CASINO manual - Theory of Condensed Matter

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where L is a cutoff length, C is the truncation order and Θ denotes the Heaviside function. Notice that the above can be easily extended to inhomogeneities other than atoms by introducing any other coordinates appearing in the potential energy into the expressions. This is not implemented in casino yet. 23.1.1 The η function Three forms of the homogeneous backflow function have been tested with casino. Following extensive tests, no advantages were found from using the rational and Gaussian forms, and only the polynomial form remains in the code, N ∑ η η ij = f(r ij ; L η ) c k rij k , (175) where N η is the expansion order and {c l } are the optimizable coefficients. We expect this simple form to be flexible and efficient. The natural polynomial basis is used to improve the speed of evaluation, and rounding errors are not likely to appear for N η + C < 20 24 . k=0 23.1.2 The µ function We use the following polynomial expansion for µ, N ∑ µ,I µ iI = f(r iI ; L µ,I ) k=0 d k,I r k iI , (176) where L µ,I is the cutoff length for ion I, N µ,I is the expansion order and {d k,I } are the optimizable coefficients. 23.1.3 The Φ and Θ functions Our choice for Φ jI i and Θ jI i are the following power expansions, N ∑ eN,I Φ jI i = f(r iI ; L Φ,I )f(r jI ; L Φ,I ) k=0 N ∑ eN,I Θ jI i = f(r iI ; L Φ,I )f(r jI ; L Φ,I ) k=0 N ∑ eN,I l=0 N ∑ eN,I l=0 N ∑ ee,I m=0 N ∑ ee,I m=0 ϕ klm,I r k iIr l jIr m ij , (177) θ klm,I r k iIr l jIr m ij , (178) where N eN,I and N ee,I are the expansion orders, L Φ,I are the truncation lengths and ϕ klm,I and θ klm,I are optimizable coefficients. 23.2 Constraints on the backflow parameters In SJ wave functions it is common practice to impose the electron–electron cusp conditions on the parameters in the Jastrow factor and the electron–nucleus ones on the orbitals in the Slater determinant. Backflow can modify the cusp conditions; we have constrained the backflow parameters so that they do not. When AE atoms are present, it can be shown that the electron–nucleus cusp conditions cannot be fulfilled unless there is no homogeneous backflow term present. However, this issue can be bypassed by smoothly truncating η(r ij ) around such nuclei. This is automatically done by casino. An additional set of constraints can be added in order to satisfy the relation ξ i ({r j }) = ∇ i Y ({r j }) , (179) where Y ({r j }) is an object called backflow potential, which appears in the derivation of backflow of Ref. [47]. This equation is already satisfied by both the electron–electron and the electron–nucleus 24 It is estimated that numerical problems arise in the evaluation of polynomials beyond order ∼ 20 when using double-precision arithmetics. More complicated polynomial forms (e.g., Chebyshev polynomials) should be used to be able to exceed this limit. 150
terms by definition, and it can be imposed on the electron–electron–nucleus backflow functions by using an appropriate set of constraints, which correspond to the ‘no-curl’ flag in the phi term in correlation.data. It is found that the ‘no-curl’ conditions do not give particularly good backflow functions; it has been left in casino as a means of drastically reducing the number of free parameters in cases where this is absolutely necessary. 23.3 Improving the nodes of Ψ T casino’s implementation of backflow has the ability to improve the nodal surface of the trial wavefunction with respect to that of the SJ form. There is an additional computational expense in going from SJ to BF, which is a factor of ∼ 2 for the AE neon atom—based on the CPU time taken to achieve a given error bar in the mean energy—in both VMC and DMC; other systems may vary, but we expect backflow to be a reasonably efficient way of reducing the fixed-node error in most cases. Plot of the wave-function nodes of a water molecule, performed by moving an electron in the molecular plane (XY ) while keeping all others fixed in an arbitrary configuration (generated using HF-VMC). 0.04 0.02 0 -1 -0.5 0 0.5 -0.02 HF SJ BF -0.04 -0.06 Plot of the HF, SJ and BF wave-functions across a node in a water molecule, corresponding to moving (x, y) from (−1, −1) to (−1, 0.5) in the previous graph. It should be noticed, however, that the application of backflow has a greater effect on VMC than on DMC. For instance, BF-VMC applied on AE neon has been found to recover a 53% of the correlation 151
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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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Page 109 and 110: 8.4.2 Generating gwfn.data files wi
Page 111 and 112: After successfully running properti
Page 113 and 114: % mv Test.FChk dna.Fchk run gaussia
Page 115 and 116: • By default, gaussian uses spin
Page 117 and 118: Routine Purpose qmc write Writes th
Page 119 and 120: not the case the defaults can be ch
Page 121 and 122: 8.10 TURBOMOLE Website: www.turbomo
Page 123 and 124: • 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: where n is the order of the expansi
Page 159 and 160: which is therefore independent of r
Page 161 and 162: where the local energy, E L , is E
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Page 165 and 166: The energy minimization method used
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Page 169 and 170: • Is emin min energy too high? Th
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Page 173 and 174: that the number of configuration mo
Page 175 and 176: • It is possible to use blip orbi
Page 177 and 178: 0.5 0.5 0.0 particle 1 det 1 : 9 or
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
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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
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The input keyword to activate the c
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To each walker r j , we assign a li
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37 Magnetic fields and the fixed-ph
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38 Analysis of the performance of C
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the orbitals, α = 2 [11]. The use
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39.2 Implementation basics and perf
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• Use ‘endif’ and ‘enddo’
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• Evidence of testing on serial a
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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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