{ typeset by foiltex · 2013-08-12 · atom, but consider variance of hellmann-feynman derivative...

– Typeset by FoilTEX – 1

QUANTUM MONTE CARLO

Forces and dynamics. Expectation values other than the energy.

Mike Towler

TCM Group, Cavendish Laboratory, University of Cambridge

QMC web page: www.tcm.phy.cam.ac.uk/∼mdt26/casino2.html

Email: [email protected]


Derivatives of the energy

E(Fi) = E(0) + Fi

[∂E

∂Fi

]Fi=0︸︷︷︸

dipole moment

+12

3∑j=1

FiFj

[∂2E

∂FiFj

]Fi=0,Fj=0︸︷︷︸

dipole polarizability tensor

+ · · ·

• Many quantities of physical interest can be formulated as an energy derivative andthus an ability to calculate derivatives accurately in QMC considerably enhancesthe scope of the method.

• Then can calculate e.g. dipole moment without need to sample pure distribution(by applying a small electric field in each direction and observing the change inthe energy).

• Perturbation could be electric field (giving dipole moment, dipole polarizabilitytensor etc..) or displacement of nuclear position (giving forces, etc.) orcombination (e.g. intensity of peaks in IR spectrum depends on change in dipolemoment corresponding to a change in geometry).

• Energy derivatives can be computed numerically (by finite differencing) oranalytically (by differentiating the appropriate energy expressions).


Forces

• Average components of force on atoms given by first derivatives of the energywith respect to nuclear displacements

FRi = −∂E0(R)∂Ri

where R represents the 3Nn nuclear coords. Ri is displacement of particularnucleus in direction of desired force component.

• Define local force as

FRi(x,R) = −∂V (x,R)∂Ri

x represents the 3Ne electronic coordinates and V the potential energy operator.

• If the Hellmann-Feynman theorem applies, then average forces can be rewrittenas the statistical average of the local force over the exact distribution Ψ2

0(x).

FRi = 〈FRi(x,R)〉Ψ20(x)

Usually it doesn’t (since things other than the potential energy depend on R).

• Forces −→ equilibrium geometries, potential energy surfaces, vibrationalfrequencies, dynamics. Useful! But forces are difficult to calculate in QMC..


Finite differences

Approximate derivative by standard two-point finite difference formula:

dE

dqi≈ E(δqi)− E(0)

δqi

• Traditional ab initio quantum chemistry methods yield approximate values ofproperties that can be determined to arbitrary precision (primarily governed bythe degree to which the wave functions/density can be converged).

• Important to distinguish between accuracy and precision. Accuracy is determinedby the approximation (Hartree-Fock, CI etc..). Monte Carlo methods are veryaccurate but their precision is limited by statistical uncertainty.

• This limited precision can lead to problems when taking finite differences. Typicalstatistical uncertainty in the derivative is 20 to 100 (or greater) times that of theuncertainty in the energies.


Finite differences and correlated sampling

• One solution to this problem is to use correlated sampling : construct Monte Carloalgorithm so that the statistical fluctuations in E(0) and E(δqi) are correlated.

• In the limiting case where the two local energies EL(x; 0) and EL(x; δqi) arerelated pointwise by a constant offset, correlated sampling of the difference wouldhave zero statistical uncertainty.

• In reality the variance of the difference increases as the perturbation increases,because the correlation between the perturbed and unperturbed systemsdecreases.

• Use VMC walk to generate ensemble of configurations X(i)k for the unperturbed

system. Then get energies of perturbed system using the same configurationsand reweighting the energies (in similar way to weighted varmin):

E(0) ≈ 1MNc

M∑i=1

Nc∑k=1

EL(X(i)k ; 0)

E(δqi) ≈∑Mi=1

∑Nck=1EL(X(i)

k ; δqi)W (X(i)k ; δqi)∑M

i=1

∑Nck=1W (X(i)

k ; δqi)

W (X(i)k ; δqi) =

Ψ2(X(i)k ; δqi)

Ψ2(X(i)k ; 0)

(Weights)


Finite differences in practice

• Accuracy and precision of the correlated sampling degrades rapidly upon increaseof the difference between the two geometries.

• This difficulty is reduced if the trial wave function for all the geometries differentfrom the reference one is reoptimized, and a “warp” coordinate transformation isused. See Filippi and Umrigar Phys. Rev. B 61, R16291 (2000).

• However, computational cost needed to compute a D-dimensional gradient is, atleast, D + 1 times the one for a single energy estimate while the cost for theHessian matrix scales as D(D + 1)/2.

Serious problem for more than two atoms!


VMC force is sum of three terms

∇REVMC =〈ΨT |∇RH|ΨT 〉〈ΨT |ΨT 〉︸︷︷︸FHFTR

+ 2〈Ψ′T |H|ΨT 〉〈ΨT |ΨT 〉

− 2EVMC〈Ψ′T |ΨT 〉〈ΨT |ΨT 〉︸︷︷︸

FΨR

+∑i

∂EVMC

∂ci

∂ci∂R︸︷︷︸

F cR

• FHFTR is the Hellmann-Feynman term.

• FΨR (the Pulay term) involves partial derivatives of ΨT with respect to the nuclear

positions acting only on the part of the trial wave function that explicitly dependson R e.g. atomic basis function centres or electron-nuclear terms in the Jastrowfactor.

Therefore best to choose basis functions that do not depend directly on R,although sometimes this is unavoidable.

• F cR accounts for the action of ∇R on the parameters that only indirectly couplewith the nuclear positions (e.g. orbital coefficients).

Expect that ∂E∂ci

derivatives in F cR term will be near-zero if wave function isoptimized through energy minimization. Up to very recently has unfortunatelybeen more usual to minimize the variance, though optimal parameters in eachcase usually pretty similar. The parameters also contain statistical noise.


Expression for the force in VMC


+ 2〈Ψ′T |H|ΨT 〉〈ΨT |ΨT 〉


FΨR

+∑i

∂EVMC

∂ci

∂ci∂R︸︷︷︸

F cR

Hellmann-Feynman theorem in Hartree-Fock/DFT

1/0/0 3/1/0 5/2/0 10/2/0 12/4/1 exact

Basis set quality (s/p/d)

-0.55

-0.5

-0.45

-0.4

Tot

al e

nerg

y (a

.u.)

Energy

0.5

0.55

0.6

0.65

Forc

e (a

.u.)

HFT forceAnalytic force

Hartree-Fock HFT and analytic derivatives for H2

+ at R=1a.u.

HFT is valid in Hartree-Fock and DFT with a plane-wave basis set (which doesn’tdepend on nuclear positions). For basis sets with explicit dependence on nuclearpositions (such as Gaussians) then the HFT is only valid in the limit of a completebasis set. Need to compute additional Pulay terms to get correct forces.


Infinite variance problem

Define the local HFT force as FHFTL,i = (−∇H)i for i = 1 . . . 3N , where

∇H =∑α,i

Zαri −Rα

|ri −Rα|3+∑α,β

ZαZβRα −Rβ

|Rα −Rβ|3

Then the variance of the local force is given by σ(FHFTL,i )2 = 〈

(FHFTL,i

)2〉−〈FHFTL,i 〉2.

Consider H atom with wave function Ψ = e−ar. Average force clearly zero foratom, but consider variance of Hellmann-Feynman derivative in x direction, i.e.

σ2 = 〈x2

r6〉 − 〈 xr3〉2. After conversion to spherical polars with x = r sin θ cos θ

σ2 =∫

sin3(θ) cos2(φ) dθdφ∫e−2ar

r2dr − · · ·

Thus the variance of the Hellmann-Feynman force is infinite!– Typeset by FoilTEX – 12

Renormalization

Assaraf and Caffarel, J. Chem. Phys. 119, 10536 (2003).

∇EVMC =1

〈ΨT |ΨT 〉〈ΨT |H ′ +

(H − EL)Ψ′TΨT

+ 2Ψ′TΨT

(EL − EVMC)|ΨT 〉

Basic idea is to construct a “renormalized” or improved observable that has thesame average as the original one but a lower variance.

• In this way, can remove the pathological part responsible for the infinite varianceof the Hellmann-Feynman estimator.

• Better in large systems since it allows simultaneous computation of all thegradient components in a single run, without the burden of optimizing D + 1wave functions.

Other Possibilities

• Filter out s-component of density (Chiesa et al., PRL, 94, 036404 (2005).)

• Use pseudopotentials that remove singularity at origin - no more infinite varianceproblem. Difficult to differentiate non-local pseudopotential operator (but thishas now been done in CASINO, see Badinski and Needs, “Accurate forces inQMC calculations with non-local pseudopotentials”, PRE 76, 036707 (2007)).


Forces in DMC

• Taking the gradient of the fixed-node DMC energy leads to additional Pulay-liketerms, and the estimate of the force is unbiased when these terms are included.The additional terms include gradients of the DMC wave function Φ which cannotbe straightforwardly evaluated.

• Can replace these terms by similar ones involving the gradient of the trial functionΨT . Though approximate, this scheme should give good results if ΨT is accurateenough.

• Can generate the “pure” distribution ΦΦ using e.g. approximate extrapolatedestimation method, future walking, or reptation MC. Claimed in literature thatHFT force calculated with the pure distribution is equal to the exact negativeDMC energy gradient, but it is clear that this cannot be correct. In fact there isan additional “nodal term” which is only zero when the nodal surface of ΨT isexact.


Forces in CASINO from version 2.1.1

Si atom in SiH H atom in SiH

• Badinski/Needs derive gradients of non-local pseudopotential operator for FPLA(standard) and SPLA (Casula) localization schmes.

• They calculate equilibrium bond lengths and vibrational frequencies in goodagreement with experiment with an accuracy comparable to or greater thanDFT-PBE, MP2, CC for the 5 molecules H2, LiH, SiH, SiH4 and GeH.

input keyword: forces– Typeset by FoilTEX – 15

Ab initio molecular dynamics

• AIMD based on DFT is an exceptionally powerful simulation tool for dynamicalstudies of several hundred atoms for times of order 100 ps (e.g. for waterproperties, biological systems, high-pressure systems, phase diagrams).

• Using AIMD many properties can be predicted for wide range of ρ, P and T . Butsome properties need accuracy beyond DFT (van der Waals’ bonded systems,transition-metal compounds, calculation of excitation energies and energy gaps,accurate energy barriers in chemical reactions, bond dissociation processes etc.etc.).




How do we improve the accuracy of these properties over DFT?




How do we improve the accuracy of these properties over DFT?

• Need to use a more accurate technique : options include QMC, GW, BSE,post-HF etc.. Good for static calculations but presumably very expensive fordynamics.

Let’s try to couple DMC with AIMD somehow..


DMC-MD work first published by these people

Jeff Grossman (Lawrence Livermore)

Lubos Mitas (North Carolina State University)

“Efficient quantum Monte Carlo energies for molecular dynamics simulations”Phys. Rev. Lett., 94, 056403 (2005).


Snapshot approach

• Take snapshots of AIMD trajectory every so often and do full DMC calculationfor each point e.g. silane molecule for T ≈ 1500K.

• Very expensive! Discrete sampling may miss important events.


QMC energies in an AIMD simulation


Is there a more efficient way to use QMC in AIMD?

In DMC, walkers evolve in 3Ne dimensions, and the wave function is sampledaccording to the imaginary time-dependent Schrodinger equation.

φ(r, τ) = exp(−τH)φ(r, τ = 0)

But can the stochastic electronic propagation in DMC be coupled to MD?

Is there a more efficient way to use QMC in AIMD?

In DMC, walkers evolve in 3Ne dimensions, and the wave function is sampledaccording to the imaginary time-dependent Schrodinger equation.

φ(r, τ) = exp(−τH)φ(r, τ = 0)

But can the stochastic electronic propagation in DMC be coupled to MD?

Analogy with DFT-MD for coupling electrons and ions:

Born-Oppenheimer dynamics

Φ = min [EKS]

MIRI = −∇IΦ

Car-Parrinello dynamics

µΨi = −∂EKS [Ψ,R]∂Ψi

−∑

ΛijΨj

MIRI = −∂EKS [Ψ,R]∂RI


Basic idea

How far does an electron movein a typical DMC time step?

Basic idea


0.01 . . . 0.1 au

Basic idea


0.01 . . . 0.1 au

How far does an ion movein a typical MD time step?

Basic idea


0.01 . . . 0.1 au


0.0001 . . . 0.001 au

Basic idea


0.01 . . . 0.1 au


0.0001 . . . 0.001 au

So, instead of discrete sampling of each point with new DMC run, calculate DMCenergies ‘on the fly’ during the dynamic simulation. Continuously update the DMCwalkers so that they correctly represent the evolving wave function.

• Evolution of both configuration spaces is coupled : as the ionic dynamicaltrajectories evolve, so does the population of DMC electrons.

• The slow evolution in AIMD is highly advantageous in DMC!


What a DMC population should look like

0-600

-595

-590

-585

-580

-575

-5700

100

150

200

250

300

POPULATION OF WALKERS

TOTAL ENERGY


Try coupled DMC-MD as described above

Oh dear!


Considerations

• Wave function changes as ions move : bias in walker distribution?

wrong wave function at the beginning of each step

Considerations



• DMC dynamics not right : problem with rejection step?

detailed balance may no longer hold

Considerations





• Dynamics of orbital occupations : orbital rotation, state crossings?

swapping during dynamics

Considerations





• Dynamics of orbital occupations : orbital rotation, state crossings?

swapping during dynamics

• Wave function changes as ions move : node crossings?

electrons can cross nodes and nodes can cross electrons


Fix the algorithm

DMC wave function bias (walker distribution)

At MD time step tMD = 0 the DMC walker distribution is f(R, τ ; H(0)). At nextMD step tMD = 1 should be sampling f(R, τ + ∆τ ; H(1)). To sample correctly,need to start with f(R, τ ; H(1)) rather than original f(R, τ ; H(0)).

Use correlated sampling to reweight each walker

The propagation of walkers is governed by the DMC approximation for the Green’sfunction which is a product of Gaussians describing diffusion and drift times aweight with an exponentiated local energy. Correlated sampling is used to correctthe Green’s function by a proper modification of the walker weights.

For each step, reweight each DMC walker by :

w(R(τ), tMD = 1) =G[R(τ)← R(τ −∆τ); H(1)

]G[R(τ)← R(τ −∆τ); H(0)

]∼=

Ψ2T

[R(τ); H(1)

]Ψ2T

[R(τ); H(0)

]e−∆τ(EL[H(1)]−EL[H(0)])/2


Fix the algorithm II

Detailed balance

To account for MD steps, need to modify Green’s function in the accept-reject step.

Orbital occupations

Possible that during dynamics the ordering of states is changed. Can detect orbitalswapping before the first DMC step of a new MD step by computing the overlapof all orbitals with one another. Easy to estimate since same set of walkers can beused to perform integral.

If unoccupied orbital swapped with occupied, simplest solution is to reconverge newset of walkers from scratch. ‘Brute force’ approach! However, this seems to happenquite rarely. For excited state of silane with near-degenerate orbitals accessiblethrough fluctuations, happened only once in the simulation. For ground state,doesn’t happen at all.

Node crossing

Eliminate walkers that cross nodes..


How many DMC steps needed per MD step?

Use large discrete sampled runs (1000 steps each) for comparison.

• As simulation progresses, 1-step CDMC energies begin to differ significantly fromdiscrete DMC. Using 3 steps corrects ‘time lag’. About an order of magnitudemore efficient than discrete sampling.

Thermal averages appear to be convergedfor N ≥ 3.


Efficiency

Perform 3-step CDMC on SinHn clusters :

Cluster AIMD time CDMC time DMC steps configs/node nodesSiH4 12 10 3 300 16Si5H12 35 45 3 300 32Si14H20 106 56 3 50 128Si35H36 310 290 3 35 256

Time for CDMC comparable to AIMD!

Grossman/Mitas also quote some interesting results:

• H2O molecule dissociation path from constrained AIMD. CDMC energetics agreewith discretely sampled result. Dissociation energy (127(2) kcal/mol) agrees withexperiment (125.9 kcal/mol). Can describe bond-breaking..

• Calculated heat of vaporization of water (from full liquid simulations of 32 watermolecules). Liquid binding energy (average CDMC energy for the 32 water systemminus 32 times the DMC energy of a single water molecule) + 1.5 kcal/molto account for the quantum motion of the H atoms. Results - LDA-PBE :Hvap = 6.2(1), CDMC : Hvap = 9.1(4), Expt. : Hvap = 9.92 kcal/mol.


Forces

So far :

• Accurate energies (at the level of fixed-node DMC).

• Can compute energy differences on the fly (couple two DMC electronic populationsto the AIMD simulation e.g. one for the ground state and one for a given excitedstate).

• Good for calculations of excited states and thermodynamic averages.

However the ionic trajectories were all generated using DFT up to now.

Is it possible to incorporate QMC forces into the simulation?

Forces

So far :

• Accurate energies (at the level of fixed-node DMC).

• Can compute energy differences on the fly (couple two DMC electronic populationsto the AIMD simulation e.g. one for the ground state and one for a given excitedstate).

• Good for calculations of excited states and thermodynamic averages.

However the ionic trajectories were all generated using DFT up to now.

Is it possible to incorporate QMC forces into the simulation?

• Using a standard method (basically finite differencing with some fancy tricks)Grossman and Mitas computed the DMC forces at each MD step for SiH4 andcompared results with LDA at around 1500K. Excellent agreement for Si and H.


Conclusions

• Grossman and Mitas have proposed a new method for coupling ab initio MD ionicsteps with stochastic DMC electronic steps to provide accurate DMC energies onthe fly.

• The technique exploits the slowness of MD evolution which enables the QMCsampling process to be updated very efficiently.

• Accurate for both thermal averages and description of energies along pathways.

• They have also carried out the first QMC/MD simulations using both forces andenergies from QMC (albeit not very good forces). Presumably we could do thisbetter now.


Running DMC-MD calculations with CASINO (new!)Script runqmcmd used to automate DMC-MD using casino and the pwscf DFT code (part of

Quantum Espresso available for free at www.quantum-espresso.org - must use version 4.3+).

The script works by repeatedly calling CASINO run scripts runpwscf and runqmc which know how

to run the two codes on any known machine. See instructions in utils/runqmcmd/README.

Setup the pwscf input (’in.pwscf’) and the CASINO input (’input’ etc. but no wave function

file) in the same directory. For the moment we assume you have an optimized Jastrow from

somewhere (this will be automated later). Have the pwscf setup as ‘calculation = ”md”’, and

‘nstep = 100’ or whatever. The runqmcmd script will then run pwscf once to generate 100

xwfn.data files, then it will run casino on each of the xwfn.data. The first will be a proper DMC

run with full equilibration (using the values of DMC EQUIL NSTEP, DMC STATS NSTEPetc.). The second and subsequent steps (with slightly different nuclear positions) will be restarts

from the previous converged config.in - each run will use new keywords DMCMD EQUIL NSTEPand DMCMD STATS NSTEP (with the number of blocks assumed to be 1. The latter values are

used if new keyword DMC MD is set to T, and they should be very small).

It is recommended that you set DMC SPACEWARPING AND DMC REWEIGHT CONF to T

in CASINO input when doing such calculations.

The calculation can be run through pwfn.data, bwfn.data or bwfn.data.b1 formats as specified

in the pw2casino.dat file (see casino and pwscf documentation).


So obviously we’ll be doing this sort of thing soon

5 MINUTE BREAK– Typeset by FoilTEX – 32

What expectation values can CASINO calculate?

• density

• spin density

• spin density matrix in non-collinear spin systems.

• pair correlation function (PCF)

• spherically-averaged pair correlation function (SPCF)

• localization tensor

• structure factor

• spherically-averaged structure factor

• one-particle density matrix (OBDM)

• two-particle density matrix (TBDM)

• condensate fraction estimator (unbiased TBDM, goes as TBDM-OBDM**2)

• dipole moment


CASINO input file (expectation value section)# EXPECTATION VALUESdensity : F #*! Accumulate densityspin_density : F #*! Accumulate spin densitiesspin_density_mat : F #*! Accumulate spin density matrixpair_corr : F #*! Accumulate recip. space PCFpair_corr_sph : F #*! Accumulate spherical PCFloc_tensor : F #*! Accumulate localization tensorstructure_factor : F #*! Accumulate structure factorstruc_factor_sph : F #*! Accumulate sph. struc. factoronep_density_mat : F #*! Accumulate 1p density matrixtwop_density_mat : F #*! Accumulate 2p density matrixcond_fraction : F #*! Accumulate cond fractiondipole_moment : F #*! Accumulate elec. dipole momentexpval_cutoff : 30.d0 hartree #*! G vector cutoff for expvalpermit_den_symm : F #*! Symmetrize QMC charge dataqmc_density_mpc : F #*! Use QMC density in MPC intint_sf : F #*! Calc ee int from strucfac


Interface to other packages

Gaussians:

Plane waves:CASINO

TCM atomic code

Numerical orbitals:

ABINIT

CASTEP

K270

GAUSSIAN94/98

CRYSTAL95/98/03

TURBOMOLE

pwfn.data

GP

PWSCF

bwfn.data

awfn.data

gwfn.data

correlation.data

mpc.data

pseudopotential file

Blips

out

input expot.data

expval.data

vmc.histdmc.histetc..


The expval.data file - headerSTART EXPVALTitleSilicon 2x2x2

File version1

Number of particle types (e.g. 2=electrons, 4=electrons+holes)2

Number of each type of particle32 32

Dimensionality3

Periodicity3

Primitive translation vectors (au)0.000000000000000 5.129662155669400 5.1296621556694005.129662155669400 0.000000000000000 5.1296621556694005.129662155669400 5.129662155669400 0.000000000000000

Multiples of primitive translation vectors2 2 2

Volume of simulation cell2159.664411362648480

Wigner-Seitz radius of simulation cell7.254437790939672

Number of available G-vector sets1


The expval.data file - G vector sets

START GVECTOR SET 1Energy cutoff (au) used to generate set30.000000000000000

Number of G-vectors in set725

Primitive reciprocal lattice vectors (au)-0.612436561756342 0.612436561756342 0.6124365617563420.612436561756342 -0.612436561756342 0.6124365617563420.612436561756342 0.612436561756342 -0.612436561756342

G-vector components Gx, Gy, Gz (au)0.000000000000000 0.000000000000000 0.0000000000000000.612436561756342 -0.612436561756342 -0.612436561756342-0.612436561756342 0.612436561756342 0.612436561756342-0.612436561756342 -0.612436561756342 0.612436561756342.......................................................-3.674619370538052 -3.674619370538052 -1.2248731235126843.674619370538052 3.674619370538052 1.224873123512684

END GVECTOR SET 1


The expval.data file - example data set

START DENSITYAccumulation carried out usingVMC

Use G-vector set1

Number of sets1

START SET 1Particle type1

Total weight10.0000000000000

Complex charge-density coefficients (real part, imaginary part)8.00000000000000 0.1.67967897738036 0..........................6.435364313328349E-004 0.6.435364313328349E-004 0.

END SET 1END DENSITY


Continuing old runs

• Set newrun to F in the input file, and rename the config.out file - containingthe final state of a previous VMC/DMC run - to config.in. Then CASINOwill read in an old expval.data file before continuing the QMC calculation.

• This can be used either to continue the accumulation of an expectation valuefrom the previous run, or to add a new expectation value to the expval.datafile.


Accumulating things in reciprocal space

For example : the DENSITY

• After each single electron move from r′ −→ r, accumulate density in reciprocalspace as ρ(G) = ρ(G) + exp(iG · r) for each G vector.

• At the end of the simulation, divide by the total weight (e.g. the number ofaccumulation steps in VMC)to get the average of each ρ(G).

• Fourier coefficients are normalized such that ρ(G = 0) is the number of electronsin the primitive cell. Get this by dividing the above average ρ(G) by the totalnumber of primitive cells in the simulation cell.

• Then the density in real space is given by

ρ(r) =nG∑i=1

ρ(G) exp(iG · r)


Accumulating things in real space

For example : pair correlation function (PCF) in 3D homogeneous, isotropic system

The PCF gαβ can be collected in bins of width ∆, giving

gαβ(rn) =ΩΩn

⟨Nnαβ

NαNβ

⟩where Nn

αβ is the number of α, β-spin pairs whose separation falls within the nthbin and Ωn is the volume of the nth bin. The volume of the nth bin is given by

Ωn =43π(n3 − (n− 1)3)∆3 = 4π(n2 − n+

13

)∆3

0 10 20 30Distance (au)

0

0.5

1

Pair

cor

rela

tion

func

tion

up-spin:up-spinup-spin:down-spindown-spin:down-spin

Homogeneous electron gas (rs=10)

NOT REBLOCKED

0 10 20 30Distance (au)

0

0.5

1Pa

ir c

orre

latio

n fu

nctio

n

up-spin:up-spinup-spin:down-spindown-spin:down-spin

Homogeneous electron gas (rs=10)

REBLOCKED


The plot expval utility

• This utility reads the expval.data file produced by CASINO, performsappropriate conversions and transformations on the data therein, and writesthe results to files visualizable by freely available standard packages such asxmgrace, gnuplot etc.

• Output is placed in lineplot.dat, 2Dplot.dat, or 3Dplot.dat depending onthe dimensionality of the plot.

• The utility takes its basic instructions from the plot expval block in input whichdefines the geometrical region over which the data will be plotted (a line, planeor volume of appropriate size). Any additional options will be selected followingdirect appeal to the user at run time.


The XMGRACE package : 1D plots

0 10 20 30 40r (a.u.)

0

10

20

30

40

Den

sity

SCF and QMC density along 111 direction bcc hydrogen crystal with large lattice constant


The plot 2D utility

CASINO 2D plot

0 5 10 15 20 25 0

5

10

15

20

25


The plot 2D utility : -surf option

-5 0

5 10

15 20

25-5 0

5 10

15 20

25

-5 0 5

10 15 20 25 30 35 40 45 50

CASINO 2D plot


Mixed and pure estimators

Calculate energy from the mixed estimator:

EDMC =∫

ΦHΨT dR∫ΦΨT dR

=∫

ΨTΦ(Ψ−1T HΨT ) dR∫

ΨTΦ dR

' 1M

∑i

EL(Ri)

• Crucial step in deriving this requires the equality of 〈ΨT |H|Φ〉 with 〈Φ|H|ΨT 〉together with the eigenvalue equation HΦ = EΦ.

• For an operator that does not obey an eigenvalue equation with respect to Φ,i.e. one that does no commute with the Hamiltonian, the mixed estimator doesnot yield the correct expectation value.

• To use DMC to estimate the expectation value of such an operator, we need tosample from the “pure” distribution :

〈A〉 =∫

ΦAΦ dR∫ΦΦ dR

Can obtain with e.g. extrapolated estimator, future walking, reptation MC.


Extrapolation of pair correlation function

0 100 200 300 400r / a.u.

0

0.5

1

1.5

2

g(r)

4 param J - VMC4 param J - DMC4 param J - Extrap.6 param J - VMC6 param J - DMC6 param J - Extrap.

Ferromagnetic WC, rs=110, N=64


Structure factor

HΨ = EΨ

Potentially very important in new approaches to calculating electron-electroninteractions with smaller finite-size effects in periodic systems.

0 1 2 3 4 5 6k

0

0.5

1

S(k)

Silicon 2x2x2 crystal - Gaussian basis


Exchange-correlation hole and the adiabatic connection

Exact relationship between the exchange-correlation energy, Exc, and the ground-state many-electron wave function Ψλ associated with different values of theCoulomb coupling constant, λ.

gαβ(r, r′) =N(N − 1)nα(r)nβ(r′)

∫ 1

0

dλ

∫dx3 · · · dxN |Ψλ(rα, r′β,x3, . . . ,xN |2

g(r, r′) =∑α,β

nα(r)nβ(r′)n(r)n(r′)

gαβ(r, r′)

ρxc(r, r′) = n(r′)[gαβ(r, r′)− 1]

exc(r) =n(r)

2

∫dr′

ρxc(r, r′)|r− r′|

Integrate exc(r) over all space to get exchange-correlation energy Exc!


Exchange-correlation hole and the adiabatic connection


Metal-insulator transitions

What is an insulator?

• Phenomenological- Has vanishing electrical conductivity in a (weak) static electrical fieldat 0K.- Supports bulk macroscopic polarization (pure ground state property!).

• Band theory (‘one-electron wave functions’)Fermi level lies within a band gap rather than within a band.

• Quantum Monte Carlo (‘many-electron wave functions’)Thought required.. [What about Kohn theory (1964) and modernreinterpretations?]

Associated questions

• What is meant by localized/delocalized electrons?

• What are strongly correlated electrons?


Band insulators

Impose PBCs : the one-electron wave functions are then Bloch functions.

Bloch functions obey BLOCH’S THEOREM:

Ψnk(r) = eik·runk(r) or Ψ(r + t) = Ψ(r)eik·t

Count states in each energy range, and classify:

Increasing pressure (for example) will change shape of bands and may lead tometal-insulator transitions of various sorts.


What is a Mott insulator?

Gebhard

• “For a Mott insulator the electron-electron interaction leads to the occurence oflocal moments. The gap in the excitation spectrum for charge excitations mayarise from the long-range order of the pre-formed moments (Mott-Heisenberginsulator) or by a quantum phase transition induced by charge and/or spincorrelations (Mott-Hubbard insulator)”

• “Mott insulating behaviour is understood as a cooperative many-electronphenomenon.. [It] cannot be understood within the framework of a single-electron theory - many-body effects must be included.”

• “..materials in which electron-electron interactions [are so] important that a naiveband structure approach will no longer be appropriate”

Mott

• “..a material that would be a metal if no moments were formed. ...depends onthe existence of moments and not on whether or not they are ordered.”

Pasternak (an experimentalist)

• “In this [W > U ] regime, the d-electron correlation collapses, giving rise to aninsulator-metal transition concurrent with a magnetic moment breakdown. Thisphenomenon is called the Mott transition”.


Linear chain of hydrogen atoms

“Consider a linear chain of hydrogen atoms with a lattice constant of 1 A. This hasone electron per atom in the conduction band and is therefore metallic. Imaginewe now dilate the lattice parameter of the crystal to 1 metre. We would agreethat at some point in this dilation process the crystal must become an insulatorbecause certainly when the atoms are 1 metre apart they are not interacting. Butband theory says that the crystal remains a metal because at all dilations the energydifference between occupied and unoccupied states remains vanishingly small. Nowlook at this thought experiment from the other way. Why is the crystal with alattice parameter of 1 metre an insulator? Because to transfer an electron from oneatom to another we have to supply an ionization energy, I to remove the electronand then we recover the electron affinity, A, when we add the electron to the neutralH atom. The energy cost in this process is U = I − A. Band theory ignores termssuch as these.”

Hubbard model

H =∑i,j tijaiσajσ + U

∑i ni↑ni↓


DFT treatment of linear hydrogen chain

Examine linear chain of H atoms in band theory

• Genuine 1-dimensional periodic boundary conditions

• High quality local (Gaussian) basis set centred on the H atoms.

Properties of isolated H atom computed with this basis

Calculated ExactTotal energy (HF) -0.499993 Ha -0.5 HaVirial coefficient 1.00007 1Hyperfine coupling constant 1419.3 MHz 1420 MHz


Band theory of linear hydrogen chain

0 2 4 6 8 10

Lattice spacing (Å)

-0.56

-0.54

-0.52

-0.5

-0.48

-0.46

Tot

al e

nerg

y (a

.u.)

(DFT band calculations with LDA exchange-correlation functional)

Metallic at all dilations

1 atom per cell

Band theory of linear hydrogen chain

0 2 4 6 8 10


-0.56

-0.54

-0.52

-0.5

-0.48

-0.46

Tot

al e

nerg

y (a

.u.)

(DFT band calculations with LDA exchange-correlation functional)

Metallic at all dilations

1 atom per cell

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

ENERGY (HARTREE)

0

50

100

150

200

DE

NS

ITY

OF

ST

AT

ES

(S

TA

TE

S/H

AR

TR

EE

/CE

LL)

a = 3Å


Correlated electron systems

”Strong Coulomb interactions” : actually statement about pair correlation function.VMCg""(r; r0; [n])r at bond centerr0 in (110) plane

** *

*

***

0.9 0.7 0.5 0.3 0.1

0

0.5

1

VMCg"#(r; r0; [n])r at bond centerr0 in (110) plane

** *

*

***

1.02 0.97 0.92 0.87 0.82

0.8

0.9

1

↑‘TRUE MANY-BODY EFFECTS’ HERE

correlated electron system - ”non-vanishing pair correlation function between ↑- and↓-electrons”strongly-correlated system - ”pair correlation functions for electrons of same spinand electrons of opposite spin are comparable in size

Correlated electron systems

”Strong Coulomb interactions” : actually statement about pair correlation function.VMCg""(r; r0; [n])r at bond centerr0 in (110) plane

** *

*

***

0.9 0.7 0.5 0.3 0.1

0

0.5

1

VMCg"#(r; r0; [n])r at bond centerr0 in (110) plane

** *

*

***

1.02 0.97 0.92 0.87 0.82

0.8

0.9

1

↑‘TRUE MANY-BODY EFFECTS’ HERE

correlated electron system - ”non-vanishing pair correlation function between ↑- and↓-electrons”strongly-correlated system - ”pair correlation functions for electrons of same spinand electrons of opposite spin are comparable in size”

So to describe strong-correlations just need to make the wave function flexibleenough for ↑- and ↓-electrons to avoid each other?


Spin polarization

Need single determinants of one-electron spin orbitals!

• Restricted form All spin orbitals are pure space-spin products of the form φnα orφnβ and are occupied singly or in pairs with a common orbital factor φn.

• Unrestricted form Spin orbitals no longer occupied in pairs but still pure space-spinproducts φnα or φnβ. Different spatial factors φn, φn for different spins.

• General unrestricted form No longer restrict to simple product form. Eachspin orbital now a 2-component complex spinor orbital: Ψ1 = φα1α + φβ1β and

Ψ2 = φα2α+ φβ2β. Non-collinear spins.

Pretty pictures of non-collinear spin states


Spin-unrestricted band treatment oflinear hydrogen chain

0 1 2 3 4 5 6


-0.56

-0.55

-0.54

-0.53

-0.52

-0.51

-0.5

-0.49

-0.48

-0.47

-0.46

Tot

al e

nerg

y (a

.u.)

ferromagneticspin-unpolarized

(LSDA functional)

METAL INSULATOR

1 atom per cell


Band structure of linear hydrogen chain

-π/2a 0 π/2a

k

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Ene

rgy

(au)

Hydrogen linear chain (a=3 Å)Band structure



-0.4 -0.3 -0.2 -0.1 0-40

-20

0

20

40

-0.4 -0.3 -0.2 -0.1 0-40

-20

0

20

40

Hydrogen linear chainDensity of states

-0.4 -0.3 -0.2 -0.1 0-40

-20

0

20

40

-0.4 -0.3 -0.2 -0.1 0-40

-20

0

20

40

-0.4 -0.3 -0.2 -0.1 0-40

-20

0

20

40

-0.4 -0.3 -0.2 -0.1 0-40

-20

0

20

40

-0.4 -0.3 -0.2 -0.1 0-40

-20

0

20

40

-0.4 -0.3 -0.2 -0.1 0

ENERGY (HARTREE)

-40

-20

0

20

40

-0.4 -0.3 -0.2 -0.1 0-40

-20

0

20

40

3 Å6 Å 2.5 Å

2.1 Å 2.05 Å 2.02 Å

2.0 Å 1.9 Å 1.5 Å



0 1 2 3 4 5 6


-0.56

-0.54

-0.52

-0.5

-0.48

Tot

al e

nerg

y (a

.u.)

ferromagneticantiferromagnetic

Total energy of a linear chain of hydrogen atoms(spin-unrestricted DFT band calculations with LSDA functional)

METAL INSULATOR

2 atoms per cell


Hubbard bands

[Mott book]

Let ψi be the many electron wave function with an extra electron on atom i. Astate in which the electron moves with wave number k can be described by themany-electron wave function: ∑

i

eikaiψi

These states form a band of energies - this is called the upper Hubbard band - (a).Similarly, the lower Hubbard band - (b) - represents a band of states in which a‘hole’ can move. Metal-insulator transition occurs when these bands overlap.


N+1/N-1 system

−2U = +

LSDA (eV) UHF (eV) B3LYP (eV)U (as above) 11.44 13.03 12.01U (Band gap) 5.30 11.99 8.56

Should be around 13 eV.


N+1 system

0 2 4 6 8 10


-0.56

-0.54

-0.52

-0.5

-0.48

Tot

al e

nerg

y pe

r at

om (

a.u.

)Total energy of N+1 hydrogen chain

10 atoms and 11 electrons per cell (LSDA)

1

356810

Non-magnetic metal

(ferro)magnetic insulator

(ferro)magnetic metal


Effect of extra electron on the density of states

-0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1

ENERGY (HARTREE)

-1000

0

1000

DE

NS

ITY

OF

ST

AT

ES

(S

TA

TE

S/H

AR

TR

EE

/CE

LL)

a = 8.5 Å

N case

up spin

down spin

Effect of extra electron on the density of states

-0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1

ENERGY (HARTREE)

-1000

0

1000

DE

NS

ITY

OF

ST

AT

ES

(S

TA

TE

S/H

AR

TR

EE

/CE

LL)

a = 8.5 Å

N case

up spin

down spin

-0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1

ENERGY (HARTREE)

-1000

0

1000

DE

NS

ITY

OF

ST

AT

ES

(S

TA

TE

S/H

AR

TR

EE

/CE

LL)

a = 8.5 ÅN+1 case

up spin

down spin


’Crystal field’ splitting of d orbitals in octahedral coordination


What do we expect to see in NiO?

Interactions

Parameterize on-site interactions in terms of U and U ′ (Coulomb interactionsbetween electrons in same (U) or different (U ′) d orbitals) and J (exchangeinteraction between same spin electrons). Augment with ∆CF i.e. crystal-fieldsplitting energy due to neighbours.

On a Ni site in NiO:

↑-spin eg electron feels: 7U ′ − 4J + ∆CF

↑-spin t2g electron feels: U + 6U ′ − 4J↓-spin eg electron feels: U + 7U ′ − 3J + ∆CF

↓-spin t2g electron feels: U + 6U ′ − 2J

Expt: U=5.8eV, J=0.67eV, U’=4.5eV, ∆CF=1.1eV

CF

up spin down spin

eg

t2g

t2g

eg

U−3J+∆ CF

2J

2J−∆


Model vs. UHF

CF

up spin down spin

eg

t2g

t2g

eg

U−3J+∆ CF

2J

2J−∆−0.5 −0.25 0 0.25 0.5 0.75 1

ENERGY relative to highest occupied level (au)

DE

NS

ITY

OF

ST

AT

ES

Ni eg

Ni t2g

total oxygen

up−spin

down−spin

Antiferromagnetic (AF2)


UHF: effect of magnetic ordering

−0.5 −0.25 0 0.25 0.5 0.75 1


DE

NS

ITY

OF

ST

AT

ES

Ni eg

Ni t2g

total oxygen

up−spin

down−spin

Antiferromagnetic (AF2)

−0.5 −0.25 0 0.25 0.5 0.75 1


DE

NS

ITY

OF

ST

AT

ES

Ni eg

Ni t2g

total oxygen

up−spin

down−spin

Ferromagnetic


UHF vs. LSDA

−0.5 −0.25 0 0.25 0.5 0.75 1


DE

NS

ITY

OF

ST

AT

ES

Ni eg

Ni t2g

total oxygen

up−spin

down−spin

UHFAF2

−0.5 −0.25 0 0.25 0.5 0.75 1


DE

NS

ITY

OF

ST

AT

ES

Ni t2g

Ni eg

total oxygen

up−spin

down−spin

LSDAAF2


UHF vs. GGA

−0.5 −0.25 0 0.25 0.5 0.75 1


DE

NS

ITY

OF

ST

AT

ES

Ni eg

Ni t2g

total oxygen

up−spin

down−spin

UHFAF2

−0.5 −0.25 0 0.25 0.5 0.75 1


DE

NS

ITY

OF

ST

AT

ES

Ni t2g

Ni eg

total oxygen

up−spin

down−spin

GGAAF2


UHF vs. B3LYP

−0.5 −0.25 0 0.25 0.5 0.75 1


DE

NS

ITY

OF

ST

AT

ES

Ni eg

Ni t2g

total oxygen

up−spin

down−spin

UHFAF2

−0.5 −0.25 0 0.25 0.5 0.75 1


DE

NS

ITY

OF

ST

AT

ES

Ni t2g

Ni eg

total oxygen

up−spin

down−spin

B3LYPAF2


Band gaps with the B3LYP functional

Material Expt. (eV) B3LYP (eV)Si 3.5 3.8Diamond 5.5 5.8GaAs 1.4 1.5ZnO 3.4 3.2Al2O3 9.0 8.5Cr2O3 3.3 3.4MgO 7.8 7.3MnO 3.6 3.8NiO 4.3 3.9TiO2 3.0 3.4FeS2 1.0 2.0ZnS 3.7 3.5


Orbital interactions

How do orbitals interact in HF approximation?

E0 =∑a 〈a|h|a〉+

∑ab 〈aa ‖ bb〉 − 〈ab ‖ ba〉

Coulomb interaction

〈aa ‖ bb〉 =∫dr1dr2 |φa(r1)|2 1

r12|φb(r2)|2

exchange interaction

〈ab ‖ ba〉 =∫dr1dr2 φ

∗a(r1)φ∗b(r1) 1

r12φb(r2)φa(r2)


Self-interaction - HF case

• Label orbitals occupied (a, b, . . .) or virtual (i, j, . . .).What is the expression for the orbital energy?

Occupiedεa = 〈φa|h|φa〉+

∑Nb=1(〈φaφa ||φbφb〉 − 〈φaφb ||φbφa〉)

Virtualεi = 〈φi|h|φi〉+

∑Nb=1(〈φiφi ‖ φbφb〉 − 〈φiφb ‖ φbφi〉)

• Sum over b is over occupied orbitals only. Therefore for the first expression only,one of the terms will cancel when b=a:

〈φaφa ‖ φaφa〉 − 〈φaφa ‖ φaφa〉 = 0

• Therefore in the HF approximation an electron does not feel its own field, sincethe self-interaction is cancelled by an equivalent term in the exchange energy.


Self-interaction - LSDA case

• LSDA exchange energy Eexchange =∫d3r εx [ρ↑(r), ρ↓(r)] and, since the mean-

field contains all the electrons, then e.g. the LSDA energy of H atom is incorrectly-0.47 (or whatever) instead of -0.5 due to ‘self-interaction’.

Implications:

• U (interpreted as the ’self-exchange’ term) equals J (the ’different orbital’exchange term).

• But U and J differ by an order of magnitude in NiO. LSDA effectively averagesthese quantities.

• Therefore additional potential U felt by unoccupied orbitals disappears, andinstead all the states are shoved up in energy by something like the average of Uand J .

• Local density theory lumps all these exchange interactions together and thusdilutes the effect of self-exchange and underestimates the driving force for theformation of a correlated state. This is the root of the difficulty of contemporarycalculations in describing strongly-correlated systems.

Note this is nothing to do with ‘naive band theory’ not containing the U term( c©Cambridge University Condensed Matter Theory course).


What to do about it?

Problem:

The effect of self-interaction and the use of simple local exchange functionals(inherent in LSDA and all GGA treatments) will often lead to the wrong groundstate in magnetic insulators and other strongly correlated materials.

Possible solutions:

• Use ’corrections’ to LDA treatment (LDA+U, SIC-LDA).

• Use unrestricted Hartree-Fock calculations.

• Use ’hybrid functionals’ in DFT containing some fraction of the non-local HFexchange (e.g. B3LYP)

• Use ’exact-exchange’ DFT treatments.

• Use the result of any of the above as a trial wave function for quantum MonteCarlo (which is self-interaction free).


Quantum Monte Carlo - NiO results

a (A) EC (eV) Eg (eV)UHF 4.26 6.2 14.2LSDA 4.09 10.96 0GGA 4.22 8.35 0B3LYP 4.22 7.8 3.9DMC(UHF) 9.40(13)DMC(B3LYP) 9.19(14) 4.3(2)Exp. 4.17 9.5(±?) 4.0–4.3

Lattice constant, a, cohesive energy, EC, and band gap, Eg, of NiO.


Optimize nodal surface:Hartree-Fock exchange weight in hybrid functionals


Superconducutivity in cuprates with QMC

Very accurate results with no parameters!


Superconducutivity in cuprates with QMC

Some lattice degrees of freedom depend strongly on the magnetic state!Spin lattice coupling removed with 25% doping.

See Lucas Wagner’s talk in ”QMC in the Apuan Alps 2013”.


Localization

Insulating state of matter is characterized by gap to low-lying excitations, but alsoby qualitative features of the ground state - which sustains macroscopic polarizationand is localized.

Kohn 1964

Localization is a property of the many-electron wave function: insulating behaviourarises whenever the ground state wave function of an extended system breaks upinto a sum of functions ΨM which are localized in essentially disconnected regionsRM of configuration space i.e.

Ψ(x1, . . . ,xN) =+∞∑

M=−∞ΨM(x1, . . . ,xN)

where for a large supercell ΨM and ΨM ′ have an exponentially small overlap forM ′ 6= M . Under such a hypothesis, Kohn proved that the dc conductivity vanishes.

Hence, electronic localization in insulators does not occur in real space (chargedensity) but in configuration space (wave function).


Many-body phase operators

• Both macroscopic polarization and electron localization are expectation values of

‘many-body phase operators’ z(α)N , where

z(x)N = 〈Ψ|ei2πL

PNi=1 xi|Ψ〉

These quantities are zero for metals!

• Ground state expectation value of the position operator in periodic boundaryconditions

〈X〉 =L

2πIm ln zN

• Phase of z(α)N used to define the macroscopic polarization of an insulator.

• Modulus of z(α)N used to define the localization tensor 〈rαrβ〉 (finite in insulators,

diverges in metals).


Interesting connections

One-particle density matrix

〈rαrβ〉 =1

2nb

∫cell

dr∫

allspace

dr′(r− r′)α(r− r′)β|P (r, r′)|2

This is second moment of (squared) density matrix in the coordinate r− r′.

Conductivity

〈rαrβ〉 =hVc

2πe2nb

∫ ∞0

dω

ωReσαβ(ω)

where σαβ is the conductivity tensor. LHS property of ground state, RHS measurableproperty related to electronic excitations =⇒ localization tensor is a measurableproperty.

“Nearsightedness”

is the fact that the density matrix P (r − r′) is short range in the variable r − r′.The localization tensor is a measure of this.


Some interesting calculations

• Surface energy of materials

• Molecular adsorption on surfaces

• Water systems

• DMC-based statistical mechanics for surface problems.


Surface energy of materials

• When macroscopic sample of a material separated into two pieces, new surfaces are created.

The work done in creating the new surfaces divided by their area is the surface formation energy,

usually denoted by σ.

• For a crystal, σ depends on orientation of surface. Surface formation energies are important

in fields as far apart as geology and fracture mechanics. Particularly important in nanoscience,

because in particles a few nm across a significant fraction of the atoms are at or near the surface.

One consequence is that the crystal structures of nano-particles are sometimes not those of the

bulk material, because it may be advantageous to adopt a less stable bulk structure if the energies

of the surfaces are lowered. For nano-particles supported on substrates, the equilibrium form of

the nano-particles is often determined by the balance between surface and interfacial energies.

• DFT has quite serious problems in predicting σ. Known that predicted σ values can depend

strongly on the exchange-correlation functional. For example, as a broad generalisation, it seems

that GGA approximations tend to give σ values that are about 30 % less than LDA values. Even

more surprising is that LDA values sometimes seem to agree better with experiment than their

GGA counterparts. This is unexpected, because common sense suggests that the formation of

a surface involves the breaking of bonds, and GGA is usually much better than LDA for bond

formation energies.

• Problem is that σ values are not easy to measure experimentally, and those that have been

measured are sometimes subject to large errors. In this unsatisfactory situation, there is an

obvious need for accurate computed benchmarks for σ, which can be used to assess both DFT

predictions and experimental values.


Surface energy of materials: example

• Can now compare with accurate quantum chemistry calculations, using ’hierarchical method’

that in some cases allows cohesive energy, equilibrium structure and elastic properties of perfect

crystals and surface formation energies of crystals to be calculated using wave function-based

quantum chemistry at the CCSD(T) level [S.J. Binnie et al., Phys. Rev. B 82, 165431 (2010).]

• Calculations done for LiH, LiF and MgO. Disagree with DFT - who’s right? Take LiH as example.

Outstanding agreement with experiment given by all-electron DMC for LiH crystal. Equilibrium

lattice parameter a0 agrees to within 10−3 A (both experiment and DMC give 4.061 A at

T = 0 K, with zero-point corrections). Cohesive energy agrees with experiment to within

20 meV per formula unit (comparable with experimental uncertainty). Hierarchical QC approach

gives similar accuracy. No adjustable parameters in either method - gives us every reason to

expect excellent values of surface formation energy.

Method σ / J m−2

DMC pseudo 0.373(3)

DMC all-elec 0.44(1)

Quantum chem (frozen core) 0.402

Quantum chem (with core) 0.434

DFT(LDA) 0.466

DFT(PBE) 0.337

DFT(rPBE) 0.272

• DMC slab calculations on LiH went up to slab

thicknesses of 6 ionic layers, with 18 ions per

layer in the repeating cell (total number of ions

per repeating cell thus went up to 108).

• Agreement between all-electron DMC and high-

level quantum chemistry extremely close, as

it should be, and the σ values give robust

benchmarks against which to judge DFT. In

LiH, LiF, and MgO, LDA performs rather well,

the PBE value is about 30 % lower and is

considerably less good, while the revised PBE

value (rPBE) is ∼ 40 % below correct value.– Typeset by FoilTEX – 88

Molecular adsorption on surfaces

• Adsorption energies of molecules on surfaces important for many reasons (gas sensing, catalysis,

corrosion, gas purification, chemical reactions in interstellar space, atmospheric processss).

• Our quantitative knowledge of adsorption energies remains poor, largely because DFT has rather

poor predictive power for the energetics of adsorption (especially when van der Waals etc.

important).

-0.5 0 0.5 1 1.5 2Dist. from Equil. / Å

-0.2

-0.1

0

0.1

0.2

Inte

ract

ion

ener

gy /

eV

PBEWCrevPBEBLYPPBE-DrevPBE-DBLYP-DDMC


Water on graphene: DFT is completely useless

2 3 4 5 6 7 8Height (Angstrom)

-200

-150

-100

-50

0

50

100

150

200

Bin

ding

Ene

rgy

(meV

)

PBELDAPBE0revPBEBLYPB3LYP


Water on benzene: DMC compared with CCSD(T)

3 4 5 6 7Height (Angstrom)

-150

-100

-50

0

50

Bin

ding

ene

rgy

(meV

)

CCSD(T)DMC


Water on graphene: DMC vs. DFT

J. Ma, A. Michaelides, D. Alf, L. Schimka, G. Kresse, E. Wang, Adsorption and diffusion of water

on graphene from first principles, Physical Review B 84, 033402 1-4 (2011).


Water systems

Comparison of DMC total energies (filled squares) with accurate quantum chemistry benchmarks at

CCSD(T) level for a sample of 50 geometries of the H2O trimer drawn from an MD simulation of

liquid water. Horizontal axis shows CCSD(T) energy, vertical axis shows deviation of DMC energy

from CCSD(T) energy. Filled circles show the same comparison for DFT(PBE).– Typeset by FoilTEX – 93

DMC-based statistical mechanics for surface problems

Combination of DMC with statistical mechanics is feasible, because Alfe et al. haverecently shown it in action for liquid and solid iron at high temperatures. [E. Solaand D. Alfe, Phys. Rev. Lett. 103, 078501 (2009).]

Alfe group now addressing new challenge of doing thermally disordered interactingmolecules on surfaces, including interactions between adsorbed molecules andthermal disorder/entropic effects. Need to combine DMC with other techniques.

Calculate e.g. :

• adsorption isotherms (coverage θ as a function of gas pressure)

• thermal desorption rates as a function of θ and temperature T .

The outcome of all this will be the first ever computations of thermodynamicfunctions of a dissociating adsorbate with full inclusion of entropic effects - at thechemical accuracy given by DMC.


General advice

Given developments underway, we believe it is now very timely for more research groups to consider

becoming involved in the QMC enterprise. But what is the right way to do this? Some suggestions.

• Start small and work upwards: Clearly, one should gain experience first with small problems (for

example, problems involving small molecules), which can easily be run on modest local resources.

Work up from there to more ambitious problems that need large machines.

• Use standard codes: A large development effort has gone into CASINO and other QMC codes,

and it makes sense to work with a code that already has a substantial publication record.

• Collaborate: Even more than DFT, there is much that you need to know about QMC before

trying calculations. No QMC code can (yet) be treated as a black box, and it is wise to learn

from experienced practitioners. Attending this school is thus a good idea.

• Choose your problem well: Just like DFT, QMC cannot solve all the world’s problems, and it

is important to play to the strengths of the techniques and to be aware of their weaknesses

(fixed-node error/pseudopotential non-locality potential weaknesses of DMC in its present form).

Our hope for the future is that more researchers will discover for themselves the possibilities offered

by QMC on machines ranging all the way from local clusters to the national and international

petascale services now becoming available.


The end of Mike’s contributions to the summer school


{ typeset by foiltex · 2013-08-12 · atom, but consider variance of hellmann-feynman derivative...

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