Calculate reaction heat from quantum mechanics

[zhouhao]

Homework Statement

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.

Homework 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:
${\hat{H}}_{H,O,e}= -\frac{{\hbar}^2}{2m_H}{\nabla}_{H}^1-\frac{{\hbar}^2}{2m_H}{\nabla}_{H}^2-\frac{{\hbar}^2}{2m_O}{\nabla}_{O}-\sum\limits_{i=1}^{10}-\frac{{\hbar}^2}{2m_e}{\nabla}_e^i-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_H^1{(H,O,e)}|}-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_H^2{(H,O,e)}|}-\sum\limits_{i=1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_O{(H,O,e)}|}+\sum\limits_{i=1}^{10}\sum\limits_{j=i+1}^{10}\frac{e^2}{|\vec r_e^i-\vec r_e^j|}+\frac{e^2}{|\vec r_H^1{(H,O,e)}-\vec r_H^2{(H,O,e)}|}+\frac{8e^2}{|\vec r_H^1{(H,O,e)}-\vec r_O{(H,O,e)}|}+\frac{8e^2}{|\vec r_H^2{(H,O,e)}-\vec r_O{(H,O,e)}|}$

The Attempt at a Solution

So,I am confused with the model of calculating reaction heat if hypothesis A is not sure.

 

[member 553083]

The quantum mechanics model of reaction process is not complete or there are some fundamental mistake?I am guessing that according to the law of conservation of energy,the reaction heat of dissociation of $H_2O$ should be equal to the reaction heat of formation of $H_2O$.