Computational Physics

5.4 Beyond DFT: one- and two-particle excitations 103 because they impose this property in their form of the energy-functional ( ...

104 Density functional theory the energy spectrum, according to the time–energy uncertainty relation. Therefore, we can no longe ...

5.4 Beyond DFT: one- and two-particle excitations 105 Such a many-body theory was formulated by Hedin in 1965 [ 28 ], for review ...

106 Density functional theory is given as the integral of the induced charge n(r′′|r)overr, weighted by|ψ(r)|^2. Putting these r ...

5.4 Beyond DFT: one- and two-particle excitations 107 The total change in the density is therefore given by n(r′)= 2 ∑ jocc. ψj∗ ...

108 Density functional theory 5.4.3 Two-particle effects A two-particle description within the many-body theory of Green’s funct ...

5.5 A density functional program for the helium atom 109 att+h/2, wherehis the time step in going fromttot+h. As the Hamiltonian ...

110 Density functional theory of using basis functions, we solve the radial Schrödinger equation directly, just as we have done ...

5.5 A density functional program for the helium atom 111 To arrive at a program which determines the spectrum for you, you must ...

112 Density functional theory the radial wave function before integrating Poisson’s equation! If you take for the normalisation ...

5.5 A density functional program for the helium atom 113 The next step is to make the program self-consistent. This is done by a ...

114 Density functional theory normalised as in(5.76), our exchange potential reads Vx(r)=− [ 3 u^2 (r) 2 π^2 r^2 ] 1 / 3 (5.85) ...

5.6 Applications and results 115 Table 5.1.Lattice constants and cohesive energies for diamond,SiandGe. Atomic units are used. L ...

116 Density functional theory with DFT. The best approach is to use many-body theories for calculating actual excitation energie ...

Exercises 117 5.2 [C] The Hartree energy EH= 1 2 ∫ d^3 rd^3 r′ n(r′)n(r) |r−r′| overestimates the classical electrostatic energy ...

118 Density functional theory Table 5.4.Parameters for correlation energy Unpolarised Polarised A 0.0311 0.01555 B −0.048 −0.026 ...

References 119 charge from the system. In practice this means that the highest level (which is level N) is not fully occupied. W ...

120 Density functional theory [6] P. Phariseau and W. M. Temmerman,The Electronic Structure of Complex Systems, NATO ASI series ...

References 121 [31] C. Cohen-Tannoudji, B. Diu, and F. Laloë,Quantum Mechanics, vols. 1 and 2. New York/Paris, John Wiley/Herman ...

6 Solving the Schrödinger equation in periodic solids In the previous chapter we encountered density functional theory (DFT) whi ...