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, the basis set, are of a known form , the basis set, are of a known form Need to determine coefficients (cNeed to determine coefficients (cm)
Wavefunctions gives probability of finding Wavefunctions gives probability of finding electrons in space (e. g. s,p,d and f orbitals)electrons in space (e. g. s,p,d and f orbitals)
Molecular orbitals are formed by linear Molecular orbitals are formed by linear combinations of atomic orbitals (LCAO)combinations of atomic orbitals (LCAO)
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Ψ(v r ) = cm ∗Φm (
v r )
m
F
∑
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Φm (v r )
Hydrogen MoleculeHydrogen Molecule
HOMOHOMO
LUMO LUMO
VBT
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ΨHOMO =1
2(φA + φB )
ΨLUMO =1
2(φA − φB )
Hydrogen MoleculeHydrogen Molecule
Bond DensityBond Density
Ab Initio/Ab Initio/DFTDFT
Complete Description!Complete Description! Generic!Generic! Major Drawbacks:Major Drawbacks:
Mathematics can be cumbersomeMathematics can be cumbersome Exact solution only for hydrogenExact solution only for hydrogen
InformaticsInformatics Approximate solution time and storage intensiveApproximate solution time and storage intensive
– Acquisition, manipulation and dissemination problemsAcquisition, manipulation and dissemination problems
Mean Field or Hartree Fock)Mean Field or Hartree Fock) Pick single electron and average influence of Pick single electron and average influence of
remaining electrons as a single force field (Vremaining electrons as a single force field (V0 external)external)
Then solve Schrodinger equation for single Then solve Schrodinger equation for single electron in presence of field (e.g. H-atom electron in presence of field (e.g. H-atom problem with extra force field)problem with extra force field)
Perform for all electrons in system Perform for all electrons in system Combine to give system wavefunction and Combine to give system wavefunction and
energy (Eenergy (E) Repeat to error tolerance (ERepeat to error tolerance (Ei+1-Ei)
RecallRecall
SchrodingerSchrodinger Equation Equation Quantum vs. ClassicalQuantum vs. Classical Born OppenheimerBorn Oppenheimer Hartree-FockHartree-Fock ( (akaaka
Each atomic orbital/basis function is Each atomic orbital/basis function is itself comprised of a set of standard itself comprised of a set of standard functionsfunctions
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Ψ(v r ) = cm ∗Φm (
v r )
m
F
∑
Φm = Cmje−ζ mj r 2
j
N
∑
STO(Slater Type Orbital):~Hydrogen Atom Solutions
GTO(Gaussian Type Orbital):More Amenable to computation
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Φm ∝−ζ mj r2
Contraction coefficient(Static for calculation)
Expansion Coefficient
Atomic OrbitalLCAO
STO vs. GTOSTO vs. GTO
GTO GTO Improper behavior Improper behavior
for small r (slope for small r (slope equals zero at equals zero at nucleus)nucleus)
Decays too quicklyDecays too quickly
Basis SetsBasis Sets
∑ Φ∗=ΨF
mmm rcr )()(
vv
∑=ΦN
jjmjm C χ
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χ j ∝ e−ζ j r2
Optimized using atomic ab initio calculations
What “we” do!!
Basis Sets Molecular Orbital
Atomic Orbital
GTO/CGTO
STO
PGTO
Gaussian Type OrbitalsGaussian Type Orbitals
PrimitivesPrimitives
Shapes typical of H-atom orbitals Shapes typical of H-atom orbitals (s,p,d etc)(s,p,d etc)
Contracted Contracted Vary only coefficients of valence Vary only coefficients of valence
(chemically interesting parts) in (chemically interesting parts) in calculationcalculation
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χζ ,n,l,m (r,θ,ϕ ) = NYl,m (θ,ϕ )r(2n−2−l )e−ζr 2
Minimum Basis Set (STO-3G)Minimum Basis Set (STO-3G)
The number of basis functions is The number of basis functions is equal to the minimum required to equal to the minimum required to accommodate the # of electrons in accommodate the # of electrons in the systemthe system H(# of basis functions=1)-1sH(# of basis functions=1)-1s Li-Ne(# of basis functions=5) Li-Ne(# of basis functions=5)
1s,2s,2p1s,2s,2pxx, 2, 2yy, 2p, 2pz z
Basis SetsBasis SetsTypes:Types: STO-nG(n=integer)-Minimal Basis SetSTO-nG(n=integer)-Minimal Basis Set
Approximates shape of STO using single Approximates shape of STO using single contraction of n- PGTOs (typically, n=3)contraction of n- PGTOs (typically, n=3)
IntuitiveIntuitive The universe is NOT spherical!!The universe is NOT spherical!!
3-21G (Split Valence Basis Sets)3-21G (Split Valence Basis Sets) Core AOs 3-PGTOsCore AOs 3-PGTOs Valence AOs with 2 contractions, one with Valence AOs with 2 contractions, one with
2 primitives and other with 1 primitive 2 primitives and other with 1 primitive
Basis SetsBasis SetsTypes:Types:
3-21G(*)-Use of d orbital functions (23-21G(*)-Use of d orbital functions (2ndnd row atoms only)-row atoms only)-ad hocad hoc
6-31G*-Use of d orbital functions for 6-31G*-Use of d orbital functions for non-H atomsnon-H atoms
6-31G**-Use of d orbital functions for H 6-31G**-Use of d orbital functions for H as wellas well
ExamplesExamples
CC STO-3G-Minimal Basis SetSTO-3G-Minimal Basis Set
3 primitive gaussians used to model each 3 primitive gaussians used to model each STO STO
Addition of higher angular Addition of higher angular momentum functions momentum functions HCNHCN
Addition of p-function to H (1s) basis Addition of p-function to H (1s) basis better represents electron density (ie sp better represents electron density (ie sp character) of HC bondcharacter) of HC bond
Diffuse functionsDiffuse functions
Addition of basis functions with small Addition of basis functions with small exponents (I.e. spatial spread is greater)exponents (I.e. spatial spread is greater) AnionsAnions RadicalsRadicals Excited StatesExcited States Van der Waals complexes (Gilbert)Van der Waals complexes (Gilbert) Ex. Benzene-Dimers (Gilbert)Ex. Benzene-Dimers (Gilbert)
Hartree-Fock limitHartree-Fock limit NOT exact solutionNOT exact solution Does not include correlationDoes not include correlation Does not include exchangeDoes not include exchange
Accounting for Electron Accounting for Electron CorrelationsCorrelations DFT(Density Functional Theory)DFT(Density Functional Theory) Moller-Plesset (Perturbation Theory)Moller-Plesset (Perturbation Theory) Configuration Interaction (Coupling Configuration Interaction (Coupling
single electron problems)single electron problems)
Computational RemindersComputational Reminders
HF typically scales NHF typically scales N44
As increase basis set size accuracy/calculation As increase basis set size accuracy/calculation time increasestime increases
ALL of these ideas apply to any program utilizing ALL of these ideas apply to any program utilizing ab initio techniques NOT just Spartan (Gilbert)ab initio techniques NOT just Spartan (Gilbert)