CASINO manual - Theory of Condensed Matter

• The effective time step, Eq. (34), is given by,
τ eff (α, m) = τ
∑
i m ip i ∆r 2 d,i
∑
i m i∆r 2 d,i
(269)
• The drift vector limiting, Eq. (35), takes the form,
ṽ i = −1 + √ 1 + 4D i a|v i | 2 τ
2a|v i | 2 v i . (270)
D i τ
• Separate Jastrow factors must be defined for the electron–electron, hole–hole and electron–hole
interactions. The general form of the cusp condition for Coulomb interactions is,
1 dΨ
Ψ dr ∣ = 2q iq j µ ij
r=0
d ± 1 , (271)
where q i and q j are the charges in units of the charge of the electron, µ ij = m i m j /(m i + m j ) is
the reduced mass and d is the dimensionality. The minus sign is used for distinguishable particles
(e.g., anti-parallel-spin electrons or electron and holes) and the plus sign for indistinguishable
particles (e.g., parallel-spin electrons).
• Backflow transformations for the pairing wave-function have to be carefully rederived.
• The kinetic energy term in the local energy is modified to include the mass,
Similarly,
and for the drift vector F i ,
N∑ N∑
K = K i = − 1 Ψ(R) −1 ∇ 2 i Ψ(R) . (272)
2m i
i=1
i=1
T i = − 1 ∇ 2 i (ln |Ψ|) = − 1 ∇ 2 i Ψ
4m i 4m i Ψ + 1 ( ) 2 ∇i Ψ
, (273)
4m i Ψ
32 Relativistic corrections to energies
F i = 1 √ 2mi
∇ i (ln |Ψ|) = 1 √ 2mi
∇ i Ψ
Ψ . (274)
Relativistic corrections to the nonrelativistic Hamiltonian can be calculated to order c −2 , where c is
the velocity of light, using first-order perturbation theory [81, 82]. This method works well for atoms
of low nuclear charge Z when the relativistic corrections are small, but is unsatisfactory when Z is
large.
In casino the perturbative relativistic corrections can be calculated for VMC or DMC 29 by setting
the relativistic flag in the input file to T. By default the relativistic corrections are not calculated.
First we consider the mass-polarization term ε 1 , which accounts for the correction due to the finite
total nuclear mass to order 1/M, where M is the total nuclear mass in a.u. (NB, this isn’t actually a
relativistic term as such.) If there is just one nucleus present then all finite-mass effects are accounted
for to O(M −1 ); otherwise some finite-mass effects are neglected. The estimator used for this term is
ε 1 = 1 ∑
v i · v j , (275)
M
i