PGOPHER: A program for simulating rotational, vibrational and electronic spectra

TL;DR: The pgopher program as discussed by the authors is a general purpose program for simulating and fitting molecular spectra, particularly the rotational structure, which can handle linear molecules, symmetric top and asymmetric top.

Abstract: The pgopher program is a general purpose program for simulating and fitting molecular spectra, particularly the rotational structure. The current version can handle linear molecules, symmetric tops and asymmetric tops and many possible transitions, both allowed and forbidden, including multiphoton and Raman spectra in addition to the common electric dipole absorptions. Many different interactions can be included in the calculation, including those arising from electron and nuclear spin, and external electric and magnetic fields. Multiple states and interactions between them can also be accounted for, limited only by available memory. Fitting of experimental data can be to line positions (in many common formats), intensities or band contours and the parameters determined can be level populations as well as rotational constants. pgopher is provided with a powerful and flexible graphical user interface to simplify many of the tasks required in simulating, understanding and fitting molecular spectra, including Fortrat diagrams and energy level plots in addition to overlaying experimental and simulated spectra. The program is open source, and can be compiled with open source tools. This paper provides a formal description of the operation of version 9.1.

Summary

1 Introduction

• Perhaps the key feature of rotationally resolved molecular spectra is the immense amount of information on the molecule and its environment that can be extracted from spectroscopic measurements.
• The necessary downside is that such informative spectra are necessarily complicated, and extracting the information can be a daunting task.
• Its current form has come about as the result of applying it to many different spectroscopic problems and it has thus become useful in a wide range of applications.
• This paper formally describes the internal structure of the program and the algorithms used; as far as possible the program tries to use standard spectroscopic notation and conventions, but there are necessary details that must be specified.

2 Overall Operation

• The underlying structural assumption is that the Hamiltonian is expressed in terms of a series of rotational constants given explicitly for each vibrational state of each electronic state included in the calculation.
• An important optional possibility is a “perturbation” object, which specifies interactions between vibronic states, and can also be used to add non-standard terms to the Hamiltonian for a vibronic state.
• In the absence of perturbations the grouping into manifolds is arbitrary, though the calculation is slightly more efficient if the number of states in any given manifold is minimized and states with no interactions with other states are in their own manifold.
• Other objects are also available for more specialised types of calculation, such as “nucleus” objects under each state which allow hyperfine structure to be simulated.
• For instructions on setting up an object tree readers should refer to the documentation supplied with the program.

2.1 Energy Levels

• The essential structure of the program involves the expansion of the wavefunction for a given rovibrational level, Ψi as a linear combination of basis states, |j>: j i ji jc (1) To calculate the coefficients, the Hamiltonian matrix is set up and diagonalized in this basis.
• This implies no true degeneracy will be encountered, though terms omitted from the Hamiltonian may mean some degeneracies are not lifted.
• More generally the Hamiltonian matrix is scanned for independent sub-blocks with no matrix elements between them as part of the process of setting it up.

2.2 Quantum Number Assignment

• For the purpose of identifying a state m, J, s, i and M are sufficient for internal processing, but other quantum numbers (such as Ω and N for linear molecules) are typically used to specify states, and PGOPHER provides these.
• Not only do these additional quantum numbers aid interpretation, but the eigenvalue number, i, is not well defined in the presence of closely spaced states as small changes in parameter values can change the state order.
• To address this problem the idea of a sub-basis is introduced, defined such that states within a given subbasis have a clear expected energy ordering, and different vibronic states are assigned to different sub-bases.
• Assigning a sub-basis to a final eigenstate is achieved by working out the fractional contribution from a given sub-basis from the sum over the sub-basis of the square of the wavefunction coefficients of the eigenstate.
• A good example of this is hyperfine structure arising from nuclear spin, for which the natural division is to put levels of different angular momentum excluding nuclear spin (J) in different sub-bases, in addition to any other separations.

2.3 Hyperfine Structure

• Hyperfine structure is handled by setting the number of active nuclei at the molecule level, which creates a corresponding set of “nucleus” objects under each state.
• The number of nuclei does not have a hard limit, though multiple nuclei can lead to rather large calculations.
• The assignment of hyperfine quantum numbers has been mentioned above; in general each different set of hyperfine quantum numbers and J is placed in a different sub-basis.
• Nuclei with similar interaction strength (i.e. similar values for the hyperfine constants) may need to use the I12 = I1 + I2 coupling scheme to avoid strong mixing between sub-bases.

2.4 Transition Moments

• Apart from transition energy, the other important ingredient in simulating transitions is the transition intensity, for which the starting point is the transition moment.
• For the most important electric dipole case k = 1 and T1(μ) is the electric dipole moment operator (expressed in a space-fixed frame) and T1(E) is the electric field.
• For magnetic interactions, these become the magnetic dipole and magnetic field, respectively.
• The transition moment is then the matrix element of the kpT term.
• Its value must be specified, along with the origin and rotational constants, as properties of the states involved when setting up the calculation.

2.5 Line Strengths

• As the equation above implies, transition moments depend on M, even in the absence of an external field, but fortunately the M dependence disappears when the intensities are summed over M.
• The second step follows because the M dependence only arises from the Wigner-Eckart theorem as in equation (12) and is independent of all the quantum numbers apart from the total angular momentum and its projection.
• If there is more than one vibronic transition moment contributing then the line strength will depend on their relative signs and magnitudes, and the Hönl-London factor is a less useful quantity.
• The definitions used here are consistent with those of Hansson and Watson.

2.6 Intensities

• Given the line strength for a transition between an upper state, u, and a lower state, l, PGOPHER offers various options (controlled by the IntensityUnits setting) for calculating line intensities.
• For electric dipole transitions the vibronic transition moments are input in units of Debye (1 Debye = 10–21/c C m) and the line strength will therefore be in units of Debye2.
• S is used in this equation, rather than Spol, as it only applies to one photon transitions, and again state degeneracies are included in S rather than in the Boltzmann factors.
• For the HF molecule which has two nonequivalent spin ½ nuclei, the nuclear spin degeneracy of (2I1+1)(2I2+1) = 4 is independent of J, and can be omitted.
• This can be important in certain circumstances, particularly if calculating thermodynamic properties from partition functions.

2.7 Population Distribution and Partition Functions

• An important aspect of intensity calculations is the population function used, and the partition function arising from it.
• Separate vibrational and spin temperatures, Tvib and Tspin, can be specified if required in addition to the main temperature, T, which is essentially a rotational temperature.
• The origin of the calculated energies depends on the Hamiltonian chosen and how the calculation is set up, and the lowest value of Ei may be significantly different from zero.
• In principle all the required vibrational or electronic states could be included in the calculation, but this could lead to an unnecessarily complicated calculation and, in addition, the required constants may not be known.
• Two additional approaches to calculating the partition function are provided.

2.8 Energy Levels in the Presence of an External Field

• The transition moments discussed above are those required when calculating energy levels in the presence of a static external electric or magnetic field.
• For the purposes of the calculation, the current implementation requires that the space-fixed axis that defines M is taken as the direction of any external fields.
• The sum is also restricted to states in the given manifold m, which may require a set-up with all states in the same manifold.
• The resulting matrices are likely to be quite large, but this size of calculation is required for exact calculations for molecules subject to fields required for molecular steering and trapping.
• To calculate transition moments in the presence of an external field the direction of the field(s) corresponding to absorbed or emitted radiation must also be specified.

3 Molecule Types

• The basis set, quantum numbers and Hamiltonian are detailed below for each of the three types of molecules covered here.
• The basis functions all involve rotation matrices to express implicitly the dependence of the rotational wavefunction on the angles between space- and molecule- fixed axis systems; the specific choice made is as described by Brown and Howard[32], which also describes the general method used to evaluate the matrix elements.
• Imaginary operators typically appear when considering mixing between vibronic states, as (for example) in Coriolis interaction between two vibrational states for which the rotational operator is xĴ , yĴ or zĴ .
• For this reason PGOPHER allows the entire wavefunction calculation to be done using complex arithmetic, controlled by the AllowComplex flag.
• There are two considerations that lead to this small difference:.

3.1 Linear Molecules

• For each vibronic state, η, the standard components of the term symbol must be specified including the overall electron spin, S and the vibronic symmetry (Σ+, Σ–, Π, … and g or u if the molecule has a centre of symmetry).
• The lowest J levels, with J < |Λ|+S, require special consideration as there are some missing values of the quantum numbers.
• The choice of omitted spin component is therefore made by considering the normal/inverted test described above, and for non-inverted states F1 is kept and the higher spin components discarded.
• The alternative labelling scheme can be forced for O2 by setting OmegaOrder to Inverted.
• This replaces even powers of N̂ with R̂ throughout, including the operators below.

3.2 Symmetric Tops

• For each vibronic state, η, the overall electron spin, S, and vibronic symmetry must be specified.
• For vibronic states that are degenerate (E symmetry) the size of the basis is typically doubled, and the component of the vibronic state is specified by an l quantum number with values ±1.
• The sub-basis mechanism can be used to assign these quantum numbers.
• (This is controlled by the LimitSearch setting for the manifold.) A1 and A2 become A+ and A– for even J, and vice versa for odd J. Similar notation is occasionally found in the literature – see Tarrago and Nhu[42] and the HITRAN database[43].

3.3 Asymmetric Tops

• The complication of degenerate vibronic states is not possible for asymmetric tops so the l quantum number is not required, but the choice of the axis used to define K is now not obvious.
• In C2v the C2zAxis setting specifies the z' axis of the point group (the C2 axis) and the C2xAxis setting specifies the x' axis of the point group which is normally chosen to be the out of plane axis in planar molecules.
• The variations are summarised in Table 3.
• An additional symmetry notation used in asymmetric tops is ee/eo/oe/oo where the first and second letters specify the whether the Ka or Kc quantum numbers are even or odd respectively.
• These can be mapped to the symmetries specified above, and are used in specifying the statistical weights – the weights are given for levels with ee, eo, oe and oo symmetry for totally symmetric vibronic levels, and the weights are adjusted as required for other vibronic symmetries.

3.4 Matrix elements and adding additional terms to the Hamiltonian

• The specific Hamiltonians described above are only sufficient where interactions between different vibronic states can be ignored, but this is often insufficient, especially for high resolution studies or states above the vibronic ground state.
• The documentation with the program details the possible operators, which include most possible combinations of angular momentum operators for each molecular type.
• The same basic implementation can be used for both vibrational and electronic interactions.
• (The reason for including this in the perturbation section is that all the standard Hamiltonians can be converted to a sum of perturbation objects).
• This is often important as, while PGOPHER is designed to conform to common usage, non-standard Hamiltonians are not unusual in the literature.

5 Fitting Spectra

• If the effective observed values are taken as the difference between the true observations, O, and the values calculated from the current parameter set, y(p) then the standard methods of linear least squares fitting[48] can then be used to estimate the changes in the parameters, Δp, that minimises the sum of residuals squared, |O–y(p+Δp)|2 : Δp = (aTa)–1aT (O–y(p)) (80).
• The singular values are the inverse of the standard deviations of the transformed parameters, p', so a small singular value implies a poorly determined parameter.
• A single fit cycle is then performed, with some of these parameters floated; if not successful an “undo fit” button is available to step the parameters back one or more fit cycles.
• An additional problem with contour fits is that a false minimum in residuals is more likely; an obvious “easy” fit is to increase the linewidth in the simulation to wash out all the structure, and more subtle problems are also possible.
• A command is therefore available to set the increments of selected parameters to a chosen fraction of the standard deviation of the parameter from the previous fit.

6 Conclusions

• The program presented is sufficiently general purpose that it will handle most requirements for simulating and fitting rotational structure in molecular spectra.
• The most significant omissions in the current version are spherical top molecules, and a general way for handling internal rotation.
• The structure of the program allows the former to be added fairly easily, but internal rotation is a more difficult problem.
• In addition to the general purpose nature of the program, a key feature is the interactive set of tools for assigning, fitting and understanding spectra.
• There are alternative approaches to assignment that have been developed recently such as genetic algorithms[54] and the systematic search over many possible assignments used in the AUTOFIT program[55].

Figures

Figure 2 Part of screen shot for simulation of multiple interacting vibronic states in S2. It

Table 1 Definition of the Representation for the axis system for asymmetric tops

Figure 3 Sample symmetry table printout for vibronic states of A1 (first half) and B1 symmetry in C2v.

Figure 1 Minimum sample set of objects for a simulation. The left side shows the generic object types, and the right shows a screenshot from the program set up to simulate the pure rotational spectrum of HF. The “v=0” object contains the rotational constants, and <v=0|T(1)|v=0> the (transition) dipole moment.

Table 3 Symmetry properties for asymmetric tops

Table 2 Phase factors applied when the PhaseAdjust setting is on.

Full text

Western, C. M. (2017). PGOPHER: A Program for Simulating
Rotational, Vibrational and Electronic Spectra.
Journal of Quantitative
Spectroscopy and Radiative Transfer
,
186
, 221-242.
https://doi.org/10.1016/j.jqsrt.2016.04.010
Peer reviewed version
License (if available):
CC BY-NC-ND
Link to published version (if available):
10.1016/j.jqsrt.2016.04.010
Link to publication record in Explore Bristol Research
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This is the author accepted manuscript (AAM). The final published version (version of record) is available online
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1
PGOPHER: A Program for Simulating Rotational, Vibrational and Electronic
Spectra.
Colin M Western
School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, United Kingdom.
C.M.Western@bristol.ac.uk
Abstract
The PGOPHER program is a general purpose program for simulating and fitting molecular
spectra, particularly the rotational structure. The current version can handle linear molecules,
symmetric tops and asymmetric tops and many possible transitions, both allowed and
forbidden, including multiphoton and Raman spectra in addition to the common electric
dipole absorptions. Many different interactions can be included in the calculation, including
those arising from electron and nuclear spin, and external electric and magnetic fields.
Multiple states and interactions between them can also be accounted for, limited only by
available memory. Fitting of experimental data can be to line positions (in many common
formats), intensities or band contours and the parameters determined can be level
populations as well as rotational constants. PGOPHER is provided with a powerful and flexible
graphical user interface to simplify many of the tasks required in simulating, understanding
and fitting molecular spectra, including Fortrat diagrams and energy level plots in addition to
overlaying experimental and simulated spectra. The program is open source, and can be
compiled with open source tools. This paper provides a formal description of the operation
of version 9.1.
Keywords
Molecular spectra; Rotational energy levels; Perturbations; Vibrational Energy Levels;
Hyperfine structure
1 Introduction
Perhaps the key feature of rotationally resolved molecular spectra is the immense
amount of information on the molecule and its environment that can be extracted from
spectroscopic measurements. The necessary downside is that such informative spectra are
necessarily complicated, and extracting the information can be a daunting task. The program
described here, PGOPHER, has been developed as a general purpose tool to assist in this task
by simulating and fitting rotational, vibrational and electronic molecular spectra. The focus is
on an interactive graphical user interface to make simulation and assignment of spectra as
easy as the underlying spectroscopy permits, but it is also available in a command line version
for use in combination with other programs. Its current form has come about as the result of
applying it to many different spectroscopic problems and it has thus become useful in a wide

2
range of applications. This ranges from simple undergraduate spectroscopy practicals where
the rotational constant of CO is determined from a traditional infrared spectrum to complex
cases involving multiple interacting rovibronic states[1], including open shell systems and
nuclear hyperfine structure. It is not the first molecular spectroscopy program to be published
– Pickett’s CALPGM suite[2] has become something of a standard and there are several others
available including ASYTOP[3], ASYROTWIN[4], SPECVIEW[5] and JB95[6]. PGOPHER aims to cover
similar ground, but in a much more general and easy-to-use way.
For many spectroscopic problems much of the required logic used in the handling of
basis sets, energy levels and transitions is independent of the molecular type, and the
program structure reflects this. An object-oriented approach is used, which allows the
molecule-specific part to be restricted to a relatively small part of the program, with much of
the program, including the user interface, written in a general way. There are thus separate
units of the program for linear molecules, symmetric tops and asymmetric tops which are
each outlined below, and these are all concerned with the rotational structure of a particular
vibronic state. A fourth unit, which calculates vibrational structure of electronic states
ignoring rotation, is also available and is covered in the on-line documentation but not
described here as it is a relatively recent addition and has a significantly different structure.
The PGOPHER program has been developed as an open source application, and the
source and executables can be freely downloaded from the website[7]; see also [8, 9] for
permanently deposited versions of the program with a doi. This paper formally describes the
internal structure of the program and the algorithms used; as far as possible the program tries
to use standard spectroscopic notation and conventions, but there are necessary details that
must be specified. Detailed instructions for running the program, and example files are
distributed with the program. The paper is specifically based on version 9.1 of the program[9];
earlier versions are broadly similar, though some features may not be available or are slightly
different, as described in the release notes. Most results are quoted without derivation; see
standard spectroscopic texts [10-17] for the cases where details are not given.
2 Overall Operation
The underlying structural assumption is that the Hamiltonian is expressed in terms of
a series of rotational constants given explicitly for each vibrational state of each electronic
state included in the calculation. The generic term vibronic state is used here, as the
calculation makes no distinction between the electronic and vibrational parts of the
wavefunction. At a minimum the information required for each vibronic state, η, will include
the symmetry, an origin for the state (the energy in the absence of rotational terms) and one
or more rotational constants. The rotational part of the Hamiltonian is taken to include some
small terms that are notionally part of the electronic Hamiltonian, including spin-orbit and
spin-spin coupling and any lifting of vibronic degeneracies, such as lambda-doubling. These
are conventionally included in the rotational Hamiltonian, and indeed accurate energy level
calculations require this.

3
To allow multiple states to be included, the calculation is set up using a series of
objects laid out in a tree structure, as shown in Figure 1. The key object is a “state” object,
which specifies the symmetry and constants of a single vibronic state. One or more of these
are grouped under “manifold” objects which are in turn grouped under “molecule” objects
which are grouped under “species” objects. The intent is that isotopically substituted variants
of a molecule are grouped under a species object so that, for example, the intensity of a
particular transition involves the product of a relative concentration (specified at the species
level) and an abundance (specified at the molecule level). “Transition moment” objects
specify the possible transitions between states which are grouped under “Transition
moments” objects which specify the connected manifolds. At the top level is a “mixture”
object, implying a mixture of several different compounds, each of which has a species object.
The “mixture” object also contains a “simulation” object that contains global parameters such
as the temperature and line width which govern the overall appearance of the simulation. A
minimal set of objects to produce a simulation is shown in Figure 1; any level other than the
top level can have multiple objects if required.
Figure 1 Minimum sample set of objects for a simulation. The left side shows the generic
object types, and the right shows a screenshot from the program set up to simulate the pure
rotational spectrum of HF. The “v=0” object contains the rotational constants, and
<v=0|T(1)|v=0> the (transition) dipole moment.
An important optional possibility is a “perturbation” object, which specifies
interactions between vibronic states, and can also be used to add non-standard terms to the
Hamiltonian for a vibronic state. The perturbation objects are placed under the manifold
containing the states involved and interacting states must therefore be in the same manifold.
The concept of a manifold of states in fact arises out of the requirement to handle
perturbations, and is used to group interacting states together. In the absence of
perturbations the grouping into manifolds is arbitrary, though the calculation is slightly more
efficient if the number of states in any given manifold is minimized and states with no
interactions with other states are in their own manifold. To give a specific example, in
simulating the B-X transition in S
2
[18], the ground electronic state shows no perturbations so
the calculation can be structured so that each vibrational level of the ground state is in its
own manifold. In contrast, the excited B state shows significant interaction with the B" state,
and one vibrational level of the B electronic state can interact with more than one vibrational
level of the B" state, so calculation of the excited states must be set up as a single manifold

4
containing several vibrational levels from both the B and B" states. A fragment of the object
tree required in this case is shown in Figure 2.
Figure 2 Part of screen shot for simulation of multiple interacting vibronic states in S
2
. It
includes 5 vibronic states (B… ) and 6 perturbations (<B…||B…>) between them.
Other objects are also available for more specialised types of calculation, such as
“nucleus” objects under each state which allow hyperfine structure to be simulated. There
are no hard limits on the number of any type of object, though the calculation will be slower
and take more memory as the number of objects is increased. The limiting step for larger
calculations is typically the matrix diagonalization step; to give an indication a model involving
~20 interacting asymmetric top states[19, 20] takes a few seconds on a current desktop
machine. PGOPHER makes use of parallel processing where possible; many of the calculations
split naturally into independent parts, making this reasonably straightforward to implement.
For example, energy levels are typically required for a range of values of total angular
momentum, but calculations for a given value of the total angular momentum can be done
independently from each other, and thus in parallel.
For instructions on setting up an object tree readers should refer to the
documentation supplied with the program. An important consideration is that while most
objects will have many possible settings, most of these can be left at the default values. A
complicated object tree can be built up easily from a simple one by copying and pasting one
or more objects. Given a correctly set up object tree, a variety of spectroscopic calculations
can then be performed, most importantly simulating spectra and comparing and fitting them
to experiment. Multiple experimental spectra can be overlaid on the simulation, with a
separate object tree used to control them. Various supporting tools are also included in the
program, such as calibrating spectra against known transitions, assigning transitions, making
energy level plots, and other tools for showing details of the calculation. The program can
directly handle experimental data and line lists in a wide variety of standard and proprietary
formats, including simple text format, JCAMP-DX [21] spectra and HITRAN[22] line lists.

Frequently asked questions

Q1. What is the limiting step for larger calculations?

The limiting step for larger calculations is typically the matrix diagonalization step; to give an indication a model involving ~20 interacting asymmetric top states[19, 20] takes a few seconds on a current desktop machine. 

Q2. What are the common quantum numbers used to specify states?

For the purpose of identifying a state m, J, s, i and M are sufficient for internalprocessing, but other quantum numbers (such as Ω and N for linear molecules) are typically used to specify states, and PGOPHER provides these. 

Q3. What is the way to identify the exact calculation performed as part of published work?

Comparing matrix elements ofdifferent Hamiltonians is an effective way of identifying the exact calculation performed as part of published work. 

Q4. What are the conventional rotational quantum numbers for asymmetric tops?

The conventional rotational quantum numbers for asymmetric tops are Ka and Kc,corresponding to |K| along the a and c axes in the prolate and oblate limits respectively. 

Q5. What is the function used to work out the expressions for the matrix elements used for each parameter?

A basic symbolic algebra package is included in the program and is used to work out and display expressions for the matrix elements used for each parameter as part of the header for each fit or line list. 

Q6. What formats can be used to handle experimental data and line lists?

The program can directly handle experimental data and line lists in a wide variety of standard and proprietary formats, including simple text format, JCAMP-DX [21] spectra and HITRAN[22] line lists. 

Q7. What is the common case where this does not hold?

The only known common case where this does not hold is for the lowest level in the X3Σ−g state of O2 which has J=0, N=1 and e symmetry and which PGOPHER will label with F1 by default but F3 is also used in the literature. 

Q8. What is the way to work on partially resolved structures?

An effective method of working on partially resolved structure is to start with a standard line position fit, and then refine with a contour fit as this gives a good starting position for the contour fit. 

Q9. What can be done to simulate a spectroscopic experiment?

Given a correctly set up object tree, a variety of spectroscopic calculations can then be performed, most importantly simulating spectra and comparing and fitting them to experiment. 

Q10. What is the kqT' expression for the spherical tens?

In this expression kqT' corresponds to the transition moment in thebody-fixed axis system, and is an integral over electronic and vibrational coordinates only, and is thus a simple numerical property of the two vibronic states involved in the transition. 

Q11. What is the energy ordering scheme used for assignment?

For all of these quantum numbers the energy ordering scheme used for assignment means the values assigned are stable even if, as will commonly be the case, one or more of them is a poor quantum number. 

Q12. What is the kqT' of the Hönl-London factor?

The Hönl-London factor contains only the rotational quantum number dependence of the transition intensity, so for simple systems the Hönl-London factor can be calculated from the above by setting the vibronic transition moment, kqT' to one. 

Q13. What is the key feature of rotationally resolved molecular spectra?

Perhaps the key feature of rotationally resolved molecular spectra is the immenseamount of information on the molecule and its environment that can be extracted from spectroscopic measurements. 

Q14. What is the function used to step the parameters back?

A single fit cycle is then performed, with some of these parameters floated; if not successful an “undo fit” button is available to step the parameters back one or more fit cycles. 

Q15. What is the symmetry notation used in asymmetric tops?

An additional symmetry notation used in asymmetric tops is ee/eo/oe/oo where thefirst and second letters specify the whether the Ka or Kc quantum numbers are even or odd respectively.