National Sun Yat-sen University Institutional Repository:Item 987654321/30415

行政院國家科學委員會專題研究計畫 成果報告

光解過程電子極化之理論及離子成像實驗研究(3/3)

計畫類別： 個別型計畫 計畫編號： NSC93-2113-M-110-001- 執行期間： 93 年 08 月 01 日至 94 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 報告類型： 完整報告 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 94 年 9 月 30 日

(二)中英文摘要及關鍵詞

關鍵詞：角動量極化，離子成像，甲基光譜學

本研究計畫利用離子成像技巧研究分子光解動態學。研究重點包括成像實驗、成像投 影理論架構發展、角動量極化與離子成像圖樣的關連以及甲基的共振增強多光子電離光 譜。

Keywords：angular momentum polarization, ion imaging, spectroscopy of the methyl radical Ion imaging techniques were employed to study molecular photodissociation dynamics. The present project focuses on the experimental endeavor on ion imagings, framework developments on ion image projections, theoretical studies on ion image patterns with angular momentum polarizations of detected species and the resonance-enhanced multiphoton ionization spectroscopy of the methyl radical.

(三)報告內容 I. 前言 自 1992 年以來，主持人在螢光成像實驗及理論研究領域，已發表 15 篇學術論文，其 中刊登於國際一流期刊的原創性論文共 11 篇。在分子光解動力學領域，主持人發展了獨 創的實驗方法，理論分析架構，產物向量對應的偵測技術與分子動力行為之間的關係，緊 密結合理論與實驗，是自成一家之言的研究工作。本期研究工作重點包括持續離子成像實 驗並逐步微調及熟練實驗技術，針對欲偵測的甲基產物，發展必要的共振增強多光子電離 光譜學理論架構，在成像投影理論架構發展亦有全新的成果。 II.研究目的 分子光解過程是眾多化學反應中最單純的，但是多原子分子的光解仍然極為複雜而難 以完整敘述其動力過程[1]。自從 70 年代採用調頻雷射偵測產物的能態佈居，角動量極化， 角分佈，速度分佈，多重解離通道分支比例等特性以來，在實驗技術上有極大進展，包括 雷射引發螢光(LIF)[2,3]，共振增強多光子電離(REMPI)[2,3]，飛秒技巧[4]，飛行時間光碎 片 平 移 光 譜 學 (TOF−PTS)[5] ， Rydberg 標 示 技 巧 [6,7](Rydberg−taggng) 及 成 像 技 術 [8,9](ion-imaging and fluorescence imaging)等有力研究工具，獲得大量極有意義的動力學資 料，對促進了解控制化學反應進行的勢能面，角動量耦合及其動力學具深遠影響。

成像技術是一種能同時提供多重動力資料的研究方法，在採用速度投射(velocity mapping)技巧[10]將飛散的產物離子(經由 REMPI 產生)重新聚焦於多通道平板偵測器 (multichannal plate detector, MCP)而大幅降低了成像的散光，從而提昇其速度解析能力，離 子成像已成為研究基本化學反應的一種重要實驗技術。本計畫就是利用我們自行建造的一 套離子成像裝置進行實驗研究，詳情請見第三節。

[參考資料]

[1] R. Schinke, Photodissociation Dynamics (Cambridge Univ. Press, Cambridge, 1993).

[2] Atomic and Molecular Beam Methods, ed. G. Scoles (Oxford Univ. Press, New York, 1998), Vol. I.

[3] The Chemical Dynamics and Kinetics of Small Radicals, eds. K. Liu and A. Wagner (World Scientific, Singapore, 1995).

[4] L. R. Khundkar and A. H. Zewail, Annu. Rev. Phys. Chem. 41, 15 (1990).

[5] A. M. Wodtke and Y. T. Lee, in Molecular Photodissociation Dynamics, eds. M. N. R. Ashfold and J. E. Baggott (The Royal Society of Chemistry, London, 1987).

[6] L. Schnieder, W. Meier, K. H. Welge, N. M. R. Ashfold, and C. M. Western, J. Chem. Phys.

92, 7027 (1990).

III. 研究方法 － 實驗部份 主持人實驗室建造的離子成像裝置及離子透鏡組裝示意圖(圖-1 及-2)如附。本計畫工作 簡述如下。 (a) 2050mm (b) (c) _(d)

Fig. 1. Side view of apparatus. This machine consists of (a)source chamber, (b) photolysis chamber, (c) TOF tube and (d) detector chamber.

Stainless rod 13.5 mm length ceramics 9.3 mm length ceramics 57.8 mm length ceramics

Fig. 2. The ion lens assembly. (a) the exploded view. (b) the assembly drawing. A 57.8 mm length ceramics is slid into the stainless rod. The other two short ceramics are also slid into the long ceramics to form the ion lens assembly. The two short ceramics are spacers to keep the predetermined separation (15 mm).

真空系統：

整個離子成像裝置包含了四個部分，分別是（A）分子束源腔室（B）光解 腔室（C）飛行時間管（D）偵測腔室。在實驗進行之前，須先以差別式抽氣法 來使這些腔室都先達到一定程度的高真空。

步驟：

1. 開啟連接至分子束源腔室的 mechanical pump (Alcatel 2033, 單相 220V)。 2. 開啟連接至光解腔室的 mechanical pump (Alcatel 2033,三相 220V)。

3. 開啟連接至偵測腔室的 mechanical pump (Varian SD450,三相 220V）。

4. 對於各腔室所連接的 Solenoid valve 及 Gate valve 等真空閥門通以~ 5 bar 的 氮氣，以利其開啟。

5. 開啟各腔室所連接的 Solenoid valve 及 Gate valve，使得各腔室都先達到 ~10-2 torr 的真空度。

6. 開啟 diffusion pump 的冷卻水開關。

7. 開啟 turbo pump 的冷卻水開關。

8. 開啟連接至分子束源腔室的 diffusion pump (Varian VHS-10），亦即，打開其加

熱器所連接的變壓器之開關。

9. 開啟連接至光解腔室的 diffusion pump (Alcatel crystal 100)。 10. 開啟連接至偵測腔室的 turbo pump (Alcatel 400 HPC)。 11. 開啟離子式真空計，用以監控各腔室的真空度。 反應物的產生： 本 實 驗 的 反 應 物 為 C6H6(benzene) ， 在 室 溫 下 為 液 態 ， 利 用 脈 衝 束 閥 (General valve）可以產生脈衝的分子束。 步驟： 1. 將 benzene(C6H6)液體置入一不鏽鋼瓶。 2. 以冷凍-抽氣-溶解(freeze-pump-thaw)的循環方式來純化 benzene(C6H6)液體。 3. 在不鏽鋼瓶的上端開口置入一鋼管（深入液面），並以此通入~1bar 的氦氣。 4. 將不鏽鋼的側端開口連接至脈衝束閥（General valve）。 5. 將脈衝束閥固定在位於分子束源腔室的一部 XYZ 三維平移台上。

雷射系統： 本實驗利用波長 248 nm 的 KrF 準分子雷射作為激發光源,同時觀測苯離子 C6H6+(1+1 MPI)以及苯光解產生的碎片離子。 步驟： 1. 藉由反射鏡將 KrF準分子雷射導引至光解腔室，使其通過 repeller 與 extractor 2. 兩塊電極板之間的中心區域。 3. 利用 iris aperture 來選取雷射束的中心部份(直徑約一公分)後再通過聚焦 鏡，使其聚焦在 repeller 與 extractor 1 兩塊電極板之間的中心區域。 時序的控制： 本實驗的脈衝束閥、KrF 雷射兩部分需要作時序的控制。 步驟：

1. 將 digital delay/pulse generator（DG 535）對脈衝束閥的 trigger 時間點訂為 T0，

trigger 的時間長度為 200 µs。

2. 將對 KrF 雷射的 trigger 對 T0延遲 1.2 ms。

離子透鏡：

本實驗所使用離子透鏡包含了四片電極板，分別為(1) repeller (2) extractor adjacent to repeller (R1) (3) extractor adjacent to ground (R2) (4) ground，它們可 以將所產生的離子加速，同時也將位於不同位置而具有相同速度的離子重新聚 焦在偵測器上。 步驟： 1. 對 repeller electrode 施加 +1000V 的高壓。 2. 對 R1 electrode 施加 +794V 的高壓。 3. 對 R2 electrode 施加 +570V 的高壓。 4. 將 ground electrode 接地。

偵測系統:

本 實 驗 所 採 用 的 偵 測 器 包 含 (1) dual microchannel plate and phosphor screen，藉著對其內的兩塊 MCP ( Vi / V0 )施加電壓，可以將離子訊號轉換成電 子訊號，所以如將 phosphor screen 用作陽極(anode)，則該訊號便可被收集到。

步驟： 1. 將偵測器的 V0輸入端接地。 2. 在偵測器的 Vi輸入端先施加 −100 V 的電壓，然後以兩分鐘增加 −100 V 的 速度，將電壓緩緩增加至−1000 V。 3. 將偵測器 Va端的輸出訊號送至電流放大器，靈敏度設定在 200nA/V 然後再 將電流放大器的輸出訊號送至示波器作觀測。

結 果 ☉ 電子偵測 ☉ 陽離子偵測 A . 不可分離之質量峰 B. 排斥板電壓 v.s. 飛行時間 C. 速度映射 ☉ 電子偵測 背景壓力 : 2 × 10 −6 Torr 排斥板電壓 : −1500 V 微孔道板電壓 (後板) : +100 V 結果如圖一所示 ☉ 陽離子偵測 A . 不可分離之質量峰 背景壓力: 2 × 10 −6 Torr 排斥板電壓: + 1000 V 微孔道板電壓 (前板) : −1000 V 結果如圖二所示 B. 排斥板電壓 v.s. 飛行時間 背景壓力: 2 × 10 −6 Torr 排斥板電壓: +1000 V, + 2500 V, + 4000 V 微孔道板電壓(前板): −1500 V 結果如圖三、四、五所示 C. 速度映射

吸引板電壓(前板): +794 V 吸引板電壓(後板): +570 V 微孔道板電壓(前板): −1500 V 結果如圖六所示 2. 硬聚焦 背景壓力: 2 × 10 −6 Torr 離子透鏡電壓 : 排斥板電壓: + 2000 V 吸引板電壓(前板): + 1504 V 吸引板電壓(後板): + 560 V 微孔道板電壓(前板): − 1500 V 結果如圖七所示

圖一 電子偵測

飛行時間 ~ 1.7 µs

封包寬度 ~ 5 µs

圖二 陽離子偵測

不可分離之質量峰

上圖 : 100 次總合平均

下圖 : 單一事件

圖三 陽離子偵測

排斥板電壓 : + 1000 V

飛行時間 ~ 13 µs

圖四 陽離子偵測

排斥板電壓 : + 2500 V

飛行時間 ~ 10 µs

圖五 陽離子偵測

排斥板電壓 : + 4000 V

飛行時間 ~ 5 µs

圖六 陽離子偵測

軟聚焦

飛行時間 ~ 10 µs

封包寬度 ~ 20 µs

圖七 陽離子偵測

硬聚焦

上圖 : 100 次總合平均

下圖 : 單一事件

飛行時間 ~ 5 µs

封包寬度 ~ 10µs

Multiphoton ionization detection of photodissociation fragments: NO from NO2

NO2 one-color experiment:

NO2 is first dissociated and the resulting NO is then ionized in a multiphoton transition. Both events take place within the same laser pulse.

NO2 + hν → NO(Χ2Π) + O + 2hν → NO(C2Π) + O+ hν → NO+ + e− + O In this case, the excess energy at 382 nm is ∼1050 cm−1, thus only the ν= 0 level of the ground state of NO is accessible.

A . NO

+

TOF profile

1. V

R

= +2000 V

TOF peak ≅ 5.66 µs

traceC : molecular beam on

traceD : molecular beam off

2. V

R

= +1000 V

3. V

R

= +500 V

TOF peak ≅ 11.1 µs

The time-of-flight t behaves as t ∝ (m/qV

R

)

1/2

, which is very

helpful for mass identification and image magnification.

B. NO

+

ion image (polarized laser)

1. Fix V

R

at +1000 V, V

E2

at 0 V, find V

E1.

Ion lens setup

Ion lens setup

V

R

= +1000 V, V

E1

= +727 V, V

E2

= 0 V.

2. Fix V

R

at +1000 V, V

E2

at +727 V, find V

E2.

Ion lens setup

Ion lens setup

V

R

= +1000 V, V

E1

= +727 V, V

E2

= +200 V.

Ion lens setup

V

R

= +1000 V, V

E1

= +727 V, V

E2

= +400 V.

When V

E2

changes from +200 V to +400 V, the middle parts

(zero-kinetic energy) of the image become line-shaped.

Velocity is not mapping.

3. Fix V

R

at +2000 V, V

E2

at 0 V, find V

E1.

Ion lens setup

Ion lens setup

V

R

= +2000 V, V

E1

= +1534 V, V

E2

= 0 V.

When V

R

= +2000, the TOF is too short, thus NO

+

cannot fly

out of the middle regions.

The resolution of the image is too low to distinguish NO with

different KER.

C. NO

+

ion image (unpolarized laser)

Ion lens setup

Ion lens setup

V

R

= +2000 V, V

E1

= +1534 V, V

E2

= 0 V.

When using the unpolarized laser, the spatial distributions of

NO

+

is more diffused, thus the resolution of the image is also

理論架構發展部分：

a. 光解過程的電子角動量極化現象

已發表論文“Electronic angular momentum polarization of atomic fragments in diatomic photodissociation: Correlation between states in the molecular and the asymptotic regions”, JCCS 49, 723 (2002)。如附錄一。

b. 甲基共振增強多光子電離光譜學(REMPI)

如果離子成像偵測的光解產物是甲基並使用 REMPI 電離此一自由基，必須對 其 REMPI 光譜學進行徹底的瞭解。主持人已發表兩篇論文，分別是：

1) “Rotational line strengths of the v2-active two-photon transitions of the methyl radical”, J. Chem. Phys. 119, 7163 (2003)。如附錄二。

2) “Resonance-enhanced multiphoton ionization spectroscopy of CH3 and CD3. Two-photon absorption selection rules and rotational line strengths of the v3- and v4-active vibronic transitions”, J. Mol. Spectrosc. 224, 145 (2004)。如附錄 三。

c. 角動量極化與離子成像圖樣的關連

在發展了 CH3自由基 REMPI 光譜學架構後，結合主持人在 J. Phys. Chem. A

結果及展望 1. 實驗部分 實驗硬體雖遭遇維修上極大困難，但均能一一克服。NO2 光解成像實驗已完 成初步結果。 2. 理論架構 均能按照構想逐步達成。

Elec tronic An gu lar Mo men tum Po lar iza tions of Atomic Frag ments in Di atomic Photodissociations: Cor re la tions be tween Elec tronic States in

the Mo lec u lar and the As ymp totic Re gions Kuo-mei Chen* ( )

De part ment of Chem is try, Na tional Sun Yat-sen Uni ver sity, Kaohsiung, Tai wan, R.O.C.

To un der stand elec tronic an gu lar mo men tum po lar iza tion phe nom ena of atomic photofragments in di -atomic photodissociations, Wigner-Witmer cor re la tion rules should be aug mented to treat these prob lems quan ti ta tively. The es sen tial step to achieve this goal is to find the trans for ma tion re la tion ships be tween ba sis func tions of Hund’s cou pling cases (a) (or (b)) and (c). Be cause the par ti tion of a united elec tronic con fig u ra -tion into two sep a rated elec tronic con fig u ra -tions is un der taken, in gen eral no an a lyt i cal trans for ma -tion can be es tab lished. To il lus trate the pres ent scheme in ac quir ing the trans for ma tion re la tion ships, the di rect photodissociation of O B2

3

( u) into O(1D) + O(3P) is ex plic itly con sid ered. The elec tronic an gu lar mo men tum po lar iza tion of ox y gen at oms in |1D, J = 2, MJ = 0 ob served in the re cent ex per i ment on O2 photodissociation

can be ra tio nal ized by this quan ti ta tive cor re la tion rule.

IN TRO DUC TION

Di atomic photodissociations that lead to open shell atomic photofragments have re ceived much at ten tion both ex per i men tally1-3 and the o ret i cally.4-7 Re cently, the an resolved elec tronic an gu lar mo men tum po lar iza tions (elec -tronic ori en ta tion and align ment)8 of atomic photofragments have been stud ied by newlydeveloped ex per i men tal tech -niques, such as ion im ag ing9-13 and la ser-detected time- flight meth ods.14-17 The ba sic frame work in an a lyz ing the photodissociation dy nam ics of di atomic mol e cules has long been es tab lished, in clud ing Wigner-Witmer cor re la tion rules18,19 _{and di atomic ba sis func tions of var i ous Hund’s cou}

-pling cases.20,21 Pre dom i nately, Freed and co-workers4,5 have de vel oped a com plete the ory to cover ev ery as pect of this in -ter est ing field. To treat the evo lu tion and dy namic cou plings of elec tronic and nu clear mo tions from the mo lec u lar re gion to the atomic as ymp totes, ex plicit trans for ma tion re la tion -ships be tween ba sis func tions of var i ous Hund’s cou pling cases have been de rived by Freed et al.4,5 _{Among them, trans}

-for ma tions be tween Hund’s case (a) (or (b)) and (c) ba sis func tions are most rel e vant, be cause case (a) ba sis func tions

Ex cept for the an gu lar mo men tum cou pling co ef fi -cients, the trans for ma tion re la tion ships be tween Hund’s case (a) and (c) ba sis func tions con tain ad di tion ally the eigen -vector co ef fi cients c a b, where de notes the pro jec tion

of the or bital an gu lar mo men tum along the internuclear axis.4,5 Al though their gen eral forms have not been worked out, it was be lieved that c a b de pends only on the sym

-me try of the atomic term lim its.4,5 Re cent ex per i men tal re -sults on elec tronic an gu lar mo men tum po lar iza tions of atomic photofragments9-17 dic tate a quan ti ta tive frame work of Wigner-Witmer cor re la tion rules such that the or i gin of the pref er en tial pop u la tion of mag netic sublevels of atomic photofragments can be elu ci dated. Hav ing this goal in mind, we will ex am ine the over lap integrals be tween the elec tronic wave func tions (spin-orbitals) of Hund’s cou pling cases (a) and (c) in this re port. Be cause the ex plicit wave func tions in the mo lec u lar re gion and in the limit of atomic as ymp totes de pend on chem i cal spe cies and their en ergy states, gen eral ex pres sions of the elec tronic over lap integrals be tween cases (a) and (c) ba sis func tions can not be ob tained. To il lus trate our scheme, a rel e vant photodissociation pro cess of O2 from

|B3 u to O(1D) + O(3P) will be ex am ined in de tail. Af ter pre

ELEC TRONIC OVER LAP IN TE GRALS

To de rive the trans for ma tion re la tion ships be tween Hund’s cases (a) and (c) ba sis func tions, a spe cific ex am ple of O2 photodissociation is con sid ered such that the re cent ex

-per i men tal re sults on aligned O(1D) atomic photofragments9 can be com pared with the pres ent work. WignerWitmer cor -re la tion rule18,19 tells us that the ex cited mo lec u lar state |B3 u cor re lates with atomic as ymp totes O(1D) + O(3P).

Hund’s case (a) ba sis func tions of the |B3

u state in the mo

-lec u lar re gion are given by21

(1-1)

(1-2) (1-3) where is the pro jec tion of S onto the z-axis of a mol e fixed frame (MFF) and DM

J

( )* is a Wigner ro ta tion ma -trix.21 Hund’s case (c) ba sis func tions in the far-nuclei re gion are given by4,5

(2) where the sub scripts a and b de note the two nu clei, re spec -tively. Thus, the trans for ma tion co ef fi cients de pend solely on the over lap integrals be tween the elec tronic wave func tions of Hund’s cases (a) and (c). The first step is to pres ent these elec tronic wave func tions in terms of their mo lec u lar and atomic spin-orbitals. Be cause the elec tronic con fig u ra tion of O at oms is p4, eight oc cu pied MOs of O2 in the B state should

be con sid ered. In Ta ble 1, ex plicit forms of |3 u , |

1

u , |

1

u

and |3

u mo lec u lar states from a

2 3 3

u g con fig u ra tion are

listed. For the | states, they sat isfy the phase con ven tion un der a re flec tion in the xz-plane of the MFF.20 Sim i larly, elec tronic wave func tions of O at oms in |1D , |3P and |1S states can be con structed by the pro jec tion op er a tor tech -niques.23 Their ex plicit forms are listed in Ta ble 2.

The next step is to ex pand each 8 8 Slat er de ter mi nant in the elec tronic wave func tions of |3 , ,

0 1

u into a

sum of prod ucts of two 4 4 de ter mi nants. The to tal num ber of those prod ucts of de ter mi nants in the sum ma tion reaches 70 (8!/4! 4! = 70). Ba sically, the eightelectron con fig u ra -tion of O B3

( ) is par ti tioned into prod ucts of two

four-724 J. Chin. Chem. Soc., Vol. 49, No. 5, 2002 Chen

1/ 2 3 * 3 * 1 , 1 2 1 , 1 ( ) , 1 ( ) , 8 J J a u M u M J _{D D} 1/ 2 3 * 3 * 1 , 1 2 1 , 1 ( ) , 1 ( ) , 8 J J a u M u M J D D 1/2 0 3 * 0 2 1 , 0 ( ) , 4 J a u M J _D 1/ 2 * * , 2 1 , , ( ) , , , ( ) , 8 J J c a b M a b M J _{J J j D J J j D}

Table 1. Electronic Wave Functions of 3 a

, 0, 1 u , a 1 _{, 0} u , 1 , 2 u and 3 , 3, 2, 1 u from a 3 3 2 g u Configurationb a

For a | + state, the upper sign of “ ” or “m” in the linear combination of Slater determinants is taken. For a | state, the lower sign of “ ” or “m” in the linear combination of Slater determinants is taken.

b

Slater determinants in Field convention (Ref. 20) are adopted. The superscripts “+” and “ ” denote = + 1 and -1,

3 1 / 2 3 1 , 1 (2) 3 3 1 1 1 1 1 1 3 3 1 1 1 1 1 1 , 0 2 3 3 1 1 1 1 1 1 3 3 1 u g g u u u g g g g g u u u g g g u g g u u u g g g g g 3 1 / 2 1 1 1 1 1 3 3 1 1 1 1 1 1 3 3 1 1 1 1 1 1 , 1 (2) 3 3 1 1 1 1 1 u u u g g g g g u u u g g g g g u u u g g g u g g u u u g g 1 1 1 3 3 1 1 1 1 1 1 , 0 2 3 3 1 1 1 1 1 1 3 3 1 1 1 1 1 1 3 3 1 1 g g g u u u g g g u g g u u u g g g g g u u u g g g g g u u 1 1 / 2 1 1 1 1 3 3 1 1 1 1 1 1 , 2 (2) 3 3 1 1 1 1 1 1 3 3 1 1 1 1 1 1 u g g g g g u u u g g g u g g u u u g g g g g u u u g g g m 1 1 / 2 3 3 , 2 (2) 3 3 1 1 1 1 1 1 3 3 1 1 1 1 1 1 , 3 3 3 1 1 1 1 1 1 , 3 3 3 1 u g g u u u g g g g g u u u g g g u g g u u u g g g u g g 3 1 / 2 3 1 / 2 1 1 1 1 1 , 2 (2) 3 3 1 1 1 1 1 1 3 3 1 1 1 1 1 1 , 2 (2) 3 3 1 1 1 1 u u u g g g u g g u u u g g g g g u u u g g g u g g u u u 3 3 1 1 3 3 1 1 1 1 1 1 , 1 3 3 1 1 1 1 1 1 , 1 3 3 1 1 1 1 1 1 g g g g g u u u g g g u g g u u u g g g u g g u u u g g g

O(p4)b atomic as ymp totes as R , where sub scripts a and b

re fer to the two nu clei, re spec tively. Those ex panded de ter -mi nants can be clas si fied ac cord ing to the nu mer i cal val ues of ( a, b) and ( a, b). In an tic i pa tion of al lowed Hund’s

case (c) ba sis func tions, only de ter mi nants with se lected com bi na tions of ( a, b) and ( a, b) from the ex pan sion of

the Slat er de ter mi nant |3 g 3 g 1 u 1 u 1 u 1 g 1 g 1 g | are listed in Ta ble3. Ex pan sion of the re main ing Slat er de ter mi nants in Ta ble 1 can be done sim i larly. When the elec tro -static in ter ac tions are cal cu lated by uti liz ing the ex panded (70 in to tal) or the orig i nal Slat er de ter mi nants, their Coulombic and ex change integrals are iden ti cal which im plies that the ex pan sion is ex act.

The fi nal stage is to com bine far-nuclei ba sis func tions

Elec tronic An gu lar Mo men tum Po lar iza tions J. Chin. Chem. Soc., Vol. 49, No. 5, 2002 725

Table 2. Wave Functions 2S1L,ML,MS from a p4Electronic

Configurationa

a

Slater determinants in Condon and Shortley convention (Ref. 23) are adopted, for example, 1+denotes an AO withml= 1

andms= . 1 1 1 / 2 1 1 / 2 1 1 / 2 1 3 3 D, 2, 0 1 1 0 0 D, 1, 0 (2) 1 1 0 1 1 1 0 1 D, 0, 0 (6) 1 0 0 1 1 0 0 1 2 1 1 1 1 D, 1, 0 (2) 1 0 1 1 1 0 1 1 D, 2, 0 0 0 1 1 P, 1, 1 1 1 0 1 P, 1 / 2 3 3 3 1 / 2 3 3 3 1 / 2 1, 0 (2) 1 1 0 1 1 1 0 1 P, 1, 1 1 1 0 1 P, 0, 1 1 0 0 1 P, 0, 0 (2) 1 0 0 1 1 0 0 1 P, 0, 1 1 0 0 1 P, 1, 1 1 0 1 1 P, 1, 0 (2) 1 0 1 1 1 0 1 1 3 1 1 / 2 P, 1, 1 1 0 1 1 S, 0, 0 (3) 1 0 0 1 1 0 0 1 1 1 1 1

Table 3. Slater Determinants with Selected Combinations of

) ,

( a b and( a, b)from the Expansion of

g g g u u u g g 3 1 1 1 1 1 1 3 ) ,

( _{a b} ( _a, _b) Slater determinanta phase

(0, 0) (0, 1) +1 1 1 +1 1 +1 +1 1 (0, 0) (1, 0) 1 +1 +1 1 +1 1 +1 1 (1, 1) (0, 1) 1 +1 +1 1 +1 1 +1 (1, 1) (1, 0) 1 +1 1 +1 1 +1 1 ( 1, 1) (0, 1) 1 +1 1 +1 1 +1 1 ( 1, 1) (1, 0) +1 1 1 2 4 5 3 6 7 8 1 2 4 6 3 5 7 8 1 2 3 7 4 5 6 8 1 2 7 8 3 4 5 6 3 4 5 7 1 2 6 8 3 4 6 7 1 2 5 8 4 5 7 8 1 2 3 6 4 6 7 8 1 2 3 5 1 2 3 5 4 6 7 8 1 2 3 6 4 5 7 8 1 2 5 8 a b a b a b a b a b a b a b a b a b a b a 3 4 6 7 1 2 6 8 3 4 5 7 3 4 5 6 1 2 7 8 3 5 7 8 1 2 4 6 3 6 7 8 1 2 4 5 4 5 6 8 1 2 3 7 1 3 4 7 2 5 6 8 1 4 7 8 2 3 5 6 2 3 4 5 1 6 7 8 2 3 4 6 1 5 7 8 2 3 7 8 1 4 5 6 b a b a b a b a b a b a b a b a b a b a b 2 4 5 8 1 3 6 7 2 4 6 8 1 3 5 7 1 3 4 5 2 6 7 8 1 3 4 6 2 5 7 8 1 3 7 8 2 4 5 6 1 4 5 8 2 3 6 7 1 4 6 8 2 3 5 7 2 3 5 8 1 4 6 7 2 3 6 8 1 4 5 7 1 4 5 7 2 3 6 8 1 4 6 7 a b a b a b a b a b a b a b a b a b a b a 2 3 5 8 2 3 5 7 1 4 6 8 2 3 6 7 1 4 5 8 2 4 5 6 1 3 7 8 2 5 7 8 1 3 4 6 2 6 7 8 1 3 4 5 1 3 5 7 2 4 6 8 1 3 6 7 2 4 5 8 b a b a b a b a b a b a b a b

Ta ble 4 can share the same set of { i}, where i = 1,2,...,8. For

ex am ple, |3P, 0, 1a = |1+0+0 1+|a = 1₄|( 3 + 8)( 1 + 9)( 2 + 10)( 5 + 6)|a. De leting 9 and 10 which are ab sent in Ta ble

3, we can ob tain a re duced form of |3P, 0, 1 a, that is,

1

4[| 1 2 3 5|a + | 1 2 3 6|a | 1 2 5 8|a | 1 2 6 8|a]. The nu

-mer i cal val ues of the elec tronic over lap integrals be tween the case (a) and the far-nuclei ba sis func tions are listed in Ta ble 6. The nor mal iza tion con stants of Slat er de ter mi nants have been prop erly in cor po rated in the cal cu la tion. It is clear that |3 u, = will evolve solely into (2)1 -1/2[|1D,0,0a|3P,0,1 b

|3P,0,1 a|1D,0,0 b] as R . For |3 u, = 0, -1 , their

transformation re la tion ships can be found eas ily by ap ply -ing the low er -ing op er a tor S- s

i i 1 8 to |3 , 1 u = R

(2)-1/2[|1D,0,0a|3P,0,1 b |3P,0,1a|1D,0,0b]. The fi nal re sults

are listed in Ta ble 7, where the con nec tions with structure level ba sis func tions are also shown. Ac cord ing to Wigner-Witmer cor re la tion rules,18,19 _{there are 90 mo lec u lar}

states that cor re late with the atomic as ymp totes 1D + 3P. Thus, Ta ble 7 is a sub set of a 90 90 trans for ma tion ma trix. On the other hand, the con ser va tion of flux on a sin gle po ten -tial en ergy curve in the ab sence of non-adiabatic cross ings dic tates that the case (c) ba sis func tions should be nor mal ized

to one, ir re spec tive of the small nu mer i cal val ues of the elec tronic over lap integrals. To sum ma rize our scheme in cal cu lat ing elec tronic over lap integrals, a flow di a gram which il -lus trates the es sen tial steps is de picted in Fig. 1. For di rect photo dissociation pro cesses of di atomic mol e cules, this scheme is use ful to find cor re la tions be tween elec tronic states in the mo lec u lar and the as ymp totic re gions.

As soon as the trans for ma tion re la tion ships be tween cases (a) and (c) ba sis func tions have been es tab lished, the scheme of Under wood and Powis22 can be fol lowed to ob tain the wave func tions of the photodissociated at oms in terms of

726 J. Chin. Chem. Soc., Vol. 49, No. 5, 2002 Chen

Table 4. Combinations of Far-nuclei Basis Functions )

,

( a b ( a, b) Far-nuclei basis function

(0, 0) (0, 1) b a P,0,1 0 , 0 , D 3 1 (0, 0) (1, 0) b a D,0,0 1 , 0 , P 1 3 (1, 1) (0, 1) b a P, 1,1 0 , 1 , D 3 1 (1, 1) (1, 0) b 1 3_{P,1,1 D, 1,0} a ( 1, 1) (0, 1) b a P,1,1 0 , 1 , D 3 1 ( 1, 1) (1, 0) b a D,1,0 1 , 1 , P 1 3

Table 5. Explicit LCAO-MO of i’s and Their Inverse Relationships LCAO-MO AO 0 0 ) 2 ( 1/2 1 b a 0 (2) ( 1 9) 2 / 1 a 0 0 ) 2 ( 1/2 2 b a 0 (2) ( 9 1) 2 / 1 b 1 1 ) 2 ( 1/2 3 _{a b} 0 (2) ( 2 10) 2 / 1 a 1 1 ) 2 ( 1/2 4 _{a b} 0 (2) (10 2) 2 / 1 b 1 1 ) 2 ( 1/2 5 _{a b} 1 _a (2) 1/2( 3 8) 1 1 ) 2 ( 1/2 6 _{a b} 1 (2) ( 3 8) 2 / 1 b 1 1 ) 2 ( 1/2 7 _{a b} 1 _a (2) 1/2( 4 11) 1 1 ) 2 ( 1/2 8 _{a b} 1 (2) ( 4 11) 2 / 1 b 0 0 ) 2 ( 1/2 9 b a 1 (2) ( 5 6) 2 / 1 a 0 0 ) 2 ( 1/2 10 _{a b} 1 (2) 1/2( 5 6) b 1 1 ) 2 ( 1/2 11 b a 1 (2) ( 7 12) 2 / 1 a 1 1 ) 2 ( 1/2 12 _{a b} 1 _b (2) 1/2( 12 7)

Table 6. Numerical Values of Electronic Overlap Integrals

Far-nuclei basis function Case (a) electronic

wave functiona D,0,0a P,0,1b 3 1 b a D,0,0 1 , 0 , P 1 3 b a P, 1,1 0 , 1 , D 3 1 b a D, 1,0 1 , 1 , P 1 3 b a P,1,1 0 , 1 , D 3 1 b a D,1,0 1 , 1 , P 1 3 1 ₄₍₁₀₅₎1/2 1 2 / 1 ) 105 ( 4 1 2 / 1 ) 35 ( 8 1 2 / 1 ) 35 ( 8 1 2 / 1 ) 35 ( 8 1 2 / 1 ) 35 ( 8 1 2 1/2 ) 105 ( 4 1 2 / 1 ) 105 ( 4 1 2 / 1 ) 35 ( 8 1 2 / 1 ) 35 ( 8 1 2 / 1 ) 35 ( 8 1 2 / 1 ) 35 ( 8 1

the photofragmental ba sis func tions in the space-fixed frame, in clud ing their or bital mo tions. We will not re peat the der i va

-tion for the case of O2 (|B3 u ) O( 1

D) + (3P) in this re port. DIS CUS SION

From Ta ble 7, it is rec og nized that max i mally aligned ox y gen at oms in the |1_D

2, 0 state are gen er ated from the di

-rect photodissociation of O2 in |B3 u state no mat ter what

the tran si tion prob a bil i ties from the ground state to the

|3 , ,

0 1

u states are. Our quan ti ta tive scheme which

trans forms Hund’s case (a) to case (c) ba sis func tions elu ci -dates the phys i cal or i gin of the re cently ob served align ment of ox y gen at oms in |1D2, 0 state.9 Any de vi a tion from the sta

-tis ti cal pop u la tion dis tri bu tion for the O(3_{P) at oms (N(|}3_P 2):

N(|3

P1): N(|3P0) = 5:3:1, cal cu lated from Ta ble 7) should re

-flect the im por tance of non-adiabatic in ter ac tions be tween the |B3

u state and the nearby re pul sive states with an as

-ymp totic limit O(3P) + O(3P). A par tial cor re la tion di a gram of ungerade states of the ex cited O2 mol e cule with the as ymp

-totic lim its O(3P) + O(3P) and O(1D) + O(3P) is de picted in Fig. 2. For internuclear dis tances greater than 5 Bohr ra dii, the dom i nant terms in the Hamiltonian are at trib uted to the quadrupole-quadrupole in ter ac tion be tween the two at oms Elec tronic An gu lar Mo men tum Po lar iza tions J. Chin. Chem. Soc., Vol. 49, No. 5, 2002 727

Table 7. Transformation Relationships between Cases (a) and (c) Electronic Wave Functions Case (c)

Case (a)

Far-nuclei basis function (2S 1L, , ) Fine-structure level basis function (2S 1LJ,MJ ) 1 , 3 u b a b a P,0,1 P,0,1 D,0,0 0 , 0 , D ) 2 ( 1/2 1 3 3 1 b a b a P ,1 P ,1 D , 0 0 , D 2 1 2 1 2 3 2 3 2 1 b a b a P,1 P,1 D , 0 0 , D₂ 3 ₁ 3 ₁ 1 ₂ 1 3 , 0 u b a b a P,0,0 P,0,0 D,0,0 0 , 0 , D ) 2 ( 1/2 1 3 3 1 b a b a 3) P ,0 D , 0 1 ( 0 , P 0 , D ) 3 1 ( 1/2 1 2 3 2 1/2 3 2 1 2 b a b a 6) P ,0 D , 0 1 ( 0 , P 0 , D ) 6 1 ( 1/2 1 2 3 0 1/2 3 0 1 2 1 , 3 u (2) 1/2 1D,0,0 _a 3P,0, 1_b 3P,0, 1_a 1D,0,0_b b a b a P, 1 P , 1 D , 0 0 , D 2 1 2 1 2 3 2 3 2 1 b a b a P, 1 P, 1 D , 0 0 , D₂ 3 ₁ 3 ₁ 1 ₂ 1

cross ing rates.25 Be cause ex cited mo lec u lar states, other than those from the four elec tronic con fig u ra tions in Fig. 2, will cor re late to higher ex cited atomic states of ox y gen (ex clud -ing the as ymp tote 1D + 3P), the en ergy ma trix diagonalization scheme with a fi nite ba sis set is jus ti fied, as long as the atomic wave func tions in the cal cu la tion of quadrupole in ter ac tion have been op ti mized. In other words, ex ten sive cal cu la tions of mo lec u lar wave func tions can be avoided. Thus, one can fo cus on the farnuclei wave func -tions which ex hibit the cor rect cor re la tion in the mo lec u lar re gion and the evo lu tion of the eigenvector co ef fi cients as a func tion of R.

A close ex am i na tion on the ex plicit form of the nuclei ba sis func tions in Ta bles 6 and 7 shows that the al -lowed com bi na tions, such as (2)-1/2[|1D,0,0 a|3P,0,1 b

|3_P,0,1

a|1D,0,0b] sat is fies the con ser va tion of in ver sion sym

-me try (g or u), re flec tion sym me try (+ or -) and the pro jec tion

of or bital ( ) and spin ( ) an gu lar momenta onto the mo lec u -lar axis. On the other hand, the for bid den com bi na tions, such as 1

2[|

1

D,1,0a|3P,-1,1 b |3P,-1,1 a|1D,1,0b |1D,-1,0a|3P,1,1b

+ |3P,1,1 a|1D,-1,0 b] must cor re late with a |3 u, =1 state

(from a u g

3 3

* con fig u ra tion, see Fig. 2) and this far-nuclei wave func tion does not meet the con ser va tion of re flec tion sym me try. Ac cord ing to Wigner-Witmer cor re la tion rules,18,19

all the com bi na tions listed in Ta ble 4 are al lowed. Thus, the en deavor in the cal cu la tion of elec tronic over lap integrals is worth while and the above re sults il lus trate the sub tle dif fer ence be tween the pres ent scheme and WignerWitmer cor re -la tion rules. In con for mity with sym me try prop er ties of the di atomic mol e cule, the pres ent treat ment ac counts for the ex -act cor re la tion be tween elec tronic states in the mo lec u lar and the as ymp totic re gions.

CON CLU SIONS

The trans for ma tion re la tion ships be tween ba sis func -tions of Hund’s cou pling cases (a) and (c) must be solved case by case. A spe cific ex am ple of the di rect photodissociation of O B2

3

( u) into O(

1

D) + O(3P) is cho sen to il lus trate our scheme. The first step is to set up the elec tronic wave func -tions in the mo lec u lar and the far-nuclei re gions. Sec ondly, Slat er de ter mi nants in these elec tronic wave func tions are ex -panded by the same set of par ti tioned ba sis func tions. Finally, the trans for ma tion re la tion ships can be found from the nu -mer i cal val ues of elec tronic over lap integrals and a min i mum ma nip u la tion of an gu lar mo men tum al ge bra. From this te dious but faultfree scheme, the elec tronic an gu lar mo men tum po lar iza tion phe nom ena of atomic photofragments in di atomic photodissociations can be elu ci dated. For curve cross -ing prob lems, those cor re lated far-nuclei wave func tions in cor rect forms pro vide a fi nite ba sis set to find the Rde pend -ence of eigenvector co ef fi cients. Con ser va tion of sym me try prop er ties from the mo lec u lar states to their as ymp totic lim its pro vides a phys i cal foun da tion of this quan ti ta tive cor re la -tion rule.

AC KNOWL EDG MENT

This re search was sup ported by the Na tional Sci ence Coun cil of the Re pub lic of China.

Re ceived July 9, 2002.

728 J. Chin. Chem. Soc., Vol. 49, No. 5, 2002 Chen

Fig. 2. A par tial cor re la tion di a gram of ungerade states of an ex cited O2 mol e cule. The elec tronic con

-fig u ra tion m _up _gq( *) of Or 2 is de noted by the

Key Words

Elec tronic an gu lar mo men tum po lar iza tions; Di atomic photodissociations; Cor re la tions be tween elec tronic states.

REF ER ENCES

1. van Brunt, R. J.; Zare, R. N. J. Chem. Phys. 1968, 48, 4304. 2. Rothe, E. W.; Krause, U.; Duren, R. Chem. Phys. Lett. 1980,

72, 100.

3. Kato, H.; Onomichi, K. J. Chem. Phys. 1985, 82, 1642. 4. Singer, S. J.; Freed, K. F.; Band, Y. B. J. Chem. Phys. 1983,

79, 6060.

5. Singer, S. J.; Freed, K. F.; Band, Y. B. Adv. Chem. Phys.

1985, 61, 1; and ref er ences therein.

6. Siebbeles, L. D. A.; Glass-Maujean, M.; Vasyutinskii, C. S.; Beswick, J. A.; Roncero, O. J. Chem. Phys. 1994, 100, 3610. 7. Rakitzis, T. P.; Zare, R. N. J. Chem. Phys. 1999, 110, 3341. 8. Greene, C. H.; Zare, R. N. Annu. Rev. Phys. Chem. 1982, 33,

119.

9. Eppink, A. T. J. B.; Parker, D. H.; Janssen, M. H. M.; Buijsse, B.; van der Zande, W. J. J. Chem. Phys. 1998, 108, 1305. 10. Bakker, B. L. G.; Eppink, A. T. J. B.; Parker, D. H.; Costen,

M. L.; Han cock, G.; Ritchie, G. A. D. Chem. Phys. Lett.

1998, 283, 319.

11. Bakker, B. L. G.; Parker, D. H.; Han cock, G.; Ritchie, G. A. D. Chem. Phys. Lett. 1998, 294, 565.

12. Samartizis, P. C.; Bakker, B. L. G.; Rakitzis, T. P.; Parker, D.

H.; Kitsopoulos, T. N. J. Chem. Phys. 1999, 110, 5201. 13. Bracker, A. S.; Wouters, E. R.; Suits, A. G.; Vasyutinskii, O.

S. J. Chem. Phys. 1999, 110, 6749.

14. Rakitzis, T. P.; Kandel, S. A.; Zare, R. N. J. Chem. Phys.

1998, 108, 8291.

15. Rakitzis, T. P.; Kandel, S. A.; Al ex an der, A. J.; Kim, Z. H.; Zare, R. N. Sci ence, 1998, 281, 1346.

16. Rakitzis, T. P.; Kandel, S. A.; Al ex an der, A. J.; Kim, Z. H.; Zare, R. N. J. Chem. Phys. 1999, 110, 3351.

17. Al ex an der, A. J.; Kim, Z. H.; Kandel, S. A.; Zare, R. N.; Rakitzis, T. P.; Asano, Y.; Yabushita, S. J. Chem. Phys. 2000,

113, 9022.

18. Wigner, E.; Witmer, E. E. Z. Phys. 1928, 51, 859.

19. Herzberg, G. Mo lec u lar Spec tra and Molecular Struc ture. I.

Spec tra of Di atomic Mol e cules; Van Nostrand: Prince ton,

1950.

20. Lefebvre-Brion, H.; Field, R. W. Per tur ba tions in the Spec -tra of Di atomic Mol e cules; Ac a demic: Or lando, 1986.

21. Zare, R. N. An gu lar Mo men tum, Un der stand ing Spa tial

Aspects in Chem is try and Phys ics; Wiley-Interscience: New

York, 1988.

22. Under wood, J. G.; Powis, I. J. Chem. Phys. 2000, 113, 7119. 23. Con don, E. U.; Shortly, G. H. The The ory of Atomic Spec tra;

Cam bridge Univ. Press: Lon don, 1953.

24. Saute, M.; Bussery, B.; Aubert-Frécon, M. Mol. Phys. 1984, 51, 1459.

25. Asano, Y.; Yabushita, S. J. Phys. Chem. A 2001, 105, 9873. 26. van Vroonhoven, M. C. G. N.; Groenenboom, G. C. J. Chem.

Phys. 2002, 116, 1954.

Rotational line strengths of the

v

2-active two-photon transitions of the methyl radical

Kuo-mei Chena)

Department of Chemistry, National Sun Yat-sen University, Kaohsiung, Taiwan, Republic of China 共Received 23 December 2002; accepted 15 July 2003兲

To extract information on the rotational population distributions of the methyl radical from photodissociation by the 2⫹1 resonance-enhanced multiphoton ionization technique, its rotational line strength formulas of the two-photon transitions have been reexamined. Symmetry-adapted rovibronic-nuclear spin wave functions of CH₃ and CD₃ in the 兩X˜2_A

2

⬙典 and 兩np2_A 2

⬙典 electronic states were utilized in the derivation. Transformation properties of the rovibronic and nuclear spin basis functions under the permutation-inversion group D_3h(M) have been employed to construct the total wave functions which follow the appropriate statistics of CH₃and CD₃, respectively. Explicit expressions of the two-photon rotational line strengths of thev2-active vibronic bands of the methyl radical were reported. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1606443兴

I. INTRODUCTION

In recent years rotationally resolved, resonance-enhanced multiphoton ionization 共REMPI兲 spectroscopy of the methyl radical1–3has been employed to study the photo-dissociation dynamics of its precursors.4 –7To extract infor-mation on the population distribution of methyl radicals in various rotational states, the two-photon absorption line strength formula of symmetric tops8 was invariably utilized in the analysis of the REMPI spectra. Because the symmetry-adapted wave functions are linear combinations of symmet-ric top rotational state vectors when nuclear spin statistical weights are taken into account,9we decide to reexamine the problem of allowed rovibronic levels of CH₃and CD₃in the ground state 兩X˜2A₂⬙典 and an excited intermediate state

兩np2_A 2

⬙典 (n⫽3 or 4兲, where these two states are connected

by a two-photon absorption process in the REMPI. In par-ticular the role of the out-of-plane bending mode v2 in the REMPI spectra of the methyl radical is explored, where the predominance of thev2 activity has been confirmed.1,4

In the next section, the transformation properties of the rovibronic and nuclear spin basis functions are examined un-der the molecular symmetry group D3h(M) of the methyl radical. Symmetry-adapted wave functions of CH3 and CD3 in various vibronic levels兩2A₂⬙,2_ij典 are presented in Sec. III. Rotational line strength formulas of thev2-active two-photon transitions of the methyl radical are derived in Sec. IV.

II. TRANSFORMATION PROPERTIES OF BASIS FUNCTIONS

In his classical work, Herzberg10 inferred the planar structure of CD₃ from the intensity alternations of the K

⫽0 sub-band in the B˜←X˜ transition around 2144 Å. The

physical reasoning which ascertained the even–odd alterna-tion of nuclear spin statistical weights for K⫽0 rotational

energy levels of an XY3 molecule with a D3h symmetry has long been established by the scheme of Wilson,11,12 where the irreducible representations of the rotational subgroup of the full molecular point group were employed to classify the energy states. Although early electron spin resonance studies13 on the methyl radical did not reach a definite con-clusion on its planarity, Hirota and co-workers14 elucidated unequivocally the D3h structure of the CH3radical based on their high-resolution infrared diode laser absorption spectros-copy of the v2 band. Additional infrared absorption studies on CD3 and the v3 band by Hirota, Sears, and their co-workers followed.15–19 High-resolution coherent Raman spectra of thev1band of CH3, which have been obtained by Nibler and co-workers,20,21were analyzed according to a pla-nar equilibrium geometry.

In the present work, the permutation and inversion op-erations of the molecular symmetry group22,23D_3h(M) will be employed to classify the energy levels of the methyl radi-cal. The symmetry operations in D3h(M) warrant the mo-lecular Hamiltonian invariant, regardless whether the poten-tial energy curve of the methyl radical versus the v2 normal coordinate has no barrier at all or displays a small potential barrier共see Fig. 1兲. In other words, D3h(M) is the molecular symmetry group of a planar or a quasiplanar configuration. The transformation properties of the basis functions are ex-amined under three symmetry operations, which are, even permutation 共123兲, odd permutation 共23兲, and the spatial in-version in the space-fixed frame 共SFF兲 E*,23 _{because the} effects executed by the remaining operations (123)* and (23)*can be obtained straightforwardly.

Because the spin–rotation interaction of the methyl radi-cal is quite small 共coupling constant ⬍0.012 cm⫺1),14 Hund’s case共b兲 basis functions24which couple rotational and electron spin angular momentum state vectors should be adapted. One has the coupled state vector兩N 1/2 JKM典 a兲_{Electronic mail: [email protected]}

兩N 1/2 JKM典⫽

兺

M_NM_S 共⫺1兲

N⫺1/2⫹M_共2J⫹1兲1/2

⫻

冉

N 1/2 J

M_N M_S ⫺M

冊

兩NKMN典兩1/2 MS典, 共1兲

where the electron spin state vector兩12MS典 is defined in the

SFF, and (_.._.._..) is a 3- j symbol.25 In the above equation, the rotational wave function of a symmetric top molecule

兩NKMN典 is a normalized Wigner rotation matrix,25where N

denotes the rotational angular momentum quantum number, and K is its projection onto the molecular z axis. From Eq.

共1兲, the transformation properties of the coupled state vector

can be proven to be9,25,26

共123兲兩N 1/2 JKM典⫽exp共i2␲K/3兲兩N 1/2 JKM典, 共2a兲

共23兲兩N 1/2 JKM典⫽共⫺1兲N⫹K_{兩N 1/2 J⫺KM典, 共2b兲}

and

E*_{兩N 1/2 JKM}典⫽共⫺1兲N⫺K_{兩N 1/2 J⫺KM典. 共2c兲}

It should be reminded that the electron spin state vector

兩1

2MS典 is invariant under E*. A detailed account which

proves Eq. 共2b兲 is given in the EPAPS.27

Because thev2mode of the methyl radical transforms as an A₂⬙irreducible representation in D3h(M), the overall sym-metry of a vibronic state兩A₂⬙, ␷₂n典 is A₂⬙when n is even, and is A1⬘when n is odd. Nuclear spin basis functions of CH3and CD3are listed in the EPAPS.

27

For references, their transfor-mation properties under the symmetry operations共123兲, 共23兲

state. Our results are listed in the supplemental tables of the EPAPS.27 For the CH3 radical which follows the Fermi– Dirac statistics, the symmetry-adapted total wave functions must transform either as an A₂⬘ or an A₂⬙ irreducible repre-sentation of D3h(M) such that they are anti-symmetric under the odd permutation共23兲. For the CD3radical which follows the Bose–Einstein statistics, the rovibronic-nuclear spin wave functions must transform either as an A₁⬘ or an A₁⬙ irreducible representation of D3h(M) such that they are sym-metric under the odd permutation 共23兲. The statistical weights of various energy levels of CH3 and CD3 which are listed in Table I, are identical to those reported in previous literatures.12

IV. TWO-PHOTON ROTATIONAL LINE STRENGTHS

Because the symmetry-adapted rovibronic-nuclear spin wave functions listed in the supplemental tables of the EPAPS27are quite complicated, one should examine the⌬K selection rule of various vibronic bands of the 兩np2A₂⬙典

←兩X˜2_A 2

⬙典(n⫽3 or 4兲 transition of the methyl radical before deriving the rotational line strength formula. Non-null two-photon dipole transition matrix elements in the molecule-fixed frame共MFF兲,␮_{f i}q⬘␮_i0q⬙determine the⌬K selection and

⌬K⫽q⬘⫹q⬙.8From the group theory, it can be proven that 具A₂⬙兩r˜q⬘兩i典具i兩r˜q⬙兩A2⬙典 is non-null when q⬘⫹q⬙⫽0. It applies to cases of vibronic bands 0₀0, 1₀1, 1₁0, 11₁, 2₀2, 2₂0, and 2₂2. Similarly, q⬘⫹q⬙ should be null for the 21

1 and 21

3

vibronic bands. On the other hand, 具A₁⬘兩r˜q⬘兩i典具i兩r˜q⬙兩A2⬙典 is null for any combination of q⬘ and q⬙ such that transitions to the 2₀1 and 2₁0 vibronic bands are forbidden. Thus, we will focus on

⬘ ⬙

FIG. 1. Potential energy curves of the methyl radical vs the out-of-plane bending normal mode Q2.共a兲 A shallow potential without a barrier, 共b兲 a

potential with a small barrier which is less than the zero-point energy. The even or odd property of each vibrational energy level under the共23兲 sym-metry operation of D3h(M) molecular symmetry group is denoted by a ‘‘⫹’’

or a ‘‘⫺’’ superscript alongside the␷2quantum number.

TABLE I. Nuclear spin statistical weights of the methyl radical.

Vibronic

state K N

Nuclear spin statistical weight

CH3 CD3 兩2_A 2 ⬙典 0 even 4 1 odd 0 10 3 p, p⫽1,2,... any N 4 11 3(2 p⫹1)⫾1 or 3(2 p⫹1)⫾2, p⫽0,1,... any N 2 8 兩2_A 1 ⬘典 0 even 0 10 odd 4 1 3 p, p⫽1,2,... any N 4 11 3(2 p⫹1)⫾1 or 3(2 p⫹1)⫾2, p⫽0,1,... any N 2 8

1 2关具N⬘1/2 J⬘K⬙M⬘兩⫺具N⬘1/2 J⬘⫺K⬙M⬙兩兴 ⫻D00 Q_关兩N⬙ 1/2 J⬙K⬙M⬙典⫹兩N⬙1/2 J⬙⫺K⬙M⬙典兴 ⫽1 2关具N⬘1/2 J⬘K⬙M⬘兩D00 Q_兩N⬙ 1/2 J⬙K⬙M⬙典 ⫺具N⬘1/2 J⬘⫺K⬙M⬘兩D₀₀Q兩N⬙1/2 J⬙⫺K⬙M⬙典兴 ⫽1 2关1⫺共⫺1兲 N⬘⫹N⬙⫹Q_兴 ⫻具N⬘1/2 J⬘K⬙M⬘兩D₀₀Q兩N⬙1/2 J⬙K⬙M⬙典 ⫽具N⬘1/2 J⬘K⬙M⬘兩D₀₀Q兩N⬙1/2 J⬙K⬙M⬙典, 共3兲 for transitions 兩np2A₂⬙典←兩X˜2A₂⬙典, K⬙⫽3,9,15,..., N⬙ is even and N⬘ is odd. This result can be rationalized from the properties of 3- j symbols; that is, (_⫺K

⬙ N⬘ 0 Q K⬙ N⬙ ) ⫽(⫺1)N⬘_⫹N⬙_⫹Q( K⬙ N⬘ 0 Q ⫺K_⬙ N⬙ )⫽⫺(_K ⬙ N⬘ 0 Q ⫺K_⬙ N⬙ ), when N⬙ is even and N⬘ is odd. As a matter of fact, the reduction to a single matrix element of D00

Q

(Q⫽0, 2) applies to any combination

of N 共even–even, odd–odd, and even–odd兲, so long as the

selection rule ⌬K⫽0 is met.

Combining all the facts, one can prove that the two-photon absorption cross section of the methyl radical is pro-portional to8 mf 0⫽4g

兺

J⬙M⬘M⬙

冏

iJ⬘

兺

q⬘q⬙ 具N⬘1/2 J⬘K⬘M⬘兩D_0q1*_⬘D1_0q*_⬙兩N⬙1/2 J⬙K⬙M⬙典共Ei0⫺␻兲⫺1␮f i q⬘_␮ i0 q⬙

冏

2 ⫽4g

兺

J⬙M⬘M⬙

冏

iJ⬘

兺

q⬘q⬙Q 共2Q⫹1兲

冉

1 1 Q 0 0 0

冊冉

1 1 Q ⫺q⬘ ⫺q⬙ 0

冊

⫻具N⬘1/2 J⬘K⬘M⬘兩D₀₀Q兩N⬙1/2 J⬙K⬙M⬙典共Ei0⫺␻兲⫺1␮f i q⬘_␮ i0 q⬙

冏

2 ⫽4g₉

兺

J⬙M⬘M⬙

冏

兺

J⬘ 具N⬘1/2 J⬘K⬘M⬘兩D00 0_兩N⬙ 1/2 J⬙K⬙M⬙典

冋

兺

i 共Ei0⫺␻兲 ⫺1_共␮ f i 0_␮ i0 0_⫺␮ f i 1_␮ i0 ⫺1_⫺␮ f i ⫺1_␮ i0 1 _兲

册

⫹

兺

J⬘ 具N⬘1/2 J⬘K⬘M⬘兩D2₀₀兩N⬙1/2 J⬙K⬙M⬙典

冋

兺

i 共Ei0⫺␻兲 ⫺1_共2␮ f i 0_␮ i0 0_⫹␮ f i 1_␮ i0 ⫺1_⫹␮ f i ⫺1_␮ i0 1 _兲

册

冏

2 , 共4兲

where g is the statistical weight of the initial state. For a REMPI probe laser with pulse durations in the nanosecond range, its coherence width cannot excite the two initial spin-rotational fine-structure levels (J⬙⫽N⬙⫾1

2) coherently. On the other hand, the predissociation-broadened final states1– 4 can be reached coherently. This is the reason why the sum-mation over J⬘ is inside the symbol of the absolute square.

From Eq.共1兲 and the manipulation of angular momen-tum algebra, we can prove that

具N⬘1/2 J⬘K⬘M⬘兩D₀₀0兩N⬙1/2 J⬙K⬙M⬙典 ⫽␦N⬘N⬙␦J⬘J⬙␦K⬘K⬙␦M⬘M⬙ 共5a兲 and 具N⬘1/2 J⬘K⬘M⬘兩D₀₀2兩N⬙1/2 J⬙K⬙M⬙典 ⫽共⫺1兲N⬘⫹N⬙⫺M⬙⫹K⬙⫹1/2_{关共2J⬘⫹1兲共2J⬙⫹1兲} ⫻共2N⬘⫹1兲共2N⬙⫹1兲兴1/2_␦ M⬘M⬙␦K⬘K⬙ ⫻

冉

_⫺MJ⬘ J⬙ 2 ⬙ M⬙ 0

冊冉

N⬘ 2 N⬙ K⬙ 0 ⫺K⬙

冊

再

J⬙ 2 J⬘

冎

where兵_.._.._..其 is a 6- j symbol.25Substituting Eq. 共5兲 into Eq.

共4兲, one gets mf 0⫽ 4g 9

冋

_J

兺

_⬙M⬙F␦N⬘N⬙⫹

兺

J⬙M⬙J₁⬘J₂⬘ 关共2J1⬘⫹1兲共2J2⬘⫹1兲兴 1/2 ⫻共2J⬙⫹1兲共2N⬘⫹1兲共2N⬙⫹1兲 ⫻

冉

J1⬘ J⬙ 2 ⫺M⬙ M⬙ 0

冊冉

J₂⬘ J⬙ 2 ⫺M⬙ M⬙ 0

冊

⫻

冉

N⬘ N⬙ 2 K⬙ ⫺K⬙ 0

冊

2

再

J⬙ 2 J₁⬘ N⬘ 1/2 N⬙

冎

⫻

再

J⬙ 2 J2⬘ N⬘ 1/2 N⬙

冎

E

册

, 共6a兲

where the orthogonality relationship

兺

_M⬙

冉

_⫺MJ⬘_⬙ MJ⬙⬙ 20

冊冉

J⬘ J⬙ 0

⫺M⬙ M⬙ 0

冊

⫽0, 共6b兲

has been employed. The two-photon transition moments E

7165 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Two-photon transitions of methyl

E⫽

冏

兺

i 共Ei0⫺␻兲 ⫺1_共2␮ f i 0_␮ i0 0_⫹␮ f i 1_␮ i0 ⫺1_⫹␮ f i ⫺1_␮ i0 1_兲

冏

2 共7a兲 and F⫽

冏

兺

i 共Ei0⫺␻兲 ⫺1_共␮ f i 0_␮ i0 0_⫺␮ f i 1_␮ i0 ⫺1_⫺␮ f i ⫺1_␮ i0 1_兲

冏

2 . 共7b兲

From the following two sum rules:25

兺

_M⬙

冉

J1⬘ J⬙ 2 ⫺M⬙ M⬙ 0

冊冉

J₂⬘ J⬙ 2 ⫺M⬙ M⬙ 0

冊

⫽ 1 5␦J1⬘J2⬘ 共8a兲 and

兺

_J⬘ 共2J⬘⫹1兲共2N⬙⫹1兲

再

J⬙ 2 J⬘ N⬘ 1/2 N⬙

冎

2 ⫽1, 共8b兲

Eq. 共6a兲 is converted to

mf 0⫽ 4g 45

冋

_J

兺

_⬙M⬙ 5F␦N⬘N⬙⫹

兺

_J⬙ 共2J⬙⫹1兲共2N⬘⫹1兲 ⫻

冉

N⬘ N⬙ 2 K⬙ ⫺K⬙ 0

冊

2 E

册

. 共9兲

Apparently, the summation over J⬙ and M⬙ is

兺

J⬙M⬙ 1⫽

兺

J⬙ 共2J⬙⫹1兲⫽2共N⬙⫹1 2兲⫹2共N⬙⫺ 1 2兲⫹2 ⫽2共2N⬙⫹1兲, 共10兲

provided that the fine-structure levels are unresolved. Thus, the final expression is

mf 0⫽ 8g 45

冋

5共2N⬙⫹1兲F␦N⬘N⬙⫹共2N⬘⫹1兲共2N⬙⫹1兲 ⫻

冉

N⬘ N⬙ 2 K⬙ ⫺K⬙ 0

冊

2 E

册

. 共11兲

For CH3in an兩X˜2A2⬙典 vibronic state共the occupation number of the v2 mode is 0,2,...兲, g is null when K⬙⫽0 and N⬙ is odd. The corresponding rotational line strength is identically zero. Similarly, mf 0 is null when CH3 is in an兩X˜2A1⬘典 vi-bronic state 共the occupation number of the v2 mode is 1, 3,...兲, where K⬙⫽0 and N⬙ is even. Furthermore, the P- and ⬙⫽0 levels are

has to be identified. For the forbidden vibronic bands 2₀1and 2₁0, it is interesting to note that their matrix elements

D₀₀Q(Q⫽0, 2) between the initial and final angular momen-tum states are identically zero.

V. CONCLUSIONS

Symmetry-adapted rovibronic-nuclear spin wave func-tions of CH3and CD3in the兩X˜2A2⬙典and兩np2A2⬙典 electronic states were constructed according to the symmetry opera-tions of the molecular symmetry group D_3h(M). The nuclear spin statistical weights of the energy levels of the methyl radical have been confirmed to be identical to those reported in previous literatures. On the other hand, the role of the out-of-plane bending modev2and the coherent excitation of the fine-structure levels in the REMPI spectra of CH3 and CD3 have been clarified. The newly derived two-photon ro-tational line strength formulas are more realistic to meet the goal in analyzing the rotationally resolved, v2-active vi-bronic bands of the REMPI spectra of the methyl radical. Thus, the present work provides a useful framework to ex-tract information on the rotational population distribution of the methyl radical and the photodissociation dynamics of its precursors, where the v2-active vibronic bands can provide new experimental evidences.

ACKNOWLEDGMENT

This research was supported by the National Science Council of the Republic of China.

1_{J. W. Hudgens, T. G. DiGiuseppe, and M. C. Lin, J. Chem. Phys. 79, 571} 共1983兲.

2_{J. F. Black and I. Powis, J. Chem. Phys. 89, 3986共1988兲.} 3

I. Powis and J. F. Black, J. Phys. Chem. 93, 2461共1989兲.

4_{R. Ogorzalek Loo, H.-P. Haerri, G. E. Hall, and P. L. Houston, J. Chem.}

Phys. 90, 4222共1989兲.

5_{D. W. Chandler, M. H. M. Janssen, S. Stolte, R. N. Strickland, J. W.}

Thoman, Jr., and D. H. Parker, J. Phys. Chem. 94, 4839共1990兲.

6

M. H. M. Janssen, D. H. Parker, G. O. Sitz, S. Stolte, and D. W. Chandler, J. Phys. Chem. 95, 8007共1991兲.

7_{D. Y. Kim, N. Brandstater, L. Pipes, T. Garner, and D. Baugh, J. Phys.}

Chem. 99, 4364共1995兲.

8

K. Chen and E. S. Yeung, J. Chem. Phys. 69, 43共1978兲.

9_{C. H. Townes and A. L. Schawlow, Microwave Spectroscopy}

共McGraw-Hill, New York, 1995兲, pp. 62–73.

10_{G. Herzberg, Proc. R. Soc. London, Ser. A 262, 291共1961兲.} 11

E. B. Wilson, J. Chem. Phys. 3, 276, 818共1935兲.

12

G. Herzberg, Molecular Spectra and Molecular Structure III. Electronic

Spectra of Polyatomic Molecules共Van Nostrand, Princeton, 1966兲, p. 94.

13_{L. Kaplan, in Free Radicals, edited by J. K. Kochi共Wiley, New York,}

1973兲, pp. 363–369, and references therein.

14

C. Yamada, E. Hirota, and K. Kawaguchi, J. Chem. Phys. 75, 5256共1981兲.

22_{H. C. Longuet-Higgins, Mol. Phys. 6, 445共1963兲.}

23_{P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, 2nd}

ed.共NRC Research Press, Ottawa, 1998兲.

24

M. H. Alexander and P. J. Dagdigian, J. Chem. Phys. 80, 4325共1984兲.

25_{R. N. Zare, Angular Momentum共Wiley, New York, 1988兲.} 26_{J. T. Hougen, J. Chem. Phys. 37, 1433共1962兲.}

27_{See EPAPS Document No. E-JCPSA6-119-005338 for supplemental}

tables and the Appendix. A direct link to this document may be found in the online article’s HTML reference section. The document may also be reached via the EPAPS homepage 共http://www.aip.org/pubservs/ epaps.html兲 or from ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more information.

7167 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Two-photon transitions of methyl

Resonance-enhanced multiphoton ionization spectroscopy of CH

3

and CD

3

. Two-photon absorption selection rules and rotational

line strengths of the m

3

- and m

4

-active vibronic transitions

Kuo-mei Chen*

Department of Chemistry, National Sun Yat-sen University, Kaohsiung, Taiwan, ROC Received 18 August 2003; in revised form 8 January 2004

Abstract

To explore the possibility of K-level resolved, 2þ 1 resonance-enhanced multiphoton ionization (REMPI) processes of the methyl radical, the two-photon absorption selection rules and rotational line strengths of the 31

0 and 410 vibronic bands of the

transitionjnp2_A00

2i j ~X2A002i (n ¼ 3 or 4) were reported. Stringent selection rules, which were imposed upon these two-photon

transitions, are the initial K00¼ 3p (p ¼ 0; 1; 2; . . .), DK ¼
2, DU ¼
3, and DN ¼ 0;
1;
2 (O, P , Q, R, and S branches). The previously assigned 22

2vibronic band of the methyl radical should be studied by the REMPI with a better spectral resolution and

analyzed by the newly derived two-photon absorption selection rules and rotational line strength formulas. Ó 2004 Elsevier Inc. All rights reserved.

Keywords: The methyl radical; Two-photon absorption; Selection rules; Rotational line strengths; m3- and m4-active vibronic bands

1. Introduction

Resonance-enhanced multiphoton ionization (RE-MPI) spectroscopy of the methyl radical [1–4] has been proven to be a powerful technique to study the photo-dissociation dynamics of its precursors [5–20]. To ex-tract information on the population distribution of methyl radicals in various rotational states, the rota-tionally resolved 00

0 vibronic band of the two-photon

transitionjnp2_A00

2i j ~X2A002i (n ¼ 3 or 4) in the REMPI

spectrum has been thoroughly investigated [2–4,8]. Be-cause the rotational constants of the ground and excited electronic states of CH3 and CD3 are quite similar [2,3],

the 00

0 band in the REMPI spectrum exhibits a

promi-nent but highly congested Q branch and partially over-lapped O, P , R, and S branches [21]. Among them only

respectively. All the other features are either overlapped spectral lines of two different branches or unresolved stacks from contributions of various K levels. In addi-tion, predissociation-broadened linewidths [2] render the K-resolved REMPI study a difficult task. Consequently, only a deductive procedure could be implemented to obtain the population and alignment parameters of the various jN ; Ki levels of the methyl photofragment [10,12] from the experimental REMPI spectra.

Eppink and Parker [16] have found m4-excited

(exci-tation of the asymmetric deformation mode) methyl radicals from photodissociation of CH3I in the A band

by the high-resolution velocity mapping study on iodine photofragments. No spectral feature can be assigned to the 40

1 vibronic band in the 2þ 1 REMPI spectra of the

j3p2_A00

2i j ~X2A002i transition of the methyl photofrag-Journal of Molecular Spectroscopy 224 (2004) 145–150