By Ilya Prigogine, Stuart A. Rice
The Advances in Chemical Physics sequence presents the chemical physics box with a discussion board for severe, authoritative reviews of advances in each sector of the self-discipline. quantity 121 includes the most recent study on polymer melts at strong surfaces, infrared lineshapes of vulnerable hydrogen bonds, ab initio quantum molecular dynamics, and plenty of different matters.
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Extra resources for Advances in Chemical Physics Vol. 121
22 The angle between ~ me1 g and ~ me2 g in different model also shown for comparison. From Ref. 39. 24 s. h. lin et al. methods of RCs and the experimental conditions are somewhat different for different groups. Thus, to examine validity of the electronic state model and the thermal expansion model presented in this work, absorption spectra experiments of each RC at various temperatures should be carefully performed. III. ELECTRON TRANSFER, ENERGY TRANSFER, AND RADIATIONLESS TRANSITIONS A. Time-Dependent Perturbation Method We shall start with the time-dependent Schro¨dinger equation [53,69–71] qÉ ^ ¼ i" HÉ h qt ð3:1Þ where the total Hamiltonian is divided into the noninteracting and the interacting parts: ^ ¼H ^0 þ H ^0 H ð3:2Þ The Schro¨dinger equation of the noninteracting part 0 qÉ ^ 0 É0n ¼ i" H h n qt ð3:3Þ is assumed to be solvable, and we can ﬁnd the set of eigenfunctions cn with eigenenergies En satisfying the time-independent Schro¨dinger equation ^ 0 cn ¼ En cn H and the time-dependent solution is expressed as itEn 0 Én ¼ cn exp À h " ð3:4Þ ð3:5Þ By using the expansion theorem, the total wave function can be expanded in the noninteracting basis set: X É¼ Cn ðtÞÉ0n ð3:6Þ n and by substituting Eq.
In this case, Eq. 19) shows that ga is related to the lifetime of an a state. Similarly, we have rab ðtÞ ¼ Ca ðtÞCbÃ ðtÞ ¼ Ca ð0ÞCbÃ ð0ÞeÀ"hðEa ÀEb Þ eÀ2ðga þgb Þ ¼ rab ð0Þeitoba eÀgab t it t ð4:20Þ where gab is called the dephasing constant and is related to ga and gb as 1 1 1 1 þ gab ¼ ðga þ gb Þ ¼ ð4:21Þ 2 2 ta tb where ta and tb represent the lifetimes. In summary, we obtain d^ r i ^ ^r ^ À À^ ¼ À ½H 0; r dt h " ð4:22Þ 47 ultrafast dynamics and spectroscopy ^ is damping operator, and where À draa drbb ¼ Àga raa ; ¼ Àgb rbb dt dt drab 1 ¼ Àðioab þ gab Þrab ; gab ¼ ðga þ gb Þ 2 dt ð4:23Þ ð4:24Þ To show the importance of the damping operator, we apply the effective Hamiltonian to optical absorption.
H. lin et al. or " m n ¼ dnn MðtÞ mn h2 " 0 0 0 0 6¼a X X aX m00 a0 Àitom00 a0 ; na rðbÞ aa Hma; m00 a0 Hm00 a0 ; m0 a e a þ dmm0 X X ðbÞ r Hm00 a0 ; na Hn0 a; m00 a0 eÀitoma; m00 a0 "h2 m00 a aa À 1 X X 0 ðbÞ r Hma; m0 b Hn0 b; na ðeÀitom0 b; na þ eÀitoma;n0 b Þ "h2 a0 b bb a0 6¼b ð4:69Þ and X m 0 n0 1 " 1" 0 0 n "m " L mn sm0 n0 ¼ ðHmm À Hnn Þsmn þ Hmn ðsnn À smm Þ h " "h þ 1 X 00 " " m0 n Þ ðHmm0 sm0 n À smm0 H h m0 " ð4:70Þ ^ ðt À tÞ can be expanded as In Eq. 62), s ^ ^ ^ ds 1 d2 s 1 dn s 2 ^ ðt À tÞ ¼ s ^ ðtÞ þ ðÀtÞ þ ðÀtÞ þ Á Á Á þ ðÀtÞn þ Á Á Á s 2 n dt 2!