## H2

Ih-jf i -2MV22 + F2,eff(R2)) (2.40b) 2,eff (R2) V2(R2) + F12 (2.41b) The result, Eq. (2.40) is known as the time-dependent mean field or time-dependent Hartree approximation. In this approximation each system is moving in the average field of the other system. At this point the two systems are treated on the quantum level, however if there is reason to believe that classical mechanics provides a good approximation for the dynamics of system 2, say, we may replace Eq. (2.40b) by its classical...

## Appendix 7A Moments of the Gaussian distribution

Consider the -dimensional Gaussian distribution (7.54). Here it is shown that this distribution satisfies (zj) mj (SzjSzk) (A)-l jk (where Sz z - (z)) (7.109) The time ordering information in (7.54) is not relevant here. Equation (7.109) obviously holds for n 1, where W(z) ce-(1 2)a(z-m) gives (z) m and (z m)2) -2(d da)(ln fC dze (1 2)az ) a-1. In the general n-variable case we introduce the characteristic function of n variables (x1, , xn), essentially the Fourier transform of the probability...

## N rr

For a continuous charge distribution p (r) the equivalent expression is Note that taking p to be a distribution of point charges, p(r) tqtS(r rt), leads to Eq. (1.210). 46 Review of some mathematical and physical subjects Another expression of the Coulomb law is the Gauss law, which states that the electric field associated with a charge distribution p (r) satisfies the relationship < j) dsE n 4n J drp(r) (1.212) In (1.212) Q denotes a volume that is enclosed by the surface S, n is a unit...

## Info

And provided that T h D b this leads to (compare to the analysis of the low-temperature behavior of Eq. (4.52)) Indeed, both the exponential temperature dependence that characterize the Orbach process and the T1 behavior associated with the Raman type process have been observed in spin lattice relaxation.10 The golden-rule rate expressions obtained and discussed above are very usefUl for many processes that involve transitions between individual levels coupled to boson fields, however there are...

## Introduction

In elementary treatments of the interaction of atoms and molecules with light, the radiation field is taken as a classical phenomenon. Its interaction with a molecule is often expressed by where f is the molecular dipole operator while E (t) is the time-dependent electric field associated with the local electromagnetic field at the position of the molecule. In fact, much can be accomplished with this approach including most applications discussed in this text. One reason to go beyond this...

## O [Ho 2YI [Vj o V [Vj Vj10173

Where Vj is a set of system operators associated with the system-bath interaction. When constructing phenomenological relaxation models one often uses this form as a way to insure positivity. It can be shown that the general Redfield equation 5 As discussed in Sections 10.4.8 and 10.4.9, these eigenstates may be defined in terms of a system Hamiltonian that contains the mean system-bath interaction. 6 Otherwise the thermal distribution is approached with respect to the exact energies that may...

## T

+ dtj dt2 & (t1)& 2(t2)) +---- our aim now is to take these averages using the statistical properties of P and to carry out the required integrations keeping only terms of order At .To this end we note that & is of the form & (t) A + B p(t) where A and B are the deterministic operators d dx(d V(x) dx) and d dx, respectively. Since p) 0 the first term in the square bracket is simply where the operator d dx is understood to operate on everything on its right. The integrand in the...

## Y

Now, if y aat constant Tc, the energy diffusion becomes faster, the well distribution is rapidly thermalized and becomes irrelevant for the crossing dynamics. This is the moderate-high friction regime discussed above. However, increasing y while maintaining a constant tc y ratio actually leads to decreasing D. Such a limit is potentially relevant the experimental parameter pertaining to liquid friction is the liquid viscosity, and highly viscous, overdamped fluids are characterized by sluggish,...

## Co

Cx(t) (x(0)x(t)) f dM eiMt 2- - (8.37) This integral is most easily done by complex integration, where the poles of the integrand are m (i 2)Y m1 with m1 Jm y2 4. It leads to Cx (t) n lR (cos Mit + sin Mit e Yt 2 for t > 0 (8.38) For t 0 we have (x2) n IR(m2YM ) 1 and using (8.30) we get mM (x2) kBT, again as expected. 8.2.4 The absorption lineshape of a harmonic oscillator The Langevin equation (8.3i), with R(t) taken to be a Gaussian random force that satisfies (R) 0 and (R(0)R(t))...

## KgasC p131

Dt 'VgaS so that the relaxation time is (r(gas))-1 -C* ddCr kgasP (13.2) When comparing this relaxation to its condensed phase counterpart one should note a technical difference between the ways relaxation rates are defined in the two phases. In contrast to the bimolecular rate coefficient kgas, in condensed environments the density is high and is not easily controlled, so the relaxation rate is conventionally defined in terms of a unimolecular rate coefficient kcond, defined from dC* dt...

## X

A 1V (n + 1) 2 m n + 1 a 1 n 2 m n 1 (2.141) Consider now solutions of the time-dependent Schr dinger equation d (x, t) i( h2 d2 1 22 w , - r h 2mdx2 +1 m x) * (2J42) Knowing the eigenfunctions and eigenvalues implies that any solution to this equation can be written in the form (2.6), with the coefficients determined from the initial condition according to cn(t0) (fn(x) (x, to). The following problem demonstrates an important property of properly chosen wavepackets of harmonic oscillator...

## N

We can use this theorem to address extensive functions of extensive variables, which are obviously homogeneous functions of order 1 in these variables, for example, the expression E(XS, XQ, ANj ) XE(S, Q, Nj ) (1.137) just says that all quantities here are proportional to the system size. Using (1.136) with n 1 then yields dE dE dE E S + Q + V Nj--(1.138) Furthermore, since at constant T and P (from (1.134)) it follows, using (1.140) and (1.141) that The result (1.142) is the Gibbs-Duhem...

## Linear Response Theory

If cause forever follows after cause Out of an old one by fixed law if atoms Do not, by swerving, cause new moves which break The Laws of fate if cause forever follows, In infinite sequence, cause where would we get This free will that we have, wrested from fate, Lucretius (c.99-c.55 bce) The way things are translated by Rolfe Humphries, Indiana University Press, 1968 Equilibrium statistical mechanics is a first principle theory whose fundamental statements are general and independent of the...

## E

Fig. 12.4 A dressed-states representation of the model of Fig. 12.1. discussed in Section 2.2, yielding the solution (for (t 0) 2) cf. Eq. (2.32)) P2(t) 1 - Pl(t) where r is the Rabi frequency, r (1 h)y (E2 - E1)2 + 4 F1212. Two facts are evident (1) The specification of the bath state v) is immaterial here, and (2) in this case we cannot speak of a rate that characterizes the 1 2 transition. The coupling to the boson bath can change this in a dramatic way because initial levels of the combined...

Eberly, Optical Resonance and Two-Level Atoms. (Wiley, New York, 1975) C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions Basic Processes and Applications (Wiley, New York, 1998). W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973). S. Mukamel, Nonlinear Optical Spectroscopy (Oxford University Press, Oxford, 1995). M. Orszag, A quantum statistical model of interacting two level systems and radiation (Worcester...

## Some observations

The problem solved above is an example of a scattering process, treated here within a one-dimensional model. Unlike bound state systems such as the harmonic oscillator of Section 2.9, in a scattering process all energies are possible and we seek a solution at a given energy E, so we do not solve an eigenvalue problem. The wavefUnction does not vanish at infinity, therefore normalization as a requirement that dx f(x) 2 1 is meaningless. Still, as discussed in Section 2.8.1, normalization is in...

## S

Where the integral is over the phase-space surface surrounding the volume u, and where dS is a surface element vector whose direction is normal to the surface in the outward direction. Using Gauss theorem to transform the surface integral into a volume integral we get (Note that Eq. (1.258)) is the multidimensional analog of Eq. (1.36)). Comparing to (1.256) and noting that the volume u is arbitrary, we find that Note that the first line of (1.259) and the way it was derived are analogous to...

## The onelevel bridge model

Further insight can be gained by considering specific cases of the transmission function T(E), Eq. (17.22). Here we consider the case where the bridge is adequately represented by a single level, 1), with energy E1 (see Fig. 17.7). In this case the matrices G(B) and f(K) K L, R are scalar functions. From Eqs (17.23) and (17.24) we get Molecular conduction R (b) L R (c) L

## C2

The function is defined as a sum of delta-functions, however for macroscopic systems this sum can be handled as a continuous function of m in the same way that the density of modes, g(M) Xj & (m Mj) is.6 Defining the coupling density by c2 (M)g(m) c)Km Mj) (6.91) Equation (6.90) can also be written as The spectral density, Eqs (6.90) and (6.92) is seen to be a weighted density of modes that includes as weights the coupling strengths c2( ). The harmonic frequencies j 6 This is a...

## Fik

( 2,1 + e - e - Hm)2 + r + rV )) 2)2 Condonapprox . X (e2,1 + e - e - Hm)2 + ((rV2) + rV )) 2)2 where E2> 1 E E . This expression is similar to Eq. (12.60),9 except that it is written for finite temperature and that the 8 functions associated with individual vibronic transitions are replaced by Lorentzian profiles with widths that depend on the excited vibronic levels. As already said, in most spectra involving large molecules in condensed phases these individual transitions cannot be...

## 1

BiR(E) is the self energy of level 11) due to its interaction with the continuum R. Similar results, with L, l replacing R, r, are obtained by inserting the solution for ci from Eq. (9.79b) into (9.79a), leading to an additional contribution BiL(E) to the self energy of this level due to its interaction with the continuum L. Using these results in (9.79a) leads to E i E i(E) Ei + Am(E) + ail(E) (9.85) Using (9.83) in (9.79b) and (9.79c) now yields c 2 C 2 _LVbii2__'Vi, l2'c l2_ (986) (E - Er)2...

## Sab

Here the scattering amplitudes associated with the different levels s add coherently to form the total scattering flux, and a picture of additive two-step processes in s out clearly does not hold. In fact, two scenarios can appear as two limiting cases of the resonance light scattering phenomenon. In one, described by (18.26), the scattering process is coherent. This coherence is expressed by the fact that the scattering amplitude, hence the scattering intensity, depends on the way by which the...

## Dt

These equations look complicated, however, in form (as opposed to in physical contents) they are very simple. We need to remember that in any representation that uses a discrete basis set p is a vector and L is a matrix. The projectors P and Q are also matrices that project on parts of the vector space (see Appendix 9A). For example, in the simplest situation where pP is just that part of p that belongs to the P space, etc. Similarly PLP (L0P 0) PLQ (0 LP0Q) , etc. (10.93) where we should keep...

## C

Equation (8.50) is an inhomogeneous differential equation for qj (t), whose solution can be written as qj (t) Qj (t) + qj (t) (8.51) Qj(t) qj0 cos (Mjt) + sin (Mjt) (8.52) 3 For the equivalent quantum mechanical derivation of the quantum Langevin equation see G. W. Ford and M. Kac, J. Stat. Phys. 46, 803 (1987) G. W. Ford, J. T. Lewis, and R. F. O'Connell, Phys. Rev. A 37, 4419 (1988). is the solution of the corresponding homogeneous equation in which qj0 and qj0 should be sampled from the...

## G

Using the Hamiltonian (10.197), the Heisenberg equations b (i h) H,b and the commutation relations (10.192), we can easily verify that Eqs (10.50) do not only stand for the averages (b) but also for the Heisenberg operators bH(t) (i H)Ht. Another form of these equations is obtained using Eq. (10.198) Equation (10.206) has the form of a classical time evolution equation of the magnetic moment associated with an orbiting charge in a magnetic field. Such a charge, circulating with an angular...

## D pN drN199

The second equality in Eq. (1.99) defines the Poisson brackets and L is called the (classical) Liouville operator. Consider next the ensemble average A(t) (A)t of the dynamical variable A. This average, a time-dependent observable, can be expressed in two ways that bring out two different, though equivalent, roles played by the function A(rN, pN). First, it is a function in phase space that gets a distinct numerical value at each phase point. Its average at time t is therefore given by A(t) I...

## Appendix 9C Resonance tunneling in three dimensions

Here we generalize the transmission problem of Section 9.5.3 to three dimensions and to many barrier states. Consider first the three-dimensional problem with a single barrier state. The barrier is taken to be rectangular and of a finite width in the transmission (x) direction, so it divides our infinite system into two semiinfinite parts, right (R) and left (L). The transmission is again assumed to result from interactions between free particle states in the L and R subspaces and a single...

## I

V snnP(n, t ) s F (s, t) (8.86) yV(n + )P(n + 1, t) sn-1nP(n, t) (8.87) If P(n, t 0) n,n0then F (s, t 0) sn0. It is easily verified by direct substitution that for this initial condition the solution of Eq. (8.85) is F(s, t) 1 + (s - 1)e-kt This will again give all the moments using Eq. (8.79). F(s, t) 1 + (s - 1)e-kt This will again give all the moments using Eq. (8.79).

## Yit

Photosynthesis that are initiated by light absorption by a chromophore site followed by energy transfer to the reaction center. It has found widespread applications, many based on fluorescence resonance energy transfer (FRET), in which the detection of fluorescence from the acceptor molecule is used to measure distances and distance distributions between fluorescent tags in proteins and other polymers. Time resolved FRET is similarly used to observe the kinetics of conformational changes in...

## L

That originates from solving hopping equations similar to (16.118) (compare Eq. (16.127)). We see that g becomes inversely proportional to the length L for long bridges, establishing a connection to the macroscopic Ohm's law. R. J. D. Miller, G. L. McLendon, A. J. Nozik, W. Schmickler, and F. Willig, Surface Electron Transfer Processes (VCH, New York, 1995). W. Schmickler, Interfacial Electrochemistry (Oxford University Press, Oxford, 1996). G. Cuniberti, G. Fagas, and K. Richter, eds,...

## Moments

Further insight into the nature of this drift-diffusion process can be obtained by considering the moments of this probability distribution. Equation (7.3) readily yields equations that describe the time evolution of these moments. Problem 7.4. Show that both sides of Eq. (7.3) yield zero when summed over all n from - to to to, while multiplying this equation by n and n2 then performing the summation lead to Assuming (n)(t 0) (n2)(t 0) 0, that is, that the particle starts its walk from the...

## Prn1 Tn2 Ti Pr2 TiPri0

Y (r - rm)S(rn - rm )< (rn-i - rm) fo - (ri - rm) x Pm(Tn + Tn-1----+ Ti), pm(Tn-1----+ Ti) , IPm(Tn-2 +-----+ Ti) , , Pm(Ti) , Pm(0) (18.93) Next we use (18.93) and (18.88) together with (r - rm)S(rn - rm) (rn-i - rm) fo - rm) (ri - rm) m (rn - r (rn-i - r) (r2 - r) (rl - - rm) (rn - r) (ri - r)p(r) (18.94) X(n)(r ri, , rn, Ti, , Tn) (rn - r) (ri - r)x(n)(r Ti, , Tn) X(n)(r n, , t ) p r)a(n) TU , xn) (18.96) where a(n)(r1, , Tn) is the single molecule response function of Eq. (18.88). Here...

## V

The conservation of Q implies that any change in Q(t) can result only from flow of Q through the boundary of volume V. Let S be the surface that encloses the volume V, and dS a vector surface element whose direction is normal to the surface in the outward direction. Denote by Jq (r, t) the flux of Q, that is, the amount of Q moving in the direction of Jq per unit time and per unit area of the surface perpendicular to Jq. The Q conservation law can then be written in the following mathematical...

## The Fokker Planck equation

In many practical situations the random process under observation is continuous in the sense that (1) the space of possible states is continuous (or it can be transformed to a continuous-like representation by a coarse-graining procedure), and (2) the change in the system state during a small time interval is small, that is, if the system is found in state x at time t then the probability to find it in state y x at time t + St vanishes when St 0.13 When these, and some other conditions detailed...

## Xoq

Is imposed by requesting a solution with the property Pss(x) > 0. Looking for a solution of the form Pss(x) f (x)e V (x) (14.49) which integrates to give the particular solution The choice of as the upper integration limit corresponds to the needed sink boundary condition, f (x ) 0, while assuming a time-independent solution in the presence of such sink is equivalent to imposing a source. Equations (14.49) and (14.51) lead to and integrating both sides from x to x xB finally yields

## E2

Equation (17.27) implies that at low bias the junction response is linear the current is proportional to the bias voltage and the proportionality coefficient is given by the conductance g, Eq. (17.28). It is given as the product of a universal constant go (1.290 x 104 )-1 (17.29) and the all-to-all transmission coefficient evaluated at the electrode's chemical potential (or, at T 0, at the electrode's Fermi energy). At finite bias one may define the voltage dependent differential conductance...

## J r r

In (3.44), the terms on the right-hand side can be viewed as the sources of the radiation field. Two sources are seen A current (moving charge) and a time variation in the magnitude of the charge. If A 0 (ground state of the radiation field) and such sources are absent, the field will remain in this ground state. Obviously there exist other states of the free radiation field, solutions of Eq. (3.44) in the absence of sources, Before considering the solutions of this equation we note that given...

## PN9rNV

5 This type of transformation is called a Legendre transform. which describe the time evolution of all coordinates and momenta in the system. In these equations rN and pN are the 3N-dimensional vectors of coordinates and momenta of the N particles. The 6N-dimensional space whose axes are these coordinates and momenta is refereed to as the phase space of the system. A phase point (rN, pN) in this space describes the instantaneous state of the system. The probability distribution function f (rN,...

## Thermal relaxation and dephasing

In the previous sections we have considered basic processes Absorption, relaxation of excited states, fluorescence, light scattering and energy transfer. We have taken into account the fact that highly excited molecular states are embedded in, and interact with, continuous manifolds of states that induce relaxation processes. Such processes affect the width of excitation spectra, the lifetimes of excited states and the yield of re-emission in the forms of fluorescence and light scattering. We...

## F x v F2

(recall that r is negative so u v + rx v r x).Now we can use the boundary condition f (x to) 1 to get F2 sjam 2n kBT. So, finally, Appendix 14B Derivation of the energy Smoluchowski equation It is convenient to use a Langevin starting point, so we begin with Eq. (14.39). We transform to action-angle variables (K, 0) Vn(K) inrn(K)Xn(K) a(K) 0 (14.125c) where, since x and v are real, x-n x* and v-n vi. The action K is related to the energy E by K 2nf v(x)dx 2n J v(K, 0) d0 d0 (14.127)

## A simple quantummechanical model for relaxation

In what follows we consider a simple quantum-mechanical model for irreversibility. In addition to providing a simple demonstration of how irreversibility arises in quantum mechanics, we will see that this model can be used as a prototype of many physical situations, showing not only the property of irreversible relaxation but also many of its observable consequences. We consider a Hamiltonian written as a sum and use the set of eigenstates of Ho as a basis. We assume that this set is given by a...

## A

If upon reactive contact, that is, when r R*, reaction occurs instantaneously with unit probability, then B* 0. The steady-state rate is then More generally, it is possible that B disappears at R* with a rate that is proportional to B*, that is, -kBA -k*B* A, that is, kB k*B* (14.113) k 1 + (4n DXA k *e-P V *) (14'U4) which yields the result (14.112) in the limit k*e-PV* to. V* is the interaction potential between the A and B species at the critical separation distance R* (on a scale where...

## Appendix 3A The radiation field and its interaction with matter

We start with the Maxwell equations, themselves a concise summary of many experimental observations. In gaussian units these are where p and J are the charge density and current density associated with free charges in the system and where the electric fields E and displacement D, and the magnetic field H and induction B are related through the polarization P (electric dipole density) and the magnetization M (magnetic dipole density) in the medium according to Equation (3.32a) is a differential...

## Appendix 8B Obtaining the Smoluchowski equation from the overdamped Langevin equation

Our starting point is Eqs (8.129) and (8.13o). It is convenient to redefine the timescale Denoting the random force on this timescale by p(t) R(t), we have (p(t1)p(t2)) 2mykBTS(t1 t2) 2kBTS(T1 t2). The new Langevin equation becomes (p) o (p(o)p(t)) 2kBTS(T) (8.177b) The friction y does not appear in these scaled equations, but any rate evaluated from this scheme will be inversely proportional to y when described on the real (i.e. unscaled) time axis. In these scaled time variable Eqs (8.131)...

## Ar t cff kkffkeiMktik alffketik 368

Consequently the electric field operator is, from Eq. (3.47) E (r, t) iYT. V ShT ff k ( kff k e-iMkt+ik* - k eMkt-ik r ) (3.69) k ff k As noted above this is the Heisenberg representation. The corresponding Schrodinger form is Finally consider the interaction between a molecule (or any system of particles) and the radiation field. A simple expression for this interaction is provided by Eq. (3.1) or, when applied to a single molecule, by the simpler version (3.24). From...

## AT a T aaT atatT 02197

Because the diagonal elements of the operators involved are zero. 1. Show that (aat)T 1 (1 - e- hm). 2. Use these results to find the thermal averages (x2)T and (p2)T, of the squared position and momentum operators. In classical mechanics a particle with total energy E cannot penetrate a spatial regions r with potential energy V (r) > E. Such a region therefore constitutes an impenetrable barrier for this particle. In quantum mechanics this is not so, and the possibility of the quantum...

## B

Where ()rev denotes a reversible process a change that is slow relative to the timescale of molecular relaxation processes, so that at each point along the way the system can be assumed to be at equilibrium. When conditions for reversibility are not satisfied, that is, when the transition from A to B is not much slower than the internal system relaxation, the system cannot be assumed in equilibrium and in particular its temperature may not be well defined during the process. Still AS SB SA is...

## C 0 dtC t Jo

We have invoked the assumption that the interaction with the thermal environment is weakto disregard the difference between 0012 and o12. The corresponding equations for a22 and a21 are again obtained by interchanging 1 2. 10.5.2 The optically driven two-level system in a thermal environment the Bloch equations Equations (10.174) and (10.175) have the same mathematical structure as Eqs (10.155) except the specification to a two-level system and the replacement of V by F. As discussed above, it...

## Cos

Taking the integration limits to infinity rests on the assumption that the integrand is well contained within the metallic band. Eqs. (17.14) can be used together with expressions (17.4) for the Fermi function and (17.13) for the function F to evaluate the rates. These expressions can be cast in alternative forms that bring out the dependence on physical parameters. First note again that as the metal electrode comes to contact with the redox solution some charge is transferred until the system...

## Cc

J Here the subscript 1 denotes the local mode and the other modes are represented by the Einstein frequency m2, of the order of the solvent Debye frequency, and we have assumed that m1 > m2. Now, if a2 was zero, we can use the procedure that leads to Eq. (13.64) to get However, having only one, or very few, local modes, the probability to match energy exactly, that is, to have m,f - Im1 for some integer l is extremely small. Instead, the largest contribution to the rate comes from a process...

## Chemical kinetics

Consider the simple first-order chemical reaction, A > B. The corresponding kinetic equation, d A-L -k (A) (A) (t) (A) (t 0)e-kt (8.82) describes the time evolution of the average number of molecules A in the system.10 Without averaging the time evolution of this number is a random process, because the moment at which a specific A molecule transforms into B is undetermined. The stochastic nature of radioactive decay, which is described by similar first-order kinetics, can be realized by...

## Contents

1 Review of some mathematical and physical subjects 3 1.1 Mathematical background 3 1.1.1 Random variables and probability distributions 3 1.1.2 Constrained extrema 6 1.1.4 Continuity equation for the flow of conserved entities 10 1.1.6 Complex integration 13 1.1.7 Laplace transform 15 1.1.8 The Schwarz inequality 16 1.2 Classical mechanics 18 1.2.1 Classical equations of motion 18 1.2.2 Phase space, the classical distribution function, and 1.4 Thermodynamics and statistical mechanics 25 1.4.2...

## D 28

54 Review of some mathematical and physical subjects K2 2nr (z++ z-)(z+ n+ + z-n ) (1.252) The solution of(1.251)that satisfies the boundary condition (x 0) and (x to) 0 is ( 5 B) Kx, that is, (x) + ( s )e KX (1.253) We have found that in an electrolyte solution the potential on the surface approaches its bulk value on a length scale k 1, known as the Debye screening length. The theory outlined above is a takeoff on the Debye Huckel theory of ionic solvation. In the electrochemistry literature...

## D an i

T (a21 CT12) k2 1CT11 + k1 2CT22 (18.45a) - (ff21 a12) + k2 1CT11 k 2a22 Kff22 (18.45b) where the second equation contains an additional damping term with damping rate K. In correspondence, Eqs (18.43b) and (18.43c) are augmented by adding the corresponding damping terms (1 2)Ka12 and (1 2)Ka21, respectively, to their right-hand sides. In this way we account for processes that destroy the system (note that a11 + a22 is no longer conserved) in the upper state, for example, by ionization or...

## D pii dPi i

JT 77 (Vl,rPr,l Vr,lPi,r) Im > Vr,ipi,r (10.73) dt dt n a d pi ,r i r, i , V x terms containing dT nEl> rPlr n(VlrPrr Vl'rPil) + non-diagonal p elements In what follows we will disregard the terms containing non-diagonal elements of p multiplying elements of V on the right-hand side of (10.74). The rationale for this approximation is that provided assumptions (1) and (2) above are valid, p remains close to the diagonal form obtained when V 0 with non-diagonal terms of order V. Below we...

## D

This result is known as the Stokes-Einstein relation. It can also be derived from elementary considerations Let a system of noninteracting charged particles be in thermal equilibrium with a uniform electric field Ex d dx in the x direction, so that the density of charged particles satisfies p(x) exp( 3 qQ(x)). In equilibrium, the diffusion flux, Ddp dx DqpEx and the drift flux, uqExp should add to zero. This yields (11.66). The diffusion coefficient and therefore the mobility are closely...

## Dan i

- 2ReR12,21 (< W12)an + 2Re 21,12( 21 )a22 + (R21,11(0) - R2,22(0))a12 - (R12,22(0) - (0))a21 (10.174a) 8 G. Lindblad, Rep. Math. Phys. 10, 393 (1976). -IM12< 712 - -F 12( 22 - 11 ) dt n + (R22,11(0) + Rl 1,22(0) - R11,11(0) - R 2,22(0) - R12,21( 12) - R21j12( 21))< 12 + (R12,12( 21) + k*21,21(m12))< 21 + (R*1,21 ( 12) - Rl2,21 ( 12) + R12,11 (0) - R21,11 (0)) < 11 + (R22,12(M21) - R11,12(M21) + R W - R12,22(0)) < 22 (10.174b) and equations for do22 dt and d< 21 dt obtained from...

## Dd2dx2 d2dy2

< r2)t < x2)t + < y2)t + < z2 )t 6Dt (1.208) This exact solution of the diffusion equation is valid only at long times because the diffusion equation itself holds for such times. The diffusion coefficient may therefore be calculated from 12 D lim -< (r(t) r(0))2) (1.209) 1.6.1 Fundamental equations of electrostatics Unless otherwise stated, we follow here and elsewhere the electrostatic system of units. The electric field at position r associated with a distribution of point charges...

## DoN N N

In the last steps we have again used the assumption that y is small so that the Lorentzian in the integrand is strongly peaked about 0. The same approximation can be applied to the second term in Eq. (9.62). Using d - 7777 2 cos(ot) e (1 2)y 111 (9.64) < n(t)) n0e-Yt + < n)T(1 - e-Yt) We have found that due to its coupling with the thermal bath our system relaxes to a final thermal equilibrium at temperature T irrespective of its initial state. The relaxation process is exponential and the...

## Dt Pq lqp lqq Pq

Equations (10.90) and (10.91) are seen to be just the equivalent set of equations for pP and pq . One word of caution is needed in the face of possible confusion To avoid too many notations it has become customary to use Pp also to denote pP, PLP also to denote LPP, etc., and to let the reader decide from the context what these structures mean. With this convention Eq. (10.94a) is written in the form d (PP5 ) - (PlP PlQ )(PP ) (10.94b) Indeed, Eqs (10.90) and (10.91) are written in this form....

## Bod

Where the terms that contribute to S are associated with the group of modes with frequencies close to oc. Equation (12.67) shows an exponential decrease (since X < 1) of the rate with increasing v o21 oc , that is, with larger electronic energy gap E21, with corrections that arise from the dependence of S on o21. v is the number of vibrational quanta that the nc high-frequency modes must accept from the electronic motion. If, for example, nc v, the most important contributions to (12.67) are...

## Cde

Fig. 12.5 The nuclear tunneling (a) and nuclear activation (b) pathways to nonradiative electronic relaxation. where E21 - 21 has been denoted AE in Fig. 12.5. For a single mode model, the analog of Eq. (12.61) is 2n V1212 A e , k1 2( 21) --2--T a ( 21 - V a) For V 1, that is, large energy gap, E2 E1 - a, the 21 dependence is given by k1 2( 21) exp(VlnX VlnV) (12.65) For Xa < 1 this function decreases exponentially or faster with the energy gap. The same observation can be made also for the...

## E mm kBT Ecoswkt kTjdg w

If all atoms are identical, the left-hand side of (6.86) is 3Nm X(0)X(t)) where X is the atomic velocity (see Eq. (4.9)). Defining g( a> ) we rewrite (6.86) in the form (X(0)X(t)) 6mN dMg(M)e'mt (6.87) which leads, by inverting the Fourier transform, to (6.84). Using Eqs (6.81) and (6.78) in (6.80) yields CAA(t) kBTJ2j cos( jt) S 2 n J

## E1 Ei2 n22

IC1 22 E Vi,1125(E1 - Ei) (9.71) This flux corresponds to the steady-state rate k JTTTJ YJ Vl,l 1KEl - Ei) (Wi p E, E1 T1 h (9.72) C112 h h This simple steady-state argument thus leads to the same golden rule rate expression, Eq. (9.25), obtained before for this model. Let us now repeat the same derivation for the slightly more complicated example described by the Hamiltonian Ho Eo 0)< 0 +E1 1)< 1 + Ei i)(i + Er r)(r (9.74)

## E2 riLriR

Where x is the chemical potential of the electrons in the electrodes. 2. On resonance, where E1 x, and in the symmetric case, r(L) r(R) (1 2)r1 the transmission coefficient is 1 irrespective of the coupling strength. Fig. 17.8 Current I (full line) and differential conduction g( ) dl d (dotted line) displayed as a function of voltage for a junction characterized by a single resonance state. Fig. 17.8 Current I (full line) and differential conduction g( ) dl d (dotted line) displayed as a...

## Ee J

Show that this energy can be written in the form and that the full solvation energy of the charge distribution associated with Di Wo -81n (1 - drD (r) (16.92) what is the significance ofthe difference WS - Wi) -8- (l - e j d Dl(r) (16.93) in (16.90) leads directly to (16.91). The total energy in an electrostatic field inside a dielectric medium is W f d3r D f d3rD2 (16.95) So the solvation energy, the energy difference between assembling a charge distribution in vacuum and inside a dielectric...

## Electron Transfer And Transmission At Moleculemetal And Moleculesemiconductor Interfaces

Our world I think is very young, Has hardly more than started some of our arts Are in the polishing stage, and some are still In the early phases of their growth we see Novel equipments on our ships, we hear New sound in our music, new philosophies Lucretius (c.99-c.55 bce) The way things are translated by Rolfe Humphries, Indiana University Press, 1968 This chapter continues our discussion of electron transfer processes, now focusing on the interface between molecular systems and solid...

## Electronic structure of solids

In addition to the thermal bath of nuclear motions, important groups of solids metals and semiconductors provide continua of electronic states that can dominate the dynamical behavior of adsorbed molecules. For example, the primary relaxation route of an electronically excited molecule positioned near a metal surface is electron and or energy transfer involving the electronic degrees of freedom in the metal. In this section we briefly outline some concepts from the electronic structure of...

## F If 00 f 02 mf 2 f 02 0189

An interesting implication of the Schwarz inequality appears in the relationship between averages and correlations involving two observables A and B. Let Pn be the probability that the system is in state n and let An and Bn be the values of these observables in this state. Then (A2> nPnA2, (B2> nPnB2, and (AB> nPnAnBn. The Schwarz inequality now implies (A2> (B2> > (AB> 2 (1.90) Indeed, Eq. (1.90) is identical to Eq. (1.80) written in the form (a a)(b b) > (a b)2 where a and b...

## Fl

Fig. 17.7 A single-level bridge between two leads at different bias potentials (a) An unbiased junction, (b) the right electrode is negatively biased, and (c) the left electrode is negatively biased. Ti(E) r L)(E) + r(R)(E) (17.35a) (E) 2xJ2 Hi,k 2S(E - Ek) 2n ( i,* 2pK(Ek)) E i Ei + Ai(E) Ei + Af (E) + Af)(E) (17.35c)

## Further Reading Chapters 7 and

Gardiner, Handbook of Stochastic Methods, 3rd edn (Springer, Berlin, 2004). R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II, 2nd edn (Springer, Berlin, 2003). H. Risken, The Fokker-Planck Equation, 2nd edn (Springer, Berlin, 1989). Z. Schuss, Theory and Applications of Stochastic Differential Equations (Wiley, New York, 1980). N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992). R. Zwanzig, Non Equilibrium Statistical Mechanics...

## Fxx0x U fxux fjtfj0x fx Xux IJt0x X

For this identity to be true we must have U (X)x U t(X) x X (2.178a) Also, since p and U commute it follows that Using Eqs (2.178) and (2.152) it is easy to show also that U (X)a iJt(X) a X U (X)atU+(X) at X Appendix 2A (see entry 6) presents a more direct proof of these equalities using operator algebra relationships obtained there. Franck-Condon factors. As an application of the raising and lowering operator formalism we next calculate the Franck-Condon factor in a model of shifted harmonic...

## H

We want to solve the time-dependent Schr dinger equation under the assumption that the system is initially in state 11). In particular, we want to evaluate the probability P1 (t) to find the system in state 1 at time t. Before setting to solve this mathematical problem, we should note that while the model is mathematically sound and the question asked is meaningful, it cannot represent a complete physical system. If the Hamiltonian was a real representation of a physical system we could never...

## H2J

VT J dteia21t i *X20 (1 2)t2Za X2ao2a(2na+1) VfJae- E'r K)2 4a> a 2 D (2na + (12.68) where Er is the reorganization energy defined by Eqs. (12.22). A simpler equation is obtained in the classical limit where na kgT hoa for all modes a, so a ksTEr h2 k 2 l ,ik e (E21 Er)2 4kBTEr (12.69) This result has the form of a thermally activated rate, with activation energy given by Problem 12.7. Show that EA, Eq. (12.70) is equal to the height of the minimum- energy crossing point of the two potential...

## Harmonic analysis

Just as a random variable is characterized by the moments of its distribution, a stochastic process is characterized by its time correlation functions of various orders. In general, there are an infinite number of such functions, however we have seen that for the important class of Gaussian processes the first moments and the two-time correlation functions, simply referred to as time correlation functions, fully characterize the process. Another way to characterize a stationary stochastic...

## Hb

The question is Should we better regard proton transfer in the same framework as electron transfer, namely solvent rearrangement as a precursor to tunneling transition, or is it better to use a description more akin to the barrier crossing reactions discussed in Chapter 14 In the first case the reaction coordinate is associated with the solvent rearrangement as was the case for electron transfer. In the second it is the position of the proton on its way across the barrier separating its two...

## Ho EaR Wa

And the parameter R is a known function of time, for example it may correspond to the distance between two molecules colliding with each other. In this respect this problem is similar to the previous one, however, the following detail characterizes the LZ problem The time dependence is such that at t 0 (say), where R(t 0) R*, the zero order energies are equal, Ea Eb, while at t TO Ea Eb is much larger than Vab . In reality the basis functions a, Wb as well as the coupling elements Vab can also...

## Bai

SBktSAik (11.54) The appearance of in in the denominators here defines the analytical properties of this function The fact that X ( ) is analytic on the upper half of the complex plane and has simple poles (associated with the spectrum of H0) on the lower half is equivalent to the casual nature of its Fourier transform the fact that it vanishes for t < 0. An interesting mathematical property follows. For any function x ( ) that is (1) analytic in the half...

## Kqh

J dtei wif-l 1 )t+ a2 2 (n1+ )e-imlt+n1e-imlt (n2 + 1)(Mif -l l) 2 (13.72) In this case the temperature dependence is expected to be much weaker than before, Eq. (13.66), because the temperature dependent factor (n2+1) is raised to a much smaller power. Moreover, the larger is the frequency of the local phonon the smaller is l, therefore, since a1 1, the larger is the rate. In the examples considered above it has been found that the principal local mode is the almost free, only slightly...

## Ii

Fig. 2.5 Tunneling through a rectangular potential barrier characterized by a width a and a height Ur. E is the energy of the tunneling particle relative to the bottom of the potential shown. other side of the barrier. This probability is expressed in terms of a transmission coefficient, a property of the barrier particle system that is defined below. Our problem is defined by the Hamiltonian V(x) (U x < < x > R ((regionI)TT IIT) (2.199) UB xL < x < xR xL + a (regions II, III)...

## Independent particles

First, consider a homogeneous system in which the molecular dipoles do not interact with each other, either directly or through their interaction with the thermal environment. This approximation becomes better for lower molecular density. In this case Hm is a sum over individual terms, HM m hm, each associated with a different molecular dipole and its thermal environment. This implies that in P (r, t) m jlm (t)S(r-rm), the different operators jlm (t) exp(ihmt K) pm exp(-ihmt K) commute with...

## Introduction To Solids And Their Interfaces

Tight-knit, must have more barbs and hooks to hold them, Must be more interwoven, like thorny branches In a closed hedgerow in this class of things We find, say, adamant, flint, iron, bronze That shrieks in protest if you try to force The stout oak door against the holding bars Lucretius (c.99-c.55 bce) The way things are' translated by Rolfe Humphries, Indiana University Press, 1968 The study of dynamics of molecular processes in condensed phases necessarily involves properties of the...

## Introduction To Stochastic Processes

The count of atoms limited, what follows Is just this kind of tossing, being tossed, Flotsam and jetsam through the seas of time, Never allowed to join in peace, to dwell In peace, estranged from amity and growth Lucretius (c.99-c.55 bce) The way things are translated by Rolfe Humphries, Indiana University Press, 1968 As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view...

## It

Where T is the temperature of the surroundings (that of the system is not well defined in such an irreversible process). Finally, the third law of thermodynamics states that the entropy of perfect crystalline substances vanishes at the absolute zero temperature. The presentation so far describes an equilibrium system in terms of the extensive variables (i.e. variables proportional to the size of the system) E, Q, S, Nj j 1, , n . The intensive (size-independent) variables P, T, j j 1, , N can...

## J Cjss Cjii

The calculation of the decay rate of 0) in this representation amounts to repeating the problem represented by the model of Fig. 9.1 and Eq. (9.2), where states 1 and i are now replaced by 0 and j , respectively. One needed element is the coupling between states 0) and j). Using (9.112) and the fact that (0 H i) 0 we find that where the constant a was introduced in Eq. (9.39d). As discussed in Section 9.1, the required decay rate is obtained under certain conditions as ( ) the imaginary part of...

## Ujz

This result can be further simplified by using the high barrier assumption, fi(V(xb) V(0)) 1, that was already recognized as a condition for meaningful unimolecular behavior with a time-independent rate constant. In this case the largest contribution to the inner integral in Eq. (14.53) comes from the neighborhood of the barrier, x xB,so exp 3 V (x) canbereplacedbyexp 3(EB- (1 2)ma> B (x- xb)2) , while the main contribution to the outer integral comes from the bottom of the well at x 0, so...

## J

Exp D (k k0) ix0(k k0) This implies that for a particle whose quantum state is (2.98), the probability to find it with momentum hk is ck I2 exP 2D2(k k0)2 Note that Eqs (2.87)-(2.90) represent a special case of these results. The time evolution that follows from Eq. (2.98) may now be found by using Eq. (2.85). In one dimension it becomes

## J0

24 Note that the existence of the 0 function is important in the identity (18.103). The inverse Fourier transform is as is easily shown by contour integration. relating the electric and displacement fields. Together with the relationship Thus, Eqs (18.98)-(18.109) provide a microscopic expression for the dielectric response function in a system of noninteracting particles s(M) 1 + 4np- dtelM(J(t) - J*(t)) (18.110) Problem 18.7. Repeat the derivation of these linear response equations taking the...

## Jfx0 j

It is convenient to use a vector notation z (z1, , zn) x (X1, , Xn) m (m1, , mn) (7.112) define y z m. Using also (7.54), the characteristic function takes the form (x) c j dze (1 2)y'A'y+ix'z c j dye Next we change variable y u + ib, where b is a constant vector to be determined below. The expression in the exponent transforms to (1 2)y A y + ix y + ix m (1 2)u A u iu - A lb +(1 2)1) A Ib +ix u x lb +ix m (7 114) Now choose b so that A b x or b A-1 x. Then the terms marked 1 in (7.114) cancel...

## Jj

The following points are noteworthy 1. The function R(t), which is mathematically identical to the variable A of Section 6.5.1,5 represents a stochastic force that acts on the system coordinate x. Its stochastic nature stems from the lack of information about qj0 and qj0. All we know about these quantities is that, since the thermal bath is assumed to remain in equilibrium throughout the process, they should be sampled from 4 Eqs. (8.52) and (8.53) imply that the initial state of the bath modes...

## Jo

X j(-Tn), j(-Tn - Tn-l), , j(-Tn - Tn-l----Tl), peq The molecular response pertaining to its optical properties is the dipole induced in the molecule, that can be calculated from In writing Eqs (l8.85)-(l8.86) we have assumed that the molecule has no permanent dipole moment, that is Tr(jipeq) 0. From (l8.84) we then find d Tn d Tn-l d TlE (t - Tn)E (t - Tn - Tn-l) X E (t - Tn-----Tl) (n)(Tl, , Tn) (l8.87) where the nth order single molecule response functions are X Tr x(0) x( Tn), K-Tn - Tn-i),...

## Jout

In (18.18) we have lumped together all the relevant continuous state manifolds that overlap with Es into the group j . In fact, the state out) formally belongs to this group as a member of the radiative continua, however, it has special status as the outgoing state of the process under discussion. Before proceeding, let us consider the expected dependence on the intensity of the incident field. The scattering process is obviously not linear in the molecule-field interaction, however, it is...

## Ka k

Which describes a system of independent harmonic modes. A mode (k, ak) of frequency ck can therefore be in any one of an infinite number of discrete states of energies hnk,ak. The degree of excitation, nk,ak, is referred to as the occupation number or number of photons in the corresponding mode. Note that (k, a k) is collection of five numbers characterizing the wavevector and polarization associated with the particular mode. The vector potential A and the fields derived from it by Eq. (3.3)...

## L1l2l3

The presence of the periodic potential U(r) has important consequences with regard to the solutions of the time-independent Schr dinger equation associated with the Hamiltonian (4.71). In particular, a fundamental property of eigenfUnctions of such a Hamiltonian is expressed by the Bloch theorem. The Bloch theorem states that the eigenfunctions of the Hamiltonian (4.71), (4.72) are products of a wave of the form (4.73) and a function that is periodic on the lattice, that is, A corollary of Eqs...

## Lattice periodicity

The geometry of a crystal is defined with respect to a given lattice by picturing the crystal as made of periodically repeating unit cells. The atomic structure within the cell is a property of the particular structure (e.g. each cell can contain one or more molecules, or several atoms arranged within the cell volume in some given way), however, the cells themselves are assigned to lattice points that determine the periodicity. This periodicity is characterized by three lattice vectors, a,, i...

## M y my 1 imy

We found that Eq. (8.23) holds, with corrections of order m y .It should be emphasized that this argument is not rigorous because the random part of F(t) is in principle fast, that is, contain Fourier components with large m. More rigorously, the transition from Eq. (8.13) to (8.21) should be regarded as coarse-graining in time to get a description in which the fast components of the random force are averaged to zero and velocity distribution is assumed to follow the remaining instantaneous...

## M0 0 M2130

90 Quantum dynamics using the time-dependent Schrodinger equation In quantum mechanics the momentum corresponds to the operatorp -ihd dx, or the position and momentum operator satisfy the familiar commutation relationship X, p ih , i (2.132) The solutions of the time-independent Schrodinger equation H E are the (orthonormal) eigenfunctions and the corresponding eigenvalues fn(x) NnHn (ax) e- l 2) ax)1 En (n + 1 2) h (2.135) where Hn( ) are the Hermit polynomials that can be obtained from Hn+1(...

## MaM2

To see the significance of this result consider a typical matrix element of this coupling between eigenstates of H0. These eigenstates may be written as n, v) n)Xv( xa ), where the elements va of the vector v denote the states of different modes a, that is, Xnv (X ) Ua Xva(Xa) are eigenstates of Hi and Xva(.xa) is the eigenfunction that corresponds to the va th state of the single harmonic mode a. A typical coupling matrix element is then (n, v V nf, V7) Vny< Xv ( Xa ) e-i(hn-hn> ) Xv' ( Xa...

## M10188

On this basis the Hamiltonian is represented by H -GB S -1 hGBz& z (10.189) z is one of the three Pauli matrices whose mutual commutation relations correspond to angular momentum algebra. The other two are x (50) , (0 0') (10.191) x, y 2i z y, z 2i x z, x 2i y (10.192) 396 Quantum mechanical density operator Note that the Pauli matrices are the matrix representations, in the basis of Eq. (10.188), of the operators defined in Eq. (10.48), provided that we denote the states +)and -) by 11) and...

## Markovian stochastic processes

The process z(t) is called Markovian if the knowledge of the value of z (say z1) at a given time (say t1) fully determines the probability of observing z at any later time Markov processes have no memory of earlier information. Newton equations describe deterministic Markovian processes by this definition, since knowledge of system state (all positions and momenta) at a given time is sufficient in order to determine it at any later time. The random walk problem discussed in Section 7.3 is an...