## 1 Introduction  
Warm dense matter (WDM) is being studied intensively through both experimental and theoretical approaches. At high temperatures and pressures, strong correlations, quantum degeneracy, and thermal excitation effects require careful analysis of the traditional theoretical models developed for liquids, gases, solids, and plasmas [1, 2, 3]. The WDM regime is often characterized by a liquid-like structure of ions, but the distances between particles are much smaller than those in traditional liquids, along with the quantum degeneracy of electrons. A heated dense substance can be generated in various ways--such as electrical explosion of conductors--in solids and metal clusters subjected to laser pulses or powerful particle beams, and in shock wave experiments [4, 5, 6]. The transport parameters of a nonideal plasma are crucial for developing models of inertial thermonuclear fusion, understanding the behavior of astrophysical objects such as white dwarfs and the internal structure of giant planets. An experimental study of WDM has been performed at laser facilities such as the National Ignition Facility (NIF) [7], the European XFEL [8], and OMEGA [9]. The interpretation of the observations from these high-pressure experiments requires theoretical models of the states of matter under extreme conditions [10; 11; 12; 13; 14; 15]. Theoretical approaches to studying heated dense matter must account for strong interparticle interactions and quantum degeneracy within the framework of quantum many-body theory. Numerous results related to heated dense substances have been obtained using quantum molecular dynamics (QMD), where the behavior of ions is governed by classical equations of motion while electrons are treated quantum mechanically through density functional theory (DFT) [16; 17; 18; 19]. In fact, DFT is the most widely used ab initio modeling method for describing the electronic structure and related physical properties of heated dense substances. DFT calculations help determine the distribution of electron density in a quantum system at equilibrium, where free energy is minimized [20; 21; 22; 23]. Furthermore, DFT can be utilized to derive various physical properties of heated dense substances, such as the density of electronic states, bulk modulus, dielectric constant, electrical conductivity, and more. Modeling the transport properties using DFT is computationally expensive. Therefore, to calculate various transport coefficients, such as ion diffusion and viscosity, across different density and temperature regimes, effective one-component plasma models are often used in conjunction with the Chapman-Enskog approach for the Coulomb logarithm or classical MD of ions [24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. In these models, the effect of electrons is incorporated into an ion-ion pair potential. The often-used ion-ion interaction model is a Yukawa-type screened potential [34; 35; 36; 37; 38; 39; 40]. More accurate potential can be generated using logarithm of the ion-ion pair correlation function [24; 25; 26; 27], e.g., from the hypernetted chain approximation. One advantage of effective one-component plasma models is their universal dependence on a few dimensionless parameters, typically describing coupling and screening, across a wide range of temperatures and densities. Among various materials, the WDM state of beryllium (Be) has attracted significant interest since Be is one of the capsule ablator materials used in inertial confinement fusion experiments. Therefore, the thermodynamic and transport properties of beryllium under the extreme conditions of high density and temperature have been the subject of active experimental and theoretical studies in recent times [7; 41; 42; 43]. Here, we present a QMD-based analysis of effective one-component plasma models, focusing on ion diffusion and the velocity autocorrelation function.