# Calculate reaction heat from quantum mechanics

1. May 2, 2017

### zhouhao

1. The problem statement, all variables and given/known data
I have some doubts about the method constructing chemical reaction process of quantum mechanics in the referencehttp://www.southampton.ac.uk/assets/centresresearch/documents/compchem/DFT_L2.pdf, for the example of $H_2O$ molecular dissociation to $H^+$ and $OH^-$ ions, time-independent Schrodinger equation was used to construct the reaction process:
${\hat{H}}_{H_2O}{\psi}_{H_2O}(\vec r_e^i)=E_{H_2O}{\psi}_{H_2O}(\vec r_e^i)$
${\hat{H}}_{H^++OH^-}{\psi}_{H^++OH^-}(\vec r_e^i)=E_{H^++OH^-}{\psi}_{H^++OH^-}(\vec r_e^i)$
Reaction heat was defined as $\Delta{H}=E_{H^++OH^-}-E_{H_2O}$
This method fixed the position of nuclears.
If I want to derive the same reaction heat as the method, but through the wavefunction of a system consist of two hydron,one oxygen and ten electrons,the initial and end wavefunction should rely on a hypothesis,which is called A:

${\psi}_{H,O,e}^1(\vec r_e^i,\vec r_H^1{(H,O,e),\vec r_H^2{(H,O,e))},\vec r_O{(H,O,e)}={\psi}_{H_2O}(\vec r_e^i)\delta^{\frac{3}{2}}({\vec r_H^1{(H,O,e)}-\vec r_H^1{(H_2O)}})\delta^{\frac{3}{2}}({\vec r_H^2{(H,O,e)}}-\vec r_H^2{(H_2O)}})\delta^{\frac{3}{2}}({\vec r_O{(H,O,e)}-\vec r_O{(H_2O)}})$

${\psi}_{H,O,e}^2(\vec r_e^i,\vec r_H^1{(H,O,e),\vec r_H^2{(H,O,e)},\vec r_O{(H,O,e))}={\psi}_{H^++OH^-}(\vec r_e^i)\delta^{\frac{3}{2}}({\vec r_H^1{(H,O,e)}-\vec r_H^1{(H^++OH^-)}})\delta^{\frac{3}{2}}({\vec r_H^2{(H,O,e)}}-\vec r_H^2{(H^++OH^-)}})\delta^{\frac{3}{2}}({\vec r_O{(H,O,e)}-\vec r_O{(H^++OH^-)}})$

Then there is:
${\hat{H}}_{H,O,e}{\psi}_{H,O,e}^1=E_{H_2O}{\psi}_{H,O,e}^1$
${\hat{H}}_{H,O,e}{\psi}_{H,O,e}^2=E_{H^++OH^-}{\psi}_{H,O,e}^2$

When the hydron oxide electronic system interfered by foreign particles and the foreign particles run away,wavefunction of the system would be changed from ${\psi}_{H,O,e}^1$ to ${\psi}_{H,O,e}^2$,and release reaction heat $\Delta{H}=E_{H^++OH^-}-E_{H_2O}$.
Problem are:
The way of constructing reaction process in reference document depend on hypothesis A,why condition A established for sure?
What kind of interference from foreign particles could cause the reaction?I mean the wavefunction depend on hypothesis A.
2. Relevant equations
${\hat{H}}_{H_2O}= \sum\limits_{i=1}^{10}-\frac{{\hbar}^2}{2m_e}{\nabla}_e^i-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_H^1{(H_2O)}|}-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_H^2{(H_2O)}|}-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_O{(H_2O)}|}+\sum\limits_{i=1}^{10}\sum\limits_{j=i+1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_e^j|}+\frac{e^2}{|\vec r_H^1{(H_2O)}-\vec r_H^2{(H_2O)}|}+\frac{8e^2}{|\vec r_H^1{(H_2O)}-\vec r_O{(H_2O)}|}+\frac{8e^2}{|\vec r_H^2{(H_2O)}-\vec r_O{(H_2O)}|}$
${\hat{H}}_{H^++OH^-}= \sum\limits_{i=1}^{10}-\frac{{\hbar}^2}{2m_e}{\nabla}_e^i-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_H^1{(H^++OH^-)}|}-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_H^2{(H^++OH^-)}|}-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_O{(H^++OH^-)}|}+\sum\limits_{i=1}^{10}\sum\limits_{j=i+1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_e^j|}+\frac{e^2}{|\vec r_H^1{(H^++OH^-)}-\vec r_H^2{(H^++OH^-)}|}+\frac{8e^2}{|\vec r_H^1{(H^++OH^-)}-\vec r_O{(H^++OH^-)}|}+\frac{8e^2}{|\vec r_H^2{(H^++OH^-)}-\vec r_O{(H^++OH^-)}|}$
The Hamilton for {H,O,electrons} system:
${\hat{H}}_{H,O,e}= -\frac{{\hbar}^2}{2m_H}{\nabla}_{H}^1-\frac{{\hbar}^2}{2m_H}{\nabla}_{H}^2-\frac{{\hbar}^2}{2m_O}{\nabla}_{O}-\sum\limits_{i=1}^{10}-\frac{{\hbar}^2}{2m_e}{\nabla}_e^i-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_H^1{(H,O,e)}|}-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_H^2{(H,O,e)}|}-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_O{(H,O,e)}|}+\sum\limits_{i=1}^{10}\sum\limits_{j=i+1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_e^j|}+\frac{e^2}{|\vec r_H^1{(H,O,e)}-\vec r_H^2{(H,O,e)}|}+\frac{8e^2}{|\vec r_H^1{(H,O,e)}-\vec r_O{(H,O,e)}|}+\frac{8e^2}{|\vec r_H^2{(H,O,e)}-\vec r_O{(H,O,e)}|}$
3. The attempt at a solution
So,I am confused with the model of calculating reaction heat if hypothesis A is not sure.

Last edited: May 2, 2017
2. May 7, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.