Computational Physics - Department of Physics

14.5 Variational Monte Carlo for atoms 471

Hydrogen-like atomic radial functions

l\n 1 2 3

0 exp(−Zr) ( 2 −r)exp(−Zr/ 2 ) ( 27 − 18 r+ 2 r^2 )exp(−Zr/ 3 )

1 rexp(−Zr/ 2 ) r( 6 −r)exp(−Zr/ 3 )

2 r^2 exp(−Zr/ 3 )

Table 14.2The first few radial functions of the hydrogen-like atoms.

We can determineβby simply requiring^3

mke^2 β

h ̄^2

= 1 (14.17)

With this choice, the constantβbecomes the famous Bohr radiusa 0 = 0. 05 nma 0 =β=
̄h^2 /mke^2. We list here the standard units used in atomic physics and molecular physics calcu-
lations. It is common to scale atomic units by settingm=e=h ̄= 4 π ε 0 = 1 , see Table 14.3. We

Atomic Units

Quantity SI Atomic unit

Electron mass,m 9. 109 · 10 −^31 kg 1

Charge,e 1. 602 · 10 −^19 C 1

Planck’s reduced constant, ̄h 1. 055 · 10 −^34 Js 1

Permittivity, 4 π ε 0 1. 113 · 10 −^10 C^2 J−^1 m−^11

Energy, 4 π εe^20 a 0 27. 211 eV 1

Length,a 0 =^4 π εme^02 h ̄^20. 529 · 10 −^10 m 1

Table 14.3Scaling from SI units to atomic units.

introduce thereafter the variableλ

λ=

mβ^2

̄h^2

E,

and insertingβand the exact energyE=E 0 /n^2 , withE 0 = 13. 6 eV, we have that

λ=−

1

2 n^2

,

nbeing the principal quantum number. The equation we are thengoing to solve numerically
is now

−

1

2

∂^2 u(ρ)

∂ ρ^2

−

u(ρ)

ρ

+

l(l+ 1 )

2 ρ^2

u(ρ)−λu(ρ) = 0 , (14.18)

with the Hamiltonian

H=−

1

2

∂^2

∂ ρ^2

−

1

ρ

+

l(l+ 1 )

2 ρ^2

.

(^3) Remember that we are free to chooseβ.