2. THE USUAL APPROACH
The possible ways are:
๏ต Take the known function ๐น and calculate the energy E
and < ๐ด >
๏ต Take the trial function ๐ and calculate average energy
<E> and < A>
๏ต Compare these results
๏ต Idea is to compare all such average properties.
๏ต We take for simplicity ๐ 2=A
๏ต Also, compute < ๐ ฮจ > |2
3. SAMPLE RESULTS
The particle in a box
๏ต The exact wave-functions that were,
1. ฮจ1 = โ
2
๐ฟ
sin
๐๐ฅ
๐ฟ
2. ฮจ2 = โ
2
๐ฟ
sin
2๐๐ฅ
๐ฟ
3. ฮจ3 = โ
2
๐ฟ
sin
3๐๐ฅ
๐ฟ
For the known functions the results are
Wave-function Exact
Energy(E)
< ๐ 2 >
ฮจ1 9.869/๐ฟ2
0.2826 ๐ฟ2
ฮจ2 39.478/๐ฟ2 0.3206 ๐ฟ2
ฮจ3 88.826/๐ฟ2 0.3277 ๐ฟ2
4. SAMPLE RESULTS
The particle in a box
The trial functions that were taken
1. ๐ 1 = N๐ฅ ๐ฟ โ ๐ฅ
2. ๐ 2 = N ๐ฅ
๐ฟ
2
โ ๐ฅ ๐ฟ โ ๐ฅ
3. ๐ 3 = N ๐ฅ
๐ฟ
3
โ ๐ฅ
2๐ฟ
3
โ ๐ฅ ๐ฟ โ ๐ฅ
๏ต For the trial functions the results are
Wave-
function
Average
Energy(<E>)
< ๐ 2>
๐1 10/๐ฟ2 0.2857 ๐ฟ2
๐2 42/๐ฟ2 0.3333 ๐ฟ2
๐3 100/๐ฟ2
0.3636 ๐ฟ2
5. PIB RESULTS
Comparison % Error in Energy % Error in โน x2โบ
ฮจ1 vs. ๐1 1.31 1.085
ฮจ2 vs. ๐2 6.003 3.81
ฮจ3 vs. ๐3 11.17 10.95
6. SAMPLE RESULTS
The harmonic oscillator
The exact wave-functions that were,
ฮจ0 = (
1
๐
) ยผ ๐โ
๐ฅ2
2
ฮจ1 = (
4
๐
) ยผ ๐ฅ๐โ
๐ฅ2
2
For the known functions the results are
Wave-function Exact Energy(E)
ฮจ0
1
ฮจ1
3
7. SAMPLE RESULTS
The harmonic oscillator
The trial functions that were taken
๐ 0 = N (A2 โ x2)
๐ 1 = N x (A2 โ x2)
For the trial functions the results are
Wave-function Average Energy(<E>)
๐0 1.1952
๐1 3.7416
9. USE OF OVERLAP: THE HYDROGEN ATOM
AND STOs
Slater Type Orbitals(STO)
1. ๐น1๐ =
1
๐
๐โ๐
2. ๐น2๐ =
1
4 2๐
(2 โ ๐)๐โ
๐
2
Advantages
๏ต Probability of finding the electron near the nucleus is faithfully
represented.
Disadvantages
๏ต Three and four center integrals cannot be performed analytically.
๏ต No radial nodes. These can be introduced by making linear
combinations of STOs
๏ต Does not ensure rapid convergence with increasing number of
functions.
10. THE HYDROGEN ATOM: GTOs
Gaussian Type Orbitals(GTO)
๏ต Introduced by Boys (1950)
๏ต ฮฑ is a constant (called exponent) that determines the size (radial extent)
of the function
๏ต The normalized 1s Gaussian-type function is,
๐1๐
๐บ๐น
(๐ผ, r) = (
2๐ผ
๐
)
3
4 ๐โ๐ผ๐2
GTOs are inferior to STOs in these ways:
๏ต GTOโs behavior near the nucleus is poorly represented. GTOs diminish
too rapidly with distance.
๏ต The โtailโ behavior is poorly represented.
Advantage
๏ต GTOs are computationally advantageous.
Therefore, we use a linear combination of GTOs to overcome
these deficiencies. Overlap is here the key.
11. LINEAR COMBINATION OF GTOs
Contracted Gaussian functions:
๏ต Need for using better basis functions
๏ต Fixed linear combinations of primitive Gaussian functions-- Contracted
Gaussian functions
๏ต We use the following basis functions for further calculation
๐1๐
๐ถ๐บ๐น
(ฮถ =1.0, STO-1G) = ๐1๐
๐บ๐น
๐ผ11
๐1๐
๐ถ๐บ๐น
(ฮถ =1.0, STO-2G) =๐12 ๐1๐
๐บ๐น
(๐ผ12) + ๐22 ๐1๐
๐บ๐น
(๐ผ22)
๐1๐
๐ถ๐บ๐น
(ฮถ =1.0, STO-3G) =๐13 ๐1๐
๐บ๐น
(๐ผ13) + ๐23 ๐1๐
๐บ๐น
(๐ผ23) + ๐33 ๐1๐
๐บ๐น
(๐ผ33)
12. THE OVERLAP BETWEEN 1s-SLATER & CGFโs
1s Basis Functions Overlap
STO-1G 0.978400
STO-2G 0.998420
STO-3G 0.999907
Standard textbook stuff
13. THE OVERLAP BETWEEN 2s-SLATER & CGFs
2s Basis Functions Overlap
STO-1G 0.89664340
STO-2G 0.94512792
STO-3G 0.95788308
Our observation
14. THE MOTIVATION: NEED FOR A
DIFFERENT APPROACH
๏ต Standard schemes require exact ๐น and E
๏ต Errors are measured โon averageโ (integration process)
๏ต Knowledge of excited ๐น is difficult to gather
๏ต Necessity of employing some kind of self-check process
๏ต No need of any knowledge of exact function or its
property
15. A SELF-ASSESSMENT POLICY
What we do actually
๏ต TISE: ฮจโณ
= (๐ โ ๐ธ)ฮจ
๏ต Assume: ๐โฒโฒ
โ (๐ โ< ๐ธ >)๐
๏ต Differentiate once more: ๐โฒโฒโฒ
โ ๐ โ< ๐ธ > ๐โฒ
+
๐โฒ๐
๏ต Check the nodes, boundaries, origin and
minima/maxima.
16.
17. SAP Contd.
๏ต Where V(x)=0, โ๐โฒโฒ
/(๐ < ๐ธ >) โ1. Not
applicable to box and any odd states for symmetric
potentials.
๏ต At nodes,
๐โฒโฒโฒ
๐โฒ(๐โ<๐ธ>)
โ 1
๏ต At boundaries, โ
๐โฒโฒโฒ
๐โฒ<๐ธ>
โ1 (only for the box
system)
๏ต At minima/maxima,
๐โฒโฒโฒ
๐ ๐โฒ โ1
๏ต Judge the goodness using these relations
18. PERFORMANCE
The PIB and harmonic oscillator
Taking the same trial functions mentioned before, for PIB,
For harmonic oscillator,
Wave-
function
โ
๐โฒโฒโฒ
๐โฒ<๐ฌ>
at node
% Error
โ
๐โฒโฒโฒ
๐โฒ<๐ฌ>
at boundaries
% Error
๐1 --- --- --- ---
๐2 0.57143 0.42857 - 0.28571 128.57
๐3 0.54 0.46 -0.54 154.00
Wave-function
โ
๐โฒโฒโฒ
๐โฒ<๐ฌ>
at node
% Error
๐ ๐ 0.2857 71.43
19. PERFORMANCE
The anharmonic oscillators
๏ต V = x2 + x4 state = even 4th lowest
Potential Basis ๐ฟ ๐๐๐ก <E> % error
in <E>
<x2> % Error
in <x2>๏ฌ1 ๏ฌ2
1 1 6
8
10
12
2.8639
3.0964
3.6172
3.6772
28.845714
28.835718
28.8353399
28.8353385
0.03598
0.00132
0.000021
0.00
2.267767
2.261797
2.261703
2.26170396
0.26807
0.00411
0.000042
0.00
Usual strategy
22. CONCLUSION
๏ต We have introduced certain criteria to judge the quality
of an optimized approximate stationary state.
๏ต The criteria do not require any knowledge of exact
function or energy.
๏ต This self-check is more useful for higher excited states.
๏ต Our equations reduce to exactness as ๐ โ ๐น.