CASINO manual - Theory of Condensed Matter

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27.1.2 ‘Standard’ QMC Let the number of electrons in our system be N. In ‘standard’ QMC the orbitals are HF or DFT eigenfunctions extending over the entire system. On the other hand, the basis functions used to represent those orbitals are usually localized functions. Each time an electron is moved, all O(N) occupied orbitals must be evaluated; however, only the O(1) basis functions in the vicinity of the electron need to be computed to evaluate each orbital. So, the time taken to carry out a configuration move of all N electrons in the system is O(N 2 ). If the orbitals are represented by extended basis functions, such as plane waves, then the time taken for a configuration move scales as O(N 3 ), because every basis function in every orbital has to be computed for every electron. 27.1.3 Localized orbitals It is easy to show that a nonsingular linear transformation of a set of orbitals can only change the normalization of the Slater determinant of those orbitals. For insulating materials there exist straightforward and efficient algorithms for determining linear transformations to highly localized sets of orbitals [70, 71]. Highly localized orbitals can be truncated to zero at a certain distance from their centres without introducing a substantial bias. Therefore, each time an electron is moved, only a few orbitals need to be evaluated; the others must be zero because the electron lies outside their truncation radii. So the number of orbitals to be computed after each electron move is O(1) [72]. The truncation of the orbitals results in small discontinuities in the Slater wave function. These are potentially serious for QMC because they result in the presence of Dirac delta functions in the kineticenergy integrand. The delta functions cannot be sampled, so their contribution to the total energy is lost. However, in practice the resulting bias is extremely small, provided the truncation radii are sufficiently large. In fact the bias can be made arbitrarily small by increasing the truncation radii. The discontinuities can be avoided by bringing the localized orbitals smoothly to zero over a thin, spherical skin. Surprisingly, it has been shown that the bias resulting from the use of smooth truncation schemes is larger than the bias that occurs if the orbitals are truncated abruptly at their cutoff radii [73]. The reason for this is that the smooth truncation schemes introduce a new, small length scale (the skin thickness) into the problem, and the local kinetic energy in the truncation region is extremely large. A time step that is physically reasonable for the system being simulated is much too large for the length-scale introduced by the truncation region. This results in large time-step bias and frequent population-explosion catastrophes. We therefore recommend truncating localized orbitals abruptly: the bsmooth input parameter should be set to F. 27.1.4 Localized bases Suppose the basis functions are zero outside fixed radii about their centres. Then the only functions that have to be evaluated when an electron is moved are those with the electron inside their radii. So the number of basis functions to be computed simply depends on the local environment of the electron and is therefore independent of system size. Gaussian basis functions can be regarded as localized if they are truncated to zero outside a certain radius. This is done by default in casino: Gaussian functions exp(−ar 2 ) are assumed to be zero when exp(−ar 2 ) < 10 −G T , where G T is the gautol input parameter. Gaussian basis sets cannot yet be used in conjunction with localized orbitals, however. Alternatively, orbitals can be represented numerically using blip functions, which also constitute a localized basis. If the orbitals are localized then the memory requirements are greatly reduced (by a factor of O(N)), because we only have to store the blip coefficients needed to evaluate the orbital within its truncation radius. 27.1.5 ‘Linear-scaling’ QMC If the numbers of orbitals and basis functions to be evaluated are both O(1) then the CPU time for a configuration move scales as O(N). This is what is meant by ‘linear-scaling QMC’. Note, however, 166
that the number of configuration moves required to achieve a given error bar on the total energy scales as O(N); hence, in practice, the CPU time for ‘linear-scaling’ QMC calculations scales as O(N 2 ). 27.2 Using CASINO to carry out ‘linear-scaling’ QMC calculations 27.2.1 Generation of Bloch orbitals At present the Bloch orbitals must be represented in a plane-wave basis. A plane-wave DFT code should be used to generate a pwfn.data file as though an ordinary QMC calculation with a plane-wave basis were to be carried out. Note that only one k point may be used: the Γ point. This is not a severe restriction since, for large systems, one would usually only carry out calculations at Γ anyway. 27.2.2 Generation of localized orbitals The localizer code should be used to carry out the linear transformation to a localized set of orbitals. The code has the following features: • localizer implements the method described in Ref. [70]. • localizer requires a pwfn.data file holding the Bloch orbitals represented in a plane-wave basis and a centres.dat file of format: Number of centres Display coefficients of linear transformation (0=NO; 1=YES) Use spherical (1) or parallelepiped (2) localization regions x,y & z coords of centres ; radius ; no. orbs on centre (up & dn) ... where ‘N’ is the total number of localization centres. If ‘iprint’ is 1 then the coefficients of the linear combination are written to stdout; otherwise they are not. ‘icut’ can take values 1 or 2, specifying that the localization regions are spherical or parallelepiped-shaped, respectively. ‘pos(i,j)’ is the ith Cartesian component of the position vector of the jth centre. ‘radius(j)’ is the cutoff radius for the jth localization centre. ‘norbs up(j)’ and ‘norbs dn(j)’ are the number of spin-up and spin-down orbitals to be localized on the jth centre, respectively. Note that if a parallelepiped-shaped localization region is used then the shape of the parallelepiped is defined by the lattice vectors, but the distance to each face is given by the cutoff radius. • The choice of localization centres requires some chemical intuition. See the discussion in Ref. [70]. At some point in the future, the optimization of the localization centres will be enabled. Note that in some highly symmetric molecules, symmetric choices of localization centres can lead to two localized orbitals being identical. This problem can be avoided by breaking the symmetry of the localization centres. • Note that the orbitals will become linearly dependent as two centres approach one another. In the limit that two centres are located in the same place, the orbitals localized on those centres will be identical. (One should instead define a single centre and increase the number of orbitals localized on that centre.) • localizer produces a pwfn.data.localized file that contains the localized orbitals represented in the same plane-wave basis as that specified in pwfn.data. • localizer can only work with Bloch orbitals at Γ. Since it is intended for use in large systems, this should not be a serious restriction. • One can only choose different numbers of localized orbitals for spin-up and spin-down electrons if pwfn.data contains spin-polarized data. 167
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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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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
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Page 139 and 140: where u k has the periodicity of th
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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
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Page 161 and 162: where the local energy, E L , is E
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Page 181 and 182: In the infinite system limit, the s
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Page 209 and 210: To each walker r j , we assign a li
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Page 215 and 216: the orbitals, α = 2 [11]. The use
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B Appendix 2: Automatic testing of
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• 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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