Computational Physics

3.2 Examples of variational calculations 35

where the second term in the square brackets is the Coulomb attraction potential of
the nucleus. The massmis the reduced mass of the proton–electron system which
is approximately equal to the electron mass. The ground state is found at energy

E=−

m


^2

(

e^2

4 π 0

) 2

≈−13.6058 eV (3.22)

and the wave function is given by

ψ(r)=

2

a^30 /^2

e−r/a^0 (3.23)

in whicha 0 is the Bohr radius,

a 0 =

4 π 0 
^2

me^2

≈0.529 18 Å. (3.24)

In computer programming, it is convenient to use units such that equations take
on a simple form, involving only coefficients of order 1. Standard units in electronic
structure physics are so-calledatomic units:the unit of distance is the Bohr radius
a 0 , masses are expressed in the electron massmeand the charge is measured in unit
charges (e). The energy is finally given in ‘hartrees’ (EH), given bymec^2 α^2 (αis
the fine-structure constant andmeis the electron mass) which is roughly equal to
27.212 eV. In these units, the Schrödinger equation for the hydrogen atom assumes
the following simple form:
[
−

1

2

∇^2 −

1

r

]

ψ(r)=Eψ(r). (3.25)

We try to approximate the ground state energy and wave function of the hydrogen
atom in a linear variational procedure. We use Gaussian basis functions which will
be discussed extensively in the next chapter (Section 4.6.2). For the ground state,
we only need angular momentuml=0 functions (s-functions), which have the
form:
χp(r)=e−αpr

2

(3.26)

centred on the nucleus (which is thus placed at the origin). We have to specify the
values of the exponentsα; these are kept fixed in our program. Optimal values for
these exponents have previously been found by solving thenonlinearvariational
problem including the linear coefficientsCpandthe exponentsα[5]. We shall use
these values of the exponents in the program:

α 1 =13.007 73

α 2 =1.962 079

α 3 =0.444 529

α 4 =0.121 949 2.

(3.27)