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)