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Basis Set Superposition Error (BSSE). A short intro
Molecular Orbitals (MOs) are linear combinations of Atomic Orbitals (AOs), which in turn are linear combinations of other functions called ‘basis functions’. A basis, or more accurately a basis set, is a collection of functions which obey a set of rules (such as being orthogonal to each other and possibly being normalized) with which all AOs are constructed, and although these are centered on each atomic nucleus, the canonical way in which they are combined yield delocalized MOs; in other words, an MO can occupy a large space spanning several atoms at once. We don’t mind this expansion across a molecule, but what about between two molecules? Calculating the interaction energy between two or more molecular fragments leads to an artificial extra–stabilization term that stems from the fact that electrons in molecule 1 can occupy AO’s (or the basis functions which form them) centered on atoms from molecule 2.
Fundamentally, the interaction energy of any A—B dimer, Eint, is calculated as the energy difference between the dimer and the separately calculated energies for each component (Equation 1).
Eint = EAB – EA – EB (1)
However the calculation of Eint by this method is highly sensitive to the choice of basis set due to the Basis Set Superposition Error (BSSE) described in the first paragraph. The BSSE is particularly troublesome when small basis sets are used, due to the poor description of dispersion interactions but treating this error by just choosing a larger basis set is seldom useful for systems of considerable sizes. The Counterpoise method is a nifty correction to equation 1, in which EA and EB are calculated with the basis set of A and B respectively, i.e., only in EAB a larger basis set (that of A and B simultaneously) is used. The Counterpoise method calculates each component with the AB basis set (Equation 2)
EintCP = EABAB – EAAB– EBAB (2)
where the superscript AB means the whole basis set is used. This is accomplished by using ‘ghost‘ atoms with no nuclei and no electrons but empty basis set functions centered on them.
In Gaussian, BSSE is calculated with the Counterpoise method developed by Boys and Simon. It requires the keyword Counterpoise=N where N is the number of fragments to be considered (for an A—B system, N=2). Each atom in the coordinates list must be specified to which fragment pertains; additionally, the charge and multiplicity for each fragment and the whole supermolecular ensemble must be specified. Follow the example of this hydrogen fluoride dimer.
%chk=HF2.chk #P opt wB97XD/6-31G(d,p) Counterpoise=2 HF dimer 0,1 0,1 0,1 H(Fragment=1) 0.00 0.00 0.00 F(Fragment=1) 0.00 0.00 0.70 H(Fragment=2) 0.00 0.00 1.00 F(Fragment=2) 0.00 0.00 1.70
For closed shell fragments the first line is straightforward but one must pay attention that the first pair of numbers in the charge multiplicity line correspond to the whole ensemble, whereas the folowing pairs correspond to each fragment in consecutive order. Fragments do not need to be specified contiguously, i.e., you don’t need to define all atoms for fragment 1 and after those the atoms for fragment 2, etc. They could be mixed and the program still assigns them correctly. Just as an example I typed wB97XD but any other method, DFT or ab initio, may be used; only semiempirical methods do not admit a BSSE calculation because they don’t make use of a basis set in the first place!
The output provides the corrected energy (in atomic units) for the whole system, as well as the BSSE correction (which added to the previous term yields the un-corrected energy of the system). Gaussian16 also provides these values in kcal/mol as ‘Complexation energies’ first raw (uncorrected) and then the corrected energy.
BSSE is always present and cannot be entirely eliminated because of the use of finite basis sets but it can be correctly dealt with if the Counterpoise method is included.
Orbital Contributions to Excited States
This is a guest post by our very own Gustavo “Gus” Mondragón whose work centers around the study of excited states chemistry of photosynthetic pigments.
When you’re calculating excited states (no matter the method you’re using, TD-DFT, CI-S(D), EOM-CCS(D)) the analysis of the orbital contributions to electronic transitions poses a challenge. In this post, I’m gonna guide you through the CI-singles excited states calculation and the analysis of the electronic transitions.
I’ll use adenine molecule for this post. After doing the corresponding geometry optimization by the method of your choice, you can do the excited states calculation. For this, I’ll use two methods: CI-Singles and TD-DFT.
The route section for the CI-Singles calculation looks as follows:
%chk=adenine.chk
%nprocshared=8
%mem=1Gb
#p CIS(NStates=10,singlets)/6-31G(d,p) geom=check guess=read scrf=(cpcm,solvent=water)
adenine excited states with CI-Singles method
0 1
--blank line--
I use the same geometry from the optimization step, and I request only for 10 singlet excited states. The CPCP implicit solvation model (solvent=water) is requested. If you want to do TD-DFT, the route section should look as follows:
%chk=adenine.chk
%nprocshared=8
%mem=1Gb
#p FUNCTIONAL/6-31G(d,p) TD(NStates=10,singlets) geom=check guess=read scrf=(cpcm,solvent=water)
adenine excited states with CI-Singles method
0 1
--blank line--
Where FUNCTIONAL is the DFT exchange-correlation functional of your choice. Here I strictly not recommend using B3LYP, but CAM-B3LYP is a noble choice to start.
Both calculations give to us the excited states information: excitation energy, oscillator strength (as f value), excitation wavelength and multiplicity:
Excitation energies and oscillator strengths:
Excited State 1: Singlet-A 6.3258 eV 196.00 nm f=0.4830 <S**2>=0.000
11 -> 39 -0.00130
11 -> 42 -0.00129
11 -> 43 0.00104
11 -> 44 -0.00256
11 -> 48 0.00129
11 -> 49 0.00307
11 -> 52 -0.00181
11 -> 53 0.00100
11 -> 57 -0.00167
11 -> 59 0.00152
11 -> 65 0.00177
The data below corresponds to all the electron transitions involved in this excited state. I have to cut all the electron transitions because there are a lot of them for all excited states. If you have done excited states calculations before, you realize that the HOMO-LUMO transition is always an important one, but not the only one to be considered. Here is when we calculate the Natural Transition Orbitals (NTO), by these orbitals we can analyze the electron transitions.
For the example, I’ll show you first the HOMO-LUMO transition in the first excited state of adenine. It appears in the long list as follows:
35 -> 36 0.65024
The 0.65024 value corresponds to the transition amplitude, but it doesn’t mean anything for excited state analysis. We must calculate the NTOs of an excited state from a new Gaussian input file, requesting from the checkpoint file we used to calculate excited states. The file looks as follows:
%Oldchk=adenine.chk
%chk=adNTO1.chk
%nproc=8
%mem=1Gb
#p SP geom=allcheck guess=(read,only) density=(Check,Transition=1) pop=(minimal,NTO,SaveNTO)
I want to say some important things right here for this last file. See that no level of theory is needed, all the calculation data is requested from the checkpoint file “adenine.chk”, and saved into the new checkpoint file “adNTO1.chk”, we must use the previous calculated density and specify the transition of interest, it means the excited state we want to analyze. As we don’t need to specify charge, multiplicity or even the comment line, this file finishes really fast.
After doing this last calculation, we use the new checkpoint file “adNTO1.chk” and we format it:
formchk -3 adNTO1.chk adNTO1.fchk
If we open this formatted checkpoint file with GaussView, chemcraft or the visualizer you want, we will see something interesting by watching he MOs diagram, as follows:
We can realize that frontier orbitals shows the same value of 0.88135, which means the real transition contribution to the first excited state. As these orbitals are contributing the most, we can plot them by using the cubegen routine:
cubegen 0 mo=homo adNTO1.fchk adHOMO.cub 0 h
This last command line is for plotting the equivalent as the HOMO orbital. If we want to plot he LUMO, just change the “homo” keyword for “lumo”, it doesn’t matter if it is written with capital letters or not.
You must realize that the Natural Transition Orbitals are quite different from Molecular Orbitals. For visual comparisson, I’ve printed also the molecular orbitals, given from the optimization and from excited states calculations, without calculating NTOs:
These are the molecular frontier orbitals, plotted with Chimera with 0.02 as the isovalue for both phase spaces:
The frontier NTOs look qualitatively the same, but that’s not necessarily always the case:
If we analyze these NTOs on a hole-electron model, the HOMO refers to the hole space and the LUMO refers to the electron space.
Maybe both orbitals look the same, but both frontier orbitals are quite different between them, and these last orbitals are the ones implied on first excited state of adenine. The electron transition will be reported as follows:
If I can do a graphic summary for this topic, it will be the next one:
NTOs analysis is useful no matter if you calculate excited states by using CIS(D), EOM-CCS(D), TD-DFT, CASSCF, or any of the excited states method of your election. These NTOs are useful for population analysis in excited states, but these calculations require another software, MultiWFN is an open-source code that allows you to do this analysis, and another one is called TheoDORE, which we’ll cover in a later post.
XVIII RMFQT
It was my distinct pleasure for me to participate in the organization of the latest edition of the Mexican Meeting on Theoretical Physical Chemistry, RMFQT which took place last week here in Toluca. With the help of the School of Chemistry from the Universidad Autónoma del Estado de México.
This year the national committee created a Lifetime Achievement Award for Dr. Annik Vivier, Dr. Carlos Bunge, and Dr. José Luis Gázquez. This recognition from our community is awarded to these fine scientists for their contributions to theoretical chemistry but also for their pioneering work in the field in Mexico. The three of them were invited to talk about any topic of their choosing, particularly, Dr. Vivier stirred the imagination of younger students by showing her pictures of the times when she used to hangout with Slater, Roothan, Löwdin, etc., it is always nice to put faces onto equations.
Continuing with a recent tradition we also had the pleasure to host three invited plenary lectures by great scientists and good friends of our community: Prof. William Tiznado (Chile), Prof. Samuel B. Trickey (USA), and Prof. Julia Contreras (France) who shared their progress on their recent work.
As I’ve abundantly pointed out in the past, the RMFQT is a joyous occasion for the Mexican theoretical community to get together with old friends and discuss very exciting research being done in our country and by our colleagues abroad. I’d like to add a big shoutout to Dr. Jacinto Sandoval-Lira for his valuable help with the organization of our event.
Useful Thermochemistry from Gaussian Calculations
Statistical Mechanics is the bridge between microscopic calculations and thermodynamics of a particle ensemble. By means of calculating a partition function divided in electronic, rotational, translational and vibrational functions, one can calculate all thermodynamic functions required to fully characterize a chemical reaction. From these functions, the vibrational contribution, together with the electronic contribution, is the key element to getting thermodynamic functions.
Calculating the Free Energy change of any given reaction is a useful approach to asses their thermodynamic feasibility. A large negative change in Free Energy when going from reagents to products makes up for a quantitative spontaneous (and exothermic) reaction, nevertheless the rate of the reaction is a different story, one that can be calculated as well.
Using the freq option in your route section for a Gaussian calculation is mandatory to ascertain the current wave function corresponds to a minimum on a potential energy hypersurface, but also yields the thermochemistry and thermodynamic values for the current structure. However, thermochemistry calculations are not restricted to minima but it can also be applied to transition states, therefore yielding a full thermodynamic characterization of a reaction mechanism.
A regular freq calculation yields the following output (all values in atomic units):
Zero-point correction= 0.176113 (Hartree/Particle) Thermal correction to Energy= 0.193290 Thermal correction to Enthalpy= 0.194235 Thermal correction to Gibbs Free Energy= 0.125894 Sum of electronic and zero-point Energies= -750.901777 Sum of electronic and thermal Energies= -750.884600 Sum of electronic and thermal Enthalpies= -750.883656 Sum of electronic and thermal Free Energies= -750.951996
For any given reaction say A+B -> C one could take the values from the last row (lets call it G) for all three components of the reaction and perform the arithmetic: DG = GC – [GA + GB], so products minus reagents.
By default, Gaussian calculates these values (from the previously mentioned partition function) using normal conditions, T = 298.15 K and P = 1 atm. For an assessment of the thermochemistry at other conditions you can include in your route section the corresponding keywords Temperature=x.x and Pressure=x.x, in Kelvin and atmospheres, respectively.
(Huge) Disclaimer: Although calculating the thermochemistry of any reaction by means of DFT calculations is a good (and potentially very useful) guide to chemical reactivity, getting quantitative results require of high accuracy methods like G3 or G4 methods, collectively known as Gn mehtods, which are composed of pre-defined stepwise calculations. The sequence of these calculations is carried out automatically; no basis set should be specified. Other high accuracy methods like CBS-QB3 or W1U can also be considered whenever Gn methods are too costly.
Using PDB files for Electronic Structure Calculations
Quick Post on preparing Gaussian input files from PDB files.
If you’re modeling biological systems chances are that, more often than not, you start by retrieving a PDB file. The Protein Data Bank is a repository for all things biochemistry – from oligo-peptides to full DNA sequences with over 140,000 available files encoding the corresponding structure obtained by various experimental means ranging from X-Ray diffraction, NMR and more recently, Cryo Electron Microscopy (CEM).
The PDB file encodes the Cartesian coordinates for each atom present in the structure as well as their in the same way molecular dynamics codes -like AMBER or GROMACS- code the parameters for a force field; this makes the PDB a natural input file for MD.
There are however some considerations to have in mind for when you need to use these coordinates in electronic structure calculations. Personally I give it a pass with OpenBabel to add (or possibly just re-add) all Hydrogen atoms with the following instruction:
$>obabel -ipdb filename.pdb -ogjf -Ofilename.gjf -h
Alternatively, you can select a pH value, say 7.5 with:
$>obabel -ipdb filename.pdb -ogjf -Ofilename.gjf -h -p7.5
You may also use the GUI if by any chance you’re working in Windows:
This sends all H atoms to the end of the atoms list. Usually for us the next step is to optimize their positions with a partial optimization at a low level of theory for which you need to use the ReadOptimize ReadOpt or RdOpt in the route section and then add the atom list at the end of the input file:
Atomic coordinates
--blank line--
noatoms atoms=H
--blank line--
Finally, visual inspection of your input structure is always helpful to find any meaningful errors, remember that PDB files come from experimental measurements which are not free of problems.
As usual thanks for reading, commenting, and sharing.
Estimation of pKa Values through Local Electrostatic Potential Calculations
Calculating the pKa value for a Brønsted acid is very hard, like really hard. A full thermodynamic cycle (fig. 1) needs to be calculated along with the high-accuracy solvation free energy for each of the species under consideration, not to mention the use of expensive methods which will be reviewed here in another post in two weeks time.
- Fig 1. Thermodynamic Cycle for the pKa calculation of any given Bronsted acid, HA
Finding descriptors that help us circumvent the need for such sophisticated calculations can help great deal in estimating the pKa value of any given acid. We’ve been interested in the reactivity of σ-hole bearing groups in the past and just like Halogen, Tetrel, Pnicogen and Chalcogen bonds, Hydrogen bonds are highly directional and their strength depends on the polarization of the O-H bond. Therefore, we suggested the use of the maximum surface electrostatic potential (VS,max) on the acid hydrogen atom of carboxylic acids as a descriptor for the strength of their interaction with water, the first step in the deprotonation process.
We selected six basis sets; five density functionals; the MP2 method for a total of thirty-six levels of theory to optimize and calculate VS,max on thirty carboxylic acids for a grand total of 1,080 wavefunctions, which were later passed onto MultiWFN (all calculations were taken with PCM = water). Correlation with the experimental pKa values showed a great correlation across the levels of theory (R2 > 0.9), except for B3LYP. Still, the best correlations were obtained with LC-wPBE/cc-pVDZ and wB97XD/cc-pVDZ. From this latter level of theory the linear correlation yielded the following equation:
pKa = -0.2185(VS,max) + 16.1879
Differences in pKa turned out to be less than 0.5 units, which is remarkable for such a straightforward method; bear in mind that calculation of full thermodynamic cycles above chemical accuracy (1.0 kcal/mol) yields pKa differences above 1.0 units.
We then took this equation for a test with 10 different carboxylic acids and the prediction had a correlation of 98% (fig. 2)
- fig 2. calculated v experimental pKa values for a test set of 10 carboxylic acids from equation above
I think this method can really catch on for a quick way to predict the pKa values of any carboxylic acid imaginable. We’re now working on the model extension to other groups (i.e. Bronsted bases) and putting together a black-box workflow so as to make it even more accessible and straightforward to use.
We’ve recently published this work in the journal Molecules, an open access publication. Thanks to Prof. Steve Scheiner for inviting us to participate in the special issue devoted to tetrel bonding. Thanks to Guillermo Caballero for the inception of this project and to Dr. Jacinto Sandoval for taking the time from his research in photosynthesis to work on this pet project of ours and of course the rest of the students (Gustavo Mondragón, Marco Diaz, Raúl Torres) whose hard work produced this work.
Dr. Gabriel Merino wins The Walter Kohn Prize 2018
Just as I was thinking about the state of Mexican scientific environment in the global scale, Prof. Dr. Gabriel Merino from CINVESTAV comes and gets this prize awarded by the International Center for Theoretical Physics (ICTP) and the Quantum ESPRESSO Foundation, showing us all that great science is possible even under pressing circumstances.
This prize is awarded biennially to a young scientist for outstanding contributions in the field of quantum-mechanical materials and molecular modeling, performed in a developing country or emerging economy,and in the case of Dr. Merino it is awarded not only for his contributions to theory and applications but also by his contributions to the prediction of novel systems that violate standard chemical paradigms, broadening the scope of concepts like aromaticity, coordination and chemical bond. The list of his contributions is very long despite his young age and there are barely any topic in chemistry or materials science that escapes his interest.
Gabriel is also one of the leading organizers of the Mexican Theoretical Physical Chemistry Meeting, an unstoppable mentor with many of his former students now leading research teams of their own. He is pretty much a force of nature.
Congratulations to Dr. Gabriel Merino, his team, CINVESTAV and thanks for being such an inspiration and a good friend at the same time.
¡Felicidades, Gabriel!
Computational Chemistry from Latin America
The video below is a sad recount of the scientific conditions in Mexico that have driven an enormous amount of brain power to other countries. Doing science is always a hard endeavour but in developing countries is also filled with so many hurdles that it makes you wonder if it is all worth the constant frustration.
That is why I think it is even more important for the Latin American community to make our science visible, and special issues like this one from the International Journal of Quantum Chemistry goes a long way in doing so. This is not the first time IJQC devotes a special issue to the Comp.Chem. done south of the proverbial border, a full issue devoted to the Mexican Physical Chemistry Meetings (RMFQT) was also published six years ago.
I believe these special issues in mainstream journals are great ways of promoting our work in a collected way that stresses our particular lines of research instead of having them spread a number of journals. Also, and I may be ostracized for this, but I think coming up with a new journal for a specific geographical community represents a lot of effort that takes an enormous amount of time to take off and thus gain visibility.
For these reasons I’ve been cooking up some ideas for the next RMFQT website. I don’t pretend to say that my colleagues need any shoutouts from my part -I could only be so lucky to produce such fine pieces of research myself- but it wouldn’t hurt to have a more established online presence as a community.
¡Viva la ciencia Latinoamericana!
The HOMO-LUMO Gap in Open Shell Calculations. Meaningful or meaningless?
The HOMO – LUMO orbitals are central to the Frontier Molecular Orbital (FMO) Theory devised by Kenichi Fukui back in the fifties. The central tenet of the FMO theory resides on the idea that most of chemical reactivity is dominated by the interaction between these orbitals in an electron donor-acceptor pair, in which the most readily available electrons of the former arise from the HOMO and will land at the LUMO in the latter. The energy difference between the HOMO and LUMO of any chemical species, known as the HOMO-LUMO gap, is a very useful quantity for describing and understanding the photochemistry and photophysics of organic molecules since most of the electronic transitions in the UV-Vis region are dominated by the electron transfer between these two frontier orbitals.
But when we talk about Frontier Orbitals we’re usually referring to their doubly occupied version; in the case of open shell calculations the electron density with α spin is separate from the one with β spin, therefore giving rise to two separate sets of singly occupied orbitals and those in turn have a α-HOMO/LUMO and β-HOMO/LUMO, although SOMO (Singly Occupied Molecular Orbital) is the preferred nomenclature. Most people will then dismiss the HOMO/LUMO question for open shell systems as meaningless because ultimately we are dealing with two different sets of molecular orbitals. Usually the approach is to work backwards when investigating the optical transitions of a, say, organic radical, e.g. by calculating the transitions with such methods like TD-DFT (Time Dependent DFT) and look to the main orbital components of each within the set of α and β densities.
To the people who have asked me this question I strongly suggest to first try Restricted Open calculations, RODFT, which pair all electrons and treat them with identical orbitals and treat the unpaired ones independently. As a consequence, RO calculations and Unrestricted calculations vary due to variational freedom. RO calculations could yield wavefunctions with small to large values of spin contamination, so beware. Or just go straight to TDDFT calculations with hybrid orbitals which include a somewhat large percentage of HF exchange and polarized basis sets, but to always compare results to experimental values, if available, since DFT based calculations are Kohn-Sham orbitals which are defined for non-interacting electrons so the energy can be biased. Performing CI or CASSCF calculations is almost always prohibitive for systems of chemical interest but of course they would be the way to go.
Calculating NMR shifts – Short and Long Ways
Nuclear Magnetic Resonance is a most powerful tool for elucidating the structure of diamagnetic compounds, which makes it practically universal for the study of organic chemistry, therefore the calculation of 1H and 13C chemical shifts, as well as coupling constants, is extremely helpful in the assignment of measured signals on a spectrum to an actual functional group.
Several packages offer an additive (group contribution) empirical approach to the calculation of chemical shifts (ChemDraw, Isis, ChemSketch, etc.) but they are usually only partially accurate for the simplest molecules and no insight is provided for the more interesting effects of long distance interactions (vide infra) so quantum mechanical calculations are really the way to go.
With Gaussian the calculation is fairly simple just use the NMR keyword in the route section in order to calculate the NMR shielding tensors for relevant nuclei. Bear in mind that an optimized structure with a large basis set is required in order to get the best results, also the use of an implicit solvation model goes a long way. The output displays the value of the total isotropic magnetic shielding for each nucleus in ppm (image taken from the Gaussian website):
Magnetic shielding (ppm): 1 C Isotropic = 57.7345 Anisotropy = 194.4092 XX= 48.4143 YX= .0000 ZX= .0000 XY= .0000 YY= -62.5514 ZY= .0000 XZ= .0000 YZ= .0000 ZZ= 187.3406 2 H Isotropic = 23.9397 Anisotropy = 5.2745 XX= 27.3287 YX= .0000 ZX= .0000 XY= .0000 YY= 24.0670 ZY= .0000 XZ= .0000 YZ= .0000 ZZ= 20.4233
Now, here is why this is the long way; in order for these values to be meaningful they need to be contrasted with a reference, which experimentally for 1H and 13C is tetramethylsilane, TMS. This means you have to perform the same calculation for TMS at -preferably- the same level of theory used for the sample and substract the corresponding values for either H or C accordingly. Only then the chemical shifts will read as something we can all remember from basic analytical chemistry class.
GaussView 6.0 provides a shortcut; open the Results menu, select NMR and in the new window there is a dropdown menu for selecting the nucleus and a second menu for selecting a reference. In the case of hydrogen the available references are TMS calculated with the HF and B3LYP methods. The SCF – GIAO plot will show the assignments to each atom, the integration simulation and a reference curve if desired.
The chemical shifts obtained this far will be a good approximation and will allow you to assign any peaks in any given spectrum but still not be completely accurate though. The reasons behind the numerical deviations from calculated and experimental values are many, from the chosen method to solvent interactions or basis set limitations, scaling factors are needed; that’s when you can ask the Cheshire Cat which way to go
If you don’t know where you are going any road will get you there.
Lewis Carroll – Alice in Wonderland
Well, not really. The Chemical Shift Repository for computed NMR scaling factors, with Coupling Constants Added Too (aka CHESHIRE CCAT) provides with straight directions on how to correct your computed NMR chemical shifts according to the level of theory without the need to calculate the NMR shielding tensor for the reference compound (usually TMS as pointed out earlier). In a nutshell, the group of Prof. Dean Tantillo (UC Davis) has collected a large number of isotropic magnetic shielding values and plotted them against experimental chemical shifts. Just go to their scaling factors page and check all their linear regressions and use the values that more closely approach to your needs, there are also all kinds of scripts and spreadsheets to make your job even easier. Of course, if you make use of their website don’t forget to give the proper credit by including these references in your paper.
We’ve recently published an interesting study in which the 1H – 19F coupling constants were calculated via the long way (I was just recently made aware of CHESHIRE CCAT by Dr. Jacinto Sandoval who knows all kinds of web resources for computational chemistry calculations) as well as their conformational dependence for some substituted 2-aza-carbazoles (fig. 1).
Journal of Molecular Structure Vol 1176, 15 January 2019, Pages 562-566
The paper is published in the Journal of Molecular Structure. In this study we used the GIAO NMR computations to assign the peaks on an otherwise cluttered spectrum in which the signals were overlapping due to conformational variations arising from the rotation of the C-C bond which re-orients the F atoms in the fluorophenyl grou from the H atom in the carbazole. After the calculations and the scans were made assigning the peaks became a straightforward task even without the use of scaling factors. We are now expanding these calculations to more complex systems and will contrast both methods in this space. Stay tuned.
