Lh[SBitlSA0 t 0 XBAt 0 t 0 1150

Use the Kubo's identity (Appendix 11A, Eq. (11.81)) with H replaced by Ho to show that Eqs (11.50) and (11.40) are equivalent XBA(t) h WBi(t), SA )o hTr om SBi(t), SA hTr SA, po,eq SBi(t) Tr po,eq y dXSAi(-ihX)SBi(t) J dX SAi(-ihX)SBi(t))o oo SA ( )SBi (t))o - SA ( )SBi (t))o (11.51) 11.2.3 Causality and the Kramers-Kronig relations The fact that the response functions x(t) in Eqs (11.19) and (11.49) vanish for t < o is physically significant It implies that an effect cannot...

Index

Page numbers in italic, e.g. 319, refer to figures. Page numbers in bold, e.g. 579, signify entries in tables. harmonic analysis 245-7 absorption lineshape 316-22, 319, 320, 340-2 spin-boson model 439-42 thermal relaxation 667 azuline in naphtaline matrix 679-82, 680, 681, 682 broadening 668-70, 669 motional narrowing 670-4 thermal effects in resonance Raman scattering 674-9, 675 electron transfer rate 579 acetonitrile electron transfer rate 579 adiabatic ionization potential 165, 166, 537...

Subjects

The lawyers plead in court or draw up briefs, The generals wage wars, the mariners Fight with their ancient enemy the wind, And I keep doing what I am doing here Try to learn about the way things are And set my findings down in Latin verse Such things as this require a basic course In fundamentals, and a long approach By various devious ways, so, all the more, I need your full attention Lucretius (c.99-c.55 bce) The way things are' translated by Rolfe Humphries, Indiana University Press, 1968....

Appendix 8C Derivation of the Fokker Planck equation from the Langevin equation

As in (8.178), the operator is of the form ( t) A + BR(t) in which A and B are deterministic operators and R(t) is a random function of known statistical properties. We can therefore proceed in exactly the same way as in Appendix 8B. In what follows we will simplify this task by noting that the right-hand side of (8.184) contains additive contributions of Newtonian and dissipative terms. The 302 Stochastic equations of motion former is just the Liouville equation The dissipative part (terms...

Info

The identity (10.100) is the Nakajima-Zwanzig equation. It describes the time evolution of the relevant part Pp(t) of the density operator. This time evolution is determined by the three terms on the right. Let us try to understand their physical contents. In what follows we refer to the relevant and irrelevant parts of the overall system as system and bath respectively. The first term, -iPcPp(t) describes the time evolution that would be observed if the system was uncoupled from the bath...

The absorption lineshape

We next consider the effect of thermal relaxation on the absorption lineshape. We start with the Bloch equations in the form (10.184), Oy - kr(oz - Oz,eq) (18.46a) n& x-- oz - kd y (18.46c) where oz,eq and kr are given by Eqs. (10.185) and (10.186), respectively. Let us assume that the system has reached a steady state under a constant field E0, and consider the rate at which it absorbs energy from the field. We can identify this rate by observing that in (18.46a) there are two kinds of...

Relaxation of a quantum harmonic oscillator

We next consider another example of quantum-mechanical relaxation. In this example an isolated harmonic mode, which is regarded as our system, is weakly coupled to an infinite bath of other harmonic modes. This example is most easily analyzed using the boson operator formalism (Section 2.9.2), with the Hamiltonian H hiM0ata + h Vjbjbj + h jtbj + u*abt (9.44) The first two terms on the right describe the system and the bath, respectively, and the last term is the system-bath interaction. This...

The work function

Chapter 17 of this text focuses on the interface between molecular systems and metals or semiconductors and in particular on electron exchange processes at such interfaces. Electron injection or removal processes into from metals and semiconductors underline many other important phenomena such as contact potentials the potential gradient formed at the contact between two different metals , thermionic emission electron ejection out of hot metals , and the photoelectric effect electron emission...

Appendix 16A Derivation of the Mulliken Hush formula

Here we present the derivation29 of the expression 16.97 that relates the coupling between two nonadiabatic electronic states a and b to the optical transition dipole between the corresponding adiabatic states 1 and 2, as described in Section 16.10. Our discussion refers to a given fixed nuclear configuration. The electron transfer reaction is assumed to take place between two states, a state a localized on the center A and a state b localized on the center B. fa and fb are the corresponding...

Appendix 7B Proof of Eqs 764 and 765

Here we prove, for a Gaussian stochastic processes z t and a general function of time x t the results 7.64 and 7.65 . Our starting point is cf. Eq. 7.63 e'T,jxjzA jT.jX zj gt - 1 2 EjEkxj Szi Szk xk 7.119 where the sums are over the n random variables. Noting that this relationship holds for any set of constants xj , we redefine these constants by setting and take the limit Atj 0 and n in the interval to lt t1 lt t2 lt lt tn. Then

V2

fn 17 -nfn-1 - -n 1fn 1 2.157 a n -n n - 1 at n Vn 1 n 1 2.158 where we have used n to denote f n. The operators ci and a are seen to have the property that when operating on an eigenfUnction of the Harmonic oscillator Hamiltonian they yield the eigenfUnction just above or below it, respectively. a and a will therefore be referred to as the harmonic oscillator raising or creation and lowering or annihilation operators, respectively.8 Equation 2.152 also leads to hence the name number operator...

Acknowledgements

In the course of writing the manuscript I have sought and received information, advice and suggestions from many colleagues. I wish to thank Yoel Calev, Graham Fleming, Michael Galperin, Eliezer Gileadi, Robin Hochstrasser, Joshua Jortner, Rafi Levine, Mark Maroncelli, Eli Pollak, Mark Ratner, Joerg Schr der, Zeev Schuss, Ohad Silbert, Jim Skinner and Alessandro Troisi for their advice and help. In particular I am obliged to Misha Galperin, Mark Ratner and Ohad Silbert for their enormous help...