Frequency-Dependent Diffusion Constant of Quantum Fluids from Path Integral Monte Carlo and Tikhonov’s Regularizing Functional Piotr Kowalczyk,* ,† Piotr A. Gauden, ‡ Artur P. Terzyk, ‡ and Sylwester Furmaniak ‡ Applied Physics, Royal Melbourne Institute of Technology UniVersity, GPO Box 2476 V, Victoria 3001, Australia and Department of Chemistry, Physicochemistry of Carbon Materials Research Group, Nicolaus Copernicus UniVersity, Gagarin St. 7, 87-100 Torun, Poland Received May 1, 2009 Abstract: We present a novel implementation of the analytic continuation of the velocity autocorrelation function method that has been developed to study the transport properties of quantum liquids at finite temperatures. To invert the ill-posed linear Fredholm integral equation of the first kind, we combine Tikhonov’s first-order regularizing functional with several methods used for automatic selection of the regularization parameter. Taking into account our results, we recommend two methods for automatic selection of the regularization parameter, namely: L-curve and quasi-optimality criterion. We found that the frequency-dependent diffusion power spectrum of normal liquid 4 He at T ) 4 K and F) 0.01873 Å -3 (F) 31.1 mmol cm -3 ) is characterized by a single asymmetric peak. The predicted self-diffusion coefficient of 4 He at this state point of 0.57-0.58 Å 2 /ps is in excellent agreement with previous works. We demonstrate that, within proposed mathematical treatment of the quantum transport at finite temperatures, the entire real-time frequency-dependent diffusion power spectrum of liquid normal 4 He, can be successfully reconstructed from the limited number of Trotter slices and without the knowledge of covariance matrix. Moreover, the small values of regularization parameters (i.e., order of 10 -7 ) indicate that the information about quantum dynamics of normal liquid 4 He can be easily withdrawn from the high quality imaginary-time correlation function collected in the standard path integral Monte Carlo simulation. Introduction The understanding of dynamics of light particles (such as proton, electron, molecular hydrogen, helium, and others) at finite temperatures is essential for the designing of enzymatic reactions, solar cells, proton conducting mem- branes, quantum sieves, membranes for hydrogen purifica- tion, etc. 1-7 Thus, the fundamental works for well-defined quantum systems at finite temperatures are necessary to validate the methods of quantum dynamics, where a satisfac- tory solution has not been found yet. Of all quantum liquids, 4 He is the most common and characteristic one. 8 It is, therefore, an ideal system for investigating the microscopic origin of quantum effects and doing comparisons between experiments and theory. 9-14 Rabani et al. 14 published the paper where they compared several techniques used for estimating the frequency-de- pendent diffusion constant in normal liquid helium above the λ transition. The techniques studied were the quantum version of mode-coupling theory (QMCT), 15-18 the numer- ical analytic continuation method (NAC), 14,19 the centroid molecular dynamics (CMD), 20-24 and the Nakayama-Makri semiclassical approach (SC). 25,26 Their study showed that all aforementioned methods yielded similar self-diffusion constants (i.e., zero time value of the velocity autocorrelation) * Corresponding author. Telephone: +61 (03) 99252571. Fax: +61 (03) 99255290. E-mail: piotr.kowalczyk@rmit.edu.au. † Royal Melbourne Institute of Technology University. ‡ Nicolaus Copernicus University. J. Chem. Theory Comput. 2009, 5, 1990–1996 1990 10.1021/ct900215q CCC: $40.75  2009 American Chemical Society Published on Web 07/16/2009 Downloaded by RMIT CTRL LIB SERIALS on August 13, 2009 Published on July 16, 2009 on http://pubs.acs.org | doi: 10.1021/ct900215q