VOLUME 87, NUMBER 26 PHYSICAL REVIEW LETTERS 24 DECEMBER 2001 Self-Consistent Mode-Coupling Theory for Self-Diffusion in Quantum Liquids David R. Reichman 1 and Eran Rabani 2 1 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138 2 School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel (Received 12 July 2001; published 10 December 2001) A closed, self-consistent equation for the velocity autocorrelation function of a quantum liquid within the framework of a quantum mode-coupling theory is derived. The solution of the quantum generalized Langevin equation requires static input which is generated by an appropriate path-integral Monte Carlo scheme. In order to assess the accuracy of our approach we have studied the self-diffusion process of liquid para-hydrogen at two thermodynamic state points. Quantitative agreement for the self-diffusion constant is obtained in comparison to experimental measurements and other theoretical predictions. DOI: 10.1103/PhysRevLett.87.265702 PACS numbers: 67.20.+k One of the long-standing problems in physics is the quantum mechanical treatment of dynamical properties in highly quantum liquids. The direct calculation of time cor- relation functions in these condensed phase systems is an extremely difficult task. This has led to a variety of dif- ferent techniques to include the effects of quantum fluctu- ations on the dynamic response in liquids. At the present time, one of the viable alternatives to the exact quantum mechanical solution is the use of techniques that are “semi- classical” in nature, namely, the dynamic response is ob- tained with the aid of classical trajectories [1]. While such techniques appear promising, technical issues have pre- vented their use in describing dynamics in realistic quan- tum liquids. Another class of methods that has been used with success in a variety of problems involves sophisticated numerical analytical continuation of exact imaginary-time path-integral Monte Carlo (PIMC) data [2]. The applica- tion of these methods to the understanding of dynamical properties in quantum liquids has so far not been com- pletely successful [3,4]. In this Letter we develop a new approach to study dynam- ical correlations in quantum liquids within the framework of a quantum mode-coupling theory [5], and focus on the study of transport properties in these systems. Our ap- proach draws upon the pioneering work of Götze and Lücke [6,7], and Sjögren and Sjölander [8–10]. We first derive a closed, self-consistent nonlinear integro-differential equa- tion for the velocity autocorrelation function. The solution to this equation requires static input which is generated by an appropriate path-integral Monte Carlo scheme [11]. The method is then applied to study the self-diffusion process of liquid para-hydrogen at two thermodynamic state points, and comparison is made with experimental measurements [12] and other theoretical predictions [13]. The derivation of the quantum generalized Langevin equation (QGLE) for the velocity autocorrelation function (VACF) follows from the work of Zwanzig [14] and Mori [15,16]. We begin with the definition of the projection op- erator, P k [17]: P k  y,··· y, y k  y k , (1) where y k  1 b ¯ h Z b ¯ h 0 dl e 2lH ye lH (2) is the Kubo transform [18] of the velocity operator y  pm of a representative liquid particle along a chosen Cartesian direction, H is the Hamiltonian of the system, b  1 KBT , and ··· denote an ensemble average. In the above equations the notation k implies that the quantity under consideration involves the Kubo transform given by Eq. (2). Using the above definitions and following the standard procedure [19] it is straightforward to show that the QGLE for the Kubo transform of the VACF, C k y t   yy k t , is given by the exact equation C k y t   2 Z t 0 dt 0 K k t 0   C k y t 2 t 0  , (3) where  C k y t   ≠C k y t ≠t , and the Kubo transform of the memory kernel, K k t , is given by the exact equation K k t   1 y, y k    y, e i12Pk L t  y k  . (4) In the above equation, L  1 ¯ h H,  is the quantum Liou- ville operator. The above expression for the memory function com- bined with Eq. (3) is simply another way of rephrasing the quantum Wigner-Liouville equation for the VACF. The difficulty of numerically solving the Wigner-Liouville equation for the positions and velocities in a many-body system is shifted to the difficulty of evaluating the memory kernel. To circumvent this difficulty we use an approximate closure for the memory kernel of the form K k t   K k f t  1 K k s t . This form of mode-coupling theory [20] has been used successfully in the study of a certain class of classical dense fluids with remarkable suc- cess in predicting various dynamical correlations in these systems [19,21,22]. Its quantum generalization has been suggested for the application of density fluctuations in superfluid liquid helium [6,7] and in liquid para-hydrogen [5]. We note that the above formal expression for the 265702-1 0031-9007 01 87(26) 265702(4)$15.00 © 2001 The American Physical Society 265702-1