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Basis Sets and Pseudopotentials

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Slater-Type Orbitals (STO’s)

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Gaussian-Type Orbitals (GTO’s)

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Contracted Basis Sets

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Contracted Basis Sets

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Even-tempered Basis Sets

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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

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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

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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

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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

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Cartesian vs. Spherical Cartesians: s – 1 function p – 3 functions d – 6 functions f – 10 functions

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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?

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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

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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)

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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

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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

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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 TCA, 99, 265 (1998)

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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

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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

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Additional Information EMSL Basis Set Exchange: 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.