CASINO manual - Theory of Condensed Matter

3. Use Eq. (260) to calculate the kinetic-energy correction.
4. As a check, calculate a αα and b αα in Eq. (262) using the values of ũ αα + ˜p αα at the two smallest
nonzero stars of G vectors. Then evaluate the kinetic-energy correction using Eq. (263). The
user can verify that the two estimates of the kinetic-energy are in reasonable agreement (the
agreement should improve as the system size is increased).
Similar steps are carried out for a 2D system, although only the ‘simple model’ is available in this
case.
In order to calculate the finite-size correction to the kinetic energy, the user should set the finite
size corr flag to T. The finite-size correction will only be calculated if the correlation.data
file contains a nonempty Jastrow factor with either p or u terms (but not u RPA terms). The finite-size
correction is displayed near the top of the out file.
Note that it is usually essential to use p terms in the Jastrow factor when calculating the kinetic-energy
correction. Even if the p term only gives a small decrease in the total energy, it is needed in order
to get the shape of the long-ranged two-body Jastrow factor correct. To generated a blank p term to
paste into the Jastrow factor in correlation.data, the make p stars utility can be used.
30 Finite-size correction to the interaction energy
An alternative to using the MPC interaction is to calculate corrections to the XC energy from an
accumulated structure factor (essentially as described in Ref. [17]). Accumulation of the structure
factor is automatically activated if finite size corr is set to T, and the finite-size correction to the
interaction energy is written out at the end of the out file.
31 Electron–hole systems
casino has the ability to include positively charged particles of variable mass (holes) in the simulation
in addition to electrons. Currently these may only be used in electron–hole phases without an external
potential, but the code needs only a few trivial changes for these things to be able to wander around
inside real crystals (useful for studying positron problems—contact Mike Towler if you want this to
be implemented).
In this section the changes required to the casino code and to the basic equations in the presence of
holes are discussed. These largely stem from the possibility of having a variable mass ratio between
the positively and negatively charged particles. The basic differences are:
• The diffusion Green’s function, Eq. (18), becomes,
(
)
G D (R ← R ′ 1
, τ) =
(4πD e τ) exp − (R e − R ′ e − 2τD e V e (R ′ e)) 2
3Ne/2 4D e τ
(
)
1
×
(4πD h τ) exp − (R h − R ′ h − 2τD hV h (R ′ h ))2 , (266)
3N h/2 4D h τ
where e and h denote electron and hole quantities, N e and N h are the numbers of electrons and
holes, the diffusion constants are defined as D e = 1/(2m e ) and D h = 1/(2m h ), where m e and
m h are the electron and hole masses in atomic units (i.e., in units of the mass of the electron).
• When particle i is moved, Eq. (22) becomes,
r i = r ′ i + χ + 2D i τv i (R ′ ), (267)
where χ is a 3D vector of normally distributed numbers with variance 2D i τ and zero mean.
• The probability of accepting this move, Eq. (26) is then,
]
p i ≃ min
[1, exp [(r ′ i − r i + τD i (v i (R ′ ) − v i (R)) · (v i (R ′ ) + v i (R))] Ψ(R)2
Ψ(R ′ ) 2 . (268)
178