Download Quantum steam tables. Free energy calculations for H2O, D2O, H2S

Quantum steam tables. Free energy calculations for H20, 0 20, H2S,
and H2Se by adaptively optimized Monte Carlo Fourier path integrals
Robert O. Topper,a) Oi Zhang, Vi-Ping Liu, and Donald G. Truhlar
Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis,
Minnesota 55455-0431
(Received 12 October 92; accepted 3 December 92)
Converged quantum mechanical vibrational-rotational partition functions and free energies
are calculated using realistic potential energy surfaces for several chalcogen dihydrides (H20,
D 20, H 2S, H 2Se) over a wide range of temperatures (600-4000 K). We employ an adaptively optimized Monte Carlo integration scheme for computing vibrational-rotational partition functions by the Fourier path-integral method. The partition functions and free energies
calculated in this way are compared to approximate calculations that assume the separation
of vibrational motions from rotational motions. In the approximate calculations, rotations are
treated as those of a classical rigid rotator, and vibrations are treated by perturbation theory
methods or by the harmonic oscillator model. We find that the perturbation theory treatments yield molecular partition functions which agree closely overall (within ~ 7%) with the
fully coupled accurate calculations, and these treatments reduce the errors by about a factor
of 2 compared to the independent-mode harmonic oscillator model (with errors of ~ 16% ).
These calculations indicate that vibrational anharmonicity and mode-mode coupling effects
are significant, but that they may be treated with useful accuracy by perturbation theory for
these molecules. The quantal free energies for gaseous water agree well with previously available approximate values for this well studied molecule, and similarly accurate values are also
presented for the less well studied D 20, H 2S, and H 2Se.
I. INTRODUCTION
In order to calculate equilibrium constants by molecular statistical mechanics and absolute reaction rates by
transition state theory, it is necessary to calculate the canonical vibrational-rotational partition function Q( T) as a
function of the temperature T from the potential energy
function for isolated reactants, products, and transition
states in the gas phase. This is directly equivalent to estimating the absolute free energy G; in particular, the relationship between the molecular partition function and the
free energy of an ideal gas of N x molecules is given in the
Born-Oppenheimer approximation byl-4
G= -NxkBT In Q(T)Qtrans(T)
Nx
.-
(1)
where kB is Boltzmann's constant, and Qtrans(T) is the
translational partition function
Qtrans(T)=it3 (NxkBT)
p. ,
(2)
it=t7T~:BT ,
(3)
with M the molecular mass and P the pressure.
Recently, we presented a new Monte Carlo methodS
based on the Fourier path-integral formalism6-11 for the
computation of molecular vibrational-rotational partition
·)Present address: Department of Chemistry, University of Rhode. Island,
Kingston, RI 02881-0801.
functions, and we demonstrated its convergence properties
by applying it to a model coupled oscillator problem5 and
to the diatomic molecule HCI described in three Cartesian
degrees of freedom. 12 In the present work, we apply the
new Fourier path-integral Monte Carlo method to the
computation of partition functions for several triatomic
molecules: H 20, D 20, H 2 S, and H 2Se. No approximate
mode decouplings are invoked and the calculations represent the result of converged quantum mechanics for the
assumed potential functions. We compare the results of
these calculations to approximate calculations based on assuming separability of the vibrational and rotational degrees of freedom combined with treating the vibrations
with harmonic and anharmonic models and the rotations
with the standard classical rigid rotator model.
Although previous studies have made systematic comparisons between accurate quantum mechanics and approximate forms for partition functions for diatomic molecules at high temperatures,12-16 as well as for free energies
for van der Waals clusters at low temperatures,lO,11 fewer
studies have been carried out for polyatomic molecules,17-19 primarily because of the lack of accurate calculations of polyatomic energy levels. 18,20 In particular, we
note that the molecular internal partition function is conventionally obtained by calculating the vibrationalrotational eigenenergies En using variational methods 2,18,21
and summing Boltzmann factors to obtain the partition
function byl-4
Q(T) =
L
exp( -f3E n) ,
(4)
n
J. Chern. Phys. 98 (6), 15 March 1993
0021-9606/93/064991-15$06.00
© 1993 American Institute of Physics
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Topper et al.: Quantum steam tables
where n designates all of the quantum numbers for each
vibration-rotation state and /3= lIk B T. If the system of
interest is studied at low temperature, an accurate calculation of the molecular zero-point energy for a small range of
rotational excitation energies, and perhaps the excitation
energies of a few low-frequency modes, may suffice to converge Eq. (4). However, if one needs to carry out calculations at higher temperatures (in the combustion regime,
e.g.), if there are low-frequency internal rotations in the
molecule, or if more than one conformer is thermally accessible, it may be very difficult to converge Eq. (4).
In those cases where Eq. (4) is difficult to converge,
one may employ one of several approximation schemes
which have been developed for partition functions. I- 5,12-17,22-27 One goal of the present paper is to use
the Fourier path-integral Monte Carlo method to test the
accuracy which may be achieved by such schemes. Notably, almost all of the approximate methods assume that the
vibrational motion is separable from the rotational motion,
but this assumption is not made in the Fourier pathintegral scheme. Another goal of the present paper is to
demonstrate the convergence properties of our Fourier
path-integral Monte Carlo algorithm for triatomic molecules. Finally, we will present computed Gibbs free energy
functions for four triatomic molecules. Note that the Gibbs
free energy for H 2Se is not tabulated in the JANAF tables 28 of thermodynamic functions, and other previous estimates are highly approximate.
Section II briefly reviews the Fourier path-integral formalism and the adaptively optimized Monte Carlo method.
Section III presents the coordinate system and details of
the parameters used in the Fourier path-integral calculations. Section IV presents the potential surfaces used in the
present study. Section V presents the approximate methods
to be compared to the accurate calculations. Section VI
presents the results of the calculations. Section VII discusses these results and Sec. Vln contains a summary bf
our conclusions.
Q(T)=J(T)~feo_
a
-eo
···feo
-eo
(fi
1=1
dXj) (
fi IT
II. THE FOURIER PATH-INTEGRAL MONTE CARLO
METHOD
Partition functions for molecular systems are calculated in the present paper by the Fourier path-integral
Monte Carlo method. We use the Feynman path-integral
treatment of quantum statistical mechanics,7 and the nuclei are treated as particles which move under the influence
of a Born-Oppenheimer potential energy surface. Under
this set of assumptions, the vibrational-rotational partition
function for a polyatomic molecule may be written as the
following path integral:
Q( T)
f f:
(7)
a j,l are the coefficients used to represent the closed paths in
a Fourier expansion, Le.,
(8)
YJ [x(s)]
!-~ f:
n
ds H[X(S),P(S)]},
(5)
where x is a set of Cartesian coordinates which locate the
nuclei in a center-of-mass frame, a is the usual rotational
symmetry number,I,3,4 and f~§[x(s)] indicates an integral over all paths, which are denoted xes) (where s is a
parameter with the units of time) that begin and end at
points x in the N-dimensional molecular configuration
space (Le., all closed paths), and H[x(s),p(s)] is the
Hamiltonian for the system. The Hamiltonian is also a
function of the associated path momenta given by pes).
There is a formal correspondence between the parameter s
in Eq. (5) and an imaginary time t that may be associated
with each path in the path-integral representation of quantum dynamics. 6
As discussed by Feynman,6,7 Miller, 8 and Freeman,
Doll, and co-workers,9-11 the paths may be expanded in
Fourier series about free particle "reference paths" with no
loss of generality. The exact partition function is then
transformed from a path integral over all closed orbits to
an infinite-dimensional Riemann integral; for a system
with N internal degrees of freedom, this yields
£ i: 2~-S(X,a)].
1=1 1=1
Here J( T) is the Jacobian of the transformation and it. is
given by
dx
xexp
daj,l)exp [-
)=1 1=1
=~
(6)
1,1
a path that starts at a point x in configuration space and
ends at the same point after the complex time interval /3-1Z,
1
S(x,a) =1i
r/3n
J0
ds V[x(s)].
(9)
The parameters aj,l are functions of the temperature and of
the reduced masses f.L j associated with the N Cartesian
variables x j in the center-of-mass coordinate system. These
parameters are given by
The function S(x,a) is an action integral corresponding to
(10)
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Topper et al.: Quantum steam tables
In order to evaluate Eq. (6), which is an infinitedimensional integral over an infinite domain, we represent
it in a form which can be evaluated by Monte Carl029
methods. This is accomplished5 by dividing the partition
function for the fully coupled system by the partition function for a particle in an N-dimensional hypersphere with
zero potential energy. For numerical work, the domain in
configuration space in Eq. (6) is truncated to the interior
of the same hypersphere, which is designated by D, and the
infinite number of Fourier coefficients is truncated to a
finite number (K) of coefficients per degree of freedom.
The molecular partition function is then given by an integral over an [N + (NXK)]-dimensional domain:
Q( T) ~ Qph( T)
(7
J
dx
D
Joo ... Joo
-00
Xexp[-S(x,a)]g(a),
-00
(11)
(12)
'rkN ) is the hypervolume of the N-dimensional hypersphere, calculated from the standard formula 3o
h
,,!{/2RN
=r[(N/2)+I] ,
(13)
where r(N) is the usual gamma function, and R is the
hyperradius of the hypersphere. The function g(a) is a
normalized (NXK)-dimensional Gaussian probability
density function in Fourier coefficient space given by
G(a)
g(a) = - - : : - - - - - : ; - - - - J~oo'" J~oo da G(a)
(14)
with
G(a) =exp { -
il /#1 2~J
ously,5 this procedure rapidly becomes inefficient as N increases because the ratio of the volume of the sphere to that
of the cube enclosing it tends rapidly towards zero as a
function of increasing N. However, we have found that for
small molecules, the computational cost of generating the
unused configuration space points is very small compared
to the cost of integrating the potential along the sampled
paths, and therefore the present procedure is useful in a
practical sense for these systems. The Monte Carlo estimate of the integral is then given by5,!2
(Q(T)n=
3
[Qph(T) h
k=!
nk
I
nk
I
ik=!
exp[-S(xik,aik)],
(16)
da
where Qph(T) is the partition function for a particle in an
N-dimensional hypersphere, i.e.,
'r(N)
4993
(15)
If K and D are sufficiently large, Eq. (11) represents
the accurate quantum mechanical result to arbitrary accuracy. We evaluate Eq. (11) as described previously.5,12 The
nuclear coordinates are sampled using adaptively optimized stratified sampling (AOSS) of strata defined by concentric hyperspheres within D, and the Fourier coefficient
degrees offreedom are sampled according to g(a) by using
the Box-Muller algorithm31 ,32 to generate an uncorre1ated
sample in the Fourier coefficient space. This combination
of techniques has been referred to as the AOSS-U method,
where U indicates that the samples are sequentially uncorrelated. s
The configuration space coordinates of each sample are
confined within the stratum boundaries by using rejection29
methods. For example, to generate samples within a hypersphere, samples are generated within an N-dimensional
hypercube just enclosing the hypersphere, and all samples
which fall outside of the hypersphere are ignored, i.e., not
evaluated or accumulated. As has been discussed previ-
where nk is the number of samples within each stratum, n
is the total number of samples, and [Qph ( T) h is given by
Eq. (12), but with the volume of the kth stratum ('r k )
substituted for 'r.
We remind the reader that, as discussed previously,5
one of the advantages of the present sampling scheme is
that it is fully uncorrelated, and so the statistical error may
be estimated reliably and in a straightforward manner. The
statistical error of the partition function of Eq. (16) is
denoted Wn and is given by
Wn
=
~ [Qph(T)]~ [(
nk- 1
£..,
k=!
-2S(x,a»
e
_( -S(x,a»2]
nk
e
nk '
(17)
In the present work, the coordinate system used to
describe the configuration space is different than the ones
used in previous work. In the following sections, we discuss
.. the coordinate system and potential surfaces used in the
present study in some detail.
III. COORDINATE TRANSFORMATIONS AND
IMPLEMENTATION OF THE FOURIER
PATH-INTEGRAL METHOD
We will consider the vibrational-rotational dynamics
of several triatomic molecules. In order to apply the Fourier path-integral method summarized above, we specify
the nuclear positions of the triatomic in a coordinate system defined such that the kinetic energy operator is written
in the form
T=~
2
N
I
j= 1
~2
Pj,
(19)
f-£ j
where N is the number of vibrational-rotational degrees of
freedom (i.e., N=6), Pj is the momentum operator for
degree of freedom j, and f-£ j is a reduced mass. The origin
of the coordinate system is fixed at the molecular center of
mass.
These requiremepts are satisfied by the usual Jacobi
coordinates. 33- 38 For a triatomic molecule (see Fig. 1) with
nuclear masses (rnA' mE' me), the transformation between
Cartesian coordinates z and Jacobi coordinates S is
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Topper et al.: Quantum steam tables
6
H=(2f-L)-1
L p;,+V(X[,,,,,X6)'
j=1
(25)
J
Using Eqs. (20)-(24), one can represent the canonical
transformation between Cartesian and mass-scaled Jacobi
coordinates as a 9 X 9 matrix T, such that
(26)
z=Tx.
In the Fourier path-integral calculations, we set X7' x8,
and X9 equal to zero and work in the remaining sixdimensional space specified by x1,,,,,x6' We draw the configuration space starting point for each sampled path uniformly from the interior of a six-dimensional hypersphere.
The Fourier coefficients for the sampled path are drawn
from a multidimensional Gaussian distribution via the
Box-Muller algorithm. 31 ,32 The matrix T is then used to
transform each configuration along each sampled path to
Cartesian coordinates, which are then used to calculate the
potential energy along the sampled paths.
The masses are taken as mH=1837.15m e, mD
= 3671.49m e, mo=29157.0me> ms=58281.6m e, and mSe
= 145679m e, where me is the atomic unit of mass (i.e., the
mass of an electron). The parameters used in the Fourier
path-integral calculations are specified below in Sec. VI.
z
x
A =O,S,Se
FIG. 1. The coordinate system used in the present study. The relationship
between the Cartesian coordinates z and Jacobi coordinates S is discussed
in the text. The inset shows the internal coordinates used to evaluate the
potential energy along each sampled path.
In this study, we use a series of potential surfaces develqpeslby Kauppi and Halonen 39 for covalently bound
triatomic molecules, and also one surface presented by
Zhao et al. 40 A single functional form is used with a different set of parameters for each case. The internal coordinates of the molecule are specified by (see Fig. 1)
SJ+6= (mAzJ+m BZJ+3 +mczJ+6)/M,
with J= 1,2,3. Here (Zl,Z2,Z3) are the Cartesian coordinates
of A with respect to a laboratory frame, (z4"",z9) are similarly defined for atoms Band e, respectively, and M is the
total mass. (S1,S2,S3) specify the location of B relative to
e, (S4,S5,S6) locate A relative to the Be center of mass,
and (S7,S8,S9) locate the molecular center of mass. In this
coordinate system, f-L j are given by
mBmC
(mB+mC) ,
f-L4 = f-L5 = f-L6
(21)
(22)
and
f-L7=f-LS=f-L9=M.
IV. MOLECULAR POTENTIAL ENERGY SURFACES
(23)
R1 = ~(Z6-Z3)2+ (Z5-Z2)2+ (Z4-Z1)2,
(28)
cP=cos-
1
[~:R~2].
(29)
Then a set of auxiliary variables (Y1'Y2,O) is defined by
Yi= 1-exp [ -a(Rj-Re)],
(30)
O=CP-CPe,
(31)
with Re the value of R j at the equilibrium configuration and
CPe the equilibrium internal bond angle. The potential energy expressed as a function of these internal coordinates
was given by Halonen and Carrington41 in the form
In order to simplify the Hamiltonian and treat all coordinates on an equal dynamical footing, we apply a transformation to mass-scaled Jacobi coordinates {x}, i.e.,
Xj= (f-L/f-L) 1I2Sj,
(27)
(32)
(24)
where f-L is the scaling mass. All physical results are independent of f-L, so we set it equal to 2500 a. u. of mass. Fixing
the origin at the molecular center of mass, the barycentric
Hamiltonian becomes
(33)
(34)
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Topper et al.: Quantum steam tables
4995
tion is to treat the vibrations as independent-normal-mode
harmonic oscillations. The partition function for this
model is given in closed form as 3,4
o
~b (T)=
(35)
For the first four cases, the functional form given in Eqs.
(27)-(35) is used with the parameters originally given by
Kauppi and Halonen for H 20, D 20, H 2S, and H 2Se, and in
the fifth and sixth cases, we studied H 20 and D 20 with a
modified set of parameters40 developed for the study of
microhydrated reactions. 40 ,42,43 The modified parameters
still provide good agreement with the experimental harmonic frequencies of the water molecule, but they also can
be used to represent accurately the properties of the
CI-(H20) and Cl-(D 20) ionic complexes. For convenience, the parameters used for all cases are summarized in
the supplementary material. 44
V. APPROXIMATE METHODS FOR PARTITION
FUNCTION CALCULATIONS
In the present work, we compare partition functions
calculated by Fourier path-integral Monte Carlo methods
to those calculated by two ways of using perturbation theory and by the harmonic approximation. The usual starting place for approximate calculations is to assume separation of the vibrational motion from the rotational motion,
and in this approximation, the partition function is a product of vibrational and rotational factors t-4
(36)
The equations in the rest of this section assume that the
molecule is nonlinear and there are no degenerate vibrations, both of which are true for the cases studied in this
paper.
Under the assumption that the rotations are those of a
classical rigid body with internuclear distances constrained
to those of the equilibrium geometry, the partition function
for a rigid polyatomic molecule with principal moments of
inertia 1A' 1 B' and 1C and symmetry number (T is given
by t-4,45
1T1/2
Q~?-(T) =~
IT
a=A,B,C
(21{3n~ )
,
(37)
where "CRR" designates the classical rigid rotator approximation. The CRR approximation to the rotational degrees
of freedom was recently tested20 against a nonrigid
quantum-mechanical treatment of the rotational degrees of
freedom for the water molecule. It was found to underestimate the partition function by 2% at 500-1000 K and
3.5% at 2000 K.
The approximate methods discussed in the present paper all assume Eqs. (36) and (37), but they use different
expressions for the vibrational partition function. The most
common approximation to the vibrational partition func-
exp( -{3fzOJ m/2)
mIlt 1-exp(-{3fzOJ m ) '
(38)
where OJ m is the vibrational frequency (in radians per second) associated with normal mode m.
Another way to approximate the vibrational partition
function is to carry out approximate calculations of the
energy levels using perturbation theory 23,46-54 and sum
their Boltzmann factors according to Eq. (4), replacing n
by the set of quantum numbers {Vi} for each vibrational
energy level. In the present paper, we carry out a fully
coupled perturbation treatment of the J=O vibrational energy levels, where J is the rotational quantum number. We
use standard methods4 6-52 to calculate the perturbation
theory vibrational energy levels using a quartic approximation to the various force fields. As usual, our perturbation
calculations include all terms up to second order in the
cubic force constants and up to first order in the quartic
force constants. We used the SURVIBTM computer program, developed by Ermler, Hsieh, and Harding,53 with
the potential energy surfaces presented in Sec. IV as input
for the computations. No corrections for Fermi or other
resonances were included.
We next describe briefly the perturbation theory calculations. We define a set of reduced (dimensionless) normal coordinates qm'
qm= (
21TCJ.1/iim) 1/2
n
Um,
.
(39)
where U m is a mass-scaled normal-mode coordinate related
to the usual 55 mass-weighted normal coordinate Qm by
(40)
and vm=OJin/21Te is the spectroscopic frequency in units of
cm- t • Note that u m in Eqs. (39) and (40) is a linear
combination of x j of Eq. (24) with unitless coefficients.
Carrying out a fourth order Taylor expansion of the potential energy function, we can write an approximate form
for the potential of a nonlinear molecule as a function of
the reduced normal mode coordinates
he
V;:::; Ve + 2!
1/2
N-3
he
+4!
N-3 _
2
he
-
m~t vmqm+ 3! m~o f mnoqmqnqo
(41 )
m,n,o,p
wher~ Ve is the energy at the eqUilibrium geometry, f mno
and f mnop are third and fourth order normal mode force
constants, respectively, and the summations are unrestricted, i.e., all sums are from 1 to N - 3. We note that Eq.
(41) uses only one of many possible conventions for the
Taylor expansion, and that care must be used when comparing to other discussions of perturbation theory for vibrational energy levels in the literature. Throughout the
present paper, we place the zero of energy at the bottom of
each potential well so that Ve=O.
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4996
et at.: Quantum steam tables
Through the use of standard second order perturbation
theory, the vibrational energy levels, i.e., the energy levels
for a molecule with rotational quantum number J=O, are
then given by
E[{vmll=Ve+Eo+hc
+hc
N-3
1)
m~1 Vm(Vm+ 2
m~n Xmn( Vm+~) (Vn+~)'
(42)
where xmn is a second order anharmonicity constant, and
Eo is a constant (i.e., independent of vibrational quantum
numbers) term which arises in the perturbation theory
treatment. Note that Eo is often neglected in spectroscopic
applications, where only energy level differences are computed; however, it is necessary in the calculation of perturbation theory energy levels for thermochemical applications, as its omission noticeably affects the absolute values
of the vibrational energy levels, and therefore the
vibrational partition functions. 23 This term depends on
the force constants, rotational constants, and Coriolis
constants23,50, 5I
-:2
1 ~ -
7 " fmmm
EoIhc =64 '"
m f mmmm-576 '"
m
1
L
~
3
---+
64
Vm
'"
m¥=n
VmVnVoJ~no
Dmno
4 m<n<o
where
Dmno= (vm+vn+vo) (vm-vn-vo) (-vm+vn-=-vo)
x (-vm-vn+vo),
~.
- (44)
a designates a principal rotational axis (A, B, or C), (;;::; is
a Coriolis constant56 in units of cm -I, and B~a) is a rotational constant for the equilibrium configuration (also in
units of cm- I ).
Given the approximate expression for the potential
function in Eq. (35), the anharmonicity constants for an
asymmetric top molecule are given as47,49,50
1
1
-
N-3
~ -2
Xmm=16fmmmm-16 .-; fmmn
_~ f-
Xmn - 4
mmnn
2
2
[8V m - 3Vn
vn(4v~-~)
]
(45)
_ " f mmo fnno
'"
4v0
o
(46)
In full vibrational perturbation theory calculations, the
vibrational partition function is evaluated by summing the
Boltzmann factors of the energy levels given in Eq. (42)
(47)
m
where m denotes the set {VI>V 2 ,V3}' All energy levels up to
the dissociation limit are included in the sum and conver-
gence is checked by repeating the calculation including
levels only up to about 80% of the dissociation limit.
We also consider another, simpler approximate
method 23 for computing vibrational partition functions. In
this approximation, only the ground vibrational state and
the fundamental excitation energies (i.e., single-quantum
excitations from the ground state) are used as input for the
calculation. Defining A.m as the fundamental excitation energy of mode m and the ground state energy as E G, the
partition function is approximated by
.n!)PT
exp ( - (3EG )
~vib (T)=n!=~[I-exp(-{3A.m)l·
(48)
Then Wand A. m are calculated by second order perturbation theory, i.e., by Eqs. (42)-(46) with VI=v2=v3=0.
This method is denoted SPT for "simple perturbation theory," because in this approximation, we use perturbation
theory only for the ground state and single-quantum
excitations-not for all of the energy levels. In contrast,
full vibrational perturbation theory, using Eq. (47) but
still assuming separation of rotations from vibrations and
still stopping at second order, will be denoted PT..
In order to obtain the force constants (and hence the
spectroscopic constants) from the various potential energy
surfag:s using SURVIBTM, the potential must be defined
on a grid for fitting purposes. In all cases, we calculate the
potential energy on a 7 X 7 X 7 grid of points which are
equally spaced along the internuclear distances (R I,R 2)
and angle (cp) (see Fig. 1) with the grid centered at the
equilibrium geometry. The grid spacing is 0.04ao for RI
and R2 and 5° for cp (where ao is the Bohr radius 57 ). We
_have.found that varying the density of grid points (i.e.,
varying the number of points over a given range of internuclear distances) from a 5 X 5 X 5 grid to an 11 X 11 X 11
grid produced very little difference in the force constants
(less than 1 cm - 1) for all of the molecules considered in
this paper. Similar convergence tests were carried out to
explore the dependence of the vibrational perturbation theory calculations on the range of internuclear distances
used, and we find that the present calculations are well
converged with respect to this range. We also calculated
the force constants with the POLYRATE program, 58 and
slightly larger differences were found in the anharmonic
force constants.
We note that Bartlett et al. 59 used the same program to
study water with ab initio electronic structure calculations.
They used a 5 X 5 X 5 grid of points with 0.03ao grid spacing for the bond length and 3° for the bond angle. When the
same grid is used for the analytic potential functions in the
present-work, we found errors less than 1 cm -1 in the
frequencies and the cubic force constants of Eq. (41) and
less than 5 cm - I in the quartic force constants. These errors, however, lead to an error in the anharmonic constants
up to 28 cm -I. We conclude that it is essential to find a
stable fit region to ensure quantitative accuracy. (Of course
it is much easier to experiment with the grid parameters
when using an analytic potential energy function, as here,
than when using ab initio electronic structure calculations.)
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Topper et al.: Quantum steam tables
4997
TABLE 1. Calculated vibrational frequencies and perturbation theory parameters for H 20 and D 20.a
H 20
(Ref. 39)C
- f
VI
-f
V2
-f
V3
Harmonic
ZPEK
D.lh
D.2h
h
D.3
Anharmonic
ZPEi
Eoi
xIIi
xlii
XI3
X22i
i
X23
X33i
H 2O
(MKH)d
3834
1647
3935
4709
3656
1596
3745
4631
3660
1594
3753
Dp
DzO
DP
DP
(KH)
(Ref. 39)
(MKH)
(Expt.)
3834
1647
3935
4709
3834
1648
3935
n.a.
2759
1208
2883
3425
3573
1589
3658
4612
3657.
1595
3756
n.a.
2667
1180
2779
3383
14.88
-63.24
-24.53
-244.21
-17.80
-20.98
-72.12
-0.16
-42.91
-16.83
-167.15
-17.72
-14.10
-49.58
H 2O
(Expt.)e
n.a.
-42.58
-15.93
-165.82
-16.81
-20.32
-47.57
2669
1179
2785
2759
1208
2883
3425
2784
1206
2889
n.a.
2623
1177
2731
3372
2668
1178
2788
n.a.
7.66
-32.21
-12.19
-126.91
-9.79
-10.90
-40.41
-0.63
-21.95
-7.90
-87.20
-9.79
-7.14
-28.16
n.a.
-22,58
-7.58
-87.15
-9.18
-10.61
-26.15
a( •.• ) indicates that the value was not reported; n.a. indicates that the value is not available.
bCalculated from the Kauppi-Halonen (KH) potential, using a normal-mode analysis to compute vm and perturbation theory to compute D. m.
cCalculated values from Ref. 39, obtained by those authors via variational methods.
dThe same as b, except that the modified Kauppi-Halonen (MKH) surface is used.
<Experimental values summarized in Refs. 41 and 48.
fHarmonic frequency (in em-I).
SIn dependent harmonic normal mode approximation to zero point energy (in em -I).
hFundamental transition frequency (in em -I).
iAnharmonic zero point energy computed via vibrational perturbation theory (in em-I).
iparameters for Eq. (44), either computed using SURVIBTM (Ref. 49) or obtained from experiment (Refs. 39-41 and 48).
VI. RESULTS
VI. A. Perturbation theory transition energies and
spectroscopic parameters
In Table I, we present harmonic frequencies and transition frequencies for H 20 and D 20 as computed from the
potential energy surfaces summarized in the previous section. We also present the spectroscopic constants obtained
from the perturbation theory calculations, as well as exper-
imental values for these constants. 39-41,48 Considering first
the H 20 and D 20 molecules, we find that the fundamentals
we have obtained from the Kauppi-Halonen surface using
vibrational perturbation theory agree well with the fundamentals computed by Kauppi and Halonen 39 using variational methods for the vibrations. However, the fundamentals we have calculated from the modified KauppiHalonen surface40 for these molecules are somewhat
TABLE II. Calculated vibrational frequencies and perturbation theory parameters for HzS and H 2Se. a
H 2S
(KH)
VI
V2
V3
Harmonic
ZPE
D.I
D.2
D.J
Anharmonic
ZPE
2719
1214
2736
3335
Eo
3.89
-23.59
-19.57
-95.01
-5.43
-21.20
-25.29
XII
xI2
XI3
X22
X23
x33
2615
1183
2628
3291
H 2S
(Ref. 39)
2615
1183
2628
H 2S
(Expt.)
H 2Se
(KH)
2722
1215
2733
n.a.
2437
1055
2452
2972
2614
1183
2629
n.a.
2343
1033
2358
2935
n.a.
-25.09
-19.69
-94.68
-5.72
-21.09
-24.00
4.80
-21.49
-17.75
-83.32
-2.18
-17.85
-21.56
H 2Se
(Ref. 39)
H 2Se
(Expt.)
2435
1054
2448
n.a.
2344
1033
2357
2344
1034
2358
n.a.
n.a.
-21.4
-17.7
-84.9
-2.4
-20.2
-21.7
'See Table I for an explanation of rows and columns.
J. Chern. Phys., Vol. 98, No.6, 15 March 1993
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Topper et at.: Quantum steam tables
4998
TABLE III. Calculated vibrational partition functions for H 2O.'
(!vl~(T)
(fv~T( T)
(1vi~( T)
(fv~T(T)
(1vi~(T)
(1vi~(T)
TCK)
(KH,MKH)b
(KH)C
(KH)d
(MKH)e
(MKH)f
(KH vs MKH)g
200
300
400
600
1000
1500
2400
4000
1.95X 10- 15
1.56 X 10- 10
4.42X 10- 8
1.27X 10- 5
1.27X 10- 3
1.4SX 10- 2
1.16x 10- 1
7.26X 10- 1
3.40 X 10- 15
2.26X 10- 10
5.85X 10- 8
1.S4x 10- 5
1.43 X 10- 3
l.59X 10- 2
1.27X 10- 1
8.26X 10- 1
3.40X 10- 15
2.26XlO- 1O
5.8SX 10- 8
1.S4X 10- 5
1.44x 10- 3
1.60X 10- 2
1.29X 10- 1
8.12X 10- 1
3.89X 10- 15
2.48 X 10- 10
6.26X 10- 8
1.61 X 10- 5
1.48X 10- 3
1.63X 10- 2
l.31X 10- 1
7.99X 10- 1
3.89X 10- 15
2.48 X 10- 10
6.26X 10- 8
1.61 X 10- 5
1.48X 10- 3
1.64X 10- 2
l.33X 10- 1
8.S0X 10:"1
14%
9%
7%
5%
3%
6%
10%
12%
'All partition functions calculated with zero of energy at the equilibrium geometry.
blndependent mode harmonic oscillator partition function from KH and MKH potential surfaces (see Tables I-II).
"Simple perturbation theory (SPT) partition function from KH potential surface.
dperturbation theory (PT) partition function from KH potential surface.
"The same as c, except for the MKH surface.
fThe same as d, except for the MKH surface.
gThe difference between vibrational partition functions on the two surfaces calculated at the PT level. Average=6%.
different from those calculated from the Kauppi-Halonen
surface; in particular, there are sizeable differences in the
Lli and Ll3 fundamentals. Although the two potential surfaces have exactly the same harmonic frequencies, the
spectroscopic constants for the two surfaces are very different.
Table II presents similar results for the H 2S and H 2Se
molecules. We see that the calculated fundamentals are in
excellent agreement with the values calculated by Kauppi
and Halonen,39 who used variational methods. The good
agreement between our calculations and the ones carried
out by Kauppi and Halonen is a good check.
VI. B. Separable vibrational and rotational partition
functions
We present vibrational partition functions for all four
molecules in Tables III-VI, using the harmonic oscillator
model, perturbation theory, and the simple perturbation
theory expressions given above, and in Table VII we give
the classical rigid rotator partition functions for all four
molecules. We see in Tables III and IV that, although the
vibrational partition functions for the two water surfaces
are identical in the harmonic oscillator approximation,
there is a quantitative difference between the partition
functions from the two surfaces at the perturbation theory
level. For H 20, the average unsigned difference is 6%, with
the largest unsigned difference (14%) at T = 200 K. Since
the perturbation theory and simple perturbation theory results are identical at this temperature, we infer that the
discrepancy is due to the difference in the perturbation
theory estimates of the zero point energy on the two surfaces. This is confirmed by a comparison of the harmonic
partition function to the one obtained by perturbation theory for each potential surface; there is a difference of 20%
between the harmonic and perturbation theory partition
functions for H 20 on the Kauppi-Halonen surface, and a
difference of 26% for H 20 on the modified KauppiHalonen surface. The effect is similar for D 20 over the
same temperature range, with an average unsigned difference of less than 4% between the two potentials at the
perturbation theory level; here again, the largest unsigned
difference between the two surfaces (8%) is at T = 200 K.
From Table V, we see that the harmonic oscillator
vibrational partition function for H 2S at T=200 K differs
by 27% from the perturbation theory partition function,
which is the largest discrepancy between the two approximations we observe in this study. In Table VI, we see that
the harmonic oscillator partition function also disagrees
significantly with the perturbation theory result for H 2Se
(by 23% at T=200 K).
TABLE IV. Calculated vibrational partition functions for D 20.'
(!vl~(T)
(fv~T(T)
(1vi~( T)
(fv~T(T)
(1vi~(T)
~(T)
TCK)
(KH,MKH)
(KH)
(KH)
(MKH)
(MKH)
(KH vs MKH)b
200
300
400
600
1000
1500
2400
4000
1.99 X 10- 11
7.36xlO- 8
4.S2XlO- 6
2.87Xl0- 4
9.09xl0- 3
6.26xlO- 2
3.74XI0- 1
2.04 X 10°
2.70X 10- 11
9.04X 10- 8
S.27XlO- 6
3.19xlO- 4
9.81 X 10- 3
6.70XIO- 2
4.01XlO- 1
2.20XIO°
2.70X 10- 11
9.04X 10- 8
S.28xlO- 6
3.20XlO- 4
9.83XIO- 3
6.74X 10- 2
4.08 X 10- 1
2.17X 10°
2.89X 10- 11
9.46 X 10- 8
S.46x 10- 6
3.28X 10- 4
9.98XlO- 3
6.82X 10- 2
4.lOX 10- 1
2.26xlO°
2.92 X 10- 11
9.51 X 10- 8
S.48xlO- 6
3.28xlO- 4
1.00 X 10- 2
6.88X 10- 2
4.19xlO- 1
2.26x 10°
8%
5%
4%
2%
2%
2%
3%
4%
aSee Table III for an explanation of columns. One hundred and fifty eight bound energy levels are used in the perturbation theory calculations.
~he difference between vibrational partition functions on the two surfaces calculated at the PT level. Average=4%.
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Topper et al.: Quantum steam tables
TABLE V. Calculated vibrational partition functions for H 2S.'
T(K)
Q;1~(T)
(fv~T(T)
{tvi~(T)
~~(T) vs {tvi~(T)b
200
300
400
600
1000
1500
2400
4000
3.82X 10- 11
1.14X 10- 7
6.26X 10- 6
3.57XlO- 4
l.04X 10- 2
6.91 X 10- 2
4.04 X 10- 1
2.18X 100
5.23XlO- 11
1.40 X 10- 7
7.33XlO- 6
3.99XlO- 4
l.13x 10- 2
7.43 X 10- 2
4.36X 10- 1
2.37XlO°
5.23X 10- 11
1.40 X 10- 7
7.33 X 10- 6
3.99XlO- 4
1.13 X 10- 2
7.47 X 10- 2
4.42 X 10- 1
2.3 XlO°
27%
19%
17%
11%
8%
7%
9%
5%
'See Table III for an explanation of unlabeled columns.
bDeviation of harmonic oscillator partition functions from perturbation
theory partition functions.
Overall, the agreement between the SPT and PT partition functions is within 1% for all four molecules over the
range 2oo<T<15oo K, with mOdestly larger differences at
2400 K and differences up to 6%-8% at 4000 K. This is a
very important conclusion because simple perturbation
theory may be applied without convergence checks with
respect to the number of energy levels.
VI. C. Fourier path-integral Monte Carlo convergence
checks
We next describe the parameters used in the Fourier
path-integral Monte Carlo calculations. In the pathintegral calculations presented here, the potential energy is
integrated along each sampled path using either 50- or
loo-point Gauss-Legendre32 quadrature. We have carried
out convergence studies for H 20 on the modified KauppiHalonen potential energy surface (described below) at
1000 K by using 105 samples and varying the number N q of
quadrature points from 50 to 75, 100, and 150 with K = 128
and 256 Fourier coefficients per degree offreedom. We find
that the results are identical to well within the statistical
error limits. We note that similar values of N q and K were
found to be adequate in our previous study of the HCI
molecule. 12 Convergence studies of H 2S and H 2Se at this
temperature led to the same conclusion, i.e., 50 quadrature
points and 128 Fourier coefficients per degree of freedom
are adequate'to achieve the same level of convergence as
that achieved for H 20 at 1000 K, and we infer that these
parameters are adequate to achieve good convergence at
higher temperatures as well.
4999
We found that 128 Fourier coefficients per degree of
freedom was sufficient to obtain good convergence (better
than 5%) at T> 1000 K. However, similar studies of Hp
. at 600' K showed that more Fourier coefficients were
needed to achieve good convergence at lower temperatures.
In Table VIII, we present the results of a convergence
study with respe~t to the number K of Fourier coefficients
at a temperature of 600 K, using 106 samples and 50
quadrature points for each calculation. We find that adequate convergence may be achieved at this temperature by
using K =256. This is also consistent with our previous
studies. 12
Several additional parameters must be specified when
using the adaptively optimized stratified sampling (AOSS)
algorithm 5,12 to sample the configuration space. The AOSS
algorithm proceeds in three stages: (1) optimization of the
"great hypersphere" which constitutes the outermost
boundary of the domain D; (2) optimization of the three
stratification boundaries; and (3) optimal sampling of the
strata defined in stage 2. In stage 1, the outer boundary
radius was optimized by allowing a maximum of 12 trial
hyperspheres with hyperradii equally spaced on the interval from 3 to 4ao for all calculations with T> 600 K; for
T = 600 K, the stage 1 result at 1000 K was used as the
hyperradius for the great hypersphere. In stage 2, 100 bins
of equal volumes within the optimized hypersphere were
sampled until 0.40n samples were accumulated, and these
were used to optimize the stratification boundaries. A sam.ple size of 0.06n was used in each round of stage 3 until n
samples were generated. By examining several test cases,
we found no appreciable dependence of our results on
small changes in these parameters; most notably, using
larger values of the outermost hyperradius did not affect
the calculations.
All error estimates are quoted at a level of two standard deviations. As the algorithm uses uncorrelated sampling in both the configuration and Fourier coefficient
spaces, the error bars are easily estimated and are expected
to be accurate estimates (to one or two significant figures)
of 95% statistical confidence intervals. Thus, since the statistical errors dominate those due to finite N q and K, there
is approximately a 95% chance that the true partition
functions lie within the quoted error estimates. We use
either 105 or 106 samples in each calculation as required to
converge the calculation to the desired precision; in particular, we used enough sampling points so that the resulting
statistical error limits are <10% of the central value for all
calculations.
TABLE VI. Calculated vibrational partition functions for H 2Se.'
T(K)
(!v!~(T)
(fv~TCT)
{tvICT)
~~(T) vs {tvICT)
200
300
400
600
1000
1500
2400
4000
5.20X 10- 10
6.51XlO- 7
2.33XlO- 5
8.79XlO- 4
1.89X 10- 2
1.11 X 10- 1
6.08XlO- 1
3.18X 100
6.7SXlO- 1O
7.7SX10- 7
2.66X 10- 5
9.64XlO- 4
2.03 X 10- 2
1.19 X 10- 1
6.S3X 10- 1
3.44X 10°
6.7SXlO- 1O
7.7SXlO- 7
2.66XlO- 5
9.6SX 10- 4
2.03 X 10- 2
1.20 X 10- 1
6.57Xl0- 1
3.2 XlO°
23%
16%
12%
9%
7%
8%
7%
1%
"See Tables III and V for an explanation of columns.
VI. D. Comparison of vibrational-rotational partition
functions
In Tables IX-XII, we present the results of the Fourier
path-integral Monte Carlo calculations and compare them
to the approximate methods discussed in Secs. V and VI B.
The top section of Table IX presents calculations for
H 20 on the Kauppi-Halonen surface. Examining first the
results which assume an approximate separation of vibrations from classical rigid rotations, we see that the harmonic osCillator/rigid rotator approximation is in quanti-
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5000
Topper et al.: Quantum steam tables
tative disagreement with both of the perturbation theory
methods, which agree with one another quantitatively.
Next examining the FPI calculations, we see that they generally agree with the perturbation theory calculation to
within the statistical error limits for T<1500 K.
For temperatures at and above 2400 K, a noticeable
discrepancy between the path-integral diculation and either of the perturbation theory results begins to appear
(i.e., the difference between the two results is outside of the
95% confidence interval). We have carried out a convergence study of the Fourier path-integral calculations on
this potential surface at 2400 K (see Table IX), and we are
confident that our answer is well converged, and therefore
the perturbation theory/classical rigid rotator model seems
to be inaccurate in this temperature regime. This discrepancy may be due to the onset of appreciable vibrationalrotational coupling, or it may be due to the simple method
we have used to model the rotations. However, the discrepancy does not seem to be due to the onset of appreciable
dissociation, as the perturbation theory calculations at
2400 K are well converged (using 91 bound energy levels
instead of using 119 levels yields a partition function at
2400 K which differs by less than 0.01 % from the larger
calculation). A similar check at 4000 K shows a 0.3%
difference between using 119 and 91 levels, so the perturbation theory results are well converged at this temperature as well.
Perturbation theory appears to be less accurate for the
modified Kauppi-Halonen surface, for which results are
presented in the bottom section of Table IX.
The calculations for D 20 on the two surfaces are presented in Table X. For this molecule, the fractional statistical errors achieved by the Fourier path-integral method
are generally smaller than for H 20. We see that the Fourier path-integral calculations are in excellent agreement
with the perturbation theory model for T<1500 K on the
Kauppi-Halonen surface, but for T>2400 K, we again see
a noticeable discrepancy between the two results. However,
the agreement between the Fourier path-integral calculations and the perturbation theory calculations on the modified Kauppi-Halonen surface is rather poor, with the perturbation theory calculations always well outside of the
95% statistical confidence limits over the entire temperature range. The perturbation calculation is converged with
respect to the number of vibrational states included in the
calculation to within 1% at 4000 K and even better at
lower temperatures. Thus the high-temperature difference
of the accurate and approximate calculations may indicate
the breakdown of the model we use for the rotational motions for this molecule or indicate that excited-state centrifugal distortions are large on this surface. Further study
is necessary to fully evaluate the causes for the discrepancy.
In Table XI, we give results for H 2S, computed using
the Kauppi-Halonen potential energy surface for this molecule. The fractional statistical errors are somewhat
smaller for this molecule than for H 20 and D 20, and the
general trends are similar, i.e., the perturbation theory/
classical rigid rotator calculations agree closely with the
TABLE VII. Calculated classical rigid rotator partition functions
Q;;'~R(T).
TCK)
200
300
400
600
1000
1500
2400
4000
D 20
H 2O
1
H 2S
1
2.351X 10
4.320x Wi
6.651X 10 1
1.222 X 102
2.629X102
4.830X102
9.775 X 102
2.103 X 103
6.189X 10
1.137X 102
1.751 X 102
3.216X 102
6.920X102
1.271 X 103
2.573X 103
5.536X 103
H 2Se
1
6.834X10
1.256 X 102
1.933 X 102
3.551X 102
7.641 X 102
1.404 X 103
2.841 X 103
6.113 X 103
9.170X10 1
1.685 X 102
2.594X102
4.765 X 102
1.025 X 103
1.883 X 103
3.812X 103
8.202 X 103
FPI calculations for T <2400 K. The convergence of the
perturbation calculation with respect to the number of levels included is better than 2 % at 4000 K.
Table XII gives results for H 2Se, again computed using
a Kauppi-Halonen potential energy surface. We see that
the fractional statistical errors achieved by the FPI method
are slightly less than those for H 2S. The discrepancy between the accurate FPI method and approximate perturbation theory calculations is noticeable (6%) at 1500 K
for H 2Se, and it is quite large (22%) at 4000 K. The
perturbation calculation is well converged (about 2% at
4000 K, and within 0.2% for lower temperatures) in the
case of H 2Se.
VI. E. Quantum free energies
The quantum partition functions presented in the previous section may be used to compute free energies for an
ideal gas. 1- 5,24,25 In the present section, we consider free
energy functions in the form tabulated in the JANAFtables. 28 In particular, we tabulate the Gibbs free energy
function called gef( T) and defined as 25
gef(T) == - (G-W)/T
=R In Qtrans(T)+R In Q(T),
(49)
==geftrans( T) + gefint ( T),
(50)
If1
,
(j(T) =Q(T)ef3
(51)
where gef( T) has been separated into translational and
internal components. Here W is the energy of the ground
TABLE VIII. Convergence of vibrational-rotational partition functions
at 600 K for H 20 on the Kauppi-Halonen surface.'
J<b
(Q(T»n±2wnC
64
128
256
512
(2.32±0.20) X 10- 3
(2.26±0.20) X 10- 3
(1.91 ±0.16) X 10- 3
(2.01 ±0.17) X 10- 3
"Fifty Gauss-Legendre quadrature points used in all calculations with 106
samples per calculation.
. i>yhe number of Fourier coefficients per degree of freedom used in calculations.
cAOSS-U Fourier path-integral Monte Carlo calculation of the coupled
partition function. The error bars for the calculations are estimated at the
95% confidence level.
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Topper et at.: Quantum steam tables
5001
TABLE IX. Calculated vibrational-rotational partition functions for H 20 on the Kauppi-Halonen surface."
(Q(T»n±2wnb
FSEc
nd
K"
N af
1.68X 103
Kauppi-Halonen surface
1.88X 10- 3
(2.0±0.2) X 10- 3
3.77X 10- 1
(3.9±0.4) X 10- 1
7.72X 10°
(7.6±0.5) X 10°
(1.34±0.6) X 102
1.26 X 102
(1.32±0.02) X 102
(1.34±0.02) X 102
(1.33±0.02) X 102
(1.89±0.06) X 103
1.71 X 103
10%
10%
7%
5%
2%
2%
2%
3%
106
105
105
105
106
106
106
105
512
128
128
128
128
256
256
128
50
100
100
100
50
50
100
100
1.97X 10- 3
3.88X 10- 1
7.88XlOo
1.28 X 102
1.74X 103
Modified Kauppi-Halonen surface
1.97 X 10- 3
(2.3±0.2) X 10- 3
3.89X 10- 1
(4.0±0.3) X 10- 1
(8.6±0.5) X 10°
7.92 X 10°
(1.38±0.06) X 102
1.30 X 102
(2.1O±0.06) X 103
1.79 X 103
9%
8%
6%
4%
3%
106
105
105
105
105
256
128
128
128
128
50
100
100
100
100
T(K)
Q!1~~~R
(fv~T ifro~R
600
1000
1500
2400
2400
2400
2400
4000
1.56x 10- 3
3.34X 10- 1
6.98XIOO
1.14 X 102
1.88X 10- 3
3.77X 10- 1
7.69XIOO
1.24X 102
1.53 X 103
600
1000
1500
2400
4000
1.56X 10- 3
3.34X 10- 1
6.98XlOo
1.14X 102
1.53X 103
(fvi~{to~R
'All partition functions calculated with zero of energy at the equilibrium geometry. The separation of rotations from vibrations is assumed and the
rotations are treated as classical and rigid. See Table III for explanations of columns 1-4.
bAOSS-U Fourier path-integral Monte Carlo calculation of coupled partition function. The error bars for the calculations are estimated at the 95%
confidence level.
<Fractional statistical errors (FSE) at the 95% confidence level.
dThe number of samples used in Fourier path-integral Monte Carlo calculations.
"The number of Fourier coefficients used in Fourier path-integral Monte Carlo calculations.
fThe number of Gauss-Legendre quadrature points used in Fourier path-integral Monte Carlo calculations.
vibrational-rotational state and Q( T) is the internal molecular partition function with the zero of energy shifted to
this level. We will discuss the free energy function of Eq.
( 49) because of its prominent role in the literature and in
summarizing experimental data. We note, however, that to
estimate thermodynamic data (such as equilibrium constants) from potential energy or electronic structure data
requires both Wand s..ef(T), i.e., it requires information
about Q( T), not just Q( T) in which W cancels out.
Equation (49) was used to estimate gef( T) in the harmonic approximation, by simple and full vibrational perturbation theory, and from Fourier path-integral Monte
£arlo calculations. W is calculated harmonically to obtain
Q( T) from Q( T) in the harmonic approximation, and it is
calculated from perturbation theory in the other cases. The
resulting gef( T) values are in Table XIII for H 20 and in
Table XIV for D 20, H 2S, and H 2Se. In the case of H 20, a
recent calculation by Martin, Fran~ois, and Gijbels 25 is
also included in the comparison. Additional values of
gef( T) from sets of standard thermochemical tables 27,60,61
are also included.
Due to its importance, the free energy function of H 20
has been studied extensively. The most accurate previously
available values are those of Martin and co-workers,21 who
used ab initio electronic structure methods 62,63 to obtain a
quartic force field for H 20; they treated rotation by an
approximate nonseparable quantum mechanical method 24
including centrifugal distortion, and they obtained the vibrational partition functions by perturbation theory.46-54
[They did not include Eo in their calculations, but it has no
effect on gef( T) because EG is substracted from G in Eq.
(49).] We find that the ab initio results of Martin and
co-workers 25 are practically identical to those we have obtained from the Kauppi-Halonen surface via well con-
TABLE X. Calculated vibrational-rotational partition functions for D 2 0 on the Kauppi-Halonen surface."
T(K)
~~~~R
(fv~TQfo~R
600
1000
1500
2400
4000
9.24XIO- 2
6.29XlOo
7.96X 10 1
9.63X102
1.13 X 104
1.03 X 10- 1
6.79X 10°
8.52X 10 1
1.03 X 103
1.22 X 104
600
1000
1500
2400
4000
9.24XIO- 2
6.29 X 10°
7.96X 10 1
9.63 X 102
1.13X 104
1.05 X 10- 1
6.91 X 10°
8.67X 10 1
1.05 X 103
1.25 X 104
{fvi~Q;,~R
FSE
n
K
Na
Kauppi-Halonen surface
1.03 X 10- 1
(1.08±0.05) X 10- 1
6.80XlOo
(6.8±0.5) X 10°
(8.3±0.5) X 10 1
8.57XIO I
( 1.08 ± 0.04 ) X 103
1.05 X 103
(1.38 ±0.03) X 104
1.20 X 104
5%
7%
6%
4%
2%
106
105
105
105
105
256
128
128
128
128
50
100
100
100
100
Modified Kauppi-Halonen surface
1.06 X 10- 1
(1.19±0.06) X 10- 1
6.93 X 10°
(7.7±0.5) X 10°
(9.5±0.5) X 10 1
8.75X 10 1
(1.17±0.04) XJ03
1.08 X 103
4
(1.49±0.03) X 104
1.25 X 10
5%
7%
5%
3%
2%
106
105
105
105
105
256
128
128
128
128
50
100
100
100
100
(Q(T)n±2wn
'See Table IX for an explanation of rows and columns.
J. Chern. Phys., Vol. 98, No.6, 15 March 1993
Downloaded 22 Jul 2010 to 128.122.88.197. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
Topper
5002
et at.: Quantum steam tables
TABLE XI. Calculated vibrational-rotational partition functions for H 2S."
T(K)
Q!f~~~R
{fv~T~:R
{ti~Q;;,~R
(Q(T)n±2wn
600
1000
ISoo
2400
4000
1.27 X 10- 1
7.9SXlO°
9.70X 10 1
1.15 X 103
1.33 X 104
1.42 X 10- 1
8.60X 10°
1.04 X 102
1.24 X 103
1.4SX 104
1.42 X 10- 1
8.61 X 10°
1.04 X 102
1.26 X 103
1.4 X 104
(1.46±0.06) X 10- 1
(9.4±0.7) X 10°
(1.11 ±0.06) X 102
(1.27 ± 0.04) X 103
( 1.67 ± 0.04 ) X 104
FSE
4%
7%
5%
3%
2%
n
6
10
lOs
lOs
lOs
lOs
K
Na
256
128
128
128
128
SO
100
100
100
100
"See Table IX for explanations of rows and columns.
verged Fourier path-integral calculations for T.;;;2400 K.
This is very encouraging for their ab initio approach.
Empirical estimations of gef( T) are taken from the
JANAF tables28 and from Gurvich et al. 60 For the tabulation of Gurvich et aI., the vibrational partition function is
obtained by a summation of the form
m
where Em is evaluated from an expression like Eq. (42),
but including more terms, e.g., the Ymna(v m+!) (V n+!) (va
+!) terms, and where the constants were obtained from
experimental vibrational and rotational spectra. Note that
Eq. (52) does not assume separable vibrational and rotation. The rotational partition functions Qrot are calculated
using a rigid asymmetric top approximation with the
Stripp-Kirkwood correction and centrifugal distortion
corrections. The JANAF 28 tables are based on a similar,
but somewhat more complicated procedure. Both treatments 28 ,60 take account of Darling-Dennison resonance.
At all the temperatures, the results from both tables agree
very well with the Fourier path-integral calculations. However, the harmonic and perturbation theory results deviate
from the other results in Table XIII at 2400 K and higher.
The error in the perturbation results may be due to the
exclusion of the effect of centrifugal distortion on the rotational partition function since similar formalisms for the
vibrational partition functions are used in the perturbation
theory and empirical calculations, and the parameters (i.e.,
the frequencies, fundamentals, etc.) are almost the same in
the perturbation theory and empirical calculations.
In the case of D 20, similar procedures as for H 20 were
used by Gurvich et al. 60 and in JANAF 28 tables. The conclusions we derive from the comparisons are also generally
similar to those for H 20. At 1500 and 2400 K, the results
from Gurvich et al. are not within the 95% confidence
range of the path-integral calculations, but if we estimate
the errors in their tabulation by interpolating their stated
errors at lower and higher temperatures, then the results
do agree with the sum of the error bars. A significant discrepancy between the path-integral results and the results
of the harmonic and the perturbation calculations occurs
only above 2400 K.
There is less data in the literature28,60,61 for the Gibbs
free energy functions for H 2S and H 2Se than for the previoustwQ cases. This is especially true in the case of H 2Se,
where no reliable estimation is available at high temperatures. Even where tabulated data exist at lower temperatures for this molecule, they were calculated from the entropy and the heat capacity (Cp ) which were derived from
molecular constants61 and are less accurate than the
present calculations since they are based on limited data.
The present results from both perturbation theories are
quite close to the path-integral calculations, while the results from the harmonic approach are not within the 95%
confidence range of the path-integral calculations at 1500
K and higher. It is reasonable to assume that the present
quantal results for H 2Se are the most accurate results available for this molecule.
VII. CONCLUSIONS
By producing accurate quantum mechanical partition
functions, including vibration-rotation coupling, modemode coupling, and anharmonicity, we have been able to
test the perturbation theory/classical rigid rotator model
for triatomics more systematically than before. We have
found that for H 20, D 20, H 2S, and H 2Se, this model is
very accurate for the calculation of vibrational-rotational
partition functions for temperatures in the range
600.;;;T.;;;1500 K.
The fact that the perturbation theory/rigid rotator
model gives such accurate values of the partition function
for H 20, D 20, H 2S, and H 2Se for the 600,;;;T,;;;1500 K
temperature range leads to the conclusion that a classical
approximation to the rotations does not produce apprecia-
TABLE XII. Calculated vibrational-rotational partition functions for H 2Se."
T(K)
~~(£~R
~r.;~~R
{tJ;Q~R
600
1000
1500
2400
4000
4.19X 10- 1
1.94 X 10 1
2.09 X 102
2.32XW
2.61 X 104
4.60XlO- 1
2.08 X 10 1
2.24X102
2.49 X 103
2.82X104
4.60xlO- 1
2.08X10 1
2.25x 102
2.5 X 103
2.6 X 104
(Q(T)n±2wn
1
(4.7±O.2) X 10(2.1 ±0.1) X 10 1
(2.4±0.1) X 102
(2.67±0.09) X 103
(3.33±0.08) X 104
FSE
n
K
Na
4%
5%
4%
3%
2%
106
256
128
128
128
128
50
100
100
100
100
lOS
lOS
lOS
105
"See Table IX for explanations of rows and columns.
J. Chem. Phys., Vol. 98, No.6, 15 March 1993
Downloaded 22 Jul 2010 to 128.122.88.197. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
Topper et al.: Quantum steam tables
5003
TABLE XIII. Quantum Gibbs free energy functions for H 2O.'
T
ge~nt(T)
geftrans( T)b
FPIc
gef(T)
FPlc,d
gefint(T)
MFGe
gef(T)
JANAF
gef(T)
(Ref. 60)
gef(T)
SPTf
gef(T)
PTg
gef(T)
HOh
138.7
149.3
157.7
167.5
178.1
40.7
47.6
53.8
63.7
76.6
179.4 :J:8:~
179.1
196.9
212.0
231.1
179.0
196.8
211.9
231.1
254.5
179.0
196.8
212.0
231.2
254.9
178.8
196.6
211.6
230.7
253.7
178.8
196.6
211.6
230.8
253.8
178.7
196.5
211.4
230.3
253.2
600
1000
1500
2400
4000
196.9:J:8:~
211.5:!:g:~
231.2:J:8:i
254.7:J:8:~
'See Eq. (51) for a definition of free energy functions. Units of gef( T) are J (mol K) -1.
'1'ranslational contribution at P= 1 bar= 100 000 Pa.
CObtained via AOSS-U Fourier path-integral Monte Carlo from the Kauppi-Halone surface.
dThe range of 95% confidence estimated from 2wn in Table I.X.
<From Ref. 25.
rObtained via simple vibrational perturbation theory from the Kauppi-Halonen surface.
'Obtained via vibrational perturbation theory from the Kauppi-Halonen surface.
hObtained via the harmonic oscillator model from the Kauppi-Ha1onen surface.
ble errors in the vibrational-rotational partition function
for these molecules in this particular temperature regime.
It should be noted that one could alternatively use more
sophisticated methods to estimate the rotational partition
function,24 but the accuracy achieved by using the classical
rigid rotator model seems adequate for thermochemical
purposes at least for the molecules studied here.
The improvement in the treatment of vibrations by
including anharmonicity through second order is very significant, while the harmonic oscillator/rigid rotator model
disagrees strongly with the Fourier path-integral calculations (by an average unsigned error of 16% overall, with
individual differences as large as 32% ), the vibrational perturbation theory/rigid rotator model is in good overall
agreement with the Fourier path-integral calculations
(with the average unsigned discrepancy being 7% overall
and with the largest individual difference being 22%). Further comparisons of this kind for a wider variety of systems
are necessary to address whether this conclusion may be
valid for other cases.
The Fourier path-integral Monte Carlo method used to
obtain the fully coupled quantal results is more slowly con-
vergent at lower temperatures, and so we used larger sample sizes at 600 K. However, the statistical properties of the
Monte Carlo method improve rapidly as the temperature is
increased, i.e., there is a decrease in the fractional statistical errors for a given sample size. As a result, in the same
temperature range where the perturbation theory/rigid rotator model appears to begin to break down, the Fourier
path-integral method performs extremely well, achieving
statistical errors on the order of 2%-3% or less, while
using as few as 105 samples. This behavior made it possible
to carry out extremely accurate calculations of the fully
coupled vibrational-rotational partition function in the
high-temperature (T;;;'2400 K) regime. It should be recognized that the present calculations, although they already yield accuracy comparable to or better than standard
thermochemical tables for interesting molecules, are still
just prototype applications of a new method. The Fourier
path-integral method can also be applied to floppy molecules, but the perturbation theory treatment is likely to
break down for such systems. We propose that it would be
very interesting to carry out studies of systems of this type
in the future. Furthermore, with improved sampling
TABLE XIV. Quantum Gibbs free energy functions for D 20, H 2S, and H 2Se. a
ge~nt(T)
D 20
H 2S
HzSe
T
geftrans ( T)
FPI
600
1000
1500
2400
4000
600
1000
1500
2400
4000
600
1000
1500
2400
4000
140.0
150.6
159.03
168.80
179.42
146.6
157.2
165.6
175.4
186.0
157.6
168.2
176.6
186.4
197.0
48.9
56.4
63.71
74.93
89.37
49.6
58.0
65.40
75.8
90.7
52.2
60.4
67.8
80.2
95.4
gef(T)
FPI
gef(T)
(JANAF)
gef(T)
(Ref. 60)
188.9:!:8:!
188.7
207.2
223.2
243.8
268.6
196.1
214.6
230.7
251.4
276.3
188.7
207.2
223.3
243.9
268.7
196.2
214.7
230.9
251.6
276.7
207.0:!:8:~
222.7:!:8:~
243.7:!:8j
268.8:!:8:~
196.2:!:8:~
215.2:!:8:~
231.0:!:8:!
251.2:!:8:~
276.7:!:g:~
209.8:!:8:l
228.6:!:8:~
245.6:!:8:~
266.6:!:g:~
292.4:!:8:~
gef(T)
(Ref. 61)
196.1
214.8
230.9
209.3
228.3
244.6
gef(T)
(PT)
gef(T)
(SPT)
gef(T)
(HO)
188.5
207.0
223.0
·243.4
267.7
196.0
214.5
230.5
251.2
275.2
209.6
228.5
245.0
266.1
290.3
188.5
207.0
223.0
243.4
267.7
195.9
214.5
230.5
251.0
275.5
209.6
228.5
245.0
266.0
291.0
188.5
206.9
222.7
243.0
267.3
195.9
214.3
230.3
250.6
274.9
209.6
228.4
244.7
265.6
290.4
aSee Table XIII for a description of columns. All calculations obtained using the appropriate Kauppi-Halonen energy surface.
J. Chern. Phys., Vol. 98, No.6, 15 March 1993
Downloaded 22 Jul 2010 to 128.122.88.197. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
5004
Topper et al.: Quantum steam tables
procedures-which are definitely possible and which will
be pursued in later work, we should be able to treat lower
temperatures and floppy molecules.
It would be interesting to compare the present directevaluation technique to indirect techniques based on parameter integration. 64 It is especially important to determine which method or combination of methods is most
efficient for more anharmonic many-dimensional systems.
nique. By this we mean that they are affordable but more
accurate than simpler theoretical models used in the past,
and they are competitive in accuracy with experimental
methods.
Because improvements in sampling efficiency are envisioned, we anticipate future applications to larger molecules as well.
ACKNOWLEDGMENTS
VIII. SUMMARY
We have carried out a series of calculations of quantum mechanical vibrational-rotational partition functions
for four triatomic molecules using Fourier path-integral
methods, fully including the effects of anharmonicity, vibrational mode-mode coupling, and vibrational-rotational
coupling on the quantum density matrix. We compared the
resulting partition functions over a range of temperatures
(600-4000 K) to approximate forms which assume the
separation of vibrational motions from rotational motions,
treating the rotations as a classical rigid rotator and the
vibrations in three different approximations: (i) the harmonic oscillator model; (ii) full second order perturbation
theory; and (iii) a recently presented "simple perturbation
theory" model, which uses the perturbation theory calculations for the zero-point energy and the fundamentals in a
functional form similar to that used for a set of independent normal modes. We find that for all the cases considered here, the harmonic model is in quantitative disagreement (as much as 30%) with the Fourier path-integral
calculations and with the perturbation theory calculations.
The two perturbation theory models are in excellent agreement with one another throughout the 600-2400 K temperature range with differences up to 6%-8% at 4000 K,
and they agree with the Fourier path-integral results
within the statistical error limits for almost all of the cases
considered up until the highest temperatures (2400 and
4000 K), at which small differences start to manifest themselves. Thus the accuracy of the perturbation theory/
classical rigid rotator treatment appears to be excellent
overall (with an average unsigned deviation of 7% from
the Fourier path-integral result) for the systems considered. Thus success of the simple perturbation theory calculation is especially encouraging since this method requires only a subset of the quartic force field and very little
computation beyond that.
For H 20, we studied the change in partition functions
when the potential modified at the cubic level in previous
work to improve the predictions for clusters is used instead
of the accurate isolated-molecule potential of Kauppi and
Halonen. The partition functions for H 20 and D 20 change
by about 10% at high temperature.
We also used the accurate and approximate partition
functions to compute free energy functions referenced to
the ground state, and we compared these to standard thermochemical tabulations, when available. The present
methods are found to yield accuracy comparable to or better than the available empirical data, so that quantum mechanical calculations must now be added to a statistical
thermodynamics toolbox as a seriously competitive tech-
We wish to thank Dr. Gregory Tawa and Dr. Xin Gui
Zhao for a number of helpful conversations. We also thank
Mr. Destin Jume11e-y-Picokens for help with the literature
search. This work was supported in part by the National
Science Foundation. Mr. Jumelle-y-Picokens was supported by the National Institutes of Health through its
Summer High School Minority Research Apprenticeship
program.
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J. Chern. Phys., Vol. 98, No.6, 15 March 1993
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