Bachelor Thesis Scheikunde
A Python approach to the calculation of electronic
transitions of PAH ions
door
Emma Carels
22 juli 2020
Studentnummer 11427523 OnderzoeksinstituutVan 't Hoff Institute for Molecular Sciences
Onderzoeksgroep Molecular Photonics Verantwoordelijk docent Dr. Annemieke Petrignani Begeleider Dr. Alessandra Candian
Samenvatting
Als we met telescopen verre sterren en sterrenstelsels bekijken valt er iets op: er mist licht. In het continue spectrum dat er van de sterren af zou moeten komen blijken banden te missen. Deze banden heten Di↵use Interstellar Bands (DIBs), en worden veroorzaakt door moleculen die in de ruimte zweven en licht absorberen. Welke moleculen dat precies zijn, is een groot mysterie. Wat wel bekend is, is dat dit grote moleculen moeten zijn die voornamelijk van koolstof gemaakt zijn, koolstof is namelijk een van de meest voorkomende elementen in de ruimte. Voorbeelden van zulke grote koolstof moleculen zijn polycyclische aromatische koolwatersto↵en, vaak afgekort tot PAKs (of PAHs in het Engels). Dit zijn moleculen die bestaan uit allemaal benzeen ringen die op verschillende manieren aan elkaar geplakt zijn. PAKs zijn er dus in heel veel verschillende vormen en kunnen heel erg groot zijn.
Doordat PAKs zo groot kunnen zijn, is het heel moeilijk om deze moleculen in het lab te bestuderen. Om alsnog meer over deze moleculen te weten te komen, kunnen we ze met de computer bestuderen. Met speciale computer programma’s kunnen we eigenschappen van moleculen voorspellen. Een van deze computer programma’s heet LCOAO, en is door Jens Spanget-Larsen gemaakt in 1983. Het programma is in een hele oude programmeertaal geschreven, en om het programma te behouden en te verbeteren voor de toekomst wordt dit programma vertaalt naar een nieuwe programmeertaal. Tegelijkertijd, worden er met de oude code nog altijd voorspellingen gedaan voor verschillende PAKs.
Abstract
When observing stars absorption bands are detected. These bands are called di↵use interstellar bands (DIBs), and the origin of many of these bands remains unsolved. Computational methods can be used to aid finding DIB carriers. The linear combination of orthogonal atomic orbitals (LCOAO) method is considered by exploring its inner workings and its electronic transition predictions for three sets radical PAH cations: the acenes, the rylenes and a diverse set of PAHs.
The acenes and the rylenes follow linear trends: the acenes folow an upward trend for the D1transition and a downward trend for the D2transition, the rylenes follow a downward trend in both the D1and the D2transitions. The diverse set of PAHs does not follow a trend for the D1and D2transitions, the structure of the PAH is of more importance to the electronic transition than the amount of rings. Most D1 D0transitions are predicted to be very weak and all fall in the IR region with the exception of perylene. Of the second transitions only naphthalene, anthracene and tetracene of the acenes, perylene , terrylene and quaterrylene of the rylenes, and pyrene, benzo[a]pyrene, and benzo[g,h,i]perylene, of the diverse PAHs are predicted to be in the visible spectrum, and are therefore possible matches to be DIB carriers.
When compared to experimental values from literature the LCOAO predictions show an error of up to 2200 cm 1(16%), and when compared to time-dependent density functional theory (TD-DFT), the LCOAO predictions were found to be less accurate than TD-DFT.
Contents
1 Introduction 5
2 Computational theories and methods 7
3 A Python implementation of the LCOAO method 10
3.1 The Fortran implementation of the LCOAO program . . . 10
3.2 Translation to a Python implementation . . . 12
3.2.1 Advantages and disadvantages of Fortran and Python . . . 12
3.2.2 Translating the subroutines and functions . . . 14
4 Predicting electronic transitions of PAH cations with LCOAO 17 4.1 Radical cations of PAHs . . . 17
4.2 The D1and D2electronic transitions of PAH ions . . . 18
4.3 Predictive power of LCOAO . . . 20
4.3.1 Comparison with experimental data . . . 20
4.3.2 Comparison with TD-DFT . . . 23
4.4 Higher excited states of phenanthrene and pyrene . . . 24
4.5 Discussion and conclusions . . . 26
5 Conclusions and outlook 26
1
Introduction
The space between stars and galaxies was thought to be empty up until 1937, when the first atomic and molecular lines were detected.1, 2 After those first lines, more than 500 of such absorption bands have been discovered, which are called the di↵use interstellar bands (DIBs). Over 200 distinct molecules have been detected in the interstellar medium (ISM), yet these DIBs still remain mostly unsolved.3In an e↵ort to further the study of DIBs, and possibly assign certain bands to specific molecules, ESO started a new initiative: The ESO Di↵use Interstellar Bands Large Exploration Survey, or EDIBLES.4Its purpose is to extend DIB surveys and make a step forward in the characterisation of physical and chemical conditions in the ISM, and then to possibly reverse-engineer properties of the molecules that cause DIBs. A visual representation of the DIBs is given in Figure 1.
Å
Å
Å)
Å
Figure 1: Di↵use interstellar bands are observed in the visible spectrum, taken from Ref.5. It stands to reason that carbon containing molecules are possible carriers of these bands, as carbon is one of the most common elements in space. In 2015 the first DIBS were assigned to a specific molecule: C+60, also known as buckminsterfullerene or buckyball.6–8Two strong bands of the excitation spectrum of C+60 were shown to coincide with two known DIBs, and later three more matches were found with DIBs. These remain the only assignments for known DIBs, but there is a common agreement that other DIBs could be caused by large carbon-based molecules like polycyclic aromatic hydrocarbons and fullerenes.9
The carbon species this report considers are polycyclic aromatic hydrocarbons, or PAHs.More specifi-cally, the radical cations of PAHs. Big molecules like PAHs can be very difficult to synthesise and are difficult to bring into the gas phase for spectroscopy. This increases the costs of researching PAHs as possible DIB carriers. In this case, computational methods can provide more insight into PAHs being possible DIB carriers. By using computational methods to predict electronic transitions of PAHs, certain PAHs can be ruled out as carriers, while others may prove promising candidates. The most widely used method for predicting is time-dependent density functional theory (TD-DFT).10TD-DFT does however have an inaccuracy of up to 15%, which is too high to clearly discern if a PAH is a possible DIB carrier. A di↵erent method has been used to compute the electronic transitions of flat neutral PAHs: the linear combination of orthogonal atomic orbitals (LCOAO) method. This method was implemented in Fortran by Jens Spanget-Larsen in 1983.11, 12This method has been used by Roeterdink et al. to an accuracy of 1% for neutral PAHs.13 This report considers how the LCOAO program is set up and if the same order of accuracy is achieved for the radical cations of PAHs as is achieved for neutral PAHs. First the workflow of the LCOAO method and some of the underlying theory of computational methods will be
explored, then the inner workings of the LCOAO program itself will be considered and the conversion to a Python implementation of the LCOAO method is discussed. Lastly, the results from computations using the LCOAO method of a group of radical PAH cations will be presented and discussed.
2
Computational theories and methods
There are many di↵erent possible computational methods, all of which depend on mostly the same theoretical principals. The computational method that was used for this paper is the LCOAO method. The steps involved in this method as well as some of the theoretical background the LCOAO method, like many other computational methods, is based on are explored in this chapter.
The steps of the procedure for a LCOAO calculation are presented in Figure 2. The first step of a calculation with the LCOAO method is to draw the molecule with Avogadro, or similar software. The coordinates are then extracted and added to an input file for Gaussian 16 to optimise the ground state geometry of the molecule. When the geometry is optimised the new coordinates are transferred to a second input file, this is the input file for the LCOAO program. The LCOAO program is then run, and an output file containing, among other things, the electronic transitions is obtained.
Draw molecule in Avogadro
Coordinates of molecule
Create input file for Gaussian 16
Geometry optimisation in Gaussian (B3LYP/6-31G(d))
Coordinates optimised groundstate geometry
Create input file for LCOAO
LCOAO Calculation
Predictions for elec-tronic transitions
Figure 2: The steps involved in executing a LCOAO calculation.
As it is with any problem within computational chemistry, the fundamental goal with these calculations is to solve the time-independent Schr¨odinger equation for a given problem or molecule:
H = E (1)
electronic Schr¨odinger equation is used:
H (r, R) = E(R) (r, R) (2)
Where R are the locations of the nuclei in molecule, and r the electronic coordinates. The Hamiltonian for the electronic Schr¨odinger equation is as follows:
H = ~ 2 2me n X i r2 i n X i N X I ZIe2 4⇡"0rIi+ 1 2 n X ij e2 4⇡"0rij (3)
Within computational chemistry there are two main paths to solving this Schr¨odinger equation: ab initio calculations, and semi-emperical calculations. Ab initio calculation is purely based on theory, where the electronic wavefunction is approximated by a quantum mechanical model, to which only the fundamental constants and atomic numbers of the nuclei are input. As only fundamental values are used for the calculation, the accuracy of the calculation depends mainly on the choice of model for the wavefunction. Due to this, calculations of larger molecules or systems can become computationally expensive, as for an accurate result a more complex model is needed. Semi-emperical methods are therefore often used for larger molecules, as these use a simpler model that is then adjusted using parameters acquired from experimental data. The LCOAO method is a semi-emperical method, that uses parameters derived from experimental data from Benzene and Naphthalene.12
The starting point of the LCOAO method is the Hartree-Fock self-consistent field method. The first step of this method is to neglect the electron-electron potential, which depends on the electron-electron separations (the third term of Equation 3), as an approximation to the true wavefunction ( ). This simplifies Equation 2 to the following:
H0 0= E0 0 H0= n X i=1
hi (4)
Where hi is the core Hamiltonian for electron i. This simplifies the calculation, as the n-electron equation can now be split into n one-electron equations and therefore the wavefunction ( 0) can also be written as a product of n one-electron wavefunctions ( 0
a(i)) with corresponding energies Ea0(Equations 5 and 6).
hi 0a(i) = Ea0 0a(i) (5) 0= n Y i 0 a(i) (6)
Each of the one-electron wavefunctions can be expressed as a linear combination of a set of basis functions ✓j: 0 a(i) = M X j=1 cja✓j (7)
The set basis functions that is used for the LCOAO method is the set of L¨owdin orthogonal atomic orbitals (OAO). By describing the wavefunctions in the set of basis functions it reduces the calculation of wavefunctions to a calculation of just the coefficients cja. In the LCOAO method, these coefficients are computed by solving the matrix equation:
Where F is the Fock matrix with the elements: Fij=
Z
✓i⇤ f✓jdr (9)
With f the Fock operator. c is an M⇥ M matrix containing the coefficients cja. S is the overlap matrix, with elements:
Sij= Z
✓i⇤ ✓jdr (10)
Lastly, " is a diagonal M⇥ M matrix containing the orbital energies "a. Equation 8 only has a non-trivial solution if it satisfies this secular equation:
det|F "aS| = 0 (11)
It is not possible to solve this equation directly, as it involves the Fock operator. Therefore, a self-consistent field approach is used to compute the coefficients cja. The LCOAO method indeed uses an SCF approach to calculate these coefficients.
3
A Python implementation of the LCOAO method
Computational calculations on the excited states of PAHs and their ionic counterparts is of great importance for solving the mystery the DIBs present. It can be used to exclude certain PAHs as possible DIB carriers on account of their transitions being outside of the visible range, as well as give an indication of the energy for the transitions. Such predictions can aid the scanning for the transitions in laboratory settings, as it can reduce the range of wavelengths that have to be scanned in search of the electronic transitions. Finding an accurate way of predicting electronic transitions of PAHs, and other molecules, can tremendously aid the search for possible DIB carriers by filtering out the least likely candidates. This chapter focuses on one such computational method for the predictions of electronic transitions, the LCOAO method and the particular workings of the Fortran implementation of it. It then goes onto working towards a Python implementation of the program, and the di↵erences, the advantages, and the disadvantages of the two programming languages.
3.1
The Fortran implementation of the LCOAO program
The LCOAO program was developed by Jens Spanget-Larsen in 1983.12 The program has had updates in 2001, 2005 and 2012. The programming language that was used was Fortran, which was developed in the 1950’s.
The LCOAO code consists of 27 subroutines, three functions and a main program: • Subroutines: – SHIFT – STORE – LOAD – PRINT – DMOM – READER – AUFBAU – SPRT1 – GMATR – SMATR – F0MATR – LOWDIN – SCF – PI – PRATP – ENERGY – PMATR – FMATR – DIAGON – IMTQL2 – SIGMA – GCCI – CCI – TRAMOM – MCDB – BTERM – MATPRT • Functions: – H – OA – OB
The order in which these subroutines are called on in the program is presented in Figure 3. MAIN READER GMATR SMATR F0MATR
LOWDIN DIAGON IMTQL2 SIGMA
SCF DIAGON IMTQL2 SIGMA
AUFBAU
FMATR PMATR
PRATP
DIAGON IMTQL2 SIGMA
SHIFT ENERGY PMATR PRATP PI GCCI SHIFT TRAMOM CCI SHIFT
DIAGON IMTQL2 SIGMA
DMOM
SPRT1
MCDB BTERM SPRT1 DIAGON IMTQL2 SIGMA
Figure 3: The order in which the di↵erent subroutines are used within the LCOAO program. The subroutines STORE and LOAD are used to store arrays in a file and to read in a stored array respectively. This is implemented to save on working memory, as this used to be very limited. PRINT, SPRT1 and MATPRT are subroutines that print matrices and plots to the output file depending on the output options selected in the input file. Furthermore, the subroutine SHIFT transposes a matrix. The functions compute integrals. Therefore, the main focus will be on the remaining 21 subroutines.
The first subroutine that is called by the main program is the subroutine READER. This is the subroutine that reads in the input file with the coordinates, but it also sets some (semi-empirical) values. These include a set op semi-empirical values for setting up the orbitals, and the number of outer shell electrons and the Slater coefficients of the first three rows of the periodic table with the exception of the noble gasses Neon and Argon. These properties of the elements are set in Fortran data arrays, with the
index corresponding to the atomic number of the element. After the subroutine READER, the subroutine GMATR is called. This subroutine computes the Gamma integral parameters and the interatomic distance matrix.
Next the overlap matrix is computed in the SMATR subroutine, which then uses the subroutine PRINT to print the overlap matrix to the output file if IPRNT (variable for selecting the amount of output) is greater than 3.
From this overlap matrix, the F0MATR subroutine computes the Fock (F0) matrix. It computes the penetration terms for one-center F0elements, the o↵ diagonal F0elements, and the diagonal F0elements. Then the program calls the subroutine LOWDIN, which converts the F0matrix into a matrix containing linear combinations of the L¨owdin OAO basis set. This subroutine also calls the subroutine DIAGON, which is an Eispack TRED2 program that converts a symmetric matrix into a tridiagonal matrix. DIAGON then calls the subroutine IMTQL2, which is an Eispack program that calculates the eigenvectors and eigenvalues of the tridiagonal matrix. This subroutine calls the subroutine SHIFT, which shifts all MO energies by a certain bias that can be adjusted in the input file.
A self-consistent field method is applied in the subroutine SCF. This subroutine calls on many other subroutines, starting with the DIAGON, IMTQL2, SIGMA subroutines. The resulting array is then transposed using the SHIFT routine, after which the subroutine ENERGY computes the H¨uckel energy and the Roothaan closed shell energy. The subroutine PMATR computes the bond orders within the molecule. Lastly, the SCF routine call on the subroutine PI, which determines the centre of charge and shifts the molecule into a charge centered coordinate system.
The program then moves on to the subroutine GCCI, which stands for Grand Canonical Configuration Intereaction. This subroutine computes the singly excited configuration, starting with transposing the array with the subroutine SHIFT, then the subroutine TRAMOM computes the transition moment between the ground state configuration and the singly excited configuration. Subroutine CCI then converts the grand canonical CI-matrix to a canonical CI-matrix, which is the transposed with the subroutine SHIFT.
The succession of the subroutines DIAGON, IMTQL2 and SIGMA is called again. Then the subroutine DMOM prints the excitation energies together with the transition moments and the oscillator strengths to the output file. Then MCDB is called, which computes the Magnetic Circular Dichroism B-terms with the help of the subroutine BTERM. Then the subroutines DIAGON, IMTQL2 and SIGMA are called again at the end of the program.
3.2
Translation to a Python implementation
3.2.1 Advantages and disadvantages of Fortran and Python
Fortran is a relatively old programming language, and therefore is very rigid in its use of memory. It blocks out the full amount of memory that is needed for the program for the full duration of the program. This means that the program is restricted to a certain memory size, which in the case of the LCOAO program translates to a limit on the size of the molecule. To allow for bigger molecules, the whole of the program would need to be adjusted to fit the new array sizes. Another problem of the Fortran implementation of the LCOAO method is that the code is specifically made for 32-bit computer, and as technology advances more and more computers will be of the 64-bit variety.
Therefore, a conversion to a python version is desirable, as python is a relatively new programming language, or more accurately, a scripting language that is still being developed. Python is widely used within the natural sciences, like physics and astronomy, on account of its large selection of libraries that
contain natural constants and (mathemtical) functions. Python is also more dynamic in its use of memory, so the program could be made to fit any size of molecule automatically. And with the vast expansion of computer memory compared to when the first version of the LCOAO Fortran program was made, the subroutines LOAD and STORE are not needed anymore to limit the amount of working memory is needed A small advantage of Fortran over Python is that Fortran is a compiled language and not a interpreted language like Python, meaning the execution of the program is in theory faster than a Python script, but with the current processor speeds of computers, the di↵erence in actual runtime will be negligible.
A part of the actual syntax of Fortran and Python is quite similar, so a Fortran code can be translated almost one-to-one. However, there are multiple fundamental di↵erences between Fortran and Python to be mindful of. Firstly the indexing of arrays in Fortran starts at 1, while Python array indexing starts at 0. Another di↵erence between Fortran lies in the iterations of do loops in Fortran and for loops in Python: a do loop in Fortran from 1 to N runs N times, while a for loop in Python from 1 to N runs N 1 times. So in the translation a do loop from 1 to N should be converted to a for loop that either goes from 1 up to N + 1, or from 0 to N .
Another di↵erence is that Python features more types of data structures than Fortran. For example, Python has dictionaries, where elements are given a label, instead of an index number. The Fortran implementation of the LCOAO method features a data array in the subroutine READER that contains important values for a number of elements. This data is split over six arrays and within each array the array index represents the atomic number of the element. In the Python implementation of the LCOAO method these data arrays have been converted to a python dictionary. The two di↵erent implementations are shown for the element carbon in Listings 1 and 2.
1 D I M E N S I O N SDAT (36) , ASDAT (36) , APDAT (36) , BDAT (36) , ZDAT (36) , GA (36) 2
3 DATA SDAT (6) , ASDAT (6) , APDAT (6) , BDAT (6) , ZDAT (6) , GA (6) 4 1 /1.625 , -14.960 , -5.805 , -17.5 ,4. ,10.93/
Listing 1: The Fortran implementation of the data arrays for carbon.
1 E l e m e n t s = { 2 6 : { 3 " Name " : " Carbon ", 4 " Symbol " : " C ", 5 " A t o m i c _ N u m b e r " : 6 , 6 " S l a t e r _ A O _ e x p " : 1.625 , 7 " ASDAT " : -14.960 , 8 " APDAT " : -5.805 , 9 " BDAT " : -17.5 , 10 " V a l e n c e _ e l e c t r o n s " : 4 , 11 " GA " : 10.93 12 } 13 }
Listing 2: The Python implementation of the data in a Python dictionary for carbon.
The advantage of using a dictionary is that more information can be added without changing the indexing, as all elements are referred to by a label. To easily access data from the dictionary a series of functions can be written to extract the correct entries. And example of such a function is presented in Listing 3.
1 def g e t _ S l a t e r _ C o e f f i c i e n t ( i ) :
2 return E l e m e n t s [ i ][" S l a t e r _ A O _ e x p "]
3.2.2 Translating the subroutines and functions
The functions H, OA, and OB were translated to Python first, as these are three independent functions. To show an example of these translations, the Fortran implementation of function OA is presented in Listing 4 and its Python implementation in Listing 6. After the translation the three Fortran functions were isolated from the LCOAO program to be able check the results from translated functions. Both the Fortran and the Python implementations of these functions were tested multiple times with a set of random numbers to ensure the same results were obtained with the Python versions. One example of such testing is shown in Listing 5 for the Fortran implementation, and in Listing 7 for the Python version.
1 F U N C T I O N OA (A , K ) 2 i m p l i c i t real*8( A -H ,O - Z ) 3 B =1./ A 4 S =1. 5 OA =1. 6 IF ( K .LT.1) GO TO 2 7 DO 1 M =1 , K 8 L =K - M +1 9 S = L * S * B 10 1 OA = OA + S 11 2 OA = OA * B *EXP( - A ) 12 RETURN 13 END
Listing 4: The Fortran implementation of function OA, which evaluates A integrals.
1 5 . 8 7 1 8 5 5 4 0 1 0 9 1 6 0 7 9 E -004 2
3 ...P r o g r a m f i n i s h e d with exit code 0 4 Press ENTER to exit c o n s o l e .
Listing 5: The result of the Fortran implementation of function OA, when run with A = 6.5 and K = 5.
1 def OA (A , K ) : 2 B = 1/ A 3 S = 1 4 oa = 1 5 6 for m in range(1 , K +1) : 7 L = K - m +1 8 S = L * S * B 9 oa = oa + S 10 oa = oa * B * np . exp ( - A ) 11 return oa
Listing 6: The translated Python implementation of function OA, which evaluates A integrals.
1 >>> OA (6.5 ,5)
2 0 . 0 0 0 5 8 7 1 8 5 5 4 0 1 0 9 1 6 0 8
Listing 7: The result of the translated Python implementation of function OA, when run with A = 6.5 and K = 5.
After the completion of the translation of the functions present in the LCOAO program, translation of the subroutines was started. In Python the subroutines are implemented as functions, as Python does not feature subroutines. The first subroutines to be translated were the ’low-level’ subroutines that do not call
on other subroutines, which can be found at the end of each row from Figure 3. Most of the subroutines were translated mostly one-to-one, with some updating to the structure. An example of such updates is that the Fortran code contains many ’IF ... GO TO ...’ statements to di↵erentiate between di↵erent cases, see Listing 8. Python does not feature such go to statements, and therefore the if statements had to be altered and often reversed, see Listing 9.
1 IF (1.GT. NH ) GO TO 16 2 DO 14 K =1 , NH 3 UIJ = U (K , I ) * U (K , J ) 4 SX = SX + UIJ * X ( K ) 5 SY = SY + UIJ * Y ( K ) 6 SZ = SZ + UIJ * Z ( K ) 7 14 C O N T I N U E 8 ... 9 16 ...
Listing 8: An example of the Fortran implementation with an ’IF ... GO TO ...’ statement, from the subroutine TRAMOM. 1 if NH >= 1: 2 for k in range(0 , NH ) : 3 Uij = U [k , I ] * U [k , J ] 4 Sx = Sx + Uij * X [ k ] 5 Sy = Sy + Uij * Y [ k ] 6 Sz = Sz + Uij * Z [ k ]
Listing 9: The translated Python implementation of Listing 8, with the condition of the if-statement reversed.
All the ’low-level’ have been fully translated, with the exception of the printing routines and the subroutine READER, which processes the input file. A start has been made on translating ’mid-level’ subroutines, of which there are only three: IMTQL2, FMATR, and BTERM. These subroutines only call on ’low-level’ subroutines. IMTQL2 was fully translated, while a beginning was made on translating the subroutine FMATR. The subroutine BTERM, and the ’high-level’ subroutines (subroutines that call on the ’mid-level’ subroutines) have not been able to be translated within the timeframe of the project. So a total of 14 of the 22 main subroutines have been fully translated, with the translation of the 15th subroutine started. A start has also been made on building an input and an output code, to replace the READER subroutine, and the three printing subroutines PRINT, SPRT1, and MATPRT. Some of the subroutines that have been translated, have been renamed after translation to provide more clarity on the purpose of the routines:
• GMATR ! Gamma Matrix • SMATR ! Overlap Matrix • F0MATR ! F0 Matrix
• DIAGON ! Tridiagonal Matrix • FMATR ! Fock Matrix • PMATR ! Bond Order Matrix • SHIFT ! Transpose
• TRAMOM ! Transition Moment • CCI ! Canonical CI Matrix
It is difficult to test the Python versions of the subroutines against their Fortran counterparts, as even the subroutines that do not call on other subroutines do depend on the other subroutines via the arrays and variables in the program. The subroutines are therefore not yet tested against their Fortran counterparts to check their correctness. A future aim would be to finish the translation of the remaining 8 subroutines, and then combine them and test the program as a whole on a set of molecules.
4
Predicting electronic transitions of PAH cations with LCOAO
The LCOAO method has already been used for predicting electronic transitions of neutral PAHs. With these calculations an accuracy of up to 1% was obtained for the prediction of the electronic transitions of regular neutral PAHs, while TD-DFT has an inaccuracy of possibly more then 10%.10, 13However, the ISM contains a mix of neutral and radical cation PAHs.14 It is therefore also of interest to study the electronic transitions of PAHs cations using the LCOAO method. This chapter introduces a selection of radical cations that have been calculated using the work flow described in chapter 2, and presents the predicted electronic transitions. Subsequently, these calculations are, where possible, compared to experimental data collected from literature to study the applicability of the LCOAO method on ions.
4.1
Radical cations of PAHs
Three sets of PAH radical cations ranging in size from 2 rings up to 15 rings were chosen, containing a variety of PAH structures, which are shown in Figure 4. The number of carbon atoms within these PAHs ranges from 10 for the 2 ring PAH napthalene, and up to 48 for the 15 ring PAH dicoronylene. The first two sets represent two di↵erent PAH families: the acenes (Figure 4, 1a-1d), which are linear and all have a zigzag outer edge, and the rylenes (Figure 4, 2a-2c), which have a predominantly armchair outer edge. The third set (Figure 4, 3-13) contains a wide variety of PAH structures and sizes, from very symmetrical to completely asymmetrical. 1a 1b 1c 1d 3 4 5 6 8 9 10 2a 2b 2c 11 12 7 13
Figure 4: The radical cations of the following PAHs were calculated: (1) Acenes: (a) Naphthalene; (b) Anthracene; (c) Tetracene; (4) Pentacene; (2) Rylenes: (a) Perylene; (b) Terrylene; (c) Quaterrylene; (3) Phenanthrene; (4) Pyrene; (75 Benzo[a]anthracene; (6) Chrysene; (7) Ovalene; (8) Triphenylene; (9) Benzo[a]pyrene; (10) Benzo[e]pyrene; (11) Benzo[ghi]perylene; (12) Hexabenzo[a,cd,fgh,jk,m,qrs]peropyrene; (13) Dicoronylene.
4.2
The D
1and D
2electronic transitions of PAH ions
The calculated electronic transitions to the first and second electronically excited states are presented in Table 1. The electronic transitions to D1cover a wide range of energies: from 940 to 14000 cm 1(or 714 nm to 11 µm), which is a range that extents from the visible spectrum up to the mid-IR. However, the transition strengths for the D1 D0are very weak, and for some ions it is zero even. Notably, there are two exceptions to this: tetracene and pentacene, two PAHs of the acenes family. The electronic transitions to D2also cover a wide range, from 5600 up to 16500 cm 1(or 606 nm to 1.8 µm), extending from the visible up to the near-IR. The transition strengths of the transitions to the second electronially excited states D2are generally stronger than the first electronic transition, so the D2 D0transition is more likely to be detected. The exceptions to this are tetracene, perylene, and dicoronylene, as these have transition strengths of (close to) zero. Also noteworthy is the very high transition strength of the D2 D0 transition of quaterrylene, which is more than an order 0f 10 higher compared to the highest D2 D0 transitions of the other ions.
Table 1: The LCOAO predictions of the first two excited states for the radical cations sorted on their number of rings.
D1 D0 D2 D0
Radical cation Molecular
Formula Rings ⌫ (cm 1) Osc. ⌫ (cm 1) Osc. Naphthalene+ C10H8 2 6632.5 0 16554.1 0.111496 Anthracene+ C14H10 3 10030.5 0 14230.9 0.216201 Phenanthrene+ C14H10 3 3686.0 0.003062 11985.1 0.132473 Pyrene+ C16H10 4 8192.5 0 13396.8 0.059131 Tetracene+ C18H12 4 12145.3 0.314481 12279.0 0 Chrysene+ C18H12 4 4554.7 0 8285.6 0.209868 Triphenylene+ C 18H12 4 942.1 0.000183 5672.7 0.053907 Benzo[a]anthracene+ C 18H12 4 5072.3 0.014697 11178.7 0.224001 Perylene+ C20H12 5 13396.2 0.006325 14818.0 0.000175 Benzo[a]pyrene+ C20H12 5 8697.7 0.008449 13003.5 0.070832 Benzo[e]pyrene+ C20H12 5 4663.0 0.008065 10799.8 0.130805 Pentacene+ C22H14 5 10414.7 0.414732 11635.9 0.057627 benzo[g,h,i]perylene+ C22H12 6 6563.8 0.006615 13112.3 0.039109 Terrylene+ C30H16 8 9642.7 0.001005 15696.9 0.082297 Ovalene+ C32H14 10 5996.3 0 10635.7 0.113218 Quaterrylene+ C40H20 11 7413.5 0.000003 13771.6 2.806838 Hexabenzo [a,cd,fgh,jk,m,qrs] peropyrene+ C 44H20 13 2453.6 0.012765 8499.2 0.076148 Dicoronylene+ C48H20 15 7008.2 0.014249 7751.8 0
Figure 5 shows the spread of the excitation energies of the D1 D0transition of the cations. The acenes and rylenes sets are denoted with triangular and square markers respectively. The third set of the diverse PAHs are denoted with round markers. The acenes exhibit an increase in D1energy as the length of the ion increases for the first 3 members. However pentacene deviates from this upward trend. The
rylenes follow a downward trend, with the energy of the transition decreasing with size. The set with diverse PAHs show no correlation between the transition energies and the number of rings present in the ion. The PAH structure has such a large influence that it conceils any expected size dependence.
Figure 5: The predicted D1 D0transitions as a function of number of rings for the three sets of PAHs: the acenes (triangles), the rylenes (squares) and the mixed set (circles).
Similarly, Figure 6 shows the spread of the second electronic transition (D2 D0). Like in the previous figure, the acenes, rylenes, and the mixed set are indicated by triangular, square, and round markers respectively. The acenes exhibit a clear decrease in D2energy as the carbon-ring length increases. While the rylenes do not show a very clear trend with the current sample size. The mixed set also shows correlation between D2energy and size of the molecule. Though the structure varies largely in this set, the size dependence on the transition frequency is strong enough to stand out. The structure dependence in the D2 D0transitions does not fully conceal the expected size dependence as it did for the transitions to the first excited state.
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Figure 6: The predicted D2 D0transitions as a function of number of rings for the three sets of PAHs: the acenes (triangles), the rylenes (squares) and the mixed set (circles).
4.3
Predictive power of LCOAO
4.3.1 Comparison with experimental data
Experimental data was collected from literature for a subset of the PAH cations that were calculated. The experimental values, together with the LCOAO predictions, of this subset are presented in Table 2. Some of the literature did not assign the electronic transitions measured, in such cases the most similar values were taken, taking into consideration of the order of magnitude of the oscillator strengths. Such cases are denoted with an asterisk (*) in Table 2.
The experimental data shows no D1 D0transitions for most of this subset of cations, which is conform to the predictions of the LCOAO, as the LCOAO predicted transition strengths that are weak or zero for these cations (Table 1). There are two exceptions to this, the LCOAO predicted weak D1 D0 transitions for perylene and terrylene, yet experimental data shows these transitions are observable. The disparity between the LCOAO predicted transition and the experimental data is of an order of between 450-760 cm 1 (or 45 - 53 nm). The D1 D0 transition of Pentacene was also observed in experimental data, this again in accordance with Table 1, as the LCOAO predicted a stronger transition. Also noteworthy is the conformity between the LCOAO prediction and the experimental data of this transition, as it is accurate within 66 cm 1(or 6 nm). Another notable observation is the absence of the D2 D0transition for Pentacene in the experimental data, as the LCOAO predicted a weak, but not zero, transition strength for this transition. Also noticeable is that the experimental data shows a D2 D0transition for Perylene, while the LCOAO predicted an oscillator strength of close to zero. The deviation of the LCOAO predictions and the experimental data for the D2 D0transition cover a range from 120 to 2210 cm 1(or 6 - 104 nm).
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Table 2: Comparison of the transition frequencies to D1 and D2 predicted by LCOAO to reported experimental values.
D1 D0 D2 D0
Cation Rings ⌫LCOAO
(cm 1) ⌫exp (cm 1) (cm 1) ⌫LCOAO (cm 1) ⌫exp (cm 1) (cm 1)
Naphtalene+ 2 6632.5 Not obs.15 - 16554.1 14909.115 1645 (11.0%)
Anthracene+ 3 10030.5 Not obs.15 - 14230.9 14109.115 121.8 (0.9 %)
Phenanthrene+ 3 3686.0 Not obs.16 - 11985.1 1113216 853.1 (7.7%)
Pyrene+ 4 8192.5 Not obs.17 - 13396.8 1277917 617.8 (4.8%)
Perylene+ 5 13396.2 1263918 757.2 (6.0%) 14818.0 1361518 1203 (8.8%)
Pentacene+ 5 10414.7 1048119 -66.3 (0.6%) 11635.9 Not obs.19
-Terrylene+ 8 9642.7 9174*20 468.7 (5.1%) 15696.9 13489*20 2207.9 (16.4%)
Ovalene+ 10 5996.3 Not obs.*20 - 10635.7 10257*20 378.7 (3.7%)
Quaterrylene+ 11 7413.5 Not obs.20 - 13771.6 1199520 1776.6 (14.8%)
* Experimental works had no electronic transition assignments, most similar values were taken.
The values from the D2 D0transitions of both the LCOAO predictions and the experimental data together with the structures of the cations are plotted in Figure 7. Furthermore, the variation between the LCOAO method and the experimental values as a function of the number of rings and as a function of the wavelength are shown in Figures 8a and 8b respectively. The set of PAHs to compare is too small the disentangle any possible trend with structure and/or size.
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Figure 7: The D2 D0transitions energies as predicted with LCOAO (blue) with reported experimental values (orange) as function of number of rings from Table 2.
* Experimental works for Terrylene and Ovalene had no electronic transition assignments, most similar values were taken.
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(a) (b)
Figure 8: The absolute deviations between LCOAO predictions and experimental values from literature, as a function of the number of rings (a) and as a function of the experimental (actual) energy.
4.3.2 Comparison with TD-DFT
As mentioned at the beginning of this chapter, the LCOAO method preforms better than TD-DFT calculations in the case of neutral PAHs. TD-DFT values for the rylenes were gathered from literature to compare the predictive power of the LCOAO method for the radical cations to that of TD-DFT. These values are listed, together with the LCOAO predictions, in Table 3. The Table also states the experimental values, and the deviations between both computational methods and the experimental values. For the rylenes the LCOAO predictions are less accurate than TD-DFT calculations.
Table 3: Comparison of the LCOAO calculated transition energies with TD-DFT calculations and their accuracy compared to reported experimental values.
Cation Transition ⌫LCOAO
(cm 1) ⌫TD DFT (cm 1) ⌫exp (cm 1) LCOAO (cm 1) TD DFT (cm 1) Perylene+ D2 D0 14818.0 13575.1 21 1361518 1203 (8.8%) 39.9(0.3%) D4 D0 16452.8 19671.121 1903621 2583.2 (13.6%) 635.1 (3.3%) Terrylene+ D2 D0 15696.9 14717.2 21 13489*18 2207.9 (16.4%) 1228.2 (9.1%) D4 D0 18698.0 15531.821 1447821 4220.0 (29.1%) 1053.8 (7.3%) Quaterrylene+ D2 D0 13771.6 13056.521 1199520 1776.6 (14.8%) 1061.5 (8.8%)
4.4
Higher excited states of phenanthrene and pyrene
For the radical cations of phenanthrene and pyrene, multiple electronic transition measurements were found, which were compared to the predictions of the LCOAO method.16 The first seven electronic transitions of Phenanthrene are listed in Table 4 and are plotted in Figure 9a, and the di↵erence between the predicted transitions and the experimental values is shown in Figure 9b. The LCOAO predictions seem to deviate more for the higher lying transitions. Notable is that the di↵erence between the LCOAO prediction and the experimental value seems to be almost constant for D4 D6.
Table 4: Calculated and experimental electronic transitions of the phenanthrene radical cation up to D7.
Transition ⌫LCOAO (cm 1) ⌫exp (cm 1) (cm 1) D1 D0 3686 Not obs.16 -D2 D0 11985.1 1113216 853.1 (7.7%) D3 D0 18840.8 1576316 3077.8 (19.5%) D4 D0 25461.9 2124516 4216.9 (19.8%) D5 D0 27708.7 2348516 4223.7 (18.0%) D6 D0 29652.7 2525216 4400.7 (17.4%) D7 D0 37573.2 2899416 8579.2 (29.6%) (a) (b)
Figure 9: Di↵erence between LCOAO and experimental reported values of the D2energies as function of a) number of rings and b) experimental transition energy.
For Pyrene the first six transitions of the LCOAO method and the experimental data are given in Table 5.17Figure 10a shows the plot of these transitions, and Figure 10b shows the deviation between the LCOAO predictions and the experimental data. Up to D3 the deviation again increases as the energy goes up. However, the deviation between the LCOAO predictions and the experimental data decreases after the D4 D0transition.
Table 5: Calculated and experimental electronic transitions of the pyrene radical cation.
Transition ⌫LCOAO (cm 1) ⌫exp (cm 1) (cm 1) D1< D0 8192.5 Not obs.17 -D2< D0 13396.8 1277917 617.8 (4.8%) D3< D0 15596.3 1401017 1586.3 (11.3%) D4< D0 24139.8 2054217 3597.8 (17.5%) D5< D0 25487.6 2253317 2954.6 (13.1%) D6< D0 29173.0 2757917 1594.0 (5.8%) (a) (b)
Figure 10: (a) The higher excited states of Pyrene, and (b) the deviation between LCOAO predictions and experimental values of Pyrene.
4.5
Discussion and conclusions
There is no general trend visible in the predicted electronic transitions of the PAH cations. It seems that the shape of the PAH cations is of more influence on the transitions than the size of the PAH, as there is a di↵erence in transition energies between the same size PAH cations.
The visible spectrum is 12500 - 25000 cm 1, so the predicted D1transitions, with the exception of Perylene, are in the IR spectrum. These transitions are therefore not candidates for DIBs, as DIBs fall in the visible spectrum. Many of the D2transitions are also in, or close to the IR region. So The second transitions that are in the IR region can also be ruled out as possible DIB carriers.
When the LCOAO values are compared to the experimental values there does not seem to be a trend in the deviation between the LCOAO predictions and the experimental values, so there is no systematical error in the calculations. Another noteworthy feature of the comparisons between LCOAO predictioms and experimental data of both the D1and D2transitions, as well as the higher excited states of Phenanthrene and Pyrene, is that the LCOAO predictions all give a wavenumber that is higher than the actual experimental value. Also the LCOAO predictions deviate as much as 2200 cm 1(16.4%) for the D2 transitions and 8600 cm 1(29.6%) for the higher excited states of Phenanthrene, though it is expected that the LCOAO predictions deviate more for higher energies. This deviation of up to 2200 cm 1and the fact that all the LCOAO predictions are above the experimental value means that for the transitions in Table 1 that are only just in the visible spectrum, might in fact also be in the IR region, and therefore are not possible DIB carriers.
From the comparison of both the LCOAO method and TD-DFT calculations to experimental values it seems that the LCOAO method is less accurate than TD-DFT in the case of the radical cations of the Rylene family. This can be the consequence of di↵erent factors. The LCOAO method is a semi-empirical calculation, where the calculation uses certain empirical parameters to better the model. In the case of the LCOAO method, these parameters were derived from experimental data of neutral Benzene and Naphthalene, and not their cationic counterparts. Also, these empirical values come from small size PAHs, but is also used for PAHs of a bigger size, so there may be some error caused by that as well. Another possible factor is that the LCOAO computes the vertical transition energy, and with that it assumes that there is no change in geometry for the excited states. As TD-DFT does take into account any geometry changes between states, it might explain the di↵erence in accuracy between the LCOAO method and TD-DFT, although more data on this topic should be collected.
5
Conclusions and outlook
The LCOAO method was first implemented in 1983 in Fortran. Fortran is a programming language that was developed in the 1950’s, and as such the programming language shows artefacts of the computer structures of that time. The memory use of Fortran is fixed, and therefore the size of the arrays is fixed, and thus the size of the molecule that can be computed is limited. Therefore, an implementation in a more recent programming language is desired. The programming language that was chosen is Python, due to its widespread use, its simplicity, and the large selection of libraries that is available. Python also features more types of data structures, like Python dictionaries, that can be useful to set up constants.
The LCOAO method is divided in a main program and a set of 27 subroutines and 3 functions. All three of the functions were successfully translated to Python and checked on their results. Two of the 27 subroutines are not needed in the Python implementation: STORE and LOAD. These only exist to spare working memory of a computer, of which there is more than enough nowadays. Three of the subroutines,
PRINT, MATPRT, and SPRT1, are to create the output file, and these have not been translated yet, as some changes to the structure of the output files may be desired. Of the remaining 22 subroutines, 14 were translated fully to Python. The remaining 8 subroutines were not translated due to time constraints.
The electronic transitions of three sets of PAH radical cations were predicted by using the Fortran implementation of the LCOAO method: the acene family, the rylene family, and a diverse set of PAHs. The acene and the rylene families show a linear dependance on their size for the electronic transitions. The acenes have an upward trend for the D1transition and a downward trend for the D2transition. The rylenes follow an downward trend for both the D1and the D2transitions. The diverse set shows no linear dependance on the size of the PAH, the structure of the PAH has a greater influence on the electronic transitions than the size.
The first electronic transitions are all, with the exception of Perylene, in the IR spectrum, so none of these first electronic transitions of the PAH cations fall in the DIB region. Many of the second electronic transitions also fall in the IR region, so these are also ruled out as possible DIB carriers. The PAHs that are predicted to have transitions in the visible spectrum are Naphthalene, Anthracene and Tetracene (D2 D0) of the Acenes, Perylene (both transitions), Terrylene and Quaterrylene (D2 D0) of the Rylenes, and Pyrene, Benzo[a]pyrene, and Benzo[g,h,i]perylene, of the diverse PAHs (D2 D0).
The LCOAO predictions were also compared to experimental data from literature. These comparisons do not lead to a systematical error, and show that the LCOAO predictions deviate up to 2200 cm 1 (16.4%). The higher excited states from Phenanthrene and Pyrene lead to errors of up to 8600 cm 1 (29.6%), but this is to be expected as predictions of higher energies often deviate more.
The LCOAO method was also compared to how TD-DFT compares to experimental values, and it seems that the LCOAO method has less accuracy compared to TD-DFT. However, a bigger set of data points should be collected to investigate further.
The inaccuracy of the LCOAO method for radical cations could have several origins. For example, the LCOAO method is a semi-empirical calculation that has parameters based on neutral Benzene and Naphthalene. Benzene and Naphthalene are relatively small compared to some of the larger PAHs considered in this study. And the parameters were taken from the neutrals, while this study calculated ions. Another aspect of the LCOAO method is that it computes the vertical transition energies, which does not take into account any geometry changes that might happen when a PAH cation is excited.
The Fortran implementation of the LCOAO method has several limitations, and although the radical PAH cations are within these limitations, the accuracy was found to be lower than TD-DFT. In the translation to Python, some of these limitations can be removed. Python is a dynamic programming language, so the arrays within the program can be made to fit the dimensions of any molecule automatically, removing the need for changing the array sizes throughout the whole of the program to accommodate a large PAH.
Another aspect of the Fortran implementation is that it only takes flat molecules as input, so in the input file the Z-coordinates should all be zero. The LCOAO method is however usable on PAHs with an interior angle, but for the input file these molecules are forced flat, and therefore the geometry is not fully representative of the actual molecule. In the Python implementation these Z-coordinates can be read in as non-zero, and then afterwards these coordinates can then simply be left out of the calculation.
An additional interesting feature was found in the Fortran LCOAO implementation, and that is the presence of semi-empirical date for the first two rows of elements from the periodic table. However, these values are not used within the program and the calculation seems to be specialised toward molecules containing only carbon and hydrogen.
to fit future computations on di↵erent species of PAHs, and even substituted PAHs. Modifications needed would be to include more coding that include all the elements that are already included within the semi-empirical parameters, instead of defaulting to carbon. Furthermore, more of the semi-empirical parameters present in the code could be optimised and divided into di↵erent sets, so that the set of parameters can be used that best fits the computed molecule (eg. a set of parameters for neutral PAHs, a set for radical cations, a set for anions, a set for substituted PAHs, etc). The set of parameters could then be chosen by including certain settings in the input file for the molecule.
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