ContractedGaussiansIndividual Gaussians are a pretty poorrepresentation of the behavior of real atomicwavefunctions, especially near nuclei;· Bundle together, or contract, severalGaussians with different exponents into onembasis function:Xu(r) =>Ca,μ g(r, a,u)α=1Onebasisfunctionm≤6;Different exponents ;Different fixed contraction coefficients;One variation coefficient
Contracted Gaussians • Individual Gaussians are a pretty poor representation of the behavior of real atomic wavefunctions, especially near nuclei; • Bundle together, or contract, several Gaussians with different exponents into one basis function: 7 • One basis function • m ≤ 6; • Different exponents ζ; • Different fixed contraction coefficients; • One variation coefficient 𝜒𝜇(𝑟) = 𝛼=1 𝑚 𝑐𝛼,𝜇 𝑔(𝑟, 𝜁𝛼,𝜇)
Basisset. A collection of exponents and contraction coefficientsdefining a basis function for an atom or atoms;. Constructed in many ways, be fitted to numericalresults of atoms and some set of properties ofmoleculesX-type GTO # X-atomic orbital Ad-type GTO can be either 6 (Cartesian) or5 (spherical)functions, depending on whether the symmetric component(x2+y2+z2) is included or removed Similarly f-type can be 7 (spherical) or 1o (Cartesian) GTOs.Gen6D; 10Fhttp://gaussian.com/gen/8
Basis set • A collection of exponents and contraction coefficients defining a basis function for an atom or atoms; • Constructed in many ways, be fitted to numerical results of atoms and some set of properties of molecules • X-type GTO ≠ X-atomic orbital – A d-type GTO can be either 6 (Cartesian) or 5 (spherical) functions, depending on whether the symmetric component (x2+y2+z2 ) is included or removed – Similarly f-type can be 7 (spherical) or 10 (Cartesian) GTOs. 8 Gen 6D; 10F http://gaussian.com/gen/
Minimalbasissets: A minimum one basis function for each occupiedatomic orbital- Hydrogen 1s = one basis function;Fluorine1s+2s+2px+2py+2pz=fivebasisfunctions; STO-3G basis set is a minimal basis set- Each basis function is a contraction of three GTOs; Also known as a 'single-7'basis set; STO-nG basis sets available for many atoms;- Good for rough and ready calculations, but notveryaccurate;9
Minimal basis sets • A minimum one basis function for each occupied atomic orbital – Hydrogen 1s = one basis function; – Fluorine 1s + 2s +2px + 2py + 2pz = five basis functions; • STO-3G basis set is a minimal basis set – Each basis function is a contraction of three GTOs; – Also known as a ‘single-ζ ’ basis set; – STO-nG basis sets available for many atoms; – Good for rough and ready calculations, but not very accurate; 9
UseminimalbasissettocalculateHFHF/STO-3G GFInput pop=full1.05H1,+0.82Fzpz0.52F2s16eVH-F bond 0.9556 AE(RHF) = -98.5728474 Hatreeμ = 1.25 Debye1.0Fzpx1.0Fzpy-12.6eVExp. dH-E = 0.917 A, μ = 1.82 D, IP:16.1 eV0.70F2pz0.53H1s+0.41F2s-15.6eVall the energy is...in the core!allthe chemistryis...inthevalence!Good core basis set lowers energybutmay not help withthe chemical0.95 F2s0.25F1s +0.15 H1spart-39.7eVBasissetshavetobalancetreatmentof both0.99Fis704.9eVHFGFlnput,Foresman,p8010
Use minimal basis set to calculate HF • HF/STO-3G GFInput pop=full • H-F bond 0.9556 Å • E(RHF) = -98.5728474 Hatree • μ = 1.25 Debye • Exp. dH-F = 0.917 Å, μ = 1.82 D, IP: 16.1 eV • all the energy is.in the core! • all the chemistry is.in the valence! • Good core basis set lowers energy but may not help with the chemical part • Basis sets have to balance treatment of both 10 GFInput, Foresman, p80
Multi-zetabasissetsMinimal basis set doesn't leave any real roomfor atomic orbitals to“breath"(expand orcontract) when forming bonds with otheratoms;Double zeta basis set with two basisfunctionsper occupied atomic orbital;More common is split valence, which is doublezeta for valence levels, single zeta for core;11
Multi-zeta basis sets • Minimal basis set doesn’t leave any real room for atomic orbitals to “breath” (expand or contract) when forming bonds with other atoms; • Double zeta basis set with two basis functions per occupied atomic orbital; • More common is split valence, which is double zeta for valence levels, single zeta for core; 11