Electronic Structures - Chemistry

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2 If YES ⇒ go to 4 , get final Energy, C and F and finish. Thus, choice of basis sets is VERY IMPORTANT ! Therefore, let me to give a little bit more information about the basis sets Ψ = |ϕ i | ϕ i = Σ C iµ χ µ χ µ - is a Atomic Orbitals (Basis sets). In the mathematical sense, the best way to present of χ µ is Slater Type of Functional (or Orbitals) – STO – which centered at atomic centers (1) Slater basis functions (STOs) STO ~ e − r H − like 1s STO ~ re − r 2 s ~ 1s H − like ≠ 2 s H −like STO 2 p x ~ xe − r = 2p x STO 2 p y ~ ye − r = 2 p y H −like STO 2 p z ~ ze − r = 2p z ζ = “exponent” H −like ~ ( 2 − 2 r)e − r However, practically it is very difficult to use STO basis sets for molecular calculations. Therefore, people came to the idea to use the Gaussian Type of Functionals (Orbitals)- GTO. Gaussian Basis Functions • Formula of Cartesian GTO: GTO ~ e − r2 s GTO ~ xe − r2 , ye − r 2 , ze − r2 p GTO ~ xy ⋅ e − r 2 , xz ⋅ e − r 2 , yz ⋅ e − r 2 , x 2 ⋅e − r2 , y 2 ⋅ e − r 2 , z 2 ⋅e − r 2 (6d) d or xy⋅ e − r 2 , xz ⋅ e − r 2 , yz ⋅e − r 2 , ( x 2 − y 2 )e − r 2 , ( 2z 2 − x 2 − y 2 )e − r 2 (5d) 6d functions contain extra ‘s’ like function. f functions : (10f), (7f) α = “exponent” • Comparison between Gaussian-type orbitals and Slater-type orbitals ξ and α are positive numbers, determining the diffuseness or “size” of function: large exp. ⇒ small functional (spatially) 1. There are two major differences between STO and GTO: • at r = 0 ⎡ dx STO ⎤ ⎡ dx GTO ⎤ ⎣ ⎢ dr ⎥ < 0 , ⎦ dr r = 0 ⎣ ⎢ ⎥ = 0 ⎦ r =0 “GTO does not satisfy the cusp condition at r=0.” 2. At r →∞ , GTO , exponential of -αr 2 decays much rapidly the STO , exponential of -αr
3 AO Slater Density Gaussian Slater Gaussian Radius Radius Thus, GTO’s are easier to use than STO, but GTO’s are NOT optimum basis functions, because they have different functional behavior compared to the known functional behavior of Atomic Orbitals (at r = 0 and r →∞). What we shall do People came to the idea to use several GTO’s to better describe the known features of STO- functionals: or STO-NG basis sets: STO ⇒ N GTO AO Slater STO-3G STO-2G STO-1G Radius Another way is to use more Gaussian Functionals to as better as possible describe the AO’s (people call this number of Gaussians as “primitive” functionals). Even if we use a large number of (primitive) GTO’s, still we can reduce the computer time dramatically compared to STO’s. But computer time can be reduced, without sacrifing the accuracy of the calculations, by combining primitive functions into several groups, called Contracted GTO’s (CGTO’s): χ i CGF = Σ d iµ P µ GTO For example, we can take TWO GTO for describing S-orbital and combine them together as: χ 1s CGF = d 1 P 1 + d 2 P 2 Since here we used TWO primitives to get χ 1s CGF , this type of basis set (orbital) called “double-zeta” basis set, and so on.
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