Introducing a new correlation functional in density functional theory

Abstract

The correlation functional holds significance in density functional theory as it addresses electron–electron interactions beyond the mean-field approximation, enhancing the accuracy of total energy calculations, electronic excitations, and the prediction of materials properties. There are several expressions to describe this energy, and each of them has a unique set of errors in calculating particular properties of materials. This work offers a new correlation functional by employing the density's dependence on ionization energy. We theoretically derived this functional and combined it with the previously reported ionization energy dependent exchange functional to investigate its effect on the total energy, bond energy, dipole moment, and zero-point energy of 62 molecules. The comparison of this new functional in respect to existing widely used correlation models including QMC, PBE, B3LYP and Chachiyo models shows how well it works in producing accurate results with minimal mean absolute error.

Introduction

Density functional theory (DFT) is an effective tool in computational physics that allows for the prediction and analysis of numerous transport and thermal properties of solids^1,2. The precision and reliability of DFT computations are greatly influenced by the choice of exchange–correlation functional. Correlation energy is a fundamental concept in quantum mechanics that accounts for the interactions between electrons in a many-electron system. It represents the additional energy required to describe the behavior of electrons beyond what can be explained by the mean-field approximation, such as the Hartree–Fock method³. It basically represents the effects of electron–electron correlation, which arises due to their mutual electrostatic repulsion. This correlation leads to complex quantum effects that cannot be represented by simple mathematical models. Various functionals have been developed over the years, each with its own unique advantages and disadvantages^4,5. While some functionals may provide higher overall accuracy, others may be better suitable for specific kinds of systems or features. To accurately correct the positive value of Homo-energy and the orbital energy of anions, for instance, the tuned range-separated DFT approximation has been introduced by satisfying the DFT-Koopmans theory based on basis set alterations and quantum approximations⁶. The choice of functional depends on the particular problem being solved, the degree of required precision, and the computation time. Fitting the results of accurate Quantum Monte Carlo (QMC) computations of the uniform electron gas (UEG) has yielded multiple equations for the correlation energy in the local density approximation (LDA). These are quite complex expressions, usually denoted by their acronyms, like VWN, VWN5, and so forth. Nevertheless, when applied to molecular computation, they yield remarkably similar results. VWN functional is defined as follows⁷:

$$E_c^{\mathit{VWN}}=\int d^{3}r(A\left\{\begin{array}{c}ln\frac{x^2}{X(x)}+\frac{2b}{Q}{tan}^{-1}\frac{Q}{2x+b}-\frac{b{x}_0}{X(x_0)}\\ [ln\frac{{(x-x_0)}^2}{X(x)}+\frac{2(b+2x_0)}{Q}{tan}^{-1}\frac{Q}{2x+b}]\\ \end{array}\right\})$$ 1

where $x=r_s^{1/2},X(x)=x^2+bx+c,Q={(4c-b^2)}^{1/2}$ and the parameters $x_0$ , b, and c are constants that depend on the context and the specific version of the VWN functional being used⁸. The error may be obtained at a relatively high level for the LDA correlation functional. An improved LDA leads to LSDA0 that is constructed to satisfy specific requirements, but remarkably agrees with favorable results for a homogeneous two-electron density in a restricted curved three-dimensional area. The correlation part of LSDA0 is expressed as follows:

$$E_c^{LSDA0}=\int d^{3}rn\frac{-b_{1c}}{1+b_{2c}{r}_s^{\frac{1}{2}}+b_{3c}{r}_s}{g}_c(\xi )$$ 2

where, $z=(n_{\uparrow }-n_{\downarrow })/(n_{\uparrow }+n_{\downarrow })$ is the relative spin polarization defined with the spin densities $n_{\uparrow }$ and $n_{\downarrow }$, $r_s$ is the Seitz radius, b is constant and the function $g_c(z)$ is 0 for z =  ± 1 in order to satisfy the one electron self-correlation constraint and 1 for z = 0^9,10. The use of Generalized Gradient Approximation (GGA) can increase accuracy. Some of GGA correlations functional are LYP, PBE, and PW91. The PW91 correlation functional has achieved widespread acceptance because it covers short- and long-range electron correlation effects and accomplishes a comparatively low Mean Absolute Error (MAE) for the overall energy for a wide range of molecules^11–13. The well-known PBE functional correlation part is given by^10,14:

$$E_c^{\mathit{PBE}}=\int n(r){\epsilon }_c^{\mathit{PBE}}(n(r))dr$$ 3

where ${\epsilon }_c^{\mathit{PBE}}(n(r))$ , is the correlation energy density. The precise mathematical expression for ${\epsilon }_c^{\mathit{PBE}}(n(r))$ is quite complex and can be found in the original publication by Pedrew, Burke, and Ernzerhof¹⁴. This functional's error is better than LDA but still higher than other functionals. The accuracy of the LYP functional combined with Becke's three-parameter exchange functional, or B3LYP, is also better than that of more basic functionals like LDA, especially for molecule systems and transition metals^15,16.

Vydrov and Van Voorhis reported that the B97M-V functional was developed for main-group thermochemistry and non-covalent interactions. These families of functions were created by Truhlar and associates, and they exhibit improved accuracy for the total energy, particularly for non-covalent interactions^17–19. For both organic and inorganic molecules, it has demonstrated remarkable performance for the total energy with low MAE values^20–22. The mathematical formulation of the B97M-V functional is extremely complex and comprises several parameters that are determined through experimentation²².

A different correlation functional that was developed by Chachiyo et al.²³ yields impressively accurate findings when combined with their exchange functional^24–26:

$$E_c=\int n{\epsilon }_c{(1+t^2)}^{\frac{h}{{\epsilon }_c}}{d}^{3}r$$ 4

where t ($t={(\frac{\pi }{3})}^{1/6}\frac{1}{4}\frac{\left|\overrightarrow{\mathrm{\nabla }}n\right|}{n^{7/6}}$) is the gradient parameter, n is the electron density, ${\epsilon }_c$ is the correlation energy density and h is a constant with a value of 0.06672632 Hartree²³. Increasing t is intended to decrease the maximum rate of energy through the gradient suppressing factor ${(1+t^2)}^{\frac{h}{{\epsilon }_c}}$ .

In this paper, we report a new correlation functional that minimizes the mean absolute error (MAE), exceeding existing methods in terms of accuracy. The MAE is a measure of the average difference between computed and experimental values and is an essential measure of a functional's performance. Specifically, we perform a detailed analysis of over 64 different molecules and evaluate their key characteristics, including bond energies, total energy, dipole moments, and zero-point energy. We derive new correlation functional analytically; taking into account the functional's dependency on both the electron density and the ionization energy. By incorporating the ionization energy into the correlation functional, we improve the accuracy. This methodology is a complementary one to our earlier research, wherein we presented a novel exchange functional that is dependent on the ionization energy²⁷. As a result, the ionization energy is included as a significant parameter in both the correlation and exchange functional in our current study, enabling a more comprehensive description of electronic interactions²⁷.

Theoretical method

Equation (4) defines the generic form of the correlation functional²³. Based on the second-order Moller–Plesset perturbation (MP2) theory, a straightforward analytical ${\epsilon }_c$ is defined in the local-density approximation as follows²⁸:

$${\epsilon }_c=aln(1+\frac{b}{r_s}+\frac{b}{r_s^2})$$ 5

where a and b are constants and $r_s$ is the Wigner–Seitz density parameter.

In this work, we used the ionization dependent density as^29,30:

$$n(r_s)\to A{r}_s^{2\beta }{e}^{-2{(2I)}^{\frac{1}{2}}{r}_s}$$ 6

where I is the ionization energy and $\beta =\frac{1}{2}\sqrt{\frac{2}{I}}-1$ . We take A = 1 into consideration based on formula 29 in reference²⁹. Next, we determine $r_s$ to be:

$$\begin{array}{c}{r}_s=-\frac{\sqrt{\frac{2}{I}}-2}{2\sqrt{2I}}W(-\frac{2\sqrt{2I}}{\sqrt{\frac{2}{I}}-2}{n}^{\frac{1}{\sqrt{\frac{2}{I}}-2}})=-\frac{\sqrt{\frac{2}{I}}-2}{2\sqrt{2I}}(-\frac{2\sqrt{2I}}{\sqrt{\frac{2}{I}}-2})n^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n)\\ =n^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n)\\ \end{array}$$ 7

We obtain the following, using Eqs. (6) and (7):

$$\begin{array}{c}\overrightarrow{\mathrm{\nabla }}n=2A\beta {(n^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n))}^{2\beta -1}{e}^{-2{(2I)}^{\frac{1}{2}}{n}^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n)}\\ -2A{(2I)}^{\frac{1}{2}}{(n^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n))}^{2\beta }{e}^{-2{(2I)}^{\frac{1}{2}}{n}^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n)}\\ \end{array}$$ 8

So:

$$\begin{array}{c}t={(\frac{\pi }{3})}^{1/6}\frac{\sqrt{a_0}}{4}{n}^{\frac{2\beta -1}{\sqrt{\frac{2}{I}}-2}-\frac{7}{6}}(2A\beta {(W(n))}^{2\beta -1}{e}^{-2{(2I)}^{\frac{1}{2}}{n}^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n)}\\ -2A{(2I)}^{\frac{1}{2}}(n^{\frac{-1}{\sqrt{\frac{2}{I}}-2}}){(W(n))}^{2\beta }{e}^{-2{(2I)}^{\frac{1}{2}}{n}^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n)})\\ \end{array}$$ 9

Remarkably, ${\epsilon }_c$ can be interpreted as an interpolation between the paramagnetic and ferromagnetic instances for spin polarization $z=\frac{n_{\alpha }-n_{\beta }}{n}$³¹:

$${\epsilon }_c(n,z)={\epsilon }_c^0+({\epsilon }_c^1-{\epsilon }_c^0)f(z)$$ 10

where $g^3(z)=1-\frac{1}{2}f(z)andg(z)=\frac{{(1+z)}^{2/3}+{(1-z)}^{2/3}}{2}$.

Thus, we rewrite Eq. (5) as:

$${\epsilon }_c=aln(1+\frac{b}{W(n)}{n}^{\frac{-1}{\sqrt{\frac{2}{I}}-2}}+\frac{b}{{(W(n))}^2}{n}^{\frac{-2}{\sqrt{\frac{2}{I}}-2}})$$ 11

Which is clearly depends on ionization energy⁸. The Lambert function³² denoted as W, is related to density n. The correlation energy when the density tends to be uniform is represented by ${\epsilon }_c$ in Eq. (5) that is associated with density n²⁸. We obtain the following by substituting Eqs. (9) and (11) in Eq. (4):

$$\begin{array}{c}{E}_c=\int naln(1+\frac{b}{W(n)}{n}^{\frac{-1}{\sqrt{\frac{2}{I}}-2}}+\frac{b}{{(W(n))}^2}{n}^{\frac{-2}{\sqrt{\frac{2}{I}}-2}})(1+({(\frac{\pi }{3})}^{1/6}\frac{\sqrt{a_0}}{4}{n}^{\frac{2\beta -1}{\sqrt{\frac{2}{I}}-2}-\frac{7}{6}}(2A\beta {(W(n))}^{2\beta -1}{e}^{-2{(2I)}^{\frac{1}{2}}{n}^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n)}\\ -2A{(2I)}^{\frac{1}{2}}(n^{\frac{-1}{\sqrt{\frac{2}{I}}-2}}){(W(n))}^{2\beta }{e}^{-2{(2I)}^{\frac{1}{2}}{n}^{\frac{1}{\sqrt{\frac{2}{I}}-2}}W(n)}{))}^2{)}^{\frac{h}{aln(1+\frac{b}{W(n)}{n}^{\frac{-1}{\sqrt{\frac{2}{I}}-2}}+\frac{b}{{(W(n))}^2}{n}^{\frac{-2}{\sqrt{\frac{2}{I}}-2}})}}{d}^{3}r\\ \end{array}$$ 12

This is our new correlation energy in its final form. In order to get the exchange–correlation functional, as previously stated, this relationship was paired with the exchange functional that obtained in our prior study²⁷, namely with:

$$\Rightarrow E_x=\int \rho {\epsilon }_x\frac{\begin{array}{c}\left(\sqrt{9{\pi }^2{\left(2,-,\sqrt{\frac{2}{I}}\right)}^{-2}+6},-,3,\pi ,{\left(2,-,\sqrt{\frac{2}{I}}\right)}^{-1}\right)x^2+({\frac{3}{4\pi }\left(\sqrt{9{\pi }^2{\left(2,-,\sqrt{\frac{2}{I}}\right)}^{-2}+6},-,3,\pi ,{\left(2,-,\sqrt{\frac{2}{I}}\right)}^{-1}\right)}^2{x}^2\\ +\left(\sqrt{9{\pi }^2{\left(2,-,\sqrt{\frac{2}{I}}\right)}^{-2}+6},-,3,\pi ,{\left(2,-,\sqrt{\frac{2}{I}}\right)}^{-1}\right)x+\frac{4\pi }{3})\left(2,-,\sqrt{\frac{2}{I}}\right)\left(L,\left(-,\alpha ,{\left(\frac{3}{4\pi }\right)}^{\frac{-3}{\sqrt{\frac{2}{I}}-2}},x^{\frac{-3}{\sqrt{\frac{2}{I}}-2}}\right),+,1\right)\end{array}}{\left(\left(\sqrt{9{\pi }^2{\left(2,-,\sqrt{\frac{2}{I}}\right)}^{-2}+6},-,3,\pi ,{\left(2,-,\sqrt{\frac{2}{I}}\right)}^{-1}\right),x,+,\frac{4\pi }{3}\right)\left(2,-,\sqrt{\frac{2}{I}}\right)\left(L,\left(-,\alpha ,{\left(\frac{3}{4\pi }\right)}^{\frac{-3}{\sqrt{\frac{2}{I}}-2}},x^{\frac{-3}{\sqrt{\frac{2}{I}}-2}}\right),+,1\right)}{d}^{3}r$$ 13

where, ${\epsilon }_x=\frac{-3\sqrt{2I}}{4\pi s}$ , the parameter $x=\frac{4\pi }{3}s$ and $s=\frac{2\sqrt{2I}}{2{(3{\pi }^2)}^{\frac{1}{3}}}{A}^{\frac{-1}{3}}{e}^{\frac{2}{3}r\sqrt{2I}}{r}^{\frac{-1}{3}\sqrt{\frac{2}{I}}+\frac{2}{3}}$.

We utilized Siam quantum software to do numerical calculations and examine the exchange–correlation functional results^33,34. Reference³⁴ was utilized while determining the values of I for atoms and molecules.

Results and discussion

In the first step, we computed the total energy for the elements in the second and third rows of the periodic table using our new exchange–correlation functional. The errors of estimated total energies are compared with the results of the PBE, Chachiyo model, our previous functional²⁷, B3lyp, and QMC in Fig. 1. The exchange functionals in Fig. 1 of "our results in previous work" and the present research are the same, but their correlation functionals differ²⁷.

Figure 1.

The total energy mean absolute error (kcal/mol) of various DFT approaches compared to the experimental values for single atoms.

The MAE of new model is 2.5 kcal/mol, as shown in Fig. 1, which is less than the MAE of all QMC, PBE, B3LYP, and Chachiyo models. The MAE of the popular and relevant PBE model is 52.6 kcal/mol, greater than the MAE of all other functionals. The mean absolute error for molecules is plotted in Fig. 2.

Figure 2.

The total energy mean absolute error (kcal/mol) of various DFT approaches compared to the experimental values for molecules that contain (A) atoms from the first and second rows of the periodic table and (B) atoms from the third row of the periodic table.

Figure 2 makes it evident that the new functional is more accurate for molecules composed of the third rows of the periodic table. This suggests that the incorporation of ionization energy into density results in a more effective correlation functional for these compounds. This outcome is consistent with earlier research³⁵ which demonstrates that the correlation energy improves as the atomic number increases. This indicates that accurate and superior DFT findings are expected for the correlation functional for elements with large atomic numbers. For the molecules under investigation, our new exchange–correlation performs exceptionally well, yielding an absolute mean error of 2.5 kcal/mol—six times more precise than the currently available QMC results³⁶. Taking into account ionization energy can modify the correlation energy by changing the electronic structure of the system and the energies of the occupied and unoccupied orbitals. Therefore, accurate prediction of ionization energies in systems with strong electron correlation (such as transition metals, alloys, or molecules with open-shell configurations) requires taking into account correlation effects beyond basic one-electron models, and vice versa. In Tables 1 and 2, we compare new correlation functional and references^34,37 derived total energy values for single atoms and 62 molecules. It's important to note that MAE for the molecules containing the first and second rows is 3.1 kcal/mol whereas for the third row is just 1.8 kcal/mol.

Table 1.

The reference and new functional total energies of single atoms (versus Hartree).

Atoms	Reference energy37	New functional estimated energy
H	−0.5000	−0.5035
He	−2.9037	−2.9099
Li	−7.4781	−7.4795
Be	−14.6674	−14.6615
B	−24.6539	−24.6511
C	−37.8450	−37.8448
N	−54.5892	−54.5895
O	−75.0673	−75.0751
F	−99.7339	−99.7440
Ne	−128.9376	−128.9437
Na	−162.2546	−162.2582
Mg	−200.0530	−200.0579
Al	−242.3460	−242.3529
Si	−289.3590	−289.3538
P	−341.2590	−341.2539
S	−398.1100	−398.1090
Cl	−460.1480	−460.1490
Ar	−527.5400	−527.5385

Table 2.

The reference and new functional estimated total energies of molecules (versus Hartree).

Molecules	Reference energy36,37	New functional estimated energy	Molecules	Reference energy34,37	New functional estimated energy	Molecules	Reference energy36,37	New functional estimated energy
Li2	−14.9951	−14.9962	CH3Cl	−500.1240	−500.1299	SiH2(3B1)	−290.5690	−290.5678
CH	−38.4788	−38.4805	CLF	−559.9820	−559.9860	Cl2	−920.3900	−920.3909
CH2 (1A1)	−39.1346	−39.1377	CSH4	−438.7120	−438.7100	Si2H6	−582.5670	−582.5682
CH2 (3B1)	−39.1484	−39.1522	HOCl	−535.9800	−535.9769	NaCl	−622.5610	−622.5626
NH	−55.2227	−55.2246	SO2	−548.6590	−548.6574	Si2	−578.8390	−578.8335
CH3	−39.8355	−39.8409	Na2	−324.5360	−324.5397	P2	−682.7040	−682.6992
NH2	−55.8794	−55.8832	LiF	−107.4340	−107.4401	S2	−796.3840	−796.3832
OH	−75.7371	−75.7430	SiH2(1A1)	−290.6020	−290.6006	PH2	−342.5040	−342.4990
CH4	−40.5158	−40.5226	N2	−109.5420	−109.5427	SiH3	−291.2210	−291.2225
H2O	−76.4383	−76.4467	HCO	−113.8570	−113.8626	CN	−92.7250	−92.7245
HF	−100.4590	−100.4663	NO	−129.9050	−129.9067	C2H2	−77.3355	−77.3373
H2	−1.1745	−1.1781	C2H4	−78.5888	−78.5958	NH3	−56.5647	−56.5702
LiH	−8.0704	−8.0732	H2CO	−114.5090	−114.5164	CO	−113.3260	−113.3300
H2S	−399.4030	−399.4056	O2	−150.3270	−150.3348	HCN	−93.4311	−93.4330
HCl	−460.8190	−460.8220	C2H6	−79.8274	−79.8377	BeCl2	−938.9860	−934.9831
PH3	−343.1460	−343.1442	F2	−199.5300	−199.5403	BeO	−89.8465	−89.8474
SiH4	−291.8740	−291.8765	H2O2	−151.5640	−151.5753	Na2S	−722.6966	−722.6910
CS	−436.2290	−436.2292	H3COH	−115.7310	−115.7407	MgF2	−399.6619	−399.6688
SiO	−364.7340	−364.7325	N2H4	−111.8780	−111.8851	Thiadiazole	−584.9234	−584.9179
SO	−473.3780	−473.3730	CO2	−188.6010	−188.6090	Thiophene	−552.9535	−552.9510
ClO	−535.3200	−535.3246	BeH	−15.2468	−15.2456

We also compared the total energy errors with our earlier research²⁷, QMC, B3LYP, Chachiyo model, and PBE in Tables 3 and 4.

Table 3.

The comparison of the new functional with other models total energy errors for single atoms.

Total energy error	Our previous functional27	New functional	QMC	B3LYP	Chachiyo	PBE
MAE (for all single atoms)	3.7	2.5	13.3	16.7	2.7	52.6
MAE (for first and second row single atoms)	4.0	2.7	6.9	11.8	3.5	25.1

The unit of every energy is kcal/mol.

Table 4.

The comparison of new functional with other models total energy errors for examined 62 molecules.

Total energy error	Our previous functional27	New functional	QMC	B3LYP	Chachiyo	PBE
MAE (for all molecules)	3.9	2.5	23.7	28.9	3.5	79.4
MAE (for molecules made up from first and second row atoms)	3.4	3.1	13.4	24.1	3.1	41

The unit of every energy is kcal/mol.

Comprehending bond energies enables scientists to forecast and interpret a range of chemical phenomena, including molecular interactions, reaction mechanism, and the stability of materials. Additionally, it can be ascertained through DFT computations, so we computed the MAE of bond energy generated by new functional and compared it with the results of QMC and our earlier research for 62 molecules in Fig. 3. Molecule bond energy is computed by us using²⁴:

$$E_b={\sum }_{A\in atoms}{E}_{\mathit{total}}^{(A)}-E_{\mathit{total}}^M$$ 14

where the total energy of atoms and molecules are described by E^(A) and E^(M). Figure 3 illustrates that, for third row molecules, our new functional performs better than QMC's results³⁶ , while QMC's results for first and second row molecules are marginally better than our functional for bond energy. Overall, for all 62 molecules, as Table 5 demonstrates, our functional performs better than the QMC and Chachiyo models.

Figure 3.

The bond energies mean absolute error (kcal/mol) obtained by various models compared to the experimental values for (A) molecules containing atoms of the first and second rows of the periodic table and (B) molecules containing atoms of the third row of the periodic table.

Table 5.

The comparison of the bond energy errors for new functional and other models.

Bond energy error	Our previous functional	New functional	QMC	Chachiyo
MAE (for all molecules)	4.4	2.5	3.1	3.5
MAE (for molecules made up from first and second row atoms)	3.3	3.1	2.9	3.1

The unit of every energy is kcal/mol.

The dipole moment is another quantity that can be evaluated using DFT and offers important details regarding the distribution of charge inside a system. The dipole moment is theoretically equivalent to $-e\int rn(r)d^{3}r$, which is directly related to density. Predicting and interpreting a wide range of chemical and physical properties, including molecule polarity, solvation effects, molecular interactions and reactivity, and the electric field response of materials, requires an understanding of dipole moments. Figure 4 shows the precision of our functional for dipole moment calculation in comparison to the B3LYP and Chachiyo models. Table 6 summarizes the dipole moment MAE.

Figure 4.

The dipole moment mean absolute error (Debye) of various DFT approaches compared to the experimental values for molecules.

Table 6.

The comparison of the dipole moment errors for new functional and other models.

Dipole moments	Our previous functional27	new functional	B3LYP	Chachiyo
ME	−0.02	−0.001	−0.02	−0.03
MAE	0.09	0.06	0.05	0.11

The unit of every dipole moments is Debye.

As demonstrated in Fig. 4 and Table 6, our new functional has a lower MAE than our earlier work, the Chachiyo model, and B3LYP.

Lastly, we have determined the zero-point energy of molecules, which takes into consideration the quantum mechanical vibrations of atoms and molecules and accounts for the lowest possible energy state of a system. To ensure accuracy in DFT calculations, zero-point energy is crucial for molecular vibrations, thermal corrections, stability, and chemical reactivity. Equation (15) can be used to evaluate it as follows³⁸:

$$E_{\mathit{zp}}=\frac{1}{2}\sum h{\upsilon }_i$$ 15

where h is the Planck constant and i represents the frequency of a particular molecule. We plotted the zero-point energy error for each of the four models indicated in Fig. 5, demonstrating that our new functional is more accurate than the others. The MAE of zero-point energy for the models mentioned is summarized in Table 7.

Figure 5.

The zero-point energy mean absolute error (kcal/mol) obtained by various DFT approaches compared to the experimental values.

Table 7.

The comparison of the zero-point energy errors for new functional and other models.

Zero point energy	Our previous functional27	New functional	B3LYP	Chachiyo
ME	0.11	0.02	−0.07	−0.003
MAE	0.11	0.11	0.16	0.12

The unit of every energy is kcal/mol.

Table 8 provides a comparison between our new functional results and the reference values of zero-point energies and dipole moments of 62 molecules^34,39–41. This article computes total energy, bond energy, and dipole moments using the QZP-gh basis set. We have also computed zero point energy using the 631-g* basis set. It is important to note that the QZP-gh basis set was selected since its error was less than that of other base sets (TZP, pcseg).

Table 8.

The references and new functional estimated dipole moments and zero-point energies of molecules^34,39–41.

Molecules	Ezp (kcal/mol)			Dipole moment (Debye)
	Reference Ezp	New functional estimated Ezp	Errors	Experimental	Our calc (Debye)	Errors
H2	6.2	6.4	0.2	0.0	0.0	0.0
LiH	2.0	2.1	0.1	5.87	5.67	−0.2
BeH	2.9	2.9	0.0	0.25	0.28	0.03
Li2	0.5	0.5	0.0	0.0	0.0	0.0
CH	4.0	4.1	0.1	1.45	1.39	−0.06
NH	4.6	4.7	0.1	1.39	1.46	0.07
NH2	11.5	12.0	0.5	1.76	1.72	−0.04
OH	5.3	5.3	0.0	1.65	1.60	−0.05
CH4	27.1	27.4	0.3	0.0	0.0	0.0
H2O	12.9	13.1	0.2	1.85	1.81	−0.04
HF	5.9	5.9	0.0	1.82	1.75	−0.07
NH3	20.6	21.2	0.6	1.47	1.53	0.06
LiF	1.3	1.4	0.1	6.28	6.15	−0.13
CN	2.9	3.0	0.1	1.14	1.23	0.09
C2H2	15.3	15.2	−0.1	0.0	0.0	0.0
CO	3.1	3.0	−0.1	0.11	0.15	0.04
HCN	8.7	9.0	0.3	2.98	2.90	−0.08
N2	3.4	3.3	−0.1	0.0	0.0	0.0
HCO	7.8	7.7	−0.1	1.39	1.39	0.0
NO	2.7	2.8	0.1	0.16	0.17	0.01
H2CO	16.1	16.4	0.3	2.32	2.22	−0.1
O2	2.3	2.2	−0.1	0.0	0.0	0.0
F2	1.3	1.4	0.1	0.0	0.0	0.0
H2O2	–	16.0	–	1.57	1.66	0.09
CH4O	–	31.5	–	1.69	1.61	-0.08
N2H4	–	32.1	–	1.75	1.90	0.15
CO2	6.2	6.1	-0.1	0.0	0.0	0.0
SiH2	7.2	7.3	0.1	0.26	0.27	0.01
H2S	9.2	9.4	0.2	0.98	1.01	0.03
HCl	4.2	4.3	0.1	1.11	1.15	0.04
PH3	14.6	14.8	0.2	0.57	0.65	0.08
SiH4	19.2	19.0	−0.2	0.0	0.0	0.0
CS	1.8	1.8	0.0	1.95	1.97	0.02
SiO	3.5	3.6	0.1	3.09	2.94	−0.15
SO	1.6	1.7	0.1	1.54	1.48	−0.06
ClO	1.2	1.2	0.0	1.3	1.38	0.08
CH3Cl	–	23.4	–	1.88	1.95	0.07
ClF	1.1	1.1	0.0	0.88	0.94	0.06
CSH4	-	28.3	–	1.66	1.7	0.04
HOCl	8.0	8.2	0.2	1.40	1.49	0.09
SO2	4.3	4.5	0.2	1.62	1.55	−0.07
Na2	0.2	0.2	0.0	0.0	0.0	0.0
NaCl	0.5	0.5	0.0	8.99	8.74	−0.25
Si2	0.7	0.7	0.0	0.0	0.0	0.0
P2	1.1	1.2	−0.1	0.0	0.0	0.0
S2	1.0	1.1	0.1	0.0	0.0	0.0
Cl2	0.8	0.7	−0.1	0.8	0.0	0.0
BeCl2	–	3.2	–	0.0	0.0	0.0
BeO	2.0	1.9	−0.1	–	6.04	–
Na2S	–	0.5	–	0.0	0.0	0.0
MgF2	2.4	2.2	−0.2	0.0	0.0	0.0
Thiadiazole	27.4	27.1	−0.3	1.57	1.87	0.3
Thiophene	43.6	43.0	−0.6	0.55	0.51	−0.04
MAE			0.11	MAE		0.06
ME			0.02	ME		−0.001

The initial circumstances, crystal structure, and atom count of the system all affect how much time elapses for each run. We compared the mean computation times for the old and new functions in the accompanying table. As the Table 9 illustrates, the average execution time for the new correlation was 54% longer than the old functional, despite Siam-Quantum software's extremely quick calculations. This demonstrates that while the total energy is predicted more precisely by the improved correlation functional, the computation time for it also increases.

Table 9.

The percentage of modifications between the old and new functional calculations.

The quantity of atoms that each molecule is composed of	The percentage of modifications
1	33
2	70
3	33
4	50
5	45
6	35
7	64
8	78
9	66
14	71

Conclusion

We proposed an accurate and straightforward correlation functional that could be utilized in DFT. After conducting detailed computations using 62 molecules and single atoms from the first, second, and third rows of the periodic table, new correlation functional demonstrates remarkable precision in estimating bond energies, total energy, dipole moments, and zero-point energies. The examined molecules' mean absolute error in total energy calculations was 2.5 kcal/mol, which outperformed the results of the Quantum Monte Carlo (QMC) and B3LYP methods. Interestingly, new functional for third-row atoms is more accurate than for first- and second-row atoms, indicating the important role of correlation energy in heavier elements. Moreover, new functional predicts single atom characteristics better than QMC and B3LYP.

Author contributions

All authors participated on the introducing new correlation functional. All authors wrote and reviewed the manuscript. Asghar Esmaeili supervised the work. Esmaeil Rahmatpour prepared Figures and Tables.

Data availability

All data generated or analyzed during this study are included in this article. If required, any data are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.