CASINO manual - Theory of Condensed Matter

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Fpure HFT is the pure DMC HFT force, Fpure P is the pure DMC pseudopotential Pulay term, and the pure DMC nodal term Fpure N is an integral over the nodal surface Γ (defined by Ψ T = 0). Where terms appear in braces, the upper one refers to the FPLA and the lower to the SPLA. The form of the nodal term in Eq. (430) is independent of the localization scheme. The nodal term involves the gradient ∇ r Φ evaluated at the nodal surface Γ. Reference [102] shows that this gradient is defined as its limit when approaching the nodal surface from within a nodal pocket (where Φ is nonzero). The derivation of the nodal term from Eq. (420) can be found in Refs. [105, 102]. Although the HFT force Fpure HFT under the FPLA can be calculated in the pure DMC method, it is not straightforward to evaluate the action of the nonlocal operator (Ŵ + ) ′ on the unknown DMC wave function Φ B in Fpure HFT under the SPLA scheme. Therefore, the following localization approximation, (Ŵ − ) ′ Φ B Φ B ≃ (Ŵ − ) ′ Ψ T Ψ T , (431) is used in the evaluation of Fpure HFT (Ψ T − Φ). under the SPLA scheme which introduces an error of first order in Another complication arises in the pure DMC nodal term F N pure in Eq. (430) because it involves the evaluation of an integral over the nodal surface. It is unclear how to evaluate such an integral in a standard DMC simulation. The following relationship suggested in Ref. [102], F N pure = 2 F N mix + O[(Ψ T − Φ) 2 ], (432) allows the approximate evaluation of F N pure as twice its mixed DMC counterpart while introducing an error of second order in (Ψ T − Φ). Since F N pure is defined as a volume integral in Eq. (425), the pure DMC nodal term F N pure can then be calculated as a volume integral in a standard DMC simulation. Equation (432) is an application of the standard extrapolation technique [11], as in this case the variational estimate of the nodal term is zero [102]. 34.5 Implementation of forces in CASINO It is well-known that the HFT estimator has an infinite variance when the bare Coulomb potential is used to describe the electron–nucleus interaction. Different routes have been proposed to address this problem. Assaraf et al. [106, 107] added a term to the HFT force which has a zero mean value but greatly reduces the variance of the estimator. Chiesa et al. [108] developed a method to filter out the part of the electron density that gives rise to the infinite variance. Using soft pseudopotentials also eliminates the infinite variance problem [101], and this method is used in the casino code. Since the atomic force equals the negative total derivative of the DMC energy with respect to a nuclear position λ, all previous force expressions involve total derivatives, in particular Ψ ′ T . Unfortunately, it is not straightforward to calculate total derivatives in VMC and DMC. A different route is to approximate all total derivatives by their partial derivatives, which introduces an error of first order in (Ψ T − Φ). We expect, however, this approximation to be rather accurate for the following reason: taking the total derivative of the VMC or DMC energy with respect to λ gives dE dλ = ∂E ∂λ + ∑ i ∂E ∂c i dc i dλ , (433) where the c i are the parameters in Ψ T and the Hamiltonian. The sum on the right-hand side of Eq. (433) is neglected when all total derivatives are replaced with partial derivatives in all previous force expressions. This is exact in VMC when the wave function is optimized using energy minimization. In DMC, we can assume that the energy is approximately minimized with respect to the c i . Therefore, we expect that the parameters c i have little effect on both the VMC and DMC energy, i.e., ∂E/∂c i is small. Therefore, neglecting the sum in Eq. (433) or, equivalently, replacing all total derivatives with partial derivatives in the expressions above, is expected to be a good approximation. We use the analytic expressions derived in Ref. [101] for evaluating the HFT force. The Pulay terms may also be evaluated using analytic expressions, but to make the code more easily adaptable to other forms of trial wave function we use a finite-difference approach. This introduces an error which is linear in the infinitesimal nuclear displacement, ∆. We find that ∆ ≈ 10 −7 Å minimizes the resulting error in the Pulay terms, which is about seven orders of magnitude smaller than the estimated values of the total forces. See Ref. [109] for more information. 200
The input keyword to activate the calculation of forces in VMC and DMC is forces. The keyword forces info may be used to generate different levels of output. The default value is 2, which is recommended for speed. When 5 is chosen, two additional estimates of the Hellmann–Feynman force are calculated, which are only useful for debugging. These two additional estimates pick either the s-component or the p-component of the pseudopotential as the local one (the default is to pick the d-component as the local one). When calculating forces in DMC, the keyword future walking must be chosen. See also Sec. 35. Forces are only implemented for the Gaussian basis set and are only properly tested for molecules. When the molecule is linear (planar), the force algorithm in casino is optimized for geometries along the x-axis (x, y plane). See also the casino output. The forces are only implemented and tested for pseudopotentials. The implementation of forces in casino should in principle also work for allelectron calculations when the zero-variance estimators are chosen. These are the estimators ‘Total Forces+ZV’ in VMC and ‘Total Forces (mixed)’ in DMC. Also, since the SPLA scheme in DMC calculations requires an additional approximation in Eq. (431), we may expect that the FPLA scheme may give better results. For some small molecules and using large Gaussian basis sets, however, we find that the two localization schemes give very similar total forces. See also Refs. [110, 109]. 34.6 Explanation of the force estimators printed by CASINO In VMC: ‘Total Force(dloc)’ corresponds to Eq. (417) and the d-channel of the pseudopotential components is chosen local (default in casino); ‘HFT Force(dloc)’ is the first term in Eq. (417); ‘Wave function Pulay term’ is the second term in Eq. (417); ‘Pseudopotential Pulay term’ corresponds to Eq. (423) calculated in VMC, should have a zero average value, and is evaluated to check whether this condition is satisfied; ‘Total Force+ZV(dloc)’ is the same as ‘Total Force(dloc)’ with the zero-variance term added to reduce the statistical error; ‘Zero-variance term’ is the zero-variance term evaluated separately and should always have a zero average value; ‘VMC NT’ is a part of a total force estimator and should be added to the DMC estimator ‘Total Force(purHFT,purNT,dloc)’; see Ref. [111]. In DMC: ‘Total Force(purHFT, mixNT,dloc)’ corresponds to Eq. (427); ‘Total Force(purHFT, purNT,dloc)’ is a total force estimator when the VMC estimator ‘VMC NT’ is added, see Ref. [111]; ‘HFT Force(pur,dloc)’ is the first term in Eq. (427); ‘Nodal Term(mix)’ is the third term in Eq. (427) calculated as twice times the mixed nodal term, as stated in Eq. (432); ‘Nodal Term(pur)’ equals the third term in Eq. (427) calculated as a pure estimator and is part of ‘Total Force(purHFT,purNT,dloc)’; ‘Pseudopotential Pulay Term(pur)’ is the second term in Eq. (427); ‘Total Force(mix,dloc)’ corresponds to Eq. (421); ‘HFT Force(mix,dloc)’ is the first term in Eq. (421). 35 The future-walking method The standard DMC algorithm generates the ‘mixed’ probability distribution Ψ T Φ which can be used to calculate unbiased estimates of an operator A that commutes with the Hamiltonian, Ĥ. If, however, the operator A does not commute with Ĥ, the ‘pure’ probability distribution ΦΦ is required to obtain unbiased estimates. The simplest method to calculate pure estimates is to use the extrapolation estimator (2 time the mixed estimate minus the variational estimate) which introduces an error in the pure estimate that is of second order in (Ψ T − Φ). Exact pure estimates can be obtained, for example, by using the future-walking (FW) method which can straightforwardly be implemented in a standard DMC algorithm and will be discussed here, or by the reptation quantum Monte Carlo method. The FW method is used in casino with the keyword future walking. The basic idea of FW is to rewrite the pure estimate of a local operator A as 〈Φ|A|Φ〉 = 〈Φ| A Φ ∑ Ψ T |Ψ T 〉〈Φ|Φ〉〈Φ 0 | Φ0 Ψ T |Ψ T 〉 ≃ j A(r j)ω j (r j ) ∑ j ω , (434) j(r j ) with weights ω j (r j ) = Φ(r j) Ψ T (r j ) . (435) For the Eq. (434) to be satisfied, A must be a local operator. Once the weights are known, the pure estimate can be calculated as an average over the local quantity A j ω j with samples drawn from the mixed distribution generated by a standard DMC simulation. We will show in the next section, that 201
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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
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• quickblock: Simple reblocking u
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• In Movie Settings choose Trajec
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the move rejection probability is h
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This leads to the single-electron d
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In the UNR [19] scheme the modified
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13.7 Evaluating expectation values
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Population-control catastrophes sho
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where u k has the periodicity of th
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Although the constraint equations a
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18 Wave-function updating Consider
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19.2 Evaluating the nonlocal pseudo
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of a unit point charge at r j in ev
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19.4.3 1D Coulomb interaction Coulo
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To investigate whether the extrapol
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correlations, such as might be enco
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Page 237 and 238: [2] V.R. Saunders, R. Dovesi, C. Ro
Page 239 and 240: [50] W. Janke, Statistical Analysis
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CASINO manual - Theory of Condensed Matter

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