Ab initio study of low-lying triplet states of the lithium dimer

Transcription

1 Spectrochimica Acta Part A 62 (2005) Ab initio study of low-lying triplet states of the lithium dimer Boris Minaev a,b, a State University of Technology, Cherkassy 18006, Ukraine b Department of Biotechnology, SCFAB, The Royal Institute of Technology, SE 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 Li 2 initiated the interest in the scattering length of two ground state lithium atoms when they approach each other as a radical pair triplet a 3 + 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 a 3 + 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, λ ss = 0.01 cm 1, which is negligible in comparison with the line-width in experimental Fourier transform spectra published so far. Similar splitting is obtained for the upper state of the g a3 + u transition. This is in agreement with experimental rovibronic analysis of the g a3 + 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 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 H 2 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 Li 2 molecule were extensively investigated [6,7,10]. Two main transitions between low-lying singlet states were studied very carefully, namely 1(A) 1 + u X1 + g and 1(B)1 u X 1 + 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 Li 2, because the transitions from the electronic ground state X 1 + g to excited triplets are Tel.: ; fax: address: boris@theochem.kth.se. 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 in the singlet state, X 1 + g, and its triplet counterpart, a3 + u state, dissociate to the same limit. In hydrogen molecule, the singlet state is bound and the triplet state is repulsive [16,18]. In Li 2, the triplet a 3 + u state is slightly bound with dissociation energy equal to 0.04 ev [3]. In spite of its instability, the u state of H 2 molecule has been known more than one century [16]; for all that the stable Li 2 analogous state is experimentally discovered just 20 years ago [13,19] /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.saa

2 B. Minaev / Spectrochimica Acta Part A 62 (2005) Fig. 1. Potential energy curves for a number of excited states of Li 2 molecule obtained by linear response calculation from the ground state MCSCF wave function. Potential energy curves for a number of low-lying states of the Li 2 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 H 2 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,inli 2 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 g b 3 u transition near 507 nm in collision-induced fluorescence 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 Li 2 has received a great attention not only because of its important role in the perturbations and accidental predissociation of the u state (A state) [3,13], but also because of its fine structure [2,23]. The crossing between 1(A) 1 + u and 1(b)3 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 Fig. 2. MCSCF potential energy curve for the three lowest states of Li 2 molecule. electronic interaction with the 1(a) 3 + u continuum (Fig. 1) [23]. This weakly bound lowest triplet state of Li 2 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 selected (1 1 + u 13 u ) mixed levels of the A b gateway, the lowest triplet state in Li 2 molecule has been studied by double-resonance methods and by the 3 g u fluorescence detection [3,13]. All triplet states of Li 2 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 weakly bound triplet state, a 3 + u, to the ground state, X1 + 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 Li 2 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 Li 2 [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 a 3 + u state of the Li 2 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

3 792 B. Minaev / Spectrochimica Acta Part A 62 (2005) from the highest bound vibrational level wave function of the a 3 + u state [3]. Spectroscopic constants of the a3 + u state have been studied by indirect spectroscopy methods [3,13]. The a 3 + u state is not of easy access by direct excitation (a 3 + u X1 + g ), since spin orbit coupling is very weak in lithium and the singlet triplet transition is rather strongly forbidden. The weakly bound a 3 + u state in Li 2 molecule, produced by the single electron excitation 2 g 2 u has a shallow minimum at r e = 4.2 Å [3]. The lowest triplet state minimum in lithium dimer is not deep (D e = 333.7cm 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, T e = 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 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 Li 2 molecule is of great interest. The singlet triplet X 1 + g a3 + u transition probability is also important for estimation of the photodissociation cross-section in the range of nm (the vertical excitation from the ground state). In the same forbidden region of the spectrum, the most intense 0,0 band of the X 1 + g 13 u transition is expected. The radiative lifetime of the a 3 + u state is unknown since neither the phosphorescent emission a 3 + u X1 + 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 Li 2 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: H 0 κ Ψ M S n = κ E n κ Ψ M S n. (1) In Eq. (1), κ = 2S + 1 is the multiplicity of the state and S is the total spin quantum number: and M S determines the projection of the total spin: S z κ Ψ M S n = M S κ Ψ M S n. (3) The response method [28 30] have been applied for the fine structure calculations of the (a)1 3 + u,(b)13 u and g triplet states of the Li 2 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: [ ] H SSC = µ 2 s i s j B g2 e r 3 3( s i r ij )( s j r ij ) i,j ij rij 5, (4) have been calculated by the recently developed code within the MCSCF technique [30]. Here, s i is a spin angular momentum of electron i, µ B is the Bohr magneton and g e 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]: H SR = γ v N S, (5) where γ v is the electronic spin rotation constant in the v vibrational quantum state, N is the rotational angular momentum and S = i s i. 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]: γ v = 2B v g. Here, B v is a rotational constant, g is the anisotropic correction to free-electron g-value: g e = The g-tensor components perpendicular to the molecular axis, g xx = g yy = g e + 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 rotational angular momentum N is also present, but in weaker extent. In that case, the spin-splittings of the rotational levels of the 3 states are given by the following equations [17]: F 1 (N) = B v N(N + 1) + (2N + 3)B v λ v (2N + 3) 2 Bv 2 + λ v 2 2λ v B v + γ v (N + 1), (6) ˆ S 2 κ Ψ M S n = S(S + 1) 2 κ Ψ M S n, (2) F 2 (N) = B v N(N + 1), (7)

4 B. Minaev / Spectrochimica Acta Part A 62 (2005) F 3 (N) = B v N(N + 1) (2N 1)B v λ v + (2N 1) 2 Bv 2 + λ v 2 2λ v B v γ v N, (8) where F 1, F 2 and F 3 refer to the J = N + 1,N and N 1 sublevels, respectively. γ v 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 = H z,z = 3 Ψ0 z H SSC 3 Ψ0 z n,s 3 Ψ0 z H SOC 2S+1 Ψ M S n 2S+1 Ψn k H SOC 3 Ψ M S 0 2S+1 E n 3. (9) E 0 In this expression, 2S+1 Ψ M S n is an eigenfunction of the nonrelativistic Schrödinger equation, Eq. (1), which is the zerothorder wave function for perturbation treatment. The indices z and M S determine the projection of the total spin and z corresponds to M S = 0 in case of diatomic molecule with a 3 state. Accounting for rovibrational states, it leads to Eqs. (6) (8). Most of the observed a 3 + u levels of the Li 2 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]: H S = D (S 2z 13 ) S2 + E(Sx 2 S2 y ). (10) This is a compact form of more general spin Hamiltonian: H S = XSx 2 YS2 y ZS2 z [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 Li 2 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 Li 2 where a strong SOC mixing between A and b excited rovibrational levels occurs [19,23]. The 1(b) 3 u state of Li 2 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 Li 2 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 M S 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]: E e = E 0 + AΛΣ (11) where E 0 is the energy of the state when the SOC is neglected, A is the SOC constant for a given multiplet state and Σ = M S 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 1cm 1 for the 1(b) 3 u state of Li 2, 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 G(3df,3pd) [37]. The first basis is rather small (11s 5p 1d [4s 3p 1d]) and includes 36 contracted functions for Li 2 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 D 2h point group [A g,b 3u,B 2u,B 1g,B 1u,B 2g,B 3g,A u ], respectively (one MO of the a g symmetry represents the g orbital together with one MO of the b 1g 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 Li 2. 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

5 794 B. Minaev / Spectrochimica Acta Part A 62 (2005) 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/ 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 Li 2 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 1 3 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 ground state X 1 + g. Three PECs for excited states a3 + u, b 3 u and 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 a 3 + u state in the CAS- 1 calculation with the 6-311G* basis set is found to be at r e = 4.31 Å; the depth is equal to D e = 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 (T e = 7966 cm 1 ) is also in fair agreement with spectroscopic result (T e = 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 CAS- 2ab do not achieve the same perfect agreement with experimental r e and ω e constants like the valence CI method with effective core polarization potential [7]. Our CAS-1 results for r e and ω e constants practically coincide with previous MC- SCF 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)1 3 + u state (dashed line) and for the g state (solid line) of Li 2 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] Electric quadrupole moment and dipole polarizability The parallel component Q zz 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 Li 2 molecule obtained by quadratic response MCSCF method. M T S,x (T z ) means the x projection of the T S transition moment from the T z spin sublevel (M S = 0). The u X1 + g (a X) transition moment is shown by solid line, the 1 3 u X 1 + g (b X) transition moments are shown by dotted (perpendicular polarization) and dashed (parallel polarization) lines.

6 B. Minaev / Spectrochimica Acta Part A 62 (2005) Table 1 Spectroscopic constants of low-lying states of 7 Li 2 molecule State Method E e r e ω e B e Q zz α zz α xx α av X 1 + g CAS-1 a b CAS-2ab c Ab initio d Ab initio e f Ab initio g Experimental h (a) 3 + u CAS-1 a CAS-2ab c Ab initio d Ab initio g Ab initio i 4.2 f Experimental h (b) 3 u CAS CAS-2a a Ab initio d Experimental h (A) 1 + u CAS CAS-2a Ab initio d Ab initio g Experimental h g CAS-1 a CAS-2b a Ab initio d Ab initio g Experimental h E e : total energy (a.u.) at the equilibrium distance r e (Å); ω e : vibrational frequency (cm 1 ); B e : rotational constant (cm 1 ); Q zz and α zz are the parallel components of the quadrupole and polarizability tensors (a.u.). The parallel (α zz ) and perpendicular (α xx ) components of the static dipole polarizability tensor provide an average value α av = 1 3 (α zz + 2α xx ). a This work; 6-311G*. b Q zz = 2Q xx. c This work; 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: Q xx = Q yy = 2 1 Q zz. Our MCSCF calculated Q zz values are in qualitative agreement with results of Schmidt- Mink 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 a 3 + 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 = 10a 0, the difference in the electric quadrupole moments is equal to 4.6ea0 2 and it is still observable at r<15a 0. 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 excitation from the ground to the A 1 + u,13 + g and a3 + u states. The former two states dissociate to the Li(2s) + Li(2p) dissociation limit with non-zero quadrupole moment [7]. The b 3 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 b 3 u state has the largest Q zz value (Table 1). The electric dipole polarizability tensor is also changed upon excitation from the ground state to the a 3 + u state (Table 1). The calculated α ii 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 α zz value upon the a 3 + u X1 + 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 g state, polarizability is getting negative because the largest contribution to α zz comes from the a 3 + u 13 + g transition with negative energy (Table 1).

7 796 B. Minaev / Spectrochimica Acta Part A 62 (2005) While α zz (1 3 + g ) is negative and large by absolute value, the α xx (1 3 + g ) component stongly depends on the CAS. In CAS-2b, the α xx (1 3 + g ) is nearly equal to zero (Table 1) in agreement with results of [39], where the perpendicular α xx 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 ( hartree) at the point r = 2.2 Å, we get α xx = 187a0 3 and α zz = 368a0 3. At the frequency of the vertical b 3 u X 1 + g transition ( hartree), the calculated dynamic polarizability is equal to α xx = 213a0 3 and α zz = 844a0 3. At the close lying point r = 2.8 Å (the frequency of the b X transition hartree), we get α xx = 224a0 3 and α zz = 1040a0 3. 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 α zz behavior. At the point r = 3.8 Å (close to the b A crossing), α zz = 4039a0 3, 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 Zero-field splitting in 3 states The spin-splitting parameter λ from Eqs. (6) (9) is equal to 1 2D, where D is the so-called zero-field splitting constant often determined by EPR and ODMR spectra of triplet organic molecules in low-temperature crystals [36]. The D parameter for the g and a3 + u states of Li 2 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. The selected (1 1 + u 13 u ) mixed level provides the A b gateway by which the triplet manifold can be accessed. The lowest triplet state in Li 2 molecule has been studied by double-resonance methods and by 3 g u fluorescence detection [3,13]. Martin et al. [41] and Linton et al. [3] have analyzed extensive spectra of the g a3 + u transition in a near-infrared region at 0.8 1µ in 7 Li 2 and 6 Li 2, respectively, at high resolution using Fourier transform spectroscopy and have determined vibrational and rotational constants of both states. The continuum emission g a3 + 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 the g a3 + 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, 1cm 1 = 29, 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 = cm 1 and 549 MHz = cm 1 for the a 3 + u and 13 + g states, respectively. The larger λ value for the g state is determined by its shorter equilibrium distance r e = 3.1 Å. Vibrational averaged λ 0 for these two states are equal to and cm 1, respectively. This means, for example, that the Ω = 0 spin sublevel is above the degenerate Ω =±1 sublevel by cm 1 in non-rotating molecule in the a 3 + 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 λ 0 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 a 3 + u state is obtained at r e = 4.2 Å in better agreement with experimental value (4.18 Å). At this point, we get D = cm 1 = MHz, whereas the CAS-1 calculation (six determinants) provides D = cm 1 = 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 = MHz. The total energy of the a 3 + u state is equal to 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 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 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 and MHz, respectively. With G(3df,3pd) basis, the CAS-2ab (core+valence electrons) calculation provides D = 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 λ 0 value is estimated to be equal cm 1. These very small spin-splittings can explain the absence of fine structure in experimental analysis of the g a3 + u band structure [3,24,41]. Linton et al. [3] have mentioned that the possible source of error in their potential constructed

8 B. Minaev / Spectrochimica Acta Part A 62 (2005) 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 a 3 + u state, Eqs. (6) (8), which was treated as if it were a singlet [3]. In the spectrum of the g a3 + 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 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 of Li 2 a 3 + u state, the error cannot exceed cm 1. Accounting the accuracy which is necessary for estimation of scattering length in Bose Einstein condensation in Li 2 [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 a 3 + 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 u of hydrogen molecule, which is non-bound but still similar to the Li 2 a state by its g u orbital nature, indicates a great difference in magnetic dipole spin spin interaction: in the H 2 molecule, SSC is about an order of magnitude stronger than in Li 2 in the range of internuclear distances r = 2 3 Å. Though the spin-splitting in the H 2 repulsive triplet state 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 u state of H 2 molecule has a relatively large SSC expectation value: λ ss = (1/2)D = 0.34 cm 1 at the ground state internuclear distance (0.74 Å) [43]. In the Li 2 a 3 + u state, the λ ss value is equal to 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 λ ss 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 u state of H 2 and Li 2 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 λ ss parameter between H 2 and Li 2. The binding 1 2 g 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 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 Spin-splitting in the b(1) 3 u state Jeng et al. [23] have studied the fine structure of the (b)1 3 u state of the Li 2 using Doppler-free polarization spectroscopy. The triplet state was accessed by excitation of the A 1 + u (v = 2,J = 33) b3 u (v = 9,N = 32,F 1,e) spin orbit perturbed levels. The remaining two fine structure levels (F 2,f and F 3,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 M J 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 H SOC (A b) = ± 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 H A 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 ± 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 Na 2 and Li 2 has already been discussed [13]. Our MCSCF response value 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 Å. For the fine structure splitting in b 3 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 ζ Li. Both of these quantities are related to the atomic SOC parameter ζ Li 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 H SOC (A b) value is unknown. Previous calculations [46] which have used only the one-electron part

9 798 B. Minaev / Spectrochimica Acta Part A 62 (2005) of the SOC operator provides much larger H SOC (A b) value, but this approximation is not reliable. Calculation of spin spin coupling in the 1 3 u state at the equilibrium point (r e = 2.64 Å) provides the following spinsplitting parameters: D = cm 1 = 1583 MHz, E = cm 1 = MHz. The calculated SOC constant at this point is 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 = cm 1, E = cm 1. Small SOC reduction is also observed ( cm 1 ). Comparison with the analysis of Jeng et al. [23] provides only qualitative agreement. The fine structure of the 1 3 u state needs further theoretical treatment The singlet triplet transition probability The electric dipole transition moments, M n X, between the excited triplet states n and the ground singlet state X of Li 2 molecule calculated by quadratic response MCSCF method are shown in Fig. 4. These results are obtained with the 6-311G** 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 M n X,x (T z ), for example, means the x projection of the n X transition moment from the T z spin sublevel (M S = 0). The u X1 + 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 M a X transition moment is equal to ea 0. This provides the oscillator strength for vertical absorption u X1 + g to be equal to f = , which makes this transition unobservable. At the upper u state, equilibrium distance the M a X transition moment is equal to ea 0, which corresponds to the a X emission rate constant s 1 ; the corresponding radiative lifetime exceeds 10 h. The b 3 u X 1 + g transition is much more intense (Fig. 4). The perpendicular component of the M b X electric dipole transition moment is also small at short distances but it is a growing function of r. The parallel component of the M b X transition moment strongly increases in the vicinity of both equilibriums in the upper and lower states ( Å, respectively). At these two points, the transition moment M b X,z is equal to ea 0 and ea 0, respectively. Both T x and T y sublevels which correspond to F 1 and F 2 states of the b 3 u triplet have equal b 3 u X 1 + g transition intensity with z-polarization. Strong increase of the M b X,z transition moment in the vicinity of the distance r = 3.5 Å clearly indicates the well-known A 1 + u b3 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 b 3 u state are pretty far away from this region, which allows a reliable estimation of the 1 3 u X 1 + g transition probability for the 0 0 band by perturbation theory. The estimated oscillator strength for the 1 3 u X 1 + g absorption is equal to 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 u and 13 + 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: λ ss = 0.01 cm 1 (minus sign means the inverted splitting: the spin sublevels M S =±1 are lower than the M S = 0 sublevel). Similar splitting is obtained for the upper state of the g a3 + 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 u of hydrogen molecule, which is unbound, but still similar to the Li 2 a 3 + u state by its g u orbital nature, indicates a great difference in spin spin coupling strength: in H 2 molecule SSC is about an order of magnitude stronger than in Li 2 molecule in the internuclear distances range 2 3 Å. Though the spin-splitting in the H 2 repulsive triplet state 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 u state of H 2 molecule has a relatively large spinsplitting: λ ss = 0.68 cm 1 at the ground state internuclear distance (0.74 Å) [43]. Inthea 3 + u (Li 2) state, the λ ss value is equal to cm 1 at the ground state internuclear distance (2.67 Å). In the second excited u triplet state of H 2 molecule (this is the e state of the 1s 3p configuration in traditional spectroscopic notations), the spin spin coupling constant is equal to λ ss = cm 1 at the equilibrium distance r e (e) = 1.07 Å, which indicates that the Rydberg state is much more diffuse and SSC is reduced. The lowlying bound g state (1s 2s configuration), which is responsible for a continuous emission g 13 + u in hydrogen discharge, has a very small SSC expectation value: λ ss = cm 1 [43]. This value is lower than the optical resolution and explains why the spin-splitting in the g state of H 2 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.

10 B. Minaev / Spectrochimica Acta Part A 62 (2005) The only state for which the fine structure has been studied before is the c 3 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 c 3 u state strongly depends on the internuclear distance, but more accurate calculations are in progress. The singlet triplet transitions intensity calculations predict that the 1 3 u X 1 + g band can be easily observed at 890 nm since it has a relatively large oscillator strength (f = 10 7 ), while the u X1 + 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. References [1] C. Bradley, C. Sackett, J. Tollett, R. Hulet, Phys. Rev. Lett. 75 (1995) [2] L. Li, A. Yiannopoulou, K. Urbanski, A. Lyyra, B. Ji, W. Stwalley, T. An, J. Chem. Phys. 105 (1996) [3] C. Linton, F. Martin, A. Ross, I. Russier, P. Crozet, L. Li, A. Lyyra, J. Mol. Spectrosc. 196 (1999) 20. [4] X. Wang, J. Magnes, A. Lyyra, F. Martin, A. Ross, P. Dove, R.L. Roy, J. Chem. Phys. 117 (2002) [5] M. Rerat, B. Bussery-Honvault, Mol. Phys. 101 (2003) 373. [6] L. Ratcliff, J. Fish, D. Konowalow, J. Mol. Spectrosc. 122 (1987) 293. [7] I. Schmidt-Mink, W. Muller, W. Meyer, Chem. Phys. 92 (1985) 263. [8] L. Li, T. An, T.-J. Wang, A. Lyyra, W. Stwalley, R. Field, R. Bernheim, J. Chem. Phys. 96 (1992) [9] A. Yiannopoulou, K. Urbanski, S. Antonova, A. Lyyra, L. Li, T. An, B. Li, W. Stwalley, J. Mol. Spectrosc. 172 (1995) 567. [10] K. Verma, M. Koch, W. Stwalley, J. Chem. Phys. 96 (1983) [11] B.F. Minaev, Dr. Sc. Thesis, N.N. Semenov Institute of Chemical Physics, Moscow, [12] F. Engleke, H. Haage, Chem. Phys. Lett. 103 (1983) 98. [13] X. Xie, R. Field, J. Chem. Phys. 83 (1985) [14] B. Minaev, H. Ågren, Adv. Quant. Chem. 40 (2001) 191. [15] S.P. McGlynn, T. Azumi, M. Kinoshita, Molecular Spectroscopy of the Triplet State, Prentice-Hall, Englewood Cliffs, NJ, [16] G. Herzberg, Molecular Spectra and Molecular Structure. Spectra of Diatomic Molecules, vol. 1, Van Nostrand Reinhold, New York, [17] H. Lefebvre-Brion, R. Field, Perturbation in the Spectra of Diatomic Molecules, Academic Press, Orlando, [18] W. Kolos, L. Wolniewisz, J. Chem. Phys. 48 (1968) [19] X. Xie, R. Field, J. Mol. Spectrosc. 117 (1986) 228. [20] D. Konowalow, P. Julienne, J. Chem. Phys. 72 (1980) [21] D. Veza, S. Milosevic, G. Pichler, Chem. Phys. Lett. 93 (1982) 401. [22] W. Preuss, G. Baumgartner, Z. Phys. A 320 (1985) 125. [23] W.-H. Jeng, X. Xie, L. Gold, R. Bernheim, J. Chem. Phys. 92 (1991) 928. [24] C. Linton, T. Murphy, F. Martin, R. Bacis, J. Verges, J. Chem. Phys. 91 (1989) [25] F. Colavecchia, J.J.P. Burke, W. Stevens, M. Salazar, G. Parker, R. Pack, J. Chem. Phys. 118 (2003) [26] M. Gutowski, J. Chem. Phys. 110 (1999) [27] H.-K. Chung, K. Kirby, J. Babb, Phys. Rev. A 63 (2001) [28] H. Ågren, O. Vahtras, B. Minaev, Adv. Quant. Chem. 27 (1996) 71. [29] K. Ruud, T. Helgaker, K. Bak, P. Jørgensen, J. Olsen, Chem. Phys. 195 (1995) 157. [30] O. Vahtras, B. Minaev, O. Loboda, H. Ågren, K. Ruud, Chem. Phys. 279 (2002) 133. [31] B. Minaev, Opt. Spectrosc. (USSR) 36 (1974) 159. [32] M. Engstrom, B. Minaev, O. Vahtras, H. Ågren, Chem. Phys. 237 (1998) 149. [33] O. Vahtras, B. Minaev, H. Ågren, Chem. Phys. 281 (1997) 186. [34] A. Yiannopoulou, L. Li, M. Li, K. Urbanski, A. Lyyra, W. Stwalley, G.-H. Jeung, J. Chem. Phys. 101 (1994) [35] T. Uzer, A. Dalgarno, Chem. Phys. 51 (1980) 271. [36] B. Minaev, Fizika Molecul, Naukova Dumka, Kiev 7 (1979) 34. [37] R. Krishnan, J. Binkley, R. Seeger, J. Pople, J. Chem. Phys. 72 (1980) 650. [38] D. Konowalow, M. Olsen, J. Chem. Phys. 71 (1979) 450. [39] M. Merawa, M. Rerat, Eur. Phys. J. D 17 (2001) 329. [40] M. Pecul, M. Jaszunski, P. Jorgensen, Mol. Phys. 98 (2000) [41] F. Martin, R. Bacis, J. Verges, C. Linton, G. Bujin, C.H. Cheng, E. Stad, Spectrochim. Acta A 44 (1988) [42] W. Zemke, W. Stwalley, J. Phys. Chem. 97 (1993) [43] B. Minaev, O. Loboda, Z. Rinkevicius, O. Vahtras, H. Ågren, Mol. Phys. 101 (2003) [44] O. Loboda, B. Minaev, O. Vahtras, B. Schimmelpfenning, H. Ågren, K. Ruud, D. Jonsson, Chem. Phys. 286 (2003) 127. [45] M. Jenkin, R. Cox, G. Hayman, Chem. Phys. Lett. 177 (1991) 272. [46] D. Cooper, J. Hutson, T. Uzer, Chem. Phys. Lett. 86 (1982) 472. [47] M. Urban, A. Sadlej, J. Chem. Phys. 103 (1995) 9692.