CASINO manual - Theory of Condensed Matter

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therefore usually the worst possible value for estimating the total energy, although it does maintain the correct symmetry of the system. Twist averaging within the canonical ensemble means taking the average of expectation values over all simulation-cell Bloch vectors k s in the first Brillouin zone of the simulation cell, i.e., over all offsets to the grid of k vectors, keeping the number of electrons fixed [75]. At the HF level, the effect of twist averaging is to replace sums over the discrete set of single-particle orbitals by integrals over the volume of reciprocal space defined by placing copies of the simulation-cell Brillouin zone around each ground-state occupied k point. The boundary of this region tends to the Fermi surface in the limit of infinite system size and the region has the correct volume at all system sizes. Twist averaging removes a large part of the error caused by the use of a discrete set of orbitals, although some residual errors remain because the shape of the Fermi surface is not quite correct. At the single-particle level this gives a small positive bias to the KE, in which shell-filling effects are still visible. 28.2 Using twisted boundary conditions in CASINO 28.2.1 Complex wave functions and the fixed-phase approximation The procedure described in Sec. 15 for constructing real orbitals cannot be carried out if orbitals at arbitrary k points are used. In general the trial wave function must be complex under twisted boundary conditions. To specify that a complex wave function is to be used in casino the complex wf keyword should be set to T. Using complex arithmetic causes casino to run more slowly, so complex wf should be set to F wherever possible. The use of a complex wave function makes very little difference to the VMC and wave-function optimization algorithms. Since the expectation values of Hermitian operators are real, only the real parts of the local-energy components are calculated and gathered. The use of a complex trial wave function necessitates the use of the fixed-phase approximation in DMC [77], in which DMC wave function is constrained to have the same phase as the trial wave function. In fact it turns out that the fixed-node algorithm needs very few modifications to allow the use of complex wave functions within the fixed-phase approximation: the real part of the drift vector is used when proposing trial electron moves; it is not possible to reject node-crossing electron moves 28 ; and, as in VMC, only the real parts of the local energies are gathered. The fixed-phase approximation is discussed at greater length in Sec. 37. 28.3 Monte Carlo twist averaging within CASINO With casino twist-averaging can be done in two ways depending on the type of system. For electron(- hole) fluid phases the procedure can be performed wholly within casino. Unfortunately this is not the case for real systems with atoms since a new xwfn.data file is required for each twist angle, and such calculations must therefore be done in conjunction with an external wave-function generating code. These two methods are described below. 28.3.1 Electron(-hole) fluid phases For electron-hole fluid phases [75] the following procedure is used. At specified intervals during the simulation, a new k-vector offset is chosen at random, then a short period of equilibration is carried out before statistics accumulation is resumed. To specify an offset k s to the grid of k vectors for an electron(-hole) fluid, use the k offset keyword in the free particles block in input. This should be followed by the Cartesian components of the offset vector. For example: %block free_particles r_s 3 dimensionality 3 cell_geometry 0.0 0.5 0.5 0.5 0.0 0.5 28 In fact rejecting moves that cross the nodal surface has almost no effect on the time-step bias in systems with real wave functions anyway. 170
0.5 0.5 0.0 particle 1 det 1 : 9 orbitals free particle 2 det 1 : 9 orbitals free k_offset 0.1 0.2 0.3 %endblock free_particles Note that ground-state calculations for electron(-hole) fluids with real wave functions must have ‘magic’ numbers of electrons, such that there are no partially filled shells of k vectors. If the wave functions is complex, however, then any number of electrons may be used. This reflects the fact that for general twisted boundary conditions each shell contains just one k vector. Anyway, to specify that twist averaging is to be used in VMC, set vmc ntwist to the required number of twist angles to sample. After each change of the k-vector offset, vmc reequil nstep VMC equilibration moves will be carried out. Note that vmc reequil nstep can be much smaller than vmc equil nstep, because the distribution of configurations obtained at different k-vector offsets should be similar. A typical value for vmc reequil nstep might be 100. To specify that twist averaging is to be used in DMC, set dmc ntwist to the desired number of twist angles to sample. In the statistics-accumulation phase of DMC the following process will occur dmc ntwist times: (i) a random k-vector offset will be chosen, (ii) dmc reequil nstep DMC equilibration moves will be carried out, but no data will be written to dmc.hist and (iii) dmc stats nstep statistics-accumulation moves will be carried out. So the total amount of data generated is dmc ntwist × dmc stats nstep. Note that the number of post-twist-change equilibration moves dmc reequil nstep can be much less than the number of equilibration moves dmc reequil nstep because the configuration distribution at one twist angle is a reasonably good approximation to the configuration distribution at another. Obviously, if twist averaging is to be used in either VMC or DMC then complex wf should be set to T. The twist-averaged HF energy of the HEG will be calculated and written to the out file for single-determinant ground-state calculations. One can check that vmc reequil nstep and dmc reequil nstep are sufficiently large by setting use jastrow, backflow and ibran to F and comparing the VMC and DMC energies with the twist-averaged HF value (they should be the same). Note that one cannot expect to find a plateau when performing reblocking analysis until the block length is greater than the number of data points between changes of twist angle, i.e., greater than vmc nstep for VMC data and dmc stats nstep for DMC data. 28.3.2 Real systems For real systems one can perform the DFT/HF wave-function generation calculations at arbitrary grids of k points, then use the wave-function converters (including blip) in the usual fashion to generate casino input files, then read in and use those wave functions in casino, provided that complex wf is set to T. This functionality is not yet available for Gaussian orbitals, however. We provide a couple of scripts which quasi-automate this procedure: twistav pwscf for use with the pwscf plane-wave DFT code, and twistav castep for use on conjunction with the castep planewave DFT code, the castep2casino utility and the blip utility. Note that these scripts just produce the correct data files for you to analyse manually; they do not analyse the results themselves. How you want to analyse the data is up to you. Do you (i) just average the QMC energies and include the random error from twist averaging with the statistical error or (ii) do you use the DFT energies to correct the QMC energies before averaging or (iii) some other way. . . As an example, in Ref. [78] option (ii) was chosen. It is not generally necessary to re-optimize the Jastrow factor and backflow functions for the different twists; just a single correlation.data file should be supplied. 28.3.3 Monte Carlo twist averaging for real systems using CASINO and CASTEP This is automated with the twistav castep utility. This script repeatedly offsets the grid of k vectors in the castep .cell file and runs castep, then castep2casino, then blip and then casino. The castep and casino output files are put in directories called twist0001, twist0002, etc. The progress of the calculations is reported in a file called STATUS. All the necessary castep and casino input files must be set up in the directory in which the calculation is run. The complete list of k vectors 171
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CASINO User’s Guide Version 2.13
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8.6 GAUSSIAN94/98/03/09 . . . . . .
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29.2 Fourier transformation of CASI
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1 Introduction casino is a computer
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3.2 Legal stuff casino is given awa
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- Grossman-Mitas DMC-DFT molecular
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Your defined setup can then be perm
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- : perform compilation (default).
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6 Introductory user’s guide: how
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ATOM BASIS TYPE The basis used to r
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ENVMC v0.60: Script to extract VMC
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Std. err. in the mean DMC energy (a
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Jastrow factor from each cycle of t
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V(x) Ψ init (x) τ{ t Ψ 0 (x) x T
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the energy becomes roughly constant
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many cores, time limits etc, which
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Set the verbosity level of the mach
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6.7 How to run coupled DFT-DMC mole
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unqmcmd --startqmc=M Start the chai
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config.out This is the name under w
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‘slater-type’: use Slater-type
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CUSTOM SPAIR DEP (Block) This input
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initial configurations come from DM
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DTDMC (Real) Time step for DMC run
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FORCES INFO (Integer) Controls the
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nucleus is assumed to lie at the or
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MOVIECELLS (Logical) If F then casi
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OPT MAXITER (Integer) Largest permi
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of G vectors and discards all those
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half a million cores, it may be des
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VM REWEIGHT (Logical) If vm reweigh
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default this is 0 hartree. It shoul
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A very detailed specification of th
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-1 0 0 1 1 1 1 1 1 1 0 2 1 0 1 2 Pa
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cusp conditions; otherwise, only th
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Detailed information about each of
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|k| (outermost). Hence one can spec
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large and the wave function is near
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R(i) in atomic units 0.000000000000
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2. The charge-density Fourier coeff
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c_767: [ 996 ] ... Compressed expan
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valence charges for each atom %%%%%
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1988). This can be found online. An
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1.3813695578425E+00 1.3957621401173
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PWSCF Method: DFT DFT Functional: u
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0.125203784849961E-01 0.13042462855
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3.3000000000000E+00 2.0000000000000
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Potential Number of grid points 200
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9. Clearly, the total number of pos
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EBEST (DMC only) Best estimate of t
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Use G-vector set 1 Number of sets 2
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1000.0 rho_a(G)*rho_b(-G) 16.0 4.12
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total weight must always be include
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The exporter does not currently wor
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8.4.2 Generating gwfn.data files wi
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After successfully running properti
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% mv Test.FChk dna.Fchk run gaussia
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• By default, gaussian uses spin
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Routine Purpose qmc write Writes th
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not the case the defaults can be ch
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8.10 TURBOMOLE Website: www.turbomo
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• crysgen06/09, crystaltoqmc: The
Page 125 and 126: • quickblock: Simple reblocking u
Page 127 and 128: • In Movie Settings choose Trajec
Page 129 and 130: the move rejection probability is h
Page 131 and 132: This leads to the single-electron d
Page 133 and 134: In the UNR [19] scheme the modified
Page 135 and 136: 13.7 Evaluating expectation values
Page 137 and 138: Population-control catastrophes sho
Page 139 and 140: where u k has the periodicity of th
Page 141 and 142: Although the constraint equations a
Page 143 and 144: 18 Wave-function updating Consider
Page 145 and 146: 19.2 Evaluating the nonlocal pseudo
Page 147 and 148: of a unit point charge at r j in ev
Page 149 and 150: 19.4.3 1D Coulomb interaction Coulo
Page 151 and 152: To investigate whether the extrapol
Page 153 and 154: correlations, such as might be enco
Page 155 and 156: where n is the order of the expansi
Page 157 and 158: terms by definition, and it can be
Page 159 and 160: which is therefore independent of r
Page 161 and 162: where the local energy, E L , is E
Page 163 and 164: Another variant of variance minimiz
Page 165 and 166: The energy minimization method used
Page 167 and 168: valid. The first problem, which ari
Page 169 and 170: • Is emin min energy too high? Th
Page 171 and 172: It may be activated by setting vmc
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Page 175: • It is possible to use blip orbi
Page 179 and 180: The clearup twistav script can be u
Page 181 and 182: In the infinite system limit, the s
Page 183 and 184: Note that this has the k −2 diver
Page 185 and 186: • The effective time step, Eq. (3
Page 187 and 188: 1 2 H He 1.00794 4.00260 3 4 5 6 7
Page 189 and 190: and the Dirac delta function is δ(
Page 191 and 192: 33.2.2 Atomic densities K eywords:
Page 193 and 194: The spin-density matrix is a two-by
Page 195 and 196: 33.3.3 Homogeneous and isotropic sy
Page 197 and 198: 33.4 Structure factor and spherical
Page 199 and 200: 33.5 One-body density matrix, two-b
Page 201 and 202: [ where the approximation ρ (1)T
Page 203 and 204: The figure shows the localization l
Page 205 and 206: with F HFT mix = − F P mix = −
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Page 209 and 210: To each walker r j , we assign a li
Page 211 and 212: 37 Magnetic fields and the fixed-ph
Page 213 and 214: 38 Analysis of the performance of C
Page 215 and 216: the orbitals, α = 2 [11]. The use
Page 217 and 218: 39.2 Implementation basics and perf
Page 219 and 220: • Use ‘endif’ and ‘enddo’
Page 221 and 222: • Evidence of testing on serial a
Page 223 and 224: B Appendix 2: Automatic testing of
Page 225 and 226: • Delete any eepot.data and densi
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* "Generic" CASINO_ARCHs are intend
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- ‘head -n 1 /etc/redhat-release
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for a single job in an ensemble of
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inary executable. CASINO does not r
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otherwise. The following tags are o
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[2] V.R. Saunders, R. Dovesi, C. Ro
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[50] W. Janke, Statistical Analysis
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CASINO manual - Theory of Condensed Matter

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