CASINO manual - Theory of Condensed Matter

• Spin-density - spin density (ATOM, PER)
• Reciprocal-space pair-correlation function - pair corr (PER, HOM)
• Spherical real-space pair-correlation function - pair corr sph (FIN, HOM)
• Structure factor - structure factor (PER, HOM)
• Spherically averaged structure factor - struc factor sph (HOM)
• One-body density matrix (OBDM) - onep density mat (HOM)
• Two-body density matrix (TBDM) - twop density mat (HOM)
• Condensate fraction: unbiased TBDM, goes as TBDM − OBDM 2 - cond fraction (HOM)
• Momentum density - mom den (HOM)
• Localization tensor - loc tensor (PER)
• Dipole moment - dipole moment (MOL)
By default these observables are not accumulated during a VMC/DMC simulation; to do this one must
set to T the corresponding input keyword (the bold terms in brackets in the above list). With the exception
of the dipole moment (for which the required information is stored in the vmc.hist/dmc.hist
file) the activation of any of the above keywords will flag the creation of an expval.data file (see Sec.
7.11) wherein the required data will be accumulated.
The data in the expval.data file is stored in independent sets corresponding to each observable. If
a data set is already present at the start of a calculation, then any newly accumulated data will be
added to the existing data. The expval.data file also contains basic information about the system,
plus all the G-vector sets necessary to represent any reciprocal-space quantities.
At the end of the calculation, the data in expval.data can usually be visualized using the plot expval
utility. The use of this program is fairly self-explanatory. Type ‘plot expval’ in any directory containing
an expval.data file, and the utility will read the data then ask you a series of questions
designed to elicit information about exactly what kind of plot you want. It will then write out the
data in a file readable by standard plotting programs such as xmgrace (for 1D data) or gnuplot (for
2D/3D data). A casino shell-script—plot 2D—is available which will call gnuplot with appropriate
arguments.
Note that, for operators that do not commute with the Hamiltonian, the error in the usual DMC
mixed estimator will be linear in the error in the wave function. However, the error in the extrapolated
estimator 2p DMC − p VMC will be quadratic in the error in the wave function (Here p VMC and p DMC
are the VMC and DMC estimates of the expectation value using the same wave function). One may
also use the future walking technique (see Sec. 35) to obtain better estimates for such observables.
After a short summary of relevant theoretical results, each expectation value will now be described in
turn.
33.1 Basics
33.1.1 Fourier transforms
Define the Fourier transform and its inverse by
˜f(G) = 1 ∫
f(r) e iG·r dr (281)
Ω Ω
f(r) = ∑ ˜f(G) e −iG·r , (282)
G
where the set of G vectors are the reciprocal lattice vectors of the simulation cell lattice. Using this
definition, the Kronecker delta is
δ G,G ′ = 1 ∫
e i(G′ −G)·r dr, (283)
Ω
182