Static Plots by {ggplot2} and {patchwork}

The plot_EPR_Specs() is the essential function for plotting individual EPR spectra as well as those within series (e.g. time series ≡ kinetic measurements). Now, by means of this function and by the {patchwork} 📦 , we can compare the EPR spectra with the basic and “advanced” normalization ➨

# plotting the spectrum with basic normalization,
# the B-axis can be presented either 
# in "mT" (default) or "G" plot title is added 
# by `ggtitle()` function from `{ggplot2}` package
tmpd.plot.norm.01 <-
  plot_EPR_Specs(data.spectra = tmpd.data.norm.01) +
  ggplot2::ggtitle("Basic Normalization")
#
# preview
tmpd.plot.norm.01

EPR spectrum of $\small \ce{TMPD^{.+}}$ in $\small 0.3\,\mathrm{M}~\ce{TBAPF6}/\ce{DMSO}$ after potentiostatic oxidative electrolysis at $\small 0.308\,\mathrm{V}$ vs Ag-quasiref. Intensity normalized by number of scans, ‘Nscans’, and by sensitivity Q-factor, ‘qValue’.

Comparison of both spectra with different normalization ➨

# spectrum with the "advanced" normalization
tmpd.plot.norm.02 <-
  plot_EPR_Specs(data.spectra = tmpd.data.norm.02) +
  ggplot2::ggtitle("Advanced Normalization")
#
# both spectra together by `{patchwork}` package
# without y/Intensity label (`ggplot2::labs(y = NULL)`) 
# because the EPR spectra will be displayed close 
# to each other in two columns
tmpd.plot.norm.0102 <-
  tmpd.plot.norm.01 +
  tmpd.plot.norm.02 +
  ggplot2::labs(y = NULL)
#
# preview of EPR spectra with different intensities
tmpd.plot.norm.0102

Comparison of $\small \ce{TMPD^{.+}}$ EPR spectra with different intensity normalization.

If one wants to save the previous Figure @ref(fig:spectral-data-norm0102-visual), the function ggplot2::ggsave() can be applied ➨

# to save the plot `tmpd.plot.norm.0102` with the size
# of (7 x 5) inches and the resolution of dpi = 200
ggplot2::ggsave("TMPD_EPR_Norm_compar.png",
  plot = tmpd.plot.norm.0102,
  width = 7,
  height = 5,
  units = "in",
  dpi = 200
)

# other image formats like `.pdf`, `.tex`,`.jpeg`, `.eps`,
# `.bmp`, `.svg` or `.tiff` can be used as well

Interactive Plots by {plotly}

In order to explore the EPR spectrum in details (to read values, zoom or save the spectrum), the function plot_EPR_Specs2D_interact() is applied (consult the corresponding examples in the function documentation). Additionally, it can be also used for a series of EPR spectra, in overlay mode, similarly to plot_EPR_Specs(). Additional interactive 3D-plotting of EPR spectra, especially those recorded as time series, like during kinetic or variable temperature experiments, are provided by the plot_EPR_Specs3D_interact() function.

Finding the g_iso-Value

The parameter is defined by the following Equation @ref(eq:gfactor):

where the h and μ_B are Plank’s and Bohr magneton constants, respectively. Therefore, it is determined only by the microwave frequency (ν) and the magnetic flux density ($\small B_0$) of the spectrum mid-point where $\small \mathrm{d}I_{\mathrm{EPR}}/\mathrm{d}B = 0$ (in the case of CW EPR spectra, see also function eval_gFactor_Spec()).

# reading the microwave frequency ("MF") from the `.par` file
tmpd.mw.freq <-
  readEPR_param_slct(tmpd.params.file,
    string = "MF",
    origin = "winepr"
  )
# B region (349.677, 350.457) mT, including 
# the B mid-point, to calculate the g-value 
# is obtained from the interactive plot above
tmpd.g.iso.spec <-
  eval_gFactor_Spec(tmpd.data.norm.02,
    nu.GHz = tmpd.mw.freq,
    B = "B_mT",
    B.unit = "mT",
    Blim = c(349.677, 350.457)
  )
#
# value preview
tmpd.g.iso.spec
#> [1] 2.00304

The above(from spectrum)-obtained g_iso-value may be compared with that computed by Density Functional Theory (DFT) quantum chemical calculations of $\small \ce{TMPD^{.+}}$, namely by PBE0/EPR-II/CPCM-DMSO//B3LYP/6-31+G(d,p) (see also Gaussian DFT Methods and Basis Sets). The extraction of g_iso-value from the Gaussian (or from the ORCA quantum chemical package) output file is provided by the eval_gFactor_QCHcomp() function ➨

# package built-in example file => 
# "TMPDAradCatEPRa.inp.log.zip"
# which is a `.zip` of the Gaussian output file
gauss.tmpd.load <- 
  load_data_example(file = "TMPDAradCatEPRa.inp.log.zip")
gauss.tmpd.output <- unzip(gauss.tmpd.load)
#
# giso-value is automatically calculated as the mean
# of its principal components (g1,g2,g3) 
# summarized in the output file
tmpd.g.iso.dft <- 
  eval_gFactor_QCHcomp(gauss.tmpd.output)
#
# preview
tmpd.g.iso.dft
#> [1] 2.00317
#
# comparison of both theoretical and experimental g-values
# within the tolerance of 1e-3, to illustrate the automatic
# comparison of two values in R language, the function 
# `all.equal()`can be applied
all.equal(tmpd.g.iso.spec, tmpd.g.iso.dft, tolerance = 1e-3)
#> [1] TRUE

The TRUE value represents a perfect agreement between the theoretically (2.00317) and the experimentally (2.00304) determined g_iso-values, thus supporting the structure of $\small \ce{TMPD^{.+}}$. Its difference from the free electron $g_{\mathrm{e}}\approx \small 2.00232$ is a consequence of spin-orbit coupling (SOC) of the unpaired electron within the radical cation. Namely, it tells how the individual atomic orbitals contribute to single occupied molecular orbital (SOMO), i.e. the orbital characterizing the unpaired electronic state. Therefore, for $\small \ce{TMPD^{.+}}$ the SOC is not so pronounced because of the small difference from g_e. However, for heavy atoms (e.g. central atoms in metal complexes) the difference is often large and indicates larger SOC than for organic radicals like for the one mentioned above (9).

Linewidth, 𝚫B_pp Determination

The linewidth in $\mathrm{m\small T}$ can be determined from several spectral lines/regions from the interactive plot above and by the eval_DeltaXpp_Spec() function ➨

# calculate Delta Bpp from several regions
# => list of three selected elements/regions
B.regions.mT <- list(
  c(349.677, 349.977),
  c(349.177, 349.492),
  c(350.822, 351.072)
)
#
# all Delta Bpp as vector, created by the iteration 
# through all elements (with the help of 
# the `sapply()` function) of `B.regions.mT`
tmpd.delta.Bpp <- sapply(
  B.regions.mT,
  function(b) {
    eval_DeltaXpp_Spec(tmpd.data.norm.02,
      xlim = b
    )
  }
)
#
# mean value of all three linewidths in mT
tmpd.delta.Bpp.mean <-
  round(mean(tmpd.delta.Bpp), digits = 2)
#
# value preview
tmpd.delta.Bpp.mean
#> [1] 0.05

Hyperfine (HF) Structure and Simulations

The HF structure represents one of the most important characteristic of EPR spectra because it reflects the distribution of unpaired electron density within the radical/paramagnetic molecule. At least some of the hyperfine splitting constants (a, HFSCs, distance between the main lines within the pattern) may be estimated from the interactive EPR spectrum above. For the subsequent simulation of the EPR spectrum these are simply converted by the convert_a_mT_2A() into hyperfine coupling constants in $\mathrm{\small{MH}z}$ ($\small A$, HFCCs) using the following relation (Equation @ref(eq:hfatoA)) ➨

# two visible distances or HFSCs (a in mT)
# from the interactive EPR spectrum above
a1.guess.mT <- abs(350.042 - 349.342)
a2.guess.mT <- abs(349.147 - 349.342)
# both values into vector
a.guess.mT <- c(a1.guess.mT, a2.guess.mT)
#
# conversion into A (in MHz) incl. g.iso value
# (`tmpd.g.iso.spec`) from the calculation above
A.guess.MHz <-
  sapply(
    a.guess.mT,
    function(x) {
      convert_a_mT_2A(
        a.mT = x,
        g.val = tmpd.g.iso.spec
      )
    }
  )
#
# `A.guess.MHz` preview
A.guess.MHz
#> [1] 19.62  5.47

According to literature (1) and the $\small \ce{TMPD^{.+}}$ structure (see also Figure @ref(fig:dft-tmpd-structure) with the DFT-optimized geometry), one can recognize three interacting groups of equivalent nuclei : $\small 2\,\times\,\ce{^{14}N}$, $\small 4\,\times\,\ce{^1H}$ (aromatic protons) and $\small 12\,\times\,\ce{^1H}$ (methyl protons). DFT calculations may provide all $\small A$ (HFCCs) as shown below.

# loading built-in image
knitr::include_graphics("TMPD_EPR_DFT_radCat_ab.png",dpi = 200)

DFT optimized structure of TMPD radical cation with atom numbering inherited from the Gaussian 16 output file.

In order to compare the theoretical and experimental HFCCs, the function rearrange_aAiso_QCHorgau() extracts the $\small A$ values from the Gaussian or ORCA output files and calculates the mean values corresponding to selected groups of equivalent nuclei which are defined by the nuclei.list.slct argument. Therefore, according to numbering shown in Figure @ref(fig:dft-tmpd-structure) ➨

# The same package built-in example file 
# (`TMPDAradCatEPRa.inp.log.zip`)
# as for the extraction of g-value is used (see above).
symmetry.As.df <-
  rearrange_aAiso_QCHorgau(gauss.tmpd.output,
    N.nuclei = 28,
    nuclei.list.slct =
      list(
        c(7, 8), # 2 x 14N
        c(13, 14, 15, 16), # 4 x 1H (aromatic)
        c(
          17, 18, 19, 20,
          21, 22, 23, 24,
          25, 26, 27, 28
      ) # 12 x 1H (methyls)
    )
  )
#
# data frame presentation by `{kableExtra}` package
kableExtra::kbl(symmetry.As.df) %>%
  kableExtra::kable_styling(
    bootstrap_options = 
      c("striped",
        "hover",
        "condensed"
  )
)

The previous analysis reveals that the highest HF-coupling/splitting constants are related to the interaction of unpaired electron with the nitrogen and methyl-proton nuclei. Therefore the highest spin density is located on the “periphery” of the $\small \ce{TMPD^{.+}}$ molecule. This may be also supported by the simulation (theoretical calculation) of the EPR spectrum (which is computed by the eval_sim_EPR_iso() function), taking into account the following parameters ➨

• position of the EPR spectrum (g_iso-value) ➨ g.iso

• spectral linewidth corresponding to Gaussian and/or Lorentzian lineshape ➨ lineGL.DeltaB

• content of the Gaussian lineshape in the overall linear combination of both Gaussian and Lorenztian theoretical spectral forms (so called pseudo-Voigt lineshape) ➨ lineG.content

• additional instrumental parameters like central field ➨ Bcf, sweep width ➨ Bsw, microwave frequency ➨ mwGHz and number of points (spectral resolution) ➨ Npoints.

Required arguments of the eval_sim_EPR_iso() function are presented after the arrows. The estimated linewidth tmpd.delta.Bpp.mean ($\small 0.5\,\mathrm{G}$, see above) as well as the HFCCs, either directly taken from the spectrum (A.guess.MHz) or from the quantum chemical calculations (symmetry.As.df) and/or from the literature (1) can be used to simulate the EPR spectrum of $\small \ce{TMPD^{.+}}$ as follows ➨

# isotropic simulation of the TMPD radical cation where
# the instrumental parameters are taken from 
# the corresponding `.par` file: `tmpd.params.file` :
sim.tmpd.iso <-
  eval_sim_EPR_iso(
    g.iso = tmpd.g.iso.spec,
    instrum.params = NULL,
    path_to_dsc_par = tmpd.params.file,
    origin = "winepr",
    B.unit = "mT",
    nuclear.system =
      list(
        list("14N", 2, 18.6), # A(2 x 14N) = 18.6 MHz
        list("1H", 4, 5.5), # A(4 x 1H) = 5.5 MHz
        list("1H", 12, 19.6) # A(12 x 1H) = 19.6 MHz
      ), 
    # Gauss (G) & Lorentz (L) linewidth (mT):
    lineGL.DeltaB = list(0.05, 0.05),
    # pseudo-Voigt line shape with 0.5 G + 0.5 L:
    lineG.content = 0.5
  )
#
# output is either interactive spectrum plot
# or list of plot and the simulated 
# spectrum data frame.
#
# preview of the data frame
head(sim.tmpd.iso$df)
#>    Bsim_G Bsim_mT dIeprSim_over_dB
#> 1 3439.17 343.917    9.6281321e-05
#> 2 3439.22 343.922    9.6721842e-05
#> 3 3439.27 343.927    9.7169711e-05
#> 4 3439.32 343.932    9.7625349e-05
#> 5 3439.37 343.937    9.8089217e-05
#> 6 3439.42 343.942    9.8561824e-05

Afterwards, the simulated EPR spectrum can be presented/compared with the experimental one by the presentEPR_Sim_Spec() function (Figure @ref(fig:tmpd-radcat-sim-expr-compar)), which is essentially based on {ggplot2} package and therefore can be combined with the additional {ggplot2} functions .

# intesinty of the simulated spectrum
# is automatically scalled onto 
# the experimental one.
present_EPR_Sim_Spec(
  data.spectr.expr = tmpd.data.norm.02,
  data.spectr.sim = sim.tmpd.iso$df,
  B.unit = "mT"
  ) +
  # `plot_theme_NoY_ticks` theme from `{eprscope}`
  plot_theme_NoY_ticks(legend.text = element_text(size = 13))

Comparison of simulated and experiental EPR spectrum of $\small \ce{TMPD^{.+}}$.

Although the simulated $\ce{TMPD^{.+}}$ EPR spectrum is quite close to the experimental one, there are still some apparent differences. Therefore, the fitting/optimization function eval_sim_EPR_isoFit() was used to get a more accurate simulated spectrum. The result may be found in Examples of the present_EPR_Sim_Spec() function.

Integrating the EPR Spectrum

Integration process in the analysis of EPR spectra is undoubtedly an important operation because it mirrors the amount of paramagnetic centers/radicals within the sample (see also quantify_EPR_Abs()). However, it is heavily dependent on the initial form of the EPR spectrum, i.e. whether it is present either in derivative (usually in CW EPR) or already integrated (usually in 1D pulsed EPR) form/shape. Hence, the spectrum has to be integrated either twice or only once, respectively. The core-function used for the integration is based on the cumulative trapezoidal approximation, available from the {pracma} 📦 pracma::cumtrapz() (10–13).

The relative error of the cumulative trapezoidal integration is minimal, usually falling into the range of $\small \langle 1-5\rangle\,\%$ or even lower depending on the spectrum resolution (11, 13). If the initial EPR spectrum displays low signal-to-noise ratio, the integral required for the quantitative analysis looses its sigmoid-shape ( see below an example of such shape). Therefore, in such case, the best option is to simulate the EPR spectrum and to evaluate the integral from that spectrum (see e.g. the quantify_EPR_Sim_series() function). The big advantage of this method consists in graphical representation of the corresponding integrals, i.e. the integration process may be visually controlled in the same manner like within the EPR spectrometer acquisitions/processing software (see also the plot_EPR_Specs_integ() function below). The integration of the EPR spectral data is essentially provided by the eval_integ_EPR_Spec() and its application to the experimental $\small \ce{TMPD^{.+}}$ spectrum is documented by the following scripts ➨

# basic integration without baseline correction
# during the first integration.
tmpd.data.norm.02.integ <-
  eval_integ_EPR_Spec(
    data.spectr = tmpd.data.norm.02,
    B = "B_mT",
    B.unit = "mT",
    sigmoid.integ = TRUE
  )
#
# integration returns the original data frame
# with the additional integrals 
# `single_Integ` + `sigmoid_Integ`
#
# preview
head(tmpd.data.norm.02.integ)
#>          B_G  dIepr_over_dB      B_mT  single_Integ sigmoid_Integ
#>        <num>          <num>     <num>         <num>         <num>
#> 1: 3439.1699 -4.3890968e-07 343.91699 2.0128171e-06 0.0000000e+00
#> 2: 3439.2200 -2.3889797e-07 343.92200 1.9958553e-06 1.0031502e-07
#> 3: 3439.2700 -3.0166941e-07 343.92700 1.9823278e-06 1.9986707e-07
#> 4: 3439.3201  3.0704314e-10 343.93201 1.9747866e-06 2.9888990e-07
#> 5: 3439.3701 -7.2531358e-07 343.93701 1.9566436e-06 3.9727197e-07
#> 6: 3439.4199 -5.6317054e-07 343.94199 1.9245572e-06 4.9392357e-07

Both integrals, within the entire $\small B$-range, can be visualized by the plot_EPR_Specs_integ() function from Visualization and Graphics family:

# both integrals are presented separately
# with their corresponding intensities automatically
# scaled onto the `y`-axis (defined by "free_y")
plot_EPR_Specs_integ(
  tmpd.data.norm.02.integ,
  slct.integs = c(
    "single_Integ",
    "sigmoid_Integ"
  ),
  B = "B_mT",
  B.unit = "mT",
  separate.integs = TRUE,
  separate.integ.scales = "free_y"
)

Single (single_Integ) and double (sigmoid_Integ) integrated EPR spectrum of $\small \ce{TMPD^{.+}}$.

Obviously, the baseline of the single and double integrated $\small \ce{TMPD^{.+}}$ EPR spectrum is somewhat distorted and must be corrected. To fix the baseline distortion, the eval_integ_EPR_Spec() function contains corresponding arguments which control the baseline approximation, under the complex peak, by a polynomial fit and provide its subtraction from the single integrated spectrum to evaluate the double/sigmoid integral more accurately ➨

# correction of the TMPD^{.+} EPR spectrum single
# integral by the 3rd degree polynomial
tmpd.data.norm.02.integCorr <-
  eval_integ_EPR_Spec(
    data.spectr = tmpd.data.norm.02,
    B = "B_mT",
    B.unit = "mT",
    correct.integ = TRUE,
    # peak region to correct the baseline:
    BpeaKlim = c(344.5, 355.5),
    # polynomial degree to fit baseline 
    # under the peak region:
    poly.degree = 3,
    sigmoid.integ = TRUE
  )
#
# preview of the integrals,
# data frame column of corrected integral
# is denoted as `single_Integ_correct`
plot_EPR_Specs_integ(
  tmpd.data.norm.02.integCorr,
  slct.integs = c(
    "single_Integ_correct",
    "sigmoid_Integ"
  ),
  B = "B_mT",
  B.unit = "mT",
  separate.integs = TRUE,
  separate.integ.scales = "free_y"
)

Corrected single and double integrated EPR spectrum of $\small \ce{TMPD^{.+}}$ (see also Figure @ref(fig:tmpd-basic-epr-integ-visual)).

Maximum value of the double/sigmoid integral 0.0024 can be used for the estimation of $\small \ce{TMPD^{.+}}$ amount within the sample.