Page 20 - 71 the abc of dft_opt

40 CHAPTER 4. TWO ELECTRONS
meaning that they are sums of terms which each depend on only one coordinate at a time:
 2 2 
1 d d
ˆ ˆ
T = −  2 + 2  V ext = v ext (x 1 ) + v ext (x 2 ) (4.3)
2 dx 1 dx 2
Chapter 4 The electron-electron repulsion operator is a two-body operator, each term depending on
two coordinates simultaneously. Note that it is this term that complicates the problem,
by coupling the two coordinates together. The interaction between two electrons in three
Two electrons dimensions is Coulombic, i.e., 1/|r − r |. This is homogeneous of degree -1 in coordinate
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scaling. However, 1/|x − x | in one-dimension is an exceedingly strong attraction, and we
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prefer to use δ(x − x ), which has the same scaling property, but is much weaker and more
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1 In this chapter, we review the traditional wavefunction picture of Schr¨odinger for two short-ranged.
electrons. We keep everything as elementary as possible, avoiding sophistry such as the We will now consider the problem of one-dimensional He as a prototype for two electron
interaction picture, second quantization, Matsubara Green’s functions, etc. These are all problems in general. The Hamiltionian is then:
valuable tools for studying advanced quantum mechanics, but are unneccessary for the 1  d 2 d 2 
ˆ
basic logic of our presentation. H = −  2 + 2  − Zδ(x 1 ) − Zδ(x 2 ) + δ(x 1 − x 2 ). (4.4)
2 dx 1 dx 2
To find an exact solution to this, we might want to solve the Schr¨odinger equation with
4.1 Antisymmetry and spin
this Hamiltonian, but to find approximate solutions, its much easier to use the variational
When there is more than one particle in the system, the Hamiltonian and the wavefunction principle: 1 2
ˆ
includes one coordinate for each particle, as in Eqn. (1.1) for the H 2 molecule. Furthermore, E = min % Ψ | H | Ψ &, dx 1 dx 2 |Ψ(x 1 , x 2 )| = 1. (4.5)
Ψ
electrons have two possible spin states, up or down, and so the wavefunction is a function both
The most naive approach might be to ignore the electron-electron repulsion altogether.
of spatial coordinates and spin coordinates. A general principle of many-electron quantum Then the differential equation decouples, and we can write:
mechanics is that the wavefunction must be antisymmetric under interchange of any two sets
of coordinates. All this is covered in elementary textbooks. For our purposes, this implies the Ψ(x 1 , x 2 ) = φ(x 1 )φ(x 2 ) (ignoring V ee ) (4.6)
2-electron wavefunction satisfies:
where both orbitals are the same, and satisfy the one-electron problem. We know from
Ψ(r 2 , σ 2 , r 1 , σ 1 ) = −Ψ(r 1 , σ 1 , r 2 , σ 2 ). (4.1) chapter 3 that this yields
√
φ(x) = Z exp(−Z|x|). (4.7)
The ground-states of our systems will be spin-singlets, meaning the spin-part of their wave-
function will be antisymmetric, while their spatial part is symmetric: This decoupling of the coordinates makes it possible to handle very large systems, since we
need only solve for one electron at a time. However, we have made a very crude approximation
Ψ(r 1 σ 1 , r 2 σ 2 ) = Ψ(r 1 , r 2 ) χ Singlet (σ 1 , σ 2 ), (4.2)
to do this. The contribution of the kinetic energy and the potential energy is then just twice
where the spatial part is symmetric under exchange of spatial coordinates and the spin part as big as in the single electron system:
1
χ Singlet (σ 1 , σ 2 ) = √ ( | ↑↓ & − | ↓↑ &) is antisymmetric. ( 2 ) 2
2 T s = 2 Z /2 V ext = 2(−Z ) V ee = 0 (4.8)
4.2 Hartree-Fock and therefore the total energy becomes
2
E = 2& 0 = −Z (4.9)
For more than one electron, the operators in the Hamiltonian depend on all the coordinates,
but in a simple way. Both the kinetic energy and the external potential are one-body operators, which gives for He (Z=2) E = −4. This is in fact lower than the true ground-state energy
1 c !2000 by Kieron Burke. All rights reserved. of this problem, because we have failed to evaluate part of the energy in the Hamiltonian.
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