Computational Physics

4.6 Basis functions 61

spirit as in the previous chapter and inSection 4.3.2can be used, that is, expanding
the spin-orbitalsψkas linear combinations of a finite number of basis statesχp:

ψk(x)=

∑M

p= 1

Cpkχp(x). (4.55)

Then(4.51)assumes a matrix form

FCk=kSCk (4.56)

whereSis the overlap matrix for the basis used.
In the next subsection we shall consider how spin and orbital parts are combined
in the basis sets and inSection 4.6.2we shall discuss the form of the orbital basis
functions.

4.6.1 Closed- and open-shell systems

In aclosed-shellsystem, the levels are occupied by two electrons with opposite spin
whereas in anopen-shellsystem there are partially filled levels containing only one
electron. If the number of electrons is even, the system does not necessarily have to
be closed-shell since there may be degenerate levels (apart from spin-degeneracy)
each containing one electron – or we might be considering an excited state in which
an electron is pushed up to a higher level. If the number of electrons is odd, the
system will always be open-shell.
Consider the addition of an electron to a closed-shell system. The new electron
will interact differently with the spin-up and -down electrons present in the system,
as exchange is felt by parallel spin pairs only. Therefore, if the levels of the system
without the extra electron are spin-up and -down degenerate, they will now split
into two sublevels with different orbital dependence, the lower sublevel having the
same spin as the new electron.^4 We see that the spin-up and -down degeneracy of
the levels of a closed-shell system is lifted in the open-shell case.
It is important to note that even when the number of electrons is even, the unres-
tricted solution may be different from the restricted one. To see this, consider again
the discussion of the hydrogen molecule in Section 4.3.1. A possible description of
the state within the unrestricted scheme is

ψ(x 1 ,x 2 )=

1

√

2

[u(r 1 −RA)α(s 1 )u(r 2 −RB)β(s 2 )

−u(r 2 −RA)α(s 2 )u(r 1 −RB)β(s 1 )]. (4.57)

(^4) An exception to this rule occurs when the Coulomb interaction between the degenerate levels and the new
electron vanishes as a result of symmetry.