**7. Conclusion and outlook**

In this chapter, we (wongky@biomaps.rutgers.edu; kiniu@alumni.cuhk.net) discuss developing the method to systematically generate quantum free-energy profiles at an *ab initio* path-integral level in molecular simulations. Since quantum free energy or partition function is a universal central quantity in thermodynamics of biology, chemistry, and physics, we anticipate our method would be very crucial in both Life and Materials Sciences and wish that it could be used by non-specialists as a black box one day.

#### **8. References**

126 Molecular Dynamics – Theoretical Developments and Applications in Nanotechnology and Energy

Finally, we make a quite interesting table (Table 4) to compare the traditional *ab initio*  molecular orbital theory for electronic structure calculations with our systematic approach for computing internuclear quantum effects. In short, the rigor and the spirit of both types of methods is the same. We first breakdown or dissect a complicated many-body problem into many one-body problems. Then we identify which one bodies are more important. Next we

In order to systematically refine a classical free-energy profile to become ultimate quantum free-energy profile, in which both electrons and nuclei are treated quantum mechanically and adiabatically, we are developing a systematic *ab initio* path-integral free-energy

energy expansion (FEE) method (Section 4) with our automated integration-free pathintegral (AIF-PI) method (Section 5.2) such that we can perform *ab initio* path-integral simulations for realistic molecular systems. The key of this combination is that first we realize the quantum partition function can be computed as a classical configuration shown

, *EWV* (46)

where *V* is the original internuclear potential and *W* is the centroid potential. So once we get the accurate value of *W* using our AIF-PI method, we can go ahead using our FEE method to systematically upgrade the level of our classical free-energy profile to an *ab initio* pathintegral level, in which zero-point energy and tunnelling effects in nuclei, and isotope effects

perturbation (FEP) in the Hamiltonian space will be performed, using the recently derived

Fig. 1. Free energy perturbation for a water molecule in the Hamiltonian space using the

approach, we combine our novel free-

method in a more effective way, the free-energy

exp . *b Es P E K ab E s e* (47)

couple back those important one bodies to systematically approach the exact.

) approach. In this

"universal" probability density function (UPDF), which is defined as follows:

expansion (SAI-PI-FEE;

could all be incorporated.

In order to rigorously validate our

universal probability density function (UPDF).

in Eq. (27), then now in Eq. (23), we treat the *E* as:

**6. Systematic** *ab initio* **path-integral free-energy expansion approach** 


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**0**

**7**

*South Africa*

**Antisymmetrized Molecular Dynamics**

One of the aims of nuclear physics studies is to establish a complete theoretical description of the structure of nuclear systems. The correct theoretical description of nuclear structure is expected to help explain and accurately predict different properties of and process in nuclei (Donnelly & Raskin, 1986). Fundamental to a complete description of nuclear structure are the wave function describing nuclear systems, the Hamiltonian describing interactions in the nucleus and electromagnetic form factors describing charge and currents distributions in the nucleus. None of these components is completely understood and, therefore, none can be completely determined for a given nuclear system, yet. As a result, theoretical models of these components, based on different approximations that are guided by experimental observations, are usually employed in the description of nuclear systems. The quality of such models is often judged by their ability to explain existing experimental observations. Parallel to the theoretical developments, the developments in experimental technologies has led not only to the availability of more precise experimental data, but also to data in kinematical regions previously not accessible. The availability of precise experimental data in a wide kinematical region allow for more accurate quantitative testing and, therefore, development of realistic theoretical models that, in turn, generate more accurate predictions of experimental

Besides the understanding of static and dynamical properties of nuclear matter, studies in nuclear physics are aimed at constructing a comprehensive description of properties of nucleon-nucleon interactions. Few-nucleon systems provide unique favourable environment for such investigations (Rampho, 2010). In theoretical investigations, the interaction models in few-nucleon systems can be treated realistically and the resulting dynamical equations can be solved directly. The formulation and solution of dynamical equations for many-body systems is, on the other hand, quite challenging. Progress towards a better understanding of the nuclear force has been made over the years. Based on the accumulated experimental nucleon-nucleon scattering data different phenomenological nucleon-nucleon interaction models have been suggested. The models are constructed by fitting the models to existing nucleon-nucleon scattering data as well as some properties of the 2H nucleus (Cottingham et al., 1973; Lagaris & Pandharipande, 1981; Machleidt et al., 1987; Nagels et al., 1978; Wiringa et al., 1984). These interaction models are known as modern or realistic nucleon-nucleon potentials and are able to explain most of the static properties of light nuclei. Since the exact form of the short-range behavior of the nucleon-nucleon interaction is not completely determined, yet, the short-range part of many of these potential models is

**1. Introduction**

outcomes (Golak et al., 2005).

**and Nuclear Structure**

Gaotsiwe J. Rampho and Sofianos A. Sofianos *Department of Physics, University of South Africa*

