Antisymmetric polarizabilities of rotational sublevels for linear molecules with a non-degenerate ground electronic state

Chemical Physics 285 (2002) 261–276 www.elsevier.com/locate/chemphys

Antisymmetric polarizabilities of rotational sublevels for linear molecules with a non-degenerate ground electronic state Ren-hui Zheng, Dong-ming Chen, Tian-jin He, Fan-chen Liu * Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China Received 3 July 2002

Abstract Using the semiclassical perturbation theory, we study the antisymmetric polarizabilities of rotational sublevels, the M-dependent splittings and shifts of the sublevels and corresponding rotational spectral lines of a linear molecule induced by a right (+) or left ()) circularly polarized resonant optical field. The deduced theoretical formulas for these quantities indicate that for rotational sublevels M of linear molecules with a non-degenerate ground electronic state, the antisymmetric polarizabilities induced by a resonant polarized optical fields exist and are the same order of magnitude of the symmetric ones. Based on them, M-dependent (both size and sign of M) splitting and shifts of the sublevels and corresponding spectral lines can be produced and the degeneracy for +M and )M sublevels can be completely broken by the circularly polarized resonant optical field. As examples, we have calculated these quantities for H2 and CO molecules. For H2 molecule, the antisymmetric and symmetric polarizabilities of the rotational magnetic sublevels in the ground electronic state X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ, which are induced by a optical field with frequency v0 resonant with 1 þ 0 0 the transition X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! B Ru ðv ¼ 0; J ¼ 2Þ, are the same order of 10 a.u.; and laser-induced Mdependent splittings and shifts for M ¼ 1; 0; 1 sublevels and the corresponding resonant Rayleigh scattering lines 1 þ 0 0 0 X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ $ B Ru ðv ¼ 0; J ¼ 2; M Þ can be the order of 0.1–1 GHz when the intensity of circularly 2 polarized optical field is 100 MW cm . Similarly the antisymmetric polarizabilities of CO molecule in the rotational magnetic sublevels X1 Rþ ðv ¼ 0; J ¼ 1; M ¼ 1Þ using IR field with the frequency v0 resonant with the transition X1 Rþ ðv ¼ 0; J ¼ 1Þ ! X1 Rþ ðv0 ¼ 1; J 0 ¼ 2Þ are the same order (10 a.u.) of the symmetric ones. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Antisymmetric polarizability; Ac Stark effect; Molecule rotation effect

1. Introduction Antisymmetric matter tensors are important and have been studied for many years in connec-

*

Corresponding author. Fax: +86-551-360-3388. E-mail address: [email protected] (F.-C. Liu).

tion with molecular light scattering and optical activity [1,2]. Antisymmetric Raman scattering for a molecule is characterized by the anomalous depolarization ratio that is conveniently expressed in terms of the rotational invariants of the squares of transition polarizability component [3]. Antisymmetric electronic and vibronic Raman transitions have been investigated in previous treatments

0301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 8 3 5 - 2

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[4–9]. Ziegler et al. [10] have demonstrated the effect of free rotation on the depolarization ratios due to resonance excitation and proved theoretically that in contrast to the classical result, when the Raman cross-sections dominated by resonance with a single vibronic band, due to the rotation effect antisymmetric transition polarizability invariants will contribute to the scattering intensities and will produce the ‘‘anomalous’’ depolarization ratio. This is purely a consequence of the rotational motion effect on the squares of the antisymmetric transition polarizability. On the other hand, about optical activity, Buckingham et al. [11,12] pointed out the rotation of the plane of polarization in the path length z is determined by the antisymmetric polarizability and is directly proportional to it. The antisymmetric polarizability is different from the antisymmetric transition polarizability invariants in two ways: (i) they have different definition; (ii) the latter is determined by square of the tensor components. Can the rotation effect produce the antisymmetric polarizability? The first motivation of this study is to explore the contribution from the rotational motion to the antisymmetric polarizabilities, which has analogy to Ziegler et al.’s rotational effect [10] but is different from Baranova and Zel’dovich’s Coriolis force effect of a molecular rotation [13]. Secondly, due to the optical Stark effect, the light shifts and splittings of the energy levels of a molecule in a laser field are proportional to the polarizabilities [14–17]. Recently, the light shifts and splittings of rotational magnetic sublevels were, respectively, investigated by molecular spectroscopy for molecular systems H2 [18,19], CO [20,21], N2 [19,20,22], C2 H2 [23], NO [24] and so on in intense non-resonant linear polarized laser fields. All of these experiments have shown M2 dependent splittings of the rotational magnetic sublevels, but it is still independent of the sign of M and has double degenerate about the rotational magnetic quantum numbers M, which indicates the non-resonant polarizabilities of rotational magnetic sublevel are symmetric. However, for an atom in circularly polarized light the light shifts of the levels depend not only on the absolute value of M but also on its sign and in the general case the sublevels different in the sign of M are separated

[14] by the circularly polarized optical field, which indicates that there are antisymmetric polarizabilities for the magnetic sublevels of an atom induced by a right or left circularly polarized optical field [25]. By analogy, we want to know whether antisymmetric polarizabilities of rotational sublevels jJMi in a circularly polarized optical field are zero or not, and whether they can produce the M-dependent light shift and splitting and remove the degenerate for +M and )M sublevels completely by a circularly polarized resonant optical field. In this report, using the semiclassical perturbation theory we study the antisymmetric polarizabilities of rovibrational sublevels jJMi and M-dependent light splitting and shifts of the sublevels and the corresponding rovibrational spectral lines of a linear molecule induced by a right (+) and left ()) circularly polarized resonant optical field. In Section 2, the theoretical formulas for these quantities have been deduced and in Section 3 taking H2 and CO as examples these quantities have been calculated. The deduced formulas and calculated results indicate that the antisymmetric polarizabilities of rovibrational sublevels jJMi of a linear molecule can be induced by a resonant (or near resonant) optical fields and have the same order of magnitude of the symmetric polarizabilities. Based on them, M-dependent (both size and sign of M) splittings and shifts of the sublevels and corresponding spectral lines (e.g., resonant Rayleigh scattering) can be produced and the degeneracy for +M and )M sublevels and the spectral lines can be completely broken by the circularly polarized resonant optical field. For example, the M-dependent splitting of the resonant Rayleigh 1 þ scattering lines X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ $ B Ru 0 0 0 ðv ¼ 0; J ¼ 2; M Þ can be the order of 0.1–1 GHz when the intensity of circularly polarized optical field is 100 MW cm2 , which can be measurable referring to the relative experiment results [18–24]. Non-zero antisymmetric polarizabilities break degenerate of rotational sublevels M in circularly polarized light completely, which may make it possible to explore the single sublevel M, i.e., Mselectivity. Even more, the antisymmetric polarizabilities can change some properties of optical Stark spectroscopy and light scattering, such as

R. Zheng et al. / Chemical Physics 285 (2002) 261–276

M-dependent laser-induced shifts, large resonant depolarization ratio of sublevels M and so on. Further, the magnetic field in the Faraday effect might be replaced by the resonant circularly polarized optical field to orient the rotational sublevels M and study the optical rotation and circular dichroism of rotational or rovibrational levels [11].

2. Theory In this section, first, we study the polarizabilities of rotational sublevels jJMi of a linear molecule, especially the antisymmetric polarizabilities induced by a right (+) or left ()) circularly polarized optical field (in Section 2.1), then based on them, investigate M-dependent splittings and shifts of energy levels and rotational spectral lines of a linear molecule in a circularly polarized optical field (in Section 2.2). These two problems can be resolved by the semiclassical perturbation theory [16,26,27] and the dressed-state theory [14,17, 20,28,29]. The former is valid in non-exact resonant case [17,20,26,27] and in this section we use the Kramers–Heisenberg–Dirac expression of molecular polarizability and the semiclassical dispersion theory of ac Stark effect on the light shifts and splittings of rotational magnetic sublevels of a linear molecule. 2.1. Antisymmetric polarizabilities induced by a resonant optical field

263

frequency, and vg~n~ is the transition frequency between the states g~ and n~. Cg~n~ is the dephasing constant (FWHH). We assume that molecular rovibronic states of a linear molecule can be written as the product of separable vibronic and rotational wave functions: j~ gi ¼ jgvijJMi;

ð2Þ

j~ ni ¼ jnv0 ijJ 0 M 0 i;

ð3Þ

g and n are the ground and excited electronic states. v, J , M are the quantum numbers for the vibrational, total angular momentum, and space fixed z-axis component of angular momentum, respectively. The unprimed and primed indices correspond to the initial, intermediate rotation– vibration quantum numbers. Rotational eigenfunctions of a linear molecule are given by properly normalized Wigner rotation matrices [31]: 1=2

jJMi ¼ ½ð2J þ 1Þ=8p2 

DJM0 :

ð4Þ

Neglecting centrifugal correction terms rotational energy Ev ðJ Þ of a linear molecule will be taken simply as Ev ðJ Þ ¼ J ðJ þ 1ÞBv ;

ð5Þ

where rotational constants Bv for the vibrational level v have its usual definition [31]:   1 Bv ¼ Be  a e v þ ; ð6Þ 2

2.1.1. M-Dependent polarizabilities for a given rotational sublevel jJMi In the Kramers–Heisenberg–Dirac sum-overall-state descriptions of molecule polarizability, the polarizability components akl of a molecule in the state g~ in the space fixed frame (SFF) is [1,2,10] X 1 h~ gjerk j~ nih~ njerl j~ gi akl ¼ þ c:c:; ð1Þ h ðvg~n~  v0  iCg~n~=2Þ n~

Be is rigid rotator rotational constant in the equilibrium position; ae is non-rigid rotator correction to Be . The polar vector components rk ðrl Þ in the space fixed framework are related to the corresponding components rq ðrq0 Þ in the molecule fixed framework (MFF) using the Wigner rotation matrices [31,32]

where c.c. is the complex conjugate term (the nonresonant term) [26,30], g~ and n~ stand for initial and intermediate rovibronic states. rk , rl are the spherical vector components of the position vectors in the space fixed frame which are complex polar for circularly polarized light. v0 is the optical

rk ¼

X q

D1 kq rq ¼

X

ð1Þkq D1kq rq :

ð7Þ

q

Using this transformation, the polar vector polarizability akl of the rotational magnetic sublevel jgvijJMi is given by

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akl

gv nv kþlqq 2 X ðMq Þnv0 ; ðMq0 Þgv hJMjD1kq jJ 0 M 0 ihJ 0 M 0 jD1lq0 jJMið1Þ ¼ þ c:c:; 0 h q;q0 ;nv0 ðvgv;nv0 þ vJ ;J 0  v0  iCJnv0 =2Þ

0

0

ð8Þ

J 0 ;M 0

vgv;nv0 is the vibronic transition energy and vJ ;J 0 is the rotational contribution to the molecular transition energy. The vibronic transition moments in the MFF are represented as gv

ðMq Þnv0 ¼ hgvjerq jnv0 i:

ð9Þ

Since different rotational sublevels jJMi of a molecule in a strong light field have different jMjdependent polarizabilities, in Eq. (8) we have not sum over M states in the ground rovibronic levels. This method avoiding the M summation has been given in theoretical papers [20,32–34] and experiment studies [18,20–22]. When v0 is resonant to a vibronic absorption band and only single intermediate vibronic state jnv0 i contributions to akl , using the integral formulas of three Wigner rotation matrix products [31], Eq. (8) becomes 1X M 0 þMþkþlqq0 gv nv0 akl ¼ ð1Þ ðMq Þnv0 ðMq0 Þgv h q;q0 J 0 ;M 0

 ð2J þ 1Þð2J 0 þ 1Þ  

J 1 J0

!

0 q 0 J0 1 J 0 q

0

0

!

J

1 J0

and a11 ðk ¼ 1; l ¼ 1Þ, which are rewritten as a0 , aþ and a , respectively in a shorthand notation for simplicity. (ii) The triangular conditions of non-zero 3j symbols    0  J 1 J0 J 1 J and 0 q0 0 0 q 0 need q ¼ q0 ¼ 0 and allow only J 0 ¼ J þ 1 (R term) and J 0 ¼ J  1 (P term) to contribute to the resonant polarizabilities, thus the polarizabilities a0 , aþ or a can be written into two terms, aR ðJ 0 ¼ J þ 1Þ and aP ðJ 0 ¼ J  1Þ terms. (iii) Finally calculating the 3j-symbol in Eq. (10), Eq. (10) for non-zero complex polar polarizabilities becomes a0; ¼ aR0; þ aP0; ;

ð11Þ

where for R term aR is aR0 ¼ ½ðJ þ 1Þ2  M 2 2fR ;

M ¼ 0; 1; . . . ; J ; ð12Þ

! aRþ ¼ ½ðJ þ 1ÞðJ þ 2Þ  Mð2J þ 3Þ þ M 2 fR ;

M k M 0 ! J0 1 J

M ¼ 0; 1; . . . ; J ;

M 0 l M 0

1

ðvgv;nv0 þ vJ ;J 0  v0  iCJnv0 =2Þ : ð10Þ

Inspection of Eq. (10) shows how the symmetry properties of the 3j symbol [31], statements of the conservation of angular momentum, formally control the resonance rotational selection rules, thus reduce the akl expression of Eq. (10) as follows: (i) From Wigner-3j symbols  0    J 1 J0 1 J J ; and M 0 l M M k M 0 k þ l ¼ 0 can be deduced which reduce the nine polarizability components akl to three non-zero components a00 ðk ¼ l ¼ 0Þ, a11 ðk ¼ 1; l ¼ 1Þ

ð13Þ

aR ¼ ½ðJ þ 1ÞðJ þ 2Þ þ Mð2J þ 3Þ þ M 2 fR ; M ¼ 0; 1; . . . ; J; ð14Þ where the parameter fR in Eqs. (12)–(14) corresponding to the vibronic transition moment is nv0 2

fR ¼

jðM0 Þgv j ; 2ð2J þ 1Þð2J þ 3ÞðDR  iCJ þ1 =2Þ

ð15Þ

CJ þ1 is the dephasing constants for the rotational states jJ þ 1i of the excited vibrational state jnv0 i. DR ¼ vgv;nv0 þ vJ ;J þ1  v0 , using Eq. (5) it gets DR ¼ vgv!nv0 þ ðBv0  Bv ÞJ 2 þ ð3Bv0  Bv ÞJ þ 2Bv0  v0 :

ð16Þ

The transition frequency vgv!nv0 , from the ground vibrational state jgvi to the excited vibrational state jnv0 i is [35]

R. Zheng et al. / Chemical Physics 285 (2002) 261–276

vgv!nv0 ¼ ðTn  Tg Þ þ x0e ðv0 þ 1=2Þ  x0e x0e ðv0 þ 1=2Þ 2

 ½xe ðv þ 1=2Þ  xe xe ðv þ 1=2Þ ;

2

ð17Þ

where Tn , Tg are the electronic energies of states jni, jgi, respectively; xe , x0e are vibration spacings; xe xe , x0e x0e are anharmonic corrections to vibrational spacing. For P term ap0; have been similarly derived and listed in Table 1(A).

265

ample for the polarizability a11 , its antisymmetric and symmetric parts are aant ¼ ð1=2Þða11  a11 Þ and asym ¼ ð1=2Þða11 þ a11 Þ, respectively. Thus the polarizability for the R terms induced by circularly polarized optical field can be expressed as aR ¼ aRsym  aRant ;

ð19Þ

aRþ

the – sign is due to reverse sign from where for the antisymmetric part of a11 to that of a11 . Two terms in Eq. (19) are

2.1.2. The antisymmetric polarizabilities for a rotational sublevel jJMi It is known that the polarizability akl can be given as the sum of the symmetric and antisymmetric parts [1,2,13]:

aRsym ¼ ðaR þ aRþ Þ=2;

ð20Þ

aRant ¼ ðaR  aRþ Þ=2:

ð21Þ

akl ¼ asym þ aant

aRsym ¼ ½ðJ þ 1ÞðJ þ 2Þ þ M 2 fR ;

1 1 ¼ ðakl þ alk Þ þ ðakl  alk Þ; 2 2

Using Eqs. (13) and (14), we can get M ¼ 0; 1; . . . ; J ;

ð18Þ

asym is the symmetric polarizability, which includes the anisotropic and isotropic parts, and aant is the antisymmetric polarizability. In our cases, for ex-

aRant ¼ Mð2J þ 3ÞfR ;

(A) P term of the polarizabilities induced by a resonant optical field M ¼ 0; 1; . . . ; ðJ  1Þ

aPþ ¼ ½ðJ  1ÞJ þ Mð2J  1Þ þ M 2 fP ;

M ¼ ðJ  2Þ; . . . ; ðJ  1Þ; J

aP ¼ ½ðJ  1ÞJ  Mð2J  1Þ þ M 2 fP ;

M ¼ J ; . . . ; ðJ  1Þ; ðJ  2Þ

0

fP ¼

M ¼ 0; 1; . . . ; J :

ð23Þ

For P term, the polarizability in Table 1(A) can be expressed as

Table 1

aP0 ¼ ½J 2  M 2 2fP ;

ð22Þ

2 jðM0 Þnv gv j 2ð2J þ 1Þð2J  1ÞðDP  iCJ 1 =2Þ

DP ¼ vgv!nv0 þ ðBv0  Bv ÞJ 2  ðB0v0 þ Bv ÞJ  v0

(B) P term of the symmetric (aPsym ) and antisymmetric (aPsym ) polarizabilities aPsym

8 < ½ðJ  1ÞJ þ M 2 fP ; ¼ ½ðJ  1ÞJ þ Mð2J  1Þ þ M 2 fP =2; : ½ðJ  1ÞJ  Mð2J  1Þ þ M 2 fP =2;

M ¼ 0; 1; . . . ; ðJ  2Þ M ¼ J  1; J M ¼ J ; ðJ  1Þ

aPant

8 < Mð2J  1ÞfP ; ¼ ½ðJ  1ÞJ þ Mð2J  1Þ þ M 2 fP =2; : ½ðJ  1ÞJ  Mð2J  1Þ þ M 2 fP =2;

M ¼ 0; 1; . . . ; ðJ  2Þ M ¼ J  1; J M ¼ J ; ðJ  1Þ

266

aP ¼ aPsym  aPant :

R. Zheng et al. / Chemical Physics 285 (2002) 261–276

ð24Þ

And the symmetric and antisymmetric polarizabilities aPsym , aPant of P terms are deduced and listed in Table 1(B). In contrast to the non-resonant case (see Appendix A), from Eqs. (19)–(24) and Table 1 the polarizability induced by resonant optical field has not only the symmetric part but also the antisymmetric part. Moreover, comparing Eq. (23) with Eq. (22), one can see that the antisymmetric part is the same order of magnitude of that of the symmetric one and for the rotational magnetic quantum states jJ ; Mi and jJ ; Mi, the polarizabilities are identical in the symmetric part and of reverse sign in the antisymmetric part. Here the appearance of non-zero antisymmetric polarizabilities for rotational sublevels M needs necessary conditions that the incident excitation should be resonant or near-resonant to the vibronic transition or rovibration transition in the ground electron state. When the incident excitation is far off-resonant, the complex set of intermediate rotational sublevels jJ 0 ; M 0 i in Eq. (8) can be sum over them and closed, which cannot produce the antisymmetric polarizability (see Appendix A). In addition, laser power should be intensive enough to make an ac Stark effect break the degeneracy between the different sublevels jJ ; Mi (see section 2.2), as the experiment studies [18–22] did. 2.2. M-Dependent splittings and shifts of energy levels and spectral lines induced by circularly polarized optical field 2.2.1. Laser-induced M-dependent splittings of energy levels for a linear molecule It is known [36] that the oscillating electric field of a light wave can cause ac Stark effect on a molecule and induce the shift of a molecular energy level ð1=2Þ~ E  ~ a~ E. When the optical field is intensive enough, each of sublevel M splits uniquely in resonant circularly polarized optical field. Taking the energy level jgvi as an example, when the frequency v0 of a right (+) or left ()) circularly polarized optical field approaches the gv ! nv0 transition, laser-induced shift of the sublevel jgvijJMi is [36]

1 ð0Þ 2 ðE Þ Re a ; ð25Þ 2h where Eð0Þ is the scalar electric field strength of the circularly polarized laser; a are the laser-induced polarizabilities a11 or a11 of sublevels jJMi and a ¼ aR þ aP . Since aR and aP terms have different frequency dependence through fR , DR and fp , Dp , respectively (see Eqs. (15) and (16) and Table 1(A)) and different resonant frequency points, when the frequency is near the aR resonant point R term will dominate the shifts. For simplicity, here we consider only the laser-induced shifts due to R term. Eq. (25) is rewritten as

DW ¼ 

DW ¼ 

1 ð0Þ 2 ðE Þ Re aR : 2h

ð26Þ

Substituting Eqs. (13)–(15) into Eq. (26), shifts of sublevels M induced by a right (+) or left ()) circularly polarized optical field are DW ¼

Vv0 ðvÞ R ðJMÞ; 2

ð27Þ

and Vv0 ðvÞ ¼

2

2  nv0  Eð0Þ ðM0 Þgv  DR

h2 ðD2R þ C2J þ1 =4Þ

;

ð28Þ

ðJ þ 1ÞðJ þ 2Þ  Mð2J þ 3Þ þ M 2 2ð2J þ 1Þð2J þ 3Þ ðJ þ 2Þ M2 þ ¼ 2ð2J þ 3Þ 2ð2J þ 1Þð2J þ 3Þ M ; M ¼ 0; 1; . . . ; J :  2ð2J þ 1Þ

R ðJMÞ ¼

ð29Þ The light shifts DW of the sublevel jgvJMi introduced by the molecular state jnv0 i can be factorised into a rotational part R ðJMÞ (describing the whole angular dependence of DW Þ and a vibrational part Vv0 ðvÞ=2 (describing the ‘‘strength’’ of the shifts DW of state jgvi induced by state jnv0 i). The rotational part R ðJMÞ consists of three terms: The first term produces the whole shift of Jstate energy levels; the second term breaks the degeneracy between the different jMj magnetic sublevels; and the third term further removes the degeneracy between the +M and )M sublevels by

R. Zheng et al. / Chemical Physics 285 (2002) 261–276

267

the circularly polarized light, referring to the references [18,20,21,23,32,37] this is a new term resulted from the antisymmetric polarizabilities induced by the right (+) or left ()) circularly polarized resonant optical field. For a circularly polarized resonant field the expressions of DW (Eqs. (27)–(29)) is resemblant to those for an atom [14,25], but for a molecule the antisymmetric polarizability of the sublevel M are the same order of magnitude as the symmetric one, while for an atom the antisymmetric polarizability are typically much smaller than the symmetric polarizability [25]. Similarly, laser-induced shifts of sublevels M due to aP term can be deduced when the light frequency is approach P term resonance.

in Eq. (1), respectively and have opposite sign of the energy denominators in each term of Eq. (1) in the resonant case. This light shift result indicates that the imaginary line width term iCg~n~=2 in the complex conjugate term of Eq. (1) should be of the opposite sign to that in the first term in Eq. (1) and associates with the convention in [26]. Thus Eq. (28) is reduced to

2.2.2. Laser-induced M-dependent splittings and shifts of molecular spectral lines The laser-induced M-dependent splittings of energy levels can be observed by spectral methods, such as the single-photon absorption spectra, resonance fluorescence spectra, resonant Raman scattering, and resonant Rayleigh scattering. Recently, resonance light scattering (RLS) is considered as both extremely sensitive and selective in probing a number of different systems [38–40]. As an example we discuss the shifts and splittings in resonance Rayleigh scattering lines. When the frequency v0 of a circularly polarized optical field approaches the gv ! nv0 transition, the laser-induced shift of Rayleigh scattering lines for the initial state jgvijJMi through the intermediate state jnv0 ijJ 0 M 0 i (abbreviated to jgvijJMi $ jnv0 i jJ 0 M 0 i from now on) is

Dv ¼ 

1 Dv ¼ ðDW 0  DW Þ; ð30Þ h where DW and DW 0 are laser-induced shifts of sublevels jgvijJMi and jnv0 ijJ 0 M 0 i, respectively. When the optical field is resonant with R term, i.e., J 0 ¼ J þ 1, and P term contribution is omitted. Based on the sum-over-states expression for the complex polarizability Eq. (1), it can be shown that the relation DW 0 ¼ DW holds for the two sublevels coupled by a resonant optical field [14]. The reason is that the polarizabilities of the ground state and the resonant excited state are dependent on the first term and its conjugate term

2

ðEð0Þ Þ Re aR : ð31Þ h From Eq. (27), we can obtain the shifts of the Rayleigh scattering lines jgvijJMi $ jev0 ijJ 0 M 0 i in a right (+) or left ()) circularly polarized optical field Dv ¼

Vv0 ðvÞR ðJMÞ ; h

M ¼ 0; 1; . . . ; J : ð32Þ

Eqs. (28), (29) and (32) indicate that the light shifts Dv are dependent on magnitude and sign of magnetic quantum number M of initial state of a linear molecule, thus in molecular spectra, for example resonant Rayleigh scattering a single rotational Rayleigh line without the optical field can be split into many M-dependent components induced by the resonant circularly polarized optical field, according to both the magnitude and the sign of M. Second-order Stark effects usually depend on M 2 , so that the levels and rotational spectral lines are separated into pairs of degenerate ( M) except for M ¼ 0, many experiment studies [18,20–24] demonstrated this results by using linearly polarized light. Here Eq. (32) shows that using resonant circularly polarized light the levels and rotational spectrum lines can be further split according to the sign of M, thus the optical Stark effects of circularly polarized resonant light removes degeneracy of M completely, just as the first-order Stark effect in a constant electric field does [41].

3. Calculations As examples, we have studied two systems of H2 and CO molecules using the deduced equations in Section 2. For H2 molecule, we have calculated the antisymmetric and symmetric polarizabilities

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R. Zheng et al. / Chemical Physics 285 (2002) 261–276

3.1. Antisymmetric polarizabilities

of the rotational magnetic sublevels in the ground electronic state X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ, which are induced by a optical field with frequency v0 resonant with the transition X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! 0 0 B1 Rþ u ðv ¼ 0; J ¼ 2Þ. Based on them, circularly polarized-laser-induced M-dependent splittings and shifts of the resonant Rayleigh scattering lines 1 þ 0 0 0 X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ $ B Ru ðv ¼ 0; J ¼ 2; M Þ have been evaluated. Similarly, we also give the antisymmetric polarizabilities of CO molecule in the rotational magnetic sublevels X1 Rþ ðv ¼ 0; J ¼ 1; M ¼ 1Þ using circularly polarized IR field with the frequency v0 resonant with the transition X1 Rþ ðv ¼ 0; J ¼ 1Þ ! X1 Rþ ðv0 ¼ 1; J 0 ¼ 2Þ. In the calculation, the value of Eð0Þ is obtained from the laser intensity I ¼ 1=2e0 cðEð0Þ Þ2 [16]. For laser intensity 100 MW cm2 , Eð0Þ is 86.6 kV cm1 . The relative molecular constants of H2 [35] and CO [42,43] are listed in Table 2. The matrix elenv0 ment ðM0 Þgv can be obtained from the corresponding oscillator strength using the formula [44] fgv!nv0 ¼

8p2 me nv0 2 vgv!nv0 jðM0 Þgv j ; h

From the above theoretical analysis, we know that under the action of the optical fields at the resonant frequency the antisymmetric polarizability of sublevels jJMi exists. We take H2 and CO as examples. Table 3 shows the symmetric and antisymmetric polarizabilities of R term and P term for rotational magnetic sublevels X1 Rþ g ðv ¼ 0; J ¼ 1; M ¼ 1Þ of H2 when the optical field v0 is 1 þ 0 resonant with X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! B Ru ðv ¼ 0; 0 J ¼ 2Þ transition. Table 4 shows the symmetric and antisymmetric polarizabilities of R term and P term for the rotational magnetic sublevels X1 Rþ ðv ¼ 0; J ¼ 1; M ¼ 1Þ of CO when the circularly polarized optical field v0 is resonant with ðv ¼ 0; J ¼ 1Þ ! ðv0 ¼ 1; J 0 ¼ 2Þ transition in the electronic ground state X1 Rþ . The results in Tables 3 and 4 indicate that the antisymmetric polarizability and the symmetric polarizability are the same order of 10 a.u. for H2 and CO. The polarizability of H2 are a little larger than those of CO because of the relative larger transition matrix element of H2 . When the laser frequencies approach the resonant frequency for R term, the polarizabilities aP of P term are less than 1% of R term for H2 (Table 2) and 7% for CO (Table 3), i.e., R term dominates the polarizability and P term has little influence on it. In the region of the laser frequency resonant with P term similar cases for P term can also appear. In order to compare the M-dependence of the polarizabilities, Table 5 shows the polarizabilities of the rotational magnetic sublevels X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ of H2 induced by an optical field v0 1 þ 0 resonant with X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! B Ru ðv ¼ 0 0; J ¼ 2Þ transition. From Table 5, the polarizabilities of rotational magnetic sublevels jJMi have

ð33Þ

where we neglect the influence of the optical field 0 and assume isoupon the matrix element ðM0 Þnv gv tropy for the transition jgvi ! jnv0 i. The oscillator 1 þ 0 strength of X1 Rþ g ðv ¼ 0Þ ! B Ru ðv ¼ 0Þ transi3 tion for H2 is 1:75  10 [35], and X1 Rþ ðv ¼ 0Þ ! X1 Rþ ðv0 ¼ 1Þ transition for CO is 1:08  105 [42]. Using Eqs. (17) and (33) and Table0 2, the nv corresponding matrix elements jðM0 Þgv j are 31 3:95  10 C m for H2 molecule and 1:99  1031 C m for CO molecule, respectively. All the line widths were assumed to have the same magnitude of C ¼ 109 Hz [35]. Here we did not take account of the influence of line shape of the incident light since the laser line width can be 1 MHz [45]. Table 2 Molecular structure constants of H2 [35] and CO [42,43] State

T0

xe

x e xe

Be

ae

fgv!ev0

X Rþ g B1 Rþ u 1 þ

0 90203.55 0

4401.21 1357.39 2169.81

121.34 20.42 13.288

60.8530 20.035 1.93128

3.0622 1.2312 0.0175

1:75  103

1

ðH2 Þ ðH2 Þ X R ðCOÞ

1:08  105

The electronic energies above the ground state T0 , vibration spacings xe , anharmonic corrections to vibrational spacing xe xe , rigid rotator rotational spacing Be , non-rigid rotator correction ae , and the oscillator strength fgv!nv0 , where gv ! nv0 represents 1 þ 0 1 þ 1 þ 0 1 X1 Rþ g ðv ¼ 0Þ ! B Ru ðv ¼ 0Þ for H2 and X R ðv ¼ 0Þ ! X R ðv ¼ 1Þ for CO. All the energy parameters are in cm .

R. Zheng et al. / Chemical Physics 285 (2002) 261–276

269

Table 3 Symmetric and antisymmetric polarizabilities (in a.u.) of R term and P term for rotational magnetic sublevels X1 Rþ g ðv ¼ 0; J ¼ 1; M ¼ 1Þ 1 þ 0 0 of H2 when the circularly polarized optical field v0 is resonant with X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! B Ru ðv ¼ 0; J ¼ 2Þ transition v0 (1015 Hz)

aRsym

aRant

aPsym

aPant

2.65944 2.65943 2.65942 2.65941 2.65939 2.65938 2.65937 2.65936 2.65935 2.65934 2.65933 2.65932 2.65931 2.65930 2.65929 2.65924 2.65923 2.65922 2.65921 2.65920 2.65919 2.65918 2.65917 2.65916 2.65915 2.65914 2.65913 2.65912

20.2941 21.6138 23.1171 24.8450 26.8522 29.2122 32.0270 35.4420 39.6722 45.0490 52.1117 61.8004 75.9141 98.3802 139.7247 )126.7049 )91.7413 )71.8989 )59.1129 )50.1876 )43.6039 )38.5471 )34.5413 )31.2897 )28.5976 )26.3320 )24.3990 )22.7305

14.4958 15.4384 16.5122 17.7465 19.1801 20.8658 22.8764 25.3157 28.3373 32.1779 37.2226 44.1432 54.2243 70.2716 99.8033 )90.5035 )65.5295 )51.3564 )42.2235 )35.8483 )31.1456 )27.5337 )24.6724 )22.3498 )20.4268 )18.8086 )17.4279 )16.2360

0.1298 0.1302 0.1306 0.1309 0.1313 0.1316 0.1320 0.1324 0.1327 0.1331 0.1335 0.1339 0.1343 0.1346 0.1350 0.1370 0.1374 0.1378 0.1382 0.1386 0.1390 0.1394 0.1398 0.1402 0.1406 0.1410 0.1414 0.1419

)0.1298 )0.1302 )0.1306 )0.1309 )0.1313 )0.1316 )0.1320 )0.1324 )0.1327 )0.1331 )0.1335 )0.1339 )0.1343 )0.1346 )0.1350 )0.1370 )0.1374 )0.1378 )0.1382 )0.1386 )0.1390 )0.1394 )0.1398 )0.1402 )0.1406 )0.1410 )0.1414 )0.1419

1 atomic unit ða:u:Þ ¼ 1:481845  1025 cm3 .

strong M-dependence due to the antisymmetric part, which may provide the possibility of M-selectivity. On the whole the calculated results indicate: (i) the antisymmetric and symmetric polarizabilities of the sublevel jJMi have the same order of magnitude; (ii) due to the contribution from the antisymmetric part the polarizabilities are M-dependence, which can break the degenerate between jJ ; þMi and jJ ; þMi states (see next section). 3.2. Shifts and splittings of resonance Rayleigh scattering lines of H2 induced by circularly polarized optical field Hara et al. [46] have measured the rotational Rayleigh–Brillouin spectrum of light scattered

from H2 at densities ranging from 0.5 to 104 amagat. At high densities a triplet structure (two Brillouin side bands and one Rayleigh band) is seen; and at very low densities the triplet structure disappears and a single broad Gaussian line remains [46]. Here we do not consider the influence of the Brillouin side bands and only discuss the single Rayleigh line in very low densities. The resonant right (+) and left ()) circularly polarized optical fields with the laser power 100 MW cm2 and the frequency v0 resonant with X1 Rþ g 0 0 ðv ¼ 0; J ¼ 1Þ ! B1 Rþ transition u ðv ¼ 0; J ¼ 2Þ of H2 molecules are assumed to illuminate H2 molecules. Only R term of the polarizability is considered when the frequency of the laser approaches the R resonant point. Using Eq. (27) and Table 2, the light shifts and splittings of rotational

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R. Zheng et al. / Chemical Physics 285 (2002) 261–276

Table 4 Symmetric and antisymmetric polarizabilities (in a.u.) of R term and P term for the rotational magnetic sublevels X1 Rþ ðv ¼ 0; J ¼ 1; M ¼ 1Þ of CO when the circularly polarized optical field v0 is resonant with ðv ¼ 0; J ¼ 1Þ ! ðv0 ¼ 1; J 0 ¼ 2Þ transition in the electronic ground state X1 Rþ v0 ð1013 Hz)

aRsym

aRant

aPsym

aPant

6.463 6.462 6.461 6.460 6.459 6.458 6.457 6.456 6.455 6.454 6.453 6.452 6.451 6.450 6.445 6.444 6.443 6.442 6.441 6.440 6.439 6.438 6.437 6.436 6.435 6.434 6.433

5.5632 5.9533 6.4022 6.9244 7.5393 8.2741 9.1675 10.2772 11.6926 13.5600 16.1372 19.9239 26.0319 37.5380 )30.9810 )22.6996 )17.9113 )14.7910 )22.5966 )10.9692 )9.7141 )8.7168 )7.9051 )7.2318 )6.6641 )6.1791 )5.7599

3.9737 4.2524 4.5730 4.9460 5.3852 5.9100 6.5482 7.3409 8.3518 9.6857 11.5266 14.2313 18.5942 26.8128 )22.1293 )16.2140 )12.7938 )10.5650 )8.9976 )7.8351 )6.9387 )6.2263 )5.6465 )5.1656 )4.7601 )4.4137 )4.1142

0.2449 0.2499 0.2552 0.2607 0.2664 0.2724 0.2786 0.2852 0.2921 0.2993 0.3068 0.3148 0.3232 0.3320 0.3847 0.3973 0.4108 0.4252 0.4406 0.4572 0.4751 0.4945 0.5155 0.5384 0.5634 0.5908 0.6211

)0.2449 )0.2499 )0.2552 )0.2607 )0.2664 )0.2724 )0.2786 )0.2852 )0.2921 )0.2993 )0.3068 )0.3148 )0.3232 )0.3320 )0.3847 )0.3973 )0.4108 )0.4252 )0.4406 )0.4572 )0.4751 )0.4945 )0.5155 )0.5384 )0.5634 )0.5908 )0.6211

1 atomic unit ða:u:Þ ¼ 1:481845  1025 cm3 .

magnetic sublevels X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ and 0 0 0 B1 Rþ ðv ¼ 0; J ¼ 2; M Þ of H2 molecule have been u calculated and sketched in Fig. 1. Based on them and Eq. (32), the light shifts and splittings of the resonant Rayleigh scatting lines X1 Rþ g ðv ¼ 0; 0 0 0 J ¼ 1; MÞ $ B1 Rþ ðv ¼ 0; J ¼ 2; M Þ have also u been evaluated and listed in Table 6 and shown in Fig. 1. In Fig. 1, the relative intensities and the light broadenings are also sketched by the bar height and width, respectively, and each of them correspond approximately to the square of the corresponding polarizability and the polarizability [41,17], respectively. From Table 6 and Fig. 1, three prominent results can be seen. (i) When modulating the frequency of the applied laser to be resonant with X1 Rþ g 0 0 ðv ¼ 0; J ¼ 1Þ ! B1 Rþ u ðv ¼ 0; J ¼ 2Þ transition,

the single rotation Rayleigh line split into three Mdependent Rayleigh components corresponding respectively to M ¼ 1, 0, )1, due to the ac Stark effect by the circularly polarized optical field. The relative ratios of the shifts of these M-dependent components are 2:6:11 (or 11:6:2) for M ¼ 1, 0, )1 in a right (+) (or left ())) circularly polarized optical field. The splitting intervals between the components for M ¼ 1, 0, )1 are the same order of magnitude of the shifts. Dye and Bischel [18] investigated experimentally the optical Stark effect of non-resonant optical field on the Raman spectra of H2 molecules in the v ¼ 1 vibrational state and measured the light shift and splitting of Q (1) rovibrational Raman transition, i.e., the shift 27 MHz GW1 cm2 for M ¼ 0 and 19 MHz GW1 cm2 for M ¼ 1. Thus M2 -dependent splittings of the rovibrational

R. Zheng et al. / Chemical Physics 285 (2002) 261–276

271

Table 5 M-dependence of the polarizabilities for the rotational magnetic sublevels X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ of H2 induced by a right (+) or left 1 þ 0 0 ()) circularly polarized optical field v0 resonant with X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! B Ru ðv ¼ 0; J ¼ 2Þ transition v0 (1015 Hz)

2.65944 2.65943 2.65942 2.65941 2.65939 2.65938 2.65937 2.65936 2.65935 2.65934 2.65933 2.65932 2.65931 2.65930 2.65929 2.65924 2.65923 2.65922 2.65921 2.65920 2.65919 2.65918 2.65917 2.65916 2.65915 2.65914 2.65913 2.65912

aþ (a.u.)

a (a.u.)

M ¼ 1

M ¼0

M ¼1

M ¼ 1

M ¼0

M ¼1

34.8 37.1 39.6 42.6 46.0 50.1 54.9 60.8 68.0 77.2 89.3 105.9 130.1 168.7 239.5 )217.2 )157.3 )123.3 )101.3 )86.0 )74.7 )66.1 )59.2 )53.6 )49.0 )45.1 )41.8 )39.0

17.3950 18.5261 19.8146 21.2957 23.0162 25.0390 27.4517 30.3788 34.0047 38.6134 44.6672 52.9718 65.0692 84.3259 119.7640 )108.6042 )78.6354 )61.6276 )50.6682 )43.0180 )37.3748 )33.0404 )29.6068 )26.8197 )24.5122 )22.5703 )20.9135 )19.4833

6.0580 6.4358 6.8660 7.3604 7.9346 8.6096 9.4146 10.3910 11.6004 13.1374 15.1560 17.9250 21.9582 28.3779 40.1914 )35.9275 )25.9371 )20.2670 )16.6131 )14.0622 )12.1803 )10.7347 )9.5894 )8.6595 )7.8895 )7.2414 )6.6883 )6.2107

6.0580 6.4358 6.8660 7.3604 7.9346 8.6096 9.4146 10.3910 11.6004 13.1374 15.1560 17.9250 21.9582 28.3779 40.1914 )35.9275 )25.9371 )20.2670 )16.6131 )14.0622 )12.1803 )10.7347 )9.5894 )8.6595 )7.8895 )7.2414 )6.6883 )6.2107

17.3950 18.5261 19.8146 21.2957 23.0162 25.0390 27.4517 30.3788 34.0047 38.6134 44.6672 52.9718 65.0692 84.3259 119.7640 )108.6042 )78.6354 )61.6276 )50.6682 )43.0180 )37.3748 )33.0404 )29.6068 )26.8197 )24.5122 )22.5703 )20.9135 )19.4833

34.8 37.1 39.6 42.6 46.0 50.1 54.9 60.8 68.0 77.2 89.3 105.9 130.1 168.7 239.5 )217.2 )157.3 )123.3 )101.3 )86.0 )74.7 )66.1 )59.2 )53.6 )49.0 )45.1 )41.8 )39.0

1 atomic unit ða:u:Þ ¼ 1:481845  1025 cm3 . aþ  a11 , a  a11 .

Raman lines have been obtained in their experiment. Here the calculated results of Fig. 1 and Table 6 indicate that by using the circularly polarized resonant optical field, M double-degeneracy of rotational sublevel M and its relative rovibrational spectrum lines can be further broken into two components for + or ) sign of M. This results from the contribution of non-zero antisymmetric polarizabilities of M states induced by the resonant optical field. (ii) Using the resonant circularly polarized optical field one not only split the Rayleigh scattering lines but also make the detect easier. Near R term resonance, in a right or left circularly polarized optical field with intensity 100 MW cm2 laser-induced shifts of Rayleigh scattering lines 1 þ 0 0 0 X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ $ B Ru ðv ¼ 0; J ¼ 2; M Þ

of H2 can be the order of 0.1–1 GHz (Table 6), i.e., 1–10 MHz MW1 cm2 . In non-resonance cases, the corresponding laser-induced shifts are the order of 10 MHz GW1 cm2 [18]. Resonant laser-induced shifts are much larger than non-resonant ones. Using relatively weaker resonant optical field large shifts of molecular energy levels can be induced. (iii) In addition, Fig. 1 also shows two rotational sublevels coupled by the resonant field shift to opposite directions by the same amount as pointed out in Section 2.2.2, which associates with the convention of the imaginary line widths of [26]. Here we have taken resonant Rayleigh scattering as an example to show and discuss the M-dependent shifts and splitting of sublevels M. However, as we have pointed out in Section

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R. Zheng et al. / Chemical Physics 285 (2002) 261–276

1 þ 0 0 0 Fig. 1. Splittings of rotational sublevels X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ and B Ru ðv ¼ 0; J ¼ 2; M Þ and shifts of Rayleigh scattering lines of these sublevels for H2 when the applied right (+) or left ()) circularly polarized optical field with intensity 100 MW cm2 at the 1 þ 0 0 frequency v0 ¼ 2:65930  1015 Hz resonant with X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! B Ru ðv ¼ 0; J ¼ 2Þ transition. The ratio of the heights of bars show the relative intensities of Rayleigh scattering M-components and the width is the relative broadenings that correspond 1 þ 0 0 approximately to the scattering intensity. The origin indicates the position of X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! B Ru ðv ¼ 0; J ¼ 2Þ transition in a zero field.

2.2.2, the foregoing laser-induced M-dependent shifts and splitting of rovibrational sublevels can be also observed by other spectral methods, such as the single photon absorption spectra, resonance fluorescence spectra, resonant Raman scattering.

3.3. Discussions: the valid of the perturbation theory Finally, we discuss the valid of the sum-over-state expression of polarizabilities used in this paper. It is known that the dispersion theory of the ac Stark effect is hold when

R. Zheng et al. / Chemical Physics 285 (2002) 261–276

273

Table 6 1 þ 0 0 0 Shifts and splittings of resonance Rayleigh scattering lines X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ $ B Ru ðv ¼ 0; J ¼ 2; M Þ induced by the resonance right (+) or left ()) circularly polarized optical field v0 for H2 v0 (105 Hz)

Dvþ (GHz)

2.65944 2.65943 2.65942 2.65941 2.65939 2.65938 2.65937 2.65936 2.65935 2.65934 2.65933 2.65932 2.65931 2.65930 2.65929 2.65924 2.65923 2.65922 2.65921 2.65920 2.65919 2.65918 2.65917 2.65916 2.65915 2.65914 2.65913 2.65912

Dv (GHz)

M ¼ 1

M ¼0

M ¼1

M ¼ 1

M ¼0

M ¼1

0.650 0.692 0.740 0.795 0.859 0.935 1.025 1.134 1.270 1.442 1.668 1.978 1.430 1.149 1.472 )4.055 )2.936 )2.301 )1.892 )1.606 )1.396 )1.234 )1.106 )1.001 )0.915 )0.843 )0.781 )0.728

0.325 0.346 0.370 0.398 0.430 0.467 0.513 0.567 0.635 0.721 0.834 0.989 1.215 1.574 2.236 )2.028 )1.468 )1.151 )0.946 )0.803 )0.698 )0.617 )0.553 )0.501 )0.458 )0.421 )0.390 )0.364

0.1083 0.1153 0.1233 0.1325 0.1432 0.1558 0.1708 0.1891 0.2116 0.2403 0.2780 0.3297 0.4050 0.5248 0.7453 )0.6759 )0.4894 )0.3835 )0.3153 )0.2677 )0.2326 )0.2056 )0.1843 )0.1669 )0.1525 )0.1405 )0.1302 )0.1213

0.1083 0.1153 0.1233 0.1325 0.1432 0.1558 0.1708 0.1891 0.2116 0.2403 0.2780 0.3297 0.4050 0.5248 0.7453 )0.6759 )0.4894 )0.3835 )0.3153 )0.2677 )0.2326 )0.2056 )0.1843 )0.1669 )0.1525 )0.1405 )0.1302 )0.1213

0.325 0.346 0.370 0.398 0.430 0.467 0.513 0.567 0.635 0.721 0.834 0.989 1.215 1.574 2.236 )2.028 )1.468 )1.151 )0.946 )0.803 )0.698 )0.617 )0.553 )0.501 )0.458 )0.421 )0.390 )0.364

0.650 0.692 0.740 0.795 0.859 0.935 1.025 1.134 1.270 1.442 1.668 1.978 2.430 3.149 4.472 )4.055 )2.936 )2.301 )1.892 )1.606 )1.396 )1.234 )1.106 )1.001 )0.915 )0.843 )0.781 )0.728

0

2 ðEð0Þ Þ2 jðM0 Þev gv j

D2R

1

[16,20], i.e., the condition is that the frequency of the laser can be controlled precisely to be a distance from aR term resonance and the intensity of the laser is not too intense. In the cases discussed in this pa0 per, we know by calculation that ðEð0Þ ÞjðM0 Þev gv j is smaller than 5  109 Hz, and DR larger than 2  1010 Hz, thus, ev0 2

2

ðEð0Þ Þ jðM0 Þgv j D2R

< 0:16;

which agree approximately with the foregoing 0 condition. Also, the value of ðEð0Þ ÞjðM0 Þev gv j is much smaller than the rotational constants Bv (Table 2),

so it is reasonable that the interaction between the molecule and the laser beam is a perturbation of the molecule rotational level.

4. Conclusions Using the semiclassical perturbation theory, we study the antisymmetric polarizabilities of rotational sublevels jJMi and the M-dependent splitting and shifts of the energy sublevels and corresponding rotational spectral lines of a linear molecule induced by a right (+) or left ()) circularly polarized resonant optical field. As examples, we have calculated these quantities for H2 and CO molecules. For H2 molecule, the antisymmetric and

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R. Zheng et al. / Chemical Physics 285 (2002) 261–276

symmetric polarizabilities of the rotational magnetic sublevels in the ground electronic state X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ, which are induced by an optical field with frequency v0 resonant with the 1 þ 0 transition X1 Rþ g ðv ¼ 0; J ¼ 1Þ ! B Ru ðv ¼ 0; 0 J ¼ 2Þ, are the same order of 10 a.u.; the original single rovibrational line in resonant Rayleigh 1 þ 0 scattering X1 Rþ g ðv ¼ 0; J ¼ 1; MÞ $ B Ru ðv ¼ 0; J 0 ¼ 2; M 0 Þ without the optical field is split into three M-dependent components corresponding respectively to M ¼ 1, 0, )1 induced by the circularly polarized resonant field; and laser-induced M-dependent splittings and shifts of the resonant Rayleigh scattering lines can be the order of 0.1–1 GHz when the intensity of circularly polarized optical field is 100 MW cm2 , which can be measurable referring to the relative experiment results [18–24]. The deduced theoretical formulas and calculated results indicate that the antisymmetric polarizabilities of rovibrational sublevels jJMi of a linear molecule can be induced by a resonant optical field and have the same order of magnitude of the symmetric polarizabilities. Based on them, Mdependent splitting and shifts of rovibrational sublevels and spectral lines can be produced and the degeneracy for +M and )M sublevels and the spectral lines can be completely broken by the circularly polarized resonant optical field.

Acknowledgements This work is supported by the National Science Foundation of China (Grant No. 20173051) and the Science Foundation of Anhui Province in China (Grant No. 01046302).

Appendix A. The polarizabilities of a linear molecule in non-resonance field If the incident excitation is non-resonance, i.e., jvgv;nv0  v0 j  vJ ;J 0 and vJ ;J 0 can be neglected, the sum over the complete set of rovibronic eigenfunctions (jJ 0 M 0 i) in Eq. (8) can be closed. The resulting product of coordinate transformation rotation matrices can be reduced to a single rotation matrix [31]

D1kq D1lq0 ¼

X

ð1Þ

qq0

ð2k 0 þ 1Þ

k 0 ;q;q0





1 1 k l

k0 q



1 q

1 q0

 k0 0 Dkqq0 : q0 ðA:1Þ

The subsequent integral of three rotation matrices is readily evaluated [29] and thus, in the non-resonance limit, Eq. (8) becomes 1X nv0 M 0 ðMq Þgv ðakl Þg ¼ nv0 ðMq0 Þgv ð1Þ ð2J þ 1Þð2k þ 1Þ h k0 ;q;q0 v0 ;q;q0



  1 1 k0 1 1 k0 k l q q q0 q0    0 J k0 J J k J  M q M 0 q0 0 1  ðvgv;nv0  v0 Þ : 

ðA:2Þ

Using the properties of the 3j symbols, q ¼ q0 ¼ 0 can be deduced from     J k0 J J k J ; and M q M 0 q0 0 and k þ l ¼ q ¼ 0 from   1 1 k0 ; k l q for k ¼ l ¼ 0 ðakl Þg corresponds to the polarizability a00 , for k ¼ 1, l ¼ 1, a11 ; and for k ¼ 1, l ¼ 1, a11 . The expressions of non-zero components of the polarizability tensors deduced from Eq. (A.2), respectively are 1 c 3M 2  J ðJ þ 1Þ ; a00 ¼  a þ 3 3 ð2J  1Þð2J þ 3Þ

ðA:3Þ

1 c 3M 2  J ðJ þ 1Þ a11 ¼ a þ ; 3 3 ð2J  1Þð2J þ 3Þ

ðA:4Þ

1 c 3M 2  J ðJ þ 1Þ a11 ¼ a þ ; 3 3 ð2J  1Þð2J þ 3Þ

ðA:5Þ

where the isotropic factor a and the anisotropy factor c are 1 Xh nv0 gv nv0 a¼ ðM1 Þgv nv0 ðM1 Þgv  ðM0 Þnv0 ðM0 Þgv h v0 i gv nv0 1 þ ðM1 Þnv0 ðM1 Þgv ðvgv;nv0  v0 Þ ; ðA:6Þ

R. Zheng et al. / Chemical Physics 285 (2002) 261–276

c¼

1X gv nv0 gv nv0 ½ðM1 Þnv0 ðM1 Þgv þ 2ðM0 Þnv0 ðM0 Þgv h v0 gv

nv0

1

þ ðM1 Þnv0 ðM1 Þgv ðvgv;nv0  v0 Þ ;

ðA:7Þ

Eqs. (A.3)–(A.5) indicate that for a linear molecule the non-resonant antisymmetric polarizability does not exist induced either by a linear polarized optical field or by a circularly polarized optical field and the polarizability is symmetric which is similar to the case of the polarizability in a static electric field [14]. The polarizabilities of states jJ ; Mi induced by a non-resonant optical field are equal, i.e., M 2 -dependence. Thus, the M degeneracy is not completely broken due to the non-resonant optical field. The optical Stark splittings of rotational Raman transition caused by the optical polarizability isotropy and anisotropy of N2 and H2 molecules in the case of a nonresonant high-intensity optical field were reported [18,20,22]. All of the forgoing experiments no one has not observed the complete separation of the rotation energy, i.e., the rotational splittings are still double degenerate about the rotational magnetic quantum numbers M for the antisymmetric polarizability is zero in non-resonant cases.

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