Analysis of Density Functional Tight Binding (DFTB) with Natural Bonding Orbitals (NBOs)

Abstract

The description of chemical bonding by the Density Functional Tight Binding (DFTB) model is analyzed using Natural Bonding Orbitals (NBOs) and compared to results from Density Functional Theory (B3LYP/aug-cc-pVTZ) calculations. Several molecular systems have been chosen to represent fairly diverse bonding scenarios that include standard covalent bonds, hypervalent interactions, multi-center bonds, metal-ligand interactions (with and without pseudo Jahn-Teller effect) and through-space donor-acceptor interactions. Overall, the results suggest that DFTB3/3OB provides physically sound descriptions for the different bonding scenarios analyzed here, as reflected by the general agreement between DFTB3 and B3LYP NBO properties, such as the nature of NBOs, the magnitude of natural charges and bond orders as well as dominant donor-acceptor interactions. The degree of ligand to metal charge transfer and the ionic nature of the pentavalent phosphate are overestimated, likely reflecting the minimal basis nature of DFTB3/3OB. Moreover, certain orbital interactions, such as geminal interactions, are observed to be grossly overestimated by DFTB3 for the hypervalent phosphate and several transition metal compounds that involve copper and nickel. The study indicates that results from NBO analysis can be instructive for identifying electronic structure descriptions at the approximate quantum mechanical level that require improvement and thus for guiding the systematic improvement of these methods.

Graphical Abstract

1. Introduction

For the study of condensed phase problems, approximate quantum mechanical (QM) methods are highly valuable since they are computationally efficient and therefore can be used in simulations that involve extensive sampling of the conformational space.^1–5 In recent years, several approximate QM methods have become popular and used in various chemical, biological and materials applications; these include most notably the Density Functional Tight Binding (DFTB) models^6–8 and recent extension of the Neglect of Diatomic Differential Overlap (NDDO) methods:⁹ PM6¹⁰/PM7,¹¹ PDDG¹² and OM2/OM3.¹³ Many of these methods have been parameterized and tested with both bonded^14,15 and non-bonded interactions,^16–19 and various corrections have been developed to reduce the errors for specific type of interactions; examples include several empirical corrections to hydrogen-bonding^20,21 and dispersion interactions,¹⁶ as well as polarization corrections.^22–24 Semi-empirical methods may also be parametrized for specific types of reactions,^25,26 such as AM1/d-PhOT²⁷ and SCC-DFTBPR²⁸ for phosphoryl transfer and phosphate hydrolysis, respectively.

To further improve the accuracy and transferability of these approximate QM methods, in addition to the popular data driven approach, it is valuable to analyze these methods from an electronic structure point of view.^29–31 Although there is no unique way to decompose the electronic energy of a molecular system, either for covalent or non-covalent interactions, it is meaningful to compare the components at two different levels of QM methods within the same framework; whether the results of the comparison leads to an apparent avenue to improve the more approximate QM method depends on the analysis framework of choice.

In this work, we analyze the DFTB approach^6,32 within the Natural Bonding Orbital (NBO) framework.^33–35 This is motivated by the following considerations. DFTB is an approximate QM method derived by expanding the total electronic energy in the Density Functional Theory framework around a reference density, which is taken to be the sum of atomic densities (in the presence of a restraining potential); depending on the order of truncation, this leads to the DFTB2³² and DFTB3³⁶ methods, respectively. The method is made computationally efficient by the use of a minimal basis set, two-center approximation to the matrix elements and simplified treatment of the second- and third-order contributions. Despite these approximations, the DFTB2 and DFTB3 methods were found to give generally reliable structures and conformational energies;⁷ for reaction energies and barriers,¹⁵ the accuracy for calibrated systems can be comparable to DFT with double-zeta plus polarization quality of basis set, making the DFTB approach attractive for studies where conformational sampling is essential.⁸ We note that Grimme and co-workers also developed tight-binding models in a similar framework but with different parameterization strategies that target mainly structures and vibrational frequencies.^37,38

Nevertheless, the DFTB approach can have significant errors for the energetics for certain covalent and non-covalent interactions. For non-covalent interactions,¹⁷ the errors in DFTB likely originate primarily from the minimal basis nature of the method and the monopole approximation, which limits its accuracy in polarization and therefore interactions such as hydrogen-bonding interactions; the use of standard GGA functional (PBE) also limits its description of dispersion interactions.¹⁶ These limitations can be alleviated by including additional corrections such as empirical dispersion¹⁶ and a polarization response contribution;^23,24 alternative approaches have also been used to reduce errors in hydrogen-bonding interactions.^20,21 Although additional work is clearly warranted to make these improvements more transferable in different environments and generally applicable to molecular simulations, it appears that non-covalent interactions can be systematically improved for DFTB to a level adequate for many applications.

For covalent interactions, however, the situation is less clear. For example, for our parameterization of DFTB3 for P,³⁹ we noted that it was di cult to obtain parameters that work simultaneously for phosphate hydrolysis reactions that involve metaphosphate and pentavalent phosphate species, which are featured in the so-called dissociative and associative pathways, respectively.⁴⁰ Since most parameters in DFTB(3) are determined by computations of atoms and small molecules (mostly diatomics for pair-wise repulsive potentials and Hamiltonian matrix elements), it was di cult to reveal the origin of the persistent errors observed for the phosphate reactions. This motivated the current work, in which we compare the electronic structures of various phosphate species at the DFTB3 level to those at the DFT (mainly B3LYP) level; we hope such a comparison will lead to more fundamental metrics that guide the improvement of DFTB3.

Another important area involves transition metal ions, which play important roles in (bio)chemistry⁴¹ and currently there is no satisfactory approximate QM methods available for these important elements.⁴² We recently started to develop DFTB3 methods for several transition metal ions, such as Zn,⁴³ Cu⁴⁴ and Ni.⁴⁵ Through careful parameterization (e.g., explicitly considering the orbital angular momentum dependence of the Hubbard parameters), we were able to capture the structural properties of many Zn/Cu/Ni complexes of biological relevance, including, for example, pseudo Jahn-Teller distortion⁴⁶ of Cu(II) and Ni(II) compounds.^45,47 Energetic properties, such as metal-ligand binding energies, however, have been more di cult to capture adequately. Although we reasoned that further improvements can be made by improving the description of ligand polarization and explicit interactions among the metal d electrons, it is valuable to better understand to what degree the current DFTB3 model describes the electronic structure of metal compounds. This constitutes the other important motivation of this study.

Regarding the approach for characterizing the electronic structure of covalent bonding, several methods are available in the literature.^48–54 We chose to employ the NBO framework^33–35 because it has been successfully employed to study bonding in many organic and metal compounds and it is a self-consistent method that does not rely on the choice of QM method. Importantly, the NBO suite of algorithms^55,56 exists as a standalone program that was developed to work with a multitude of electronic structure packages, thus requiring a minimal amount of information from a QM calculation as input; this makes the analysis of DFTB3 results in the NBO framework particularly straightforward.

In the following, we first briefly review the key steps in a NBO analysis, to make it clear what information is needed from a DFTB(3) calculation; we summarize explicitly the modifications we introduced to CHARMM⁵⁷ and DFTB+⁵⁸ packages that enable NBO analysis of DFTB calculations using these programs. Next, we present NBO analysis for several representative phosphate species and discuss the similarities and differences between DFTB3 and DFT results; whenever useful, we also include results from PM6,¹⁰ a widely used NDDO approach. To supplement these bonding analyses, we next turn to results for donor-acceptor interactions in a series of phosphates and for the n → π* interactions in peptides; these topics represent biochemical examples for which NBO analyses were proven insightful. As a last set of examples, we analyze the electronic structure of several Cu and Ni compounds, include several that exhibit pseudo Jahn-Teller distortions, focusing again on the similarities and differences between DFTB3 and DFT descriptions. Finally, we end with concluding remarks.

2. Computational Methods

2.1. A brief review of NBO analysis

NBO analysis is a self-consistent theoretical framework for analyzing the bonding nature of molecules^{34,35,59–61} and materials.⁶² It develops a set of mathematical algorithms to analyze the electronic wavefunction (or the one-particle density matrix) of a molecule in terms of localized Lewis-like chemical bonds. The key starting point of NBO analysis is the determination of natural atomic orbitals (NAOs),⁶³ which are linear combinations of atomic orbitals that feature the maximal occupations on each atom in the molecular environment. The NAOs are distinguished by their intra- and inter-atomic orthonormality, and are obtained by a series of orbital transformations that rely on the “occupancy-weighted symmetric orthogonalization” procedure;⁶³ the multi-step transformation leads to a transformation matrix, T_NAO, that defines the NAOs in terms of the original AO basis. The one-particle density matrix in the original AO basis can then be transformed into the NAO basis,

$$P^{NAO}=T_{NAO}{P}^{AO}{T}_{NAO}^{†},$$ (1)

which was used to define the natural population analysis,⁶³ whose results have been shown to be insensitive to the size of the basis set, in contrast to, for example, the Mulliken population analysis.⁶⁴

By diagonalizing the atomic blocks of P^NAO, one first identifies one-center orbitals with nearly full (n_i ≈ 2) occupancy; these are core or lone pair orbitals. The density of these one-center orbitals are then removed from the corresponding atomic blocks of P^NAO, leading to the “depleted” density matrix (blocks),³³ ${\tilde{P}}^{NAO}$; alternatively, this can be done by strict orthogonal projection.⁵⁶ Two-center bond searches are then performed for the atom-pair (AB) blocks of the depleted/projected density matrix,

$${\tilde{P}}^{(AB)}=\left[\begin{array}{cc}{\tilde{P}}_{AA}^{NAO}& {\tilde{P}}_{AB}^{NAO}\\ {\tilde{P}}_{BA}^{NAO}& {\tilde{P}}_{BB}^{NAO}.\end{array}\right]$$ (2)

The eigenvectors with high occupancy correspond to the A-B bonding NBOs,⁶⁵ which also help define the natural hybrid orbitals (NHOs)³³ that are localized on each atom; in practice, an additional step is taken to ensure orthogonality of the NHOs to the one-center NBOs. If an insufficient number of two-center bonding orbitals are observed, it is necessary to examine three-atom blocks of the (depleted) density matrix to identify possible three-center bonds; for cases of multi-center bonding one may request analysis of n-atom blocks with the MCB keyword, or one could additionally request resonance NBOs introduced in NBO7.⁵⁶ The NBO search identifies the set of Lewis type (“occupied”,“lone pair”) NBOs $\left({\Omega }_i^{(L)}\right)$ and computes the total density of these orbitals, ${\rho }^{(L)}={\sum }_i{n}_i{\left|{\Omega }_i^{(L)}\right|}^2$; the remaining non-Lewis type NBOs $\left({\Omega }_j^{(NL)}\right)$ reflect the “error” of idealized natural Lewis description of the bonding structure. The interaction between the Lewis (donor) and non-Lewis (acceptor) orbitals characterizes the degree of breakdown of the localized Lewis representation; such interaction is reported in NBO analysis through a second-order perturbation approximation,

$$\Delta E_{ij}^{(2)}=n_i\frac{{\left|\langle {\Omega }_i^{(L)}\left|\widehat{F}\right|{\Omega }_j^{(NL)}\rangle \right|}^2}{{ϵ}_j-{ϵ}_i},$$ (3)

in which n_i is the occupation number of ${\Omega }_i^{(L)}$, and ϵ’s are orbital energies. The list of large $E_{ij}^{(2)}$ values provides a self-consistent framework for identifying key “charge transfer”, “conjugation” or “resonance” effects that stabilize the molecule. For cases that exhibit significant occupancy of non-Lewis NBOs, it is useful to use multiple “resonance Lewis structures” to represent the total density matrix through the natural resonance theory (NRT),⁶⁶

$$P=\sum _{\alpha }{w}_{\alpha }{P}_{\alpha }^{(L)},$$ (4)

in which w_α is the weight of the α-th Lewis resonance structure.

2.2. NBO analysis for DFTB3 using CHARMM and DFTB+

As the brief summary of NBO analysis indicated, the sets of orbital transformations between AO, NAO and NBO require only the one-particle density matrix, P^AO, and the overlap matrix associated with the AO basis, S^AO; evaluation of the donor-acceptor interaction via the second-order perturbation expression (Eq. 3) further requires the Fock (Kohn-Sham) matrix in the AO basis, F^AO. Therefore, NBO analysis can be readily conducted for DFTB calculations using the NBO program^55,56 once the three matrices (P^AO, S^AO, F^AO) are available as the NBO.47 file. We have modified both CHARMM^57,67 and DFTB+⁵⁸ codes to print these matrices in the proper format required by NBO; in addition, structural and AO basis information is also recorded to prepare the entire NBO.47 file in an automated fashion. For plotting the NBOs for DFTB calculations, NBO output and additional information regarding the Slater atomic basis functions are fed into waveplot to generate the corresponding cube files, which can be used by other programs such as VMD⁶⁸ for three-dimensional plots in different representations. Scripts for performing NBO analysis and plotting the NBOs for DFTB calculations are available from the authors upon request.

Due to its computational efficiency, DFTB is used extensively in QM/MM based molecular dynamics simulations,^7,8 thus the modifications we have made in CHARMM could in principle be used to analyze the fluctuations of bonding properties (e.g., bond order and donor-acceptor interactions) more in-depth using NBO; such analyses might be relevant to the discussion of dynamical effects in enzyme catalysis,⁶⁹ for example. Work along this line, however, is beyond the scope of this study.

2.3. Computational details

The DFTB calculations here are conducted with the DFTB3³⁶/3OB parameter set^{39,43–45,70} using the CHARMM⁵⁷ and DFTB+⁵⁸ programs. For comparison, DFT calculations at the B3LYP/aug-cc-pVTZ⁷¹ level are conducted using Gaussian09;⁷² for selected cases, PM6 calculations are also carried out for comparison. Most calculations are done in the gas phase; for the phosphates discussed in Sect.3.2.2, which are highly charged, continuum solvent models are used, which is PCM⁷³ for DFT and Poisson-Boltzmann⁷⁴ (though with a salt concentration of 0 M) for DFTB3.⁷⁵ Structures are optimized at the respective levels of theory. All NBO analyses are done with the latest NBO^55,56 package.

3. Results and Discussion

Before discussing larger and more complex molecules, we show several NBOs for two small molecules, water and ethane, in Fig. 1, to provide a baseline comparison between DFTB3/3OB and DFT (B3LYP/aug-cc-pVTZ). Evidently, the NBOs at the two levels of theory are very similar, in terms of both hybridization and polarization. The comparison also makes clear that a variation of ~0.5 in the degree of hybridization (e.g., sp^2.77 vs. sp^3.24 for the O-H bonding NBO in water or sp^2.81 vs. sp^2.35 for the C-C bonding NBO in ethane) implies only minor difference in the nature of bonding orbitals.

Figure 1:

Comparison of NBOs for simple molecules at DFTB3/3OB and B3LYP/aug-cc-pVTZ levels: (a) H₂O and (b) C₂H₆.

3.1. Bonding NBOs of Phosphates

In this section, we examine the bonding NBOs for phosphate species relevant to phosphoryl transfer reactions⁴⁰ with different coordinate environments of the phosphorus center: a tetra-coordinated phosphate diester, [PO(OH)(OMe)₂], a tri-coordinated metaphosphate, [PO₃]⁻, which is involved in the dissociative mechanism, and a penta-valent intermediate, [PO(OH)₂(OMe)₂]⁻, which is involved in the associative mechanism.

For the tetra-coordinated phosphate diester, as seen in Table 1, the DFTB3/3OB and B3LYP results are generally very consistent in terms of the nature of the bonding NBOs; the P hybrid bonding orbital is of sp³ nature in all P-O bonds except in the bond with the otherwise uncoordinated oxygen; for the latter nominal P-O double bond (as reflected by the natural bond order in Table 2), the P hybrid bonding orbital is of sp² nature at both DFTB3 and B3LYP levels. We note that there is little involvement of phosphorus d orbitals in the bonding NBOs at both DFTB3 and B3LYP levels. The degree of polarization (i.e., weights of P and O hybrid atomic orbitals) in the bonding NBOs is generally consistent between DFTB3 and B3LYP, which is also manifested in the natural charges and natural bond orders shown in Table 2; in general, the magnitude of natural charges is lower at the DFTB3 level, and the nominal P-O double bond has a lower bond order at the DFTB3 level (1.43) than B3LYP (1.69). Considering the significant difference in the size of basis set in DFTB3 and B3LYP/aug-cc-pVTZ calculations, the agreement between the NBO results is quite encouraging.

Table 1:

Natural Bonding Orbitals for P-O bonds and Selected Donor-Acceptor Interactions (in kcal/mol) in a Protonated Di-methyl Phosphate^a from DFTB3/3OB and B3LYP/aug-cc-pVTZ Calculations

	DFTB3		B3LYP
Bond	P	O	P	O
P-O	0.60 sp2.14d0.00	0.80 sp2.94	0.51 sp2.07d0.04	0.86 sp2.18
P-OH	0.53 sp2.41d0.02	0.85 sp3.22	0.47 sp3.34d0.10	0.88 sp2.47
P-OMe	0.54 sp3.28d0.02	0.84 sp2.81	0.47 sp3.29d0.10	0.89 sp2.60
P-OMe	0.54 sp3.47d0.02	0.84 sp3.17	0.46 sp3.31d0.11	0.89 sp2.79
$n_O\to {\sigma }_{P-O_{Me}}^{*}$	19.2		25.1
$n_O\to {\sigma }_{P-O_H}^{*}$	17.9		21.7
$n_O\to {\sigma }_{P-O_{Me}}^{*}$	8.4		15.3
$n_{O_{Me}}\to {\sigma }_{P-O_{Me}}^{*}$	8.1		9.4
$n_{O_{Me}}\to {\sigma }_{P-O_{Me}}^{*}$	10.0		11.7

^a.O_Me indicates an oxygen bonded to a methyl group; O_H is protonated; O is deprotonated.

Table 2:

Natural Charges and Bond Orders at DFTB3/3OB and B3LYP/aug-cc-pVTZ Levels for Selected Atoms^a in Different Phosphate Molecules

Atom/Bond	Metaphosphate	Di-methyl Phosphate	Pentavalent Intermediate
P	+2.35 (0.10)b/+1.98 (0.12)	+2.54 (0.10)/+2.02 (0.12)	+2.50 (0.09)/+2.24 (0.10)
O	−1.12/−0.99	−1.07/−0.95	−1.20/−1.14
OH	–	−0.99/−0.77	−1.06/−0.98
OMe	–	−0.83/−0.60	−0.84/−0.68
P-O	1.33/1.33	1.69/1.43	1.48/1.06
P-OH	–	0.72/0.77	0.74/0.69
P-OMe	–	0.77/0.84	0.71/0.70

^a.O stands for the deprotonated oxygen, O_H is protonated oxygen and O_Me is the oxygen bonded to a methyl group. Average results are shown for chemically identical atoms.

^b.The values in parentheses are the populations for the d orbitals.

For the metaphosphate, due to the highly symmetric nature of the molecular ion, the bonding NBOs at the DFTB3 and B3LYP levels are consistently formed by sp² P and sp³ O hybrid atomic orbitals (Fig. 2a). The degrees of polarization are comparable at the two levels of theory, as also reflected by the natural charges in Table 2. Again, the P d orbitals contribute about 0.1 electrons to the natural charge of P at both DFTB3 and B3LYP levels.

Figure 2:

Comparison of NBOs for phosphate molecules at DFTB3/3OB and B3LYP/aug-cc-pVTZ levels: (a) metaphosphate ${\text{PO}}_3^{-}$ and (b) a pentavalent phosphate species (see Table 3).

For the penta-valent intermediate, as shown in Table 3, there are larger differences between DFTB3 and B3LYP results (see also Fig. 2b), although the general trends are still consistent. The nominal P-O double bond (see the natural bond order in Table 2) involves sp² P orbital with relatively small degree of d involvement. For all other nominal P-O single bonds, the involvement of d is substantially higher than in the tetra-coordinated phosphate ester; in two cases, the fractions of p and d P orbitals differ somewhat between DFTB3 and B3LYP. Nevertheless, the contribution from P d orbitals to the natural charge is still rather small and about 0.10 at both DFTB3 and B3LYP levels. Again, the degrees of polarization in the bonding NBOs are consistent at DFTB3 and B3LYP levels, as also reflected by the natural charges and natural bond orders in Table 2; the largest difference is the bond order for the nominal P-O double bond, which is 1.48 for B3LYP and 1.06 for DFTB3. Note that the natural bond orders for other “single” P-O bonds are consistently between 0.7–0.8 for both the tetra-coordinated phosphate diester and the penta-valent intermediate, regardless of the level of theory.

Table 3:

Natural Bonding Orbitals for P-O bonds and Selected Donor-Acceptor Interactions (in kcal/mol) in a Pentavalent Phosphate Diester^a [PO(OH)₂(OMe)₂] from DFTB3/3OB and B3LYP/aug-cc-pVTZ Calculations

	DFTB3		B3LYP
Bond	P	O	P	O
P-O	0.56 sp2.29d0.22	0.83 sp3.29	0.48 sp2.06d0.40	0.86 sp2.29
P-OH	0.35 sp3.70d2.98	0.94 sp5.61	0.37 sp3.48d2.36	0.94 sp3.71
P-OH	0.48 sp3.19d0.70	0.88 sp4.19	0.41 sp2.32d1.54	0.91 sp2.77
P-OMe	0.42 sp3.06d1.76	0.91 sp3.96	0.39 sp2.83d1.79	0.92 sp3.08
P-OMe	0.49 sp3.28d0.53	0.87 sp3.52	0.40 sp2.49d1.62	0.92 sp2.65
${\sigma }_{P-O_{Me}}\to {\sigma }_{P-O}^{*}$	34.1 (63.8)b		18.7
${\sigma }_{P-O_{Me}}\to {\sigma }_{P-O_H}^{*}$	74.1 (185.1)		42.9
${\sigma }_{P-O_{Me}}\to {\sigma }_{P-O_{Me}}^{*}$	44.4 (106.0)		24.5
${\sigma }_{P-O_{Me}}\to {\sigma }_{P-O_H}^{*}$	52.0 (130.0)		24.5
${\sigma }_{P-O}\to {\sigma }_{P-O_{Me}}^{*}$	41.3 (68.7)		24.1
${\sigma }_{P-O}\to {\sigma }_{P-O_H}^{*}$	44.4 (80.5)		27.5
${\sigma }_{P-O_H}\to {\sigma }_{P-O_{Me}}^{*}$	60.0 (140.3)		38.8
${\sigma }_{P-O_H}\to {\sigma }_{P-O}^{*}$	31.1 (57.8)		18.2
${\sigma }_{P-O_H}\to {\sigma }_{P-O_{Me}}^{*}$	42.3 (94.9)		23.5
${\sigma }_{P-O_H}\to {\sigma }_{P-O_H}^{*}$	50.0 (112.2)		26.3
${\sigma }_{P-O_{Me}}\to {\sigma }_{P-O_{Me}}^{*}$	45.1 (97.4)		26.7
${\sigma }_{P-O_{Me}}\to {\sigma }_{P-O_H}^{*}$	51.3 (111.3)		27.7
${\sigma }_{P-O_H}\to {\sigma }_{P-O_{Me}}^{*}$	57.6 (132.8)		30.6
${\sigma }_{P-O_H}\to {\sigma }_{P-O_H}^{*}$	62.2 (150.5)		33.7

^a.O_Me indicates an oxygen bonded to a methyl group; O_H is protonated; O is deprotonated.

^b.Note that all strong donor-acceptor interactions identified are geminal interactions between neighboring bonds that are nearly perpendicular to each other. Values in parentheses are computed at the PM6 level.

In short, comparison of the bonding NBOs in the three phosphate species suggests that despite the use of a minimal basis set, the DFTB3 description of covalent bonding in these molecules agrees well with B3LYP using a much larger basis set (aug-cc-pVTZ), including the degree of bond polarization and therefore the natural charges and natural bond orders. Although the contribution of d occupation to the natural charge of P is very modest (~0.1), the d orbitals appear to play important polarization roles in the bonding NBOs in the penta-valent intermediate. Therefore, the inclusion of d orbitals in the parameterization of DFTB3/3OB for phosphorus likely contributes to the overall encouraging performance of the method for phosphate compounds³⁹ as compared to traditional NDDO methods such as PM6.

3.2. Donor-acceptor Interactions

In this subsection, we analyze donor-acceptor interactions from NBO analysis, which provides a more nuanced evaluation of electronic structure at the DFTB3 level as compared to DFT (B3LYP) with a large basis set (aug-cc-pVTZ). We again start with the three small phosphate species, then move on to a series of “high energy” phosphate compounds, whose hydrolysis free energies were correlated to specific donor-acceptor interactions in the NBO framework;^76,77 we conclude this subsection with a discussion of n → π* interaction, which has been proposed to provide substantial stabilization to specific peptide structure.^78–83

3.2.1. Orbital Interactions in Phosphates

For the tetra-coordinated phosphate di-ester, Table 1 lists several dominant donor-acceptor interactions, which range from ~10 to ~25 kcal/mol in magnitude; they involve charge transfer/delocalization from the oxygen lone pairs to the anti-bonding P-O orbitals. The significant populations of the anti-bonding P-O orbitals are consistent with the decrease of P-O bond orders from unity to ~0.7–0.8 as shown in Table 2. At the DFTB3/3OB level, the magnitudes of orbital interactions are generally similar, with DFTB3 interactions being slightly weaker than B3LYP.

In the penta-valent intermediate, the situation is different and reveals a potential limitation of the current DFTB3 model. Table 3 lists dominant interactions, which are geminal interactions between a bonding NBO and the anti-bonding NBO of a neighboring P-O bond. The magnitude of these interactions can be as large as 40 kcal/mol, which is likely due to the small bond angles between neighboring P-O bonds; indeed, all large geminal interactions shown in Table 3 are between bonds that form a nearly 90° angle. Interestingly, while these interactions are also highlighted at the DFTB3/3OB level, the magnitudes are consistently overestimated by almost a factor of two. Curiously, the trend of significantly overestimated geminal interaction is not unique to DFTB3; as shown in Table 3, PM6 overestimates these orbital interactions even more dramatically, almost by a factor of 4 or larger. Since germinal interactions tend to correlate with the ionic character of neighboring bonds,³⁴ the overestimated geminal interactions at the DFTB3 level suggest that the penta-valent phosphate compound has more ionic character with DFTB3 than with B3LYP; this is confirmed by a NRT analysis (see Fig. 3), which finds that the ionic resonance structures constitute more than 55% weight at the DFTB3 level, in contrast to ~30% for B3LYP.

Figure 3:

Weights of dominant ionic resonance structures of the penta-valent phosphate species from NRT analysis. Values without and with parentheses are B3LYP and DFTB3/3OB results, respectively.

To explore whether the significant overestimation of geminal interactions is unique to the hypervalent phosphate compound, in Fig. 4, we examine the geminal interaction and its bond-angle dependence in di-methyl phosphate ester and cyclopropane and compare the results at DFTB3, PM6 and B3LYP levels. For both molecules, the magnitude of geminal interactions is substantially lower than those observed for the penta-valent phosphate intermediate, which is consistent with the idea that the latter hypervalent species features considerably stronger ionic character. Nevertheless, the trends observed for the more covalent di-methyl phosphate ester and cyclopropane are informative. As shown in Fig. 4, although the potential energy curves are very similar, especially between DFTB3 and B3LYP, the geminal interactions can differ notably. At small bond angles, the geminal interactions are the strongest in magnitude at the B3LYP level, while both PM6 and DFTB3 values are slightly lower. As the bond angle expands, PM6 remains close to B3LYP, while the DFTB3 value increases significantly for both cases studied here.

Figure 4:

Comparison of geminal interactions at DFTB3/3OB, PM6 and B3LYP/aug-cc-pVTZ levels in (a) di-methyl phosphate ester and (b) cyclopropane. For each panel, the left plot is the potential energy as a function of the bending angle (in degrees), and the right plot is the second-order geminal interaction as a function of the bending angle.

With the second-order perturbation expression, the geminal interaction between NBOs σ_AX and ${\sigma }_{AY}^{*}$ can be written as the following,

$$\Delta E_{gem}^{(2)}=-2\frac{{\langle {\sigma }_{AX}\left|\widehat{F}\right|{\sigma }_{AY}^{*}\rangle }^2}{{ϵ}_{{\sigma }_{AY}^{*}}-{ϵ}_{{\sigma }_{AX}}}.$$ (5)

Further writing the NBOs in terms of the NHOs, σ_AX = c_Ah_A +c_Xh_X, ${\sigma }_{AY}^{*}=c_Y{h}_{A^{\prime }}-c_{A^{\prime }}{h}_Y$, we see that the geminal interaction has contributions from one-center, non-bonded and cross-bonded matrix elements,

$$\langle {\sigma }_{AX}\left|\widehat{F}\right|{\sigma }_{AY}^{*}\rangle =c_A{c}_Y{F}_{A{A}^{\prime }}-c_{A^{\prime }}{c}_X{F}_{XY}+\left(c_X{c}_Y{F}_{A^{\prime }X}-c_A{c}_{A^{\prime }}{F}_{AY}\right).$$ (6)

Considering the relatively large distance between X and Y, the magnitude of the matrix element F_XY is expected to be small, especially at expanded X-A-Y angles. Therefore, it is likely that the trend of the geminal interaction is dictated by the one-center matrix element and its bond angle dependence. Comparison of the matrix element F_AA′ at DFTB3 and B3LYP levels indeed finds that the DFTB3 value increases more rapidly as a function of the X-A-Y bond angle; for example, for the di-methyl phosphate ester, F_AA′ at the DFTB3 level is almost twice as large as the B3LYP value at the bond angle of 140°. More systematic analysis is warranted, but the current result highlights that notable errors may exist in even one-center matrix elements for DFTB3/3OB.

3.2.2. Anomeric Effects in High-energy Phosphates

In a series of studies,^76,77 Ruben and co-workers analyzed the electronic structure of a series of “high energy” phosphate compounds and aimed to identify features that correlate with the magnitude of the hydrolysis free energy and destabilization of the scissile P-O bond (Fig. 5a). It was observed that the charge transfer from the oxygen lone-pair to the anti-bonding orbital of the scissile P-O bond is a good indicator. Indeed, the sum of computed second-order “anomeric” energies from all three antiperiplanar phosphoryl oxygen atoms was found to correlate strongly (R = 0.90) with the optimized equilibrium length for the scissile P-O bond, which varied more than 0.1 Å among the ten compounds studied, as well as with (R = −0.93) the experimentally determined hydrolysis free energy. Analysis of σ* (O − P) and σ(O − P) occupancies indicated that the anomeric charge transfer effect correlated most strongly with the strength of the O-P bond, as compared to the “electrostatic dipole effect” associated with a decrease of σ(O − P) population (see Fig. 5a).

Figure 5:

Anomeric effects and destabilization of the scissile P-O bond in a series of “high energy” phosphate species analyzed in Ref. 76. (a) Effects that potentially contribute to the destabilization of the scissile P-O bond (adapted from Ref.⁷⁶); (b) Comparison of absolute anomeric interactions (summed over contributions from all three antiperiplanar phosphoryl oxygen atoms) at DFTB3/3OB, PM6 and B3LYP/aug-cc-pVTZ levels (with either a PCM or PB implicit solvent model, see Method) and correlation with the optimized scissile P-O bond distance.

In Fig. 5b, we confirm with B3LYP/aug-cc-pVTZ/PCM calculations that the sum of anomeric energies indeed correlate well with the optimized scissile P-O bond distance. In fact, similarly good correlations are also observed with single point PM6/PCM and DFTB3/PB calculations. There is considerable difference, however, among the computed second-order anomeric interactions. While DFTB3/3OB underestimates the magnitude of the interactions, PM6 overestimates the interactions to a similar degree. Therefore, similar to the observation made in the last subsection, while DFTB3 provides a qualitatively similar description for the orbital interactions, there can be significant deviation from B3LYP calculations with a large basis set for quantitative results.

3.2.3. n → π*Interaction in Peptides

We conclude this subsection by a brief analysis of n → π* interactions, which have been proposed to be involved in the stabilization of specific structures in peptides and proteins.^78,83 Specifically, we study the set of compounds analyzed both computationally and experimentally in Ref. 78. As summarized in Fig. 6, the impact of the n → π* interaction between the two carbonyl groups is manifested by the preference to the trans over the cis conformer. With an electron-donating substitution for R₁, one expects a stronger contribution of the n → π* interaction and therefore a larger preference to the trans conformer; the degree of stabilization of the trans conformer also depends on the endo/exo pucker of the pyrrolidine ring.⁷⁸

Figure 6:

Several L-proline methyl esters (R₁ =H, CH₃; R₂=H) used to probe the importance of n → π* interaction,⁷⁸ which exists in the trans conformer as illustrated for the case of R₁=CH₃, R₂=H.

As shown in Table 4, the B3LYP calculations indeed capture the expected trends, and the magnitude of the second-order n → π* interaction ranges from 0.3 to 1.3 kcal/mol, thus relatively weak but not negligible. At the DFTB3 level, the encouraging observation is that the preference to the trans conformer, especially with the methyl substitution for R₁, is well captured. The magnitude of the second-order n → π* interaction is even smaller with DFTB3, with the largest value being ~0.5 kcal/mol. At the PM6 level, the computed second-order n → π* interactions are similar in magnitude as DFTB3, although the relative conformational energies are not well described. Considering that many factors contribute to the energy landscape of peptides and proteins, it is di cult to argue that DFTB3 provides the qualitatively correct description for molecules studied here solely because of the n → π* interaction, which is even weaker at the DFTB3 level than B3LYP. Nevertheless, it is conceivable that NBO analysis may provide useful insights in cases where larger discrepancy is observed in local conformational preferences relative to higher level calculations,⁸⁴ thus being complementary to other efforts for improving non-covalent interactions at the DFTB3 level for condensed phase applications.^17,19,24

Table 4:

Isomerization Energy and Second-order n → π* Interactions (in kcal/mol) for several L-proline methyl esters (see Fig. 6) from DFTB3/3OB, PM6 and B3LYP/aug-cc-pVTZ Calculations^a

	B3LYP		DFTB3/3OB		PM6
Moleculeb	ΔE	$\left|E_{n\to {\pi }^{*}}^{(2)}\right|$	ΔE	$\left|E_{n\to {\pi }^{*}}^{(2)}\right|$	ΔE	$\left|E_{n\to {\pi }^{*}}^{(2)}\right|$
M1 (endo)	−0.6	0.3	−0.8	0.1	−0.4	0.1
M1 (exo)	−1.0	0.6	−0.4	0.2	0.1	0.2
M2 (endo)	−1.3	0.4	−1.1	0.1	−0.1	0.2
M2 (exo)	−1.5	1.3	−0.9	0.5	0.8	0.5

^a.ΔE is the energy difference between the cis and trans conformers, and a negative value favors the trans conformer. Structures are optimized at the respective level of theory.

^b.See Fig. 6, M1: R₁=H, R₂=H; M2: R₁=CH₃, R₂=H. endo/exo specifies the pucker of the pyrrolidine ring.

3.3. Metal-Ligand Interactions

3.3.1. Bonding of Cu carbonyl compounds

The bonding of metal carbonyl cations has previously been studied by Frenking et al. for a number of transition metals at the MP2 level.⁸⁵ In the case of Cu⁺, the metal charge in the monocarbonyl was observed to be reduced by only 0.07 e and the bonding between Cu and the CO ligand is mainly electrostatic in nature. The donation of electron density from the ligands to Cu increases fourfold in the di-, tri-, and tetra-carbonyls, leading to weaker Coulombic contributions to the M-CO bonding.

As seen in Table 5, the natural charge of the metal in [Cu(CO)₁]⁺ with B3LYP indicates a donation of 0.09 e from the CO ligand to the metal whereas the DFTB3 natural charge shows a more significant donation of 0.20 e. As the number of ligands increases, the trend observed by Frenking et al.⁸⁵ is observed for B3LYP and DFTB3, with DFTB3 consistently showing a lower positive charge on the metal center. The natural charge of oxygen under B3LYP is negative as expected by the CO dipole, while the DFTB3 charge is close to zero but positive. The impact of the positive charge of oxygen with DFTB3 is that carbon has a smaller charge when treated with DFTB3 compared to B3LYP. The difference in charges is readily explained by examining the natural charges of an isolated CO molecule, for which B3LYP has a larger charge of +0.49 e on C as compared to a charge of +0.11 e with DFTB3. Therefore, DFTB3 underestimates the relative polarity of C and O, leading to an unexpected positive charge on oxygen when CO coordinates to Cu. For comparison, the natural charges of [Ni(CO)₃]²⁺, which shows large structural deviations between B3LYP/aug-cc-pVTZ and DFTB3/3OB⁴⁵ (further discussed below), are shown in Fig. 7a. Similar to [Cu(CO)_n]⁺, oxygen takes a positive charge with DFTB3, indicating a potential area of improvement. For additional comparison, the natural charges in [Cu(CH₃)₂]⁻ and [Cu(CH₃)₂]⁰, which are Cu(I) and Cu(II) compounds, respectively, shown in Fig. 7b reveal that DFTB3 reproduces the correct charge sign for the ${\text{CH}}_3^{-}$ ligand, albeit with a smaller magnitude.

Table 5:

Natural Charges, Bond Orders, and Donor-Acceptor Interactions (in kcal/mol) at DFTB3/3OB and B3LYP/aug-cc-pVTZ Levels^a for [Cu(CO)_n]⁺

Atom/Bond	[Cu(CO)1]+	[Cu(CO)2]+	[Cu(CO)3]+	[Cu(CO)4]+	CO
Cu	+0.91 (+0.80)	+0.67 (+0.59)	+0.71 (+0.64)	+0.70 (+0.64)
C	+0.37 (+0.15)	+0.45 (+0.17)	+0.42 (+0.10)	+0.41 (+0.08)	+0.49 (+0.11)
O	−0.28 (+0.05)	−0.29 (+0.04)	−0.32 (+0.02)	−0.34 (+0.01)	−0.49 (−0.11)
Cu-C	1.14 (0.96)	0.54 (0.60)	0.38 (0.46)	0.26 (0.30)
O-C	2.86 (2.88)	2.96 (2.90)	2.95 (2.86)	2.96 (2.92)
$n_{Cu}\to {\pi }_{CO}^{*}$	5.8 (5.5)	0.7 (1.2)	1.5–3.5	0.8–1.5
			(0.8–3.5)	(1.2–1.5)
${\sigma }_{CuC}\to {\sigma }_{CO}^{*}$	2.0 (42.0)	1.3 (35.1)	1.3 (35.5)	1.4 (−)
$n_{C^{\prime }}\to {\sigma }_{CuC}^{*}$	–	94.5 (291.0)	65.1 (18.8)	59.7 (12.9)

^a.DFTB3 natural charges, bond orders, and interaction energies are shown in parenthesis.

Figure 7:

Natural Charges for (a) [Ni(CO)₃]²⁺ (for which DFTB3/3OB and B3LYP leads to a D_3h and C_2v structure, respectively⁴⁵) and (b) [Cu(CH₃)₂]^−/0. In panel (b), the three figures are for optimized $\text{Cu}{\left({\text{CH}}_3\right)}_2^{-}$, optimized Cu(CH₃)₂, and optimized Cu(CH₃)₂ with the linear constraint, respectively. B3LYP structures are used in the calculations, and DFTB3/3OB gives very similar [Cu(CH₃)₂]^−/0 structures. Values without and with parenthesis are B3LYP/aug-cc-pVTZ and DFTB3/3OB natural charges, respectively.

Despite the differences in natural charges, the DFTB3 natural bond orders for Cu-C and C-O show good agreement with B3LYP. It should be noted that NBO favors a 3-center C:-Cu-:C hyperbond description for n = 2 − 4 rather than yielding multiple Cu-C orbitals, leading to fractional Cu-C bond orders (see Table 5). The hybridizations of the natural bonding orbitals for Cu-C bonds, shown in Table 6, indicate agreement between DFTB3 and B3LYP in terms of polarization of the NBOs. The hybrid on Cu is largely of s character for all metal carbonyls under both methods due to Cu⁺ having a filled d shell, making the vacant 4s orbital the acceptor for electron density from C. The hybrid on C is of sp^0.50 character for B3LYP in all cases, while for DFTB3, the hybrid is closer to sp character for n = 1 − 3. For the isolated CO molecule, the C lone pair has a sp^0.32 and sp^0.38 hybridization for B3LYP and DFTB3, respectively. Thus, the deviation from the idealized sp hybridization for B3LYP is due to the polar nature of CO, which DFTB3 fails to capture adequately for CO bonded to transition metals due likely to the minimal basis nature. For [Cu(CO)₄]⁺ the hybridization of C agrees well in both methods.

Table 6:

Natural bonding orbitals for Cu-C bonds in [Cu(CO)_n]⁺.

	DFTB3		B3LYP
n	Cu	C	Cu	C
1	0.37 sp0.04d0.10	0.93 sp0.90	0.28 sp0.01d0.03	0.96 sp0.59
2	0.39 sp0.06d0.20	0.92 sp0.84	0.35 sp0.00d0.06	0.94 sp0.55
3	0.32 sp0.18d0.05	0.95 sp0.83	0.31 sp0.02d0.02	0.95 sp0.52
4	0.27 sp0.29d0.00	0.96 sp0.45	0.29 sp0.03d0.00	0.96 sp0.50

The bottom of Table 5 lists key donor-acceptor interactions in the copper carbonyls. Due to the high electrostatic nature of the Cu-C bonds, π-backbonding is not expected to be significant. Indeed, the π-backbonding in DFTB3 and B3LYP is about 5 kcal/mol in [Cu(CO)₁]⁺, which corresponds to the highest CuC bond order of the copper carbonyls. The backbonding contribution drops to ~1–3 kcal/mol as the number of CO ligands increases. DFTB3 captures this trend well. When it comes to the geminal ${\sigma }_{CuC}\to {\sigma }_{CO}^{*}$ interaction, B3LYP and DFTB3 differ greatly, with DFTB3 giving interaction energies of 35–40 kcal/mol while B3LYP gives energies in the range of 1–2 kcal/mol. This is demonstrated in Fig. 8a, where there is clearly better overlap of the σ_CuC and ${\sigma }_{CO}^{*}$ DFTB3 orbitals in large part due to the sp hybrid on carbon. In the case of n = 4, this geminal interaction is not present for DFTB3 due to the better agreement of the hybrid on carbon between DFTB3 and B3LYP. One additional interaction of interest is the frontbonding $n_{C^{\prime }}\to {\sigma }_{CuC}^{*}$ interaction in which the lone pair on C’ is spatially oriented for overlapping with a Cu-C antibonding orbital. In the case of [Cu(CO)₂]⁺, DFTB3 highly overestimates this interaction, giving an interaction energy that is nearly three times larger than with B3LYP. The situation is reversed for [Cu(CO)₃]⁺ and [Cu(CO)₄]⁺, however, with DFTB3 giving an energy value that is over four times smaller than the B3LYP value. The difference is seen by the hybridization (and therefore, orientation) of the C lone pair, which is sp^2.69 for DFTB3 in [Cu(CO)₂]⁺ compared to sp^0.76 for B3LYP, allowing the lone pair to have better overlap with the Cu-C antibonding orbital in DFTB3. The hybridization of the C lone pairs in [Cu(CO)₃]⁺ is sp^0.40 and sp^0.71 for DFTB3 and B3LYP, respectively. Thus, the DFTB3 carbon lone pairs have more s character and have lower overlap with the Cu-C antibonding orbital. Similarly, for [Cu(CO)₄]⁺, the lone pair has more s character with DFTB3 (sp^0.41) than with B3LYP (sp^0.69). The consequence of the hybridizations of the carbon lone pairs is illustrated in Fig. 8b for [Cu(CO)₂]⁺ and [Cu(CO)₃]⁺.

Figure 8:

NBO description of donor-acceptor interactions at the DFTB3/3OB and B3LYP/aug-cc-pVTZ levels: (a) ${\sigma }_{CuC}\to {\sigma }_{CO}^{*}$ in [CuCO]⁺, and (b) $n_C\to {\sigma }_{CuC}^{*}$ in [Cu(CO)₂]⁺ (top) and [Cu(CO)₃]⁺ (bottom).

3.3.2. Pseudo Jahn-Teller Distortions

One particular question of interest is how DFTB3 can describe the electronic structure of (pseudo) Jahn-Teller distortions.⁴⁶ Recently, NBO has been used to describe the geometric distortion in Si₂H₄⁸⁶ and in tetrahydridodimetallenes.⁸⁷ In these studies, visual rationalizations for stabilization as well as donor-acceptor interactions have been considered to provide rationalizations for the geometric distortions. Thus, it is informative to compare DFTB3 and B3LYP for several metal compounds in which a pseudo Jahn-Teller distortion is observed.

For a simple example, we consider the case of the Cu(II) molecule Cu(CH₃)₂, which exhibits a distortion from the linear structure of the anionic Cu(I) analogue. This distortion is illustrated in Fig. 7b. DFTB3 correctly captures the geometric features of this distortion.⁴⁴ To assess how DFTB3 describes the pseudo Jahn-Teller distortion, we perform an NBO analysis at the B3LYP and DFTB3 levels in a linear, pre-distortion geometry and in the optimized bent geometry of Cu(CH₃)₂. An open-shell NBO analysis is used in these cases, which describes the molecule in terms of “different Lewis structures for different spins”,⁶¹ which leads to different NBOs for each spin. Under a non-relativistic approximation, the Lewis structure corresponding to the α spin does not have a direct spin-dependent interaction with the Lewis structure corresponding to the β spin and vice versa, but the final geometry is expected to reflect a compromise between the bonding pattern of both Lewis structures. We note that the traditional filling of orbitals following Hund’s rules equates the α system to that of anionic [Cu(CH₃)₂]⁻, which is expected to be linear. Thus, the donor-acceptor interactions of the α NBOs shown in Table 7 show that the $n_C\to {\sigma }_{CuC}^{*}$ interaction is stronger by 8 kcal/mol under B3LYP and DFTB3; this result seemingly favors the linear structure (however, see below). In terms of the absolute magnitude, DFTB3 underestimates these interactions by about 10 kcal/mol.

Table 7:

Donor-Acceptor Interactions (in kcal/mol) Between Natural Bonding Orbitals in Cu(CH₃)₂.

Spin	Interaction	Geometrya	DFTB3	B3LYP
α	$n_C\to {\sigma }_{Cu{C}^{\prime }}^{*}$	linear	30.8	40.6
		bent	22.2	32.5
β	$n_C\to l{v}_{Cu}$	linear	-	107.7
		bent	343.6	150.2
	$n_C\to {\sigma }_{Cu{C}^{\prime }}^{*}$	linear	32.6	0.1
		bent	6.8	0.1

^a.For structures, see Fig. 7b.

The preference for the bent structure is more clearly exhibited by analyzing the β system. Whereas the α system is consistent in all four cases (the Lewis structures are shown in Fig. 9), giving one Cu-C bond and assigning a lone pair on the second carbon, the β system gives several different Lewis structures. For the bent geometry with B3LYP and with DFTB3, NBO analysis yields a natural Lewis structure with two Cu-C bonds. For the linear structure with B3LYP, NBO analysis yields a three center C-Cu-C bond in the Lewis structure, while the NBO analysis of the DFTB3 data gives one Cu-C bond and a lone pair on the other C. The DFTB3 β natural Lewis structure is fundamentally similar to that of the α Lewis structure, save for an empty valence NBO on copper that corresponds to an electron hole, denoted as lone valence (lv) below. To allow for a more direct comparison with the α structure, a directed NBO search is used for the β Lewis structures using the $CHOOSE keyword from the DFTB3 linear NBO calculation. In three of the cases, namely linear B3LYP, bent B3LYP, and bent DFTB3, a strong n_C → lv_Cu interaction arises in the order of hundreds of kcal/mol (see Table 7). The magnitude of this interaction is justified because in these three cases, the NBO analysis favors a Cu-C bond, as shown by the preferred Lewis structures. With B3LYP, the n_C → lv_Cu interaction is stronger by over 40 kcal/mol in the bent structure and it overpowers the $n_C\to {\sigma }_{CuC}^{*}$ interaction that would favor the linear structure in the α system. The large value of the interaction in the bent DFTB3 calculation is due to the small energy difference between the acceptor and donor NBOs (cf. eq. 3).

Figure 9:

α and β Lewis structures obtained by NBO analysis of Cu(CH₃)₂. A dashed bond indicates a three center bond, and an empty circle indicates a lone valence (lv), i.e., a hole, on the metal center.

One final interaction of interest is the geminal $n_C\to {\sigma }_{CuC}^{*}$ interaction. While one would expect this interaction to be strongly favored by the bent structure, it is negligible for the bent and linear structures with B3LYP. On the other hand, DFTB3 shows a strong interaction in both cases, with the linear structure giving an interaction on the order of 30 kcal/mol and the bent structure showing an interaction of about 6 kcal/mol. The discrepancy between B3LYP and DFTB3 is due to the Fock matrix elements. While the $n_C\to {\sigma }_{Cu{C}^{\prime }}^{*}$ interaction is stronger for the linear structure, it is highly compensated by the magnitude of the n_C → lv_Cu interaction in the bent structure, resulting in DFTB3 correctly favoring the distorted structure.

There are certain cases, particularly for nickel, for which DFTB3 gives rather different structures compared to B3LYP.⁴⁵ As an initial exploration, we consider [Ni(CO)₃]²⁺, for which DFTB3 gives the D_3h structure on the left of Fig. 7a with equivalent 120°C-Ni-C angles, while B3LYP gives the pseudo Jahn-Teller distorted C_2v structure on the right of Fig. 7a with two equivalent CO ligands separated by a 157.5° angle. To understand the nature for the preference of DFTB3 for the symmetric structure, NBO analysis is conducted with DFTB3 and B3LYP on both the symmetric and distorted structures. To aid with notation, the equivalent carbons in the C_2v structure will be denoted as C_a and the third carbon will be denoted as C_b (see Fig. 7a). The NBOs obtained for [Ni(CO)₃]²⁺ are listed in Table 8. First, as the α-Lewis structure corresponds to that of a filled d-shell configuration, we note the unsurprisingly strong similarities of the α-spin orbitals between DFTB3 and B3LYP. In the symmetric D_3h structure, one σ_NiC NBO is recovered that forms two three-center hyberbonds that involve the lone pair on each of the other carbon atoms. In the distorted C_2v structure, one σ_CuCa NBO is recovered that forms a three-center hyberbond with the other C_a atom. The Ni hybrid that makes up the bonding Ni-C NBO is almost a pure s orbital, indicating that bonding to the CO ligands is possible due to the empty Ni s orbital. For both geometries of [Ni(CO)₃]²⁺, the DFTB3 Ni hybrid that bonds to carbon has slightly more p-character than the corresponding B3LYP hybrid, leading to a better directionality to carbon and a weaker sideways $n_C\to {\sigma }_{Ni{C}^{\prime }}^{*}$ interaction. The relevant donor-acceptor interactions shown in Table 9 indicate that the $n_C\to {\sigma }_{Ni{C}^{\prime }}^{*}$ interaction is more than twice as strong with B3LYP than with DFTB3. While this stands in contrast with the angle dependence for geminal interactions discussed in Fig. 4 and Eq. 6, there is no one-center matrix element F_AA′ in this case due to interaction with the lone pair n_C and the interaction depends on the matrix element F_CNi. This highlights a difference in the way DFTB3 and B3LYP estimate geminal interactions and ionic destabilizing interactions. Nonetheless, the hybrid coefficients for the Ni-C bonding NBOs match almost perfectly and the sp character of the carbon hybrids is similar to that of the closed-shell [Cu(CO)_n]⁺ molecules.

Table 8:

Natural Bonding Orbitals for Ni-C Bonds in [Ni(CO)₃]^{2+ a}

			DFTB3		B3LYP
Symmetry	spin	Bondb	Ni	C	Ni	C
D3h	α	Ni-C	0.32 sp0.18	0.94 sp0.80	0.33 sp0.03d0.01	0.94 sp0.59
		C lp		1.00 sp0.40		1.00 sp0.74
	β	Ni-C	0.60 sp0.05d3.25	0.80 sp1.28	0.37 sp0.04d0.90	0.93 sp0.63
					(0.45 sp0.05d2.71)	(0.89 sp0.69)
		Ni lv	1.00 sp0.01d0.94		1.00 sp0.26d2.00
		C lp		1.00 sp6.24		1.00 sp0.93
C2v	α	Ni-Ca	0.35 sp0.12d0.04	0.94 sp0.80	0.35 sp0.02d0.01	0.94 sp0.59
		Ca lp		1.00 sp0.41		1.00 sp0.50
		Cb lp		1.00 sp0.39		1.00 sp0.75
	β	Ni-Ca	0.58 sp0.06d2.53	0.81 sp1.25	0.44 sp0.02d1.31	0.90 sp0.66
		Ni-Cb	0.55 sp0.05d1.72	0.83 sp1.16	0.46 sp0.03d1.86	0.89 sp0.69
		Ni lv	1.00 sp0.04d1.98		1.00 sp0.07d1.85
		Ca lp		1.00 sp6.84		1.00 sp0.94

^a.For structures and labels for the carbon atoms, see Fig. 7a.

^b.lp: lone pair; lv: lone valence (an electron hole).

Table 9:

Donor-Acceptor Interactions (in kcal/mol) Between Natural Bonding Orbitals^a in [Ni(CO)₃]²⁺

Spin	Interaction	Symmetry	DFTB3	B3LYP
α	$n_C\to {\sigma }_{Ni{C}^{\prime }}^{*}$	D3h	12.8 (×2)	35.9 (×2)
		C2v	18.3 (C=Ca)	44.3 (C=Ca)
			9.1 (C=Cb)	20.8 (C=Cb)
β	$n_{C_a}\to {\sigma }_{Ni{C}^{\prime }}^{*}$	D3h	5.0 (×2)	44.5, 7.8
		C2v	27.5 (C’=Ca)	66.2 (C’=Ca)
			2.9 (C’=Cb)	4.3 (C’=Cb)
	$n_{C_a}\to l{v}_{Ni}$	D3h	65.3	2.3
		C2v	48.5	1.0

^a.Note that DFTB3 allows fractional occupancy and has five equivalent β Ni d orbitals, while B3LYP has three occupied and two empty β Ni d orbitals. For structures and labels for the carbon atoms, see Fig. 7a.

There is a larger difference between DFTB3 and B3LYP NBOs for the β-Lewis structure, which corresponds to a partially-filled valence shell on nickel. Notably, for the symmetric D_3h structure that is obtained with DFTB3, DFTB3 gives two equivalent Ni-C NBOs while B3LYP gives two Ni-C NBOs with distinctly different hybridizations for the hybrid on Ni (one of ~sd hybridization and another of ~sd³ hybridization). Additionally, the β-Lewis structure features a nickel lone valence NBO that corresponds to an electron hole on nickel. Going from the D_3h structure to the distorted C_2v structure, there is a significant change in the d-character of the Ni hybrid that forms the Ni-C NBO from sd³ hybridization to ~sd² in DFTB3, while for B3LYP there is a compromise of hybridization between the two Ni-C NBOs, with the Ni-C_a hybrid increasing in d-character slightly and the Ni-C_b hybrid decreasing in d-character from ~sd³ to sd². Additionally, the lone pair on carbon has a large p-character for DFTB3, but it has the expected sp hybridization for B3LYP. Similar to the α interaction energies, the relative value of the $n_{C_a}\to {\sigma }_{Ni{C}^{\prime }}^{*}$ interaction remains higher for B3LYP. As expected, the $n_{C_a}\to {\sigma }_{Ni{C}_a^{\prime }}^{*}\left({\text{C}}_a\ne {\text{C}}_a^{\prime }\right)$ interaction is stronger than the $n_{C_a}\to {\sigma }_{Ni{C}_b}^{*}$ interaction in DFTB3 and B3LYP.

Table 9 also includes the interactions between the lone pair on carbon and the lone valence NBO on nickel. Although the hybridization of the nickel lone valence NBO matches between DFTB3 and B3LYP, the magnitude of the interaction with the n_Ca is very large for DFTB3 compared to B3LYP. The reason for the strong interaction is visually explained by the NBOs shown in Fig. 10. Evidently, the lone valence NBO is oriented toward the carbon lone pair in DFTB3, resulting in the strong donor-acceptor interaction. More importantly, the lone valence NBO seemingly influences the preference for the C_2v structure to optimize back to D_3h with DFTB3, as suggested by the direction of the hybrid compared to the direction of the Ni-C bond. The orientation of the nickel lone valence NBO in DFTB3 compared to B3LYP can be attributed to the combination of NAOs that make up the hybrid. For B3LYP, the lone valence hybrids have the form,

$${\text{D}}_{3h}:h_{N{i}_{lv}}=0.681\left(3d_{x^2-y^2}\right)+0.397(4s)+0.374(5s)-0.345\left(3d_{z^2}\right)+0.280\left(4p_z\right),$$

$${\text{C}}_{2v}:h_{N{i}_{lv}}=0.764\left(3d_{z^2}\right)+0.421(4s)+0.391(5s),$$

whereas for DFTB3, the hybrids are,

$${\text{D}}_{3h}:h_{N{i}_{lv}}=0.717(4s)+0.595\left(3d_{xy}\right)+0.3436\left(3d_{z^2}\right),$$

$${\text{C}}_{2v}:h_{N{i}_{lv}}=0.752\left(3d_{xy}\right)+0.576(4s)+0.278\left(3d_{z^2}\right).$$

Figure 10:

Comparison of the shape of the nickel lone valence hybrid in [Ni(CO)₃]²⁺ with DFTB3/3OB and B3LYP calculations, which favor D_3h and C_2v symmetry, respectively,⁴⁵ as shown in Fig. 7a.

Thus, the B3LYP lone valence is decidedly oriented in the z direction, whereas the DFTB3 lone valence is oriented between the x and y axes. This indicates that more attention has to be paid to the way the d orbitals in nickel are used for bonding. We note that DFTB3 allows for non-integer occupations of orbitals and splits the three β electrons evenly among the five d atomic orbitals, leading to an equal treatment of the three CO ligands. By contrast, in B3LYP three of the d atomic orbitals take one β electron and two are left vacant, leading to two of the three CO ligands to be treated equally. Overall, as summarized in Table 9, the sum of the donor-acceptor energies is greater for the C_2v structure with B3LYP. If we ignore the interaction with the “lone valence” on nickel, then it would seem that DFTB3 also favors the C_2v structure; however, the n_Ca → lv_Ni interaction strongly favors the D_3h over the C_2v geometry.

4. Concluding Remarks

In many chemical and biological applications, approximate QM methods still hold unique promise due to their computational efficiency. Therefore, in addition to enhancing accuracy of approximate QM methods via innovative parameterization³⁸ and correction²¹ (e.g., based on machine learning^88,89), there has been a resurgent interest in better understanding the impact of various approximations on computed energetics and properties.^9,30,31 As an effort along this line, here we compare DFTB and DFT in the framework of Natural Bonding Orbital (NBO) analysis, so as to understand to what degree DFTB captures the key electronic structure features of common chemical bonding scenarios. Several molecular systems have been chosen to represent fairly diverse bonding scenarios that include standard covalent bonds, hypervalent interactions, multi-center bonds, metal-ligand interactions and through-space donor-acceptor interactions. With such analysis, we hope to establish whether DFTB captures specific effects (e.g., pseudo Jahn-Teller distortion) with the proper physical reason, and to identify situations that current DFTB models (e.g., DFTB3/3OB) systematically exhibit significant errors and therefore require improvement.

Overall, the current results suggest that DFTB3/3OB provides physically sound descriptions for the different bonding scenarios analyzed here. This is reflected by the general agreement between DFTB3 and DFT (B3LYP with a large basis set) in terms of the nature of NBOs, as represented by the coefficients and hybridization types of natural hybrid orbitals on different atoms, the natural charges and natural bond orders. In favorable cases, consistent trends in dominant donor-acceptor interactions are also observed and provide support for DFTB3’s ability to capture effects such as pseudo Jahn-Teller distortion in [Cu(II)(CH₃)₂] and destabilization of scissile P-O bond in a series of “high energy” phosphates, “for the right reason”. These observations highlight that effective minimal basis approaches can be developed for describing rather complex bonding situation, as also supported by the development of polarizable atomic orbitals⁹⁰ and intrinsic atomic orbitals.⁵²

On the other hand, we do observe significant differences between DFTB3 and B3LYP for several NBO properties. For example, the bond order for the nominal P-O double bond in the pentavalent phosphate is substantially underestimated at the DFTB3 level. The degree of charge transfer from ligands to the metal ion is often overestimated at the DFTB3 level, leading to smaller positive charge on the metal ion; similar observation was made in a previous DFTB/MM study of solvated Na⁺/K⁺/Ca²⁺ ions in water.⁹¹ These errors arise due largely to the minimal basis nature of DFTB3, and it is of interest to see if they can be improved through including polarization corrections^22–24 that are self-consistent with the DFTB density determination. Furthermore, geminal interactions in the hypervalent phosphate species and metal-ligand compounds are grossly overestimated at the DFTB3 level, sometimes by an order of magnitude. These differences are correlated with discrepancies in energetics and structures between DFTB3 and DFT calculations. For the hypervalent phosphate species, the overestimated geminal interactions at the DFTB3 level imply a higher ionic character, which might be related to the larger error in energy observed for the pentavalent intermediate in phosphate hydrolysis reaction.³⁹ For [Ni(CO)₃]²⁺, the discrepancies in key orbital interactions might explain the distinct preference of DFTB3 and B3LYP to different structures.⁴⁵ Whether the observed differences in the orbital interactions are due mainly to errors in the pre-computed Hamiltonian matrix elements or to the neglect of three-center contributions in the current formulation of DFTB remain to be further analyzed. The finding that some metal-ligand NBOs at the DFTB3 level feature somewhat different hybridization types compared to B3LYP suggests that, in addition to improved description among d electrons,⁴⁵ care also needs to be exercised for the properties (e.g., eigenvalues⁴⁴) of atomic orbital basis for the DFTB parameterization of transition metal ions. Therefore, our study highlights that results from NBO analysis can be instructive in guiding the improvement of approximate QM methods such as DFTB to ensure a physically sound description of chemical bonding, which is required for accuracy and transferability.