Merge pull request #149 from AdityaSavara/relative-formation-energies

Relative formation energies

docs/source/tutorials/generating_an_input_file.rst

@@ -46,9 +46,9 @@ of the header and first few lines are provided below:
 ::
 
     surface_name    site_name   species_name  formation_energy    bulk_structure  frequencies reference
-    None            gas         CH4             0                   None            []          By Definition
-    None            gas         H2O             0                   None            []          By Definition
-    None            gas         H2              0                   None            [4401]      By Definition
+    None            gas         CH4             0                   None            []          Defined as part of reference state to have formation_energy of 0
+    None            gas         H2O             0                   None            []          Defined as part of reference state to have formation_energy of 0
+    None            gas         H2              0                   None            [4401]      Defined as part of reference state to have formation_energy of 0
     None            gas         CO              2.74                None            [2170]      Energy Environ. Sci., 3, 1311-1315 (2010)
     Pt              211         CO              1.113               fcc             []          J. Phys. Chem. C, 113 (24), 10548-10553 (2009)
 
@@ -134,13 +134,17 @@ formation\_energy
 ^^^^^^^^^^^^^^^^^
 
 This is the core of the input file since it defines the energetics of
-the system. It should be the "generalized formation energy" (see
+the system. It should be  a "relative free energy of formation" (see
 :ref:`Formation Energy Approach <formation_energy>`) of the "species\_name" on the "surface\_name" and
-"site\_name". These energies are usually very hard to come by, and must be
-computed by an electronic structure method such as DFT, or in some cases they
-can be measured experimentally. It is extremely important that all energies
-share a
-common thermodynamic reservoir for each atomic constituent (see :ref:`Formation
+"site\_name". A "relative free energy of formation" is different from a standard free energy of formation
+in that a "standard free energy of formation" uses pure elements as the reference states.
+In contrast, a "relative free energy of formation" can include heteroatomic molecules (such as CO)
+within the reference state, such that the relative energies of molecules in the reference state are set equal to 0.
+For species on surfaces, formation energies are often not available experimentally, and thus
+there is great value in computing them by an electronic structure method such as DFT. For gas phase species,
+experimental values are more accurate and can be used directly or as a correction for DFT values.
+To perform thermodynamic calculations, it is generally necessary that all energies
+share a common thermodynamic reference state for the species in the reaction mechanism. (see :ref:`Formation
 Energy Approach <formation_energy>`).
 
 frequencies
@@ -201,21 +205,45 @@ Formation Energy Approach
 ~~~~~~~~~~~~~~~~~~~~~~~~~
 
 One key point for generating input files is that the energies are
-computed as a "generalized formation energy" relative to a *common
-reference*:
+computed as a "relative free energies of formation" relative to a *common
+reference* state. This relative free energy of formation is provided by:
 
-:math:`E_i = U_i - \sum_j (n_j R_j)`
+:math:`G_i = H_i - T*S_i - \sum_j (n_j R_j)`
 
-where :math:`E_i` is the "generalized formation energy" of species :math:`i`, :math:`U_i` is the
-raw/DFT energy of species :math:`i`, :math:`nj` is the number of atomic species :math:`j` in :math:`i`,
-and :math:`\left|R_j\right|` is the reference energy of that atomic species. Mathematically
+:math:`G_i` is the "relative Gibbs free energy of formation" of species :math:`i` .
+:math:`H_i` is the  enthalpy of species :math:`i` (see further below).
+:math:`S_i` is the  absolute entropy of species :math:`i` .
+:math:`nj` is the number of atomic species :math:`j` in :math:`i`,
+and :math:`\left|R_j\right|` is the reference Gibbs free energy of that atomic species. Mathematically
 this looks a little confusing (especially with such crude notation) but
-in practice it is pretty easy. For example, say we want to find the
+in practice it is pretty easy.
+The general principle is similar to https://en.wikipedia.org/wiki/Born%E2%80%93Haber_cycle 
+and https://en.wikipedia.org/wiki/Hess%27s_law
+For this type of input file of CatMAP, we will
+be specifying the 0K electronic contribution to the relative formation energies,
+and these do not have a temperature depndence. How other terms become included will be explained later.
+
+In practice, today, the value for :math:`H_i` is generally approximated as being equal to the electronic energy.
+In this case, the equation becomes 
+
+:math:`G_i = U_i - T*S_i - \sum_j (n_j R_j)`
+
+Where :math:`U_i` is the raw/DFT energy of species :math:`i`, 
+
+Here, we will provide a simple example of how to provide the electronic energies information
+that CatMAP needs to calculate this contribution for the relative free energy of formation.
+We  will use the variable :math:`E_i` to emphasize that this term, based on `U_i`, represents
+the electronic energy contribution for :math:`G_i` . 
+
+In CatMAP, the the Zero Point Energy (ZPE) correction terms should not be included in `U_i`, 
+The ZPE correction terms for `U_i` will be added later by CatMAP based on
+the vibrational frequencies provided.  The :math:`-T*S_i` term will also be added later by CatMAP
+
+Let's look at an example. Say we want to find the
 energy of gas-phase CO relative to carbon (C) in methane (|CH4|), oxygen
 (O) in |H2O|, and hydrogen (H) in molecular hydrogen (|H2|). We first
 compute the reference energies (:math:`\left|R_j\right|`) for each atomic species:
 
-
 .. math::
 
     R_{\rm{H}} &= 0.5(U_{\rm{H}_2}) \\
@@ -225,12 +253,12 @@ compute the reference energies (:math:`\left|R_j\right|`) for each atomic specie
 (where again U is a "raw" energy from an ab-initio calculation, or a
 "regular" formation energy from NIST).
 
-Now we can compute the "generalized formation energy" of CO as:
+Now we can compute the electronic contribution to the relative formation energy of CO as:
 
 :math:`E_{\rm{CO}} = U_{\rm{CO}} - R_{\rm{C}} - R_{\rm{O}}`
 
 In the case where CO is adsorbed to a surface, say Pt(211), we can
-compute a "generalized" formation energy relative to the clean surface:
+compute the electronic contribution to the relative formation energy relative to the clean surface:
 
 :math:`E_{{\rm{CO}}*@{\rm{Pt}}(211)} = U_{{\rm{Pt}}(211)+{\rm{CO}}*} - U_{{\rm{Pt}}(211)} - R_{\rm{C}} - R_{\rm{O}}`
 
@@ -248,7 +276,7 @@ Then one could compute the barrier for :math:`{\rm{C-O}}` dissociation as:
 If this still doesn't make sense, try working through the
 `example <#example>`__ below.
 
-In principle the choice of reference species is arbitrary since the
+In principle the choice of reference states is arbitrary since the
 reference energies :math:`|R_j|` cancel out in any relative quantities. However, in
 many cases it is necessary to use some correction scheme for the
 gas-phase energies if they are poorly described by the level of theory
@@ -259,11 +287,53 @@ by DFT, so it would not make sense to use these to compute the reference
 energies :math:`|R_j|`.
 
 It is also worth re-iterating that the *same reference energies* :math:`|R_j|` *must
-be used for all energies in a given input file*. One can usually see
-which gas-phase species are used as references since their formation
-energies will be 0 by definition (see :ref:`overview <overview>`).
+be used for all energies in a given input file*. The best practice
+is to first set any pure element reactants in the system as having relative free energies of formation of 0
+and then to add in gases with one additional element as having relative free energies of formation of 0.
+Finally, other species (the remaining species with elements already used) will have relative free energies of formation defined based on these reference states
+as well as the math:`U_i` (or math:`U_i - T*S_i` ) computed values for these other / remaining species.
+When looking at an input file that has been created correctly, the gas-phase species that were used 
+as part of the reference state are easy to recognize since their relative formation
+energies will be set to 0. (see :ref:`overview <overview>`).
+
+Formation Energy Approach and Temperature Dependence
+^^^^^^^^^^^
+
+The below paragraphs are for informational purposes only.
+
+As noted above, a free energy of formation has several terms:
+
+:math:`G_i = H_i - T*S_i - \sum_j (n_j R_j)`
+
+In general, for formation energies, the Temperature and Pressure and any other quantities used for defining the 
+reference states should be reported in the manuscript (whether using a relative formation energy or a standard formation energy).
+The temperature dependence and entropy contributions are handled elsewhere in CatMAP,
+and are thus not included in the "formation_energy" field of the input file.
+
+The best practice and state of the art today is to include the entropy of formation, :math:`S_i` when calculating :math:`G_i`
+The value from :math:`T*S_i`  (and the values within :math:`\left|R_j\right|` ) will then include the values
+for the entropy contribtuions calculated at a given temperature based on the partition functions
+for vibrations, rotations, and the Sackur-Tetrode equation. The Sackur-Tetrode equation includes
+both the translational partition function contribution and a quantum configurational term.
+(The Sackur-Tetrode equation is often referred to as simply the "translational entropy", which can be misleading).
+
+Computational calculation of the entropy contribution to :math:`G_i` has a significant computational expense (because it
+requires more than single point calculations), and many studies do not require that level of accuracy 
+since for many systems changes in :math:`U_i` affect the chemistry and kinetics more than changes in :math:`S_i`
+
+When the term :math:`T*S_i` is approximated as sufficiently insignificant, the equation reduces to:
+
+:math:`G_i = U_i - \sum_j (n_j R_j)`
+
+However, CatMAP supports an inclusion of an approximate entropy, and use of this feature is encouraged.
+CatMAP uses existing codes to calculate the entropy contributions of 
+for vibrational stretching modes and gas phase translations from statistical mechanics,
+and thus we include stretching mode vibrational frequencies near the end of the example below.
+
+CatMAP's built-in thermo corrections will then use the frequencies to add in ZPE + enthalpy + entropy to complete `G_i`.
+In the future, even more accurate entropy terms may be included, but present day
+studies are adequately served by using stretching mode contributions for adsorbates.
 
-.. _example:
 
 Example
 -------