Quantum mode-coupling theory: Formulation and applications to normal and supercooled quantum liquids

27 Feb 2005 19:42 AR AR241-PC56-06.tex XMLPublishSM(2004/02/24) P1: JRX10.1146/annurev.physchem.56.092503.141138

Annu. Rev. Phys. Chem. 2005. 56:157–85doi: 10.1146/annurev.physchem.56.092503.141138

Copyright c© 2005 by Annual Reviews. All rights reservedFirst published online as a Review in Advance on November 19, 2004

QUANTUM MODE-COUPLING THEORY:Formulation and Applications to Normaland Supercooled Quantum Liquids

Eran Rabani1 and David R. Reichman2

1School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University,Tel Aviv 69978, Israel; email: [email protected] of Chemistry and Chemical Biology, Harvard University, Cambridge,MA 02138; email: (see Acknowledgments)

Key Words quantum dynamics, normal and supercooled quantum liquids, pathintegral Monte Carlo

Abstract We review our recent efforts to formulate and study a mode-couplingapproach to real-time dynamic fluctuations in quantum liquids. Comparison is madebetween the theory and recent neutron scattering experiments performed on liquidortho-deuterium and para-hydrogen. We discuss extensions of the theory to super-cooled and glassy states where quantum fluctuations compete with thermal fluctuations.Experimental scenarios for quantum glassy liquids are briefly discussed.

1. INTRODUCTION

The development of molecular hydrodynamic and mode-coupling theories of clas-sical liquids began in the 1960s and continues to this day (1–4). Current theories ofboth single-particle and collective dynamics are quite sophisticated, allowing fora treatment of both short- and long-time dynamics in a semiquantitative way. Fordense liquids, such theories are quite mature (2–4), whereas the theories describingsupercooled liquids and glasses are somewhat more primitive (5–7).

Although molecular hydrodynamic theories of classical fluids may be comparedto classical molecular dynamics simulations that yield essentially exact dynamicalinformation for a given form of the potential energy interactions between parti-cles, no similar comparison may be made for quantum fluids. The fact that exactreal-time simulation of quantum dynamics is not possible means that even thedevelopment of approximate numerical approaches for the simulation of quan-tum dynamics is quite a challenging task. Currently, semiclassical (8–23), path-integral centroid (24–28), and analytic continuation methods (29–38) are state ofthe art. On the other hand, theories of quantum liquids that promise quantitativecomparison with experiment and simulation over a broad frequency range are

0066-426X/05/0505-0157$20.00 157

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158 RABANI REICHMAN

almost nonexistent. In this article, we outline and review our approach to such atheory.

Our viewpoint unifies a numerically exact path-integral treatment of quantumstatics (39, 40) with a combined kinetic and mode-coupling treatment of quan-tum dynamics (41, 42). Recent experiments probing single-particle and collectivefluctuations in simple quantum liquids such as liquid para-hydrogen and liquidortho-deuterium allow for nontrivial tests of the new theory (43–49). Comparisonwith experiment suggests that even though our approach is theoretical in nature,it is competitive with the most sophisticated simulation approaches in terms ofaccuracy over the full frequency range relevant for experiments.

For classical liquids, the description of supercooled and glassy states remainsa challenge (6, 50, 51). Even the numerical simulation of deeply supercooledliquids is not feasible due to the dramatic slowing of dynamical fluctuations nearthe glass transition (52–55). The understanding of the similar slowing that mayoccur in quantum systems is virtually unexplored and constitutes a frontier areaof research (56–60). Indeed, understanding how glassiness may modify a widerange of electronic and magnetic phenomena in strongly correlated systems isan area that demands attention and will not be discussed here. We focus on theinterplay between simple nuclear quantum fluctuations and glassy behavior in amanner that will allow for the description of form factors and localization lengths(61). Our approach may be considered an analog of the treatment of mean-fieldquantum spin-glasses (57, 58). The brief description of our approach will suggesthow interesting open questions, such as quantum aging and exotic glass meltingscenarios, may be treated.

2. SELF-CONSISTENT QUANTUM MODE-COUPLINGTHEORY

In this section we review the quantum mode-coupling theory (QMCT) we havedeveloped to study time-dependent correlations in quantum liquids (43–49). Ourapproach draws upon the pioneering work of Gotze & Lucke (41, 42), and Sjogren& Sjolander (62–64), and is based on augmenting an exact closed, self-consistentquantum generalized Langevin equation (QGLE) for the dynamical variable ofinterest, and introducing a suitable approximation to the memory function of theQGLE. The solution of the QGLE requires as input static, equilibrium informationthat can be generated using an appropriate path integral Monte Carlo (PIMC)scheme (65).

We focus in this section on the quantum mode-coupling formulation for theintermediate scattering function describing collective density fluctuations in neatquantum liquids. We also describe the PIMC scheme that we have developed toobtain the proper static input required for the solution of quantum mode-couplingequations. We note that the approach we review is not limited to this correlationfunction, and may be applied to other dynamical properties in dense quantumliquids (44–47).

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QUANTUM MODE-COUPLING THEORY 159

The quantum mode-coupling equations for the intermediate scattering functionhave a very similar structure to their classical counterparts (2–4). However, theydescribe the time evolution of the Kubo transform of the corresponding correlationfunctions, and therefore are fully quantum mechanical in nature. The classicallimit of these equations (including the approximate memory kernels) can easily beobtained by taking the limit h → 0.

Although other formulations are plausible, we have shown that the approachbased on the Kubo transform is the most convenient one (43–49). In particular,its combination with the PIMC method described in Section 2.3 is the most sta-ble computationally. To obtain the corresponding real-time quantum correlationfunction, the standard relation in frequency space must be applied (66):

S(q, ω) = βhω

2

[coth

(βhω

2

)+ 1

]Sκ (q, ω), 1.

where Sκ (q, ω) is the Fourier transform of the Kubo transform of the intermediatescattering function, Fκ (q, t), given by

Sκ (q, ω) = 1√2π

∞∫−∞

dteiωt Fκ (q, t), 2.

and S(q, ω) is the dynamic structure factor (also the corresponding Fourier trans-form of the non-Kubo intermediate scattering function).

2.1. Quantum Generalized Langevin Equation

The derivation of the QGLE for the intermediate scattering function follows fromthe work of Zwanzig (67) and Mori (68, 69). We begin with the definition of twodynamical variables, the quantum collective density operator,

ρq =N∑

α=1

eiq·rα , 3.

and the longitudinal current operator,

jq = 1

2m|q|N∑

α=1

[(q · pα)eiqrα + eiqrα (pα · q)

], 4.

where rα is the position vector operator of particleα with a conjugate momentum pα

and mass m, and N is the total number of particles. The quantum collective densityoperator and the longitudinal current operator satisfy the continuity equation, ˙ρq =iq jq, where the dot denotes a time derivative, i.e., ˙ρq = i

h [H , ρq].Next we define the projection operator, Pκ (70):

Pκ = 〈A†, · · ·〉〈A†, Aκ〉 Aκ , 5.

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160 RABANI REICHMAN

where the vector operator A combines the two dynamical variables, ρq and jq, toform a row vector, A = (ρq, jq). In the above equation the notation κ implies thatthe quantity under consideration involves the Kubo transform given by (66)

Aκ = 1

βh

βh∫0

dλe−λH AeλH , 6.

where H is the Hamiltonian of the system, β = 1kBT is the inverse temperature,

and 〈· · ·〉 denotes a quantum mechanical ensemble average.Following the standard procedure (1, 67–69) and using the projection opera-

tor defined above, the time evolution of the Kubo transform of the intermediatescattering function Fκ (q, t) = 1

N 〈ρ†q, ρ

κq (t)〉 is given by the exact equation

d2 Fκ (q, t)

dt2+ ω2

κ (q)Fκ (q, t) +t∫

0

dt ′K κ (q, t − t ′)d Fκ (q, t ′)

dt ′ = 0. 7.

In the above equation ω2κ (q) is the Kubo transform of the frequency factor given

by (for n = 1)

ω2nκ (q) = 1

Sκ (q)

⟨dnρ†

q

dtn,

dnρκq

dtn

⟩. 8.

Sκ (q) = Fκ (q, 0) = 1N 〈ρ†

q, ρκq (0)〉 is the Kubo transform of the static structure

factor. The Kubo transform of the memory kernel appearing in Equation 7 is relatedto the Kubo transform of the random force and is formally given by

K κ (q, t) = 1

N J κ (q)

⟨R†

q, ei(1−Pκ )Lt Rκq

⟩, 9.

where the random force operator is given by

Rq = d jqdt

− i |q| J κ (q)

Sκ (q)ρq. 10.

In the above equations, J κ (q) = 1N 〈 j†

q , jκq (0)〉 is the Kubo transform of the zero-

time longitudinal current correlation function, and L = 1h [H , · · ·] is the quantum

Liouville operator.

2.2. Quantum Mode-Coupling Approximation

The above QGLE for the Kubo transform of the intermediate scattering functioncombined with the expression for its memory kernel is simply another way torephrase the quantum Wigner-Liouville equation for the dynamical variable ρq(t).The difficulty of numerically solving the Wigner-Liouville equation for a many-body system is shifted to the difficulty of evaluating the memory kernel itself. Toovercome this difficulty we have developed a theory to calculate K κ (q, t) that is

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based on a combination of kinetic and mode-coupling theories. We use an approx-imate form for the memory kernel (6) given by K κ (q, t) ≈ K κ

b (q, t) + K κmct (q, t).

This form has been applied successfully to a great number of physically inter-esting classical problems including the study of density and current fluctuations(2–4), solvation and relaxation dynamics (71–73), reaction dynamics (74, 75), andnonlinear spectroscopy (76–78) in classical liquids.

2.2.1. BINARY PORTION We now discuss the quantum-mechanical generalizationof the above approximation that has been suggested by Gotze and Lucke (41,42) to study density fluctuations in superfluid helium. To obtain the quantumbinary portion, K κ

b (q, t), and the quantum mode-coupling portion, K κmct (q, t), of

the memory kernel, we follow the standard procedure, with the projection operatorand memory kernel given by Equations 5 and 9, respectively. The fast-decayingbinary term is determined from a short-time expansion of the exact Kubo transformof the memory function, and is given by

K κb (q, t) = K κ (q, 0) f [t/τ (q)], 11.

where the lifetime is given by

τ (q) = [−K κ (q, 0)/2K κ (q, 0)]−1/2. 12.

In the above equations K κ (q, 0) and K κ (q, 0) are the zero- and second-timemoments of the memory kernel, and are given by the exact relations,

K κ (q, 0) = ω4κ (q)

ω2κ (q)

− ω2κ (q) 13.

and

K κ (q, 0) = −ω6κ (q)/ω2

κ (q) + [ω4

κ (q)/ω2κ (q)

]2, 14.

where ω2nκ (q) is given by Equation 8. The shape of the function, f (x), in Equation

11 can be approximated by a Gaussian exp(−x2) or sech2(x). Both forms havebeen used in the study of classical liquids (2) and are exact to second order in time.

In practice we often use a simpler expression for the lifetime given in terms ofK κ (q, 0) by the following approximation (79):

τ κ (q) = [γ K κ (q, 0)]−1/2, 15.

where γ is an empirical constant. This approximation involves only the first twomoments ω2

κ (q) and ω4κ (q), and not ω6

κ (q). Although the calculation of ω6κ (q) is

possible, it involves a higher order derivative of the interaction potential and thusbecomes a tedious task for the path integral Monte Carlo technique. Interestingly,we find that the approximation to the lifetime given by Equation 12 is accurate towithin 5% for classical fluids for γ = 3/4.

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162 RABANI REICHMAN

2.2.2. MODE-COUPLING PORTION The slowly decaying mode-coupling portion ofthe memory kernel, K κ

mct (q, t), must be obtained from a quantum mode-couplingapproach. The basic idea behind this approach is that the random force projectedcorrelation function, which determines the memory kernel for the intermediatescattering function [cf. Equation 9], decays at intermediate and long times predom-inantly into modes that are associated with quasi-conserved dynamical variables.It is reasonable to assume that the decay of the memory kernel at long times willbe governed by those modes that have the longest relaxation time. The slow decayis basically attributed to couplings between wavevector-dependent density modesof the form

Bk,q−k = ρkρq−k, 16.

where translational invariance of the system implies that the only combinationof densities whose inner product with a dynamical variable of wavevector −q isnonzero is ρk and ρq−k.

The simplest way to extract the slow mode-coupling portion of the memory ker-nel is to introduce another projection operator, Pκ

m(q), which projects any variableonto the subspace spanned by Bk,q−k, and is given by

Pκm(q) = 1

2

∑k

Bκk,q−k〈B†

k,q−k, · · ·〉⟨B†

k,q−k, Bκk,q−k

⟩−1, 17.

where Bκk,q−k = ρκ

k ρκq−k. The factor of 1

2 in Equation 17 ensures that Pκm(q)2 =

Pκm(q). The first approximation made by mode-coupling theory replaces the pro-

jected time evolution operator, ei(1−Pκ )Lt , by its projection onto the subspacespanned by Bk,q−k. Under this approximation the slow mode-coupling portionof the memory kernel reads

K κmct (q, t) = 1

N J κ (q)

⟨R†

q,Pκm(q)eiLtPκ

m(q)Rκq

⟩. 18.

The second approximation involves the factorization of four-point density cor-relations into a product of two-point density correlation (80, 81). The four-pointdensity correlation arises from terms likePκ

m(q)Rκq in Equation 18. Under these two

approximations the slow mode-coupling portion of the memory kernel is given by

K κmct (q, t) = 2

(2π )3n J κ (q)

∫dkV κ (q, k)V κ (q, k)

× [Fκ (k, t)Fκ (|q − k|, t) − Fκ

b (k, t)Fκb (|q − k|, t)

], 19.

where n is the liquid number density. As can be seen, the above expression forthe mode-coupling portion of the memory kernel couples different density modes.The strength of coupling between the modes is determined by the vertices V κ (q, k)and V κ (q, k). As will become clear below, this coupling is essential to properlydescribe the long-time phenomena of quantum liquids.

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The binary term of the Kubo transform of the intermediate scattering functionappearing in the above equation, Fκ

b (q, t), is obtained from a short time expansionof Fκ (q, t) similar to the expansion used for the binary term of K κ (q, t), and isgiven by

Fκb (q, t) = Fκ (q, 0) exp

[−1

2ω2

κ (q)t2

]. 20.

The subtraction of the product of terms in Equation 19 involving Fκb (q, t) pre-

vents over-counting the total memory kernel at short times (as the binary portion ofthe memory kernel is exact to second order in time). In other words, the subtractionensures that the even time moments of the total memory kernel are exact to forthorder in time.

Finally, the vertices in Equation 19 are given by

V κ (q, k) = 1

2N

[ ⟨ρ

†kρ

†q−k,

ddt jκ

q

⟩Sκ (k)Sκ (|q − k|) − i |q| J κ (q)

Sκ (q)

⟨ρ

†kρ

†q−k, ρ

κq

⟩Sκ (k)Sκ (|q − k|)

]21.

and

V κ (q, k) = 1

2N

[ ⟨ddt j†

q , ρκk ρκ

q−k

⟩Sκ (k)Sκ (|q − k|) − i |q| J κ (q)

Sκ (q)

⟨ρ†

qρκk ρκ

q−k

⟩Sκ (k)Sκ (|q − k|)

]. 22.

The second vertex involves a double Kubo transform (82). These vertices in-clude a three-point static correlation of density modes and longitudinal currentmodes. In the applications reported below we have calculated them using the fol-lowing approximations for the three-point correlation functions. For 〈ρ†

qρκk ρκ

q−k〉the (Kubo) convolution approximation has been developed (83), leading to

1

N

⟨ρ†

qρκk ρκ

q−k

⟩ ≈ S(k)Sκ (|q − k|)Sκ (q), 23.

where S(k) is the quantum mechanical static structure factor (non-Kubo trans-formed version). For 〈 d

dt j†q , ρκ

k ρκq−k〉 we have used the fact that the Kubo transform

J κ (q) can be approximated in many cases by kBT/m (m is the mass of the particleand T is the temperature), within an error that is less than 1% for the relevant qvalues studied in this work. Based on this fact, we approximate 〈 d

dt j†q , ρκ

k ρκq−k〉 by

1

N

⟨d

dtj†q , ρκ

k ρκq−k

⟩≈ −kBT

mq

[q · kSκ (|q − k|) + (q − k) · qSκ (k)

]. 24.

The combination of these approximations leads to a simplified vertex given by

V κ (q, k)† = V κ (q, k) ≈ ikBT

2mq

(q · kSκ (k)

+ (q − k) · qSκ (|q − k|) − q2 S(k)

Sκ (k)

). 25.

At high values of k the vertex should decay to zero, whereas the approximationfails to do so. Hence in the applications discussed below we have employed a

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164 RABANI&squf;REICHMAN

cutoff to overcome the shortcoming of our approximation. The choice of cutoff isdiscussed below.

In summary, to obtain the Kubo transform of the intermediate scattering func-tion, the frequency factor ω2

κ (q) and the memory kernel K κ (q, t) are required asinput. Because the memory kernel depends on Fκ (q, t), the QGLE for the interme-diate scattering function (Equation 7) must be solved self-consistently. The time-independent terms in the memory kernel and the frequency factor can be obtainedfrom static equilibrium input. However, these terms involve thermal averages overoperators that combine positions and momenta of all particles. In the subsection be-low we review the PIMC technique suitable for thermal averages of such operators.

2.3. Path Integral Monte Carlo Scheme

In this subsection we describe the PIMC scheme we have developed that is suitablefor the calculation of the time-independent terms needed as input for the memorykernel and the frequency factor. These static terms involve thermal averages overoperators that combine positions and momenta of all particles and thus special caremust be taken. As noted by Schulman the calculational rules for such averages canbe delicate (84).

For the sake of simplicity and clarity we describe the method for the Kubotransform of a general average of the form

CκOO = ⟨

OαOκα′⟩ = 1

βh

βh∫0

dλCOO (λ) = 1

βhQ

βh∫0

dλTr e−(β−λ)H Oαe−λH Oα′ ,

26.where Q = Tr exp(−β H ) is the partition function and the operator Oα is a functionof positions of all particles and momentum of particle α and is of the general form

Oα = [pαG(r) + G(r)pα] , 27.

where G(r) is an arbitrary well-behaved complex function, and r ≡ r1 · · · rN is ashorthand notation for the position vectors of all particles. The derivation of theKubo transform of an average that contains higher powers of momentum can beobtained following similar lines given below (65).

Using the coordinate representation the trace in Equation 26 can be written as

COO (λ) = 1

Q

∫drdr′〈r|e−(β−λ)H O†

αe−λH |r′〉〈r′|Oα′ |r〉. 28.

The interval β can be discretized into P Trotter slices of size ε = β/P , suchthat λ ≡ λ j = ( j − 1)ε, where j is the index of the slice. By inserting a completeset of states between the short imaginary time propagators, it is easy to show that

COO (λ j ) =∫

dr1 · · · drP+1 Pj (r1, · · · , rP+1)

× 〈rP− j+1|Oα|rP− j+2〉〈rP+1|Oα′ |r1〉, 29.

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where the open chain probability, Pj (r1, · · · , rP+1), is given by

Pj (r1, · · · , rP+1) = 1

Q

P∏s =P− j+1

〈rs |e−ε H |rs+1〉. 30.

Using the coordinate representation of the matrix element of the operator Oα ,

〈r′|pαG(r) + G(r)pα|r〉 = i[G(r)∇r′

αδ(r − r′) − G(r′)∇rα

δ(r − r′)], 31.

and the well known relation∫dr f (r)∇rα

[δ(r)] = −∫

drδ(r)∇rα[ f (r)], 32.

it can be shown that

COO (λ j ) = −∫

dr1 · · · drP+1δ(r1 − rP+1)δ(rP− j+2 − rP− j+1)

×[G†(rP− j+2)∇rP− j+1

α− G†(rP− j+1)∇rP− j+2

α

]×

[G(r1)∇rP+1

α′ − G(rP+1)∇r1α′

]Pj (r1, · · · , rP+1). 33.

Note that the pairs G(r) and ∇rαhave different arguments in Equation 31, which

is reflected also in the different imaginary time-slice of G(r) and ∇rαin Equation

33. This guarantees that higher derivatives of the function G(r) are not required;however, the more important consequence is that this computational scheme ismore stable and thus more accurate (65). The final step involves the differentiationof Pj (r1, · · · , rP+1) and the integration over the two delta functions in Equation33, which leads to

COO (λ j ) = − 1

4ε2

∫dr1 · · · drP P(r1, · · · , rP )

G†(rP− j+1)G(r1)(rP− j+2α − rP− j

α

)(r2α′ − rP

α′), 34.

for j = 2 · · · P − 1. In the above equation a second-order Trotter split for theshort imaginary time propagators was used and only the term in 1/ε was kept.P(r1, · · · , rP ) is the regular sampling function used in the standard cyclic PIMCmethod (with rP+1 ≡ r1 and r0 ≡ r1).

3. NORMAL QUANTUM LIQUIDS

In order to assess the accuracy of the proposed QMCT it is necessary to compare theprediction of the theory to other theoretical/computational methods and to experi-mental results. Here we provide a review of results obtained using the QMCT forcollective density fluctuations in dense liquids, including liquid ortho-deuterium

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166 RABANI REICHMAN

and liquid para-hydrogen. Because exact solvable models for realistic quantumliquids do not exist, we have selected a set of simple liquids to test our theory.Several reasons lead us to study these systems. First, despite the fact that theseliquids may be treated as Boltzmann particles without the complexity of numer-ically treating particle statistics (85, 86), they still exhibit some of the hallmarksof highly quantum liquids. Recent theoretical (43–49, 88–97; and P.J. Rossky,private communication) and experimental studies (89, 90, 99–104) on liquid deu-terium/hydrogen show that these dense liquids are characterized by quantum dy-namical susceptibilities that are not reproducible using classical theories. In fact,para-hydrogen in the classical system is a nondiffusive ordered solid (88). Fur-thermore, simple models describing the pair interaction potential for these liquidsare readily available (105, 106). Finally, a large body of literature is available, andtherefore it is possible to compare the prediction of the present theory with othercomputational/experimental results.

To further motivate the use of the QMCT for liquid ortho-deuterium and forliquid para-hydrogen we have compared the potential of mean force (PMF) forthese quantum liquids to the PMF for a classical liquid metal under dense liquidconditions. The results are shown in Figure 1. It is well known that for systemswith a soft PMF, such as liquid metals (107, 108), a classical version of the mode-coupling theory (MCT) provides a quantitative description of density fluctuations(2–4). Despite the fact that the interaction potential of the quantum liquids is very

Figure 1 Comparison of the potential of mean force for quantum and classical liquids.The solid lines are the results for ortho-deuterium and para-hydrogen. Dashed linesand dotted lines are the results for a classical liquid lithium and Lennard-Jones (LJ)fluid, respectively. Also shown is the bare pair-potential for the quantum liquid (105).

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different from that of a liquid metal, the PMF for the quantum liquids is soft,and resembles that of the classical liquid metal. Also shown is the PMF for aclassical Lennard-Jones (LJ) fluid, which is steeper and less harmonic around thefirst minimum. This comparison provides anecdotal evidence that QMCT may infact be more accurate than MCT for classical liquids.

3.1. Technical Simulations Details

The solution of the QGLE for the intermediate scattering function describingcollective density fluctuations requires as input the Kubo transform of the staticstructure factor Sκ (q) and the static structure factor itself S(q), the Kubo transformof the zero-time longitudinal current correlation function J κ (q), and the frequencyfactors ω2n

κ (q). To obtain these static quantities we have performed PIMC simu-lations for liquid deuterium/hydrogen using the NVT ensemble with 256 particlesinteracting via the Silvera-Goldman potential (105, 106), where the entire moleculeis described as a spherical particle, so the potential depends only on the radial dis-tance between particles. In the path integral simulation the imaginary time intervalwas discretized into P Trotter slices of size ε = β/P , with P = 20 and P = 50 forliquid ortho-deuterium and liquid para-hydrogen, respectively. A primitive splitfor each imaginary time slice was used, and the staging algorithm (109) combinedwith a centroid move was employed for the Monte Carlo moves. Approximately 107

Monte Carlo passes were made; each pass consisted of attempting staging and cen-troid moves in all atoms and for all the beads that were staged. The acceptance ratiosfor staging and centroid moves were approximately ∼1/4 and ∼1/2, respectively.

The lifetime τ κ (q), the zero value of the memory kernel K κ (q, 0), and the fre-quency factor ω2

κ (q) for all liquids studied are shown in Figure 2. These quantitiesalong with the static structure factor (shown for each liquid below) were usedto generate the vertex V κ (q, k) and the memory kernel K κ (q, t) ≈ K κ

b (q, t) +K κ

mct (q, t) needed for the solution of the QGLE for Fκ (q, t). To obtain K κmct (q, t)

we solved Equation 19 with a cutoff in |q − k| to overcome the divergent behaviormentioned above (see Equation 25). The choice of this cutoff is straightforwardgiven that the approximate vertex decays to zero at intermediated values of k beforethe unphysical divergent behavior steps in. We have used |q − k|cut = 5.66 A−1

and |q − k|cut = 4.73 A−1 for ortho-deuterium and para-hydrogen, respectively.Because the memory kernel depends on the value of the Fκ (q, t) itself, the solu-tion must be obtained self-consistently. The initial guess for the memory kernelwas taken to be equal to the fast binary portion (cf. Equation 20). The integro-differential equations were solved using a fifth-order Gear predictor-corrector al-gorithm (110). Typically, less than 10 iterations were required to converge thecorrelation function, with an average error smaller than 10−8 percent.

3.2. Liquid Ortho-Deuterium

The first test of the QMCT was done for liquid ortho-deuterium at T = 20.7Kand ρ = 0.0254 A−3. More details on the calculations can be found in Reference

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168 RABANI REICHMAN

Figure 2 Plots of the lifetime τ κ (q) (lower panel), the zero-time value of the memorykernel K κ (q, 0) (middle panel), and the frequency factor ω2

κ (q) (upper panel) for liquidortho-deuterium (solid line) and liquid para-hydrogen (dotted line).

(49). In Figure 3 we plot the dynamic structure factor, S(q, ω), calculated fromthe QMCT. The results are compared to the experiments of Mukherjee et al. (102)for several values of q (the theoretical values of q are slightly different from theexperiments because of the limitations associated with the constant volume simu-lation approach). The results shown are normalized such that

∫ ∞−∞ dωS(q, ω) = 1

for each q .The overall agreement between the QMCT and the experiments is very good.

In particular the QMCT captures the position of both the low- and high-intensitypeaks and their width for nearly all wavevectors shown. The high-intensity peakat finite frequency gives rise to pronounced oscillatory behavior in the intermedi-ate scattering function (see Figure 4 below), signifying the existence of collectivecoherent excitations in liquid ortho-deuterium. As the value of q approaches qmax

[qmax ≈ 2 A−1 is the value of q where S(q) reaches its first maximum] the finitefrequency peak disappears. The fact that for limiting cases the QMCT somewhat

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Figure 3 (Lower panels) A plot of the dynamic structure factor of liquid ortho-deuterium for several values of q (in units of A−1). The thick solid lines and dashedlines are Quantum Mode-Coupling Theory (QMCT) and Maximum Entropy (MaxEnt)results, respectively. The thin solid lines are the results of Quantum Viscoelastic Model(QVM) (47), and open circles are the experimental (Exp.) results (102). (Upper panel)A plot of the static structure factor of liquid ortho-deuterium computed from the PathIntegral Monte Carlo simulations.

underestimates the width of the low-intensity peak may be attributed to the broad-ening associated with the instrumental resolution (102).

Several approximations were introduced to derive the working equations withinthe QMCT. In order to test the accuracy of the approximation for the vertexgiven by Equation 25, we have replaced it with its classical limit (2). The re-sults for the low-frequency part of S(q, ω) (not shown) are in poor agreementwith the experiments. In particular the low-intensity peak is absent at interme-diate values of q , and the higher frequency peak is somewhat shifted. Thus, aproper description of the vertex given by the approximation in Equation 25 is

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170 RABANI REICHMAN

Figure 4 Plots of the Kubo transform of the intermediate scattering function (leftpanels), and the Kubo transform of the quantum binary and mode-coupling portionsof the memory kernel (right panels) for liquid ortho-deuterium. The values of q in-dicated in the panels are in units of A−1. QMCT, Quantum Mode-Coupling Theory;MaxEnt, Maximum Entropy.

necessary for a quantitative description of the dynamics at intermediate and longtimes.

In Figure 3, we also compare the results obtained from the QMCT to resultsobtained from a quantum viscoelastic model (43) and to results obtained from anumerical analytic continuation approach based on the maximum entropy method(33, 37). The quantum viscoelastic model (QVM) is similar in spirit to the presentapproach. It is based on the same QGLE developed in Section 2. However, thequantum viscoelastic approximation to the memory kernel assumes a single re-laxation time given by a simple exponential decay of the memory kernel for eachq value. As a result, the QVM completely neglects couplings between differentdensity modes (43, 47). In comparison to the QMCT results, the QVM fails toreproduce the low-intensity peak observed in S(q, ω). This peak can be associatedwith the long-time relaxation dynamics of density fluctuations. Thus, the inclusionof the quantum mode-coupling portion to the memory kernel that couples differentdensity modes is important to properly describe long-time phenomena.

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Turning to the results obtained from a numerical analytic continuation approachbased on the maximum entropy (MaxEnt) method, it is clear why MaxEnt failsto provide a quantitative description of the density fluctuations in liquid ortho-deuterium. It is well known that the MaxEnt approach fails when several nearlyoverlapping frequency scales arise in a problem. This is clearly the case when theMaxEnt approach predicts a single frequency peak instead of two, at a positionthat is approximately the average position of the two experimental peaks. Onlywhen the dynamics are characterized by a single relaxation time, like the case atqmax, does the MaxEnt approach provide quantitative results. On the other hand,MaxEnt provides quantitative results for the short-time dynamics as reflected inthe width of the finite frequency peak in S(q, ω) (see also the discussion of Figure4 below).

In Figure 4, we show the results for the Kubo transform of the intermediatescattering function for liquid ortho-deuterium obtained from the QMCT. The re-sults are compared with the MaxEnt method for several values of q. The rightpanels of Figure 4 show the binary and mode-coupling portions of the memorykernel obtained from the QMCT for the same values of q. The MaxEnt resultswere generated from S(q, ω) by taking the Fourier transform of S(q, ω) dividedby the proper frequency factor to “Kubo” the real-time correlation function.

A large discrepancy between the QMCT and MaxEnt results for the Kubotransform of the intermediate scattering function at intermediate and long timesfor all values of q ≤ qmax is observed. This is expected because of the pooragreement between the two approaches in the frequency domain. The agreementbetween the QMCT, which is exact to order t6, and the MaxEnt results at shorttimes is expected. However, the MaxEnt result fails to reproduce the proper os-cillatory behavior observed in Fκ (q, t) that gives rise to a peak in S(q, ω) at afinite frequency (see Figure 3 above). Common to both approaches is that as qapproaches qmax the decay rate of the Kubo transform of the intermediate scatter-ing function decreases, giving rise to a quantum mechanical de Gennes narrowingof the dynamic structure factor. In addition, we observe an increase in the de-cay rate of Fκ (q, t) at high values of q, where the two approaches yield nearlyidentical results on a sub-picosecond timescale. The agreement between the twoapproaches at high values of q is not surprising, as the decay of the intermediatescattering function is rapid, on timescales accessible to the analytic continuationapproach.

In the right panels of Figure 4 we plot binary and mode-coupling portions of thememory kernels for different q values. A clear separation of timescales betweenthe decay of the fast binary portion of the memory kernel and that of the slowmode-coupling portion is evident. This separation of timescales is at the heart ofthe quantum mode-coupling approximation. We find that the contribution of thequantum mode-coupling portion of the memory kernel is significant for all valuesof q below qmax, whereas at the highest value of q shown, the contribution ofK κ

mct (q, t) is negligible. This is consistent with results obtained for classical densefluids (2–4), signifying the need to include the long-time portion of the memorykernel for a quantitative description of the dynamics at all times.

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172 RABANI REICHMAN

3.3. Liquid Para-Hydrogen

In many ways liquid para-hydrogen is very similar to liquid ortho-deuterium.Although the interaction potential is the same for both liquids within the Born-Oppenheimer approximation (for zero molecular angular momentum), the phasediagram is somewhat different because of the lower mass of para-hydrogen. There-fore, we expect that the accuracy of the QMCT will be similar to that obtainedfor ortho-deuterium. As will become clear below, this is not the case. Althoughqualitative features are captured by QMCT and to some extent by the QVM (andnot by the MaxEnt analytic continuation approach), the agreement between theexperiments and the theory is far from the quantitative agreement observed forliquid ortho-deuterium.

In Figure 5, we show the results for the Kubo transform of the intermediatescattering function for liquid para-hydrogen at T = 14K and ρ = 0.0235 A−3.This density was chosen to be the average density under zero pressure (111). In

Figure 5 Plots of the Kubo transform of the intermediate scattering function (leftpanels), and the Kubo transform of the quantum binary and mode-coupling portions ofthe memory kernel (right panels) for liquid para-hydrogen. The values of q indicatedin the panels are in units of A−1. QMCT, Quantum Mode-Coupling Theory; MaxEnt,Maximum Entropy.

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the left panels of Figure 5 we compare the results obtained from the QMCT withthose obtained from the MaxEnt method for several values of q. The right panelsof Figure 5 show the binary and mode-coupling portions of the memory kernelobtained from the QMCT for the same values of q. The agreement between theQMCT and the MaxEnt results is good at short times. Both approaches capturethe quantum mechanical de Gennes narrowing associated with the decrease inthe decay rate of the Kubo transform of the intermediate scattering function as qapproaches qmax. In addition both capture the increase in the decay rate of Fκ (q, t)at high values of q , where the two yield nearly identical results. However, similar tothe case of liquid ortho-deuterium, at intermediate and long times, when q ≤ qmax,the results deviate markedly from each other.

The QMCT results show pronounced oscillatory behavior in Fκ (q, t) that givesrise to two peaks in S(q, ω) at a finite frequency as can be seen in Figure 6.Only one peak is observed in the MaxEnt approach. Similar peaks were alsoobserved experimentally by Bermejo et al. (103) and using the centroid moleculardynamics (CMD) (26) approach by Kinugawa (92). As can be seen in Figure 6,the experimental result for the higher frequency peak is slightly shifted to lowerfrequencies with a smaller amplitude.

The discrepancy between the QMCT and the experiments is not completelyclear. The fact that the QMCT theory provides a quantitative description for densityfluctuations in liquid ortho-deuterium and to some extent fails for liquid para-hydrogen is surprising and will remain an open question for future investigation.As mentioned above, the two liquids share many common features. Their densitiesand melting temperatures are comparable and the interparticle interactions areidentical within the Born-Oppenheimer approximation. Thus, we expected theQMCT to provide a quantitative description of collective density fluctuations inliquid para-hydrogen as well.

One missing factor is the instrument response function which we have notincluded. One would expect that this would lower the amplitude of the finitefrequency peaks that we have calculated. It is also possible that several approxi-mations incurred by our approach led to these differences. The position of the finitefrequency peak depends mainly on the value of ω2

κ (q), a static quantity that hasbeen determined exactly by the PIMC method. To test the effects of the memorykernel on the position of the peak, we have modified the lifetime by changing γ

between 1/4 and 3 (the results reported above assume γ = 3/4). The results forthe position of the finite frequency peak and its width depend weakly on the valueof γ . We have also found that the level of accuracy made for the approximate ver-tex given by Equation 25 is similar to that made for liquid ortho-deuterium. It isinteresting to note that the discrepancy of peak location is similar to that producedby the CMD result (92). However, the use of CMD for this problem is unjustifiedbecause the density operator is not a linear function of phase space coordinates.

Finally, we note that the agreement between the QMCT and the MaxEnt methodsis excellent for self-transport in para-hydrogen (87), whereas the results for thedynamic structure factor plotted in Figure 6 show significant deviations betweenthe two approaches. Most likely the discrepancy comes from multiple frequency

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174 RABANI REICHMAN

Figure 6 (Lower panels) A plot of the dynamic structure factor of liquid para-hydrogen for several values of q (in units of A−1). The thick solid lines and dashedlines are Quantum Mode-Coupling Theory (QMCT) and Maximum Entropy (MaxEnt)results, respectively. The thin solid lines are the results of Quantum Viscoelastic Model(QVM) (43), and open circles are the experimental (Exp.) results (103). (Upper panel)A plot of the static structure factor of liquid para-hydrogen computed from Path IntegralMonte Carlo simulations.

peaks in S(q, ω). MaxEnt is nearly quantitative as long as S(q, ω) is characterizedby a single peak, namely at high q values where the decay of Fκ (q, t) is on asub-picosecond timescale, and at q values near qmax.

4. SUPERCOOLED QUANTUM LIQUIDS

When a liquid is rapidly cooled below its melting temperature, the system entersa glassy phase, where the relaxation is characterized by very slow dynamics, typ-ically given by a stretched exponential form (50). The crossover temperature for

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such a behavior for most glass-forming materials—such as polymer glasses, spinglasses, and structural glasses—occurs at a temperature where quantum fluctua-tions are negligible. Several exceptions, when quantum fluctuations are comparableto thermal fluctuations, have been studied, including the so-called Coulomb glass(112, 113) and quantum spin glasses (57–60).

In this section we discuss the properties of a class of a quantum glass-formingliquid, namely, that of a structural quantum glass. Several important open questionsrelated to the dynamical and thermodynamical properties of these systems arise.First, is it possible to form a structural glass where quantum fluctuations are com-parable to thermal fluctuations? Can a structural supercooled quantum liquid meltupon cooling because of a transition to the superfluid phase? Are the dynamicalsignatures of a structural quantum glass similar to those of its classical counterpart?

In principle, our quantum mode-coupling theory can also be used to study arange of problems related to the dynamics of supercooled quantum systems. In or-der to study such interesting problems, and to provide answers to some of the abovequestions, we must further develop our theoretical apparatus. So far we have fo-cused on the dynamical properties of neat fluids (where the entire molecule was de-scribed as a spherical particle). Such liquids do not show glassy behavior due to thefact that even on the timescale of computer simulations, when the liquid is rapidlycooled below its melting temperature, the system simply crystallizes (114). Thisis also true for quantum liquids such as those studied in the previous section (61).

Thus in order to study the dynamical behavior of supercooled quantum liq-uids it is necessary to generalize the quantum mode-coupling theory describedabove to the case of mixtures of solvents. A generic model of LJ binary mixtures,sometimes referred to as the Kob-Andersen model, has been studied in detail forclassical glass-forming liquids (115–118). In this section a quantum mechanicalgeneralization of the Kob-Andersen model is described.

Before proceeding, we would like to point out one possible experimental re-alization of a structural quantum glass: a mixture of ortho-deuterium and para-hydrogen (61). Indeed, preliminary results described elsewhere (61) indicate that itis possible to supercool a quantum mixture of ortho-deuterium and para-hydrogen,whereas a similar classical system crystallizes. Furthermore, the onset temperaturefor slow relaxation occurs at temperatures higher than the predicted transition tothe superfluid phase (86, 119).

4.1. Quantum Mode-Coupling Theory for Binary Mixtures

We consider a two-component system of quantum particles with concentrationsx j = N j/N , where N j is the number of particles for species j = 1, 2. Similar tothe case of neat fluids, we focus on density fluctuations described by the partialintermediate scattering functions

Fκi j (q, t) = 1

N

⟨ρ

†i (q), ρκ

j (q, t)⟩, 35.

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176 RABANI REICHMAN

where ρ j (q) = ∑N j

α=1 eiq·rα is the density operator at wavevector q for species j .Following the standard projection operator procedure described in Section 2, thetime evolution of the Kubo transform of the partial intermediate scattering functionis given by the exact matrix equation

d2Fκ (q, t)

dt2+ Ω2

κ (q)Fκ (q, t) +t∫

0

dt ′Kκ (q, t − t ′)dFκ (q, t ′)

dt ′ = 0, 36.

where Fκ (q, t) is a 2 × 2 matrix with components Fκi j (q, t). In the above equation

Ω2nκ (q) = J2n

κ (q)[Sκ (q)]−1 (for n = 1) is a 2 × 2 matrix; [Sκ ]i j = Fκi j (q, 0) is the

Kubo transform of the partial static structure factor, and [J 2nκ ]i j = 〈 dn ρ

†i (q)

dtn ,dn ρκ

j (q)

dtn 〉.Following the lines sketched for a neat fluid in Section 2 the memory kernel

matrix Kκ (q, t) ≈ Kκb (q, t)+Kκ

mct (q, t) is approximated by a sum of two matrices,a fast-decaying binary portion and a slower decaying mode-coupling portion. Theformer is obtained from a short-time expansion of the exact memory kernel, andis given by

Kκb (q, t) = Kκ (q, 0) exp

− 3

4Kκ (q, 0)t2

, 37.

where Kκ (q, 0) = Ω4κ (q)[Ω2

κ (q)]−1 − Ω2κ (q). The matrix elements of the mode-

coupling portion of the memory kernel are given by

[K κ

mct (q, t)]

i j ≈ kBT

2mi x j (2π )3n

∑α,β

∑α′,β ′

∫dkV κ

iα,β(q, k)V κjα′β ′ (q, k)

× [Fκ

αα′ (k, t)Fκββ ′ (|q − k|, t) −Fκ

b,αα′ (k, t)Fκb,ββ ′ (|q − k|, t)

],

38.

where the elements of the vertex matrix are given by

V κjαβ(q, k) ≈ q · k

q

[S−1

κ (k)]

jαδ jβ + (q − k) · qq

[S−1

κ (|q − k|)] jβδ jα

− q

x j[S(k)]αβ

[S−1

κ (k)]αβ

δ jαδ jβ. 39.

In the above equation, [S−1κ (q)]i j is the i j element of the inverse matrix of Sκ (q)

and δi j is Kronecker’s delta function. As in the case of a neat fluid, the binary termof the Kubo transform of the partial intermediate scattering function appearing inEquation 38, Fκ

b (q, t), is obtained from a short-time expansion of Fκ (q, t), and isgiven by

Fκb (q, t) = Fκ (q, 0) exp

(−1

2Ω2†

κ (q)t2

). 40.

The symbol † in the above equation designates the transpose of the matrixΩ2

κ (q).

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4.2. Density Fluctuations in Supercooled QuantumBinary Mixtures

We focus on a 80:20 mixture (x1 = 0.8 and x2 = 0.2) of LJ particles with a massequal to that of a hydrogen molecule. The pair interaction potential is given byVi j = 4εi j [(σi j/r )12 − (σi j/r )6], i, j = 1, 2. The parameters for the interactionsbetween particles 1 were chosen to mimic that of the Silvera-Goldman para-hydrogen pair-potential; ε11 = 30 K and σ11 = 3 A. For particles 2 we tookε22 = 0.5ε11 and σ22 = 0.88σ11, and the cross terms were ε12 = 1.5ε11 andσ12 = 0.8σ11. The results reported below were obtained for a liquid numberdensity that is somewhat above the triple-point density of liquid para-hydrogen(n = 0.8/σ 3

11 and 0.9/σ 311), and for a range of temperatures above and below the

melting temperature of the mixture ( 13 ≤ kBT

ε11≤ 1). These densities were chosen

to ensure that the onset temperature of slow relaxation is low enough to overlapwith the onset of quantum fluctuations.

To obtain the static quantities required for the solution of the QGLE for thebinary mixtures we have performed PIMC simulations using the NVT ensemblewith N = 256 particles. In the path integral simulation the imaginary time intervalwas discretized into P Trotter slices of size ε = β/P , with P = 100 (this value wassufficient to converge the results at the lowest temperature studied). The stagingalgorithm (109) combined with a centroid move was employed for the Monte Carlomoves. The number of beads that were staged was 50 above kBT/ε11 = 1

2 and 25below this temperature. The PIMC simulations were started from the face-centeredcubic (fcc) lattice configuration at the highest temperature studied (kBT/ε11 = 1.0).The number of Monte Carlo passes at each temperature was sufficient to equilibratethe system at that given temperature. The final configuration at each temperaturewas used as the initial configuration for the next temperature.

In Figure 7 we show the partial static structure factors Si j (q) for the two reduceddensities studied (n = 0.8/σ 3

11 and n = 0.9/σ 311) and for selected temperatures.

The interval range shown is the relevant one for the mode-coupling calculationsreported below. Similar to the classical case (117), the dependence of the partialstructure factors on temperature is very smooth. As the temperature decreases (ordensity increases) the height of the first peak in Si j (q) slightly increases.

The Kubo transformed partial intermediate scattering functions for n = 0.8/σ 311

and n = 0.9/σ 311 are plotted in Figures 8 and 9, respectively. The results for each

component shown are plotted at qmax for that component, namely, at σ11qmax = 6.42and σ11qmax = 4.96 for the 11 and 22 components, respectively. Time is measured

in units of√

mσ 211/48ε11.

The short-time decay of the partial intermediate scattering functions, whichis weakly temperature dependent for both densities, can be approximated by aquadratic decay given by Equation 40. This short-time decay is dominated byballistic motion. At intermediate times, a small shoulder begins to form. Thisshoulder is absent in normal liquids. (See Figures 4 and 5 for the normal liquidcase). It becomes more pronounced as the temperature is lowered. At long times,

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Figure 7 Partial structure factor as a function of q. (Left panels) n = 0.8/σ 311 and

kBT/ε11 = 0.4, 0.5 and 0.67. (Right panels) n = 0.9/σ 311 and kBT/ε11 = 0.67, 0.83

and 1.0. The higher first peak of the partial structure factor corresponds to the lowertemperature for each panel, respectively.

the decay of the partial intermediate scattering functions is characterized by asingle relaxation time that can be fit to a simple exponential form.

The two-step relaxation and the appearance of the shoulder are typical of clas-sical supercooled liquids (116). The standard interpretation of this for classicalsystems is that at intermediate times, the presence of the shoulder is associatedwith motion of particles inside the cage (known as the β relaxation), whereas thelonger time decay is associated with hopping of particles outside their cage (knownas the α relaxation). Our results suggest that this interpretation may also hold forsupercooled quantum liquids.

In order to provide a more conclusive picture of slowing down in supercooledquantum liquids we must solve the quantum mode-coupling equations at lower tem-peratures. However, several approximations built into the quantum mode-couplingtheory limit the application to a relatively high-temperature supercooled state. Themajor approximation involves the simplified version of the quantum vertex givenby Equation 39, which is based on the (Kubo) convolution approximation, andleads to unphysical divergence at high values of q. It is sufficient for the descrip-tion of density fluctuations in normal quantum liquids where a cutoff in q providesquantitative results. For the deep supercooled state, couplings to higher q valuesare necessary to properly describe the slowing down of dynamical susceptibilities.

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Figure 8 Semi-log plots of the partial intermediate scattering function (left panels)and log-log plots of the partial memory kernel (right panels) at q = qmax for eachcomponent, at kBT/ε11 = 0.4, 0.5 and 0.67. Slower relaxation corresponds to lowertemperature.

However, at these higher q values the divergent behavior of the simplified vertex in-tervenes and prohibits an accurate calculation of the dynamic response of the liquid.

Let us discuss the results shown in the right panels of Figures 8 and 9, wherewe plot the Kubo transformed partial memory kernel at the same value of q usedfor the partial intermediate scattering function. The decay of the memory kernelis characterized by two relaxation regimes. At short times the quadratic decayobserved is dominated by the binary portion and at long times the decay is givenby a simple exponential form. As the temperature is lowered, the relaxation ofthe memory kernels becomes slower, in agreement with the results for the partialintermediate scattering function.

5. CONCLUSIONS

We have developed a self-consistent quantum mode-coupling theory to study dy-namical correlations in normal and supercooled quantum liquids. The power of thepresent theory is that it uses a short-time expansion of the memory kernel of theQGLE combined with an approximate, physically motivated, long-time approxi-mation to obtain dynamical information at all times. The calculation of the timecorrelation function of interest is accomplished in three steps:

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180 RABANI REICHMAN

Figure 9 Semi-log plots of the partial intermediate scattering function (left panels)and log-log plots of the partial memory kernel (right panels) at q = qmax for eachcomponent at kBT/ε11 = 0.67, 0.83 and 1.0. Slower relaxation corresponds to lowertemperature.

1. Formulation of the exact quantum equations of motion for the dynamicalvariable of interest: Using the projection operator technique of Zwanzig andMori, an exact QGLE for the Kubo transform of the correlation function ofinterest was derived. This QGLE rephrases the quantum Wigner-Liouvilleequation for the dynamical variable of interest, and the difficulty to nu-merically solve the Wigner-Liouville equation is shifted to the difficulty ofevaluating the quantum memory kernel in the QGLE.

2. Approximation to the memory kernel: To overcome this difficulty, we havedeveloped a suitable approximation for the memory kernel modeled on aclassical mode-coupling approach. Three approximations were introduced:First, an approximate closure for the memory kernel given by a sum over thefast quantum binary portion and the slow quantum mode-coupling portionhas been used. Next, the exact projected dynamics of the memory kernelwere replaced with an approximate unprojected form. Finally, we have re-placed four-point density correlations with a product of two-point densitycorrelations in the memory kernel.

3. Solution of the equation of motion: The solution of the QGLE combined withthe approximate form for the memory function requires static equilibriuminput (the short-time moments of the memory kernel, the frequency factor,and the vertices). These properties involve thermal averages over operators

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that combine position and momenta of all particles. To obtain these averageswe have developed a PIMC scheme that is suitable for thermal averages ofsuch operators for a many-body system.

The main limitation of our approach is that, at the present time, the approxima-tions made to the memory kernel of the QGLE have not been justified mathemat-ically. Thus, the accuracy of the approximations can only be tested for classicalliquids, or compared to experiments or other approximate methods, such as thesemiclassical-, centroid-, or analytic continuation–based approaches. We have ap-plied our approach to study both single and collective dynamical susceptibilities innormal and supercooled quantum liquids. In this review, however, we have focusedon the more interesting problem of collective density fluctuations.

The first set of applications describe density fluctuations in normal ortho-deuterium and para-hydrogen. We have compared the results obtained from QMCTwith those from experiments, the numerical MaxEnt analytic continuation ap-proach, and QVM. The results obtained using the QMCT for collective densityfluctuations are in excellent agreement with the experiments for liquid ortho-deuterium. The two other theoretical/numerical approaches deviate from the ex-periments for nearly the entire relevant wavevector range. Failure of the MaxEntresult was attributed to the presence of two peaks in the dynamic structure factor.The fact that our QMCT captures the position of both the low- and high-intensitypeaks in the dynamic structure factor and their width is an important result, asour previous studies in which we invoked a single relaxation time for the memorykernel of the QGLE (QVM) failed to reproduce the low-frequency peak. Thus, it isthe self-consistent treatment of the mode-coupling portion of the memory kernelas well as the proper treatment of the vertex that account for a correct descriptionof the intermediate- and long-time dynamics.

The agreement between the QMCT and experiments for liquid para-hydrogenis not perfect. However, the results obtained from the QMCT are far superior incomparison to the two other theoretical/numerical approaches considered. Thediscrepancy between the QMCT and the experiments for liquid para-hydrogenis not completely clear. It is possible that, along with inaccuracy incurred by ourapproach, there may be some difficulties in extracting the experimental response. Itis interesting to note that the discrepancy of peak location for the dynamic structurefactor is similar to that produced by the centroid molecular dynamics result.

We have also applied the QMCT to study density fluctuations in supercooledquantum liquids. Here, only predictions of the QMCT can be made, as experimen-tal results are not yet available, and all other computational methods are limitedto equilibrium short-time dynamics, far from the relevant timescale for slow dy-namics expected for supercooled quantum liquids. Our study has addressed sev-eral fundamental open questions regarding the dynamics and thermodynamics ofstructural quantum glass-forming liquids. Our preliminary results indicate that it ispossible to supercool a quantum mixture of ortho-deuterium and para-hydrogen,whereas a similar classical system crystallizes. Furthermore, there are indicationsthat the onset temperature for slow relaxation occurs at temperatures higher than the

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182 RABANI REICHMAN

predicted superfluid transition for this mixture. We have also considered a quantummechanical Kob-Andersen model, and generalized the QMCT of neat fluids to thebinary mixture case. Generally, the dynamical properties of the quantum structuralglass-forming liquid are similar to their classical counterparts.

Unlike other techniques, our approach is a theory and thus provides additionalinsight into the dynamics of normal and supercooled quantum liquids. This factimplies several additional advantages. First, the method requires no computation ofsemiclassical trajectories of any kind, thus offering a more efficient (and perhapsmore accurate) numerical scheme to approximate quantum mechanical correla-tion functions. Second, it could be used to study a range of problems related to theslow dynamics of out-of-equilibrium quantum systems, whereas other numericalapproaches are limited to short-time dynamics and/or at equilibrium. In addition,the QMCT may be applied to general liquid state correlation functions, as longas a reasonable approximation to the memory function of interest is used. Fur-thermore, situations where the static distribution is not described by Boltzmannstatistics can easily be handled within the framework of the quantum hydrody-namic approach developed here, as the additional complication of proper particlestatistics may be absorbed into the PIMC calculation of the static input. Anotherimportant advantage of the present approach is that the high temperature/classicallimits can easily be obtained by taking the limit h → 0 in the QGLE and in thememory kernel. In this limit, the Kubo transform of any operator is replaced withthe corresponding classical variable, and the equations of motion reduce to theirclassical mode-coupling counterparts. Finally, because the starting point of themethods used in this work is the exact QGLE, improvements may be made withmore sophisticated approximations to the memory function.

ACKNOWLEDGMENTS

We would like to thank Hans C. Andersen, Bruce J. Berne, Giulio Biroli, Leticia F.Cugliandolo, Kunimasa Miyazaki, and Irwin Oppenheim for very useful discus-sions. This work was supported by The Israel Science Foundation (grant number31/02-1 to E.R.). DR Reichman is an Alfred P. Sloan Foundation Fellow and aCamille Dreyfus Teacher-Scholar. New address for DR Reichman after August1, 2004: Department of Chemistry, Columbia University, New York, NY 10027;[email protected].

The Annual Review of Physical Chemistry is online athttp://physchem.annualreviews.org

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March 15, 2005 20:18 Annual Reviews AR241-FM

Annual Review of Physical ChemistryVolume 56, 2005

CONTENTS

QUANTUM CHAOS MEETS COHERENT CONTROL, Jiangbin Gongand Paul Brumer 1

FEMTOSECOND LASER PHOTOELECTRON SPECTROSCOPY ON ATOMSAND SMALL MOLECULES: PROTOTYPE STUDIES IN QUANTUMCONTROL, M. Wollenhaupt, V. Engel, and T. Baumert 25

NONSTATISTICAL DYNAMICS IN THERMAL REACTIONS OFPOLYATOMIC MOLECULES, Barry K. Carpenter 57

RYDBERG WAVEPACKETS IN MOLECULES: FROM OBSERVATIONTO CONTROL, H.H. Fielding 91

ELECTRON INJECTION AT DYE-SENSITIZED SEMICONDUCTORELECTRODES, David F. Watson and Gerald J. Meyer 119

QUANTUM MODE-COUPLING THEORY: FORMULATION ANDAPPLICATIONS TO NORMAL AND SUPERCOOLED QUANTUM LIQUIDS,Eran Rabani and David R. Reichman 157

QUANTUM MECHANICS OF DISSIPATIVE SYSTEMS, YiJing Yanand RuiXue Xu 187

PROBING TRANSIENT MOLECULAR STRUCTURES IN PHOTOCHEMICALPROCESSES USING LASER-INITIATED TIME-RESOLVED X-RAYABSORPTION SPECTROSCOPY, Lin X. Chen 221

SEMICLASSICAL INITIAL VALUE TREATMENTS OF ATOMS ANDMOLECULES, Kenneth G. Kay 255

VIBRATIONAL AUTOIONIZATION IN POLYATOMIC MOLECULES,S.T. Pratt 281

DETECTING MICRODOMAINS IN INTACT CELL MEMBRANES,B. Christoffer Lagerholm, Gabriel E. Weinreb, Ken Jacobson,and Nancy L. Thompson 309

ULTRAFAST CHEMISTRY: USING TIME-RESOLVED VIBRATIONALSPECTROSCOPY FOR INTERROGATION OF STRUCTURAL DYNAMICS,Erik T.J. Nibbering, Henk Fidder, and Ehud Pines 337

MICROFLUIDIC TOOLS FOR STUDYING THE SPECIFIC BINDING,ADSORPTION, AND DISPLACEMENT OF PROTEINS AT INTERFACES,Matthew A. Holden and Paul S. Cremer 369

ix

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March 15, 2005 20:18 Annual Reviews AR241-FM

x CONTENTS

AB INITIO QUANTUM CHEMICAL AND MIXED QUANTUMMECHANICS/MOLECULAR MECHANICS (QM/MM) METHODS FORSTUDYING ENZYMATIC CATALYSIS, Richard A. Friesnerand Victor Guallar 389

FOURIER TRANSFORM INFRARED VIBRATIONAL SPECTROSCOPICIMAGING: INTEGRATING MICROSCOPY AND MOLECULARRECOGNITION, Ira W. Levin and Rohit Bhargava 429

TRANSPORT SPECTROSCOPY OF CHEMICAL NANOSTRUCTURES:THE CASE OF METALLIC SINGLE-WALLED CARBON NANOTUBES,Wenjie Liang, Marc Bockrath, and Hongkun Park 475

ULTRAFAST ELECTRON TRANSFER AT THEMOLECULE-SEMICONDUCTOR NANOPARTICLE INTERFACE,Neil A. Anderson and Tianquan Lian 491

HEAT CAPACITY IN PROTEINS, Ninad V. Prabhu and Kim A. Sharp 521

METAL TO INSULATOR TRANSITIONS IN CLUSTERS, Bernd von Issendorffand Ori Cheshnovsky 549

TIME-RESOLVED SPECTROSCOPY OF ORGANIC DENDRIMERS ANDBRANCHED CHROMOPHORES, T. Goodson III 581

INDEXESSubject Index 605Cumulative Index of Contributing Authors, Volumes 52–56 631Cumulative Index of Chapter Titles, Volumes 52–56 633

ERRATAAn online log of corrections to Annual Review of Physical Chemistrychapters may be found at http://physchem.annualreviews.org/errata.shtml

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Quantum mode-coupling theory: Formulation and applications to normal and supercooled quantum liquids

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Transcript of Quantum mode-coupling theory: Formulation and applications to normal and supercooled quantum liquids