National Sun Yat-sen University Institutional Repository:Item 987654321/30924

行政院國家科學委員會專題研究計畫 期中進度報告

成像技巧研究氣相及凝態系統動態行為(1/3)

計畫類別： 個別型計畫 計畫編號： NSC94-2113-M-110-012- 執行期間： 94 年 08 月 01 日至 95 年 07 月 31 日 執行單位： 國立中山大學化學系(所) 計畫主持人： 陳國美 共同主持人： 裴呈志 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢

中 華 民 國 95 年 5 月 12 日

(二)計畫中、英文摘要及關鍵字 關鍵字:螢光成像，離子成像，光解過程，同調及非同調角動量極化，切片式螢光成像顯微 術 本計畫利用螢光成像及離子成像技巧研究分子光解過程的動態學與光碎片同調及非同調角 動量極化，及螢光成像切片技術研究細胞的新陳代謝作用。申請人提出全新的切片式螢光 成像顯微術以增進螢光成像方法的能量解析度，並對離子成像決定角動量極化參量提出全 新的實驗方法。 計畫英文摘要。

Keywords: fluorescence imaging, ion imaging, photodissociation process, coherent and incoherent angular momentum polarization, fluorescence imaging by sliced microscopy (FISM)

Fluorescence imaging and ion imaging techniques will be employed to study molecular photodissociation dynamics, and the determination of the coherent and incoherent angular momentum polarizations of photofragments. A slicing fluorescence imaging method will be developed to study the metabolism of cells in vivo. A brand new scheme which utilizes a sliced microscopy is proposed to improve the energy resolution of fluorescence imaging techniques. To determine angular momentum polarization parameters, a new experimental recipe will be employed to extract these dynamical information.

(三)報告內容 執行成果 1. 採購一套 5 ns 短脈衝 Nd/YAG 雷射系統(1064, 532, 355 及 266 nm 輸出)，(72 萬元，中 山大學支助22 萬元)，已完成驗收，並利用此套雷射執行實驗工作中。 2. 螢光成像實驗 已執行過ICN 及 NO2光解之螢光成像實驗，正改進實驗條件及技巧中，見附件一。 3. 離子成像實驗 正執行NO2光解之離子成像實驗，初步成果見附件2。

4. 理論架構發展

已 完 成 ICN 光 解 之 量 子 理 論 之 第 一 步 探 討 。 論 文 初 稿 “Quantum theory of ICN photodissociation: Density matrices of photofragments from a parallel transitions”已送 J. Chem. Phys.審查。目前正繼續遠距離 I 及 CN 的雙極-四極及四極-四極交互作用計算， 以決定 Ω′ =1分子態與漸近態之間的關聯。Ω′=0至Ω′=±1之間的非緩變躍遷過程。 ICN 在垂直躍遷下的光解理論，及 ICN 在平行垂直同時躍遷並有非緩變躍遷過程存在下 的完整面貌，此一系列的工作將對三原子線型分子的光解量子理論建立完整架構。後續 的LIF 及 REMPI 偵測理論推導將直接與實驗結果掛鉤。

附件一。螢光成像實驗

本實驗室已測試過利用FISM 對 ICN 的 308 nm 光解，執行 CN 產物在 388.34 nm LIF 之偵 測。初步結果簡述如次。

I. 交叉 pump-probe 雷射實驗裝置如下圖所示：

eye piece probe

laser objective

lens for high power laser focusing 15 mm f. l. photolysis laser

CCD camera

II. 首先利用交叉 pump (focused) - probed (unfocused)雷射測試，其螢光成像結果如下:

III. 利用交叉 pump-probe (both focused)雷射測試，無法成功獲得螢光成像。在實驗室現有 平移台(手動式)無法精確重疊雷射束焦點的條件下，看不到是正常的。 IV. 為了測試焦點處有無訊號，我們利用同向 pump-probe 雷射測試，裝置示意圖如下： 388.54 nm 308 nm dichroic mirror

x 5 high power laser focusing lens, 15 mm f. l. flow cell microscope CCD camera

V. 同向雷射測試結果如下：

CN fluorescence image under a co-propagating laser beams configuration (CCD gate is delayed 70 ns with respect to the probe laser pulse and the gate width is 10 ns)

beam waist (focal point) 的大小約為 0.3 mm。成像(單發所得的結果)變寬的原因可由下 圖所示瞭解。

pump laser 的波形(前者)居然有兩個峰長達 70 ns，probe laser(後者)的脈衝寬度也有 30 ns。所以上圖的成像是將 pump 及 probe 在前端重合再延遲 70 ns 開 CCD 10 ns 所得的 結果。我們觀測到最少在30 ns 中所產生的 CN 產物，所以成像只見到變寬而沒有觀察 到兩塊飛散的區域。但基本上，在重疊焦點處(308 及 388 nm 產生的焦點非常接近)是 可以在單發時觀測到訊號。

如果有2 ns pump 及 probe lasers，及µm3大小的重疊焦點(已有 2 µm spot size, 15 mm f. l. lens for high laser focusing, Optics for Research)，依照目前測試結果執行 FISM 是確實可行的。ICN 在 308 nm 的吸收強度只有峰值(246 nm)的 1/10，CN recoil velocity 在峰值的光解也會由1800 m/s (308 nm 光解)增加，可在較早時間觀測，其螢光強度也 會成指數函數增加。 總結來說，高品質聚焦鏡頭(現有的)可提供µm 大小的焦點，短脈衝(2 ns) pump 及 probe lasers 可保證在 2 ns 內產生激發態產物，如此 CCD 可在更早的時序(30 ~ 50 ns) 開閘2 ns 觀測到螢光成像，而 Newton 球殼層厚度可控制在µm 尺度。觀測 100 µm 半 徑的成像，可達成能量解析度改善的目標。值得注意的是雷射能量要集中在ns 尺度而 其時間跳動(jitter)應小於 1 ns，焦點空間跳動也應小於± ±1 µm，確實需要購置品質較 佳之pump 及 probe lasers。目前採購之 Nd/YAG 雷射可解決部分問題。

附件二。離子成像實驗

研究NO2 光解。REMPI 偵測 NO 產物並利用離子成像技巧執行工作。

One-color experiment :

NO

2

is first dissociated and the resulting NO is then

ionized in a multiphoton transition ((2+1) REMPI).

Both events take place within the same laser pulse.

NO

₂

+

h

ν

→ NO(

X

2

Π

) + O +

2

h

ν

→NO(

_C

2

Π

_{) + O +}

_{h →}

ν

_NO

+

₊

_e

−

_{+ O.}

)

0

,

1

(

δ

band system at ~ 365 nm:

Π

2

C

(

v

′

=

1

) ←

X

2

Π

(v

′′

=

0

) transition.

A . NO

+

TOF profile

V

R

= + 2000 V

B. NO

+

ion image

Ion lens setup

Two-color experiment:

The first laser is used to dissociate the NO

2

molecules, while the second, delayed by 100 ns, is

used to ionize the NO(

_X

2

Π

_{) fragment in a (1+1)}

REMPI

through the A

2

Σ

+

state.

The first is a frequency-tripled Nd/YAG laser (at

355 nm). The second is a frequency doubled

(β-BaBO

₃

) XeCl-pumped dye laser (at ∼226 nm).

A . NO

+

TOF profile

trace D: probe laser only

trace C: both laser on

B. (1+1) REMPI spectrum

NO resulting from the dissociation of NO

2

in the

C. NO

+

ion image

(1)

laser delay time (relative to gas pulse)

(a) 400µs (b) 450µs (c) 500µs

When the molecules expand supersonically in a molecular beam,

clusters as well as monomers are produced and with adiabatic

cooling of molecules cluster formation cannot be completely

suppressed.

The formation of the clusters is greatest in the middle of the

molecular beam pulse, while monomers dominate in the leading

edge of the beam pulse.

Signals due to clusters have a low kinetic energy release and show

up as“blobs”in the middle of the image.

Problems due to cluster formation are avoided by firing the lasers in

the early part of the molecular beam pulse.

(2)

ion lens voltage

(a) VR = +2000V (2) VR = +1000V

The radius R in the 3D fragment distribution behaves as

, where N a constant magnification factor, T is

the kinetic energy of the fragment, q fragment charge, V

1/2

N[T/(qVR)]

R

=

R

repeller

voltage.

Particles having the same kinetic energy appear at the same radius

in the image, independent of the mass of the particle.

附件三

Quantum theory of ICN photodissociation: Density matrices of

photofragments from a parallel transition

Kuo-mei Chena), Kuang-chien Chen, and Tsung-hang Yang

Department of Chemistry, National Sun Yat-sen University, Kaohsiung, Taiwan, Republic of China

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

A quantum treatment on ICN photodissociation from an initial parallel transition to the asymptote ) 0 0 (Ω′= ←Ω ′′= , I(2 _1/2) 2 1

₎

(

Σ J′M′N

〉

+ P CN + is presented. Density matrices of both photofragments are derived and explicit expressions of the state multipoles in terms of the angular momentum coupling coefficients and the rotation-bending factors have been obtained. The present theoretical framework provides a foundation to study photofragments with non-null electronic and/or spin angular momenta. To investigate the angular momentum polarizations phenomena, these density matrices can play a prime role in laser-based detections of state-selected photofragments.

I. INTRODUCTION

To perceive the nature of photodissociation processes of linear triatomic molecules, Morse and Freed1 have developed a quantum framework which takes the dynamic axis switching effect into account. Their work was highlighted by introducing the rotation-bending factors to predict the rotational state population distribution and the state-resolved anisotropy parameters. To analyze the abundant experimental results on photofragments with non-null electronic (including spin) angular momenta from the photodissociation of triatomic molecules, the classical work by Morse and Freed should be slightly modified to account for additional degrees of freedom.

In a quantum treatment on H2O photodissociation, Balint-Kurti2 has established the

complete basis functions of both the ground and the asymptotic states, including the electronic and spin angular momentum state vectors of OH and H photofragments. A unique feature in this theme is that the ground state basis functions are defined in the Jacobi coordinate system, which is the natural choice to describe the scattering event of photofragments in the repulsive excited state. In other words, Balint-Kurti had to expand an initial state vector in terms of a large number of radial and angular basis functions defined in the Jacobi coordinate system. Although the calculation of the rotation-bending factors was avoided in his approach, additional complications could arise when trying to establish the legitimate ground state wave function.

In a seminal exposition, Dixon3 has presented a bipolar harmonics formalism to elucidate

the vector correlation between photofragment rotational and translational motions. The present work focuses on a theoretical framework of ICN photodissociation that leads to a quantum version of Dixon’s theory. The link between bipolar moments3 and the quantum mechanical coupling coefficients (angular momentum and rotation-bending) is sought in our treatment. In the past, Chen and Yeung4 have demonstrated that the density matrix theory is a powerful technique to deal with the complicated two-photon dissociation process of diatomic molecules. From their results, it is evident that a quantum mechanical framework parallel to the semiclassical Dixons’s theory should be based on a density matrix formalism. Similar approaches that exploit the language of density matrix in the treatment of photodissociation processes have been adopted by Baugh and co-workers,5 as well as in the work by Underwood and Powis6 in more recent studies.

Much experimental studies on ICN photodissociation in the A-band absorption have been implemented, including the rotational alignment measurements of CN by polarized laser-induced fluorescence (LIF) spectroscopy,7,8 the population determination of CN fine-structure components,9 the orientation measurements of CN by circularly polarized LIF spectroscopy,10,11 and the vector correlation measurements by high-resolution Doppler spectroscopy.12-15 Thus, a complete treatment on ICN photodissociation which considers all the angular momenta involved in the excitation photon, the precursor, and photofragments is essential to rationalize these experimental data.

It is well-known that there are at least three interacting potential energy surfaces

contributing to the A-band absorption of ICN.13,14,16-21 In the present work, we only consider a parallel transition Ω′=0 ← Ω ′′=0 induced by a plane-polarized photolysis laser and the evolution of the excited state density matrix to those of the photofragments. The corresponding asymptotic limit of the Ω′=0 excited state is

)

(

1₂ 2 / 1 2 , CN ) (

I P + Σ+ J′M′N

〉

,13,14,16-21 where Hund’s case (b) basis functions for CN will be utilized. The nonadiabatic transition from Ω′=0 to Ω′=±1 is not considered in the present treatment. The case of the coherent perpendicular transition, Ω′=±1 ← Ω ′′=0 of ICN will be published elsewhere.

In Sec. II, the density matrix of ICN in an excited state Ω′=0 is derived. The density matrices of CN and I photofragments are presented in the following two sections. For a primitive determination of state multipoles by the translational spectroscopic techniques, we will provide explicit formulae of the angular distribution and anisotropy parameters of CN and I photofragments in Sec. V.

II. DENSITY MATRIX OF ICN EXCITED STATE Ω′=0

Because ICN exhibits a large spin-orbit interaction, the only conserved internal angular momentum is the projection of the total electronic (orbital + spin) angular momentum

along the molecular axis. Various electronic states are designated by this quantum number throughout the present work. To illustrate the relationship between reference systems of the bound and the repulsive states, a schematic diagram is depicted in Fig.

22 ) (=Λ+Σ Ω Ω 5

1, where an instantaneous configuration of ICN that executes a small amplitude bending motion is chosen. For an unpolarized ensemble of ICN molecules, the ground state density matrix23 ρ_g can be expressed as

=ρ_g (Q g CN ρ C = + k J

}

{

[ ] 0, 0, , , ) , ( _{1 2} 1

∑

′′ − _{Ω ′′= ′′ ′′ ′′ Ω ′′= ′′ ′′ ′′} ′′ ′′ ′′ M g Q Q ν K ν K J J K M J K M ρ , (1)

where ρ 1,Q2) is the density matrix of the two stretching vibrational motions, ν,K ′′ is a two-dimensional isotropic harmonic oscillator state vector with a projection K ′′ along the molecular axis.1,24 J′′K′′M ′′ in the above equation is the well-known symmetric top state vector22 and is given by ) ( )

8 1 2 ( 1/2 * 2 αβγ π J K M D J M K J′′ ′′ ′′ = ′′+ ′′_{′′ ′′} 2 / 1 ) 1 2 ( ] [J ≡ J + , where is a Wigner rotation matrix. The notation will be adopted in this work.

)

αβγ

J M

D ′′_′*_{′K ′′}(

It is anticipated from Dixon’s theory3 that the density matrix of a selected fragment, for example , should exhibit the form

) , ( ) , ( ) , , , ( ) ( 1 00 ( ) ( ) CN CN Q

[

k J l l T J J T l l k q k q kqll J ′ ′ ′ ′ ′ ′ ₋ ′ ′

∑

ρ ρ ρ

]

2 1 1 _{( , ) ( , )} 0 2 ) , , , , ( _{1 2} 1 2 ( ) ( 2) 2 0

(

)

∑

′ ′ − ′ ′ ′ − ′ ′ q kl l k q k q J J T l l T q q k k l l J k k ρ , (2)

where C is a constant, ρ_CN(Q′₁) is the vibrational density matrix of CN with a normal mode , and are state multipoles

1 Q′ (⋅ ⋅_{⋅ ⋅} 0 0 ρ ) ⋅ ⋅ 2 0 ρ 23

allowed in excitation by plane polarized photons, and is a 3-j symbol.22 In the above equation, is an irreducible tensor operator and is defined by ) (k q T 23 6

M J M J q M M k J J k J J T M M M J k q ′ ′ =

∑

− ′ ₋ ′ _{′ −} ′ ′ ′ ′ − ′ ] [ ) 1 ( ) , (

(

)

) ( . (3)

In addition, J ′M represents a ket vector of CN in a Hund’s case (b) coupling scheme22 and is given by

)

(

2 1 CN 2 1 _{( 1) [ ]} , 2 1 M M M J N J MN J M J S N M M M N S N − ′ ′ − ∑ = ′ ∑ = ′ + +

∑

− + CN 2 1 S N M NM × , (4) where NM_N and CN 2 1 S

M are the rotational and the spin state vectors of CN photofragments, respectively.

The l-th partial wave25 of the interfragmental orbital motion is represented by lµ and

) , ( ) ( ) 4 ( π 1/2 θ_SF φ_SF µ _il _{jl kR Ylµ} l = . (5)

The approximate radial wave function in the form of a spherical Bessel function , corresponds to an identity scattering matrix S in Pack’s theory

) (kR

j_l

26

on atom-diatomic molecule scattering problem. By such an approximation which neglects the inelastic scattering between I and CN, we can focus on the angular momentum correlation between the orbital and intrinsic angular momenta of photofragments. For a refined treatment, one should follow Pack’s space-fixed scattering theory26 to calculate S.

In latter developments, explicit expressions of both ρ_CN and ρ_I are derived and state

multipoles in terms of the angular momentum coupling coefficients and the rotation-bending factors are given. Thus, a quantum version of Dixon’s semiclassical results, for example Eq. (2), will be confirmed in the present work. Incidentally, all the basis functions of the ground state and the asymptotic limit are defined by Eqs. (1) to (5).

For a parallel transition induced by a plane polarized photolysis laser, the excited state density matrix

) 0 (∆Ω= e ρ of ICN is given by27,28 ) ( ˆ⋅r ⋅r =ε ρ ε ρ_{e g} † , ) , ; , ( , , ) ( ) ( ) , ; , ( 2 1 2 1 2 1 2 2 2 1 1 1 ) ( 1 1 2 1

∑

′ ′ ′ ′Ω′Ω′ ′ ′ Ω′ ′ ′ Ω′ ′′ ′′ ′ ′ = K K J Jkq k q k q e CN Q R K K T J K J K R Q Q Q F ρ ρ ν ν ρ (6)

where εˆ⋅r is a transition dipole operator and F(Q₁,Q₂;Q₁′,R) is a Franck-Condon factor between the two electronic states.29,30 The state multipole eρ_qk is given by

)

(

2 1 2 1 1 2 1 1 1 ) 1 ( ] [ ] [J k _MJ J_M k_q M M M q M J k q e − ′ − ′ ′ ′ − ′′ =

∑

′ ′ ′′ + ′ − ′ − ρ

)

ˆ

(

0

0

ˆ

,

_{2 2 2} 2

′

′

′

⋅

r

Ω′′

=

′′

′′

′′

Ω′′

=

′′

′′

′′

⋅

r

Ω′

×

J

K

M

ε

,

J

K

M

,

J

K

M

ε

† 1 1 1 1

,

J

′

K

′

M

′

Ω′

. (7)

In the case of photolysis by plane polarized photons, _t

t t r D r ( )~ ˆ⋅ = ₀ =

∑

1₀* αβγ ε r , where r~ is _t

a spherical dipole operator in a molecule-fixed frame (MFF). It is straightforward to prove that

M

K

J

,

M

K

J

′

′

′

⋅

Ω′′

=

′′

′′

′′

Ω′

₂

,

_{2 2 2}

ε

ˆ

r

0

K K K M K K J J M M J J J J

m

− ′′− ′′+ ′ ′′ ′_{′ −} ′′ _′′ ′_{′′ −} ′′_′′ Ω′ ′ ′′ = 2 _{,0 ,} 2 2 2 1 ||

(

)

(

₀

)

_{2 2} 1 0 1 ] ][ [ ) 1 ( δ δ ,

(8) 8

where the parallel transition moment

m

_|| in the MFF is defined by the matrix element 0

~ 0 ₀

|| = Ω′= r Ω ′′=

m

. From Eqs. (6)-(8), we can prove by the angular momentum algebraic manipulations27 that the excited ρ_e has the following form, that is,

K K R Q R Q Q Q F _CN e = ( , ; ′, )

m

( 1′) ( ) , ′′ , ′′ 2 || 1 2 1 ρ ρ ν ν ρ

]

[

1 1 1 1 1 1 2 1 _{0, 0,} 0 1 ] [ 3 1

_∑

(

)

_∑

′ ′ ′ ′′ ′ = Ω′ ′ ′′ ′ = Ω′ ′′ − ′′ ′′ ′ ′′    × J M M K J M K J K K J J J       ′′ ′ ′ ′′ − ′′ ′′ ′ ′′ − ′′ ′′ ′ ′ ′ ′′ − +

∑

′ ′ ′ + ′ + ′′ J J J K K J J K K J J J J J J J J J J 2 1 2 1 2 1 2 / 1 1 1 2 0 1 0 1 ] ][ ][ [ ) 1 ( ) 3 10 (

(

)(

)

2 1 2 1

}

]

[

1 1 2 1 1 1 2 1 _{0, 0,} 0 2 ) 1 ( 1 1

(

)

_{J K M J K M} M M J J M M _{Ω′= ′ ′′ ′ Ω′= ′ ′′ ′} ′ − ′ ′ ′ − ×

∑

′ ′ ,

(9) where

_{

. .

}

is a 6-j symbol. . . . . M 22

This is consistent with the fact that only population (the first sum over ₁′) and alignment (the second sum over M ′ ) can be created in an ensemble after ₁ absorbing plane polarized photons.31 The only difference between the bound-bound and the bound-continuum transitions is that a sum over the allowed J ′ and ₁ should be taken in the latter case.

) 1 , ( 2′ =J′′ J′′± J

III. DENSITY MATRIX OF CN PHOTOFRAGMENT

The density matrix of photofragments ρ_FRG is imposed by the dynamic evolution of

e

ρ on the excited state potential energy surface. Quantum mechanically, ρ_FRG is given simply by

H Hρe

=

FRG

ρ ,

(10)

where the excited state potential operator should be utilized in _{H . Obviously, it is a difficult} quantum chemistry problem. To avoid the insurmountable barrier in calculations, it is noticed that _{H is totally symmetric and the rank of a tensor operator is preserved under H . Thus,} one can rely on the conservation of the good quantum number Ω′ to judge whether the matrix elements of _{H are null or not.}

From Eqs. (9) and (10), one should examine the tensor operators

∑

′ ′ ′′ ′ = Ω′ ′ ′′ ′ = Ω′ 1 1 1 1 1 0, , 0 M M K J M K J H H and

(

)

0 2 ) 1 ( 1 1 2 1 1 1 M M J J M M ′ − ′ ′ ′ −

∑

′ ′ H H Ω′=0,J1′K′′M1′ Ω′=0,J2′K′′M1′

× . Because the rank of a tensor operator is preserved under _{H , the above tensor operators can be expanded in terms of tensor operators of the total} electronic angular momenta of I and CN, the rotational tensor operator, and the density matrix of the ground electronic state of CN. For example, the zeroth rank tensor operator should be 22

∑

Ω′= ′ ′′ ′ Ω′= ′ ′′ ′ i M M K J M K J _H H 0, 1 1 0, 1 1

∑

′ ′ ′′ ′ ′ ′′ ′ = = = 1

]

[

{

0,0 0,0 1 1 1 1 M M K J M K J j j a CN CN 2 , 1 , 0 ) ( ) ( 1 ) , ( ) 1 , 1 ( ] [ ) 1 (− = =

}

∑ ∑ + + + = − − −

∑

q k k q k q k q k _{k C T j j T J J , (11)} where j=j_I +j_CN and 2 1 CN I = j =

j represent the asymptotic limit from a Ω′=0 state. It is proven in the appendix that all the C_k,s in Eq. (11) vanish. Similar results can be proven

to hold in the case of the second rank tensor operator. Thus, the remaining expansion coefficient a can be determined by

0 , 0 0 0 0 0 Ω′= Ω′= = = = j , j a _{H H} . (12) 0 , 0 =

j can be transformed into state vectors defined in the MFF and     _{− − −} = = = = − MF CN, 2 1 2 1 MF I, 2 1 2 1 MF CN, 2 1 2 1 MF I, 2 1 2 1 2 / 1 MF (2) 0 , 0 0 , 0 j j . The matrix

element Ω′=0_H j =0,0 is easily proven to be

    _{Ω′= − − Ω′= −} = = = Ω′ − MF 2 1 2 1 2 1 2 1 MF 2 1 2 1 2 1 2 1 2 / 1 , 0 , 0 ) 2 ( 0 , 0 0_{H j H H} . (13)

From the reflection symmetry properties of the half-integer angular momentum state vector,22 one gets MF 2 1 2 1 2 1 2 1 MF 2 1 2 1 2 1 2 1 , 0 , 0 − = Ω′= − = Ω′ _H σ_{v H} MF 2 1 2 1 2 1 2 1 _, 0 ) 1 ( ) 1 ( 2 1 2 1 2 1 2 1 − = Ω′ − − = + − _H MF 2 1 2 1 2 1 2 1 , 0 − = Ω′ − = _H . (14) Thus, A j 1/2 MF 2 1 2 1 2 1 2 1 2 / 1 2 , 0 2 0 , 0 0 = = Ω′= − ≡ = Ω′ _{H H} (15-1) and 11

∑

′ ′ ′′ ′ = Ω′ ′ ′′ ′ = Ω′ 1 1 1 1 1 0 0 M M K J , M K J , _H H CN CN 1 1 1 1 2 0 , 0 0 , 0 2

[

]

1 ∑ ∑ ′ ′′ ′ ′ ′′ ′ = = = + + ′

∑

M M K J M K J j j A   _{− − + − −} = CN 2 1 2 1 CN 2 1 2 1 I 2 1 2 1 I 2 1 2 1 CN 2 1 2 1 CN 2 1 2 1 I 2 1 2 1 I 2 1 2 1 2 A

]

CN 2 1 2 1 CN 2 1 2 1 I 2 1 2 1 I 2 1 2 1 CN 2 1 2 1 CN 2 1 2 1 I 2 1 2 1 I 2 1 2 1 _{− − − − −} − CN CN 1 1 1 1 1

]

[

′ ′′ ′ ′ ′′ ′ ∑ ∑ × + + ′

∑

M M K J M K J .

(15-2)

If the CN photofragments are the only species to be monitored (not a state selected coincidence experiment), one has to take the trace of ρ_FRG with respect to the space, and the density matrix of

I j CN ρ is given by CN 2 1 2 1 ) 0 ( 0 1 CN 2 2 1 2 1 2 / 1 CN =(2) F(Q ,Q ;Q′,R)

m

|| A ρ (Q′)ρ(R)ν,K′′ ν,K′′ T ( ) ρ

]

[

1 1 1 1 1 1 2 1 CN CN 0 1 ] [ 3 1

_∑

(

)

_∑

′ ′ + + _{′ ′′ ′ ′ ′′ ′} ′′ − ′′ ′′ ′ ′′    ∑ ∑ × M J M K J M K J K K J J J       ′′ ′ ′ ′′ − ′′ ′′ ′ ′′ − ′′ ′′ ′ ′ ′ ′′ − +

∑

′ ′ ′ + ′ + ′′ J J J K K J J K K J J J J J J J J J J 2 1 2 1 2 1 2 / 1 1 1 2 0 1 0 1 ] ][ ][ [ ) 1 ( ) 3 10 (

(

)(

)

2 1 2 1    ′ ′′ ′ ′ ′′ ′ ′ − ′ ′ ′ − ×

∑

′ ′

]

[

1 1 2 1 1 1 2 1

)

(

₀2 ) 1 ( 1 1 _{J K M J K M} M M J J M M .

(16) CN

ρ can be expanded further in terms of tensor operators of CN rotational angular momentum N and partial waves of interfragmental orbital motions. To derive the expansion,

we need relationships1,22 that connect Wigner rotation matrices, CN rotational wave functions, and orbital angular momentum wave functions. They are:

)

(

)

(

4 3 1 1 2 1 1 3 1 2 / 1 2 1 1 4 3 2 1 ] [ ) 8 ( K m m J l N M m m J l N J M K J m mm m ′′ ′ ′ ′ ′ = ′ ′′ ′ π −

∑

) ( ) ( 4 2 3 1 αβγ αβγ l m m N m m D D × , (17-1) ) ( ) ( ) ( 5 3 5 5 1 3 1 αβγ αβγ φ N m m m N m m N m m D d D =

∑

′ ′ ′ , (17-2) and ) ( ) ( ) , , ( ) ( 6 4 6 7 7 6 7 2 4 2 αβγ φSF θSF χSF θ φ l m m l m m m m l m m l m m D d d D =

∑

.

(17-3) After a lengthy but straightforward manipulation of angular momentum algebra, the final results are:

]

[

1 1 1 1 1 , , ) (

∑

′ ′ ′′ ′ ′ ′′ ′ ′′ ′′ M M K J M K J K K R ν ν ρ

∑

′ − − − + + ′ + ′ _{′′ ′ ′′ ′ ′}       ′ ′ ′ ′ − = kqNll q N l J _{Z K N l J Z K N l J} J l l k N N l l N J ] [ ] [ ] [ ] ( , , , , ) ( , , , , ) [ ) 1 ( 2 _{1 1} 1 1 1 2 6 1 1 ν ν ) , ( ) , ( ( ) ) ( _{N N T l l} T_qk k_q ′ × ₋

(18-1) and

]

[

1 1 1 2 1 1 1 1 2 1 0 2 ) 1 ( , , ) (

∑

(

)

′ ′ _{′ ′′ ′ ′ ′′ ′} ′ − ′ ′ ′ − ′′ ′′ M M M K J M K J M M J J K K R ν ν ρ

)

(

₀2 ] ][ [ ] [ ] [ ] [ ] [ ] [ ) 1 ( 2 ₁ 3 ₂ 3 2 1 1 _{1 2} 1 2 2 1 2 q q k k k k l l N J J q k k l Nl l l J − ′ ′ ′ − =

∑

′ − − − ′ + + ′ 13

) , ( ) , ( ) , , , , ( ) , , , , ( 2 ) ( ) ( 2 1 2 1 2 1 2 1 N N T l l T J l N K Z J l N K Z J l N J l N k k k q k q ′ ′ ′ ′′ ′ ′′         ′ ′ ′ × ν ν ₋ , (18-2) where is a 9-j symbol. . . . . . . . . .       22

The Morse-Freed rotation-bending factor1 in the above

equation is defined by . ) ( ) ( ) ( 0 ) , , , , ( 1 , 0 0 1 1

(

)

η δ ηδ ρδ ψ δ δ ν d d _ν d m m J l N J l N K Z l_{m K}J _m K m ′′ ′ − ′′ ∞

∫

∑

₋ ′ = ′ ′′ (19)

Readers interested in the numerical calculations of Z-functions should consult the work by Morse and Freed.1

From the transformation relationships between the coupled and the uncoupled basis functions,32 we can prove that

CN CN CN 2 1 2 1 ) 0 ( 0 ) ( ) ( ) , (N N T ∑+ ∑+ T_qk ) , ( ] [ ) ( ) 1 ( ( ) 2 1 2 2 1 1/2 2 1 J J T N N k J J J _qk J N k J ′ ′       ′ ′ ′ − =

∑

′ + + + ′ . (20)

From Eqs. (16), (18) and (20), the final expression of ρ_CN is

∑

′′ ′ + + ′ + ′ + + ′ − ′ ′ = kq l lJJ N q k l J J CN Q A R Q Q Q F

m

1 1 2 1 ) 1 ( ) ( ) , ; , ( 2 _{1 2 1 ||}2 2 ₁

{

CN ρ ρ       ′ ′       ′ ′ ′′ − ′′ ′′ ′ ′ ′ ′ ′′ × − − − 2 1 1 2 1 1 1 2 2 6 1 0 1 ] [ ] [ ] [ ] [ ] ][ [ 3 1

(

)

N N k J J J l l k N N K K J J l l N J J J ) , ( ) , ( ) , , , , ( ) , , , , ( K N l J₁ Z K N l J₁ T( ) J J T( ) l l Z ′′ ′ ′′ ′ ′ _qk ′ ′ k_q ′ × ν ν ₋ 14

1 1 2 2 4 2 4 1 2 1 2 / 1 ] [ ] [ ] [ ] [ ] [ ] ][ ][ ][ [ ) 3 10 ( ) 1 ( 2 1 2 1 1 1 2 1 _{− − −} ′ ′ ′ ′ + + ′ + + ′ + ′′ + + ′ ′ ′ ′ ′ ′′ − +

∑

k k J J J J N l l q k k l lJJ J N k N l l J J J         ′ ′ ′       ′ ′       ′′ ′ ′ − ′′ − ′′ ′′ ′ ′′ − ′′ ′′ ′ × 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 1 0 2 0 1 0 1

)(

)(

)

(

J l N J l N k k N N k J J J J J q q k k K K J J K K J J

}

) , ( ) , ( ) , , , , ( ) , , , , ( K N l J₁ Z K N l J₂ T( 1) J J T( 2) l l Z ′′ ′ ′′ ′ ′ _qk ′ ′ k_q ′ × ν ν ₋ . (21)

In this quantum version of Dixon’s bipolar harmonics formalism,3 explicit expressions of state multipoles (a quantum analogy of Dixon’s bipolar moments) are obtained in terms of the angular momentum coupling coefficients and the rotation-bending factors. The anticipation envisaged from Dixon’s theory is thus confirmed. In Dixon’s bipolar harmonics formalism,3 it has been indicated that the k indices in the bipolar moments should be restricted to even values by symmetry constraints. In the case of monitoring CN photofragments, this statement is true because the reflection invariance of the electronic degrees of freedom with respect to the scattering plane is retained, after the trace of ρ_FRG is taken to remove the space. Judging from the fact that the zeroth rank tensor operator

I J CN 2 1 2 1 ) 0 ( 0 ( ) T is reflection invariant and all the state multipoles in Eq. (21) are real numbers, one can safely conclude that k indices in ρ_CN must be even integers, according to density matrix theory.23

IV. DENSITY MATRIX OF I PHOTOFRAGMENT

To derive the density matrix of the I photofragment, one has to take the trace of the complete density matrix with respect to all the allowed states in the ∑+ ′ ′

〉

2 1

,JMN space of

the CN photofragment. The starting point is to cast j =0,0 j =0,0 in Eq. (15-2) into a decoupled representation and its explicit form is given by32

[

₂1 ₂1 CN ) 1 ( 0 I 2 1 2 1 ) 1 ( 0 CN 2 1 2 1 ) 0 ( 0 I 2 1 2 1 ) 0 ( 0 2 1 ) ( ) ( ) ( ) ( 0 , 0 0 , 0 j T T T T j = = = −

]

CN 2 1 2 1 ) 1 ( 1 I 2 1 2 1 ) 1 ( 1 CN 2 1 2 1 ) 1 ( 1 I 2 1 2 1 ) 1 ( 1 ( ) T ( ) T ( ) T ( ) T ₋ + ₋ + . (22)

The spin-rotation interaction of the CN radical dictates the coupling N+j_CN =J′_CN such that its basis functions are represented by the Hund’s case (b) coupling scheme. When the spin tensor operators of CN in j =0,0 j =0,0 , ( ) (K _{J J} q ′

(Eq. (22)) are coupled to T one should fabricate tensor operators T to take this spin-rotation interaction into account. For example, one of those tensor operators in the complete density matrix,

), , ( ) (k _{N N} q ) ′ ) , ( ) ( ) (1₂1_{2 I 0}(1) 1₂1_{2 CN} ( ) ) 1 ( 0 T T N N T _qk − can be coupled to32 ) , ( ) ( ) (₂1 ₂1 _{I 0}(1) ₂1 ₂1 _CN ( ) ) 1 ( 0 T T N N T _qk − ) , ( 1 0 1 ] ][ ][ [ 3 ) 1 ( ) ( ( ) 2 1 2 1 2 2 / 1 1 I 2 1 2 1 ) 1 ( 0

(

)

T J J K J J k N N q q K k J K k T _qK K J q k _{′ ′}         ′ ′ − ′ − − =

∑

′ + + . (23)

When the trace is taken with respect to the J′_CN space, it is easily proven that

and the net result is

0 0 ) ( ] [ ) , ( TrT_qK J′ J′ = J′δ_K δ_q 0 ) ( 1 ] [ ) 1 ( ₀(1) ₂1 ₂1 _I 2 1 2 1 2

]

[

2 1 =       ′ ′ −

∑

′ + − ′ T J N N J J N J

, where a sum rule of 6-j symbols has been

utilized.33 Similarly, the contribution of Tr

[

(T₁(1)(1₂₂1)_IT₋(₁1)(₂1 ₂1)_CN +T₋(₁1)(1₂ 1₂)_IT₁(1)(1₂ 1₂)_CN)

CN

]

) , ( ) ( J′

×T_qk N N to ρ_I is null. The only non-null term is

∑

′ − − _′ J T J N ₂1 _I 2 1 ) 0 ( 0 2 1 2 / 1 ) ( ] [ ] [ ) 2 (

after the trace is taken. Thus, we can prove from Eqs. (9), (10), (15), (18), (22) and the above result that the density matrix of the I photofragment is given by

1′, R 1 I =2F(Q ,Q ρ 2 1 _J

∑ ∑

2 2 2;Q )

m

|| A 2 1 2 1 2 3 4 6 1 2 / 1 )] , , , , ( [ 0 1 ] [ ] [ ] [ ] ][ [ ) 2 ( 3 1

(

)

1 J l N K Z K K J J J l N J J N J Nl ′ ′′ ′′ − ′′ ′′ ′ ′ ′ ′′    × ± = ′ ′ − − − _ν ) , ( ) ( _{I 0}(0) 2 1 2 1 ) 0 ( 0 T l l T ×

∑

∑

± = ′ ′ ′ ′ − − − + + ′′ _{′′ ′ ′ ′ ′} − + 2 1 2 1 2 1 1 4 4 2 4 1 2 / 1 ] [ ] [ ] [ ] [ ] [ ] ][ [ ) 1 ( ) 3 1 ( N J J J Nll l N J _{J J J N l l J}       ′ ′ ′       ′′ ′ ′ ′′ − ′′ ′′ ′ ′′ − ′′ ′′ ′ × N J J l l J J J K K J J K K J J 1 2 2 1 2 1 1 1 2 2 0 1 0 1

)(

)

(

). , ( ) ( ) , , , , ( ) , , , , ( K N l J₁ Z K N l J₁ T₀(0) 1₂ 1_{2 I}T₀(2) l l Z ′′ ′ ′′ ′ ′ ′ × ν ν (24)

For an unpolarized ensemble of ICN molecules photolyzed by the plane polarized photons, iodine photofragments in the state are unpolarized from a parallel transition to a single

repulsive state, according to the above equation.

2 / 1 2 P 0 = Ω′

V. ANGULAR DISTRIBUTION AND ANISOTROPY PARAMETERS

A primitive determination of the state multipoles in Eqs. (21) and (24) is a translational spectroscopic measurement of angular distributions of photofragments and their anisotropy parameters. Assuming the particle-counting detector is polarization insensitive (no state

selection), one can express the efficiency operators in the space-fixed frame as4,34,35 ) , ( ) ( 0 0 0 ] ][ ][ ][ [ ) 1 ( _{0 0}(0) CN

(

)

D T J J k l l k l l J _qk kq l l J q l _{′ ′} ′ _{′ ′} − =

∑

′ ′ + _φθχ

ε

†_T †

(25-1) ) , ( ) (k _{l l} q ′ − and I 2 1 2 1 ) 0 ( 0 0 2 / 1 I ( 1) 2 [ ][ ][ ]

(

_{0 0 0}

)

D ( )T ( ) k l l k l l _qk kq l l q l _φθχ

ε

=

∑

− ′ ′ ′ + _† T † .

(25-2) ) , ( ) (k _{l l} q ′ −

The angular distribution of CN photofragments (no state selection) is

) ( Tr ) 8 ( ) ( 2 1 _{CN CN} CN θ = π − ρ

ε

I 2 1 2 1 2 4 2 6 1 ₀ [ ( , , , , )] 1 ] [ ] [ ] [ ] ][ [ 3 1

(

)

2 1 1

{

Z K N l J K K J J l N J J J C N J J Nl ′ ′′ ′′ − ′′ ′′ ′ ′ ′ ′′ =

∑ ∑

± = ′ ′ − − _ν

)

(

₀1 ] [ ] [ ] [ ] ][ [ ) 3 10 ( ) 1 ( 2 4 2 4 1 4 1 2 / 1 2 1 2 1 K K J J N J J J J N J N J l Nl J J ′′ − ′′ ′′ ′ ′ ′ ′ ′′ − +

∑

∑

± = ′ + ′′ ′ ′ ′ 1 2 2 1 2 1 1 2 2 0 0 0 2 0 1

)(

)

(

_    ′ ′ ′       ′′ ′ ′ ′ ′′ − ′′ ′′ ′ × N J J l l J J J l l K K J J , ) (cos ) , , , , ( ) , , , , (ν K N l J₁ Z ν K N l J₂ P₂ θ

}

Z ′′ ′ ′′ ′ ′ × (26)

where P₂(cosθ) is a second-order Legendre polynomial and θ is the angle between the recoil direction and the Z-axis of the space-fixed frame (the polarization direction of the photolysis photons). In deriving the above equation, we have employed the orthogonality relationship Tr[T_q(k)(J,J)T_q(_′k′)(J,J)†]=δ_kk′δ_qq′. Eq. (26) can be cast into the well-known angular distribution function for one-photon photodissociation processes, that is,

)] (cos 1 [ 4 ) ( β ₂ θ π σ θ P

I = + , where σ is the absorption cross section and β is an anisotropy parameter. Explicit expression of β_CN can be obtained from Eq. (26) easily. According to Eq. (26), the population distribution of CN photofragments in ∑+;N±₂1,M′N ₂1

〉

can be represented as 6 1′] [N ′′, J 4 [ N]− , Jl, ′ J 2 1 2 1 2 4 2 1 )] , , , ( [ 0 1 ] [ ] ][ [ ) 1 ( 2 ) (

(

)

1 J l N K Z K K J J l J N N P Nl J ′ ′′ − ′′ ′′ ′ ′′ + = + − − ′

∑

ν (27-1) and 2 1 2 1 2 6 1 2 1 )] , , ( [ 0 1 ] [ ] ][ [ 2 ) (

(

)

1 N K Z K K J J l J J N N P Nl J ′ ′′ ′′ − ′′ ′′ ′ ′ ′′ = − − ′

∑

ν . (27-2)

The population ratio of the two fine-structure components F₁ and F₂ ( )

2 1

± = N 22

is just the ratio of their statistical weighting factors. Any deviation from this predicted value must be attributed to the simultaneous parallel and perpendicular transitions of ICN and the nonadiabatic crossing of wave packets to a different potential energy surface.

Similarly, the angular distribution of I photofragments can be obtained from Eqs. (24) and (25-2). It turns out that I_I(θ) is identical to I_CN(θ) (Eq. (26)) as expected. For a characteristic distribution of kinetic energy release of photofragments at a given photolysis wavelength, I_I(θ)=I_CN(θ) infers that a coincident pair of I and CN shares an identical angular distribution, in coping with the conservation of linear momentum and energy.

For laser-based detection schemes, for example LIF, resonance-enhanced multiphoton

ionization, or imaging techniques, Eqs. (21) and (24) provide a loaded arsenal to tackle all the possible alternatives. Density matrix theory is perfectly suitable to treat the subsequent photon excitation or ionization processes.

VI. CONCLUSIONS

A quantum framework that analyzes the photofragmentation processes of a real system, specifically the ICN photodissociation, has been implemented. In the present work, we focus on the parallel transition to a single excited state of ICN. Density matrix theory has been employed to follow the dynamics from the initial excitation of ICN to its asymptotic limit CN ) 0 (∆Ω= ) ( I 2P_1/2 + ) ,

(∑+ J′M′N ₂1

〉

. Density matrices of both photofragments are derived and explicit expressions of the state multipoles in terms of the angular momentum coupling coefficients and the rotation-bending factors have been obtained. Our results are in total conformity with Dixon’s semiclassical treatment based on bipolar harmonics formalism. For laser-based detection of state-selected photofragments, the presently derived density matrices are extremely useful as a guideline in initiating new experiments, especially in angle-resolved measurements. From this quantum treatment on photodissociation processes of polyatomic molecules, it is evident that dynamics of all the correlated angular momenta should be taken into account in future studies.

ACKNOWLEDGMENTS

One of the authors (KMC) acknowledges useful comments from Professor K. F. Freed on the transformation relationships between rotational and orbital angular momentum wave functions. This research was supported by the National Science Council of the Republic of China.

APPENDIX: PROOF OF NULL CONTRIBUTION OF T_q(k)(j =1, j=1)T₋(_qk)(J,J) TO

H Hρe

In this appendix, we will prove that all the C in Eq. (11) are null. First, the contribution of s , k

]

1′ M

[

) 1 , 1 ( _{1 1 1} ) 0 ( 0 = =

∑

′ ′′ ′ ′ ′′ i M K J M K J j j T to HρeH is examined. In the MFF, T₀(0)(j =1, j =1) is MF MF 2 / 1 3 1_{) 1 1} ( m m m

∑

. From Eq. (11), it can be proven that

]

[

0 0 ( 1, 1) Tr ₀(0) 0 = Ω′= Ω′= T j = j = C _{H H} MF MF 2 / 1 3 1 10 0 0 10 ) ( _HΩ′= Ω′= _H = 2 MF 2 1 2 1 2 1 2 1 MF 2 1 2 1 2 1 2 1 2 / 1 3 1 2 1 , 0 , 0 ) ( Ω′= − 〉 + Ω′= − 〉 = _{H H} = 0,

(A1)

where Eq. (14) has been utilized.

Secondly, the contribution of T₀(1)(j=1,j =1)T₀(1)(J,J) to _Hρ_eH is examined. If is transformed to the MFF and the Wigner rotation matrices are purposely combined to form the tensor operator

) 1 , 1 ( ) 1 ( 0 j = j = T 1 1 1 1K M J K M

J′ ′′ ′ ′ ′′ ′ , one can prove that

) , ( ) 1 , 1 ( ₀(1) ) 1 ( 0 j j T J J T = = 2 1 2 1 1 2 2 1 1 2 1 2 1

(

)(

)(

1

)

0 1 0 1 1 1 ] [ ] [ 3 ) 1 ( 2 1 2 1 2 1 M m m J J m m J J m m J J q q m m m m J ′ − ′ − − ′ − =

∑

+ − − 1 1 1 1 MF 2 MF 1 2 1 1 1 1 _{1 1} 1

)

(

)

(

q q J K M J K M K K q J J K K q J J _{′ ′′ ′ ′ ′′ ′} ′′ − ′ ′′ − ′ × . (A2)

When the orthogonality relationship of the Wigner rotation matrices is recalled,22 the special choice which combines the rotational tensor operator in the form J₁′K′′M₁′ J₁′K′′M₁′ is justified. From Eqs. (11) and (A1), one gets

]

[

0, 0, ( 1, 1) ( , ) Tr 31/2 _{1 1 1 1 0}(1) ₀(1) 1 1 J J T j j T M K J M K J C M = = ′ ′′ ′ = Ω′ ′ ′′ ′ = Ω′ =

_∑

′H H 2 1 2 1 1 2 2 1 1 2 1 2 2 / 3

(

)(

)(

1

)

0 1 0 1 1 1 ] [ ] [ ) 1 ( 3 2 1 1 2 1 M m m J J m m J J m m J J m m M m m J ′ − ′ − − ′ − =

∑

′ − − 0 10 0 0 10 0 1 MF MF 2 1

)

(

_{′′ −} ′ _′′ Ω′= Ω′= 〉 = × _{H H} K K J J . (A3)

Similarly, one can prove that the contribution of to the second rank tensor operator

) , ( ) 1 , 1 ( ₀(1) ) 1 ( 0 j j T J J T = = H 1 1 1 1 2 1 0 2 ) 1 (

(

)

1 1 _{K M} M M J J M M _{′ ′′ ′} ′ − ′ HΩ′=0,J1′K′′M1′ Ω′=0,J ′ ′ −

∑

′ ′ is

also null. Identical procedures can be employed to confirm that in Eq. (11) is null. Thus, the only non-null expansion coefficient a is given by

2 C 2 2 A . 22

FIGURE CAPTION

FIG. 1. A schematic diagram of the reference coordinate systems of ICN. Euler angles which connect the space-fixed frame with the molecular axis of ICN at the equilibrium configuration, the CN axis, and the recoil direction Rˆ are (α,β,γ), (α′,β′,γ′), and (φSF,θSF,χSF), respectively.

1

M. D. Morse and K. F. Freed, J. Chem. Phys. 74, 4395 (1981).

2

G. G. Balint-Kurti, J. Chem. Phys. 84, 4443 (1986).

3

R. N. Dixon, J. Chem. Phys. 85, 1866 (1986).

4

K. Chen and E. S. Yeung, J. Chem. Phys. 72, 4723 (1980).

5

C. D. Fuglesang, D. A. Baugh, and L. C. Pipes, J. Chem. Phys. 105, 9796 (1996).

6

J. G. Underwood and I. Powis, J. Chem. Phys. 113, 7119 (2000).

7

G. E. Hall, N. Sivakumar, and P. L. Houston, J. Chem. Phys. 84, 2120 (1986).

8

M. A. O’Halloran, H. Joswig, and R. N. Zare, J. Chem. Phys. 87, 303 (1987).

9

H. Joswig, M. A. O’Halloran, R. N. Zare, and M. S. Child, Faraday Discuss. Chem. Soc. 82,

79 (1986).

10

E. Hasselbrink, J. R. Waldeck, and R, N. Zare, Chem. Phys. 126, 191 (1988).

11

J. F. Black, E. Hasselbrink, J. R. Waldeck, and R. N. Zare, Mol. Phys. 71, 1143 (1990).

12

I. Nadler, D. Mahgerefteh, H. Reisler, and C. Wittig, J. Chem. Phys. 82, 3885 (1985).

13

J. F. Black, J. R. Waldeck, and R. N. Zare, J. Chem. Phys. 92, 3519 (1990).

14

J. F. Black, J. Chem. Phys. 98, 6853 (1993).

15

S. W. North, J. Mueller, and G. E. Hall, Chem. Phys. Lett. 276, 103 (1997).

16

S. Yabushita and K. Morokuma, Chem. Phys. Lett. 175, 518 (1990).

17

H. Guo and G. C. Schatz, J. Chem. Phys. 92, 1634 (1990).

18

Y. Wang and C. X. W. Qian, J. Chem. Phys. 100, 2707 (1994).

19

Y. Amatatsu, S. Yabushita, and K. Morokuma, J. Chem. Phys. 100, 4894 (1994).

20

J. M. Bowman, R. C. Mayrhofer, and Y. Amatatsu, J. Chem. Phys. 101, 9469 (1994).

21

J. Qian, D. J. Tannor, Y. Amatatsu, and K. Morokuma, J. Chem. Phys. 101, 9597 (1994).

22

R. N. Zare, Angular Momentum (Wiley, New York, 1988).

23

K. Blum, Density Matrix Theory and Applications, 2nd ed. (Plenum, New York, 1996).

24

C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover, New York, 1975), p. 32

25

L. D. Landau, and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon, Oxford, 1977).

26

R. T. Pack, J. Chem. Phys. 60, 633 (1974).

27

K. Chen and E. S. Yeung, J. Chem. Phys. 70, 1312 (1979).

28

K. Chen and J. Chang, J. Phys. Chem. 101, 2525 (1997).

29

Y. B. Band and K. F. Freed, J. Chem. Phys. 68, 1292 (1978).

30

Y. B. Band, M. D. Morse, and K. F. Freed, J. Chem. Phys. 68, 2702 (1978).

31

A. J. Orr-Ewing and R. N. Zare, Annu. Rev. Phys. Chem. 45, 315 (1994).

32

U. Fano and G. Racah, Irreducible Tensorial Sets (Academic, New York, 1959).

33

D. M. Brink and G. R. Satchler, Angular Momentum (Clarendon, Oxford, 1968).

34

S. Devons and L. J. B. Goldfarb, in Handbuch der Physik, ed. S. Flügge (Springer, Berlin, 1957), Vol. 42.

35

D. A. Case, G. M. McClelland, and D. R. Herschbach, Mol. Phys. 35, 541 (1978).

α

N

φ

R

I

_φ

,

)

(θ

SF SF ,

(β )

α

θ

C

δ

',

(β

'

)

CM

CN