Effective rotational Hamiltonian for molecules with internal rotors: Principles, theory, applications and experiences

Journal of Molecular Spectroscopy 278 (2012) 52–67

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Feature Article

Effective rotational Hamiltonian for molecules with internal rotors: Principles, theory, applications and experiences Peter Groner ⇑ Department of Chemistry, University of Missouri – Kansas City, 5100 Rockhill Rd., Kansas City, MO 64110, USA

a r t i c l e

i n f o

Article history: Received 31 March 2012 In revised form 5 June 2012 Available online 15 June 2012 Keywords: Effective Hamiltonian Rotational spectroscopy Large-amplitude motion Internal rotation

a b s t r a c t The effective rotational Hamiltonian for molecules with one or two periodic large-amplitude motions has had considerable success in modeling rotational spectra over the past 15 years. This article spells out the basic premises and summarizes the theory of this Hamiltonian which is implemented in the program code called ERHAM. The essential features of the code are described. The modular approach to define tunneling parameters (and implicitly the Hamiltonian) makes the code very flexible. Published applications of ERHAM (including comparisons of results between ERHAM and other program codes) are reviewed and the limitations of ERHAM are discussed. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction During the past two decades, significant progress has been made in analyzing rotational spectra of molecules with large amplitude motions (LAMs) and fitting them to approximately experimental uncertainties. The progress is particularly true for molecules with internal rotation(s). Several methods have been developed that are reasonably successful in fitting small and large data sets of molecules with up to two internal rotors with transition frequencies from the microwave to the terahertz regions. Kleiner [1] has recently reviewed the theory for different models of molecules with one methyl internal rotor for high-resolution spectroscopy applications and outlined principles and methods of different freely available computer programs for such models. Her emphasis has been on the comparison of methods and results between such codes and the programs of the BELGI family [2,3]. She also classified the program codes in terms of method and whether the least-squares fits are performed on a single vibrational state (‘‘local’’) or on a number of such states simultaneously (approximately ‘‘global’’). Program JB95, developed by Plusquellic, Pratt and coworkers [4,5], is based on the principal axes method (PAM) with perturbation theory. The code of the SPFIT/SPCAT package by Pickett [6] with the IAMCALC front end has been ascribed by Kleiner [1] to the rho axis method (RAM). The BELGI programs for molecules with one internal rotor with Cs [2] or C1 [3] point group symmetry at equilibrium are based entirely on the RAM. They can be used for the vibrational ground and several excited states of the periodic LAM simultaneously and are therefore global codes. Be⇑ Fax: +1 816 235 5502. E-mail address: [email protected] 0022-2852/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jms.2012.06.006

cause the next group of programs use at least part of the Hamiltonian in q-axis systems and transform results to the principal axis system for the final analysis, they have been classified by Kleiner [1] as combined axis methods (CAMs). Woods’ programs for one or two methyl internal rotors [7,8] have been called traditionally to be based on the internal axis method (IAM) but for which RAM (or CAM) seems to be a more appropriate label. These programs have been used in the past by many research groups; some investigators have modified and improved the codes incrementally over the years. For example, such a program has been used to analyze millimeter wave data of acetone [9]. This improvement process has culminated in program XIAM by Hartwig and Dreizler [10] which treats molecules with one, two or three threefold internal rotors. The effective rotational Hamiltonian by Groner [11] (in the remainder of this article often referred to as ‘‘Paper I’’) implemented in the ERHAM code looks like another step in the same direction (incidentally, like Woods’ original code, it does not calculate barriers to internal rotation). Of all these programs, ERHAM and the theory behind it seem to be the most mysterious. Kleiner [1] assigned ERHAM to the group of programs that use the CAM method; this classification makes sense because ERHAM uses, like Woods’ programs and their successors, a RAM type set-up for parts of the theory and transforms results of this set-up to a reference axes system which may be the principal axes system. Because ERHAM does not actually solve the internal rotation Hamiltonian like PAM, IAM, RAM or even CAM, ERHAM should be an acronym by itself which can be interpreted as Effective Rotational HAMiltonian or Effective Rotational HAmiltonian Method. In the 1980s, Hougen and coworkers had published a series of papers exploring the use of phenomenological Hamiltonians to assign, fit and predict rotational spectra of a number of molecules

P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

with several large-amplitude internal motions. During this period, they had developed such Hamiltonians based on the molecular symmetry groups for hydrazine, NH2ANH2 [12], the water dimer, (H2O)2 [13], and several isotopologues of methanamine, CH3NH2, [14]. In the papers on methanamine, IAM-type models and extended molecular symmetry groups [15] have been used to derive expressions for matrix elements; all these matrix elements are remarkably similar to Eq. (3–27) in Lin and Swalen’s review [16] for the K-dependence of the internal rotation energy in the IAM treatment of internal rotation. The similarity inspired a quest to derive such matrix elements rigorously from the theory for a molecule with one internal rotor. The results of this endeavor [17] include: (i) Not only the internal rotation energy depends periodically on the quantity r  qK (explanation of symbols and definitions follow later) but also the eigenfunctions. (ii) The eigenfunctions can be expanded as Fourier series. (iii) The coefficients of the series are functions of the internal rotation variable; for eigenfunctions of states sufficiently below the barrier to internal rotation, the coefficient functions have their probability densities centered in different wells of the potential function and are therefore called localized functions. (iv) All matrix elements of the complete Hamiltonian can be expanded as Fourier series because they are also periodic in r  qK. The refinement and extension of the theory to molecules with two internal rotors has lead to the effective rotational Hamiltonian for molecules with two periodic LAMs and to the program now called ERHAM. Since its development [11], the program has been used successfully to analyze spectra of molecules with one or two internal rotors to mostly experimental uncertainty. It has been used for molecules like dimethyl ether [18], acetone [19] and methyl formate [20] that have been detected in interstellar clouds by radio astronomers. They are very interested in accurate predictions of line positions and intensities to help them distinguish the dense and intense spectra of these and other ‘‘weeds’’ from the less intense spectra of species yet to be identified. Other molecules like ethyl methyl ether [21], methyl carbamate [22], pyruvic acid [23] and pyruvonitrile [24] are of interest to them because they are likely candidates to be detected in similar environments with a new generation of telescopes. The use of ERHAM has also been instrumental, together with the availability of broadband millimeter to submillimeter wave spectra, for the first assignments of rotational transitions in the excited states of both torsional modes of acetone [25,26]. In another case, the program has been used to extract information about the splitting of torsional sublevels that has been used subsequently with data from the far infrared spectrum to determine the torsional potential function of 3-methyl-1,2butadiene (also called 1,1-dimethyl allene) [27]. Very recently, ERHAM has been used analyze the Fourier transform microwave spectrum of the complex of CHClF2 with H2O, in which the water molecule is the internal rotor [28]. This article has been written to inform readers and current and future users about various aspects of the theory and the program. The remainder of the article is organized as follows: Section 2 begins with a statement of the goals of theory and program and the emphasis on the importance of the periodic properties to the realization of the goals and on a necessary transformation. Molecular models and molecular symmetry groups covered by theory and program are described in the second part of the section. Section 3 contains a summary of the essential aspects of the theory including the effects of symmetry. Section 4 begins with the description of the modular approach to define the Hamiltonian by specifying the tunneling coefficients. Short subsections address the calculation of energy levels and the least-squares fit, the prediction of transitions and the use of the program for molecules with just one periodic LAM. After a discussion of the output of derived internal rotation parameters, the section concludes with a small

53

example of the initial steps necessary to obtain a satisfactory fit. Section 5 contains a list of publications in which ERHAM was used to analyze rotational spectra (with some observations by this author), a number of known limitations and disadvantages and a summary of other users’ comments and comparisons between the results of ERHAM and other programs. 2. Scope of theory and program 2.1. Goals and essential features of the method The effective rotational Hamiltonian for asymmetric rotor molecules with up to two periodic large-amplitude internal motions (LAMs) has been developed to predict and fit rotational spectra of such molecules 1. 2. 3. 4. 5.

up to large rotational quantum numbers, quickly, precisely and efficiently, in any vibrational state, for any internal rotor periodicity, for a variety of systems with different molecular symmetries, 6. and without the need to solve the internal rotation Hamiltonian itself. A secondary purpose has been data reduction in preparation for eventual determination of barriers to internal rotation e.g. reducing the original experimental rotational spectrum to spectroscopic parameters. Such an effective Hamiltonian can be applied without knowledge of the potential surface of the LAMs; the only requirement is knowledge of the symmetry of the system. The essential symmetry of such systems is caused by the periodicities of the LAMs; therefore, the theory is based on these periodic motions, prototypes of which are internal rotations. In the q-axis method (RAM) [1], the internal rotation Hamiltonian for an n-fold periodic internal rotor is written in terms of the internal rotation coordinate s, the conjugate momentum ps, and the component of the overall angular momentum Pz parallel to the q-vector as

HT ¼ Fðps  qP z Þ2 þ VðsÞ

ð1Þ

with F as the internal rotation constant, q as the length of the q-vector, and the potential

VðsÞ ¼ Vðs þ 2p=nÞ:

ð2Þ

In the q-axis system, it is diagonal in K, the quantum number of the projection of the overall angular momentum J onto the q-axis. For each K, it has n sets of solutions, one for each value of the symmetry number r = 0, 1, 2, . . . , n  1 which characterizes the free internal rotor basis functions

jjri ¼ ð2pÞ1=2 eiðnjþrÞs :

ð3Þ

The partition into such sets is due to the periodic properties of the Hamiltonian. Another important consequence of these properties is that the eigenvalues of the Hamiltonian (1) can be written as a Fourier series

Ev rK ¼

X

ev q cosð2pqðr  qKÞ=nÞ:

ð4Þ

q¼0

In this equation, v is the torsional (vibrational) quantum number, evq are the Fourier coefficients. Eq. (4) is a generalization of Lin and Swalen’s Eq. (3–27) in [16] that goes back to Itoh [29], but the periodic dependence of EvrK on the product qK has been demonstrated by Koehler and Dennison [30].

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P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

The effective rotational Hamiltonian is based on a thorough investigation of the periodicity of the LAMs. Not only the energy eigenvalues of the RAM internal rotation Hamiltonian (1) are periodic in (r  qK) but also the eigenfunctions. As a result, all expectation values (matrix elements) of any operator involving the variable s depend periodically on the projection quantum number K. Therefore, the effective rotational Hamiltonian can be set up by using this periodicity without the need to actually solve the internal rotation Hamiltonian similar to the effective or phenomenological Hamiltonians used by Ohashi, Hougen and coworkers. [31–33,12–14]. The application of the RAM Hamiltonian is straightforward if a molecule has just one internal rotor. In the presence of two or more periodic LAMs, one has to choose a suitable axes system since the complete Hamiltonian matrix must be set up in a single axes system. The periodic nature of the solutions of each periodic LAM, in particular the eigenfunctions, can still be used if they are transformed to a reference axes system which may or may not be the principal axes system. The transform of the (diagonal) matrix elements of the internal rotation energy from the q-axis systems of each rotor into the reference axes system has been implemented in programs by Woods [8] and Vacherand et al. [9]. However, this approach fails to take into account any kinetic or potential interactions between the periodic LAMs. Hartwig and Dreizler (XIAM) [10] considered some top-top interactions by transforming the matrices of the operators ps  qPz from the q-axis system to the principal axes system before multiplying them as needed. In the effective rotational Hamiltonian, a different transformation that includes all top-top interactions (although the internal rotation Hamiltonian is not actually solved, see above) has been developed after a careful analysis; this transformation becomes more involved with additional internal rotors. 2.2. Models treated by the program and their symmetries The effective Hamiltonian has been derived and can be used for a number of different models. They are characterized by the number of periodic LAMs and whether they are equivalent or not, and by additional symmetry operations that may be present. Systematic investigations of molecular symmetry groups of molecules with methyl internal rotors have been published in the past for molecules with two methyl group internal rotors based on the rotation internal rotation Hamiltonian [34] and the isometric group [35]. More recent applications have derived molecular symmetry groups in terms of permutation inversion operations on a case-by-case basis. Molecular symmetry groups of asymmetric rotor molecules with three internal rotors have been derived systematically in terms of permutation inversion and isometric groups [36]; those with one or two periodic LAMs are just special cases of those with three LAMs. Paper I introducing the effective rotational Hamiltonian [11] summarizes results from molecular symmetry groups, and the theory and the program code use relations derived from group theory throughout. The different cases treated in that paper, their symmetry groups, and sample molecules are listed in Table 1. The notation for the symmetry groups used in the table is based on the fact that, for these particular LAM problems, the molecular symmetry group is isomorphic to a semi-direct product of an invariant ‘‘permutation subgroup’’ and a ‘‘frame subgroup’’ [37]. The ‘‘permutation subgroup’’ itself is a cyclic group of order n, Cn, or a direct product of cyclic groups, C n1  C n2  C n3  . . . The ‘‘frame subgroup’’ is isomorphic to a point group. For the molecular models treated in this article, that point group is the symmetry group of the equilibrium structure. A prime (0 ) is added to the notation of the point group if the LAMs are not equivalent. Rovibronic energy levels are split by the motions of periodic LAMs. The kth LAM, periodic in 2p/nk, causes a splitting into nk

Table 1 Molecular symmetry groups of molecules with one or two periodic LAMs. Number of LAMs

Model Symmetry group

Shorthand Order Exampleb notationa of group

1

C1 Cs

[n]C1 [n]Cs

Cn^C1 Cn^Cs

n 2n

CH3CHFCl CH3OH

2 (non-equivalent) C1(n) Cs(n)

(Cn  Cm)^C1 [n1n2]C 01 (Cn  Cm)^Cs [n1n2]C 0s

n1n2 CH3OOCD3 2n1n2 CH3OCD3

2 (equivalent)

(Cn  Cn)^Cs (Cn  Cn)^Cs (Cn  Cn)^Ci (Cn  Cn)^C2v (Cn  Cn)^C2h

2n2 2n2 2n2 4n2 4n2

C2(e) Cs(e) Ci(e) C2v(e) C2h(e)

[nn]C2 [nn]Cs [nn]Ci [nn]C2v [nn]C2h

CH3OOCH3 CH3CHFCH3 ? CH3OCH3 t-CH3CH@CHCH3

a Notation analogous to [36]. The direct product of the ‘‘permutation subgroup’’ is symbolized by the square bracket containing the orders of the cyclic groups. The point group symbol represents the ‘‘frame subgroup’’. Presence or absence of a prime (0 ) indicate that the LAMs are non-equivalent or equivalent, respectively. b Examples for n1 = n2 = n = 3.

(torsional) sublevels, some of which may be degenerate. The sublevels may be distinguished by the different values of the symmetry numbers rk (modulo nk) = 0, 1 ,. . ., ,nk  1. For one periodic LAM with n = 3, r is 0, 1, or 2. The conventional notation for these sublevels is A for r = 0, and E for the degenerate pair of r = 1 and r = 2 (equivalent to r = 1). For two periodic LAMs, there are n1n2 sublevels which may be labeled by the sets of symmetry numbers (r1r2). These sets are also the best labels for the irreducible representations of the symmetry group [n1n2]C 01 because they can be generated with the basis jr1 r2 i ¼ ð2pÞ1 eiðr1 s1 þr2 s2 Þ . In other words, the eigenfunctions of the torsional sublevels labeled by (r1r2) transform like the irreducible representations (r1r2) of this group. The irreducible representations are all one-dimensional; some are real, the rest of them form separately degenerate complex conjugate pairs: (r1r2) and (r1, r2) = (n1  r1, n2  r2). This means that, in the absence of external electric or magnetic fields, the energy levels belonging to complex conjugate pairs are identical and therefore degenerate. In the symmetry group [n1n2]C 0s , the number and dimensions of the irreducible representations are different but the number and labels of distinct sublevels are the same as for [n1n2]C 01 . If the periodic LAMs are equivalent (when n1 = n2 = n and at least one of the extra symmetry operators induces an interchange of the LAM coordinates), additional degeneracies among the (r1r2) substates occur. This reduces the number of distinct sublevels but increases the degeneracies of some of them. Table 2 indicates how many distinct sublevels occur for given n1 and n2 for non-equivalent LAMs with groups [n1n2]G0 and given n for equivalent LAMs with groups [nn]G. (See Section 4.1 for the allowed combination of symmetry numbers.)

Table 2 Numbers of distinct torsional sublevels and their degeneracies for the groups [n1n2]G0 for non-equivalent LAMs and for the groups [nn]G for equivalent LAMs. Symmetry groups

Degeneracy of sublevels 1 2

[n1n2]G0 n1, n2 both odd n1 + n2 odd n1, n2 both even

1 (n1n2  1)/2 2 (n1n2  2)/2 4 (n1n2  4)/2

[nn]G n = odd n = even

1 n1 2 n1

4

Sublevels Distinct total sublevels

n1n2 n1n2 n1n2 (n  1)2/4 n2 n(n  2)/4 n2

(n1n2 + 1)/2 (n1n2 + 2)/2 (n1n2 + 4)/2 (n + 1)2/4 n(n + 2)/4 + 1

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P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

The Hamiltonian H is split into three parts:

3. Summary of theory

H ¼ T þ V ¼ HR þ HI þ HRI ; ~ þ DqFDq ~ þ DqFq ~ 0 þ q0 FDq ~ ÞP; HR ¼ PðA

3.1. Basics The variables of the periodic internal motions, s1 and s2, both have domains from 0 to 2p. Because of the existence of n1 or n2 indistinguishable conformations within the domains of the variables, molecular properties (like the potential function) are periodic in 2p/n1 or 2p/n2, respectively. The effective potential energy V(s1, s2) for molecules with two periodic LAMs is real, can be written as a two-dimensional Fourier series, and includes all potential interaction terms between the LAMs. Paper I uses the term ‘‘effective potential’’ without going into detail. The effective potential of the LAMs consists of the ‘‘true’’ potential function modified by two contributions. The first addition is the usually small pseudopotential arising from the dependence of the moment of inertia tensor (or rather of its inverse, the l-tensor) on the LAM coordinates [38]. The second addition originates in vibrational effects. It is assumed that the Hamiltonian of the infinitesimal vibrational coordinates has been solved as a function of the slowly varying LAM coordinates s1 and s2. For each vibrational state, this adds a different vibrational contribution to the effective potential of the LAMs [38]. Furthermore, for all other quantities of interest like the l-tensor, the expectation values over the vibrational coordinates in that vibrational state must be used before the LAM Hamiltonian is solved. This means that the Hamiltonian H above is an effective Hamiltonian for a particular vibrational state characterized by the set of vibrational quantum numbers v. The kinetic energy T for overall rotation and internal motions in an arbitrary molecular frame-fixed center-of-mass axes system has the form

T¼



~_ ~ s x

 I x ~Ixs

Ixs



Is

x s_

 ð5Þ

where x is the column vector of the components of the overall angular velocity, and s_ is the two-dimensional column vector of the velocities of the internal motions. Ix is the regular 3  3 moment of inertia tensor for overall rotation, Is is the 2  2 matrix containing the moments of inertia of the internal motions on the diagonal, and Ixs is the 3  2 matrix containing the cross terms between overall and internal velocities. The tilde denotes transposed vectors or matrices. The submatrices Ix, Is and Ixs are in general functions of both internal variables. Expressing the kinetic energy in terms of the conjugate momenta, one obtains

   ~ qF   A þ qFq P ~ p ~ T¼ P : ~ p Fq F

ð6Þ

P is the vector of the components of the overall angular momentum, p contains the internal angular momenta, and 2

A ¼ ðh =8p2 ÞðIx Þ1 ; 1

q ¼ ðIx Þ Ixs ; 2 ~ Ix qÞ1 : F ¼ ðh =8p2 ÞðIs  q

ð6aÞ ð6bÞ ð6cÞ

A, F, and q have the same dimensions as Ix, Is and Ixs, respectively, and they too are functions of the variables s1 and s2. At this point, q is replaced by

q ¼ q0 þ Dq

ð7Þ

where q0 is the constant mean and Dq contains the variable-dependent contributions.

~ q0 ÞFðp  q ~P ~ 0 PÞ þ V; H I ¼ ðp ~ qFp þ p ~ FDq ~ P: HRI ¼ ½PD

ð8aÞ ð8bÞ ð8cÞ ð8dÞ

It is understood, that H may contain operators (not shown) of higher order in overall or internal angular momenta. The overall order of internal and overall angular momenta is even. Every single term in the operator H consists in general of a product of two operators Rl and Tl

H¼

X

Rl T l :

ð9Þ

l

The rotational operator Rl itself is some product of the components of the overall angular momentum P whereas the ‘‘torsional’’ operator Tl contains components of the internal angular momentum p and/or functions of the LAM coordinates. The operator HR can be written in cylindrical tensor form [39] as

HR ¼

X kpr

1 ½Pk Ppz ; Prþ þ Pr þ Bkpr 2ð1 þ d0r Þ

ð10Þ

where k, p and r are even, [a, b]+ is the anticommutator ab + ba, and the coefficients Bkpr correspond to the functions Tl. The essential idea is: The effective rotational Hamiltonian for molecules with two periodic LAMs is set up in the basis

jJKMv r1 r2 i ¼ jJKMijv r1 ðKÞr2 ðsKÞi

ð11Þ

where jJKMi is the usual symmetric rotor basis for overall rotation, v represents the set of vibrational quantum numbers (including those for torsional vibrations), r1 and r2 are the symmetry numbers referring to the underlying free internal rotor basis, and s = ±1 depends on the relative orientation of the internal rotation axes with respect to the reference axes (its use is explained later). The internal motion basis jv r1 ðKÞr2 ðsKÞi is derived from the solutions of HI (Eq. (8c)) as shown later. The general matrix elements of H in the basis jJKM v r1 r2 i are therefore given by

hJK 0 M v 0 r1 r2 jHjJKMv r1 r2 i ¼

X 0 hJK MjRl jJKMi l

 hv 0 r1 ðK 0 Þr2 ðsK 0 ÞjT l jv r1 ðKÞr2 ðsKÞi: ð12Þ An effective rotational Hamiltonian could be obtained from this equation if all matrix elements off-diagonal in v were neglected and if the matrix elements of Tl on the right were treated as adjustable parameters. Obviously, such an effective Hamiltonian would have way too many parameters. Because the solutions of HI have unique properties due to its periodicity, it is possible to reduce the number of fitting parameters of such a Hamiltonian significantly. These properties make it possible to expand the function jv r1 ðKÞr2 ðsKÞi (identical to UðK;sKÞ v r1 r2 ðs1 ; s2 Þ in paper I) in a Fourier series which may be rewritten in the form

jv r1 ðKÞr2 ðsKÞi ¼ ðN 1 N2 Þ1=2 

NX 1 1N 2 1 X

ei2pðq1 ðr1 q1 KÞ=n1 þq2 ðr2 q2 sKÞ=n2 Þ jv q1 q2 ðK; sÞi

q1 ¼0 q2 ¼0

ð13Þ 0 k

where qk is the length of the vector q and the coefficients are the functions jv q1 q2 ðK; sÞi. (Details about these functions and the number of terms in the series, N1N2, are given later). As a consequence, all matrix elements of Tl in Eq. (12) become similar Fourier series whose coefficients are integrals expressed in the basis jv q1 q2 ðK; sÞi. Because the periodic properties of the Hamiltonian lead to

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P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

hv 0 q01 q02 ðK 0 ; sÞjT l jv q1 q2 ðK; sÞi ¼ hv 0 00ð0ÞjT l jv ; q1  q01 ; q2  q02 ; ðK  K 0 ; sÞi;

ð14Þ

the integrals depend only on the differences K  K 0 ; q1  q01 ; q2  q02 rather than on K 0 ; K; q01 ; q1 ; q02 or q2 individually. If the Fourier series converge rapidly, all matrix elements of Tl depend only on a small number of adjustable parameters.

The next step is the basis transformation from (15) to the new basis (11) that contains the eigenfunctions of HI(K, sK) which form a subset of the solutions of (21a). The new basis is orthonormal with respect to v and K. The transformed matrix elements of HR are given by Eq. (12). For the matrix elements of HI one obtains

hJK 0 M v 0 r1 r2 jHI jJKM v r1 r2 i ¼

XX K1

Y K 0 K ðK 1 ; K 2 Þ

K2

 hv 0 r1 ðK 0 Þr2 ðsK 0 ÞjHI ðK 1 ; K 2 Þjv r1 ðKÞr2 ðsKÞi:

3.2. Outline of derivation

ð23Þ ðK 1 ;K 2 Þ

To arrive at the desired results alluded to in Section 3.1, including the final form of the matrix elements in the basis (11), the Hamiltonian has been set up first in the basis

jJKMj1 r1 j2 r2 i ¼ jJKMijj1 r1 ijj2 r2 i

ð15Þ

with jjk rk i as a free internal rotor basis (3). This is done to derive the correct transformation between the different q-axis systems and the reference axes system which may be the principal axes system. The matrix elements of HR are then given by 0

hv 0 r1 ðK 0 Þr2 ðsK 0 ÞjHI ðK 1 ; K 2 Þjv r1 ðKÞr2 ðsKÞi XX ¼ hv 0 r1 ðK 0 Þr2 ðsK 0 Þjv 000 r1 ðK 1 Þr2 ðK 2 Þi v 000 v 00

 hv 000 r1 ðK 1 Þr2 ðK 2 ÞjHI ðK 1 ; K 2 Þjv 00 r1 ðK 1 Þr2 ðK 2 Þi  hv 00 r1 ðK 1 Þr2 ðK 2 Þjv r1 ðKÞr2 ðsKÞi:

0

hJK 0 Mj1 r1 j2 r2 jHR jJKMj1 r1 j2 r2 i X 0 0 0 ¼ hJK MjRl jJKMihj1 r1 j r2 jT l jj1 r1 j2 r2 i:

ð16Þ

l

The matrix elements of HI are more complicated since this operator contains both overall and internal angular momentum operators like

X

p0k ¼ pk 

qak P a :

ð17Þ

a

X jJKMj1 r1 j2 r2 i ¼ jJK k MihJK k MjJKMijj1 r1 j2 r2 i

ð18Þ

Kk

where the zk-axis of the basis jJK k Mi is parallel to the q-axis of the kth internal rotor. In such a system, the operator p0k (17) becomes

pk  qk Pzk

Only v 00 and v 000 are summation indices. The central bracket in the ðK ;K Þ double sum reduces to dv 00 v 000 Ev 001r1 r2 2 and the complete matrix element for HI in the final basis is obtained as

hJK 0 M v 0 r1 r2 jHI jJKM v r1 r2 i ¼ 

X v

00

0

hJK 0 Mj1 r1 j2 r2 jpk 

K1

qak Pa jJKMj1 r1 j2 r2 i

Kk 0

0

 hj1 r1 j2 r2 jpk  qk K k jj1 r1 j2 r2 i:

ðK ;K Þ

hv 0 r1 ðK 0 Þr2 ðsK 0 Þjv 00 r1 ðK 1 Þr2 ðK 2 ÞiEv 001r1 r2 2 ð25Þ

The brackets on the right may be approximated by

hv 0 r1 ðK 0 Þr2 ðsK 0 Þjv 00 r1 ðK 1 Þr2 ðK 2 Þi  dv 0 v 00 ;

ð26aÞ

hv 00 r1 ðK 1 Þr2 ðK 2 Þjv r1 ðKÞr2 ðsKÞi  dv 00 v :

ð26bÞ

For

ð27Þ

ð19Þ

hJK 0 M v 0 r1 r2 jHI jJKM v r1 r2 i  dv 0 v

XX Y K 0 K ðK 1 ; K 2 Þ K1

Taking into account that HI contains combinations of the operators p0k such as p01 p02 þ p02 p01 , the transformation is a little more involved and the matrix elements of HI become eventually



K2

NX 1 1N 2 1 X

ei2pðq1 ðr1 q1 K 1 Þ=n1 þq2 ðr2 q2 K 2 Þ=n2 Þ ev q1 q2 :

q1 ¼0 q2 ¼0

ð28Þ

0

hJK 0 Mj1 r1 j2 r2 jHI jJKMj1 r1 j2 r2 i XX 0 0 ¼ Y K 0 K ðK 1 ; K 2 Þhj1 r1 j2 r2 jHI ðK 1 ; K 2 Þjj1 r1 j2 r2 i K1

Y K 0 K ðK 1 ; K 2 Þ

K2

the approximations are exact. If not all conditions expressed by (27) are fulfilled, the delta symbols in (26a) and (26b) are usually very good approximations. The final form of the matrix elements of HI in the basis (11) is therefore

X

a X 0 hJK MjJK k MihJK k MjJKMi

0

XX

K ¼ K 0 ¼ K 1 ¼ sK 2 :

and its matrix elements are therefore 0

ð24Þ

 hv 00 r1 ðK 1 Þr2 ðK 2 Þjv r1 ðKÞr2 ðsKÞi:

The basis can be written alternatively as

¼

Obviously, the matrix element of HI(K1, K2) in Eq. (23) is not Ev r1 r2 because jv r1 ðKÞr2 ðsKÞi is only an eigenfunction of HI(K1, K2) if K = K0 = K1 = sK2. However, the bracket on the right in (23) may be expanded as follows:

ð20Þ

K2

jv r1 ðK 1 Þr2 ðK 2 Þi ¼ ur1 r2 ðN1 N2 Þ1=2

with

HI ðK 1 ; K 2 Þ ¼

2 X 2 X ðpk  qk K k ÞF ðklÞ ðpl  ql K l Þ þ Vðs1 ; s2 Þ;

ð21aÞ

Y K 0 K ðK 1 ; K 2 Þ ¼ ð1=2Þ½hJK 0 MjJK 1 MihJK 1 MjJK 2 MihJK 2 MjJKMi þ hJK 0 MjJK 2 MihJK 2 MjJK 1 MihJK 1 MjJKMi:

ð21bÞ

Eigenvalues NX 1 1N 2 1 X

ei2pðq1 ðr1 q1 K 1 Þ=n1 þq2 ðr2 q2 K 2 Þ=n2 Þ ev q1 q2



NX 1 1N 2 1 X

1 ;K 2 Þ ei2pðq1 r1 =n1 þq2 r2 Þ=n2 Þ #ðK v q1 q2 ðs1 ; s2 Þ: ð29Þ

q1 ¼0 q2 ¼0

k¼1 l¼1

1 ;K 2 Þ EðK v r1 r2 ¼

The matrix elements of HRI (8d) will be treated later. In paper I, the eigenfunctions of HI(K1,K2), |vr1(K1)r2(K2)i, have been expanded in the Fourier series:

ð22Þ

q1 ¼0 q2 ¼0

and eigenfunctions of HI(K1, K2) in the basis jj1 r1 ijj2 r2 i and their properties have been derived in the appendix of paper I.

The complex phase factor /r1 r2 can be chosen to be 1 for the applications of the effective Hamiltonian discussed in this paper and Nk = mknk where mk is the integer expansion factor of the domain 1 ;K 2 Þ of sk [17,15]. The functions #ðK v q1 q2 ðs1 ; s2 Þ can be defined in terms of |vr1(K1)r2(K2)i with rational rk ¼ r0k =mk (integer r0k ). However, only functions |vr1(K)r2(sK)i with integer rk from all solutions of HI(K1, K2) are allowed in the basis (11). The functions with extended 1 ;K 2 Þ domains (e.g. #ðK v q1 q2 ðs1 ; s2 Þ and |vr1(K1)r2(K2)i with rational rk) are periodic in 2pmk, and they are orthogonal over this extended domain. The alternate expansion given in Eq. (13) can be derived from

57

P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

the results in paper I. Accordingly, the functions of the basis |vq1q2(K, s)i in (13) become

jv q1 q2 ðK; sÞi ¼

#ð0;0Þ v q1 q2 ð 1 ;

s s2 Þe

iKðq1 s1 þsq2 s2 Þ

ð30Þ

:

At first glance, the right side does not look to be periodic in 2pmk even though #ð0;0Þ v q1 q2 ðs1 ; s2 Þ is. However, there is no contradiction if the expansion factors mk are chosen to make qkmk integer even 1 ;K 2 Þ numbers (or very nearly so) [15]. The functions #ðK v q1 q2 ðs1 ; s2 Þ have another remarkable property, namely ðK ;K 2 Þ

1 1 ;K 2 Þ #ðK v q1 q2 ðs1 ; s2 Þ ¼ #v 00

ðs1  2pq1 =n1 ; s2  2pq2 =n2 Þ:

ð31Þ

In other words, the functions with non-zero q1 and q2 are just copies of the function with q1 = q2 = 0 shifted by integer multiples of 2p/nk. For a torsional state sufficiently below the potential barrier, the 1 ;K 2 Þ probability density of the function #ðK v q1 q2 ðs1 ; s2 Þ is concentrated within the well of the periodic potential centered at sk = 2pqk/nk. For this reason, they have been called localized functions (paper I, [11]) and they correspond to the ‘‘framework functions’’ in phenomenological Hamiltonians [31–33]. The matrix elements of Tl appearing in Eq. (12) are now expressed as Fourier series whose coefficients are essentially integrals in the basis |vq1q2(K,s)i (30). With (13), the matrix elements of the operator Tl which does not contain any of the internal angular momentum operators pk becomes

hv 0 r1 ðK 0 Þr2 ðsK 0 ÞjT l jv r1 ðKÞr2 ðsKÞi ¼

0

0

ðKK ;sðKK ÞÞ 1 q2

X qq0 T lv 0 v q

ð32Þ

X q1 q2 ¼ e ðj;sjÞ T lv 0 v q q 1 2

ipjðq1 q1 =n1 þsq2 q2 =n2 Þ

¼e

ð33aÞ

;

hv 00ð0ÞjT l jv q1 q2 ðj; sÞi: 0

ð33bÞ

ðj;sjÞ

The Fourier coefficients T lv 0 v q q are called tunneling coefficients be1 2 cause they are matrix elements in the basis of the localized functions which themselves are identical except for the shifts of their origins. Therefore, the matrix elements of HR are given by

hJK 0 Mv 0 r1 r2 jHR jJKMv r1 r2 i ¼

X

JK 0 M Rl jJKMi

l



NX 1 1N 2 1 X q1 ¼0 q2 ¼0

ðj;sjÞ

X q 1 q 2 T lv 0 v q

1 q2

ð34Þ

T l ¼ ðtak pk þ pk t ak Þ with tak as function of s1 and s2 with the proper periodicity and their matrix elements have been derived in paper I as

hv 0 r1 ðK 0 Þr2 ðsK 0 Þjtak pk þ pk t ak jv r1 ðKÞr2 ðsKÞi NX 1 1N 2 1 X

ðj;sjÞ

ðj;sjÞ

X q1 q2 ½T av 0 v q1 q2 þ ðK þ K 0 ÞT av 0 v q1 q2 

ð35Þ

q1 ¼0 q2 ¼0

with j = K  K0 . The first coefficient in the brackets is given by ðj;sjÞ

T av 0 v q1 q2 ¼ eipjðq1 q1 =n1 þsq2 q2 =n2 Þ  hv 0 00ð0Þjt ak eijðq1 s1 þsq2 s2 Þ pk þ pk tak eijðq1 s1 þsq2 s2 Þ jv q1 q2 ð0Þi

X q1 q2

X  0 ðj;sjÞ ½ JK M P a jJKM iT av 0 v q1 q2 a

q1 ¼0 q2 ¼0

ðj;sjÞ þ JK M Pz Pa þ P a Pz jJKM iT av 0 v q1 q2  

0

ð38Þ

The second terms in the square bracket add small contributions to the matrix elements of HR but the first term contributes something new: rotational matrix elements of angular momentum operators of odd order multiplied by tunneling coefficients. 3.3. Important results The complete matrix element of H in the basis (11) is therefore to a good approximation

hJK 0 M v 0 r1 r2 jHjJKMv r1 r2 i ¼

X

q1 ¼0 q2 ¼0

l

þ dv 0 v

NX 1 1N 2 1 X ðj:sjÞ JK 0 M Rl jJKM i X q 1 q 2 T lv 0 v q q

ð36Þ

NX 1 1N 2 1 XX X Y K 0 K ðK 1 ; K 2 Þ ei2pðq1 ðr1 q1 K 1 Þ=n1 þq2 ðr2 q2 K 2 Þ=n2 Þ ev q1 q2 K1

K2

q1 ¼0 q2 ¼0

ð39Þ In this form, the overall angular momentum operators Rl now include those of odd order because of the contributions from HRI. The relations

T lv 0 v q

1 q2

ðj:sjÞ 1 q2

¼ xT lv 0 v q

ðj:sjÞ

¼ xT lvv 0 ;q

1 ;q2

ðj:sjÞ 1 ;q2

¼ T lvv 0 ;q

one obtains eventually for the matrix elements of HRI

ð40aÞ ð40bÞ

where x is +1 or 1 if the rotational angular momentum operator Rl is of even or odd order, respectively, reduce the number of independent parameters somewhat. It needs to emphasized that the energy tunneling parameters ev q1 q2 are real, but the tunneling parameters ðj:sjÞ T lv 0 v q q in general are complex. They are real (for x = +1) or purely 1 2 imaginary (for x = 1) when j is 0. Also, it has been shown in paper I that for j = ±1

hJ; K  j; MjPy jJKMi ¼ jihJ; K  j; MjPx jJKMi;

ð41Þ

and therefore,

hJ; K j; M v r1 r2 jðPx Pz þ Pz Px ÞT xz þ ðPy Pz þ Pz Py ÞT yz jJKMv r1 r2 i XX ðj;sjÞ ðj;sjÞ X qq0 ðT xzqq0  jiT yzqq0 ÞX qq0 : ¼ hJ; K  j; MjPx Pz þ P z Px jJKMi q

q0

As a consequence, if tunneling parameters are indeed complex, all tunneling matrix elements of operators of odd order in Py can be neðj;sjÞ glected because the real part of T xzqq0 and the imaginary part of ðj;sjÞ T yzqq0 cannot be distinguished. Of course, this holds also for the ðj;sjÞ ðj;sjÞ imaginary part of the T xzqq0 and the real part of T yzqq0 . Notice that only one approximation has been made: the approximation of the integrals (26a) and (26b) by the Kronecker symbols. The effective Hamiltonian is obtained by neglecting matrix elements that are not diagonal in v. By rewriting the double sums and using the relations between the tunneling coefficients, the general matrix element of the operator H may be written as (dropping the symbols for the vibrational states v from the tunneling coefficients)

hJK 0 M v r1 r2 jHjJKM v r1 r2 i X 0 XX Y K prime K ðK 1 ;K 2 ÞR0 þ hJK MjRl jJKMifxþ C l ðTÞ þ ix Sl ðTÞg; ¼ K1

K2

l

ð43Þ

ðj;sjÞ

whereas T av 0 v q1 q2 is obtained by replacing Tl in Eq. (33b) by ðdk1 þ sdk2 Þqk t ak : Using also

hJK 0 MjPa jJKMiðK þ K 0 Þ ¼ hJK 0 MjPz Pa þ Pa Pz jJKMi;

1 2

ð42Þ

with j = K  K0 . The individual operators Tl in HRI (8d) have the form

¼

NX 1 1N 2 1 X

ev q1 q2 ¼ ev q1 q2 ¼ ev ;q1 ;q2 ¼ ev ;q1 ;q2

where ipðq1 ð2r1 q1 ðKþK 0 ÞÞ=n1 þq2 ð2r2 sq2 ðKþK 0 ÞÞ=n2 Þ

¼

ðj:sjÞ

NX 1 1N 2 1 X q1 ¼0 q2 ¼0

hJK 0 M v 0 r1 r2 jHRI jJKM v r1 r2 i

ð37Þ

where

X

R0 ¼ e00 þ 2

q>0

( C 0qq eqq þ C 0qq eqq þ

) q1 h i X C 0qq0 eqq0 þ C 0q0 q eq0 q q0 ¼qþ1

ð43aÞ

58

P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

C 0qq0 ¼ cosð2pðqðr1  q1 K 1 Þ=n1 þ q0 ðr2  q2 K 2 Þ=n2 ÞÞ;

ð43bÞ

x ¼ ð1  xÞ=2;

ð43cÞ

operators of even or odd order, only cosine or sine terms, respectively, appear in the Fourier expansion. 3.4. Symmetry considerations

( X ðjÞ ðj;sjÞ ðj;sjÞ C qq T lqq þ C qq T lqq C l ðTÞ ¼ T l00 þ 2 q>0 q1 h i X ðj;sjÞ ðj;sjÞ C qq0 T lqq0 þ C q0 q T lq0 q

þ

The final form of the matrix elements of the effective rotational Hamiltonian needs to be modified if other symmetry operators besides those expressing the periodicities of the LAMs are present. If we number (for two threefold rotors) just the atoms of the first internal rotor as 1, 2, 3 and those of the second one as 4, 5, 6, the generating PI operators of the LAMs are (123) and (456), respectively, and the molecular symmetry group in the notation used in Section 2.2 is [33]C 01 . If any of the additional operators A = (1)(23)(4)(56)⁄, B = (14)(25)(36), C = AB = (14)(26)(35)⁄ or all of them are present, the symmetry groups are with A: [33]C 0s , with B: [33]C2, with C: [33]Cs or [33]Ci, with A, B, and C: [33]C2v or [33]C2h. The operators A, B, or/and C generate additional relationships between matrix elements, eigenvalues, eigenfunctions, localized functions and tunneling coefficients which are listed in Table II of paper I. They cause a decrease in the number of the independent tunneling coefficients and the Fourier series must be rewritten to reflect that fact. The transformation properties of the operators Tl play a critical role here. Because none of the symmetry operators induce a change of the reference axes (such as an exchange of the x and y reference axes), any operator RlTl must be invariant under any symmetry operation. Therefore, the transformation properties of Tl are the same as those of the operators Rl (e.g. they have the same characters). The characters of the

) ð43dÞ

q0 ¼qþ1

( X ðj;sjÞ ðj;sjÞ Sqq T lqq þ Sqq T lqq Sl ðTÞ ¼ 2 q>0 q1 h X

þ

q0 ¼qþ1

ðj;sjÞ

Sqq0 T lqq0

ðj;sjÞ

þ Sq0 q T lq0 q

i

) ð43eÞ

j ¼ K  K 0;

ð43fÞ

aqq0 ¼ pðqð2r1  q1 ðK þ K 0 ÞÞ=n1 þ q0 ð2r2  sq2 ðK þ K 0 ÞÞ=n2 Þ; ð43gÞ C qq0 ¼ cos aqq0 ;

Sqq0 ¼ sin aqq0 :

ð43hÞ

This is the final form for the matrix elements of the effective rotational Hamiltonian if the molecule allows no symmetry elements other than the periodicity of the LAMs. The relations between the tunneling coefficients are such that, for overall angular momentum

Table 3 The symmetry of the model defined by the ISCD and NC parameters.a ISCD -5⁄ -4⁄ -3⁄ -2 -1 0⁄ 1 2 3

NC

+1 -1 ±1 ±1 +1 -1 +1 -1

4⁄ a

Symmetry group

Additional information

[nn]C2h [nn]C2h [nn]C2v [nn]C2v [nn]C2v [n1n2]C 0s [n1n2]C 0s [n1n2]C 01 [nn]C2 [nn]C2 [nn]Cs [nn]Cs [nn]Ci

equivalent LAMs, C2 axis is z equivalent LAMs, C2 axis is x or y equivalent LAMs, both internal rotor axes in rxy equivalent LAMs, C2 axis is z equivalent LAMs, C2 axis is x or y non-equivalent LAMs, both internal rotor axes in rxz or ryz non-equivalent LAMs, both internal rotor axes in rxy non-equivalent LAMs equivalent LAMs, C2 axis is z equivalent LAMs, C2 axis is x or y equivalent LAMs, ryz or rxz equivalent LAMs, rxy equivalent LAMs

An asterisk denotes values of ISCD that have not yet been used.

Table 4 Parameter values at several stages of the initial LS fits for CH380Se80SeCH3.

q b

a A (MHz) B (MHz) C (MHz) DJ (kHz) DJK (kHz DK (kHz) dJ (kHz) dK (kHz) e10 (MHz) [A  (B + C)/2]10 (kHz) Frequencies Parameters s (weighted) a

Ref. [41].

Initial set

End of fit 3

End of fit 8

Final fita

0.031219 20.0 20.5 8163.74148 2816.50266 2570.37485 1.7724 6.1676 28.276 0.48183 15.58 22.53

0.031219 20.0 20.5 5156.05907 1481.09175 1420.18064 0.2651 1.6500 28.276 0.37111 15.58185 51.56

0.015230 35.376 39.099 5156.15976 1481.14445 1420.13032 0.3889 0.4778 5.0135 0.07272 18.99 119.50

0.013930(49) 36.554(52) 39.093(14) 5156.15730(41) 1481.13557(56) 1420.13993(60) 0.3720(12) 0.6352(93) 4.198(84) 0.05621(90) 14.59(28) 132.61(55) 4.07(15) 98 13 0.323

59 2.46  106

59 7 81.7

59 12 1.38

59

P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67 Table 5 Applications of ERHAM to molecules with one internal rotor.

a b c

Molecule

Groupa

Settingb

Statesc

Reference

NH2COO-CH3 Methyl carbamate H13COO-CH3, H12COO-CH3 Methyl formate

[3]Cs [3]Cs

ab ab

CH3-COCOOH Pyruvic acid CH3-COCN Pyruvonitrile CF3(CF2)3O-CH3 Methyl perfluorobutyl ether (CF3)2CFCF2OCH3 methyl perfluoroisobutyl ether sym-CH2DCO-CH3, sym-CHD2CO-CD3 Acetone-d1, -d5 CHClF2-H2O Chlorodifluoromethane-water

[3]Cs [3]Cs [3]C1 [3]C1 [3]Cs [2]Cs

ab ab

G G T(m18, 2m18) G T(m23, 2m23) V(m24, 2m24) G T(m18) V(m23) G G G G

[22] [20] [42] [23] [24] [43] [43] [44] [28]

ab ab

Molecular symmetry group. The notation ‘‘ab’’ indicates that the internal rotation axis is in the ab principal plane. G = ground state, T = torsional excited state, V = excited states of other vibrations.

operators Rl and Tl are denoted k, l and m under the operators A, B, and C, respectively. Table V in paper I summarizes these characters for a series of rotational operators for three different orientations of the reference axes. It includes the appropriate values of the parameter s occurring for the first time in Eq. (11) above; if the product of the direction cosines of the internal rotation axes with respect to the reference axis z is positive or negative, s is +1 or 1, respectively. The symmetry adapted expressions can be derived from the information given in a series of tables. Tables III and IV of paper I list the applicable Fourier series of the expansions of the matrix elements for non-equivalent and equivalent rotors, respectively. The symbols x± are defined in (43c). The symbols k±, l± and m± are defined in the same way. The expansions in terms of the tunneling coefficients are denoted by Cl(T) or Sl(T) but since the tunneling coefficients are complex, e.g. ðj;sjÞ

T lqq0

ðj;sjÞ

¼ Rlqq0

ðj;sjÞ

þ iIlqq0 ;

ð44Þ

the notation Cl(R), Cl(I), Sl(R), or Sl(I) indicates whether the series in question involves the real or imaginary parts of the coefficients T. An example for the use of these equations is given in Section 4.6. 4. Structure of the ERHAM program 4.1. Modular approach to define Hamiltonian The effective rotational Hamiltonian for one or two periodic LAMs has been implemented in the Fortran program ERHAM for the cases listed in Table 1. The program is available on the PROSPE web site maintained by Kisiel [40]. It consists of the three principal parts for input, least-squares (LS) fit of spectroscopic parameters to observed transition frequencies, and prediction of transition frequencies with relative intensities. The data for both the LS fit (which is performed first) and the prediction are entered in the input section. The LS fit is performed if (i) the number of iterations requested is positive, (ii) at least one spectroscopic parameter is declared variable, and (iii) at least one transition with non-zero weight is defined. Predictions are made if the lower limit of the range for the rotational quantum number J is less than or equal to the upper limit. The codes for the fit and the prediction share the set-up of the Hamiltonian matrix and the calculation of eigenvalues and eigenvectors. The program may be used for up to six different non-interacting vibrational states with common parameters for the q-vectors (e.g. qk, bk, and ak); if desired, some of the other spectroscopic parameters may be shared by several vibrational states. The essential spectroscopic constants for the calculations are polar coordinates of the end points of the q-vectors (e.g. qk, bk, and ak) as well as, for each vibrational state, the rotational constants and the first energy tunneling parameters, e01 and e10. For

relative intensities, the temperature and the components of the electric dipole moment are required. Control parameters determine the periodicities of the LAMs (nk), the symmetry group and the orientation of the reference axes (which may be the principal axes) and, for the prediction, the ranges of the rotational quantum number J and the transition frequencies. Each spectroscopic parameter (except qk, bk, and ak and the rotational constants) may be entered as a tunneling parameter which is defined by the integers q, q0 , x, j, k, p and a real value. The first four integer parameters are based on the notation introduced for the tunneling parameters in Section 3. The last three integers are related to the exponents of the angular momentum operators in Eq. (10). Specifically, k and p are the exponents of P and Pz in Eq. (10) whereas |j| corresponds to r in the same equation. At the same time, j ‘‘measures’’ the distance of a matrix element from the main diagonal in the Hamiltonian matrix. However, the sign of j has a different meaning in the program from that in the notation for the tunneling coefficients (where it means below or above the main diagonal). In the program it is used define the real or imaginary parts of the complex coefficient. For coefficients with x = +1 (Rl of even order in the overall angular momentum), j > 0 and j < 0 denote the real and imaginary parts, respectively; in contrast, if x = 1, j > 0 and j < 0 refer to the imaginary and real parts, respectively. This modular approach to define tunneling coefficients provides great flexibility that does not restrict a user to the limited set of operators included in most program codes. The appendix contains additional information about the tunneling parameter input with the specific codes for many of them, lists the admissible values of q and q0 and which ones to choose first, and contains some remarks about the notation for tunneling parameters. ðjÞ It is emphasized here that the coefficients T l00 are ‘‘non-tunneling’’ because they are independent of the subscripts q and q0 and they contribute equally to all torsional substates. They refer to the hypothetically unsplit state and have no effect on the torsional splitting. The rotational and centrifugal distortion constants belong to this category as well as e00. Because the rotational constants themselves cannot be coded as such with the modular approach described above (only linear combinations of them), rotational constants and distortion constants up to the sextic order (in the A-reduction, Ir-representation [39]) are hard-coded into the program. It is easy to add distortion constants of higher order as necessary or to switch to the S-reduction by defining the necessary parameters with the tunneling parameter input (paying attention to the sign differences). The coefficients e00 cannot be determined from rotational transitions within a vibrational state alone; rotationally resolved vibration– rotation spectra would be necessary to determine (relative) e00 values. Unfortunately, the program version on the PROSPE web site cannot be used for that purpose.

60

P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

ERHAM performs several tests on the input data to prevent contradictions between them. Sometimes, contradicting information is just ignored: in these cases, the echo output of the input data indicates that the program has made changes to the input. In other cases, a message alerts the user that contradicting or inconsistent information has been provided. The parameter ISCD that selects one of the models listed in Table 1 is the most important one in this context. For instance, if ISCD = 2 for the model with C2 symmetry at equilibrium, the program does not require data for the second internal rotor, e.g. n2, q2, b2, and a2. Also, it just ignores tunneling coefficient input for T qq0 or eqq0 if q0 > q because T qq0 and T q0 q are related for equivalent periodic LAMs. The second important control parameter is NC which is called direction cosine parameter in the input description of the program. It corresponds to s in the outline of theory (Section 3) and controls the setting of the reference axes. Its allowed values are +1 and 1. The value of s is not critical for non-equivalent rotors; however, it must be taken into account when conclusions about the geometrical settings of the internal rotation axes are drawn. A summary of the models and settings in terms of the parameters ISCD and NC is given in Table 3. The program also performs tests to ensure that only combinations of the symmetry numbers (r1r2) are accepted that correspond to a specific algorithm. For non-equivalent periodic LAMs, r1 is any integer 0 6 r1 6 n1/2. For r1 = 0 and r1 = n1/2, r2 is any integer 0 6 r2 6 n2/2; otherwise, 0 6 r2 6 n2  1. For equivalent periodic LAMs, r1 6 r2 applies in addition to the rules for nonequivalent LAMs. Messages alert the user that erroneous input is ignored when the definition of the torsional substates for the predictions or for transitions submitted for the least-squares fit does not correspond to these rules.

4.2. Calculation of energy levels and least-squares fit For each torsional substate identified by the combination of the symmetry numbers (r1r2) of each vibrational state at any value of the rotational quantum number J, the full (2J + 1) by (2J + 1) block of the complex Hamiltonian matrix is set up in the basis |JKMvr1r2i as described in Section 3. For some torsional substates, this matrix may be factored into smaller diagonal sub-blocks by symmetrization. Diagonalization of each complex sub-block is achieved by a Housholder transformation to a real tridiagonal matrix whose eigenvalues are obtained by bisection. The eigenvectors, obtained by reverse iteration, are transformed back into the |JKMvr1r2i basis for the calculation of derivatives or relative intensities. The selected spectroscopic constants are adjusted in a standard non-linear weighted least-squares fit using singular value decomposition to fit the observed transition frequencies. The transitions are identified by the symmetry numbers rk and, for the upper and lower energy levels, the rotational quantum numbers J and the labels of the energy level N in the stack of levels ordered by increasing energy. The weights of the transition frequencies are the inverse squares of the supplied experimental uncertainties. A transition receives zero weight when its experimental uncertainty is entered as 0. It is also possible to define blends of several transitions assigned to the same observed frequency; a blend is defined as a weighted linear combination of transition frequencies assigned to the frequency of the observed blend. ERHAM is fast because it does not have to solve the internal rotation Hamiltonian and the largest matrices that are diagonalized have the dimensions (2J + 1) by (2J + 1). It is somewhat slower than a program for an asymmetric rotor without internal rotors, because it needs to diagonalize such matrices for each of the distinct torsional substates, and symmetrization into smaller submatrices is only possible for some of them.

4.3. Prediction of spectra If the program is directed to do so, transitions with frequencies, line strengths S, and relative intensities based on S, spin weights, the values of the components of the electric dipole moments and temperature are predicted. For each vibrational state, the predictions are restricted by its own range of the rotational quantum number J, the provided frequency limits and the minimum line strength. At the end of the prediction for a given vibrational state, they are ordered by the frequencies of the transitions. The labels of the predicted transitions include the symmetry numbers rk and, for both the upper and lower energy levels, the rotational quantum numbers J and the labels of the energy levels N in the stack of levels ordered by increasing energy. This is the identification of the transitions required for the LS fit. For certain torsional sublevels, some N labels in the list of transitions carry a minus sign. That same sign is also printed in the list of energy levels right after the value of the corresponding energy. If minus signs appear at all for certain torsional sublevels, all energy levels of the E block in a Wang-type asymmetric rotor basis carry them. In addition, all energy levels of one of the other Wang-type blocks (E+, O+ or O) carry the minus signs as well, depending on the torsional sublevel and the point group of the equilibrium geometry. The list of ordered frequencies includes Ka  Kc labeling. It is based on the assumption that the energy levels are ordered in all torsional substates like for an asymmetric rotor. This assumption may lead to incorrect labels in some torsional substates. In some cases, the sign of the N label may be used to check the Ka  Kc labeling. The program prints line strengths S for all components of the electric dipole moments. However, the relative intensity takes into account only those line strengths for which the corresponding dipole component is non-zero. 4.4. Application of ERHAM to problems with a single internal rotor The program makes no distinction between molecules with one or two internal rotors. One has to provide data for the ‘‘unused’’ internal rotor. Ideally, qk, bk, and ak should be set to zero, nk to 1 (see also the next section). Only ISCD = 1 (if internal rotor is in the xz or yz symmetry plane), ISCD = 0 (if internal rotor is in the xy symmetry plane) or ISCD = +1 (no symmetry plane) are meaningful. Depending on which internal rotor is used, only tunneling coefficients with q0 = 0 or q = 0 should be defined (see section H of the instructions available with the program). The program does not check whether the input file contains data that are not compatible with the specifications described in this paragraph. The q-axis system can be used if bk, and ak of the internal rotor are set to 0. The parameters Dab (or Dac or Dbc or all of them) can be defined with tunneling parameter input. For all three of them, use q = q0 = k = 0 and x = +1; for Dab, Dac, or Dbc, use j = 1, 1, or 2, and p = 1, 1, or 0, respectively. Like in the BELGI programs [1] which are set up entirely in the q-axis system, parameters printed for rotational and centrifugal distortion constants, should all carry the subscript ‘‘RAM’’ in such a case, and transformation to principal axis values can be achieved by the methods described by Kleiner [1]. 4.5. Complimentary output of various internal rotation parameters Between the least-squares fit and the prediction of transitions, the program runs through a short section that provides additional information on internal rotation parameters. Among them are the moments of inertia of the internal rotors, the reduced moments of inertia, the angles defining the direction cosines of the internal rotation axes, the angle x between the internal rotation axes, the

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F-numbers (F1, F2, F12), and the energy differences of the torsional substates with respect to the totally symmetric state (00). These quantities are derived from the q-vector parameters, the rotational constants, and the energy tunneling coefficients eqq0 . The section of the program is intended to provide a quick estimate of these parameters. Ideally, one should use equilibrium rotational constants or at least ground state rotational constants. However, the program uses just the rotational constants of the first vibrational state defined in the program regardless of whether it is the ground state or one of the excited states. This creates some variability in the results. Also, if each vibrational state is fit independently, the q-vector parameters themselves may exhibit some variation which, of course, contributes to the variations of the results from this section. An example can be found in the data in Table 2 in the paper for pyruvonitrile by Kras´nicki et al. [24]. Moreover, most of these quantities depend heavily on the q-vector data. If the polar coordinates of the q-vectors are ill-defined (e.g. if the splitting is very small), all quantities except for the energy differences are prone to substantial errors. Unfortunately, the program may crash in this section if some quantities that should be 0 happen to acquire a very small negative value due to numerical noise and we know what happens when a program tries to calculate a square root of a negative number. A similar problem arises when the argument of cyclometric functions (like arc cos) is absolutely larger than 1 due to numerical noise. In applications to a single internal rotor, it is therefore necessary to supply very small values for q, b, and a of the ‘‘unused’’ internal rotor to make it through this section to the prediction. 4.6. A case study: CH380Se80SeCH3 The initial series of least-squares fits for CH380Se80SeCH3 may serve as an example for the use of the program. The equilibrium geometry has C2 symmetry; therefore, its symmetry group is [33]C2, and the b principal inertial axis coincides with the C2 symmetry axis. The input parameters ISCD = 2 and NC = 1 define this setting. The internal rotors are defined by the parameters n1 (called N1 in the program – see instruction file on PROSPE web site [40]) is set to 3, and by initial values for q1, b1, and a1 (RHO1, BETA1 and ALPHA1 in the program, respectively). The complete Fourier series expansions of the internal rotation energy and of the coefficients of the rotational operators in the Hamiltonian were obtained from the information in Tables IV and V in paper I as follows: Internal rotation energy:

( ) q1 X 0 X 0 0 0 0 R0 ¼ e00 þ 2 C qq eqq þ C qq eqq þ ðC qq0 þ C q0 q Þeqq C 0qq0 ¼ cosð2pðqðr1  q1 K 1 Þ=n þ q0 ðr2  q2 K 2 Þ=nÞÞ P2 ; P2z ðx ¼ þ1;

ð45aÞ

q0 ¼qþ1

q>0

ð45bÞ

l ¼ þ1; j ¼ 0Þ :

Rl ¼ C l1 ðRÞ ¼ Rl00 þ 2

( ) q1 X X C qq Rlqq þ C qq Rlqq þ ðC qq0 þ C q0 q ÞRlqq0 q>0

q0 ¼qþ1

ð46Þ P2x  P 2y ðx ¼ þ1;

l ¼ þ1; j ¼ 2Þ :

Rl ¼ C l1 ðRÞ þ iC l2 ðIÞ

( ) q1 X X ¼ Rl00 þ 2 C qq Rlqq þ C qq Rlqq þ ðC qq0 þ C q0 q ÞRlqq0 q>0 q1 X X ðC qq0  C q0 q ÞIlqq0 þ i2 q>0 q0 ¼qþ1

q0 ¼qþ1

ð47Þ

Py P z þ Pz Py ðx ¼ þ1;

l ¼ þ1; j ¼ 1Þ :

Rl ¼ C l1 ðRÞ þ iC l2 ðIÞ ¼ Rl00 þ 2

( ) q1 X X C qq Rlqq þ C qq Rlqq þ ðC qq0 þ C q0 q ÞRlqq0 q0 ¼qþ1

q>0

þ i2

X

q1 X

ðC qq0  C q0 q ÞIlqq0

ð48Þ

q>0 q0 ¼qþ1

Px ðx ¼ 1;

l ¼ þ1; j ¼ 1Þ :

Rl ¼ iSl4 ðRÞ  Sl3 ðIÞ

( ) q1 X X Sqq Rlqq þ ðSqq0  Sq0 q ÞRlqq0 ¼ i2 q0 ¼qþ1

q>0

( ) q1 X X 0 0 0 2 Sqq Ilqq þ ðSqq þ Sq q ÞIlqq Pz ðx ¼ 1;

ð49Þ

q0 ¼qþ1

q>0

l ¼ 1; j ¼ 0Þ :

Rl ¼ Sl4 ðIÞ ¼ 2

( ) q1 X X Sqq Ilqq þ ðSqq0  Sq0 q ÞIlqq0 q>0

ð50Þ

q0 ¼qþ1

C qq0 ¼ cosðpðqð2r1  qðK þ K 0 ÞÞ þ q0 ð2r2 þ qðK þ K 0 ÞÞÞ=nÞ Sqq0 ¼ sinðpðqð2r1  qðK þ K 0 ÞÞ þ q0 ð2r2 þ qðK þ K 0 ÞÞÞ=nÞ

ð51aÞ ð51bÞ

Instead of the operator PyPz + PzPy which has the correct symmetry, one could have chosen PxPz + PzPx with x = +1, l = 1, j = 1, and Rxz = iCl1(Ixz) + Cl2(Rxz). Owing to the factor ji in the relation between the matrix elements of Px and Py (41), the Fourier series of the matrix elements of PxPz + PzPx has exactly the same form as in (48) because only the real quantities Rxz–jIyz and Ixz–jRyz are determinable but not Rxz, Iyz, Ixz and Ryz individually due to (42). Assigned transition frequencies from molecular beam Fourier transform microwave spectra were obtained together with assignments and rotational constants from an asymmetric rotor fit. The rotational transitions were split into quartets spread over a range from 1 to about 30 MHz. The first input file contained a complete set of rotational and quartic distortion constants (Ir-representation, A-reduction [39]), the first energy tunneling coefficient e10 and a value for q1 from a different molecule, with guesses for the other coordinates of the q-vector (b1 and a1). Only 59 out of a total of 98 frequencies from 25 rotational transitions initially had non-zero weights, mostly because some splittings were very small or because the assignments of some torsional substates were tentative. The first two fits were performed to determine approximate rotational and distortion constants. Since DK and dK were not well determined in the first fit, they were kept constant during the second fit whose results were used as starting parameters for the third fit which included e10 as an adjustable parameter. By this time, the initial weighted standard deviation s of 2.46  106 had decreased via 646 (fit 1) and 570 (fit 2) to 82 (fit 3). The parameters from fit 3 were used as input for the subsequent fits described below. After 5 iterations in fit 4 (with two additional adjustable parameters, q1 and b1), s reached 45. After correction of a typographical error for one frequency by 1 MHz, fit 5 had s = 29. At the end of fit 6 which included also a1 as a variable with s = 13, the assignments of the (11) and (12) components of one transition had to be interchanged; thus fit 7 reached s = 11. In fit 8, all quartic distortion constant constants were declared variable; the standard deviation s reached 1.4. The published final fit [41] included all 98 frequencies with one additional tunneling parameter, [A  (B + C)/2]10, and had a weighted standard deviation s of 0.32. Since the estimated experimental uncertainty was 4 kHz, this means the

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unweighted standard deviation was 1.3 kHz. Table 4 contains more information on the initial least-squares fits. Lessons learned from this example: for a state with small splittings for the rotational transitions, one only needs reasonable estimates of the q-vector parameters, good guesses of rotational and centrifugal distortion constants, and a guess for e10 (and e01 for non-equivalent rotors) to get started IF the assignments of the transitions are correct. To guess initial values for e10 and e01 successfully, one should take into account that they are negative or positive if the vibrational quantum numbers of the torsional motions are even or odd, respectively, because the order of A and E levels for 3-fold internal rotors switches from one vibrational level to the next. However, interactions with other vibrations may render this rule of thumb invalid. The magnitudes of e10 and e01 increase with the magnitude of the difference between the A and E levels. The splittings increase with these magnitudes for similar values of q1 and q2; however, an increase in q1 and/or q2 from one molecule to another also increases the size of the splittings. If the magnitudes of e10 and e01 are less than about 100 MHz, only very few other tunneling coefficients seem to be necessary. The first ones to try are tunneling coefficients for the rotational constants, or the next order of the tunneling energy coefficients e20, e02, e11 or e1–1. For large data sets with high quantum numbers J and Ka, it may be necessary to introduce tunneling coefficients for distortion constants. The introduction of such tunneling coefficients often causes significant changes of other parameters. Tunneling coefficients of rotational operators of odd order are due to the variability of the q-vectors. Only in a few cases have they brought significant improvements in LS fits. The same is true for tunneling coefficients of operators like PxPz + PzPx if the reference axes are principal axes. 5. Applications, limitations and comparisons 5.1. One-rotor applications Program ERHAM has been used to fit spectra of several molecules with one internal rotor (Table 5). All have a rotating methyl group except for the complex CHClF2 with water where water is the internal rotor with n = 2. With two exceptions without a symmetry plane, all have the internal rotor axis in the ab symmetry plane.

In the papers on methyl carbamate [22] and methyl formate ([19,42], the least-squares fits include a few thousand transitions from the microwave to the submillimeter wave region for the vibrational ground state (GS) or, for methyl formate [42], for the first two torsional excited states (TESs) which have been fit independently from each other and from the ground state. For pyruvic acid [23] and pyruvonitrile [24], the analysis has included transitions in a similar frequency range for the ground state and a number of vibrational excited states of the methyl torsions and skeletal torsional or bending modes; all excited states of both molecules have been fitted independently from the ground state. The other molecules listed in Table 5 have been investigated by Fourier transform microwave spectroscopy only. The ethers studied by Grubbs and Cooke [43] have no symmetry plane at equilibrium (which is evident from the data in the supplementary material but not from the data reported in the paper itself). ERHAM has also been used [44] to re-analyze the microwave and millimeter wave data reported for symmetric acetone-d1 and -d5 [45]. For the molecules mentioned in this paragraph, the standard deviations in the least-squares fits reached generally about experimental precision for the vibrational ground states (with some variations). For vibrational excited states, standard deviations were not quite as good but still substantially better than what can be achieved with some other programs (see Section 5.4). The CHClF2AH2O complex has the internal rotor axis in the ab symmetry plane [28]. A few facts about this species are noteworthy: (i) This is the first molecule analyzed by ERHAM whose molecular properties are periodic in 2p/2 as opposed to 2p/3. (ii) According to the analysis, the lowest internal rotation substate in the vibrational ground state is not the totally symmetric sublevel with r = 0 (A state) but rather the r = 1 sublevel (B state) because the first tunneling coefficient of the energy is positive. In addition, the low J transitions of the r = 1 state are observed at higher frequencies in the pulsed molecular beam FT microwave spectrum than those of the r = 0 state. (iii) The first tunneling coefficient of the linear combination (B + C)/2 was extremely important for the successful fit and relatively much larger ( 200 ppm of (B + C)/2) than the values obtained for other molecules analyzed so far with this method (a few ppm). The size of this coefficient is responsible for the fact that the transitions of the r = 1 sublevel occur at higher frequencies than those of r = 0. It is assumed that the size of this coefficient is due the extreme floppiness of the complex and the

Table 6 Applications of ERHAM to molecules with two internal rotors. Molecule

Groupa

Settingb

Statesc

Reference

CH3AOACH3 dimethyl ether

[33]C2v

ab, b

CH3AC(O)ACH3 acetone

[33]C2v

ab, b

CH3AC(O)ACH3 acetone (potential function) CH3A13C(O)ACH3 acetone CH3AC(O)ACH3 acetone CH3ACH2ACH3 propane CH3AOC(O)OACH3 dimethyl carbonate CH3AC(@C@CH2)ACH3 3-methyl-1,2-butadiene CH3ACH2OACH3 ethyl methyl ether CH3A80Se80SeACH3 dimethyl diselenide CH3A78Se80SeACH3 dimethyl diselenide CH3ASO4ACH3 dimethyl sulfate CH3ASO4A13CH3 dimethyl sulfate CH3AC(O)OC(@CH2)ACH3 isopropenyl acetate

[33]C2v [33]C 0s [33]C2v [33]C2v [33]C2v [33]C 0s [33]C2 [33]C 01 [33]C2 [33]C 01 [33]C 01

ab, ab ab, ab, ab, ab b

G T global G G G T(m12) T(m17) d G T(m12) G G G T(m14, m27) G G T(m14) global G G G G G G

[11] [18] [46] [19] [25] [26] [49] [44] [44] [47] [53] [27] [21] [41] [41] [48] [48] [54]

13

a b c d

b

b b b a

Molecular symmetry group. ‘‘ab’’ indicates that the internal rotation axes are in the ab principal plane; ‘‘b’’ or ‘‘a’’ indicate the C2 symmetry axis. G = ground state, T = torsional excited state, ‘‘global’’ indicates a global fit with common q-vector parameters for all states. The correct label for this vibrational mode is m24 according to Ref. [50].

P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

significant structural changes due to relaxation during the periodic internal motion.

63

wave regions. The interested reader is referred to the paper on the acetone spectrum of m17 (aka m24) for further details [26]. 5.3. Limitations

5.2. Two-rotor applications More applications of ERHAM have been published for molecules with two methyl group internal rotors (Table 6). One half of the publications have the subject of dimethyl ether or acetone; both molecules have an equilibrium structure with C2v symmetry. Other molecules with the same symmetry include propane, dimethyl carbonate and 3-methyl-1,2-butadiene (1,1-dimethyl allene). Molecules investigated with lower symmetry are ethyl methyl ether as well as dimethyl diselenide and dimethyl sulfate, and isopropenyl acetate. About 350 frequencies of the ground state together with 20–30 frequencies for each of five torsional excited states of dimethyl ether (symmetry group [33]C2v) have been used in the ‘‘demonstration of principle’’ application of ERHAM [11]. Subsequent extensions of the analysis of the ground state include over 1600 [18] and 6800 frequencies [46] from the microwave to the terahertz regions. The data sets for each of the ground and two torsional excited states of acetone include between 600 and 1000 frequencies [21,25–26] from 8 to 360 GHz. About 1000 transitions in the same frequency range have been analyzed for the ground state of ethyl methyl ether ([33]C 0s ) [21]. The study of the dimethyl carbonate spectrum from 8 to 350 GHz contains about 270 assigned transitions. The investigation of the rotational spectrum of propane in the ground and two torsional excited states [47] includes frequencies from the microwave to terahertz regions, but it is not clear from the paper how many transitions were included in the analysis. The other applications of ERHAM to molecules with two methyl internal rotors are based on microwave data only (between about 30 and 250 transitions each). In dimethyl allene ([33]C2v), the C2 axis coincides with the a principal inertial axis in contrast to the other molecules with the same symmetry group where the C2 axis coincides with the b axis. For this molecule, microwave data from the literature have been analyzed to extract the energy differences between the torsional sublevels to provide additional information to far infrared data in a study of the torsional potential function. The paper on dimethyl diselenide [41] reports results for two symmetric ([33]C2) and seven asymmetric ([33]C 01 ) isotopologues. Two symmetric ([33]C2) and three asymmetric ([33]C 01 ) isotopologues of dimethyl sulfate have been studied in natural abundance [48]. Isopropenyl acetate also has the symmetry group [33]C 01 because its equilibrium structure has C1 point group symmetry. In another application of ERHAM, preliminary data from the first torsional excited state of acetone have been used to reevaluate the torsional potential function of acetone [49]. As for molecules with just one internal rotor, the standard deviations of the LS fits are generally of the order of the experimental uncertainties for the vibrational ground states. However, the standard deviations for the analyses of the torsional excited states of acetone fall short of that goal, particularly for the m17 state (whose correct label should be m24 [50]). There are two possible causes: (i) The torsional splittings of the rotational transitions are so large (because the states are rather close to the barrier of internal rotation) that the Fourier series expansions of the matrix elements converge very slowly (see also Section 5.3 on the limitations of ERHAM). (ii) The assigned transition frequencies do not carry enough information about the asymmetry splitting of the asymmetric rotor because most of the assigned transitions are Ka or Kc degenerate. Only a sufficient number of well resolved transitions with unambiguous assignments at microwave frequencies will help to alleviate this problem. Both excited state spectra have been assigned almost exclusively in the millimeter and submillimeter

As described in the previous sections, ERHAM is fast and generally precise. But there are limitations besides not yet discovered potential programming errors. The first limitation is that the program does not provide barriers to internal rotation. This is obvious from the description above since the internal rotation Hamiltonian is not solved in program. However, barriers can be calculated with an appropriate program from the F constants and the splitting of the torsional sublevels which can be obtained from the tunneling coefficients eqq0 with the appropriate equation for R0 (Table 3 or Table 4 of [11]) by setting J = K1 = K2 = 0. The initial aims of the effective Hamiltonian and the program were efficient and fast fitting and prediction of spectra of molecules with two periodic LAMs and data reduction to determine energy differences between torsional substates. Such differences carry invaluable information about torsional states that can be used alone or together with wavenumbers of direct torsional transitions from far-infrared and Raman spectra to determine torsional potential functions for such molecules. ERHAM has been used for that purpose in two examples mentioned in the previous section [27,49]. Determinations of two-dimensional potential functions from spectroscopic data [35,51,52] are extremely challenging, much more than the average analysis of the rotational spectra of such molecules with ERHAM and, in the author’s experience, are subject to very nonlinear and ill-conditioned least-squares problems. For that reason, ERHAM was designed to determine energy differences between torsional sublevels but not barrier information. The determination of the potential barriers for molecules with just one periodic LAM is much simpler, of course, and using ERHAM for such molecules was rather an afterthought than a goal. The effectiveness of ERHAM is expected to reach its limit for systems with very low barriers or for internal rotation states near or above the effective barrier to internal rotation. As eigenstates approach the barrier and eventually surpass it, the Fourier expansion of the matrix elements (39) will not converge as quickly because the ‘‘localized’’ functions are no longer very well localized. This means that too many tunneling coefficients are necessary to achieve a satisfactory parameterization of the Hamiltonian. For the molecules studied so far, this has not been a major problem although the 2nd torsional state of acetone m17 (aka m24) [26] had relatively large residuals. With the potential and the F constants given in [49], m17 is 225 cm1 above the potential minimum whereas the effective barrier is 251 cm1. But, in this case, a lack of a sufficient number of transitions with resolved asymmetric rotor splittings also poses a major problem to stability of the least-squares fit. Another problem area is the labeling of the rotational energy levels. As described in Section 4.3, a rather simplistic scheme is used to determine Ka, Kc labels. (Future revisions may provide something better). This scheme may break down on three occasions. In the first case, accidentally nearly degenerate levels with the same Ka or Kc may be switched due to numerical noise. In some cases, the minus sign printed for some energy levels may help to alleviate this problem. In a second case, rotation-internal rotation interactions may cause shifts of energy levels that render the asymmetric rotor labeling incorrect. In this case, it may happen that, for instance, some b-type transitions suddenly are labeled as c-type transitions (and vice versa); such c-type transitions have been called ‘‘false’’ forbidden transitions in some publications [21]. In the third case, wavefunctions belonging to different Wang subblocks may be mixed due to the interactions. In such a case, the intensity of an asymmetric rotor transition may be split among

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usually two transitions (example: a b-type transition has a reduced intensity and a ‘‘true’’ forbidden c-type transition is present, usually near the allowed b-type transition) [21]. A global analysis of torsional ground and excited states similar to the BELGI programs developed by Kleiner and coworkers [1] is impossible at present. Currently, ERHAM allows a global fit only in the sense that several torsional states can be fit together to a common set of the q-vector parameters where each state has its own set of rotational and distortion constants and tunneling coefficients. But since other interactions of the Coriolis or Fermi type between vibrational states cannot be taken into account, the term ‘‘global’’ for an analysis is clearly restricted compared to the BELGI code. 5.4. Comparisons with other programs Other investigators have compared results obtained with different freely available programs that fit and predict spectra of molecules with internal rotors. Comparisons have been performed for the programs SPFIT, developed by Pickett [6], XIAM by Hartwig [10], BELGI by Kleiner and coworkers (see [1] for detailed references) and ERHAM. 5.4.1. SPFIT–ERHAM comparisons Endres et al. [46] observed that their very large data set for the ground state of dimethyl ether could be fit almost equally well with both programs with comparable numbers of fitting parameters (reduced standard deviations 0.99 for ERHAM and 1.5 for SPFIT). Kisiel et al. [23] noted in their study of pyruvic acid that both programs produced comparable fits for the ground state although with fewer parameters in the case of ERHAM. The disadvantage of ERHAM is the limited experience in the selection of the most effective parameters [23]. However, they stated that ERHAM is clearly superior to analyze the methyl and skeletal torsional excited states with a smaller number of parameters. The emphasis in the comparison of the results from SPFIT and ERHAM for propane by Drouin et al. [47] seems to have been on testing recent modifications to SPFIT in an effort to achieve similar results with both programs. As it turned out, the modifications to SPFIT were successful for the applications to propane at the slight cost of a moderately larger set of fitting parameters as can be concluded from the results presented in the tables. 5.4.2. XIAM–ERHAM comparisons Kisiel and coworkers compared the results obtained by these two programs for pyruvic acid [23] and pyruvonitrile [24]. They noted that ERHAM produces generally the best quality fit and is successful in fitting experimental data particularly for vibrational exited states where the limited set of distortion constants in XIAM is no longer able to accommodate higher order interactions. They also stated that ERHAM allowed to expand the assignment of Estate lines significantly and to account for practically all confidently assigned lines of pyruvonitrile [24]. Lovas et al. [53] in their work on dimethyl carbonate noted the advantages of ERHAM over XIAM: calculations for high rotational quantum numbers up to J = 120, availability of weighted LS fits, fitting blends as weighted averages, and lower weighted standard deviation. Favero et al. [48] stated that XIAM and ERHAM performed similarly well for the ground states of dimethyl sulfate isotopologues and that internal rotation parameters derived from the results from the two methods were close to each other. On the other hand, they observed that ‘‘ERHAM requires some expertise in choosing the best set of parameters’’, and that ERHAM does not calculate the value of the potential barrier directly (they used another program to determine the barrier from ERHAM results). Nguyen and Stahl [54] observed that XIAM, due to its limitations, could not deal

satisfactorily in a single fit with the large splitting caused by the acetyl CH3 group and the narrow splitting due to the CH3 group in the isopropenyl fragment (standard deviation 75 kHz). However, ERHAM reached a standard deviation of 2.3 kHz which is very close to the experimental uncertainty. They also noted with regret the lack of barrier information. 5.4.3. BELGI–ERHAM comparisons Some results from these two programs have been compared (Methyl carbamate, methyl formate) but unfortunately never with identical data sets. Ilyushin et al. [55] who used a new version of BELGI originally written for internal rotors with a sixfold potential did fit RAM parameters to more than 10 000 lines observed and assigned in the ground state and in m18 of methyl formate. They emphasized that Maeda et al. [42] in the analysis of m18 with ERHAM had to exclude about half of the 2600 assigned frequencies with high J and Ka from the least-squares fit in order to get satisfactory convergence and an acceptable standard deviation. 6. Conclusions and outlook Overall, the effective rotational Hamiltonian for molecules with one or two periodic LAMs has significantly advanced the fitting and simulation of rotational spectra of such molecules, particularly for molecules with two internal rotors. It has enabled the first assignment of transitions in the torsional excited states of acetone, and it has been used for a molecule whose internal rotor has a twofold barrier. Two observations made during the review of papers that used ERHAM for the analysis may help us to understand the potential and the limitations of the program better. 1. In Sections 5.4, results obtained by different program codes have been compared for molecules with identical data sets. In some cases, ERHAM performed only slightly better than SPFIT or XIAM. In other cases, the results of ERHAM were clearly superior. The obvious question is: Why? A possible answer is suggested by a comparison of the energy tunneling coefficient e10 which is the dominant term that determines the energy differences between the torsional substates: For the examples in Section 5.4 with comparable fits between ERHAM and SPFIT or XIAM, |e10| is between 0.37 and 250 MHz, whereas for the other cases |e10| is significantly larger between 746 and 7116 MHz. (The absolutely largest e10 so far has been obtained for the 2m18 state of methyl formate with 74 500 MHz.) In other words, ERHAM produces better results than SPFIT or XIAM when the tunneling splitting is larger, perhaps because it is more flexible in the choice of fitting parameters. 2. The comment by Ilyushin et al. [55] about the ERHAM analysis of the m18 state of methyl formate [42] (see also Section 5.4) seems to point directly at the expected limitation of ERHAM for states ‘‘close’’ to the potential barrier. It is probable, that the assigned but excluded transitions in the m18 state of methyl formate can be fit satisfactorily with ERHAM; but this will require a much larger set of tunneling parameters. In that light, the difficulties encountered during the fit of the m17 (aka m24) state of acetone [26] may look like they are caused by the same problem. We may need many additional tunneling parameters for a satisfactory fit, but we don’t have all the information yet to determine them. In the case of acetone, it is also obvious that Coriolis interaction between the torsional states m12 and m17 (aka m24) probably causes additional problems. What kind of developments could enhance the versatility of the effective rotational Hamiltonian for two periodic LAMs and ERHAM? The ability to fit and predict transitions between different

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P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

vibrational states would make it possible to use the program for the analysis of vibration–rotation spectra in the infrared and far infrared. An extension that allows interactions between different vibrational states might help to make progress in the analysis of m12 and m17 (aka m24) of acetone. Both of these developments are in the testing stage. Another expansion should include the possibility to analyze spectra of molecules like gauche ethanol, or even the gauche ethanol interacting with trans ethanol. And: Could we apply the ERHAM method to molecules with more than two periodic LAMs? The principles are the same. The major modification of ERHAM would involve the transformation of the q-axis energies to the reference axes system (the equivalent of the matrix Y KK 0 (21b)) and the specifications for the different molecular symmetry groups like those for molecules with three periodic LAMs [36].

Appendix A. Notation and usage of spectroscopic parameters Table A1 contains a list of the Watson-type Bkpr coefficients of order up to k + p + r = 8, their corresponding angular momentum operators, and the integers x, j, k, and p necessary to define them in the tunneling parameter input. A small selection of coefficients with odd p or r = |j| that are not allowed in a Watson Hamiltonian are listed at the end of the table; some of them are allowed in a qaxis system (example: Dab) whereas tunneling components of some are allowed even in a principal axis system. Also listed are the definitions of the linear combinations of rotational constants and the quartic and sextic centrifugal distortion constants in terms of the Bkpr coefficients. Footnotes explain which coefficients are allowed in the A- or S-reductions of the Hamiltonian. Additional

Table A1 Integers x, j, k, and p in tunneling parameter input and suggested notations (B-notation, A-reduction, or S-reduction). Operator

x

j

k

p

Coefficient (A-reduction)

B200 B020

P2

1 1

0 0

2 0

0 2

(B + C)/2 = B200 A  (B + C)/2 = B020

(B + C)/2 = B200 A  (B + C)/2 = B020

B002

P 2þ þ P 2 P4

1

2

0

0

(B  C)/4 = B002

(B  C)/4 = B002

1 1

0 0

4 2

0 2

DJ = B400 DJK = B220

DJ = B400d DJK = B220d

P 2z

B400 B220

P 2 P 2z

Coefficient (S-reduction)

B040

P 4z

1

0

0

4

DK = B040

DK = B040d

B202

P 2 ðP 2þ þ P 2 Þ

1

2

2

0

dJ = B202

d1 = B202d

B022

a

B004

b

½P 2z ; P 2þ þ P 4þ þ P 4 6

P 2 þ =2

e

dK = B022

e

1

2

0

2

1

4

0

0

P 4 P 2z

1 1

0 0

6 4

0 2

UJ = B600 UJK = B420

HJ = B600d HJK = B420d

B240

P 2 P 4z

1

0

2

4

P 6z

1

0

0

6

UKJ = B240 UK = B060

HKJ = B240d

B060 B402

4

1

2

4

0

2

2

2

1

2

0

4

uJ = B402 uJKe = B222 uKe = B042

h1 = B402d

1 1

4

2

0

1

6

0

0

B600 B420

P

B222a B042a B204b B006b B800 B620

P 6 P 2z

B260 B080 B602 B422

a

B242a B062a B404

P ðP 2þ þ P 2 Þ P 2 ½P 2z ; P 2þ þ P 2 þ =2 ½P 4z ; P 2þ þ P 2 þ =2 P 2 ðP 4þ þ P 4 Þ P 6þ þ P 6 8 P

B440

b

B206b B008b i

B011 B011 i B002 i B010i B001i B001 i a

c

Coefficient (Bkpr notation)

P 4 P 4z P 2 P 6z P 8z P 6 ðP 2þ þ P 2 Þ P 4 ½P 2z ; P 2þ þ P 2 þ =2 P 2 ½P 4z ; P 2þ þ P 2 þ =2 ½P 6z ; P 2þ þ P 2 þ =2 P 4 ðP 4þ þ P 4 Þ P 2 ðP 6þ þ P 6 Þ P 8þ þ P 8 [Pz, P+ + P]+/2 [Pz, P+  P]+/2 P 2þ  P 2 Pz P+ + P P+  P

d2 = B004f

HK = B060d e e

h2 = B204f h3 = B006f

1 1

0 0

8 6

0 2

f f

f

1

0

4

4

f

f

1

0

2

6

f

f f f

1

0

0

8

f

1

2

6

0

f

f

1

2

4

2

f

1

2

2

4

f

1

2

0

6

f

1

4

4

0

f

1

6

2

0

f

1

8

0

0

f

1 1 1

1 1 2

0 0 0

1 1 0

Dabf,g,h Dacf,g,h Dbcf,g,h

1 1 1

0 1 1

0 0 0

1 0 0

gaf,g gbf,g gcf,g

Dabf,g,h Dacf,g,h Dbcf,g,h gaf,g gbf,g gcf,g

Not to be used in S-reduction. Not to be used in A-reduction. The (non-tunneling) rotational constants and the quartic and sextic centrifugal distortion constants are explicitly defined in the program for the A-reduction in the Irrepresentation. d S-reduction: For DJ, DJK, DK, d1, HJ, HJK, HKJ, HK, h1, the explicitly defined parameters of the A-reduction can be used if the parameters marked with footnote e are constant and 0; however, the sign must be switched for d1. e S-reduction: the explicitly defined parameter of the A-reduction must be constant and 0. f This parameter must be defined using the tunneling parameter input. g In a principal axis system, only tunneling components are allowed. h For one internal rotor in the q-axis system: the q-axis angles b and a must be constant and 0. i Coefficients with odd numbered subscripts are not allowed in a Watson Hamiltonian. The asterisk refers to the imaginary (when x = 1) or real (when x = 1) component of the complex tunneling coefficient (see Section 4.1). Higher order coefficients of this type may be generated by increasing k and/or p in increments of 2 (reducibility or completeness has not been studied for these coefficients). b

c

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P. Groner / Journal of Molecular Spectroscopy 278 (2012) 52–67

Table A2 Tunneling parameter notation.a sq = |q| + |q0 |

Parameter Two periodic LAMs

A-reduction

S-reduction

q

q0

x

j

k

p

1 0 2 0 1 1 3 0

0 1 0 2 1 1 0 3

1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

a

e10 e01 e20 e02 e1–1 e11 e30 e03

1 1 2 2 2 2 3 3

etc. ½B200 qq0

|q| + |q0 |

½ðB þ CÞ=2qq0

½ðB þ CÞ=2qq0

q

q0

1

0

2

½B020 qq0

|q| + |q0 |

½A  ðB þ CÞ=2qq0

½A  ðB þ CÞ=2qq0

q

q0

1

0

0

2

½B002 qq0

|q| + |q0 |

½ðB  CÞ=4qq0

½ðB  CÞ=4qq0

q

q0

1

2

0

0

½B400 qq0

|q| + |q0 |

½DJ qq0

½DJ qq0

q

q0

1

0

4

0

½B220 qq0

|q| + |q0 |

½DJK qq0

½DJK qq0

q

q0

1

0

2

2

½B040 qq0

|q| + |q0 |

½DK qq0

½DK qq0

q

q0

1

0

0

4

½B202 qq0

|q| + |q0 |

½dJ qq0

½d1 qq0

q

q0

1

2

2

0

½B022 qq0

|q| + |q0 |

½dK qq0

½B004 qq0

|q| + |q0 |

½B011 qq0

|q| + |q0 |

½B002 qq0 ½B010 qq0

q

q0

1

2

0

2

½d2 qq0

q

q0

1

4

0

0

½Dab qq0

½Dab qq0

q

q0

1

1

0

1

|q| + |q0 |

½Dbc qq0

½Dbc qq0

q

q0

1

2

0

0

|q| + |q0 |

½g a qq0

½g a qq0

q

q0

1

0

0

1

1 2 3

0 0 0

1 1 1

0 0 0

0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1

0 0 2 0 0 0 2 2 4 1 2

2 0 0 4 2 0 2 0 0 0 0

0 2 0 0 2 4 0 2 0 1 0

0

1

0

0

1

One periodic LAMb,c

e1 e2 e3

1 2 3

etc.

a

[B200]q [B020]q [B002]q [B400]q [B220]q [B040]q [B202]q [B022]q [B004]q [B011]q ½B002 q

q q q q q q q q q q q

[(B + C)/2]q [A  (B + C)/2]q [(B  C)/4]q [DJ]q [DJK]q [DK]q [dJ]q [dK]q [Dab]q [Dbc]q

[d2]q [Dab]q [Dbc]q

q q q q q q q q q q q

[B010]q

q

[ga]q

[ga]q

q

[(B + C)/2]q [A  (B + C)/2]q [(B  C)/4]q [DJ]q [DJK]q [DK]q [d1]q

0

For ‘‘non-tunneling’’ coefficients, the square brackets and the subscript qq may be omitted. For ‘‘non-tunneling’’ coefficients, the square brackets and the subscript q may be omitted. The columns headed by q and q0 may be interchanged in the tunneling parameter input if r1 and r2 are switched in the transition frequency input and the order of the qvectors is reversed. b

c

information is provided on the explicitly defined parameters in the A-reduction and how the S-reduced Hamiltonian can be defined. Table A2 contains information about the tunneling parameters, specifically about the subscripts q and q0 . ‘‘Non-tunneling’’ tunneling parameters where q = q0 = 0 affect all torsional substates equally; they determine energy levels and transition frequencies of the hypothetically unsplit state. The splittings of energy levels and frequencies are determined by the proper tunneling parameters where the sum sq = |q| + |q0 | > 0. For one periodic LAM, parameters T qq0 with q P 0 and q0 = 0 (or q = 0 and q0 P 0) are allowed. For two equivalent periodic LAMs, the conditions are q P 0 and q 6 q0 6 q. For two non-equivalent periodic LAMs, parameters T qq0 and, unless q = |q0 |, also T q0 q with q P 0 and q 6 q0 6 q are allowed. The importance and magnitude of a tunneling coefficient are approximately determined by sq. Coefficients with a larger sq are less important and have smaller absolute values than coefficients with a smaller sq. (Exception: if the reference axes are the principal axes, certain ‘‘non-tunneling’’ coefficients like Dab are 0 by symmetry or because of the choice of the reduction of the Hamiltonian.) Therefore, it is generally inappropriate to try to improve a fit by using a tunneling parameter, say [B420]20, unless the ‘‘non-

tunneling’’ [B420]00 = B420 and [B420]10 (with a smaller sq) are already among the fitting parameters and determinable. In a similar vein, a tunneling parameter of an S-reduction distortion constant should not be used if all other parameters belong to the Areduction. Table A2 lists examples of the notation for the tunneling parameters, this time with the complete set of integers for the tunneling parameter input. The notation with Bkpr in square brackets which carry the q or qq0 subscripts introduced originally is the preferred notation. The notation based on the A- or S-reduction symbols in square brackets is probably more appealing to the average spectroscopist. However, there is a pitfall: in all papers published so far with tunneling coefficients of quartic centrifugal distortion constants determined by ERHAM, the signs of these coefficients should have been opposite to the signs printed by the program if the coefficients were reported as e.g. ½DJK qq0 . References [1] I. Kleiner, J. Mol. Spectrosc. 260 (2010) 1–18. [2] J.T. Hougen, I. Kleiner, M. Godefroid, J. Mol. Spectrosc. 163 (1994) 559–586.

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