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Dunning_BasisSets_HF - Solution of the Electronic...

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Solution of the Electronic Schrödinger Equation Using Basis Sets to Solve the Hartree- Fock Equations
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Using Basis Sets to Solve the Hartree-Fock Equations Comments on Hartree-Fock Theory ± Reasonable Description of Electronic Structure ± Most atoms ± Many molecules near their equilibrium geometries (although there are exceptions) ± Basis for More Accurate Calculations ± HF is zero-order wave function used in many more accurate models of the electronic structure of atoms and molecules that include electron correlation Configuration interaction Perturbation theory Coupled cluster theory
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Using Basis Sets to Solve the Hartree-Fock Equations Hartree-Fock Wave Function and Energy ± e (1,2,3,..., N e ) = ˆ A ² a (1) ² b (2) ² c (3) ± ² n ( N e ) = 1 N e ! ² a (1) ² a (2) ± ² a ( N e ) ² b (1) ² b (2) ± ² b ( N e ) ² ² ³ ² ² n (1) ² n (2) ± ² n ( N e ) Just to review, the Hartree-Fock (HF) wave function for N e electrons is where is a spin orbital, i.e. , a combination of a spatial function and a spin function. The energy of the HF wave function is: ± i E = h i i i = 1 N e ± + (J i j ² K i j ) j = i + 1 N e ± i = 1 N e ± = h i i i = 1 N e ± + 1 2 (J i j ² K i j ) j = 1 N e ± i = 1 N e ±
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Using Basis Sets to Solve the Hartree-Fock Equations Hartree-Fock Equations ˆ f HF ± i = [ ˆ h + ( ˆ J k ² ˆ K k ) k = 1 N e ³ ] ± i = ´ i i ± i The optimum orbitals to use in the HF wave function are the solutions of the HF equations where ˆ J k = dr 2 ± ² k * (2) ² k (2) r 12 ˆ K k = dr 2 ± ² k * (2) ˆ p 12 ² k (2) r 12 ( is the permutation operator) ˆ p 12 ˆ h = ± 1 2 ² 1 2 ± Z I r I 1 I = 1 N N ³ One electron operator: (kinetic energy plus nuclear attraction) Coulomb operator Exchange operator
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Using Basis Sets to Solve the Hartree-Fock Equations RHF Equations for Closed Shell Singlet States For a singlet state with all of the orbitals doubly occupied, N e = 2 n , and the HF wave function is 1 ± e RHF (1,2,3,..., N e ) = ˆ A ² a ² a ² b ² b ± ² n ² n ³´³´ ± ³´ ˆ f RHF ± i = ˆ h + (2 ˆ J k ² ˆ K k ) k = 1 n ³ ´ µ · ¸ ¹ ± i = º i ± i The HF equations are These are the closed shell Restricted Hartree-Fock (RHF) wave function and equations. For atoms and diatomic molecules, the integro-differential RHF equations can be solved by finite difference or finite element methods. For polyatomic molecules, these approaches pose significant mathematical problems and the equations have traditionally been solved using basis set expansion methods.
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Using Basis Sets to Solve the Hartree-Fock Equations Basis Set Expansion Method The orbital ± i in the RHF equation is a function of a set of one-electron variables, ( x 1 , y 1 , z 1 ), and thus it can be represented by an expansion in a basis set: ± i = ² ³ c ³ i ³ = 1 N bf ´ where ± ² is also a function of ( x 1 , y 1 , z 1 ).
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