First-principles studies of the structural, elastic and optical properties of non-centrosymmetric cyclophosphates

Using density functional theory methods using gradient, hybrid, short- and long-range hybrid functionals, including taking into account the dispersion correction, in the basis of localized atomic orbitals of the CRYSTAL package, calculations of the crystal and electronic structure, elastic, piezoelectric, linear and nonlinear optical properties of hexagonal KMg(PO$_3$)$_3$, KCa(PO$_3$)$_3$, RbCd(PO$_3$)$_3$, trigonal KZn(PO$_3$)$_3$, RbZn(PO$_3$)$_3$, tetragonal K$_2$Sr(PO$_3$)$_4$ cyclophosphates. It has been shown that in hexagonal phosphates, fluorine and oxygen atoms form [P$_3$O$_9$] rings, in tetragonal phosphates – [P$_4$O$_{12}$], in trigonal phosphates – [P$_3$O$_9$] trimers, united through zinc atoms into hexagonal rings around K(Rb) atoms. Band structures and partial densities of electronic states were calculated, and the nature of valence and unoccupied states was determined. Elastic constants and moduli were calculated and conclusions were drawn about the plasticity or fragility of materials, and from the components of the piezotensor about their mechanoelectric properties. The coefficients of second harmonic generation and birefringence were obtained and the use as nonlinear optical materials was assessed.