Computational Physics

Table 12.1.

Variational Monte Carlo energies.

Harmonic oscillator

Hydrogen atom

Helium atom

α

〈E

〉

var

(〈

E〉

)

Ev

var

(E

)v

α

〈E

〉

var

(〈

E〉

)α

〈E

〉

var

(〈

E〉

)

0.4

0.5124

(^1

)

0.02521

(^5

)

0.5125

0.0253125

0.8

−

0.4796(2)

0.0243(6)

0.05

−

2.8713

(^4

)

0.1749

(^2

)

0.45

0.50276

(^4

)

0.00556

(^2

)

0.50278

0.00557

0.9

−

0.4949(1)

0.0078(2)

0.075

−

2.8753

(^4

)

0.1531

(^2

)

1/2

1/2

0

1/2

0

1.0

−

1/2

0

0.10

−

2.8770

(^3

)

0.1360

(^2

)

0.55

0.50232

(^6

)

0.00454

(^1

)(

1 )

0.5022727

0.0045558

1.1

−

0.4951(2)

0.0121(4)

0.125

−

2.8780

(^4

)

0.1223

(^2

)

0.6

0.5084

(^1

)

0.0168

(^4

)

0.508333

0.0168056

1.2

−

0.4801(3)

0.058(2)

0.15

−

2.8778

(^3

)

0.1114

(^2

)

0.175

−

2.8781

(^3

)

0.1028

(^2

)

0.20

−

2.8767

(^4

)

0.0968

(^2

)

0.25

−

2.8746

(^10

)

0.0883

(^2

)

VMC energies are given for the harmonic oscillator, the hydrogen atom and the helium atom for various values of the variationalparameters. In each case, 400 walkers have been used and 30 000 displacements per walker were attempted. The first 4000 of thesewere removed from the data to ensure equilibrium. The expectation value

〈E

〉of the ground state energy is given, together with the

variance in this quantity, var

(〈

E

〉)

. For the harmonic oscillator, the analytical values for the energies and variance are also given (

Ev

and var

(E

)v

).