《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）6-BasisSet

Lasttime1.Born-Oppenheimerapproximation;2. Linear Combination Atomic Orbitals3. Semi-empirical methods;4.Open-shell systems;5. SCF details2

Last time 2 1. Born-Oppenheimer approximation; 2. Linear Combination Atomic Orbitals; 3. Semi-empirical methods; 4. Open-shell systems; 5. SCF details

ContentsBasis set1.Gaussian type orbitals2. Pople basis sets3.Dunning basis set4.Basis set effect5.Pseudopotentials or EffectiveCorePotentials6. Population analysisJensen, chp 5Foresman, chp 10m

Contents 3 Basis set 1. Gaussian type orbitals 2. Pople basis sets 3. Dunning basis set 4. Basis set effect 5. Pseudopotentials or Effective Core Potentials 6. Population analysis Jensen, chp 5 Foresman, chp 10

Gaussianbasis setsSeek basis functions to make integrals calculatedconveniently. Slater functions (e- (r), correct functional form nearnucleus, but many-center integrals cannot becalculated analyticallyGaussian functions (e-r^2)- STOs could be approximatedas linear combinations ofGaussian-typeorbitals (GTOs)instead (Frank Boys);-Formolecules,writemolecularorbitalsinexpansionsintermsofGaussianatomic-likefunctionscenteredontheatoms;- Gaussian basis sets (John Pople)4

Gaussian basis sets • Seek basis functions to make integrals calculated conveniently • Slater functions (e- ζr ), correct functional form near nucleus, but many-center integrals cannot be calculated analytically • Gaussian functions (e-ζr^2) – STOs could be approximated as linear combinations of Gaussian-type orbitals (GTOs) instead (Frank Boys); – For molecules, write molecular orbitals in expansions in terms of Gaussian atomic-like functions centered on the atoms; – Gaussian basis sets (John Pople) 4

PrimitiveGaussianPrimitiveGaussianisoneGaussianbasisfunction(GTO);exponent andan angularpart given bysome productof x, y, and z;3/4.(8)i+j+ki!j!k!,2.7[(2i) (2i)1 (2k)lxiy/ 2ke-r2g(r,,i,j,k) = i=j= k = O, a single, spherical s-type primitiveGaussian;(i, j, k) = (1,0,0), (0,1,0), and (0,0,1), a set of threep-type primitive Gaussians;( determines how fast the basis function decays awayfromtheatom;.Big ( = fast decay = function close to nucleus;5

Primitive Gaussian • Primitive Gaussian is one Gaussian basis function (GTO); • exponent ζ and an angular part given by some product of x, y, and z; 5 • i = j = k = 0, a single, spherical s-type primitive Gaussian; • (i, j, k) = (1,0,0), (0,1,0), and (0,0,1), a set of three p-type primitive Gaussians; • ζ determines how fast the basis function decays away from the atom; • Big ζ = fast decay = function close to nucleus; 𝑔 𝑟, 𝜁, 𝑖,𝑗, 𝑘 = ( 2𝜁 𝜋 ) 3/4 [ (8𝜁) 𝑖+𝑗+𝑘 𝑖!𝑗! 𝑘! 2𝑖 ! 2𝑗 ! 2𝑘 ! ]𝑥 𝑖𝑦 𝑗 𝑧 𝑘𝑒 −𝜁𝑟 2

SlaterandGaussianbasisfunctions0.90.80.7apn!0.60.500.40.30.20.103o2r (a.u.)Behavior of e where x = r (solid line, STO) and x = r2 (dashed line, GTO)A fixed linear combination of Gaussians to form amore suitable basis function0.6m0.5STO(r) =Cα GTO(r,Sα)α=10.2236r(a.u.)

Slater and Gaussian basis functions 6 • A fixed linear combination of Gaussians to form a more suitable basis function 𝑆𝑇𝑂(𝑟) = ෍ 𝛼=1 𝑚 𝑐𝛼 𝐺𝑇𝑂(𝑟, 𝜁𝛼)

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