Matrix Hartree-Fock Equations for a Closed Shell System

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1 atix Hatee-Fock Equations fo a Closed Shell System A single deteminant wavefunction fo a system containing an even numbe of electon N) consists of N/ spatial obitals, each occupied with an α & β spin has the fom ψ,, 3,, N) = A ϕ ) α) ϕ ) β) ϕ 3) α3) ϕ 4) β4) ϕ N ) α N ) ϕ N) β N) The total electonic enegy is N/ N/ N/ N/ i i i) E = ψ H ψ = f + J K Whee the one electon tems ae i= i, nuclei Zk i = ϕi i k= Rk f ϕ the coulomb, Ji exchange Ki tems ae * * Ji = ϕi ) ϕi) g, ) ϕ ) ϕ ) dv ) dv ) * * Ki = ϕi ) ϕ ) g, ) ϕi ) ϕ ) dv ) dv ) whee g, ) =. The optimal obitals obey the Hatee-Fock equations whee whee Fϕ = εϕ i i i N / = ) F = f + U = f + J K J. F. Haison Januay 005

2 f = nuclei k= Zk R k = ) ϕ ),) ϕ ) = ) ϕ),) J dv g dv g K ) ),) = dv ϕ g Pϕ) χ µ µ= To solve the Hatee-Fock equations we intoduce an basis of known functions { } which ae usually atomic like obitals hosted by the vaious nuclei in the system wite ϕ = χ c = χc i µ µ i i µ = whee we define the ow column vectos χ = χχ χ3 χ ) The Hatee-Fock equations become c i c i ci = c i χµ µ i = εi χµ µ i µ = µ = F c c * if one multiplies by χ µ integates one obtains Fνµ cµ i = εi µν cµ i µ = µ = o F c = ε c i i i whee we define the Fock F ) ovelap ) matices as J. F. Haison Januay 005

3 F = χ F χ = F ) µν µ ν µν = χ χ = ) µν µ ν µν Note that one can define a matix ccc c ) wite the eiegnvalue poblem as c = as a ow vecto of column vectos 3 whee Fc = cε ε ) = i i i δ ε The Fock matix is composed of one two electon tems n / ) F = χ f + U χ = χ f + J K χ µν µ ν µ ν = The one electon tem, χµ f χ ν, is easily calculated wheeas the two electon tems, χµ U χ ν ae moe complicated as they depend on the expansion coefficients c i which ae to be detemined. Intoducing the basis function expansion into J & K gives = ) = J dv) g,) ϕ c c dv) χ ) g,) χ ) λ ρ λ ρ λ= ρ= K dv) ϕ g,) P ϕ c c dv) χ ) g,) P χ ) = ) ) = λ ρ λ ρ λ= ρ= so ) J K = c c dv) χ ) g,) P χ ) λ ρ λ ρ λ= ρ= The two electon tem becomes J. F. Haison Januay 005 3

4 n/ ) χ U χ = c c dv) χ ) χ ) g,) P χ ) χ ) µ ν λ ρ µ λ ν ρ = λ= ρ= Since the only tem in the sum that depends on is the poduct of the expansion coefficents we can intechange the ode of the summations wite ) χ U χ = P dv) χ ) χ ) g,) P χ ) χ ) µ ν λρ µ λ ν ρ λ= ρ= whee the matix P has the elements P ) λρ n / = P =c c λρ λ ρ = is called the Density matix. The integal is often witten in a moe compact fom as ) µλ g,) P ) νρ = dv) χ ) χ ) g,) P χ ) χ ) = µ U ν = Pλρ λ= ρ= µ λ ν ρ χ χ The elements of the Fock matix ae then µν = µν + λρ λ= ρ= F f P To solve fo c & ε one fist calculates all of the equied integals, µν,f µν,& then fom an estimate of P. One then foms the Fock matix solves the equations Fc = cε. The calculated vectos in c ae then used to calculate a new P which is compaed to the initial estimate. They ae almost always diffeent so one foms a new Fock matix with the new P, calculates an updated P with the esulting vectos in c compaes the two. If they diffe one epeats the iteations until the input output P ae identical to some ageed on accuacy. The final c & ε ae the molecula obital coefficients one electon enegies. Once one has the molecula obital coefficients one can calculate the total enegy as an expectation value of the Hamiltonian Ĥ augmented with the nuclea epulsion enegy V NN J. F. Haison Januay 005 4

5 N/ N/ Nuc ) E = ψ H ψ + V = f + J K + NN i i i i= i, K< This may be simplified consideably by noting that ZK Z R K N/ N/ N/ f f c c = ϕ ϕ = µ f ν = P f = tacepf i i i µ i ν i µν µν i= i= i= µ = ν = µ = ν= In a simila fashion we find N/ i i ) = J K P P µν λρ i, µ = λ= ρ= ν= The electonic enegy becomes Pµν fµν + Pµν Pλρ = Pµν fµν + Pλρ µ = ν = µ = λ= ρ= ν = µ = ν= λ= ρ= Which fom the definition of the Fock matix given above we ecognize as µ = ν= + ) = ) Pµν fµν Fµν tacep f +F So to obtain the electonic enegy we take the conveged Fock matix, add the one electon opeato matix, multiply the sum by the conveged density matix fom the tace. The total enegy is then Nuc = ψ ψ + NN = P f +F ) + E H V tace K< ZK Z R K J. F. Haison Januay 005 5