Quantum Monte Carlo for Atoms, Molecules and Solids
W. A. Lester, Jr.
Department of Chemistry, University of California, Berkeley, California
Lubos Mitas
Department of Physics, North Carolina State University, Raleigh,
North Carolina 27695
Brian Hammond
Microsoft Corporation, 45 Liberty Boulevard, Great Valley Corp
Ctr., Malvern, Pennsylvania 19355
Abstract
The quantum Monte Carlo (QMC) method has become an increasingly important method
for the solution of the stationary Schrödinger equation for atoms, molecules and solids.
The method has been established as an approach of high accuracy that scales better with
system size than other ab initio methods. Further, the method as typically implemented,
takes full advantage of parallel computing systems. These aspects for electronic structure
will be described as well as recent applications that demonstrate the breadth of
application of the method. In addition, frontier areas for further theoretical development
are indicated.
I.
Introduction
In the past decade the field of computational science, the ability to simulate physical processes based on
fundamental physical laws, has undergone rapid growth. This progress has been achieved not only due to
the exponential increase in computing power now available but also because of new generation of
simulation algorithms. However, each new generation of high-performance computing platforms presents
a renewed challenge to the developers of computational methods to take full advantage of the underlying
computer architecture. In many cases the original computational method has not been well suited to both
reduce the time to solution for a given physical system, and, more importantly to increase the accuracy of
the simulation.
The ab initio simulation of the electronic structure of atoms, molecules and solids has greatly benefited
from an enormous increase in computing power, but it has also remained bound for the most part by
approximations originally created to make the computations possible. The most popular of these
approximations are those based on basis set expansions that have their origin with either the Hartree-Fock
or Kohn-Sham equations. These methods have presented a number of challenges to the use of massively
parallel computers as well as well-known limitations in their accuracy.
A method that shows great promise both in its inherent accuracy as well as its ability to make efficient use
of modern computing power is quantum Monte Carlo (QMC). QMC is a broad term that can refer to a
number of techniques that use stochastic methods to simulate quantum systems. For electronic structure
theory we are referring to the simulation of the electronic Schrödinger equation by the use of random
numbers to sample the electronic wave function and its properties. For reviews since 2001, see AspuruGuzik 1, Aspuru-Guzik2, Anderson 1, Towler1. QMC has consistently shown the ability to recover 90-95
% of the electron correlation energy (the difference between the Hartree-Fock and exact energies), even in
cases that are poorly described by the Hartree-Fock method even for large systems with hundreds of
electrons. In a few-electron systems, the method is capable to solve the stationary Schrodinger problem
virtually exactly.
Until recently the success of the QMC method was limited to simulation and prediction of properties of
small molecules consisting of low atomic weight atoms, owing to the amount of computer time required
to reduce statistical error to physically significant levels. Great strides in methods combined with QMC’s
inherent parallelism have removed these limits and made possible the accurate simulation of much larger
molecules as well as solids [ref]. Advances in the method have widened the application of QMC to many
electronic properties of these systems besides the energy. One of the greatest strengths of the approach
lies in the ability to numerically sample quantum operators and wave functions that cannot be analytically
integrated or represented. This capability has enabled the QMC method to make significant contributions
to fundamental understanding of the electronic wave function and its properties.
The QMC method has grown in interest for the study of a wide range of physical systems including
atomic, molecular, condensed-matter, and nuclear. In this paper, we shall not address the latter area [ref],
which is mentioned only to indicate the broad range of phenomena accessible by the method. The general
applicability of the method has led us to limit consideration in this article to the electronic structure of the
aforementioned systems due to space limitations and author interest. Path integral MC (PIMC) will not be
discussed nor shall approaches for the determination of the internal energy of molecular systems
including clusters [ref.]
Certain properties make QMC particularly attractive for the treatment of electronic structure. These
include limited dependence on basis set which is a feature that dominates all other ab initio methods, high
accuracy of energy calculations that are typically as accurate as other state-of-the-art ab initio methods
[refs], and dependence of computation on system size N or scaling [refs], as well as the ease of adaptation
of QMC computer programs to parallel computers [refs].
This paper is organized as follows. Section II will summarize the main variants of QMC applied to atoms,
molecules, and solids. These methods are variational Monte Carlo (VMC) and diffusion Monte Carlo
(DMC). Section III turns to other issues that play an important role in the development of QMC. Section
IV briefly summarizes selected applications that provide an indication of the breadth and level of
accuracy for systems studied to date. Finally, Section V points to various new methods that hold promise
for advancing the state-of-the-art.
II.
QMC Methods
2
The QMC methods to be discussed solve the non-relativistic, time-independent, fixed-nuclei, electronic
Schrödinger equation,
(1)
where
is the molecular Hamiltonian. The Hamiltonian contains the kinetic and potential energy
operators for the electrons,
where
is the Lapacian operator for the kinetic energy,
(2)
and
is the potential energy operator, which for a system comprised of
nuclear charges
electrons and
nuclei with
and no external fields is,
(3)
where r is the distance between the particles. Atomic units have been used, in which the mass and the
charge of the electron are set to unity.
To solve this time-independent equation using Monte Carlo, we start with the time-dependent
Schrödinger equation written in imaginary time by substituting τ = it,
(4)
that has the formal solution
(5)
The quantity ET is an energy offset chosen to reduce the impact of the exponential factor. As τ → ∞ this
equation (2.5) converges exponentially to the lowest electronic state with non-zero Ck , which is usually
the ground state Φ0. Equation (2.4) above has been written with the Hamiltonian expanded to show that it
has the form of a diffusion equation (the kinetic energy term) and a rate equation (with “rate constant” 𝒱
– ET). Both these processes can be simulated using Monte Carlo sampling methods to propagate the
solution to large imaginary time. (Note: since τ is imaginary time, no description of real time dynamics is
possible; see, however, ref.[Mitas-Grossman].)
The simulation is performed by constructing by choosing an initial ensemble of pseudo particles, each of
which represents the positions of all the electrons in the system – R={x1, x2, …, xn}. Each of these
electronic configurations then undergoes a random walk in imaginary time until sufficient time has
elapsed for the excited states to decay away. The resulting density distribution is proportional to Φ0. Such
a simulation, however, poses two serious problems. First, to simulate a diffusion equation one requires the
distribution to represent a positive density, whereas for Fermion systems Φ0 necessarily changes sign.
3
Second, the rate equation depends on the potential energy which varies widely and diverges when two
particles approach, which will cause the Monte Carlo simulation to be unstable.
Both these issues can be dealt with by the introduction of importance sampling. Choose a trial function,
ΨT, which is a good approximation to the desired state Φ0. Define a new function,
,
and then rewrite the imaginary time dependant Schrödinger equation in terms of this new function:
(6)
where
of large
, and
, and
. The quantity
is the local energy of
introduce the term
is a vector field pointing towards regions
. The effect of importance sampling is twofold: first to
which is a directed drift moving the system to areas of large
,
and second, the rate term now depends on the local energy which will be much smoother than the
potential energy for good trial function choices. It can be seen from the form of the local energy, as
becomes a constant, namely
the ground state energy.
Once
has converged to
, expectation values of Φ0 may be measured by continuing the Monte
Carlo simulation and sampling various operators, the most important of these being the Hamiltonian. The
energy of
is obtained from the local energy as follows,
(7)
The last step is obtained from the Hermitian property of ℋ. One difficulty imposed by importance
sampling is that all properties involving an operator which does not commute with the Hamiltonian
will result in a mixed estimate,
several methods to extract the pure
rather than the “pure” estimate
. There are
distribution needed for operators that do not commute with the
Hamiltonian from the mixed distribution that will be addressed below.
Simulating the importance sampled Schrödinger equation is formally the same as before, with the
diffusion process now having a directed drift, and the new “rate constant” being
The sampling
procedure can be performed in a number of ways. Below we present the two QMC approaches that
dominate interest and have been used for most calculations: Variational Monte Carlo (VMC) and
Diffusion Monte Carlo (DMC). VMC is a method to sample from the
distribution yielding the energy
and other properties of the trial function. DMC is a projector method which samples from the mixed
distribution and in principle yields exact energies and properties.
A.
VMC
4
The VMC method exploits the QMC formalism above to evaluate the expectation values of a known trial
function, ΨT, rather than the ground state Φ0 [Reynolds3]. This is useful when the function ΨT cannot be
integrated analytically or by other numerical methods. The method is especially valuable for investigating
novel wave function forms. VMC is also used as a precursor to DMC as a method to optimize
and to
generate initial walker distributions.
The importance sampled Schrödinger equation without the local-energy dependent term is a Fokker-Plank
equation with solution
,
(8)
One can generate an ensemble of pseudo particles {Ri} via the related Langevin Equation, which can be
integrated over a small time step , to give a prescription for moving a pseudo particle with coordinates
R to a new position R′,
(9)
Here D is the diffusion constant (1/2) and χ is a Gaussian random variable with zero mean and variance of
2Dδτ. Once the ensemble has converged to the distribution
properties such as the local energy can be
sampled. In practice Metropolis sampling is introduced to further speed convergence and eliminate any
dependence on the time step.
Trial functions for VMC can be constructed in many ways and specifics are treated separately below.
B.
DMC
The DMC method is a projector approach in which a stochastic imaginary-time evolution is used to
improve a starting trial wave function. The governing equations can readily be obtained by multiplying
the time-dependent Schrödinger Equation in imaginary time, Eq. (4), by a known approximate wave
function T. One defines a new distribution given as the product of T and the unknown exact wave
function, . The resultant equation has the form of a diffusion equation with drift, provided
is positive
definite. In general the exact  and the approximate T will differ in various regions and their product
will not be positive, therefore one imposes the fixed-node boundary condition which results in a positive
distribution
. The wave function is an approximation to Φ in that the nodes of are the same
as ΨT.
There are two primary difficulties to be overcome to simulate the particles, i.e., the electrons in imaginary
time. These are: how to create a stochastic process that generates the proper distribution, and how to
obtain the appropriate Fermion ground state, not the lowest energy solution which would be the Boson
ground state.
1.
The Short Time Approximation to the Green's Function
5
In order to create an ensemble of pseudo (imaginary time) particles with the distribution
we need a
method similar to that in VMC that moves the particles of the system through a series of time steps. This
aim can be achieved in a manner similar to VMC, by integrating the Schrödinger equation to obtain the
function f at a time τ+δτ from the f at time τ,
(10)
The function
is the Green's function of the Hamiltonian, formally related by
.
The exact Green's function is a solution to the Schrödinger equation, and is therefore unknown in general.
However, we can approximate the Green's function as
which is exact in the limit
that δτ →0 (since 𝒯 and 𝒱 do not commute). For the importance sampled equation the kinetic energy term
is modified to include the drift term, so that expression for the short time approximate Green's function is
(11)
This is now composed of a gaussian probability distribution function with a mean that drifts with velocity
and spreads with time as
, and a rate term which grows or shrinks depending on the value of
relative to the fixed quantity
. Hence this Green's function has the expected behavior of a diffusion-like
term multiplied by branching term.
2.
The DMC walk
There are numerous methods to carry out the simulation (as well as other possible short-time
approximations), but one of the simplest method is as follows:
(a) Create an initial ensemble of walkers,
, typically the result of a VMC simulation;
(b) For each walker, each electron (with coordinates
) will diffuse and drift one at a time for a time
step δτ by,
(12)
where χ is a pseudo random number distributed according to the Gaussian part of
;
(c) If the electron crosses a node of the trial function the move is rejected and go back to (b) for the
next electron.
(d) To insure detailed balance, accept the move of the electron with probability
where
6
(13)
(e) Return to step (b), moving all the electrons in this walker. Compute the local energy and other
quantities of interest.
(f) Calculate the multiplicity M (the branching probability) for this walker from the rate term in the
Green’s function,
(14)
The quantity δτa is the effective time step of the move, taking into account the rejection of some
moves. The effective time of the move is calculated from the mean-squared distance of move,
which would be
without rejection. With rejection they move
and
so one can compute,
.
(15)
The walker is then replicated or destroyed based on M. To do so create the integer
where ξ is a random number between zero and one, if
then make
copies, otherwise the multiplicity is zero and the walker is removed from the ensemble. (In some
implementations the ratio
is kept as a weight for the walker, and in the Pure DMC method
M is kept as a weight and the ensemble size is kept fixed.)
(g) Weight the local energy and other quantities of interest by M.
(h) Return to step (b) for the remaining walkers.
(i) Compute the ensemble averages and continue to the next time step. At suitable intervals one can
update the trial energy ET to regulate the size of the ensemble. It may also be necessary to
“renormalize” the ensemble when it grows to large or too small. This can be done by randomly
deleting or replicating walkers. Care must be taken as renormalization can cause an energy bias.
(j) At the end of the run the final energy and properties are computed and various statistical methods
can be applied to determine the variance of the mean values.
(k) Repeat for several values of δτ . Because the simulation is only exact in the limit
it is
necessary to either extrapolate to zero time or demonstrate that the bias is smaller than the
statistical uncertainty of the result.
3.
The variational principle in DMC
The DMC method is fundamentally a ground state method. No bounds exist similar to the variational
principle or variance minimization for excited states. Nevertheless, excited states that are the lowest of a
given symmetry are routinely addressed [refs.], and it has been shown that excited states of the same
symmetry as a lower state can be computed with the fixed node method with excellent results [ref.].
C.
Auxiliary Field QMC
7
Recent studies by Zhang and collaborators show favorable results using a real phase form of AFQMC. In
this approach, …[with refs.]
D.
Trial Wave Functions
1.
Form of the Trial Function
The types of wave functions that have been used in VMC and DMC begin with those of basis set quantum
chemistry, referring primarily to Hartree-Fock (HF) and various multi-configuration formalisms. The
latter include multi-configuration self-consistent-field (MCSCF) and configuration interaction (CI). In
addition, various density functional theory (DFT) wave functions have been used, in particular, a number
of generalized gradient forms [refs.]. There has been some use of Moeller-Plesset perturbation theory
functions, but state-of-the-art coupled cluster forms have not been used as trial wave functions owing to
the large number of operations required to evaluate such functions the numerous times required by QMC.
The trial wave functions play two important roles in the QMC calculations. The first is to avoid the
fermion sign problem by use of the fixed-node approximation. The second is efficiency: the statistical
fluctuations are significantly reduced by increasing trial function accuracy. This is practically very
important since the efficiency gain can reach two to three orders of magnitude. For cutting-edge
applications that involve large numbers of electrons or high statistical accuracy, trial function quality is
crucial and determines whether such can calculations are feasible.
To satisfy both these requirements trial functions one chooses the following general form:
Eq.
where
is an antisymmetric wave function and the second term is a symmetric factor which
includes explicit electron-electron correlations, i.e., terms explicitly dependent on interparticle
coordinates. Focusing first on the correlation function one can expand
in functions of two- and
three- (an so forth) particle interactions,
Eq.
where
describes correlations between electrons i and j, and
describes the dominant
3-particle correlations of the type electron-electron-ion where I labels ions – to date higher order
correlations have not be explored. The correlation functions
and
are further expanded in appropriate
basis sets with expansion coefficients chosen to satisfy know analytic behavior of the exact wave function
(typically the two-particle cusp conditions, satisfaction of which will keep the local energy bounded when
two particles meet) and then optimized by VMC methods. A simple form of correlation function is the
Pade-Jastrow (often referred to as the Jastrow factor) form
8
Note that by including explicit correlation into the trial wave function we are able recover a large fraction
of the correlation energy at the VMC level and significantly reduce the variance of the local energy by
satisfying the electron-electron cusp conditions.
In recent years, considerable effort has been invested in improvements of the antisymmetric function,
because it determines the fermion nodes and therefore controls the fixed-node bias. The simplest
antisymmetric wave function is a Slater determinant of spin orbitals. For QMC trial functions it has been
shown [ref] that this determinant can be factored into a product of spin-up and spin-down Slater
determinants of one-particle orbitals,
or, more generally, a linear combination of such products,
.
A single determinantal product multiplied by a Jastrow factor is often called a Slater-Jastrow wave
function. The one-electron orbitals from various methods have been tested and utilized in such wave
functions. Most popular have been Hartree-Fock (HF) and post-HF orbitals such as those from multiconfiguration self-consistent-field (MCSCF) and configuration interaction (CI) approaches. Orbitals from
density functional theory (DFT) have also been employed, in particular, a number of generalized gradient
and hybrid (with mixed exact Fock exchange) have been explored [Wagner1,Wagner2]. In addition, there
has been some use of orbitals generated from Moeller-Plesset perturbation theory, as well as direct
optimization of the one-electron orbitals together with other variational parameters as will be mentioned
below.
Most recently, pair-orbitals (geminals) have been introduced into QMC. This direction was pioneered in
the work of Sorella’s group [Casula1, Casula2] by employing the antisymmetrized product of pairing
orbitals which is known in condensed matter physics as Bardeen-Cooper-Schrieffer (BCS) wave function,
given as,
where
is the singlet pair orbital assuming that the total spin of the system is zero. Although the form
can be generalized to include spin polarized states, for the fully spin-polarized state the wave function
becomes uncorrelated like in HF. A more general form has been employed by Bajdich and coworkers
[Bajdich1, Bajdich2]
]
which has a form of a pfaffian [ref] of the pair spin-orbitals. By assigning electron spins one can write the
9
form as a pfaffian with pair singlet, triplet and unpaired spatial orbitals and thus describe correlation of
any spin state on the same systematic pairing level as given by
Psi_PF = pf [matrix of chi^^, phi^v,chivv etc equation (5) in Bajdich1]
The pair orbitals are expanded in an appropriate one-particle basis as given by
(Brian please retype the expressions for phi and chi as given below)
phi(i,j) = sum k>=l c_kl[gamma_k (i) gamma_l (j) + gamma_l(i)gamma_k(j)]
chi(i,j)= sum k>l c_kl[gamma_k (i) gamma_l (j) – gamma_l(i)gamma_k(j)]
For calculations with a sizable number of electrons it is important that the pfaffian can be evaluated by
algorithms which are analogous to Gaussian elimination for determinants and have the same
computational scaling. This proved to be quite successful and single pfaffians lead to systematically
improved results for the recovered correlation energy as has been tested on the first row atoms and
molecules. Further improvements include expansion in pfaffians which are much more compact than
expansions in determinants typically decreasing the number of terms by an order of magnitude while
recovering 98-99 % of the correlation energy for the first row systems [Bajdich 2]. Further research in this
direction is in progress.
Less traditional forms : transcorrelated constructions etc.
2.
Trial wave function optimization
Optimization of trial wave functions in QMC is a complex and challenging task due to the overall
nonlinearity of the optimization and stochastic integral evaluation. Any optimization algorithm used must
deal with multiple issues of the statistical "noise" of the quantities being optimized (typically the local
energy or its variance), potential linear dependencies in the parameters since they often constitute an over
complete set, and the problems of local minima. Many of these problems have been effectively resolved
in recent years and currently tens or hundreds parameters can be optimized with good efficiency
[Umrigar1, Umrigar2, Rappe]. These developments have enabled the concurrent optimization of Jastrow
parameters, determinantal expansion coefficients, and one-particle orbitals for the C2 molecule leading to
an accuracy of 0.XXXX eV, { equal or better than experiment?}. Very recently, of the order of ten of
thousands parameters have been effectively optimized in simulations of liquid hydrogen [Sorella1].
3.
Evaluation of the trial function: an approach to Linear Scaling
10
One of the most computationally intensive steps in the QMC algorithm is the evaluation of trial function,
and specifically the Slater determinant. For increasingly larger molecules, the number of operations
needed to evaluate the determinant and hence the energy at each point of a QMC simulation increases
where N is the rank of the determinant. For large systems the determinant becomes increasingly sparse
because the interaction between electrons diminishes exponentially with distance [Maslen], and several
approaches have been introduced to take advantage of this property to reduce the scaling of the
computational effort to a linear function of N.
In various implementations a number of steps have been taken in order to obtain near-linear scaling
[Manten, Williamson]. In particular, wave function evaluation in the QMC code Zori follows the
procedure of Aspuru-Guzik et al. [Aspuru-Guzik3] in which a sparse representation of the Slater matrix is
used leading to evaluation of only the non-zero elements of the wave. This approach, combined with the
use of localized orbitals in a Slater-type-orbital basis set, significantly extends the size of molecule that
can be treated with the QMC method – in this case the largest system contained 390 electrons. Tests
confirm that this approach leads to linear scaling of Slater determinant evaluation with system size, see
[Lester2]. With this approach, the calculation of excitation energies scales as N2.
E.
Pseudopotentials and Effective Core Potentials
Pseudopotentials, or Effective Core Potentials (ECPs) as they are often referred to in quantum chemical
literature, are important for increasing efficiency of calculations for heavier atoms. As was shown
previously [Ceperley, Hammond2] the cost of calculations grows proportionally to
, where Z is the
atomic number. Clearly, such Z dependence makes calculations of heavy atom systems computationally
demanding and inefficient with most of the computer time used for sampling large energy fluctuations in
the atomic core region, which has relatively little affect on the chemistry. Despite this shortcoming allelectron calculations have been carried out for systems containing atoms as large as Xe [Towler]. When
the important chemistry and physics occurs in the valence space, it is possible to replace the atomic core
with accurate ECPs. The construction of quality ECPs is therefore of prime importance, and often
electrons occupying the shell below the valence electrons are included in the valence space to take into
account "core polarization" effects, e.g. including the 3s and 3p electrons in the valence shell for the 3dtrasition metals. The ability of QMC to include ECPs has induced further developments in ECPs, such as
new ECP forms with smooth and bounded behavior at the nucleus [Greef, Lester, Needs, Filippi].
Evaluation of ECPs in the VMC method is straightforward, however DMC and other exact QMC methods
are more difficult. The ECP is in the form of a non-local integral operator, hence the value of the ECP at a
single point depends on the entire wave function. In DMC the wave function is unknown, and so we must
evaluate the ECP using
rather than , introducing a so called “localization approximation”
[Christiansen, Reynolds, Mitas]. This approximation enabled a number of important calculations,
demonstrating that with accurate trial functions the bias is rather small (proportional to the square of the
trial function error). A less desirable property of the localization approximation is that the resulting fixednode energy is not an upper bound to the exact energy, and the form of ECP introduced large energy
fluctuations. Based on QMC lattice model developments, [Bemmel] showed how to construct an upper
bound with the use of ECPs. The latter approach has been implemented in fixed-node DMC [Casula3]
and resulted in improved local energy fluctuations and stability of the calculations, allowing the use of
less accurate trial functions and larger time steps.
11
F.
Calculations of excited states
There is no difficulty in carrying out QMC calculations of excited states that are the lowest state of a
given symmetry. One simply requires a trial wave function of the appropriate symmetry. Excited states of
the same symmetry as a lower state are more difficult, in principle, owing to the absence of a theorem that
insures that the calculation evolves to the appropriate excited state. Early calculations of the E state of H2
confirmed the capability of the DMC method to accurately calculate the energy of the state for which a
small MCSCF trial wave function was used. [Grimes].
Rather challenging calculations of excited states for small hydrocarbon system have been carried out in
Filippi’s group [Filippi3]. The difficulty comes from the fact that these systems exhibit a number of
excited states which are competitive in energy and sensitive to the treatment of electron correlation. Using
careful optimization of the variational wave functions for both ground states and excitations the authors
were able to identify the correct excitations within the given symmetry sectors and relative differences.
G.
Computation of properties other than the energy: moments, forces,
equilibrium geometries, transition states,.[Caffarel1,2,3, Rappe2, Schuetz,
Liu, Hammond1, Liu] .
Calculations of properties that do not commute with Hamiltonian are more difficult to carry out with
DMC methods because the distribution is the mixed distribution
, not the pure fixed-node solution.
The calculation of pure expectation values requires sampling from the pure distribution which has been
addressed by Barnett, et al. [Barnett1, Barnett2] in studies of QMC expectation values of moments, and
by by Rothstein, et al. [Rothsteina,b, Caffarela] for differential operators and relativistic corrections. All
such “pure” methods require the computation of weighting factors (either explicitly or by random walk)
which correct for the mixed distribution. This adds significant computation above and beyond energy, and
adds variance to the final result.
A second difficulty in computing properties other than the energy is that their estimators can have
substantially larger fluctuations. The use of importance sampling can in principle reduce the variance of
the local energy to zero for the case where the trial function is exact trial. This zero-variance principle
does not hold for other properties, e.g. the dipole moment operator, and the variance is particularly
significant for calculations of atomic forces because the estimators involve derivatives of the total energy.
Various aspects of these complications have been tackled in recent years. As suggested recently, the
variance issue can adressed by modifying the estimator so that the fluctuations decrease significantly
without biasing the resulting estimate [Caffarel1].
??(need more on this or leave out) The issue of divergences of potentials and associated large fluctuations
has been addressed by Chiesa et al [Chiesa] by filtering out the corresponding terms and thereby
decreasing the corresponding variance.
The mixed estimator issue has been also studied and a promising method was introduced by Moroni and
Baroni [Moroni] called reptation Monte Carlo. The approach eliminates the branching step from DMC
and enables one to compute pure estimators effectively using forward walking (or, more precisely,
backward averaging) without carrying the weights. To date the method has not been efficient for large
12
systems, however, recent developments [Moroni, private] indicate that the method can be improved to
remove this shortcoming.
Another approach that has yielded very good results is the phaseless auxiliary field quantum Mont Carlo
method [Zhang1]. Except for special cases, two-body interactions require auxiliary fields that are
complex. As a result, the single-particle orbitals become complex, and the MC averaging over auxiliary
field configurations becomes an integration over complex variables in many dimensions which leads to a
phase problem that defeats the algebraic scaling of Monte Carlo and makes the method scale
exponentially. This is analogous to but more severe than the sign problem with real auxiliary fields or in
real-space methods. The sign problem in the complex plane is eliminated by projection onto the real axis
introducing thus an approximation analogous to fixed-node approximation in the usual DMC method.
The new approach enables the use of any one-particle basis and projects out the ground state by random
walks in the space of Slater determinants. Some of the less desireable features of this method is the fact
that the energy is not necessarily variational and the amount of correlation energy is limited by the size of
the used basis. Nevertheless, excellent results have been obtained with the method using a plane-wave
basis and nonlocal pseudopotentials [Zhang2, Zhang3].
[LM-what additional difficulties did you have in mind?]
H.
Beyond the Fixed-Node approximation
Exciting progress has been achieved in understanding the fundamental approximation of the DMC
method, namely, the fixed-node bias. In particular, a conjecture by Ceperley [ref] that the fermion node
hypersurface divides the space of electron configurations into the minimal number of two domains for
non-degenerate ground states has been actually explicitly demonstrated for a range of non-interacting
systems and for important trial functions such as those based on pairing orbitals in BCS and pfaffian
forms. This work shows that the topologies of the nodal surfaces are generically (i.e., almost always)
rather simple except possibly for complicated cases of quantum phase transitions and large-scale/infinite
degeneracies or nonlocal interactions. This research prompted further investigations into the accurate
shapes of the nodal surfaces that are important for minimization of fixed-node errors using, for example,
multi-pfaffian expansions [Bajdich1,2]. This line of research has also opened a new direction which
introduces concepts of topology and quantum geometry into electronic structure theory. Although much
needs to be done, further research based on ideas well-established in field theory and quantum statistical
mechanics is anticipated to influence both fundamental understanding of quantum systems and practical
applications.
I.
Coupling of QMC and molecular dynamics (MD) approaches.
The first attempt to combine the strength of QMC with ab intio MD methods was a study by Grossman
and Mitas [Grossman2]. This effort was based on efficient calculations of DMC energies along the
nuclear trajectories generated by the ab initio DFT/MD method. The key gain was obtained by updating
the walker distribution and DMC trial wave function along the nuclear paths. Nuclei in ab initio MD
require very small timestep displacements that are of the order of 0.001-0.0001 a.u., which is sufficient to
provide an opportunity for the electronic walker distribution to relax in a few DMC steps, because DMC
step is typically one to two orders of magnitude larger (0.1 – 0.01 a.u.). The authors found that about three
13
DMC steps are enough to update the wave function for each step in nuclear position and the QMC part of
the calculation increased the overall computation by only a factor of two. Further, they found that
although the detailed wave function information at any given step is determined by the walker distribution
so that the error bars are significant at any single point in the MD simulation, over the period of the entire
MD run they were able to obtain thermodynamic averages with very small error bars. One of the key
results was an estimation of the heat of water evaporation, a dynamical and finite temperature quantity,
that was found to be 6.5XXX kcal/mol, in excellent agreement with the experimental value of 6.1
kcal/mol (the DFT estimate is too large by about 30 %). Very recent MD/QMC calculations were carried
out by Sorella et al. [Sorella1] for hydrogen systems included also forces from QMC in the study of liquid
hydrogen and its properties at varying thermodynamical conditions.
TOTALLY NEW DIRECTION – Christov, combines QMC with Bohmian dynamics, all in real time!
Only 1-D application to date. Possible some mention, what do you think?
III.
Applications
Following is a list of calculations with brief comments that provide insight on the types of DMC
calculations that have been carried out to date. The list is biased towards calculations of the energy for
which DMC has been found to yield highly accurate results, although findings regarding other properties
are given.
A.
Molecular systems involving only first row atoms
ETHANE
A further validation of the DMC method for small molecules was recently reported in the form of the
singlet-triplet splitting in ethane. This property was computed using a large basis set with the CCSD(T)
method and compared the result of a DMC calculation with a Slater-Jastrow trial function. Excellent
agreement between the methods was obtained. [Dixon].
OZONE-I’ll have to double check these results by others (???)
O4 – there is a recent PRL by Caffarel and collaborators that looks interesting.
The O4 Molecule
The O4 species has relevance for atmospheric processes. In this regard the barrier to dissociation and the
heat of formation have particular importance. Owing to its small size the expectation is that state-of-theart computational approaches would provide a full understanding of the electronic properties of the
system, but complications due to the open-shell nature of O2, a proper description to bond breaking, and
the complicated spin recoupliongs that occur in this system present essential complications.. A recent
DMC study has addressed these difficulties in describing the passage from a singlet O4 reactant to two
ground state O2 triplet products [Caffarel5].
The authors have carefully addressed the issue of accurate differences of total energies for
14
barriers, enthalphies, affinities, etc. required for these calculations, an issue which pervades all
electronic structure methods. In this study, the exceptional step was taken of varying the
multireference character of the nodes that made possible very high accuracy for the energy
differences.
C20 - there is the comparison of results that Peter Taylor and
Porphyrin – large system; singlet-triplet and singlet-singlet energy differences (Bill ?)
Free Base Porphyrin
Accurate DMC calculations of allowed and nonallowed transitions in porphyrin have been reported
[Aspuru-Guzik4]. The vertical transition between the ground state singlet and the second excited state
singlet as well as the adiabatic transition between the ground state and the lowest triplet state were
computed for this 162-electron system. The DMC results were found to be in excellent agreement with
experiment.
Photosynthesis study – size systems that are being treated that no other ab initio approach can address.
Lubos – what studies in condensed matter, as well as your work on atoms and molecules?
1.
JCG’s G1 benchmark study
A systematic study of QMC performance on the G1 set of molecules has been carried out by Grossman
[Grossman1]. He used the simplest QMC single determinant Slater-Jastrow wave functions and found
mean error/max error …. in comparison with CCSD(T) … . The largest error was found for P_2
molecule and further improvements of the trial function using multi-reference MCSCF antisymmetric part
decreased the error to ….. This provided an important systematic insight into the QMC performance at
that time. It showed great potential of QMC method but also pointed out that in difficult cases the simple
Slater-Jastrow wave function might not be sufficient for desired accuracy. It pointed out a clear need for
improvements of trial wave functions and directed the effort towards overcoming the limits of simple
wave functions.
B.
Transition Metal Systems
Significant progress has been achieved with DMC in the calculation of transition metal systems since the
pioneering studies of Fe [Mitas1], of CuSin[Ovcharenko], and CO adsorption on a Cr(110) surface [El
Akramine]. Recent developments include all-electron FN-DMC calculations of the low-lying electronic
states of Cu and its cation. The states considered were the most relevant for organometallic chemistry of
Cu-containing systems, namely, the 2S, 2D, and 2P electronic states and the 1S ground state of Cu+. The
FN-DMC results presented in this work provide, to the best of our knowledge, the most accurate
nonrelativistic all-electron correlation energies for the lowest-lying states of copper and its cation. To
compare results to experimental data the authors included relativistic contributions for all states through
numerical Dirac-Fock calculations, which for Cu provide almost the entire relativistic effects. It was
15
found that the fixed-node errors using HF nodes for the lowest transition energies and the first ionization
potential of the atom cancel out within statistical fluctuations. The overall accuracy achieved with DMC
for the nonrelativistic correlation energy (statistical fluctuations of about 1600 cm–1 and near cancelation
of fixed-node errors) is good enough to reproduce the experimental spectrum when relativistic effects are
included. These results illustrated that, despite the presence of the large statistical fluctuations associated
with core electrons, accurate all-electron FN-DMC calculations for transition metals are presently feasible
using extensive, but accessible computer resources.
For transition-metal-containing molecules, recent developments include calculations of transition metaloxygen molecules such as TiO, MnO for binding energies, equilibrium bond lengths and dipole moments
[Wagner 1, Wagner2]. Calculations of energy related quantities showed the strength of QMC quite
clearly since with the simplest Slater-Jastrow trial functions one could get results on par with quite large
CCSD(T) calculations and within a few percent from experiments or better. In addition, calculations of
bond-breaking is not much more complicated than in the equilibrium unlike in CCSD(T) method where
special care has to be taken to obtain reliable results for such cases. The dipole moments showed
somewhat larger discrepancies for CrO and MnO, even when taking into account less realiable
experimental data. Dipole moments are very sensitive to the contribution of higher excitations, while
energy is less so, and show the missing pieces in the trial wave functions and points out a direction for
improvements. Very recently, large-scale QMC calculations has been carried out of FeO solid at high
pressures providing equilibrium properties such as equilibrium lattice constant, bulk modulus, cohesive
energy of 9.66(6) eV per FeO (experiment 9.7(1) eV per FeO), all in excellent agreement with experiment
[Kolorenc]. In addition, the calculations have produces also very good estimation of the band gap and,
finally, enabled to construct equations of state and estimation of transition pressure into another phase at
the lower end of available experimental data. Clearly, this is an important accomplishment showing that
calculations of complicated systems with several hundreds of valence electrons are possible and will
become more routine with the onset of the next generation of parallel architectures and multi-core
processors.
C.
Solids
D.
Weak interactions
Noteworthy DMC studies of weakly bound molecules have established the capability of FNDMC to yield
highly accurate interaction energies [Anderson2, Luechow1]. It was shown that FNDMC does not suffer
from basis set superposition error owing to the convergence of monomer(s) with basis set, unlike ab initio
post-Hartree-Fock methods. However, it was found for large systems that special attention must be paid
to the ability of the atomic orbital basis set to describe the asymptotic behavior of the wave function to
avoid sampling errors. Accurate results were reported for water, ammonia, and two benzene dimers – one
T shaped and the other displaced parallel to the first.
Software Packages
Some of the QMC methods matured and are implemented in several existing codes and packages which
are available for use by communities at large. We will mention the packages which are most familiar to
us: CASINO [NeedsCASINO], QMCPACK [KimQMCPACK], QWalk [WagnerQWALK] and CHAMP
16
[CHAMP]. QMCPACK and QWalk are distributed under Open Source licences.
Acknowledgments
WAL was supported by the Director, Office of Science, Office of Basic Energy Sciences, Chemical
Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098, and by
the CREST Program of the U. S. National Science Foundation. LM is supported by NSF/CMG grant
EAR-0530110 and by the DOE grant DE-FG05-08OR23337 and some of the projects mentioned in the
paper have been carried out using INCITE and NCMS allocations at DOE ORNL computing facilities.
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19
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