zhouhao
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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 Schrödinger 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.
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