**2. The generator coordinate Hartree-Fock method and the construction of basis sets**

In this section, we provide a brief history of the scientific scene that gave rise to the GCHF method and the atmosphere in which it has developed. We will also make a presentation of the GCHF method as strategy for building extended and contracted basis sets as well as the procedure used to evaluate their quality and the selection of polarization and diffuse functions used in calculations of perovskites.

### **2.1. Brief history of scientific scene and atmosphere for the development of GCHF method**

In 1957, the generator coordinate (GC) method for the nuclear bound state [4] was introduced in literature. According to this method, the variational trial function is written as an integral transform over a nucleonic wave function and a weight function depending on a parameter (GC), i.e., *f*(α). In this way, the common variational principle, δE/δα=0 (where E is the total energy of system and α is the GC), a priori, was made more powerful with the requirement δE/δ*f*α, leading to an integral equation. Most applications in nuclear physics relied on the Gaussian overlap approximation (GOA). Although there were some attempts in numerical solution [5], the discussion of the GC method against the background of the Fredholm theory of linear integral equations was reported in literature [6, 7]. Probably the first application of GC to electrons in a molecular system (hydrogen molecule) was reported in literature in the second half of the 1970s [8]. After, the GC method was applied to several model problems, including the He atom, with special emphasis on various aspects of the discretization technique [9]. Besides, further developments in discretization techniques were reported in literature [10, 11].

On the other hand, in the late 1960s, the integral method for atomic and molecular systems was introduced in the literature, closely related to GC [12, 13]. In these applications, explicit forms were chosen for *f*(α) (as the delta function) leading to a variational treatment for the integration limits. Extensive bibliography on this method was found in literature [14].

The GCHF method was introduced in 1986 [3], and one of the first applications was in the generation of Gaussian- and Slater-type orbitals (GTO and STO) universal basis sets [15–17]. The GCHF method was used to build contracted GTF (Gaussian Type-Function) basis sets for the first- and second-row atoms which were applied in calculations of various properties at the HF, CISD (configuration interaction with single and double excitations), and MP2 (Möller-Plesset perturbation theory to second order) levels for a group of neutral and charged diatomic species [18, 19]. Also in the 1990s, efforts were concentrated on the development of GCHF formalism for molecular systems and the first applications have been focused on building basis for H2, N2, and Li2 [20] and LiH, CO, and BF [21]. Applying GCHF basis sets for calculation of properties of polyatomic systems with the first application being concerned with the study of electronic properties and IR spectrum of high tridymite began in the second half of the 1990s [22]. First-principles (ab initio) calculations of electron affinities of enolates were also per‐ formed [23]. Theoretical interpretation of IR spectrum of hexaaquachromium (III) ion, tetraoxochromium (IV) ion, and tetraoxochromium (VI) ion [24] and theoretical interpretation of the Raman spectrum [25] and the vibrational structure of hexaaquaaluminum (III) ion [26] were also conducted with GCHF basis. Process of adsorption of sulfur on platinum (2 0 0) surface [27], infrared spectrum of isonicotinamide [28], and transition metal complexes [29– 32] were also studied with GCHF basis sets.

#### **2.2. Construction of extended and contracted basis sets for calculations in perovskites**

procedure used to evaluate their quality and the selection of polarization and diffuse functions

In 1957, the generator coordinate (GC) method for the nuclear bound state [4] was introduced in literature. According to this method, the variational trial function is written as an integral transform over a nucleonic wave function and a weight function depending on a parameter (GC), i.e., *f*(α). In this way, the common variational principle, δE/δα=0 (where E is the total energy of system and α is the GC), a priori, was made more powerful with the requirement δE/δ*f*α, leading to an integral equation. Most applications in nuclear physics relied on the Gaussian overlap approximation (GOA). Although there were some attempts in numerical solution [5], the discussion of the GC method against the background of the Fredholm theory of linear integral equations was reported in literature [6, 7]. Probably the first application of GC to electrons in a molecular system (hydrogen molecule) was reported in literature in the second half of the 1970s [8]. After, the GC method was applied to several model problems, including the He atom, with special emphasis on various aspects of the discretization technique [9]. Besides, further developments in discretization techniques were reported in literature [10,

On the other hand, in the late 1960s, the integral method for atomic and molecular systems was introduced in the literature, closely related to GC [12, 13]. In these applications, explicit forms were chosen for *f*(α) (as the delta function) leading to a variational treatment for the integration limits. Extensive bibliography on this method was found in literature [14].

The GCHF method was introduced in 1986 [3], and one of the first applications was in the generation of Gaussian- and Slater-type orbitals (GTO and STO) universal basis sets [15–17]. The GCHF method was used to build contracted GTF (Gaussian Type-Function) basis sets for the first- and second-row atoms which were applied in calculations of various properties at the HF, CISD (configuration interaction with single and double excitations), and MP2 (Möller-Plesset perturbation theory to second order) levels for a group of neutral and charged diatomic species [18, 19]. Also in the 1990s, efforts were concentrated on the development of GCHF formalism for molecular systems and the first applications have been focused on building basis for H2, N2, and Li2 [20] and LiH, CO, and BF [21]. Applying GCHF basis sets for calculation of properties of polyatomic systems with the first application being concerned with the study of electronic properties and IR spectrum of high tridymite began in the second half of the 1990s [22]. First-principles (ab initio) calculations of electron affinities of enolates were also per‐ formed [23]. Theoretical interpretation of IR spectrum of hexaaquachromium (III) ion, tetraoxochromium (IV) ion, and tetraoxochromium (VI) ion [24] and theoretical interpretation of the Raman spectrum [25] and the vibrational structure of hexaaquaaluminum (III) ion [26] were also conducted with GCHF basis. Process of adsorption of sulfur on platinum (2 0 0) surface [27], infrared spectrum of isonicotinamide [28], and transition metal complexes [29–

**2.1. Brief history of scientific scene and atmosphere for the development of GCHF**

used in calculations of perovskites.

32] were also studied with GCHF basis sets.

**method**

66 Piezoelectric Materials

11].

The GCHF approach is based in choosing the one-electron functions as the continuous superposition:

$$\log\_i(\mathbf{l}) = \begin{bmatrix} \Psi\_i(\mathbf{l}, \mathbf{a}) \mathbf{f}\_i(\mathbf{a}) \text{ dai} = \mathbf{l}, \mathbf{2}, \mathbf{3}, \dots \mathbf{n} \end{bmatrix} \tag{1}$$

where ψ<sup>i</sup> are the generator functions (GTOs for the case of perovskites) and *f*<sup>i</sup> are the weight functions (WFs) and ü is the GC.

The φ<sup>i</sup> are then employed to build a Slater determinant for the multi-electronic wave function and minimizing the total energy with respect to *f*<sup>i</sup> (α), which arrives to the HF-Griffin-Wheeler (HFGW) equations:

$$\begin{cases} \mathsf{F}(\alpha, \beta) \cdot \varepsilon \mathsf{S}(\alpha, \beta) \mathsf{I} f(\beta) \mathsf{d}\beta = \mathsf{O} & \mathsf{i} = \mathsf{I}, \ \mathsf{Z}, \\ \mathsf{3}, \ldots, \mathsf{m} \end{cases} \tag{2}$$

where üi are the HF eigenvalues and the Fock kernels, F(α,β) and S(α,β) are defined in Refs. [3, 16].

The HFGW equations are integrated numerically through discretization with a technique that preserves the integral character of the GCHF method, i.e., integral discretization (ID). The ID technique is implemented with a relabeling of the GC space [15], i.e.,

(3)

with *A* is a scaling parameter numerically determined. For perovskites *A* = 6.0.

The new GC Ω space is discretized, by symmetry, in an equally space mesh formed by ü values so that

$$
\Delta \Omega = \Omega\_{\text{min}} + (\text{k} + \text{l}) \Delta \Omega \quad \text{k} = 1, 2, 3, \dots \text{N} \tag{4}
$$

In Eq. (4), N corresponds to the number of discretization points defining the basis set size, Ωmin is the initial point, and ΔΩ is the increment.

The values of Ωmin (lowest value) and the highest value Ωmax=Ωmin+(N − 1)ΔΩ are chosen in order to adequately encompass the integration range of *f*(Ω). This is visualized by drawing the WFs from preliminary calculations with arbitrary discretization parameters. To illustrate the application of the GCHF method in the choice of basis sets for perovskites, we refer to our publication that investigates the piezoelectricity in LaFeO3 [33]. **Figure 1** shows the respective 2s, 3p, and 5d weight functions for O (3 P), Fe (5 F), and La (2 D) atoms.

**Figure 1.** 2s, 3p, and 5d WFs for O (3 P), Fe (5 D), and La (2 D) atoms obtained with (20s14p), (30s19p13d), and (31s23p18d0 Gaussian) basis sets, respectively. Reproduction authorized by authors [33].

In the solution of the discretization of Eq. (2), the (22s14p), (30s19p13d), and (32s24p117d) GTOs basis sets were used to O (3 P), Fe (5 D), and La (2 D) atoms, respectively, as defined by the mesh of Eq. (3). The values of Ωmin and Ωmax were selected in order to satisfy the relevant integration range of each WF atom. In **Table 1**, the discretization parameters (which define the exponents) for the built basis sets are shown.


a The scaling parameter used for s, p, and d symmetries for all atoms studied is equal to *A* = 6.0. Reproduction authorized by authors [33].

**Table 1.** Discretization parameters (which define the exponents) for O (3 P), Fe(5 D), and La(2 D) atomsa .

For perovskite calculations, a basis set is usually developed in three stages: (1) construction of the extended basis set in atomic calculations, (2) construction of the contracted basis set using a segmented or general scheme (at this stage, due to increased availability of software access, we use the segmented contraction scheme), and (3) addition of supplementary functions of diffuse and polarization character. The first stage is usually a straightforward task. To illustrate the second step, we consider the segmented contraction [31] of basis sets to O (3 F), Fe (5 D), and La (2 D) atoms in LaFeO3 [33]. The 20s14p basis set to O atom was contracted to 7s6p as follows: 16, 1, 1, 1, 1/9, 1, 1, 1. For Fe atom, the 30s19p13d GTO basis set was contracted to 13s8p6d according to 14, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1/12, 1, 1, 1, 1, 1, 1, 1/8, 1, 1, 1, 1, 1. For La atom, the 32s24p17d GTO basis set was contracted according to 13, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1/10, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1/10, 2, 1, 1, 1, 1, 1.

### **2.3. Quality evaluation of the basis sets in perovskite calculations**

**Figure 1.** 2s, 3p, and 5d WFs for O (3

68 Piezoelectric Materials

GTOs basis sets were used to O (3

a

La (2

authorized by authors [33].

exponents) for the built basis sets are shown.

**Table 1.** Discretization parameters (which define the exponents) for O (3

1/10, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1/10, 2, 1, 1, 1, 1, 1.

P), Fe (5

P), Fe (5

**Symmetry O Fe La**

The scaling parameter used for s, p, and d symmetries for all atoms studied is equal to *A* = 6.0. Reproduction

the second step, we consider the segmented contraction [31] of basis sets to O (3

For perovskite calculations, a basis set is usually developed in three stages: (1) construction of the extended basis set in atomic calculations, (2) construction of the contracted basis set using a segmented or general scheme (at this stage, due to increased availability of software access, we use the segmented contraction scheme), and (3) addition of supplementary functions of diffuse and polarization character. The first stage is usually a straightforward task. To illustrate

D) atoms in LaFeO3 [33]. The 20s14p basis set to O atom was contracted to 7s6p as follows: 16, 1, 1, 1, 1/9, 1, 1, 1. For Fe atom, the 30s19p13d GTO basis set was contracted to 13s8p6d according to 14, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1/12, 1, 1, 1, 1, 1, 1, 1/8, 1, 1, 1, 1, 1. For La atom, the 32s24p17d GTO basis set was contracted according to 13, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1,

(31s23p18d0 Gaussian) basis sets, respectively. Reproduction authorized by authors [33].

D), and La (2

In the solution of the discretization of Eq. (2), the (22s14p), (30s19p13d), and (32s24p117d)

D), and La (2

mesh of Eq. (3). The values of Ωmin and Ωmax were selected in order to satisfy the relevant integration range of each WF atom. In **Table 1**, the discretization parameters (which define the

**Ωmin ΔΩ** *N* **Ωmin ΔΩ** *N* **Ωmin ΔΩ** *N*

P), Fe(5

D), and La(2

D) atomsa .

F), Fe (5

D), and

s −0.3969 0.122 22 −0.5998 0.114 30 −0.6304 0.115 32 p −0.4387 0.119 14 −0.2758 0.110 19 −0.3938 0.104 24 d − – – −0.3771 0.117 13 −0.4532 0.113 17

D) atoms obtained with (20s14p), (30s19p13d), and

D) atoms, respectively, as defined by the

In order to evaluate the quality of the contracted GTO basis sets in perovskite studies, the calculations of total energy, the highest occupied molecular orbital (HOMO) energy and the one level below to highest occupied molecular orbital (HOMO-1) energies for perov‐ skites fragments at the HF level [35], are performed and the results are compared with those obtained from the extended GTOs basis sets. For LaFeO3, the <sup>2</sup> FeO+1 and <sup>1</sup> LaO+1 frag‐ ments were studied. Comparison of the calculated values with the contracted and extended GTOs basis sets, respectively, shows differences of 0.2534 and 0.2827 hartree for the total energy and 8.8 × 10−4 and 1.26 × 10−3 hartree for the energy orbital. These values show a very good quality of the contracted GTOs basis sets for the study of properties of perov‐ skite LaFeO3 [33].

#### **2.4. Supplementation of the basis sets with polarization and diffuse functions for calculations in perovskites**

In order to better describe the properties of perovskite systems in the implementation of ab initio calculations, the inclusion of polarization functions in GTO basis sets is necessary. A methodology that has been a good strategy in the choice of polarization function for con‐ tracted GTOs bases sets is to extract the polarization function from the own Gaussian prim‐ itive basis set using successive calculations for the [ABO3]2 fragment for different primitive functions, taking into account the minimum energy criterion. For the [LaFeO3]2 fragment, the polarization function was included in the contracted GTO basis set for the O atom, i.e., αd = 0.30029 [33].

The role of the basis set is a crucial point in ab initio calculations of systems containing transition metals, since the description of the metal atom's configuration in complex is different from neutral state. In our studies with perovskites, the adequate diffuse functions for supple‐ mentation of contracted GTOs basis sets have been selected via one of the following methods: (1) the exponents of the basis sets are ranked according to magnitude and plotted on a logarithmic scale with equally spaced abscissas. Then the extrapolation of curve was done to smaller values of exponents, thereby obtaining exponents for a diffuse function [36] and (2) using the total energy optimization of the ground-state anions of the metals present in perovskite structure [37].

For LaFeO3, the adequate diffuse functions were chosen using the first methodology described, i.e., for the contracted GTO basis set of Fe and La atoms, the diffuse functions are, respectively, αs = 0.0138038, αp = 0.1000000, and αd = 0.054954; αs = 0.0125892, αp = 0.0446683, and αd = 0.0112201.

The results obtained in the study of the perovskite LaFeO3 with contracted GTO ba‐ sis sets constructed with GCHF method strategy are well documented in literature [33].
