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Natural bond orbital (NBO) analysis[1, 2] has been implemented in a succession of program versions [most recently NBO 5.0 and derivative revisions1 (“NBO5”)] that have achieved widespread usage in the chemical community. However, as NBO applications spread to additional electronic structure system (ESS) host programs and a broader range of bonding phenomena, certain limitations of the NBO5-level program structure, output conventions, and search algorithms have become apparent. The newly released NBO 6.0 (“NBO6”) removes many such limitations and incorporates new analysis options that significantly extend the range of chemical applicability.
A primary limitation of earlier versions stems from the requirement that ESS and NBO source code be compiled and linked to form an integrated binary executable program for interactive options (e.g., $DEL-deletions, NEDA, NCS, NJC, etc.). Such linked source-code integration presents legal, commercial, and technical obstacles that limit interactive ESS/NBO implementations, particularly when ESS source code distribution is restricted. In such cases, only unlinked stand-alone GenNBO versions, such as GenNBO 5.0W for PC-Windows users, were available to perform noninteractive options using archive (FILE47) input from a detached host ESS.
A second limitation arises from certain implicit assumptions of pre-NBO6 output labeling that (although generally consistent for near-equilibrium ground state species) become increasingly problematic for the exotic bonding motifs of strongly delocalized or excited-state species. The potential inconsistencies arise from details of NBO search algorithms that can lead to suboptimal natural Lewis structure (NLS) assignment and related irregularities in $CHOOSE, natural resonance theory (NRT), and other options.
As described below, removal of these limitations in NBO 6.0 opens the way to many new ESS/NBO6 partnerships as well as a variety of new analysis options that significantly enhance the robustness and reliability of NBO-based description. Despite the many underlying changes, accomplished NBO users should notice few changes in the way NBO analysis is performed or presented in user output, thereby allowing easy migration to powerful new ESS/NBO6 capabilities as NBO6-compatible ESS versions become available.
Link-Free ESS/NBO6 Interactivity
Unlike previous NBO versions, NBO 6.0 is designed to work in a tandem host-client “binary pair” relationship with a host ESS binary program, using a special message-passing protocol that requires no compilation or linking. In principle, an NBO6-compatible ESS binary will automatically launch the NBO6 binary to perform complex interactive tasks that formerly required an integrated ESS/NBO binary program (compiled and linked from component source code). The underlying structural changes are all but transparent to the end-user. However, the new link-free programming model significantly simplifies creation of ESS/NBO6 capabilities through simple “load and go” installation, averting the many pitfalls of acquiring, compiling, and linking source programs from different providers.
ESS/NBO6 message-passing works via a socket-pair communication protocol that passes all data as character strings through a small C-code interface (included with NBO 6.0 binary or source distributions). On the host side, the NBO6-compatible ESS distribution includes the necessary message-passing capability as appropriately modified from template code provided by the NBO development team. On the client side, the NBO 6.0 distribution includes the necessary script/batch file to set the NBOEXE environmental variable that enables host-ESS and client-NBO6 binaries to “find” one another and cooperatively perform interactive tasks in all common Windows, Mac, Linux, and Unix environments. For parallelized cluster configurations, interprogram communications are managed by the ESS master process while slave processes await the completion of NBO6 tasks.
From the user's viewpoint, for noninteractive implementations the stand-alone GenNBO6 program of NBO 6.0 appears to function exactly like its predecessors. As before, GenNBO6 reads the FILE47 archive file (produced by many ESS programs) and performs the tasks as requested in the $NBO keylist, without ever needing to recalculate the wavefunction. Structurally, however, GenNBO6 is again a tandem “GEN/NBO6” pairing composed of host gennbo.exe and client nbo6.exe programs that cooperate to process the $NBO keylist requests of the input FILE47 file (directly analogous to how the ESS/NBO6 tandem pair processes the ESS input file). Although NBO 6.0 can only work interactively with more recent “NBO6-compatible” ESS host programs, stand-alone GenNBO6 offers full backward compatibility with previous GenNBO applications.
Improved Labeling Conventions and Search Algorithms for Excited-State and Other Exotic Species
Previous NBO versions relied on a simple “starring” convention to distinguish low-occupancy non-Lewis (NL) NBOs (e.g., antibond “BD*” or Rydberg-type “RY*”) from high-occupancy Lewis-type (L) NBOs (e.g., core “CR”, lone pair “LP”, or bond “BD”) in program output. However, this labeling convention potentially conflicts with established chemical usage in which a star (*) denotes out-of-phase (“antibonding”) symmetry with respect to an inversion center or reflection plane. In such cases, previous NBO versions recognized the anomaly of a “high-occupancy antibonding” NBO by issuing a warning message and changing the label from “BD” to “BD*”. But this in turn leads to errors in assessing the NL-error (“rho* value”) of the associated Lewis structure, and thereby to grossly suboptimal NLS assignments for such species. Such relabeling difficulties of the default NBO search were compounded by inconsistency with the somewhat simplified procedures used in alternative $CHOOSE or NRT searches. Although uncommon in familiar near-equilibrium ground-state species, such conflicts become increasingly vexing in excited-state and delocalized multicenter bonding motifs.
NBO 6.0 abandons any such presumed association of “*” with “NL.” Instead, L-type vs. NL-type NBOs are clearly delineated in separate output sections, and strict accounting of L/NL character is maintained internally. The order of NBOs within each section is also altered to more closely reflect the order in which NBOs are determined (1c < 2c < … for L-type; 2c < 3c< … < 1c for NL-type), according to multicenter character and occupancy. The former 1c “LP*” label is replaced by “LV” (for “lone valency” or “lone vacancy”) for the distinctive unfilled valence-type nonbonding NBO of hypovalent borane-like species. For multicenter NBOs, asterisks are retained only to designate out-of-phase symmetry [as in “BD*” (2c antibonds) or “3C*” (strongest 3c antibonding character)], regardless of occupancy. In common cases, the most noticeable difference is that extravalent Rydberg-type NBOs are now given “RY” rather than “RY*” labels.
The NBO 6.0 labeling changes and improved consistency between default NBO, $CHOOSE, and NRT structure searches are accompanied by a number of subtle algorithmic improvements. These primarily affect the basic multicenter sequential NLS search procedure to remove former “loopholes” and maintain stricter control of orthonormality (a signature of NAO/NBO-based analysis methods). Two such algorithmic changes are particularly noteworthy:
The NLS search formerly proceeded through a coarse grid of pair-occupancy thresholds (0.10e-decrements from starting 1.90e threshold) that sometimes allowed a superior NLS to be “overlooked.” Each search threshold also required additional cycles (permutational reorderings) to guard against possible dependencies on atom numbering in the coarse-grid search. NBO 6.0 now carefully examines the density matrix for specific occupancy thresholds at which degenerate NLS-switchover could possibly occur (in effect, using an “infinitely fine” grid search that is independent of atom numbering) and returns the best NLS at each such threshold. As usual, the bonding pattern of lowest overall rho(NL) value is returned as the final NLS, whose optimal character is now guaranteed at the theorem level.
The sequential multicenter 1c, 2c, … NBO search formerly involved a simplified “depletion” procedure to remove 1c contributions from the density matrix prior to the 2c search, with slight overlap errors subsequently removed by occupancy-weighted symmetric orthogonalization. In NBO 6.0, however, this approximate procedure is replaced by strict orthogonal projection (projective annihilation) to remove possible m-center overlap-mixing artifacts with fewer-center NBOs. This has barely perceptible numerical effect on NBO descriptors (typically in the 4–5th decimal place), but provides an additional tier of protection against insidious “illusions of overlap” that affect other analysis methods.
In addition, the multicenter NBO search is now automatically extended to m = 3, and the former 3CBOND and 3CHB keywords become superfluous. Evidence (if present) of 3c/2e bridge-bonding2 or 3c/4e hyperbonding3 is automatically reported in default NBO 6.0 output.
New Analysis Options and Illustrative Applications
In addition to many small algorithmic improvements in former keyword options (including the NBCP keyword that was recently documented), NBO 6.0 offers a variety of new analysis options that reflect the broadened perspective and extensible interactivity made possible by its deep structural changes. In this section, we briefly illustrate the “look and feel” of default NBO6 output as well as applications of newer keyword options. The examples are drawn primarily from the domain of weak metallic interactions that challenge localized description. In each case, we make consistent use of B3LYP/6-311++G** level of theory, with sample I/O truncated to the principal NBOs of chemical interest.
Default NBO6 search output
As a simple example of default NBO6 output, we consider the closed-shell linear triatomic BeLi2 species with numbering and geometry as shown below:
Results of the default NBO search are displayed in I/O-1, truncated to show only two of the 59 RY-type NBOs and omit the NAO expansion coefficients (essentially, 1.0000 for the 1s NAO at each center) for CR-type NBOs 1–3. The Lewis-type output displays the 2s(Be) nonbonding (LP) NBO 4 and the remarkable “long antibond” (BD*) NBO 5, both strongly delocalized (occupancies 1.7105, 1.7043, respectively). As described elsewhere, the “Li∧Li′” longbond is of paradoxical antibonding ( *LiLi′) phase pattern, with participating 2s(Li), 2s(Li′) NAOs exhibiting no appreciable direct “bonding overlap” (see Fig. 1). I/O-1 also displays the leading few NL NBOs (neglecting all but two of the 59 RY-type orbitals), particularly the valence-type in-phase (BD) longbond, NBO 6, and extravalent (RY) 2p(Be), NBO 7, both manifesting significant occupancy (0.2796, 0.2955). The paradoxical phase inversions, reciprocal delocalizations, and other exotic features of 3c/4e metallic longbonding phenomena raised significant issues in NBO5-level output, but are now handled smoothly by NBO 6.0.
Natural Coulombic energy analysis
A crude picture of classical-type electrostatic interactions can be based on the concept of effective atomic point charges QA that interact according to the classical law of Coulomb electrostatics,
where RAB is the interatomic distance between nuclei A, B. The classical Coulomb formula (1) is highly questionable in the short-range domain of quantal exchange interactions, but may nevertheless provide a useful estimate of electrostatic effects when RAB separations sufficiently exceed the sum of atomic van der Waals radii. When eq. (1) is evaluated with natural atomic charges, the formula defines what may be called the “Natural Coulomb Electrostatics” (ENCE) potential energy for the species and geometry in question.
The NCE keyword provides the NPA-based evaluation of Coulomb electrostatic potential energy ENCE. The geometrical variations of ENCE provide simple estimates of electrostatic contributions to intra- or intermolecular interaction energy that can be compared with independently estimated values of steric and donor-acceptor4 contributions. Such simplified estimates complement the more sophisticated dissection of interaction energy in NEDA-based energy decomposition analysis.
An issue in all such energy decompositions is the coupling between localized L and delocalized NL contributions to perceived classical-like electrostatic or steric components. For ENCE, one can quantitatively assess such L/NL coupling by comparing the actual NPA charge QA with that (QA(L)) for the ideally localized NLS in which each L-type NBO has exact double-occupancy. This allows one to separate the L-type charge distribution of an idealized Lewis structural model from the NL-type “charge transfer” delocalizations of nonclassical origin. The NCE analysis module displays such L/NL contributions to ENCE for each atom pair, providing a direct estimate of resonance-type CT corrections to classical-like Coulomb electrostatics. For open shells, an additional spin-charge NCE table shows the distinct α-NCE and β-NCE contributions of each spin set, again emphasizing the limitations of naive classical interpretations.
NCE analysis is requested by simply including the “NCE” keyword in the $NBO keylist. As a simple example, we consider square-planar PtH42−, a transition metal complex ion whose MO vs. NBO characterization has been previously described,5 but whose properties might be alternatively interpreted in terms of a purely ionic [Pt2+(H−)4] or “ion-dipole” (crossed H−···Pt-H interactions) model. I/O-2 shows NCE output for this species, with default NLS corresponding to the PtH2(H−)2 description of three interacting molecular units:
As shown in the first output table, the formal QH(L) = −1.0000 on each hydride ion is considerably reduced by the NL charge shift (0.6404) that leads to final NPA assigned charge, QH = −0.3596. The charge on the formally neutral PtH2 unit is similarly shifted to −1.2808 by powerful resonance delocalizations.
The second output table shows the corresponding potential energy values (relative to infinitely separated atoms and ions) for the idealized vs. actual charge assignments. These exhibit the profound effects of resonance charge shifts on perceived assessments of electrostatic effects. For example, the “ion-dipole” (H− ···Pt-H2) potential energy between units 1 and 2 might be variably assigned values ranging from −0.04175 (26.20 kcal/mol attractive) to 0.11186 (70.19 kcal/mol repulsive) depending on how atomic charges are chosen (QA(L), QA(L+NL), or something in-between). Understanding deeper aspects of L/NL contributions to charge distributions can warn against the superficiality of fitting actual binding potentials to quasiclassical formulas such as eq. (1).
Natural cluster unit analysis
For chemical theorists, principal attention often focuses on the strong quantum covalency forces leading to molecule formation,
However, from a broader biochemical or materials perspective, primary interest shifts to weaker forces of aggregation leading to supramolecular clustering and eventual condensation into macroscopic phases, namely,
The successive supramolecular clustering processes (2b) involve a range of H-bonding and other (so-called) “noncovalent forces” that are active in the domain of near-ambient terrestrial conditions, including weak dispersion forces of London and Casimir–Polder type.6 These forces lead finally to aggregated liquid and solid phases of the macroscopic world for even the most weakly interacting subspecies. Deeper understanding of the complex processes in (2b) rests on identifying the intrinsic cluster “units” or “building blocks” that underlie each aggregation step, as well as the nature of intercluster forces between such units.
To address broader aspects of sequence (2a,b), we can envision a formal interaction parameter τNCU that varies continuously from the weakest dispersion-type interactions to the powerful exchange-type forces of chemical bonding. Each range of τNCU values leads to characteristic “natural cluster units” (NCUs) that are intrinsic building blocks of that range. The NCU module determines the numerical τNCU transition values and associated NCU building blocks that characterize a chosen system of nuclei and electrons in specified nuclear geometry.
Mathematically, the τNCU interaction parameter can be expressed as a dimensionless “strength” of interaction between atoms A and B,
evaluated as the matrix norm of off-diagonal couplings between corresponding atomic blocks of the NAO density matrix, namely,
Here, “Tr” denotes the matrix trace and DA, DB are density matrix blocks for atoms A, B with corresponding electronic populations nA,nB. Alternatively, the τNCU(A,B) value can be related to the square-root of the NAO-based Wiberg bond index, which represents a type of “bond order” between atoms A, B. Conceptually, the τNCU coupling-strength parameter can also be pictured as an “effective temperature” that leads to complete atomic dissociation at sufficiently high values, or complete condensation at sufficiently low values, but with distinctive alternative NCU cluster patterns at characteristic τNCU transition values.
For a chosen interaction strength parameter τNCU, each pair of atoms A, B can be considered to form a connective “link” if, and only if,
Each contiguously linked network of atoms thereby identifies a distinct NCU for the chosen τNCU value, analogs to the usual identification of NBO-linked networks as “molecular units.” In the appropriate range of τNCU values, the NCU assignments will agree with NBO-based molecular unit assignments. However, alternative NCU patterns are generally found in other τNCU ranges.
For a chosen input system of nuclei and electrons, the NCU module displays the list of distinct τNCU transition values and associated NCU clustering patterns for all possible interaction strengths, 0 ≤ τNCU ≤ 1. As shown in eq. (3b), NCU analysis depends only on NAO-based definitions of underlying constituent atoms, with no other dependence on NBO/NRT-based description of intra- or intermolecular interactions. Nevertheless, NBO analysis of individual NCU species and their mutual interactions should serve to further illuminate the binding energetics of the composite system.
As a simple illustration of NCU analysis, let us consider a system of nine lithium atoms (Li9) in various isomeric arrangements as pictured in Figures 2a–2c. The figure depicts three stationary points of the B3LYP/6-311++G** potential energy surface, (a) the equilibrium staggered (“stg”) isomer, (b) a higher-energy “bcc” structure (transition species: three imaginary frequencies) that resembles the unit cell of metallic lithium, and (c) a still higher-energy linear (“lin”) structure (transition species: two degenerate imaginary frequencies), with other symmetry, structural, and numbering details as shown in the figure caption. Table 1 summarizes energies and standard-state free energies of these species with respect to various atomic, ionic, and diatomic dissociation products.
Table 1. Total energies E (with standard-state free energies G(0) for stable equilibrium species; a.u.) and corresponding atomization energies ΔEatom, ΔG(0)atom, (kcal/mol; per-atom) for various Lin species
The NCU break-up sequence for each isomeric Li9 species can be depicted as shown below (leading in each case to fully dissociated Li monomers),
The principal NCU species (and associated τNCU values) of each sequence are depicted graphically in Figures 3a–3c. I/O-3 shows sample NCU output for the complex four-step Li9 (lin) break-up sequence in (5c). In this case [reading the sequence (5c) from left to right, or Fig. 3c from bottom to top], the composite Li9 first fragments to a central Li5 pentamer flanked by a terminal molecular Li2 unit on each end (at τNCU = 0.480), then to a second Li2 molecule on each end flanking a central Li monomer (at τNCU = 0.610), then to a single Li2 molecule on each end flanking five dissociated Li monomers (at τNCU = 0.680), and finally to complete dissociation to monomers (at τNCU = 0.830).
Only the highest-energy Li9 (lin) motif of Figure 3c exhibits “expected” diatomic Li2 molecules as NCU building blocks. As shown in Figure 3a, the equilibrium Li9 (stg) structure is formed from a square-pyramidal Li5 (C4v) core NCU, with each triangular face capped by a weakly bound Li atom. The corresponding Li9 (bcc) sequence in (5b) may seem to involve standard diatomic Li2 molecules as NCU building blocks, but closer inspection (Fig. 3b) shows that each Li2 NCU is actually of Li∧Li′ long-bonded character, connecting diagonally opposite corners of the cluster cube “across” the central Li monomer.
These simple examples serve to illustrate the richness of NCU structural bonding motifs that may be expected in still larger clusters or extended crystalline systems, extending the boundaries of “bonding interactions” beyond the limit (2a) of molecule formation.
General 1e properties (PROP) analysis
The link-free connectivity of NBO 6.0 to a greater variety of host ESS programs implies that NBO analysis can now be accessed for a broader range of specialized properties that are provided by specific ESS hosts. The PROP module of NBO 6.0 makes such analyses accessible for any possible 1-electron property that can be evaluated from the electron density or 1-electron density operator. Such properties include kinetic energy (KINETIC), nuclear-electron potential energy (V), dipole moment (DIPOLE), Fock/Kohn–Sham 1e total energy (F), overlap (S), and electron density (DM), which are commonly included in the FILE47 archive file for input to a stand-alone GenNBO program. However, through the interactive ESS/NBO6 interface, a virtually unlimited set of additional 1e properties becomes accessible to PROP keyword analysis.
As indicated by examples given above, the PROP keyword must include an identifying label (id_label) of the chosen property PROP=id_label (e.g., PROP = KINETIC). The proper “id_label” is matched to .47 file input (e.g., $KINETIC keylist) or provided by the host ESS program through ESS/NBO6 message-passing protocol. Consult the ESS program documentation for available properties and labels from each host.
For any chosen property, the format of PROP output resembles that of DIPOLE or NJC output. The overall expectation value of the property is expressed as a sum of Lewis and NL contributions, with subsidiary correlation correction for self-consistent field (post-SCF) methods.
The PROP keyword can be illustrated for the kinetic energy operator of the BeLi2 species discussed above, whose PROP output segment is shown in I/O-4.
As shown in I/O-4, the total kinetic energy (14.7415 a.u.) is significantly reduced (ca. 27.42 kcal/mol) by NL-type resonance delocalization effects. As expected, kinetic energy is dominated by well-localized core NBOs 1-3, which exhibit only small NL-corrections [ca. 1.2 kcal/mol (0.01%)]. Instead, surprisingly large (ca. 30%!) NL-type shifts are seen in both valence-type NBOs 4, 5, leading in the former case (the 2sBe lone pair) to 75.30 kcal/mol reduction and in the latter case (the Li∧Li′ BD* “long antibond”) to 46.62 increase in kinetic energy, a significant net contribution to overall resonance stabilization in this species.
The dramatic resonance-type kinetic energy reduction in NBO 4 is evidently associated with effective “box enlargement” as the central donor 2sBe spreads (delocalizes) into acceptor NBO 6, the in-phase combination of widely separated 2sLi orbitals. Conversely, the back-donation from NBO 5 into the central 2pBe acceptor NBO 7 acts in the opposite sense (but with weaker magnitude, presumably reflecting 2sBe vs. 2pBe kinetic energy difference). Higher levels of correlated theory would also indicate the unusually large “e-corr” corrections that are associated with such longbond delocalizations. Consistent with other segments of NBO analysis, the PROP = KINETIC keyword indicates the highly unusual aspects of BeLi2 binding compared to conventional covalent or ionic bonding limits.
Other new keywords and features
Brief descriptions (without illustrative applications) are also provided below for various new keyword options, extensions of older options, or other improved features of NBO 6.0:
MATRIX keyword: Powerful extensions of older operator-matrix and transformation-matrix output are now, respectively, achieved by general commands of the form MATRIX = <OP/BAS> or MATRIX = <BAS1/BAS2>, where the bracket-list entries specify any operator “OP” and basis set label “BAS” recognized by the ESS/NBO6 combo. For example, “MATRIX = <F/PNLMO>” provides the Fock matrix (F) in the PNLMO basis, an “unusual” combination not allowed in older keyword syntax.
XMOL keyword: Molecular geometry can now be output for graphical display by XMOL and other standard viewers.
Local NRT options: NRT analysis algorithms have been significantly improved, including new “local” options of the form “NRT=<atom-1, atom-2, … atom-n>.” Such commands restrict the NRT search to subgroups (resonance units) of atoms, allowing “divide and conquer” strategies for systems with too many allyl- or Kekulé-type resonating groups for conventional treatment.
NRTCML keyword: This keyword converts NRT output to the popular CML (chemical markup language) format for applications such as ChemDraw or MarvinView for producing journal-quality images of resonance structures.
NRTMOL keyword: This keyword produces a valid MDL (Molfile) format file for more general cheminformatics applications. The Molfile can be input to OpenBabel for conversion to a wide variety of alternative formats including SMILES structural specification.
Full natural orbital (NO) support: Supported natural-type orbital basis sets now include fully delocalized (n-center Löwdin-type) natural orbitals (NO) for correlated levels of theory. The delocalized NOs supplement localized NAOs, NHOs, NBOs semilocalized NLMOs, delocalized CMOs, and preorthogonal counterparts (PNAOs, PNHOs, …) in checkpointing and MATRIX keyword options.
Natural Bond Critical Point (NBCP) extensions: Algorithmic improvements and new guided options (“NBCP=WBI”) now make NBCP searches more robust and efficient for larger molecules.
Higher angular symmetries: Full support for orbital basis functions of “pure” h-type (l = 5) and i-type (l = 6) angular symmetry is now included.
Natural chemical shielding (NCS) improvements: A more flexible input format has been adopted, and persistent bugs in the G09/NBO5 “MO” option have been corrected to allow NICS-type jobs to be run successfully. The FILE47 archive file for any Gaussian NMR job now automatically includes the necessary field-dependent derivatives as keylists that allow all NCS options to be performed in stand-alone GenNBO mode.
Convenient binary distributions: Although source distributions are still available, the NBO 6.0 program is primarily intended for distribution as precompiled binary .exe files, ready to run in a broad variety of popular OS/hardware environments.
Other small algorithmic tweaks have improved program reliability and consistency throughout the code, as well as removing archaic programming constructs that are inconsistently supported in modern compilers. NBO 6.0 is therefore considered to be the secure new foundation for all planned future extensions of NBO-based methodology.
The authors thank Steve Baker (Indiana State Univ.) for assistance in socket-pair coding, website forum construction, and other aspects of NBO6 site management. The authors also thank ESS program developers for cooperative assistance in building individual ESS/NBO6 interfaces, including Todd Martinez (TeraChem), Mike Frisch (Gaussian), Mike Schmidt (GAMESS), Hans-Joachim Werner (Molpro), and others to follow.