Lecture 30: Linear Variation Theory The material in this lecture covers the following in Atkins.
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Transcript of Lecture 30: Linear Variation Theory The material in this lecture covers the following in Atkins.
Lecture 30: Linear Variation Theory The material in this lecture covers the following in Atkins.
14.7 Heteronuclear diatomic molecules (c) The variation principle
Lecture on-line Linear Variation Theory (PowerPoint) Linear Variation Theory (PDF)
Handout for this lecture
The Linear Variation Method
We have a Hamiltonian H with the eigenfunctions ψna nd eigenvalues En give n by t he SWE
Hψn = Enψn
Let u s l ook a t th e groundstat e ψ1 with t he energ y E1 .
W e would lik e to fi nd a wavefuncti on Φ1 fo r which the
energy
W = ∫ Φ
1
∗H Φ
1d τ / ∫ Φ
1
∗Φ
1d τ
i s clos e to E1
The linear variation method
We write Φ1 as a ttriri aall wavefuncti on in te rms of a
line ar combinati on of kno wn function s {fi }
tha t depen ds on th e same variable s a s ψ1 a nd hav e the
sa meboundary conditions
Φ1 = ∑j=1
j=n Cjfj
Or
Φ1 = C1f1 +C2f2 +....+Cjfj + Cnfn
The linear variation method
We shall now vary all the coefficients {Cj ,j=1,n}
in such a way that W has the smallest possible
value.
That is ,we shall find the absolute minima of the
function W(C1,C2,C3,.....,Ci,..Cn).
Let the values of the coefficients {C1,C2,C3,.....,Ci,..Cn}
at the minimum be given by
{C11 ,C
12,C
13, ...,C
1j ,..C
1n }
The linear variation method
What do we know about this particular coefficients ?
We know that if we look at the derivatives of W
δ W
δ Ci
= W’(C1,C2,..... ,Cn)
The n ’W (C11 ,C
12,C
13, ...,C
1j ,..C
1n ) = 0
W e shall us e thi s fac t to fi nd { C11 ,C
12,C
13, ...,C
1j ,..C
1n}
The linear variation method
First let us substitute the expression for Φ1 i nto the
expressi on fo r W. The denomenato r of I1 is
⌡⎮⌠
Φ1 ∗ Φ1 dτ = ⌡⎮⌠
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n Cj*fj*
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑k=1
k=n Ckfk
W e shal l assume tha t a ll functi ons {fj j=1,n} ar e rea l and
tha t al l coefficient s {Cj=1,n} are real
The linear variation method
After multiplication of the two parantheses
⌡⎮⌠
Φ1 ∗ Φ1 dτ =∑j=1
j=n ∑k=1
k=n Cj Ck ⌡
⎮⌠
fjfkdτ
Le t usintroduce : ⌡⎮⌠
fjfkdτ = Sjk
Thu s : ⌡⎮⌠
Φ1 ∗ Φ1 dτ =∑j=1
j=n ∑k=1
k=n Cj Ck Sjk
The linear variation method
For the numerator in the expression for I1 we have
⌡⎮⌠
Φ1∗ H Φ1dτ = ⌡⎮⌠
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n Cj*fj* H
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑k=1
k=n Ckfk
O r aft er multiplicati on ofparanthesis
⌡⎮⌠
Φ1∗ H Φ1dτ = =∑j=1
j=n ∑k=1
k=n Cj Ck ⌡
⎮⌠
fj H fkdτ
The linear variation method
We shall now introduce
⌡⎮⌠
fj H fkdτ = Hjk
We know that H is Hermetian
⌡⎮⌠
fj* H fkdτ = ⌡⎮⌠
fk (H fj)*dτ (1)
We shall also assume that it is real H = (H )*.
Thus since {fi=1,n} are real functions it follows from (1)
Hjk = Hkj
The linear variation method
We have
⌡⎮⌠
Φ1∗ H Φ1dτ = =∑j=1
j=n ∑k=1
k=n Cj Ck Hkj
We can now write W =
∑j=1
j=n ∑k=1
k=n Cj Ck Hkj
∑j=1
j=n ∑k=1
k=n Cj Ck Skj
The linear variation method
Or
W⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j Ck Skj = ∑
j=1
j=n ∑k=1
k=n C j Ck Hkj
It is important t o observ e tha (t C1,C2,..,Ci,.. Cn) are
independe nt variables
W e shal l no w differentiat e wi th respect t o one of
them,say C i , on bothside s of th e equation.
The linear variation method
We have
δ W
δ Ci ⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j Ck Skj + W
δδCi
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j Ck Skj =
δ
δ Ci j = 1
j = n
∑
k = 1
k = n
∑ Cj
Ck
Hkj
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
Let u s no w loo k a t δ
δCi⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j Ck Skj =
The linear variation method
Since we differentiate a sum by differentiating
each term from rules for differentiating a product
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n
δδCi( C j Ck Skj ) =
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n [
δC jδCiCk Skj +
δCk δCi Cj Skj]
Since (C1,C2,..,Ci,.. Cn) ar e independe nt variables
δCjδCi = 0 i f ≠i j
δCjδCi = 1 i f i= j ;
δCjδCi = δij
The linear variation method
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n [δjiCk Skj +δkiCj Skj] = ∑
k=1
k=n CkSik + ∑
j=1
j=n CjSjk
= ∑k=1
k=n CkSik + ∑
k=1
k=n CkSki = 2 ∑
k=1
k=n CkSik
Since : Sik = ⌡⎮⌠
fi fk dτ = ⌡⎮⌠
fk fi dτ = Ski
Thus δ
δCi⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n Cj Ck Skj = 2 ∑
k=1
k=n CkSik
The linear variation method
Now by replacing Skj with Hkj
δδCi
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j Ck Hkj = 2 ∑
k=1
k=n CkHik
Thu s from: δ W
δ Ci ⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j Ck Skj
+ WδδCi
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j Ck Skj = W
δδCi
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j CkH kj
The linear variation method
we get by substitution
δ W
δ Ci
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑j=1
j=n ∑k=1
k=n C j Ck Skj + 2 W
⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑k=1
k=n CkSik =
2 ∑k=1
k=n CkHik
This equation is satisfie d fo r al {l Cj ,j=1,n}. The optimal
set for whic h W i s a t aminimu mmus t i n additi on satisfy
δ W
δ Ci
= 0
The linear variation method
They must thus satisfy
2 W⎝⎜⎜⎜⎛
⎠⎟⎟⎟⎞
∑k=1
k=n CkSik = 2 ∑
k=1
k=n CkHik
Or b y combini ng terms: ∑k=1
k=n Ck [ Hik –2WSi k ] = 0
writte n out
C1 [ Hi1 –2WSi1 ]+C2 [ Hi2 –2WSi2 ] ,.Cn [ Hin –2WSi n ]
= 0
The linear variation method
We have obtained this set of equations from
δ W
δ Ci
= 0
Howev er i= 1,2,3,.. ..,nW e ca n a s a consequenc e obtai n th e se t of nequations
∑k=1
k=n Ck [ Hik -WSi k ] = 0 =i 1,n
The linear variation method
C1[ H11 -WS11 ]+C2 [ H12 - WS12 ] ,...Cn [ H1n - WS1n ]
= 0
C1[ H21 - WS21 ]+C2 [ H22 - WS22 ] ,...Cn [ H2n - WS2n ]
= 0
C1[ H31 -WS31 ]+C2 [ H32 -WS32 ] ,...Cn [ H3n -WS3n ] =
0
.......
C1 [ Hi1 -WSi1 ] +C2 [ Hi2 -WSi2 ] ,...Cn [ Hin -WSin ] = 0
C1 [ Hn1 -WSn1 ] +C2 [ Hn2 -WSn2 ] ,...Cn [ Hnn -WSnn ]
= 0
This is a set of n homogeneous equations
The linear variation method
This set of equations has only non-trivial solutions
provided that the secular determinant
[ H11 -WS11 ] [ H12 - WS12 ] [ H1n - WS1n ]
[ H21 - WS21 ] [ H22 - WS22 ] [ H2n - WS2n ]
[ H31 -WS31 ] [ H32 -WS32 ] [ H3n -WS3n ] = 0
.......
[ Hi1 -WSi1 ] [ Hi2 -WSi2 ] [ Hin -WSin ]
....
[ Hn1 -WSn1 ] [ Hn2 -WSn2 ] [ Hnn -WSnn ]
The linear variation method
By expanding out the determinant we will obtain an n-
order polynomial in W.This polynomial has n-roots
I[1]1 <I
[2]1 < I
[3]1 < .....,<I
[i]1 <..I
[n]1
The lowest root I[1]1
is the approximation to the
actual groundstate
energy E1E
1
E
E
3
E
i
E
n
I
1
[1]
I
1
[2]
I
1
[3]
I
1
[i]
I
1
[n]
The linear variation method
Further I[i]1 is an approximation to Ei. In all cases I
[i]1
larger than or equal Ei. Having obtained the roots we
can now find {Ci,i=1,n) by substituting into the set of
equationsC1[ H11 -WS11 ]+C2 [ H12 - WS12 ] ,...Cn [ H1n - WS1n ] = 0
C1[ H21 - WS21 ]+C2 [ H22 - WS22 ] ,...Cn [ H2n - WS2n ] = 0
..
C1 [ Hi1 -WSi1 ] +C2 [ Hi2 -WSi2 ] ,...Cn [ Hin -WSin ] = 0
....
C1 [ Hn1 -I1Sn1 ] +C2 [ Hn2 -I1Sn2 ] ,...Cn [ Hnn -I1Snn ] = 0
where W = I[i]1 i=1,n
The linear variation method
What you should learn from this lecture1. You should understand that in linear variationtheory the trial wavefunction is written as a linear combination of KNOWN functionswhere the relative contribution from each functionis optimized.
2. You should know how the set of linear equationsare generated and why the secular determinant must bezero and how this is used to determine theenergies.
3. You should be able to derive the equation in the case where n=2