Accessible approaches for vibrational zero point energy calculation of organoboron compounds

Highlights

• Vibrational zero-point energies of organoboron compounds are estimated by two empirical approaches.

• The estimated ZPEs are compared with those obtained by HF/6-31G* method.

• The empirical approaches employed constitute reliable methods to estimate ZPEs.

Abstract

In this paper, two empirical approaches were used to calculate vibrational zero-point energies (ZPEs) of organoboron compounds. Firstly, the contribution of boron atom has been determined and incorporated in the Schulman and Disch empirical formula. For the second approach, the bond contributions of BH, BC, BO, BCl, BN, BF and BB were determined and introduced to our empirical formula already established. A comparison of results obtained for 120 organoboron compounds with the reported available values and those obtained using HF/6-31G* method, indicates the reliability of the empirical employed approaches to reproduce vibrational zero-point energies of organoboron compounds.

Introduction

Organoboron compounds have been widely used in the fields of materials chemistry, pharmaceutical sciences, and the synthetic organic chemistry as important reagents for a range of transition-metal catalyzed processes [[1], [2], [3], [4]]. They have found many applications in recent years as a catalyst for industrially important reactions such as amidation, and as radical precursors [5]. However, thermochemistry data are either unavailable or available but for simpler organoboron compounds [6,7]. To solve this problem, recourse to the methods of quantum chemistry or to empirical rules appears necessary. Currently, for the accurate determination of thermochemical data for small and medium molecules more sophisticated methods can be used, such as CBS-QB3 [8,9], Gn [10] and the ccCA approach [11,12]. The limiting factor for the thermochemical accuracy could be the nuclear movement rather than the electronic structure when these approaches are extended to large systems [13,14]. In the formulas of statistical mechanics, the vibrational zero-point energy (ZPE or ZPVE) constitutes the most significant term for the correction of the total energy of the molecules. ZPE can be approximated by the harmonic formula:

$$ZPE=\frac{R}{2}\sum _{i-l}^{3N-6}\left(\frac{hc}{k}\right)v_i$$

Where the sum runs for the 3N-6 normal frequencies (ν_i) in the case of a nonlinear molecule of N atoms, R is gas constant, k is Boltzmann constant, c is speed of light and h is Planck constant.

Thus, for a more precise calculation of the enthalpies of formation, a reliable determination of vibrational zero point energy appears necessary. For small molecules [15,16], the knowledge of all vibration normal modes by Infrared and Raman spectroscopic methods allows the experimental determination of this quantity. However, for medium and large systems this determination presents experimental difficulties due to the existence of harmonics and combination frequencies in molecular spectra. Also, this experimental determination becomes even more difficult for the hazardous compounds which can be difficult to handle in air [17,18]. Otherwise, vibrational ZPE can be obtained by computing molecular vibrational frequencies using quantum chemistry methods [[19], [20], [21], [22], [23]]. However, such calculations are not usable by the chemist and technologist in the daily practice on the one hand and can be very demanding regarding computer time and disk space on the other. Moreover, the theoretical vibrational frequencies are generally overestimated. A major source of this disagreement is the neglect of anharmonicity effects in the theoretical treatment [24]. To solve this problem, we use an empirical correction factor which depends on the method, the basis set, the property to be estimated (ZPVE, H_vib(T), S_vib (T), …) and sometimes on the nature of the vibrations (low-frequency vibrations, high-frequency vibrations) [25]. For this reason, several researchers have sought to develop empirical rules making it is possible to determine vibrational ZPE. Fortunately, vibrational zero point energy can be expressible with a good approximation by the rules of additivity. Theoretical foundation for this additivity, which has a long history, has been established [[26], [27], [28]]. In principle, the property of a molecule is linked to structurally dependent parameters whose determination is a function of the frequency groups representing the molecule and their contributions.

Thus, in recent decades, several empirical rules [[29], [30], [31], [32], [33]] have been developed to estimate ZPEs of molecules without using quantum chemistry methods. These additivity rules are classified into two families: methods based on the atomic contributions and methods based on the contributions of bonds or groupings. The former take into account each of the atoms present in a given molecule. Thus, the studied property is calculated as a sum of atomic contributions. For the second strategy, the molecule is divided into different fragments (bonds or groups). When a fragment is present in one molecule or another, the value of its contribution is conserved.

Using the first type of approach based on the additivity of atomic contributions, Flanigan et al. [29] have used the simple empirical relationship for C_nH_m hydrocarbons:

ZPE = 2n +7 m (kcal/mol)

Using the least squares method, Schulman and Disch changed this relation [34] in 1985. They determined the contributions of the hydrogen and carbon atoms for hydrocarbons, and deduced the following relationship:

ZPE = 3.88n +7.12 m – 6.19 (kcal/mol)

This last formula has been extended to polyatomic molecular systems containing nitrogen, oxygen, chlorine, fluorine [35], bromine, sulfur [36] and silicon [37]. Recently [38], we have also determined the atom contribution of the phosphorus atom to obtain the ZPEs of organophophorus compounds.

The equation previously quoted takes now the form:

$$\begin{array}{cc}\hfill \text{ZPE}=\sum _i^N{N}_i{X}_i-6.19\hfill & \hfill \left(\text{kcal/mol}\right)\hfill \end{array}$$

with N the number of kinds of atom in the molecule, N_i the number of atoms of type i and X_i the increment of the atom i.

In this background, a similar linear relationship relating the molecular stoichiometry and the ZPVE for molecules containing H, C, O, N, F and Cl was presented by Grice and Politzer [15].

In 2003, Ruzsinszky et al. [39] investigated how the partial charges computed within density functional theory (DFT) would correlate with ZPVE values. Their results show that atomic partial charges can be employed for a rapid estimation of ZPVEs. However, this method still requires to quantum chemical calculations to estimate the ZPEs of the molecules.

In the case of hydrocarbons, Fliszar et al. [22] found that the sum ZPE + H (T) -H (0) obeys certain simple additivity rules. They proved that the sum correlates with the structural characteristics of the molecule, such as the number of atoms and the degree of branching.

For the second type of approach, Pitzer [40] determined empirically thermodynamic functions for gaseous hydrocarbons, and he assigned an empirical value for each mode of vibration. In a later study on n-paraffins, Cottrell [41] saw that ZPE increases gradually by successive additions of the methylene (CH₂) groups. A decade later, Pitzer and Catalano [42] used 17.7 kcal/mol per CH₂ unit to estimate vibrational zero-point energy of these compounds.

Bernstein published few papers [[43], [44], [45]] about empirical calculations of vibrational zero-point energy of ethylene, halomethanes, haloethylenes, isotopes of methane, and benzene, taking into consideration the contributions of the internal coordinates, and the interactions between them. A few years later, Fujimoto and Shingu [46] identified three empirical parameters to calculate ZPEs of hydrocarbons with a similar accuracy to that of Bernstein, which are the contributions of carbon chain, CH and CC bonds.

In 1980, relying on a system of harmonic oscillators Oi and co-workers [[26], [27], [28]] have described the theoretical foundation showing the additivity of vibrational zero point energy.

In 2001, the by relating vibrational zero point energy to nature and type of bonds forming a molecule we contributed to this bibliography through an original empirical relationship [47] that allows the calculation of ZPEs of organic compounds. The empirical rule found is written as follows:

$$\begin{array}{cc}\hfill \text{ZPE}\left(\text{emp}\text{.}\right)=\sum _{\text{i}}^{\text{P}}{\text{N}}_{\text{i}}\times B{\text{C}}_{\text{i}}\text{- 2}\text{.09}\hfill & \hfill \left(\text{kcal/mol}\right)\hfill \end{array}$$

with P the number of bonds in the molecule, N_i the number of bonds of type i and BC_i the contribution of the bond i to the ZPE.

This equation has been used to calculate the ZPEs of several organic derivatives belonging to different categories of compounds and containing various bonds such as CH, NH, OH, SH, CO, CC, CN, CS, NN, CF, CCl, CC, CN, CO, CS, CC and CN. A few years later, we determined the contribution of the CBr bond [48] and incorporated it into our empirical formula to calculate vibrational ZPEs of bromo compounds. For various molecules containing this bond, the ZPEs calculated correlate well with experimental values. In addition, we were able to expand the application of this empirical formula to organosilicon compounds. The bond contributions of SiC, SiH, SiO, SiCl and SiSi were determined [49]. For more than 90 organosilicon compounds containing these bonds, the results obtained are in good agreement with experimental available values. The computed ZPEs were compared with the results obtained by Schulman and Disch's extended empirical formula and with the scaled values deducted of semi-empirical AM1 and DFT (B3LYP/6-31G*) methods, in all cases with satisfactory results.

Moreover, we have extended the field of application of this empirical model to organophosphorus compounds (III) [38], the bond contributions of PF, PC, PH, PCl, PS, PN and PO were determined and incorporated them into our empirical formula. Comparison of the ZPEs estimated for 101 organophosphorus compounds (III) with the reported values and with those calculated by ab initio (HF/6-31G*) and DFT (B3LYP/6-31G*) methods, shows the reliability of the used empirical formula.

More lately, the calculation of vibrational zero-point energies (ZPEs) of organophosphorus (V) compounds was reported [50]. The bond contributions of PO and PS have been determined and integrated in our empirical rule to evaluate vibrational zero point energies of the investigated compounds. ZPEs for more than 80 organophosphorus compounds (V) containing these bonds coincide with the reported available values. Furthermore, the comparison of these results with those estimated by a similar empirical formula and by quantum chemistry methods (HF/6-31G*, B3LYP/6-31G*), indicates the reliability of our empirical model.

In the present paper and following our previous work concerning the estimation of vibrational zero-point energy, we have used two empirical approaches to calculate the vibrational zero point energies (ZPE) of organoboron compounds. In the first, the contribution of boron atom was determined and incorporated into the empirical formula of Schulman and Disch. For the second approach, the bond contribution of BH, BC, BO, BCl, BN, BF and BB were determined and introduced into our empirical formula already established. The results obtained by the two approaches are compared with the reported available values and with those obtained using the Hartree-Fock method (HF / 6-31G *).