In addition to many small algorithmic improvements in former keyword options (including the NBCP keyword that was recently documented[10]), *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,[11] 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 BeLi_{2} 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,[12] 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[12] 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 *Q*_{A} that interact according to the classical law of Coulomb electrostatics,

- (1)

where *R*_{AB} 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 *R*_{AB} 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” (*E*_{NCE}) potential energy[13] for the species and geometry in question.

The NCE keyword provides the NPA-based evaluation of Coulomb electrostatic potential energy *E*_{NCE}. The geometrical variations of *E*_{NCE} provide simple estimates of electrostatic contributions to intra- or intermolecular interaction energy that can be compared with independently estimated values of steric[14] and donor-acceptor4 contributions. Such simplified estimates complement the more sophisticated dissection of interaction energy in NEDA-based energy decomposition analysis.[15]

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 *E*_{NCE}, one can quantitatively assess such L/NL coupling by comparing the actual NPA charge *Q*_{A} with that (*Q*_{A}^{(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 *E*_{NCE} 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 PtH_{4}^{2−}, 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 [Pt^{2+}(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 PtH_{2}(H^{−})_{2} description of three interacting molecular units:

As shown in the first output table, the formal *Q*_{H}^{(L)} = −1.0000 on each hydride ion is considerably reduced by the NL charge shift (0.6404) that leads to final NPA assigned charge, *Q*_{H} = −0.3596. The charge on the formally neutral PtH_{2} 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-H_{2}) 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 (*Q*_{A}^{(L)}, *Q*_{A}^{(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,

- (2a)

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,

- (2b)

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 τ_{NCU} that varies continuously from the weakest dispersion-type interactions to the powerful exchange-type forces of chemical bonding. Each range of τ_{NCU} values leads to characteristic “natural cluster units” (NCUs) that are intrinsic building blocks of that range. The NCU module determines the numerical τ_{NCU} transition values and associated NCU building blocks that characterize a chosen system of nuclei and electrons in specified nuclear geometry.

Mathematically, the τ_{NCU} interaction parameter can be expressed as a dimensionless “strength” of interaction between atoms A and B,

- (3a)

evaluated as the matrix norm of off-diagonal couplings between corresponding atomic blocks of the NAO density matrix, namely,

- (3b)

Here, “Tr” denotes the matrix trace and D_{A}, D_{B} are density matrix blocks for atoms A, B with corresponding electronic populations *n*_{A,}*n*_{B}. Alternatively, the τ_{NCU}(A,B) value can be related to the square-root of the NAO-based Wiberg bond index,[16] which represents a type of “bond order” between atoms A, B. Conceptually, the τ_{NCU} 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 τ_{NCU} transition values.

For a chosen interaction strength parameter τ_{NCU}, each pair of atoms A, B can be considered to form a connective “link” if, and only if,

- (4)

Each contiguously linked network of atoms thereby identifies a distinct NCU for the chosen τ_{NCU} value, analogs to the usual identification of NBO-linked networks as “molecular units.” In the appropriate range of τ_{NCU} values, the NCU assignments will agree with NBO-based molecular unit assignments. However, alternative NCU patterns are generally found in other τ_{NCU} ranges.

For a chosen input system of nuclei and electrons, the NCU module displays the list of distinct τ_{NCU} transition values and associated NCU clustering patterns for all possible interaction strengths, 0 ≤ τ_{NCU} ≤ 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 (Li_{9}) 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 Δ*E*_{atom}, Δ*G*^{(0)}_{atom}, (kcal/mol; per-atom) for various Li_{n} speciesSpecies | *E* | *G*^{(0)} | Δ*E*_{atom} | Δ*G*^{(0)}_{atom} |
---|

Li | −7.491333 | −7.504735 | 0.0 | 0.0 |

Li^{−} | −7.511841 | −7.524589 | 0.0 | 0.0 |

Li_{2} | −15.015837 | −15.033765 | −10.40 | −4.84 |

Li_{3}^{−} | −22.559306 | −22.581148 | −13.55 | −1.54 |

Li_{9} (*S*_{8}) | −67.706497 | −67.736528 | −19.84 | −1.06 |

Li_{9} (*D*_{4h}) | −67.686793 | – | −18.46 | – |

Li_{9} (*D*_{∞h}) | −67.582456 | – | −11.19 | – |

The NCU break-up sequence for each isomeric Li_{9} species can be depicted as shown below (leading in each case to fully dissociated Li monomers),

- (5a)

- (5b)

- (5c)

The principal NCU species (and associated τ_{NCU} values) of each sequence are depicted graphically in Figures 3a–3c. I/O-3 shows sample NCU output for the complex four-step Li_{9} (*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 Li_{9} first fragments to a central Li_{5} pentamer flanked by a terminal molecular Li_{2} unit on each end (at τ_{NCU} = 0.480), then to a second Li_{2} molecule on each end flanking a central Li monomer (at τ_{NCU} = 0.610), then to a single Li_{2} molecule on each end flanking five dissociated Li monomers (at τ_{NCU} = 0.680), and finally to complete dissociation to monomers (at τ_{NCU} = 0.830).

Only the highest-energy Li_{9} (*lin*) motif of Figure 3c exhibits “expected” diatomic Li_{2} molecules as NCU building blocks. As shown in Figure 3a, the equilibrium Li_{9} (*stg*) structure is formed from a square-pyramidal Li_{5} (*C*_{4v}) core NCU, with each triangular face capped by a weakly bound Li atom. The corresponding Li_{9} (*bcc*) sequence in (5b) may seem to involve standard diatomic Li_{2} molecules as NCU building blocks, but closer inspection (Fig. 3b) shows that each Li_{2} 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 BeLi_{2} 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 2s_{Be} 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 2s_{Be} spreads (delocalizes) into acceptor NBO 6, the in-phase combination of widely separated 2s_{Li} orbitals. Conversely, the back-donation from NBO 5 into the central 2p_{Be} acceptor NBO 7 acts in the opposite sense (but with weaker magnitude, presumably reflecting 2s_{Be} vs. 2p_{Be} kinetic energy difference). Higher levels of correlated theory would also indicate the unusually large “e-corr” corrections that are associated with such longbond delocalizations.[12] Consistent with other segments of NBO analysis, the PROP = KINETIC keyword indicates the highly unusual aspects of BeLi_{2} binding compared to conventional covalent or ionic bonding limits.