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Energy decomposition analysis

Advanced Review

Lili Zhao, Moritz von Hopffgarten, Diego M. Andrada, Gernot Frenking

Published Online: Nov 10 2017

DOI: 10.1002/wcms.1345

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The energy decomposition analysis (EDA) is a powerful method for a quantitative interpretation of chemical bonds in terms of three major components. The instantaneous interaction energy ΔEint between two fragments A and B in a molecule A–B is partitioned in three terms, namely (1) the quasiclassical electrostatic interaction ΔEelstat between the fragments; (2) the repulsive exchange (Pauli) interaction ΔEPauli between electrons of the two fragments having the same spin, and (3) the orbital (covalent) interaction ΔEorb which comes from the orbital relaxation and the orbital mixing between the fragments. The latter term can be decomposed into contributions of orbitals with different symmetry which makes it possible to distinguish between σ, π, and δ bonding. After a short introduction into the theoretical background of the EDA we present illustrative examples of main group and transition metal chemistry. The results show that the EDA terms can be interpreted in chemically meaningful way thus providing a bridge between quantum chemical calculations and heuristic bonding models of traditional chemistry. The extension to the EDA–Natural Orbitals for Chemical Valence (NOCV) method makes it possible to breakdown the orbital term ΔEorb into pairwise orbital contributions of the interacting fragments. The method provides a bridge between MO correlations diagrams and pairwise orbital interactions, which have been shown in the past to correlate with the structures and reactivities of molecules. There is a link between frontier orbital theory and orbital symmetry rules and the quantitative charge‐ and energy partitioning scheme that is provided by the EDA–NOCV terms. The strength of the pairwise orbital interactions can quantitatively be estimated and the associated change in the electronic structure can be visualized by plotting the deformation densities. This article is categorized under: Structure and Mechanism > Molecular Structures Electronic Structure Theory > Density Functional Theory Computer and Information Science > Visualization

First column: Plot of the deformation densities Δρ_1–4 with associated stabilization energies ΔE_1–4 of the four most important orbital interactions in B₂(NHC^Me)₂. The color code for the charge flow is red→light blue. Third and fourth column: Plot of the interacting donor and acceptor orbitals and calculated eigenvalues ε of (NHC^Me)₂ and (¹Σ_g⁺) B₂. Second column: Resulting MOs of the complex B₂(NHC^Me)₂. Figure taken from Ref 172.

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(a) Calculated geometry at BP86/TZVPP of the complex B₂(NHC^Me)₂ with the most important bond lengths in Å. Experimental values of the analogous complex B₂(NHC^Dip)₂ are shown in parentheses. Schematic MO diagram of B₂ in (b) the X³Σ_g⁻ ground state; (c) (3)¹Σ_g⁺ excited state. (d) Schematic view of out‐of‐phase (+,−) and in‐phase (+,+) σ donation of the ligand orbitals into the vacant 1σ_u und 2σ_g MOs of B₂. (e) Calculated excitation energy ΔE for X³Σ_g⁻→(3)¹Σ_g⁺ of B₂ and bond dissociation energy D_e of B₂(NHC^Me)₂. Figure taken from Ref 172.

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Plot of the deformation densities Δρ of the most important orbital interactions in (a) C(PPh₃)₂ and (b) C(NHC^Me)₂ between C in the excited¹D electronic reference state and the ligands L₂ in the frozen geometry of the molecule. Associated stabilization energies ΔE₁ − ΔE₃ are given in kcal/mol (see also Table ). The eigenvalues |υ₁| − |υ₃| indicate the size of the charge deformation. The color code of the charge flow is red→blue. Figure taken from Ref 161.

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Sketch of the bonding situation in carbones CL₂. (a) Lewis structure. (b) Donation of the lone pair electrons of L into vacant orbitals of carbon. The in‐phase(+,+) combination of the lone‐pair electrons donates charge into the sp.‐hybridized σ‐AO of C while the out‐of‐phase (+,−) combination of the lone‐pairs donates charge into the 2p_π║ AO of carbon.

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(a)–(d). Plot of deformation densities Δρ₁−Δρ₄ (isocontour 0.003 a.u.) of the pairwise orbital interactions in [B(CO)₂]⁻ together with the associated interaction energies ΔE_n and charge eigenvalues |υ_n| (in e). The charge flow is red→blue. The charge eigenvalues υ give the amount of donated/accepted electronic charge. (e) Schematic representation of the orbitals involved in the OC→B⁽⁻⁾←CO σ donation and the OC←B⁽⁻⁾→CO π‐backdonation. Only one component of the latter is shown. Figure taken from Ref 142.

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Plot of deformation densities Δρ of the pairwise orbital interactions and the associated interaction energies ΔE_orb between CO and BeO moieties in OCBeO and OCBeCO₃. The direction of the charge flow is red→blue. (a) σ donation OC→BeO. (b) and (c) π backdonation OC←BeO. (d) σ donation OC→BeCO₃. (e) and (f) π backdonation OC←BeCO₃. Figure taken from Ref 98.

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Optimized geometries at M06‐2X/def2‐TZVPP, MP2/cc‐pVTZ (in parentheses) and CCSD(T)/cc‐pVTZ [in brackets]. The bond lengths and angles are in [Å] and [°], respectively. Calculated bond dissociation energies (D_e) for the C─Be and O─Be bond are given in kcal/mol. Theoretical CCSD(T)/cc‐pVTZ (and experimental) stretching frequencies ν(C─O) and their shift Δν(C─O) wrt free CO (cm⁻¹). Figure taken from Ref 98.

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Schematic representation of the six sd⁵ hybrid orbitals. Figure taken from Ref 87.

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MO correlation diagram between Mo and (ZnH)₁₂ in [Mo(ZnH)₁₂]. Each fragment in its septet state. Figure taken from Ref 11.

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X‐ray structure of [Mo(ZnCp*)₃(ZnMe)₁₂]. Mo red, Zn green, C gray. Figure taken from Ref 87.

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MO correlation diagram between a d⁸ transition metal with the electronic configuration (a_1g)²(e_2g)⁶(e_1g)⁰ and a cyclic 12π aromatic sandwich ligand. Shapes of the orbitals have been taken from the Bz₂ ligand. The orbitals of (Cp₂)²⁻ look very similar. Figure taken from Ref 78.

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(a) Schematic representation of the tautomers ‘normal’ N‐heterocyclic carbene (nNHC), ‘abnormal’ N‐heterocyclic carbene (aNHC) and imidazol (IMID). (b) Schematic representation of the complexes of the above ligands with W(CO)₅.

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Schematic representations of interactions according to the Dewar–Chatt–Duncanson model. (a) and (b): σ donation and π‐back donation in an interaction between a transition metal (TM) and a carbonyl (CO); (c) and (d): σ donation and π backdonation in an interaction between a TM and a C─C double‐ or triple bond; (e) and (f): additional possible contributions of π donation and δ backdonation in the interaction between a TM and a C─C‐triple bond; (g) and (h): σ donation and π backdonation in singlet carbene complexes.

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(a) Equilibrium geometries of benzene and cyclobutadiene and distorted geometries which were used in the EDA calculations in Ref 65. (b) Schematic representation of the two fragments (colored in red and blue) which were used for the EDA calculations in Ref 65.

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Fragments which are used for the EDA (a) of benzene and (b) 1,3,5,7‐octatetraene. D, doublet; OS, open shell singlet.

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Breakdown of the actual donor orbital Ψ₂ and acceptor orbital Ψ₋₂ that are associated with the deformation density Δρ(2) in the transition state for H₂ addition to the amido digermyne 1Ge into the most important MOs of the fragments. The mixing coefficients that are given in parentheses show that the donor orbital Ψ₁ is mainly composed of the HOMO of 1Ge and the acceptor orbital Ψ₋₁ is mainly composed of the LUMO of H₂. The contributions of further three MOs of the fragments are much smaller.

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Breakdown of the actual donor orbital Ψ₁ and acceptor orbital Ψ₋₁ that are associated with the deformation density Δρ(1) in the transition state for H₂ addition to the amido digermyne 1Ge into the most important MOs of the fragments. The mixing coefficients that are given in parentheses show that the donor orbital Ψ₁ is mainly composed of the HOMO of H₂ and the acceptor orbital Ψ₋₁ is mainly composed of the LUMO of 1Ge. The contributions of further four MOs of the fragments are much smaller.

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Top: Calculated transition state for H₂ addition to the amido digermyne 1Ge. Middle row: Schematic representation of the orbital interaction between the σ(H₂) MO and the LUMO of 1Ge. Shape of the HOMO of H₂ and LUMO of 1Ge. Plot of the deformation density Δρ(1) which is associated with the strongest orbital interaction ΔE₁ between H₂ and 1Ge in the transition state. Bottom row: Schematic representation of the orbital interaction between the HOMO of 1Ge and the σ(H₂)* MO of H₂. Shape of the HOMO of HOMO of 1Ge and LUMO of H₂. Plot of the deformation density Δρ(2) which is associated with the second strongest orbital interaction ΔE₂ between H₂ and 1Ge in the transition state. The color code for the charge flow is red→blue. Figure taken from Ref 177.

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Energy decomposition analysis
Moritz von Hopffgarten,Gernot Frenking
Published Online: June 28 2011
DOI: 10.1002/wcms.71

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