**11.** *Ab initio* **method description**

The UHF method in the MOLCAO approximation serves as the basis of the method of calculation. The molecular orbital (MO) (*φ<sup>i</sup>* ) can be expanded in the atomic orbital (AO) χ*μ*(**r**) basis set

$$\varphi\_i(\mathbf{r}) = \sum\_{\mu}^{C\mu} \mathcal{X}\_{\mu}(\mathbf{r}).\tag{26}$$

Here *μ* denotes the number of AO. Matrix *C* is obtained by solving the Hartree-Fock matrix equation

$$FC = ESC.$$

The roots of this secular equation lie in the molecular energies *Ei* of electrons, where *F* stands for the Fockian matrix and *S* for the matrix of overlap integrals:

$$\mathbf{S}\_{\mu\nu} = \left[ \mathbf{\mathcal{X}}\_{\mu} \mathbf{\mathcal{X}}\_{\nu} \mathbf{d} \mathbf{r} \right] \tag{28}$$

The coefficients *Ciμ* allowed us to find the density matrix of the electron distribution:

$$P\_{\mu\nu} = 2\sum\_{i=1}^{N/2} C\_{i\mu} C\_{i\nu},\tag{29}$$

where *N* is the number of electrons, and the sum is over all occupied MO. We can find the bond order between the A and B atoms,

$$P\_{AB} - \sum\_{\mu \le A} \sum\_{\nu \le B} P\_{\mu\nu} \tag{30}$$

and the charge of atoms,

authors [39] had measured X-ray photoelectron emission spectra (XPS) of the SbSI crystals of valence bands (VB) and core levels (CL). XPS results showed the splitting of the CL. However,

The UHF method in the MOLCAO approximation serves as the basis of the method of

( ) ( ). *Ci*

m

 c <sup>=</sup> åm m*r r*

Here *μ* denotes the number of AO. Matrix *C* is obtained by solving the Hartree-Fock matrix

*FC ESC* = .

*S d* mn

The coefficients *Ciμ* allowed us to find the density matrix of the electron distribution:

*A B AB <sup>P</sup> <sup>P</sup>*

m n

mn<sup>=</sup> å <sup>=</sup>

bond order between the A and B atoms,

/ 2 <sup>1</sup> 2 , *<sup>N</sup> i i <sup>i</sup> P CC*

m n

> mn

where *N* is the number of electrons, and the sum is over all occupied MO. We can find the

 m n = ò c c*r.*

*i*

j

The roots of this secular equation lie in the molecular energies *Ei*

for the Fockian matrix and *S* for the matrix of overlap integrals:

) can be expanded in the atomic orbital (AO) χ*μ*(**r**)

(26)

(27)

(28)

of electrons, where *F* stands

(29)

Î Î <sup>=</sup>å å (30)

they have not explained theoretically the XPS spectra of the VB.

**11.** *Ab initio* **method description**

calculation. The molecular orbital (MO) (*φ<sup>i</sup>*

basis set

106 Piezoelectric Materials

equation

$$\mathcal{P}\_A = \mathcal{Z}\_A - \sum\_{\mu \in A} P\_{\mu \mu} \tag{31}$$

The calculations were performed by the GAMESS program [33]. Atomic coordinates have been used from the work of Lukaszewicz et al. [17].
