Advances in Chemical Physics, Vol. 121 by Ilya Prigogine, Stuart A. Rice

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 zone of the self-discipline. quantity 121 includes the newest learn on polymer melts at good surfaces, infrared lineshapes of susceptible hydrogen bonds, ab initio quantum molecular dynamics, and plenty of different matters.

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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 ɼ Cn ðtÞÉ0n ð3:6Þ n and by substituting Eq.

47) can be written as ^ ¼ Trb ð^ rÞ s ð4:48Þ ^ to pick up the matrix element rma;na , that We shall use the projection operator D is, ^ r; ^1 ¼ D^ r ^ r ^2 ¼ ð1 À DÞ^ r ð4:49Þ and ^1 þ r ^2 ¼ r ^ r ð4:50Þ 50 s. h. lin et al. where 0 0 0 0 a ;n b Dm ¼ daa0 dmm0 dnn0 dbb0 dab ma;nb ð4:51Þ Notice that ^ rÞma; nb ¼ ðD^ XX m0 a0 n0 b0 0 0 0 0 a ;n b 0 Dm ma; nb rm0 a0 ; n0 b ¼ dab rma; nb ð4:52Þ Applying the Laplace transformation ^ðpÞ ¼ r ð1 ^ðtÞ dt eÀpt r ð4:53Þ 0 we obtain ^rðpÞ ^ð0Þ ¼ ÀiL^ p^ rðpÞ À r ^ L^ ^r1 ðpÞ À iD ^ L^ ^r2 ðpÞ ^1 ð0Þ ¼ ÀiD p^ r1 ðpÞ À r ð4:54Þ ð4:55Þ and ^ L^ ^r1 ð pÞ À ið1 À DÞ ^ L^ ^r2 ðpÞ ^2 ð0Þ ¼ Àið1 À DÞ p^ r2 ð pÞ À r ð4:56Þ ^2 ðpÞ yields Eliminating r ^ L^ ^r1 ð pÞ À iD ^L ^ ^1 ð0Þ ¼ ÀiD p^ r1 ðpÞ À r 1 ^ ^ ð0Þ À MðpÞ^ r r1 ðpÞ ^ L ^ 2 p þ ið1 À DÞ ð4:57Þ ^ where MðpÞ denotes the memory kernel ^ ^L ^ MðpÞ ¼D 1 ^ L ^ ð1 À DÞ ^ L ^ p þ ið1 À DÞ ð4:58Þ It follows that d^ r1 ^ L ^ ^ L^ ^r1 À iD ^ Le ^ Àitð1ÀDÞ ^2 ð0Þ À r ¼ ÀiD dt ðt ^ r1 ðt À tÞ dtMðtÞ^ ð4:59Þ 0 where ^ L ^ ^ ^ Le ^ Àitð1ÀDÞ ^ L ^ MðtÞ ¼D ð1 À DÞ ð4:60Þ ultrafast dynamics and spectroscopy 51 An important case is ðbÞ rma;na ¼ smn raa ð4:61Þ ðbÞ where raa represents the equilibrium distribution.

RELAXATION DYNAMICS OF A SYSTEM IN A HEAT BATH A. Density Matrix Method We shall start with the definition of density matrix [82–84]. For this purpose, we consider a two-state system. According to the expansion theorem we have É ¼ Ca ðtÞua þ Cb ðtÞub ¼ Ca0 ðtÞÉ0a ðtÞ þ Cb0 ðtÞÉ0b ðtÞ ð4:1Þ where   it É0a ðtÞ ¼ exp À Ea ua ; h "   it É0b ðtÞ ¼ exp À Eb ub "h ð4:2Þ ^ of the system are defined as follows: The matrix elements of the density matrix r raa ¼ Ca ðtÞCaà ðtÞ ¼ jCa ðtÞj2 ¼ jCa0 ðtÞj2 ð4:3Þ rbb ¼ Cb ðtÞCbà ðtÞ ¼ jCb ðtÞj2 ¼ jCb0 ðtÞj2 ð4:4Þ rab ¼ Ca0 ðtÞCb0 ðtÞà eÀ"hðEb ÀEa Þ ¼ rÃba ð4:5Þ and it It follows that  ^¼ r raa raa rab rbb  ð4:6Þ From Eqs.

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