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 rule^18,19 tells us that the ex cited mo lec u lar state -trix.²¹ Hund’s case (c) ba sis func tions in the far-nuclei re gion are given by^4,5

(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}_{u g}³ 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 _u, 0, into a1 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 combination of Slater determinants is taken. For a | state, the lower sign of “ ” or “_m” in the linear combination of Slater determinants is taken.

bSlater determinants in Field convention (Ref. 20) are adopted.

The superscripts “+” and “ ” denote = + 1 and -1,

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 Configuration^a

aSlater determinants in Condon and Shortley convention (Ref.

23) are adopted, for example, 1⁺denotes an AO withml= 1

Table 3. Slater Determinants with Selected Combinations of

)

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

Table 5. Explicit LCAO-MO of _i’s and Their Inverse Relationships

Table 6. Numerical Values of Electronic Overlap Integrals

Far-nuclei basis function

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(¹D) + (³P) in this re port.