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Natural Bond Orbital analysis: Significance of stabilization energy determined by 2nd order perturbation
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Michael Wallis Adkins
Natural Bond Orbital analysis: Significance of stabilization energy determined by 2nd order perturbation
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.
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.
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:
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.
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.
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.
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:
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.
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:
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:
Responding to a couple of the specific questions at the end of the post:
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.
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:
From Valency and Bonding, p228, this favorable hyperconjugative delocalization is reflected in $E(2)$ values of greater magnitude:
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.
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.
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:
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:
Responding to a couple of the specific questions at the end of the post:
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.
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:
From Valency and Bonding, p228, this favorable hyperconjugative delocalization is reflected in $E(2)$ values of greater magnitude:
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.
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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