Quantum monte carlo methods for constrained systems

Quantum Monte Carlo Methods for Constrained Systems

Sarah Wolf,[a] Emanuele Curotto,*[a] and Massimo Mella[b]

The torsional ground state for ethane, the torsional, rotational,

and mixed torsional and rotational ground state of propane are

computed with a version of diffusion Monte Carlo adapted to

handle the geometric complexity of curved spaces such as the

Ramachandra space. The quantum NVT ensemble average for

the mixed torsional and rotational degrees of freedom of pro-

pane is computed, using a version of Monte Carlo path integral,

also adapted to handle curved spaces. These three problems

are selected to demonstrate the generality and the applicability

of the approaches described. The spaces of coordinates can be

best constructed from the parameters of continuous Lie groups,

and alternative methods based on vector spaces, where

extended Lagrangian terms would be too cumbersome to

implement. We note that the geometric coupling between the

torsions and the rotations of propane produces a substantial

effect on the ground state energy of propane, and that the

quantum effects on the energy of propane are quite large even

well above room temperature. VC 2014 Wiley Periodicals, Inc.

DOI: 10.1002/qua.24647

Introduction

The time scale problem is a familiar one to quantum chemists.

Rooted into our common basic training, the Born Oppen-

heimer approximation is the essential tool needed to unravel

fundamental problems that would otherwise be intractable. In

its most familiar form, it allows one to separate the electronic

degrees of freedom, associated with a set of much lighter

bodies, from the heavier nuclear degrees of freedom. With

some notable exceptions of marked nonadiabatic behavior,

the unraveling of which remains an very active field of

research, the Born Oppenheimer approximation yields very

accurate results, and it is often used to build accurate models

for the phenomenological potential energy surface felt by the

nuclei. The latter set of bodies is traditionally treated by classi-

cal mechanics. Generally, at sufficiently elevated temperatures,

for sufficiently heavy nuclei, and for sufficiently harmonic inter-

actions, classical mechanics provides reasonably accurate

answers. In this article, however, we depart from the tradi-

tional theme and explore the time scale problem one encoun-

ters when by necessity the quantum laws of motion are

applied to the nuclei themselves. This implies the nuclei are

light, the temperature is too low, and the interactions are

highly anharmonic. There is a vast number of fundamental

problems in chemical physics, where all these conditions take

place at the same time, and insight into these is fundamental

to a myriad of disciplines ranging from astrophysics to compu-

tational biology. More specifically, the set of problems that

have preoccupied our two groups intensely for a number of

years are the theoretical estimation of physical properties of

molecular clusters. Clusters in general and molecular clusters

in particular are models of condensed matter that can be

studied both theoretically and experimentally. Insight gained

from these investigations has already created a vast improve-

ment in our understanding of the complicated phenomena

that take place in the assembly process, thermodynamic stabil-

ity as function of size of condensed matter, microsolvation,

adsorption versus absorption, the effects of surface tension

just to name a few. At the temperature and pressure condi-

tions that produce stable molecular aggregates, nuclei of ele-

ments in the first three periods are sufficiently light to

produce significant quantum effects, whereas at the same

time, the weak interaction between the molecules in the

aggregate are far from harmonic. More importantly for the dis-

cussion in this article, most intramolecular degrees of freedom

are associated with relatively deeper dissociation energies, as

in the case of stretching modes, and relatively stiff force con-

stants compared to the intermolecular degrees of freedom.

Atomistic quantum simulation of molecular aggregates is ren-

dered either particularly challenging or practically intractable

by the large difference in the time scales.

The classical equations of motion satisfied by a typical

molecular aggregate are stiff. A small time step is required to

sample properly the high frequency dynamics, whereas a long

time scale is required to capture the effects on the system

from the set of lower frequency events. Additionally, this prob-

lem exacerbates the lack of ergodicity and the occurrence of

rare events when exploring thermodynamic properties in the

classical limit with random walks. Therefore, judicious use of

holonomic constraints is routine in classical simulations of

molecular clusters. The typical outcome for thermodynamic

properties is a significant reduction in the statistical error,

[a] S. Wolf, E. Curotto

Department of Chemistry and Physics, Arcadia University, Glenside, Penn-

sylvania, 19038-3295

E-mail [email protected]

[b] Massimo Mella

Dipartimento di Scienza ed Alta Tecnologia, via Valleggio 11, Universit�a

degli studi dell’Insubria, 22100, Como, Italy

Contract grant sponsor: ACS (Petroleum Research Fund; E.C.);

contract grant number: 48146-B6.

Contract grant sponsor: Universit�a degli Studi dell’Insubria via the fund

Fondi di Ateneo per la Ricerca (FAR; M.M).

VC 2014 Wiley Periodicals, Inc.

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making simulations more efficient. In quantum simulations,

there is an added benefit gained by constraining stiff modes, at

least for the two stochastic approaches we discuss in this

article. The convergence properties of the diffusion Monte

Carlo[1–12] (DMC) and the imaginary time path integral[13–31]

(MCPI) are greatly enhanced by constraining high frequency modes.

In a recent article,[25] we investigate a simple one-

dimensional (1D) harmonic chain with 1000 particles. All par-

ticles have the same mass, and every particle is connected to

two neighboring particles on a line with one stiff harmonic

spring on one side and a one soft one on the other side. The

simple harmonic chain model constructed this way mimics a

set of condensed molecules. Using analytical solutions of the

imaginary time path integral at finite Trotter number, we com-

pare the convergence of the analytical solutions of the path

integral expression for the heat capacity as a function of tem-

perature and Trotter number. Our results show that the adia-

batic approximation is accurate for temperatures below

1

10ffiffiffi2p

b�hxh

; (1)

where xh is the smallest of the high frequency set. Below

this temperature, a fully flexible simulation is highly inefficient

as it requires many hundreds of time slices to converge, where

the constrained simulations require less than 20 slices to con-

verge for most of the temperature range. Consequently, the

efficiency gains produced by performing the MCPI simulation

with constraints are massive. Similar gains are quantifiable for

DMC.[1] In Figure 1, we show the estimate of the ground state

energy of a particle of unit mass in a harmonic potential V5k

x2=2 with k510 a.u. as a function of simulation time for

various values of the step size Ds. This is the parameter that

has to be systematically reduced until convergence is achieved

in DMC simulations.

To properly interpret the result in Figure 1, it is important to

keep in mind that a similar computation with k51 a.u. is fully

converged with a Ds50:02 a.u. (the largest value of Ds in the

set of simulations represented). It is evident that an increase in

frequency by a mere factor of 3.2 requires a DMC step smaller

than a factor of 10. As is also evidenced in Figure 1, DMC sim-

ulations must also converge with respect to simulation pseu-

dotime (i.e., the number of DMC steps, on the x-axis in Fig. 1)

and a smaller Ds increases proportionally the total number of

DMC steps required to reach the asymptotic distribution, that

is, the ground state wavefunction in this case. When DMC is

used to simulate systems with multiple time scales, the stiffer

mode demands a small Ds, while the softer one demands a

relative longer time scale to reach convergence, and conse-

quently a much greater number of steps. Therefore, when in a

system, the subdivision of stiff and soft modes is clear, and

the adiabatic approximation is expected to work well, it is

highly advantageous to use the proper holonomic constraints,

and the efficiency gained can be on the same order of magni-

tude as what we measure in MCPI simulations.

However, constraining intermolecular degrees of freedom, like

bond stretches, bond angles, and the like, create curved spaces

and special techniques have to be developed to handle the

geometric complexity of these. The purpose of this article is to

review some of the recent advances in the form of algorithms

specifically designed to carry out quantum Monte Carlo simula-

tions in curved spaces, and to demonstrate their powers. In the

methodology section, we review briefly the basic objects

needed for the geometry and the dynamics in general non-

Figure 1. Ground state energy of a relatively stiff monodimensional harmonic oscillator for various values of the DMC step size. The population size is

maintained around 105 replicas. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Euclidean spaces. Then, we develop the general theory for DMC

and MCPI in non-Euclidean spaces. Our results section contains

selected numerical examples and in the conclusions section, we

discuss a set of directions that our group is currently undertak-

ing to further develop our tools and continue to explore the

rich and important field of condensed molecular matter.

Methodology

The methods discussed in this article are based on four fields of

modern applied mathematics and computation theory. These are

differential geometry,[32,33] Lie algebra and Lie groups,[34] the

Feynman path integral,[35–38] and Stochastic Calculus.[39] In this

article, combinations of these theories are applied to quantum

Monte Carlo simulations.[40–44] The books cited are excellent sour-

ces for those readers that wish to deepen their own understand-

ing in these rich theoretical areas. Before working through the

main theory for diffusion and path integral simulations in non-

Euclidean spaces, we briefly introduce some of the fundamentals

of differential geometry to clarify the notation used in the article.

It is important to keep in mind that even Euclidean spaces can

be mapped with curvilinear coordinates, and these give rise to

the same tensorial objects introduced in the next section. Curvi-

linear coordinates are often chosen with the eventual purpose of

exploiting symmetries (as e.g., an isotropic potential in R3), that

eventually are used to identify cyclic coordinates, and simplify

the treatment of systems. As the formalism of Euclidean spaces

mapped with curvilinear coordinates takes us more than half

way there, we start our discussion from curved spaces. However,

there are numerous kinds of curved spaces, and the methods in

this article have been developed to work only in orientable Rie-

mann spaces with a positive definite signature, no torsion (vide

infra), and that can be represented with a single coordinate map

except for a set of points with zero measure.[39] In this article, we

use the symbol Rn to notate an Euclidean space mapped with

an orthogonal set of Cartesian coordinates.

Non-Euclidean spaces

Most n-dimensional spaces used in molecular physics can be

derived with the aid of coordinate maps,

U : x0 x (2)

a set of n algebraic (typically nonlinear) equations that trans-

form a set of coordinates x to a set x0 in one or two main

ways:

i. They are subspaces of Rn1c with the introduction of c

holonomic constraints, such as, for example, the two-

sphere (S2) mapped from R3 with polar angles h;/,

h5tan 21 x21y2ð Þ1=2

z; (3)

/5tan 21 y

x; (4)

subject to the constraint x21y21z25r25 constant.

ii. They are the spaces of parameters of some continuous

Lie group.

We have specifically chosen examples that are best derived by

the second method to demonstrate the power of the

approaches contained in this article, and these are presented

in detail in the results section. In all cases, the spaces must be

sufficiently smooth and analytical, meaning the set of coordi-

nates that make up the map U are everywhere continuous,

have a finite first derivative and a symmetric Hessian derivative@

@xl0@

@xm0U5@l0@m0U5@m0@l0U; (5)

at every point in the space. These features allow one to

expand the coordinate map locally, that is, in an infinitesimal

neighborhood of a mapped point, into a linear “Cartesian-like”

map. Locally flat spaces of this nature are known as Riemann

spaces and the local Cartesian-like coordinates are known as

Riemann coordinates. The machinery of the chain rule is

implemented in the analysis of the geometric features of the

space, and a powerful notation using Greek subscripts and

superscripts is commonly introduced to handle tensorial

objects and their transformation properties. For example, com-

mon vectors are represented by their components. For exam-

ple, Va is the component along the unit vector ea that form

the basis for the vector space. The placement of the Greek

index indicates the contravariant manner by which compo-

nents of vectors transform upon coordinate changes xl ! xl0 ,

Vl05@xl0

@xlVl: (6)

The notation in the last equation also includes an implied

sum from one to the number of dimensions of the space

every time the same Greek letter occurs in an expression in

both the lower and upper position. One Greek index in the

lower position indicates the covariant way in which compo-

nents of one-form (such as derivatives of scalar functions @rf )

transform. The inverse of the Jacobian matrix elements in Eq.

(6) are used instead. The process of categorizing covariant and

contravariant behavior is generalized to ðn; n0Þ tensors with n

indexes in the contravariant position and n0 in the covariant

position.

A particularly useful (0,2) tensor, or two-form, is the metric

tensor,

gl0;m05@xm

@xm0@xl

@xl0glm: (7)

This entity contains all the geometric information about the

space. For example, the dot product in a non-Euclidean space

is defined using the metric tensor

A � B5glmAlBm5AmBm; (8)

where the right-hand most expression demonstrates how the

metric tensor is used to lower the index of a vectorial quantity

A and produce its one-form equivalent.



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Am5glmAl:

In a similar manner, the inverse of the metric tensor glm can

be used to raise indexes. Other useful information such as con-

nections between the degrees of freedom of the space, and the

curvature of the space can be extracted from the derivatives of

the metric tensor. A space with a symmetric tensor glm5gml is a

torsion free space, and a space that produces a positive definite

metric tensor is a space with positive definite signature. In the

molecular physics literature, the Hessian metric tensor (always

symmetric and positive definite) is commonly known as the

mass matrix, and this object additionally contains information

about the effective mass of the system. The Hessian metric ten-

sor is normally generated with Eq. (7) by transforming from a

Cartesian space, where it is represented as a diagonal matrix

with masses along the diagonal. The Hessian metric tensor is an

essential object for the development of dynamic theories in

manifolds. The classical Lagrangian of a system in non-Euclidean

space, for example, is written as follows

L51

2glm _xl _x m2V: (9)

The derivation of the laws of motion follows the same varia-

tional homotopy calculus as in Rn. The end result is the cele-

brated Euler–Lagrange equations, and these have exactly the

same expression in all Riemann spaces, Euclidean or non-

Euclidean alike. One interesting feature of the metric tensor is

that the coordinates chosen to map a given space may not be

orthogonal, and this is reflected by the coupling matrix ele-

ment between the velocities _xl and _x m. Therefore, the laws of

motion, classical or otherwise, couple the two degrees of free-

dom. To distinguish the latter types of couplings from those

that occur, when the potential energy term contains product

terms like xlxm we call them “geometric couplings,” while the

traditional “dynamic couplings” are those produced by the

potential energy. In most systems of interest, degrees of free-

dom are coupled in both manners, making it difficult to

extract the subtle effects that one kind of coupling has on

simulated physical properties relative to the other. Both types

of coupling terms can be safely ignored whenever the differ-

ence in time scale between the two degrees of freedom is

such that an adiabatic approximation is accurate. However,

when the time scales are similar, significant contributions to

the physical properties of the system can be expected.

DMC in non-Euclidean spaces

We now proceed to the task of deriving the diffusion equation

in general Riemann spaces. Fick’s first law of diffusion defines

the flux vector J as the derivative of Pðx; tÞ, the scalar probabil-

ity of finding a diffusion particle between x and x1dV,

Jm52D@mP; (10)

where D is the diffusion coefficient, and x represents the set

of generalized coordinates of the n-dimensional Riemann

space M. dV is the differential volume element in M,

dV5ffiffiffiffiffiffijgj

pdx1 � dx2 � �� dxn; (11)

whereffiffiffiffiffiffijgj

pis the square root of the determinant of the met-

ric tensor, and the symbol � notates the wedge product. The

square root of the determinant of the metric tensor makes the

totally antisymmetric density dV invariant under change of

variables and is known as the Jacobian.

The reader will have noticed that Fick’s first law produces a

one form for the flux. Fick’s second law is a statement regard-

ing the conservation of flux across an oriented boundary in

M. In integral form, for a source and sink free conditions it

translates to a statement of conservation of probability,

ðM

div J dm1

ðM

@P

@tdm50: (12)

The proper divergence expression for a (1,0) vector that sat-

isfies the conservation laws in integral form is

div X51ffiffiffiffiffiffijgj

p @m

ffiffiffiffiffiffijgj

pXm

� �: (13)

The divergence takes this form because the derivative of non-

scalar quantities in Riemann spaces does not transform cova-

riantly. Rather, the covariant derivative of a (1,0) vector is given by,

@lXm 1 CmlkXk: (14)

One defines the Christoffell connections of the second kind

for Riemann spaces (Cmlk), by forcing the expression in Eq. (14)

to transform as a (1,1) tensor. Furthermore, it can be shown

that the Christoffell connections can be evaluated using the

following symmetric derivative of the metric tensor

Cmlb5

1

2gmq @lgbq1@bgql2@qglb� �

: (15)

Equation (15) is obtained by taking the derivative of glm and

forcing it to transform as a (0,3) tensor under change of coordi-

nates.[32,33] The expression in Eq. (13) is the trace of the object

in Eq. (14), as notated by the repeated index in the upper and

lower position. Additionally, as the flux is a (0,1) vector, its index

is raised using the inverse of the metric tensor,

div J51ffiffiffiffiffiffijgj

p @m


pglmJl

� �: (16)

Finally, inserting Fick’s first law, Eq. (10), on the right to elim-

inate J gives,

div J52D1ffiffiffiffiffiffijgj

p @m


pglm@lP

� �: (17)

The operator on the right-hand side of Eq. (17)

1ffiffiffiffiffiffijgj

p @m


pglm@l5DLB ; (18)

is known as the Laplace–Beltrami operator.



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The time-dependent Schr€odinger equation, in imaginary

time (t5isÞ is isomorphic to the diffusion equation, with a

source-sink term V2Vref where Vref is simply a guess for the

ground state energy of the system.[1–12] In M it reads,

2�h2

2DLB w1 V2Vrefð Þw5�h

@w@s

: (19)

This differential equation can be simulated in general Rie-

mann spaces [6–11] as well as its counterpart in Rn. The proof

of this statement begins with the local expansion of the map

U at a point x0 up the linear term. The resulting map, valid in

the neighborhood of x0, is well defined at every point in M by

the requirements for U that have been elaborated earlier.

Then, the partial derivatives in Eq. (7) are constant (i.e., eval-

uated at x0). Consequently, the metric tensor is constant in the

infinitesimal neighborhood of x0 as well, and the Laplace–Bel-

trami operator takes a relatively simple form,

DLB ! glm x0ð Þ@m@l: (20)

From the local diffusion equation

�h@w@s! 2

�h2glm x0ð Þ2

@m@lw1 V2Vrefð Þw; (21)

one can write a solution which is useful for the DMC

approach. Branching is carried out the same way as in Rn,

whereas the solution of the source-sink free diffusion equation

can be used to sample random numbers that have �hDsglm xkð Þas the correlation matrix.

ffiffiffiffiffiffiffiffiffiffiffiffiffijgj

2p�hDs

rexp 2

1

2�hDsglm xkð ÞDxlDxm

� �: (22)

The random numbers (Dx) in Eq. (22), are used to simulate

diffusion for a particle from xk to xk11, where xk is the point

from which the map is expanded, rather than x0. We have

explored several methods to sample the distribution in Eq.

(22). The most general approach is to use the Cholesky

decomposition of the correlation matrix, and the Box–Muller

algorithm[45] together. The Box–Muller algorithm is used to

generate n values for the set Dy, distributed as Gaussian varia-

bles with zero mean, no correlation, and with unit variance.

Then, the Cholesky decomposition of the correlation matrix

�hDsglm xkð Þ5LLT ; (23)

produces the proper transformation from the Dy uncorrelated

set to the correlated one,[46]

Dx5LDy: (24)

Second-order DMC on the two-sphere

The diffusion procedure simulated using the aforementioned

process is convergent to first-order in Ds, the time step, for the

most general case. This situation can be improved in the special

case where a non-Euclidean space has a constant metric tensor,

as a modified branching procedure can be used to accelerate the

convergence of the DMC method up to a third-order behavior.[5]

This approach relies on the possibility of decomposing a diffu-

sion step in two consecutive ones with half Ds either exactly

(e.g., in Rn), or within a well-specified order of error in Ds.

In the case of the diffusion equation on the two-sphere, S2, a

particular choice of coordinate involving the geodesic distance

allows one to implement a perturbation treatment[47] leading to

a short time approximation of the diffusion Green’s function that

is a third-order in Ds and fourth in the ratio between the diffu-

sion displacement along a great circle and the sphere radius.

Such approximation has been tested, and confirmed to have a

cumulative second-order error in Ds, when computing dynami-

cal observables such as the average diffused angle along a great

circle after a chosen elapsed time nstep Ds (nstep � 102100).[11]

The need for testing the diffusive dynamics directly, rather than

the steady state estimate of a few observables as commonly

done while testing DMC, is rooted into the fact that the s!1distribution for the diffusion in a curved space is a constant.

Coupled with the commonly used symmetric splitting of the

branching step based on the average of prediffusion and postdif-

fusion potential energy, the approximated Green’s function deliv-

ers a robust total second-order in the energy observables, which

can be improved by means of an a posteriori extrapolation.[4] Addi-

tional usage of the algorithm just discussed, beside the initial

application to O2Hen systems,[11] has indicated that its perform-

ance does not deteriorate even when dealing with highly quan-

tum objects such as para H2 adsorbed on ammonia clusters.[48]

Albeit not strictly necessary in the applications so far pre-

sented, importance sampling introduced in the branching

step, or simulated with a drifting term via optimized guiding

functions is possible in non-Euclidean spaces as well.[8] The

gain in efficiency is comparable[2,3] to that obtained in Rn.

Feynman path integral in non-Euclidean spaces

The Feynman quantization for Riemann spaces was first formu-

lated by DeWitt.[49] Its formalism is based on the following

construct for the matrix element hxi; tijxi21; ti21 i of the time

evolution operator,

hxi; tijxi21; ti21i51

2pi�h

d=2

g21=4i D1=2 xi; tijxi21; ti21ð Þg21=4

i21

3 expi

�hS xi; tijxi21; ti21ð Þ

� �;

(25)

where, gi is the determinant of the metric tensor evaluated at

xi 2M. The action S xi; tijxi21; ti21ð Þ is defined as the time inte-

gral of the Lagrangian, and D xi; tijxi21; ti21ð Þ is the Van Vleck

determinant,

Dlm52@2S

@xli @xm

i21

: (26)

Equations (25) and (26) pertain to one of the N time inter-

vals used to subdivide the path t0; x0 ! tN; xN. These equations

can be used to formulate the regular time sliced approach.



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If the action S, and the Van Vleck determinant is expanded

about xi21; ti21 (the so called prepoint expansion), up to first-

order in ti2ti21ð Þ one obtains an approximate expression for

hxi; tijxi21; ti21 i,

hxi; tijxi21; ti21i �N

2pi ti2ti21ð Þ�h

� �n=2

g1=2i

3expi

2�h ti2ti21ð Þ glm xli 2xl

i21

� �xm

i 2xmi21

� �2

i

�hti2ti21ð ÞVeff

� �:

(27)

The effective potential energy in this expression contains a

quantum correction,[49]

Veff 5V2�h2 ti2ti21ð Þ

6R; (28)

where R is the Riemann–Cartan curvature scalar a contrac-

tion[32,33] of the Riemann curvature tensor,

Rqrlm5@lC

qmr2@mC

qlr1Cq

lkCkmr2Cq

mkCklr: (29)

The Riemann curvature tensor contracts to the Ricci tensor,

Rlm5Rklkm; (30)

and the Riemann–Cartan curvature scalar is obtained by con-

tracting the Ricci tensor with the inverse of the metric tensor,

R5glmRlm: (31)

The resolution of the identity in curved spaces can be rigor-

ously applied to derive the path integral expression using the

same limiting procedure as in Euclidean spaces mapped by Car-

tesian coordinates. However, the short time approximation has

to be obtained by expanding the propagator into a power series

and keeping all the terms that are first-order in Dt. This pro-

duces the correction term in Eq. (28). The term proportional to

the Riemann curvature scalar results from the proper transforma-

tion of the path integral measure.[39] Unless the Riemann curva-

ture scalar is constant, it cannot be omitted from the expression

of the path integral or the latter will not converge. Problems

that require curved spaces with a nonuniform Riemann curva-

ture can be quantized in a number of equally valid ways.[17,49]

However, for the ellipsoids of inertia, and the Ramachandra

spaces we explore in this work, the Riemann curvature is con-

stant and these additional difficulties can be safely ignored. Fur-

thermore, it is important to realize that as the Riemann

curvature is a scalar quantity, it is inherently independent of the

coordinate map chosen to reach points in these spaces.

Path integral simulations of molecular aggregates do not

abound in the chemical physics literature.[13–27] It is clear that

the primitive approach is rather limited and that methods with

accelerated convergence would be necessary.[28–31] Accelerating

the convergence of the traditional time-sliced path integral sim-

ulations require either a first- or a second-order derivative of

the potential. The gains obtained by reducing the number of

slices N, must be weighted against the additional cost for

computing the derivatives. In multidimensional applications,

especially when the potential energy models are sophisticated,

the computation of the gradient, or the Hessian can in fact off-

set the efficiency gained using a second-order method. It is the

combination of these factors that made the reweighted random

series, coupled with the finite difference estimators, such power-

ful contributions[30] even in Rn. These approaches have been

recently extended to simulations in Riemann spaces.[23–27]

With the reweighted random series, one gains quadratic

convergence and statistically stable estimators without the

need to evaluate any derivatives of V. For the reweighted Fou-

rier–Wiener path integral in imaginary time, one begins by

redefining the random path with k0

m > km terms,

xl uð Þ5xl0 1r

Xkm

k51

alk Kk uð Þ1r

Xk0mk5km11

alk

~Kk uð Þ; (32)

where

r5�hb1=2; b5 kBTð Þ21; u5t

b�h: (33)

and constructs the functions ~Kk uð Þ so that the partial averag-

ing expansion about the core path derived from Eq. (32) is

equal to the same derived with the infinite series. The require-

ment for ~Kk uð Þ is easily derived.[30]

Xk0mk5km11

~K2

k uð Þ5u 12uð Þ2Xkm

k51

K2k uð Þ: (34)

For the Fourier–Wiener path integral, the path functions are

Kk uð Þ5ffiffiffiffiffi2

p2

rsin kpuð Þ

k; (35)

~Kk uð Þ5f uð Þffiffiffiffiffi2

p2

rsin kpuð Þ

k; (36)

and

f uð Þ5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu 12uð Þ2

Pkm

k512

p2k2 sin 2 kpuð ÞPk0mk5km

2p2k2 sin 2 kpuð Þ

vuut : (37)

Equation (32) is inserted into Eq. (27) with

i ti2ti21ð Þ5 b�h

N: (38)

Then, for a N point quadrature, and an n-dimensional

Riemann space the density matrix qRW becomes,

qRW x; x0;bð Þ5 1

2p

Nn=2

�h2b� �2n=2

JKðd a½ �r


pexp 2b

ð1

0

duU x uð Þð Þ� �

;

(39)

where JK is the Jacobian of the transformation xi ! ai , and



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U x uð Þð Þ5 1

2glm _xl _x m1Veff : (40)

This algorithm, complete with the numerical difference esti-

mators for the energy and heat capacity

hEib5n

2b1

@

@bbð1

0

duU

� � �; (41)

CV

kB5

n

21

n2

41nb

@

@bbð1

0

duU

� � �1

b2 @

@bbð1

0

duU

� �� 2* +

2b2 @2

@b2bð1

0

duU

� � �2

2n

22b

@

@bbð1

0

duU

� � �� 2

;

(42)

produces the desired convergence properties of the energy

and heat capacity. The derivatives,

@

@bbð1

0

duU

� �; (43)

are evaluated numerically.

Results

Ramachandra spaces

We have selected two molecular systems that are small

enough to generate reproducible results in a short amount of

time, but complicated enough to demonstrate the power and

the applicability of the methods reviewed in the article. These

are ethane and propane. In both the cases, it is more difficult

to define a map using the first approach discussed earlier, that

is, explicitly write down the equations of constraints. Rather,

the use of Lie groups of continuous transformations creates a

systematic approach to construct Cartesian coordinates from a

well defined and physically meaningful parameter space, and

this can be generalized and automated to produce linear

chains of any dimension.

The process we use to construct the coordinate map for a

linear saturated hydrocarbon CnH2n12,

U : R3 3n12ð Þ ! Tn21 � I3

for ethane (n 5 2) and propane (n 5 3) is an adaptation of

the algorithm outlined in Patriciu et al.[50] The vector space

Tn21 � I3 is the Cartesian product of the Ramachandra space

of torsions and the inertia ellipsoid for the rotational degrees

of freedom. The latter is typically mapped with three Eulerian

angles using the Rz /ð ÞRx hð ÞRz wð Þ convention, whereas Rama-

chandra’s space is typically mapped with n 2 1 dihedral angles.

We first build a body-fixed Cartesian coordinate xðBF Þ represen-

tation, and then rotate it into a space-fixed representation,

using the following procedure:

1. Carbon atom 1 is placed at the origin, xðCÞ1 5 0; 0; 0ð Þ.

2. Carbon atom 2 is placed along the z-axis, its coordinates

are given by the vector xðCÞ2 5rC ez5 0; 0; rCð Þ where ez is

the unit vector along z, and the carbon–carbon average

distance is rC52:9101 bohr.

3. The position of the other carbon atoms is obtained by

rotating and translating the coordinates of carbon 2,

xðCÞk125rC Rz /1ð ÞRx p2cð Þ � � � Rz /kð ÞRx p2cð Þez1x

ðCÞk11;

where, k varies from 1 to n 22, the angle c52tan 21ffiffiffi2p

is the

tetrahedral angle, and /k are the dihedral angles with /n50.

4. The coordinates of the hydrogen atoms are constructed

by performing a similar set of rotations and translations

to the vector rHez, where the carbon–hydrogen average

distance is rH52:0598 bohr. The first hydrogen atom on

carbon 1 depends on /1,

xðH1Þ1 5rHRz /1ð ÞRx cð Þez:

The other two hydrogen atoms are found simply by changing

the angle of rotation about the z-axis to /112p=3 and

/114p=3. For the hydrogen atoms on carbon atom k52 to

n22, the first set of rotation angles about z are augmented by

2p=3 and 4p=3, whereas the angle about x is always p2c.

xðH1Þk 5rHRz /1ð ÞRx p2cð Þ � � � Rz /k211

2p3

Rx p2cð Þez1x

ðCÞk ;

xðH2Þk 5rHRz /1ð ÞRx p2cð Þ � � � Rz /k211

4p3

Rx p2cð Þez1x

ðCÞk :

For carbon n 2 1, the last rotation about z is by 12p=3 for

one hydrogen atom and 14p=3 for the second hydrogen

atom. For carbon n, the last rotation about z is by /n21 for

one hydrogen atom, by /n2112p=3, and by /n2114p=3,

respectively, for the other two.

5. The coordinates constructed in the previous four steps

are translated to the center of mass

xðBF Þ5x2xCM :

For ethane and propane, the center of mass is independent of

the dihedral angles, and this fact simplifies the analysis.

6. The coordinate of each atom are operated upon by the

rotations about the three Euler angles,

xðSF Þ5Rz /ð ÞRx hð ÞRz wð ÞxðBF Þ;

where, clearly, the body-fixed configuration only depends on

the torsion angle(s), whereas the space-fixed coordinate set is

a function of possibly all n 1 2 variables.

The expressions for the body-fixed coordinates contain several

fixed parameters, the dihedral angle c, the carbon–hydrogen

distance rH, the carbon–carbon distance rC, and a Lie group

parameter space of all permissible values of the dihedral

angles /k . Equation (7) is used to transform a 3 3n12ð Þ33

3n12ð Þ diagonal matrix with the proper masses along the

diagonal that correspond to the element associated with the



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particular body-fixed Cartesian coordinate. If we arrange the

variables so that x15/1; x25/2; . . ., up to n 2 1 and the last

three variables for /; h, and w, then the lowest 333 block of

the metric tensor has a standard general form[44] that depend

on trigonometric functions of the Euler angles and elements

of the inertia tensor,

glm5 Clm� �

j0j00

Xn

i51

mix3 i21ð Þ1j0

BFð Þ x3 i21ð Þ1j00

BFð Þ : (44)

n � l; m � n12. The symbol Clm represents a set of nine dis-

tinct two-forms that operate in R3 associated with the body-

fixed Cartesian coordinates for atom i,

Clm� �

j0j005djj000 @lR� �j

j0 @mR½ �j000

j00 : (45)

Expressions for these 333 tensorial elements can be

obtained readily.[44]

DMC simulations of ethane and propane

For ethane, it is possible to obtain relatively simple analytical

expressions for the body-fixed frame Cartesian coordinates as

the center of mass obtained by the procedure in the previous

section is simply rC=2. From these, it is straightforward to

obtain analytical expressions for the inertia and the metric ten-

sor. Ethane is a prolate top, and its inertia tensor is independ-

ent of /.

Ixx5Iyy51

2mcr2

c 1mH 6 rHcos c21

2rC

2

13r2Hsin 2c

( ); (46)

Izz52lR2e : (47)

where

lR2e53mHr2

Hsin 2c: (48)

The nonzero matrix elements of the metric are shown later.

Only the elements on the main diagonal and above are dis-

played as the metric tensor is symmetric.

g115lR2e (49)

g125lR2e cos h (50)

g145lR2e (51)

g225Ixx sin 2h12lR2e cos 2h; (52)

g2452lR2e cos h; (53)

g335Ixx ; (54)

g4452lR2e : (55)

The symmetry Ixx5Iyy has been used. We verify these expres-

sions with mathematica,[51] and we obtain the same result,

although the full simplification algorithm does not yield these

expressions directly, rather we obtain expressions containing

the masses of the two elements and the parameters rH and rC.

Nevertheless, it is straightforward to insert Eqs. (46) and (47)

into Eq. (49) through (55) and verify the equivalences.

For propane, the center of mass is independent of the two

torsion angles /1;/2, however, the expressions are rather com-

plex, and inspection of analytic expressions is substantially less

insightful. A similar observation is made for the inertia tensor.

Propane is an asymmetric rotor, and the elements of the iner-

tia tensor are constants, but in the body-fixed frame we have

defined earlier the inertia tensor is not diagonal, rather a rela-

tively small nonzero Iyz term in present. Therefore, the expres-

sions contained in Eq. (44) do not simplify significantly. It is

instructive to inspect the nonzero structure of the 535 matrix

representation of the metric tensor nonetheless.

gT2�I3

lm 5gT2

lm gClm

gClm gI3

lm

0@

1A (56)

The top 232 block related to the 2D Ramachandra space,

gT2

lm , is diagonal, and the two diagonal elements have the

same expression,

g115g225lR2e ; (57)

where the right-hand side is the same as in ethane, given in

Eq. (48). This fact implies that the two torsional degrees of

freedom are not coupled with one another geometrically. As

in ethane, the rotational degrees of freedom couple geometri-

cally with the torsional ones. In the coupling block gClm, the

expressions connecting /1 and the two Euler angles / and w,

are identical to those in Eqs. (50) and (51), and the element

connecting /1 with h vanishes. However, for the second-

torsional degree of freedom the coupling elements are more

complex and all three Euler angles contribute nonvanishing

coupling elements with /2.

g2351

3lR2

e (58)

g2452ffiffiffi2p

3lR2

e (59)

g2551

3lR2

e cos h22ffiffiffi2p

cos /sin hn o

: (60)

The lower 333 block gI3

lm

� �contains functions of the two

Euler angles / and h, as in the most general case, and is inde-

pendent of the two torsion angles. Therefore, from all these

expressions, we note that Christoffel connection coefficients

between the torsions and the rotations vanish. Nonvanishing

Christoffel connection coefficients are another potential source

of geometric couplings for more subtle but potentially signifi-

cant effects. The Christoffel connection coefficients are present

in the classical equations of motion derived from the general

Lagrangian in Eq. (9).

For the ensuing analysis, it is critical to realize that for pro-

pane, none of the degrees of freedom are dynamically

coupled, as the potential energy for propane is simply a

Fourier expansions[52] for the torsions and an external field,



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V5V3 111

2cos 3/11

1

2cos 3/21cos h

; (61)

with V354:78083 mhartree as for ethane. The last term on the

right-hand side is introduced to mimic the typical level of hin-

drance to rotation a propane molecule experiences on a sur-

face. As a result, it is possible to gauge directly the effects

that geometrical couplings have on the ground state energy

and other important physical properties by considering both

the sets of degrees of freedom separately, and then, compare

the results with a simulation that includes all five degrees of

freedom at once. We could have used ethane for this investi-

gation, but we selected propane because the analysis we pres-

ent here would be very cumbersome to carry out with the

method of extended Lagrangian,[53] or with vector spaces.[54]

For ethane, the simulation in the torsional space alone pro-

vides valuable insight and a convenient mean of comparing

our DMC simulations results with a vector space computation

of the ground state energy and wavefunction. For ethane, we

use the following potential energy model[52]

V51

2V3 11cos 3/ð Þ; (62)

with V354:78083 mhartree.

The results of four DMC simulations are graphed in Figure 2.

We use the basic DMC algorithm,[1] amended only by the pro-

cedure for the diffusion steps, as elaborated by the discussion

surrounding Eqs. (23) and (24). The population size is controlled

by the usual feedback mechanism,[1] and maintained around

105 replica. At each value of Ds, we run 23 independent

simulations allowing 105 steps to reach equilibrium, and col-

lecting averages for another 105 steps. Using the standard

deviation in the mean from the 23 independent simulations,

we construct the error bars shown in Figure 2. For ethane

[panel(d)], we compare the ground state energy obtained

from the DMC simulations over the potential surface in Eq.

(62) against the lowest eigenvalue obtained by a vector space

diagonalization. To carry out the latter, we construct a

representation of the Hamiltonian in the vector space

wm51ffiffiffiffiffiffi2pp exp im/ð Þ; m50;61;62; . . .

and use diagonalization to find the energy and eigenvectors

of the system. The ground state is triply degenerate and with

lR2e 5g11520; 787:24 a.u. we obtain 0.4947397(7) mhartree

with 601 basis sets as its energy. The error indicated in the

energy is from the basis set convergence, and we estimated it

from a separate computation using 801 basis sets. The ethane

molecule in T1 is isomorphic to a particle in a ring with a

mass of 3mH and a radius equal to rHsin cT . A step size of Ds� 60 a.u. is sufficient to converge to the correct answer. The

ground state energy for the ethane molecule in the space

fixed frame is 0.494(7) mhartrees in excellent agreement with

the vector space result. The statistical error is also relatively

small compared to typical unguided DMC simulations.

The graph in Figure 2a contains the outcome of the same

simulation for the propane molecule in the body-fixed frame,

namely in the 2D toroid space T2. The potential energy model

used is in Eq. (61) but for this simulation the h dependent

term is left out. A step size of 50 a.u. produces a ground state

Figure 2. Convergence of the ground state energy for propane and ethane is shown by graphing the ground state energy estimate versus the step size

used in the DMC simulation. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]



http://wileyonlinelibrary.com

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energy of ET2 50:989ð2Þ mhartree. The figure in parenthesis is

uncertain as the result of statistical fluctuations. The diagonal-

ization results is obtained simply by doubling the ground state

energy of ethane (0.989479402 mhartree), and the two esti-

mates agree quantitatively with one another. We then perform

two other sets of simulations for propane, using the same

parameters, and varying only the step size. One simulation is

performed by holding the two torsions rigidly at their mini-

mum energy values (staggered), and using only the V3cos hterm from Eq. (61). The other, is performed in the 5D space

T2 � I3. The convergence profiles with respect to Ds are in

Figure 2b and 2c, respectively.

From all the data in Figure 2, we make several observations.

First, we note a convergence behavior for ethane in T1 and

propane in T2 to an order in Ds grater than linear. The other

two simulations of propane converge linearly instead. The rea-

sons for the observed enhanced convergence behavior are as

follows. For all our simulations, we make use of a second-

order branching expression,[4]

wi5exp 21

2V x0i� �

1V xið Þ22Vref

� �Ds

� �: (63)

The metric tensor in the torsional subspaces for ethane and

propane is diagonal and independent of the values of the

dihedral angles, therefore, a number of higher order terms

that enter generally in the expression of the Green function

propagator are absent. The homotopy of the space (the ring is

a multiply connected set of spaces) and the boundary condi-

tions that the propagator must satisfy, may contribute addi-

tional linear terms in general, but these are clearly too small

to be observed in ethane and propane. All these conditions

create the observed nonlinear behavior of the ground state

energy with respect to Ds. Furthermore, systems with larger

number of torsions create more complicated expressions for

the metric tensor in Tn, and this object need not be constant.

Therefore, in general the procedure outlined in this article is

expected to converge linearly in a Ramachandra space of arbi-

trary dimension.

Second, the solution of the Schr€odinger equation by vector

space and diagonalization is considerably more involved for

the propane molecule in both the I3 and the T2 � I3 spaces.

For the 3D case, the asymmetry of the inertia tensor alone

requires a perturbation approach to handle the free rotation

part, together with the Lanczos algorithm, and other sparse

matrix strategies to handle angular momentum states greater

than l > 14. The coupling block gClm has to be included in the

pentadimensional treatment, and its contribution can only be

handled by perturbation methods. Unfortunately, the relatively

large size of the perturbation created by the gClm block is such

that the pentadimensional propane problem is particularly

challenging from the vector space prospective, whereas its

DMC and MCPI simulations are not any more involved than

the 3D and 2D counterparts.

Last, we are able to quantify the excess energy caused by

the coupling block gClm from the three ground state energies

of propane,

Eexcess 5ET2�I3 2ET2 2EI3 (64)

Using ET2�I3 51:269ð9Þ, EI3 50:118ð6Þ, and ET2 50:989ð2Þmhartree, we obtain an excess energy equal to Eexcess 50:162ð1Þ mhartree. This value is clearly statistically significant and

quite substantial (12.8% of ET2�I3 ). Remarkably, the geometric

coupling affect the estimate of the ground state energy with-

out contributing any explicit term in the estimator of the

property as this is obtained from the population average

potential energy, and the degrees of freedom in question are

not dynamically coupled. Rather, the geometric coupling terms

in this example affect the equilibrium population distribution

via the correlation terms included in the sampling of the diffu-

sion steps according to the procedure in Eqs. (22–24).

MCPI simulations of propane

For the stochastic evaluation of the Feynman path integral, there

are difficulties in using angular variables. For instance, the ran-

dom series expansion of the Brownian bridge becomes much

more complicated to implement into an algorithm when peri-

odic boundary conditions are used on the values of the angles.

In trying to use the Euler angle h in conjunction with Eq. (32),

the random variables (the coefficients of the path) must be con-

strained themselves to values that produce the correct range for

the angle h. Fortunately, we have found a set of coordinates

that can map spaces like the ellipsoid of inertia I3, or the space

T2 � I3 for the example problem. These are the stereographic

projections. The toroid spaces created by the dihedral angles

can be easily remapped by defining nl, as follows,

nl52 cos /l

12sin /l(65)

for all the n 2 1 dihedral angles. The transformation of varia-

bles from a dihedral angle /l to the projection nl is identical

to that used for the particle in a ring of unit radius,[44] and for

propane, with a mass equal to 3mHrHsin 2c. The expressions in

Eq. (65) follow from straightforward trigonometric identities.

The transformation of variables from Euler angles to projec-

tions is slightly more involved.[44] One begins by defining a 4D

space of quaternions, constrained to the surface of a three-

sphere with unit radius, that is, q1ð Þ21 q2ð Þ21 q3ð Þ21 q4ð Þ251.

The conversion map from Euler angles to quaternion coordi-

nates that satisfies this equation of constraint is,

q15cosh2

cosw1/

2

; (66)

q25sinh2

cosw2/

2

; (67)

q35sinh2

sinw2/

2

; (68)

q45cosh2

sinw1/

2

: (69)

Congruently with this result, we have shown that the one-

sphere result in Eq. (65) can be generalized to a n-sphere



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space or unit radius, defined as dlmqlqm51, where dlm is the

usual Kronecker delta and the sum is from 1 to n 1 1. The n

stereographic projections are computed directly from,

nl52ql

12qn11l51; 2; . . . ; n (70)

If we define hll0 as the partial derivatives involved in the

map of I3,

dh5h11dn11h1

2dn21h13dn3; (71)

d/5h21dn11h2

2dn21h23dn3; (72)

dw5h31dn11h3

2dn21h33dn3: (73)

and we compute the equivalent quantities from Eq. (65), we

have all the elements for the transformation of the metric ten-

sor in Eq. (7). Explicit expressions for the nine hll0 , we define

here can be found,[44]

h1152

16d2n1

d1 r14ð Þ ; h125

2d5n2

d1d2 r14ð Þ ; h135

2d5n3

d1d2 r14ð Þ ;

(74)

h215h3

15d5

d4; h2

258n1n2

d41

n3

d3; h2

358n1n3

d42

n2

d3; (75)

h325

8n1n2

d42

n3

d3; h3

358n1n3

d41

n2

d3: (76)

where the auxiliary expressions d1 through d5 are defined as

d15

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16 n1� �2

1 r24ð Þ2q

; (77)

d25

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 n1� �2

q; (78)

d35 d2� �2

; (79)

d4516 n1� �2

1 r24ð Þ25 d1� �2

: (80)

d554 2 n1� �2

2 r24ð Þh i

; (81)

All these equations can be coded readily, and the computa-

tion of the metric tensor of propane in T2 � I3 is fast. There-

fore, the technical problems surrounding Eq. (32) for the MCPI

simulations of propane in T2 � I3 are eliminated, if we per-

form our random walk with the stereographic projections and

the auxiliary random path coefficients. The results of several

MCPI simulations on the pentadimensional propane molecule

in an external field are presented in Figure 3.

The quantum NVT ensemble average of the total energy in

hartree [cf. Eq. (41)] is graphed as a function of b51=kBT for

several values of the core coefficients km defined in Eq. (32).

The parameter k0m regulates the length of the tail used to

accelerate the convergence of the energy and heat capacity

estimators to third-order. The nonlinear convergence pattern

for hEib is evidenced by the data graphed in Figure 3, when

comparing the relative distance between the tree sets of simu-

lations. More importantly, we find that the quantum effects on

the equilibrium thermodynamic properties of propane are

quite large over a vast range of temperatures. At room tem-

perature, for example, b � 1050 hartree21, the km 5 10 simula-

tion estimates an energy (17.39 mhartree) three times larger

than the classical one (5.48 mhartree). The data in Figure 3

demonstrates that the classical limit is being reached asymp-

totically as b! 0, and that the km 5 4 simulation is essentially

converged for b < 250 hartree21, while the km 5 10 simulation

Figure 3. Reweighted random series simulation of propane in T2 � I3 mapped with stereographic projection coordinates. The average energy as a function

of b51=kBT is graphed for several values of km [cf. Eqs. (32–42)].



http://q-chem.org/


(white squares) is converged for b > 2000 a.u. In fact, a

km 5 14 simulation (black triangles) is statistically indistinguish-

able from the km 5 10 simulation (white squares) in the entire

range of b in Figure 3. However, the quantum effects at b550

hartree21 (� 6300 K) are still visible on the scale of the graph.

This observation alone demonstrates the importance of includ-

ing quantum effects even when studying the gaseous state of

propane.

Conclusions

In this article, we review the details of two powerful quantum

Monte Carlo techniques (DMC and MCPI) that our groups have

adapted to generic manifolds. These adapted techniques find

important applications in molecular physics, whenever it is sen-

sible to separate time scales. The advantage of constraining

high frequency degrees of freedom manifests itself in the con-

vergence properties with respect to the imaginary time step of

the methods, and the gains in efficiency can be massive. We

have chosen two examples from molecular physics where the

need for a curved space is evident. Both ethane and propane

have stretching and bending modes with frequencies that are

4–5 times larger than those associated with the torsional

degree of freedom, therefore, isolating the internal and external

rotations from the rest of the normal modes is a natural choice.

Assuming the potential energy of a system under investigation

is known accurately, DMC and MCPI permit to introduce con-

trollable approximations. For example, the finiteness in the time

step in DMC, related to the Trotter number for MCPI, can be

systematically varied until convergence is confirmed. A similar

statement applies for DMC and MCPI in manifolds, whereby

one essentially introduces one additional approximation that

can be carefully controlled, namely the adiabatic approximation.

For instance, in the range of temperatures considered in Figure

3, there are intervals where the bending and the stretching

modes are no longer predominantly in the ground state[26] and

these will contribute to the energy profile in Figure 3 on the

left side. However, for these higher frequency degrees of free-

dom, a harmonic approximation[55,56] is valid for a good range

of values of b, and this approximation too can be systematically

corrected. Alternatively, and for even higher temperatures, path

integrals treated using molecular dynamics and multiple time

stepping techniques[57] can be used to simulate with greater

efficiency. Notice that the range of temperatures where the

approaches proposed in this work and in Ref. 57] may be both

efficiently applied is expected to be rather narrow, as the adia-

batic approximation is valid only when temperatures are low

with respect to bending and stretching frequencies. The latter

situation requires many “slices” in the ring polymers to correctly

describe the associated degrees of freedom with the approach

in Ref. 57], possibly making any comparison based on relative

efficiency unfair. Also, the range of temperatures where the adi-

abatic approximation is valid, if at all, depends on the system,

however, the number of problems in molecular physics that can

be addressed with the two methods we have developed is truly

vast given the large number of small molecules like ethane and

propane, and the plethora of possible applications in academic

and industrial endeavors. For instance, let us mention the simu-

lation of condensation/evaporation for gases used in the refrig-

eration technology, as well as in the study of stability and

structure of aerosols. Although we have not “canned” our

approaches into black box methods, it seems feasible at the

moment. Besides, our experience has contributed to develop

the idea that facing the grand challenges of introducing quan-

tum effects in material science simulations will require juxtapos-

ing the use of holonomic constraints and multiple time

stepping techniques. This is the task, we are currently facing

while working on simulating ammonia clusters[10] and droplets.

The molecule of propane is simulated using a combination of

curved spaces, the ellipsoid of inertia to treat its rotations rela-

tive to the laboratory frame, and the 2D toroid to handle the

two torsions. The internal coordinate map is best generated

from the parameter set of the continuous Lie groups of rota-

tions and translations that are used to generate the configura-

tion in Cartesian coordinates as elaborated in the results

section. Using holonomic constraints is significantly more chal-

lenging for this particular problem given the number of con-

straining conditions. For propane, for example, one would have

to write a total of 28 equations, and many of these are not triv-

ial to express, especially those involving the angles. Therefore,

the method of extended Lagrangian[53] is more challenging to

implement. Also challenging for propane is the treatment of

the pentadimensional space by vector space methods.[54] The

impact that the geometric couplings between the torsions and

the rotations has on the ground state energy is substantial for

propane, and these terms cannot be ignored when constructing

a vector space representation of the Hamiltonian operator.

However, projecting the Laplace–Beltrami operator in a suitably

chosen vector space is a nontrivial task for the asymmetric ellip-

soid of inertia space alone, and the aforementioned coupling

terms complicate the treatment substantially.

We have demonstrated that DMC and MCPI can be carried

out in Riemann spaces with positive signature, by simply intro-

ducing the proper expressions of the metric tensor, which is a

unit matrix only in flat spaces mapped by Cartesian coordinates.

However, simulations in manifolds force one to introduce the

Jacobian factors in Eq. (11) as well. In MCPI, the Jacobian has to

be included in the move strategy to maintain detailed balance.

Additionally, the transformation of the Wiener measure[49] intro-

duces a quantum correction to the potential energy in the form

of the curvature scalar [cf. Eqs. (27) and (28)]. In DMC, the diffu-

sion and branching process do not require explicitly Jacobian

factors. Nevertheless, the interpretation of the population distri-

bution for the unguided algorithm, for example, is that it

approaches asymptotically the ground state wavefunction w0,

and in turn this object is related to a probability distribution Pxð Þ to locate a particle between x and x1dV, that is,

P xð Þ5 g1=2jw0j2Ð

Mg1=2jw0j

2dV: (82)

The Jacobian factor in Ramachandra’s spaces is still a matter

of controversy, and it is this topic that motivated the work of

Patriciu et al.[50] It is worth discussing the issue at this point.



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In their book,[43] Frenkel and Smit argue that the outcome of a

simulation carried out with rigid modes should be the same as

the result of a simulations where the high frequency degrees

of freedom are treated with stiff springs. There is a fundamen-

tal difference between the spaces these two systems require.

The former is typically a curved Riemann manifold M with all

the geometric intricacies that have been discussed in this arti-

cle. The latter is a curvilinear remapping of R3n !M�N for

n atoms with 3n-Cartesian coordinates. The metric tensors and

the Jacobi factors gM� �

, and gM�N� �

associated with the two

maps are inherently different. For DMC simulations, the differ-

ence between the two types of spaces only affects properties

derived from the integral of the wavefunction, it does not

impact, for example, the ground state energy. In MCPI simula-

tions, however, the resolution of the identity is needed to

derive the density matrix expression and this gives rise to the

Jacobian factors in Eq. (11) and the Riemann curvature sca-

lar[44] in Eq. (28). We have recently shown[44] that the ratio

det gM�N� �1=2

det gMð Þ1=2(83)

and the curvature is generally a constant for all the types of

ellipsoids of inertia, for the n2 sphere and for the monodi-

mensional toroid. Therefore, the simulations in these types of

spaces are in fact equivalent to those predicted by simulations

with stiff modes in place of rigid constraints.[43] For the Rama-

chandra spaces the issue remains, even though we do calcu-

late the Riemann curvature scalar for both ethane and

propane and we find that it is constant. Patriciu et al. have

argued in their work that simulations in the Ramachandra

space should have a unit Jacobian factor.[50] They arrive at

such conclusion by comparing the statistics of classical tor-

sional Monte Carlo simulations with those obtained by

dynamic simulations where the constraints are replaced by stiff

bond lengths and bond angles. However, this argument does

not take into consideration how quantum simulations differ in

the two types of spaces, and the controversy is far from being

put to rest. It is also clear that it would be difficult to resolve

the matter experimentally as the zero-point energy contrib-

uted by the five degrees of freedom in propane is the same

regardless of the space chosen M or M�N. Furthermore, we

have made the general assertion that geometric and dynamic

couplings can be neglected, whenever the adiabatic approxi-

mation is accurate between the stiff and the soft modes. With

the tools discussed in this article one can, at least in principle,

carefully verify such statements. In fact, a systematic study of

this nature to provide general guidelines useful for the com-

munity is a goal for our group in the near future.

The tools we have been adapting for applications in con-

densed molecular matter with the adiabatic approximations

will continue to need refinement, and we take this opportu-

nity to describe some of the future directions for further

developments in this area. These endeavors are occupying our

groups at the time of this writing. Higher order convergence

for MCPI in manifolds have been developed, and we have

implemented them for the propane problem in this article.

However, the second-order methods for DMC with respect to

the time interval Ds are only available for Rn and for the two-

sphere S2. By simply expanding the map to second-order one

generates all the terms in the Laplace–Beltrami operator that

are missing in Eq. (20), and these can be incorporated into a

second-order diffusion branching scheme in two ways: (a) by

treating the terms neglected in the first-order scheme as drift-

ing terms and (b) by making use of Ito–Taylor expansions for

the metric tensor and its Cholesky decomposition.

Another area that is under development currently in our

laboratories is the adaptation of the ring polymer dynamics

approach[58–66] to curved spaces. Briefly, in curved spaces one

can write an expression for the Ring-polymer Hamiltonian for

a n-bead system, where n is now the Trotter number,

Hn5Xn

j51

1

2glmpljpmj1

1

2b2n�h2

Xn

j51

glm xlj 2xl

j21

� �xm

j 2xmj21

� �

1Xn

j51

V xj

� �;

(84)

and where the second term on the right is the Matzubara har-

monic term. The subscript j on each coordinate is used to

identify each “bead.” We have recently developed an integra-

tor in M using the variational principle but again avoiding the

explicit use of holonomic constraints expressions, eliminating

the need to use extended Lagrangian.[67,68] Rather one begins

by setting to zero the variation of the action,

dðLdt (85)

where L is as in Eq. (9). When the integral is replaced by its

discretized numerical equivalent, we obtain the discretized

form of the equation of motion,

plk51

2glmk

xmk112xm

k

Dt

1

1

2glmk11

xmk112xm

k

Dt

2Dt

4@lgrmk

xrk112xr

k

Dt

xm

k112xmk

Dt

1

Dt

2@lVk

(86)

and

plk1151

2glmk

xmk112xm

k

Dt

1

1

2glmk11

xmk112xm

k

Dt

1Dt

4@lgrmk11

xrk112xr

k

Dt

xm

k112xmk

Dt

2

Dt

2@lVk11;

(87)

where in Eqs. (86) and (87) the notation xlk and glmk is an

abbreviation of the respective quantities at time t5tk . Equa-

tion (86) represents a set of n independent coupled nonlinear

algebraic equations. The root xlk11 depends, in general, on the

values of all the other roots, and the solution for such nonlin-

ear coupled system can only be found iteratively by evaluating

the metric tensor at an initial guess for xlk11 and then solving

for the latter from Eq. (86). Once the procedure is iterated to

self consistency, the solution is used to compute the updates

in momenta in Eq. (87). We are currently testing this numerical



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integrator in ellipsoids of inertia and we measure second-order

convergence in the position as well as the energy. More

importantly, we observe no significant drift in the energy over

very long time scales. The details of this work will be pub-

lished elsewhere.

Last, this work on the smallest examples of Ramachandra’s

spaces leaves us with a multitude of interesting questions and

the answers to which can be potentially transforming for the

community interested in protein modeling. Specifically, how

do the quantum effects and the coupling between hindered

rotation and torsion depend on the size of the molecule? One

would expect the effects to get smaller as the mass associated

with individual torsions grows. However, in the Ramachandra

spaces, concerted motion of sets of torsional degrees of free-

dom take the system from one set of minima to another, and

the effective mass involved does not necessarily need to be

large. Quantum methods like DMC adapted for torsional

spaces can yield not only the correct ground state but can

also alleviate some of the difficulties regarding the lack of con-

vergence to the ergodic limit for the state of the art stochastic

methods available today.[69–89] This problem has plagued the

work in the Ramachandra spaces from the outset. We have

found that DMC alleviates quasiergodicity, and that for suffi-

ciently large systems DMC alone can become trapped in local

minima and become nonergodic. However, our groups have

already found ways to improve the convergence of DMC

toward the ergodic sampling limit.[12]

Keywords: quantum Monte Carlo � Holonomic con-

straints � Ramachandra space � diffusion Monte Carlo � path

integral Monte Carlo

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Received: 19 November 2013Revised: 31 January 2014Accepted: 4 February 2014Published online 28 February 2014



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Quantum monte carlo methods for constrained systems

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