universidade de coimbra faculdade de ciˆencias e tecnologia · universidade de coimbra faculdade...

UNIVERSIDADE DE COIMBRA

Faculdade de Ciencias e Tecnologia

Departamento de Quımica

Accurate ab initio-based doublemany-body expansion potentialenergy surfaces and dynamics for

sulfur-hydrogen molecules

Yu Zhi Song

COIMBRA

2011

UNIVERSIDADE DE COIMBRA

Faculdade de Ciencias e Tecnologia

Departamento de Quımica

Accurate ab initio-based doublemany-body expansion potentialenergy surfaces and dynamics for

sulfur-hydrogen molecules

Dissertation presented for fulfillment

of the requirements for the degree of

“Doutor em Ciencias, especialidade em

Quımica Teorica”

Yu Zhi Song

COIMBRA2011

Dedicated to my family

Acknowledgments

First and foremost, I would like to express my sincere thanks to my supervisor,

Professor Antonio J. C. Varandas, for his inspirational instructions, patient guid-

ance and invaluable encouragement throughout my PhD study at the Department

of Chemistry, University of Coimbra, Portugal. I greatly admire and appreciate

his comprehensive knowledge of Theoretical Chemistry and perseverance attitude

toward scientific research. His effort, knowledge and attitude lead me step by step

to advance in my study, and finally make this dissertation possible.

Besides, I would like to thank all the members of the Theoretical and Com-

putational Chemistry (T&CC) Group. They have created a friendly and warm

atmosphere. Especially, I acknowledge Dr. Sergio P. J. Rodrigues, Dr. Pedro

J. S. B. Caridade and Dr. Luıs A. Poveda for their instructions, discussions

and comments. Much thanks should also be given to all the other members in

our group: Yongqing Li, Vinıcius C. Mota, Luıs P. Viegas, Breno R. L. Galvao,

Maikel B. Furones, Jing Li, Biplab Sarkar, Angel C. G. Fontes, Alexander Alijah,

and Flavia Rolim.

I deeply express my thanks to my MSc supervisor, Professor Chuankui Wang,

for his supervision and guidance during my MSc study at Shandong Normal

University. I take this opportunity to thank Professor Qingtian Meng and Shenglu

Lin. I managed to join the T&CC group with their recommendation. I am also

grateful to Professor Keli Han for his help during my study in Dalian.

I wish to thank the financial support from Fundacao para a Ciencia e a Tec-

nologia, Portugal, with the reference SFRH/BD/28069/2006.

Finally, in particular, I would like to thank my parents, my wife and my

daughter for their unlimited love, understanding, encouragement and countless

support all these years.

Contents

Acknowledgments v

Foreword 1

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

I Theoretical framework 7

1 Concept of potential energy surface 9

1.1 Adiabatic representation . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . 12

1.3 Crossing of adiabatic potentials . . . . . . . . . . . . . . . . . . . 13

1.4 Features of potential energy surface . . . . . . . . . . . . . . . . . 14

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Calculation and representation of potential energy surface 17

2.1 Hartree-Fock theory . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Derivation of the expectation value . . . . . . . . . . . . . 18

2.1.2 Derivation of the Hartree-Fock equation . . . . . . . . . . 21

2.2 Koopmans’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Configuration interaction method . . . . . . . . . . . . . . . . . . 24

2.4 Multiconfigurational SCF method . . . . . . . . . . . . . . . . . . 26

2.5 Multireference CI method . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Møller-Plesset perturbation theory . . . . . . . . . . . . . . . . . 30

2.7 Coupled cluster theory . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

viii Contents

2.8.1 Slater and Gaussian type orbitals . . . . . . . . . . . . . . 34

2.8.2 Classification of basis sets . . . . . . . . . . . . . . . . . . 36

2.8.3 Basis set superposition error . . . . . . . . . . . . . . . . . 37

2.9 Semiempirical correction of ab initio energies . . . . . . . . . . . . 38

2.9.1 Scaling the external correlation energy . . . . . . . . . . . 38

2.9.2 Extrapolation to complete basis set limit . . . . . . . . . . 39

2.10 Analytical representation of potential energy surface . . . . . . . . 42

2.10.1 The many-body expansion method . . . . . . . . . . . . . 43

2.10.2 The double many-body expansion method . . . . . . . . . 44

2.10.3 Approximate single-sheeted representation . . . . . . . . . 45

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Exploring PESs via dynamics calculations 57

3.1 Quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 The QCT method . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1 Unimolecular decomposition . . . . . . . . . . . . . . . . . 61

3.2.2 Bimolecular reaction . . . . . . . . . . . . . . . . . . . . . 61

3.3 Excitation function and rate constant . . . . . . . . . . . . . . . . 64

3.3.1 Reaction with barrier . . . . . . . . . . . . . . . . . . . . . 64

3.3.2 Barrier-free reaction . . . . . . . . . . . . . . . . . . . . . 65

3.4 Electronic degeneracy factor . . . . . . . . . . . . . . . . . . . . . 66

3.5 Products properties from QCT runs . . . . . . . . . . . . . . . . . 67

3.5.1 Relative velocity and translational energy . . . . . . . . . 68

3.5.2 Velocity scattering angle . . . . . . . . . . . . . . . . . . . 69

3.5.3 Internal energy . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5.4 Rotational angular momentum . . . . . . . . . . . . . . . . 70

3.5.5 Rotational and vibrational energies . . . . . . . . . . . . . 70

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

II Case Studies 77

4 CR-CC and MRCI(Q) studies for representative cuts of H2S 79

Contents ix

A comparison of single-reference coupled-cluster and multi-reference

configuration interaction methods for representative cuts of the

H2S(1A′) potential energy surface . . . . . . . . . . . . . . . . . . 81

5 Accurate DMBE/CBS PES for ground-state H2S 105

Accurate ab initio double many-body expansion potential energy surface

for ground-state H2S by extrapolation to the complete basis set limit107

Supporting Information . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6 Accurate DMBE/SEC PES for ground-state H2S 141

Potential energy surface for ground-state H2S via scaling of the external

correlation, comparison with extrapolation to complete basis set

limit, and use in reaction dynamics . . . . . . . . . . . . . . . . . 143


7 Accurate DMBE/CBS PES for ground-state HS2 173

Accurate DMBE potential energy surface for ground-state HS2 based

on ab initio data extrapolated to the complete basis set limit . . . 175


8 Conclusions and outlook 211

Mathematical appendices 213

Foreword

Atmospheric sulfur chemistry has played a significant role in the early atmosphere

of Earth [1, 2]. In particular, independent isotope fractionation studies are pro-

viding new insight into our understanding of the role that sulfur played in the

early Earth atmosphere [3–6]. Recent studies are also revealing that sulfur chem-

istry is important in the chemical evolution of the atmospheres of giant planets

such as Jupiter [13] and also in the atmosphere of ancient Mars [14]. Reduced

sulfur-containing molecules also show their importance in biochemistry [7–10] and

combustion chemistry [11, 12]. One of the major sulfur-bearing species present

in the atmosphere of the large planets is H2S. The earliest laboratory studies of

the photochemistry of H2S had HS as a major species resulting from the photo-

chemistry [15–17]. Secondary photodissociation of HS radicals has been observed

to produce S atoms [18]. Due to the important role that H2S plays in the various

areas of chemistry, it received much theoretical [19–22] and experimental [23–25]

consideration over the years. Moreover, the HS2 radical plays an important role

in a variety of environments, notably combustion and the oxidation of reduced

forms of sulfur [11, 12, 26, 27]. Amounts of investigation [28–34] have been carried

on HS2 both experimentally and theoretically since Porter [15] first proposed that

the HS2 radical was produced during the photolysis of HSSH. Thus, the model-

ing of accurate global potential energy surfaces (PESs) of H2S and HS2 molecular

systems, combined with dynamics studies, may enhance the understanding of the

atmospheric sulfur chemistry.

The PES of a molecule is a function of the relative positions of the nuclei

whose description is justified within the Born-Oppenheimer [35] separation. An

analytical representation of the PES is achieved using different formalisms, such

as the double many-body expansion (DMBE) method [36–38]. The latter consist

2 Foreword

of expanding the potential energy function of a given molecular system in terms of

the potential energies of its fragments. Information about a PES can be obtained

both from the analysis of experimental data and from ab initio calculations. At

present, robust theoretical frameworks and computational resources make possi-

ble to extensively explore the configuration space with the aim of constructing

accurate and global ab initio-based PESs.

The main goal of the present doctoral thesis is the construction of DMBE

PESs for the electronic ground-state H2S and HS2 molecular system, as well

as the studies of structure, energetics, and spectroscopy. The obtained PESs are

also used for exploratory quasi-classical trajectory calculations of the thermal rate

constants and cross sections of gas-phase reactions. The present PESs for H2S

and HS2 can be employed as building-blocks of DMBE PESs of larger molecular

systems, such as SH3 and H2S2, which contain the mentioned triatoms.

This thesis is divided in two parts. The first part concerns with the theo-

retical framework, while the case studies are presented in the second part. In

the first part, Chapter 1 presents the concept of PES. Chapter 2 gives a survey

of the ab initio methods and the formalisms used to construct analytical repre-

sentations of PES, while Chapter 3 deals with methods here employed to study

dynamics properties using the obtained PESs. In Chapter 4, we compared the re-

sults of the conventional CCSD, CCSD(T), the renormalized CR-CCSD(T), CR-

CCSD(TQ), CR-CC(2,3) and CR-CC(2,3)+Q calculations with the MRCI(Q)

results for the three important cuts of the H2S(X 1A′) PES. In Chapter 5, a

DMBE/CBS PES is reported for H2S(X 1A′) on the basis of a least-squares fit to

MRCI(Q)/AV(T,Q)Z energies which are extrapolated to the complete basis-set

(CBS) limit. While, a DMBE/SEC PES is presented in Chapter 6, which is ob-

tained from a least-squares fit to MRCI(Q)/AVQZ energies which are semiempir-

ically corrected by the DMBE scaled external correlation (DMBE-SEC) method.

Quasiclassical trajectory studies have been carried out on both PESs. In Chapter

7, a DMBE/CBS PES is reported for HS2(X 2A′′

) on the basis of a least-squares

fit to MRCI(Q)/AV(T,Q)dZ extrapolated to the CBS limit. Finally, the main

achievements are summarized and further possible applications are outlined in

Chapter 8.

Foreword 3

Bibliography

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[2] K. S. Habicht, M. Gade, B. Thamdrup, P. Berg and D. E. Canfield, Science

298, 2372 (2002).

[3] U. H. Wiechert, Science 298, 2341 (2002).

[4] J. Farquhar, B. A. Wing, K. D. McKeegan, J. W. Harris, P. Cartigny and

M. H. Thiemens, Science 298, 2369 (2002).

[5] J. Savarino, A. Romero, J. Cole-Dai, S. Bekki and M. H. Thiemens, Geophys.

Res. Lett. 30, 2131 (2003).

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Mundy, Astrophys. J. 428, 680 (1994).

[7] P. C. Jocelyn, Biochemistry of the SH-Group, (Academic Press, London,

New York, 1972).

[8] K. Abe and H. Kimura, J. Neurosci. 16, 1066 (1996).

[9] M. Whiteman, N. S. Cheung, Y.-Z. Zhu, S. H. Chu, J. L. Siau, B. S. Wong,

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794 (2004).

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[12] K. Sendt, M. Jazbec and B. S. Haynes, Proc. Combust. Inst. 29, 2439 (2002).

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4 Foreword

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[23] S.-H. Lee and K. Liu, Chem. Phys. Lett. 290, 323 (1998).

[24] J. D. Cox, D. D. Wagman and V. A. Medvedev, CODATA Keyvalues for

Thermodynamic (Hemispher, New York, 1984).

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Foreword 5

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A. Riganelli (Springer, Berlin, 2000), vol. 75, pp. 33–56.

Part I

Theoretical framework

Chapter 1

Concept of potential energysurface

Potential energy surfaces (PESs) play an important role in the application of

electronic structure methods to the study of molecular structures, properties and

reactivities [1–5]. The concept of a PES is a consequence of the separation of

the nuclear and electronic motions as proposed by Born-Oppenheimer approx-

imation [6]. The PES is a hyper surface defined by the potential energy of a

collection of atoms over all possible atomic arrangements [7], which has 3N − 6

coordinate dimensions, where N is the number of atoms (N ≥ 3). More detailed

discussion on PES can be found elsewhere [2–4, 8–10]. In the following, we review

the main ideas related to molecular PES.

1.1 Adiabatic representation

Given a molecular system, the stationary Schrodinger equation is written as:

Htot(R, r)Ψtot(R, r) = EtotΨtot(R, r), (1.1)

where Ψtot(R, r) and Etot are the eigenfunctions and eigenvalues of the molecular

system. Htot(R, r) is total electron-nuclei Hamiltonian, which can be written as

Htot(R, r) = Tn(R) + He(R, r) (1.2)

where Tn represents nuclear kinetic operator, He is the electronic Hamiltonian.

R and r are the nuclear and electron coordinates respectively. The electronic

10 Concept of potential energy surface

Hamiltonian, depending also on nuclear coordinates, can be written as

He(R, r) = Te(r) + Vee(r) + Ven(R, r) + Vnn(R) (1.3)

with Te being the electrons kinetic energy operator, Ven includes all electron-

nucleus interactions and Vnn stands for nuclei-nuclei interactions.

For a system with N nuclei and ne electrons, the above presented terms are

given by (using atomic units [11])

Tn = −N∑

k

(1

2Mk

)∇2

k ≡ ∇2n (1.4)

where Mk is the mass of the kth nucleus. We have here introduced the symbol of

∇2n, which includes the mass dependence, sign and summation.

Te = −1

2

ne∑

i

∇2i (1.5)

Vee =1

2

ne∑

i6=j

1

rij

(1.6)

Ven = −N∑

k

ne∑

i

Zk

|Rk − ri|(1.7)

Vnn =1

2

N∑

k 6=k′

ZkZk′

Rkk′

(1.8)

where Zk is the charge number of the kth nucleus, rij = |ri − rj|, and Rkk′ =

|Rk − Rk′|.Assume for the moment that all nuclei were fixed in the space, the motion of

the electrons would be governed by the electronic Schrodinger equation:

He(R, r)φi(R, r) = Ei(R)φi(R, r) (1.9)

where φi(R, r) and Ei(R) are the adiabatic eigenfunctions and eigenvalues of the

electrons with the fixed nuclear coordinates R as parameters, for a given ith

electronic state. The adiabatic eigenfunctions can be chosen to be orthogonal

and normalized (orthonormal) complete basis set

δij =

∫φ∗

i (R, r)φj(R, r)dR =

{1; i = j0; i 6= j

(1.10)

1.1 Adiabatic representation 11

The total wave function can then be written as an expansion in the com-

plete set of electronic adiabatic eigenfunctions [8, 12, 13], with the expansion

coefficients being functions of the nuclear coordinates.

Ψtot(R, r) =∞∑

i

ψi(R)φi(R, r) (1.11)

where ψi(R) is the nuclear wave function in the adiabatic representation. Sub-

stituting (1.11) into (1.1), making use of the expressions of the terms in the total

Hamiltonian Htot(R, r) as described above, the following coupled equations are

obtained,

∞∑

i

[Tn(R) + He(R, r)

]ψi(R)φi(R, r) = Etot(R, r)

∞∑

i

ψi(R)φi(R, r) (1.12)

considering the fact that φi(R, r) are the eigenfunctions of the electronic Schrodinger

equation (1.9) and orthonormal. If we multiply φ∗j(R, r) to (1.12) and integrate

over all the electron coordinates, the right term of (1.12) can be reduced to

∞∑

i

∫φ∗

j (R, r)Etot(R, r)ψi(R)φi(R, r)dr = Etot(R, r)ψj(R) (1.13)

while the second term in the left of (1.12) can be simplified as

∞∑

i

∫φ∗

j(R, r)He(R, r)ψi(R)φi(R, r)dr = Ej(R)ψj(R) (1.14)

Substituting (1.13), (1.14) and (1.4) into (1.12), one can obtain that (the coordi-

nate dependence is omitted for simplicity)

∞∑

i

∫φ∗

j Tnψiφidr + Ejψj = Etotψj

∞∑

i

∫φ∗

j∇2nψiφidr + Ejψj = Etotψj

∞∑

i

∫φ∗

j

[φi∇2

nψi + ψi∇2nφi

+2(∇nφi)(∇nψi)

]dr + Ejψj = Etotψj

∇2nψj + Ejψj +

∞∑

i

Λjiψj = Etotψj (1.15)


where Λji are the elements of the coupling matrix operator Λ, which arises from

the action of the nuclear kinetic energy operator Tn on the electron wavefunction

φi(R, r), given by:

Λji = 2Fji · ∇n +Gji (1.16)

where

Fji =

∫φ∗

j∇nφidr (1.17)

and

Gji =

∫φ∗

j∇2nφidr (1.18)

are the first- and second-order non-adiabatic coupling elements, which are respon-

sible for non-adiabatic transitions. The direct calculation of the nonadiabatic cou-

pling matrix is usually a very difficult task in quantum chemistry. However, what

makes the adiabatic representation so powerful is the use of adiabatic approxi-

mation [14] in which the off-diagonal couplings Λji(i 6= j) are discarded. This

approximation is based on the rationale that the nuclear mass is much larger

than the electron mass, and therefore the nuclei move much slower than the elec-

trons. Thus the nuclear kinetic energies are generally much smaller than those of

electrons and consequently the nonadiabatic coupling matrices in (1.15), which

result from nuclear motions, are generally small. Thus, we obtain the adiabatic

approximation for the nuclear wavefunction

(Tn + Ej(R) + Λjj

)ψj(R) = Etotψj(R) (1.19)

where Λjj = 2Fjj · ∇n +Gjj is the diagonal correction.

1.2 Born-Oppenheimer approximation

In the Born-Oppenheimer approximation (BOA) [6, 15], the diagonal correction

term Λjj in (1.19) is neglected, as it is smaller than Ej(R) by a factor roughly

equal to the ratio of the electronic and nuclear masses, which is usually very small

(even for a H atom, the ratio is ∼ 5 × 10−4). Thus, (1.19) takes the following

form, where the electronic energy plays the role of a potential energy.

(Tn + Ej(R)

)ψj(R) = Etotψj(R) (1.20)

1.3 Crossing of adiabatic potentials 13

Until now, we achieved a complete separation of electronic motion from that

of nuclei in the adiabatic BOA, by first solving for the electronic eigenvalues

Ej(R) at given nuclear geometries and then the nuclear dynamics problem for the

nuclei that move on a PES [16], which is a solution to the electronic Schrodinger

equation (1.9). The general criterion for the validity of the adiabatic BOA is that

the nuclear kinetic energy be small relative the energy gaps between electronic

states such that the nuclear motion does not cause transitions between electronic

states.

1.3 Crossing of adiabatic potentials

The adiabatic potentials Ej(R) can sometimes cross or come near each other at

some nuclear configurations. This corresponds to the case of degeneracy or quasi

degeneracy of electronic states. In order for adiabatic potentials to cross, certain

conditions must be satisfied. Usually, the crossing of adiabatic potentials that

belong to the same electronic symmetry can only occur at nuclear configurations

that correspond to certain symmetries of the molecular configuration. This does

not apply to adiabatic states that have different symmetries [4]. The following

is a simple heuristic derivation of the condition for the crossing of two adiabatic

potential curves which are of the same symmetry.

For a coupled two state problem, let ψ1 and ψ2 be the wavefunctions of two

electronic states which have the same symmetry and spin. We assume that these

two wave functions can be written as a linear combination of two orthonormal

basis functions ψa and ψb. In that case, the energies of the two states are the

eigenvalues of the 2 × 2 hermitian matrix

(Haa Hab

Hab Hbb

)(1.21)

By diagonalizing the matrix operator H , we obtain the adiabatic potentials

V± =Haa +Hbb

2± 1

2

√(Haa −Hbb)2 + 4|Hab|2 (1.22)

and the gap between the two adiabatic potentials is given by

∆V = V+ − V− =√

(Haa −Hbb)2 + 4|Hab|2 (1.23)


For two adiabatic potentials to cross, the two positive terms in (1.23) must satisfy

the following equations simultaneously

{Haa(R) = Hbb(R)Hab(R) = 0

(1.24)

If ψa and ψb have different symmetries then Hab will be zero for all values of R.

In that case there may be a point or points at which (1.24) is satisfied, i.e. the

energies of two states are equal. These points will then be crossing points of the

potential energy curves [2, 17, 18].

1.4 Features of potential energy surface

The stationary points are the most important features of a PES, which have zero

gradient components (∂V/∂ρk = 0). These stationary points can be of several

types, depending on the second derivatives of the potential energy. The second

order derivatives of the potential energy at a stationary point can be expressed,

in terms of the internal coordinates ρi, as the (3N−6)× (3N−6) Hessian matrix

with elements defined by

∂2V/∂ρi∂ρj (1.25)

If all eigenvalues of the Hessian matrix are positive, the stationary point is a

minimum on the surface, at which an infinitesimal step in any direction leads

to an increase in potential energy. The minimum may correspond to reactants,

products or intermediates. Likewisely, maxima on the surface has all the negative

eigenvalues of the Hessian matrix, at which an infinitesimal step in any direction

leads to a decrease in potential. However, maxima are not generally of special

physical significance.

Saddle points are of particular interest in chemical kinetics because they lie

on the paths between points on the surface identified with reactant molecules

and points on the surface identified with product molecules. A saddle point is

the highest point on the path which involves the lowest increase in the potential

on passing from reactants to products. Stationary points also exist which have

more than one negative curvature along the principal axes. If there are k negative

second order derivative, then it is called k-th order saddle point. However, these

1.4 Bibliography 15

do not generally have any special kinetic significance because once a transition

state has been located, it should be verified that it indeed connects the desired

minima. At the saddle point, the vibrational normal coordinate associated with

the imaginary frequency is the reaction coordinate, and an inspection of the

corresponding atomic motion may be a strong indication that it is the correct

transition state.

Many methods are developed to locate stationary points on PESs, such as

Refs. [19–24] and the references cited therein.

Bibliography

[1] A. J. C. Varandas, Conical Intersections Electronic Structure, Dynamics and

Spectroscopy (World Scientific, 2004), chap. Modeling and interpolation of

global multi-sheeted potential energy surfaces, p. 205.

[2] J. N. Murrell, S. Carter, S. C. Farantos, P. Huxley, and A. J. C. Varandas,

Molecular Potential Energy Functions (Wiley, Chichester, 1984).



[4] J. Z. H. Zhang, Theory and Applications of Quantum Molecular Dynamics

(World Scientific, Singapore, 1999).

[5] A. J. C. Varandas, Int. Rev. Phys. Chem. 19, 199 (2000).


[7] C. J. Cramer, Essentials of Computational Chemistry: Theories and Models

(John Wiley & Sons, 2004).

[8] A. S. Davidov, Quantum Mechanics (Pergamon, Oxford, 1965).

[9] H. Eyring and S. H. Lin, in Physical Chemistry, An Advanced Treatise, Vol.

VIA, Kinetics of Gas Reactions, edited by H. Eyring, D. Henderson, and

W. Jost (Academic, New York, 1974), p. 121.


[10] R. Jaquet, Potential Energy Surfaces (Springer, 1999), chap. Interpolation

and fitting of potential energy surfaces, Lecture Notes in Chemistry.

[11] M. Piris, Fısica Cuantica (Editorial ISCTN, La Habana, 1999).

[12] F. Jensen, Introduction to Computational Chemistry (John Wiley & Sons,

2007).

[13] A. Messiah, Quantum Mechenics (Wiley, New York, 1966).

[14] M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon

Press, Oxford, 1954).

[15] A. C. Hurley, Introduction to the Electron Theory of Small Molecules (Aca-

demic Press: London, 1976).

[16] N. C. Handy and A. M. Lee, Chem. Phys. Lett. 252, 425 (1996).

[17] C. A. Mead, J. Chem. Phys. 70, 2276 (1982).

[18] H. C. Longuet-Higgins, Proc. R. Soc. Ser. A 344, 147 (1975).

[19] H. B. Schlegel, Ab Initio Meth. Quant. Chem. I, 249 (1987).

[20] T. Schlick, Rev. Comput. Chem 3, 1 (1992).

[21] M. L. McKee and M. Page, Rev. Comput. Chem 4, 35 (1993).

[22] R. Fletcher, Practical Methods of Optimization (Wiley, Chichester, 1987).

[23] J. D. Head, B. Weiner, and M. C. Zerner, Int. J. Quantum Chem. 33, 177

(1988).

[24] K. Bondensgard and F. Jensen, J. Chem. Phys. 104, 8025 (1996).

Chapter 2

Calculation and representation ofpotential energy surface

The potential energy surface (PES), which is indeed a function obtained by fitting

to the ab initio energies, should describe the molecular energy as the internuclear

distance is changing continuously. Ab initio energies used to map a PES can be

gathered by solving the electronic problem represented in (1.9). Methods aimed

at solving (1.9) are broadly referred to the electronic structure calculation [1] and

significant advances have been made over many years [2] in the accurate ab initio

evaluation of the molecular energy within the Born-Oppenheimer approximation

(BOA) [3, 4]. The most common type of ab initio calculation is the Hartree-Fock

(HF) calculation [5, 6]. At higher levels of approximation, the quality of the wave

function is improved, so as to yield more and more elaborate solutions.

A large number of ab initio methods are available in many package program

to perform high level electronic structure calculations, such as Gaussian03 [7],

GAMESS [8] and Molpro [9]. In the following sections, a brief discussion of ab

initio methods, adopted for the calculation of PESs is presented.

2.1 Hartree-Fock theory

This section is divided into two parts. In the first part, the formula for the

expectation value E = 〈ΦA|He|ΦA〉 is deduced for the case in which ΦA is a

Slater determinant. While in the second part we derive the HF equation by

minimizing the expectation value E. Detailed treatments of the derivation of the

18 Calculation and representation of potential energy surface

HF equations may be found in Refs. [5, 6].

2.1.1 Derivation of the expectation value

HF theory is one of the simplest approximate theories for solving the many-body

Hamiltonian. For a molecular system consisting of N electrons and Nn nuclei,

the electronic Hamiltonian for a fixed nuclear configuration can be written as

He =

N∑

i

h(i) +1

2

∑

i6=j

1

rij(2.1)

where the second term represents the electron-electron interaction Vee and the

one electron Hamiltonian is given by:

h(i) = −1

2∇2

i +Nn∑

k

Zk

rik(2.2)

where rik = |ri − Rk|, rij = |ri − rj| and Zk is the electric charge of the kth

nucleus. Thus, (1.9) becomes:

(N∑

i

h(i) +1

2

∑

i6=j

1

rij

)φn(r) = Enφn(r) (2.3)

where the dependence of h(i), En and φn on R has been omitted for clarity and

nuclear-nuclear interactions have also been excluded.

It is not feasible to solve (2.3) exactly, as it is a complex many body problem.

Thus, usually further approximations are needed. One of the most important ap-

proximations in solving the electron problem is the HF approximation. The basic

idea of the HF approximation is as follows. It is well known that we can get the

exact solution of the electronic problem for the simplest atom, hydrogen, which

has only one electron. We imagine that if we add another electron to hydrogen, to

obtain H−. Assuming that the electrons do not interact with each other (i.e., that

Vee = 0), then the Hamiltonian would be separable. Thus, the total electronic

wavefunction φ(r1; r2) describing the motions of the two electrons can be written

as the product of two hydrogen atom wavefunctions, ψH(r1)ψH(r2). In the same

way, for the general problem, the electron wavefunction φ is approximated by the

2.1 Hartree-Fock theory 19

symmetrized Hartree product of one-electron spin orbitals

Φ =

N∏

k=1

ψk(k) = ψ1(1)ψ2(2) · · ·ψN(N) (2.4)

where the spin orbital ψk(k) = φkχs is defined as the product of the spatial

wavefunction φk(r) and the spin wavefunction χs of the kth electron. The spin

orbital ψk(k) is chosen to be orthonormal 〈ψk|ψk′〉 = δkk′.

In order to produce the antisymmetry property of electron, an antisymmetry

operator A needs to act on the spin orbital

ΦA = A

N∏

k=1

ψk(k) (2.5)

where the antisymmetry operator A takes the following form

A =1√N !

∑(−1)αP (2.6)

where P is the permutation operator and the summation is over all the electron

permutations. A fulfills the following relationships:

A2 =√N !A, A† = A, A(

1

rij) = (

1

rij)A, Ah(i) = h(i)A (2.7)

Thus, (2.5) can be written in the form of Slater determinant:

ΦA =1√N !

∣∣∣∣∣∣∣∣∣∣∣∣

ψ1(1) ψ1(2) · · · ψ1(N)ψ2(1) ψ2(2) · · · ψ2(N)

· · · ·· · · ·· · · ·

ψN (1) ψN (2) · · · ψN(N)

∣∣∣∣∣∣∣∣∣∣∣∣

=1√N !

det[ψ1ψ2 · · ·ψN ] (2.8)

The expectation value of the Hamiltonian in the HF approximation can be written


as (using (2.5) and (2.7))

E = 〈ΦA|He|ΦA〉 = 〈AN∏

k=1

ψk(k)|He|AN∏

k=1

ψk(k)〉

=√n!〈

N∏

k=1

ψk(k)|He|AN∏

k=1

ψk(k)〉

=∑

α

(−1)α〈N∏

k=1

ψk(k)|He|PN∏

k=1

ψk(k)〉 (2.9)

For the one-electron operator, all matrix elements involving a permutation op-

erator gives zero, since all the spin orbitals ψk are normalized. Thus, only the

identity operator can give a non-zero contribution. For coordinate i this yields a

matrix element over orbital i.

εi = 〈N∏

k=1

ψk(k)|hi|PN∏

k=1

ψk(k)〉 = 〈i|hi|i〉 (2.10)

For the two-electron operator, only the identity and Pij operators can give nonzero

contributions. A three-electron permutation will again give at least one overlap

integral between two different spin orbitals, which will be zero.

〈ΦA|1

rij|ΦA〉 = 〈ψi(i)ψj(j)|

1 − Pij

rij|ψi(i)ψj(j)〉

= Jij −Kij (2.11)

The Jij matrix element is called a Coulomb integral, which is the classical elec-

trostatic energy. It represents the classical repulsion between two charge distri-

butions described by ψ2i (i) and ψ2

j (j). Jij matrix element is written as

Jij = 〈ψi(i)ψj(j)|1

rij|ψi(i)ψj(j)〉 = 〈ij| 1

rij|ij〉 (2.12)

The Kij matrix element is called an exchange integral, and has no classical anal-

ogy. It is a consequence of the fermionic character of the electrons.

Kij = 〈ψi(i)ψj(j)|Pij

rij|ψi(i)ψj(j)〉 = 〈ij| 1

rij|ji〉 (2.13)

2.1 Hartree-Fock theory 21

Substituting (2.10) – (2.13) into (2.9), one can obtain the expectation value of

the Hamiltonian

E =

N∑

i

hi +1

2

N∑

ij

[Jij −Kij] (2.14)

This is the desired expression for E in terms of integrals over the spin orbital ψi

for a single determinant wave function.

2.1.2 Derivation of the Hartree-Fock equation

For the purpose of deriving the variation of the expectation value E, it is conve-

nient to express the energy in terms of Coulomb (J) and exchange (K) operators.

Ji(1)|ψj(1)〉 = 〈i| 1

r12|i〉|ψj(1)〉 =

[∫ψ∗

i (2)1

r12ψi(2)dτ2

]ψj(1) (2.15)

Ki(1)|ψj(1)〉 = 〈i| P12

r12|j〉|ψj(1)〉 =

[∫ψ∗

i (2)1

r12ψj(2)dτ2

]ψi(1) (2.16)

Substituting (2.15) and (2.16), the expectation value in (2.14) becomes

E =

N∑

i

〈ψi|hi|ψi〉 +1

2

∑

ij

(〈ψj |Ji|ψj〉 − 〈ψj |Ki|ψj〉) (2.17)

The wave function that makes the energy a minimum or at least stationary can

be determined by minimizing the expectation value in (2.17) with respect to

variation of one-electron spin orbitals δψk, with the Lagrange multipliers λij [10]

and the orthogonality condition. We define the Lagrange function as

L = E −∑

ij

λij(〈ψi|ψi〉 − δij) (2.18)

The Lagrange function is stationary with respect to an orbital variation

δL = δE −∑

ij

λij(〈δψi|ψi〉 + 〈ψi|δψi〉) = 0 (2.19)


where the variation of the energy is given by

δE =

N∑

i

(〈δψi|hi|ψi〉 + 〈ψi|hi|δψi〉) +

∑

ij

(〈δψj |Ji −Ki|ψj〉 + 〈ψi|Jj −Kj |δψi〉)

=N∑

i

(〈δψi|Fi|δψi〉 + 〈ψi|Fi|δψi〉) (2.20)

where Fi is the Fock operator written as

Fi = hi +

N∑

i

(Ji −Ki) (2.21)

Making use of (2.19) to (2.21) and 〈ψ|δψ〉 = 〈δψ|ψ〉∗ and 〈ψ|F |δψ〉 = 〈δψ|F |ψ〉∗,it is not difficult to derive the following HF equation

Fiψi =N∑

j

λijψj (2.22)

The equations above may be simplified by choosing a unitary transformation that

makes the matrix of Lagrange multipliers diagonal (i.e. λij = 0 and λii = ǫi).

These spin orbitals ψ′ are called canonical spin orbitals and can be constructed

from a unitary transformation of ψ.

ψ′i =

∑

j

ψiUji (2.23)

where Uij is the matrix of the unitary transformation. The antisymmetrized

wavefunction ΦA is invariant with respect to any unitary transformation of spin

orbitals, since

Φ′A = det|Ψ′| = det|U†ΨU| = det|Ψ| = ΦA (2.24)

The HF equation of (2.22) then forms a set of pseudo-eigenvalue equations

Fiψ′i =

N∑

i

ǫiψ′i (2.25)

2.2 Koopmans’ theorem 23

A set of functions that are a solution to (2.25) are called self-consistent field

(SCF) orbitals. The orbital energy ǫi derived from the above equations is

ǫi = 〈ψ′i|ǫi|ψ′

i〉 = hi +

N∑

j

(Jij −Kij) (2.26)

The total energy can be written either as (2.9) or in terms of HF orbital energies

ǫi

E =N∑

i

ǫi −1

2

N∑

ij

(Jij −Kij) (2.27)

The total electronic energy is not simply a sum of HF orbital energies. The Fock

operator contains terms describing the repulsion to all other electrons and the

sum over spin orbital energies, therefore counts the electron-electron repulsion

twice which must be corrected for.

Although HF theory often gives useful and even accurate results for quantities

like equilibrium geometries of molecules, it neglects correlation between electrons

by assuming a single-determinant form for the wavefunction. The electrons are

subject to an average potential arising from the other electrons, which neglects

the instantaneous or correlated motions of electrons. It is useful to define the

difference between the exact energy of the electron system and HF energy as

electron correlation energy.

2.2 Koopmans’ theorem

It is still possible to relate ǫi to physical measurements, although the fact that

the total energy is not given by the sum of HF orbital energies. If certain assump-

tions are made, we are able to equate orbital energies with molecular ionization

energies or electron affinities. This identification is related to a theorem due to

Koopmans [11]. For a neutral molecular containing N electrons, the total energy

in the HF approximation is given by (2.27) and we write it again here.

ENg =

N∑

i

ǫi −1

2

N∑

ij

(Jij −Kij) (2.28)


If one electron is removed from the kth orbital, the remaining N − 1 electrons

remain unchanged, the HF energy for this N − 1 electron system is given by

EN−1k =

N∑

i6=k

ǫi −1

2

N∑

i6=k,j 6=k

(Jij −Kij)

= ENg − hk −

1

2

N∑

i

(Jik −Kik) − 1

2

N∑

j

(Jkj −Kkj)

= ENg −

[hk +

N∑

j

(Jkj −Kkj)

](2.29)

because Jik = Jki and Kik = Kki. Subtracting the two total energies given by

(2.28) and (2.29), we can obtain the ionization energy (IE) [12] from the kth

orbital

IE = ENg − EN−1

k = hk +

N∑

j

(Jkj −Kkj) = εk (2.30)

As seen from (2.27), this is exactly the orbital energy εk. Similarly, the elec-

tron affinity (EA) of a neutral molecule is given as the orbital energy of the

corresponding anion, i.e. the energy for adding an extra electron to the sth

unoccupied orbital

EA = EN+1s −EN

g = εs (2.31)

(2.30) and (2.31) are the result known as Koopmans’ theorem [11], which gives

physical meaning to orbital energies and thus a means of calculation approximate

ionization energies and electron affinities. However, the theorem is very approxi-

mate [13]. First, the Koopermans’ theorem assumes that spin orbitals are frozen

after losing or adding an electron. In reality, the spin orbitals will relax and the

optimized orbitals will be different from the original ones after losing or adding an

electron. Secondly, the Koopermans’ theorem is based on the HF approximation

and neglects electron correlations.

2.3 Configuration interaction method

Configuration interaction (CI) [14] is one of the most general ways to improve

upon HF theory by adding a description of the correlations between electron

2.3 Configuration interaction method 25

motions. CI uses a variational wave function that is a linear combination [15] of

configuration state functions (CSFs) built from HF spin orbitals

Φ =∑

k

ckΦk (2.32)

In order to keep track of all the possible HF orbitals, we often write the

ground-state HF wave function as Φ0, the Slater determinant with an electron

“excited” from the ith occupied orbital to the ath unoccupied orbital as Φai , the

“doubly-excited” Slater determinants as Φabij , etc.. With this notation, we can

rewrite the wave function in (2.32) as

Φ = c0Φ0 +

N∑

i=1

M∑

a=N+1

cai Φai +

N∑

i>j=1

M∑

a>b=N+1

cabij Φab

ij + · · · (2.33)

where N is the number of electrons, so M is the total number of the HF orbitals.

Sometimes, it is helpful to abbreviate the indices on the Slater determinants

by introducing vectors, i and a, whose components are the orbitals from which an

electron is removed (i1, i2, . . .) and the orbitals to which it is excited (a1, a2, . . .),

respectively. By convention, the (2.33) is written as the form

Φ = c[1,2,...N ][1,2,...N ]Φ

[1,2,...N ][1,2,...N ] +

N∑

i1=1

M∑

a=N+1

c[a1,i2,...iN ][i1,i2,...iN ] Φ

[a1,i2,...iN ][i1,i2,...iN ]

+N∑

i1>i2=1

M∑

a1>a2=N+1

c[a1,a2,...iN ][i1,i2,...iN ] Φ

[a1,a2,...iN ][i1,i2,...iN ] + · · · =

∑

i,a

cai Φai (2.34)

To compute the configuration interaction wave function, we start with the

Shrodinger equation HΦ = EΦ. We then left multiply a Slater determinant Φai

to the both sides of the Shrodinger equation

Φai HΦ = Φa

iEΦ (2.35)

Substituting the (2.34) into the above equation, one gets

Φai H∑

j,b

cbjΦbj = Φa

iE∑

j,b

cbjΦbj (2.36)

and integrate the (2.36), we can obtain∑

j,b

〈Φai |H|Φb

j〉cbj = E∑

j,b

〈Φai |Φb

j〉cbj = E∑

j,b

δijδabcbj = Ecai (2.37)


If we define the Hamiltonian matrix as Ha,b

i,j = 〈Φai |H|Φb

j〉, then the CI procedure

leads to a general matrix eigenvalue equation and (2.37) becomes

∑

j,b

Ha,b

i,j cbj = Ecai (2.38)

The solution of the CI procedure are the eigenvalues E and their corresponding

eigenvectors cai .

Configuration interaction calculations are classified by the number of excita-

tions used to make each determinant. When we truncate at zeroth-order, we have

the HF approximation [5, 6]. At first order, we have only one electron has been

moved for each determinant, it is called a Configuration Interaction with Single

excitation (CIS) [1]. CIS calculations give an approximation to the excited states

of the molecule, but do not change the ground state energy. At second order, we

have Configuration Interaction with Single and Double excitation (CISD) [1, 16]

yielding a ground-state energy that has been corrected for correlation. Triple-

excitation (CISDT) [1] and quadruple-excitation(CISDTQ) [1] calculations are

done only when very high accuracy results are desired. When we include all

possible excitations, we say that we are doing a Full Configuration Interaction

calculation, which is called Full CI (FCI) [15]. The number of all the possible ex-

citations is given by the number of determinants in an FCI wave function, which

has the following expression [12]

Ntot =

(M

N

)=

M !

N !(M −N)!(2.39)

For sufficiently large M , the FCI calculation will give an essentially exact result.

However, full CI calculations are rarely done due to the immense amount of

computer power required. For most applications, the CIS and CISD can give

good description of the electronic correlation energy.

2.4 Multiconfigurational SCF method

The Multiconfigurational Self Consistent Field (MCSCF) [17–19] method can be

considered a combination between the CI method (where the molecular orbitals

are not varied but the expansion of the wave functions) and the HF approximation

2.5 Multireference CI method 27

(where there is only one determinant but the molecular orbitals are varied). Si-

multaneous optimization of two sets of parameters is a difficult nonlinear problem,

and in practice severely restricts the length of the MCSCF expansions relative to

those of CI wave functions. Then, a compromise appears between generation of a

configuration space sufficiently flexible to describe the molecular system and the

number of variables to be computationally tractable.

A successful approach to select the MCSCF configurations is to partition

the molecular orbital space into three subspaces, containing inactive, active and

virtual (or unoccupied) orbitals respectively, which is known as the complete

active space self-consistent field (CASSCF) method [18, 19]. Typically, the core

orbitals of the system are treated as inactive and the valence orbitals as active.

Thus, the complete active space (CAS) consists in all configurations obtained

by distributing the valence electrons in all possible ways in the active orbitals,

keeping the core orbitals doubled occupied in all configurations, which is usually

called full valence complete active space (FVCAS) [19]. The configuration so

obtained is often referred to as reference configuration and the corresponding

space spanned is called the reference space.

In a CASSCF wave function, a part of the electronic correlation is covered,

called static or nondynamical correlation which arise from the strong interaction

between configurations nearly degenerated and is unrelated to the instantaneous

repulsion between the electrons. This last energy contribution constitutes the dy-

namical correlation energy. For high accuracy treatment of dynamical correlation,

additional calculations must be carried out based on the initial MCSCF method,

such as multirefernce CI (MRCI) method [20–24], which has been extensively

employed in this thesis.

2.5 Multireference CI method

The multireference configuration interaction (MRCI) method is a powerful one

to calculate accurate PESs. The general form of the MRCI method is MRCISD,

which includes only all the single and double excitations, i.e., neglects configura-

tions with more than two electrons in external orbitals. Its wavefunction can be


written as [25]

|Ψ〉 =∑

I

CI |ΨI〉 +∑

S

∑

a

CSa |Ψa

S〉 +∑

P

∑

ab

CPab|Ψab

P 〉 (2.40)

where a and b refer to external orbitals, i.e., those not occupied in the reference

configurations, I denotes an orbital configuration with N electrons in the internal

orbital space while S and P denote internal N − 1 and N − 2 electronic hole

states [25–27]. |ΨI〉, |ΨaS〉 and |Ψab

P 〉 are internal, singly external and doubly

external configurations containing 0, 1, 2 occupied external orbitals, respectively.

Since there are usually many more external orbitals than internal ones, the

double external CSF’s |ΨabP 〉 are the most numerous in (2.40). If this number

is denoted as NP and the number of external orbitals denoted as N , then the

number of operations per iteration is proportional to NPN4 + Nx

pN3 [25], where

1 < x < 2. For this reason, it is difficult to perform uncontracted MRCI calcu-

lations with large reference configuration spaces (and large basis sets) which is

generated by two electron excitations from each individual reference configura-

tion. In order to reduce the computational effort, different contraction schemes

have been proposed [25, 28–31]. In the hybrid internally contracted MRCI (ICM-

RCI) [23, 25, 32], the internal configurations Ψklij and singly external configura-

tions Ψkaij are not contracted, while the doubly external configurations Ψab

ij are

contracted.

Using the configuration basis defined above, the total wavefunction maybe

written as [33]

|Ψ〉 =∑

I

CI |ΨI〉 +∑

S

∑

a

CSa |Ψa

S〉 +∑

ω=±1

∑

ab

∑

t≥u

Ctu,ωab |Ψab

tu,ω〉 (2.41)

where Ctu,ωab = ωCtu,ω

ba and the internally contracted doubly external configurations

are defined as

|Ψabtu,ω〉 =

1

2

(Eat,bu + ωEbt,au

)|Ψref〉 (2.42)

where ω = 1 for external singlet pairs and ω = −1 for external triplet pairs, and

|Ψref〉 is a reference wavefunction, which may be composed of many configurations

|Ψref〉 =∑

R

αR|ΨR〉 (2.43)

2.5 Multireference CI method 29

The internally contracted configurations |Ψabtu,w〉 can be expanded in terms of

the set of standard uncontracted doubly external CSFs |Ψabp 〉 according to

|Ψabtu,ω〉 =

∑

P

〈ΨabP |Ψab

tu,ω〉|ΨabP 〉 (2.44)

where the contraction coefficients are given by

〈ΨabP |Ψab

tu,ω〉 =1

2

∑

R

αR〈ΨabP |Eat,bu + ωEbt,au|ΨR〉 (2.45)

showing that these configurations are obtained by contracting different internal

states.

The configurations in (2.42) can be orthogonalized using the overlap matrix

S(ω) with its elements given by

S(ω)tu,rs = 〈Ψref |Etr,us + ωEts,ur|Ψref〉 (2.46)

so that the new orthonormal basis is obtained as

|ΨabD,ω〉 =

∑

t≥u

T(ω)D,tu|Ψab

tu,ω〉 T(ω) = (S(ω))−1/2 (2.47)

Substituting (2.47) into (2.41), the wavefunction becomes

|Ψ〉 =∑

I

CI |ΨI〉 +∑

S

∑

a

CSa |Ψa

S〉 +∑

ω=±1

∑

ab

∑

D

CD,ωab |Ψab

D,ω〉 (2.48)

This representation of the wavefunction is equivalent to (2.41), with D denotes

the orthogonalized internally contracted N − 2 electron states. The Hamiltonian

matrix can be diagonalized by using the popular procedure of Davidson [34–36],

which relies upon the formation of residual vectors that can then be used to

generate an updated vector of CI expansion coefficients. The residual vector can

be expressed as

〈ΨabD,ω|H −E|Ψ〉 =

{1

2

[GD,ω + ω(GD,ω)†

]−ECD,ω

}

ab

(2.49)

〈Ψas |H −E|Ψ〉 = (gs − ECs)a (2.50)

〈Ψas |H − E|Ψ〉 = gI −ECI (2.51)

The explicit formulas for the quantities GD,ω, gs and gI can be found in Refs. [23,

34, 37], which are calculated using an efficient direct CI method [34, 38].


2.6 Møller-Plesset perturbation theory

In order to apply perturbation theory [39] to the calculation of the correlation

energy, the unperturbed Hamiltonian must be selected. The most common choice

is to take this as a sum over Fock operators [12], leading to Møller Plesset (MP)

perturbation theory [40–42], within which the unperturbed Hamiltonian is written

as

H0 =

N∑

i=1

Fi =

N∑

i=1

(hi +

N∑

j=1

gij

)=

N∑

i=1

hi +

N∑

i=1

N∑

j=1

gij (2.52)

where we write gij = (Jij − Kij) for simplicity. The perturbed Hamiltonian can

be expressed as

H =

N∑

i=1

hi +

N∑

i=1

N∑

j>i

gij (2.53)

So, we can write the perturbation as the difference between the perturbed and

unperturbed Hamiltonians.

H ′ = H − H0 =

N∑

i=1

N∑

j>i

gij −N∑

i=1

N∑

j=1

gij = −1

2

N∑

i=1

N∑

j=1

gij (2.54)

which is the difference between the instantaneous and average electron-electron

interaction. This perturbation is sometimes called the fluctuation potential as one

imagine that it measures the deviation from the mean of the electron interaction.

The zeroth-order wavefunction is the HF determinant, and the zeroth-order

energy is just a sum of one electron energies of the occupied spin orbitals

E(0) = 〈Ψ0|H0|Ψ0〉 = 〈Ψ0|N∑

i=1

Fi|Ψ0〉 =

N∑

i=1

ǫi (2.55)

The first order correction to the energy is the average of the perturbation over

the unperturbed wavefunction, which is given by

E(1) = 〈Ψ0|H ′|Ψ0〉 = −1

2

N∑

i=1

N∑

j=1

〈Ψ0|gij|Ψ0〉 = −1

2(Jij −Kij) (2.56)

Comparing (2.56) with the expression for the total energy in (2.27), it is seen

that the first-order energy (sum of E0 and E1) is exactly the HF energy. The

2.7 Coupled cluster theory 31

total energies MPn up to nth order can be written as

MP0 = E(0) =N∑

i=1

ǫi

MP1 = E(0) + E(1) = EHF (2.57)

The second order correction to the ground state energy depends on the first

order correction to the wavefunction. This in turn depends on matrix elements

of the perturbation operator between the unperturbed ground and all possible

excited state of H0. The detailed discussion can be found in Refs. [12, 40], and

we only write the formula of the second order correction here

E(2) =

occ∑

i<j

vir∑

a<b

(〈φiφj|φaφb〉 − 〈φiφj|φbφa〉)εi + εj − εa − εb

(2.58)

where i and j are occupied orbitals, a and b are virtual orbitals.

2.7 Coupled cluster theory

Since its introduction into quantum chemistry in the late 1960s by Czek and

Paldus [43–45], coupled cluster (CC) theory has been widely used for the ap-

proximate solution of the electronic Schrodinger equation and the prediction of

molecular properties. The wavefunction of the CC theory is written as an expo-

nential ansatz:

ΨCC = eTΦ0 (2.59)

where Φ0 is a Slater determinant usually constructed from HF molecular orbitals,

the exponential operator eT may be expanded in a power series as

eT = 1 + T +1

2T2 +

1

6T3 + · · · =

∞∑

k=0

1

k!Tk (2.60)

with the excitation operator defined by

T = T1 + T2 + T3 + · · · + TN (2.61)


where the Ti operator acting on an HF reference wavefunction Φ0 generates all

ith excited Slater determinants

T1Φ0 =occ∑

i

vir∑

a

tai Φai

T2Φ0 =

occ∑

i<j

vir∑

a<b

tabij Φab

ij (2.62)

where the expansion coefficients t are referred to as amplitudes, i and j are indices

for the occupied orbitals and a and b are for the virtual orbitals.

With the CC wavefunction in (2.59), the Schrodinger equation becomes

HeT|Φ0〉 = EeT|Φ0〉 (2.63)

The standard formulation of CC theory is to proceed by projecting the coupled

cluster Schrodinger equation onto the reference wavefunction. One may multiply

this equation from the left by Φ0 and integrate to obtain the expression for the

CC energy

〈Φ0|HeT|Φ0〉 = ECC〈Φ0|EeT|Φ0〉 = ECC (2.64)

If all cluster operators up to TN are included in T, all possible excited deter-

minants are generated and the CC wavefunction is equivalent to full CI. This is

impossible for all but the smallest systems [12]. The truncations must be done to

the T operator. How severe the approximation depends on how many terms are

included in T. Including only the T1 operator does not give any improvement

over HF, since matrix elements between the HF and singly excited states are

zero. The lowest level of approximation is therefore T = T2, which is referred to

as CC Doubles (CCD) [46, 47], and a more complete model is referred to as CC

Singles and Doubles (CCSD) [46, 48] with T = T1 + T2, which involve a compu-

tational effort that scales as n2on

4u [49–52] (no and nu are the numbers of occupied

and unoccupied orbitals in a molecular basis set). The CCSDT [53] model with

T = T1 + T2 + T3, which iteratively treats the third-order excitations, involves

a computational effort that scales as n3on

5u [49–52]. Thus, it can consequently

only be used for small systems. Alternatively, the triples contribution may be

evaluated by perturbation theory and added to the CCSD results. The most

practical and sufficiently accurate approach to this problem is CCSD(T) [54],

2.7 Coupled cluster theory 33

where the effect of triple excitations is estimated through perturbation theory

with a non-iterative cost scaling as n3on

4u. Higher order hybrid methods such as

CCSD(TQ) [55–57], where the connected quadruples contribution is estimated by

fifth-order perturbation theory, are also possible, but they are again so demanding

that they can only be used for small systems [55, 56].

It is well known that the standard single reference CC methods, such as

CCSD(T), fail when applied to biradicals, bond breaking, and other situations

involving large nondynamic correlation effects [49, 50, 52, 58, 59]. A few attempts

have been made in recent years to address this question. One of them are the

methods belong to a family of completely renormalized (CR) CC approaches de-

veloped at Michigan State University [60–63] and incorporated in the GAMESS

package [8]. In analogy to CCSD(T), all renormalized CC methods, including

the CR-CCSD(T) [64–66], CR-CC(2,3) [61–63, 67], CR-CCSD(TQ) [64–66], CR-

CC(2,3)+Q [68–71] approaches, are based on an idea of adding non-iterative a

posteriori corrections due to higher-than-doubly excited cluster to CCSD energy

[triples in the CR-CCSD(T) and CR-CC(2,3) cases, and triples and quadruples

in the CR-CCSD(TQ) and CR-CC(2,3)+Q cases]. One of the advantages of

the renormalized CC approaches is their ability to improve the poor CCSD(T)

results in multi-reference situations involving bond breaking and biradicals, with-

out making the calculations considerably more expensive and without using the

multideterminantal reference wave functions [60, 61, 72–84]. Indeed, the most

expensive steps of the CR-CCSD(T) and CR-CC(2,3) approaches, in which one

corrects the CCSD energy for the effects of triply excited clusters, scale as n2on

4u

in the iterative CCSD part and 2n3on

4u in the non-iterative part related to the

calculations of the relevant triples corrections. For comparison, the computer

costs of determining the triples correction of CCSD(T) scale as n3on

4u. The CR-

CCSD(TQ) and CR-CC(2,3)+Q methods are more expensive, since, in addition

to the n2on

4u steps of CCSD and 2n3

on4u steps of the triples corrections, one needs

the 2n2on

5u steps to calculate the corrections due to quadruples, but even the

most demanding 2n2on

5u steps of CR-CCSD(TQ) and CR-CC(2,3)+Q are much

less expensive than the iterative steps related to the full inclusion of triples and

quadruples(n3on

5uand n4

on6u, respectively).


2.8 Basis sets

Basis sets are the foundation of modern electronic structure theory. Efficient

quantum chemical calculations on general molecules would not be possible with-

out basis sets. When molecular calculations are performed, it is common to build

the molecular orbitals as a linear combination of atomic orbitals (LCAO-MO),

centered at each atomic nucleus within the molecule

ψi =n∑

µ=1

cµiχµ (2.65)

where the ψi is the ith molecular orbital, cµi are the coefficients of the linear

combination, χµ is the µth atomic basis set orbital, and n is the total num-

ber of the atomic orbitals. Initially, these atomic orbitals were typically Slater

orbitals, which corresponded to a set of functions which decayed exponentially

with distance from the nuclei. Later, it was realized that these Slater-type orbitals

(STOs) [85] could in turn be approximated as linear combinations of Gaussian

orbitals instead. Today, there are hundreds of basis sets composed of Gaussian-

type orbitals (GTOs) [86, 87]. The brief ideas on the types of basis sets are given

in this section.

2.8.1 Slater and Gaussian type orbitals

Slater-type orbitals (STOs) are functions used as atomic orbitals in the LCAO-

MO method. They are named after the physicist John C. Slater, who introduced

them in 1930 [85], which have the following functional form

χζnlm(r, θ, φ) = Nrn−1e−ζrY m

l (θ, φ) (2.66)

where N is a normalization constant written as

N = (2ζ)n√

(2ζ)/(2n)! (2.67)

where n is a natural number that plays the role of principal quantum number, r

is the distance of the electron from the atomic nucleus, and ζ = (Z−s)/n (where

Z is the atomic number and s is a screening constant) is a constant related to

the effective charge of the nucleus, the nuclear charge being partly shielded by

2.8 Basis sets 35

electrons. It is common to use the spherical harmonics Y ml (r) depending on the

polar coordinates of the position vector r as the angular part of the Slater orbital.

The exponential dependence ensures a fairly rapid convergence with increasing

numbers of functions. However, the calculation of three- and four-center two-

electron integrals cannot be performed analytically [12]. Thus, STOs are only

used for atomic and diatomic systems where high accuracy is required, and in

semi-empirical methods where all three- and four-center integrals are neglected.

To speed up molecular integral evaluation, GTOs were first proposed by

Boys [86] in 1950, which can be written in terms of polar or Cartesian coor-

dinates as

χζnlm(r, θ, φ) = Nr2n−2−le−ζr2

Y ml (θ, φ)

χζlxly lz

(r, θ, φ) = Nxlxylyzlze−ζr2

(2.68)

The sum of the exponents of the Cartesian coordinates, l = lx + ly + lz, is used

to mark functions as s-type (l=0), p-type (l=1), d-type (l=2), and so on. The

main difference to the STOs is that the variable r in the exponential function is

squared. Thus, Gaussian Product Theorem [88] guarantees that the product of

two GTOs centered on two different atoms is a finite sum of Gaussians centered on

a point along the axis connecting them. In this manner, four-center integrals can

be reduced to finite sums of two-center integrals, and in a next step to finite sums

of one-center integrals. This speeds up by 4–5 orders of magnitude compared to

Slater orbitals more than outweighs the extra cost entailed by the larger number

of basis functions generally required in a Gaussian calculation.

However, GTOs have two major problems compared with STOs. First, GTOs

do not have a cusp at r = 0. The other problem is that GTOs fall off too rapidly

for large r. It is known from (2.68) that GTOs with large α are very much

concentrated around the origin (in the limit of infinite α they tend to a Dirac

delta function) and can mimic the correct cusp, while GTOs with small α are

very diffuse (spread out) and can describe the behavior of a molecular orbital

(MO) at large r. Thus, these problems can be corrected by linear combinations

of GTOs, which leads to the introduction of contracted sets of primitive GTOs


(CGTOs) [89, 90], which take the following formation

χν =M∑

ν=1

dµνgν(αν) (2.69)

where M is the length of the contraction, the gν ’s are primitive Gaussians func-

tions, dµν and is contraction coefficients which can be determined by least-square

fits to accurate atomic orbital or by minimization of the total HF energy.

2.8.2 Classification of basis sets

Once the type of function (STO or GTO) is selected, the next step is to choose

the number of functions to be used. The smallest number of functions possible,

to contain all the electrons in a neutral atom, is called minimum basis set. Thus,

for hydrogen (and helium) it means a single s-function, for the first row elements

of the periodic table, it requires two s-functions (1s, 2s) and one set of p-functions

(2px, 2py, 2pz), and so on. The next improvement is to double all basis functions,

producing a Double zeta (DZ) type basis, which employs two s-functions for

hydrogen (1s and 1s′), four s and two p-functions for the elements on the first

row, and so on. One can also go further to Triple Zeta (TZ), Quadruple Zeta

(QZ), Quintuple Zeta (5Z) and so on.

Often it takes too much effort to calculate a DZ for every orbital. Instead,

many scientists simplify matters by calculating a DZ only for the valence orbital.

Since the inner-shell electrons are not as vital to the calculation, they are de-

scribed with a single Slater Orbital. This method is called a split-valence basis

set. The n − ijG or n − ijkG split-valence basis sets are due to Pople and co-

workers [91–93], where n is the number of primitives summed to describe the

inner shells, ij or ijk is the number of primitives for contractions in the valence

shell. In the 6 − 31G basis, for example, the core orbitals are described by a

contraction of six GTOs, whereas the inner part of the valence orbitals is a con-

traction of three GTOs and the outer part of the valence is represented by one

GTOs. Including polarization functions give more angular freedom so that the

basis is able to represent bond angles more accurately, especially in strained ring

molecules. For example, 6−31G∗∗ basis set, the first asterisk indicate the addition

of d functions on the non-hydrogen atoms and the second asterisk a p function

2.8 Basis sets 37

on hydrogen atom [94]. Another type of functions can be added to the basis for

a good description of the wavefunction far from the nucleus are the diffuse func-

tions, which are additional GTOs with small exponents. The diffuse functions

are usually indicated with a notation “+”. For example, in the 6-31++G basis

set, the ”++” means the addition of a set of s and p function to the heavy atoms,

while an additional s diffuse GTO for hydrogen.

For correlated calculations, the basis set requirements are different and more

demanding since we must then describe the polarizations of the charge distri-

bution and also provide an orbital space suitable for recovering correlation ef-

fects. For this purpose, the correlation consistent basis sets are very suited,

which is usually denoted as cc-pVXZ [95, 96]. The “cc” denotes that this is a

correlation-consistent basis, meaning that the functions were optimized for best

performance with correlated calculations. The “p” denotes that polarization func-

tions are included on all atoms. The“VXZ” stands for valence with the cardinal

number X = D, T,Q, . . . indicate double-, triple- or quadruple-zeta respectively.

The inclusion of diffuse functions, which can improve the flexibility in the outer

valence region, leads to the augmented correlation-consistent basis sets aug-cc-

pVXZ [95, 96], where one set of diffuse functions is added in cc-pVXZ basis.

2.8.3 Basis set superposition error

In quantum chemistry, calculations of molecular properties are all done using a

finite set of basis functions, which may lead to an important phenomenon referred

as the basis-set superposition error (BSSE) [97, 98]. As the atoms of interacting

molecules (or of different parts of the same molecule) approach one another, their

basis functions overlap. Each monomer ”borrows” functions from other nearby

components, effectively increasing its basis set and improving the calculation of

derived properties such as energy. If the total energy is minimized as a function of

the system geometry, the short-range energies from the mixed basis sets must be

compared with the long-range energies from the unmixed sets, and this mismatch

introduces an error. Other than using infinite basis sets, the counterpoise method

(CP) [99] is used to account for the BSSE. In the CP method, the BSSE is

calculated by re-performing all the calculations using the mixed basis sets, and

the error is then subtracted a posterior from the uncorrected energy. On the other


hand, the BSSE can be corrected by scaling [100] or extrapolating [101–104] the

ab initio energies to the complete basis set limit as discussed in the next section.

2.9 Semiempirical correction of ab initio ener-

gies

2.9.1 Scaling the external correlation energy

The truncate CI wave function lacks of size-extensivity, as it does not include

as much of the dynamical or external electron correlation effects. A method to

incorporate semiempirically the external valence correlation energy was proposed

by Brown and Truhlar [105]. In such approach the non-dynamical (static) or

internal correlation energy is obtained by an MCSCF calculation and the part

of external valence correlation energy by an MRCISD calculation based on the

MCSCF wave functions as references. Then, it is assumed that the MRCISD

includes a constant (geometry independent) fraction F of the external valence

correlation energy, which can be extrapolated with the formula

ESEC (R) = EMCSCF (R) +EMRCISD (R) − EMCSCF (R)

F(2.70)

where ESEC (R) denotes the scaled external correlation (SEC) energy, and the

empirical factor F is chosen for diatomics to reproduce a bond energy, and for

systems with more atoms chosen to reproduce more than one bond energy in an

average sense [105].

As pointed out by these authors [105], the SEC method requires a large enough

MCSCF calculation and one-electron basis sets, in order to include dominant

geometry dependent internal correlation effects and an appreciable fraction of

the external valence correlation energy.

By including information relative to experimental dissociation energies, the

SEC method attempts to account for the incompleteness of the one-electron basis

set [106]. MRCISD calculation based on large enough basis set contributes to

minimize undesirable BSSE, which may be corrected subsequently by scaling the

external correlation energy.

Varandas [106] suggested a generalization of the SEC method by noticing the

conceptual relationship between it and the double many-body expansion (DMBE)

2.9 Semiempirical correction of ab initio energies 39

method [107]. In fact, in the DMBE scheme each n-body potential energy term is

partitioned into extended-Hartree-Fock (internal correlation) and dynamic corre-

lation (external correlation) parts. In his proposal, denoted as DMBE-SEC [106],

this author writes the total interaction energy, relative to infinitely separated

atoms in the appropriate electronic states, in the form

V (R) = VMCSCF (R) + VSEC (R) (2.71)

where

VMCSCF (R) =∑

V(2)AB,MCSCF (RAB) +

∑V

(3)ABC,MCSCF (RAB, RBC, RAC) + . . . (2.72)

VSEC (R) =∑

V(2)AB,SEC (RAB) +

∑V

(3)ABC,SEC (RAB, RBC, RAC) + . . . (2.73)

and the summations run over the subcluster of atoms (dimers, trimers, ...) which

compose the molecule.

The scaled external correlation energy component for the n-th terms is given

by

V(n)AB...,SEC =

V(n)AB...,MRCISD − V

(n)AB...,MCSCF

F (n)AB

(2.74)

where F (n)AB... is the n-body geometry independent scaling factor.

As in the original SEC method, optimal values for two-body factors F (2)AB are

chosen to reproduce experimental dissociation energies, a criterion which may

be adopted for higher-order terms if accurate dissociation energies exist for the

relevance subsystems [106]. For the triatomic case a good guess for F (3)ABC can be

the average of the two-body factors

F (3)ABC =

1

3

[F (2)

AB + F (2)BC + F (2)

AC

](2.75)

Improved agreement with experiment and best theoretical estimates, is ob-

tained when ab initio energies are corrected with the DMBE–SEC method. Par-

ticularly important, for dynamics calculations, are the correct exothermicities for

all arrangement channels, exhibited by the DMBE–SEC potential surfaces [106].

2.9.2 Extrapolation to complete basis set limit

The large majority of electronic structure calculations employ an expansion of

the orbitals in a basis set, almost always of the Gaussian type and located at the


nuclear positions. The basis set incompleteness is one of the factors that limits

the ultimate accuracy. For small molecule, a significant enhancement to progress

in electronic structure calculations has become possible with the introduction of

correlation-consistent basis sets developed by Dunning [95]. The most common

family of Dunning basis sets are denoted as cc-pVXZ, especially the augmented

ones (aug-cc-pVXZ). More recently there has been much interest in exploiting

the systematic behavior of these basis sets with respect to the cardinal number

X by carrying out calculations for several values of X and extrapolating to the

complete basis set (CBS) limit. Much effort has been put onto the extrapolation

schemes to obtain the molecular energy at the CBS limit at a computational

cost as low as possible [96, 108–119]. In most recent articles [110–114], Varandas

proposed a practical scheme for extrapolating electronic energies calculated with

correlation consistent basis sets of Dunning. The MRCI(Q) electronic energy is

best treated in split form by writing [110]

EX(R) = ECAS(R) + Edc(R) (2.76)

where R specifies the three-dimensional vector of space coordinate, the subscript

X indicates that the energy has been calculated in the AVXZ basis and the su-

perscripts CAS and dc stand for complete-active space and dynamical correlation

energies, respectively. For CAS (uncorrelated in the sense of lacking dynamical

correlation energies), several schemes have been advanced [108–112]. To extrap-

olate Hartree-Fock energies using AVTZ and AVQZ basis sets, the most reliable

protocol is possibly the one due to Karton and Martin [108] denoted as KM(T,Q).

Our past experience [111] with AV5Z and AV6Z energies suggests that the same

protocol can be successfully utilized with the CAS energy component, hence we

will adopt the KM(T,Q) protocol in the present work. This assumes the form

ECASX (R) = ECAS

∞ (R) +B/Xα (2.77)

where α = 5.34 is an effective decay exponent.

To extrapolate the dynamic correlation (dc) energy, the uniform singlet- and

triplet-pair extrapolation (USTE) scheme [110] developed by Varandas has shown

great promise in extrapolating from (T,Q) cardinal-number pairs, which assumes

2.9 Semiempirical correction of ab initio energies 41

the following formation

EdcX (R) = Edc

∞(R) +A3

(X + α)3+

A5

(X + α)5(2.78)

with A5 being determined by the auxiliary relation

A5 = A5(0) + cA5/43 (2.79)

where A5(0) = 0.0037685459Eh, c = −1.17847713E−5/4h , and α = −3/8, with

E∞ and A3 are determined from a fit to the dc energies with the AVTdZ and

AVQdZ basis sets [110].

The computational cost can be further reduced by obtaining the dynamical

correlation energy at MRCI(Q)/AVQZ basis set via correlation scaling (CS) at

one [112] or more [113] pivotal geometries, heretofore denoted as CSN where N

is the number of pivots. It is the joint use of the CSN and USTE(T,Q) methods.

The full approach, hereafter called CSN/USTE(T,Q), involves six basic steps:

(a) calculation of the PES at Nt points with the X−2 and X−1 basis sets using

MRCI(Q), and N(= 1 − 4) such calculations with the target basis set of rank X

at the pivots; (b) calculation of all the considered geometries of the PES at the

CAS level with the target X basis set level; (c) extrapolation to the CBS limit

(i.e., X = ∞) of the CAS energies using (2.77); (d) prediction by extrapolation to

X = ∞ of the dc energy at the pivots by using (2.78); (e) prediction of the CBS

dc energies of the remaining Nt −N points by CS [112, 113] using the X = D, T

and CBS dc energies at the pivots; (f) calculation of the full CBS PES by adding

the CAS/CBS and extrapolated dynamical correlation energies from steps (c)–

(e). In case of single pivot, which is denoted by CS1/USTE(T,Q) method [112],

assumes the form

Edc(R)=χ∞,3(R)Edc3 (R) (2.80)

where the scaling function χ assumes the form

χ∞,3(R)=1 +S3,2(R) − 1

S3,2(Re) − 1[S∞,3(Re) − 1] (2.81)

where Re indicates the pivotal geometry, and

Sm,n(R)=Edc

m (R)

Edcn (R)

(2.82)


Thus, CBS extrapolation via (2.80)–(2.82) utilizes the correlation energies calcu-

lated with AVDZ and AVTZ basis sets using a single pivotal geometry at which

the dc energy has also been calculated with the AVQZ basis set.

2.10 Analytical representation of potential en-

ergy surface

Once enough information on the potential energy of the system under study has

been gathered, it is necessary to present it in a realistic global functional form, in

order to carry out dynamics studies. Nevertheless, a choice of suitable function

is not an easy task. A successful representation of a global PES for dynamical

calculations should satisfy certain criteria, as discussed by Wright and Gray [120]

and remarked by Varandas [121]:

1. “It should accurately characterize the asymptotic reactant and product

molecules (or more generally any fragment of the full system).”

2. “It should have correct symmetry properties of the system.”

3. “It should represent the true PES in interaction regions for which experi-

mental or non-empirical theoretical data are available (including, in prin-

ciple, the very short-range and long-range regions associated with various

asymptotic channels [122]).”

4. “It should behave in a physically reasonable manner in those parts of the

interaction region for which no experimental or theoretical data are avail-

able.”

5. “It should smoothly connect the asymptotic and interaction region in a

physically reasonable way.”

6. “The function and its derivatives should have as simple an algebraic form

as possible consistent with the desired quality of the fit.”

7. “It should require as small a number of data points as possible to achieve

an accurate fit.”

2.10 Analytical representation of potential energy surface 43

8. “It should converge to the true surface as more data become available.”

9. “It should indicate where it is most meaningful to compute the data points.”

10. “It should have a minimal amount of ad hoc or ’patched up character’.”

Criteria from 1 to 5 must be obeyed in order to obtain reasonable results in

subsequent calculations using the function. Criteria from 6 to 10 are desirable

for practical reasons. Finding a function that meets these criteria requires skill

and experience, and considerable amount of patience [123].

Methods to construct analytical PESs have been developed for many years.

Among them are the semiempirical London-Eyring-Polanyi-Sato (LEPS) [124–

126] and diatomics-in-molecules (DIM) [127–129] methods, which use theoretical

and experimental information to fit a functional form derived from simple molecu-

lar orbital theory. A more general approach is the many-body expansion (MBE)

method developed by Murrel and co-workers [130–132], which proposed to de-

scribe the total interaction of the polyatomic system by adding all the many-body

interactions of each fragment. The PESs discussed in the present thesis are rep-

resented using an improved version of MBE due to Varandas [100, 107, 122, 133]:

the double many-body expansion (DMBE) method, which consists in partitioning

each n-body contribution in short-range and long-rang parts. The more detailed

discussion of MBE and DMBE methods is given in the following sections.

2.10.1 The many-body expansion method

The many-body expansion (MBE) for a single-valued PES of an N -atomic system

is written as [131]:

VABC···N(R) =∑

V(1)A +

∑V

(2)AB (RAB) +

∑V

(3)ABC(RAB, RAC, RBC) + · · ·

+V(N)ABC···N(R) (2.83)

where V(1)A is the energy of the atom A, and the summation runs over all the

one-body terms. If the reference energy is taken as the energy of all the atoms

in their ground states, then V(1)A will be zero. V

(2)AB (RAB) is a two-body energy

term, depending on the distance separating the two atoms, and which goes to

zero as RAB tends to infinity. V(3)ABC(RAB, RAC, RBC) is a three-body energy which


depends on the three distances of the triangle ABC. The last term in the expan-

sion V(N)ABC···N(R) is the n-body energy. It depends, as the total potential function,

on the 3N − 6 internal coordinates.The MBE function is designed to satisfy all

dissociation limits, and it also provides a strategy for building up PESs of larger

polyatomic systems. Once the potentials of all the fragments are deduced, they

can be used in all the polyatomics containing such fragments.

2.10.2 The double many-body expansion method

MBE method is proposed to provide an analytical representation of PESs for all

possible configurations of the system. Then, its functional form must properly

reproduce all the regions, from short range interactions to long range ones. How-

ever, the method fails in keeping only one function to reproduce both ranges.

Thus, the idea of splitting each many-body terms into two parts arises. In such

spirit, Varandas [100, 107, 122, 133] extended the many-body expansion to the

double many-body expansion (DMBE) in which each many-body term is splitted

into two parts: one accounting for the long range or dynamical correlation (dc)

energy and the other describing the short range or extended-Hartree-Fock (EHF)

energies.

V (RN) =N∑

n=1

∑

Rn⊂RN

[V

(n)EHF(Rn) + V n

dc(Rn)]

(2.84)

where, Rn denotes any set of n(n−1)/2 coordinates of the fragment containing n

atoms, which is a subset of RN ≡ [R1, R2, . . .RN(N−1)/2], and last sum is carried

out over all such subsets.

In a series of papers Varandas and coworkers [107, 122, 134–137] proposed

general expressions for the n-body dynamic correlation energy term, to repro-

duce the proper anisotropy and asymptotic behavior of the PES for the entire

configuration space (for details see cases studies). An important result refers to

the introduction of an universal charge-overlap damping function to account for

the damping of the dispersion coefficients for intermediate and small interatomic

separations [134].

Extended Hartree-Fock approximate correlation energy for two- and three-

body interactions (EHFACE2 and EHFACE3) models have been proposed [135],

2.10 Analytical representation of potential energy surface 45

from simple, yet reliable, physical motivated forms. To represent the global short-

range energy for two-body potentials, a screened extended-Rydberg form (with an

extra R−1 term) can be adapted, which reproduces the exact ZAZB/R behavior

at the united atom limit (EHFACE2U model [137]). In turn, the three-body

EHF potential is represented by the following three-body distributed-polynomial

form [138]

V(3)EHF (R) =

m∑

i=1

P (i) (Q1, Q2, Q3)

3∏

j=1

{1 − tanh

[γ

(i)j

(Rj − Ri,ref

j

)]}(2.85)

where P (i) (Q1, Q2, Q3) are polynomials in the symmetric coordinates {Qk}, ex-

pressed as combinations of the internuclear coordinates {Rj}, which transform

as irreducible representations of the permutation group of the molecule [132]. In

turn, Ri,refj represents a convenient reference geometry to which the i-th com-

ponent of (2.85) is referred to [138]. If extensive ab initio data is available, the

optimized coefficients for the two-body and three-body terms can be obtained

using linear or non-linear least-square fits [139].

2.10.3 Approximate single-sheeted representation

When two or more potential energy surfaces cross, an exact treatment demands

a multi-sheeted representation of the PES. Such representation of the potential

can be expressed as the lowest eigenvalues of a square matrix of order equal to

the number of states involved [140]. Thus, the elements of the diabatic potential

matrix can be written as many-body expansions or double-many body expansions

involving the appropriate electronic states of the fragments.

However, for many situations an approximated single-sheeted representation

can provide a good analytical form for dynamical purposes. For example, if a

crossing between states is present, it can be avoided in such a way that the po-

tential function smooths the region around the intersection point. If the crossing

is located well above the dissociation channels or the stationary points of the

molecule, it is expected to have a minor influence in the dynamics of the system.

In the present work, DMBE potential energy surfaces for the systems H2S (1A′)

and HS2 (2A′′) were calibrated following this procedure. Further dynamics calcu-

lations using the obtained DMBE-PES for H2S were carried out for the reaction


S (1D) + H2

(X 3Σ+

g

)→ SH (X 2Π)+ H (2S) showing the reliability of the approx-

imated single-sheeted form.

A single-sheeted representation uses switching functions to account for the

presence of different states of fragments for different regions of the configuration

space. The switching function was firstly applied by Murrel and Carter [141]

to construct a MBE PES for ground-state H2O. These authors introduced a

switching one-body term for oxygen, allowing that in the PES the atomic state

O (1D) is connected for the channel H2

(X1Σ+

g

)+ O (1D) and disconnected for

the other dissociation limits. A smooth description of the PES which accounted

for such a behavior of the oxygen atomic state, is warrant employing a switching

function in the form

f (x) =1

2[1 − tanh (αx)] (2.86)

which has the limits unity as x→ −∞ and zero as x→ +∞.

By choosing the variable x as

x = nρ3 − ρ1 − ρ2 (2.87)

where ρi = Ri − R0i are the displacements of the internuclear distances from a

reference structure (R3 the H–H distance, and R1, R2 the OH distance), then it

is easy to see that x takes the limit −∞ for dissociation into H2 + O and +∞ for

dissociation to OH + H provided n ≥ 2.

Although the function (2.86) shows a proper behavior for short and interme-

diate distances of the triatomic, it cannot reach a unique value at the three-atom

limit [141]. In fact, even for large H–H separation, the function (2.86) switches

from 0 to 1 when oxygen moves far away from the diatomic. To correct such

unphysical behavior, Varandas and Poveda [142] have suggested an improved

switching function in their work on DMBE PES of NH2(2A′′). The same func-

tional form is employed in the construction of an approximated single-valued

DMBE PES for H2S (X 1A′), which can be written as

V (R) = V(1)

S(1D)f(R) +3∑

i=1

V (2)(Ri) + V (3)(R) (2.88)

where V(1)

S(1D) represents the energy difference between the 1D and 3P states of

atomic sulfur, f(R) is the switching function used to warrant the correct behav-

ior at the H2(X1Σ+

g ) + S(1D) and SH(X 2Π) + H(2S) dissociation limits, while

2.10 Bibliography 47

V (2)(Ri) and V (3)(R) represent the two-body and three-body energy terms re-

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Chapter 3

Exploring PESs via dynamicscalculations

Reaction dynamics deals with the intra- and inter-molecular motions that char-

acterize the elementary act of a chemical reaction. It also deals with the quantum

states of the reactants and products. Once the electronic problem is solved, result-

ing in an appropriate representation of the potential energy function as referred

in the previous part, a chemical reaction may be understood as the motion of

atomic nuclei through such a potential. Thus, classical or quantum mechanical

methods can be used to characterize the chemical reaction. For the study of reac-

tions presented in this thesis, a quasiclassical trajectory (QCT) method [1–4] was

used. The basis of QCT as well as some features of molecular reaction dynamics

are briefly reviewed in this chapter.

3.1 Quantum dynamics

The exact way to treat the dynamics of molecular collisions is to use quantum

scattering methods. A consistent and comprehensive treatment of quantum scat-

tering can be found in a recent book by Zhang [5]. The motion of the atomic

nuclei is given by the time-dependent Schrodinger equation (TDSE)

ih∂

∂tΨ(R, t) = HΨ(R, t)

= TnucΨ(R, t) + E(R)Ψ(R, t) (3.1)

58 Exploring PESs via dynamics calculations

where Tnuc is the operator for the kinetic energy of the nuclei, E(R) is the elec-

tronic energy, i.e., the potential energy surface, and Ψ(R, t) the wave function as

a function of all the nuclear coordinates R. The TDSE is a first-order differential

equation in time, and the time evolution is given once the initial state is specified.

Assume that the initial state is given by Ψ(R, t0), then

Ψ(R, t) = U(t, t0)Ψ(R, t) (3.2)

where the propagator U(t, t0) satisfies the initial condition U(t0, t0) = 1 and is

given by

U(t, t0) = e−iH(t−t0)/h

≡ 1 − iH(t− t0)/h− H2(t− t0)2/2h2 + · · · (3.3)

that is the exponential of an operator is defined by its (formal) Taylor expansion.

(3.2) is seen to be a solution by direct substitution into (3.1). This method

for solving the TDSE is known as the time-dependent wave packet (TDWP)

approach, also known as the grid method, which involves the following steps: (i)

representing the initial wave function Ψ(R, t0) on a finite grid, (ii) propagating

the wave function from time t0 to time t, till the end of the dynamical event and

(iii) analyzing the final wave function for reaction attributes.

The initial wave function is discretised on a finite grid, the size of which should

be large enough to contain all asymptotic channels, and care should be taken to

avoid the wave function reaching grid boundaries during time evolution. Once

the initial wave function is chosen, the next step is its propagation in time. This

involves the calculation of the HΨ(R, t) as well as the time derivative of the wave

function. First let us consider the evaluation of the action of the Hamiltonian

operator on the wave function. The evaluation of HΨ(R, t) consists of two parts:

TnucΨ(R, t) and E(R)Ψ(R, t). The latter is obtained by a simple, multiplication

of Ψ(R, t) at each grid point by E(R). Evaluation of TnucΨ(R, t) is a bottleneck

in all time-dependent quantal calculations as the kinetic energy operator Tnuc is

nonlocal in the coordinate representation. The introduction of the fast Fourier

transform (FFT) method by Feit et al. [6–8] and Kosloff et al. [9] for computing

the action of the kinetic energy part of the Hamiltonian on the wave function

was a significant development in the area of time-dependent quantum mechanical

3.2 The QCT method 59

(TDQM) calculations. Alternatively, the discrete variable representation (DVR)

method [10–15] was originally introduced by Light and coworkers [13, 14], can be

used for calculating the second derivative of the wave function. Computationally,

the DVR method scales as N2 compared to N lnN of the FFT method [16, 17],

with N the number of grid points.

Once an appropriate spatial discretisation scheme is chosen, and the initial

wave function on the grid is set up, the next step in the TDQM approach is

the time propagation of the wave function. In doing this, the action of the time

propagator on Ψ(R, t) has to be evaluated. A number of methods to carry out

this forward in time wave function propagation have been developed, basically:

The second-order differencing scheme [18], the split-operator method [19, 20],

the Chebyshev polynomial expansion method [21–23], and the Lanczos recursion

scheme [24–26].

The most active development in the field of quantum scattering on bimolecu-

lar chemical reactions in the gas phase has been in the theory and applications of

TDQM methods, where the wave packet methodology is establishing as a major

technique in the field [27]. An important advantage of the wave packet approach is

that by an appropriate choice of a single initial wave packet and time evolution,

one can obtain scattering information at a number of energies. Various meth-

ods [28–32] have been developed for obtaining energy- as well as state-resolved

reaction probabilities or cross sections from the scattered wave packet. The basic

idea behind all these methods is to project the outgoing wave packet on to the

asymptotic fragment states and use a time-energy or a coordinate-momentum

Fourier transform to obtain the scattering amplitude as a function of energy.

3.2 The QCT method

Classical trajectories are the limits of high particle masses and high energies of

quantum-mechanical scattering process [3, 33]. They are used when dealing with

a molecular process in all the complexity and reality. They provide a feasible

connection between experimental observations and the interaction potential of

the atoms. When using classical trajectories one question arises: are chemical

reactions close to classical simplicity or do they require the detailed attention of


quantum considerations? The answer is that we usually think of these processes

as classical ones, with quantum corrections required under certain conditions

[34]. A qualitative argument is that de Broglie wavelength are short enough and

that so far has not been shown that tunneling corrections are very important to

classical interpretations [3]. Besides, as remarked in a series of works by Karplus et

al. [2, 35, 36], in a classical and quantum treatment of the same molecular system,

no significant differences have been obtained. Of course, some discrepancies might

appear for low translational energy processes when quantum effects are expected

to be significant [3].

In a classical trajectory study, the motions of the individual atoms are simu-

lated by solving Hamilton’s or Newton’s equations of motion expressed in terms

of the coordinates q and momentum p of the system. In the Hamilton for-

mulation [37], propagation is done by numerical integration of the first-order

differentialdqidt

=∂H (q,p)

∂pi

,dpi

dt= −∂H (q,p)

∂qi(3.4)

where the Hamiltonian function of the system (H) is the sum of the kinetic

Tkin(p,q) and potential Vpot(q) energies:

H (q,p) = Tkin (q,p) + Vpot (q) (3.5)

The potential energy function Vpot(q) is the already mentioned PES, which is

represented by an analytic DMBE function. Hamilton’s equations (3.4) are solved

numerically and numerous algorithms have been developed for this task [3]. When

a set of trajectories is completed, the final values of momenta and coordinates are

transformed into quantities, like reaction rate constants. A significant aspect of a

trajectory simulation is the choice of the initial coordinates and momenta. These

initial conditions are chosen such that results from an ensemble of trajectories

may be compared with experiment and theory and be used for predictions about

the system’s molecular dynamics. Monte Carlo methods are commonly used [1,

2, 4] for sampling appropriate distributions of initial values of coordinates and

momenta.

As was mentioned above, a dynamical study of a molecular collision can be car-

ried out by means of classical equations. However, once configurations of the sep-

arated reagents are described by their vibrational and rotational (ro-vibrational)


quantum states, initial conditions of the collision should be generated account-

ing for them. This is the idea of QCT method [35]: to solve classical equations

of motion considering the initial conditions of the reactants according to their

quantum states. Similarly, the states of the product molecules can be assigned

by determining the quantum numbers describing the best their ro-vibrational

motion.

3.2.1 Unimolecular decomposition

In the unimolecular dissociation, a molecule is prepared in a vibrational-rotational

excited state A∗ above the unimolecular threshold from which the molecule has

a probability to dissociated to products. Assuming that the system is initially

excited with a microcanonical ensemble and its intramolecular dynamics is er-

godic [38–40], the probability of decomposition per unit time will be [41–43]

P (t) = k (E) exp [−k (E) t] (3.6)

given equal probability during any time interval for reaction to occur. k(E) is the

classical microcanonical unimolecular rate constant, which is expressed as [43]

k(E) =N(E)

hρ(E)(3.7)

where N(e) is the sum of states at the transition state for the decomposition

and ρ(E) is the density of states for A∗. According to the classical/quantum

correspondence principle [44, 45], the classical and quantum k(E) become equiv-

alent at high energies. However, for E near the unimolecular threshold E0, the

classical k(E) may be significantly larger than the quantum k(E), since classical

mechanics allows the transition state to be crossed and products to be performed

without the presence of zero-point energy [45].

3.2.2 Bimolecular reaction

For bimolecular reaction, let us consider two reactant molecules A and B, ap-

proaching with a relative velocity vrel (with module vrel), which may be oriented

such that the reactants approach head-on (along a line connecting the center of


masses) or with a glancing blow collision. The difference between these two en-

counters is quantified by the impact parameter of the collision b, which is defined

as the distance of closest approach of the reactants in the absence of any interac-

tions between them. Thus, head-on collision occurs when b=0, and b>0 stands

for oblique direction or glancing blow collision. The maximum value of b which

leads to reaction is called maximum impact parameter, bmax. Beyond bmax, the

collisions are so glancing that probability of reaction is vanishingly small.

A measure of the effective collision area is given by the cross section. The

cross section for the reaction between A and B to form products:

A+B → products (3.8)

may be expressed as σR (vrel, ν, J) [46], where ν and J denote the vibrational

and rotational quantum numbers of the reactants respectively. Assuming Boltz-

mann distributions of vibrational-rotational levels specified by temperature T ,

the reactive Boltzmann-average cross-section can be obtained as

σr(Etr, T ) =∑

v

∑

J

σR(vrel, v, J)Pv(T )PJ(T ) (3.9)

where Pv(T ) and PJ(T ) are the normalized Boltzmann distributions of the vi-

brational and rotational quantum numbers of the reactants respectively.

Multiplying σ(Etr;T ) by the relative velocity vrel and integrating over the

Boltzmann distribution one gets the bimolecular thermal rates constant:

k (T ) =

∫ ∞

0

vrelσr (vrel;T )P (vrel;T ) dvrel (3.10)

Inserting the Maxwell-Boltzmann distribution for P (vrel;T ) into (3.10) and

introducing the translational energy by the relation Etr = µABv2rel/2, the thermal

rate constant can be written as

k (T ) =

(8kBT

πµ

)1/2

〈σr (Etr)〉 (3.11)

where the average cross section for temperature T will be

〈σr (Etr)〉 =

∫ ∞

0

σr (Etr)Etr

(kBT )2 e−Etr/kTdEtr (3.12)


Then, the integral (3.12) can be evaluated by sampling randomly the transla-

tional energy Etr by the von Neumann rejection method [45] or by the cumulative

distribution function (CDF) [3]

Etr = −kBT ln(ξ

(1)tr ξ

(2)tr

)(3.13)

where ξ(1)tr and ξ

(2)tr are independent random numbers.

In turn, a simple expression for the reaction cross section can be derived from

the classical mechanical expression for this quantity [47]

σr =

∫ bmax

0

Pr (b) 2πbdb (3.14)

where b is the collision impact parameter, bmax is its largest value that leads to

reaction and Pr (b) is the so-called opacity function given the impact parameter

distribution.

From (3.14) it is derived [1] that

σr = 〈Pr (b)〉πb2max (3.15)

Random values of b between 0 and bmax may be sampled with the CDF:

ξ =

∫ b

0

P (b) db (3.16)

where ξ is a random number. Then, the average reaction probability is 〈Pr (b)〉 =

Nr/N , where N is the total number of trajectories and Nr the subset of N

representing the number of reactive trajectories. By substituting in (3.15), the

reaction cross-section is [4]

σr =Nr

Nπb2max (3.17)

In the same way, as the translational energies are randomly sampled, the

bimolecular rate constant in (3.11) may be expressed as [4]

k (T ) =

(8kBT

πµ

)1/2Nr

Nπb2max (3.18)


3.3 Excitation function and rate constant

Molecular beam experiments provide high initial collision energy resolution [48].

That is why they are often employed to measure the translational energy depen-

dence of the reaction cross section (excitation function). Much of the interesting

information about an elementary chemical reaction can be summarized in such

a function [49]. Besides, it is also needed to calculate the rate constant for spe-

cific ro-vibrational states of the reactants. Once its value is obtained for a given

translational energy, some models are used to represent it.

3.3.1 Reaction with barrier

Based on the fitting of available data, LeRoy [49] proposed some particular mod-

els:

Class I reactions

σ(Etr) =

{C(Etr − Eth

tr )ne−m(Etr−Ethtr

) Etr ≥ Ethtr

0 Etr < 0(3.19)

where m,n ≥ 0. Those functions increase from 0 at Etr = Ethtr , the exponential

term causes the excitation function to pass through a maximum as the energy in-

crease. Such a dependence describe properly the excitation functions for neutral-

neutral reactions. The H + SO2 reaction studied by Ballester et al. [50] properly

fit to this model.

By substituting (3.19) into (3.12), an analytical expression for the rate con-

stant is obtained:

k(T ) = C

(8kBT

πµ

)1/2(kBT )ne−Eth

tr/kBT

(1 +mkBT )n+2×

×[Γ(n + 2) + Γ(n+ 1)

(1 +mkBT )Ethtr

kBT

](3.20)

where Γ is the Gamma function, see appendix.

Class II reactions

σ(Etr) =

{C(Etr−Eth

tr)n

Etre−m(Etr−Eth

tr) Etr ≥ Eth

tr

0 Etr < 0(3.21)

3.3 Excitation function and rate constant 65

these functions are very similar to the previous one, however they include the

excitation function for the collision of hard spheres which requires a critical energy

Ethtr [48]. This excitation function yields to a rate constant:

k(T ) = C

(8kBT

πµ

)1/2(kBT )n−1Γ(n+ 1)e−Eth/kBT

(1 +mkBT )n+1(3.22)

Class III reactions

σ(Etr) =

{CEn

tr Etr ≥ Ethtr

0 Etr < 0(3.23)

This type of functions applies for collisions between low energy ions and polariz-

able molecules [49]. For these functions, the rate constant becomes:

k(T ) = C

(8kBT

πµ

)1/2

(kBT )n[Γ(n+ 2) − P (n+ 2, Ethtr /kBT )] (3.24)

being P the incomplete Gamma function, see appendix.

3.3.2 Barrier-free reaction

In the collision of two particles (with masses m1 and m2) interacting along the

centers of mass line, the two-body problem can be simplified into a one-body

problem. There, a particle of mass µ (µ=m1m2/(m1 +m2)) moves under the in-

fluence of an effective potential (Veff) given by the sum of the interaction between

both particles and a centrifugal potential [51].

For reactions which proceed through an attractive potential energy surface,

without a barrier (capture-like), the centrifugal barrier on the effective potential

Veff may still prevent reaction. To obtain a simple model of such a kind of collision,

structureless reactants will be assumed. Considering also a long-range attractive

potential in the form:

V (R) = −Cn

Rn(3.25)

where Cn and n are parameters depending on the interaction type, with n = 3

when there are dipole-dipole like, n = 4 for quadrupole-dipole and so on [52,

53]. The distance between reactants is represented by R. Of course the above

assumption is a large simplification of the problem as in real collisions we deal

with reactants having different electric multipoles and also their values can change


as the reaction proceeds. However, these effects are supposed to be included in

the values on n and Cn with some intermediate values, not corresponding exactly

to any specific multipole interaction, but to a mixture of them.

The effective potential becomes:

Veff(R) = Etrb2

R2− Cn

Rn(3.26)

where b is the impact parameter. Veff(R) has a maximum value at R = R0:

R0 =

(nCn

2Etrb2

)1/(n−2)

(3.27)

With the condition that the translational energy must equal the maximum value

of the effective potential for b=bmax, the excitation function then becomes:

σ(Etr) = πb2max = nπ(n− 2)(2−n)/n

(Cn

2Etr

)2/n

(3.28)

By substituting the previous expression into (3.12), the rate constant is obtained

as:

k(T ) = 2nπ(n− 2)(2−n)/n

(2

πµ

)1/2(Cn

2

)2/n

Γ

(2n− 2

n

)(kBT )(n−4)/2n (3.29)

Even when this result was obtained for a simplified model of interaction, it fits

particularly well the radical-radical reactions [54].

3.4 Electronic degeneracy factor

In calculating cross sections and rate constants for molecular collision processes,

one must consider all the possible potential energy surfaces upon which collision

occur [55–60]. As early as 1936, it was pointed out by Ravinowich [61], that

theoretically calculated rate constants differ in a factor from experimental re-

sults. This factor depends upon the electronic degeneracy of the involved species.

Bunker and Davidson [62, 63] remarked the role of such a factor. In the work

of Truhlar [55] the proper inclusion of the electronic degeneracy was presented

while Muckerman and Newton [56] pointed out its dependence on temperature.

Main ideas of the degeneracy factor are briefly presented in the following.

3.5 Products properties from QCT runs 67

In some collision processes (e.g. He + Ne), both collision partners are nonde-

generate [55]. In some other systems (e.g., H + H2), both the separated collision

partners and the lowest energy state have the same degeneracy g (g = 2). It is a

good approximation to consider that the internuclear motion is governed by one

PES, corresponding to the lowest energy electronic state of the system. For most

collision problems, however, one must consider more than one electronic state:

e.g., I(2P3/2) has g = 4 so the collision partners in I + I have g = 16. However,

the ground state of I2 is non-degenerate. Coupling between the 16 states of I2

is expected at large internuclear distances where the states are nearly degener-

ate. In the absence of a detailed treatment of this non-adiabatic coupling it is

reasonable to use Born-Oppenheimer approximation at all internuclear distances.

In this approximation each collision occurs on one potential energy surface, but

only 1/16 of the collisions occur in the ground state surface [55, 62–64].

Thus, when comparing rate constants with experimental values, an electronic

degeneracy factor [56, 57]

ge(T ) =gcomp

greact1greact2

(3.30)

should be included. The numerator denotes the degeneracy of the whole molecular

system and the denominator accounts for the degeneracies of the reactants. Note

that these factors must include the dependence on temperature of spin orbit

splitting. In this way, the rate constant in (3.18) will be expressed as

k (T ) = ge (T )

(8kBT

πµ

)1/2Nr

Nπb2max (3.31)

where ge (T ) is the temperature-dependent electronic degeneracy factor [55, 56,

63, 65], introduced to account for the probability of a collision occurring on a

particular surface.

3.5 Products properties from QCT runs

In QCT calculations, the end point of a trajectory occurs when it enters a region

of phase space designated as reactants or products space [4, 66–68]. Once the

product molecules have been determined by testing interatomic distances using


geometric and energetic criteria, it can be determined whether the molecules are

in bound, quasi-bound or dissociative states.

In the chemical reaction:

A + B → C + D (3.32)

the properties with interest are commonly: the C + D relative translational en-

ergy, the C and D vibrational and rotational energies and the scattering angle

between the initial A + B and the final C + D relative velocity vectors. These

properties are calculated from space-fixed Cartesian coordinates and momenta at

the termination of a classical trajectory. The procedures here described are incor-

porated in the general chemical dynamics program VENUS [69] used to calculate

the trajectories for the reactions studied in this thesis.

3.5.1 Relative velocity and translational energy

The product relative velocity is the difference between the velocities of the centers

of mass of C and D. For example for the x component of the center of mass

position and velocity of product D is given by:

XD =

nD∑

i=1

mixi/MD , XD =

nD∑

i=1

mixi/MD (3.33)

where the sum is over nD, the number of atoms in D, mi are the masses and xi

are the x coordinates of the atoms. MD is the mas of D, upper case variables

identify the center of mass position and velocity. The product relative velocity is

the time derivative of the relative coordinate:

R = RD − RC

= (XD −XC)i + (YD − YC)j + (ZD − ZC)k (3.34)

= Rxi +Ryj +Rzk

R = Rxi + Ryj + Rzk

where i, j,k are the unitary vectors in the x,y,z directions respectively. The pro-

duct translational energy is:

Erel =µCDR · R

2(3.35)

3.5 Products properties from QCT runs 69

where µCD = MCMD/(MC + MD) is the CD reduced mass. Erel may also be

written as the sum of the relative translational energy along the line of centers

C − D and the energy of the orbital (angular) motion:

Erel =µCDR

2

2+

l2

2µCDR2(3.36)

being R the module of the velocity along line of centers (radial velocity), and R

the distance between them:

R = (R · R)1/2 , R =RxRx +RyRy +RzRz

R(3.37)

l is the orbital angular momentum (and l its module):

l = µCDR × R = lxi + lyj + lzk (3.38)

3.5.2 Velocity scattering angle

The velocity scattering angle θv is the angle between the relative velocity vector

for the reactants R0 and the product’s relative velocity vector R, given by:

θv = cos−1

(R · R0

RR0

)(3.39)

3.5.3 Internal energy

To calculate the internal rotational and vibrational energy of the products requires

the coordinates and velocities of each atom of the molecule in the center of mass

frame of the molecule:

x′i = xi −XD , x′i = xi − XD , i = 1, nD (3.40)

the internal energy of D is:

ED = TD + VD (3.41)

where TD and VD are the kinetic and vibrational energies of D respectively. VD is

determined from the potential energy function and TD is given by:

TD =

nD∑

i=1

mi(x2i + y2

i + z2i )

2(3.42)


3.5.4 Rotational angular momentum

The rotational angular momentum j of the product molecule D is the sum of the

angular momentum ji of the individual atoms of D relative to its center of mass:

jD =

nD∑

i=1

ji = jxi + jyj + jzk (3.43)

the atomic angular momentum is given by:

ji = mir′i × r′i (3.44)

The total angular momentum of the C + D products is the vector sum:

L = l + jC + jD (3.45)

3.5.5 Rotational and vibrational energies

If the product correspond to a diatomic species, same procedure as previously

described in equations (3.35-3.38) can be used. The internal energy TD of a

diatomic molecule 1-2, can be written:

TD =µ12r

2

2+

j2

2µ12r2(3.46)

where µ12 is the reduced mass of D, r is the 1-2 bond length. Similar expressions

than (3.35-3.38) are used for r and r. The rotational quantum number J for D

is found from the expression:

j =√J(J + 1)h (3.47)

Since calculation is classical, non-integer values are obtained for J ; then, rounding

is often used.

The vibrational quantum number is obtained with help of semi-classical quan-

tization condition [70, p71]:∮prdr = (n +

1

2)2πh (3.48)

where the momentum pr = µr and the cyclic integral denotes integration over

one orbit. From the equations (3.41) and (3.46) pr is given by:

pr =

[2µ12

(ED − j2

2µ12r2− VD(r)

)]1/2

(3.49)

3.5 Bibliography 71

as for J , non-integer values of n are often obtained.

If D is a polyatomic species it is not a simple to calculate rotational and

vibrational quantum numbers [4]. Semi-classical quantization can be used as

in case of diatomic molecules, presented above. However, mostly because of

the multidimensional character, such a task is tedious. As a result most of the

semi-classical quantization has been limited to triatomics. So far, there is not a

general form to calculate both rotational and vibrational quantum numbers from

its Cartesian coordinates [4].

It is always possible to calculate the average vibrational and rotational energies

of a polyatomic product:

ED = 〈EvibD 〉 + 〈Erot

D 〉 (3.50)

Because of the ro-vibrational coupling the vibrational and rotational energies

of D, EvibD and Erot

D , will fluctuate as the molecule vibrates. An instantaneous

rotational energy for D may be calculated from:

ErotD =

1

2ωD · jD (3.51)

jD has been defined in (3.43) and ωD is the angular velocity of D.

The average rotational energy is computed by averaging over the longest vi-

brational period of the product. Then, by means of equation (3.51), the average

vibrational energy can also be obtained.

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Part II

Case Studies

Chapter 4

CR-CC and MRCI(Q) studies forrepresentative cuts of H2S

J. Mol. Struct. Theochem 859, 22-29 (2008).

A comparison of single-reference coupled-clusterand multi-reference configuration interactionmethods for representative cuts of the H2S(1A′)potential energy surface

Y. Z. Song, A. Kinal, P.J.S.B. Caridade, P. Piecuch and A.J.C. Varandas

Departamento de Quımica, Universidade de Coimbra

3004-535 Coimbra Codex, Portugal.

(Received: May 24, 2007; accepted: February 26, 2008)

Abstract

Three cuts of the H2S(1A′) potential energy surface, which correspond to the dissocia-

tion of a single S–H bond [cut (i)], the simultaneous dissociation of both S–H bonds [cut

(ii)], and the C2v dissociation pathway leading to H2(X1Σ+

g ) and S(3p4 1D) [cut (iii)],

are examined with the conventional and completely renormalized (CR) coupled-cluster

(CC) methods and the multi-reference configuration interaction approach [MRCI(Q)].

The size extensive CR-CC method with singles, doubles, and non-iterative triples,

termed CR-CC(2,3), provides the results of the MRCI(Q) quality for cuts (i) and (iii).

To obtain a similar quality for cut (ii), the CR-CC(2,3) energy must be corrected for

the effect of quadruply excited clusters.

82 Y. Z. Song, A. Kinal, P.J.S.B. Caridade, P. Piecuch and A.J.C. Varandas

1 Introduction

Atmospheric sulfur plays a major role in environmental studies, particularly

in areas such as acid rain, air pollution, and global climate change, with the

S(3P, 1D) + H2 reaction receiving much theoretical and experimental considera-

tion over the years. Amongst the many experiments that have been performed

are those by Lee and Liu,1 who carried out molecular beam experiments to mea-

sure the integral cross sections and vibrational state-resolved differential cross

sections in collisions of S(1D) with H2, D2, and HD. Equilibrium geometries

and dissociation energies of H2S have been studied experimentally as well.2–4

There have been several theoretical investigations of the S(1D) + H21, 5–9 and

S(3P ) + H28, 10 reactions. In particular, Zyubin et al.6 studied the dynamics of

the S(1D) + H2/D2 reactions using an ab initio potential energy surface (PES)

calculated at the multi-reference configuration interaction (MRCI) level with a

multi-configuration self-consistent field (MCSCF) reference wave function and a

number of correlation-consistent basis sets ,11, 12 including the simplified variant

of the aug-cc-pVQZ basis set, abbreviated as pvqz′+. Ho et al.7 studied the

S(1D) + H2 reaction using a PES obtained by an improved interpolation of the

MRCI data of Zyubin et al.6 on a three-dimensional regular grid in Jacobi co-

ordinates. Both calculations6, 7 indicate a barrier-free insertion pathway along

T-shaped geometries.

In this paper, we examine representative cuts of the H2S(1A′) PES using a

number of cost-effective, conventional as well as completely renormalized (CR),

single-reference coupled-cluster (CC) methods, as recently incorporated 13, 14 in

the GAMESS package,15 and the internally contracted variant of the MRCI ap-

proach including the quasi-degenerate Davidson correction [MRCI(Q)],16, 17 as

implemented in MOLPRO.18 Amongst the former methods, we consider the ba-

sic CCSD (CC singles and doubles)19–21 approach and the widely used CCSD(T)

approximation,22 in which a non-iterative, quasi-perturbative correction due to

triply excited clusters is added to the CCSD energy, as well as the completely

renormalized extensions of CCSD(T), including the original CR-CCSD(T) ap-

proach23–25 and its improved, size extensive generalization termed CR-CC(2,3)26–28

(see Refs. [29–33], for selected reviews). As in the case of CCSD(T), the CR-

J. Mol. Struct. Theochem 859, 22-29 (2008). 83

CCSD(T) and CR-CC(2,3) energies are calculated by adding the a posteriori

corrections due to triply excited clusters to the CCSD energy, and the differ-

ence only is in the equations defining the triples corrections, which in the case

of CR-CCSD(T) and CR-CC(2,3) are derived from the asymmetric energy ex-

pressions defining the method of moments of CC equations.23–26, 29–33 Since the

triples levels of the single-reference CC and CR-CC theories may not be suffi-

cient to handle the fragmentation of H2S into non-interacting atoms, which one

of the cuts of the H2S(1A′) PES examined in this work leads to, in addition

to the CCSD(T), CR-CCSD(T) and CR-CC(2,3) approaches, we consider the

CR-CCSD(TQ) method,23–25 in which the suitably renormalized non-iterative

correction due to a combined effect of triply and quadruply excited clusters is

added to the CCSD energy, as well as the augmented variant of CR-CC(2,3),

termed CR-CC(2,3)+Q, in which the CR-CC(2,3) approach is corrected for the

dominant quadruples effects by adding the difference of the CR-CCSD(TQ) and

CR-CCSD(T) energies to the CR-CC(2,3) energy (cf., e.g., Refs. [33–36]).

The common characteristics of the CC and CR-CC approaches examined in

this work, which make them particularly appealing in the context of laborious

and repetitive single-point calculations that are needed to generate PESs for dy-

namical and spectroscopic studies, is the ease-of-use, related to the fact that these

are all single-reference methods, and the relatively low computer costs, which do

not exceed the iterative n2on

4u and non-iterative n3

on4u or n2

on5u steps (no and nu

are the numbers of occupied and unoccupied orbitals in a molecular basis set

used in correlated calculations). The potential problem that all single-reference

methods face is the difficulty with describing bond breaking situations (cf., e.g.,

Refs. [13, 14, 23–34, 37], and references therein), which traditionally require a

multi-reference description represented in this study by the MRCI(Q) approach.

Thus, one of the main objectives of this work is to establish the minimum level

of the single-reference CC theory that would be appropriate for examining the

H2S(1A′) PES, so that we could either eliminate the need for laborious MRCI(Q)

calculations or, with the help of the energy switching/morphing38 procedures,

reduce the usage of MRCI(Q) to a minimum.

In order to address the above objective and examine the relative performance

of the CC, CR-CC, and MRCI(Q) approaches in calculations for H2S, which is


a system whose dynamics and spectroscopy we would like to investigate in the

future, the following three important cuts of the H2S(1A′) PES are considered in

this study: (i) the dissociation of a single S–H bond which correlates with the

H(1s 2S) + SH(X 2Π) asymptote, (ii) the simultaneous, C2v-symmetric, disso-

ciation of both S–H bonds correlating with the 2H(1s 2S) + S(3p4 3P ) channel,

and (iii) the C2v-symmetric, minimum energy dissociation pathway leading to

H2(X 1Σ+g ) and S(3p4 1D) obtained with MRCI(Q). By probing the above three

cuts of the H2S(1A′) surface, which correspond to different types of bond breaking

situations in H2S and widely varying energy regions, and by comparing various CC

and CR-CC results with the reliable MRCI(Q) data, we can determine what level

of single-reference CC theory is capable of producing the results of the MRCI(Q)

or similar quality. In order to make sure that our conclusions are not affected

by a particular choice of a basis set, the CC, CR-CC, and MRCI(Q) calculations

are performed with a number of basis sets belonging to the correlation-consistent

aug-cc-pCVXZ variety.11, 12, 39 Although for the general purpose use in dynamics

applications, one might also wish to consider the S(3P ) + H2 reaction and the

corresponding triplet surface, that in itself would be an independent and labori-

ous study, which we plan to pursue in a future work. The present paper focuses

on the lowest-energy singlet surface.

2 Computational details

The MRCI(Q) approach of Refs. [16, 17], implemented in MOLPRO18 and ex-

ploited in this work, is based on the usual idea of generating all single and double

excitations from the multi-dimensional reference space (MRCISD), which is com-

bined with the internal contraction scheme that reduces the huge dimensionality

of the resulting CI eigenvalue problem to manageable sizes and with the a pos-

teriori quasi-degenerate Davidson corrections that take care of the higher-order

excitations neglected at the MRCISD level. The MRCI(Q) calculations reported

in this work were carried out using the multi-determinantal reference function

obtained in the single-root complete-active-space self-consistent-field (CASSCF)

calculations. The active space used in the CASSCF and subsequent MRCI(Q)

calculations consisted of valence orbitals that correlate with the 1s shells of the hy-


drogen atoms and the 3s and 3p shells of the sulfur. Since core electrons affect the

energy differences between different points on the PES (cf., e.g., Refs. [40, 41]), all

electrons were correlated in the MRCI(Q) calculations, i.e., the single and double

excitations from both the core and valence orbitals in each reference determinant

were allowed in the relevant CI wave function expansions.

For consistency, all electrons were also correlated in the single-reference CC

and CR-CC calculations, which used the spin- and symmetry-adapted restricted

Hartree-Fock (RHF) determinant as a reference and which were performed using

the original computer programs developed at Michigan State University13, 14, 26, 28

and incorporated into the GAMESS package.15 As explained in the Introduction,

in addition to the conventional CCSD and CCSD(T) approximations, in this

study we used a few methods that belong to a family of renormalized CC ap-

proaches13, 14, 23–28 (cf. Refs. [29–33] for reviews).

In analogy to CCSD(T), all renormalized CC methods, including the CR-

CCSD(T),23–25 CR-CC(2,3),14, 26–28 CR-CCSD(TQ),23–25 and CR-CC(2,3)+Q33–36

approaches used in this work, are based on an idea of adding non-iterative a poste-

riori corrections due to higher–than–doubly excited clusters to the CCSD energy

[triples in the CR-CCSD(T) and CR-CC(2,3) cases, and triples and quadruples

in the CR-CCSD(TQ) and CR-CC(2,3)+Q cases]. One of the main advantages

of the renormalized CC approaches is their ability to improve the poor CCSD(T)

results in multi-reference situations involving bond breaking and biradicals, with-

out making the calculations considerably more expensive and without using the

multi-determinantal reference wave functions (cf., e.g., Refs. [13, 14, 23–37, 42–53]

for selected examples). Indeed, the most expensive steps of the CR-CCSD(T)23–25

and CR-CC(2,3)14, 26–28 approaches scale as n2on

4u in the iterative CCSD part and

2n3on

4u in the non-iterative part related to the calculations of the relevant triples

corrections. For comparison, the computer costs of determining the triples correc-

tion of CCSD(T) scale as n3on

4u. The CR-CCSD(TQ)23–25 and CR-CC(2,3)+Q33–36

methods, in which the suitably renormalized non-iterative corrections due to

triply and quadruply excited clusters are added to the CCSD energy, are more

expensive, since, in addition to the n2on

4u steps of CCSD and 2n3

on4u steps of the

triples corrections, one needs the 2n2on

5u steps to calculate the corrections due

to quadruples, but even the most demanding 2n2on

5u steps of CR-CCSD(TQ) and


CR-CC(2,3)+Q are much less expensive than the iterative steps related to the full

inclusion of triply and quadruply excited clusters (n3on

5u and n4

on6u, respectively).

One can propose a few variants of the CR-CC(2,3), CR-CCSD(TQ), and CR-

CC(2,3)+Q approaches. In this work, we focus on the two basic variants of

the CR-CC(2,3) approach, namely, the full CR-CC(2,3) method, as described in

Refs. [14, 26–28], and the simplest CR-CC(2,3),A variant, in which the diago-

nal matrix elements of the similarity-transformed Hamiltonian of CCSD involv-

ing triply excited determinants, 〈Φabcijk |H(CCSD)|Φabc

ijk〉 that enter the CR-CC(2,3)

triples correction are replaced by the spin-orbital energy differences character-

izing triple excitations, (ǫa + ǫb + ǫc − ǫi − ǫj − ǫk), where ǫp are the usual

spin-orbital energies (see, e.g., Ref. [28]; particularly Eq. (23)). It is interest-

ing to examine the full implementation of CR-CC(2,3) as well as its simplified

CR-CC(2,3),A version, since the CR-CC(2,3),A approach is equivalent to the

CCSD(2)T method of Ref. [48], which also aims at improving the performance

of CCSD(T) at larger internuclear separations when the RHF orbitals used in

this study are employed. The CR-CCSD(TQ) calculations were performed us-

ing the most complete variant ‘b’ of this approach [CR-CCSD(TQ),b], intro-

duced in Ref. [25] and reviewed in Ref. [29, 31] (see Eqs. (87)-(94) in Ref. [29] or

Eqs. (62), (65) and (68) in Ref. [31]). The same variant ‘b’ of CR-CCSD(TQ) is

used to correct the CR-CC(2,3) results for the effect of quadruples through the

CR-CC(2,3)+Q method. The CR-CC(2,3)+Q energy is calculated in as {CR-

CC(2,3) + [CR-CCSD(TQ),b - CR-CCSD(T)]}, where CR-CC(2,3) is the full

CR-CC(2,3) defined in Refs. [14, 26–28] and where we use the energy difference

[CR-CCSD(TQ),b - CR-CCSD(T)] to estimate the effect of quadruply excited

clusters, neglected in the CR-CC(2,3) calculations.

The biggest advantage of the CR-CC methods, when compared to multi-

reference approaches, is their black-box character. In contrast to the MRCI(Q)

and other multi-reference calculations, one does not have to choose active orbitals

or perform additional operations, such as internal contractions of numerous con-

figuration state functions (CSFs) that the MRCISD algorithms produce, to carry

out the CR-CC calculations. The CR-CC methods, particularly CR-CCSD(T)

and CR-CC(2,3), are also cost effective when compared to the MRCISD-based

approaches. For example, the CPU operation count of a typical MRCISD calcu-


lation scales as ∼ MkMRCIn2on

4u, where M is the number of the reference CSFs

and kMRCI is the number of MRCISD iterations required to achieve convergence.

For comparison, the CPU operation count of the CR-CCSD(T) and CR-CC(2,3)

calculations scales as ∼ τkCCSDn2on

4u + 2n3

on4u, where τ = 1 for CR-CCSD(T),

τ = 2 for CR-CC(2,3) [which requires to converge the standard and then left

CCSD equations14, 26–28], and kCCSD is the number of CCSD iterations required

to achieve convergence (we have assumed that no ≪ nu). Assuming that kCCSD

is similar to kMRCI and that M ≫ 1, which is often the case, the CR-CCSD(T)

and CR-CC(2,3) calculations are less expensive than the MRCISD calculations

as long as 2no < kCCSD(M − τ) ≈ kMRCIM . This condition is satisfied in the

calculations performed in this work, where no = 9, M = 65(assuming the Cs

symmetry), and kCCSD ≈ kMRCI ≈ 10 − 15 if the convergence threshold is set

at 10−7 hartree. The internal contractions of CSFs and other commonly used

procedures, such as selection thresholds for the dominant configurations, reduce

the costs of MRCISD calculations, but, ultimately, the CR-CCSD(T) and CR-

CC(2,3) methods can certainly be regarded as cost-effective compared to MRCI

techniques. After all, one could also try to reduce costs of the CR-CCSD(T), CR-

CC(2,3), and other CR-CC calculations by using selection thresholds for cluster

amplitudes and higher-than-double excitations. The CR-CCSD(TQ) and CR-

CC(2,3)+Q methods are more expensive than the CR-CCSD(T) and CR-CC(2,3)

approaches, but they can also be less expensive than the MRCISD calculations

as long as 2nu < kCCSDM (again, we have assumed that kCCSD ≈ kMRCI, M ≫ 1,

and no ≪ nu). In our case, nu = 45, 112, and 217 for the aug-cc-pCVXZ basis

sets with X = 2 − 4, respectively, used in this study, so that the CPU operation

counts characterizing the CR-CCSD(TQ) and CR-CC(2,3)+Q calculations for

H2S are smaller than those of the corresponding MRCISD calculations, although

the benefits of using the CR-CCSD(TQ) and CR-CC(2,3)+Q approaches, when

compared to MRCI techniques, in calculations with very large basis sets, where

nu can be very large, are less obvious. We have to point out though that the CR-

CCSD(TQ) and CR-CC(2,3)+Q methods use a single Hartree–Fock determinant

as a reference, whereas all MRCI methods rely on multi-determinantal references

and the idea of selecting appropriate active orbitals, which is not always obvi-

ous. We should also mention that as all CC approaches, the CR-CCSD(TQ) and


CR-CC(2,3)+Q methods offer a significantly better treatment of dynamical cor-

relation effects, when compared to MRCI. The above discussion of the relative

computer costs, combined with the black-box character of the CR-CC approaches,

make the CR-CC methods an attractive alternative to multi-reference techniques.

All calculations employed the standard aug-cc-pCVXZ basis sets, with X =

D,T,Q,11, 12, 39 in which additional tight functions are added to the valence basis

sets of the aug-cc-pVXZ quality to improve the description of core and core-

valence correlation effects, as given in Ref. [54]. To facilitate our presentation, we

use a simplified notation in which we refer to the aug-cc-pCVXZ basis set via the

cardinal number X (X = 2 for aug-cc-pCVDZ, X = 3 for aug-cc-pCVTZ, X = 4

for aug-cc-pCVQZ).

3 Results and discussion

To explore the benefits of exploiting the renormalized CC methods in studies

of the H2S(1A′) PES, particularly those offered by the CR-CC(2,3) and the

CR-CC(2,3)+Q method, as compared to the MRCI(Q) approach which pro-

vides an accurate representation of the global PES of H2S, we performed the

CCSD, CCSD(T), CR-CCSD(T), CR-CC(2,3),A≡CCSD(2)T , CR-CC(2,3), CR-

CCSD(TQ)≡CR-CCSD(TQ),b, CR-CC(2,3)+Q, and MRCI(Q) calculations for

cuts (i) to (iii) of the global H2S PES mentioned in the Introduction. Specif-

ically, in cut (i), one of the S–H bonds and the H–S–H angle were kept fixed

at the equilibrium values taken from Ref. [2] (Re = 1.3356 A and αe = 92.12

degree, respectively), while the other S–H bond length was varied from R = Re

to R = 5Re, which is essentially equivalent to the single-bond dissociation of H2S

into H(1s 2S)+SH(X 2Π). In the case of cut (ii), the H–S–H angle was kept fixed

at its equilibrium value, while the two S–H bonds were symmetrically stretched,

from R = Re to R = 5Re, i.e., until the C2v-symmetry-preserving dissociation of

H2S into 2H(1s 2S)+S(3p4 3P ) occurs. Finally, in the C2v-symmetric cut (iii), we

followed the approximate minimum energy path toward the dissociation of H2S

into H2(X1Σ+

g ) and S(3p4 1D), as defined by the coordinate Y that measures the

distance between the S nucleus and the line connecting both H nuclei, with the

H–S–H angle α optimized at each value of Y ranging from 0.8 A to 2.0 A. The


optimization of the α angle for each value of Y along this pathway was performed

at the MRCI(Q)/X=4 level. The equilibrium values of Y and α are Ye = 0.9268

A and αe = 92.12 degree.

The CC, CR-CC, and MRCI(Q) results for the three PES cuts (i)–(iii) and

three aug-cc-pCVXZ basis sets examined in this work are summarized in Ta-

bles 1–3. The corresponding total electronic energies at the equilibrium geometry

are given in Table 4. In each case, the energy is calculated relative to the corre-

sponding energy at the minimum taken from Ref. [2], as [E −E(Re, αe)] for cuts

(i) and (ii), and [E − E(Ye, αe)] for cut (iii). The single S–H bond dissociation

pathway of cut (i) is characterized by the lowest energies, which do not exceed

40,000 cm−1 (∼ 32, 000 − 34, 000 cm−1 if the most accurate CR-CC(2,3)+Q and

MRCI(Q) approaches are exploited). The C2v-symmetric minimum energy disso-

ciation path leading to H2(X 1Σ+g ) + S(3p4 1D) defining cut (iii) goes to similar

energies, while the much higher energies, on the order of ∼ 60, 000−70, 000 cm−1

(∼ 60, 000− 65, 000 cm−1 in the CR-CC(2,3)+Q and MRCI(Q) calculations) are

reached for the simultaneous dissociation of both S–H bonds defining cut (ii).

To facilitate the analysis, we show in Figure 1 the differences ∆E between the

CC/CR-CC and the corresponding MRCI(Q) energies, defined for each method

as [E − E(Re, αe)] ([E − E(Ye, αe)] for cut (iii)). As in Tables 1–3, this is done

for each of the three aug-cc-pCVXZ basis sets used in our calculations.

The last Table 4 compares the approximate dissociation energies correspond-

ing to cuts (i) (the H2S → H + SH dissociation) and (ii) (the H2S → 2H + S

dissociation) calculated from the various ab initio approaches examined in this

work with the corresponding experimental and other theoretical values. For each

electronic structure method, the corresponding dissociation energy is defined as a

difference between the energy at largest internuclear separation in a given cut and

the corresponding energy at the equilibrium geometry used in this work, taken

from Ref. [2]. We should note that for some of the CC/CR-CC methods, includ-

ing the CCSD(T) approach for cuts (i) and (ii), and the CCSD, CR-CCSD(T),

CR-CC(2,3),A, and CR-CC(2,3) approaches for cut (ii), the calculated dissoci-

ation energies are meaningless, since the corresponding potential functions have

well-pronounced humps at larger S–H separations and are, therefore, useless for

the dissociation energy calculations. However, the CCSD, CR-CCSD(T), CR-


CC(2,3),A, CR-CC(2,3), CR-CCSD(TQ), and CR-CC(2,3)+Q dissociation ener-

gies obtained for cut (i) and the CR-CCSD(TQ) and CR-CC(2,3)+Q dissociation

energies obtained for cut (ii) are quite meaningful, since the corresponding poten-

tial energy curves along the relevant bond breaking coordinates behave in at least

qualitatively correct manner. It is clear from Table 5 that for both types of frag-

mentations of H2S, represented by cuts (i) and (ii), the approximate dissociation

energies resulting from the CR-CC(2,3)+Q calculations show very small differ-

ences with the corresponding MRCI(Q) and experimental values, being within

the accuracy expected for the best theoretical values currently available in the

literature [∼ 1 kcal/mol for cut (i) and ∼ 2 − 5 kcal/mol for cut (ii)]. Table 5

shows that the CR-CC(2,3) dissociation energy obtained for cut (i) is in excellent

agreement with the MRCI(Q), experimental, and other theoretical data as well.

This can be understood, since this is the case of the H2S → H + SH single-bond

dissociation, which can be well described at the triples level of the CC theory

if the triply excited clusters are incorporated in the CC calculations in a proper

manner, as is done in the CR-CC(2,3) and other CR-CC approaches. The stan-

dard way of incorporating triples via the CCSD(T) theory is clearly insufficient,

producing humps on the CCSD(T) PES at larger S–H separations of cuts (i) and

(ii) (see Tables 1 and 2, and the discussion below). This is reflected in the poor

and apparently meaningless dissociation energies resulting from the CCSD(T)

calculations, even for the single-bond dissociation corresponding to the ‘easier’

cut (i) (see Table 5).

Tables 1–3 and Figure 1 shows the challenges facing the single-reference CC

methods when describing PESs along bond breaking coordinates, while emphasize

the usefulness of the three selected dissociation pathways in examining the relative

performance of the CC/CR-CC and MRCI(Q) methods, since cuts (i)–(iii) probe

different types of bond stretching/breaking situations. They also enable us to

demonstrate that the single-reference CR-CC methods, particularly CR-CC(2,3)

and CR-CC(2,3)+Q, can provide reliable information about the PES of H2S.

We begin with the discussion of the CR-CC(2,3) and CR-CC(2,3)+Q results

obtained for cut (i). Not surprisingly, the CCSD approach is qualitatively correct

in this case. Cut (i) corresponds to single-bond breaking, which is, in a zero-

order approximation, a two-electron process. In consequence, the CCSD approach


Table

1.

Dis

soci

atio

nof

asi

ngl

eS

–Hb

ond

inH

2S

[into

H(1s

2S

)+

SH

(X2Π

);cu

t(i)

].

CC

SD

CC

SD

(T)

CR

-CC

SD

(T)

CR

-CC

(2,3

),A

a)

RX

=2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

1.25R

e65

1871

8972

9363

1469

4070

3463

6370

1271

1163

4469

7770

721.

50R

e17

006

1797

418

193

1642

017

280

1747

616

581

1750

317

711

1652

517

404

1759

81.

75R

e25

210

2655

526

883

2393

825

077

2535

824

351

2562

225

929

2423

225

412

2569

02.

00R

e30

408

3213

032

556

2808

029

438

2977

628

983

3058

330

971

2877

930

224

3056

22.

50R

e34

984

3719

537

586

2981

931

180

3158

432

435

3438

434

765

3206

433

731

3403

63.

00R

e36

014

3839

839

021

2897

830

162

3046

532

906

3493

935

449

3246

234

159

3455

54.

00R

e36

485

3898

039

634

2777

228

728

2896

532

892

3494

635

459

3241

034

097

3448

75.

00R

e36

564

3908

339

742

2739

928

264

2848

132

799

3483

835

347

3231

833

989

3437

3

NP

E47

5958

8862

1243

2048

4949

7213

9120

2922

0610

3913

5713

51

CR

-CC

(2,3

)C

R-C

CS

D(T

Q)b)

CR

-CC

(2,3

)+Q

c)M

RC

I(Q

)

RX

=2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

1.25R

e63

2169

5170

5263

4169

9070

8962

9969

2970

3063

0469

6970

811.

50R

e16

429

1731

117

529

1652

317

446

1765

516

371

1725

417

473

1635

117

247

1746

21.

75R

e23

989

2517

125

480

2424

825

518

2582

723

886

2506

825

379

2381

024

939

2522

32.

00R

e28

369

2981

730

191

2883

430

433

3082

528

220

2966

830

045

2807

129

367

2968

62.

50R

e31

410

3309

533

476

3223

934

187

3457

531

214

3289

933

286

3098

532

366

3271

13.

00R

e31

746

3349

033

948

3271

534

749

3526

531

554

3329

933

764

3148

232

867

3321

34.

00R

e31

746

3345

133

931

3272

934

788

3531

031

583

3329

333

781

3158

332

968

3331

25.

00R

e31

710

3341

633

866

3265

134

699

3521

631

562

3327

733

735

3159

132

975

3331

8

NP

E40

874

779

412

1718

6120

4425

857

362

60

00

a)

Equ

ival

ent

toth

eC

CS

D(2

) Tm

eth

od

ofR

ef.

[48]

.b)

Th

eC

R-C

CS

D(T

Q),

bap

pro

ach

ofR

ef.

[25]

.c)

CR

-CC

(2,3

)+Q

=C

R-C

C(2,3

)+

[CR

-CC

SD

(TQ

),b−

CR

-CC

SD

(T)]

.


produces errors relative to MRCI(Q) that monotonically increase with the S–H

distance R. The most accurate non-iterative treatment of triply excited clusters

offered by the CR-CC(2,3) approach is sufficient in this case to produce a highly

accurate description. Indeed, the CR-CC(2,3) method reduces the 4,973, 6108,

and 6,424 cm−1 maximum errors in the CCSD/X= 2−4 results for cut (i), relative

to MRCI(Q), to 425, 729, and 765 cm−1, respectively (see Table 1 and Figure 1).

The CR-CC(2,3)+Q approach reduces these maximum errors even further, to 229,

533, and 575 cm−1, respectively. These are impressive improvements if we realize

the serious difficulties the traditional CC approaches, such as CCSD(T), have in

describing the PES of H2S, and the fact that the energies characterizing cut (i)

exceed 30, 000 cm−1 as the S–H bond is significantly stretched. The differences

between the CR-CC(2,3) or CR-CC(2,3)+Q and MRCI(Q) energies on the order

of a few hundred cm−1 at larger S–H separations are within the accuracy of the

MRCI(Q) calculation, so that we can regard the CR-CC(2,3), CR-CC(2,3)+Q,

and MRCI(Q) results as essentially equivalent. This is emphasized by the small

non-parallelity errors (NPEs) relative to MRCI(Q) characterizing the CR-CC(2,3)

and CR-CC(2,3)+Q data in Table 1 (NPE is defined as the difference between

the maximum and minimum signed errors relative to MRCI(Q) along a PES cut).

The situation created by cut (ii) is entirely different and more challenging.

In this case, the CCSD approach is no longer qualitatively correct, producing a

large, unphysical hump in the region of the intermediate R values. This can be

understood if we realize that cut (ii) corresponds to a double S–H dissociation,

which is, in a zero-order description, a four-electron process that requires the

explicit inclusion of the triply as well as quadruply excited clusters. The proper

treatment of these clusters offered by the CR-CC(2,3)+Q method leads to a

highly accurate description of cut (ii), which can compete with the MRCI(Q)

results (see Table 2 and Figure 1). For example, the CR-CC(2,3)+Q approach

reduces the 5989 and 8402 cm−1 errors relative to MRCI(Q) in the CCSD results

at R = 2Re and R = 3Re obtained with the X = 4 basis set, where the MRCI(Q)

energies become as high as 58,056 and 63,560 cm−1, respectively, to as little as 580

and 684 cm−1, while providing a smooth description of cut (ii), with the energies

monotonically increasing R. The relative small NPE values relative to MRCI(Q)

characterizing the CR-CC(2,3)+Q results, which range, depending on the basis


Table

2.C

2v-s

ym

met

ric

dou

ble

dis

soci

atio

nof

H2S

[into

2H(1s

2S

)+

S(3p4

3P

);

cut

(ii)

].

CC

SD

CC

SD

(T)

CR

-CC

SD

(T)

CR

-CC

(2,3

),A

a)

RX

=2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

1.25R

e12

902

1428

214

493

1247

113

759

1395

012

581

1391

614

114

1253

513

839

1402

91.

50R

e33

610

3560

336

056

3234

534

119

3452

632

735

3463

635

066

3257

534

387

3479

01.

75R

e49

671

5247

153

159

4687

149

280

4988

047

971

5063

151

278

4752

050

003

5059

52.

00R

e59

378

6312

264

045

5394

757

080

5785

956

656

6019

561

053

5549

258

767

5954

02.

50R

e64

981

7028

871

610

5046

654

578

5555

560

458

6533

366

515

5670

060

860

6185

23.

00R

e64

759

7052

171

962

4335

146

654

4738

959

496

6462

065

859

5472

158

675

5960

94.

00R

e64

202

7005

671

523

3750

639

800

4024

558

436

6351

864

750

5324

757

029

5792

55.

00R

e64

009

6986

371

327

3603

737

971

3833

158

036

6307

164

289

5275

156

466

5735

0N

PE

5528

7717

8246

2396

025

165

2559

539

0631

2835

3080

2778

7378

43

CR

-CC

(2,3

)C

R-C

CS

D(T

Q)b)

CR

-CC

(2,3

)+Q

c)M

RC

I(Q

)

RX

=2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

1.25R

e12

477

1376

913

979

1253

213

867

1406

712

429

1372

113

931

1244

613

813

1403

71.

50R

e32

352

3415

934

594

3260

234

503

3493

532

219

3402

634

463

3218

934

050

3450

01.

75R

e46

998

4949

150

111

4776

150

406

5105

746

787

4926

749

890

4664

549

044

4966

32.

00R

e54

587

5792

358

746

5668

860

098

6094

454

620

5782

658

636

5448

557

328

5805

62.

50R

e55

199

5945

260

569

6389

767

941

6899

658

638

6205

963

050

5899

762

102

6290

83.

00R

e53

309

5740

958

440

6623

970

527

7166

460

052

6331

664

244

5961

762

747

6356

04.

00R

e52

231

5610

057

277

6714

971

657

7285

660

945

6423

965

384

5975

462

886

6369

75.

00R

e52

034

5579

856

790

6717

371

724

7293

261

171

6445

165

433

5977

162

900

6370

9N

PE

8090

7697

7609

7316

8770

9193

1759

1643

1830

00

0

a)

Equ

ival

ent

toth

eC

CS

D(2

) Tm

eth

od

ofR

ef.

48.

b)T

he

CR

-CC

SD

(TQ

),b

app

roac

hof

Ref

.25

.c)

CR

-CC

(2,3

)+Q

=C

R-C

C(2,3

)+

[CR

-CC

SD

(TQ

),b−

CR

-CC

SD

(T)]

.


set, between 1643 and 1830 cm−1, where the MRCI(Q) energies at larger S–H

separations are on the order of 60,000 cm−1, show that the CR-CC(2,3)+Q and

MRCI(Q) PESs are very similar to each other in the overall characteristics. The

inclusion of triply excited clusters alone, even when this is done via the best non-

iterative triples CR-CC(2,3) approximation, is not sufficient when we go beyond

the R = 2Re value of the S–H distance defining cut (ii). One needs the quadruply

excited clusters as well, as a comparison of the CR-CC(2,3), CR-CC(2,3)+Q, and

MRCI(Q) results in Table 2 and Figure 1 clearly indicates. On the other hand, if

we limit ourselves to the R ≤ 2Re region of cut (ii), where quadruples are of lesser

importance, the CR-CC(2,3) approach is sufficient, providing the energies which

are at most a few hundred cm−1 above the corresponding MRCI(Q) energies.

The CCSD approach is also erratic in the case of cut (iii), in which both S–H

bonds have to be somewhat stretched during the formation of the H2(X1Σ+

g )

and S(3p4 1D) products, although the errors in the CCSD energies relative to

the corresponding MRCI(Q) data are not nearly as large in this case as for the

other two cuts. One of the reasons is that the minimum energy path defining

the cut (iii) leads to the formation of the closed-shell H2S molecule, which is

a two-electron system, for which CCSD is exact, and the open-shell, but still

singlet S(3p4 1D) atom. In consequence, the CR-CC(2,3) or CR-CC(2,3)+Q

methods, in which one adds non-iterative corrections due to triply or triply and

quadruply excited clusters to the already reasonable CCSD data, lead to the

highly accurate description of cut (iii), which almost perfectly agrees with the

MRCI(Q) data. The differences between the CR-CC(2,3) or CR-CC(2,3)+Q and

MRCI(Q) energies along cut (iii) are on the order of 10 cm−1 in the entire range of

Y values examined in this work. The above discussion confirms the known fact

that one has to go beyond the basic CCSD approximation and account for the

dominant higher–than–doubly excited clusters to obtain an accurate description

of the PES. This is certainly true when bonds are stretched or broken, since one

needs triples or triples and quadruples to recover large non-dynamical correlation

effects, which in the single-reference CC theory have to be treated dynamically.

This is also true in the vicinity of the equilibrium geometry, where one needs to go

beyond doubles in CC calculations to obtain an accurate description of dynamical

correlation effects. The above analysis also emphasizes the excellent performance


Table

3.C

2v

dis

soci

atio

nof

H2S

into

H2(X

1Σ

+ g)

and

S(3p4

1D

)[c

ut

(iii

)].

CC

SD

CC

SD

(T)

CR

-CC

SD

(T)

CR

-CC

(2,3

),A

a)

YX

=2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

0.80

1037

978

951

1059

1018

993

1055

1008

983

1057

1013

987

0.90

1235

2911

4742

1244

3912

4540

1.00

241

411

432

190

359

378

201

371

391

196

366

385

1.10

1809

2183

2236

1677

2029

2074

1705

2068

2115

1695

2051

2097

1.20

4667

5274

5358

4424

4981

5052

4477

5055

5130

4457

5025

5096

1.50

1820

019

139

1929

817

559

1842

318

549

1770

518

609

1874

517

654

1853

818

667

1.75

2627

526

983

2716

826

058

2671

126

873

2611

926

801

2696

926

095

2675

826

922

2.00

3067

431

390

3165

430

648

3133

131

587

3067

731

383

3164

330

668

3135

731

614

NP

E19

0820

0120

1714

6215

5116

7715

5617

2218

1715

2117

1018

03

CR

-CC

(2,3

)C

R-C

CS

D(T

Q)b)

CR

-CC

(2,3

)+Q

c)M

RC

I(Q

)

YX

=2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

X=

2X

=3X

=4

0.80

1062

1019

994

1056

1012

986

1063

1022

998

1053

978

939

0.90

1247

4211

4540

1248

4311

4031

1.00

189

359

379

195

367

387

183

354

374

191

366

388

1.10

1675

2032

2083

1691

2053

2100

1660

2017

2068

1670

2025

2073

1.20

4423

4990

5071

4449

5027

5101

4395

4961

5043

4408

4978

5055

1.50

1757

918

475

1862

817

625

1853

518

672

1749

918

400

1855

617

504

1843

118

581

1.75

2605

526

712

2688

726

083

2676

426

935

2601

926

675

2685

226

023

2672

026

898

2.00

3063

431

297

3155

530

659

3136

131

624

3061

531

275

3153

630

586

3128

331

549

NP

E14

6716

2016

8815

0416

6317

6914

2515

9516

710

00

a)

Equ

ival

ent

toth

eC

CS

D(2

) Tm

eth

od

ofR

ef.

48.

b)T

he

CR

-CC

SD

(TQ

),b

app

roac

hof

Ref

.25

.c)

CR

-CC

(2,3

)+Q

=C

R-C

C(2,3

)+

[CR

-CC

SD

(TQ

),b−

CR

-CC

SD

(T)]

.


Table 4. Total electronic potential energy (in hartree) of H2S for the aug-cc-pCVXZ basis sets with X = 2 − 4, at its equilibrium geometry.

Method X=2 X=3 X=4

CCSD -399.0903427 -399.2653018 -399.3360413CCSD(T) -399.0962865 -399.2765937 -399.3489478CR-CCSD(T) -399.0954187 -399.2748652 -399.3470188CR-CC(2,3),A -399.0957269 -399.2755743 -399.3478581CR-cc(2,3) -399.0966205 -399.2767368 -399.3488560CR-CCSD(TQ) -399.0959706 -399.2755432 -399.3476998CR-CC(2,3)+Q -399.0971724 -399.2774147 -399.3495370MRCI(Q) -399.0943527 -399.2693303 -399.3394693

of the CR-CC(2,3) approach for cuts (i) and (iii) and of the CR-CC(2,3)+Q

method for cuts (i)–(iii). The CR-CC(2,3) and CR-CC(2,3)+Q methods represent

the non-traditional ways of incorporating triples or triples and quadruples in the

CC calculations. The traditional approach to the incorporation of triply excited

clusters in CC theory is offered by CCSD(T). It is interesting to examine what

types of accuracies are offered by the CCSD(T) approach when compared to its

CR-CC(2,3) analog.

As shown in Figure 1 (see, also, Tables 1–3), the CCSD(T) approach produces

very small differences with MRCI(Q), when the stretches of the S–H bonds are

small. For example, in the case of cut (i) the differences between the CCSD(T)/X=

4 and MRCI(Q)/X = 4 energies do not exceed 135 cm−1 when R does not

exceed 2Re. For cut (ii), the differences between the CCSD(T)/X = 4 and

MRCI(Q)/X=4 energies do not exceed 217 cm−1 in the R ≤ 2Re region, although

the positive energy difference between CCSD(T)/X = 4 and MRCI(Q)/X = 4 at

R = 1.75Re of 217 cm−1 becomes negative (-197 cm−1) at R = 2Re. This is a

signature of the failure of CCSD(T) in the R > 2Re region. Indeed, the small

positive differences between the CCSD(T)/X = 4 and MRCI(Q)/X = 4 energies

in the R < 2Re region of cut (ii) grow to -7353 cm−1 at R = 2.5Re, -16,171

cm−1 at R = 3Re, and -25,378 cm−1 at R = 5Re. This should be compared to

the 142, 684, and 1724 cm−1 differences between the CR-CC(2,3)+Q/X=4 and

MRCI(Q)/X = 4 energies at the same values of R. A very similar behavior is

observed for other basis sets and CCSD(T) also fails at larger S–H separations of


Table 5. Dissociation energies, in kcal mol−1, for cuts (i) and (ii) in the presentwork.

Cut (i) Cut (ii)

X=2 X=3 X=4 X=2 X=3 X=4

CCSD 104.54 111.74 113.63 183.01∗ 199.75∗ 203.94∗

CCSD(T) 78.34∗ 80.81∗ 81.43∗ 103.04∗ 108.56∗ 109.59∗

CR-CCSD(T) 93.78 99.61 101.06 165.93∗ 180.33∗ 183.81∗

CR-CC(2,3),A 92.40 97.18 98.28 150.82∗ 161.44∗ 163.97∗

CR-CC(2,3) 90.66 95.54 96.83 148.77∗ 159.53∗ 162.37∗

CR-CCSD(TQ) 93.35 99.21 100.69 192.06 205.07 208.52

CR-CC(2,3)+Q 90.24 95.14 96.45 174.90 184.27 187.08

MRCI(Q) 90.32 94.28 95.26 170.89 179.84 182.15

exp. 95.2a), 95.6b) 182.3c), 184.6b)

Other (theoretical) 94.6d), 94.7e) 180.1d), 179.8e)

Each CC, CR-CC, and MRCI(Q) value is obtained by forming a differencebetween the energy at largest internuclear separation in a given cut and thecorresponding energy at the equilibrium geometry taken from Ref. 2. Thedissociation energies that are meaningless due to the pathological behavior of agiven method at larger internuclear separations are marked by an asterisk.a) From Ref. 4.b) From Ref. 3.c) From Ref. 2.d) From Ref. 6, using seven active orbitals for the CASSCF and MRCI, and apvqz′+ basis set.e) From Ref. 6, using eight active orbitals for the CASSCF and MRCI, and apvqz′+ basis set.

cut (i) (see Tables 1 and 2, and Figure 1). For example, the small, 90 cm−1, dif-

ference between the CCSD(T)/X = 4 and MRCI(Q)/X = 4 energies at R = 2Re

grows to -1127 cm−1 at R = 2.5Re, -2748 cm−1 at R = 3Re, and -4837 cm−1 at

R = 5Re. Again, the CR-CC(2,3) and CR-CC(2,3)+Q approaches behave much

better, giving the much smaller errors of 765, 735, and 548 cm−1 in the CR-

CC(2,3)/X = 4 case and 575, 551, and 417 cm−1 in the CR-CC(2,3)+Q/X = 4

case at R = 2.5Re, 3Re, and 5Re, respectively. The only cut, for which CCSD(T)


works well, is cut (iii), where, in analogy to CR-CC(2,3), the differences between

the CCSD(T) and MRCI(Q) energies are on the order of 10 cm−1 in the entire

Y = 0.8 − 2.0 A region.

The approximate variant of CR-CC(2,3), termed CR-CC(2,3),A, which, as

stated earlier, is equivalent to the CCSD(2)T method of Ref. [48], and the CR-

CCSD(T) approach of Refs. [23–25], which also aim at improving the CCSD(T)

results are larger internuclear separations via non-iterative corrections due to

triples added to the CCSD energy, are not as effective as the full CR-CC(2,3)

method. Similarly, the CR-CC(2,3)+Q approach is more effective than the orig-

inal CR-CCSD(TQ),b method of Ref. [25]. For example, in the case of cut (i)

the relatively small errors in the CR-CC(2,3)/X=4 energies relative to the cor-

responding MRCI(Q)/X = 4 energies at R = 2Re, 3Re, and 5Re of 505, 735,

and 548 cm−1, respectively, are considerably smaller than the analogous errors

obtained with the CR-CC(2,3),A and CR-CCSD(T) approaches, which give 876,

1342, and 1055 cm−1 in the CR-CC(2,3),A case and 1285, 2236, and 2029 cm−1

in the CR-CCSD(T) case. The 2888, 8104, and 9223 cm−1 errors relative to

MRCI(Q) obtained with CR-CCSD(TQ)/X = 4 [i.e., CR-CCSD(TQ),b/X = 4]

method at R = 2Re, 3Re, and 5Re for cut (ii) reduce to 580, 684, and 1724 cm−1,

respectively, when the CR-CC(2,3)+Q/X = 4 method is employed (see Tables 1

and 2, and Figure 1). Similar improvements in the CR-CCSD(T), CR-CC(2,3),A,

and CR-CCSD(TQ) results offered by the CR-CC(2,3) and CR-CC(2,3)+Q ap-

proaches are observed for other basis sets. The CR-CCSD(T), CR-CC(2,3),A,

and CR-CCSD(TQ) methods improve the poor description of cuts (i) and (ii) in

regions of larger S–H distances by the CCSD(T) approach, but they are not as

effective as the CR-CC(2,3) and CR-CC(2,3)+Q approximations, The only cut

where the CR-CCSD(T), CR-CC(2,3),A, and CR-CCSD(TQ) methods are more

or less as effective as the CR-CC(2,3) and CR-CC(2,3)+Q approaches is cut (iii),

although even in that case the CR-CC(2,3) and CR-CC(2,3)+Q methods are

somewhat more accurate.

4 Summary and concluding remarks

We have compared the results of the conventional CCSD and CCSD(T), and


−7

.0

−3

.5

0.0

3.5

7.0

1 2

3 4

5

R/R

e

X=4

Ι

−7

.0

−3

.5

0.0

3.5

7.0

10−3

∆E/cm−1

X=3

Ι

−7

.0

−3

.5

0.0

3.5

7.0

X=2

Ι

−3

0

−1

50

15

12

34

5

R/R

e

X=4

ΙΙ

−3

0

−1

50

15

X=3

ΙΙ

−3

0

−1

50

15

X=2

ΙΙ

−2

−1012 0

.51

1.5

2

Y/Å

X=4

ΙΙΙ

CC

SD

CC

SD

(T)

CR

−C

CS

D(T

)C

R−

CC

(2,3

),A

CR

−C

C(2

,3)

CR

−C

CS

D(T

Q),

BC

R−

CC

(2,3

)+Q

−2

−1012

X=3

ΙΙΙ

−2

−1012

X=2

ΙΙΙ

Fig

ure

1.

Diff

eren

ces

∆E

bet

wee

nth

eC

C/C

R-C

Cen

ergi

es,

calc

ula

ted

rela

tive

toth

eir

equ

ilib

riu

mva

lues

([E

−E

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or[E

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(Ye,α

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),an

dth

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rres

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din

gM

RC

I(Q

)re

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ergy

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able

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ree

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tle

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S.

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and

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),w

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his

ad

isso

ciat

ion

ofa

sin

gle

S-H

bon

din

SH

2in

toH

(1s

2S

)+S

H(X

2Π

);II

rep

rese

nts

theC

2v-s

ym

met

ric

dou

ble

dis

soci

atio

nof

SH

2in

to2H

(1s

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)+

S(3p4

3P

)[c

ut

(ii)

];II

Ire

pre

sents

the

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dis

soci

atio

nof

SH

2in

toH

2(X

1Σ

+ g)

+S

(3p4

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)[c

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)].X

isa

card

inal

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mb

erd

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ing

the

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the

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esar

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1.

100Y. Z. Song, A. Kinal, P.J.S.B. Caridade, P. Piecuch and A.J.C. Varandas

the renormalized renormalized CR-CCSD(T), CR-CCSD(TQ), CR-CC(2,3), and

CR-CC(2,3)+Q calculations with the MRCI(Q) results for the three important

cuts of the H2S(1A′) PES, including the dissociation of a single S–H bond, which

correlates with the H(1s 2S) + SH(X 2Π) asymptote, the simultaneous dissoci-

ation of both S–H bonds, which leads to the 2H(1s 2S) + S(3p4 3P ) products,

and the C2v-symmetric minimum energy dissociation path into H2(X1Σ+

g ) and

S(3p4 1D). We have found that all renormalized CC methods reduce the failures

of the conventional CCSD and CCSD(T) approaches in the bond stretching re-

gions of the H2S potential, with the CR-CC(2,3) and CR-CC(2,3)+Q methods

being most effective in this regard. The size extensive CR-CC(2,3) method em-

ploying the standard RHF reference provides the results of the MRCI(Q) quality

for the dissociation of a single S–H bond and for the C2v-symmetric dissocia-

tion of H2S into H2(X 1Σ+g ) + S(3p4 1D). The CR-CC(2,3) approach corrected

for the effect of quadruply excited clusters via the CR-CC(2,3)+Q method im-

proves these results even further, while producing a highly accurate description

of the simultaneous dissociation of both S–H bonds that can compete with the

high-quality data obtained in the CASSCF-based MRCI(Q) calculations. At the

same time, the CR-CC(2,3) approach is as accurate as CCSD(T) in the equi-

librium region, with CR-CC(2,3)+Q providing additional small improvements.

The fact that the CR-CC(2,3)+Q approach works well in the outer regions of the

challenging cut (ii) is most encouraging, since it is difficult to obtain reliable in-

formation about the energetics of such regions, particularly with single-reference

methods. In the PES regions relevant to reaction dynamical applications, where

the outer part of path (ii) is energetically not accessible, the relative performance

of CR-CC(2,3)+Q is even better.

Acknowledgments

This work has the supported of Fundacao para a Ciencia e Tecnologia, Portugal,

and the U.S. Department of Energy.


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[54] Basis sets were obtained from the Extensible Computational Chemistry Envi-

ronment Basis Set Database, Version 02/02/06, as developed and distributed

by the Molecular Science Computing Facility, Environmental and Molecu-

lar Sciences Laboratory which is part of the Pacific Northwest Laboratory,

P.O. Box 999, Richland, Washington 99352, USA, and funded by the U.S.

Department of Energy. The Pacific Northwest Laboratory is a multiprogram

laboratory operated by Battelle Memorial Institute for the U.S. Department

of Energy under contract DE-AC06-76RLO 1830. Contact Karen Schuchardt

for further information.

Chapter 5

Accurate DMBE/CBS PES forground-state H2S

J. Chem. Phys 130, 134317-134326 (2009).

Accurate ab initio double many-body expansionpotential energy surface for ground-state H2S byextrapolation to the complete basis set limit

Y. Z. Song and A.J.C. Varandas



(Received: January 15, 2009; accepted: March 2, 2009)

Abstract

A single-sheeted potential energy surface is reported for the ground-state H2S by fit-

ting accurate multireference configuration interaction energies calculated using aug-cc-

pVTZ and aug-cc-pVQZ basis sets with extrapolation of the electron correlation energy

to the complete basis set limit, plus extrapolation to the complete basis set limit of

the complete-active-space self-consistent field energy. A switching function has been

used to warrant the correct behavior at the H2(X1Σ+

g )+S(1D) and SH(X 2Π)+H(2S)

dissociation limits. The topographical features of the novel global potential energy sur-

face are examined in details, with the former being used for exploratory quasi-classical

trajectory calculations of the thermal rate constants for the S(1D)+H2, S(1D)+D2,

and S(1D)+HD reactions at room temperature. A comparison with other available

potential energy surfaces as well as kinetics data is also provided.

108 Y.Z. Song and A.J.C. Varandas

1 Introduction

Recently, the research on the potential energy surface (PES) and dynamics of H2S

has been the subject of considerable experimental and theoretical work. Among

the many experiments that have been reported are those by Lee and Liu1, 2 who

employed a Doppler-selected time-of-flight technique in combination with crossed

molecular beam experiment to measure integral cross sections and vibrational

state-resolved differential cross sections in collisions of S(1D) with H2 reactions

at translational energies from 0.6 to 6 kcal/mol. Accurate equilibrium geometry

and local PES have also been explored by a large number of experiment studies.3–6

Theoretically, much research has been carried out on the PES and dynam-

ics for the reaction S(1D) + H2.7–13 In particular, Zyubin et al.7 studied the

dynamics of the S(1D) + H2/D2 reactions using an ab initio PES calculated at

the multi-reference configuration interaction14 (MRCI) level with a multiconfig-

uration self-consistent field (MCSCF) reference wave function and a number of

correlation-consistent basis sets,15, 16 including the simplified variant of the aug-

cc-pVQZ basis set (such basis are generally denoted as AVXZ, with X being

usually referred to as the basis-set cardinal number), commonly abbreviated as

pvqz′+. In turn, Ho et al.10 studied the S(1D) + H2 reaction using a PES ob-

tained by an improved interpolation of the MRCI data of Zyubin et al.7 on a

three-dimensional regular grid in Jacobi coordinates. Both calculations7, 10 indi-

cate a barrier-free insertion pathway along T-shaped geometries. Recently, we

also obtained a quite reliable global PES for H2S(1A′)13 based on double many-

body expansion (DMBE) 17–19 theory, which is calibrated using 1972 MRCI points

based on the full valence complete active space (FVCAS) (Ref. 20) reference func-

tion and the AVQZ15, 21 basis set. As usual in our group, the MRCI energies cal-

culated in this way have been subsequently corrected semiempirically using the

double many-body expansion-scaled external correlation (DMBE-SEC) (Ref. 22)

method to mimic the complete basis set plus full configuration interaction limit.

The PES so obtained (hereinafter referred to as DMBE/SEC) shows the correct

long range behavior and provides a realistic representation of the PES features

at all interatomic separations.

However, the PESs mentioned above depend on the one-electron basis set

J. Chem. Phys 130, 134317-134326 (2009). 109

used for the calculation or contain some empiricism. To obtain a high-accuracy

PES using only conventional ab initio calculations, one must extrapolate to the

one-electron complete basis set (CBS) limit.23–25 Much effort has been put to

devise efficient, yet simple, extrapolation schemes.26–34 In particular, Varandas31

suggested a dual-level approach referred to as uniform singlet- and triplet-pair

extrapolation (USTE) protocol, which has been now amply tested. In particular,

the USTE scheme has been shown to extrapolate the full correlation energies

for seven closed-shell systems CH2(1A1), H2O, HF, N2, CO, Ne, and F2 at their

equilibrium geometries with a similar, often better, accordance with the target

results than the traditional and rather successful Klopper’s extrapolations pro-

tocol.28 It has also been shown35 to compare favorably with the akin numeri-

cal scheme devised by Schwenke.36 Such good performance in comparison with

the best estimates of the extrapolated energies has been observed even when

the extrapolation employs a pair of correlation-consistent basis sets with cardi-

nal numbers as low as (D, T ).37 Of particular relevance for the present work is

Varandas’ extrapolation32 of the potential energy curve of CO(A 1Π) where an

excellent agreement with experiment is obtained, especially for the minimum well

depth and location of equilibrium CO. We follow here a similar route, with our

results corroborating the findings reported in Ref. 32, which suggest that our

strategy33, 38, 39 can provide a general approach for accurate potentials of larger

dimensionality at costs that can be drastically smaller than the traditional ones.

The major goal of the current work is therefore to obtain a high quality PES of

H2S(X 1A′) via extrapolation of the electron correlation energy to the CBS limit

plus extrapolation to the CBS limit of the complete-active-space self-consistent

field energy. For this, we employ a generalization31, 32 of the scheme proposed by

Karton and Martin30 for the Hartree-Fock energies, while the dynamical correla-

tion is extrapolated using the USTE (Ref. 31) protocol. This will be done using

two relatively inexpensive basis sets for the title system, namely, the AVTZ and

AVQZ ones, although as noted above the approach has shown to yield promising

results even when employing more modest correlation consistent basis sets.

The paper is organized as follows. Section 2 describes the ab initio calculations

carried out in the present work, while the extrapolation procedure is described

in Section 3. The analytical modeling of the PES is presented in Section 4.


Specifically, Section 4.1 focuses on the switching function formalism, Section 4.2

on the two-body energy terms, Section 4.3 on the three-body ones. The main

topographical features of the DMBE/CBS PES are discussed in Section 5. In

Section 6, a comparison of the quasiclassical trajectory (QCT) calculations for

the S(1D)+H2 thermal rate constant and its various isotopomers at T = 300 K

with the experimental and the other theoretical results is also given. Section 7

gathers the concluding remarks.

2 Ab initio calculations

All electronic structure calculations have been carried out at the MRCI (Ref. 14)

[including the popular quasidegenerate Davidson correction, MRCI(Q)] level us-

ing the FVCAS (Ref. 20) wave function as reference. The aug-cc-pVTZ (AVTZ)

and aug-cc-pVQZ (AVQZ) basis sets of Dunning15, 21 have been employed40 and

the calculations carried out using the MOLPRO (Ref. 41) package. The core

orbitals have been kept frozen in all calculations, with the result of this being

briefly commented later. A total of 1984 grid points have been chosen to map the

PES over the S−H2 region defined by 1.4 ≤ RH2/a0 ≤ 3.4, 1.0 ≤ rS−H2

/a0 ≤ 10.0

and 0.0 ≤ γ/ deg ≤ 90. For the H − SH interactions, a grid defined by 2.0 ≤RSH/a0 ≤ 3.6, 1.0 ≤ rH−SH/a0 ≤ 10.0 and 0.0 ≤ γ/deg ≤ 180 has been chosen.

For both channels, r, R and γ are the atom-diatom Jacobi coordinates.

Test MRCI(Q) calculations have also been performed using the standard aug-

cc-pVXZ basis set plus core-polarization high-exponent d functions (AVXdZ) as

recommended42, 43 for compounds containing second-row atoms such as the title

one. It should be noted at this point that a detailed study44 on the importance

of higher order corrections has shown that the inclusion of core-valence effects

into the ab initio calculation has a very large effect on the predicted vibrational

spectroscopy when the surface is calculated at the coupled cluster level of theory.

Such an effect is, however, partially canceled when relativistic corrections are

also included. Since we wanted to keep the calculations tractable, valence-only

calculations have in this case too been performed. Such results will also be

discussed in Section 4.

J. Chem. Phys 130, 134317-134326 (2009). 111

3 Extrapolation to CBS limit

The MRCI(Q) electronic energy is best treated in split form by writing31

EX = ECASX + Edc

X (1)

where the subscript X indicates that the energy has been calculated in the AVXZ

basis and the superscripts CAS and dc stand for complete-active-space and dy-

namical correlation energies, respectively. For CAS (uncorrelated in the sense

of lacking dynamical correlation) energies, several schemes have been advanced

(Refs. 30–32, and references therein). To extrapolate Hartree–Fock energies using

AVTZ and AVQZ basis sets, the best available protocol is possibly the one due to

Karton and Martin30 denoted as KM(T,Q). Our past experience with AV5Z and

AV6Z energies [hereinafter denoted (5,6)] suggests that the same protocol can be

successfully utilized with the CAS energy component, hence we will adopt the

KM(T,Q) protocol in the present work. This assumes the form

ECASX = ECAS

∞ +B/Xα (2)

where α = 5.34 is an effective decay exponent and ECAS∞ is the energy when

X → ∞.

To extrapolate the dynamic correlation energy in MRCI calculation, we have

been successfully using the USTE31 protocol (see also Ref. 26),

EdcX = Edc

∞ +A3

(X + α)3+

A5

(X + α)5(3)


A5 = A5(0) + cA5/43 (4)

where E∞, A5(0), A3, c and α are parameters. By fixing α, A5(0), and c from

other criteria (entirely ab initio), Eq. (3) can be transformed into an (E∞, A3)

two-parameter rule.31 Indeed, using the USTE model in Eqs (3) and (4), it

has been shown31, 35 that both the full correlation in systems studied by the

popular single-reference Møller-Plesset (MP2) and coupled cluster [CCSD and

CCSD(T)] methods as well as its dynamical part in MRCI(Q) calculations can


be accurately extrapolated to the CBS limit. In particular, for the dynamical

correlation of 24 systems studied31 using MRCI(Q) method, the optimum values

of the “universal-like” parameters were found to be A5(0) = 0.003 768 545 9 and

c= −1.178 477 13 E−5/4h , with α= −3/8. Most significant, the USTE extrapo-

lation scheme has been shown to yield accurate extrapolated CBS energies even

when the extrapolation has been carried out from the cheapest (D, T ) pair.31, 37

Thus, we will utilize the USTE model,31 as shown in Eq. (4), to CBS extrapolate

the dynamical correlation energies for the title system.

4 Analytic modeling of the CBS data with DMBE theory

The total DMBE/CBS interaction energy is written as

V (R) = V(1)

S(1D)f(R) +

3∑

i=1

V (2)(Ri) + V (3)(R) (5)

where the first term is one-body energy term that represents the energy difference

between the 1D and 3P states of atomic sulfur once extrapolated to the CBS

limit [V(1)

S(1D) = 0.040 616 9 Eh] and R = R1, R2, R3 is the collective variable ,of

all internuclear distances. This should be compared with the results obtained

by Heinenann et al.45 from CISD+Q calculations, V(1)S(1D) =0.042 261 7 Eh, which

is only ∼ 1.0 kcal/mol larger than our result. In turn, f(R) is the switching

function used to warrant the correct behavior at the H2(X1Σ+

g ) + S(1D) and

SH(X 2Π) + H(2S) dissociation limits, while V (2)(Ri) and V (3)(R) represent the

two-body and three-body energy terms respectively. The following sections give

the details of analytical forms employed to represent the switching function, two-

body and three-body energy terms, and the extrapolation scheme used in this

work.

4.1 One-body switching function

The one-body switching function form assumes the form46

h(R1) =1

4

2∑

i=1

{1 − tanh[αi (R1 − Ri01 ) + βi (R1 − Ri1

1 )3]} (6)

J. Chem. Phys 130, 134317-134326 (2009). 113

0

1

2

3

4

5

102 V

(1) S(1 D

) h(R

1) S(1D)

S(3P)

-3

0

3

0 2 4 6 8 10erro

r/cm

-1

R1/a0

102 V

(1) S(1 D

) f(R

)

g(r1) h(R1)

0

2

4

6

8

R1/a0

04

812

16

r1/a0

024

Figure 1. Switching function used to model the single-sheeted H2S completebasis set limit (CBS) potential energy surface. Shown in the left panel are thefit of the h(R1) switching form to the ab initio points calculated for S + H2

configuration as a function of H–H distance (R1), and the differences betweenh(R1) and the ab initio points. Shown in the right-hand side panel is a perspectiveview of the global switching function.

where R1 represents the H–H distance, and αi, βi (i = 1, 2), Ri01 , and Ri1

1 are

parameters to be calibrated from a least-squares fit to an extra of 11 CBS points

that control the S(1D) − S(3P ) decay as the H–H distance increases for S + H2

isosceles configurations (see the left-hand-side panel of Figure 1). The differences

between the switching function and the ab initio points used in the fit are less

than 3 cm−1, which warrants the correct energetics at the S(1D) + H2(X1Σ+

g )

asymptote.

In order to get a smooth three-body energy term, we follow Ref. 46 and

multiply Eq. (6) by an amplitude function that annihilates Eq. (6) at short-range

regions (short S − H2 distances):

g(r1) =1

2{1 + tanh[α(r1 − r0

1)]} (7)

where r1 is the distance of the S atom to the center of mass of H2. As a result,

the final switching function assumes the form

f(R) = g(r1)h(R1) (8)


with the parameters of g(r1) being chosen such as to warrant that its main effect

occurs for S − H2 distances larger than 8 a0 or so (see the right-hand-side panel

of Figure 1). All of the numerical values of all parameters in Eq. (8) are collected

in Table 1 of the supplementary material.47

4.2 Two-body energy terms

The diatomic potential energy curves of H2(X 1Σ+g ) and SH(X 2Π), which show

the correct behavior at both the asymptotic limits R → 0 and R → ∞, have

been modeled using the extended Hartree–Fock approximate correlation energy

method, including the united atom limit48 (EHFACE2U). Thus, they assume the

following form18, 48

V (2) = V(2)EHF(R) + V

(2)dc (R) (9)

where V(2)EHF(R) and V

(2)dc (R) denote the extended Hartree–Fock and dynamical

correlation parts of the potential energy and the upper right-hand-side index

stands for two body. The latter term is modeled by49

V(2)dc (R) = −

∑

n

Cnχn(R)R−n (10)

with the damping functions for the dispersion coefficients assuming the form

χn(R) = [1 − exp(−AnR/ρ− BnR2/ρ2)]n (11)

In turn, An and Bn in Eq. (10) are auxiliary functions18, 50 defined by

An = α0n−α1 (12)

Bn = β0exp(−β1n) (13)

where α0, β0, α1 and β1 are universal dimensionless parameters for all isotropic

interactions: α0 = 16.36606, α1 = 0.70172, β0 = 17.19338 and β1 = 0.09574.

Moreover, ρ is a scaling parameter defined by ρ = 5.5 + 1.25R0, where R0 =

2(〈r2A〉1/2 + 〈r2

B〉1/2) is the LeRoy51 parameter, and 〈r2A〉 and 〈r2

B〉 are the ex-

pectation value of squared radii for the outermost electrons in atom A and B,

respectively.

J. Chem. Phys 130, 134317-134326 (2009). 115

-0.2

-0.1

0.0

V/E

h

-4

0

4

0 2 4 6 8

err

or/

cm-1

R/a0

10-2

10-1

100

101

102

103

SH(X 2Π)

Figure 2. Potential energy curve for SH(X 2Π). The circles indicate the ab

initio potential energies extrapolated to complete basis set limit.

The EHF-type energy term in Eq. (9)is written as

V(2)EHF(R) = −D

R

(1 +

3∑

i=1

airi

)exp(−γ r) + χexc(R)V asym

exc (R) (14)

where

γ = γ0[1 + γ1tan(γ2r)] (15)

r = R − Re is the displacement from the equilibrium diatomic geometry; D,

ai(i = 1, · · · , n) and γi(i = 0, 1, 2) in Eq. (14) are adjustable parameters to be

obtained as described elsewhere.18, 48 χexc is the damping function, which is

approximated by χ6(R). V asymexc (which assumes to zero for SH) represents the

asymptotic exchange energy, which assumes the general form

V asymexc = AReα(1 +

∑

i=1

aiRi)exp(−γR) (16)

Here, we employ the accurate EHFACE2U potential energy curve of ground-state

H2(X 1Σ+g ) reported in Ref. 52, and the ground–state SH(X 2Π) is obtained by


0

2

4

6

8

10

0 2 4 6 8 10

10-1

C6/

Eha6 0

R/a0

C26

C06

0 2 4 6 8 10

C26

C06

0

5

10

15

20

25

10-2

C8/

Eha8 0

C48

C28

C08

C48

C28

C08

0

2

4

6

8

10-4

C10

/Eha10 0

C010

S−H2

C010

H−SH

Figure 3. Dispersion coefficients for the atom-diatom asymptotic channels ofH2S as a function of the corresponding internuclear distance of diatom.

least-squares fit to MRCI(Q) energies calculated using AVTZ and AVQZ basis

sets, once extrapolated to the CBS limit. All parameters are numerically de-

fined in Table 2 of the supplementary material.47 Since the potential curve of

H2(X 1Σ+g ) has been examined in detail elsewhere,52 only the SH(X 2Π) potential

energy curve is shown in Fig. 2. As seen, the modeled potential mimics accurately

the ab initio energies, with the maximum error being smaller than 4 cm−1.

4.3 Three-body energy terms

4.3.1 Three-body dynamical correlation energy

The three-body dynamical correlation energy assumes the usual form of a sum-

mation in inverse powers of the fragment separation distances:52

V(3)dc = −

3∑

i=1

∑

n

fi(R)χn(ri)C(i)n (Ri, θi)r

−ni (17)

where the first summation includes all atom-diatom interactions (i ≡ A − BC).

Ri is the diatomic internuclear distance, ri is the separation between atom A and

J. Chem. Phys 130, 134317-134326 (2009). 117

the center-of-mass of the BC diatomic internuclear coordinate, and θi is the angle

between these two vectors (see Figure 1 of Ref. 53). fi = 12{1 − tanh[ξ(ηRi −

Rj − Rk)]} is a convenient switching function; corresponding expressions apply

to Rj , Rk, fj , and fk. Following the recent work on NH2,46 we have fixed η= 6

and ξ = 1.0 a−10 . χn(ri) is the damping function, which still takes the forms in

Eq. (11), but replace Ri by the center-of-mass separation for the relevant atom-

diatom channel, ri.

The atom-diatom dispersion coefficients in Eq. (17) is given by

C(i)n =

∑

L

CLnPL(cosθi) (18)

where PL(cosθi) denotes the L-th term of Legendre polynomial expansion and

CLn is the associated expansion coefficient. The expansion in Eq. (18) has been

truncated by considering only the coefficients C06 , C

26 , C

08 , C

28 , C

48 , and C0

10; all

other coefficients have been assumed to make a negligible contribution, and hence

neglected. To estimate the dispersion coefficients, we utilized the generalized

Slater-Kirkwood approximation54 and dipolar polarizabilities calculated in the

present work at the MRCI/AVQZ level.

As usual, the atom-diatom dispersion coefficients so calculated for a set of

nuclear distances have then been fitted to the form

CL,A−BCn (R) = CL,AB

n + CL,ACn +DM(1 +

3∑

i=1

airi)exp(−

3∑

i=1

biri) (19)

where r=R − RM is the displacement relative to the position of the maximum

and b1 = a1. CL,ABn (L = 0), the atom-atom dispersion coefficients, are given in

Table 2 of the supplementary material. Similarly, the least-squares parameters

DM , ai, and bi are collected in Table 3 of the same supplementary information.

In turn, the internuclear dependence are displayed in Fig. 3. Note that, for R=0,

the isotropic component of the dispersion coefficient is fixed at the corresponding

value in the A–X pair, where X represents the united atom of BC at the limit of

a vanishingly small internuclear separation.

As pointed out elsewhere,52 Eq. (17) causes an overestimation of the dynamical

correlation energy at the atom-diatom dissociation channel. To correct such a

behavior, we have multiplied the two-body dynamical correlation energy for i-

pair by Πj 6=i(1 − fj), correspondingly for channels j and k. This ensures that


Table 1. Stratified rmsds of DMBE/CBS potential energy surface.

Energy Na rmsda N b>rmsd

10 147 0.062 920 181 0.185 2030 223 0.279 3740 269 0.371 5450 310 0.434 7160 365 0.631 7370 442 0.647 11080 502 0.809 10190 591 0.816 98100 972 0.817 175150 1645 0.941 315200 1861 0.936 362400 1924 0.952 370600 1956 0.954 379800 1970 0.965 3781000 1977 0.969 3791500 1984 0.971 378

a In kcal mol−1.b Number of points in the indicated energy range.c Number of points with an energy deviation larger than the rmsd.

the only two-body contribution at the ith channel is that of BC. No attempt

has been made to calculate the dispersion coefficients at the CBS extrapolated

MRCI(Q)/(T,Q) level. Instead, we assumed the MRCI/AVQZ values reported

in Ref. 13. Indeed, previous work55 on molecular polarizabilities has shown that

accurate estimates of such properties should be obtained at this level of theory,

thus dispensing a substantial amount of computational labor. It should be noted

that this does not have any effect whatsoever on the calculated total molecular

energies but only on their partition into the EHF and dc components, i.e., it

only facilitates the modeling by avoiding the above-mentioned calculations. The

relevant numerical data necessary to define the dynamical correlation are given

in Table 3 of the supplementary material.

J. Chem. Phys 130, 134317-134326 (2009). 119

4.3.2 Three-body extended Hartree-Fock energy

By removing, for a given triatomic geometry, the sum of the one-body and two-

body energy terms from the corresponding DMBE/CBS interaction energies in

Eq. (8), which was defined with respect to the infinitely separated ground-state

atoms, one obtains the total three-body energy. Then, by subtracting the three-

body dynamical correlation contribution Eq. (17) from the total three-body en-

ergy that is calculated in that way, one obtains the three-body extended Hartree-

Fock energy. This following three-body EHF energy can be represented by the

following three-body distributed-polynomial56 form

V(3)EHF =

3∑

j=1

P j(Q1, Q2, Q3) ×3∏

i=1

{1 − tanh[γji (Ri − Rj,ref

i )]} (20)

where P j(Q1, Q2, Q3) is the j-th polynomial up to six-order for j = 1, 2 and

second order for j=3. As usual, we obtain the reference geometries Rj,refi by first

assuming their values to coincide with bond distances of the associated stationary

points. Subsequently, we relax this condition via a trial-and-error least-squares

fitting procedure. Similarly, the nonlinear range-determining parameters γji have

been optimized in this way. The complete set of parameters amounts to a total

of 107 linear coefficients ci, 9 nonlinear coefficients γji , and 9 reference geometries

Rj,refi . All the numerical values of the least-squares parameters are gathered in

Table 4–6 of the supplementary material.47 Table 1 shows the stratified root-

mean-squared deviations (rmsd) values of the final potential energy surface with

respect to all the fitted ab initio energies. As shown in Table 1, a total of 1984

CBS points [as obtained from the corresponding CBS/MRCI(Q)/(T,Q) energies]

have been used for the calibration procedure, thus covering a range up to ∼1500 kcal mol−1 above the H2S global minimum. The fit shows the total root

mean square derivation is rmsd=0.97 kcal mol−1.

5 Features of the DMBE/CBS potential energy surface

Figures 4 - 8 illustrate the major topographical features of the H2S DMBE/CBS

PES reported in the present work. A characterization of their attributes (geom-

etry, energy, and vibrational frequencies) is given in Table 2 and 3. The results


Table 2. Stationary points of H2S(1A′) ground state PES (harmonic frequenciesin cm−1).

Feature R1 R2 R3 E/Eha ∆V b ωsym ωasym ωbend

Global minimumAVTdZ c 3.6548 2.5321 2.5321 -0.2866 -98.11 2705 2727 1194AVQdZ d 3.6562 2.5302 2.5302 -0.2905 -99.11 2704 2727 1192CBSd

e 3.6580 2.5296 2.5297 -0.2929 -99.68 2704 2727 1190AVTZ(Q) f 3.6636 2.5375 2.5375 -0.2849 -97.15 2722 2743 1191AVQZ(Q) g 3.6606 2.5329 2.5329 -0.2895 -98.55 2708 2731 1190CBS h 3.6609 2.5315 2.5315 -0.2921 -99.30 2705 2728 1188DMBE/CBS i 3.6615 2.5320 2.5320 -0.2921 -99.30 2665 2691 1199DMBE-SEC j 3.6593 2.5295 2.5295 -0.2892 -99.04 2701 2726 1191DMBE/SEC k 3.6623 2.5295 2.5295 -0.2892 -99.04 2643 2684 1147ab initio l 3.6728 2.5409 2.5409 -98.70 2683 2696 1183RKHS m 3.6481 2.5293 2.5293 -0.2867 -98.57 2709 2777 1200exp. n 3.6142 2.5096 2.5096 -99.10 2615 2626 1183

H · · ·S · · ·H transition stateDMBE/CBS i 4.9720 2.4860 2.4860 -0.1839 -31.40 3527 2973 1551iDMBE/SEC k 4.9094 2.4547 2.4547 -0.1785 -29.57 3151 3186 1510i

H − S · · ·H transition stateDMBE/CBS i 6.3311 2.5470 3.7841 -0.1219 7.50 2444 1388i 939iDMBE/SEC k 6.3389 2.5624 3.7765 -0.1164 9.39 2477 1444i 1123i

S − H · · ·H transition stateDMBE/CBS i 2.0723 2.7223 4.7946 -0.1246 5.81 1410 1839i 960iDMBE/SEC k 2.0665 2.6758 4.7423 -0.1217 6.07 1462 1618i 919i

a Energy relative to three-atom limit S + H + H.b Relative to the S(1D) + H2 asymptote (in kcal/mol).c Fitted to a dense grid of MRCI(Q)/AVTdZ points.d Fitted to a dense grid of MRCI(Q)/AVQdZ points.e Fitted to a dense grid extrapolated CBS/MRCI(Q)/AV(T,Q)dZ points.f Fitted to a dense grid of MRCI(Q)/AVTZ points.g Fitted to a dense grid of MRCI(Q)/AVQZ points.h Fitted to a dense grid of CBS/MRCI(Q)/AV(T,Q)Z points.i From global DMBE/CBS PES.j Fitted to a dense grid of MRCI/AVQZ points, which is using the DMBE-SECmethod.k From global DMBE/SEC PES (Ref. 13).l ab initio calculation from Ref. 7.m RKHS PES (Ref. 10).n Experimental values (Ref. 5).

J. Chem. Phys 130, 134317-134326 (2009). 121

-14

-10

-6

-2

2

1 2 3 4 5 6

102 V

/Eh

R1/a0

S(3P) + 2H

CBSDMBE-SECDMBE/CBSDMBE/SEC

RKHS

H H

S(1D)

R1

r1 = 20a0

Figure 4. Comparison of the H2 potential energy curve including the one bodyterm referring to the S(1D) − S(3P ) excitation energy. The solid line shows thepresent CBS potential energy surface. The reference energy refers to S(3P)+H+H.

obtained from other PESs and some spectroscopic properties are also included in

these two tables for comparison. The global minimum for the H2S ground state

fitted to the dense grid of MRCI(Q)/AVTZ and AVQZ energies are −0.2849Eh

and −0.2895Eh, respectively, while our DMBE/CBS PES predicts a value of

−0.2921Eh; all energies are relative to the three atom dissociation limit. Clearly,

such values follow the expected trend upon extrapolation to the complete basis set

limit. Also shown are the properties of the global minimum calculated using the

AVXdZ basis set, which is seen to have a lower energy than the one obtained with

the AVXZ basis set. As for the harmonic frequencies, these are predicted from our

DMBE/CBS PES to be 2665 cm−1, 2691 cm−1, and 1199 cm−1, whereas the cor-

responding values calculated from a fit to a dense grid of CBS/MRCI(Q)/AVXZ

points near the equilibrium geometry are 2705 cm−1, 2728 cm−1, and 1188cm−1.

Clearly, the DMBE/CBS values show excellent agreement with the ones based on

the polynomial fit. In turn, the frequencies calculated by us using the Ho et al.10

most recently RKHS PES are 2709 cm−1, 2777 cm−1, and 1200 cm−1, while the

experimental values from Ref. 5 are 2615 cm−1, 2626 cm−1, and 1183 cm−1, re-

spectively. Thus, our results are in good agreement both with the experimental5


−0.30

−0.25

−0.20

−0.15

V/E

h AVTdZAVQdZ

CBSdCBS

DMBE−SECDMBE/CBSDMBE/SEC

RKHS

0

1

20 40 60 80 100 120 140 160 180

CB

S−C

BS

d/m

Eh

α /deg

H H

S

RoptSH Ropt

SHα

Figure 5. Showing in the upper panel is the optimized C2v bending curve ofH2S as a function of bending angle, the bond distances of SH are optimized foreach bending angle. Showing in the lower panel is the difference between ab

initio points calculated at AV(T,Q)Z basis sets, which are then extrapolated tocomplete basis set, respectively.

and other theoretical work.7, 10

The geometric parameter for bent and linear form of H2S at various levels

of calculation are shown in Table 3. It is found that the global minimum from

our DMBE/CBS PES is located at R1 = 1.9376 A, R2 = R3 = 1.3399 A, while

the results from our DMBE/SEC form are R1 = 1.9380 A, R2 =R3 = 1.3386 A,

both in good agreement with each other. Comparing the barrier with the pre-

viously known experimental and theoretical values (the corresponding references

are listed in the right column of Table 3), it also shows that our DMBE/CBS

PES gives a good description of this important attribute for studying chemi-

cal reaction on the title system. Moreover, Table 2 shows that the features of

the other transition states in the DMBE/CBS PES are in quite good agreement

with the results from our previous DMBE/SEC PES, although the DMBE/CBS

PES predicts a slightly deeper well depth. It is also worth noting that the well

depth of SH· · ·H transition state is slightly deeper than the well depth of the

J. Chem. Phys 130, 134317-134326 (2009). 123

HS· · ·H transition state, a result also in agreement with the one extracted from

our DMBE/SEC (Ref. 13) PES.

Figure 4 shows the H2 potential energy curve (including the one-body term re-

ferring to the energy difference between the 1D and 3P electronic states of atomic

sulfur). Shown by the dotted line is the H2 curve obtained by RKHS (Ref. 10)

PES, with the corresponding dissociation energy is De =−0.129 614 36Eh. This

may be compared with the value of De =−0.13136974Eh from our DMBE/SEC

PES, where the dynamical correlation energy has been corrected by scaling us-

ing the DMBE-SEC method, thus about 1.1 kcal/mol lower. In turn, the H2 +

S dissociation energy of H2 from the DMBE/CBS PES (solid line) is De =

−0.133 857 84Eh, thus 1.56 kcal/mol lower than our DMBE/SEC result. In the

long-range regions, where H2S species dissociates to three atom limit, S(3P ) +

H(2S) + H(2S), the three curves run nearly parallel to each other and show the

correct behavior. The open circles indicate the ab initio CBS/MRCI(Q) points,

clearly showing the solid line provides a high-accuracy fit to the ab initio data.

Figure 5 compares the optimized C2v bending curves for DMBE/SEC and

RKHS PES with the one obtained from the DMBE/CBS PES here reported.

Note that the latter mimics well the ab initio CBS/MRCI(Q)/(T,Q) points shown

by the open circles (these correspond to energies computed for an optimized

bond length at each value of the valence 6 HSH angle). We observe that the

DMBE/CBS PES predicts a lower well depth than the RKHS and DMBE/SEC

PESs, a trend clearly visible from Figure 5. This is not unexpected since the

Davidson correction itself for a fixed basis set has yielded slightly more attractive

energies than the DMBE-SEC method for the NH2 system.46 Table 3 gathers

the extensive work that has been done on the geometric parameters and barrier

height of linear (HSH 6 HSH = 180o). Tarczay et al.57 expected that rSH in

the linear form of H2S is shorter than the equilibrium rSH bond distance, which

is confirmed at all high levels of theory applied in their work. Nevertheless,

some of the other published PESs for H2S in Table 3 lack this feature.58–62 In

turn, the barriers for linearity calculated from these PESs58–63 are ranging from

18 792 cm−1 (Ref. 61) to 31 326 cm−1 (Ref. 62), thus showing rather big difference.

The bond distance of linear HSH from our DMBE/CBS PES is rSH = 1.3155 A,

which is also shorter than the equilibrium rSH = 1.3399 A; and the barrier to


Table 3. Geometric parameters (in A) of the bent and linear H2S forms as wellas the barrier (in cm−1 ) to linearity.

Bent form Linear form

Level re(SH) re(HH) re(SH) Barrier Reference

DMBE/CBS 1.3399 1.9376 1.3207 23753 This workDMBE/SEC 1.3386 1.9380 1.2988 24296 Ref. 13RKHS 1.3384 1.9304 1.2855 25059 Ref. 10aug-cc-pVTZ CCSD(T)−all 1.3375 1.9244 1.3166 24268 Ref. 57KJ 1.3366 1.9266 1.3605 20867 Ref. 58SCZWHR 1.3376 1.9298 1.3397 22588 Ref. 59BZWRR CEPA 1.3355 1.9253 1.4486 23311 Ref. 60PJT 1.3360 1.9274 1.3636 18792 Ref. 61HC 1.3356 1.9234 1.3731 31326 Ref. 62KH 1.3356 1.9234 1.3321 29498 Ref. 63Exp. 1.3356 1.9233 Ref. 4

aCCSD(T) stands for coupled cluster singles and doubles with perturbative triplescorrection (all electrons have been correlated), while CEPA refers to Meyers cou-pled electron pair approximation (CEPA), which explicitly includes single anddouble substitutions with respect to a closed-shell HartreeFock determinant andan approximate treatment of the most important higher substitutions. If notindicated otherwise in text, the remaining acronyms identify the authors by thefirst initial of their names.

linearity is 23 753 cm−1. This which corresponds to the H − S − H transition

state located at R1 = 4.9720 a0, R2 = R3 = 2.4860 a0 and the frequencies are

ω1(S − H)symm = 3527 cm−1, ω2(S − H)asym = 2973 cm−1, ω3(bend) = 1551i cm−1;

see Table 2. We should emphasize the ab initio points have been highly weighted

in the least-square fitting procedure such as to warrant an accurate description

of the topographical features of PES at the relevant regions.

Figure 6 shows energy contours for S atom moving around H2 ground-state

diatomic whose bond length is fixed at its equilibrium geometry rHH = 1.401 bohr.

The corresponding plot for H atom moving around SH diatom with its bond

distance fixed at rSH = 2.537 bohr is presented in Figure 8. Both of the plots show

smooth behavior at short and long range regions. Of course, this also implies

J. Chem. Phys 130, 134317-134326 (2009). 125

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

y/a 0

x/a0

20

20

19

19

18

18

16

16

16

H

15 15

14

14

14

13

13 13

12

H

12

12

12

11

11 11

10

10

10

9

9

8

8

7

7

6

6

5

5 4 3 2

13

13

13 13 13

12

12

12

12

12

12

11

11

11

11

11

11

10

10

10

10

10

10

9

9

9 9 9

9 8

8

8

8

8

8

8

8

8

8

7

7

7

7

7

7

7

7

7

7

6

6

6

6

6

6

6

6

5

5

5

5

5

5

5 5

5

5

4 4

4

4 4

4

4

4

4 4

4

3

3

3

3

3

3

3

3

3

3

2

2

2

2

2

2

2

2

2

1

1

1 1

1

1

1

1

1

1

Figure 6. Contour plot for S atom moving around a fixed H2 diatomic inequilibrium geometry RH2

= 1.401 a0, which lies along the X-axis with the centerof the bond fixed at the origin. Contour are equally spaced by 0.005Eh, startingat −0.1935Eh. The dashed area are contours equally spaced by −0.00007Eh,starting at −0.13395Eh.

-20

-15

-10

-5

0

5

10

15

2 4 6 8 10

Vn/k

cal

mo

l-1

RS-H2/a0

V2

V0

(a)

RKHSDMBE/SECDMBE/CBS

-0.8

-0.4

0.0

0.4

0.8

4 6 8 10 12 14

V2

V0

(b)

Figure 7. Isotropic (V0) and leading anisotropic (V2) components of SH2 in-teraction potential energy, with the diatomic molecule fixed at the equilibriumgeometry. The axes in both panels are in the same units.


0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

y/a 0

x/a0

35

34

34

33

33

32

32

31

31

S 30

30

30 29 29

29 29

28

28

H

28 27

27

27

27

27

27

26

26

26

26

26

25

25

25

25

25

24

24

24

24

24

23

23

23

23

23

22

22

22

22

22

21 21

21

21

20

20

20

20

19

19

19

19

18

18

18

18

17

17

17

16

16

16

15

15 15

14

14

14

13

13

13

12

12

11

11

11

10

10

9

9

8

8

8

7

7

6

6

5

4

4

3

2

6 6

6

6

6

6

5 5

5

5

5

5

5

4

4

4

4

4

4

4

3

3

3

3

3

3

3

3

2 2

2

2 2

2

2

2

2

2

2

1

1

1

1

1

Figure 8. Contour plot for H atom moving around a fixed SH diatomic RSH =2.537 a0, which lies along the X axis with the center of the bond fixed at theorigin. Contour are equally spaced by 0.0055Eh, starting at −0.2865Eh. Thedashed lines are contours equally spaced by −0.0005Eh, starting at −0.1395Eh.

that any crossing seam between two sheets of the same spin-spatial symmetry

has been replaced by an avoided-crossing seam due to the single-sheeted nature

of the representation. This is the case with the Σ/Π conical intersection arising

for collinear SHH geometries (see maximumlike topography at the bottom right-

hand-side of Figure 8). Hopefully, this will show no significant impact on the

dynamics of reactions involving the title system, which to our knowledge has been

studied thus far with only single-sheeted forms like the one reported here. Such

an issue requires, of course, confirmation via a more detailed analysis including

topological effects or preferably nonadiabatic dynamics on a multisheeted form.

Two important quantities in the study of S + H2 scattering processes are the

spherically averaged isotropic (V0) and leading anisotropic (V2) potentials, which

are shown in panels (a) and (b) of Figure 7. Specifically, how close on the average

the atom and molecule can approach each other is determined by the magnitude

of V0, while the magnitude of V2 indicates whether or not the molecule prefers

to orient its axis along the direction of the incoming atom: a negative value

favors the collinear approach while a positive value favors the approach through

an isosceles triangular geometry. As Figure 7 shows, there is no barrier in V2, and

V2 is always positive, which corroborates the fact that the interaction favors the

J. Chem. Phys 130, 134317-134326 (2009). 127

Table 4. QCT thermal rate constants (in 10−10cm3 s−1) at T = 300 K for theS(1D) + H2,D2,HD reactions. The intermolecular and intramolecular isotope ef-fects, Γinter and Γintra, are defined as kSH+H/kSD+D and kSH+D/kSD+H, respectively.

SH + H SD + D SH + D SD + H Γinter Γintra

DMBE/CBS 0.68 0.72 0.30 0.42 0.94 0.71DMBE/SEC 1.22 0.97 0.53 0.55 1.26 0.96QCTa 1.30 0.94 0.51 0.55 1.38 0.93Lin et al.b 1.51 1.03 0.39 0.86 1.46 0.45Chang et al.c 6.10 4.27 1.90 2.99 1.43 0.64Exp. 2.10d − − − − 1.0 − 1.1e

1.39 ± 0.07f

a Ref. 8, b Ref. 65, c Ref. 66, d Ref. 67, e Ref. 68, f Ref. 1.

insertion of the S atom perpendicularly to the H2 molecule. As it is expected, the

minimum of V0 from the present DMBE/CBS PES is about 2 kcal mol−1 lower

than that from previous DMBE/SEC and RKHS PESs, as we would expected

from their attributes (Table 2).

6 Exploratory dynamics studies on the DMBE/CBS po-

tential energy surface

Although we plan to run detailed calculations on the dynamics and kinetics of

the title species, we have here run for testing purposes calculations of the ther-

mal rate constant for the S(1D)+H2(ν = 0, j = 0), S(1D)+D2(ν = 0, j = 0) and

S(1D)+HD(ν = 0, j = 0) reactions at T = 300 K. The VENUS96 (Ref. 64) com-

puter code has been utilized using batches of 5000 trajectories. An integration

step size of 1.5 × 10−16s has been chosen such as to warrant conservation of the

total energy to better than one part in 103. As usual, all trajectories started at

a distance between the incoming atom and the center-of-mass of the diatom of 9

A, a value sufficiently large to make the interaction energy essentially negligible.

The results are collected in Table 4 along with the intermolecular (Γinter) and in-

tramolecular (Γintra) isotope effects.8 Also gathered is the available experimental

and theoretical data. Note that the QCT thermal rate constants listed in Table 4


have been divided by 5 to account for the multisurface factor, since 1D state of

S atom is five-fold degenerate.

As mentioned in the Ref. 8, 65, 66, the ordering for the isotope systems is

kH2> kHD > kD2

, a conclusion similar to the one obtained in the present work.

As can be seen, the QCT thermal rate constant for the reaction S(1D) + H2

is calculated to be 0.68 × 10−10cm3 s−1, which is in reasonably good agreement

with the value from our previous DMBE/SEC PES 1.22 × 10−10cm3 s−1 and the

experimental data 2.1 × 10−10cm3 s−1 by Black and Junsinski.67 In addition,

the intramolecular isotope effect Γintra = 0.71 is in excellent agreement with the

recent experiment result Γintra = 1.39 ± 0.07 from Lee et al.1 Unfortunately, no

experimental data on the intermolecular isotope effect is available that would

help confirm the good performance of our calculations in reproducing the kinetic

isotope effect for the title reaction. Yet, they match well the results from other

theoretical work.8, 65, 66

7 Concluding remarks

A global single-sheeted potential energy surface has been reported for the ground

electronic state of hydrogen sulfide on the basis of a least-squares fit to nearly

two thousand MRCI(Q) energies calculated using AVTZ and AVQZ basis sets

subsequently extrapolated to the CBS limit. The various topographical features

of the novel PES obtained via an analytical fit with DMBE theory have been

carefully examined and compared with other realistic PESs as well as experimen-

tal results available in the literature. Based on such features, it is concluded

that an accurate extrapolation to the CBS limit of the CASSCF and dynamical

correlation energies, and hence of the H2S PES, has been achieved. In summary,

the DMBE/CBS PES reported in the present work provides an accurate global

fit to all such calculated ab initio energy points. Reaction probabilities at room

temperature employing the novel DMBE/CBS PES have also been calculated for

the S(1D)+H2 and its isotopic variants S(1D)+D2 and S(1D)+HD. The results

have shown good agreement with the ones obtained from other theoretical work

and with available experimental data.

J. Chem. Phys 130, 134317-134326 (2009). 129

Acknowledgments

This work has the support of Fundacao para a Ciencia e a Tecnologia, Portu-

gal (Contract Nos. POCI/QUI/60501/ 2004 and REEQ/128/QUI/2005) under

the auspices of POCI 2010 of Quadro Comunitrio de Apoio III co-financed by

FEDER.

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J. Chem. Phys 130, 134317-134326 (2009): SI.

Accurate ab initio double many-body expansionpotential energy surface for ground-state H2S byextrapolation to the complete basis set limit




(Received: January 15, 2009; accepted: March 2, 2009)


Table 1. Parameters in the switching functions of Eq. (6) to (8).

α1/a−10 0.85996982

α2/a−10 0.76175120

β1/a−30 0.77391740

β2/a−30 0.11487678

R101 /a0 3.14122827

R111 /a0 4.80494587

R201 /a0 4.07465909

R20/a0

1 5.75513968α/a−1

0 0.75r01/a0 5.50

Table 2. Parameters of two-body potential energy curves.

SH(X2Π) H2(X1Σ+g )

Re/a0 2.537 1.401D/Eh 0.310424 0.22979439a1/a0

−1 1.714397 1.82027480a2/a0

−2 0.489462 0.52437767a3/a0

−3 0.288355 0.36999610γ0/a0

−1 1.232777 1.094670γ1/a0

−1 4.950683 1.009737γ2/a0

−1 0.033818 0.235856

A/Eha0−eα -0.8205

a1/a0−1 0

α 2.5γ/a0

−1 2.0R0/a0 7.9652 6.9282C6/Eha0

−6 34.49 6.499C8/Eha0

−8 896.5 124.4C10/Eha0

−10 26332.1 3286.0C11/Eha0

−11 -3475C12/Eha0

−12 121,500C13/Eha0

−13 -291,400C14/Eha0

−14 6,061,000C15/Eha0

−15 -23,050,000C16/Eha0

−16 393,800,00

J. Chem. Phys 130, 134317-134326 (2009): SI. 137

Table

3.

Nu

mer

ical

valu

es(i

nat

omic

un

it)

ofth

ep

aram

eter

sin

Eq.

(18)

.

C0 6(R

)C

2 6(R

)C

0 8(R

)C

2 8(R

)C

4 8(R

)C

0 10(R

)

S−

H2

RM/a

03.

4158

3.20

383.

4069

3.22

663.

1853

3.40

10D

M/E

h15

.944

510

.536

151

1.36

225

808.

1457

56.7

931

2049

3.48

71a

1/a

−1

01.

2043

4183

0.49

0821

601.

2527

8782

1.09

7666

061.

0249

1625

0.12

7035

07a

2/a

−2

00.

3754

8701

0.01

9991

300.

4230

4258

0.40

7342

290.

3508

0748

-0.3

5822

586

a3/a

−3

00.

0472

4979

-0.0

1036

219

0.05

0989

720.

0507

5238

0.03

9618

980.

1591

3341

b 2/a

−2

00.

2071

0333

0.23

3175

350.

1953

8314

0.19

1466

120.

5033

9247

0.26

8506

83b 3/a

−3

02.

5×

10−

91.

8×

10−

94.

8×

10−

97.

6×

10−

91.

1×

10−

80.

0309

6264

H−

SH

RM/a

04.

2552

4.12

934.

2825

4.14

924.

1292

4.29

95D

M/E

h6.

7494

5.00

6019

0.02

0248

4.71

6731

.299

869

07.4

344

a1/a

−1

01.

2620

0585

0.47

3135

191.

2999

1575

0.88

2280

151.

1292

0709

1.32

7548

12a

2/a

−2

00.

4588

1086

0.01

4027

260.

4590

7555

0.23

6561

950.

3700

7487

0.45

9505

77a

3/a

−3

00.

0582

7349

0.01

1494

780.

0598

1692

0.02

2045

870.

0264

9191

0.06

3739

75b 2/a

−2

00.

2824

1957

0.29

9644

260.

3689

5540

50.

2858

2916

0.61

0370

300.

4408

1555

b 3/a

−3

00.

0156

7011

1.1×

10−

90.

0301

9273

4.1×

10−

95.

7×

10−

90.

0408

4231


Table

4.

Nu

merical

values

ofth

eco

efficien

tsof

the

first

poly

nom

ialP

(1)

inE

q.

(20).

γ11 /a

−1

0=

0.50γ

12 /a−

10

=0.70

γ13 /a

−1

0=

0.70

R1,r

ef1

/a0

=1.4

R1,r

ef2

/a0

=4.0

R1,r

ef3

/a0

=4.0

c1 /a

00=

7.8901627605c2 /a

−1

0=

−0.9565229866

c3 /a

−1

0=

0.5544109970c4 /a

−2

0=

1.3424341746c5 /a

−2

0=

1.1811165158c6 /a

−2

0=

−1.0962601413

c7 /a

−2

0=

0.4588492482c8 /a

−3

0=

−0.1601152303

c9 /a

−3

0=

−0.2945529462

c10 /a

−3

0=

−0.0296734570

c11 /a

−3

0=

0.1046895874c12 /a

−3

0=

0.0511805767c13 /a

−3

0=

−0.0839322415

c14 /a

−4

0=

0.0827547462c15 /a

−4

0=

0.0196868279c16 /a

−4

0=

0.1048173372c17 /a

−4

0=

−0.0537538691

c18 /a

−4

0=

−0.1371734600

c19 /a

−4

0=

0.0095792999c20 /a

−4

0=

0.0907280738c21 /a

−4

0=

0.0832075281c22 /a

−4

0=

0.0123621556c23 /a

−5

0=

−0.0150525206

c24 /a

−5

0=

0.0130963449c25 /a

−4

0=

−0.0342313537

c26 /a

−5

0=

0.0343300371c27 /a

−5

0=

−0.0119826823

c28 /a

−4

0=

−0.0076204364

c29 /a

−5

0=

−0.0129176244

c30 /a

−5

0=

0.0182736716c31 /a

−4

0=

−0.0211688967

c32 /a

−5

0=

0.0042346530c33 /a

−5

0=

−0.0154459320

c34 /a

−4

0=

0.0140877616c35 /a

−6

0=

0.0017288556c36 /a

−6

0=

−0.0011452291

c37 /a

−6

0=

0.0000832818c38 /a

−6

0=

−0.0028620844

c39 /a

−6

0=

−0.0000180890

c40 /a

−6

0=

0.0016703671c41 /a

−6

0=

0.0012084930c42 /a

−6

0=

0.0036267421c43 /a

−6

0=

0.0008211044c44 /a

−6

0=

−0.0054797863

c45 /a

−6

0=

−0.0004348137

c46 /a

−6

0=

−0.0024786636

c47 /a

−6

0=

0.0029533592c48 /a

−6

0=

−0.0007092122

c49 /a

−6

0=

0.0033317750c50 /a

−6

0=

0.0015208972

J. Chem. Phys 130, 134317-134326 (2009): SI. 139

Table

5.

Nu

mer

ical

valu

esof

the

coeffi

cien

tsof

the

seco

nd

pol

yn

omia

lP

(2)

inE

q.

(20)

.

γ2 1/a

−1

0=

1.0

γ2 2/a

−1

0=

0.6

γ2 3/a

−1

0=

0.6

R2,r

ef1

/a0

=4.

4R

2,r

ef2

/a0

=2.

2R

2,r

ef3

/a0

=2.

2

c 1/a

0 0=

−11.4

1804

9741

c 2/a

−1

0=

−0.

5040

2858

53c 3/a

−1

0=

−3.

2306

5781

12c 4/a

−2

0=

−2.

8666

3115

74c 5/a

−2

0=

−2.

6802

7195

53c 6/a

−2

0=

−3.

9401

5189

42c 7/a

−2

0=

0.96

6809

9212

c 8/a

−3

0=

−0.

1264

2091

76c 9/a

−3

0=

−0.

7426

3481

20c 1

0/a

−3

0=

−0.

0102

5223

92c 1

1/a

−3

0=

−0.

9738

9601

38c 1

2/a

−3

0=

0.72

8821

1207

c 13/a

−3

0=

−0.

3926

2512

13c 1

4/a

−4

0=

−0.

2006

9513

85c 1

5/a

−4

0=

−0.

3362

8899

72c 1

6/a

−4

0=

−0.

0459

0292

64c 1

7/a

−4

0=

0.10

8259

7465

c 18/a

−4

0=

−0.

4704

3395

28c 1

9/a

−4

0=

0.19

1586

4659

c 20/a

−4

0=

−0.

5648

8118

01c 2

1/a

−4

0=

0.13

7234

2759

c 22/a

−4

0=

−0.

0228

4919

80c 2

3/a

−5

0=

0.00

4404

6458

c 24/a

−5

0=

−0.

0962

7459

24c 2

5/a

−4

0=

−0.

0540

9118

81c 2

6/a

−5

0=

−0.

0371

6189

32c 2

7/a

−5

0=

0.05

0896

7168

c 28/a

−4

0=

−0.

0747

6424

26c 2

9/a

−5

0=

0.09

2402

7522

c 30/a

−5

0=

−0.

1543

2424

46c 3

1/a

−4

0=

0.04

0889

6239

c 32/a

−5

0=

0.06

8705

9401

c 33/a

−5

0=

0.03

6233

6716

c 34/a

−4

0=

−0.

0248

6780

09c 3

5/a

−6

0=

−0.

0021

6931

88c 3

6/a

−6

0=

−0.

0009

0689

45c 3

7/a

−6

0=

−0.

0188

4869

69c 3

8/a

−6

0=

−0.

0104

3412

01c 3

9/a

−6

0=

0.00

9171

7419

c 40/a

−6

0=

0.00

0670

2945

c 41/a

−6

0=

0.00

0313

7173

c 42/a

−6

0=

−0.

0023

0441

73c 4

3/a

−6

0=

0.01

0311

6039

c 44/a

−6

0=

−0.

0247

2837

16c 4

5/a

−6

0=

0.00

3300

3087

c 46/a

−6

0=

0.01

0441

2240

c 47/a

−6

0=

−0.

0149

4154

96c 4

8/a

−6

0=

−0.

0060

3354

90c 4

9/a

−6

0=

0.00

9908

7989

c 50/a

−6

0=

−0.

0034

2496

09


Table

6.

Nu

merical

values

ofth

eco

efficien

tsof

the

second

poly

nom

ialP

(3)

inE

q.

(20).

γ31 /a

−1

0=

1.5γ

32 /a−

10

=0.5

γ33 /a

−1

0=

0.5

R3,r

ef1

/a0

=4.9

R3,r

ef2

/a0

=2.6

R3,r

ef3

/a0

=2.6

c1 /a

00=

−0.7148006960

c2 /a

−1

0=

−0.0117713764

c3 /a

−1

0=

−0.1831504257

c4 /a

−2

0=

−0.2253366894

c5 /a

−2

0=

−0.2831064857

c6 /a

−2

0=

−0.6714854715

c7 /a

−2

0=

0.2559035104

Chapter 6

Accurate DMBE/SEC PES forground-state H2S

J. Phys. Chem. A 113, 9213-9219 (2009).

Potential energy surface for ground-state H2S viascaling of the external correlation, comparison withextrapolation to complete basis set limit, and use inreaction dynamics

Y. Z. Song, P. J. S. B. Caridade and A.J.C. Varandas



(Received: April 24, 2009; Revised Manuscript Received: June 26, 2009)

Abstract

A global double many-body expansion potential energy surface is reported for the

electronic ground state of H2S by fitting accurate ab initio energies calculated at the

multireference configuration interaction level with the aug-cc-pVQZ basis set, after

slightly correcting semiempirically the dynamical correlation by the double many-body

expansion-scaled external correlation method. The function so obtained has been com-

pared in detail with a potential energy surface of the same type recently reported (Song,

Y. Z.; Varandas, A. J. C. J. Chem. Phys. 2009, 130, 134317.) by extrapolating the

calculated raw energies to the complete basis set limit, eschewing any use of information

alien to ab initio theory. The new potential energy surface is also used for studying the

dynamics and kinetics of the S(1D)+H2/D2/HD reactions.

144 Y. Z. Song, P. J. S. B. Caridade and A. J. C. Varandas

1 Introduction

The S(1D,3 P )+H2 reaction and its isotopic variants have been the subject of con-

siderable theoretical and experimental work due to its major role in environmental

issues, particularly in areas such as acid rain, air pollution, and global climate

change. Specially, the reactions of S(1D) with H2 and its isotopic are known to

proceed via an insertion path way, as demonstrated in many experimental ob-

servations1–4 and theoretical work.5–12 In a series of experiments, Inagaki et al.1

measured doppler profiles of H and D atoms from the reaction S(1D) + HD by a

laser-induced fluorescence technique with a vacuum ultraviolet laser. They have

observed an isotopic channel branching ratio of φ(SD + H)/φ(SH + D) is mea-

sured to be 0.9±0.1 for the reaction of S(1D) + HD at average collision energy of

1.2 kcal mol−1. Such a measured branching ratio and translational energy release

suggest that the reaction proceeds by insertion via formation of a long-lived com-

plex. Lee et al.2–4 measured the integral cross sections (ICSs) and vibrational

state-resolved differential cross sections (DCSs) for S(1D) + H2/D2/HD reactions

at several collision energies, which showed the ICSs decay monotonically with the

collision energy.

Theoretically, much research has been explored on the potential energy sur-

face (PES) and dynamics for the reaction S(1D) + H2.5–13 Specially, Zyubin et

al.6 obtained the electronic ground-state PES by fitting to a grid of over 2000

points based on the reproducing kernel Hilbert space (RKHS) approach and a

many-body expansion14 of the energy. The results indicate a barrierless insertion

pathway along the T-shaped geometry and an 8 kcal/mol barrier for abstrac-

tion along a collinear path. Subsequently, Chao et al.7 reported an extensive

quasiclassical trajectory (QCT) study of the S(1D) + H2/D2/HD reactions using

the PES of Zyubin et al.,6 which qualitatively reproduced the nearly symmet-

ric forward/backward DCSs, the monotonically decaying ICSs, and the product

internal state distributions observed in the experiment.2–4 Later, Ho et al.8 pro-

vided a new interpolation of the ab initio data of Zyubin et al.6 to obtain an

improved PES by fitting the same set of ab initio data, which also indicates a

barrierless insertion path along the T-shape geometry. Recently, Lin et al.12 car-

ried out quantum statistical and wave packet studies of the title reaction. The

J. Phys. Chem. A 113, 9213-9219 (2009). 145

total ICSs have been predicted to decay monotonically with the collision energy

thus supporting a barrierless insertion mechanism. Most recently,15 we reported

a PES for ground-state H2S by fitting accurate ab initio energies calculated using

Dunning’s16, 17 aug-cc-pVTZ and aug-cc-pVQZ (simply, AVTZ and AVQZ) basis

sets via extrapolation of the electron correlation energy to the complete basis set

limit (CBS) plus extrapolation to CBS of the complete-active-space self-consistent

field energy (this PES hereafter denoted as DMBE/CBS). Exploratory dynamics

calculation on the DMBE/CBS PES led to a prediction of 0.71 for the intramolec-

ular isotope effect (Γintra = kSD+H/kSH+D) for S(1D) + HD, which reproduces the

recent experiment result of Γintra = 1.39 ± 0.07 by Lee and Liu.2

In this work, we report a realistic global PES for H2S(1A′) based on DMBE18–22

theory that is calibrated from 1972 ab initio points that were calculated at the

multireference configuration interaction (MRCI)23 level, using the full valence

complete active space (FVCAS)24 reference with the AVQZ basis set. The cal-

culated ab initio energies are then corrected semiempirically using the double

many-body expansion-scaled external correlation method (DMBE-SEC)25 to ex-

trapolate to the limit of a one-electron CBS and full CI expansion and are subse-

quently modeled using DMBE theory. As usual, the resulting PES (DMBE/SEC)

shows the correct long-range behavior at all dissociation channels, while providing

an accurate fit of the calculated data at all separations.

This paper is organized as follows. Section 2 reports the ab initio calcula-

tions, and Section 3 reports the formalism used for the analytical modeling. The

discussion of its major topographical features is in Section 4, while Section 5

probes its dynamics performance when used to calculate thermal rate constants

and vibrational state-resolved ICSs. The concluding remarks are in Section 6.

2 Ab initio calculations and scaling of the external corre-

lation

The ab initio calculations have been carried out at the MRCI23 level using the

FVCAS24 wave function as reference and Dunning’s16, 17 AVQZ basis set. All cal-

culations have been performed with the Molpro26 package for electronic structure

calculation. A grid of 1972 raw ab initio points have been chosen to map the PES


over the S−H2 region defined by 1.4 ≤ RH2/a0 ≤ 3.4, 1.0 ≤ rS−H2

/a0 ≤ 10.0 and

0.0 ≤ γ/ deg ≤ 90. For the H−SH interactions, a grid defined by 2.0 ≤ RSH/a0 ≤3.6, 1.0 ≤ rH−SH/a0 ≤ 10.0 and 0.0 ≤ γ/deg ≤ 180 has been chosen. As usual, r,

R and γ are the atom-diatom Jacobi coordinates for relevant channel.

The raw ab initio energies calculated above have been subsequently corrected

semiempirically with the DMBE-SEC25 method such as to account for electronic

excitations beyond singles and doubles and, most importantly, for the incom-

pleteness of the basis set. The total DMBE-SEC interaction energy assumes the

form

V (R)=VFV CAS(R) + VSEC(R) (1)

where

VFV CAS(R)=∑

AB

V(2)AB,FV CAS(RAB) + V

(3)ABC,FV CAS(RAB, RBC, RAC) (2)

VSEC(R)=∑

AB

V(2)AB,SEC(RAB) + V

(3)ABC,SEC(RAB, RBC, RAC) (3)

where R = {RAB, RBC, RAC} being a collective variable of all internuclear dis-

tances. Explicitly, the expansion of the terms in Eq. (3) assume the form:

V(2)AB,SEC(RAB)=

V(2)AB,FV CAS−CISD(RAB) − V

(2)AB,FV CAS(RAB)

F(2)AB

(4)

V(3)ABC,SEC(R)=

V(2)AB,FV CAS−CISD(R) − V

(3)ABC,FV CAS(R)

F(3)ABC

(5)

Following the previous work,25 F(2)AB in Eq. (4) is chosen to reproduce the bond

dissociation energy of the corresponding AB diatomic, while F(3)ABC in Eq. (5) is

estimated as the average of the three two-body F−factors. For the AVQZ basis

set, such a procedure yields F(2)HH =0.9773, F

(2)SH =0.8877, and F

(3)SHH =0.9176.

3 Double many-body expansion representation

Within the framework of DMBE theory,18–22 the single-sheeted PES for H2S(1A′)

assumes the form

V (R)=V(1)

S(1D)f(R) +

3∑

i=1

[V

(2)EHF(Ri) + V

(2)dc (Ri)

]+ V

(3)EHF(R) + V

(3)dc (R) (6)

J. Phys. Chem. A 113, 9213-9219 (2009). 147

where V(1)S(1D) represents the energy difference between the 1D and 3P states of

atomic sulfur: V(1)S(1D) =0.0431045Eh, f(R) is the switching function used to war-

rant the correct behavior at the H2(X 1Σ+g ) + S(1D) and SH(X 2Π) + H(2S) dis-

sociation limits. In turn, the two-body and three-body energy terms are splitted

into extended Hartree-Fock (EHF) and dynamical correlation (dc) contributions.

Because the formalism is close to the one used for the DMBE/CBS PES,15 only

a brief sketch will be presented here (see also Refs. 27, 28).

3.1 Dissociation scheme and one-body switching function

The title system has the following dissociation scheme:

H2S(1A′

) → H2 (X 1Σ+g ) + S(1D) (7)

→ SH (X 2Π) + H( 2S) (8)

Because SH (X 2Π) dissociates to ground-state atoms S(3P ) and H(2S), it is nec-

essary to introduce a function to remove the S(1D) state from this channel. Fol-

lowing Ref. 27, this is accomplished by using the switching function

f(R)=g(r1)h(R1) (9)

with the parameters of g(r1) being chosen such as to warrant that its main effect

occurs for S − H2 distances larger than 8 a0 or so (see the right-hand-side panel

of Figure 1 in the Supporting Information). In turn, the parameters in h(R1) are

calibrated from a least-squares fit to an extra of 10 AVQZ points that control

the S(1D) − S(3P ) decay with a growing H − H distance (see the left-hand-side

panel of Figure 1 in the Supporting Information). All parameters in Eq. (9) are

numerically defined in Table 1 of the Supporting Information.


The potential energy curves of the diatomic fragments have been modeled with

the extended Hartree-Fock approximate correlation energy method including the

united atom limit (EHFACE2U),29 which shows the correct behavior at the

asymptotes R → 0 and R → ∞. Specifically, the EHF energy part assumes


the form

V(2)EHF(R) = −D

R

(1 +

3∑

i=1

airi

)exp(−γ r) + χexc(R)V asym

exc (R) (10)

where

γ = γ0[1 + γ1tan(γ2r)] (11)


ai(i = 1, · · · , n) and γi(i = 0, 1, 2) in Eq. (14) are adjustable parameters to

be obtained as described elsewhere.19, 29 χexc is the damping function, which is

approximated by χ6(R). V asymexc (which assumes to be zero for SH) represents the

asymptotic exchange energy, which assumes the general form

V asymexc = AReα(1 +

∑

i=1

aiRi)exp(−γR) (12)

In turn, the dc part is written as30

Vdc(R)=−∑

n=6,8,10

Cnχn(R)R−n (13)

where Cn are dispersion coefficients and χn are damping functions. For H2 (X 1Σ+g ),

we will utilize the accurate potential energy curve of Ref. 28, while SH (X 2Π)

is modeled from our own ab initio energies and the experimental dissociation

energy.31, 32 The relevant numerical data are gathered in Table 2 of the Support-

ing Information. Because the H2 (X 1Σ+g ) potential function is examined in detail

elsewhere,28 Figure 2 of Supporting Information illustrates only SH (X 2Π), which

is seen to mimic accurately the calculated ab initio energies.

3.3 Three-body energy terms

3.3.1 Three-body dc energy

The three-body dc energy assumes the usual form of a summation in inverse

powers of the fragment separation distances28

V(3)dc =−

∑

i

∑

n


−ni (14)

J. Phys. Chem. A 113, 9213-9219 (2009). 149

Table 1. Stratified Root-Mean-Square Deviations of DMBE/SEC PES.

Energya) N b) rmsda) Nc)>rmsd

10 41 0.096 720 55 0.125 1330 85 0.385 1040 115 0.458 1650 145 0.567 2360 185 0.572 3280 306 0.636 67

100 767 0.601 146150 1474 0.741 319200 1769 0.768 386250 1799 0.778 395500 1877 0.817 410

1000 1942 0.840 4351500 1959 0.855 4402000 1963 0.855 4422500 1972 0.862 444

a) The units of energy and rmsd are kcal mol−1.b) Number of points in the indicated energy range.c) Number of points with an energy deviation larger than the rmsd.

where the first summation runs over all atom-diatom interactions (e.g., i ≡A − BC), and fi(R)= 1

2{1− tanh[ξ(ηRi −Rj −Rk)]} are damping function.28 In

turn, Ri is the diatomic internuclear distance for the i-pair, ri is the correspond-

ing atom-diatom (center-of-mass) separation, and θi is the angle between −→r i and−→R i (see Figure 1 of Ref. 33). Following the recent work,27 we have fixed η = 6

and ξ = 1.0 a−10 . χn(ri). All of the numerical values of parameters in Eq. (14)

are collected in Table 3 of the Supporting Information, while their internuclear

dependence are displayed in Figure 3 of the Supporting Information.

3.4 Three-body EHF energy

For a given triatomic geometry, the total three-body energy is obtained by sub-

tracting the sum of the one- and two-body energies from the corresponding


Table 2. Stationary Points of H2S DMBE/SEC PES a).

Feature R1/a0 R2/a0 R3/a0 E/Ehb) ∆V c) ω1 ω2 ω3

global minimumDMBE/SEC d) 3.6623 2.5295 2.5295 -0.2892 -99.04 2643 2684 1147Expt e) 3.6142 2.5096 2.5096 -99.10 2615 2626 1183ab initio f) 3.6728 2.5409 2.5409 -98.70 2683 2696 1183RKHS g) 3.6481 2.5293 2.5293 -0.2867 -98.57 2709 2777 1200DMBE/CBS h) 3.6615 2.5320 2.5320 -0.2921 -99.30 2665 2691 1199

stationary points d)

H − S − H 4.9094 2.4547 2.4547 -0.1785 -29.57 3151 3186 1510iH − S · · ·H 6.3389 2.5624 3.7765 -0.1164 9.39 2477 1444i 1123iS − H · · ·H 2.0665 2.6758 4.7423 -0.1217 6.07 1462 1618i 919i

a)Harmonic frequencies in cm−1.b)Energy relative to the three-atom limit S + H + H.c)Relative to the S(1D) + H2 asymptote (in kcal mol−1).d)This work.e)Experimental values.42f)Ab initio calculation from Ref. 6.g)Calculated using RKHS PES.8h)Calculated using DMBE/CBS PES.15

DMBE-SEC interaction energies in Eq. (6). Then, by removing the three-body dc

energy part described in Eq. (14) from the total three-body energy, the three-body

EHF energy is obtained. This is finally modeled by using three-body distributed-

polynomial34 form

V(3)EHF =

2∑

j=1

P j(Q1, Q2, Q3) ×3∏

i=1

{1 − tanh

[γj

i (Ri − Rj,refi )

]}(15)

where P j(Q1, Q2, Q3)(j = 1, 2) is a polynomial up to six-order in the popular D3h

symmetry coordinates14, 35, 36

As usual, we obtain the reference geometries Rj,refi by first assuming their

values to coincide with bond distances of the associated stationary points. Sub-

sequently, we relax this condition via a trial-and-error least-squares fitting pro-

cedure. Similarly, the nonlinear range-determining parameters γji have been op-

J. Phys. Chem. A 113, 9213-9219 (2009). 151

timized in this way. The complete set of parameters amounts to a total of 100

linear coefficients ci, six nonlinear coefficients γji , and six reference geometries

Rj,refi . All the numerical values of the least-squares parameters are gathered in

Table 4 and 5 of the Supporting Information. Table 1 shows the stratified root-

mean-squared deviations (rmsd) values of the final PES with respect to all the

fitted ab initio energies. A total of 1972 points covering a range of energy up

to ∼ 2500 kcal mol−1 above the H2S global minimum, have been used for the

calibration procedure, with the total rmsd is 0.862 kcal mol−1.

4 Features of the DMBE/SEC PES

The approximate minimum energy path of the DMBE/SEC PES is displayed in

Figure 1 as a function of r, which measures the distance between the S atom

and the center of HH diatom, with the bond length of HH being optimized at

each value of r. Also shown for a comparison in this figure is the corresponding

path of the accurate DMBE/CBS PES recently reported.15 The first visible fea-

ture is absence of a barrier for perpendicular insertion of S(1D) atom into HH

diatom. Also apparent is the parallelism between the minimum energy paths of

the DMBE/SEC and DMBE/CBS PESs. This is quite pleasing, since they corre-

spond to optimized paths and the pragmatic DMBE-SEC method only corrects

the dc at the equilibrium geometry of the relevant fragments. A similar remark

can be made from Figure 2, which shows the optimized C2v bending curve of H2S

as a function of bending angle, with the bond distance of SH optimized at each

bending angle. Note that the barrier to linearity calculated from our DMBE/SEC

PES is 24296 cm−1, thus, only 28 cm−1 larger than the value of 24268 cm−1 calcu-

lated by Tarczay et al.37 It may also be compared with the value of 23753 cm−1,

which predicted to be only 543 cm−1 higher.

The near parallel behavior of DMBE/SEC and DMBE/CBS PESs is high-

lighted in the bottom panels of Figure 1 and 2, with the DMBE/CBS predicting

a slightly deeper well depth than the DMBE/SEC PES. As noted above, this may

largely be due to the fact that the DMBE-SEC method employs a single constant

scaling factor (approximated by an average of three diatom scaling factors) for

all the points calculated with AVQZ basis set, have not included the Davidson


−0.30

−0.25

−0.20

−0.15

−0.10

V/E

h

DMBE/SECDMBE/CBS

0

2

4

0 2 4 6 8

∆V

/kca

l mol

−1

r/a0

H H

S

RoptHH

r

Figure 1. Approximate minimum energy path as a function of r (distancebetween the S atom and the center of HH diatom), with the HH bond lengthoptimized at each value of r.

−0.30

−0.25

−0.20

−0.15

V/E

h

DMBE/SECDMBE/CBS

0.00

2.50

5.00

20 40 60 80 100 120 140 160 180

∆V

/kca

l mol

−1

α /deg

H H

S

RoptSH Ropt

SHα

Figure 2. Optimized C2v bending curve of H2S as a function of bending angle,with the bond distance of SH optimized for each bending angle.

J. Phys. Chem. A 113, 9213-9219 (2009). 153

0

2

4

6

8

0 2 4 6 8

y/a 0

x/a0

42

41

41

40

40

38

38

37

37

36

35

35

34

34

33

32

32

31

30

30

29

28

28

28

27

27

26

26

26

25

25

24

24

24

23

22

22

21

21

20

19

19

18

17

17

16

15

15

14

13

12

12

11

10 9

8 7

5 4 2

H H

S

x

y

Figure 3. Contour plot for a C2v insertion of S atom into H2. Contours areequally spaced by 0.0075Eh, starting at −0.280Eh. The dashed areas are contoursequally spaced by 0.004Eh, starting at −0.0095Eh.

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

R3/

a 0

R2/a0

25

25 25

24

24 24

23

23

22

22

21

21

21

20

20

20 20

19

19

18

18

18 18

17

16

16

16 16

15

14

14

14 14

13

13

13 12

12

12

11

11

11 11

10

10

10 10

9

9

9

8

8

8

8

7

7 7 5

2

H S HR2 R3

Figure 4. Contour plot for bond stretching in linear H − S − H. Contours areequally spaced by 0.007Eh, starting at −0.175Eh.


correction. It may also be due to the fact that the MRCI energies utilized to

calibrate the DMBE/SEC form have not included the Davidson correction. This

suggests that the DMBE/SEC scheme slightly underestimates such a popular

correction, a finding also supported from our recent work on the NH2 system.27

Quantitatively, the energy difference at the global minimum in Figure 1 and 2 is

∼ 2 kcal mol−1, with the well depth calculated relative to the three-atom dissoci-

ation limit being −0.2892Eh and −0.2921Eh, respectively, for the DMBE/SEC

and DMBE/CBS PESs. Thus, the well depth at equilibrium H2S is enhanced by

∼ 0.0038Eh(∼ 2.38 kcal mol−1) by the Davidson correction. In fact, if this is not

included for the AVTZ and AVQZ energies, the well depth extrapolated to CBS

limit is 0.2921 − 0.0038 = 0.2883Eh, thus only ∼ 0.56 kcal mol−1 smaller than

the well depth of the DMBE/SEC PES. The result also corroborates the high

reliability and consistency of the CBS and DMBE-SEC methods.

The characterization of global minimum and other stationary points (geome-

try, energy and vibrational frequencies) is shown in Table 2. The global minimum

is located at R1 = 3.6623a0 and R2 = R3 = 2.5295a0, which shows a maximum

deviation of only 0.0025a0 for SH bond length (R2 and R3), when compared with

results for the DMBE/CBS PES (R1 =3.6615a0 and R2 =R3 = 2.5320a0).

Figures 3–7 shows the major topographical features of the H2S DMBE/SEC

PES reported in the present work. The salient features are some of the most

relevant stationary points for the title system. They also illustrate its smooth

and correct behavior over the whole configuration space, including the long-range

regions, clearly an asset of DMBE theory. Besides the global minimum at rS−H2≈

1.8a0 and rHH ≈ 3.6a0, Figure 3 also shows that the sulfur atom approaches H2

from large atom-diatom separations along T-shaped geometries via a barrierless

process, thus agreeing with previous findings for the DMBE/CBS PES15 and the

recent theoretical work of Zyubin et al.6 and Ho et al.8

Figure 4 shows a contour plot for linear H–S–H stretch. The notable feature

from this plot is the existence of a H–S–H linear saddle point located at rSH =

2.4547a0 with an energy of 69.47 kcal mol−1 above the global minimum of H2S

but still 29.57 kcal mol−1 below the energy of the S(1D) + H2 asymptote. This

compares well with the DMBE/CBS PES, where the saddle point is predicted

to occur at rSH = 2.4860a0 with an energy of 67.90 kcal mol−1 above the global

J. Phys. Chem. A 113, 9213-9219 (2009). 155

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

R2/

a 0

R1/a0

16 16

15

15

15 15

14

14 14

14

14

13

13

13 13

12

12

12

12 12

11

11

10

10 10

9 9

9

9

9

8

8

7

7 7

6 6

6

6

6

5

5

4

4

4

4 4

3

3

3

3

3

2

2

2 2

1 1

S H HR2 R1

Figure 5. Contour plot for bond stretching in S−H−H collinear configuration.Contours equally spaced by 0.01Eh, starting at −0.135Eh. The dashed area arecontours equally spaced by 0.0025Eh, starting at −0.00425Eh.

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

y/a 0

x/a0

20

19

19

18

18

17

17

16

16

H

15

15

14

14

14

13

13

13 13

H

12

12 12

11 11

11

10 10

10

9

9

8

8

7

7

6

6

5

5

4 3 2

16

16

16

16

16

15

15

15

15

15

14

14

14

14

14

13

13

13

13

13

13

12

12

12

12

12 12

11

11

11

11

11

10

10

10

10

10

10

9 9

9

9

9

9

8

8

8

8 8

8

7

7

7

7

7

7

7

7

7

6

6

6

6

6

6

6

6

5

5 5

5

5

5

5

5

5

5

5

5

4

4

4

4

4

4

4

4

4

3

3

3

3

3

3

3

2

2 2

2

2

2

2

1 1

1

1

1

1

1

Figure 6. Contour plot for S atom moving around a fixed H2 diatom in equilib-rium geometry RH2

=1.401 a0, which lies along the X-axis with the center of thebond fixed at the origin. Contour are equally spaced by 0.0045Eh, starting at−0.189Eh. The dashed area are contours equally spaced by −0.00008Eh, startingat −0.13155Eh.


0

1

2

3

4

5

6

7

-8 -6 -4 -2 0 2 4 6

y/a 0

x/a0

35

34

33

32

31

31 30

30 29

S 28

28 27

27

27

26

26

26

25

H

25

24

24

24

24

23

23

23

23

22

22

22

21

21 21

20

20

19

19

19

18

18

18 17

17

17

16

16

15

15

15

14

14 13

13

12

11

11

10 9

8

7

6

5

4

2

1

6

6

6

6

6

5 5

5

5

5 4

4

4

4

4

4

3

3

3

3

3

3

2

2

2

2

2

1

1

1

1

1

1

1

Figure 7. Contour plot for H atom moving around a fixed SH diatom with thebond length fixed at RSH =2.534 a0, which lies along the X axis with the center ofthe bond fixed at the origin. Contour are equally spaced by 0.0055Eh, startingat −0.2855Eh. The dashed lines are contours equally spaced by −0.00045Eh,starting at −0.13935Eh.

minimum and 31.40 kcal mol−1 below the reactant asymptote. Also visible in this

Figure is an H− S · · ·H saddle point located at R2 =2.5624 a0 and R3 =3.7765 a0

with an energy of 9.39 kcal mol−1 higher than the S(1D) + H2; see Table 2.

The major feature of the DMBE/SEC PES for collinear S–H–H are illustrated

in Figure 5. As seen, the collinear saddle point is found to have a geometry with

rSH =2.6758 a0 and rHH =2.0665 a0, and a barrier height of 6.07 kcal mol−1. This

compares with the homologous values of rSH = 2.7223 a0 and rHH = 2.0723 a0,

and 5.81 kcal mol−1 for the DMBE/CBS PES. Although the DMBE/CBS PES

predicts somewhat deeper global minimum and smaller barrier heights for the

saddle points than the DMBE/SEC PES, their location is very similar with a

maximum deviation of 0.063 a0 for the HH bond length of the H–S–H linear

saddle point.

Figure 6 shows energy contours for S atom moving around ground-state H2

whose bond length is fixed at its equilibrium geometry of rHH = 1.401 a0. The

corresponding plot for H atom moving around SH diatom with its bond distance

fixed at rSH =2.534 a0 is shown in Figure 7. The two plots clearly show a smooth

behavior both at short- and long-range regions. Another important aspect of

J. Phys. Chem. A 113, 9213-9219 (2009). 157

10-11

10-10

10-9

200 600 1000 1400 1800 2200 2600 3000 3400

k(T

) / c

m3 s-1

T / K

Chang et al.

exp.Lin & Guo

S+H2 ➝ SH+HS+D2 ➝ SD+D

S+HD ➝ SD+HS+HD ➝ SH+D

S+HD➝ (SD+H)&(SH+D)

Figure 8. Temperature dependence of thermal rate constants for S(1D) +H2/D2/HD reactions. Note that, the dotted line shows the thermal rate con-stants for S(1D) + HD reactions with the product both SD + H and SH + D.

Figure 7 is the existence of Σ/Π conical intersection for collinear S − H − H

geometries, since H2S may dissociate to SH(X 2Π) + H(2S) and SH(A 2Σ+) +

H(2S). Similarly to DMBE/CBS, the DMBE/SEC PES cannot describe the

crossing due to being single-sheeted. The full implications of this on dynamics of

the title reaction cannot be anticipated, although available results suggest that

they are probably minor.

5 Dynamics of S(1D) + H2/D2/HD Reactions

5.1 Thermal rate coefficients

This section presents the results of dynamics calculations using the DMBE/SEC

function described in the present work. First, we report QCT calculations of the

thermal rate coefficients (or constants) for the S(1D)+H2/D2/HD reactions over

the temperature range 300 K to 3000 K by running a total of 5000 trajectories

per temperature with an adapted version of the VENUS9638 code. For this,

the rovibrational state of the H2 molecule has been sampled according to the


Table 3. Least-Squares Parameters in Eq. (16) for S+H2/D2/HD Reaction RateConstants.

A/cm3 s−1 K−n n B/K

S + H2 → SH + H 3.08 × 10−11 0.25 13.20S + D2 → SD + D 3.71 × 10−11 0.18 12.44S + HD → SH + D 1.19 × 10−11 0.27 13.12S + HD → SD + H 2.06 × 10−11 0.17 12.48

procedure of Ref. 39 but with the rovibrational partition function weighted for

the ortho-para symmetry of the hydrogen molecule. An integration step size of

1.5 × 10−16s has been chosen such as to warrant conservation of the total energy

to better than one part in 103. The trajectories are started at an atom-diatom

distance of 9 A, a value sufficiently large to make the interaction energy essentially

negligible.

To model the temperature dependence of the calculated rate constants, a

three-parameter Arrhenius equation has been utilized

k(T )=AT nexp(−BT

) (16)

with the parameters being numerically defined in Table 3. Figure 8 illustrates

the temperature dependence of thermal rate coefficients for the various isotopic

reactions here studied. They are found to be ordered for the various isotopes

as kH2> kHD > kD2

, at all temperatures. For S(1D)+H2 reaction at 300 K, the

rate constant is found to be 1.22 × 10−10cm3 s−1, thus, is in reasonably good

agreement with the experiment40 (2.1 × 10−10cm3 s−1) and the quantum sta-

tistical result (1.51 × 10−10cm3 s−1 for para-H2; 1.48 × 10−10cm3 s−1 for ortho-

H2) of Lin et al.12 The thermal rate constants for S(1D) + HD reaction with

the product SH + D and SD + H are calculated to be 0.53 × 10−10cm3 s−1 and

0.55× 10−10cm3 s−1, respectively. Thus, the intramolecular isotope effect defined

is Γintra = kSH+D/kSD+H = 0.96, which is slightly larger than the value of 0.71 ob-

tained from DMBE/CBS PES,15 but is close to the Lee and Liu2, 3 experimental

result of 1.39 ± 0.07 and the value of (0.9 ± 0.1) observed by Inagaki et al.1 and

J. Phys. Chem. A 113, 9213-9219 (2009). 159

Table 4. ICS σ (in A2) for the S(1D) + H2(ν = 0, j = 0) Reaction at 2.24 and3.96 kcal mol−1 Collision Energya).

ν ′ branchingtotal ν ′ = 0 ν ′ = 1 ratio

j=0 j=1 j=0 j=1 j=0 j=1 j=0 j=1

Ec =2.24 kcal mol−1

MGB(DMBE/SEC) 26.09 26.15 20.69 20.14 5.40 6.01 0.26 0.30QM(RKHS)/b) 27.21 27.42 24.17 23.63 3.04 3.79 0.13 0.16HB(RKHS)b) 24.28 25.70 20.40 21.09 3.88 4.61 0.19 0.22GB(RKHS)b) 22.85 24.45 20.88 21.55 1.97 2.90 0.09 0.13


MGB(DMBE/SEC) 24.16 23.76 18.02 17.55 6.14 6.21 0.34 0.35HB(RKHS)b) 22.06 22.14 17.45 17.30 4.61 4.84 0.26 0.22GB(RKHS)b) 21.76 21.99 18.01 17.88 3.74 4.11 0.21 0.23

a) The vibrational branching ratio is defined as σ(ν ′=1)/σ(ν ′ =0).b) From Ref. 10.

the QCT result of 0.93 of Banares et al.11

5.2 ICSs

By running batches of 105 trajectories at collision energies of 2.24 and 3.96

kcal mol−1, vibrational state-resolved ICSs have also been calculated for the S(1D)+

H2(ν = 0, j= 0, 1) reactions. The calculated total and ν ′ state-resolved ICSs are

gathered in Table 4. At the collision energy 2.24 kcal mol−1, the total ICSs ob-

tained from our DMBE/SEC PES are found in good overall agreement with the re-

sult calculated by Banares et al.10 with quantum mechanical, QCT histogramatic

binning and QCT Gaussian-weighted binning methods using the RKHS PES of

Ho et al.8 denoted as QM(RKHS), HB(RKHS), and GB(RKHS), respectively.

Perhaps, more interesting is a comparison between the vibrational state-resolved

ICSs. These have been calculated here with the momentum Gaussian-binning

method,41 MGB(DMBE/SEC).

Figure 9 shows the MGB(DMBE/SEC) product rotational distributions for


0

1

2

1 4 7 10 13 16 19 22 25

cros

s se

ctio

n/Å

2

rotational quantum number j´

v´=0

v´=1

j=0


HB (RKHS)MGB (DMBE/SEC)

1 4 7 10 13 16 19 22 25

v´=0

v´=1

j=1

Figure 9. Vibrational state-resolved ICSs calculated for S(1D ) + H2(ν = 0, j=0, 1) reaction at Ec =2.24 kcal mol−1. Left panel: j=0. Right panel: j=1.

0

1

2

1 4 7 10 13 16 19 22 25

cros

s se

ctio

n/Å

2

rotational quantum number j´

v´=0

v´=1

j=0


HB (RKHS)MGB (DMBE/SEC)

1 4 7 10 13 16 19 22 25

v´=0

v´=1

j=1

Figure 10. Vibrational state-resolved ICSs calculated for S(1D ) + H2(ν=0, j=1) reaction at Ec =3.96 kcal mol−1. Left panel: j=0. Right panel: j=1.

S(1D)+H2(ν=0, j=0, 1) reactions calculated at a collision energy of 2.24 kcal mol−1.

Also shown by dashed line are the HB(RKHS) results of Banares et al.10 The

MGB(DMBE/SEC) distribution is seen to peak at j′ = 11 and j′ = 13 for initial

rotational states of j=0 and j=1, respectively, when the product is SH(ν ′ =0).

In turn, for the product SH(ν ′ = 1), the corresponding rotational distributions

peak at j′=6. As Figure 10 shows, the results are in good agreement with those

of Banares et al.10 for both collisional energies.


A global single-sheeted DMBE/SEC PES has been reported for the ground state

of H2S based on a least-squares fit to a set of high level AVQZ ab initio energies

J. Phys. Chem. A 113, 9213-9219 (2009). 161

that have been corrected by the DMBE-SEC method. The various topographical

features of the novel PES have been examined in detail and compared with the

DMBE/CBS PES and other PESs, as well as experimental results available in

the literature. The accuracy and consistency of the DMBE-SEC approach have

also been confirmed by comparing the corrected energies with those obtained

from CBS extrapolation to the one-electron CBS limit. Finally, the QCT ther-

mal rate constants calculated with the DMBE/SEC PES for S(1D)+H2/D2/HD

reactions have been shown to be in good agreement with available experimen-

tal and theoretical data and so did the vibrational state-resolved ICSs for the

S(1D) + H2(ν=0, j=0, 1) reactions. On the basis of the above, the DMBE/SEC

PES here reported may be recommended for dynamics studies of any type.

Having reported15 another PES for the title system out of the same raw ab

initio energies, one may wonder about their relative merits. As far as the accu-

racy of the fits is concerned, they can hardly be discriminated since they have

rather similar rmsd. Of course, the DMBE/CBS PES has been constructed in

a purely ab initio fashion, whereas the DMBE/SEC one here reported entails a

small degree of empiricism via scaling of the external (or dynamical) correlation.

The fact that they are so similar can therefore be regarded as an asset on the

consistency of both schemes. Regarding the performance of the two PESs against

experimental data, an answer must await until an extensive dynamics analysis

that goes beyond the rate constant data is reported for both. However, even then,

any superior agreement of one against the other must be qualified as discarding

nonadiabatic effects due to the single-sheeted nature of both PESs. Such studies

are currently in progress.

Acknowledgments

This work has been supported by the Fundacao para a Ciencia e Tecnologia,

Portugal.

Supporting Information Available:

Tables showing parameters and numerical values and figures showing the switch-

ing function used to model the single-sheeted H2S DMBE/SEC PES, EHFACE2U


PEC for SH(X 2Π), dispersion coefficients for the atom-diatom asymptotic chan-

nels of H2S, and contour plot for bond stretching. This material is available free

of charge via the Internet at http://pubs.acs.org.

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J. Phys. Chem. A 113, 9213-9219 (2009): SI.

Potential energy surface for ground-state H2S viascaling of the external correlation, comparison withextrapolation to complete basis set limit, and use inreaction dynamics

Y. Z. Song, P. J. S. B. Caridade and A.J.C. Varandas



(Received: April 24, 2009; Revised Manuscript Received: June 26, 2009)

166 Y. Z. Song, P. J. S. B. Caridade and A.J.C. Varandas

Table 1. Parameters in the switching functions of Eq. (8).

α1/a−10 0.416369

α2/a−10 0.724019

β1/a−30 1.084970

β2/a−30 0.114421

R101 /a0 1.13386

R111 /a0 4.84622

R201 /a0 3.92365

R201 /a0 5.90712

α/a−10 0.75

r01/a0 5.5


SH(X 2Π) H2(X 1Σ+g )

Re/a0 2.534 1.401D/Eh 0.307638 0.22979439a1/a0

−1 1.714093 1.82027480a2/a0

−2 0.470893 0.52437767a3/a0

−3 0.292038 0.36999610γ0/a0

−1 1.226740 1.094670γ1/a0

−1 6.136070 1.009737γ2/a0

−1 0.028129 0.235856

A/Eha0−eα -0.8205

a1/a0−1 0

α 2.5γ/a0

−1 2.0R0/a0 7.9652 6.9282C6/Eha0

−6 34.49 6.499C8/Eha0

−8 896.5 124.4C10/Eha0

−10 26332.1 3286.0C11/Eha0

−11 -3475C12/Eha0

−12 121,500C13/Eha0

−13 -291,400C14/Eha0

−14 6,061,000C15/Eha0

−15 -23,050,000C16/Eha0

−16 393,800,00

J. Phys. Chem. A 113, 9213-9219 (2009): SI. 167

Table

3.

Nu

mer

ical

valu

es(i

nat

omic

un

it)

ofth

ep

aram

eter

sin

Eq.

(18)

.

C0 6(R

)C

2 6(R

)C

0 8(R

)C

2 8(R

)C

4 8(R

)C

0 10(R

)

S−

H2

RM/a

03.

4158

3.20

383.

4069

3.22

663.

1853

3.40

10D

M/E

h15

.944

510

.536

151

1.36

225

808.

1457

56.7

931

2049

3.48

71a

1/a

−1

01.

2043

4183

0.49

0821

601.

2527

8782

1.09

7666

061.

0249

1625

0.12

7035

07a

2/a

−2

00.

3754

8701

0.01

9991

300.

4230

4258

0.40

7342

290.

3508

0748

-0.3

5822

586

a3/a

−3

00.

0472

4979

-0.0

1036

219

0.05

0989

720.

0507

5238

0.03

9618

980.

1591

3341

b 2/a

−2

00.

2071

0333

0.23

3175

350.

1953

8314

0.19

1466

120.

5033

9247

0.26

8506

83b 3/a

−3

02.

5×

10−

91.

8×

10−

94.

8×

10−

97.

6×

10−

91.

1×

10−

80.

0309

6264

H−

SH

RM/a

04.

2552

4.12

934.

2825

4.14

924.

1292

4.29

95D

M/E

h6.

7494

5.00

6019

0.02

0248

4.71

6731

.299

869

07.4

344

a1/a

−1

01.

2620

0585

0.47

3135

191.

2999

1575

0.88

2280

151.

1292

0709

1.32

7548

12a

2/a

−2

00.

4588

1086

0.01

4027

260.

4590

7555

0.23

6561

950.

3700

7487

0.45

9505

77a

3/a

−3

00.

0582

7349

0.01

1494

780.

0598

1692

0.02

2045

870.

0264

9191

0.06

3739

75b 2/a

−2

00.

2824

1957

0.29

9644

260.

3689

5540

50.

2858

2916

0.61

0370

300.

4408

1555

b 3/a

−3

00.

0156

7011

1.1×

10−

90.

0301

9273

4.1×

10−

95.

7×

10−

90.

0408

4231


Table

4.

Nu

merical

values

ofth

eco

efficien

tsin

the

poly

nom

ialP

(1)

ofE

q.

(19).

γ11 /a

−1

0=

0.30γ

12 /a−

10

=0.70

γ13 /a

−1

0=

0.70

R1,ref

1/a

0=

1.40R

1,ref

2/a

0=

4.00R

1,ref

3/a

0=

4.00

c1 /a

00=

0.1863809740c2 /a

−1

0=

0.0732702281c3 /a

−1

0=

0.6404797902c4 /a

−2

0=

0.0446227567c5 /a

−2

0=

0.0685991247c6 /a

−2

0=−

0.2569878613c7 /a

−2

0=

0.0578809025c8 /a

−3

0=

0.0059786913c9 /a

−3

0=−

0.0450976941c10 /a

−3

0=−

0.0180934342c11 /a

−3

0=

0.1111309577c12 /a

−3

0=

0.0508468820c13 /a

−3

0=

0.0457526472c14 /a

−4

0=

0.0101461079c15 /a

−4

0=

0.0214942720c16 /a

−4

0=

0.0209654604c17 /a

−4

0=

0.0234285683c18 /a

−4

0=−

0.0507736835c19 /a

−4

0=

0.0191879595c20 /a

−4

0=−

0.0151803777c21 /a

−4

0=

0.0343793193c22 /a

−4

0=

0.0126054717c23 /a

−5

0=−

0.0009143720c24 /a

−5

0=−

0.0019551634c25 /a

−4

0=−

0.0060493317c26 /a

−5

0=

0.0092731812c27 /a

−5

0=−

0.0014011668c28 /a

−4

0=

0.0017759506c29 /a

−5

0=−

0.0073031290c30 /a

−5

0=

0.0051938362c31 /a

−4

0=−

0.0168584075c32 /a

−5

0=−

0.0037464038c33 /a

−5

0=−

0.0003256735c34 /a

−4

0=

0.0071751313c35 /a

−6

0=

0.0001634043c36 /a

−6

0=

0.0009391217c37 /a

−6

0=−

0.0003775879c38 /a

−6

0=−

0.0011477968c39 /a

−6

0=

0.0006535595c40 /a

−6

0=

0.0003599987c41 /a

−6

0=

0.0005556464c42 /a

−6

0=−

0.0005017138c43 /a

−6

0=

0.0001666837c44 /a

−6

0=−

0.0006301966c45 /a

−6

0=

0.0010048876c46 /a

−6

0=

0.0010231005c47 /a

−6

0=

0.0012287487c48 /a

−6

0=

0.0001456064c49 /a

−6

0=

0.0010567549c50 /a

−6

0=

0.0012160573

J. Phys. Chem. A 113, 9213-9219 (2009): SI. 169

Table

5.

Nu

mer

ical

valu

esof

the

coeffi

cien

tsin

the

pol

yn

omia

lP

(2)

ofE

q.

(19)

.

γ2 1/a

−1

0=

0.45

γ2 2/a

−1

0=

0.75

γ2 3/a

−1

0=

0.75

R2,r

ef1

/a0

=4.

40R

2,r

ef2

/a0

=2.

20R

2,r

ef3

/a0

=2.

20

c 1/a

0 0=−

10.5

5179

1893

c 2/a

−1

0=−

1.93

3556

7478

c 3/a

−1

0=−

0.42

2796

7767

c 4/a

−2

0=−

1.91

7374

9076

c 5/a

−2

0=−

1.21

9547

9297

c 6/a

−2

0=

0.96

8277

1418

c 7/a

−2

0=−

1.00

3677

6328

c 8/a

−3

0=−

0.24

1506

2961

c 9/a

−3

0=−

0.34

3373

3884

c 10/a

−3

0=

0.10

0351

7286

c 11/a

−3

0=

0.16

4477

0879

c 12/a

−3

0=−

0.00

5045

8404

c 13/a

−3

0=

0.34

7417

7500

c 14/a

−4

0=−

0.08

6127

2640

c 15/a

−4

0=

0.17

4221

0263

c 16/a

−4

0=

0.14

9584

7251

c 17/a

−4

0=

0.07

1660

0644

c 18/a

−4

0=

0.04

8298

3891

c 19/a

−4

0=

0.00

4358

7882

c 20/a

−4

0=−

0.34

5394

3295

c 21/a

−4

0=−

0.01

7444

2602

c 22/a

−4

0=−

0.23

1391

5973

c 23/a

−5

0=

0.00

3758

3849

c 24/a

−5

0=

0.06

5797

0147

c 25/a

−4

0=−

0.00

3883

0682

c 26/a

−5

0=

0.05

2003

3708

c 27/a

−5

0=

0.03

4394

8252

c 28/a

−4

0=−

0.03

1810

0340

c 29/a

−5

0=

0.01

1500

1166

c 30/a

−5

0=−

0.02

0508

5674

c 31/a

−4

0=−

0.04

6825

5826

c 32/a

−5

0=

0.02

7096

5389

c 33/a

−5

0=

0.07

1985

5268

c 34/a

−4

0=−

0.00

4714

7170

c 35/a

−6

0=

0.00

0248

1213

c 36/a

−6

0=

0.00

0512

6647

c 37/a

−6

0=−

0.01

2247

4116

c 38/a

−6

0=−

0.00

3092

2963

c 39/a

−6

0=−

0.00

8254

9333

c 40/a

−6

0=

0.00

0692

9704

c 41/a

−6

0=

0.00

0602

0904

c 42/a

−6

0=−

0.00

1313

6727

c 43/a

−6

0=

0.00

0493

5302

c 44/a

−6

0=

0.01

0407

9458

c 45/a

−6

0=

0.01

4433

3366

c 46/a

−6

0=−

0.00

1797

6912

c 47/a

−6

0=

0.00

4889

3114

c 48/a

−6

0=

0.00

9229

7798

c 49/a

−6

0=

0.00

6241

8890

c 50/a

−6

0=−

0.00

7515

7117


0

1

2

3

4

5

0 2 4 6 8 10

102 V

(1) S(1 D

) h(R

1)

R1/a0

S(1D)

S(3P)

102 V

(1) S(1 D

) f(R

)

g(r1) h(R1)

0

1

2

3

4

5

6

R1/a0

02

46

810

1214

16

r1/a0

0

2

4

Figure 1. Switching function used to model the single-sheeted H2S DMBE/SECPES. Shown in the left panel is the fit of the h(R1) switching form to the ab initio

points calculated for S + H2 configuration as a function of H–H distance (R1).Shown in the right-hand side panel is a perspective view of the global switchingfunction.

-0.20

-0.15

-0.10

-0.05

0.00

0 2 4 6 8

V/E

h

R/a0

10-2

10-1

100

101

102

103

SH(X 2Π)

Figure 2. EHFACE2U PEC for SH (X 2Π). The solid dots indicate the ab initio

energies calculated at MRCI(Q)/AVQZ level.

J. Phys. Chem. A 113, 9213-9219 (2009): SI. 171

0

2

4

6

8

10

0 2 4 6 8 10

10-1

C6/

Eha6 0

R/a0

C26

C06

0 2 4 6 8 10

C26

C06

0

5

10

15

20

25

10-2

C8/

Eha8 0

C48

C28

C08

C48

C28

C08

0

2

4

6

8

10-4

C10

/Eha10 0

C010

S−H2

C010

H−SH

Figure 3. Dispersion coefficients for the atom-diatom asymptotic channels ofH2S as a function of the corresponding internuclear distance of diatom.

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

R3/

a 0

R2/a0

19

19

19

18

18

18

17

17

17

17

16

16

16

16

15

15

15

15

14

14

14

14

13

13

13

13

12

12

12

12

11

11

11

11

10

10

10

9

9

8

8

7

7

6

6

5

4

3

2

H H

SR2 R3α

Figure 4. Contour plot for H−S−H bond stretching with the bond angle (6 HSH)fixed at 92.75o. Contours equally spaced by 0.015Eh, starting at −0.285Eh.

Chapter 7

Accurate DMBE/CBS PES forground-state HS2

J. Phys. Chem. A XXX, XXXX-XXXX (2011).

Accurate DMBE potential energy surface forground-state HS2 based on ab initio dataextrapolated to the complete basis set limit




(Received: XXXX XX, 2011; Revised Manuscript Received: XXXX XX, 2011)

Abstract

A double many-body expansion potential energy surface is reported for the electronic

ground state of HS2 by fitting accurate multireference configuration interaction energies

calculated with aug-cc-pVTdZ and aug-cc-pVQdZ basis sets upon separate extrapola-

tion of the complete-active-space self-consistent field and dynamical correlation com-

ponents of the total energy to the complete basis set limit. The major topographical

features of the potential energy surface are examined in detail, and the model function

used for a thermalized calculation of the rate constants for the S + SH → H + S2 reac-

tion at 298 and 400K. A value of (1.44 ± 0.06) × 10−11cm3 s−1 at 298K is obtained,

providing perhaps the most reliable estimate of the rate constant known thus far for

such a reaction.


1 Introduction

The HS2 radical plays an important role in a variety of environments, notably

in combustion and the oxidation of reduced forms of sulfur.1–4 A vast amount

of investigation has been carried on HS2 both experimentally and theoretically

since the pioneering work by Porter5 who has first proposed that the HS2 radical

was produced during the photolysis of H2S2.

The first millimeter-wave spectra of HS2 reported by Yamamoto and Saito6

provided the first experimental information about the ground state structure of

HS2. Isoniemi et al.7 has subsequently investigated the infrared spectroscopy of

the HS2 radical in an Ar matrix following the 266 nm photolysis of H2S2. The

absorption bands for two vibrational motions of HS2 radical, namely the H-S

stretch and the HSS bend, are observed to be 2463 and 903 cm−1. In a recent

experiment, Ashworth and Fink8 have recorded the chemiluminescence spectrum

of the HS2 radical with a high-resolution Fouriertransform spectrometer. The

overview spectrum in the region between 4000 cm−1 and 9000 cm−1 has been

analyzed and the so obtained vibrational parameters presented.

Out of a vast theoretical work, Sannigrahi et al.9 pioneered the ab initio

calculations on the ground and excited states of HS2, which provided both struc-

tural and vibrational information. Owens et al.10 investigated HX2 radicals

(X = Al, Si, P, and S) using coupled cluster theory, CCSD(T), having reported

the equilibrium structure and vibrational frequencies at the global minimum.

Later, Denis11 reported more accurate structural and thermodynamic properties

of both ground and excited states of HS2 and HS+2 at the CCSD(T) and B3LYP

density functional levels of theory. Moreover, the structural and energetic prop-

erties of ground state HS2 and HSS → SSH transition state have been examined

using the CCSD(T) method by Francisco,12 who estimated the energy change

for the isomerization reaction to be of 31.7 ± 1 kcal mol−1. Quite recently, Peter-

son et al.13 calculated the equilibrium geometry of the ground and first excited

electronic states of HS2 with highly correlated coupled cluster methods followed

by basis set extrapolation. The centrifugal distortion constants, harmonic fre-

quencies, and vibration-rotation coupling constants have then been calculated

for both electronic states of HS2 and DS2 using accurate three-dimensional, near-

J. Phys. Chem. A XXX, XXXX-XXXX (2011). 177

equilibrium potential energy and dipole moment functions.

All the above studies have focused mainly on the structural and spectroscopic

constants of the HS2 radical at its global minimum. Indeed, as far as we are

aware, no work toward obtaining a global potential energy surface (PES) for

ground-state HS2(X 2A′′) has been reported thus far. In this work, we present

a realistic global PES for HS2(X 2A′′) based on double many-body expansion

(DMBE)14–17 theory by accurately fitting to the ab initio physically motivated

forms to the calculated ab initio energies once extrapolated to the complete basis

set (CBS) limit.

The paper is organized as follows. Section 2 describes the ab initio calculations

employed in the present work, while Section 3 provides a survey of the procedure

utilized to CBS extrapolate the calculated energies. The analytical modeling of

the DMBE/CBS PES is presented in Section 4. The major topographical features

of the resulting PES are discussed in section 5. Section 6 gathers the concluding

remarks.

2 Ab initio calculations

The ab initio calculations are carried out at the multi-reference configuration

interaction level18, 19 of theory, including the popular quasi-degenerate David-

son correction for quadruple excitations [MRCI(Q)],20 using the full-valence-

complete-active space (CASSCF21) wave function as reference (abbreviated as

CAS). All calculations are performed using the MOLPRO22 package. The stan-

dard aug-cc-pVXZ (AVXZ) basis set of Dunning23, 24 plus core-polarization high-

exponent d functions (AVXdZ)25 has been used for the S atom and AVXZ for

the H atoms, with X = T,Q. A grid of 1601 ab initio points have been cho-

sen to map the PES over the H − S2 region defined by 3.0 ≤ RS2/a0 ≤ 4.5,

2 ≤ rH−S2/a0 ≤ 10 and 0 ≤ γ/deg ≤ 90. For the S − SH interactions, a grid

defined by 2 ≤ RSH/a0 ≤ 4, 2 ≤ rS−SH/a0 ≤ 10 and 0 ≤ γ/deg ≤ 180 has been

chosen. For both channels, r, R and γ are the atom-diatom Jacobi coordinates.


3 Extrapolation to CBS limit

The ab initio energies calculated in this way have been subsequently extrapolated

to CBS limit. To perform the extrapolation, the MRCI(Q) energy is treated in

split form by writing26

EX(R) = ECASX (R) + Edc

X (R) (1)

where the subscript X indicates that the energy has been calculated in the

AVXdZ basis and the superscripts CAS and dc stand for complete-active-space

and dynamical correlation energies, respectively. Note that all extrapolations

are carried out pointwise, and hence, the vector R of the nuclear geometrical

coordinates will be omitted for simplicity.

To extrapolate the CAS (uncorrelated in the sense of lacking dynamical corre-

lation) energies, we have adopted the two-point extrapolation protocol proposed

by Karton and Martin (KM):27

ECASX = ECAS

∞ +B/Xα (2)

where α is an effective decay exponent. Being a two-parameter protocol (ECAS∞ , B),

a minimum of two raw energies will be required for the extrapolation. Specially,

Eq. (2) will be calibrated from the CAS/AV(T,Q)dZ energy pairs using a value

of α=5.34; note that this value has been found optimal when extrapolating HF

energies to the CBS limit, and has been suggested26 to be valid also for the CAS

energy.

To extrapolate the dynamic correlation (dc) energy, we utilize our own uni-

form singlet- and triplet-electron pair (USTE26, 28) scheme, which has already

been successfully utilized to construct global DMBE/CBS PESs of H2S(X 1A′)29

and NH2(1 2A′)30 (and partly also for ground-state H2O31 as well as jointly with

correlation scaling for the quartet ground-state of N332). It assumes the form.

EdcX = Edc

∞ +A3

(X + α)3+

A5

(X + α)5(3)


A5 = A5(0) + cA5/43 (4)


with A5(0) = 0.0037685459Eh, c = −1.17847713E−5/4h , and α = −3/8 are the

universal-type parameters. Thus, E∞ and A3 are the unknown to be determined

from a fit to the dc energies calculated with the AVTdZ and AVQdZ basis sets

[USTE(T,Q)]. Note that the USTE model suitably calibrated for a specific theory

has been shown26 to yield both the CBS extrapolated full correlation in systems

studied that by the popular single-reference Møller-Plesset (MP2) and coupled

cluster [CCSD and CCSD(T)] methods as well as its dynamical part in MRCI(Q)

calculations in very good agreement with the best available estimates. Since the

method has been described in detail in various papers ever since its proposal,

we refer the reader to Ref. [33] from which other references can be obtained by

cross-referencing.

4 Single-sheeted DMBE potential energy surface

Within the framework of DMBE14–17 theory, the single-sheeted PES is written as

V (R)=3∑

i=1

[V

(2)EHF(Ri) + V

(2)dc (Ri)

]+ V

(3)dc (R) + V

(3)ele (R) + V

(3)EHF(R) (5)

where R = {R1, R2, R3} is a collective diatomic internuclear separation of tri-

atomic, and the three-body electrostatic term (only present in the presence of

overlapping polarizable species) has been separated from the corresponding EHF

energy for clarity. Thus, the two-body energy terms are split into two contribu-

tions: the extended Hartree- Fock (EHF) and dynamical correlation (dc) energies.

Although a similar partition applies to all other n-body energy terms, the electro-

static (ele) long-range contribution has been separated for clarity as noted above,

since it varies at long-range with a form akin to three-body dynamical correlation

terms. The following subsections give a detailed description of the two-body and

three-body energy terms employed in Eq. (5).


The diatomic potential energy curves of S2(X3Σ−

g ) and SH(X 2Π), have been

calibrated by fitting ab initio energies extrapolated as described in the previous


Table

1.

Equ

ilibriu

mgeom

etries(b

ohr),

vib

rational

frequ

encies

(cm−

1)an

dd

issociation

energies

(kcalmol −

1)for

SH

and

S2 .

SH

(X2Π

)S

2 (X3Σ

−g)

Re

ωe

D0

De

Re

ωe

D0

De

∆E

a

MR

CI(Q

)/AVD

dZ

b2.5615

2676.477.43

81.263.6465

689.885.60

86.595.33

MR

CI(Q

)/AVT

dZ

b2.5380

2687.881.67

85.513.6106

707.794.47

95.469.95

MR

CI(Q

)/AVQ

dZ

b2.5361

2694.483.03

86.883.5937

719.898.29

99.3212.44

MR

CI(Q

)/AV

5dZ

b2.5351

2694.183.42

87.273.5881

722.4100.21

101.2413.97

DM

BE

/CB

SP

ES

c2.5354

2697.583.91

87.773.5841

731.0100.90

101.9514.18

CC

SD

(T)/A

VT

dZ

d2.5388

2688.681.49

85.333.6022

714.494.26

95.289.95

CC

SD

(T)/A

VQ

dZ

d2.5371

2696.382.79

86.643.5850

726.198.29

99.3312.69

CC

SD

(T)/A

V5d

Zd

2.53622696.7

83.1987.05

3.5793728.7

99.86100.90

13.85C

BS

d2.5358

2697.783.58

87.443.5742

731.5101.34

102.3914.95

CB

S+

CV

d2.5328

2701.283.70

87.563.5669

734.2101.67

102.7215.16

CB

S+

CV

+T

d2.5336

2696.983.75

87.613.5672

734.4101.44

102.4914.88

CB

S+

T/T

Zd

2.53342697.6

83.7087.56

3.5667735.1

101.05102.10

14.54C

BS

e3.5733

723.3E

xp

t. f2.5339

2695.883.50±

0.787.35±

0.73.5701

725.7100.76±

0.02101.80±

0.0214.45±

0.72E

xp

t. g101.89±

0.007a

∆E

=D

e (S2 )−

De (S

H).

bT

his

work

.O

btain

edfrom

poten

tialen

ergycu

rvesfi

ttedto

MR

CI(Q

)/AVX

dZ

(X=D,T,Q,5)

energies.

cT

his

work

.O

btain

edfrom

DM

BE

/CB

SP

ES

.d

Ref.

[13],th

ed

issociation

energies

arecalcu

latedbyD

e=D

0+ω

e /2.e

Ref.

[34],th

eC

BS

value

ofR

e (S2 )

iscalcu

latedu

sing

the

CC

SD

T/A

V(Q,5)d

Zen

ergies,w

hile

ωe (S

2 )is

calculated

usin

gth

eC

CS

D(T

)/AV

(D,T

)dZ

energies.

fR

ef.[35,

36].g

Ref.

[37].


SH(X 2Π)

10-2

10-1

100

101

102

103

S2(X 3∑−g)

-0.20

-0.15

-0.10

-0.05

0.00

0 2 4 6 8

V/E

h

R/a0

0 2 4 6 8

Figure 1. EHFACE2U potential energy curves for SH (X 2Π) and S2 (X 3Σ−g ).

The circles indicate the ab initio energies extrapolated to CBS limit and the linesthe EHFACE2U values.

section, having been fitted to an extended Hartree-Fock approximate correlation

energy curve, including the united atom limit38 (EHFACE2U).

The dc energy term assume the following form15, 38

V(2)dc (R)=−

∑

n=6,8,10

Cnχn(R)R−n (6)

with the damping functions for the dispersion coefficients assuming the form

χn(R)=[1 − exp(−AnR/ρ−BnR2/ρ2)]n (7)

where An =α0n−α1 and Bn =β0exp(−β1n) in Eq. (6) are auxiliary functions.15, 17

α0, β0, α1 and β1 are universal dimensionless parameters for all isotropic interac-

tions: α0 =16.36606, α1 =0.70172, β0 =17.19338 and β1 =0.09574. Moreover, ρ is


a scaling parameter defined by ρ/a0 =5.5+1.25R0, where R0 =2(〈r2X〉1/2+〈r2

Y 〉1/2)

is the LeRoy39 parameter, and 〈r2X〉 is the expectation value of squared radius for

the outermost electron in atom X (X=A,B).

The exponential decaying portion of the EHF-type energy term is written as

V(2)EHF(R) = −D

R

(1 +

5∑

i=1

airi

)exp(−γ r) (8)

where

γ = γ0[1 + γ1tan(γ2r)] (9)


ai(i = 1, · · · , n) and γi in Eq. (8) are adjustable parameters to be obtained as

described elsewhere.15, 38

The numerical values of all the parameters for both diatomic potentials are

gathered in Table 1 of the Supporting Information, while their internuclear depen-

dences are shown in Figure 1. As seen, the modeled potentials mimic accurately

the ab initio energies. Equilibrium geometry, vibrational frequencies and dissocia-

tion energy are collected in Table 1. For both S2 and SH, the dissociation energies

increase monotonously as the size of basis sets increasing from AVDdZ to AV5dZ

and the DMBE/CBS PES gives the deepest well depth. Comparing with the CBS

results by Peterson et al.13 which are extrapolated using CCSD(T)/AV(Q, 5)dZ

energies, our results for the SH diatomic predict a difference of 0.0004 a0 in the

equilibrium geometry, give vibrational frequencies 0.2 cm−1 smaller and dissoci-

ation energies only 0.33 kcal mol−1 smaller. For the S2 diatom, the variations

are 0.0099 a0, 0.5 cm−1 and 0.44 kcal mol−1, respectively. This may be considered

quite good since we have avoided any expensive MRCI(Q)/AV5dZ calculations.

Comparing with the experimental values, our results also provide an excellent

agreement. Table 1 also shows the difference of SH and S2 dissociation energies

[De(S2)−De(SH)]. The first observation goes to the AVDdZ result which as might

be expected is poor, with the AVTdZ result still giving an error of more than

4 kcal mol−1 when comparing with the experimental value. Naturally, the AVQdZ

and AV5dZ results show enhanced agreement with experiment, with differences of

2.01 kcal mol−1 and 0.48 kcal mol−1, respectively. Notably, the DMBE/CBS PES

predicts a value of 14.18 kcal mol−1, which differs by 0.77 kcal mol−1 from the CBS


results by Peterson et al.13 and 0.27 kcal mol−1 from the experimental value.

4.2 Three-Body Energy Terms

4.2.1 Three-body Dynamical Correlation Energy

The three-body dynamical correlation energy assumes the usual form of a sum-

mation in inverse powers of the fragment separation distances:40

V(3)dc = −

3∑

i=1

∑

n


−ni (10)

where the first summation includes all atom-diatom interactions (i ≡ A − BC).

Ri is the diatomic internuclear distance, ri is the separation between atom A and

the center-of-mass of the BC diatomic internuclear coordinate, and θi is the angle

between these two vectors (see Figure 1 of Ref. 41). fi = 12{1 − tanh[ξ(ηRi −

Rj − Rk)]} is a convenient switching function, where we have fixed η = 6 and

ξ = 0.6 a−10 ; corresponding expressions apply to Rj , Rk, fj, and fk. χn(ri) is

the damping function, which still takes the forms in Eq. (7), but replace R by

the center-of-mass separation for the relevant atom-diatom channel. The atom-

diatom dispersion coefficients in Eq. (10) is given by

C(i)n =

∑

L

CLnPL(cosθi) (11)

where PL(cosθi) denotes the L-th term of Legendre polynomial expansion and CLn

is the associated expansion coefficient. The expansion in Eq. (11) has been trun-

cated by considering only the coefficients C06 , C

26 , C

08 , C

28 , C

48 , and C0

10; all other

coefficients have been assumed to make a negligible contribution, and hence ne-

glected. To estimate the dispersion coefficients, we have utilized the generalized

Slater-Kirkwood approximation.42 As usual, the atom-diatom dispersion coeffi-

cients so calculated for a set of nuclear distances have then been fitted to the

form

CL,A−BCn (R) = CL,AB

n + CL,ACn +DM(1 +

3∑

i=1

airi)exp(−

3∑

i=1

biri) (12)

where r = R − RM is the displacement relative to the position of the maximum

and b1 = a1. CL,ABn , for L = 0, are the corresponding atom-atom dispersion


coefficients(for L 6= 0, CL,ABn = 0). The least-squares parameters that result

from such fits are collected in Table 2 of the Supporting Information, while their

internuclear dependences are displayed in Figure 1 of the Supporting Information.

Note that, for R = 0, the isotropic component of the dispersion coefficient is fixed

at the corresponding value in the A–X pair, where X represents the united atom

of BC at the limit of a vanishingly small internuclear separation. As pointed

out elsewhere,40 Eq. (10) causes an overestimation of the dynamical correlation

energy at the atom-diatom dissociation channel. To correct such a behavior, we

have multiplied the two-body dynamical correlation energy for i-pair by Πj 6=i(1−fj), correspondingly for channels j and k. This ensures that the only two-body

contribution at the i-th channel is that of BC.

4.2.2 Three-body electrostatic energy

Since the H atom has spherical symmetry, the long-range electrostatic potential

terms of HS2 only arise from the interaction of the permanent quadrupole moment

of the sulfur atom with the permanent dipole and quadruple moments of SH

diatom. Following the previous work43–45 , the electrostatic energy is written as

V(3)ele = f(R){C4(R, r)ADQ(θa, θ, φab))r−4 + C5(R, r)AQQ(θa, θ, φab))r−5} (13)

where the f(R), R, r and θ have the same meaning as in Section 4.2.1, the

θa is the angle that defines the atomic quadrupole orientation, and φab is the

corresponding dihedral angle. The coefficients C4(R, r) and C5(R, r) are given by

C4(R, r) =3

2QSDSH(R)χ4(r)

C5(R, r) =3

4QSQSH(R)χ5(r) (14)

where the DSH(R) and QSH(R) are the permanent electric dipole and quadru-

ple moments of SH, and QS is the quadruple moment of the sulfur atom. The

functional form of the angular variations of ADQ and AQQ take the expressions em-

ployed in previous work?, 45, 46 based on the classical-optimized-quadruple (COQ)

model.47–51

The analytical expression for the SH dipole has been obtained by fitting our


Table 2. Numerical values, for SH dipole and quadrupole moments.

DSH QSH

Q∞/ea20 1.014

M6/ea80 20000

Rref/a0 2.2198 3.7669DM 0.308922 ea0 0.716170 ea2

0

a1/a−10 -0.090708 0.088913

a2/a−20 -0.158775 -0.337353

a3/a−30 0.038302 0.266863

b1/a−10 -0.090708 -0.208205

b2/a−20 -0.064710 0.508490

b3/a−30 0.062764 0.112271

0.0

0.1

0.2

0.3

0.4

D/e

a 0

DSH

-1

0

1

2

3

0 1 2 3 4 5 6 7 8 9

Q/e

a 02

R/a0

QSH

Figure 2. Variation of SH dipole and quadrupole moments with internucleardistance.


own ab initio results to the form52

D(R) = DM(1 +

3∑

i=1

ai ri) exp(−

3∑

i=1

bi ri) (15)

where r = R−Rref and Rref is the reference distance corresponding to the maxi-

mum in the D(R) curve, and b1 ≡ a1. In turn, the variation of the SH quadrupole

moment with the internuclear distance has been fitted to the following model51

Q(R) = DM(1 +

3∑

i=1

ai ri) exp(−

3∑

i=1

bi ri) +Q∞ + χ8(R)

M6

R6(16)

where r = R − Rref with Rref being the reference distance corresponding to the

maximum in the Q(R) curve. Q∞ is the value of the separated-atoms quadrupole

limit. The parameters in Eq. (15) and (16) are collected in Table 2, while their

graphical view of the modeled functions can be seen in Figure 2.

4.2.3 Three-body extended Hartree-Fock energy

By removing, for a given triatomic geometry, the sum of the two-body energy

terms from the corresponding DMBE interaction energies Eq. (5), which was de-

fined with respect to the infinitely separated ground-state atoms, one obtains

the total three-body energy. Then by subtracting the three-body dynamical cor-

relation contribution Eq. (10) and the three-body electrostatic energy Eq. (13)

from the total three-body energy, one obtains the three-body extended Hartree-

Fock energy. This can be represented by the following three-body distributed-

polynomial45 form

V(3)EHF =

3∑

j=1

P j(Q1, Q2, Q3) ×3∏

i=1

{1 − tanh[γji (Ri − Rj,ref

i )]} (17)

where P j(Q1, Q2, Q3) is the j-th polynomial up to six-order in the symmetry

coordinates. As usual, we obtain the reference geometries Rj,refi by first assuming

their values to coincide with bond distances of the associated stationary points.

Subsequently, we relax this condition via a trial-and-error least-squares fitting

procedure. Similarly, the nonlinear range-determining parameters γji have been

optimized in this way. The complete set of parameters amounts to a total of 150


Table 3. Accumulated (acc) and stratum (strat) root-mean-square deviations(kcal mol−1) of the DMBE potential energy surface.

Energy Na max. devb rmsd N c>rmsd

acc strat acc strat acc strat acc strat acc strat

10 0–10 51 51 0.288 0.288 0.120 0.120 15 1520 10–20 95 44 0.677 0.677 0.231 0.313 28 1530 20–30 151 56 0.915 0.915 0.332 0.455 47 2340 30–40 263 112 2.284 2.284 0.462 0.594 67 3150 40–50 350 87 2.284 2.027 0.548 0.750 97 2460 50–60 448 98 2.918 2.918 0.587 0.709 121 3070 60–70 618 170 3.425 3.425 0.586 0.584 154 3380 70–80 795 177 3.425 3.270 0.577 0.541 190 4090 80–90 917 122 3.425 3.058 0.599 0.731 225 37100 90–100 1011 94 3.425 3.128 0.617 0.766 253 27150 100–150 1388 377 3.671 3.671 0.651 0.734 318 73200 150–200 1451 63 3.848 3.848 0.712 1.529 325 23250 200–250 1488 37 3.848 3.742 0.745 1.570 329 12500 250–500 1591 103 4.419 4.419 0.851 1.777 315 311000 500–1000 1601 10 4.419 1.926 0.853 1.151 322 5

a Number of calculated DMBE/CBS points up to the indicated energy range.b Maximum deviation up to indicated energy range. c Number of calculatedDMBE/CBS points with an energy deviation larger than the rmsd.

linear coefficients ci, 9 nonlinear coefficients γji , and 9 reference geometries Rj,ref

i .

All the numerical values of the least-squares parameters are gathered in Table 3

and 4 of the Supporting Information. Table 3 shows the partial and accumulated

stratified root-mean-squared deviations (rmsd) of the final DMBE/CBS PES with

respect to all the fitted ab initio energies. As shown in Table 3, a total of 1601

points have been used for the calibration procedure, with the energies covering a

range up to 1000 kcal mol−1 above the HS2 global minimum. The fit shows the

total root mean square derivation is 0.853 kcal mol−1.

5 Features of the potential energy surface

Table 4 gathers the relative energies of the present DMBE/CBS PES. The re-

sults carried out by Peterson et al.13 at the CCSD(T)/AVXdZ level and those


Table 4. Relative energetics for HS2 DMBE/CBS PES.

Relative energeticsa

Level of theory De(H + S2) De(S + SH) De(H + 2S) De([HSS → SSH]6=)

DMBE/CBS PESb 60.95 75.13 162.90 33.63CCSD(T)/AVTdZc 60.01 69.97 155.30CCSD(T)/AVQdZc 60.63 73.31 159.96CCSD(T)/AV5dZc 60.77 74.63 161.67CBSc 60.88 75.90 163.27CBS+CVc 60.71 76.07 163.61CBS+CV+Tc 61.16 76.04 163.65CCSD(T)/AVTZd 55.1 67.4 30.5CCSD(T)/AVQZd 55.8 71.3 31.2CCSD(T)/AV5Zd 55.9 73.5 31.6CBSd 56.2 74.1 31.7 ± 1

a The units of energy is kcal mol−1.b This work.c Ref. 13.d Ref. 12.

1

2

3

4

5

6

7

8

2 3 4 5 6 7 8

R2/

a 0

R1/a0

24

67

8

9

9 10

11

11

11

12

12

13

13

13

14

1414

15

15

15

16

17

1717

18

18

18

19

19

21

2121

22

22 22

23

23

23

H

S S

R2

R1

α

Figure 3. Contour plot for bond stretching in H− S− S, keeping the HSS anglefixed at 101.96 o. Contours equally spaced by 0.01Eh, starting at −0.259Eh.


by Francisco12 at the CCSD(T)/AVXZ level are also gathered in this table for

comparison. The energies of H + S2 relative to the HS2(X2A′′) global minimum

calculated by Francisco at CCSD(T)/AVXZ (X = T,Q, 5) are 55.1 kcal mol−1,

55.8 kcal mol−1, 55.9 kcal mol−1, and the CBS limit gives a value of 56.2 kcal mol−1,

thus ∼ 4.9 kcal mol−1 smaller than the CCSD(T)/AVXdZ (X = T,Q, 5) val-

ues reported by Peterson et al.13 For the relative energies of S + SH, Fran-

cisco12 predicts values ∼ 2.0 kcal mol−1 smaller than those of Peterson et al.13

This suggests that one needs to include the core-polarization high-exponent d

functions (AVXdZ) as recommended53, 54 for compounds containing second-row

atoms, such as the title one. The relative energies for H + S2 and S + SH cal-

culated from the present DMBE/CBS PES, which are extrapolated to CBS uti-

lizing the MRCI(Q)/AV(T,Q)dZ scheme described above are predicted to be of

60.95 kcal mol−1 and 75.13 kcal mol−1, showing differences of 0.07 kcal mol−1 and

0.77 kcal mol−1 relative to the values of Peterson et al.,13 respectively. In turn,

the well depth of the HS2 global minimum predicted from our DMBE/CBS PES

is 162.90 kcal mol−1, which agrees well with the CBS result (163.27 kcal mol−1)

by Peterson et al.13 Indeed, the difference is of only 0.37 kcal mol−1 despite the

fact that smaller basis sets have here been utilized. The calculated hydrogen

atom exchange barrier for the HSS → SSH reaction is 33.63 kcal mol−1, which is

∼ 1.93 kcal mol−1 lower than the CBS result (31.7 ± 1.0 kcal mol−1) suggested by

Francisco.12

Table 5 compares the attributes of the stationary points (geometry, energy,

and vibrational frequencies) of the HS2 DMBE/CBS PES with the other the-

oretical and experimental results. The global minimum for the HS2 ground

state from our DMBE/CBS PES is predicted to be located at RSS = 3.7106 a0,

RSH = 2.5306 a0, and 6 HSS = 101.96o, and the result from our fit to dense grid

CBS/MRCI(Q)/AV(T,Q)dZ points gives thatR1 = 3.7072 a0, R2 = 2.5439 a0,

and 6 HSS = 102.37o, thus differing by 0.0034 a0, 0.0133 a0, and 0.41o. The

results calculated by Peterson et al.13 which are extrapolated to CBS limit

using CCSD(T)/AV(Q, 5)dZ energies are R1 = 3.7099 a0, R2 = 2.5509 a0, and

6 HSS = 101.50o and the corresponding experimental55 values areRSS = 3.7044 a0,

RSH = 2.5555 a0, and 6 HSS = 101.74o, with the differences from the result of our


Table

5.

Com

parison

ofstation

aryp

oints

ofH

S2

DM

BE

/CB

SP

ES

a.

RSS /a

0R

SH/a

06

HS

S/o

E/E

hω

1 (S−

S)

ω2 (b

end

)ω

3 (S−

H)

Glob

alm

inim

um

ab

initio

b3.7072

2.5439102.37

-0.2596552

9092488

DM

BE

/CB

SP

ES

c3.7106

2.5306101.96

-0.2596588

8732597

Exp

.3.7044

d2.5555

d)

101.74d

596f,

595g,

600h

934e,

892f,

904g

2463e,

2688f

CC

SD

(T)/cc-p

VT

dZ

i3.7426

2.5489101.40

592913

2607B

3LY

P/6-311+

G(3d

f,2p)i

3.73562.5610

102.13586

9102556

CC

SD

(T)/A

VT

Zj

3.76622.5549

100.90585

9042607

CC

SD

(T)/A

VQ

Zj

3.73602.5530

101.30598

9102604

CC

SD

(T)/A

V5Z

j3.7190

2.5511101.40

CB

Sk

3.70992.5509

101.50607

9192600

HS

2isom

erizationab

initio

b3.9091

2.937248.12

-0.20611345i

5481935

DM

BE

/CB

SP

ES

c3.9047

2.939848.39

-0.20601577i

5621886

CC

SD

(T)/A

VT

Zj

3.95332.9347

47.701208i

5381919

CC

SD

(T)/A

VQ

Zj

3.92312.9310

48.001275i

5481933

CC

SD

(T)/A

V5Z

j3.9042

2.929148.20

T-sh

aped

H−

S2

structu

reD

MB

E/C

BS

PE

Sc

3.61464.2960

65.12-0.1500

700994i

1074i

S−

H−

Ssad

dle

poin

tD

MB

E/C

BS

PE

Sc

5.87522.9376

0-0.1165

250690i

1750i

aH

armon

icfreq

uen

ciesare

incm

−1.

bF

ittedto

ad

ense

gridof

CB

S/M

RC

I(Q)/A

V(T,Q

)dZ

poin

ts.c

Th

isw

ork.

d

Exp

erimen

tvalu

es. 55

eE

xp

erimen

tvalu

es. 7f

Exp

erimen

tvalu

es. 8g

Exp

erimen

tvalu

es. 56

hE

xp

erimen

tvalu

es. 57

i

Ref.

11.j

Ref.

12.k

Ref.

13.T

he

CB

Slim

itw

asob

tained

by

extrap

olationof

the

AVQ

dZ

and

AV

5dZ

CC

SD

(T)

correlationen

ergies.


1

2

3

4

5

6

7

8

2 3 4 5 6 7 8

R2/

a 0

R1/a0

1

1

2

34

4

45

6

6 6

7

7

7

8

8

8

9

910

11

11 11

12

12

12

13

13

13

14

14

15

15

H S SR2 R1

Figure 4. Contour plot for bond stretching in S− S−H collinear configuration.Contours equally spaced by 0.0085Eh, starting at −0.16Eh.

DMBE/CBS PES being (0.0007 a0, 0.0203 a0, 0.46o) and (0.0064 a0, 0.0249 a0, 0.22o)

in the above order. As for the harmonic frequencies, the DMBE/CBS PES from

the present work predicts values of 588, 873, and 2597 cm−1 (SS stretch, HSS

bending and SH stretch, respectively), with the results from the fit to the cal-

culated dense grid of points being 552, 909, and 2488 cm−1. The most recent

experimental values given by Ashworth et al.8 are 596, 892, and 2688 cm−1. The

corresponding results reported by Peterson et al.13 who have extrapolated to

CBS using CCSD(T)/AV(Q, 5)dZ energies are 607, 919, and 2600 cm−1. Clearly,

the differences from that of our DMBE/CBS PES are quite small, amounting to

(19, 47, 3) cm−1 in the above order. As seen in Table 5, the results from our

DMBE/CBS PES are also in good agreement with other available experimental

and theoretical values.

Figure 3 to 8 illustrate the main topographical features of the HS2 DMBE/CBS

PES. Figure 3 shows energy contours for SS and SH stretching with the HSS angle

kept fixed at the corresponding equilibrium value, while Figure 4 shows contours

for the SS and SH stretching at the S − S − H collinear configuration. In turn,


1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

R3/

a 0

R2/a0

1

2

2

2

3

33

4

4

4

5

5

5

6

66

7

8

8

8

8

9

9

9

10

10 10

11

11

11

12

12

13

14

14 14

15

1515

S H SR2 R3

Figure 5. Contour plot for bond stretching in S − H − S linear configuration.Contours equally spaced by 0.0085Eh, starting at −0.138Eh.

0

2

4

6

8

2 4 6 8

y/a 0

x/a0

2

4

5

6 7

7

7

8

8

9

910

10

10

11

11

12

12

12

13

13

13

14

15

15

15

16

16

16

18

18

19

19

19

20

21

21

S S

H

x

y

Figure 6. Contour plot for a C2v insertion of H atom into S2 diatom. Contoursequally spaced by 0.0085Eh, starting at −0.21Eh.


0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

y/a 0

x/a0

S S

3

3

4

45

5

6

67

7

7

8

88

99

910

10 10

11

11

1111

12

12

12

12

12

13

1313

13

14

14

14

14

15

161617

17

1818

19

20

20

21 2222

23

24

24 25

1

1

1

1

1

1

1

1

1

2

2

22

2

2

2

2

2

2

3

3

333

3

3

33

4

4

44

4

4

44

5

5

5

5

55

5

6

6

6

6

6

6

66

7

7

7

7

7

77

8

8

8

8

8

8

8

9

9

9

9

99

10

10

10

10

10 10

11

11

11

11

1111

12

1212

12

12

12

131313

13

13 13

14

14

14

14 14

14

15

15

15

15

1515

16

16

16

16

16

16

17

17

17

17

17

17

18

1818

18

18 18

19

19

19

19 19

19

202020

20 20

21

2121

21

21

21

22

22

22

22

22

22

23

2323

23

23

23

2424

24

2424

252525

25

25

25

26

2626

26

26

26

272727

27 27

2828

28

28 28

29

29

29

29

29

3030

3030

30

Figure 7. Contour plot for H atom moving around a fixed SS diatom with thebond length fixed at RSS =3.5841 a0, which lies along the X axis with the centerof the bond fixed at the origin. Contour are equally spaced by 0.009Eh, startingat −0.258Eh. The dashed lines are contours equally spaced by −0.00008Eh,starting at −0.1625Eh.

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

y/a 0

x/a0

S H

34

45

5

66 77

8

8

8

9

99

1010

10

11

11

11

12

12

12

13

13 13

14

14

14 14

15

15

15

15

15

1616

16

16

1617

17

17

18

18

18

19

19

20

20

21

21222323 24

24

25

25

25

26

1

1 1

1

1

1

1

1

2

2

2

2

2

2

2

3

3

3

3

3

3

3

4

4 4

4

44

4

4

5

5

55

55

5

5

6 6

6

6

6

6

67 7

7

7

8

88

8

8

8

8

9

9

9

9

9

9

10

10

10

1010

10

10

11

1111

11

11

11

11

11

12 12

12

12

12

12 13 13

13

13

13

13

14

14

14

14

14

14

1515

1515

15

1616

1616

16

16

17 17

17

17

17

17

18

18

Figure 8. Contour plot for S atom moving around a fixed SH diatom with thebond length fixed at RSH =2.5354 a0, which lies along the X axis with the centerof the bond fixed at the origin. Contour are equally spaced by 0.008Eh, startingat −0.2594Eh. The dashed lines are contours equally spaced by −0.0002Eh,starting at −0.1402Eh.


Figure 5 shows a contour plot for the stretching of the two SH bonds at a S−H−S

linear configuration. The notable features from Figure 5 are the two equivalent

asymmetric hydrogen bonded minima which are separated by a barrier lying half

way between them. The location of the barrier is found to be RSS =5.8752 a0 and

RSH = 2.9376 a0, with the harmonic frequencies at the top being 250, 690i, and

1750i cm−1, respectively.

Figure 6 shows a contour plot for the C2v insertion of H into the S2 diatomic.

The important features from this Figure are the saddle point structure for the

HSS → SSH isomerization (see text before) and the barrier for the H + S2 reac-

tion. The saddle point for isomerization is found to be located at RSS =3.9047 a0,

RSH =2.9398 a0, and 6 HSS=48.39o, with the barrier height lying at −0.2060Eh,

and the harmonic frequencies being 1577i, 562, and 1886 cm−1. Comparing

with the results of Francisco,12 the agreement is good. The barrier is located

at RSS = 3.6146 a0 and RSH = 4.2960 a0, with the well depth being −0.1500Eh.

The vibrational frequencies are also gathered in Table 5.

Figure 7 shows a contour plot for a H atom moving around a SS diatom

with the bond length fixed at RSS = 3.5841 a0, which lies along the X-axis with

the center of the bond fixed at the origin. The two salient features are the deep

minima connected by a C2v saddle point which allows scrambling of the two sulfur

atoms. Also visible along the C2v line is a closed contour apparently showing a

maximum. It is indeed a conical intersection that is undescribable within the

present single-sheeted DMBE formalism. In turn, Figure 8 shows a contour plot

for S atom moving around a fixed SH diatom with the bond length fixed at

RSH =2.5354 a0, which lies along the X-axis with the center of the bond fixed at

the origin. The two plots clearly reveal a smooth behavior both at short and long

range regions.

Shown in Figures 9 and 10 are the spherically averaged isotropic (V0) and lead-

ing anisotropic potentials for H + S2(V2) and S + SH(V1, V2) scattering processes

with the diatom fixed at its equilibrium geometry. Note that the magnitude of

the isotropic average potential V0 determines how close on average the atom and

molecule can approach each other, while the sign of V2 indicates whether or not

the molecule prefers to orient its axis along the direction of the incoming atom:

a negative value favors the collinear approach while a positive value favors the


-30

-20

-10

0

10

3 4 5 6 7 8 9 10

Vn/k

cal

mo

l-1

RH-SS/a0

V2V0

Figure 9. Isotropic (V0) and leading anisotropic (V2) components of H − S2

interaction potential energy, with the diatom fixed at its equilibrium geometry.

-40

-20

0

20

40

3 4 5 6 7 8 9 10

Vn/k

cal m

ol-1

RS-SH/a0

V2

V0 V1

Figure 10. Isotropic (V0) and leading anisotropic (V1, V2) components of S−SHinteraction potential energy, with the diatom fixed at its equilibrium geometry.

approach through an isosceles triangular geometry. For the S − SH interaction,

there is a balance which makes a prediction not so readly done. Specifically, it is

shown in Figure 10 that there is no barrier in the spherically averaged isotropic

term (V0) while the anisotropic (V2) component is non-negative at the regions of


Table 6. QCT thermal rate constants (in 10−11cm3 s−1) at 298 K for S + SHreaction.

T/K rate constant

DMBE/CBS PES 298 1.44 ± 0.06a

Exp. 295 < 0.498b

295 4.0c

300 4.5d

a This work. b Ref. [59]. c Ref. [60]. d Ref. [58].

interest. This is partly compensated with a strongly attractive V1 contribution

from intermediate up to long-range interaction regimes. From their balance and

the fact that the reaction is exoergic, it turns out that the S + SH reaction is

much easier to occur than the H + S2 reverse reaction. The present result there-

fore supports that of Mihelcic and Schindler,58 who have shown the reaction of

SH radicals with S atom to form S2 and H to occur with a fast rate. In fact,

the experimental findings of Porter5 also support that the origin of the HS2 rad-

ical stems from the reaction of an S atom with a SH radical. Indeed, the recent

theoretical calculations by Francisco12 also seem to support the same conclusion.

Finally, preliminary rate constant calculations have been carried out for the

reaction S + SH → H + S2 by running quasi-classical trajectories (QCT) on the

PES of the present work at two temperatures, 298 and 400 K. A total of 5000

trajectories per temperature has been employed, with an integration step size

chosen to be 1.5 × 10−16s such as to warrant conservation of the total energy to

better than one part in 103. The trajectories have been started at a distance

between the incoming atom and the center-of-mass of the diatom of 9 A, a value

considered sufficiently large to make the interaction energy negligible.

The thermal rate constant for the formation of S + SH assumes the general

form

k(T ) = ge(T )

(8kBT

πµS+SH

)1/2Nr

Nπb2max (18)

with the estimated error of the rate constant being given by ∆k(T ) = k(T )[(N −Nr)/NNr]

1/2; T is the temperature, kB the Boltzmann constant, bmax the maxi-


1.300

1.375

1.450

1.525

250 300 350 400 450

k(T

)×1

0-11 cm

3 s-1

T

k(T )T=298 KT=400 K

Figure 11. Temperature dependence of the rate constant predicted by Eq. (18)for the reaction S + SH → H + S2. Indicated by the symbols are the QCT resultsat T = 298 K and T = 400 K, jointly with the associated error bars.

mum impact parameter, Nr the number of reactive trajectories in a total of N ,

µS+SH the reduced mass of the reactants, and ge(T ) the electronic degeneracy

factor61, 62 which assumes the form:

ge(T )=2

[5+3 exp(−569.83/T )+exp(−825.34/T )]×[2+2 exp(−542.36/T )](19)

where 2 in the numerator is the degeneracy of the HS2(X 2A′′) PES and the

denominator corresponds to the electronic partition function of the reagents: the

first part is the electronic degeneracy of S(3P ), which splits into 3P2,3P1 and 3P0

with the energy gaps63 569.83 K and 825.34 K; the second part is the degeneracy of

SH(2Π) with the two (Π) levels split into 2Π1/2−2 Π3/2, which has an energy gap36

of 542.36 K and 2 is here because each of these two states is doubly degenerate.

The result at T = 298 K is compared with the available experimental data

in Table 6. The thermalized rate constant is predicted to be of (1.44 ± 0.06) ×10−11cm3 s−1, thus about 2-3 times smaller than two of the reported experimental

values: 4.0 × 10−11cm3 s−160 and 4.5 × 10−11cm3 s−1.58 In turn, the experimental


result of 0.498 × 10−11cm3 s−1 by Nicholas et al.59 underestimates our predic-

tion by about a similar factor as above. The temperature dependence of the

rate constant predicted by Eq. (18) is depicted in Figure 11. Also shown are

the actually calculated QCT results at T = 298 and 400K, as well as the associ-

ated error bars. Not surprisingly, the rate constant value of 1.45 × 10−11cm3 s−1

predicted by Eq. (18) at T = 400 K slightly overestimates our test calculation of

1.37 ± 0.05 × 10−11cm3 s−1 at 400 K, since the former is based on the average

velocity for 298 K. In fact, a decreasing trend with temperature of the rate con-

stant is to be expected since the title reaction occurs on a barrier-free PES. A

full detailed analysis of the dynamics and kinetics will be reported elsewhere.


We have reported a global DMBE/CBS PES for the electronic ground state of

HS2, on the basis of fitting ab initio energies extrapolated to CBS limit. The

USTE(T,Q) extrapolation scheme is employed to warrant CBS-limit accuracy

even though the calculations employed relatively inexpensive basis sets. As shown

above, the DMBE/CBS PES describes all major topographical features of the HS2

PES but those forbidden within the employed single-sheeted approach. Indeed,

a comparison of its attributes with experimental and other accurate theoretical

values shows quite a good agreement. This and the results of preliminary rate

constant calculations clearly commends the use of the current PES for more

detailed adiabatic dynamics studies of the title reaction.

Acknowledgments

This work has the support of Fundacao para a Ciencia e a Tecnologia, Portugal.

Supporting Information Available

Tables of parameters in the two-body energy curves, numerical values of pa-

rameter in Eqs. (12) and (17), and plots of dispersion coefficients for the atom-

diatom channels. This material is available free of charge via the Internet at

http://pubs.acs.org.


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J. Phys. Chem. A XXX, XXXX-XXXX (2011).

Accurate DMBE potential energy surface for HS2

electronic ground state extrapolated to completebasis set limit




(Received: XXXX XX, 2011; Revised Manuscript Received: XXXX XX, 2011)



SH(X 2Π) S2(X 3Σ−g )

Re/a0 2.5354 3.5841D/Eh 0.31033012 0.51296953a1/a0

−1 1.67410759 1.68786700a2/a0

−2 0.74475240 0.76016264a3/a0

−3 0.82143960 0.79472876a4/a0

−4 0.16174114 0.22948982a5/a0

−5 0.14533900 0.21061607γ0/a0

−1 1.18919093 1.31585703γ1/a0

−1 1.17024046 8.33849756γ2/a0

−1 0.38831750 0.04799128R0/a0 7.9652 9.0601C6/Eha0

−6 34.49 107.21C8/Eha0

−8 896.50 3192.10C10/Eha0

−10 26332.10 124580.72

0

5

10

15

20

0 2 4 6 8 10

10-1

C6/

Eha6 0

R/a0

C26

C06

0 2 4 6 8 10

C26

C06

0

10

20

30

40

50

10-2

C8/

Eha8 0

C48

C28

C08

C48

C28

C08

5

10

15

20

10-4

C10

/Eha10 0

C010

H−S2 C0

10

S−SH

Figure 1. Dispersion coefficients for the atom-diatom asymptotic channels ofHS2 as a function of the corresponding internuclear distance of diatom.


Table

2.

Nu

mer

ical

valu

es(i

nat

omic

un

it)

ofth

ep

aram

eter

sin

Eq.

(14)

.

C0 6(R

)C

2 6(R

)C

0 8(R

)C

2 8(R

)C

4 8(R

)C

0 10(R

)

H−

SS

RM/a

04.

9214

84.

8288

4.82

814.

7760

4.76

414.

7679

DM/E

h84

.081

3511

.532

224

68.6

146

1391

.818

911

8.39

8395

075.

4990

a1/a

−1

00.

8698

9697

0.58

8852

400.

9847

6098

1.08

8608

681.

3315

9042

0.51

2014

06a

2/a

−2

00.

0661

6242

0.10

2858

810.

2599

7124

0.43

4145

830.

5936

4574

0.02

9643

41a

3/a

−3

0-0

.047

1507

-0.0

4222

080.

0012

5114

0.05

2250

700.

0757

4099

-0.0

3658

25b 2/a

−2

00.

2645

6931

0.21

1651

260.

2437

0685

0.16

5096

260.

2627

8623

0.14

2068

99b 3/a

−3

09.

8×

10−

10

2.1×

10−

91.

2×

10−

34.

7×

10−

98.

0×

10−

31.

3×

10−

9

S−

SH

RM/a

04.

2469

4.11

674.

2335

4.12

194.

1072

4.22

47D

M/E

h16

3.77

3117

.314

4894

.710

916

49.1

790

101.

2788

1929

39.9

868

a1/a

−1

01.

2763

3859

0.66

2190

941.

1643

9838

0.89

9914

181.

1962

5539

1.18

7934

69a

2/a

−2

00.

4758

1996

0.07

2810

760.

2002

2457

0.24

0216

160.

4374

2452

0.43

0479

89a

3/a

−3

00.

0624

5822

-0.0

0474

620.

0017

1755

0.02

0416

940.

0439

6150

0.05

3520

10b 2/a

−2

00.

2679

3696

0.25

1449

190.

5539

0644

0.26

1295

560.

5905

7178

0.20

6632

70b 3/a

−3

08.

6×

10−

39.

7×

10−

11

9.1×

10−

27.

1×

10−

97.

2×

10−

95.

5×

10−

9


Table 3. Parameters and reference geometries used in the extended Hartree-Fockenergy in Eq. (19).

Coefficients P (1) P (2) P (3)

γ(j)1 /a−1

0 0.7 0.9 1.2

γ(j)2 /a−1

0 0.8 0.4 1.5

γ(j)3 /a−1

0 0.8 0.4 1.5

R(j),ref1 /a0 3.5 4.0 4.5

R(j),ref2 /a0 4.5 3.5 3.0

R(j),ref3 /a0 4.5 3.5 3.0

Table 4. Numerical values of coefficients used in the extended Hartree-Fockenergy in Eq. (19).


C1/a00 2.5656891858 −4.4018970272 −0.1748674046

C2/a−10 0.4946249706 0.6763409428 −0.2042577133

C3/a−10 −0.9923218194 −1.0192058982 0.0302456578

C4/a−20 0.5281076194 −0.5098085680 0.0514657112

C5/a−20 0.5636820135 −0.9372251860 −0.2060555907

C6/a−20 −0.5390556668 −0.1017160894 0.0853983662

C7/a−20 −0.0504258496 1.0288374169 0.1274739429

C8/a−30 0.0887837218 0.0606399233 0.0105592497

C9/a−30 0.2126692282 0.1730796087 −0.1778279268

C10/a−30 0.0176805850 −0.0595788635 0.0232538867

C11/a−30 −0.2451313631 −0.2163583219 0.0180208419

C12/a−30 −0.0126519598 −0.3378078095 0.1703284237

C13/a−30 −0.0962712282 −0.3194255698 0.0457613843

C14/a−40 0.0103214956 0.0815374630 0.0908085603

C15/a−40 −0.0050098585 0.1552637987 0.1010176946

C16/a−40 0.0010540376 −0.0439988101 −0.0253305727

C17/a−40 −0.0639567633 0.0446206639 0.1435340476

C18/a−40 −0.1181074157 0.3273683014 −0.0987451222

C19/a−40 −0.0564017773 −0.1033405174 0.3595532144

C20/a−40 −0.1283670191 0.1285696988 0.1939413026

C21/a−40 0.0079676106 −0.0399035035 0.0524821818

C22/a−40 0.0095769087 0.0248924324 0.0470161325

C23/a−50 0.0116493778 −0.0242003565 0.0114364333


Table 4. Continue


C24/a−50 0.0060833990 0.0058941977 0.1205887776

C25/a−50 0.0028253004 −0.0067499030 −0.0245638837

C26/a−50 −0.0132788415 0.0264087708 0.1485535677

C27/a−50 −0.0059048074 −0.0088883544 −0.0032306277

C28/a−50 −0.0081473976 −0.0499356499 −0.0150918965

C29/a−50 −0.0054726725 −0.0201440657 0.2034233803

C30/a−50 −0.0038568823 0.0471539114 0.0851029758

C31/a−50 0.0272467784 0.0258224852 0.0412572170

C32/a−50 0.0189176130 0.0083891864 −0.0092047391

C33/a−50 0.0149410485 −0.0354861918 −0.0235335967

C34/a−50 −0.0008307622 0.0008626675 0.0092821262

C35/a−60 −0.0007601549 0.0022808809 −0.0022078766

C36/a−60 −0.0021310571 0.0014257881 0.0179511439

C37/a−60 0.0031426954 0.0068931434 −0.0019840124

C38/a−60 0.0022159664 0.0050743053 0.0459817788

C39/a−60 −0.0026153532 0.0027061724 0.0003750257

C40/a−60 −0.0014385966 −0.0009279219 0.0092078922

C41/a−60 0.0004026307 0.0002131393 −0.0052059461

C42/a−60 −0.0020704122 0.0044064586 −0.0132840747

C43/a−60 −0.0015348145 −0.0020707846 0.0478452266

C44/a−60 −0.0043293107 0.0083085287 0.0248669465

C45/a−60 0.0143612048 0.0082398092 0.0084507522

C46/a−60 0.0054149621 −0.0068855815 −0.0001962011

C47/a−60 0.0022354201 0.0022151274 −0.0329637753

C48/a−60 0.0024867614 −0.0031146835 −0.0146158653

C49/a−60 −0.0000415079 0.0000954224 −0.0001198254

C50/a−60 −0.0007761159 0.0033918702 0.0114931122

Chapter 8

Conclusions and outlook

In the present thesis we reported a series of published results, mainly focused on

the ab initio calculation and modeling of DMBE PESs for the electronic ground-

state H2S and HS2. We have compared the results of the conventional CCSD and

CCSD(T), and the renormalized CR-CCSD(T), CR-CCSD(TQ), CR-CC(2,3),

and CR-CC(2,3)+Q calculations with the MRCI(Q) results for the three impor-

tant cuts of the H2S(1A′) PES. We have found that all renormalized CC methods

reduce the failures of the conventional CCSD and CCSD(T) approaches in the

bond stretching regions of the H2S potential, with the CR-CC(2,3) and CR-

CC(2,3)+Q methods being most effective in this regard.

A global single-sheeted DMBE/CBS PES has been reported for the electronic

ground state of hydrogen sulfide on the basis of a least-squares fit to MRCI(Q)

energies calculated using AVTZ and AVQZ basis sets subsequently extrapolated

to the CBS limit. A global single-sheeted DMBE/SEC PES has also been reported

based on a least-squares fit to MRCI(Q) energies calculated using AVQZ basis sets

subsequently corrected by the DMBE-SEC method. The various topographical

features of the novel DMBE/CBS and DMBE/SEC PESs have been examined in

detail and compared with the other PESs, as well as experiments available in the

literature. The accuracy and consistency of the DMBE-SEC approach has also

been confirmed by comparing the corrected energies with those obtained from

CBS extrapolation. Quasiclassical trajectory studies are also carried out on the

thermal rate constants for S(1D)+H2/D2/HD reactions, which have been shown

to be in good agreement with available experimental and theoretical data, and so

did the vibrational state-resolved ICSs for the S(1D)+H2(ν=0, j=0, 1) reactions.

212 Conclusions and outlook

We have reported a global DMBE/CBS PES for the electronic ground-state

HS2, on the basis of fitting MRCI(Q)/AV(T,Q)dZ energies which are extrapo-

lated to CBS limit. The USTE(T,Q) extrapolation scheme is employed to war-

rant the accuracy of the CBS limit. The DMBE/CBS PES describes all major

topographical features of HS2 PES. Indeed, a comparison of its attributes with

experimental and other accurate theoretical values shows quite a good agreement.

This and the results of preliminary rate constant calculations for S+SH → H+S2

reaction clearly commends the use of the current PES for more detailed adiabatic

dynamics studies of the title reaction.

The PESs reported in the present thesis give a good description of H2S and

HS2 molecular systems. They can be further used to study other reactive pro-

cesses. Finally, they may enable the construction of larger polyatomic DMBE

PES in which H2S and/or HS2 are contained, such as SH3 and H2S2.

Mathematical appendices

A Linear least-squares

Linear least-squares [1, 2] is one of the most commonly used methods in numerical

computation, which is to fit a set of data points (xi, yi) to a linear combination

of any M specified functions of x

y (x) =

M∑

k=1

akXk (x) (A 1)

where X1(x), . . . , XM(x) are arbitrary fixed functions of x, called the basis func-

tions. A set of best-fit parameters correspond to a minimum of the merit function

χ2 =

N∑

i=1

[yi −

∑Mk=1 akXk (xi)

σi

]2

(A 2)

where σi is the measurement error (standard deviation) of the ith data point,

The minimum of (A 2) occurs when the derivative of χ2 with respect to all

the M parameters ak vanishes, i.e.

0 =

N∑

i=1

1

σ2i

[yi −

M∑

j=1

ajXj (xi)

]Xk (xi) k = 1, . . . ,M (A 3)

Interchanging the order of summations, we can write (A 3) as the matrix

equation

Λ · a = β (A 4)

where

Λkj =N∑

i=1

Xj (xi)Xk (xi)

σ2i

, βk =N∑

i=1

yiXk (xi)

σ2i

(A 5)

and a is the column vector of the adjustable parameters.

214 Mathematical appendix

(A 3) and (A 4) are called the normal equations of the least squares problem.

Their solutions can be obtained by using the Gauss-Jordan elimination method,

which consists in looking for the column vector a by applying row operations on

the augmented matrix Λ|c to transform the matrix Λ into diagonal form.

A more general procedure to minimize (A 1), preventing no solution of (A

4) due to singularity in Λ, is to use the singular value decomposition (SVD)

technique. Such a method is based on a theorem which states that any N ×M

matrix A, no matter how singular the matrix is, can be factorized in the form

A = U ·W · VT (A 6)

where U is an N ×M orthogonal matrix, W is a M ×M diagonal matrix with

positive or zero elements (the singular values of A) and VT is the transpose of

the M ×M orthogonal matrix V.

Defining the following matrix and column vector

Aij =Xj (xi)

σiand bi =

yi

σi(A 7)

(A 1) can be written as

χ2 = |A · a − b|2 (A 8)

Then, the solution of the least-squares problem in (A 8) can be written as

a =M∑

i=1

(U(i)· bωi

)V(i) (A 9)

with the variance in the estimate of a parameter aj is given by

σ2j (aj) =

M∑

i=1

(Vji

ωi

)2

(A 10)

References

[1] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Nu-

merical Recipes in Fortran: the Art of Scientific Computing (Cambridge

University Press, New York, 1992).

[2] R. A. Horn, and Ch. R. Johnson, Matrix Analysis (Cambridge University

Press, Cambridge, 1999)

Mathematical appendix 215

B Gamma Function

The Gamma function is defined1 by the integral

Γ(z) ≡∫ ∞

0

tz−1e−tdt (B 1)

When the argument z is an integer, the Gamma function can be written in the

form of a factorial function:

Γ(n+ 1) = n! (B 2)

Gamma function satisfies recurrence relation:

Γ(z + 1) = zΓ(z) (B 3)

The natural logarithm of the Gamma function is implemented in the gammln

function from Numerical recipes.

The Incomplete Gamma Functionis defined by:

P (a, x) ≡ γ(a, x)

Γ(a)≡ 1

Γ(a)

∫ x

0

ta−1e−tdt, (a > 0) (B 4)

It has the limiting values

P (a, 0) = 0 and P (a,∞) = 1 (B 5)

The complement Q(a, x) is:

Q(a, x) ≡ 1 − P (a, x) ≡ 1

Γ(a)

∫ ∞

x

ta−1e−tdt, (a > 0) (B 6)

Functions gammp and gammq from Numerical recipes provides P and Q functions

respectively.

1All definitions and properties from “Numerical Recipes in Fortran ’77”

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