Computational Physics

4 The Hartree–Fock method 4.1 Introduction Here and in the following chapter we treat two different approaches to the many- elec ...

44 The Hartree–Fock method approach. There exist several methods that improve on the approximations made in the HF method. The H ...

4.2 The Born–Oppenheimer approximation and the IP method 45 Therefore, important approximations must be made, and a first step c ...

46 The Hartree–Fock method The resulting uncoupled orindependent-particle(IP) Hamiltonian has the form HIP= ∑N i= 1 [ p^2 i 2 m ...

4.3 The helium atom 47 Schrödinger equation and we are left with:^2 [ − 1 2 ∇ 12 − 1 2 ∇ 22 − 2 r 1 − 2 r 2 + 1 |r 1 −r 2 | ] φ( ...

48 The Hartree–Fock method The neglect of correlations sometimes leads to unphysical results. An example is found in the dissoci ...

4.3 The helium atom 49 on the left hand side we recognise the potential resulting from a charge distribution caused by all the e ...

50 The Hartree–Fock method overr 1 leads to ∑ pq ( hpq+ ∑ rs CrCsQprqs ) Cq=E′ ∑ pq SpqCq (4.14) with hpq= 〈 χp ∣∣ ∣ ∣− 1 2 ∇^2 ...

4.3 The helium atom 51 The program is constructed as follows. First, the 4×4 matriceshpq,Spqand the 4× 4 × 4 ×4 arrayQprqsare c ...

52 The Hartree–Fock method 4.4 Many-electron systems and the Slater determinant In the helium problem, we could make use of the ...

4.4 Many-electron systems and the Slater determinant 53 states: (x 1 ,...,xN)=ψ 1 (x 1 )···ψN(xN). (4.24) The one-electron stat ...

54 The Hartree–Fock method To find the probability of finding two electrons at positionsr 1 andr 2 ,we must sum over the spin va ...

4.5 Self-consistency and exchange: Hartree–Fock theory 55 . . ... . . .. . .. (a) (b) (c) Figure 4.1. The Hartree–Fock spectrum. ...

56 The Hartree–Fock method are filled. Of course, it is not clear a priori that the lowest energy of the system is found by fill ...

4.5 Self-consistency and exchange: Hartree–Fock theory 57 therefore calculate the expectation value of the energy for an arbitra ...

58 The Hartree–Fock method We now define the operators Jk(x)ψ(x)= ∫ ψk∗(x′) 1 r 12 ψk(x′)ψ(x)dx′ and (4.39a) Kk(x)ψ(x)= ∫ ψk∗(x′ ...

4.5 Self-consistency and exchange: Hartree–Fock theory 59 where in the second step the following symmetry property of the two-el ...

60 The Hartree–Fock method The resulting statesψ′kthen form an orthonormal set, satisfying(4.50)with (^) kl= ∑ lm UkmmUml†. (4. ...

4.6 Basis functions 61 spirit as in the previous chapter and inSection 4.3.2can be used, that is, expanding the spin-orbitalsψka ...

62 The Hartree–Fock method This describes two electrons located at different nuclei, which is correct for large nuclear separati ...