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On the zero temperature limit of the Kubo-transformed quantum time correlation functionLisandro Hernández de la Peñaa

a Department of Chemistry & Biochemistry, Kettering University, 1700 University Avenue,Flint, Michigan, USAAccepted author version posted online: 14 Jun 2013.Published online: 08 Jul 2013.

To cite this article: Lisandro Hernández de la Peña (2014) On the zero temperature limit of the Kubo-transformed quantumtime correlation function, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 112:7,929-936, DOI: 10.1080/00268976.2013.812755

To link to this article: http://dx.doi.org/10.1080/00268976.2013.812755

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Molecular Physics, 2014Vol. 112, No. 7, 929–936, http://dx.doi.org/10.1080/00268976.2013.812755

RESEARCH ARTICLE

On the zero temperature limit of the Kubo-transformed quantum time correlation function

Lisandro Hernandez de la Pena∗

Department of Chemistry & Biochemistry, Kettering University, 1700 University Avenue, Flint, Michigan, USA

(Received 20 April 2013; final version received 3 June 2013)

The zero temperature limit of several quantum time correlation functions is analysed. It is shown that while the canonical quan-tum time correlation function retains the full dynamical information as temperature approaches zero, the Kubo-transformedand the thermally symmetrised quantum time correlation functions lose all dynamical information at this limit. This is shownto be a consequence of the projection onto the ground state, via the limiting process of the quantities B and A(t), eithertogether as a product, or separately. Although these findings would seem to suggest that finite-temperature methods com-monly used to estimate Kubo correlation functions would be incapable of retaining any ground state dynamics, we proposea route for recovering in principle all dynamical information at the ground state. It is first shown that the usual frequencyspace relation between canonical and Kubo correlation functions also holds for microcanonical time correlation functions.Since the Kubo-transformed microcanonical correlation function can be obtained from the usual finite-temperature functionby including a projection onto the corresponding microcanonical ensemble, finite-temperature methods, properly modifiedto incorporate such a constraint, can be used to capture full quantum dynamics at any arbitrary energy state, including theground state. This approach is illustrated with the application of centroid dynamics to the ground state dynamics of theharmonic oscillator.

Keywords: quantum dynamics; time correlation function; ground state dynamics; path integral centroid variables

1. Introduction

Time correlation functions are central to the descriptionof dynamical processes [1,2]. It is well known that, forinstance, the time dependent response of a system to aweak external field can be expressed in terms of the timecorrelation function of a dynamical property of the systemas prescribed by linear response theory [3,4]. In this context,a variety of kinetic coefficients can be computed in terms oftime correlation functions, including diffusion coefficients,relaxation times, and chemical reaction rate constants, etc.

The calculation of time correlation functions in classicalsystems at finite temperature can be carried out straightfor-wardly through the solution of the inherent Hamiltonianor stochastic dynamics of the system. In contrast, the cal-culation of quantum time correlation functions is seriouslycomplicated by the simultaneous presence of real and imag-inary (i.e. thermal) times [5]. The standard object for theanalysis of quantum dynamics is the quantum time corre-lation function,

C(t) ≡ 1

ZTr

[e−βH BA(t)

], (1)

which we refer to as canonical or standard time corre-lation function. In this equation, H denotes the manybody Hamiltonian, Z is the canonical partition function,

∗Email: lhernand@kettering.edu

β = 1/kBT is the inverse temperature in energy units, andA(t) = e

i�

H t Ae− i�

H t is an observable A at time t in theHeisenberg representation.

Since the computation of C(t) is particularly problem-atic [6], especially because it’s a complex-valued function,several modern techniques [7–13] aim at the computation(often approximate) of the Kubo-transformed quantum timecorrelation function, CK(t), given by

CK (t) ≡ 1

Zβ

∫ β

0dλ Tr

[e−(β−λ)H Be−λH A(t)

], (2)

which is known to be a real function. The rationale is that,although the symmetry properties of a Kubo-transformedfunction have close resemblance to the corresponding clas-sical time correlation function [13] (making it amenable toseveral approximations), it contains the same informationas the quantum time correlation function.

In this work, we show that, while in the zero tempera-ture limit the usual quantum time correlation function re-tains the full dynamical information, the Kubo-transformedquantum time correlation function looses all dynamical in-formation at this limit. We argue that since the thermallysymmetrised quantum time correlation function also losesall dynamical information and yields the same constantvalue as the Kubo-transformed function, this overall result

C© 2013 Taylor & Francis

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cannot be due to the classical symmetries satisfied by theKubo-transformed function. It is instead a consequence ofthe projection onto the ground state, via the limiting pro-cess, of the quantities B and A(t), either together as anoperator product, or separately.

These findings would seem to suggest that finite-temperature methods commonly used to estimate Kubocorrelation functions would be incapable of retaining anyground state dynamics. However, we propose a route forrecovering in principle all dynamical information at theground state by means of a modified version of the usualfinite-temperature Kubo methods. It is first shown that theusual frequency space relation between canonical and Kubocorrelation functions also holds for microcanonical timecorrelation functions. Then, since the Kubo-transformedmicrocanonical correlation function can be obtained fromthe usual finite-temperature function by including a pro-jection onto the corresponding microcanonical ensemble,finite-temperature methods, incorporating the appropriateenergy constraint, can be used to capture the full quan-tum dynamics of any arbitrary energy state, including theground state. This approach is illustrated with the applica-tion of centroid dynamics to the ground state dynamics ofthe harmonic oscillator.

The paper is organised as follows. The zero temper-ature limit of several quantum time correlation functionsand its physical interpretation is presented in Section 2.The relation between the standard and Kubo-transformedmicrocanonical correlation functions is discussed in Sec-tion 3. An illustration of how to compute the microcanoni-cal Kubo-transformed correlation functions using centroiddynamics is given in Section 4. Finally, in Section 5, ourconclusions are presented.

2. Quantum correlation functionsat zero temperature

2.1. The zero temperature limit

Denoting E0 as the ground state energy of the system,we can write the quantum time correlation function inEquation (1) as the following:

limβ→+∞

C(t)

= limβ→+∞

1

Ze−βE0

∑n

e−β(En−E0)〈n|BA(t)|n〉

= 〈0|BA(t)|0〉, (3)

where we used the fact that limβ→+∞ Z ≡ limβ→+∞Tr[e−βH ] ≈ e−βE0 , and the trace operation has been per-formed in the energy eigenstate basis. This result implies,as expected, that there is a dynamics driven by the matrixelement of the observable between the ground state and ev-ery other eigenstate of the system. For simplicity, here we

consider cases with a unique ground state, but it is straight-forward to extend the formulation to cases with degenerateground states.

The zero temperature limit of the Kubo-transformedquantum time correlation function can be handled inan analogous fashion. Again, expressing the Kubo-transformed function in the energy basis we have

limβ→+∞

CK (t)

= limβ→+∞

1

Ze−βE0

∑n,n′

e−β(En−E0)〈n|B|n′〉〈n′|A(t)|n〉

×[

1

β

∫ β

0dλ e−λ(En′−En)

]

= limβ→+∞

1

Ze−βE0

∑n,n′

e−β(En−E0)〈n|B|n′〉〈n′|A(t)|n〉

×[

1 − e−β(En′ −En)

β(En′ − En)

]. (4)

An elementary analysis shows that for n = n′ the factorin square brackets in the last line from above is unity forall values of β. (Alternatively, it is clear from the integralevaluation in square brackets in the second line when n =n′.) Thus, the double sum from the above evaluation can bedivided into two terms, according to the following operatorequation:

∑n,n′ = ∑

n=n′ +∑n�=n′ .

The second term with∑

n�=n′ can be written as thefollowing:

limβ→+∞

1

Ze−βE0

∑n�=n′

〈n|B|n′〉〈n′|A(t)|n〉

×[e−β(En−E0) − e−β(En′ −E0)

β(En′ − En)

]= 0. (5)

The factor in the square brackets vanishes because the nu-merator is always finite (with each exponential term beingeither 1 or 0) whereas the denominator tends to infinity.

Therefore, only the first term with n = n′ remains,

limβ→+∞

CK (t)

= limβ→+∞

1

Ze−βE0

∑n

e−β(En−E0)〈n|B|n〉〈n|A(t)|n〉

= 〈0|B|0〉〈0|A(t)|0〉= 〈0|B|0〉〈0|A|0〉. (6)

Since the Kubo-transformed function becomes a constant,all dynamical information is lost at this limit. Note that, inparticular, the Kubo-transformed function could be zero.

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2.2. Physical interpretation

It is convenient to note that C(t) is related in frequency spaceto the (imaginary part of the) system’s response, χ ′′(t), viathe Fluctuation–Dissipation theorem [14]

χ ′′(w) = 1

2�

[1 − e−β�ω

]C(ω), (7)

and in the zero temperature limit they become essentiallythe same function, i.e.

χ ′′(w) ={

0 if ω = 0C∞(ω) if ω �= 0,

(8)

where

C∞(ω) =∫

dt e−iωtC∞(t)

= 2π∑

n

〈0|B|n〉〈n|A|0〉δ(

ω−En − E0

�

). (9)

Here, the tilde of a function is used to indicate that it hasbeen Fourier transformed, and the subscript ∞ denotes thatthe β → +∞ limit has been taken.

The Kubo-transformed function, on the other hand, isrelated to the response function in a classical-like manner

χ ′′(ω) = βω

2CK (ω), (10)

and the response function diverges linearly, in general, at thezero temperature limit. One might argue that this classicalsymmetry is responsible for the loss of information at thezero temperature limit (or quantum limit), after all, inverseFourier transforming of the relationship above leads to

i∂CK (t)

∂t= 2

βχ ′′(t), (11)

which implies that CK(t) must be a constant at the zerotemperature limit as β approaches +∞.

In order to show that the result found for the CK(t)is not a consequence of the classical-like behaviour ofthe Kubo-transformed function, we analyse the thermallysymmetrised quantum time correlation function, G(t). Thisfunction is defined as [15]

G(t) ≡ 1

ZTr

[e− β

2 H Be− β2 H A(t)

], (12)

and is related in frequency space to the response functionthrough

χ ′′(w) = 1

�sinh

(β�ω

2

)G(ω). (13)

It contains the same quantum dynamical information asC(t) or CK(t). Furthermore, similar to what happens withthe Fluctuation–Dissipation theorem, the expression abovetransforms into Equation (11) in the � → 0 limit. Thezero temperature limit of the symmetrised quantum timecorrelation function is

limβ→+∞

G(t)

= limβ→+∞

1

Ze−βE0

∑n,n′

〈n|B|n′〉〈n′|A(t)|n〉e−β(En+En′

2 −E0)

= 〈0|B|0〉〈0|A|0〉. (14)

Note that the exponential inside the summation in the sec-ond line from the above evaluation becomes unity onlywhen the exponent is exactly zero, i.e. n = n′ = 0. It isapparent here that the operators B with A(t) are separatelyprojected onto the ground state.

In general, one can define the following correlator

Cλ(t) ≡ 1

ZTr

[e−(β−λ)H Be−λH A(t)

], (15)

which satisfies

Cλ(t) =⎧⎨⎩

C(t) if λ = 0G(t) if λ = β/2C(−t) if λ = β.

(16)

The Kubo-transformed function is the mean value of thecorrelator in the sense that

CK (t) = 1

β

∫ β

0dλ Cλ(t). (17)

Note that except for the relationships established by Equa-tions (16) and (17), the correlator cannot be linked to theresponse function. The zero temperature limit for the cor-relator is given by

limβ→+∞

Cλ(t) =⎧⎨⎩

〈0|BA(t)|0〉 if λ = 0〈0|B|0〉〈0|A|0〉 if 0 < λ < β

〈0|AB(−t)|0〉 if λ = β,

(18)

which shows that the dynamics is retained only locally in λ.For the Kubo-transformed function, the averaging processindicated in Equation (17) eliminates the dynamics due tothe zero measure associated with any local value of λ.

3. Quantum dynamics in the microcanonicalensemble

In general, one can analyse the dynamics in the micro-canonical ensemble at some specified energy value E. Let’s

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define the normalisation parameter

= Tr[δ(E − H )e−βH ]

= e−βE∑

n

δ(E − En) = e−βE(E) (19)

where (E) ≡ is the microcanonical partition function ordensity of states. Then, consider the following correlationfunction of a Kubo type

DK (t) ≡ 1

∫ β

0

dλ

βTr

[δ(E − H )e−(β−λ)H Be−λH A(t)

]= 1

∑n

′ 〈n|∫ β

0

dλ

βB(−iλ�)A(t)|n〉

= 1

∑n

′DKn (t), (20)

where the sum above is restricted, in a macroscopic sense,to states satisfying E < En < E + �E where ordinarily�E � E (or alternatively, in the microscopic sense, overdegenerate states). There are obviously of such states,and this expression represents a microcanonical average.

In terms of energy eigenstates and using properties ofthe trace operation, the expression for DK

n (t) defined abovebecomes,

DKn (t) =

∑n′

〈n|B|n′〉〈n′|A(t)|n〉 1

β

∫ β

0dλ e−λ(E′

n−En)

=∑n′

〈n|B|n′〉〈n′|A(t)|n〉[

1 − e−β(En′ −En)

β(En′ − En)

]. (21)

The Fourier transform of this expression leads to

DKn (ω) =

∫dt

2πe−iωtDK

n (t)

=∫

dt

2πe−iωt

∑n′

〈n|B|n′〉〈n′|A|n〉eiEn′ −En

�t

×[

1 − e−β(En′ −En)

β(En′ − En)

]

=∑n′

〈n|B|n′〉〈n′|A|n〉δ(

ω − En′ − En

�

)

×[

1 − e−β(En′ −En)

β(En′ − En)

]. (22)

This last result suggests the use of the frequency factor,F(ω) = β�ω/(1 − e−β�ω), that allows one to perform the in-verse Kubo transform on DK

n (t). Indeed, the multiplicationof DK

n (ω) by F(ω) and the inverse Fourier transformation

yields the following correlation function,∫dω eiωtF (ω)DK

n (ω)

=∑n′

∫dω eiωt 〈n|B|n′〉〈n′|A|n〉δ

(ω − En′ − En

�

)

×[

1 − e−β(En′ −En)

β(En′ − En)

]β�ω

1 − e−β�ω

=∑n′

ei(

En′ −En

�

)t 〈n|B|n′〉〈n′|A|n〉

= 〈n|BA(t)|n〉 = Dn(t), (23)

where the properties of the delta function have been used.Since this analysis holds for every DK

n (t) in Equation (20),and due to the fact that Fourier transformation is a lin-ear operation, we find that the following frequency spacerelationship holds

D(w) = F (ω)DK (w), (24)

where D(w) is the Fourier transformed of the microcanon-ical time correlation function defined as

D(t) = 1

∑n

′Dn(t) = 1

Tr[δ(E − H )BA(t)]. (25)

The frequency factor F(ω) is interestingly the usual oneused in the inversion of Kubo-transformed correlation func-tions CK(t) [7,8,16,17], and it is often derived in the contextof the canonical ensemble [18]. However, the fact that theleft-hand side of Equation (24) is independent of β indi-cates that this frequency relationship holds in the absenceof thermal equilibrium. As will be seen explicitly in thefollowing section, the canonical and microcanonical Kubocorrelations CK(t) and DK(t) have very different properties.

It is worth noting that the analysis above is consistentwith the discussion in the preceding section. Indeed, to takethe zero temperature limit of Equation (21), we can dividethe sum there into ground and excited states contributions(∑

n = ∑n = 0 + ∑

n > 0), and with an analysis similar tothe one made earlier in Equation (4) we find,

limβ→+∞

CK0 (t) = 〈0|B|0〉〈0|A(t)|0〉

+ limβ→+∞

∑n>0

〈0|B|n〉〈n|A(t)|0〉

×[

1 − e−β(En−E0)

β(En − E0)

]. (26)

The second term of the above expression is zero in theβ → +∞ limit. One therefore obtains the result in Equa-tion (6), i.e. limβ→+∞ CK

0 (t) = 〈0|B|0〉〈0|A|0〉. The use ofthe ground state density matrix represents an alternativeway of deriving that result.

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Molecular Physics 933

However, if one does not take the zero temperature limitin Equation (21), the discussion here suggests an approachto obtain ground state correlation functions from meth-ods that yield Kubo-transformed correlation functions. In-deed, if one performs the inverse Kubo transform of Equa-tion (21), one can recover the time correlation functionof Equation (23). Thus, the dynamics at any energy level,including the ground state, can be cast in the form of amicrocanonical Kubo-transformed correlation function.

This result has important implications: It suggests a wayto obtain ground state correlation functions from methodsdesigned to yield finite temperature Kubo-transformed cor-relation functions. Such methods include centroid molecu-lar dynamics (CMD) [7,8,16,19] and ring polymer molecu-lar dynamics (RPMD) [13,17,20], etc. These methods, how-ever, must be properly adapted to account for the groundstate projection operator.

4. Illustration: the harmonic oscillator

In this section we illustrate how the ideas and results dis-cussed this far apply to the harmonic oscillator. The micro-canonical time correlation function of position operators atsome energy E is first derived, and the ground state correla-tion function immediately follows. It will be apparent thatthis last result can be also found by applying a zero tempera-ture limiting operation to the standard quantum correlationfunction.

Kubo transforming the microcanonical correlationfunction, by using the procedure described in the preced-ing section, leads to a general expression from where theground state Kubo function follows. It is automatically evi-dent that this last result cannot be found by means of a zerotemperature limiting operation of the ordinary Kubo func-tion, which has already been shown in Section 2 to containno dynamics.

It is subsequently discussed how to use centroid dy-namics to obtain the ground state Kubo function for thisparticular system. The strategy there consists of redefin-ing the centroid density and centroid symbols in order toaccount for the ground state projection.

4.1. Quantum time correlation functionsand ground state dynamics

The position–position microcanonical quantum time corre-lation function for the harmonic oscillator can be written inthe following form:

Dxx(t) = 1

Tr

[δ(E − H )x(t)x

]= �

2mω0

∑n,n′

eit(En′−En)/�δ(E − En)

× [nδ(n′ + 1 − n) + n′δ(n + 1 − n′)

]

= �

2mω0

[eiω0t

∑n

nδ(E − En)

+ e−iω0t∑n′

n′δ(E − En′−1)

]. (27)

However, since

∑n

nδ(E − En) = 1

�ω0

(E

�ω0− 1

2

), (28)

and

=∑

n

δ(E − En) = 1

�ω0, (29)

one immediately obtains the general result

Dxx(t) = �

2mω0

[2E

�ω0cos(ω0t) − i sin(ω0t)

]. (30)

For the ground state, this expression reduces to

Dxx(t) = �

2mω0[cos(ω0t) − i sin(ω0t)] , (31)

which coincides, as expected, with the zero temperaturelimit of the canonical correlation function that is given bythe standard result

Cxx(t) = 1

ZTr

[e−βH x(t)x

]= �

2mω0

[coth

(β�ω0

2

)cos(ω0t) − i sin(ω0t)

].

(32)

The Kubo-transformed of the microcanonical correla-tion function in Equation (30) can be found using the fre-quency space relation discussed in the previous section andyields

DKxx(t) = sinh(βω0�/2)

mω20β

[2E

ω0�cos

(tω0 − iβω0�

2

)

− i sin

(tω0 − iβω0�

2

) ]. (33)

All the dynamics is equivalently expressed in this form forany energy level. In particular, for the ground state, thisexpression reduces to

DKxx(t) =

(1 − e−ω0β�

)2mω2

0β[cos(ω0t) − i sin(ω0t)] . (34)

It is important to note that, in contrast to the result in Equa-tion (31), this last expression in Equation (34) cannot be

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found by taking the zero temperature limit of the Kubo-transformed quantum time correlation function. In fact, thestandard Kubo-transformed function is simply given by

CKxx(t) = 1

Zβ

∫ β

0dλ Tr

[e−(β−λ)H xe−λH x(t)

]= 1

mω20β

cos(ω0t), (35)

and goes to zero as β goes to infinity, in agreement with thediscussion in Section 2. Note that the Kubo function CK

xx(t)is identical to the classical correlation function as can beverified by direct calculation [or by taking the classical limitβ� → 0 in Equation (32)]. Furthermore, in general, beyondthe harmonic case, the standard Kubo-transformed functionCK

xx(t) cannot reduce to the microcanonical Kubo functionDK

xx(t), by means of any kind of limiting process becausethe former is a real function while the latter is a complexvalued function in time.

In practice, one expects to be able to estimate Kubofunctions like the one in Equation (34) by some availablemeans and after inverse Kubo transformation recovers itsequivalent form given in Equation (31). The next subsectionillustrates how to obtain in particular Equation (34), how-ever, it must be clear that this strategy could be generallyused to obtain the dynamics at any energy state.

4.2. Centroid dynamics with groundstate projection

We now show that the ground state Kubo-transformed cor-relation function in Equation (34), can be obtained by meansof some type of phase space average of certain centroidsymbols. This is precisely the strategy used in centroid dy-namics and serves as a motivation in what follows.

In centroid dynamics, the usual Kubo-transformed timecorrelation function of Equation (35) is computed by [7,8]

CKxx(t) = 1

Z

∫ ∫dxcdpc

2π�ρc(xc, pc)xcxc(t). (36)

In this equation, ρc(xc, pc) = Tr [ϕ(xc, pc)] is the centroiddensity, and

ϕ(xc, pc) = �

2π

∫ ∞

−∞dζ

∫ ∞

−∞dηeiζ (xc−x)+iη(pc−p)−βH

(37)

is the centroid phase space representation of the Boltzmannoperator. The quantity xc(t) is the time dependent centroidsymbol of position defined according to

xc(t) = Tr[δc(xc, pc)x(t)

], (38)

where δc(xc, pc) = ϕ(xc, pc)/ρc(xc, pc) is the so-calledquasi-density operator.

Consider now the following correlation function

�Kxx(t) = 1

∫ ∫dxcdpc

2π��0(xc, pc)xcX0(t), (39)

which is yet to be determined. In view of Equation (19), oneanticipates that = e−β�ω0/2. The phase space functions�0(xc, pc) and X0(t) involve some type of ground stateprojection (which justifies the use of the subscript ‘0’ intheir notation) and are defined below.

Let’s define the ground state density in centroid phasespace �0(xc, pc) as

�0(xc, pc) = Tr [ϕ(xc, pc)|0〉〈0|]=

∫dx

∫dx ′ψ0(x)〈x|ϕ(xc, pc)|x ′〉ψ∗

0 (x ′)

= βω0�

(1 + α) sinh(βω0�/2)exp

[−

(β + 2

(1 + α)ω0�

)

×(

p2c

2m+ 1

2mω2

0x2c

)]. (40)

This function is a two-dimensional Gaussian in (xc, pc)phase space and is required for the average defined in Equa-tion (39). It is straightforward to show that the phase spaceaverage of this function yields precisely the parameter ,which is simply a normalisation constant.

The final expression in the equation above is obtainedby performing the double integral in the second line ofEquation (40) using the ground state harmonic oscillatorwave function

ψ0(x) =(mω0

π�

)1/4exp

[−mω0

2�x2

], (41)

the general form of the position matrix element [7]

〈x|ϕ(xc, pc)|x ′〉 = exp

[− β

2m

(pc − im

β�(x − x ′)

)2]

×〈x|ϕ(xc)|x ′〉, (42)

and the harmonic oscillator expression [7]

〈x|ϕ(xc)|x ′〉 = ω0�β/2

sinh(ω0�β/2)

√mω0

π�α

× exp

[−βmω2

0

2x2

c −mω0

�α

(x+x ′

2−xc

)2

−(

mω0α

4�+ m

2β�2

)(x − x ′)2

], (43)

where

α = coth

(ω0�β

2

)− 2

ω0�β. (44)

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Molecular Physics 935

In analogy with the quasi-density operator used in cen-troid dynamics, we define a ground state projected quasi-density operator as

ξ0 = ϕ(xc, pc)|0〉〈0|�0(xc, pc)

, (45)

properly normalised with the ground state distribution de-fined above. Then, the trace of the product of this operatorand the position operator in the Heisenberg representationleads to a time dependent ground state position symbol,which can be expressed as a function of xc and pc accordingto

X0(t) = Tr[ξ0(xc, pc)x(t)

]= 1

(1 + α)

(xc + i

pc

mω0

)e−iω0t . (46)

The second line in this expression is somewhat labo-rious. The operator ξ0(xc, pc) in position representation isgiven by

〈x|ξ0(xc, pc)|x ′〉 = 1

�0(xc, pc)

(mω0

π�

)1/2

×∫

dy exp[−mω0

2�(y2 + x ′2)

]〈x|ϕ(xc, pc)|y〉

=√

mω0

π�exp

[− 2

�ω0

(1

4mω2

0

(x2 + x ′2)

− (pc − imω0xc)2

2m(1 + α)2− ω0x(ipc + mω0xc)

1 + α

)], (47)

and, at t = 0, we find

X0(0) =∫

dxx〈x|ξ0(xc, pc)|x〉

= 1

(1 + α)

(xc + i

pc

mω0

)(48)

The time dependent symbol requires the inclusion of theharmonic oscillator real time propagator [21]

〈x|e−iH t/�|x ′〉 =(

mω0

2πi� sin(ω0t)

)1/2

exp

{imω0

2� sin(ω0t)

× [cos(ω0t)(x2 + x ′2) − 2xx ′]}

(49)

which, after somewhat lengthy but straightforward algebra,yields Equation (46) by means of

X0(t) =∫

dx

∫dx ′

∫dx ′′〈x|ξ0(xc, pc)|x ′〉

〈x ′|eiH t/�|x ′′〉x ′′〈x ′′|e−iH t/�|x〉. (50)

A simpler derivation of Equation (46) is given byL. Hernandez de la Pena (in preparation).

Having the explicit forms of �0(xc, pc) and X0(t) givenin Equations (40) and (46), it is straightforward to showthat the phase space integration suggested in Equation (39)yields

�Kxx(t) = 1

2mω20β

(1 − e−ω0β�

)e−iω0t , (51)

which is precisely the ground state Kubo-transformed cor-relation function DK

xx(t) of Equation (34). As noted in theprevious section, inverse Kubo transformation of this func-tion leads to the microcanonical ground state position cor-relation function of the harmonic oscillator:

Dxx(t) = 1

Tr

[δ

(�ω0

2− H

)xx(t)

]. (52)

We have therefore shown that the full dynamical informa-tion has been recovered for this simple system by meansof a phase space average of centroid symbols in a mannerthat resembles the usual approach in centroid dynamics. Adistinctive feature is that one of the centroid symbols andthe centroid distribution require the inclusion of an energyprojection operator in its definition.

5. Conclusions

In this paper we have analysed the zero temperature limit ofthe Kubo-transformed quantum time correlation function.The lack of dynamical information at this limit is not a con-sequence of its classical-like dynamical symmetry. The zerotemperature limit is essentially the ground state projectionof a relevant quantity. Two different entities are howeverprojected in the different time correlation functions. In thecase of the standard quantum time correlation function,the product BA(t) is projected onto the ground state. Inthe Kubo-transformed function and the symmetrised quan-tum time correlation functions, the operators B with A(t)are separately projected onto the ground state eliminatingall the dynamics. The analysis presented here is general andderive from the formal properties of quantum time correla-tion functions.

It would seem, as a consequence of these facts, thatone would not be able to use finite-temperature Kubo meth-ods to study ground state dynamics. However, we haveshown that the dynamics at any energy state can be cast inthe form of a microcanonical Kubo-transformed correla-tion function by using the same frequency relationship thatlinks the canonical and the Kubo-transformed correlationfunctions. This result has important implications. In par-ticular, it suggests a way to obtain ground state correlationfunctions from methods designed to yield finite temperatureKubo-transformed correlation functions. These methods in-clude centroid molecular dynamics [7,8,16,19], ring poly-mer molecular dynamics [13,17,20], semiclassical methods[22] and thermal Gaussian molecular dynamics [23,24].

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936 L.H. de la Pena

As an illustration, we analysed the ground state dynam-ics of the harmonic oscillator by means of a centroid dynam-ics that includes a ground state constraint. We showed thatproceeding in a manner analogous to the route followed fordefining path integral centroid variables, it is possible to de-fine a ground state projected density and a ground state pro-jected centroid symbol. A proper phase space average of theproduct of this ground state symbol and the standard cen-troid symbol yields the microcanonical Kubo-transformedtime correlation function. While this illustration is sufficientto convey the essence of the strategy proposed, very robustmethods for addressing the use of projection operators incentroid dynamics exist. For example, the CMD approachcombined with projection operators has been successfullyderived for the case of bosonic and fermionic exchange[9,25,26] and inversion symmetries [10]. The argumentsgiven in those applications hold perfectly well for a pro-jection onto any subset of energy eigenstates. For the casediscussed in this work, one can follow the procedure of Refs.[9,10] to write a ground state projected Kubo-transformedcorrelation function. The approach in that work relies onthe redefinition of the quasi-density operator proposed byJang and Voth [7] in a manner consistent with the propersymmetries of the system [9,10].

It is worth noting that Ramırez and co-workers haveanalysed the zero temperature limit of the centroid densityand found that it should be a delta function centred at themean value of the centroid variable [27]. However, the anal-ysis carried out in this work allows one to go beyond thatequilibrium analysis and study dynamical properties.

A possible way to obtain the ground state projection isthe use of the path integral ground state (PIGS) approach[28–31] combined with projected centroid dynamics tech-niques [9,25,26,31]. Future work will focus on the combi-nation of CMD and RPMD calculations with such groundstate projection methods.

AcknowledgementsThe author would like to thank Prof. P.-N. Roy for several usefuldiscussions. This work has been partially funded by a start-uppackage provided to the author by Kettering University.

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