### 1. Development of mathematical and computational techniques for ab initio calculation of properties of atoms, molecules, solids.

- The determination of the properties of atoms, molecules and solids is one of the peak issues of contemporary research, since their theoretical calculation can reduce considerably the number of the necessary experiments, resulting thus in the reduction of the production cost of new products (solid state devices, pharmaceuticals, chemicals). For this purpose it is necessary to determine the energy eigenstates. However, since the exact determination of the eigenstates of the Hamiltonian of complex systems is impossible, one has to resume to approximations. These methods are based on theorems concerning these systems. One such theorem, on which the Density Functional Theory (DFT) Approximations are founded, is that of Hohenberg and Kohn, which shows that there is one to one correspondence between electron density and ground state. Our group has showed a corresponding theorem for excited states (Density Functional Theory for Atoms and Molecules, p. 206, R.G. Parr and Weitao Yang, Oxford University Press, N.York, 1989). The excited state DFT is based on this theorem.
- We developed the approximate method of direct mapping, where one uses an explicit form of the effective potential instead of deriving it indirectly through the density, which is an unknown physical quantity has to be determined by the calculation. (J. Chem. Phys. 125, 234111 (2006). With this approach one avoids the self-consistency problem and therefore the time needed for the calculations is significantly reduced.
- We developed a mehtodology for analyzing an urestricted Slater determinant in terms of eigenstates of
**S**The Hamiltonian of a many electrn system, in the absence of non-uniform magnetic field, commutes with^{2}.**S**so there are common eigenstates of these two operators. These eigenstates in the general case are not single Slater determinants but linear combination of Slater determinants. Thus, using the Unrestricted HF as the approximate wavefunction we can determine a better ground state but also excited states (Molecular Physics, 109, 11,1495 (2011))..^{2} - Excites states: We developed an approach, which is based on the Hartree-Fock method, does not require time consuming calculations and can give a good approximation to the excitation energies. The methodology is based on the fact that the subspaces of the occupied and virtual orbitals are mutual orthogonal. We evaluated the excited state wavefunctions for the series of atoms from Li to Mg.