Sum Rule Constraints and the Quality of Approximate Kubo

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Sum Rule Constraints and the Quality of Approximate Kubo Transformed Correlation Functions Lisandro Hernandez de la Pena J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b07624 • Publication Date (Web): 19 Jan 2016 Downloaded from http://pubs.acs.org on January 27, 2016

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The Journal of Physical Chemistry

Sum Rule Constraints and the Quality of Approximate Kubo Transformed Correlation Functions Lisandro Hernández de la Peña

∗

Department of Chemistry & Biochemistry, Kettering University, Flint, MI, 48439 E-mail: [email protected]

Abstract In this work, a general protocol for evaluating the quality of approximate Kubo correlation functions of non-trivial systems in many dimensions is discussed. We rst note that the generalized deconvolution of the Kubo transformed correlation function onto a time correlation function at a given value of τ in imaginary time, such that 0 < τ < β¯ h, leads to a series of sum rules applicable to the n-th derivative of the

Kubo function and whose iterative extension allows to link derivatives of dierent order in the corresponding correlation functions. We focus on the case when τ = β¯h/2 for which all deconvolution kernels become real valued functions and their asymptotic behavior at long times exhibit a polynomial divergence. It is then shown that thermally symmetrized static averages, and the averages of the corresponding time derivatives, are ideally suited to investigate the quality of approximate Kubo correlation functions at successively larger (and up to arbitrarily long) times. This overall strategy is illustrated analytically for a harmonic system, and numerically for a multidimensional double-well potential and a Lennard-Jones uid. The analysis includes an assessment of RPMD

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position autocorrelation results as a function of the number of dimensions in a doublewell potential and of the RPMD velocity autocorrelation function of liquid neon at 30 K.

Introduction Time correlation functions play a central role in non-equilibrium statistical mechanics as they allow us to compute kinetic parameters such as diusion coecients, viscosities, rates constants, etc.

1,2

Although classical time correlation functions can be computed straight-

forwardly, the corresponding correlation functions of systems where quantum behavior is signicant are particularly challenging due to the intractability of the underlying quantum dynamics of a multidimensional system.

3,4

Since the Kubo transformed quantum time correlation function is the most classical of all quantum correlation functions, last 25 years.

5

it has become a main target in this area during the

Approximations to the Kubo function can be obtained through a variety

of methods including centroid molecular dynamics, (RPMD),

12

semiclassical theory,

from vibrational spectroscopy rate constants,

20

17

13

and others,

1416

611

ring polymer molecular dynamics

with a wide scope of applications ranging

to molecular diusion in uids,

electron solvation,

21

18,19

transition state theory

etc.

An assessment of the quality of approximate Kubo transformed correlation functions is generally a dicult but necessary task.

Since purely imaginary correlation functions can

be straightforwardly computed using standard path integral methods, they are often used to test their real time counterparts after undergoing a wick rotation from real to imaginary time.

4

The imaginary-time mean square displacement, for example, is related to the Fourier

transform of the real time velocity autocorrelation function via a double-sided Laplace transform which places this later quantity in imaginary time.

22

The correlation functions involving

other operators can also be analyzed in this context but is generally not as simple. The comparison in imaginary time introduces, in addition, certain ambiguities regarding the actual

2


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errors in real time precisely due to the wick rotation procedure. Sum rule constraints provide a powerful alternative to evaluating the accuracy of real time quantum dynamics in multidimensional systems. Static canonical averages, for example, have been used to analyze the quality of Kubo correlation functions but with limited success.

23,24

This is due to the specic characteristics of the associated kernel which exhibit a logarithmic divergence at zero and a fast decay with time allowing only to test the short time behavior of the correlation function. The advantage of this strategy however, relies in that no additional manipulation of the correlation function is required as the analysis remains in real time. In this work, a general protocol for evaluating the quality of approximate Kubo correlation functions of non-trivial systems in many dimensions is discussed.

This approach

is motivated by the generalized deconvolution of the Kubo transformed correlation function onto a time correlation function at a given value of

0 < τ < β¯ h.

τ

in imaginary time, such that

An initial analysis of this kind, in the context of the sum rules enforced by the

maximum entropy analytic continuation method, has been presented in Ref. 25. These sum rules are further investigated here while highlighting their validity up to the

n-th

deriva-

tive of the Kubo function as well as their iterative extension to link derivatives of dierent order in the corresponding correlation functions, provided that some additional (although mild) conditions are satised. We further focus on the case when

τ = β¯ h/2

for which all

deconvolution kernels become real valued functions and their asymptotic behavior at long times exhibit a polynomial divergence. It is then shown that thermally symmetrized static averages, and the averages of the corresponding time derivatives, are ideally suited to investigate the quality of approximate Kubo correlation functions at successively larger (and up to arbitrarily long) times. This overall strategy is illustrated analytically for a harmonic system, and numerically for a multidimensional double-well potential and a Lennard-Jones uid. The numerical examples, in particular, analyze the quality of RPMD results of position autocorrelation functions in a double-well potential system as a function of dimension, and the RPMD velocity autocorrelation function of liquid neon at 20 K.

3



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The paper is organized as follows. The general deconvolution of the Kubo function and resulting constraints are discussed rst. The particular form of these expressions for the case of the thermally symmetrized correlation function is analyzed subsequently.

This overall

approach is then illustrated with a few representative systems. And nally, we present our conclusions on the main ndings in this work and indicate several aspects worth pursuing in the future.

Deconvolution of the Kubo function and generalized constraints Consider the general correlation function

CAB (t + iλ¯ h) =

i 1 h −β Hˆ ˆ ˆ Tr e AB(t + iλ¯ h) , Z

(1)

where

2 ˆ = pˆ + V (ˆ H x), 2m

(2)

is the Hamiltonian operator dening the quantum system, consisting of a quantum particle of mass

Z

m

in an external potential

ˆ ˆ − i Ht ˆ ˆ = e h¯i Ht V (ˆ x), B(t) Be h¯

is the canonical partition function, and

units. Meanwhile the parameter

λ

β = 1/kB T

is an observable

B

at time

t,

is the inverse temperature in energy

is bounded to the closed interval

[0, β].

Note, in particular, that Eq. (1) transforms into the standard or canonical correlation function

CAB (t) =

i 1 hˆˆ ˆ Tr AB(t)e−β H , Z

4


(3)

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or the thermally symmetrized quantum time correlation function

i 1 h ˆ −β H/2 ˆ ˆ ˆ Tr A e B(t) e−β H/2 , Z

GAB (t) ≡

for

λ = 0

or

λ = β/2,

hermitian operators

Aˆ

respectively.

and

ˆ, B

K CAB (t)

26

(4)

The Kubo transformed correlation function of two

on the other hand, is dened by

1 = Zβ

Z

β

h i ˆ ˆ −λH ˆ ˆ dλ Tr e−(β−λ)H Ae B(t) .

Working in the energy representation is not dicult to show that Kubo transformed correlation function

(5)

0

K (t) CAB

CAB (t + iλ¯h)

are related in frequency space by

β¯hωe−(λ−β/2)¯hω ˜ K CÃB (λ; ω) = C (ω) 2 sinh(β¯hω/2) AB where

K (ω) are CÃB (λ; ω) and CÃB

and the

the Fourier transform of

(6)

K (t), CAB (t + iλ¯h) and CAB

respec-

tively.

Sum rules for Provided that

0

Sum Rule Constraints and the Quality of Approximate Kubo

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