1549380323-Statistical Mechanics Theory and Molecular Simulation

544 Quantum time-dependent statistical mechanics ofRfion the frequencyω. When the integral in the third line of eqn. (14.2.42) i ...

Frequency spectra 545 Writing theδ-function as an integral, eqn. (14.3.3) becomes R(ω) = 1 ̄h^2 |F(ω)|^2 ∫∞ −∞ dt ∑ i,f wiei(Ef− ...

546 Quantum time-dependent statistical mechanics Although quantum time correlation functions possess many of the same properties ...

Frequency spectra 547 event by a factor of exp(−β ̄hω) in a canonical distribution, for which the probability of finding the sys ...

548 Quantum time-dependent statistical mechanics Finally, substituting eqn. (14.3.21) into eqn. (14.3.18) and the resultinto eqn ...

Examples of frequency spectra 549 (Shankar, 1994), where the action of ˆaand ˆa†on an energy eigenstate of the oscillator is aˆ| ...

550 Quantum time-dependent statistical mechanics for approximating the quantum position autocorrelation function of an anharmoni ...

Quantum linear response theory 551 As a specific example of an infrared spectrum, we show, in Fig. 14.5(a), computed IR spectra ...

552 Quantum time-dependent statistical mechanics to the eigenstates ofHˆ 0. This approach, known asquantum linear response theor ...

Quantum linear response theory 553 where〈Aˆ〉is the equilibrium ensemble average ofAˆ. When eqn. (14.5.6) is substituted into eqn ...

554 Quantum time-dependent statistical mechanics to make contact with the treatment of Section 14.3, we sett 0 =−∞in eqn. (14.5. ...

Quantum linear response theory 555 By decomposing the susceptibility into its real and imaginary parts, we can relate it directl ...

556 Quantum time-dependent statistical mechanics Substituting the definitions ofR(ω) andR(−ω) from eqns. (14.3.7) and (14.3.15) ...

Approximations 557 = i ̄h [ Tr ( ρˆ 0 AˆBˆ(t) ) −Tr ( ρˆ 0 Bˆ(t)Aˆ )] = i ̄h 〈 AˆBˆ(t)−Bˆ(t)Aˆ 〉 =− i ̄h 〈 [Bˆ(t),Aˆ] 〉 =−ΦBA(t) ...

558 Quantum time-dependent statistical mechanics Let us begin with a standard nonsymmetrized time correlation function defined b ...

Approximations 559 x x’ x’’ x x’ (a) (b) Fig. 14.6(a) Diagram of the real- and imaginary-time paths for the correlation function ...

560 Quantum time-dependent statistical mechanics GAB(t) = 1 Q(N,V,T) ∫ dx〈x|Aˆ(ˆx)ei Hˆτc∗/ ̄hˆ B(ˆx)e−i Hˆτc/ ̄h |x〉 (14.6.7) = ...

Approximations 561 novel Monte-Carlo schemes to be devised for computing this function (Jadhao and Makri, 2008). The second alte ...

562 Quantum time-dependent statistical mechanics transform (see Appendix D). Unfortunately, the reverse process, transforming fr ...

Approximations 563 U 0 (xc) =− 1 β ln {( 2 πβ ̄h^2 m ) 1 / 2 ∮ Dx(τ)δ(x 0 [x(τ)]−xc)e−S[x(τ)]/ ̄h } , (14.6.18) wherex 0 [x(τ)] ...