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PREFACE: I am no expert on this topic. My questions at the bottom may be off base. I have some experience with symmetry-adapted perturbation theory (SAPT) when it comes to analyzing intermolecular interactions. The total interaction energy of a system can be quantified in four different energy components including exchange, induction, electrostatics, and dispersion.

Natural Bond Orbital (NBO) analyses translate the complex quantum-mechanical wavefunction into a more tangible Lewis-dot-like formalism (natural Lewis structure). The introduction of the following series of equations has been largely influenced by the presentation of Weinhold and Landis (2012).

Important Terms:

  1. Lewis-type NBOs - filled 'donor' orbitals (e.g. $\sigma_{AB}, \sigma_{CD}, \cdots$)
  2. Non-Lewis-type NBOs - vacant 'acceptor' orbitals (e.g. $\sigma_{AB}^{\ast}, \sigma_{CD}^{\ast}, \cdots$)

Lewis and non-Lewis NBOs mix:

Consider a wavefunction, $\Psi$, comprised of two components, a known natural Lewis structure wavefunction, $\Psi^{(L)}$, and a non-Lewis correction wavefunction, $\Psi^{(NL)}$, such that

$$\Psi = \Psi^{(L)} + \Psi^{(NL)}$$

The natural Lewis structure wavefunction is simply a fictitious representation of an exact physical system where resonance effects are non-existent.

Now imagine a corresponding Hamiltonian, $\hat{\mathcal{H}}$, which can also be treated as an additive scheme with corresponding components to each piece of the wavefunction:

$$\hat{\mathcal{H}} = \hat{\mathcal{H}}^{(L)} + \hat{\mathcal{H}}^{(NL)}$$

Given these two equations and the form of the Schrödinger Equation, $\hat{\mathcal{H}}\Psi=E\Psi$, there must also exist a similar construction of the energy of the system, $E$, such that

$$E = E^{(L)} + E^{(NL)}$$

These three equations provide the foundation for a "systematic 'perturbation theory' analysis", where $\hat{\mathcal{H}}^{(L)}$ is the unperturbed Hamiltonian with a corresponding (known) eigenfunction, $\Psi^{(L)}$, and energy eigenvalue $E^{(L)}$. Note that $\Psi^{(L)}$ is not only known, but it is related to an 'idealized single reference picture' in which each Lewis-type NBO has exact double occupancy for closed shell systems.

As with Hartree-Fock, the limit of a single-reference method can be reached by an infinite expansion of a basis ($\Omega)$ in the one-electron eigenvalue equation. For the natural Lewis structure, this can be written as

$$h^{(0)} \Omega_i^{(L)} = \varepsilon_i^{(L)}\Omega_i^{(L)} \qquad \mathrm{where~~} i = 1,2,\cdots,n$$

where $\varepsilon_i^{(L)}$ are orbital energies of Lewis-type NBOs. Because $h^{(0)}$ also contains the non-Lewis contributions such that

$$h^{(0)} \Omega_j^{(NL)} = \varepsilon_j^{(NL)}\Omega_j^{(NL)} \qquad \mathrm{where~~} j = n+1,\cdots$$

Because natural Lewis structures contain zero resonance, the donor (Lewis-type NBOs) and acceptor (non-Lewis-type NBOs) orbitals do not interact due to their orthogonal nature

$$\int \Omega_i^{(L)\ast}h^{(0)}\Omega_j^{(NL)}d\tau=0 \qquad \mathrm{for~all~~} i,j$$

However, a real-world $1e^-$ Hamiltonian operator, $F$ (i.e. Fock operator, Kohn-Sham operator, etc.), will have donor-acceptor interactions such that

$$F_{ij} = \int \Omega_i^{(L)\ast}F\Omega_j^{(NL)}d\tau\neq0$$

leading to mixing (delocalization) which connects the non-Lewis type NBOs with Lewis-type NBOs.

Second-Order Perturbative Treatment of Delocalizations

This mixing of the donor and acceptor orbitals can be treated with second-order perturbation theory. Even though $\Psi^{(L)}$ ignores interactions with non-Lewis acceptors, it contains more than 99% of the total electron density, $\rho_L$, offering a good starting point for an unperturbed wavefunction.

The mixing of donors and acceptors lead to an overall energy lowering ("stabilization"), a quantum mechanical phenomenon. Consider two interacting orbitals, as schematized below. One is doubly-occupied with a lone electron pair, $n$, and the other is an unoccupied antibonding orbital (here a $\pi^{\ast}$ orbital).

$\hskip1in$ Donor-acceptor orbital mixing

The overlap of the lone electron pair, $n$, with the vacant antibonding orbital, $\pi^{\ast}$ causes an change in energy of the lower occupied orbital, $x$. This change in energy between the non-mixing orbital with $n$ and the mixing orbital $x$ (denoted as $\Delta E$) is referred to as 'stabilization energy' of the electron delocalization. This energy (in kcal mol$^{-1}$) is determined via the following equation:

$$\Delta E_{ij}^{(2)} = \frac{-q_i \left| F_{ij} \right|^2}{\left(\varepsilon_j^{(NL)}-\varepsilon_i^{(L)}\right)}$$

where $\varepsilon_j^{(NL)}$ is the energy of the non-Lewis NBO (i.e. $\pi^{\ast}$), $\varepsilon_i^{(L)}$ is the energy of the orbital occupied by $n$, and $q_i$ is the occupancy of the orbital ($q=2$ in the above figure). The 'stabilization energy' $\Delta E_{ij}^{(2)}$ as determined by second-order perturbation treatments is commonly abbreviated as $E(2)$.

Recap:

So we have introduced (albeit very poorly) the concept of a separable wavefunction into natural Lewis and non-Lewis type formalisms. These two states mix, allowing for delocalization of electrons, a phenomenon which leads to lower energy states. This energy lowering is commonly called 'stabilization energy'.

Physical Significance?

Unfortunately, after such an introduction, I am still at a loss as to what all of this really means. As a theoretical chemist (I'm a chemist, theoretically speaking...), I am interested in the physical significance of this type of quantity [$E(2)$]. However, I am unsure how significant this type of value truly is. Do these delocalizations (energy stabilizations) lead to an overall lower energy of the system? Would a system with more of these delocalizations be expected to lie lower in energy than a comparable system without delocalizations? Can that type of correlation be made?

I guess I just do not see any usefulness in computing these $E(2)$ values for any reason because, while it may tell you that particular donor-acceptor interactions lead to this energy lowering, it doesn't offer much more than that. What am I missing here?

TL;DR - What can I use $\mathbf{E(2)}$ values for?

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    $\begingroup$ I think you are missing that the number behind $E(2)$ can tell you a lot about the energetic cost when breaking resonance, which in turn affects reactions in which the molecule analysed takes part. $\endgroup$ – tschoppi May 9 '14 at 16:14
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    $\begingroup$ This value gives you a hint on how "accurate" your Lewis structure is. The smaller it is, the better it will be described by Lewis. As being theoretically a chemist myself, this is the only thing I could cook up in an instant. Please always keep in mind, that NBO is an Analysis only and it is strongly dependent on what you want to find. ({default} NBO for $\ce{H2O}$ gives a sp- and p-lone-pair orbital...) But I like your question very much, if I would have time... $\endgroup$ – Martin - マーチン May 12 '14 at 12:03
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    $\begingroup$ @Martin I don't question its usefulness as a way to qualitatively interpret the wavefunction into something more tangible that most people will understand. I simply find its non-ability to be a quantitative and convergent approximation to be a complete turn off for me. I've seen it used in the literature plenty of times to quantitatively describe something but it cannot really be used for that. I'd rather rely on something that gives you can indication as to how 'correct' something is (i.e. convergent methods). *sigh $\endgroup$ – LordStryker May 12 '14 at 15:47
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    $\begingroup$ If you are looking for physical significance I suggest you consider whether it is measurable, I.e., does a Hermitian operator exist which will give you E (2). $\endgroup$ – Deathbreath May 17 '14 at 14:10
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    $\begingroup$ @Deathbreath The associated operator is indeed Hermetian. However, this is not really the 'problem'. The problem lies with approximating the wavefunction as an additive scheme involving 'Lewis' and 'Non-Lewis' components. This approximation is largely unphysical if I recall correctly. $\endgroup$ – LordStryker May 18 '14 at 3:01
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TL;DR: Lewis $\to$ Non-Lewis $\mathbf{E(2)}$ values have no direct physical significance, are intrinsically un-measurable, and serve only to quantify the extent to which the "real" wavefunction for a system deviates from the fictional idealized Lewis-structure wavefunction.

$E(2)$ values do, however, correlate with a variety of trends in chemical bonding and reactivity, and thus can be helpful in interpreting experimental or computational data or in highlighting potentially interesting lines of inquiry.


From a comment:

LordStryker: This approximation is largely unphysical if I recall correctly.

You do recall correctly. I'm reading Weinhold's Valency and Bonding (1st ed.) currently, and the first chapter is peppered with definitions of artificial Hamiltonians and Fock operators. For example, the first example he gives is of a Hamiltonian operator for independent (not field-averaged!) electrons:

$$ \hat h = \hat t\!_\mathrm e + \hat v_\mathrm{ne}, $$

where $\hat t\!_\mathrm e$ and $\hat v_\mathrm{ne}$ are the kinetic energy and nucleus-electron interaction operators, and the electron-electron interaction operator $\hat v_\mathrm{ee}$ is completely absent! If I'm reading the text correctly, the proper Hartree-Fock Hamiltonian is then defined in terms of $\hat h$ and a "perturbation" operator that is essentially just the missing electron-electron interaction term:

$$ \hat H^\mathrm{\small HF} = \hat h + \hat H^\mathrm{\small (pert)} \equiv \hat h + \hat v_\mathrm{ee} $$

I don't think he actually ever uses these definitions in the course of developing the NBO methodology, but it's instructive that this is the first example he chose in defining the perturbation approach that underlies it. To the best of my ability to determine:

The core of NBO analysis is founded upon selecting non-physical, but chemically intuitive, reference states, and quantifying the extent of the departure from these fictional references that is required in order to reach the "real" wavefunction of interest.

In other words, Martin is exactly right:

This value gives you a hint on how "accurate" your Lewis structure is. The smaller it is, the better it will be described by Lewis.


Responding to a couple of the specific questions at the end of the post:

Q: Do these delocalizations (energy stabilizations) lead to an overall lower energy of the system?

Absolutely, when compared to the fictitious idealized-Lewis reference. In his 2012 book, Discovering Chemistry with Natural Bond Orbitals (the one cited at the top of the original post), Weinhold illustrates quite explicitly how coercing the wavefunction not to exploit these delocalizations leads to appreciably more-positive energies. Again, though, the calculations carried out with these delocalizations prohibited are entirely unphysical, so it's not as though a system "knows it should delocalize as much as it can to get more stable"—any real system will already intrinsically exhibit all available delocalization that provides increased energetic stability.

Q: Would a system with more of these delocalizations be expected to lie lower in energy than a comparable system without delocalizations?

Yes. This is how NBO explains the spatial patterns of, e.g., hyperconjugation. In the Wikipedia article on the topic, for example, the role of hyperconjugation in establishing the energetic favorability of the staggered conformation of ethane is discussed. Per the below figure (public domain), the staggered conformation allows electrons in a given $\ce{C-H}$ $\sigma$-bonding orbital to delocalize into the $\sigma^*$-antibonding orbital of a parallel $\ce{C-H}$ bond on the vicinal carbon:

staggered ethane hyperconjugated orbitals

From Valency and Bonding, p228, this favorable hyperconjugative delocalization is reflected in $E(2)$ values of greater magnitude:

Second-order perturbative estimates [$E(2)$ values] indicate that each trans-like donor-acceptor [vicinal $\sigma \to \sigma^*$] interaction stabilizes the [staggered-conformation] molecule by $\pu{2.58kcal mol^-1}$, compared with only $\pu{0.89kcal mol^-1}$ for the cis-like interactions [in the eclipsed geometry]. The smaller gauche-like stabilizations ($\pu{0.20kcal mol^-1}$ at $60^\circ$ in the staggered conformer, $\pu{0.70kcal mol^-1}$ at $120^\circ$ in the eclipsed conformer) diminish the difference somewhat, but still preserve a significant hyperconjugative advantage for the staggered conformer.

So, while it's experimentally impossible to quantitatively "measure the energy" of an ethane molecule in which hyperconjugative delocalization is forbidden to occur, the $E(2)$ values provide support for the qualitative argument of hyperconjugation as a significant element of the preference of ethane for the staggered conformation. The relative (in)stability of various chemical features found across a wide range of systems can be examined by calculating judiciously selected $E(2)$ values.

Q: So say a model dimer system has an $E(2)$ of $\pu{-10kcal mol^-1}$ for a particular intermolecular interaction and has an overall electronic binding energy of $\pu{-20kcal mol^-1}$. Does that mean that half of the binding energy is due to this $E(2)$ value? Can that type of correlation even be made? (from this comment)

I agree with tschoppi: Based on my reading, yes, I think Weinhold would make exactly this kind of argument. I am ill-equipped to discuss in detail the validity of such an argument, however—though I think that there is at least qualitative, maybe semi-quantitative, value to it.

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