the journal of chemical thermodynamics volume 73 issue 2014 [doi 10.1016_j.jct.2013.12.018] irikura,...

7/26/2019 The Journal of Chemical Thermodynamics Volume 73 Issue 2014 [Doi 10.1016_j.jct.2013.12.018] Irikura, Karl K. --

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Anharmonic partition functions for polyatomic thermochemistry

Karl K. Irikura

Chemical Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8320, USA

a r t i c l e i n f o

Article history:

Available online 4 January 2014

Dedicated to the memory of the lateProfessor Manuel Ribeiro da Silva

Keywords:

Anharmonicity

Density of states

Partition function

Quantum chemistry

Thermochemistry

a b s t r a c t

In quantum chemistry, the computation of anharmonic vibrational frequencies, instead of only harmonic

frequencies, is now convenient and increasingly common. For thermochemistry, one ramification is that

vibrational partition functions may be constructed using anharmonic manifolds. Practical issues are dis-cussed here, and a few examples are provided. Evaluating the accuracy of the results is hindered by the

lack of experimental data for ideal-gas heat capacities at high temperature.

Published by Elsevier Ltd.

1. Introduction

Quantum chemistry is increasingly important in thermochem-

istry. Advances in electronic structure theory now allow precise

predictions (that is, uncertainties of only a few kJ mol

1

) to bemade on molecules as large as guanine and tetracene [1], in the

ideal-gas limit. At the small-molecule extreme, precision for small

clusters of helium is high enough to serve as a physical measure-

ment standard [2]. Although special problems are encountered

for molecules with unusual electronic structure, such as

metal-centered radicals, for typical organic compounds the

accuracy bottleneck has become the contributions of molecular

vibrations.

The vibrational zero-point energy (ZPE) contributes already at

zero temperature. ZPE is a large quantity, on the order of 20 n

kJ mol1 for typical organic compounds, where n is the number

of atoms in the molecule. It is manifested in the atomization en-

thalpy of a molecule, because atoms contain no ZPE. Since atoms

are a common, if not optimal, choice of thermochemical reference

state in quantum calculations, errors in the ZPE can be important.

However, because ZPE cannot be measured (or even defined)

experimentally, its accuracy is problematic to assess[3].In conven-

tional quantum thermochemistry, vibrations are computed in the

harmonic approximation. Thus, ZPE is taken as one-half the sum

of the frequencies. It is often multiplied by an empirical scaling

factor to compensate for bias [4].

Besides ZPE, the other major contribution of vibrations is to

temperature-dependent thermodynamic functions, by way of the

molecular partition function. Standard practice is to make the sim-

plifying approximations that electronic, vibrational, rotational, and

translational degrees of freedom are separable and non-interact-

ing, that the molecular geometry is independent of rotational exci-

tation, and that vibrations are harmonic. This is well-known as therigid-rotor/harmonic-oscillator (RRHO) approximation. Sometimes

the vibrational frequencies are scaled by empirical factors before

computing the thermodynamic functions[5].Unlike the ZPE, ther-

modynamic functions can be measured experimentally. Thus, it is

possible to assess the accuracy of theoretical predictions, at least in

principle.

Deficiencies of the harmonic-oscillator model are well-known, if

seldom addressed in practice. Torsional motion, such as exhibited

by methyl groups, is not harmonic. The classic work by Pitzer[6]

continues to guide development in this area [7]. Other large-

amplitude vibrations, such as inversions and ring-puckering, are

equally problematic [8]. Even for relatively rigid molecules,

vibrations are anharmonic. In the simplest anharmonic model,

vibrational energy levels are conventionally described by two-

mode anharmonicity constants (xij) according to the approximate

equation(1). The xiare harmonic constants, thexijare anharmonicconstants, vi is the quantum number for the ith vibrational mode,

and the sums are over all the vibrational modes in the molecule

Ev1;v2;. . . ;vN Xi

xi vi1

2

Xi

Xj6i

xij vi1

2

vj

1

2

: 1

Simple anharmonic calculations are increasingly available as

keyword options in popular quantum chemistry software.

Although such calculations require far more computer time than

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J. Chem. Thermodynamics 73 (2014) 183189

Contents lists available at ScienceDirect

J. Chem. Thermodynamics

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harmonic calculations, they require little or no additional effort by

the user. Thus, one may anticipate that their popularity will in-

crease. The output of such quantum chemistry programs typically

includes a table of the spectroscopic constants for use in equation

(1). The purpose of the present work is to examine the effect upon

thermodynamic functions of including simple vibrational anhar-

monicity as obtained from such programs. The resulting model

may be called the rigid-rotor/anharmonic-oscillator (RRAO)approximation.

Only small molecules were considered for investigation here.

Larger molecules [9] were avoided for three reasons. First, the

anharmonic theory included in the quantum software is second-

order vibrational perturbation theory (VPT2), which presumes that

there are only minor deviations from a harmonic potential energy

function. Internal rotation and ring puckering, which are common

in large molecules, have multiple minima that cannot be described

by small adjustments to a harmonic potential. The VPT2 model is

expected to give poor results in such situations. Furthermore, it

is doubtful that such a spectrum can be fitted by equation (1).

Second, the equations of VPT2 feature many denominators that

are energy differences between vibrational levels. Pathological

results may occur when these energy differences approach zero,

which is increasingly likely in larger molecules. Third, low-quality

electronic structure theory produces anharmonic spectra of

dubious quality[10]. Consequently, high-quality calculations are

desired here, which are too expensive for large molecules.

2. Computational methods[11]

2.1. Difficulties with standard formalism

Unfortunately, it is not straightforward to compute anharmonic

partition functions. Harmonic partition functions can be written in

simple closed form, as shown in textbooks on statistical thermody-

namics. Uncoupled anharmonic oscillators can be handled by ana-

lytical approximations[12]or by efficient methods of enumeration

[13], but the levels described by equation (1)must be enumerated

individually. This presents a practical problem because there are

billions of levels even for small molecules. Another complication

is that the anharmonic constants are often negative-valued.

Eventually, the energy given by equation(1)decreases as the quan-

tum numbers increase, which is physically unreasonable (although

in some cases it can be interpreted as bond dissociation). Thus, the

vibrational manifold must be truncated wherever this occurs. This

truncation may lead to physically unreasonable behavior, as exem-

plified below. The molecules in the present study are small enough

to be tractable using a simple Monte Carlo method to estimate the

anharmonic density of states (DOS), starting from a set of harmonic

and anharmonic constants. A more efficient method, based

upon the Wang-Landau classical algorithm [14], has been

described and developed by Basire et al. [15]and by Nguyen andBarker[16].

2.2. Density of states (DOS)

For computational efficiency, energy levels are grouped in en-

ergy bins (width =DE= 2 cm1) instead of listing them individu-

ally. The number of states in the ith bin is the density of states

times the width of the bin, qEiDE. Thus, the vibrational partitionfunction is expressed by equation(2). In practice, the sum cannot

run to infinite energy and is truncated at some cutoff energy, Emax,

so that the sum stops atimax Emax=DE. In other words, the DOS is

taken to be zero at energies above the cutoff. This is in contrast to

the RRHO, in which the DOS increases without bound and the

vibrational partition function is summed analytically to E 1

QvibT Xi

qEiDE expEi=kT: 2

For small problems the density of states, qE, may be evaluatedexactly within the model by exhaustive enumeration of vibrational

levels (often called direct counting). However, thermodynamic

functions converge slowly with the cutoff energy. This requires a

large value ofEmax, which makes enumeration impractically slow.

An approximate Monte Carlo method is described below.Low-lying states contribute most to the partition function. Thus,

the most reliable DOS is obtained by direct enumeration up to

some practical energy, followed by an estimate up to the cutoff en-

ergy,Emax, at which the DOS is truncated. Emaxmay be increased to

verify that it is high enough for thermodynamic functions to be

converged at the temperatures of interest. The Monte Carlo esti-

mate may be improved either by increasing the number of samples

or, equivalently, by constructing it as a composite of estimations

with increasing values ofEmax. This is advantageous because the

computational efficiency decreases as Emax increases.

For ThF4, enumeration was used for energies up to 7000 cm1

followed by Monte Carlo estimation to 15,000 cm1 and then again

to 50,000 cm1. For CF3Br, enumeration was used up to

20,000 cm1

followed by Monte Carlo estimation to 30,000 cm1

and then again to 50,000 cm1. For C2H4 (ethylene), enumeration

was used up to 30,000 cm1 followed by Monte Carlo estimation

to 50,000 cm1. For C2H4O (ethylene oxide), enumeration was used

up to 25,000 cm1 followed by Monte Carlo estimation up to

50,000 cm1.

For reference, harmonic (all x ij= 0) and uncoupled anharmonic

(xij= 0 fori j) DOSs were computed using the SteinRabinovitch

algorithm [13] with a bin size of 2 cm1. The accuracy of this

algorithm is limited only by the accumulation of rounding errors

associated with the energy binning.

2.3. Monte Carlo estimation

Each vibrational energy level is specified uniquely by a set ofvibrational quantum numbers, which may be represented as a

point in quantum-number space. We estimate the DOS by Monte

Carlo sampling within an appropriate, fixed volume in quantum-

number space. If a large number of points is sampled (Ntot), and

Ni of those points have energies within the bin Ei, then the DOS

is approximated by equation(3), where Vtot is the volume of the

sampling region. If the volume or the dimensionality is too large,

this strategy is uselessly inefficient, but it is adequate for the

molecules under consideration here. The present procedure has

similarities with that by Barker [17] but was developed

independently.

qEiDE Ni=NtotVtot: 3

The sampling volume is defined by a set of maximum values ofthe quantum numbers. It is represented by inequality(4), wherebiis the boundary (maximum) value for the (non-negative) quantum

number vi. Geometrically, the sampling volume is a right-hypertri-

angle in quantum-number space with a vertex at the origin and a

vertex on the ith axis at bi. Its volume is given by equation (5),

where D is the dimensionality, that is, the number of vibrational

modes. Note that using a rectangular sampling volume would de-

crease computational efficiency by a large factor of (D!). Given a

cutoff energy (Emax), each boundary valuebiis initially determined

from equation(1)while holding vj= 0 (j i). The initial value ofbiis the smaller of: (1) the value ofvifor whichE=Emax; (2) the value

ofvifor which @E=@vi 0. Because many anharmonic constants are

negative-valued, regions of the upper surface of the volume may be

convex instead of flat. To compensate, an inflation factor (/),

184 K.K. Irikura / J. Chem. Thermodynamics 73 (2014) 183189


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greater than 1 but typically less than 2, is applied to all boundary

values to ensure that the sampling volume includes all points of

interest. The value of/ may be increased to verify that the sam-

pling volume is large enough, but if/ is too large the acceptance

ratio (that is, the computational efficiency) will be impractically

small. Finally, each bi is increased by 1 so that the desired range

of values will be covered after truncating to integers.

Xi

vi=bi 6 1; 4

Vtot D!1Yi

bi: 5

Uniform sampling over the hyper-triangular volume starts with

a set of D random floating-point variables, ui, each taken from a

rectangular distribution over the conventional interval [0, 1). They

are mapped onto floating-point values,yi, using the transformation

in eq. (6) (see Appendix for derivation), starting with i= 1 and

working upward. Truncating gives integer values for the quantum

numbers, vi= [yi] 6 bi. A point is rejected if its energy exceeds Emaxor if@E=@vj 6 0 for any j.

yi bi1 1 ui1=D1i

Yj


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oxide. The VPT2//CCSD(T)/cc-pVTZ result is higher than the exper-

imental value by 280 cm1. This appears to be a failure of VPT2,

since the harmonic calculation (x14) is only 28 cm1 higher that

the experimental fundamental. Such failures are usually attribut-

able to near-resonances and are sensitive to the rather arbitrary

choice of associated thresholds[22]. In this case, most of the x i,14are positive-valued and several are implausibly large (seeSupple-

mentary material). Note that eq.(1)implies that the fundamental

frequency, mj, for mode j has the value given below.

mj xj1

2Xij

xij 2xjj: 10

As pointed out by a reviewer, it is standard practice among

spectroscopists to perform a de-perturbation analysis to address

pathologies such as that observed for m14for ethylene oxide. How-

ever, no such analysis was done here, for two reasons. First, such

analysis is beyond the skills of non-specialists (including the

present author), who rely upon the quantum chemistry software.

Second, this failure for such a simple molecule is a warning that

VPT2 is questionable, or at least premature, for general application

in thermochemistry.

3.3. ZPE

Anharmonic, uncoupled-anharmonic (also called diagonal-anharmonic), and harmonic values are presented in table 3. The

anharmonic contribution to ZPE is fairly small for these four mol-

ecules, and the off-diagonal contribution is nearly negligible.

3.4. Density of states (DOS)

Figure 2compares DOSs from harmonic (xij= 0), diagonal (xij= 0

fori j), and anharmonic models for ThF4. At low energy, the re-

sults conform to expectations; the diagonal-anharmonic DOS is

slightly larger than the harmonic DOS and the fully anharmonic

DOS is much larger. Above 17,000 cm1, however, the anhar-

monic DOS is the smallest of the three models, even declining in

magnitude above 35,000 cm1. The off-diagonal anharmonicities

have a major effect. For CF3Br the situation is the same qualita-tively, but the crossover occurs near 26,000 cm1 and the decline

begins near 90,000 cm1. For C2H4 the harmonic and uncoupled-

anharmonic models give nearly superimposable DOSs at high en-

ergy. The anharmonic DOS is larger by an increasing factor until

200,000 cm1, where it begins a decline. It crosses the harmonic

and uncoupled DOSs around 240,000 cm1. For C2H4O, the three

models give very similar results (figure 3). The anharmonic DOS

is larger until it crosses the other DOSs near 230,000 cm1. Beyond

that, it is nearly constant until declining gently from around

350,000 cm1. For a thermochemical context, note that

100,000 cm1 1200 kJ mol1 12.4 eV, larger than nearly any

bond dissociation energy and most ionization energies.

There are few measurements of DOS in gas-phase molecules. No

data are available for comparison with the present calculations.However, for formaldehyde (CH2O), is has been reported that the

RRHO and RRAO models predict DOSs that are too small by factors

of 11 and 6, respectively[23].

3.5. Thermodynamic functions

The values of the thermodynamic functions depend upon the

cutoff energy (Emax) used in the vibrational partition function. A

representative example is shown in figure 4. To obtain converged

results, the required cutoff energy is much larger than kT(in wave

number units k 0.695 cm-1 K1). A higher cutoff energy is

needed for functions at higher temperature. Moreover, the re-

quired cutoff is roughly proportional to the number of atoms in

the molecule. All thermodynamic functions in this study were con-verged by choosing a high cutoff energy ofEmax= 50,000 cm1. The

heat capacity (Cp) is more sensitive toEmaxthan the entropy (S) and

the enthalpy content (DT0Hvib), presumably because it depends

upon the second derivative of the partition function instead of only

the first derivative. The same convergence study for the harmonic

partition functions shows the same qualitative behavior, and con-

verges properly to the analytical harmonic results.

TABLE 1

Lowest and highest harmonic frequencies and anharmonicity constant of largest

magnitude (constants not scaled).

xlow/cm1 xhigh/cm

1 Largest xij/cm1

ThF4 110.3 cm1 594.6 2.26

CF3Br 308.2 1218.6 88.01

C2H4 823.4 3248.7 68.03

C2H4O 816.3 3210.8 106.40

TABLE 2

Vibrational fundamentals from anharmonic theory and from experiment (Refs. [35],

[36],[37], and[38]for ThF4, CF3Br, C2H4, and C2H4O, respectively).

Theory/cm1 Expt/cm1

ThF4, m1 (a1) 589 618 (est.)

m2 (e) 112 121 (est.)

m3 (t2) 532 520m4 (t2) 107 116

CF3Br, m1(a1) 1079 1089

m2 (a1) 761 760

m3 (a1) 360 349

m4 (e) 1193 1210

m5 (e) 545 547

m6 (e) 306 306

C2H4, m1 (ag) 3010 3026

m2 (ag) 1635 1623

m3 (ag) 1342 1342

m4 (au) 1025 1023

m5 (b1g) 3077 3103

m6 (b1g) 1222 1236

m7 (b1u) 945 949

m8 (b2g) 943 943

m9 (b2u) 3096 3106

m10 (b2u) 822 826m11 (b3u) 2997 2989

m12 (b3u) 1435 1444

C2H4O, m1 (a1) 2976 3005

m2 (a1) 1493 1497

m3 (a1) 1261 1270

m4 (a1) 1129 1120

m5 (a1) 878 877

m6 (a2) 3065 3050 to 3073

m7 (a2) 1177 1043 to 1046

m8 (a2) 1090 807 to 851; 1020

m9 (b1) 2985 2978

m10 (b1) 1490 1470

m11 (b1) 1206 1151

m12 (b1) 894 822

m13 (b2) 3184 3065

m14 (b2) 1427 1147

m15 (b2) 853 808

TABLE 3

Vibrational zero-point energies (unscaled) under different approximations.

ThF4 CF3Br C2H4 C2H4O

Harmonic/cm1 1384.0 3191.4 11,157.1 12,640.6

Uncoupled/cm1 1382.9 3197.4 11,142.4 12,625.1

Anharmonic/cm1 1375.8 3177.3 11,007.5 12,667.6

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4. Discussion

Explicitly considering first-order anharmonicity, as done here,

has variable effects upon computed ideal-gas thermodynamic

functions. The RRAO and RRHO thermodynamic functions (S, Cp,

andDT0Hvib) are tabulated and summarized graphically in theSup-

plementary materialfor the four molecules considered here. Below

1000 K, the anharmonic values are larger than the harmonicvalues, except in the case of C2H4O. Among the four molecules,

the extreme examples are C2H4O (ethylene oxide) and ThF4. For

C2H4O, the anharmonic DOS is nearly identical with the harmonic

DOS (seefigure 3). Consequently, the thermodynamic functions are

nearly equal. The largest difference is for Cp, shown infigure 5. In

contrast, the anharmonic DOS for ThF4 (figure 2) displays trunca-

tion effects as discussed above. The largest difference is again for

Cp. As shown in figure 6, the anharmonic Cp(T) rises faster than

the harmonic function, reaches a maximum near T =700 K, and

then declines sharply. Like the decline in the DOS, this does not ap-

pear physically reasonable. However, an alternative, if unlikely,

interpretation is possible. If the DOS turnover corresponds to disso-

ciation of the molecule, rather than being an artifact of the simple

first-order anharmonic approximation (equation (1)), then the

decline in Cp for ThF4 molecules does not reflect what would be

measured experimentally for a sample of ThF4 together with its

equilibrated dissociation products.

Harmonic vibrational frequencies are often multiplied by

empirical scaling factors to correct for anharmonicity and for other

deficiencies. Scaling factors have been optimized for thermody-

namic functions[2427]. Unfortunately, because of the lack of data

[24],the experimental benchmarks in such exercises are usually

not from experimental measurements, but from RRHO calculations

based upon experimental vibrational frequencies. Below, we con-

sider whether the results of the present RRAO calculations can be

recovered by using scaled frequencies in the RRHO model.

For each combination of one molecule, one thermodynamic

function, and one temperature, we seek a frequency scaling factor

that causes the RRHO value of the thermodynamic function toequal the RRAO value. A scaling factor can always be found to make

the entropies match, but not necessarily forCpor for DT0H(see Sup-

plementary material for details). The resulting scaling factors are

listed in the Supplementary Material. No single value of the scaling

factor can make the RRHO results conform to the RRAO results,

either at all temperatures for a single thermodynamic function or

for all thermodynamic functions at a single temperature. This is

the expected behavior, but not often acknowledged.

FIGURE 2. Densities of states for ThF4 from harmonic, uncoupled anharmonic, and

coupled anharmonic models.

FIGURE 3. Densities of states for C2H4O (ethylene oxide) from harmonic, uncoupled

anharmonic, and coupled anharmonic models.

FIGURE 4. Ideal-gas heat capacity (Cp) of C2H4O (ethylene oxide) from the RRAO

model, as a function of cutoff (i.e., truncation) energy in the vibrational partition

function. From bottom to top: T/K = 298.15, 400, 500, 700, 900, 1200, 1500, and2000.

FIGURE 5. Ideal-gas heat capacity (Cp) of C2H4O (ethylene oxide) from the RRAOand RRHO models.

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An important question is whether the anharmonic constants,

gained at such great computational expense, yield thermodynamic

functions that are in better agreement with experiment. As shown

infigure 6, and in the Supplementary Material, the largest differ-ence from harmonic theory is for heat capacity (Cp) and at high

temperature. Unfortunately, experimental data are unavailable at

high temperatures. Comparisons with available data, as shown

below, are inconclusive.

For ThF4, a comparison in the reference[21]was made for the

entropy at T =1150 K. The experimental value [28] is

483 4 J K1 mol1. The anharmonic value was 514 J K1

mol1, in worse agreement than the harmonic value of 490 J K1

mol1. The present results are essentially the same, as expected

because the spectroscopic constants from the reference[21]were

used here.

For CF3Br, no experimental data appear to be available at any

temperature despite the technological importance of this

compound.For C2H4, the ideal-gas entropy at T = 169.4 K has been reported

as S = (198.2 0.4) J K-1 mol1 [29]. The present computations

yield anharmonic and harmonic values of (198.2 and 198.1)

J K1 mol1, respectively. These values agree equally well with

the experimental value. For Cp, several measurements have been

made at temperatures up to 473 K [30], but many of these

publications are difficult to obtain. Published values, covering the

temperature range from 178 K to 473 K, are compared with the

calculations infigure 7. The experimental data by Burcik et al., by

Eucken and Parts, and by Haas and Stegeman were taken from

the reference [31] (uncertainties not reported). The data by

Senftleben[32]are higher, for the real gas, and not included in this

comparison. Although the highest-temperature datum appears to

agree better with the RRAO model than with the RRHO model,the weaker agreement at low temperatures suggests that this is

fortuitous.

For C2H4O, the ideal-gas entropy at T =283.66 K has been re-

ported as S= 240.1 J K-1 mol1 (no uncertainty reported) [33].

The RRAO and RRHO models yield (240.1 and 240.2) J K1 mol1,

respectively. These values agree equally well with the experimen-

tal value. For heat capacity, the models are compared in figure 8

with experimental ideal-gas data by Hurly[34]. The RRHO model

is markedly (2 times) closer to experiment than the more sophis-

ticated RRAO model, although both underestimate the heat capac-

ity. As mentioned above, the anharmonic constants associated with

mode 14 are suspect. If those constants are arbitrarily divided by

10 in the RRAO model, the resulting Cp(T) values are 2 times clo-

ser to experiment than the RRHO, but such an arbitrary procedurehas little predictive value.

5. Conclusions

High-temperature experimental measurements of ideal-gas

heat capacities are needed to test the theory more stringently.

However, data below T =500 K suggest that including first-order

vibrational anharmonicity, by means of eq. (1), does not improve

upon the simple harmonic-oscillator model, at least for the small,

rigid molecules studied here. Failures of VPT2 theory can cause

serious errors in vibrational constants, as found here for ethylene

oxide. In the regime of non-physical, negative energy increments,

truncation of the density of states (DOS) leads to implausible re-

sults, as found here for thorium tetrafluoride. An alternative

approximation for the DOS is needed at these higher energies.

Off-diagonal anharmonicity contributes negligibly to ZPE for these

four molecules, but can be important for thermodynamic

functions.

Appendix A

Size of Monte Carlo sampling volume

Equation (5) may be proved by induction on D. For D= 1, the

length of the line segment is V1=b1. For D= 2, the area of theright-triangle isV2=b1b2/2. The volume of the right-hyper-triangle

FIGURE 6. Ideal-gas heat capacity (Cp) of ThF4 from the RRAO and RRHO models. FIGURE 7. Ideal-gas heat capacity (Cp) of C2H4 from the RRAO and RRHO models

and from experimental measurements.

FIGURE 8. Ideal-gas heat capacity (Cp) of C2H4O from the RRAO and RRHO models

and from experimental measurements.

188 K.K. Irikura / J. Chem. Thermodynamics 73 (2014) 183189


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in (D+ 1)-space is the integral (equation(A.1)) over 0; bD1 of the

slice of areaVD, whose verticesaidepend upon the integration var-

iableyD1 by way of equality(4), as shown explicitly in eq. (A.2).

Substitution and integration complete the proof, equation(A.3).

VD1

Z bD10

VDyD1dyD1 1

D!

Z bD10

YDi1

ai dyD1; A:1

ai bi 1 yD1bD1

; A:2

VD1 1

D!

YDi1

bi

Z bD10

1 yD1bD1

DdyD1

1

D 1!

YD1i1

bi: A:3

Probability distribution function

To build a sampling method starting from a rectangular distri-

bution, we want the probability distribution function to satisfy

equation (A.4), where u is from the rectangular distribution. We

want the probability of choosing yD to equal the fraction of the

lower-dimensional right-hypertriangle that constitutes the slicealongy D, as shown in equation(A.5). Combining these two equa-

tions gives an expression for du=dyDthat can be integrated to yield

equation (A.6). To have correspondence between the intervals

u 2 0; 1 and yD 2 0; bD, the integration constant must be c= 1.

Rearranging to expressyD in terms ofu gives equation(A.7).

PyDdyD 1du; A:4

PyDdyD 1

VDVD1 1

yDbD

D1dyD; A:5

u c 1 yDbD

D; A:6

yD bD1 1 u1=D: A:7

Once a value is selected for yD, it constrains the value for yD1because of inequality(4), and so forth for the subsequent y i. This

sequence of constraints is captured by the continued product in

equation (6). The bi may be sampled in any order; the ordering

of the indices in equation(6)is for convenience.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in

the online version, at http://dx.doi.org/10.1016/j.jct.2013.12.018.

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JCT 13-419

K.K. Irikura / J. Chem. Thermodynamics 73 (2014) 183189 189
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