Basis Sets: advantages
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1 Basis Sets: advantages Analogy with ordinary vector analysis One refers to a vector in R 3 using its projections on a given basis set v v =(v 1 ,v 2 ) e 1 e 2 One speaks about: Vector analysis: QM: COORDINATES COEFFICIENTS COORD. SYSTEM BASIS FUNCTIONS v = c i e i i=1 3 ∑ Basis Sets: example LCAO MO’s Atomic basis function χ AO : p z orbital on every carbon atom χ AO (r ) = Nze −ζ r φ 1 MO (r ) = N ' χ AO (r − R I ) i=1 6 ∑ Ψ 1 MO (r ) = φ 1 MO φ 2 MO φ 3 MO
Transcript of Basis Sets: advantages
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Basis Sets: advantages
Analogy with ordinary vector analysis One refers to a vector in R3 using its projections on a given basis set
v
v =(v1,v2) e1
e2
One speaks about: Vector analysis: QM: COORDINATES COEFFICIENTS COORD. SYSTEM BASIS FUNCTIONS
v = ciei
i=1
3
∑
Basis Sets: example
LCAO
MO’s Atomic basis function χAO : pz orbital on every carbon atom
χ AO (r) = Nze−ζr
φ1MO (r) = N ' χ AO (r − RI )
i=1
6
∑
Ψ1MO (r) = φ1
MOφ2MOφ3
MO
2
Basis Sets: AO vs MO
LCAO (linear combination of atomic orbitals) approximation: MO’s can be constructed from AO’s
An “orbital” is a one-electron quantity
Basis functions, ,are represented by atom-centered Gaussians in most quantum chemistry programs (why?)
Some older programs used Slater functions
Physicists like plane wave basis sets
( , )basisnr Rφ
1( , ) ( , ), :
( ) ( , ) , :
AO basisn n i i n i
iMO AOm mn n n nm
n
r R c r R c fix coefficients
r D r R D computed
χ φ
χ=
=
Φ =
∑
∑
Basis Sets: functional forms
Slide adapted from lecture of David Sherill
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Basis Sets: functional forms
Slide adapted from lecture of David Sherill
Basis Sets: functional forms
n: length of contraction ci,ζi fixed (determined once)
Slide adapted from lecture of David Sherill
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Basis Sets: functional forms
Basis Sets: Gaussian product theorem
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Basis Sets: types
Slide adapted from lecture of David Sherill
Basis Sets: types
Slide adapted from lecture of David Sherill
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Basis Sets: types - polarization
Slide adapted from lecture of David Sherill
Basis Sets: types – diffuse functions
Slide adapted from lecture of David Sherill
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Basis Sets: types
( )2, ,
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1( , , ) i
a b c
rS a b ci
ix y z N c x y z e ζφ −
=
= ∑Slide adapted from lecture of David Sherill
(3s)->1s
Basis Sets: types
Slide adapted from lecture of David Sherill
(3s/3s3p)->1s/1s1p
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Basis Sets: types
Slide adapted from lecture of David Sherill
Basis Sets: types
Slide adapted from lecture of David Sherill
(6s/4s4p)->1s
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Basis Sets: types
Slide adapted from lecture of David Sherill
Basis Sets: accuracy
VDZ VTZ VQZ V5Z V6Z VDZ VTZ VQZ V5Z V6Z
Slide adapted from lecture of David Sherill
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Basis Sets: drawbacks
Slide adapted from lecture of Stefan Grimme
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Slide adapted from lecture of Stefan Grimme
Slide adapted from lecture of Stefan Grimme
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Slide adapted from lecture of Stefan Grimme
Slide adapted from lecture of Stefan Grimme
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Basis Sets: size
Slide adapted from lecture of David Sherill
