1. Introduction
To describe the rotation and vibration of a well-bent triatomic molecule, such as H
S, with a large, non-accessible barrier to linearity (of 24,423
cm
−1 [
1]), a molecular kinetic energy operator (KEO) in terms of the 3 vibrational (e.g., two stretching and one bending) and three rotational coordinates (usually Euler angles) [
2] is constructed. The three vibrational modes correspond to the standard bent-molecule methodology that is based on the
vibrational modes. We traditionally start from selecting a zero-order Hamiltonian
=
+
appropriate for a simplified molecule with the rotation-vibration interaction neglected.
is the Hamiltonian for the vibrational motion, obtained by entirely neglecting rotation in the complete Hamiltonian, while
is the Hamiltonian for the molecule rotating rigidly in its equilibrium configuration. The eigenfunctions of
are products
×
of an eigenfunction
of
and an eigenfunction
of
, and these products are used as basis functions for solving the Schrödinger problem for the complete rotating and vibrating molecule, which is done either by perturbation theory or in a variational approach.
For the description of a linear triatomic molecule, for example CO
(and actually also H
O), whose potential energy surface would allow for the linear configuration to become feasible during (typical) vibration, we cannot proceed as described for H
S above, as the corresponding KEO becomes singular at linear molecular geometries. The reason is that, in the linear equilibrium configuration, the molecule cannot rotate about its own axis (systematically called
a, the axis of least moment of inertia [
2]). The corresponding rotation is undefined and, thus, leads to a singularity in the KEO. Another way of expressing this is to say that, for a linear molecule, we cannot initially neglect the interaction between the bending motion and the rotation about the
a axis, since this rotation only takes place when the molecule bends out of the linear equilibrium configuration, as discussed extensively in Ref. [
3]. Therefore, in considering a simplified molecule with the rotation-vibration interaction neglected with the Hamiltonian
=
+
, we cannot practically include the rotation about the
a axis in
, and this operator now only describes the so-called end-over-end rotation about the
b and
c axes [
2] in the equilibrium configuration. The
b and
c axes [
2] are both perpendicular to the molecular axis
a.
In the literature, there are two main approaches for resolving the singularity issue. The traditional textbook approach is to treat linear molecules as systems with
vibrational modes (four in the case of triatomics) with only two rotational degrees of freedom. This approach is usually associated with the normal coordinates as vibrational degrees of freedom. The kinetic energy operator is not singular in such a representation, which is the main point of the, so-called, “(
)” methodology [
4]. This method is very popular in quantum-chemistry calculations and it has also been used in ro-vibrational, variational calculations (for example, by Carter and Handy [
5], Mátyus et al. [
6], Bowman et al. [
7], Schroeder and Sebald [
8]). The main disadvantages of the
approach are (i) the normal coordinates are not very chemically intuitive when describing molecular potential energy surfaces (PES), so that, consequently, (ii) the convergence of the calculations that correspond to higher vibrational excitations is slow and, finally, (iii) this approach is not well defined for quasi-linear molecules (molecules with a bent equilibrium structure and a low, penetrable barrier to linearity), which is, when the linear configuration is accessible on the time scale of a typical experiment). A neat feature of the “(
)” approach, which is relevant to this work, is that the corresponding vibrational basis functions, as well as the rotational (rigid-rotor) basis functions, transform according to the infinite symmetry groups
C or
D, which are not only an appropriate choice for linear molecules, but also give access to powerful symmetry group (or representation-theory) methods. As far as the ro-vibrational calculations are concerned, the more appropriate symmetry groups for description of the zero-order solutions of
and
, (i.e., for the classification of the physically correct nuclear-rotation-vibrational states) are the finite groups
C(M) or
D(M). Ref. [
2] describes the extensions to the infinitely large extended molecular groups
C (EM) and
D (EM) [
9] isomorphic to the point groups
C and
D [
2], respectively, which we discuss further below. The symmetry group methods have proved to be important when solving the corresponding nuclear motion problems, see e.g., Chubb et al. [
10].
The disadvantages of the linear
approach motivated the development of an alternative treatment for all chain molecules (linear, quasi-linear, and bent) 50 years ago [
11] (also see Refs. [
12,
13,
14,
15]), based on
vibrational coordinates and three rotational (Euler) angles, which is more typical for bent systems. The KEO is singular at the linear configuration (as well as the zero-order Hamiltonians) and, therefore, requires specially designed basis functions to resolve this singularity. The
vibrational coordinates have the advantage of being chemically intuitive when describing the molecular PES.
The main trick to resolve the singularity in the KEO is to construct the corresponding bending basis functions to have the correct asymptotic behaviour when the molecule approaches the linear configuration; see, for example, the discussion by Jensen [
13]. This is achieved by including the rotation about the
a axis in
for a linear molecule. The corresponding ro-vibrational basis set becomes dependent on the projections of the vibrational and rotational angular momenta (in units of
ℏ) onto the molecular axis
a (
k and
ℓ, respectively), subject to the constraint
[
4,
16,
17,
18]. Among the popular choices of the bending basis functions with correct behaviour are the associated Legendre polynomials [
12,
19]
, with
v and
ℓ as the corresponding vibrational (bending) quantum numbers.
The vibrational basis functions are effectively wavefunctions of a bent molecule and, therefore, they do not reflect the symmetry properties of a linear molecule, C or D. Indeed, for a centrosymmetric molecule, such as CO they transform as D(M) (isomorphic to C) and only span four irreducible representations (irreps), , , , and . For a non-centrosymmetric linear molecule, such as HCN, they transform as C(M) (isomorphic to C) and they span two irreducible representations, and . The vibrational basis set is orthogonal in ℓ due to the constraint .
This should be compared to the infinitely versatile symmetries that were obtained when the same molecule is treated as linear via
D (EM) or
C (EM) [
9] with their much more detailed options for classification of the rotational and vibrational wavefunctions. In these groups, the vibrational and rotational basis functions span irreducible representations of
C or
D and transform as
,
,
,
, ... for
ℓ= 0, 1, 2, 3, ... and
k= 0, 1, 2, 3, respectively. Moreover, their intrinsic properties offer efficient tools of associated irreducible representations of molecular symmetry groups when constructing the Hamiltonian matrix elements or building symmetry adapted ro-vibrational basis functions, which are not available for the bent molecule approaches.
These rich properties of the infinite symmetry groups have motivated us to extend the molecular symmetry D(M) to an infinite group with the group structure that is similar to D (or, analogously, extend C(M) to an infinite group with a group structure that is similar to C for a non-centro-symmetric linear molecule). The corresponding irreducible representations , , , ... (ℓ > 0, or k > 0, respectively) are doubly degenerate and, therefore, are not directly suitable for this purpose. Therefore, the aim of these extensions is to obtain a more flexible classification of the rotational and vibrational wavefunctions than that provided by D(M) and C(M), allowing for non-degenerate irreps, as required for bent molecules. The corresponding extended symmetries do not necessarily have to have any physical meaning or to be connected with energy conservation, as in the case of traditional "true” molecular symmetry groups. Instead, a group is constructed in such a way, so that the irreps have properties that are formalised from the outset, rather than the behaviour of the irreps being a result of the symmetry of the molecule itself. Thus, any "artificial” group, constructed in this way, which satisfies the standard group theorems and fulfills the purpose of being directly associated with the classification will do. The approach described here, then, is, in effect, a reversal of the standard procedure where the properties of the basis set are based on the symmetry group of the molecule. Here, we design the symmetry group to fulfill the required structure of the ro-vibrational basis functions and the matrix elements of the Hamiltonian.
To this end, in the present work we introduce a formally infinite molecular symmetry (MS) group by extending D(M) [C(M)] with some artificial irreps with the four [two] ‘physical’ irreps , , and [ and ] retained. The group is referred to as D(AEM) [C(AEM)], where AEM stands for ‘artificial extended molecular’ (group). We will also introduce a finite analogy D(AEM) [C(AEM)], where n is depends on the value of , and use it to classify the vibrational and rotational basis functions of the bent ( approach) similar to the classification used for linear, D[C] (EM)-based systems.
The new symmetry group
D(AEM) [
C(AEM)] has been implemented in the variational program
TROVE (Theoretical ROVibrational Energies) [
20,
21], with the aim to facilitate the computation of vibrational energies and wavefunctions for linear molecules.
TROVE is a general, efficient computer program for simulating, by variational methods, hot rotation-vibration spectra of small- to medium-sized polyatomic molecules of arbitrary structure. It has been applied to a large number of polyatomic species [
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41] with various symmetry types [
2], such as
C,
C,
C(M),
D(M),
T(M),
G etc.
TROVE is one of the main tools of the ExoMol project [
42,
43]. Updates of
TROVE have been recently reported in Refs. [
19,
43,
44], with the most recent extension [
19] being the implementation in
TROVE of the
methodology with a singularity resolution based on the inclusion of interaction between the bending motions and the rotation about the
a axis. Additionally, the adaptation of linear-molecule symmetry that was described in Ref. [
10] was implemented. The theory and programming work reported in Ref. [
19] and in the present work have already been employed for generating hot molecular line lists for SiO
[
45] and CO
[
46] as well as to produce a quadrupole spectrum of H
O [
47].
TROVE uses an automatic approach for constructing a symmetry-adapted basis set to be used in setting up a matrix representation of the molecular rotation-vibration Hamiltonian [
48], in the present case for
D(AEM) [
C(AEM)].
1.1. Symmetry Properties of the Linear Molecule () Basis Functions: D(EM)
In order to compute the
vibrational energies for a linear molecule, we solve the Schrödinger equation for the Hamiltonian
, which involves the bending and stretching motions and, as explained in Introduction, the rotational motion about the
a axis. In general, an
N-atomic chain molecule has
stretching vibrations, one for each bond, and
bending vibrations, one at each “inner” nucleus; there is no bending motion at the two “terminal” nuclei. It is well known (see, for example, Ref. [
2]) that, if we transform
to depend on normal coordinates
for non-degenerate vibrations and
for doubly degenerate vibrations, and we express the Born–Oppenheimer potential energy function in
in the harmonic approximation, i.e., as a second-order polynomial in the normal coordinates
Q, the eigenfunctions of
are products of harmonic oscillator functions
Each of the
stretching vibrations is described by a one-dimensional harmonic oscillator eigenfunction
(
i= 1, 2, 3, ...,
), and each of the
bending vibrations is described by a two-dimensional harmonic oscillator eigenfunction
(
j=
N,
, ...,
). Here,
is the corresponding vibrational angular momentum quantum number and
and
are the degenerate vibrational modes that are often represented by polar coordinates
and
via (
=
(see Chapter 11.3 of Ref. [
2]). These two-dimensional (2D) bending functions
transform according to the irreps of the MS group
D (EM) [
9] (see
Table 1), with
identifying the corresponding irrep
,
,
, …, for
, 1, 2, …, respectively, and
spanning
. When combined with the rotational, rigid-rotor, basis function
, the constraint
is used, where
ℓ is the total vibrational angular momentum (in units of
ℏ) of the product-type vibrational basis. Here,
J is the total angular momentum,
is the projection of the angular momentum on the molecular
z (=
a) axis and
is the projection on the laboratory-fixed
Z axis. The rotational basis functions
transform according to the operations of
D (EM), with the similar mapping between the irreps and the rotational quantum number
k, as found for
ℓ:
(
),
(
),
(
), etc. The correlation between the
D (EM) [
9] irreps and the quantum numbers
ℓ,
k is very useful, as it enables the use of powerful group symmetry tools for classifying and even constructing the ro-vibrational (bending) basis functions, with the vibrational basis function in Equation (
1) now given by:
The two bending modes
and
have identical harmonic vibrational wavenumbers,
=
. As discussed above, both the vibrational
and rotational
basis functions of the linear molecule CO
are described by the irreps of
D (EM), and they are fully defined by the corresponding values of
and
k shown in
Table 1. The
ℓ-dependent vibrational basis functions span the
g-type and
u-type representations, while the
k-dependent rotational functions can only be of
g (i.e., do not change upon the
transformation). These functions are linked via the condition
. The ro-vibrational eigenfunctions of a linear XY
type molecule can only transform as one of the four irreps of
D(M) that are listed in
Table 2 (in fact, due to the zero-nuclear spins of C and O, the ro-vibrational wavefunctions of CO
can only span two representations,
and
) . Therefore, physically meaningful combinations of
and rotational
from their direct products can only span these four irreps. These properties can be efficiently explored by the efficient symmetry group algebra via projection and symmetry reduction techniques, as was shown, e.g., in Chubb et al. [
10].
1.2. Symmetry Properties of the Bent Basis Functions
In the
(bent molecule) framework applied to a linear molecule, the initial treatment is based on the ‘zero-order’ Hamiltonian
where, now,
describes the end-over-end rotation only while the rotation about the
a axis is included in
, since
a-axis rotation only takes place when the molecule bends out of the linear equilibrium configuration. For a linear centrosymmetric triatomic molecule, a vibrational basis function is given by
where
measures the bending displacement from the linear equilibrium configuration and the stretching vibrations are described by the bond lengths
and
. The bending function
shown in Equation (
3) plays the central role in the
formalism via two main properties: (i) it is constructed to possess the correct dependence at
to resolve the singularity of the KEO [
13,
19]:
and (ii) it is linked to the rotational basis functions
via
as a general property of the basis sets of linear polyatomic molecules with the
Sayvetz condition k =
ℓ =
[
16] (also see Ref. [
10]).
The harmonic-oscillator basis functions
in Equation (
2), used in the
approach, are orthogonal because they are eigenfunctions for the vibrational angular momentum operator
with different eigenvalues
. In the calculations described here for triatomic molecules, the bending basis functions
in Equation (
3) (which only depend on
and not on
) are typically generated by Numerov–Cooley numerical integration [
49,
50,
51], for example, Section 6 of Jensen [
13]) with the value of
k =
entering as a parameter. Two such functions with different values of
, but the same values of
are orthogonal, the orthogonality resulting from integration over
. In order to generate basis functions for the
calculation, we multiply the vibrational basis functions of Equation (
3) by the
-dependent functions
and the resulting complete basis functions
are generally orthogonal, the orthogonality being manifested by the integration over
for functions with different values of
and over
for functions with the same value of
, but different values of
. On a practical level, it is discussed for a triatomic molecule in Section 6 of Jensen [
13] that
and it is recognised that the factor
is also contained in
, where it is written as
. This identity is taken into account when forming a harmonic-oscillator/rigid-rotor basis set for diagonalizing
; the harmonic-oscillator functions in Equation (
2) are not multiplied by the complete
, but only by the
-dependent functions
from Equation (
5).
The purely vibrational part of the linear-molecule Hamiltonian,
, commutes with
and, so,
(in general
ℓ) is a good quantum number for its eigenfunctions; they can be labelled by
(
ℓ). The part of the kinetic energy operator that describes end-over-end rotation about the
b and
c axes, however, mixes basis functions with different
ℓ-values; it gives rise to non-vanishing off-diagonal matrix elements with
≤ 2. It is interesting to note that, in the case of the
linear molecule approach (
Section 1.1), the orthogonality between basis functions with different
ℓ values is automatically satisfied due to different symmetries (irreps) that correspond to different values of
ℓ. The symmetry properties of the wavefunctions of linear molecules under the
(bent) approach are much less informative than those under the
approach. In the case of a centrosymmetric triatomic, the
approach employs
D(M) with four irreps (see
Table 2) in contrast to the irreps of the infinite group
D (EM). Moreover, the bending wavefunctions
can only be of
symmetry of
D(M). Yet, the property of orthogonality between basis functions with different
ℓ-values is very similar to the orthogonality that arises from different irreps being generated by the wavefunctions in Equation (
2) in the
treatment.
In the present work, we investigate whether the
D(M) symmetry description of the basis functions in Equation (
3), naturally arising for the
case, can be extended in such a manner that it becomes similar to the much more detailed description, in terms of
D (EM), obtained for the
case. In particular, we would like a description in which basis functions with different
ℓ-values span different irreps so that we can take advantage of the specific properties of the linear-molecule basis set functions. For example, the vibrational Hamiltonian matrix, constructed in an extended-symmetry-adapted basis set, will automatically be block diagonal in
ℓ and thus its construction and diagonalization can be optimally cost-effective. There exists no “true” molecular symmetry group, based on permutation-inversion symmetry, with
ℓ-dependent irreps. Thus, our theory does not automatically yield such irreps. However, in the following, we show that is possible to introduce an “artificial” symmetry group that is based on the property of cyclic groups of prime-order that produces the desired
ℓ-dependent irreps. By analogy to the extended group
D (EM), the new artificially extended group will be referenced as
D(AEM), where AEM stands for ‘artificial extended molecular’.
1.3. “Artificial” Molecular Group Symmetry D(AEM) for Centrosymmetric Triatomic Molecules
We define the finite artificial extended MS group
D(AEM), as follows
where
Z is the cyclic group of order 2 and, therefore, it consists of the set
with addition modulo 2. The integer
n depends on the value of
. Taking the limit
defines the infinite
D(AEM) group. Note that this is a countable infinity, as opposed to the group
D (EM). The vibrational
and rotational
basis set functions of differing
ℓ (or
k) values are assigned to different irreps of this group.
The group
D(AEM) must fulfil certain conditions necessary for our purposes. First, all irreps should be one-dimensional and, for simplicity, real. The former condition is so that each irrep of
D(AEM) is correlated with one irrep of
D(M). The irreps of
D(AEM) will be labelled as
C irreps with an extra superscript (see
Table 2), e.g.,
. The bending function
or rotational function
, if they transform as irrep
in
C, would be assigned to
or
, respectively. For example, a vibrational function with
and transforming as
in
C would be assigned the symmetry
in the
D(AEM).
From D(AEM) and D(AEM), we select four elements and match each with a C element. Subsequently, the characters of those elements for each 0-superscripted irrep should be the same as the corresponding C irrep. When combining a bending function that transforms as with a rotational function which transforms as , their product should transform as for some if . If , then they should transform as . For example, should be .
We illustrate these rules with the group
D(AEM) in
Table 3 for the characters and
Table 4 for the multiplication table of the irreps.
These properties are intended to allow us to assign each vibrational basis function to an irrep, which depends on ℓ, as discussed above. It will also combine k and ℓ basis set functions correctly, so that the condition of can be imposed by giving an irrep with superscript ≠ 0 a statistical weight factor of 0.
In variational calculations utilising finite basis sets, it is practical to use the finite analogy
D(AEM) of the infinite group
D, see Chubb et al. [
10]. Accordingly, the practical procedure introduced and developed here involves
D(AEM), rather than
D(AEM). Here
n is some integer value defined to cover the highest excitations of
in the bending
and rotational
functions.
To show that
D(AEM) has the required properties, and it is effectively the only group that does, first note that, since the irreps are one-dimensional, the group is abelian, and, thus, by the fundamental theorem of finite abelian groups [
52], can be expressed as a direct product of cyclic groups of prime order, i.e.,
where
is prime and the
consists of the set
with addition modulo
for
. However, the cyclic group
has a representation where element
i is associated with
, which is only real for all
i if
, thus our group can only be of the form
for some
n. For a given
(or the same
), we must have at least
representations (four for each
ℓ including
.
D(AEM) has
representations, hence
where ⌈⌉ rounds up to the nearest integer. In order to show that the irrep product properties hold, first consider
Z. It has character table shown in
Table 5.
In the following, we omit labelling the elements and the irreps, and simply write the character table as a matrix. Then
Z is written
The character table of a direct product of two groups can then be found by taking an outer product of a matrix. This is defined by
Now,
C =
Z⊗
Z. Hence,
D(AEM) works for
, where we label the 0-superscripted irreps of
D(AEM) in the same order as
Table 3. We call the
D(AEM) character table matrix
G, i.e.,
and say that this has the property that
in that for any irreps of
and
of
D(AEM), the result of
is another irrep in
D(AEM). If we now consider the case
, we have the character matrix
This is a more succinct way to write
Table 3. The first row containing four irreps is 0-superscripted, labelled as in
Table 3. The next four are 1-superscripted, in the same order as the corresponding 0-superscripted irreps, and so on. With this construction, for any two irreps of
and
of
D(AEM), if they have the same superscript,
will cancel the −s of the
Gs and the result will be an irrep of the 0-superscripted block. Moreover, this will be the same irrep one would obtain had one used the
C group. If the irreps do not have the same superscript, then we will have
type multiplication whose result will be an irrep not in the 0-superscripted block. We can see that this easily generalises to any
. This concludes the definition of
D(AEM).
Table 6 describes how the characters for
D(AEM) are defined for an arbitrary
n, although, in practice, one would use the outer product formulation to build the character table.
The introduced structure is similar to that of Extended molecular symmetry groups
G(EM), which are used in the ro-vibrational problems of non-rigid molecules, such as hydrogen peroxide (H
2O
2) [
34] and ethane (C
2H
6) [
53].
In these cases, the group structure is
while
D(AEM) is given by
which clarifies how this sort of construction would work for any arbitrary group
G:
The basis functions for these types of molecules also contain artificial, unphysical irreps. For example, the rotational or vibrational wavefunctions of
G(EM), the MS group of ethane C
H
, can transform as odd or even under
, labelled as
s and
d, respectively. Likewise, the full rotation-vibrational wavefunction can only be of
s type and the artificial
d types must be eliminated [
53].
The artificial symmetry group that has been introduced here has been implemented in the existing structure of the TROVE program, as illustrated below.
2. Example of D(AEM) in Ro-Vibrational Calculations of CO Energies with TROVE
Let us consider a ro-vibrational basis set of a centrosymmetric linear triatomic (CO
) in a symmetry adapted form (compare to Equation (
3)) and a bisecting embedding [
19]:
where
s are the irreps,
,
is are function labels,
and
are the equivalent stretches, and
with
as an inter-bond angle.
Assuming a small rotational problem with
, we need
, and, hence,
(Equation (
9)). There are
irreps in the group
D(AEM), but our basis functions can be only one of (for
)
irreps, and the remainder is not utilised. However, they are still necessary when combining basis functions, as was illustrated in
Table 4.
The procedure works as follows: initially, the reduced Hamiltonian matrix (see
Appendix A for details) for the bending motion is manually made block diagonal in
ℓ and the whole matrix is diagonalized. In principle, TROVE could treat these blocks separately and diagonalize each individually. However, in practice, the dimension is small enough that diagonalizing the entire matrix is computationally acceptable.
In order to obtain symmetrized (in
D(AEM)) eigenfunctions for both the bending and stretching Hamiltonians, the standard TROVE symmetrisation procedure is applied, as described in [
48]. The resulting irreps are the 0-superscripted ones of
D(AEM) and they correspond to the
C(M) ones. To correctly symmetrise this way, the operation
, using the notation of
Table 3, has the same behaviour on the coordinates as operation
.
Once 0-superscripted irreps are obtained, the rotational and bending wavefunctions are reassigned to the appropriate k- or ℓ-superscripted irrep.
In our example, . The same polyad value is used for the symmetrized basis set.
Table 7 lists a few vibrational energies with their symmetry assignment in
D(AEM). The irreps that are generated by the bending basis functions can only be
, and they are assigned to
for a given
ℓ. In this case,
. The stretching functions are all assigned as a 0-superscripted irrep. This ensures that, when they are combined with the bending function, the “base” irrep obtained (i.e., the letter and subscript of the irrep label) would have been the same if we had used the
C group. The only 0-superscripted irrep is the one where
, as expected, and only this would be retained from this set.
Table 8 shows the
and
states for the
case. Here, we see that
k ranges from 0 to 2, and the rotational irreps are superscripted accordingly. The combined stretching and bending irreps are the same as their corresponding rotational irrep and, thus, the full functions are of type
or
. It should be noted that our assignment in
Table 7 and
Table 8 of
K,
,
,
ℓ, and
is based on the largest basis set contribution approach and, therefore, can deviate from the experimental assignment of these states as formed from the corresponding analysis using effective Hamiltonian solutions.
3. Conclusions
The artificial symmetry group D(AEM) introduced here was designed to take advantage of the detailed symmetry information that is available in the linear-molecule treatment, even when the bent-molecule approach is used. The construction of D(AEM) is based on the cyclic group of order 2. D(AEM) is shown to have all the properties required: basis functions with different ℓ or k functions (vibrational or rotational, respectively) generate different irreducible representations of D(AEM). This is a natural property for the linear molecule treatment and we can now also employ it for quasi-linear or bent molecules treated in the approach, but using non-degenerate irreps that are required for bent systems.
In this work, we showed that it is possible to build a symmetry group to satisfy a set of criteria required to simplify the basis set and Hamiltonian matrix elements for variational calculations. This is opposite to the usual practice, where the symmetry group, to a large extent, is dictated by the physics of the molecule itself, which, in turn, defines the computational approach. Our main criterion here is to associate the projections ℓ and k with one-dimensional (1D) irreducible representations with the goal of applying a general symmetry-adapted variational approach to the ro-vibrational calculations of a linear triatomic molecule. Thus, the very specific constraints and associated k-structured block-diagonal Hamiltonian matrix, which are typical for the linear molecule calculations, have been formulated in terms of irreducible representation algebra, as implemented in TROVE.
The authors of this paper originally tried an alternative approach that would make use of the Extended Molecular Symmetry (EMS) Groups
C (EMS) and
D(EMS) of Chapter 17 of Ref. [
2]. As discussed above, these groups are obtained by, among other changes, extending the MS groups
C(M) and
C(M) [
2] by artificial operations, such as
, a rotation of an arbitrary angle
about the
a axis. In the resulting EMS groups
C(EM) and
D(EM) the vibrational basis function
generates a
,
,
,
.... Similarly,
generates a
,
,
,
… symmetry for
k= 0, 1, 2, 3, … See Tables 17-1 and 17-2 of Ref. [
2]. The introduction of
C(EM) and
D (EM) was, in principle, motivated by the same ideas that inspired the present work. These groups are more closely related to actual symmetries than the results presented here, since they utilise the near-symmetry of arbitrary rotations about the
a axis; the ideas of the present work are more abstract.
Despite those benefits, the present approach was taken, as it proved challenging to practically utilize C(EM) and D(EM). Moreover, although more physically relevant, the C(EM) and D(EM) have the disadvantage that they involve the degenerate irreps , , .... The double degeneracy is a simple consequence of the fact that any rotation about the a axis exists in two equivalent forms, which we can call clockwise and anti-clockwise, respectively. The 2 × 2 transformation matrices of these irreps are not uniquely determined and more complicated to apply than the characters of non-degenerate representations.