For Molecular Clusters: Many-Body Energy Decomposition Analysis =================================================================== The procedure of doing many-body energy decomposition analysis (MB-EDA) can be found in :doc:`eg-h2o6` and :doc:`eg-mc6h6`. Here we will give some theoretical arguments. TSO-EDA ------------------------ TSO-EDA is based on target state optimization self-consistent field method (`J. Chem. Theory Comput. 2023, 19, 1777 `_) and decomposes the total interaction energy into five terms, i.e. electrostatic, exchange, polarization, charge transfer, and dispersion energies. The sum of electrostattic and exchange energy is the Heitler-London term (`Phys. Chem. Chem. Phys. 2024, 26, 17549 `_): .. figure:: pics/eda-2.jpg Here: - Ectrostatic term: Represents the semiclassical Coulombic interaction of charged particles from different monomers; - Exchange term: Represents quantum effect due to the antisymmetric character of the electronic wave function and the satisfaction of the Pauli exclusion principle; - Polarization term: Represents the polarization of the electron density of one monomer by the presence of other monomer; - Charge transfer term: Represents the charge transfer between monomers; - Dispersion term: Represents the dispersion interaction between monomers. The BSSE effect is included in charge transfer term. All above terms can be found in Qbics output. Many-Body TSO-EDA ---------------------- This method is developed in `Phys. Chem. Chem. Phys. 2024, 26, 17549 `_ and is used to analyze the many-body effects in a molecular cluster. The total interaction energy is decomposed into 2-body, 3-body, and higher-order terms, like this: .. math:: \Delta E^{\text{int}} = \frac{1}{2!} \sum_{I_1 \neq I_2}^{N} \Delta E_{I_1 I_2}^{(2)} +\frac{1}{3!} \sum_{I_1 \neq I_2 \neq I_3}^{N} \Delta E_{I_1 I_2 I_3}^{(3)} + \cdots + \frac{1}{n!} \sum_{I_1 \neq \cdots \neq I_n}^{N} \Delta E_{I_1 \cdots I_n}^{(n)} + \cdots + \Delta E_{I_1 \cdots I_N}^{(N)} \equiv \sum_{n=2}^{N} \Delta E^{(n)} The terms higher than :math:`\Delta E^{(2)}` is the many-body interaction term. Usually the most important one is the three-body effect :math:`\Delta E^{(3)}`, the effects of which can be decomposed into three ones: - :math:`\Delta E^{(3)} < 0`: Indicate a **cooperative effect** of the monomers in a cluster. The many-body interaction is stabilizing the cluster. This is often seen in hydrogen bonding clusters, like water clusters. - :math:`\Delta E^{(3)} > 0`: Indicate an **anti-cooperative effect** of the monomers in a cluster. The many-body interaction is destabilizing the cluster. This is often seen in a cluster of charged species, like ionic liquid clusters. Also see below. - :math:`\Delta E^{(3)} \approx 0`: Indicate a **non-cooperative effect** of the monomers in a cluster. There is little many-body interaction in the cluster. This is often seen in a cluster of molecules without charges or hydrogen bonds. Each order can be decomposed into electrostatic, exchange, polarization, charge transfer, and dispersion terms: .. math:: \Delta E_X^{(n)} = \Delta E_X^{(n)\text{-el}} + \Delta E_X^{(n)\text{-ex/xc}} + \Delta E_X^{(n)\text{-pl}} + \Delta E_X^{(n)\text{-ct}} + \Delta E_X^{(n)\text{-disp}} Usually, electrostatic and exchange terms are highly additive, while polarization and charge transfer terms are non-additive. The dispersion term is always additive. Input Examples -------------------- MB-EDA needs to be calculated with Qbics, which can be obtained here: ``_. The details of compilation can be found here: ``_. For EDA keyword, please refer to: ``_. Example: EDA for GeH\ :sub:`3`\ F-NCH Complex by B3LYP-D3BJ/def2-SVP ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For the complex GeH\ :sub:`3`\ F-NCH, we can do EDA calculation by the following input: .. code-block:: bash :linenos: :caption: eda-1.inp mol Ge 0.00000000 0.00221863 -0.79935317 H 0.00000000 1.48645043 -0.40384625 H 1.28514604 -0.74161126 -0.40477816 H -1.28514603 -0.74161126 -0.40477816 F 0.00000000 0.00108752 -2.56116087 C 0.00000000 -0.00225138 3.35662076 H 0.00000000 -0.00220444 4.43604901 N 0.00000000 -0.00207825 2.20326200 end basis def2-svp end scf charge 0 spin2p1 1 type U # For EDA calculations, this must be added explicitly. end grimmedisp type bj end eda type tso # You can also change it to: gks frag 0 1 1-5 # Define GeH3F. frag 0 1 6-8 # Define HCN. end task eda b3lyp end The atom indices are shown below: .. figure:: pics/basinfo-1.jpg The results are: .. code-block:: bash :linenos: :caption: eda-tso.out WITH BSSE correction: Electrostatic interaction energy: -4.98 kcal/mol Exchange-correlation interaction energy: 4.22 kcal/mol Polarization interaction energy: -0.62 kcal/mol Charge transfer interaction energy: -1.31 kcal/mol Grimme's dispersion interaction: -1.58 kcal/mol ---------------------------------------------------------------- Total interaction energy: -4.27 kcal/mol We can see that this complex is stabilized by ``Electrostatic interaction eneregy``, which is compatible with the chemical intuition that it is stabilized by sigma-hole. Example: MB-EDA for Molecular Cluster (NH\ :sub:`4`:sup:`+`)\ :sub:`2`\ (H\ :sub:`2`\ SO\ :sub:`4`)(HSO\ :sub:`4`:sup:`-`)\ :sub:`2` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The title cluster is composed of two NH\ :sub:`4`:sup:`+` cations, one H\ :sub:`2`\ SO\ :sub:`4` molecules, and two HSO\ :sub:`4`:sup:`-` anions. This cluster is used in the study of atmopheric chemistry.We can do MB-EDA calculation by the following input: .. code-block:: bash :linenos: :caption: eda-2.inp basis def2-svp end scf charge 0 # Total charge. spin2p1 1 type U end grimmedisp type bj end eda type mb_tso mb_level 4 frag +1 1 1-5 # NH4+ frag +1 1 6-10 # NH4+ frag 0 1 11-17 # H2SO4 frag -1 1 18-23 # HSO4- frag -1 1 24-29 # HSO4- end mol N 0.13124700 -1.86033100 -1.49054300 H -0.68471400 -1.96085700 -0.84840100 H 0.16284500 -2.63375000 -2.14527600 H -0.00155300 -0.97157900 -1.98611500 H 1.02982000 -1.79400200 -0.97437700 N -1.89606400 2.02266900 1.95536400 H -2.33766600 1.07911300 1.78190600 H -1.20423600 1.92734800 2.69193100 H -1.40455300 2.34660500 1.08417600 H -2.60508400 2.69280200 2.23215700 S 3.40269500 -0.73966700 0.43845300 O 4.56636300 -1.26003200 1.03924300 O 2.66268100 -1.55477200 -0.49575900 O 2.42657400 -0.30120000 1.56959800 O 3.78755300 0.58843400 -0.27018200 H 2.99297400 1.01172300 -0.68798600 H 1.56137200 -0.00498400 1.17228000 S -3.05756300 -0.82805000 0.17173500 O -2.21824200 -1.98280100 -0.09354400 O -3.00471800 -0.39464500 1.56194000 O -2.90973700 0.26502300 -0.77053900 O -4.55472700 -1.30387800 -0.08712900 H -4.73898300 -2.05912100 0.48328300 H -1.51856700 0.72871100 -1.53329100 S 0.24159900 1.52238500 -0.65825900 O -0.59336700 0.90131200 -1.85962900 O 1.55183900 1.72430300 -1.22252400 O -0.45708500 2.72297400 -0.23978600 O 0.20716100 0.49864900 0.39894800 end task eda b3lyp end The atom indices are shown below, which is used to define the fragments ``frag``: .. figure:: pics/eda-1.jpg The results are: .. code-block:: bash :linenos: :caption: eda-2.out Table 5. Summary (kcal/mol). --------------------------------------------------------------------------------------------------------------------------------- Interactions delE_el delE_xc delE_pl delE_ct delE_bsse delE_disp delE_tot --------------------------------------------------------------------------------------------------------------------------------- SUM of 2-body -3.51853640E+02 1.04231044E+02 -5.50310217E+01 -9.70290703E+01 4.35696096E+01 -1.98079059E+01 -3.75920984E+02 SUM of 3-body 1.72107484E-09 1.45358519E+00 2.82254671E+01 2.81735373E+01 -1.24450163E+01 1.01531078E-02 4.54177264E+01 SUM of 4-body -4.14670076E-09 1.70212282E-02 -2.25462767E+00 -5.15894065E+00 1.89758583E+00 2.28017787E-05 -5.49893846E+00 Remain 7.51748885E-09 -8.58522126E-04 6.43113307E-02 4.30266615E-01 -1.22896862E-01 5.97720460E-07 3.70823167E-01 --------------------------------------------------------------------------------------------------------------------------------- SUM -3.51853640E+02 1.05700792E+02 -2.89958710E+01 -7.35842070E+01 3.28992822E+01 -1.97977294E+01 -3.35631373E+02 --------------------------------------------------------------------------------------------------------------------------------- We can see that the total interaction energy is -335.63 kcal/mol, which is decomposed into 2-body, 3-body, 4-body, and remaining terms. The 2-body term is the most important one (-375.92 kcal/mol), while the 3-body term is also significant, but **anti-cooperative** (destablizing the complex) (+45.42 kcal/mol). The 4-body term is small (-5.50 kcal/mol, slightly cooperative). The remaining term (sum of 5- and 6-body) is very small (+0.37 kcal/mol) and can be ignored. We can also see that the electrostatic and exchange energy are highly **additive**, while the polarization and charge-transfer energy are **non-additive**. For different kinds of clusters, the 3-body effects (many-body interactions) can be quite different, see `Phys. Chem. Chem. Phys. 2024, 26, 17549 `_ for more information.