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№1 слайд![Basis Sets and](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img0.jpg)
Содержание слайда: Basis Sets and Pseudopotentials
№2 слайд![Slater-Type Orbitals STO s](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img1.jpg)
Содержание слайда: Slater-Type Orbitals (STO’s)
№3 слайд![Gaussian-Type Orbitals GTO s](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img2.jpg)
Содержание слайда: Gaussian-Type Orbitals (GTO’s)
№4 слайд![Contracted Basis Sets](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img3.jpg)
Содержание слайда: Contracted Basis Sets
№5 слайд![Contracted Basis Sets](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img4.jpg)
Содержание слайда: Contracted Basis Sets
№6 слайд![](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img5.jpg)
№7 слайд![Even-tempered Basis Sets](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img6.jpg)
Содержание слайда: Even-tempered Basis Sets
№8 слайд![Well-tempered Basis Sets , ,](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img7.jpg)
Содержание слайда: Well-tempered Basis Sets
α, β, γ, and δ are parameters optimized to minimize the SCF
energy
Exponents are shared for s, p, d, etc. functions
№9 слайд![](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img8.jpg)
№10 слайд![Plane Wave Basis Sets Used to](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img9.jpg)
Содержание слайда: Plane Wave Basis Sets
Used to model infinite systems (e.g. metals, crystals, etc.)
In infinite systems, molecular orbitals become bands
Electrons in bands can be described by a basis set of plane waves of the form
The wave vector k in a plane wave function is similar to the orbital exponent in a Gaussian function
Basis set size is related to the size of the unit cell rather than the number of atoms
№11 слайд![Polarization Functions](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img10.jpg)
Содержание слайда: Polarization Functions
Similar exponent as valence function
Higher angular momentum (l+1)
Uncontracted Gaussian (coefficient=1)
Introduces flexibility in the wave function
by making it directional
Important for modeling chemical bonds
№12 слайд![Diffuse Functions Smaller](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img11.jpg)
Содержание слайда: Diffuse Functions
Smaller exponent than valence functions
(larger spatial extent)
Same angular momentum as valence
functions
Uncontracted Gaussian (coefficient=1)
Useful for modeling anions, excited states and weak (e.g., van der Waals) interactions
№13 слайд![Cartesian vs. Spherical](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img12.jpg)
Содержание слайда: Cartesian vs. Spherical
Cartesians:
s – 1 function
p – 3 functions
d – 6 functions
f – 10 functions
№14 слайд![Cartesian vs. Spherical](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img13.jpg)
Содержание слайда: Cartesian vs. Spherical
Suppose we calculated the energy of HCl using a cc-pVDZ basis set using Cartesians then again using sphericals.
Which calculation produces the lower energy? Why?
№15 слайд![Pople Basis Sets Optimized](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img14.jpg)
Содержание слайда: Pople Basis Sets
Optimized using Hartree-Fock
Names have the form
k-nlm++G** or k-nlmG(…)
k is the number of contracted Gaussians used for core
orbitals
nl indicate a split valence
nlm indicate a triple split valence
+ indicates diffuse functions on heavy atoms
++ indicates diffuse functions on heavy atoms and hydrogens
№16 слайд![Pople Basis Sets Examples -](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img15.jpg)
Содержание слайда: Pople Basis Sets
Examples:
6-31G Three contracted Gaussians for the core with the valence represented by three contracted Gaussians and one
primitive Gaussian
6-31G* Same basis set with a polarizing function added
6-31G(d) Same as 6-31G*
6-31G** Polarizing functions added to hydrogen and heavy atoms
6-31G(d,p) Same as 6-31G**
6-31++G 6-31G basis set with diffuse functions on hydrogen and
heavy atoms
The ** notation is confusing and not used for larger basis sets:
6-311++G(3df, 2pd)
№17 слайд![Dunning Correlatoin](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img16.jpg)
Содержание слайда: Dunning Correlatoin Consistent Basis Sets
Optimized using a correlated method (CIS, CISD, etc.)
Names have the form
aug-cc-pVnZ-dk
“aug” denotes diffuse functions (optional)
“cc” means “correlation consistent”
“p” indicates polarization functions
“VnZ” means “valence n zeta” where n is the number of functions used to describe a valence orbital
“dk” indicates that the basis set was optimized for relativistic calculations
Very useful for correlated calculations, poor for HF
Size of basis increases rapidly with n
№18 слайд![Dunning Basis Sets Examples](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img17.jpg)
Содержание слайда: Dunning Basis Sets
Examples:
cc-pVDZ Double zeta with polarization
aug-cc-pVTZ Triple zeta with polarization and
diffuse functions
cc-pV5Z-dk Quintuple zeta with polarization optimized for relativistic effects
№19 слайд![Extrapolate to complete basis](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img18.jpg)
Содержание слайда: Extrapolate to complete basis set limit
Most useful for electron correlation methods
P(lmax) = P(CBS) + A( lmax)-3
P(n) = P(CBS) + A( n)-3
n refers to cc basis set level: for for DZ, 3 for TZ, etc.
Best to use TZP and better
http://molecularmodelingbasics.blogspot.dk/2012/06/complete-basis-set-limit-extrapolation.html
TCA, 99, 265 (1998)
№20 слайд![Basis Set Superposition Error](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img19.jpg)
Содержание слайда: Basis Set Superposition Error
Occurs when a basis function centered at one nucleus contributes the the electron density around another nucleus
Artificially lowers the total energy
Frequently occurs when using an unnecessarily large basis set (e.g. diffuse functions for a cation)
Can be corrected for using the counterpoise correction.
- Counterpoise usually overcorrects
- Better to use a larger basis set
№21 слайд![Counterpoise Correction E A](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img20.jpg)
Содержание слайда: Counterpoise Correction
E(A)ab is the energy of fragment A with the basis functions for A+B
E(A)a is the energy of fragment A with the basis functions centered on fragment A
E(B)ab and E(B)b are similarly defined
№22 слайд![Additional Information EMSL](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img21.jpg)
Содержание слайда: Additional Information
EMSL Basis Set Exchange:
https://bse.pnl.gov/bse/portal
Further reading:
Davidson, E. R.; Feller, D. Chem. Rev. 1986, 86, 681-696.
Jensen, F. “Introduction to Computational Chemistry”, 2nd
ed., Wiley, 2009, Chapter 5.
№23 слайд![Effective Core Potentials](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img22.jpg)
Содержание слайда: Effective Core Potentials (ECPs) and Model Core Potentials (MCPs)
№24 слайд![Frozen Core Approximation](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img23.jpg)
Содержание слайда: Frozen Core Approximation
№25 слайд![Pseudopotentials - ECPs](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img24.jpg)
Содержание слайда: Pseudopotentials - ECPs
№26 слайд![Shape Consistent ECPs](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img25.jpg)
Содержание слайда: Shape Consistent ECPs
№27 слайд![Energy Consistent ECPs](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img26.jpg)
Содержание слайда: Energy Consistent ECPs
№28 слайд![Pseudo-orbitals](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img27.jpg)
Содержание слайда: Pseudo-orbitals
№29 слайд![Large and Small Core ECPs](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img28.jpg)
Содержание слайда: Large and Small Core ECPs
№30 слайд![Pseudopotentials - MCPs](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img29.jpg)
Содержание слайда: Pseudopotentials - MCPs
№31 слайд![MCP Formulation](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img30.jpg)
Содержание слайда: MCP Formulation
№32 слайд![-electron Hamiltonian](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img31.jpg)
Содержание слайда: 1-electron Hamiltonian
№33 слайд![MCP Nuclear Attraction](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img32.jpg)
Содержание слайда: MCP Nuclear Attraction
№34 слайд![MCP vs. ECP](/documents_6/6fbae40096641e011f6a6ad0f6e152f3/img33.jpg)
Содержание слайда: MCP vs. ECP