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Topic: Adding and multiplying wave functions  (Read 11464 times)

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Offline ddue2

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Adding and multiplying wave functions
« on: January 28, 2012, 06:03:19 PM »
Sometimes a new wave function is obtain by summing two other wave functions (e.g. for H2+ wave function = constant * (1sA + 1sB) where 1sA and 1sB are wave functions for an electron in a 1s orbital on hydrogen A or B) and sometimes by taking a product of two wave functions (e.g. H2 wave function = constant * 1sA(1) * 1sB(2) where 1 and 2 stand for electron 1 and 2 on hydrogen A or B.
When are new wave functions obtained by addition and when are they multiplication?

Offline juanrga

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Re: Adding and multiplying wave functions
« Reply #1 on: January 29, 2012, 09:14:04 AM »
Sometimes a new wave function is obtain by summing two other wave functions (e.g. for H2+ wave function = constant * (1sA + 1sB) where 1sA and 1sB are wave functions for an electron in a 1s orbital on hydrogen A or B) and sometimes by taking a product of two wave functions (e.g. H2 wave function = constant * 1sA(1) * 1sB(2) where 1 and 2 stand for electron 1 and 2 on hydrogen A or B.
When are new wave functions obtained by addition and when are they multiplication?

You are comparing one electron case with a two electron case.

For an electron in a molecule (e.g, H2+) the stationary wavefunction ## \Phi(x) ##, where ## x ## is the coordinate of the electron can be initially developed as a combination of atomic wavefunctions such as 1sA and 1sB in your example.

For two electrons in a molecule (e.g, H2) the stationary wavefunction ## \Phi(x_1,x_2) ##, where ## x_j ## is the coordinate of the electron j can be initially developed as a product of atomic wavefunctions such as 1sA(1) (which is a function of ## x_1 ##) and 1sB(2) (which is a function of ## x_2 ##) in your example.
« Last Edit: April 03, 2012, 09:00:39 AM by Borek »
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Offline ddue2

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Re: Adding and multiplying wave functions
« Reply #2 on: January 29, 2012, 10:30:24 PM »
Thank you for the reply.
« Last Edit: January 29, 2012, 10:48:06 PM by ddue2 »

Offline ddue2

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Re: Adding and multiplying wave functions
« Reply #3 on: January 29, 2012, 10:46:26 PM »
Here is a follow-up question.
For one of the N - H bonds in NH3 the wave function can be written as psi = 1sHa(1)sp13(2) + 1sHa(2)sp13(1).  This is for two electrons in one MO.  What is the rationale for taking products of wave functions and then adding them? Is each of the products the same as allowing an electron in each AO and the sum is to allow them to exchange places?

Offline juanrga

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Re: Adding and multiplying wave functions
« Reply #4 on: January 31, 2012, 09:33:47 AM »
Here is a follow-up question.
For one of the N - H bonds in NH3 the wave function can be written as psi = 1sHa(1)sp13(2) + 1sHa(2)sp13(1).  This is for two electrons in one MO.  What is the rationale for taking products of wave functions and then adding them? Is each of the products the same as allowing an electron in each AO and the sum is to allow them to exchange places?

Imagine that ##H_A## is the Hamiltonian of system A and ##H_B## the Hamiltonian of system B. The time-independent Schrödinger equations are

$$ H_A \Psi(x_A) = E_A \Psi(x_A) $$

$$ H_B \Psi(x_B) = E_B \Psi(x_B) $$

with ## E_A ## and ## E_B ## the respective energies. Multiplying the first equation by a function of coordinates ##x_B## and the second by a function of coordinates ##x_A## does not change anything because ## H_A ## only works on ## x_A ## and ## H_B ## only does on ## x_B ##. I.e.,

$$ H_i \Psi(x_i)\Psi(x_j) = \Psi(x_j) H_i \Psi(x_i)$$

Now consider a total Hamiltonian ## H_{AB} = H_A + H_B ## and use the above equations, you obtain

$$ H_{AB} \Psi(x_A)\Psi(x_B) =  ( E_A + E_B ) \Psi(x_A)\Psi(x_B) $$

I.e. the function for the total system is a product ## \Psi(x_A,x_B) = \Psi(x_A)\Psi(x_B) ##

Mathematically, this method is named separation of variables.

You can repeat this for the NH3 and obtain a product ## \Psi(x_1,x_2) = \Psi(x_1)\Psi(x_2) ## for the MO. Notice that up to this point we are considering a product of functions but not saying what functions.

One possibility is ## \Psi'(x_1,x_2) = 1\mathrm{s}(x_1)\mathrm{sp}^3(x_2) ## and other ## \Psi''(x_1,x_2) = \mathrm{sp}^3(x_1)1\mathrm{s}(x_2) ##. Mathematically, a linear combination of two special solutions of a differential equation give a general solution of the equation. That is the reason which you sum both to obtain a more general solution.

Physically, you can interpret this sum as that a given electron (e.g. 1) is neither in atom N nor in atom H but in a superposition.
« Last Edit: April 03, 2012, 09:01:26 AM by Borek »
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