490 14 Quantum Monte Carlo Methods
Since the potential is symmetric with respect to the interchange ofR→−Randr→−r
it means that the probability for the electron to move from one proton to the other must be
equal in both directions. We can say that the electron sharesit’s time between both protons.
With this caveat, we can now construct a model for simulatingthis molecule. Since we have
only one elctron, we could assume that in the limitR→∞, i.e., when the distance between
the two protons is large, the electron is essentially bound to only one of the protons. This
should correspond to a hydrogen atom. As a trial wave function, we could therefore use the
electronic wave function for the ground state of hydrogen, namely
ψ 100 (r) =(
1
πa^30) 1 / 2
e−r/a^0. (14.35)Since we do not know exactly where the electron is, we have to allow for the possibility that
the electron can be coupled to one of the two protons. This form includes the ’cusp’-condition
discussed in the previous section. We define thence two hydrogen wave functions
ψ 1 (r,R) =(
1
πa^30) 1 / 2
e−|r−R/^2 |/a^0 , (14.36)and
ψ 2 (r,R) =(
1
πa^30) 1 / 2
e−|r+R/^2 |/a^0. (14.37)Based on these two wave functions, which represent where theelectron can be, we attempt
at the following linear combination
ψ±(r,R) =C±(ψ 1 (r,R)±ψ 2 (r,R)), (14.38)withC±a constant. Based on this discussion, we add a second electron in order to simulate
the H 2 molecule. That is the topic for project 14.3.
14.3.The H 2 molecule consists of two protons and two electrons with a ground state energy
E=− 1. 17460 a.u. and equilibrium distance between the two hydrogen atoms ofr 0 = 1. 40 Bohr
radii. We define our systems using the following variables. Origo is chosen to be halfway
between the two protons. The distance from proton 1 is definedas−R/ 2 whereas proton 2
has a distanceR/ 2. Calculations are performed for fixed distancesRbetween the two protons.
Electron 1 has a distancer 1 from the chose origo, while electron 2 has a distancer 2. The
kinetic energy operator becomes then
−∇^21
2
−
∇^22
2
. (14.39)
The distance between the two electrons isr 12 =|r 1 −r 2 |. The repulsion between the two elec-
trons results in a potential energy term given by
+^1
r 12. (14.40)
In a similar way we obtain a repulsive contribution from the interaction between the two
protons given by
+^1
|R|, (14.41)
whereRis the distance between the two protons. To obtain the final potential energy we
need to include the attraction the electrons feel from the protons. To model this, we need
to define the distance between the electrons and the two protons. If we model this along a