Introduction

The Potential Energy Surface (PES) of a molecular system is a core concept of atomic and molecular physics. A PES shows how potential energy varies with respect to the system’s geometric degrees of freedom. It governs every atomic nucleus’s dynamics, dictating how they vibrate, rotate and translate in space [1]. It can also describe reaction paths [2] and regions of possible stable and unstable molecular structures, as well as be used in the calculation of several spectroscopic [3] and thermodynamic [4] properties of a molecular system.

In the context of weakly bounded systems, such as those in the interstellar medium, properties derived from PES can be more easily compared to their corresponding experimental results [5]. Therefore, it is crucial to study this type of systems, especially when applied to astronomical sciences.

Given the great importance of a PES in studying a molecular system, an accurate description of this surface is key to deriving important properties that can be compared with experimental results [2,3,4]. For this matter, the description of PESs for small molecules can be done using post-Hartree-Fock methods such as n th-order Møller-Plesset Perturbation Theory (MPn) [6,7,8,9] or Coupled Cluster Theory (CCSD(T)) [9, 10], along with large basis sets such the Dunning basis [11]. The data acquired is then fitted to an analytical form in order to study molecular dynamical properties of the system [6].

Such methods require more time and computational resources to evaluate the energy corrections of post-HF, even when considering small molecules. For instance, when studying collision dynamics of molecules such as propylene oxide [9, 12], the concept of PES becomes less effective due to the increased dimensionality of the hypersurface. As such, a full description is compromised and, instead, the PES is restricted to only some degrees of freedom.

On the other hand, Kohn-Sham (KS) Density Functional Theory (DFT) adopts a different approach by considering the energies and operators as functionals of the electron density \(\rho (\vec {r})\) [13]. This approach reduces the dimensionality of the problem from 3N variables to just 3 (N being the number of electrons) and makes calculations faster [14]. Despite the fact that some exchange-correlation functionals describing optical and electronic properties of several molecular systems [15,16,17,18] successfully, there is an absence of appropriate functionals that can reproduce the theoretically accurate MPn results for PESs [10].

The exchange-correlation functional \(E_{xc}[\rho (\vec {r})]\), a key component of KS-DFT, is often approximated using a plethora of different methods [19]. As an example of the latter, a particular class known as long-range corrected (LRC) hybrid density functionals, introduces a range separation parameter ω that splits and modulates short- and long-range interactions as follows [20]:

$$ \frac{1}{r} = \underbrace{\frac{\text{erfc}(\omega r)}{r}}_{SR} + \underbrace{\frac{\text{erf}(\omega r)}{r}}_{LR}, $$
(1)

The long-range parameter ω modulates the distance where the range separation occurs. The lower the value of ω, the farther this separation occurs. Usually, LR interactions are modeled by HF exchange and correlation, while the hybrid, semi-local exchange-correlation energies model the SR interactions [21]. The substitution indicated in (1) is done directly in the KS hamiltonian.

This class of functionals initially gained notoriety for its implementation in correcting the HOMO-LUMO gap calculations in organic materials [18]. Their respective default ω values were determined by optimizing each LRC functional against a training set of different molecular systems in order to obtain more accurate thermochemical and kinetic properties in mind [20, 22].

Since then, ω optimization has been used for calculating optical properties [15,16,17] and binding energies [23] of molecular systems. However, in principle, the ω parameter could be tuned to fit an existing MPn reference PES via optimization algorithms.

The present work aims to analyze ω optimization as an alternative solution for the construction and description of PESs for weakly bound systems. We compare the results using both the default and optimized parameters with the reference calculations using MP4, with the goal of obtaining results similar to those of perturbative methods while requiring less computational time.

Methodology

System geometry

The system of interest for this work consists of a hydrogen peroxide molecule (H2O2) and a krypton atom (Kr), as shown in Fig. 1. Not only is H2O2 a simple molecule, but it is also the smallest molecule in nature to present chirality [8, 24]. Besides, its PESs are already well characterized and adjusted to an analytical form [6].

Fig. 1
figure 1

Studied system comprised of a Kr atom (above, in blue) and a H2O2 molecule (below), where the oxygen atoms are in red and the hydrogen atoms are in gray. The fixed parameters D, d, χ are shown, as well as the variables R and 𝜃

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We performed a rigid scan on the system using the Gaussian 16 software [25], where the fixed parameters were the O − O and H − O distances D = 1.450,Ä and d = 0.966,Ä, respectively, as well as the angle of incidence, set to 90o. This was done to reduce the system’s dimensionality and make comparisons with previous studies [6].

Only the dihedral angle 𝜃, defined as the angle between the two planes defined by the H − O − O bonds, and the distance R between the Kr atom and the O − O bond’s midpoint were the variables of interest of our PES scan. These variables’ values ranged between 0o and 360o with a step of 10o, and 3 Ä and 5 Ä with a step of 0.1 Ä, respectively.

The reference method used was the MP4 method, using the aug-cc-pVTZ basis set. As is typical with this basis set, we used the counterpoise correction to avoid the basis set superposition error. This combination of method and basis set has a good track record in producing reliable PESs [6,7,8].

The DFT calculations also used the same basis sets, and the ω B97XD [20], the LC-BLYP [26], and the LC-ω PBE [22] functionals were tentatively optimized. These functionals were chosen for their previous use in a related study [10].

ω optimization procedure

For the optimization procedure to be computationally viable, an important assumption was made: for each LRC-DFT functional, there is a unique optimal, geometry-independent long-range parameter ω such that the PES scanned using it has the closest possible qualitative and quantitative properties to the reference PES.

The upside of this approach is that, once the ω parameter is optimized, no further MP4 calculations are necessary. Therefore, one uses only DFT calculations with that optimized ω value for every new point outside the reference range.

We computed the PES with an LRC-DFT functional and a specific ω value. The same nuclear configurations were used for MP4 and DFT PESs. Once the scan is completed, the energy values of each and every nuclear configuration were compared with the corresponding reference values via a predetermined error metric.

This way, the idea of optimizing functionals is, therefore, translated into minimizing the discrepancy Δ between the reference and DFT PESs. In order to perform this minimization, we used the Nelder-Mead [27] algorithm for its simplicity and ease of implementation. In the present work, we used the Mean Square Error (MSE), which returns the root mean squared of the energy difference:

$$ {\Delta}_{MSE} = \sqrt{\frac{1}{MN}{\sum}_{i=1}^{N}{\sum}_{j=1}^{M} (E_{ij}^{MP4} - E_{ij}^{DFT})^{2}}, $$
(2)

where N = 21 is the number of radial points and M = 36 is the number of angular points.

For the ω optimization scheme described above, a Python program was developed using the numpy [28] and scipy [29] libraries. Additionally, we used the matplotlib [30] library for the plots, and the os library to integrate the Python program with the Linux operating system’s shell and the Gaussian 16 software [25]. All programs and scripts used here are available publically in an online repository.

Results and discussion

MP4 Theory

By the end of MP4 calculations, the PES in Fig. 2 was obtained. In order to facilitate comparisons, the remainder of the graphs shown in this article will consist of cross-sections of the PES taken along the R axis, with 𝜃 = 0o fixed, and the 𝜃 axis, with R = 3.5Ä fixed. We will compare the minima of both curves with their respective DFT counterparts, as these points are relevant in several applications of a PES [3, 6].

Fig. 2
figure 2

PES of the H2O2–Kr system with MP4/aug-cc-pVTZ level

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Typical behavior for interatomic potentials can be readily observed in the radial curve: the energy rapidly increases as the nuclei approach each other and, when they are far apart, the energy approaches a fixed energy value (which was taken to be 0 for 𝜃 = 0o, like all other potential surfaces raised in this work). Between these asymptotic regions, an equilibrium position at Re = 3.667Ä can be found, whose associated energy is De = 363.768 cm− 1. These values were estimated using an interpolated univariate spline algorithm used for constructing the PEC plots.

For the angular section, the results obtained also agree with similar results in the literature [6]. It is a symmetric, periodic curve with two prominent energy barriers at the cis (𝜃 = 0°) and trans (𝜃 = 180°) configurations of the hydrogen peroxide molecule. Also, two potential wells are present at 𝜃 = 100°and 𝜃 = 260°.

Optimized DFT

ω B97xd

After carrying out the electronic structure calculations using the ω B97xd functional with the default ω = 0.20 Bohr− 1 and optimized ω = 0.25 Bohr− 1 value for the same set of nuclear configurations as the reference PES, it is possible to compare both the optimized (continuous lines) and non-optimized (dashed lines) PECs present in Fig. 3, as well as the reference PECs (in black).

Fig. 3
figure 3

Radial (above) and angular (below) potential energy curves of the H2O2–Kr system with ω B97xd/aug-cc-pVTZ calculation level with and without ω optimization via MSE minimization compared with the MP4/aug-cc-pVTZ level PEC

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Overall, there is a very noticeable change in behavior on the optimized radial curve compared to the non-optimized one, especially in the short-range, with very minimal deviation past Re. In Table 1, the equilibrium positions Re and dissociation energies De of the radial PECs are compared. As expected, the optimized curve’s value of Re is closer to the reference value than the non-optimized one.

Table 1 Equilibrium positions Re and dissociation energies De from the radial PEC (with fixed 𝜃 = 0°), as well as energy barriers and wells for the angular PEC (with fixed R = 3.5Ä) and Mean Square Error (MSE) obtained using ω B97xd/aug-cc-pVTZ level calculation before and after optimization, along with their respective reference values
Full size table

In general, the non-optimized curve has a shallower well compared to the reference curve, which is expected for LRC functionals [10]. This underbinding of the system was corrected during ω optimization. Likewise, the angular curve was perfectly fit around the cis configuration. On the other hand, around the trans barrier, the absolute value of the energy difference was slightly more significant between the reference curve and the optimized curve than the non-optimized curve. These results are more quantitatively displayed in Table 1.

LC-BLYP

By repeating the same procedures of the previous functional, it is possible to compare both the optimized ω = 1.35 Bohr− 1 and non-optimized ω = 0.47 Bohr− 1 PECs for the LC-BLYP functional, both of which are present in Figure S1 as well as the reference PECs.

Once again, an improved description of the radial potential can be noted, especially around the minimum at R = 3.6Ä. For example, comparing the equilibrium positions Re and dissociation energies De of the three curves in Table S1, the discrepancy in both Re and De of the optimized curve were lowered significantly in comparison to the non-optimized one.

Also, with the values in Table S1, we noticed the behavior of the angular curve improved noticeably, adjusting itself perfectly around the cis configuration. However, for points around the trans configuration, the optimized curve approached the non-optimized curve’s behavior much more than the reference one.

LC-ω PBE

By repeating once more the same procedures of the previous functional, it is possible to compare both the optimized ω = 0.55 Bohr− 1 and non-optimized ω = 0.6 Bohr− 1 PECs for the LC-BLYP functional, both of which are present in Figure S2 together with the reference PECs.

This functional had much worse performance on the radial curve, staying almost the same as its non-optimized counterpart, as we compare the equilibrium positions and dissociation energies in Table S2. Similarly, comparing the results in Table S2, the angular curve had unsatisfactory results in the cis configuration. However, the trans configuration was well adjusted.

Out of the three functionals analyzed, this one developed poorly, and its use in optimized LRC-DFT is impractical. This low performance is due to the LC-ω PBE functional not fully contemplating the dispersion energy, which is heavily accounted for in this weakly bound system [8].

Conclusions

Expecting to find alternative methods for obtaining PESs, an MP4/aug-cc-pVTZ level PES was constructed for reference, and an optimization program for long-range parameter tuning of range-separated DFT functionals was developed in order to reproduce MP4 level results.

In general, our results show that ω optimization can improve DFT results, provided a single ω value can optimize an entire PES. The functionals had an acceptable overall performance: in particular, ω B97xd had the best fit to the reference PES after the optimization. The non-optimized ω B97xd underbound the system, and that underbinding was corrected during ω optimization.

LC-BLYP had a mildly successful optimization, coming in second place. Unlike ω B97xd and other LRC functionals [10], it overbound the system prior to the procedure, and afterwards, it was partially corrected. Unfortunately, LC-ω PBE showed poor results in comparison. This was due to the absence of dispersion corrections in this functional, which are heavily accounted for in weakly bound systems [8].

Qualitatively speaking, the computational time and effort of the ω optimization were sufficient to recommend its usage. Even though some applications require more precise and accurate results that justify the longer, more expensive calculations of post-HF methods, the optimized LRC-DFT can still be viable for on-the-fly calculations involving points near the minima, provided the optimization is done a priori.

This consideration, along with lower computational resource usage, promotes optimized LRC-DFT as a more viable alternative to regular DFT, expanding the range of possible methods for PES descriptions of weakly bound systems. On the other hand, this research paves the way for different functionals, minimization algorithms, and molecular systems to be studied in the future under the same goal of long-range optimization for PES descriptions.