1 Introduction

The density functional theory (DFT) is one of the most popular methods for quantum chemical calculation. The total energy of DFT is expressed as a functional of electron density [1]. Because the exact functional form is unknown, approximate functionals are generally used. Despite the widespread use of DFT, the approximate functionals are limited by inaccurate definitions of chemical phenomena, such as reaction barriers [2, 3], band gap [4], polarizability [5], and charge transfer [6]. The causes of these errors have been discussed in terms of the self-interaction error (SIE), which stems from the exchange term of the approximate functional [7]. More inclusively, the definition of the delocalization error is proposed [8, 9].

The delocalization error occurs in the fractional occupation number (FON) states [8, 10]. The ground-state energy obtained by the exact functional varies piecewise linearly with respect to the FON electrons between two consecutive integers. However, the approximate functional deviates from the linearity. For example, the total energy change dependent on electron number with FON states becomes a concave curve in the highest occupied molecular orbital (HOMO) [10].

Several researchers have developed methods for correcting the delocalization error or SIE [7, 11,12,13,14,15]; however, this aspect continues to be challenging in DFT [16]. Global and range-separated hybrid functionals, which mix certain ratios of Hartree–Fock exchange (HFx) into exchange functional, reduce these errors to a certain extent. The long-range corrected (LC-) DFT [17, 18], which calculates exchange energy as long-range HFx and short-range exchange functional, indicates linear dependence of total energy on change of the occupation number of HOMO [19]. For the description of the linearity in core orbitals, the importance of short-range HFx is revealed by the LCgau scheme [20] and core-valence-Rydberg functional [21, 22].

The accurate description of linearity is one measure of evaluating the performance of functionals. Orbital-specific hybrid functionals are constructed by imposing the linearity condition for orbitals of FON states, which reasonably reproduces the ionization potential and excitation energies involving core, valence, and Rydberg orbitals [23,24,25,26]. The localized orbital scaling correction was developed by enforcing the linearity condition for the relation between energy and electron number and correctly described phenomena, such as polarizability, molecular dissociation, and band gap [14, 27,28,29,30].

When calculating heavy-element compounds, relativistic effects need to be considered. The fundamental theory for incorporating relativistic effects into quantum chemistry is the use of the four-component (4c) Hamiltonian, which provides accurate results through explicit treatment of both positive- and negative-energy states. The 4c Hamiltonian consists of a one-electron Dirac Hamiltonian [31] and two-electron interaction operator, such as Coulomb [32], Coulomb–Gaunt [33], and Coulomb–Breit [34] Hamiltonians. The operators and wavefunction have large and small components.

The two-component (2c) Hamiltonian, which eliminates small components or separates the positive- and negative-energy states, is another approach toward relativistic quantum chemistry. For the one-electron system, various 2c Hamiltonians, such as the Foldy–Woutheysen Hamiltonian [35], Douglas–Kroll–Hess Hamiltonian [36,37,38,39,40], regular approximation [41, 42], and normalized elimination of the small component method [43] were proposed. Particularly, the exact two-component [44,45,46,47,48,49,50] or infinite-order two-component (IOTC) [51] Hamiltonian, which decouples the positive- and negative energy states by unitary transformation, is equivalent to one-electron Dirac Hamiltonian under the no-pair approximation [52], that is, in the case that does not consider the explicit positron–electron coupling. As a result of the transformation of the one-electron Hamiltonian, the wave function is transformed into 2c framework. Evaluating the expectation values of the untransformed operator with transformed wave function yields inaccurate results, called the picture change error (PCE) [53]. Certain 2c Hamiltonians are extended to a many-electron system including the picture change correction (PCC) of the two-electron operator [54,55,56,57,58]. We also developed accurate and efficient theories based on the IOTC Hamiltonian [59,60,61,62,63,64,65]. We particularly pointed out that the PCC of electron density in DFT is as inevitable as the transformation of one- and two-electron Hamiltonians for reproducing the accuracy of 4c theory [62, 63]. Recently, the extension of two-electron IOTC Hamiltonian to LC-DFT was proposed for the PCC of both electron density and range-separated two-electron term.

In this study, we examine the PCE of FON states in the relativistic theory based on the spin-free IOTC Hamiltonian. The remainder of this paper is organized as follows. In Sect. 2, the linearity condition for FON states and the picture change (PC)-corrected DFT as the theoretical background are discussed. Section 3 explains the computational details. The results and discussion are presented in Sect. 4 followed by the conclusion in Sect. 5.

2 Theoretical background

2.1 Linearity condition for FON states

The energy of the exact DFT with respect to FON electron is expressed as follows:

$$\begin{gathered} E\left( {N + \Delta n} \right) = \;\left( {1 - \Delta n} \right)E\left( N \right) + \Delta nE\left( {N + 1} \right) \\ = \left( {E\left( {N + 1} \right) - E\left( N \right)} \right)\Delta n + E\left( N \right), \\ \end{gathered}$$
(1)

where N is a positive integer, Δn is a fractional number (0 ≤ Δn ≤ 1), and E(N) is the ground-state energy of an N-electron system [66]. The curve of E with respect to Δn becomes straight, which is termed the linearity condition for total energies.

In Kohn–Sham (KS) DFT [67], the energy is expressed as follows:

$$E^{{{\text{KS}}}} = \sum\limits_{i} {f_{i} \left\langle {\varphi_{i} } \right|\hat{h}\left| {\varphi_{i} } \right\rangle } + \int {\int {d{\mathbf{r}}_{1} d{\mathbf{r}}_{2} \frac{{\rho \left( {{\mathbf{r}}_{1} } \right)\rho \left( {{\mathbf{r}}_{2} } \right)}}{{\left| {{\mathbf{r}}_{1} - {\mathbf{r}}_{2} } \right|}}} } + E_{{{\text{XC}}}} \left[ \rho \right],$$
(2)

with the occupation number fi of the i-th KS orbital φi, one-electron Hamiltonian h, and electron density ρ. The first term denotes the kinetic energy and interaction between electron density and external potential. The second term is the Coulomb interaction of electron density. The third term represents the exchange–correlation term, which is a functional of ρ. ρ is calculated as the expectation value of delta operator δ:

$$\begin{gathered} \rho \left( {\mathbf{r}} \right) = \sum\limits_{i} {f_{i} \left\langle {\varphi_{i} } \right|\delta \left| {\varphi_{i} } \right\rangle } \\ = \sum\limits_{i} {f_{i} \left| {\varphi_{i} } \right|^{2} } . \\ \end{gathered}$$
(3)

Introducing φi into Eq. (2), the second term is evaluated by the expectation value of 1/rij, \(\sum\limits_{i > j} {\left\langle {\varphi_{i} \varphi_{j} } \right|{1 \mathord{\left/ {\vphantom {1 {r_{ij} }}} \right. \kern-0pt} {r_{ij} }}\left| {\varphi_{i} \varphi_{j} } \right\rangle }\).

According to Janak’s theorem [68], orbital energy εi, which is the eigenvalue of φi, is adherent to the following expression:

$$\frac{\partial E}{{\partial f_{i} }} = \varepsilon_{i} .$$
(4)

The HOMO energy is equivalent to the negative value of the first ionization potential [69]. The differentiation of E by fi in the range of 0–1 corresponds to the orbital energy of the i-th orbital, i.e. constant value:

$$\left. {\frac{\partial E}{{\partial f_{i} }}} \right|_{{0 \le f_{i} \le 1}} = \varepsilon_{i} {\text{ = const}}{.}$$
(5)

Equation (5) leads the following expression:

$$\left. {\frac{{\partial^{2} E}}{{\partial f_{i}^{2} }}} \right|_{{0 \le f_{i} \le 1}} = \left. {\frac{{\partial \varepsilon_{i} }}{{\partial f_{i} }}} \right|_{{0 \le f_{i} \le 1}} = 0,$$
(6)

indicating that the orbital energy is constant for the change of occupation number of the corresponding orbital. This is the linearity condition for orbital energies.

2.2 Picture-change-corrected density functional theory

In this subsection, the PC-corrected DFT based on the IOTC Hamiltonian is briefly discussed. In the IOTC theory, the one-electron 4c Dirac Hamiltonian \({\varvec{H}}_{4}^{{\text{D}}}\) is completely block-diagonalized by unitary transformation as follows:

$${\varvec{U}}^{\dag } {\varvec{H}}_{4}^{{\text{D}}} {\varvec{U}} = \left( {\begin{array}{*{20}c} {{\varvec{h}}_{2}^{ + } } & {{\mathbf{0}}_{2} } \\ {{\mathbf{0}}_{2} } & {{\varvec{h}}_{2}^{ - } } \\ \end{array} } \right),$$
(7)

where U is unitary transformation for one-particle and \({\varvec{h}}_{2}^{ + }\) and \({\varvec{h}}_{2}^{ - }\) represent one-particle positive-and negative-energy states, respectively [51]. Along this unitary transformation into 2c formalism, the wavefunction has a different picture from those of the 4c framework. Therefore, transformed operators must be adopted for the calculation of expectation values.

The extension of IOTC Hamiltonian to many-electron system is accomplished by unitary transformation of the one-electron Hamiltonian and two-electron interaction, simultaneously. Here, the two-electron PCEs are corrected by the unitary transformation of the two-electron operator. In this study, we adopt the Coulomb interaction 1/rij as the two-electron interaction. In the IOTC Hamiltonian for the two-electron term [57], the approximation that treats unitary transformation for numerous particles U(i, j,…) as the direct product of one-particle unitary transformation U(i), is introduced:

$${\varvec{U}}\left( {i,j, \cdots } \right) \approx {\varvec{U}}\left( i \right) \otimes {\varvec{U}}\left( j \right) \otimes \cdots .$$
(8)

The electronic Hamiltonian consists of positive-energy parts of one- and two-electron operators as follows:

$${\varvec{H}}_{2}^{{{\text{elec}}}} = \sum\limits_{i} {{\varvec{h}}_{2}^{ + } \left( i \right)} + \sum\limits_{i} {{\varvec{g}}_{2}^{ + + } \left( {i,j} \right)} ,$$
(9)

where \({\varvec{g}}_{2}^{ + + }\) represents the two-particle positive-energy state.

When applying IOTC Hamiltonian to DFT [62, 63], the PC of δ is required because electron density is represented as the expectation value of δ shown in Eq. (3). The unitary transformation of δ is as follows:

$${\varvec{U}}^{\dag } {\varvec{\delta}}_{4} {\varvec{U}} = \left( {\begin{array}{*{20}c} {{\varvec{\delta}}_{2}^{ + } } & {{\mathbf{0}}_{2} } \\ {{\mathbf{0}}_{2} } & {{\varvec{\delta}}_{2}^{ - } } \\ \end{array} } \right),$$
(10)

where \({\varvec{\delta}}_{2}^{ + }\) and \({\varvec{\delta}}_{2}^{ - }\) represent positive and negative energy parts, respectively. The electron density of the IOTC Hamiltonian ρ+ is calculated using \({\varvec{\delta}}_{2}^{ + }\),

$$\rho^{ + } = \sum\limits_{i} {f_{i} \left\langle {\varphi_{i}^{ + } } \right.\left| {{\varvec{\delta}}_{2}^{ + } } \right|\left. {\varphi_{i}^{ + } } \right\rangle } ,$$
(11)

where \(\varphi_{i}^{ + }\) is the 2c spinor. The total energy of DFT based on the IOTC Hamiltonian is expressed as follows:

$$\begin{aligned} E^{{{\text{IOTC}}}} & = \sum\limits_{i} {f_{i} \left\langle {\varphi_{i}^{ + } } \right|{\varvec{h}}_{2}^{ + } \left( i \right)\left| {\varphi_{i}^{ + } } \right\rangle } + \int {\int {d{\mathbf{r}}_{i} d{\mathbf{r}}_{j} \rho^{ + } \left( {{\mathbf{r}}_{i} } \right)\rho^{ + } \left( {{\mathbf{r}}_{j} } \right){\varvec{g}}_{2}^{ + + } \left( {i,j} \right)} } + E_{{{\text{XC}}}} \left[ {\rho^{ + } } \right] \\ & = \sum\limits_{i} {f_{i} \left\langle {\varphi_{i}^{ + } } \right|{\varvec{h}}_{2}^{ + } \left( i \right)\left| {\varphi_{i}^{ + } } \right\rangle } + \sum\limits_{i > j} {f_{i} f_{j} \left\langle {\varphi_{i}^{ + } \varphi_{j}^{ + } } \right|{\varvec{g}}_{2}^{ + + } \left( {i,j} \right)\left| {\varphi_{i}^{ + } \varphi_{j}^{ + } } \right\rangle } + E_{{{\text{XC}}}} \left[ {\rho^{ + } } \right]. \\ \end{aligned}$$
(12)

3 Computational details

Noble gas atoms (Ne, Ar, Kr, Xe, and Rn) were numerically analyzed. Sapporo-TZP-2012 + d basis sets [70] for Ne, Ar, and Sapporo-DKH3-TZP-2012 + d basis sets [71] for Kr, Xe, and Rn were applied in an uncontracted manner.

The exchange–correlation functionals used in this study were 100% Becke’s exchange (B88) [72] with Lee–Yang–Parr (LYP) [73] correlation (BLYP), Becke’s half-and-half exchange (50% B88 + 50% HFx) with LYP correlation (BHHLYP) [74], 100% HFx with LYP correlation (HFLYP), and LC-BLYP with range-separation parameter 0.47 [3]. The modified ratio of B88 and HFx, namely, 40% B88 + 60% HFx, 30% B88 + 70% HFx, 20% B88 + 80% HFx, and 10% B88 + 90% HFx, were also employed in conjunction with LYP correlation.

For the spin-free IOTC calculation, the one-electron spin-free IOTC Hamiltonian with two-electron Coulomb operator (1eIOTC) and one- and two-electron spin-free IOTC Hamiltonian (1e2eIOTC) were used along with the PC-corrected (ρ+) and -uncorrected electron density, which correspond to Eqs. (11) and (3), respectively. FON energy was calculated self-consistently by varying the occupation numbers of HOMO or 1s orbital. All the calculations were performed with the modified version of the GAMESS program [75, 76].

4 Results and discussion

4.1 Picture change effects on delocalization error

In this subsection, we examine the picture change effects on the delocalization error. Figures 1, 2, 3 and 4 show the total energy deviation ΔE,

$$\Delta E = E\left( {N + \Delta n} \right) - \left[ {\left\{ {1 - \left| {\Delta n} \right|} \right\}E\left( N \right) + \left| {\Delta n} \right|E\left( {N - 1} \right)} \right],$$
(12)

with respect to Δn. According to the linearity condition, ΔE becomes 0 regardless of Δn in the exact energy.

Fig. 1
figure 1

Total energy deviation from ideal energy of FON states of HOMO in a Ne, b Ar, c Kr, d Xe, and e Rn atoms obtained by BLYP functionals with several relativistic treatments. Right panels show the enlarged view near the minima

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Fig. 2
figure 2

Total energy deviation from ideal energy of FON states of 1s orbital in a Ne, b Ar, c Kr, d Xe, and e Rn atoms obtained by BLYP functionals with several relativistic treatments. Right panels show the enlarged view near the minima

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Fig. 3
figure 3

Total energy deviation from ideal energy of FON states of HOMO a Ne, b Ar, c Kr, d Xe, and e Rn atoms obtained by HFLYP functionals with several relativistic treatments. Right panels show the enlarged view near the minima

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Fig. 4
figure 4

Total energy deviation from ideal energy of FON states of 1s in a Ne, b Ar, c Kr, d Xe, and e Rn atoms obtained by HFLYP functionals with several relativistic treatments. Right panels show the enlarged view near the minima

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Figure 1 provides the HOMO result of Ne, Ar, Kr, Xe, and Rn atoms obtained by the BLYP functional. Nonrelativistic Hamiltonian, 1eIOTC, 1eIOTC with ρ+, 1e2eIOTC, and 1e2eIOTC with ρ+ are compared. The right panels show the enlarged view near the minima. All elements and relativistic treatments show the concave curves. Comparing elements, the absolute value of ΔE is largest in Ne and smaller in Ar, Kr, Xe, and Rn in that order. Lighter elements have a larger delocalization error. In the results of Ne and Ar, the relativistic treatments indicate close values to nonrelativistic treatments. The relativistic treatments estimate lower ΔE of Kr, Xe, and Rn than the nonrelativistic treatment. The differences among 1eIOTC, 1eIOTC with ρ+, 1e2eIOTC, and 1e2eIOTC with ρ+ are hardly identified. PCEs of two-electron and density operators rarely affect the FON states of HOMO, which are over stabilized by one-electron relativistic effects.

Figure 2 presents the results for 1s orbital. Contrary to the case of HOMO shown in Fig. 1, the absolute value of ΔE is largest in Rn and becomes relatively small in Xe, Kr, Ar, and Ne in that order. Heavier elements are limited from larger delocalization error in the FON states of 1s orbital. Comparing nonrelativistic and relativistic treatments, the differences are found in Ar, Kr, Xe, and Rn, which are minor in Ar and obvious for other elements in the order of magnitude Kr < Xe < Rn. The PCEs of two-electron and density operators are also large. 1eIOTC (1eIOTC with ρ+) yield lower ΔE than that of 1e2eIOTC (1e2eIOTCwith ρ+). In terms of ρ+, the ΔE obtained by 1eIOTC (1e2eIOTC) is higher than that obtained by 1eIOTC with ρ+ (1e2eIOTC with ρ+). For example, the ΔE of Rn at Δn = 0.5 is − 155.6, − 161.6, − 137.6, and − 143.7 eV for 1eIOTC, 1eIOTC with ρ+, 1e2eIOTC, and 1e2eIOTC with ρ+. The PCE of two-electron and density operator overestimates and underestimates delocalization error, respectively. The results of a comparative analysis of the difference owing to the PC of two-electron and density operators show that the PCEs of two-electron operators (~ 16 eV in Rn at Δn = 0.5) are larger than those of the density operator (~ 6 eV in Rn at Δn = 0.5).

Figures 3 and 4 show the results obtained by the HFLYP functional for HOMO and 1s orbital, respectively. Contrary to Figs. 1 and 2, all curves are convex. HFx destabilizes the FON states. These errors are referred to as a localization error, which is derived from the lack of electron correlation in HFx [8]. The absolute values of ΔE are smaller than those of BLYP. Electron correlation in the exchange term is smaller than the error derived from the inexact form of exchange functional. In the results of HOMO shown in Fig. 3, the localization error for the lighter elements increases. The differences between nonrelativistic and relativistic treatments, which is slightly observed in Rn, are small. The PCE values of two-electron and density operators are also small in all elements.

Opposite to the HOMO, the localization errors of 1s orbital shown in Fig. 4 are larger in heavier elements. The effects of relativistic treatments are apparent in Ar, Kr, Xe, and Rn. The PCE of the two-electron operator, which overestimates the localization error, is clear in Xe and Rn. In contrast to the BLYP in Fig. 2, the differences due to ρ+ are so small as to be invisible. The exchange part of HFLYP is not affected by the PC of density operator because it does not include electron density. Although the correlation part is influenced by the PC of the density operator, the correlation energy is considerably smaller than the exchange energy. Therefore, the PCE of the density operator for HFLYP is small.

Owing to the error cancellation of concave and convex characteristics of BLYP and HFLYP functionals, the curvature of ΔE becomes smaller when B88 exchange and HFx are mixed as shown in Fig. S1 in Supplementary Information.

4.2 Orbital energy dependence of FON

This subsection examines the behaviors of orbital energy with respect to FON. Figures 5, 6, 7 and 8 demonstrate the orbital energies with respect to FON electron Δn. The results of the lightest and heaviest elements, Ne and Rn, are presented here. The slopes of the graphs of orbital energy with respect to Δn calculated using the least square method are tabulated in Table 1, 2, 3, and 4. The results of other elements, Ar, Kr, and Xe, are provided in Figs. S2–S7 and Tables S1–S5 in the Supplementary Information.

Fig. 5
figure 5

Orbital energy changes of HOMO, εHOMO [eV] with respect to FON, Δn in Ne atom. Relativistic treatments are a nonrelativistic, b 1eIOTC, c 1eIOTC with ρ+, d 1e2eIOTC, and e 1e2eIOTC with ρ+

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Fig. 6
figure 6

Orbital energy changes of 1s orbital, ε1s [eV] with respect to FON, Δn in Ne atom. Relativistic treatments are a nonrelativistic, b 1eIOTC, c 1eIOTC with ρ+, d1e2eIOTC, and e 1e2eIOTC with ρ+

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Fig. 7
figure 7

Orbital energy changes of HOMO, εHOMO [eV] with respect to FON, Δn in Rn atom. Relativistic treatments are a nonrelativistic, b 1eIOTC, c 1eIOTC with ρ+, d 1e2eIOTC, and e 1e2eIOTC with ρ+

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Fig. 8
figure 8

Orbital energy changes of 1s, ε1s [eV] with respect to FON, Δn in Rn atom. Relativistic treatments are a nonrelativistic, b 1eIOTC, c 1eIOTC with ρ+, d 1e2eIOTC, and e 1e2eIOTC with ρ+

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Table 1 Slopes of orbital energy change with respect to FON electron [eV] in HOMO of Ne atoms in Fig. 5
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Table 2 Slopes of orbital energy change with respect to FON electron [eV] in 1s orbital of Ne atoms in Fig. 6
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Table 3 Slopes of orbital energy change with respect to FON electron [eV] in HOMO of Rn atoms in Fig. 7
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Table 4 Slopes of orbital energy change with respect to FON electron [eV] in 1s orbital of Rn atoms in Fig. 8
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Figure 5 shows the results for the HOMO of Ne. The corresponding slopes are shown in Table 1. The orbital energies εHOMO of BLYP monotonically increase as Δn increases. Upon adding HFx, the slope of εHOMO versus Δn graph becomes smaller. In the results of HFLYP, εHOMO decreases as Δn increases. Moreover, 30%B88 + 70%HFx + LYP shows the smallest absolute value of the slope, indicating the smallest FON dependence of εHOMO. The slope of LC-BLYP is similar to that of BHHLYP. Nonrelativistic and relativistic treatments exhibit the above tendency and quantitatively similar orbital energies. One-electron relativistic effects and the PCEs of two-electron and density operators are small in the HOMO of the Ne atom.

Figure 6 shows the results for 1s orbital of Ne. The corresponding slopes are shown in Table 2. The qualitative tendency of the orbital energies ε1s with respect to Δn is similar to that of εHOMO: ε1s increases in BLYP and decreases in HFLYP as Δn increases and the combinations of B88 and HFx lie between BLYP and HFLYP. Moreover, 40%B88 + 60%HFx + LYP indicates the smallest absolute value of the slope, namely the smallest FON dependence of ε1s. The magnitude of the slope is larger than that of the HOMO, indicating that ΔE of 1s orbital in Fig. 2a are larger than that of HOMO in Fig. 1(a). The slope of LC-BLYP is larger than that of BHHLYP and closer to BLYP. As mentioned in previous studies [20,21,22], the short-range HFx is important in inner shell orbitals. The relativistic effects and PCEs are small, as in the results of HOMO.

Figure 7 shows the results for the HOMO of Rn. The corresponding slopes are shown in Table 3. The qualitative trends are similar to the results for HOMO of Ne, with the exception of the results of LC-BLYP. Among the combination of B88 and HFx, 10%B88 + 90%HFx + LYP indicates the smallest FON dependence of εHOMO. εHOMO obtained by LC-BLYP, whose graph shows smaller slope than that of 10%B88 + 90%HFx + LYP, is almost independent from Δn. As in the case of Ne, the relativistic effects are small.

Figure 8 shows the results for 1s of Rn. The corresponding slopes are shown in Table 3. The qualitative trends are similar to the results for 1s orbital of Ne. Unlike Figs. 5, 6 and 7, the differences caused by the nonrelativistic and relativistic treatments are pronounced in the 1s orbital of Rn. The orbital energy changes of relativistic treatments are steeper than that of nonrelativistic Hamiltonian: the absolute values of the slopes obtained by relativistic treatments are apparently larger than those by nonrelativistic Hamiltonian. The changes of ε1s (in eV) along the increase of FON, 0 < Δn < 1 obtained by BLYP are − 88,176.8 to − 87,267.6 in nonrelativistic Hamiltonian, − 99,454.4 to − 98,240.7 in 1eIOTC, − 99,397.0 to − 98,129.5 in 1eIOTC with ρ+, − 99,671.6 to − 98,601.7 in 1e2eIOTC, and − 99,613.6 to − 98,489.3 in 1e2eIOTC with ρ+. The range of ε1s in 1eIOTC is approximately 11,000 eV lower than that of nonrelativistic Hamiltonian. Comparing 1eIOTC with 1e2eIOTC, the range of the ε1s of 1e2eIOTC is more than 300 eV lower than that of 1eIOTC. Using ρ+ makes the range of ε1s higher. The PCEs of two-electron and density operators on the values of ε1s cannot be disregarded while the relativistic effects are dominated by the one-electron term. The linearity of orbital energy is also influenced by PCE. The functional that yields the smallest slope of ε1s versus Δn graph is 20%B88 + 80%HFx + LYP for nonrelativistic Hamiltonian and 1e2eIOTC and 10%B88 + 90%HFx + LYP for other relativistic treatments.

5 Conclusion

In this study, we examined the PCEs in the FON states of HOMO and 1s orbitals of noble gas atoms in the framework of the spin-free IOTC Hamiltonian. Calculations of the delocalization error revealed that the FON electron states were over stabilized by relativistic treatments. The PCEs of two-electron and density operators were remarkable in the core region of heavy elements: the former and the latter overestimated and underestimated the delocalization error, respectively. Corresponding to these results of total energies, the values of orbital energies and the slope of their changes to FON in core region of heavy elements were affected by PCEs. The PCEs of two-electron and density operators should be corrected when considering the linearity condition of total and orbital energies.