Computational Physics

Exercises 83 The bosons interact via aδ-function potential: H=− ∑N i= 1 ∂^2 ∂xi^2 +g ∑N i>j δ(xi−xj). This means that the par ...

84 The Hartree–Fock method 4.7 Consider a Slater determinant constructed from spin-orbitalsψk,k=1,...,N.A unitary transformation ...

Exercises 85 The matrix to be diagonalised is given by Fpq=hpq+ ∑ rs CrCsQprqs with Qprqs=〈pr|g|qs〉= ∫ d^3 r 1 d^3 r 2 φp(r 1 )φ ...

86 The Hartree–Fock method (a) Suppose the basis containsMbasis functions, at least how many of these matrix elements must be ca ...

References 87 mentioned in Section 4.3.1, the Hartree approach is good enough for the ground state of a two-electron system beca ...

88 The Hartree–Fock method [8] W. J. Hehre, L. Radom, P. R. Schleyer, and J. A. Pople,Ab Initio Molecular Orbital Theory.New Yor ...

5 Density functional theory 5.1 Introduction In the previous chapter we saw how the many-electron problem can be treated in the ...

90 Density functional theory 5.1.1 Density functional theory: physical picture In density functional theory, an effective indepe ...

5.1 Introduction 91 charge density. This again contrasts with Hartree–Fock where the one-electron spin- orbitals have a definite ...

92 Density functional theory wave functionswhich are consistent with the densityn(r), we can write E[n]=min |n 〈|H|〉 (5.6) a ...

5.1 Introduction 93 electrons can be solved for trivially, and we can use this to tackle the problem of interacting electrons. I ...

94 Density functional theory Hartree energy respectively. It is important to note that we have made no approx- imations so far b ...

5.2 The local density approximation 95 forExc, as the exact form of this functional is unknown, and our approximation might be b ...

96 Density functional theory which we have already encountered at the end ofSection 4.5.1. The value for the constant is found a ...

5.3 Exchange and correlation: a closer look 97 5.3 Exchange and correlation: a closer look 5.3.1 The adiabatic theorem and the n ...

98 Density functional theory where we have used the definition Fλ[n]=min ψ|n 〈|H0,λ|〉 (5.31) The minimisation is carried out o ...

5.3 Exchange and correlation: a closer look 99 the full many-body wave function into account: it is the exchange-only part of th ...

100 Density functional theory which follows directly from the normalisation of the wave function. Furthermore, ∫ n(x)dx=N (5.46) ...

5.4 Beyond DFT: one- and two-particle excitations 101 been carried out by several groups, and some well known functionals are th ...

102 Density functional theory this to be electron numberN) of a neutral system is moved very far away from all the nuclei in the ...