The phase transitions in finite complex molecular systems, i.e. the transition from a stable 3D molecular structure to a random coil state or vice versa (also known as folding process), has a long standing history of investigation. The phase transition of this nature occur or can be expected in many different complex molecular systems and in nano objects, such as polypeptides, proteins, polymers, DNA, fullerenes, nanotubes (see Fig. 1). They can be understood as a first order phase transitions, which are characterized by a rapid growth of the system free energy at a certain temperature. As a result, the heat capacity of the system as a function of temperature acquires a sharp maximum at the phase transition temperature.

**Figure 1.** *Examples of finite complex molecular systems experiencing phase transitions.*

*Ab initio* approach for studying finite phase transitions

We have developed a novel *ab initio* theoretical method for the phase transition description of the mentioned molecular systems [Yakubovich, Solov'yov, Solov'yov, Greiner, 2006-2009]. In particular, it was demonstrated that in polypeptides one can identify specific twisting degrees of freedom responsible for the folding dynamics of the amino acid chain (see Fig. 2).

**Figure 2.** *Dihedral angles φ and ψ used for characterization of the secondary structure of a polypeptide chain. The dihedral angle χ _{i} characterizes the rotation of the side radical along the C_{i}^{α}-C_{i}^{β} bond.*

The essential domain of the potential energy surface (PES) of polypeptides with respect to these degrees of freedom can be calculated and thoroughly analysed on the basis of *ab initio* methods such as density functional theory or Hartree-Fock method (see Fig. 3).

**Figure 3.** *PESs for different amino acids of alanine polypeptide consisting of 21 amino acids calculated as the function of twisting dihedral angles φ and ψ in: (a) second alanine, (b) third alanine, (c) fourth alanine (d) fifth alanine and (e) tenth alanine. Amino acids are numbered starting from the NH _{2} terminal of the polypeptide. Energies are given with respect to the lowest energy minimum of the PES in eV.*

This knowledge is sufficient for the construction of the partition function of a polypeptide chain and thus for the development of its complete thermodynamic description, which includes calculation of all essential thermodynamic variables and characteristics, e.g. heat capacity, PT temperature, free energy etc. The method has been proved to be applicable for the description of the PT in polyalanine of different length by the comparison of the theory predictions with the results of several independent experiments and with the results of molecular dynamics simulations [Yakubovich, Solov'yov, Solov'yov, Greiner, 2006-2009].

Comparison of the results of this method with the results of molecular dynamics simulations (see Fig. 4) allows one to establish the accuracy of the new approach for molecular systems of relatively small size and then to extend the description to the larger molecular objects, which is especially essential and interesting in those cases when molecular dynamics simulations are hardly possible because of computer power limitations.

**Figure 4.** *Dependencies of the heat capacity on temperature calculated for the alanine polypeptides consisting of 21, 30, 40, 50 and 100 amino acids. The results obtained using the statistical approach are shown with the thick solid line, while the results of MD simulations are shown with the thin solid line. C _{300} denotes the heat capacity at 300 K.*