz =
844a03 . At the close lying point r = 2.8 Å (the frequency of
the b–X transition 0.0528 hartree), we get αxx = 224a03 and
αzz = 1040a03 . These results are in a good agreement with
dynamic polarizability calculations in terms of the Cauchy
moments [40].
At higher frequency, where the A–X transition occurs,
there is a cusp in the αzz behavior. At the point r = 3.8 Å
(close to the b–A crossing), αzz = −4039a03 , so the parallel component of dynamic polarizability goes to −∞ at the
A–X resonance frequency. Thus, the quadratic response calculations of the frequency dependent dynamic polarizability
provide physically reasonable criteria for reliable response
theory application to the S–T transition intensity.
The MCSCF wave functions are pretty flexible at all levels of approximation; thus, the spin-splitting parameters and
the singlet–triplet transition probabilities indicate good convergence in respect to basis set and CAS enlargement. These
properties are more accurate than the total energy predictions.
3.3. Zero-field splitting in 3 states
The spin-splitting parameter λ from Eqs. (6)–(9) is equal to
1
2 D, where D is the so-called zero-field splitting constant of-
ten determined by EPR and ODMR spectra of triplet organic
molecules in low-temperature crystals [36]. The D parameter
3 +
for the 13 +
g and a u states of Li2 molecule is presented
in Fig. 3 as a function of internuclear Li–Li distance. The
D value is given in resonance frequency units (MHz), since
it can be measured only by magnetic resonance methods in
molecular beam.
3
The selected (11 +
u –1 u ) mixed level provides the A–
b gateway by which the triplet manifold can be accessed.
The lowest triplet state in Li2 molecule has been studied
by double-resonance methods and by 3 g → 13 +
u fluorescence detection [3,13]. Martin et al. [41] and Linton et
3 +
al. [3] have analyzed extensive spectra of the 13 +
g –a u
7
transition in a near-infrared region at 0.8–1µ in Li2 and
6 Li , respectively, at high resolution using Fourier trans2
form spectroscopy and have determined vibrational and rotational constants of both states. The continuum emission
3 +
13 +
g → a u in a near-infrared region with the radiative
lifetime of 62 ns of v = 0 and the peak at 1.3µ has been
simulated by ab initio calculations in an attempt to establish an excimer laser [6,20]. In spite of the great interest to
3 +
the 13 +
g –a u spectra, the spin-splitting of both states is
unknown so far.
Our MCSCF CAS-1 calculations indicate that D and λ
values for these states are very small for optical experimental determination (Fig. 3, 1 cm−1 = 29,979.64 MHz).
The D values are negative for the both states in the whole
range of internuclear distances and converge to a proper
zero limit. Though the |D| value increases with shorter internuclear distances it is still quite small. At the equilibrium distance λ is equal to −318 MHz = −0.0106 cm−1 and
3 +
−549 MHz = −0.0183 cm−1 for the a3 +
u and 1 g states,
3
+
respectively. The larger |λ| value for the 1 g state is determined by its shorter equilibrium distance re = 3.1 Å. Vibrational averaged λ0 for these two states are equal to −0.011 and
−0.019 cm−1 , respectively. This means, for example, that the
Ω = 0 spin sublevel is above the degenerate Ω = ±1 sublevel by 0.022 cm−1 in non-rotating molecule in the a3 +
u
state. Small deviations from monotonous behavior at short
internuclear distances in Fig. 3 are connected with Rydberg
character admixture and do not influence the λ0 values.
Extension of CAS and basis set leads to slightly smaller
spin-splitting in both states. In CAS-2a (211,017 determinants) calculation, the equilibrium point for the a3 +
u state is
obtained at re = 4.2 Å in better agreement with experimental
value (4.18 Å). At this point, we get D = −0.02146 cm−1 =
−643.4 MHz, whereas the CAS-1 calculation (six determinants) provides D = −0.02199 cm−1 = −659.4 MHz.
It is interesting to note that full CI calculation in 6-311G**
basis set (4,658,402 determinants) gives almost the same
value D = −659.31 MHz. The total energy of the a3 +
u state
is equal to −14.89558 a.u. at this point. Comparison with
Table 1 indicates a close agreement with the CAS-2ab result. All electrons calculation with CAS [5, 2, 2, 0, 5, 2, 2, 0],
which includes only 59,036 determinants, provides D =
−662 MHz; the total energy is almost the same and is equal
to −14.89535 a.u.
Stronger dependence of the D value on the CAS and basis set extension has been found for the upper triplet state.
For the 13 +
g state, comparison of different CAS and basis set extension has been performed at the point 3.2 Å. The
CAS-1 and CAS-2a results with 6-311G* basis set for the D
value are equal to −1088.1 and −993.7 MHz, respectively.
With 6-311++G(3df,3pd) basis, the CAS-2ab (core+valence
electrons) calculation provides D = −918.7 MHz. Thus, the
15% reduction from the simplest calculation is obtained in
the most accurate method. Accounting the same slope of the
curve like in Fig. 3, the λ0 value is estimated to be equal
−0.0156 cm−1 .
These very small spin-splittings can explain the absence
3 +
of fine structure in experimental analysis of the 13 +
g –a u
band structure [3,24,41]. Linton et al. [3] have mentioned
that the possible source of error in their potential constructed
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
by the Rydberg–Klein–Rees method [17] can lie in the fact
that all the experimental data were combined into a single
fit, even though the spectra of different N quantum numbers
originated from different spin components of the a3 +
u state,
Eqs. (6)–(8), which was treated as if it were a singlet [3]. In
3 +
the spectrum of the 13 +
g –a u transition, there was no evidence of spin-splitting; the only hint of triplet structure was
a slight broadening of all lines in the Fourier transform spectrum but the spin sublevels were not resolved [3,24,41,42].
This appears that the spin-splitting is less than the line-width
in the Fourier transform spectra (about 0.1–0.05 cm−1 [24]),
which is in agreement with our calculations.
From analysis of results with different CAS and basis sets
and from comparison with similar calculations [30,43,44],
one can conclude that the error of about few percents is expected for spin–spin coupling expectation value; in the case
−1
of Li2 a3 +
u state, the error cannot exceed 0.001 cm . Accounting the accuracy which is necessary for estimation of
scattering length in Bose–Einstein condensation in Li2 [3,5]
our spin-splitting calculation can be reasonable. The SOC
contribution to zero-field splitting is found completely negligible near the equilibrium point of the a3 +
u state.
It is interesting to compare spin–spin coupling strength in
the lowest triplet states of lithium and hydrogen molecules.
Comparison with the triplet state 13 +
u of hydrogen
molecule, which is non-bound but still similar to the Li2 a
state by its ␴g ␴u orbital nature, indicates a great difference
in magnetic dipole spin–spin interaction: in the H2 molecule,
SSC is about an order of magnitude stronger than in Li2 in
the range of internuclear distances r = 2–3 Å. Though the
spin-splitting in the H2 repulsive triplet state 13 +
u cannot be
observed, it is still an important theoretical criterion of diffuse
character of the wave function, because it can be compared
with the zero-field splitting parameters in other molecules.
The 13 +
u state of H2 molecule has a relatively large SSC
expectation value: λss = (1/2)D = 0.34 cm−1 at the ground
state internuclear distance (0.74 Å) [43]. In the Li2 a3 +
u
state, the λss value is equal to −0.018 cm−1 at the ground
state internuclear distance (2.67 Å). If one compare the spinsplitting at the same internuclear distance, r = 2 Å, in both
molecules one gets 0.12 and −0.02 cm−1 , respectively. The
spin–spin interaction is approaching zero at long distances,
but still there is a big difference in λss parameter of both
molecules in the range of chemical bonding and in the intermediate region of internuclear distances around the equilibrium. This means that an average separation between two
non-paired electron spins in the triplet 13 +
u state of H2 and
Li2 molecules is very different. The result seams to be quite
natural, since the valence electron in Li is screened by the
core and is far away from the nucleus. At the same time, a
simple comparison of the 1s|r−3 |1s and 2s|r −3 |2s values
in both atoms cannot provide explanation of the great difference in the λss parameter between H2 and Li2 . The binding
1␴2g electron density is concentrated between the nuclei while
the antibonding 1␴u electron density is delocalized in the
outer region. The spin–spin interaction of the two unpaired
797
electrons in the triplet 1␴g 1␴u state is determined in a great
extent by this peculiarity of the orbital structure in comparison with the triplet 2␴g 2␴u state, not only by the atomic
radial distribution.
3.4. Spin-splitting in the b(1)3 u state
Jeng et al. [23] have studied the fine structure of
the (b)13 u state of the Li2 using Doppler-free polarization spectroscopy. The triplet state was accessed by ex3
citation of the A1 +
u (v = 2, J = ∼33)–b u (v = 9, N =
32, F1 , e) spin–orbit perturbed levels. The remaining two fine
structure levels (F2 , f and F3 , e) were excited by using the
Zeeman interaction to mix levels of different J. Two polarization geometries were used to study separately different
groups of MJ levels in magnetic fields up to 3 kG.
The spin–spin and spin–rotation fine structure constants
were determined by a fit to the Zeeman splitting pattern by
using the Hund’s case (b) symmetrized basis functions. The
SOC contribution to the fine structure is too small in the high
J levels in order to be found accurately enough from experimental deperturbation analysis. The electronic SOC perturbation for the interaction between the singlet and triplet states
was determined to be HSOC (A–b) = 0.1172 ± 0.0008 cm−1
[45]. From the Zeeman effect in Doppler-free polarization
spectra [23], the electronic part of the SOC matrix element is
obtained by dividing the observed HA–b by the calculated vibrational (0.38) and rotational (0.489) overlap integrals. This
result for the SOC integral at the A–b crossing from Zeeman
study is in a good agreement with deperturbed separation between the A and b levels obtained by Xie and Field [13] using
line shape analysis (0.114 ± 0.006 cm−1 ). The SOC matrix
element between A–b states is smaller than those observed in
sodium dimer and a comparison between the values in Na2
and Li2 has already been discussed [13].
Our MCSCF response value 0.0658 cm−1 at the point of
the A–b crossing is about 40% too small. This result does not
depend on the basis set and CAS extension or on the internuclear distance in the range 3.2–3.6 Å. For the fine structure
splitting in b3 u state, the molecular spin–orbit contribution
could not be extracted from the experimental Zeeman effect
study [23]. This is not surprising since the spin–orbit interaction in Li, is very small, and its contribution to the fine structure splitting is further suppressed for high rotational level
(N = 32) by a factor of N −1 [23]. Jeng et al. tried to estimate
the fine structure spin–orbit constant from the SOC matrix element between the A and b states using a simple one-electron
approximation and effective atomic spin–orbit parameter ζLi .
Both of these quantities are related to the atomic SOC parameter ζLi in the single configuration approximation [17]. This
approximation is rather crude when such small SOC integrals
are considered; both one- and two-electron part of the SOC
operator are dependent on the state structure and internuclear
distance. The reason for the discrepancy between our result
and the experimental HSOC (A–b) value is unknown. Previous
calculations [46] which have used only the one-electron part
798
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
of the SOC operator provides much larger HSOC (A–b) value,
but this approximation is not reliable.
Calculation of spin–spin coupling in the 13 u state at the
equilibrium point (re = 2.64 Å) provides the following spinsplitting parameters: D = −0.0528 cm−1 = −1583 MHz,
E = −0.0047 cm−1 = −142.5 MHz. The calculated SOC
constant at this point is 0.1356 cm−1 . The absolute value
of D decreases with r and that of E increases slightly;
for example, at r = 3.6 Å, we get D = −0.0259 cm−1 ,
E = −0.0054 cm−1 . Small SOC reduction is also observed
(0.1287 cm−1 ). Comparison with the analysis of Jeng et al.
[23] provides only qualitative agreement. The fine structure
of the 13 u state needs further theoretical treatment.
3.5. The singlet–triplet transition probability
The electric dipole transition moments, Mn–X , between
the excited triplet states n and the ground singlet state X of Li2
molecule calculated by quadratic response MCSCF method
are shown in Fig. 4. These results are obtained with the 6311G** basis and CAS-1. The CAS-2ab calculation provides
almost the same values evaluated at the control points 2.7
and 4.2 Å. The notation Mn–X,x (T z ), for example, means the
x projection of the n–X transition moment from the T z spin
sublevel (MS = 0).
1 +
The 13 +
u –X g (a–X) transition is polarized perpendicular to the molecular axis and is found to be extremely weak
(the electric dipole transition moment is shown by solid line
in Fig. 4). At the ground state, equilibrium distance the Ma–X
transition moment is equal to 1.1 × 10−5 ea0 . This provides
1 +
the oscillator strength for vertical absorption 13 +
u ←X g
−14
to be equal to f = 4 × 10 , which makes this transition unobservable. At the upper 13 +
u state, equilibrium distance the
Ma–X transition moment is equal to 4 × 10−6 ea0 , which corresponds to the a → X emission rate constant 9 × 10−7 s−1 ;
the corresponding radiative lifetime exceeds 10 h.
The b3 u –X1 +
g transition is much more intense (Fig. 4).
The perpendicular component of the Mb–X electric dipole
transition moment is also small at short distances but it
is a growing function of r. The parallel component of the
Mb–X transition moment strongly increases in the vicinity
of both equilibriums in the upper and lower states (2.6–2.67
Å, respectively). At these two points, the transition moment
Mb–X,z is equal to 0.0011ea0 and 0.0013ea0 , respectively.
Both T x and T y sublevels which correspond to F1 and F2
states of the b3 u triplet have equal b3 u –X1 +
g transition
intensity with z-polarization. Strong increase of the Mb–X,z
transition moment in the vicinity of the distance r = 3.5 Å
3
clearly indicates the well-known A1 +
u –b u crossing. Because of SOC mixing between these two states at the crossing
point, perturbation theory cannot be applied in this region.
Fortunately, equilibrium points of the ground state and of the
b3 u state are pretty far away from this region, which allows
a reliable estimation of the 13 u –X1 +
g transition probability for the 0–0 band by perturbation theory. The estimated
oscillator strength for the 13 u ← X1 +
g absorption is equal
to 9.9 × 10−8 . Thus, this transition can be observed as a weak
tail in the long-wave range (890 nm) by usual spectroscopic
means.
4. Conclusions
Two low-lying triplet states of the lithium molecule of the
3 +
13 +
u and 1 g symmetry are calculated by MCSCF and
response techniques with account of spin–spin coupling. The
second-order spin–orbit coupling contribution to the zerofield splitting of these states is found to be negligible. The
D parameters (traditionally used in EPR spectroscopy) are
presented in the form of the internuclear distance-dependent
functions (D = 2λ).
The 1(a)3 +
u state is weakly bound and has a negligible spin-splitting: λss = −0.01 cm−1 (minus sign means the
inverted splitting: the spin sublevels MS = ±1 are lower
than the MS = 0 sublevel). Similar splitting is obtained for
3 +
the upper state of the 13 +
g –a u transition. This is in
agreement with the fact that in experimental analysis the
triplet structure was not resolved in this transition [3,24].
Comparison with the lowest triplet state 13 +
u of hydrogen molecule, which is unbound, but still similar to the
Li2 a3 +
u state by its ␴g ␴u orbital nature, indicates a great
difference in spin–spin coupling strength: in H2 molecule
SSC is about an order of magnitude stronger than in Li2
molecule in the internuclear distances range 2–3 Å. Though
the spin-splitting in the H2 repulsive triplet state 13 +
u cannot be observed, it is still an important theoretical criterion
of diffuse character of the wave function, because it can be
compared with the zero-field splitting parameters in other
molecules.
The 13 +
u state of H2 molecule has a relatively large spinsplitting: λss = −0.68 cm−1 at the ground state internuclear
distance (0.74 Å) [43]. In the a3 +
u (Li2 ) state, the λss value
is equal to −0.018 cm−1 at the ground state internuclear distance (2.67 Å). In the second excited 23 +
u triplet state of H2
molecule (this is the e state of the 1s␴ 3p␴ configuration in
traditional spectroscopic notations), the spin–spin coupling
constant is equal to λss = −0.082 cm−1 at the equilibrium
distance re (e) = 1.07 Å, which indicates that the Rydberg
state is much more diffuse and SSC is reduced. The lowlying bound 13 +
g state (1s␴ 2s␴ configuration), which is
3 +
responsible for a continuous emission 13 +
g → 1 u in hydrogen discharge, has a very small SSC expectation value:
λss = −0.006 cm−1 [43]. This value is lower than the optical
resolution and explains why the spin-splitting in the 13 +
g
state of H2 molecule has not been observed. Only this Rydberg state of hydrogen molecule is similar to the 3 valence
terms of lithium dimer in their negligible spin-splitting pattern. Comparison of the triplet valence states indicates that the
spin–spin coupling in hydrogen molecule is much stronger
than in lithium dimer.
B. Minaev / Spectrochimica Acta Part A 62 (2005) 790–799
The only state for which the fine structure has been studied
before is the c3 u metastable triplet. In the present calculations, the SOC contribution has been found larger than the
SSC expectation value in qualitative agreement with previous results. We have found that the D value in the c3 u state
strongly depends on the internuclear distance, but more accurate calculations are in progress.
The singlet–triplet transitions intensity calculations predict that the 13 u ← X1 +
g band can be easily observed
at 890 nm since it has a relatively large oscillator strength
1 +
(f = 10−7 ), while the 13 +
u –X g transition is unobservable because of negligible intensity.
Acknowledgments
This work was supported by the Wenner-Grenn Foundation. Collaboration with Olov Vahtras and Hans Ågren is
greatly appreciated.
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