CASINO manual - Theory of Condensed Matter

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one band is partly filled, while in non-metallic systems all the bands are fully occupied or empty. Exceptions arise in some systems in which the band is partially filled but the Coulomb interaction between the electrons is sufficiently strong that the electrons are localized. An alternative view to band theory would help to distinguish the character of the system. It is argued by Kohn [89] that the system is insulating as a result of wave function localization in the configuration space. The development of the Berry-phase theory of polarization [90, 91, 92, 93, 94] further advanced Kohn’s idea and provided tools to measure the localization of the electrons and the polarization. This Berry-phase approach solves the problem that the polarization is ill defined in an extended system by computing the quantity directly from the many-body wave function instead of the electron positions. Souza and Martin [93] provided the expressions for the localization tensor and the polarization in terms of a many-body Wannier wave function. This wave function is linked to Kohn’s wave function: for an insulating system it is localized in configuration space in the thermodynamic limit. Souza and Martin [93] also showed that the localization tensor is related to a frequency integral of the conductivity. The conductivity formula implies that for a system with non-vanishing conductivity the localization tensor is infinite, otherwise it has a finite value. In [95], the off-diagonal elements of the localization tensor are used to calculate the DC conductivity in the transverse direction for a quantum hall fluid. The localization tensor and polarization are written in terms of a many body operator 〉 Z (α) N 〈Ψ|e = iGα·X(Rm) |Ψ , (409) where G α is a simulation cell reciprocal lattice vector and X is the sum of electron positions ∑ n i=1 r i of a configuration R m . The Berry-phase polarization is given by P α = N V 〈r α〉 c , (410) where 〈r α 〉 c is the expectation value of the electron distribution given by 〈r α 〉 c = 1 NG α IlogZ (α) N , (411) the equation (410) measures the polarization current of the system in response to an adiabatic change of the Hamiltonian by approximating the Coulomb interaction as a first-order perturbation. The localization tensor can be interpreted as a measure of the quadratic spread of a charge distribution. This gives an indication of how well the electrons are localized in the simulation cell according to the wave function. This is written as where N is the number of electrons in the simulation cell. The off-diagonal elements of the localization tensor are defined by 〈 r 2 α 〉c = −1 NG 2 log |Z (α) N |2 , (412) α 〈r α r β 〉 c = −1 log |Z(α) N ||Z(β) N | NG α G β |Z (αβ) N | , (413) where Z (αβ) N is defined as 〈 Ψ|e −i(Gα−G β)·X(R m) |Ψ 〉 . In QMC, the localization tensor and polarization are calculated from Z N with the periodic boundary conditions imposed. This is done by summing the electron positions of each configuration R m to calculate e −iGα·X(Rm) . This is then averaged over configurations generated by VMC or DMC to give mean ¯z N = 1 M∑ Z N (R m ) , (414) M m=1 to ensure (414) tends to (409) and the statistical error is small, a large number of steps must be taken to sample the configuration. The localization tensor diverges when Z N is zero. Numerically, Z N would never be zero, but could become small so that the localization tensor becomes very large when approaching metal-insulator transition from the insulating side [96]. As the difference between ¯z and Z N (R m ) becomes large when approaching the divergence, it is often useful to examine the error bar in Z N to determine when the localization tensor diverges. 196
The figure shows the localization length of a linear chain of antiferromagnetic hydrogen atoms as a function of lattice spacing. The localization length has a maximum at 1.4 Å, indicating that the chain becomes metallic at around this lattice spacing. 10 Localization length (Bohr^2) 8 6 4 2 0 0 1 2 3 4 5 Lattice spacing (Å) 33.8 Dipole moment (molecules only) K eyword: dipole moment The electric dipole moment of a finite set of point charges of charge q i and position vector r i is defined to be p = ∑ q i r i . (415) i It is easy to show that the value of p is independent of the origin if the overall system is electrically neutral. We may use QMC to estimate the mean dipole moment of a molecule by averaging p over the set of configurations generated by the VMC or DMC algorithms. The mean value of |p| 2 may be determined in a similar fashion. This will be done if the dipole moment keyword is set to T in the input file. The raw data will be written to the vmc.hist or dmc.hist file (not the expval.data file!) and the error bars on components of the dipole moment should be evaluated using the reblock utility. Note that the casino reblock utility reports only the components and not the magnitude of the dipole moment in order to allow the user to decide how to deal with the symmetry. Suppose that symmetry dictates the dipole moment will point in the x direction. The y and z components should be zero, but there will be some noise when they are evaluated in QMC. If you work out |p| = √ |p x | 2 + |p y | 2 + |p z | 2 then you will get something larger than |p x |, tending to |p x | in the limit of perfect sampling (i.e., a biased estimate with finite sampling). You will also get larger error bars on |p| than |p| x . For an example application to the water molecule see J. Chem. Phys. 127, 124306 (2007). 34 Atomic forces The total atomic force is defined as the negative energy gradient with respect to the atomic position within the Born-Oppenheimer approximation, where the atomic positions are treated as parameters 197
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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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Page 239 and 240: [50] W. Janke, Statistical Analysis
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CASINO manual - Theory of Condensed Matter

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