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### Topic: general wave function of a molecule  (Read 4934 times)

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##### general wave function of a molecule
« on: December 02, 2015, 01:09:53 PM »
Hi!!

a) : At lesson I don't understand very well how is the general correct form for a Wave Function ψ of a molecule using the lcao method(ie using the MO)...

b)  Also, the professor said that for a multi-electron system the general wave function ψ=ψ(1,2,3...n) can be factored in ψ=Φ(1)Φ(2)Φ(3)...*Φ(n)...where Φ(i) is the i-esimal wave function that describes the i-esimal electron in this system.

c) another consideration that we have done at lesson is that: for a  multi-electron wave function ψ(1,2...n) i have to consider also the SPIN part and the  complete spin-orbital wave function  must  observe the   antisymmetric principle

So considering all these key points how can I imagine the general form of a  wave function  ψ for a multi-electron system??

For example the molecule of B_2  has 10 electrons whose collocation in MO's are:

1sσ)2
1sσ*)2
2sσ2
2sσ*2
2pxπ1
2pyπ1

So the system B_2 con be described by the using of 6 MO,each of these can be found through the LCAO method:

1sσ=  1s_A+1s_B]  (where A,B are the two different atoms of Boron)
1sσ= 1s_A-1s_B

2sσ= c_32s_A+2s_B
ECC...

The correct form  of these MO  actually   requires the Variational method  in order to find the correct value of each (coefficients c_i)OA that is involved in the LCAO ( for every MO):

Ψ= c_(1A) 1s_A + c_(1B) 1s_B + c_(2A) 2s_A + c_(2B) 2s_B  + c_(3A) 2px_A + c_(3B) 2px_B + ...
I have 10 OA involved in the LCAO so i'll get 10 MO,each one with some particular value of the  coefficients c_i:
Φ_1 =
Φ_2
.
.
.
Φ_10

Now that i know the numeric expression of these 10 MO Φ_i how can write the general wave function of the system B_2
As a prodcut of these  MO??

Ψ=Φ_1*Φ_2*Φ_3*...*Φ_10

I  don't think  this is correct ...because i'm not considering yet:

a) the spin parts ( because if the MO Φ_i contains two electrons I think I have to introduce the spin part α /β for the wave function Φ_i --- > Φ_i(1,2 electron)=Φ_i(1,2)*ω_i(α,β) =Φ_i(1)*Φ_i(2) *α(1)*β(2)+  Φ_i(1)*Φ_i(2) *α(2)*β(1)  )

b)but the last expression is not correct yet, because must be osbserved, for the total wave function ψ,  the  antisymmetric principle( so I  think I have to  write the  SLATER DETERMINANTAL WAVE FUNCtION)...

Someone can tell me if my assumptions are correct...and how is  the correct general form for the wave function ψ(1,2....10) of B_2 atom?? (or how concepts i have to consider in order to write it in the correct form)

Thanks VERY much!!

#### Enthalpy

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##### Re: general wave function of a molecule
« Reply #1 on: December 03, 2015, 02:33:14 PM »
b) Did he?

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##### Re: general wave function of a molecule
« Reply #2 on: December 04, 2015, 03:48:42 PM »
b) Did he?

he said that is the Hartree-Fock approximation...is wrong?

#### morten925

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##### Re: general wave function of a molecule
« Reply #3 on: December 05, 2015, 03:23:07 PM »
b) Did he?

he said that is the Hartree-Fock approximation...is wrong?

I don't think it's wrong but I would use the terminology "the Orbital Approximation" instead. The essence of the HF approximation is that each N-1 electron "feels" the mean Coulombic potential energy field created by N electrons, i.e. no electron-electron interaction, which should be equivalent to the Orbital Approximation. Post HF methods take into account electron-electron interaction like a full CI expansion and MP2. I recently posted a question about density functional theory by the way.

#### Irlanur

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##### Re: general wave function of a molecule
« Reply #4 on: December 06, 2015, 03:09:28 PM »
Quote
i.e. no electron-electron interaction
HF does not take into account any instantaneous CORRELATION, of course there is some electron-electron interaction in HF.

#### morten925

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##### Re: general wave function of a molecule
« Reply #5 on: December 06, 2015, 07:17:07 PM »
Quote
i.e. no electron-electron interaction
HF does not take into account any instantaneous CORRELATION, of course there is some electron-electron interaction in HF.

Nevertheless, not direct e-e interaction

#### pm133

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##### Re: general wave function of a molecule
« Reply #6 on: January 10, 2016, 05:47:40 AM »
b) Did he?

he said that is the Hartree-Fock approximation...is wrong?

He's either wrong or you've misunderstood.
He has shown you the Hartree Product form of a wavefunction.

#### pm133

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##### Re: general wave function of a molecule
« Reply #7 on: January 10, 2016, 05:54:05 AM »
OK so modifying posts on this forum doesn't work as I would have expected it.

I'll try again.

Part b) of the original post was the Hartree Product form of a wavefunction.
It is not the Hartree Fock approximation.

Sorry for the utter mess of posts